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abstract: 'We present a new algorithm for identifying the transition and emission probabilities of a hidden Markov model (HMM) from the emitted data. Expectation-maximization becomes computationally prohibitive for long observation records, which are often required for identification. The new algorithm is particularly suitable for cases where the available sample size is large enough to accurately estimate second-order output probabilities, but not higher-order ones. We show that if one is only able to obtain a reliable estimate of the pairwise co-occurrence probabilities of the emissions, it is still possible to uniquely identify the HMM if the emission probability is *sufficiently scattered*. We apply our method to hidden topic Markov modeling, and demonstrate that we can learn topics with higher quality if documents are modeled as observations of HMMs sharing the same emission (topic) probability, compared to the simple but widely used bag-of-words model.'
bibliography:
- 'hmm\_refs.bib'
title: |
**Learning Hidden Markov Models from Pairwise Co-occurrences\
with Application to Topic Modeling**
---
Introduction
============
Hidden Markov models (HMMs) are widely used in machine learning when the data samples are time *dependent*, for example in speech recognition, language processing, and video analysis. The graphical model of a HMM is shown in Figure \[fig:hmm\]. HMM models a (time-dependent) sequence of data $\{Y_t\}_{t=0}^T$ as indirect observations of an underlying Markov chain $\{X_t\}_{t=0}^T$ which is not available to us. Homogeneous HMMs are parsimonious models, in the sense that they are fully characterized by the transition probability $\Pr[X_{t+1}|X_{t}]$ and the emission probability $\Pr[Y_t|X_t]$ even though the size of the given data $\{Y_t\}_{t=0}^T$ can be very large.
Consider a homogeneous HMM such that:
- a latent variable $X_t$ can take $K$ possible outcomes $x_1,...,x_K$;
- an ambient variable $Y_t$ can take $N$ possible outcomes $y_1,...,y_N$.
Recall that [@rabiner1986introduction; @Ghahramani2001]:
- Given both $\{X_t\}_{t=0}^T$ and $\{Y_t\}_{t=0}^T$, the complete joint probability factors, and we can easily estimate the transition probability $\Pr[X_{t+1}|X_{t}]$ and the emission probability $\Pr[Y_t|X_t]$.
- Given only $\{Y_t\}_{t=0}^T$, but assuming we know the underlying transition and emission probabilities, we can calculate the observation likelihood using the forward algorithm, estimate the most likely hidden sequence using the Viterbi algorithm, and compute the posterior probability of the hidden states using the forward-backward algorithm.
The most natural problem setting, however, is when neither the hidden state sequence nor the underlying probabilities are known to us—we only have access to a sequence of observations, and our job is to reveal the HMM structure, characterized by the transition matrix $\Pr[X_{t+1}|X_{t}]$ and the emission probability $\Pr[Y_t|X_t]$ from the set of observations $\{Y_t\}_{t=0}^T$.
Related work {#sec:related}
------------
The traditional way of learning a HMM from $\{Y_t\}_{t=0}^T$ is via expectation-maximization (EM) [@rabiner1986introduction], in which the expectation step is performed by calling the forward-backward algorithm. This specific instance of EM is also called the Baum-Welch algorithm [@baum1970maximization; @Ghahramani2001]. However, the complexity of Baum-Welch is prohibitive when $T$ is relatively large—the complexity of the forward-backward algorithm is ${\mathcal{O}}(K^2T)$, but EM converges slowly, so the forward-backward algorithm must be called many times. This is a critical issue, because a HMM can only be learned with high accuracy when the number of observation samples $T$ is large enough.
One way of designing scalable algorithms for learning HMMs is to work with sufficient statistics—a summary of the given observation sequence, whose size does not grow with $T$. Throughout this paper we assume that the HMM process is stationary (time-invariant), which is true almost surely if the underlying Markov process is ergodic and the process has been going on for a reasonable amount of time. With $T$ large enough, we can accurately estimate the co-occurrence probability between two consecutive emissions $\Pr[Y_t,Y_{t+1}]$. According to the graphical model shown in Figure \[fig:hmm\], it is easy to see that given the value of $X_t$, $Y_t$ is conditionally independent of all the other variables, leading to the factorization $$\begin{aligned}
\label{eq:hmm_fac2}
\Pr[Y_t,Y_{t+1}]
= \sum_{k,j=1}^{K}&\Pr[Y_t|X_t=x_k]\Pr[Y_{t+1}|X_{t+1}=x_j]\Pr[X_t=x_k,X_{t+1}=x_j]\end{aligned}$$ Let ${\boldsymbol{\varOmega}}\in{\mathbf{R}}^{N\times N}$, ${\boldsymbol{M}}\in{\mathbf{R}}^{N\times K}$, and ${\boldsymbol{\varTheta}}\in{\mathbf{R}}^{K\times K}$, with their elements defined as $$\begin{aligned}
\varOmega_{n\ell} &= \Pr[Y_t=y_n,Y_{t+1}=y_\ell],\\
M_{nk} &= \Pr[Y_t=y_n|X_t=x_k],\\
\varTheta_{kj} &= \Pr[X_t=x_k,X_{t+1}=x_j].\end{aligned}$$ Then, equations can be written compactly as $$\begin{aligned}
\label{eq:MTM}
{\boldsymbol{\varOmega}}= {\boldsymbol{M}}{\boldsymbol{\varTheta}}{\boldsymbol{M}}^{{\!\top\!}}.\end{aligned}$$ Noticing that $\eqref{eq:MTM}$ is a nonnegative matrix tri-factorization with a number of inconsequential constraints for ${\boldsymbol{M}}$ and ${\boldsymbol{\varTheta}}$ to properly represent probabilities, @Vanluyten2008 [@Lakshminarayanan2010; @Cybenko2011] proposed using nonnegative matrix factorization (NMF) to estimate the HMM probabilities. However, NMF-based methods have a serious shortcoming in this context: the tri-factorization is in general not unique, because it is fairly easy to find a nonsingular matrix ${\boldsymbol{Q}}$ such that both ${\boldsymbol{M}}{\boldsymbol{Q}}\geq0$ and ${\boldsymbol{Q}}^{-1}{\boldsymbol{\varTheta}}{\boldsymbol{Q}}^{-{{\!\top\!}}}\geq0$, and then $\widetilde{{\boldsymbol{M}}}={\boldsymbol{M}}{\boldsymbol{Q}}$ and $\widetilde{{\boldsymbol{\varTheta}}}={\boldsymbol{Q}}^{-1}{\boldsymbol{\varTheta}}{\boldsymbol{Q}}^{-{{\!\top\!}}}$ are equally good solutions in terms of reconstructing the co-occurrence matrix ${\boldsymbol{\varOmega}}$. When we use $({\boldsymbol{M}},{\boldsymbol{\varTheta}})$ and $(\widetilde{{\boldsymbol{M}}},\widetilde{{\boldsymbol{\varTheta}}})$ to perform HMM inference, such as estimating hidden states or predicting new emissions, the two models often yield completely different results, unless ${\boldsymbol{Q}}$ is a permutation matrix.
A number of works propose to use *tensor* methods to overcome the identifiability issue. Instead of working with the pairwise co-occurrence probabilities, they start by estimating the joint probabilities of three consecutive observations $\Pr[Y_{t-1},Y_t,Y_{t+1}]$. Noticing that these three random variables are conditionally independent given $X_t$, the triple-occurrence probability factors into $$\begin{aligned}
\Pr[Y_{t-1},Y_t,Y_{t+1}] = \sum_{k=1}^{K}\Pr[X_t=x_k]\Pr[Y_{t-1}|X_t=x_k]\Pr[Y_{t}|X_t=x_k]\Pr[Y_{t+1}|X_t=x_k],\end{aligned}$$ which admits a tensor *canonical polyadic decomposition* (CPD) model [@Hsu2009; @Anandkumar2012; @Anandkumar2014]. Assuming $K\leq N$, the CPD is essentially unique if two of the three factor matrices have full column rank, and the other one is not rank one [@harshman1970foundations]; in the context of HMMs, this is equivalent to assuming ${\boldsymbol{M}}$ and ${\boldsymbol{\varTheta}}$ both have linearly independent columns, which is a relatively mild condition. The CPD is known to be unique under much more relaxed conditions [@sidiropoulos2017tensor], but in order to uniquely retrieve the transition probability using the relationship $$\Pr[Y_{t+1}|X_{t}] = \sum_{j=1}^{K}
\Pr[Y_{t+1}|X_{t+1}\!=\!x_j]\Pr[X_{t+1}\!=\!x_j|X_t],$$ $K\leq N$ is actually the best we can achieve using triple-occurrences without making further assumptions. [^1] A salient feature in this case is that if the triple-occurrence probability $\Pr[Y_{t-1},Y_t,Y_{t+1}]$ is exactly given (meaning the rank of the triple-occurrence tensor is indeed smaller than $N$), the CPD can be efficiently calculated using generalized eigendecomposition and related algebraic methods [@sanchez1990tensorial; @leurgans1993decomposition; @DeLathauwer2004a]. These methods do not work well, however, when the low-rank tensor is perturbed; e.g., due to insufficient mixing / sample averaging of the triple occurrence probabilities.
It is also possible to handle cases where $K>N$. The key observation is that, given $X_t$, $Y_t$ is conditionally independent of $Y_{t-1},...,Y_{t-\tau}$ and $Y_{t+1},...,Y_{t+\tau}$. Then, grouping $Y_{t-1},...,Y_{t-\tau}$ into a single categorical variable taking $N^\tau$ possible outcomes, and $Y_{t+1},...,Y_{t+\tau}$ into another one, we can construct a much bigger tensor of size $N^\tau\times N^\tau\times N$, and then uniquely identify the underlying HMM structure with $K\gg N$ as long as certain linear independence requirements are satisfied for the conditional distribution of the *grouped* variables [@Allman2009; @bhaskara14a; @Huang2016; @Sharan2017]. It is intuitively clear that for fixed $N$, we need a much larger realization length $T$ in order to accurately estimate $(2\tau+1)$-occurrence probabilities as $\tau$ grows, which is the price we need to pay for learning a HMM with a larger number of hidden states.
This paper
----------
The focus of this paper is on cases where $K \leq N$, and $T$ is large enough to obtain accurate estimate of $\Pr[Y_t,Y_{t+1}]$, but not large enough to accurately estimate triple or higher-order occurrence probabilities. We [*prove*]{} that it is actually possible to recover the latent structure of an HMM only from pairwise co-occurrence probabilities $\Pr[Y_t,Y_{t+1}]$, provided that the underlying emission probability $\Pr[Y_t|X_t]$ is *sufficiently scattered*. Compared to the existing NMF-based HMM learning approaches, our formulation employs a different (determinant-based) criterion to ensure identifiability of the HMM parameters. Our matrix factorization approach resolves cases that cannot be handled by tensor methods, namely when $T$ is insufficient to estimate third-order probabilities, under an additional condition that is quite mild: that the emission probability matrix ${\boldsymbol{M}}$ must be *sufficiently scattered*, rather than simply full column-rank.
We apply our method to hidden topic Markov modeling (HTMM) [@gruber2007hidden], in which case the number of hidden states (topics) is indeed much smaller than the number of ambient states (words). HTMM goes beyond the simple and widely used bag-of-words model by assuming that (ordered) words in a document are emitted from a hidden topic sequence that evolves according to a Markov model. We show improved performance on real data when using this simple and intuitive model to take word ordering into account when learning topics, which also benefits from our identifiability guaranteed matrix factorization method.
As an illustrative example, we showcase the inferred topic of each word in a news article (removing stop words) in Figure \[fig:topic\], taken from the Reuters21578 data set obtained at [@reuters21578]. As we can see, HTMM gets much more consistent and smooth inferred topics compared to that obtained from a bag-of-words model (cf. supplementary material for details). This result agrees with human understanding.
Second-order vs. Third-order Learning
=====================================
We start by arguing that for the same observation data $\{Y_t\}_{t=0}^T$, the estimate of the pairwise co-occurrence probability $\Pr[Y_t,Y_{t+1}]$ is always more accurate than that of the triple co-occurrence probability $\Pr[Y_{t-1},Y_t,Y_{t+1}]$.
Let us first explicitly describe the estimator we use for these probabilities. For each observation $Y_t$, we define a coordinate vector ${\boldsymbol{\psi}}_t\in{\mathbf{R}}^K$, and ${\boldsymbol{\psi}}_t=\bm{e}_k$ if $Y_t=y_k$. The natural estimator for the pairwise co-occurrence probability matrix ${\boldsymbol{\varOmega}}$ is $$\label{eq:omega2}
\widehat{{\boldsymbol{\varOmega}}} = \frac{1}{T}\sum_{t=0}^{T-1}{\boldsymbol{\psi}}_t{\boldsymbol{\psi}}_{t+1}^{{\!\top\!}},$$ and similarly for the triple co-occurrence probability $\underline{{\boldsymbol{\varOmega}}_3}$ $$\label{eq:omega3}
\widehat{\underline{{\boldsymbol{\varOmega}}_3}} = \frac{1}{T-1}\sum_{t=1}^{T-1}{\boldsymbol{\psi}}_{t-1}\circ{\boldsymbol{\psi}}_t\circ{\boldsymbol{\psi}}_{t+1},$$ where $\circ$ denotes vector outer-product. [^2]
The first observation is that both $\widehat{{\boldsymbol{\varOmega}}}$ and $\widehat{\underline{{\boldsymbol{\varOmega}}_3}}$ are unbiased estimators: Obviously $\operatorname{E}({\boldsymbol{\psi}}_t{\boldsymbol{\psi}}_{t+1}^{{\!\top\!}})={\boldsymbol{\varOmega}}$ and likewise for the triple-occurrences, and taking their averages does not change the expectation. However, the individual terms in the summation are not independent of each other, making it hard to determine how fast estimates converge to their expectation. The state-of-the-art concentration result for HMMs [@kontorovich2006measure] states that for any 1-Lipschitz function $f$ $$\Pr[|f(\{Y_t\})-\operatorname{E}f(\{Y_t\})|>\epsilon]
\leq 2\exp\left(-T\epsilon^2/c\right),$$ where $c$ is a constant that only depends on the specific HMM structure but not on the function $f$ (cf. [@kontorovich2006measure] for details). Taking $f$ as any entry in $\widehat{{\boldsymbol{\varOmega}}}$ or $\widehat{\underline{{\boldsymbol{\varOmega}}_3}}$, we can check that indeed it is 1-Lipschitz, meaning as $T$ goes to infinity, both estimators converge to their expectation with negligible fluctuations.
We now prove that for a given set of observations $\{Y_t\}_{t=0}^T$, $\widehat{{\boldsymbol{\varOmega}}}$ is always going to be more accurate than $\widehat{\underline{{\boldsymbol{\varOmega}}_3}}$. Since both of them represent probabilities, we use two common metrics to measure the differences between the estimators and their expectations, the Kullback-Leibler divergence $D_\text{KL}(\cdot)$ and the total-variation difference $D_\text{TV}(\cdot)$.
\[ppst:2>3\] Let $\widehat{{\boldsymbol{\varOmega}}}$ and $\widehat{\underline{{\boldsymbol{\varOmega}}_3}}$ be obtained from the same set of observations $\{Y_t\}_{t=0}^T$, we have that $$\begin{aligned}
D_\textup{KL}(\widehat{{\boldsymbol{\varOmega}}}\|{\boldsymbol{\varOmega}}) &\leq
D_\textup{KL}(\widehat{\underline{{\boldsymbol{\varOmega}}_3}}\|\underline{{\boldsymbol{\varOmega}}_3})
\qquad\text{and}\\
D_\textup{TV}(\widehat{{\boldsymbol{\varOmega}}}\|{\boldsymbol{\varOmega}}) &\leq
D_\textup{TV}(\widehat{\underline{{\boldsymbol{\varOmega}}_3}}\|\underline{{\boldsymbol{\varOmega}}_3}).\end{aligned}$$
The proof of Proposition \[ppst:2>3\] is relegated to the supplementary material.
Identifiability of HMMs from Pairwise Co-occurrence Probabilities
=================================================================
The arguments made in the previous section motivate going back to matrix factorization methods for learning a HMM when the realization length $T$ is not large enough to obtain accurate estimates of triple co-occurrence probabilities. As we have explained in §\[sec:related\], the co-occurrence probability matrix ${\boldsymbol{\varOmega}}$ admits a nonnegative matrix tri-factorization model . There are a number of additional equality constraints. Columns of ${\boldsymbol{M}}$ represent conditional distributions, so ${\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{M}}={\boldsymbol{\mathit{1}}}^{{\!\top\!}}$. Matrix ${\boldsymbol{\varTheta}}$ represents the joint distribution between two consecutive Markovian variables, therefore ${\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}=1$. Furthermore, we have that ${\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}$ and ${\boldsymbol{\varTheta}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}$ represent $\Pr[X_t]$ and $\Pr[X_{t+1}]$ respectively, and since we assume that the Markov chain is stationary, they are the same, i.e., ${\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}={\boldsymbol{\varTheta}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}$. Notice that this does not imply that ${\boldsymbol{\varTheta}}$ is symmetric, and in fact it is often not symmetric.
@huang2016nips considered a factorization model similar to in a different context, and showed that identifiability can be achieved under a reasonable assumption called *sufficiently scattered*, defined as follows.
\[def:suf\_scat\] Let $\operatorname{cone}({\boldsymbol{M}}^{{\!\top\!}})^*$ denote the polyhedral cone $\{{\boldsymbol{x}}:{\boldsymbol{M}}{\boldsymbol{x}}\geq 0\}$, and ${\mathcal{C}}$ denote the elliptical cone $\{{\boldsymbol{x}}:\|{\boldsymbol{x}}\|\leq{\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{x}}\}$. Matrix ${\boldsymbol{M}}$ is called **sufficiently scattered** if it satisfies that: (i) $\operatorname{cone}({\boldsymbol{M}}^{{\!\top\!}})^*\subseteq{\mathcal{C}}$, and (ii) $\operatorname{cone}({{\boldsymbol{M}}}^{{\!\top\!}})^\ast\cap{\rm bd}{\mathcal{C}}=\{\lambda {\bm e}_k:\lambda\geq 0,k=1,...,K\}$, where ${\rm bd}{\mathcal{C}}$ denotes the boundary of ${\mathcal{C}}$, $\{{\boldsymbol{x}}:\|{\boldsymbol{x}}\|={\boldsymbol{\mathit{1}}}^{{{\!\top\!}}}{\boldsymbol{x}}\}$.
The sufficiently scattered condition was first proposed in [@huang2014tsp] to establish uniqueness conditions for the widely used *nonnegative matrix factorization* (NMF). For the NMF model ${\boldsymbol{\varOmega}}=\bm{WH}^{{\!\top\!}}$, if both $\bm{W}$ and $\bm{H}$ are sufficiently scattered, then the nonnegative decomposition is unique up to column permutation and scaling. Follow up work strengthened and extended the identifiability results based on this geometry inspired condition [@fu2015bss; @huang2016nips; @fu2017spl]. A similar tri-factorization model was considered in [@huang2016nips] in the context of bag-of-words topic modeling, and it was shown that among all feasible solutions of , if we find one that minimizes $|\det{\boldsymbol{\varTheta}}|$, then it recovers the ground-truth latent factors ${\boldsymbol{M}}$ and ${\boldsymbol{\varTheta}}$, assuming the ground-truth ${\boldsymbol{M}}$ is sufficiently scattered. In our present context, we therefore propose the following problem formulation:
\[prob:main\] $$\begin{aligned}
\operatorname*{\textrm{minimize}}_{{\boldsymbol{\varTheta}},{\boldsymbol{M}}}~~~ & |\det{\boldsymbol{\varTheta}}| \\
\textrm{subject to}~~~ & {\boldsymbol{\varOmega}}={\boldsymbol{M}}{\boldsymbol{\varTheta}}{\boldsymbol{M}}^{{\!\top\!}}, \label{eq:Omega}\\
& {\boldsymbol{\varTheta}}\geq0, {\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}={\boldsymbol{\varTheta}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}, {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}=1, \label{eq:Theta}\\
& {\boldsymbol{M}}\geq0, {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{M}}={\boldsymbol{\mathit{1}}}^{{\!\top\!}}. \end{aligned}$$
Regarding Problem , we have the following identifiability result.
\[thm:unique\] [@huang2016nips] Suppose ${\boldsymbol{\varOmega}}$ is constructed as ${\boldsymbol{\varOmega}}={\boldsymbol{M}}_\natural{\boldsymbol{\varTheta}}_\natural{\boldsymbol{M}}_\natural^{{\!\top\!}}$, where ${\boldsymbol{M}}_\natural$ and ${\boldsymbol{\varTheta}}_\natural$ satisfy the constraints in , and in addition (i) $\operatorname{rank}({\boldsymbol{\varTheta}}_\natural)=K$ and (ii) ${\boldsymbol{M}}_\natural$ is . Let $({\boldsymbol{M}}_\star,{\boldsymbol{\varTheta}}_\star)$ be an optimal solution for , then there must exist a permutation matrix ${\boldsymbol{\varPi}}\in{\mathbf{R}}^{K\times K}$ such that $${\boldsymbol{M}}_\natural = {\boldsymbol{M}}_\star{\boldsymbol{\varPi}}, \qquad
{\boldsymbol{\varTheta}}_\natural = {\boldsymbol{\varPi}}^{{\!\top\!}}{\boldsymbol{\varTheta}}_\star{\boldsymbol{\varPi}}.$$
One may notice that in [@huang2016nips], there are no constraints on the core matrix ${\boldsymbol{\varTheta}}$ as we do in . In terms of identifiability, it is easy to see that if the ground-truth ${\boldsymbol{\varTheta}}_\natural$ satisfies , solving even without will produce a solution ${\boldsymbol{\varTheta}}_\star$ that satisfies , thanks to uniqueness. In practice when we are given a less accurate ${\boldsymbol{\varOmega}}$, such “redundant” information will help us improve the estimation error, but that goes beyond identifiability consederations.
[.3]{}
[.3]{}
[.3]{}
The proof of Theorem \[thm:unique\] is referred to [@huang2016nips]. Here we provide some insights on this geometry-inspired sufficiently scattered condition, and discuss why it is a reasonable (and thus practical) assumption. The notation $\operatorname{cone}({\boldsymbol{M}}^{{\!\top\!}})^*=\{{\boldsymbol{x}}:{\boldsymbol{M}}{\boldsymbol{x}}\geq0\}$ comes from the convention in convex analysis that it is the *dual cone* of the conical hull of the row vectors of ${\boldsymbol{M}}$, i.e., $\operatorname{cone}({\boldsymbol{M}}^{{\!\top\!}})=\{{\boldsymbol{M}}^{{\!\top\!}}\bm{\alpha}:\bm{\alpha}\geq0\}$. Similarly, we can derive that the dual cone of ${\mathcal{C}}$ is ${\mathcal{C}}^*=\{{\boldsymbol{x}}:\|{\boldsymbol{x}}\|\leq{\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{x}}/\sqrt{K-1}\}$. A useful property of the dual cone is that for two convex cones $\mathcal{A}$ and $\mathcal{B}$, $\mathcal{A}\subseteq\mathcal{B}$ iff $\mathcal{B}^*\subseteq\mathcal{A}^*$. Therefore, the first requirement of sufficiently scattered in Definition \[def:suf\_scat\] equivalently means $${\mathcal{C}}^*\subseteq\operatorname{cone}({\boldsymbol{M}}^{{\!\top\!}}).$$ We give a geometric illustration of the sufficiently scattered condition in Figure \[fig:geo-scattered\] for $K=3$, and we focus on the 2-dimensional plane ${\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{x}}=1$. The intersection between this plane and the nonnegative orthant is the probability simplex, which is the triangle in Figure \[fig:geo-scattered\]. The outer circle represents ${\mathcal{C}}$, and the inner circle represents ${\mathcal{C}}^*$, again intersecting with the plane, respectively. The rows of ${\boldsymbol{M}}$ are scaled to sum up to one, and they are represented by black dots in Figure \[fig:geo-scattered\]. Their conical hull is represented by the shaded region. The polygon with dashed lines represents the dual of $\operatorname{cone}({\boldsymbol{M}}^{{\!\top\!}})$, which is indeed a subset of ${\mathcal{C}}$, and touches the boundary of ${\mathcal{C}}$ only at the coordinate vectors.
Figure \[fig:geo-separable\] shows a special case of sufficiently scattered called *separability*, which first appeared in [@donoho2004does] also to establish uniqueness of NMF. In this case, all the coordinate vectors appear in rows of ${\boldsymbol{M}}$, therefore $\operatorname{cone}({\boldsymbol{M}})$ equals the nonnegative orthant. It makes sense that this condition makes the identification problem easier, but it is also a very restrictive assumption. The sufficiently scattered condition, on the other hand, only requires that the shaded region contains the inner circle, as shown in Figure \[fig:geo-scattered\]. Intuitively this requires that the rows of ${\boldsymbol{M}}$ be “well scattered” in the probability simplex, but not to the extent of “separable”. Separability-based HMM identification has been considered in [@Barlier2015; @Glaude2015]. However, the way they construct second-order statistics is very different from ours. Figure \[fig:geo-no\_id\] shows a case where ${\boldsymbol{M}}$ is not sufficiently scattered, and it also happens to be a case where ${\boldsymbol{M}}$ is not identifiable.
As we can see, the elliptical cone ${\mathcal{C}}^*$ is tangent to all the facets of the nonnegative orthant. As a result, for ${\boldsymbol{M}}$ to be sufficiently scattered, it is necessary that there are enough rows of ${\boldsymbol{M}}$ lie on the boundary of the nonnegative orthant, i.e., ${\boldsymbol{M}}$ is relatively sparse. Specifically, if ${\boldsymbol{M}}$ is sufficiently scattered, then each column of ${\boldsymbol{M}}$ contains at least $K-1$ zeros [@huang2014tsp]. This is a very important insight, as exactly checking whether a matrix is sufficiently scattered may be computationally hard. In the present paper we further show the following result.
\[ppst:volume\] The ratio between the volume of the hyperball obtained by intersecting ${\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{x}}=1$ and ${\mathcal{C}}^*$ and the probability simplex is $$\label{eq:ratio}
\frac{1}{\sqrt{\pi K}}\left(\frac{4\pi}{K(K-1)}\right)^{\frac{K-1}{2}}
\Gamma\left(\frac{K}{2}\right).$$
The proof is given in the supplementary material. As $K$ grows larger, the volume ratio goes to zero at a super-exponential decay rate. This implies that the volume of the inner sphere quickly becomes negligible compared to the volume of the probability simplex, as $K$ becomes moderately large. The take home point is that, for a practical choice of $K$, say $K\geq10$, as long as ${\boldsymbol{M}}$ satisfies that each column contains at least $K$ zeros, and the positions of the zeros appear relatively random, it is very likely that it is sufficiently scattered, and thus can be uniquely recovered via solving .
Algorithm
=========
Our identifiability analysis based on the sufficiently scattered condition poses an interesting non-convex optimization problem . First of all, the given co-occurrence probability ${\boldsymbol{\varOmega}}$ may not be exact, therefore it may not be a good idea to put as a hard constraint. For algorithm design, we propose the following modification to problem . $$\begin{aligned}
\label{prob:alg}
\operatorname*{\textrm{minimize}}_{{\boldsymbol{\varTheta}},{\boldsymbol{M}}}~~~ &
\sum_{n,\ell=1}^{N}-\varOmega_{n\ell}\log\!\!\sum_{k,j=1}^{K}\!\!M_{nk}\varTheta_{kj}M_{\ell j}
+ \lambda|\det{\boldsymbol{\varTheta}}| \nonumber\\
\textrm{subject to}~~~ & {\boldsymbol{M}}\geq0, {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{M}}={\boldsymbol{\mathit{1}}}^{{\!\top\!}}, \\
& {\boldsymbol{\varTheta}}\geq0, {\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}={\boldsymbol{\varTheta}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}, {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}=1. \nonumber\end{aligned}$$ In the loss function of , the first term is the Kullback-Leibler distance between the empirical probability ${\boldsymbol{\varOmega}}$ and the parameterized version ${\boldsymbol{M}}{\boldsymbol{\varTheta}}{\boldsymbol{M}}^{{\!\top\!}}$ (ignoring a constant), and the second term is our identifiability-driven regularization. We need to tune the parameter $\lambda$ to yield good estimation results. However, intuitively we should use a $\lambda$ with a relatively small value. Suppose ${\boldsymbol{\varOmega}}$ is sufficiently accurate, then the priority is to minimize the difference between ${\boldsymbol{\varOmega}}$ and ${\boldsymbol{M}}{\boldsymbol{\varTheta}}{\boldsymbol{M}}^{{\!\top\!}}$; when there exist equally good fits, then the second term comes into play and helps us pick out a solution that is *sufficiently scattered*.
Noticing that the constraints of are all convex, but not the loss function, we propose to design an iterative algorithm to solve using successive convex approximation. At iteration $r$ when the updates are ${\boldsymbol{\varTheta}}^r$ and ${\boldsymbol{M}}^r$, we define $$\begin{aligned}
\label{eq:posterior}
\varPi_{n\ell kj}^r = M_{nk}^r\varTheta_{kj}^rM_{\ell j}^r \bigg/
\sum_{\kappa,\iota=1}^{K}M_{n\kappa}^r\varTheta_{\kappa\iota}^rM_{\ell\iota}^r.\end{aligned}$$ Obviously, $\varPi_{n\ell kj}^r\geq0$ and $\sum_{k,j=1}^{K}\varPi_{n\ell kj}^r=1$, which defines a probability distribution for fixed $n$ and $\ell$. Using Jensen’s inequality [@jensen1906fonctions], we have that $$\begin{aligned}
\label{eq:upper1}
-\varOmega_{n\ell}\log\sum_{k,j=1}^{K}M_{nk}\varTheta_{kj}M_{\ell j} \leq \sum_{k,j=1}^{K}-\varOmega_{n\ell}\varPi_{n\ell kj}^r
\left(\log M_{nk} + \log\varTheta_{kj} + \log M_{\ell j} - \log\varPi_{n\ell kj}^r\right)\end{aligned}$$ which defines a convex and locally tight upperbound for the first term in the loss function of . Regarding the second term in the loss of , we propose to simply take the linear approximation $$\begin{aligned}
\label{eq:approx2}
|\!\det\!{\boldsymbol{\varTheta}}| \approx |\!\det\!{\boldsymbol{\varTheta}}^r| +
|\!\det\!{\boldsymbol{\varTheta}}^r|\operatorname{Tr}\!\left( ({\boldsymbol{\varTheta}}^r)^{\!-\!1\!}({\boldsymbol{\varTheta}}\!-\!{\boldsymbol{\varTheta}}^r) \right) \end{aligned}$$
Combining and , our successive convex approximation algorithm tries to solve the following convex problem at iteration $r$: $$\begin{aligned}
\label{prob:iter}
\operatorname*{\textrm{minimize}}_{{\boldsymbol{\varTheta}},{\boldsymbol{M}}}~~ & \sum_{n,\ell=1}^{N}\sum_{k,j=1}^{K}
-\varOmega_{n\ell}\varPi_{n\ell kj}^r \left(\log M_{nk} + \log M_{\ell j} + \log\varTheta_{kj} \right)
+ \lambda\sum_{k,j=1}^{K}\varXi_{kj}^r\varTheta_{kj} \\
\textrm{subject to}~~ & {\boldsymbol{M}}\geq0, {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{M}}={\boldsymbol{\mathit{1}}}^{{\!\top\!}}, \nonumber \\
& {\boldsymbol{\varTheta}}\geq0, {\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}={\boldsymbol{\varTheta}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}, {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}=1, \nonumber\end{aligned}$$ where we define $\bm{\varXi}^r=|\det{\boldsymbol{\varTheta}}^r|({\boldsymbol{\varTheta}}^r)^{-{{\!\top\!}}}$. Problem decouples with respect to ${\boldsymbol{M}}$ and ${\boldsymbol{\varTheta}}$, so we can work out their updates individually.
The update of ${\boldsymbol{M}}$ admits a simple closed form solution, which can be derived via checking the KKT conditions. We denote the dual variable corresponding to ${\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{M}}={\boldsymbol{\mathit{1}}}^{{\!\top\!}}$ as ${\boldsymbol{\mu}}\in{\mathbf{R}}^K$. Setting the gradient of the Lagrangian with respect to $M_{nk}$ equal to zero, we have $$M_{nk} =
\sum_{\ell=1}^N\sum_{j=1}^{K}\left(\varOmega_{n\ell}\varPi_{n\ell kj}^r+\varOmega_{\ell n}\varPi_{\ell njk}^r\right) \bigg/ \mu_k$$ and ${\boldsymbol{\mu}}$ should be chosen so that the constraint ${\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{M}}={\boldsymbol{\mathit{1}}}^{{\!\top\!}}$ is satisfied, which amounts to a simple re-scaling.
The update of ${\boldsymbol{\varTheta}}$ is not as simple as a closed form expression, but it can still be obtained very efficiently. Noticing that the nonnegativity constraint is implicitly implied by the individual $\log$ functions in the loss function, we propose to solve it using Newton’s method with equality constraints [@boyd2004convex §10.2]. Although Newton’s method requires solving a linear system of equations with $K^2$ number of variables in each iteration, there is special structure we can exploit to reduce the per-iteration complexity down to $\mathcal{O}(K^3)$: The Hessian of the loss function of is diagonal, and the linear equality constraints are highly structured; using block elimination [@boyd2004convex §10.4.2], we ultimately only need to solve a positive definite linear system with $K$ variables. Together with the quadratic convergence rate of Newton’s method, the complexity of updating ${\boldsymbol{\varTheta}}$ is $\mathcal{O}(K^3\log\log\frac{1}{\varepsilon})$, where $\varepsilon$ is the desired accuracy for the ${\boldsymbol{\varTheta}}$ update. Noticing that the complexity of a naive implementation of Newton’s method would be $\mathcal{O}(K^6\log\log\frac{1}{\varepsilon})$, the difference is big for moderately large $K$. The in-line implementation of this tailored Newton’s method <span style="font-variant:small-caps;">ThetaUpdate</span> and the detailed derivation can be found in the supplementary material.
$\lambda>0$ initialize ${\boldsymbol{M}}$ using [@huang2016nips] initialize ${\boldsymbol{\varTheta}}\leftarrow \frac{1}{K(K+1)}({\boldsymbol{I}}+{\boldsymbol{\mathit{1}}}\!{\boldsymbol{\mathit{1}}}^{{\!\top\!}})$ $\widetilde{{\boldsymbol{\varOmega}}}\leftarrow{\boldsymbol{\varOmega}}\big/{\boldsymbol{M}}{\boldsymbol{\varTheta}}{\boldsymbol{M}}^{{{\!\top\!}}}$ $\widetilde{{\boldsymbol{M}}}\leftarrow {\boldsymbol{M}}\ast \left( \widetilde{{\boldsymbol{\varOmega}}}{\boldsymbol{M}}{\boldsymbol{\varTheta}}^{{{\!\top\!}}} + \widetilde{{\boldsymbol{\varOmega}}}^{{\!\top\!}}{\boldsymbol{M}}{\boldsymbol{\varTheta}}\right)$ $\widetilde{{\boldsymbol{\varTheta}}}\leftarrow{\boldsymbol{M}}^{{{\!\top\!}}}\widetilde{{\boldsymbol{\varOmega}}}{\boldsymbol{M}}$ $\widetilde{{\boldsymbol{M}}}\leftarrow \widetilde{{\boldsymbol{M}}} \operatorname{Diag}({\boldsymbol{\mathit{1}}}^{{\!\top\!}}\widetilde{{\boldsymbol{M}}})^{-1}$ $\widetilde{{\boldsymbol{\varTheta}}}\leftarrow$ <span style="font-variant:small-caps;">ThetaUpdate</span> $({\boldsymbol{M}},{\boldsymbol{\varTheta}})\leftarrow$ Amijo line search between $({\boldsymbol{M}},{\boldsymbol{\varTheta}})$ and $(\widetilde{{\boldsymbol{M}}},\widetilde{{\boldsymbol{\varTheta}}})$ ${\boldsymbol{M}}$ and ${\boldsymbol{\varTheta}}$
The entire proposed algorithm to solve Problem is summarized in Algorithm \[alg:hmm\_id\]. Notice that there is an additional line-search step to ensure decrease of the loss function. The constraint set of is convex, so the line-search step will not incur infeasibility. Computationally, we find that any operation that involves $\varPi_{n\ell kj}^r$ can be carried out succinctly by defining the intermediate matrix $\widetilde{{\boldsymbol{\varOmega}}}={\boldsymbol{\varOmega}}/{\boldsymbol{M}}{\boldsymbol{\varTheta}}{\boldsymbol{M}}^{{{\!\top\!}}}$, where “$/$” denotes element-wise division between two matrices of the same size. The per-iteration complexity of Algorithm \[alg:hmm\_id\] is completely dominated by the operations that involve computing with $\widetilde{{\boldsymbol{\varOmega}}}$, notably comparing with that of <span style="font-variant:small-caps;">Theta-Update</span>. In terms of initialization, which is important since we are optimizing a non-convex problem, we propose to use the method by @huang2016nips to obtain an initialization for ${\boldsymbol{M}}$; for ${\boldsymbol{\varTheta}}$, it is best if we start with a feasible point (so that the Newton’s iterates will remain feasible), and a simple choice is scaling the matrix ${\boldsymbol{I}}+{\boldsymbol{\mathit{1}}}\!{\boldsymbol{\mathit{1}}}^{{\!\top\!}}$ to sum up to one. Finally, we show that this algorithm converges to a stationary point of Problem , with proof relegated to the supplementary material based on [@razaviyayn2013unified].
Assume <span style="font-variant:small-caps;">ThetaUpdate</span> solves Problem with respect to ${\boldsymbol{\varTheta}}$ exactly, then Algorithm \[alg:hmm\_id\] converges to a stationary point of Problem .
![The total variation difference between the ground truth and estimated transition probability (top) and emission probability (bottom). The total variation difference of the emission probabilities is calculated as $\frac{1}{2K}\|{\boldsymbol{M}}_\natural-{\boldsymbol{M}}_\star\|_1$, since each column of the matrices indicates a (conditional) probability, and the total variation difference is equal to one half of the $L_1$-norm; and similarly for that of the transition probabilities after rescaling the rows of ${\boldsymbol{\varOmega}}_\natural$ and ${\boldsymbol{\varOmega}}_\star$ to sum up to one. The result is averaged over 10 random problem instances.[]{data-label="fig:sim_exp1"}](sim_exp1){width="\linewidth"}
Validation on Synthetic Data
============================
In this section we validate the identifiability performance on synthetic data. In this case, the underlying transition and emission probabilities are generated synthetically, and we compare them with the estimated ones to evaluate performance. The simulations are conducted in MATLAB using the HMM toolbox, which includes functions to generate observation sequences given transition and emission probabilities, as well as an implementation of the Baum-Welch algorithm [@baum1970maximization], i.e., the EM algorithm, to estimate the transition and emission probabilities using the observations. Unfortunately, even for some moderate problem sizes we considered, the streamlined MATLAB implementation of the Baum-Welch algorithm was not able to execute within reasonable amount of time, so its performance is not included here. For the baselines, we compare with the plain NMF approach using multiplicative update [@Vanluyten2008] and the tensor CPD approach [@Sharan2017] using simultaneous diagonalization with Tensorlab [@tensorlab3.0]. Since we work with empirical distributions instead of exact probabilities, the result of the simultaneous diagonalization is not going to be optimal. We therefore use it to initialize the EM algorithm for fitting a nonnegative tensor factorization with KL divergence loss [@shashanka2008probabilistic] for refinement.
We focus on the cases when the number of hidden states $K$ is smaller than the number observed states $N$. As we explained in the introduction, even for this seemingly easier case, it is not known that we can guarantee unique recovery of the HMM parameters *just from the pair-wise co-occurrence probability*. What is known is that the tensor CPD approach is able to guarantee identifiability given exact triple-occurrence probability. We will demonstrate in this section that it is much harder to obtain accurate triple-occurrence probability comparing with the co-occurrence probability. As a result, if the sufficiently scattered assumption holds for the emission probability, the estimated parameters obtained from our method are always more accurate than those obtained from tensor CPD.
Fixing $N=100$ and $K=20$, the transition probabilities are synthetically generated from a random exponential matrix of size $K\times K$ followed by row-normalization; for the emission probabilities, approximately 50% of the entries in the $N\times K$ random exponential matrices are set to zero before normalizing the columns, which is shown to satisfy the sufficiently scattered condition with very high probability [@huang2015principled]. We let the number of HMM realizations go from $10^6$ to $10^8$, and compare the estimation error for the transition matrix and emission matrix by the aforementioned methods. We show the total variation distance between the ground truth probabilities $\Pr[X_{t+1}|X_t]$ and $\Pr[Y_t|X_t]$ and their estimations $\widehat{\Pr}[X_{t+1}|X_t]$ and $\widehat{\Pr}[Y_t|X_t]$ using various methods. The result is shown in Figure \[fig:sim\_exp1\]. As we can see, the proposed method indeed works best, obtaining almost perfect recovery when sample size is above $10^8$. The CPD based method does not work as well since it cannot obtain accurate estimates of the third-order statistics that it needs. Initialized by CPD, EM improves from CPD but the performance is still far away from the proposed method. NMF is not working well since it does not have identifiability in this case.
Application: Hidden Topic Markov Model
======================================
Analyzing text data is one of the core application domains of machine learning. There are two prevailing approaches to model text data. The classical bag-of-words model assumes that each word is *independently* drawn from certain multinomial distributions. These distributions are different across documents, but can be efficiently summarized by a small number of *topics*, again mathematically modeled as distributions over words; this task is widely known as *topic modeling* [@hofmann2001unsupervised; @blei2003latent]. However, it is obvious that the bag-of-words representation is oversimplified. The $n$-gram model, on the other hand, assumes that words are conditionally dependent up to a window-length of $n$. This seems to be a much more realistic model, although the choice of $n$ is totally unclear, and is often dictated by memory and computational limitations in practice—since the size of the joint distribution grows exponentially with $n$. What is more, it is somewhat difficult to extract “topics” from this model, despite some preliminary attempts [@wallach2006topic; @wang2007topical].
We propose to model a document as the realization of a HMM, in which the topics are hidden states emitting words, and the states are evolving according to a Markov chain, hence the name *hidden topic Markov model* (HTMM). For a set of documents, this means we are working with a *collection* of HMMs. Similar to other topic modeling works, we assume that the topic matrix is shared among all documents, meaning all the given HMMs share the same emission probability. For the bag-of-words model, each document has a specific topic distribution ${\boldsymbol{p}}_d$, whereas for our model, each document has its own *topic transition probability* ${\boldsymbol{\varTheta}}_d$; as per our previous discussion, the row-sum and column-sum of ${\boldsymbol{\varTheta}}_d$ are the same, which are also the topic probability for the specific document. The difference is the Markovian assumption on the topics rather than the over-simplifying independence assumption.
We can see some immediate advantages for the HTMM. Since the Markovian assumption is only imposed on the topics, which are not exposed to us, the observations (words) are not independent from each other, which agrees with our intuition. On the other hand, we now understand that although word dependencies exist for a wide neighborhood, we only need to work with pair-wise co-occurrence probabilities, or 2-grams. This releases us from picking a window length $n$ in the $n$-gram model, while maintaining dependencies between words well beyond a neighborhood of $n$ words. It also includes the bag-of-words assumption as a special case: If the topics of the words are indeed independent, this just means that the transition probability has the special form ${\boldsymbol{\mathit{1}}}{\boldsymbol{p}}_d^{{\!\top\!}}$. The closest work to ours is by @gruber2007hidden, which is also termed hidden topic Markov model. However, they make a simplifying assumption that the transition probability takes the form $(1-\epsilon){\boldsymbol{I}}+ \epsilon{\boldsymbol{\mathit{1}}}{\boldsymbol{p}}_d^{{\!\top\!}}$, meaning the topic of the word is either the same as the previous one, or independently drawn from ${\boldsymbol{p}}_d$. Both models are special cases of our general HTMM.
In order to learn the shared topic matrix ${\boldsymbol{M}}$, we can use the co-occurrence statistics for the entire corpus: Denote the co-occurrence statistics for the $d$-th document as ${\boldsymbol{\varOmega}}_d$, then $\operatorname{E}{\boldsymbol{\varOmega}}_d = {\boldsymbol{M}}{\boldsymbol{\varTheta}}_d{\boldsymbol{M}}^{{\!\top\!}}$; consequently $${\boldsymbol{\varOmega}}= \frac{1}{\sum_{d=1}^{D}L_d}\sum_{d=1}^{D}L_d{\boldsymbol{\varOmega}}_d,$$ which is an unbiased estimator for $${\boldsymbol{M}}{\boldsymbol{\varTheta}}{\boldsymbol{M}}^{{\!\top\!}}= \frac{1}{\sum_{d=1}^{D}L_d}\sum_{d=1}^{D}L_d{\boldsymbol{M}}{\boldsymbol{\varTheta}}_d{\boldsymbol{M}}^{{\!\top\!}},$$ where $L_d$ is the length of the $d$-th document and ${\boldsymbol{\varTheta}}$ is conceptually a weighted average of all the topic-transition matrices. Then we may apply Algorithm \[alg:hmm\_id\] to learn the topic matrix. We illustrate the performance of our HTMM by comparing it to three popular bag-of-words topic modeling approaches: pLSA [@hofmann2001unsupervised], LDA [@blei2003latent], and FastAnchor [@arora2013practical], which guarantees identifiability if every topic has a characteristic *anchor word*. Our HTMM model guarantees identifiability if the topic matrix is *sufficiently scattered*, which is a more relaxed condition than the anchor word one. On the Reuters21578 data set obtained at [@reuters21578], we use the raw document to construct the word co-occurrence statistics, as well as bag-of-words representations for each document for the baseline algorithms. We use the version in which the stop-words have been removed, which makes the HTMM model more plausible since any syntactic dependencies have been removed, leaving only semantic dependencies. The vocabulary size of Reuters21578 is around $200,000$, making any method relying on triple-occurrences impossible to implement, and that is why tensor-based methods are not compared here.
![Coherence of the topics.[]{data-label="fig:reuters_coh"}](reuters_coh){width="0.55\linewidth"}
Because of page limitations, we only show the quality of the topics learned by various methods in terms of coherence. Simply put, a higher coherence means more meaningful topics, and the concrete definition can be found in [@arora2013practical] and in the supplementary material. In Figure \[fig:reuters\_coh\], we can see that for different number of topics we tried on the entire dataset, HTMM consistently produces topics with the highest coherence. Additional evaluations can be found in the supplementary material.
Conclusion
==========
We presented an algorithm for learning hidden Markov models in an unsupervised setting, i.e., using only a sequence of observations. Our approach is guaranteed to uniquely recover the ground-truth HMM structure using only pairwise co-occurrence probabilities of the observations, under the assumption that the emission probability is *sufficiently scattered*. Unlike EM, the complexity of the proposed algorithm does not grow with the length of the observation sequence. Compared to tensor-based methods for HMM learning, our approach only requires reliable estimates of pairwise co-occurrence probabilities, which are easier to obtain. We applied our method to topic modeling, assuming each document is a realization of a HMM rather than a simpler bag-of-words model, and obtained improved topic coherence results. We refer the reader to the supplementary material for detailed proofs of the propositions and additional experimental results.
Appendix {#appendix .unnumbered}
========
Proof of Proposition 1
======================
For categorical probabilities $\bm{p}$ and $\bm{q}$, their Kullback-Leiber divergence is defined as $$D_\textup{KL}(\bm{p}\|\bm{q}) = \sum_{n=1}^{N} p_n\log\frac{p_n}{q_n},$$ and their total variation distance is defined as $$D_\textup{TV}(\bm{p}\|\bm{q}) = \frac{1}{2}\sum_{n=1}^{N} |p_n-q_n|.$$
The key to prove Proposition 1 is the fact that the cooccurrence probability ${\boldsymbol{\varOmega}}$ can be obtained by marginalizing $X_{t-1}$ in the triple-occurrence probability $\underline{{\boldsymbol{\varOmega}}_3}$, i.e., $${\boldsymbol{\varOmega}}(i,j) = \sum_{n=1}^{N}\underline{{\boldsymbol{\varOmega}}_3}(n,i,j).$$ Similarly, this holds for the cumulative estimates described in §2 of the main paper as well, $$\widehat{{\boldsymbol{\varOmega}}}(i,j) = \sum_{n=1}^{N}\widehat{\underline{{\boldsymbol{\varOmega}}_3}}(n,i,j).$$
Using the log sum inequality, we have that $${\boldsymbol{\varOmega}}(i,j)\log\frac{{\boldsymbol{\varOmega}}(i,j)}{\widehat{{\boldsymbol{\varOmega}}}(i,j)} \leq
\sum_{n=1}^{N}\underline{{\boldsymbol{\varOmega}}_3}(n,i,j)
\log\frac{\underline{{\boldsymbol{\varOmega}}_3}(n,i,j)}{\widehat{\underline{{\boldsymbol{\varOmega}}_3}}(n,i,j)}.$$ Summing both sides over $i$ and $j$, we result in $$D_\textup{KL}(\widehat{{\boldsymbol{\varOmega}}}\|{\boldsymbol{\varOmega}}) \leq
D_\textup{KL}(\widehat{\underline{{\boldsymbol{\varOmega}}_3}}\|\underline{{\boldsymbol{\varOmega}}_3})$$
Using Hölder’s inequality with $L_1$-norm and $L_\infty$-norm, we have that $$|{\boldsymbol{\varOmega}}(i,j)-\widehat{{\boldsymbol{\varOmega}}}(i,j)| \leq
\sum_{n=1}^{N}|\underline{{\boldsymbol{\varOmega}}_3}(n,i,j) - \widehat{\underline{{\boldsymbol{\varOmega}}_3}}(n,i,j)|.$$ Summing both sides over $i$ and $j$ and then dividing by 2, we obtain $$D_\textup{TV}(\widehat{{\boldsymbol{\varOmega}}}\|{\boldsymbol{\varOmega}}) \leq
D_\textup{TV}(\widehat{\underline{{\boldsymbol{\varOmega}}_3}}\|\underline{{\boldsymbol{\varOmega}}_3})$$ **Q.E.D.**
Proof of Proposition 2
======================
The volume of a hyper-ball in ${\mathbf{R}}^{n}$ with radius $R$ is $$\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}R^{n}.$$ The elliptical cone ${\mathcal{C}}^*=\{ {\boldsymbol{x}}:\|{\boldsymbol{x}}\|\leq{\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{x}}/\sqrt{K-1} \}$ intersecting with the hyperplane ${\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{x}}=1$ is a hyperball in ${\mathbf{R}}^{K-1}$ with radius $\sqrt{\frac{1}{K(K-1)}}$. Therefore, the volume of the inner-ball is $$V_b = \frac{\pi^{\frac{K-1}{2}}}{\Gamma(\frac{K+1}{2})}(K(K-1))^{-\frac{K-1}{2}}.$$
The nonnegative orthan intersecting with ${\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{x}}=1$ is a regular simplex in ${\mathbf{R}}^{K-1}$ with side length $\sqrt{2}$. Its volume is $$V_s = \frac{\sqrt{K}}{(K-1)!} = \frac{\sqrt{K}}{\Gamma(K)}.$$
Their ratio is $$\begin{aligned}
\frac{V_b}{V_s}
&= \frac{\frac{\pi^{\frac{K-1}{2}}}{\Gamma(\frac{K+1}{2})}(K(K-1))^{-\frac{K-1}{2}}}
{\frac{\sqrt{K}}{\Gamma(K)}} \\
&= \frac{1}{\sqrt{K}}\left(\frac{\pi}{K(K-1)}\right)^{\frac{K-1}{2}}
\frac{\Gamma(K)}{\Gamma(\frac{K+1}{2})}\\
&= \frac{1}{\sqrt{K}}\left(\frac{\pi}{K(K-1)}\right)^{\frac{K-1}{2}}
\frac{\Gamma(\frac{K}{2})}{2^{1-K}\sqrt{\pi}}\\
&= \frac{1}{\sqrt{\pi K}}\left(\frac{4\pi}{K(K-1)}\right)^{\frac{K-1}{2}}
\Gamma\left(\frac{K}{2}\right)\end{aligned}$$ **Q.E.D.**
This function of volume ratio is plotted in Figure \[fig:vol\_ratio\]. As we can see, as $K$ increases, the volume ratio indeed goes to zero at a super-exponential rate.
![The volume ratio between the hyperball obtained by intersecting $\mathcal{C}$ and the hyperplane ${\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{x}}=1$ and the probability simplex, as $K$ increases.[]{data-label="fig:vol_ratio"}](vol_ratio){width=".5\linewidth"}
Derivation of <span style="font-variant:small-caps;">ThetaUpdate</span>
=======================================================================
It is described in [@boyd2004convex §10.2] that for solving a convex equality constrained problem $$\begin{aligned}
\operatorname*{\textrm{minimize}}_{x}~ & f(x) \\
\text{subject to}~ & Ax=b\end{aligned}$$ using the Newton’s method, we start at a feasible point $x$, and the iterative update takes the form $x\leftarrow x - \alpha\Delta_\text{nt}x$, where the Newton direction is calculated from solving the KKT system $$\begin{bmatrix}
\nabla^2f(x) & A^{{\!\top\!}}~ \\ A & 0
\end{bmatrix}
\begin{bmatrix}
\Delta_\text{nt}x \\ d
\end{bmatrix} =
\begin{bmatrix}
-\nabla f(x) \\ 0
\end{bmatrix}.$$
Assuming $\nabla^2f(x)\succ0$ and $A$ has full row rank, then the KKT system can be solved via elimination, as described in [@boyd2004convex Algorithm 10.3]. Suppose $A\in{\mathbf{R}}^{m\times n}$, if $\nabla^2f(x)$ is diagonal, the cost of calculating $\Delta_\text{nt}x$ is dominated by forming and inverting the matrix $ADA^{{\!\top\!}}$ with $D$ being diagonal.
Now we follow the steps of [@boyd2004convex Algorithm 10.3] to derive explicit Newton iterates for solving (11). First, we re-write the part of (11) that involve ${\boldsymbol{\varTheta}}$ here: $$\begin{aligned}
\operatorname*{\textrm{minimize}}_{{\boldsymbol{\varTheta}}>0}~~ & \sum_{n,\ell=1}^{N}\sum_{k,j=1}^{K}
-\varOmega_{n\ell}\varPi_{n\ell kj}^r \log\varTheta_{kj} + \lambda\sum_{k,j=1}^{K}\varXi_{kj}^r\varTheta_{kj} \\
\textrm{subject to}~~ & {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}=1, {\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}={\boldsymbol{\varTheta}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}.\end{aligned}$$
Let ${\boldsymbol{\theta}}=\operatorname{vec}({\boldsymbol{\varTheta}})$, then equality constraint has the form $\bm{A}{\boldsymbol{\theta}}=\bm{b}$ where $$\bm{A} = \begin{bmatrix}
{\boldsymbol{\mathit{1}}}^{{\!\top\!}}\otimes{\boldsymbol{\mathit{1}}}^{{\!\top\!}}\\
{\boldsymbol{\mathit{1}}}^{{\!\top\!}}\otimes{\boldsymbol{I}}- {\boldsymbol{I}}\otimes{\boldsymbol{\mathit{1}}}^{{\!\top\!}}\end{bmatrix}.$$ Matrix $\bm{A}$ does not have full row rank, because the last row of $\bm{A}$ is implied by the rest. Therefore, we can discard the last equality constraint. We will keep it when calculating matrix multiplications for simpler expression, and discard the corresponding entry or column/row for other operations.
Obviously $\bm{A\theta}$ has the form $$\bm{A\theta} =
\begin{bmatrix}
{\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}\\
{\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}- {\boldsymbol{\varTheta}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}\end{bmatrix},$$ which costs $\mathcal{O}(K^2)$ flops. For a slightly more complicated multiplication $$\bm{A}\operatorname{Diag}({\boldsymbol{\theta}})\bm{A}^{{\!\top\!}}= \begin{bmatrix}
{\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}& {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{R}}^{{\!\top\!}}- {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{\varTheta}}\\
~{\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}- {\boldsymbol{\varTheta}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}& \operatorname{Diag}({\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}+ {\boldsymbol{\varTheta}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}) - {\boldsymbol{\varTheta}}- {\boldsymbol{\varTheta}}^{{\!\top\!}}~
\end{bmatrix},$$ which also costs $\mathcal{O}(K^2)$ flops to compute. For $[~d_0~\bm{d}^{{\!\top\!}}~]^{{\!\top\!}}\in{\mathbf{R}}^{K+1}$, $$\bm{A}^{{\!\top\!}}[~d_0~\bm{d}^{{\!\top\!}}~]^{{\!\top\!}}=
\operatorname{vec}\left(d_0{\boldsymbol{\mathit{1}}}{\boldsymbol{\mathit{1}}}^{{\!\top\!}}+ \bm{d}{\boldsymbol{\mathit{1}}}^{{\!\top\!}}- {\boldsymbol{\mathit{1}}}\bm{d}^{{\!\top\!}}\right).$$
At point ${\boldsymbol{\theta}}$, the negative gradient is $-\nabla f({\boldsymbol{\theta}}) = \operatorname{vec}(\bm{G})$ where $$G_{kj} = \frac{\sum_{n,\ell=1}^{N}\varOmega_{n\ell}\varPi_{n\ell kj}^r}{\varTheta_{kj}}
- \lambda\varXi_{kj}^r,$$ and the inverse of the Hessian $\left( \nabla^2f({\boldsymbol{\theta}}) \right)^{-1} = \operatorname{Diag}(\operatorname{vec}(\bm{R}))$ where $$R_{kj} = \frac{\varTheta_{kj}^2}
{\sum_{n,\ell=1}^{N}\varOmega_{n\ell}\varPi_{n\ell kj}^r}.$$
Let $${\boldsymbol{H}}= \begin{bmatrix}
{\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{R}}{\boldsymbol{\mathit{1}}}& {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{R}}^{{\!\top\!}}- {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{R}}\\
~{\boldsymbol{R}}{\boldsymbol{\mathit{1}}}- {\boldsymbol{R}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}& \operatorname{Diag}({\boldsymbol{R}}{\boldsymbol{\mathit{1}}}+ {\boldsymbol{R}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}) - {\boldsymbol{R}}- {\boldsymbol{R}}^{{\!\top\!}}~
\end{bmatrix}$$ and then delete the last column and row of ${\boldsymbol{H}}$, and $$S_{kj} = R_{kj} G_{kj}$$ $${\boldsymbol{g}}= \begin{bmatrix}
{\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{S}}{\boldsymbol{\mathit{1}}}\\ ~{\boldsymbol{S}}{\boldsymbol{\mathit{1}}}- {\boldsymbol{S}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}~
\end{bmatrix}$$ and then delete the last entry of ${\boldsymbol{g}}$. We can first solve for $\bm{d}$ by $$\bm{d} = {\boldsymbol{H}}^{-1}{\boldsymbol{g}}= [~d_0~\widetilde{\bm{d}}^{~{{\!\top\!}}}~]^{{\!\top\!}}.$$ Then we append a zero at the end of $\bm{d}$ and define $$[~\bm{d}^{{\!\top\!}}~0~]^{{\!\top\!}}\rightarrow \bm{d} = [~d_0~\widetilde{\bm{d}}^{\,{{\!\top\!}}}~]^{{\!\top\!}}.$$
The Newton direction $\Delta_\text{nt}{\boldsymbol{\theta}}$ can then be obtained via $$\Delta_\text{nt}{\boldsymbol{\theta}}= \left(\nabla^2f({\boldsymbol{\theta}})\right)^{-1}
\left( \bm{A}^{{\!\top\!}}\bm{d} + \nabla f({\boldsymbol{\theta}}) \right).$$
In matrix form, it is equivalent to $$\Delta_\text{nt}{\boldsymbol{\varTheta}}= {\boldsymbol{R}}\ast \left(d_0{\boldsymbol{\mathit{1}}}\!{\boldsymbol{\mathit{1}}}^{{\!\top\!}}+ \widetilde{\bm{d}}{\boldsymbol{\mathit{1}}}^{{\!\top\!}}- {\boldsymbol{\mathit{1}}}\widetilde{\bm{d}}^{~{{\!\top\!}}} - \bm{G}\right).$$
The in-line implementation of <span style="font-variant:small-caps;">ThetaUpdate</span> is given here.
${\boldsymbol{\varTheta}},\widetilde{{\boldsymbol{\varTheta}}},\lambda,\rho$ ${\boldsymbol{\varXi}}\leftarrow|\det{\boldsymbol{\varTheta}}|{\boldsymbol{\varTheta}}^{-{{\!\top\!}}}$ ${\boldsymbol{G}}\leftarrow \widetilde{{\boldsymbol{\varTheta}}}\big/{\boldsymbol{\varTheta}}- \lambda{\boldsymbol{\varXi}}$ ${\boldsymbol{R}}\leftarrow {\boldsymbol{\varTheta}}\ast{\boldsymbol{\varTheta}}\big/\widetilde{{\boldsymbol{\varTheta}}}$ $\displaystyle {\boldsymbol{H}}\leftarrow \begin{bmatrix}
{\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{R}}{\boldsymbol{\mathit{1}}}& {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{R}}^{{\!\top\!}}\!-\! {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{R}}\\
~{\boldsymbol{R}}{\boldsymbol{\mathit{1}}}\!-\! {\boldsymbol{R}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}& \operatorname{Diag}({\boldsymbol{R}}{\boldsymbol{\mathit{1}}}\!+\! {\boldsymbol{R}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}) \!-\! {\boldsymbol{R}}\!-\! {\boldsymbol{R}}^{{\!\top\!}}~
\end{bmatrix}$ delete the last column and row of ${\boldsymbol{H}}$ $\displaystyle {\boldsymbol{g}}\leftarrow\begin{bmatrix}
{\boldsymbol{\mathit{1}}}^{{\!\top\!}}({\boldsymbol{R}}\ast{\boldsymbol{G}}){\boldsymbol{\mathit{1}}}\\ ~({\boldsymbol{R}}\ast{\boldsymbol{G}}){\boldsymbol{\mathit{1}}}- ({\boldsymbol{R}}\ast{\boldsymbol{G}})^{{\!\top\!}}{\boldsymbol{\mathit{1}}}~
\end{bmatrix}$ delete the last entry of ${\boldsymbol{g}}$ $\bm{d} \leftarrow {\boldsymbol{H}}^{-1}{\boldsymbol{g}}$ $[~d_0~\widetilde{\bm{d}}^{~{{\!\top\!}}}~]^{{\!\top\!}}\leftarrow[~\bm{d}^{{\!\top\!}}~0~]^{{\!\top\!}}$ $
\Delta_\text{nt}{\boldsymbol{\varTheta}}= {\boldsymbol{R}}\ast \left(d_0{\boldsymbol{\mathit{1}}}\!{\boldsymbol{\mathit{1}}}^{{\!\top\!}}+ \widetilde{\bm{d}}{\boldsymbol{\mathit{1}}}^{{\!\top\!}}- {\boldsymbol{\mathit{1}}}\widetilde{\bm{d}}^{~{{\!\top\!}}} - \bm{G}\right)
$ ${\boldsymbol{\varTheta}}\leftarrow{\boldsymbol{\varTheta}}-\Delta_\text{nt}{\boldsymbol{\varTheta}}$ ${\boldsymbol{\varTheta}}$
Proof of Proposition 3
======================
The form of Algorithm 1 falls exactly into the framework of block successive convex approximation (BSCA) algorithm proposed by [@razaviyayn2013unified] with only one block of variables. Invoking [@razaviyayn2013unified Theorem 4], we have that every limit point of Algorithm 1 is a stationary point of Problem (7). Additionally, since the constraint set of Problem (7) is compact, *any* sub-sequence has a limit point, which is also a stationary point. This proves that Algorithm 1 converges to a stationary point of Problem (7). **Q.E.D.**
Additional Synthetic Experiments
================================
In this section we conduct a similar synthetic experiment to identify HMM parameters, but with a much smaller problem size, so that we can include the classical Baum-Welch algorithm [@baum1970maximization] as another baseline. Fixing $N=16$ and $K=4$, the transition probabilities are synthetically generated from a random exponential matrix of size $K\times K$ followed by row-normalization; for the emission probabilities, the top $K\times K$ part of the $N\times K$ random exponential matrices are set to be the identity matrix before column normalization, so that it is guaranteed to be sufficiently scattered. We let the number of HMM realizations go from $10^3$ to $10^5$, and compare the estimation error for the transition matrix and emission matrix by the aforementioned methods. We show the total variation distance between the ground truth probabilities $\Pr[X_{t+1}|X_t]$ and $\Pr[Y_t|X_t]$ and their estimations $\widehat{\Pr}[X_{t+1}|X_t]$ and $\widehat{\Pr}[Y_t|X_t]$ using various methods. The result is shown in Figure \[fig:suppl\_small\].
Similar to the experiment shown in the main paper, the proposed method works the best in terms of estimating the HMM parameters, without sacrificing too much computational times. Much to one’s surprise, the Baum-Welch algorithm is not working very well in terms of estimation error. This is possibly because we limit the maximum number of EM iterations to be 500 (default setting of the MATLAB implementation), which may not be enough for convergence. What is expected is that the computational time of Baum-Welch grows linearly with respect to the length of the HMM observations, while other methods are independent from it.
![The total variation difference between the ground truth and estimated transition probability (left) and emission probability (middle), and the elapsed time (right) for $N=16$ and $K=4$. The total variation difference of the emission probabilities is calculated as $\frac{1}{2K}\|{\boldsymbol{M}}_\natural-{\boldsymbol{M}}_\star\|_1$, since each column of the matrices indicates a (conditional) probability, and the total variation difference is equal to one half of the $L_1$-norm; and similarly for that of the transition probabilities after rescaling the rows of ${\boldsymbol{\varOmega}}_\natural$ and ${\boldsymbol{\varOmega}}_\star$ to sum up to one. The result is averaged over 10 random problem instances.[]{data-label="fig:suppl_small"}](suppl_small){width="1.3\linewidth"}
![The elapsed time for the synthetic experiment with $N=100$ and $K=20$ as in the main paper.[]{data-label="fig:suppl_time"}](suppl_time){width=".5\linewidth"}
An interesting remark is that when $T=12,800$, the per-iteration elapsed time of Baum-Welch is approximately 1 second. Recall that each iteration of Baum-Welch calls for the forward-backward algorithm, with complexity $\mathcal{O}(K^2T)$. This means for the problem size considered in the main paper, each iteration of Baum-Welch takes approximately 4 minutes to 7 hours, depending on the realization length. This is clearly not feasible in practice.
We also present the elapsed time of the four algorithms excluding the Baum-Welch algorithm for the case considered in the main paper, i.e., $N=100$ and $K=20$. Similar to the timing result shown in Figure \[fig:suppl\_small\], the proposed method takes the longest time compared to the other three, but not significantly; also recall that the propose method works considerably better in terms of estimation accuracy.
Additional HTMM Evaluations
===========================
In the main body of the paper we showed that HTMM is able to learn topics with higher quality using pairwise word cooccurrences. The quality of topics is evaluated using coherence, which is defined as follows. For each topic, a set of words $\mathcal{V}$ is picked (here we pick the top 20 words with the highest probability of appearing). We calculate the number of documents each word $v_1$ appears $\text{freq}(v_1)$ and the number of documents two words $v_1$ and $v_2$ both appear $\text{freq}(v_1,v_2)$. The coherence of that topic is calculated as $$\sum_{v_1,v_2\in\mathcal{V}} \log\left(\frac{\text{freq}(v_1,v_2)+\epsilon}{\text{freq}(v_1)}\right) .$$ The intuition is that if both $v_1$ and $v_2$ both have high probability of appearing in a topic, then they have high probability of co-occurring in a document as well; hence a higher value of coherence indicates a more indicative topic. The coherence of the individual topics are then averaged to get the coherence for the entire topic matrix.
![The perplexity of different models as number of topics $K$ increases.[]{data-label="fig:perp"}](reuters_perp){width=".5\linewidth"}
Here we show some more evaluation results. Using the learned topic matrix, we can see how it fits the data directly from *perplexity*, defined as [@blei2003latent] $$\exp\left( -\frac{\sum_d\log p(\text{doc}_d)}{\sum_d L_d} \right).$$ A smaller perplexity means the probability model fits the data better. As seen in Figure \[fig:perp\], HTMM gives the smallest perplexity. Notice that since HTMM takes word ordering into account, it is not fair for the other methods to take the bag-of-words representation of the documents. The bag-of-words model is essentially multinomial, whose pdf includes a scaling factor $\frac{n!}{n_1!...n_K!}$ for different combinations of observation orderings. In our case this factor is not included since we *do* know the word ordering in each document. For HTMM the log-likelihood is calculated efficiently using the forward algorithm.
This result is not surprising. Even using the same topic matrix, a bag-of-words model tries to find a $K$-dimensional representation for each document, whereas HTMM looks for a $K^2$-dimension representation. One may wonder if it is causing over-fitting, but we argue that it is not. First of all, we have see that in terms of coherence, HTMM learns a topic matrix with higher quality. For learning feature representations for each document, we showcase the following result. Once we have the topic-word probabilities and topic weights or topic transition probability, we can infer the underlying topic for each word. For bag-of-words models, each word only has one most probable topic in a document, no matter where it appears. For HTMM, once we learn the transition and emission probability, the topic of each word can be optimally estimated using the Viterbi algorithm. For one specific news article from the Reuters21578 data set, the topic inference given by pLSA is:
The word topic inference given by HTMM is:
As we can see, HTMM gets much more consistent and smooth inferred topics, which agrees with human understandings.
Learning HMMs from Triple-occurrences
=====================================
Finally, we show a stronger identifiability result for learning HMMs using triple-occurrence probabilities.
Consider a HMM with $K$ hidden states and $N$ observable states. Suppose the emission probability $\Pr[Y_t|X_t]$ is generic (meaning probabilities not satisfying this condition form a set with Lebesgue measure zero), the transition probabilities $\Pr[X_{t+1}|X_t]$ are linearly independent from each other, and each conditional probability $\Pr[X_{t+1}|X_t]$ contains no more than $N/2$ nonzeros. Then this HMM can be uniquely identified from its triple-occurrence probability $\Pr[Y_{t-1},Y_t,Y_{t+1}]$, up to permutation of the hidden states, for $K\leq \frac{N^2}{16}$.
It is clear that identifiability holds when $K\leq N$, so we focus on the case that $N<K\leq \frac{N^2}{16}$.
As we explained in §1.1, the triple-occurrence probability can be factored into $$\Pr[Y_{t-1},Y_t,Y_{t+1}] = \sum_{k=1}^{K}\Pr[X_t=x_k]\Pr[Y_{t-1}|X_t=x_k]
\Pr[Y_{t}|X_t=x_k]\Pr[Y_{t+1}|X_t=x_k].$$ Using tensor notations, this is equivalent to $$\underline{{\boldsymbol{\varOmega}}_3} = {\text{\textlbrackdbl}\bm{p};\bm{L},{\boldsymbol{M}},\bm{N}\text{\textrbrackdbl}},$$ where $$\begin{aligned}
p_k & = \Pr[X_t=x_k], \\
L_{nk} &= \Pr[Y_{t-1}=y_n|X_t=x_k], \\
N_{nk} &= \Pr[Y_{t+1}=y_n|X_t=x_k].\end{aligned}$$ Let $\overline{{\boldsymbol{\varTheta}}}$ denote the row scaled version of ${\boldsymbol{\varTheta}}$ so that each row sums to one, then $\overline{{\boldsymbol{\varTheta}}}$ denotes the transition probability. Then we have $$\label{eq:L=MT}
\bm{L}={\boldsymbol{M}}\overline{{\boldsymbol{\varTheta}}}^{{\!\top\!}}.$$
Since ${\boldsymbol{M}}$ is generic and $\overline{{\boldsymbol{\varTheta}}}$ is full rank, both $\bm{L}$ and $\bm{N}$ are generic as well. The latest tensor identifiability result by @chiantini2012generic [Theorem 1.1] shows that for a $N\times N\times N$ tensor with generic factors, the CPD $\underline{{\boldsymbol{\varOmega}}_3} = {\text{\textlbrackdbl}\bm{p};\bm{L},{\boldsymbol{M}},\bm{N}\text{\textrbrackdbl}}$ is essentially unique if $$K\leq 2^{ 2\lfloor\log_2N\rfloor - 2 },$$ or with a slightly worse bound $$K\leq \frac{N^2}{16}.$$
This does not mean that any non-singular $\overline{{\boldsymbol{\varTheta}}}$ can be uniquely recovered in this case. Equation is under-determined. A natural assumption to achieve identifiability is that each row of $\overline{{\boldsymbol{\varTheta}}}$, i.e., each conditional transition probability $\Pr[X_{t+1}|X_t]$, can take at most $N/2$ non-zeros. In the context of HMM, this means that at a particular hidden state, there are only a few possible states for the next step, which is very reasonable. For a generic ${\boldsymbol{M}}$, $$\operatorname{spark}({\boldsymbol{M}}) = \operatorname{krank}({\boldsymbol{M}}) + 1 = N+1.$$ @Donoho2003 showed that for such a ${\boldsymbol{M}}$, and a vector $\bm{\theta}$ with at most $N/2$ nonzeros, $\bm{\theta}$ is the unique solution with at most $N/2$ nonzeros that satisfies ${\boldsymbol{M}}\bm{\theta} = \bm{\ell}$. Therefore, if we seek for the sparsest solution to the linear equation , we can uniquely recover ${\boldsymbol{\varTheta}}$ as well.
[^1]: In the supplementary material, we prove that if the emission probability is *generic* and the transition probability is *sparse*, the HMM can be uniquely identified from triple-occurrence probability for using the latest tensor identifiability result [@chiantini2012generic].
[^2]: In some literature $\circ$ is written as the Kronecker product $\otimes$. Strictly speaking, the Kronecker product of three vectors is a very long vector, not a three-way array. For this reason, we chose to use $\circ$ instead of $\otimes$.
| 0 |
---
abstract: 'We study the properties of $g^{1}$, the first excited state of the gluon in representative variants of the Randall Sundrum model with the Standard Model fields in the bulk. We find that measurements of the coupling to light quarks (from the inclusive cross-section for $pp\to g^{1} \to t\bar t$), the coupling to bottom quarks (from the rate of $pp\to g^{1} b$), as well as the overall width, can provide powerful discriminants between the models. In models with large brane kinetic terms, the $g^1$ resonance can even potentially be discovered decaying into dijets against the large QCD background. We also derive bounds based on existing Tevatron searches for resonant $t \bar{t}$ production and find that they require $M_{g^{1}} \gtrsim 950$ GeV. In addition we explore the pattern of interference between the $g^1$ signal and the non-resonant SM background, defining an asymmetry parameter for the invariant mass distribution. The interference probes the relative signs of the couplings of the $g^{1}$ to light quark pairs and to $t\bar t$, and thus provides an indication that the top is localized on the other side of the extra dimension from the light quarks, as is typical in the RS framework.'
author:
- |
[Ben Lillie$^{b, c}$, Jing Shu$^{a, b, c}$, Tim M.P. Tait$^{c}$]{}\
\
\
title: 'Kaluza-Klein Gluons as a Diagnostic of Warped Models '
---
ANL-HEP-PR-07-40\
EFI/07-18\
Introduction {#sec:intro}
============
The large hierarchy between the Planck scale where quantum gravity effects are important, and the scale where the electroweak symmetry is broken, drives the wealth of models at the electroweak scale, and motivates the Large Hadron Collider (LHC) experiments. While weakly coupled supersymmetry remains a leading candidate to stabilize the hierarchy, the Randall-Sundrum (RS) models of a warped extra dimension [@Randall:1999vf] have recently emerged as a fascinating alternative, which may be connected to string landscape solutions of the cosmological constant problem [@jmr], and possess an interesting four dimensional dual interpretation in terms of the composite states of a strongly coupled nearly conformal field theory (CFT) [@Arkani-Hamed:2000ds].
The original RS model had all of the Standard Model confined to the IR brane (appearing as composites in the dual description). However, the RS solution to the hierarchy problem requires only the Higgs to be localized at the IR boundary, and there are compelling reasons to consider most of the SM might actually lie near the UV brane (and thus mostly fundamental with respect to the CFT in the dual description). Theories with the SM in the bulk can incorporate Grand Unification of couplings [@Randall:2001gb], motivate the flavor hierarchy of fermion masses [@Huber:2000ie], and incorporate a dark matter candidate [@Agashe:2004ci]. However, such theories face significant challenge from precision electroweak observables [@Davoudiasl:1999tf], requiring specific features [@Agashe:2003zs; @Davoudiasl:2002ua; @Agashe:2006at] in order to remain natural.
At the LHC, production of colored states is usually dominant, and the Kaluza-Klein (KK) excitations of the gluons are particularly attractive, because they are singly produced and thus have larger rates than the KK quarks. Thus, they are usually considered to be likely to be the first signs of warped physics, and the first excitation of the gluon ($g^1$) the state for which we will have the most statistics available in order to unravel the details of the underlying theory. They are the natural place to explore whether or not we can use LHC data to determine which particular detailed RS model has been realized in nature, and which parameters describe it. Recently, significant work has begun on some of the simplest RS constructions to study the production and decay of the first KK mode of the gluon, in order to determine the reach of the LHC to discover RS through its detection [@Agashe:2006hk].
While the KK gluon is the most promising avenue to discover RS, it is nevertheless challenging. The coupling to the light quarks that are the primary constituent of the proton, while characterized by the strong coupling, are somewhat suppressed by the localization of the light fermions close to the UV boundary (in the CFT language, the light fermions are largely fundamental fields and couple to the gluon largely through a small mixing with CFT states). This leads to somewhat smaller production cross sections than are typical of QCD. The decay of the gluon is expected to be predominantly into top quarks, a consequence of the large top mass, which necessitates that top is itself located close to the IR brane (mostly composite). The tops are produced from a very heavy resonance, and are highly boosted, which makes it experimentally challenging to reconstruct them from the large QCD backgrounds [@Agashe:2006hk].
In this article, we explore several more of the commonly considered theories which attempt to render RS consistent with precision electroweak data. We consider the model with a simple $SU(2)$ [@Agashe:2003zs] custodial symmetry (already studied before [@Agashe:2006hk]) as a beginning, and also consider models with large brane kinetic terms [@Davoudiasl:2002ua] or an expanded custodial symmetry which protects the $Z$-$b$-$\bar{b}$ vertex from large corrections [@Agashe:2006at; @Contino:2006qr; @Carena:2007ua] in order to characterize the difference in the properties of the first KK mode of the gluon in each case. We find that there are general features which can discriminate between the cases, and thus that the specific realization of the RS model leaves an imprint in the properties of the KK gluon.
The article is laid out as follows: in section \[sec:models\], we review the specific details of the models under consideration. In Section \[sec:xsec\] we show the $g^1$ production rates and decay properties and show how the strong coupling can lead to interesting finite width effects in section \[sec:interfere\]. Section \[sec:conclusions\] contains our conclusions.
Models {#sec:models}
======
The Basic RS Model with the SM in the Bulk
------------------------------------------
The basic RS model is a slice of AdS$_5$ with the background metric $$ds ^2 = \Big( \frac{z_h}{ z}\Big)^2 \left[ \eta_{ \mu \nu }
d x ^{\mu } d x^{ \nu } + ( d z ) ^2 \right] , \label{metric-conf}$$ with curvature $\kappa = 1/z_h \lesssim M_{Pl}$. $x^\mu$ are the coordinates of the four large dimensions, $z$ parameterizes the coordinate along the extra dimension, and $\eta_{\mu \nu} = Diag(-, +, +, +)$ is the four-dimensional metric. Greek letters denote the four large dimensions $0,1,2,3$ and capital roman letters include the fifth dimension as well. The UV boundary is at $z_h = 1/ \kappa$ where the scale factor $(z_h / z)^2 = 1$ and the IR boundary is at $z_v \sim 1 / {\rm TeV}$, as motivated by the hierarchy problem.
We are particularly interested in a model where all Standard Model (SM) fields, except perhaps the Higgs, propagate in the entire 5-d spacetime, and will be primarily concerned with the gluon and colored fermion fields. The action for the gauge fields and fermions is, S = d\^5x {- F\_[MN]{}\^[a]{}F\^[MN a]{} + i \^ e\_\^[M]{} D\_M + i c } . \[eq:action0\] where $\Gamma^{\dot{M}}$ are the 5d ($4 \times 4$) Dirac matrices, $e_{\dot{M}}^{M}$ is the veilbein, $a$ is an adjoint gauge index and $c$ parameterizes the magnitude of a bulk mass for the fermion in units of the curvature.
We work in a unitary gauge $A_5 = 0$, and decompose the 5d fields in KK modes, A\^a\_(x,z) = \_n A\^[a(n)]{}\_(x) g\^[(n)]{}(z) ,\
\_[L,R]{} (x,z) = (z)\^[3/2]{} \_n \^[n]{}\_[L,R]{}(x) \^[(n)]{}\_[L,R]{} (z) . The wave functions are given by combinations of Bessel functions g\^[(n)]{}(z) & = & N\_n z . \[eq:KKgluon\] with normalization factor $N_n$ and admixture controlled by $b_n$. The mass spectrum is controlled by the boundary conditions, with the masses satisfying, b\_n = - = - . \[eq:b0\] For an unbroken gauge group, there is a zero mode with wave function $g^0(z) = 1 / \sqrt{L}$, $L = 1/k \log z_v / z_h$. Of particular note for the following is the fact that the light KK states have most of their support close to the IR boundary.
The physics of bulk fermions was worked out in [@Grossman:1999ra]. The spectrum depends sensitively on the bulk mass term $c$. To remove unwanted light degrees of freedom, we impose the boundary conditions such that either the right- or the left-chiral zero mass component is projected out. The KK states form left- and right-chiral pairs whose wave functions are also Bessel functions, \^[(n)]{}\_ (z) = [N]{}\_n (z) , \[eq:fermion\] where $-~(+)$ are for the right- (left-) chiral modes, and the masses are determined by imposing the equality, \_n & = & = , and ${\cal N}_n$ is a normalization factor. The zero mode wave functions are, \^[(0)]{}(z) = [N]{}\_0 ( z )\^[1/2- c\_ ]{} . These wavefunctions assume the right-handed zero mode is the one allowed by the boundary conditions; the left-handed case is given by $c \rightarrow -c$. The zero mode is exponentially peaked toward the UV boundary for $c < -1/2$ and toward the IR for $c > -1/2$. To avoid confusion, we adopt a notation where $c$’s explicitly refer to right-chiral fields, so the left-chiral fermions should be understood to actually have $-c$ as their mass parameter.
Assuming $\mathcal{O}(1)$ 5D Yukawa couplings, the hierarchy in the effective 4D Yukawa couplings can be motivated by the exponential suppression of the wave functions at the IR boundary for order one differences in $c$. In particular, one cannot allow strong suppression of the top quark wave functions on the IR boundary, because to reproduce the observed top mass one would have to adjust the 5d Yukawa coupling to be too strong to have a perturbative description. There is further motivation from precision electroweak data [@Agashe:2003zs; @Davoudiasl:2002ua], which prefers the light fermion mass parameters to be close to $-1/2$ (including the left-handed top, as it comes along with the left-handed bottom, leading to tension in the choice of $c$ for $Q_3$) in order to cancel the leading contribution to the $S$ parameter from the weak boson KK modes. With this setup, and the additional suppression of the $T$ parameter from a custodial $SU(2)$, masses of the KK modes of around $3$ TeV are roughly consistent with precision measurements.
Specifically, we consider $c_{t_R} \sim 0$, $c_{Q_{3L}} \sim 0.4$, and all others $c_f \lesssim -0.5$. As we will see shortly, the physics we study does not depend strongly on $c$ once it is $< -1/2$, so the specific values for light fermions are not important. The choices of $c$ specify the fermion zero mode wave functions, and we compute the couplings of the first KK gluon to the fermion zero modes as the integral over the wave functions. The light quarks all have very similar couplings of roughly $g_{f} \simeq -g_S / 5$, the third family left-handed quarks $g_{Q3} \simeq g_S$, and the right-handed top quark $g_{t} \simeq 4 g_S$, where $g_S$ is the strong coupling constant which characterizes the coupling of the gluon zero mode.
IR Brane Kinetic Terms
----------------------
An alternate way to render precision electroweak data consistent with low KK mode masses is to include large-ish kinetic terms for the gauge fields on the IR brane [@Davoudiasl:2002ua]. Such terms repell the KK mode wave functions from the brane, and have a large effect on the phenomenology of the KK modes [@Carena:2002me]. Brane terms are a class of higher dimensional operators of the 5d theory, - d\^5x { F\_[MN]{}\^[a]{}F\^[MN a]{}} 2 r\_[IR]{} ( z - z\_v ) \[eq:actionbk\] and will be induced by the orbifold boundary conditions and localized fields [@Georgi:2000ks]. Their magnitude $r_{IR}$ is a free parameter of the effective theory. While the size of the IR boundary kinetic term for the gluon is not closely connected to the quality of the electroweak fit, one would expect that if the UV physics is such that there are large IR kinetic terms for the electroweak bosons, such terms are probably also large for the gluon as well. Thus, if one could infer the presence of large gluon terms, it would at least suggest that a similar term is present in the electroweak sector, and responsible for the success of the SM in explaining the electroweak fit.
The IR boundary kinetic term does not affect the form of the bulk wave functions, Eq. (\[eq:KKgluon\]). The boundary conditions become b = - = - , \[eq:eigenmassbk\] indicating different admixture of the Bessel functions $J_1$ and $Y_1$ in the solutions. While there is no analytic solution for the masses, they may be easily obtained numerically. In Figure \[fig:bkm\], we show the variation of the first KK mode gluon mass and coupling to UV-localized states as a function of the magnitude of the IR brane term $\kappa r_{IR}$. In Figure \[fig:gffbkcoup\], we show the dependence of the coupling on $c$ for a few different choices of $\kappa r_{IR}$. The inclusion of the boundary terms ameliorates the strongest constraints from precision electroweak data, and opens up considerably more freedom to choose the fermion $c$’s. However, in computing properties below, we imagine a situation in which the $c$’s are as in the $SU(2)$ custodial version outlined above (for example, to explain the flavor hierarchies), with large contributions to the electroweak $T$-parameter controlled by the IR boundary kinetic terms.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[fig:bkm\]The 1st KK gluon mass in units of $1/z_v$ and coupling of the first KK gluon to a fermion zero mode localized at UV brane as a function of brane kinetic term $\kappa r_{IR}$.](g1mass.pdf "fig:") ![\[fig:bkm\]The 1st KK gluon mass in units of $1/z_v$ and coupling of the first KK gluon to a fermion zero mode localized at UV brane as a function of brane kinetic term $\kappa r_{IR}$.](g1ffbk.pdf "fig:")
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[fig:gffbkcoup\] Coupling of the first KK gluon (with respect to the zero mode gluon coupling) with $\kappa r_{IR} = 0,1,5,10,20$ (descending) to a fermion zero mode as a function of bulk mass parameter $c$. ](g1ffc.pdf)
Holographic Higgs with Expanded Custodial Symmetry
--------------------------------------------------
The models with a custodial $SU(2)$ symmetry or large IR boundary kinetic terms (combined with the choices of the $c$’s motivated above) continue to be challenged by the large top mass, which we saw did not allow $Q_3$ to be pushed quite as far away as was optimal for the lighter fermions. This results in corrections to the $Z$-$b_L$-$\bar{b}_L$ coupling compared to those of light fermions which are slightly too large for the experimental errors, and push in a direction unhelpful for $A^{FB}_b$ [@Choudhury:2001hs].
In [@Agashe:2006at], it was noticed that a subgroup of the custodial symmetry can protect the $Z$-$b_L$-$\bar{b}_L$ coupling, provided the third generation doublet is embedded in a representation for which the $SU(2)_L$ and $SU(2)_R$ representations (and the third component of each) are the same. This implies that to better fit $Z$-$b_L$-$\bar{b}_L$, we expand $Q_3$ into a bi-doublet under ($SU(2)_L$, $SU(2)_R$). The unwanted additional fermions in the bi-doublet are removed from the zero mode spectrum by adjusting their boundary conditions. Having promoted $Q_3$ to a bi-doublet, we recover freedom to consider the $c$ parameter for $Q_3$ very different from $-1/2$.
In order to have a specific framework, we analyze the model of gauge-Higgs unification [@Carena:2007ua] (similar to an earlier model [@Agashe:2004rs]) in which the allowed parameter space is analyzed in great detail [@Carena:2006bn], reproducing light fermion masses and mixings, and demanding consistency with flavor-changing neutral currents induced by the KK modes of the gauge bosons. While some of the features are particular to the gauge-Higgs unified model and the mechanism by which it realizes fermion masses and mixings, some of the most important features are fairly generic to models in which an expanded custodial symmetry is protecting $Z$-$b_L$-$\bar{b}_L$.
The bulk gauge symmetry is $SO(5) \times U(1)_X$, broken by boundary conditions to $SU(2)_L \times SU(2)_R \times U(1)_X$ on the IR boundary, and to the Standard Model $SU(2)_L \times U(1)_{Y}$ gauge group on the UV brane [@Agashe:2004rs]. The $U(1)_{X}$ charges are adjusted so as to recover the correct hypercharges, where $Y/2 = T^{3}_{R} + Q_{X}$ with $T^3_{R}$ the third $SU(2)_{R}$ generator and $Q_{X}$ the $U(1)_{X}$ charge. As motivated above, we wish $Q_3$ to be part of a bi-doublet, and an economical choice is to embed it in a ${\bf5}_{2/3}$ of $SO(5)$ (the subscript refers to the $U(1)_X$ charge). As discussed in [@Carena:2006bn], it is preferable to place $t_R$ in a seperate ${\bf 5}_{2/3}$ to avoid large negative corrections to the $T$ parameter. $b_R$ is part of a ${\bf 10}_{2/3}$, allowing for the bottom Yukawa coupling, and the first and second generations are replicas of this structure in order to generate CKM mixing in a straight-forward way. Enhanced coupling to bottom quarks is also potentially a signal of RS attempts to explain the observed deviation in $A_{FB}^b$ [@Djouadi:2006rk].
The scan over parameters of [@Carena:2007ua] prefers that the quarks and leptons of the first two generations are localized close to the Planck boundary in order to suppress flavor changing neutral currents. The expanded custodial symmetry, combined with relatively light KK modes for the $Q_3$ custodial partners, is so efficient at suppressing contributions to the $T$ parameter, that it reduces some of the usual SM top contribution, and can result in $T$ large and negative, in conflict with the electroweak fit [@lepewwg]. To ameliorate this new concern, the freedom to consider $Q_3$ closer to the IR boundary (compensated by moving $t_R$ somewhat away from it) is crucial, allowing $c_{Q_{3L}} \sim 0.2$, $c_{t_R} \sim -0.49$, and $c_{f} \lesssim -0.5$, for which the couplings to the first KK mode of the gluon are approximately $g_f \sim -g_S /5$, $g_{t} \sim 0.07 g_S$, and $g_{Q3} \sim 2.76 g_S$.
This model generically leads to very light KK quarks, the lightest of which are the $SO(5)$ bi-doublet partners of the right-handed up-type quarks of the first two generations $u_i$ (by virtue of the choice of $c$ for the two light generations) [@Carena:2007ua]. Each generation contains $$\begin{aligned}
\begin{array}{c}
\begin{array}{ccccccc}
~ & ~ & Q^{i}_{2R} &=& \begin{pmatrix}
\chi^{u_{i}}_{2R}(+,-) & q^{\prime {u_{i}}}_R(+,-) \\
\chi^{d_{i}}_{2R}(+,-) & q^{\prime {d_{i}}}_R(+,-) \end{pmatrix},
\end{array}
\end{array}
\label{multiplets}\end{aligned}$$ along with their $(-,+)$ left-handed Dirac partners. The $(\pm,\,\mp)$ refers to their boundary conditions on the (UV, IR) boundaries, and do not lead to zero modes (as desired), and modify the equation which determines their masses and admixture of Bessel functions. For the right-handed $(+,-)$ states this leads to, $$\begin{aligned}
\beta_n & = &
\frac{ J_{ |c-1/2| } \left( m_n z_v\right) }
{ Y_{|c-1/2| } \left( m_n z_v \right) }
= \frac{ J_{|c-1/2| \mp 1} \left( m_n z_h \right) }
{ Y_{ |c-1/2| \mp 1} \left( m_n z_h \right) } ~,
\label{keyequation}\end{aligned}$$ with upper(lower) signs for $c>-1/2$ ($c<-1/2$). The left-handed $(-,+)$ states satisfy, $$\begin{aligned}
\beta_n & = &
\frac{ J_{ |c+1/2| } \left( m_n z_h \right) }
{ Y_{|c+1/2| } \left( m_n z_h \right) }
= \frac{ J_{|c+1/2| \mp 1} \left( m_n z_v \right) }
{ Y_{ |c+1/2| \mp 1} \left( m_n z_v \right) } ~.
\label{keyequation2}\end{aligned}$$
Armed with these wave functions, we compute the coupling of these potentially light first KK modes of the custodial partners to the first KK mode of the gluon. The results for both chiralities are presented in Figure \[fig:gffmpcoup\], and indicate that one chirality is always very strongly coupled, $g \sim 6 g_S$, irrespective of the value of $c$.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[fig:gffmpcoup\] Coupling of the First KK mode of the gluon to the light KK modes of the custodial partners of the right-handed up-type quark as a function of the bulk mass parameter $c$. The left panel shows the left-handed coupling whereas the right panel the right-handed coupling.](g1ffl.pdf "fig:") ![\[fig:gffmpcoup\] Coupling of the First KK mode of the gluon to the light KK modes of the custodial partners of the right-handed up-type quark as a function of the bulk mass parameter $c$. The left panel shows the left-handed coupling whereas the right panel the right-handed coupling.](g1ffr.pdf "fig:")
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
A Warped Higgsless Model
------------------------
A final variant of the warped theory has no Higgs, and breaks the electroweak symmetry by boundary conditions [@Csaki:2003dt]. The need for the KK modes of the weak vector bosons to unitarize $WW$ scattering implies that the scale of KK mode masses is at most several hundred GeV, whereas the need to be consistent with precision electroweak data and realize a large top mass requires [@Cacciapaglia:2006mz][^1] g\_[t]{} = 2.5 g\_S , g\_[Q3]{} = 2 g\_S , g\_[b]{} = -0.32 g\_S ,\
g\_[[other RH]{}]{} = -0.33 g\_S , g\_[[other LH]{}]{} = 0.15 g\_S . \[eq:hlfermion\] We see that the basic trend is very similar to the other RS models we consider. The main distinguishing feature is the fact that the mass of the KK gluon must be less in order for the Higgsless model to remain consistent with perturbative unitarity.
Production and Decay {#sec:xsec}
====================
The details of production of KK gluons at the LHC will depend on how they couple to the relevant partons at LHC energies, and these differences will give us a powerful way to discriminate between models. Note that the vertex with two gluons and a KK gluon is zero at tree-level, meaning that the dominant production mode is $q\bar q$ annihilation. As is well-known, in the standard RS framework the KK gluon coupling to all fermions aside from $t_R$ is suppressed. As we saw above, models with brane kinetic terms can increase couplings to UV-localized fields, which increases the rate and affect the branching ratios. In addition, the models with custodial symmetry have a large coupling to $Q_3$, turning on a new production mode from bottom fusion, but have a smaller branching ratio because the decay into the custodial partner KK modes may compete with top. In Figure \[fig:xsec\], we plot the cross section, calculated at leading order by MADGRAPH [@Alwall:2007st], $pp \rightarrow g^1$ for $\sqrt{S} = 14$ TeV, as a function of $g^1$ mass for the models considered above, including the channels initiated by light quark fusion and bottom fusion.
![\[fig:xsec\] Cross section for $pp \rightarrow g^1$ at the LHC, for standard RS with the SM in the bulk ($\kappa r_{IR}=0$), three models with large brane kinetic terms ($\kappa r_{IR} = 5, 10, 20$) and the model with a larger custodial symmetry, in the cases when $N=0$ or $1$, of the additional KK custodial partner quarks are light enough that $g^1$ can decay into them.](tt.pdf)
![\[fig:bxsec\] Cross section for $pp \rightarrow b g^1$ at the LHC, for standard RS with the SM in the bulk ($\kappa r_{IR}=0$), a model with a large brane kinetic term ($\kappa r_{IR} = 5$) and the model with a larger custodial symmetry, in the cases when $0$ ($1$) of the additional KK custodial partner quarks are light enough that $g^1$ can decay into them.](btt.pdf)
As indicated above, models with the extra custodial symmetry to protect the $Z$-$b_L$-$b_L$ coupling from large corrections have considerably more freedom to locate $Q_3$ closer to the IR brane, and considerations of the $T$ parameter prefer to do so. This enhances the $g^1$ coupling to left-handed bottoms (up to about $3 g_S$) and results in large production from bottom quark fusion, as shown in Figure \[fig:xsec\]. It would be useful to be able to discern that the increase over the expected production rate in the standard bulk SM RS picture is because of this enhancement of the coupling to bottom (which would be suggestive of the expanded custodial symmetry), as opposed to a straight enhancement of the coupling to all light quarks (which would be more suggestive of a large kinetic term on the IR boundary). One could study the rapidity distribution of the $g^1$ itself (as reflected in the final state top pair distribution). The fact that both $b$ and $\bar{b}$ are sea quarks would imply a more central rapidity distribution than would result from $q$ and $\bar{q}$, because $q$ as a valence quark will tend to carry more momentum than $\bar{q}$. However, the $g^1$ rapidity distribution is only modestly sensitive to the initial state, and is also sensitive to the $g^1$ mass and width. Thus, we turn to a more straight-forward measure of the contribution of $b \bar{b}$ to $g^1$ production [^2] which is to compare the rate of $g^1 \rightarrow t \bar{t}$ to $b g^1 \rightarrow b t \bar{t}$. In Figure \[fig:bxsec\], we present these rates for standard RS, the model with $\kappa r_{IR}=5$, and the model with $Q_3$ localized around the IR brane. We find that as expected, the rate for the model with custodial symmetry is enhanced by the large bottom coupling by about an order of magnitude. In addition, the model with IR boundary kinetic terms shows a rate which is suppressed by a factor of about five, because while the boundary kinetic term slightly enhances the coupling of the UV-localized $b_R$, it more dramatically suppresses the coupling to the IR-localized $b_L$ (c.f. Figure \[fig:gffbkcoup\]). Ultimately, one must include the SM background and detector efficiencies for a specific decay channel of $g^1$. As a step in this direction, in Figure \[fig:diffcrosplot\] we plot the differential cross-section for both the $pp \to t\bar t$ and $pp \to b t\bar t$ signals and SM backgrounds with respect to the $t \bar{t}$ invariant mass, in the standard RS model and one with a larger custodial symmetry. In both cases, for $M_{g^1} = 2$ TeV, a peak is visible above the SM background, and the size of $g^1 b$ production relative to $g^1$ production discriminates between the two models.
![\[fig:diffcrosplot\] Invariant mass distribution of $t\bar t$ in the standard RS model ($\rm{SU(2)_V}$ custodial symmetry) and the model with a larger $\rm{O(3)}$ custodial symmetry in $pp \to t\bar t$ and $pp \to b t\bar t$ respectively. ](diffcros.pdf)
The width of $g^1$ is strongly dominated by the states close to the IR brane to which it couples strongly. Generically, the partial width into $f \bar{f}$ for which the left- and right-chiral interactions with $g^1$ are $g_L$ and $g_R$ is given by, $$\begin{aligned}
\Gamma_{G1 \rightarrow f \bar{f}} & = & \frac{1}{48 \pi M_{g^1}}
\sqrt{1 - \frac{4 m^2_f}{M^2_{g^1}}}
\left[ \left( g_L^2 + g_R^2 \right) \left(M_{g^1}^2-m_f^2 \right)
+ 6 g_{L} g_{R} m_f^2 \right] \nonumber \\
& \simeq & \frac{M_{g^1}}{48 \pi} \left( g_L^2 + g_R^2 \right) ~,
\label{decayequation}\end{aligned}$$ where the final approximation holds in the limit $M_{g^1} \gg m_f$. Decays to top quarks are always important, because either $t_R$ or $Q_3$ must be IR-localized to realize the large top mass. In addition, when the custodial partner KK quarks are light enough for $g^1$ to decay into them, they will also take a substantial fraction of the branching ratio, because they are also IR-localized and have large coupling. The IR boundary kinetic terms can suppress the coupling to top, and enhance the decay into light quarks. In Table \[tab:brs\] we list the branching ratios into top quarks, bottom quarks, light quarks (jets) and exotic quarks in several different RS models. The total width also sensitively depends on the couplings, and how many custodial partners are available as decay modes. The width is generally large, owing to the strong couplings present, and it may be possible to reconstruct it from the final state invariant mass distributions, which would also allow one to use it as an additional source of information. The final column of Table \[tab:brs\] shows the total width $\Gamma_{g^1} / M_{g^1}$ for each model. Variations are typically around $5\%$, with the exception of the model with an extra custodial partner, whose very strong coupling has a big effect on the width. In fact, allowing too many additional custodial partners will rapidly drive $\Gamma_{g^1} \gtrsim M_{g^1}$, an indication of a break-down of perturbation theory. From Eq. (\[decayequation\]), we can infer that there can be at most four new custodial quarks whose masses are less than $M_{g^1} / 2$.
Model top quarks bottom quarks light quarks custodial partners $\Gamma_{g^1}/M_{g^1}$
---------------------- ---------------- ------------------- ------------------ ------------------------ ----------------------------
Basic RS 92.6% 5.7% 1.7% 0.14
$\kappa r_{IR} = 5$ 2.6% 13.2% 84.2% 0.11
$\kappa r_{IR} = 20$ 7.8% 15.1% 77.1% 0.05
$O(3)$, $N=0$ 48.8% 49.0% 2.0% 0.11
$O(3)$, $N=1$ 14.6% 14.6% 0.6% 70.2% 0.40
: \[tab:brs\] The branching ratios of $g^1$ into tops, bottoms, light quarks (jets), and custodial partners, as well as the total width $\Gamma_{g^1} / M_{g^1}$, for several different RS scenarios in the limit $M_{g^1} \gg m_f$.
![\[fig:tevatron\] Cross section for $p \bar{p} \rightarrow g^1 \rightarrow t \bar{t}$ at the Tevatron as a function of the mass of $g^1$, compared with the CDF exclusion curve. The mass of custodial partners is 360GeV.](Tevatron.pdf)
In models with large boundary kinetic terms, $g^1$ primarily decays into light quarks, swamping the decay into tops, and its over-all width becomes much narrower. This fact, combined with the enhancement of $g^1$ production, allows for the possibility that one could discover $g^1$ in the dijet mode, against the large QCD background. To explore this possibility, in Figure \[fig:jetplot\] we plot the invariant mass distribution of QCD dijets (with rough acceptance cuts $|\eta| <1.0$ and $p_T > 20$ GeV to reduce the SM background). For $M_{g^1} = 2$ or $3$ TeV, we can reconstruct a peak against the dijet background with ample statistics. Based on the size of the signal and background, we estimate that one could potentially discover $g^1$ even if its mass is larger than 4 TeV in such models.
![\[fig:jetplot\] Invariant mass distribution of QCD dijets coming from the KK gluon resonance in models with large brane kinetic term ($\kappa r_{IR} = 20$), and the SM prediction. The cuts $p_T >20$GeV, $|\eta| <1.0$, and invariant mass $>1$TeV are applied. ](JJ1.pdf)
The highly chiral nature of the couplings of $g^1$ to top, bottom, or the custodial partners may be visible as an observable [@Agashe:2006hk]. The top final state is particularly promising, because the left-handed nature of the $W$-$t$-$b$ interaction implies that the top decay automatically analyzes its production polarization. For example, the standard RS scenario has about $95\%$ decays into right-polarized tops, whereas the model with $\kappa r_{IR} = 10$ has roughly equal decays into left- and right-polarized tops, and the model with expanded custodial symmetry with $Q_3$ localized at the IR brane has about $99\%$ decays into left-polarized tops.
Finally, given the large cross-sections, it is natural to ask what the current bounds from the Tevataron on anomalous top production imply for the KK gluon mass. A recent analysis from CDF [@tevbound] has set bounds on narrow resonances in the $t\bar t$ invariant mass spectrum. While the analysis does not strictly apply in this case, since the KK gluon is wider than the machine resolution, the actual bound will be close to that quoted in the analysis. We have plotted this in Fig. \[fig:tevatron\], along with representative cross-sections from the models under investigation here. Note that this excludes Higgsless models with KK masses below about 850 GeV, and that includes the region favored by unitarity in $WW$ scattering.
Interference {#sec:interfere}
============
There is an intriguing feature of the fermion couplings to $g^1$: the sign of the coupling depends on the sign of the $g^1$ wave function close to where the fermion is localized. As a KK mode, the $g^1$ wave function contains a node, and changes sign from one side of the extra dimension to the other. As a result the UV fermions have a minus sign relative to the zero mode gluon coupling, while the IR fermions have a plus sign. This sign should be visible in the interference between $s$-channel gluon and KK-gluon production of $t\bar t$, as illustrated in Fig. \[fig:feyn\].
![\[fig:feyn\] Graphs that interfere allowing measurement of the sign of the light quark coupling. ](feyn.pdf)
![\[fig:intplot\] Invariant mass distribution of $pp\to t\bar t$ in models with positive and negative coupling to light fermions, along with the SM prediction. ](int.pdf)
To quantify this effect we propose an asymmetry parameter $A_{i}$. This parameter should be positive or negative depending on the sign of the light quark coupling and be zero in the Standard Model. We accomplish this with the definition $$\begin{gathered}
A_{i} = -
\frac{\int dm (\frac{d\sigma}{dm} -
\frac{d\sigma}{dm}_{\rm SM})*\Theta(m - M_{g^1})}
{\int dm |\frac{d\sigma}{dm} - \frac{d\sigma}{dm}_{SM}|}.\end{gathered}$$ Here $m$ is the invariant mass in the $t\bar t$ distribution and $M_{g^1}$ is the center of the resonance. The logic of this choice is that: [*i.*]{} The SM contribution is subtracted to determine if the interference is positive or negative; [*ii.*]{} the sign of the interference changes as the resonance is crossed, hence the $\Theta$-function; [*iii.*]{} As is well-known, a positive sign will produce negative interference below the resonance and positive above due to the sign of the resonance propagator $1 / (s - M_{g^1}^2)$, hence the overall minus sign. With this definition the sign of $A_{i}$ will be that of the light quark coupling.
The normalization of the data with respect to the SM calculation is problematic. Since the resonance will result in a much larger overall cross-section, one should not normalize to the total number of events. We choose to normalize to the lowest-mass bin used in calculating the asymmetry, which allows extraction of the normalization from data, while retaining all available information in the region near the resonance.
We present values of $A_{i}$ for several masses in the basic RS model in Table \[tab:asym\]. We also show the value obtained by switching the sign of the light quark coupling. We have included a crude estimate of the smearing by shifting the value of the top and anti-top 4-momentum by a gaussian random number with width given by the ATLAS jet resolution. Since the uncertainty in top reconstruction will be dominated by the jet uncertainty this gives the correct order-of-magnitude for the smearing; we leave more refined estimates for future work. We find that the smearing makes little difference, as the resonance width is larger than the detector resolution. The results in Table \[tab:asym\] indicate that if a resonance is observed in $t\bar t$ production, $A_{i}$ is a promising variable to extract information about the underlying theory.
$g^{(1)}$ Mass plus minus
---------------- ---------- -----------
2 TeV 0.57 -0.44
3 TeV 0.54 -0.28
4 TeV 0.52 -0.16
: \[tab:asym\] Asymmetry parameter $A_{i}$ for $t\bar t$ resonances with negative (corresponding to basic RS) and positive light quark couplings.
Conclusions {#sec:conclusions}
===========
We have investigated the structure of the KK gluon resonance in several variants of the RS model. We find that this structure contains information that will help to distinguish between models even in the absence of data from the electroweak sector. The width and branching ratios will constrain the location of the fermion zero-modes as well as the number of light KK modes into which the KK gluon can decay. In addition, the ratio of cross-sections for producing the $g^{1}$ directly and in association with a $b$-jet will give specific information about the localization of the third generation quarks. Specifically, a large coupling to $b \bar b$ will prefer a model where the $Z\to b\bar b$ vertex is protected by an extended custodial symmetry. In some models, with large boundary kinetic terms, the $g^1$ can primarily decay into dijets, and it seems promising that in such models one can discern $g^1$ against the large QCD background up to masses somewhat larger than 4 TeV.
Finally, we find that the relative sign of the coupling to light quarks and to tops can be measured in the interference with $s$-channel gluon exchange. This provides an important consistancy check on the overall picture of the fermion geography and the mechanism by which flavor hierarchies are realized in the fermion Yukawa couplings.
The discovery of $g^1$ is an important first step in the discovery of RS, and further observables such as its production rate, associated rate with bottom quarks, total width and branching ratios, and interference with SM $t \bar{t}$ production, can yield information about the nature of the the RS construction, and the parameters which describe it.
[**Acknowledgements** ]{}
The authors would like to thank Erik Brubaker, Giacomo Cacciapaglia, Tao Han, Gregory House, Tom LeCompte, David Mckeen, Arjun Menon, Jose Santiago, and Carlos Wagner for helpful discussions. Research at Argonne National Laboratory is supported in part by the Department of Energy under contract DE-AC02-06CH11357.
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[^1]: We would like to thank Giacomo Cacciapaglia for assisting us with determining these couplings for the model of [@Cacciapaglia:2006mz]
[^2]: We are grateful to Tao Han for this suggestion.
| 0 |
---
abstract: 'We theoretically propose a method for on-demand generation of traveling Schrödinger cat states, namely, quantum superpositions of distinct coherent states of traveling fields. This method is based on deterministic generation of intracavity cat states using a Kerr-nonlinear parametric oscillator (KPO) via quantum adiabatic evolution. We show that the cat states generated inside a KPO can be released into an output mode by controlling the parametric pump amplitude dynamically. We further show that the quality of the traveling cat states can be improved by using a shortcut-to-adiabaticity technique.'
author:
- 'Hayato Goto,$^1$ Zhirong Lin,$^2$ Tsuyoshi Yamamoto,$^{2,3}$ and Yasunobu Nakamura$^{2,4}$'
title: 'On-Demand Generation of Traveling Cat States Using a Parametric Oscillator'
---
Introduction
============
Quantum superposition is one of the most strange and intriguing concepts in quantum mechanics and a useful resource for quantum information processing. Superpositions of macroscopically distinct states are often referred to as Schrödinger cat states, or cat states for short, named after Schrödinger’s famous gedankenexperiment with a cat in a superposition of alive and dead states [@Haroche2013a; @Wineland2013a]. In quantum optics, superpositions of distinct coherent states are called cat states [@Leonhardt], because coherent states are often regarded as the “most classical" states of light.
Such cat states have been generated experimentally by various approaches. In the optical regime, cat states of small size, which are sometimes called Schrödinger kittens, have been generated by subtracting one photon from squeezed states of light [@Ourjoumtsev2006a; @Wakui2007a]. Optical cat states of a little larger size have been generated by other methods [@Ourjoumtsev2007a; @Sychev2017a]. Note that these optical cat states are of *traveling* fields and generated *probabilistically*.
In the microwave regime, cat states of larger size have been generated experimentally [@Deleglise2008a; @Vlastakis2013a; @Leghtas2015a; @Touzard2018a]. The generation using Rydberg atoms [@Deleglise2008a] is heralded by measurement results of the atomic states, where the parity, ‘even’ or ‘odd’, of the cat state is determined randomly according to the measurement results. *On-demand* generations of microwave cat states have been demonstrated using superconducting circuits by two different approaches, one of which is based on conditional operations using a superconducting quantum bit (qubit) [@Vlastakis2013a] and the other is based on two-photon driving and two-photon loss larger than one-photon loss [@Leghtas2015a; @Touzard2018a]. By extending the former approach to a two-cavity case, entangled coherent states in two cavities have been observed experimentally [@Wang2016a]. Note that these microwave cat states are confined *inside* cavities. Recently, the cat state generated inside a cavity has been released by controlling the output coupling rate of the cavity using four-wave mixing in the qubit [@Pfaff2017a]. To the best of our knowledge, only this experiment has demonstrated *on-demand* generation of *traveling* cat states.
In this paper, we propose a simple alternative method for on-demand generation of traveling cat states. Our method is based on a recent theoretical result that a Kerr-nonlinear parametric oscillator (KPO) can generate intracavity cat states deterministically via quantum adiabatic evolution [@Goto2016a; @Puri2017a]. (The KPO has recently attracted attention for its application to quantum computing [@Goto2016a; @Puri2017a; @Goto2016b; @Nigg2017a; @Puri2017b].) In the previous work, the KPO is assumed to be lossless in ideal cases, and therefore the cat states are confined inside the KPO. In the present work, we theoretically investigate a coupled system of a KPO and a one-dimensional system (output mode). It turns out that the cat states generated inside a KPO can be released into the output mode by controlling the parametric pump amplitude dynamically, while the output coupling rate is constant. Hence by using a KPO, we can generate *traveling* cat states *on demand* without controlling the output coupling rate.
Model
=====
![Coupled system of a KPO and an output mode. The KPO is implemented by a Josephson parametric oscillator (JPO) [@Lin2014a] with a relatively large Kerr effect. The JPO is capacitively coupled to a transmission line for the output mode. Traveling cat state is generated by controlling the pump field dynamically.[]{data-label="fig-system"}](system.eps){width="8cm"}
The coupled system of a KPO and an output mode is depicted in Fig. \[fig-system\], where a superconducting-circuit implementation is supposed [@Nigg2017a; @Puri2017b; @Lin2014a]. In a frame rotating at half the pump frequency, $\omega_p /2$, of the parametric pumping and in the rotating-wave approximation, the system is modeled by the following Hamiltonian [@Goto2016a; @Puri2017a; @Gardiner1985a; @Milburn; @Duan2003a; @Goto2005a] (we use the units $\hbar = v_p =1$, where $v_p$ is the phase velocity of the electromagnetic fields in the output mode): $$\begin{aligned}
H(t)
&=
H_{\mathrm{KPO}}(t)+H_{\mathrm{out}}+H_{c},
\label{eq-H}
\\
H_{\mathrm{KPO}}(t)
&=
\frac{p(t)}{2}
\left(
a^{\dagger 2} + a^2
\right)
-
\frac{K}{2} a^{\dagger 2} a^2
+
\Delta a^{\dagger} a,
\label{eq-HKPO}
\\
H_{\mathrm{out}}
&=
\int_{-\infty}^{\infty} \! \omega b^{\dagger} (\omega ) b(\omega ) \, d\omega,
\label{eq-Hout}
\\
H_c
&=
i
\sqrt{\frac{\kappa_{\mathrm{ex}}}{2\pi}}
\int_{-\infty}^{\infty} \!
\left[
b^{\dagger} (\omega ) a - a^{\dagger} b(\omega )
\right] d\omega,
\label{eq-Hc}\end{aligned}$$ where $a^{\dagger}$ and $a$ are the creation and annihilation operators for the KPO, $p(t)$ is the time-dependent pump amplitude, $K$ is the magnitude of the Kerr coefficient [@comment-Kerr], $\Delta=\omega_{\mathrm{KPO}}-\omega_p/2$ is the detuning frequency ($\omega_{\mathrm{KPO}}$ is the one-photon resonance frequency of the KPO), $b^{\dagger} (\omega)$ and $b(\omega)$ are the creation and annihilation operators for photons of frequency (or wave number) $\omega_p/2+\omega$ in the output mode, and $\kappa_{\mathrm{ex}}$ is the energy decay rate of the KPO due to its coupling to the output mode. Here we assume no internal loss of the KPO, which is discussed later. Hereafter, we consider the resonance case ($\Delta=0$).
If the KPO is a closed system (${\kappa_{\mathrm{ex}}=0}$), a cat state ${|\alpha_0 \rangle + |{-\alpha_0} \rangle}$ can be generated from the vacuum state $|0\rangle$ via quantum adiabatic evolution by gradually increasing $p(t)$ from zero to ${p_0=K \alpha_0^2}$ [@Goto2016a; @Puri2017a]. When the KPO is coupled to the output mode, the photons inside the KPO will leak to the output mode. As a result, the entanglement between the KPO and the output mode arises during the generation. Moreover, the decay of the KPO due to the leak may degrade the adiabatic cat-state generation. Thus, it is not obvious whether or not we can generate a traveling cat state using the KPO.
Proposed method
===============
Our idea is based on the fact that any quantum state inside a *linear* cavity results in a traveling pulse in the same quantum state through the leak to the output mode, where the pulse shape is exponential corresponding to the exponential decay [@Eichler2011a]. This property of linear cavities removes the concern with the entanglement. The issue with the decay can also be solved by generating a cat state faster than the decay, which is possible if $K$ is much larger than $\kappa_{\mathrm{ex}}$. The remaining problem is that the KPO has a large Kerr effect, namely, it is not a linear cavity.
Our solution is to switch off the parametric pumping as ${p(t) \propto \exp (-\kappa_{\mathrm{ex}} t)}$ after the cat-state preparation. Then the Kerr term and the pumping term are cancelled out each other, and hence the KPO can be regarded as a linear cavity. This is confirmed as follows. Suppose that at time $t_0$, the KPO is prepared in a cat state ${|\alpha_0 \rangle + |{-\alpha_0} \rangle}$, where $\displaystyle {\alpha_0 = \sqrt{p_0/K}}$ and ${p_0 = p(t_0)}$. Since $H_{\mathrm{KPO}}(t)$ in Eq. (\[eq-HKPO\]) is rewritten as $\displaystyle {H_{\mathrm{KPO}}(t) = -\frac{K}{2}
\! \left( a^{\dagger 2} - \frac{p(t)}{K} \right)
\! \left( a^{2} - \frac{p(t)}{K} \right)}$ by dropping a c-number term, ${H_{\mathrm{KPO}}(t_0) |{\pm \alpha_0} \rangle \propto \alpha_0^2 - p_0/K = 0}$, where ${a |{\pm \alpha_0} \rangle = \pm \alpha_0 |{\pm \alpha_0} \rangle}$ [@Leonhardt]. Thus at $t_0$, the amplitude starts decreasing as $\displaystyle {\pm \alpha (t) = \pm \alpha_0 e^{-\kappa_{\tiny{\mbox{ex}}} (t-t_0)/2}}$ because of the external coupling. If we set the pump amplitude as $p(t) = p_0 e^{-\kappa_{\tiny{\mbox{ex}}} (t-t_0)}$, $H_{\mathrm{KPO}}(t) |{\pm \alpha (t)} \rangle \propto \alpha (t)^2 - p(t)/K = 0$, that is, the Kerr term and the pumping term are cancelled out at any time $t \ge t_0$. Thus, the KPO behaves like a linear cavity during the release of the cat state, and therefore the cat state prepared inside the KPO is faithfully released as a traveling cat state.
Numerical simulation
====================
To examine the above method quantitatively, we numerically solve the Schrödinger equation with the Hamiltonian in Eq. (\[eq-H\]) and evaluate the fidelity between the output pulse and an ideal cat state. An approach to the numerical simulation is based on the discretization of frequency $\omega$ (or wave number $\omega /v_p$) [@Duan2003a; @Goto2005a]. In the present case, however, the photon number in the one-dimensional system is larger than one, and consequently such discretization results in a complicated system of differential equations. In this work, we instead discretize the position, which results in simpler equations as shown below.
We first introduce the annihilation operator with respect to the position $z$: $$\begin{aligned}
\tilde{b}(z)
&=
\frac{1}{\sqrt{2\pi}}
\int_{-\infty}^{\infty} \!
b(\omega) e^{i \omega z} \, d\omega.\end{aligned}$$ Then, we move to the interaction picture with the unitary operator $U(t)=e^{-iH_{\mathrm{out}}t}$ as follows: $$\begin{aligned}
\tilde{b}_I(z,t)
&=
U^{\dagger}(t) \tilde{b}(z) U(t)
=
\tilde{b}(z-t),
\nonumber
\\
H_I(t)
&=
U^{\dagger}(t) \left[ H_{\mathrm{KPO}}(t)
+ H_c \right] U(t)
\nonumber
\\
&=
H_{\mathrm{KPO}}(t)
+
i \sqrt{\kappa_{\mathrm{ex}}}
\left[
\tilde{b}^{\dagger}(-t) a
-
a^{\dagger} \tilde{b}(-t)
\right].
\nonumber\end{aligned}$$ A pulse-mode operator for the interval $[0,T]$ is defined as $$\begin{aligned}
b_p = \int_0^{T} \! f_p(z) \tilde{b}_I(z,T) \, dz
=
\int_0^{T} \! f_p(T - z) \tilde{b}(-z) \, dz,
\nonumber\end{aligned}$$ where $f_p(z)$ is the normalized envelope function for the output pulse satisfying $\displaystyle \int_{0}^T \! |f_p(z)|^2 dz =1$. In the present work, we define $f_p(z)$ as ${f_p(z) \propto \sqrt{\langle \tilde{b}^{\dagger}_I(z,T) \tilde{b}_I(z,T) \rangle}}$, where $\langle \tilde{b}^{\dagger}_I(z,T) \tilde{b}_I(z,T) \rangle$ is the spatial distribution of photons in the output mode at the final time $T$. Note that in experiments, we can find such $f_p(z)$ from the measurement of the output power from the KPO.
Next, we divide the interval $[0,T]$ into $J$ small intervals $[z_{j-1}, z_{j}]$ ($j=1, 2, \ldots, J$), where the intervals $\Delta z_j = z_{j}-z_{j-1}$ are set to small values [@supplement]. Then, $\tilde{b}(z)$ and $f_p(z)$ are discretized as follows ($z\in [z_{j-1}, z_{j}]$): $$\begin{aligned}
\tilde{b}(-z) \sqrt{\Delta z_j}
\to \tilde{b}_j,~
f_p(T-z) \sqrt{\Delta z_j}
\to f_j.\end{aligned}$$ Then, the commutation relation $[\tilde{b}(z),\tilde{b}^{\dagger}(z')]=\delta (z-z')$ becomes $[\tilde{b}_j, \tilde{b}^{\dagger}_l]=\delta_{j,l}$ and the normalization condition $\displaystyle \int_{0}^T \! |f_p(z)|^2 dz =1$ becomes $\displaystyle \sum_{j=1}^J |f_j|^2 =1$. By transforming the integration with respect to $z$ to the summation with respect to $j$, we obtain $$\begin{aligned}
b_p = \sum_{j=1}^{J} f_j \tilde{b}_j,~
f_j =
\sqrt{\frac{\langle \tilde{b}_j^{\dagger} \tilde{b}_j \rangle}
{\sum_{l=1}^{J} \langle \tilde{b}_l^{\dagger} \tilde{b}_l \rangle}}.
\label{eq-bp}\end{aligned}$$
The Hamiltonian at time $t \in [z_{j-1}, z_j]$ is given by $$\begin{aligned}
H_I(t)
&=
H_{\mathrm{KPO}}(t)
+
i \sqrt{\kappa_{\mathrm{ex}}}
\left[
\tilde{b}_j^{\dagger} a
-
a^{\dagger} \tilde{b}_j
\right].
\label{eq-HI}\end{aligned}$$ We numerically solve the Schrödinger equation with the Hamiltonian in Eq. (\[eq-HI\]) [@supplement]. Since the Hamiltonian includes only one of $\{ \tilde{b}_j \}$, the corresponding Schrödinger equation is simple. In the present work, we investigate the cases where $\kappa_{\mathrm{ex}}=0.2K$.
As explained above, $p(t)$ should satisfy the following two conditions: $p(t)$ is increased fast enough to adiabatically generate a cat state inside the KPO before the decay spoils it; after that, $p(t)$ is decreased as $\displaystyle {p(t) \propto \exp (-\kappa_{\mathrm{ex}} t)}$ so that the Kerr term and the pumping term are cancelled out. To satisfy these conditions simultaneously, we define $p(t)$ as the output of the fourth order low-pass filter (LPF) [@supplement; @Wiseman] with the input ${p_{\mathrm{in}}(t)=K A_p \exp (-\kappa_{\mathrm{ex}} t)}$. The dimensionless parameter $A_p$ is used for tuning the photon number of the traveling cat state. We set the photon number to about 2, which is large enough for the two coherent states being distinct but small enough to solve the Schrödinger equation numerically. The bandwidth, $B$, of the LPF [@supplement; @Wiseman] is set to $B=0.5K$ between $\kappa_{\mathrm{ex}}$ and $K$. The final time $T$ is set such that the final photon number, $n_{\mathrm{in}}$, in the KPO is less than $10^{-3}$ (see Table \[table-result\]).
We obtain the density operator, $\rho$, describing the quantum state of the output pulse using the moments $M_{m,n}=\langle b_p^{\dagger m} b_p^n \rangle$ with $b_p$ in Eq. (\[eq-bp\]). The density matrix with respect to the Fock states is given by the following formula [@Eichler2012a]: $$\begin{aligned}
\rho_{m,n}
= \frac{1}{\sqrt{m!n!}}
\sum_{l=0}^{\infty} \frac{(-1)^l}{l!} M_{n+l,m+l}.
\label{eq-rho}\end{aligned}$$ Using the density matrix, we calculate the corresponding Wigner function $\displaystyle {W(\beta ) =
\frac{2}{\pi} \mbox{Tr} \! \left[ D(-\beta) \rho D(\beta) P \right]}$ [@Leonhardt; @Deleglise2008a; @Goto2016a], where $\displaystyle {D(\beta)=\exp \! \left( \beta b_p^{\dagger} - \beta^* b_p \right)}$ (the asterisk denotes complex conjugation) and $\displaystyle {P=\exp \! \left( i\pi b_p^{\dagger} b_p \right)}$.
The results of the numerical simulation are shown in Fig. \[fig-result\](a) and the first row of Table \[table-result\]. Table \[table-result\] also provides the setting of the simulation. Figure \[fig-result\](a) shows that the photon number in the KPO varies in a similar manner to the pump amplitude, as expected. The Wigner function for the output pulse in Fig. \[fig-result\](a) clearly shows the interference fringe, which is the evidence for the quantum superposition of the two coherent states, that is, the output pulse is in a cat state. (The tilt of the Wigner function, which corresponds to $\theta_{\mathrm{cat}}$ in Table \[table-result\], is due to the residual Kerr effect.) As shown in Table \[table-result\], the maximum fidelity between this output state and an ideal cat state is 0.962. Thus, the present method works successfully as expected.
![Simulation results of traveling cat-state generation using a KPO. Left: Time evolutions of the pump amplitude $p(t)$ and the expectation value of the photon number, $\langle a^{\dagger} a \rangle$, in the KPO. Right: Wigner function, $W(\beta )$, of the output pulse. See Table \[table-result\] for the settings of (a)–(d).[]{data-label="fig-result"}](result.eps){width="15cm"}
Fidelity $~~\beta_{\mathrm{cat}}^2~~$ $~~\theta_{\mathrm{cat}}/\pi~~$ $n_{\mathrm{in}}$ $~~K I_t~~$ Shortcut $~~\kappa_{\mathrm{ex}}/K~~$ $~~B/K~~$ $~~A_p~~$ $~~KT~~$ $~~J~~$
------------------------ ------------ ------------------------------ --------------------------------- --------------------- ------------- ------------ ------------------------------ ----------- ----------- ---------- ---------
Fig. \[fig-result\](a) 0.962 2.01 0.03 $6.2\times 10^{-4}$ 9.63 Unused 0.2 0.5 2.45 50 80
Fig. \[fig-result\](b) 0.930 1.96 0.02 $6.1\times 10^{-4}$ 9.64 Unused 0.2 1.0 2.15 45 80
Fig. \[fig-result\](c) 0.983 2.03 0.02 $6.4\times 10^{-4}$ 9.63 Used 0.2 0.5 2.50 50 80
Fig. \[fig-result\](d) 0.993 2.02 0.01 $6.4\times 10^{-4}$ 9.60 Used 0.2 1.0 2.25 45 80
: Results and setting of numerical simulations. Fidelity: maximum fidelity between the output state and the ideal cat state with two parameters: $\displaystyle (|\beta_{\mathrm{cat}} e^{i\theta_{\tiny{\mbox{cat}}}} \rangle
+|{-\beta_{\mathrm{cat}} e^{i\theta_{\tiny{\mbox{cat}}}}} \rangle )
/\sqrt{2(1+e^{-2\beta_{\tiny{\mbox{cat}}}^2})}$, where $\beta_{\mathrm{cat}}$ and $\theta_{\mathrm{cat}}$ are the magnitude and phase of the cat-state amplitude, respectively. $\beta_{\mathrm{cat}}^2$ and $\theta_{\mathrm{cat}}/\pi$: values maximizing the fidelity. $\beta_{\mathrm{cat}}^2$ is close to 2, which means that the photon number of the output pulse is about 2. $n_{\mathrm{in}}$: average photon number in the KPO at the final time $T$. $I_t$: time integral of the average photon number in the KPO, that is, $I_t=\int_0^T \! \langle a^{\dagger} a \rangle \, dt$. In all the cases, ${I_t \approx 10 K^{-1}=2\kappa_{\mathrm{ex}}^{-1}}$, because ${\kappa_{\mathrm{ex}} I_t}$ corresponds to the photon number of the output pulse.
\[table-result\]
Improvement by shortcut to adiabaticity
=======================================
The imperfection of the generated traveling cat state may be partially due to the leak during the initial cat-state preparation. We can speed up the preparation by setting the LPF bandwidth $B$ to a larger value, e.g., $B=K$. Then, however, nonadiabatic effects degrade the cat state. The simulation results for $B=K$ are shown in Fig. \[fig-result\](b) and the second row of Table \[table-result\]. The oscillation of $\langle a^{\dagger} a \rangle$ in Fig. \[fig-result\](b) is due to the nonadiabatic effects. Consequently, the maximum fidelity between the output state and an ideal cat state decreases to 0.930 (see Table \[table-result\]).
To mitigate the nonadiabatic effects, we can use the technique called *shortcut to adiabaticity* [@Puri2017a; @Demirplak2003a; @Campo2013a]. To maintain quantum adiabatic evolution, the shortcut-to-adiabaticity technique introduces the following counterdiabatic Hamiltonian [@Demirplak2003a; @Campo2013a]: $$\begin{aligned}
H_{\mathrm{counter}} =
i \sum_n |\dot{\phi}_n \rangle \langle \phi_n|,\end{aligned}$$ where $|\phi_n \rangle$ is the $n$-th instantaneous eigenstate of the slowly varying Hamiltonian and the dot denotes the time derivative.
In the present case, the counterdiabatic Hamiltonian is approximately given by [@supplement; @comment-shortcut] $$\begin{aligned}
H_{\mathrm{counter}} (t)
&=
i \frac{p'(t)}{2} \left( a^{\dagger 2} - a^2 \right),
\label{eq-counter}
\\
p' (t)
&=
\frac{\dot{p}(t)}{p(t)} \tanh \! \frac{p(t)}{K}.
\label{eq-pi}\end{aligned}$$ The physical meaning of the counterdiabatic Hamiltonian is to add the imaginary pump amplitude $p'(t)$ to the real one $p(t)$. This is experimentally possible by controlling the phase of the pump field.
The simulation results with the shortcut-to-adiabaticity technique are shown in Figs. \[fig-result\](c) and \[fig-result\](d) and the third and fourth rows of Table \[table-result\]. As shown in Fig. \[fig-result\](d), the oscillation of $\langle a^{\dagger} a \rangle$ does not occur even when $B=K$, unlike Fig. \[fig-result\](b). This demonstrates that the shortcut-to-adiabaticity technique works successfully. The fidelity is improved for both $B=0.5K$ and $K$. Contrary to the results without the shortcut-to-adiabaticity technique, the fidelity is higher for larger $B$, and the corresponding infidelity is lower than 1% when $B=K$ (See Table \[table-result\]). Thus, the shortcut-to-adiabaticity technique can significantly improve the quality of the traveling cat state.
Internal loss
=============
So far, internal loss of the KPO has not been taken into account. However, any actual devices have internal loss, and it degrades the coherence of the cat states. Here we briefly examine the effect of the internal loss in the superconducting-circuit implementation of the KPO [@Nigg2017a; @Puri2017b; @Lin2014a].
Assuming that $K/(2\pi)=10$ MHz and $\omega_{\mathrm{KPO}}/(2\pi)=10$ GHz as typical values, $\kappa_{\mathrm{ex}}=0.2K$ in the present simulations corresponds to the external quality factor, $Q_{\mathrm{ex}}=\omega_{\mathrm{KPO}}/\kappa_{\mathrm{ex}}$, of $5\times 10^3$. On the other hand, the probability of losing a photon inside the KPO, which gives an upper bound on the infidelity due to the internal loss, is approximately given by $\kappa_{\mathrm{in}} I_t$, where $\kappa_\mathrm{in}$ is the internal-loss rate and ${I_t=\int_0^T \! \langle a^{\dagger} a \rangle \, dt}$. In Table \[table-result\], ${I_t \approx 10 K^{-1}=2\kappa_{\mathrm{ex}}^{-1}}$ in all the cases. Thus, in order to have an intra-KPO photon-loss probability below, e.g., 10%, the required condition is $\kappa_{\mathrm{in}} I_t \approx 10 \kappa_{\mathrm{in}}/K \leq 0.1$, which is equivalent to the internal quality factor of $Q_{\mathrm{in}} \geq 10^5$. These values of $Q_{\mathrm{ex}}$ and $Q_{\mathrm{in}}$ seem feasible with current technologies.
Conclusion
==========
We have shown that the cat states deterministically generated inside a KPO via quantum adiabatic evolution can be released into an output mode by controlling the pump amplitude properly. Thus, on-demand generation of traveling cat states can be realized using a KPO. We have further shown that a shortcut-to-adiabaticity technique, where the phase of the pump field is controlled dynamically in time, can improve the quality of the traveling cat state significantly. The traveling cat states generated by a KPO can be directly observed by, e.g., homodyne or heterodyne detection of the output field. Thus, this can be used for experimentally demonstrating the ability of a KPO to generate cat states deterministically.
Acknowledgments {#acknowledgments .unnumbered}
===============
HG thanks Kazuki Koshino for his suggestion. This work was supported by JST ERATO (Grant No. JPMJER1601).
Numerical simulation method
===========================
In the simulation presented in the main text, we truncate the photon number in the output mode at 6, which is sufficiently large to express a cat state with average photon number of 2. Then, the state vector $|\psi \rangle$ describing the coupled system is represented as follows: $$\begin{aligned}
|\psi \rangle
&=
\sum_{n=0}^{N_0} \psi_0 (n) |n\rangle |0\rangle
+
\sum_{n=0}^{N_1} \sum_{j_1=1}^{J} \psi_1 (n,j_1) |n\rangle |j_1\rangle
+
\sum_{n=0}^{N_2} \sum_{j_1=1}^{J} \sum_{j_2=1}^{j_1} \psi_2 (n,j_1,j_2) |n\rangle |j_1,j_2\rangle
\nonumber
\\
&+
\sum_{n=0}^{N_3} \sum_{j_1=1}^{J} \sum_{j_2=1}^{j_1} \sum_{j_3=1}^{j_2}
\psi_3 (n,j_1,j_2,j_3) |n\rangle |j_1,j_2,j_3\rangle
+
\sum_{n=0}^{N_4} \sum_{j_1=1}^{J} \sum_{j_2=1}^{j_1} \sum_{j_3=1}^{j_2} \sum_{j_4=1}^{j_3}
\psi_4 (n,j_1,j_2,j_3,j_4) |n\rangle |j_1,j_2,j_3,j_4\rangle
\nonumber
\\
&+
\sum_{n=0}^{N_5} \sum_{j_1=1}^{J} \sum_{j_2=1}^{j_1} \sum_{j_3=1}^{j_2} \sum_{j_4=1}^{j_3} \sum_{j_5=1}^{j_4}
\psi_5 (n,j_1,j_2,j_3,j_4,j_5) |n\rangle |j_1,j_2,j_3,j_4,j_5\rangle
\nonumber
\\
&+
\sum_{n=0}^{N_6} \sum_{j_1=1}^{J} \sum_{j_2=1}^{j_1} \sum_{j_3=1}^{j_2} \sum_{j_4=1}^{j_3} \sum_{j_5=1}^{j_4} \sum_{j_6=1}^{j_5}
\psi_6 (n,j_1,j_2,j_3,j_4,j_5,j_6) |n\rangle |j_1,j_2,j_3,j_4,j_5,j_6\rangle,
\label{eq-psi}\end{aligned}$$ where the first and second ket vectors represent the Fock states of the KPO and the output mode, respectively, and $N_l$ ($l=0, 1, \ldots , 6$) is the number at which the photon number in the KPO is truncated when the photon number in the output mode is $l$. In the present simulations, we set $N_0=N_1=N_2=6$, $N_3=5$, $N_4=4$, $N_5=3$, and $N_6=2$. The second ket vector is defined with the creation operators, e.g., as follows: $$\begin{aligned}
|j_1,j_2,j_3\rangle
=\mathcal{N}(j_1,j_2,j_3) \tilde{b}^{\dagger}_{j_1} \tilde{b}^{\dagger}_{j_2} \tilde{b}^{\dagger}_{j_3} |0\rangle,\end{aligned}$$ where the normalization factor $\mathcal{N}$ is defined as $$\begin{aligned}
\mathcal{N}(j_1,j_2,j_3)
=
\left\{
\begin{matrix}
1 & \cdots & j_1>j_2>j_3, \\
1/\sqrt{2!} & \cdots & j_1=j_2>j_3, \\
1/\sqrt{2!} & \cdots & j_1>j_2=j_3, \\
1/\sqrt{3!} & \cdots & j_1=j_2=j_3. \\
\end{matrix}
\right.\end{aligned}$$ Using this representation, we numerically solved the Schrödinger equation with the Hamiltonian in Eq. (8) in the main text. Since the Hamiltonian includes only one of $\{ \tilde{b}_j \}$, the corresponding Schrödinger equation is simple. Moreover, at time $t \in [z_{j-1}, z_j]$, $|\psi \rangle$ includes only the output-mode photons satisfying $j_1 \le j$ in Eq. (\[eq-psi\]). This enables fast implementation of the simulation.
Since the photon number in the KPO is large (small) in the first (second) half of the whole process, we set correspondingly the intervals $\Delta z_j$ to small (large) values. More concretely, we set $\Delta z_j=(T/2)/(4J/5)$ for $j=1, 2, \cdots, 4J/5$ and $\Delta z_j=(T/2)/(J/5)$ for $j=4J/5+1, 4J/5+2, \cdots, J$. (So we set $J$ to multiples of 5.) We use the fourth-order Runge-Kutta method for numerically solving the Schrödinger equation, where the time steps are set to about $0.1K^{-1}$. These time steps are defined by dividing $\Delta z_j$ by appropriate integers.
As mentioned in the main text, We set $J=80$ in the present simulations. The following results show that this value of $J$ is sufficiently large. Figure \[fig-J\] shows the $J$ dependence of the final photon number, $n_{\mathrm{out}}=\sum_{j=1}^J \langle \tilde{b}_j^{\dagger} \tilde{b}_j \rangle$, in the output mode. These data, the circles in Fig. \[fig-J\], are well fitted with $n_0 - b/J$, the solid lines in Fig. \[fig-J\], where $n_0$ and $b$ are the fitting parameters. This is natural because the position discretization is the first-order approximation with respect to $J^{-1}$. The dashed lines in Fig. \[fig-J\] show $n_{\mathrm{out}}=n_0$, which are the estimated values of $n_{\mathrm{out}}$ in the limit $J \to \infty$. From the small discrepancies between the data and $n_0$, indicated by arrows in Fig. \[fig-J\], the numerical errors due to finite $J$ are estimated to be less than 1%. This indicates that $J$ is sufficiently large.
![$J$ dependence of the final photon number, $n_{\mathrm{out}}=\sum_{j=1}^J \langle \tilde{b}_j^{\dagger} \tilde{b}_j \rangle$, in the output mode. (a)–(d) correspond to Figs. 2(a)–2(d), respectively, in the main text. Circles represent simulation results. Solid lines are fitted curves with $n_{\mathrm{out}}(J) = n_0 - b/J$, where $n_0$ and $b$ are the fitting parameters. Horizontal dashed lines represent $n_{\mathrm{out}}(J) = n_0$. Arrows indicate the discrepancies between the data and $n_0$.[]{data-label="fig-J"}](J.eps){width="12cm"}
Pulse shape control using low-pass filters
==========================================
In the simulation presented in the main text, we define the pulse shape of the pump amplitude $p(t)$ as the output of the fourth-order low-pass filter (LPF) \[29\] with the input ${p_{\mathrm{in}}(t)=K A_p \exp (-\kappa_{\scriptsize{\mbox{ex}}} t)}$. The input-output relation of a LPF is given by $p_{\mathrm{out}}(t)
=\int_0^t \! B e^{-B(t-s)} p_{\mathrm{in}}(s) \, ds$ \[29\], where $B$ is the bandwidth of the LPF and $p_{\mathrm{in}}(s)=0$ (${s<0}$) is assumed. Note that $\dot{p}_{\mathrm{out}}(t)
=-B [p_{\mathrm{out}}(t) - p_{\mathrm{in}}(t)]$, where the dot denotes the time derivative. Thus, we can calculate the output of the LPF by numerically solving this differential equation. The $n$-th order LPF is defined as the output of the LPF the input of which is the output of the $(n-1)$-th order LPF.
Shortcut to adiabaticity for KPO {#sec-shortcut}
================================
Here we derive the approximate counterdiabatic Hamiltonian given by Eqs. (11) and (12) in the main text and provide numerical evidence for its validity.
First, using the completeness relation $\displaystyle \sum_n |\phi_n \rangle \langle \phi_n |=I$ ($I$ is the identity operator), the counterdiabatic Hamiltonian in Eq. (10) in the main text is rewritten as follows: $$\begin{aligned}
H_{\mathrm{counter}} =
\frac{i}{2} \sum_n \left(
|\dot{\phi}_n \rangle \langle \phi_n| - |\phi_n \rangle \langle \dot{\phi}_n|
\right).
\label{eq-Hcounter0}\end{aligned}$$
Among $\{ |\phi_n \rangle \}$, we are interested only in the following even cat state: $$\begin{aligned}
|{C_+ (p(t))} \rangle =
\frac{\left| {\sqrt{p(t)/K}} \right\rangle + \left| {-\sqrt{p(t)/K}} \right\rangle}{\sqrt{2(1+e^{-2p(t)/K})}}.\end{aligned}$$ Note that $\displaystyle H_{\scriptsize{\mbox{KPO}}}(t) |C_+ (p(t)) \rangle = \frac{p(t)^2}{2K} |C_+ (p(t)) \rangle$, and therefore the even cat state is one of the energy eigenstates. Disregarding the energy eigenstates other than $|C_+ \rangle$, the counterdiabatic Hamiltonian in Eq. (\[eq-Hcounter0\]) is approximately given by $$\begin{aligned}
H_{\mathrm{counter}} \approx
\frac{i}{2} \left(
|\dot{C}_+ \rangle \langle C_+| - |C_+ \rangle \langle \dot{C}_+|
\right).
\label{eq-Hcounter}\end{aligned}$$
Using the odd cat state $$\begin{aligned}
|C_- (p(t)) \rangle =
\frac{\left| {\sqrt{p(t)/K}} \right\rangle - \left| {-\sqrt{p(t)/K}} \right\rangle}{\sqrt{2(1-e^{-2p(t)/K})}},\end{aligned}$$ $|\dot{C}_+ \rangle$ becomes $$\begin{aligned}
|\dot{C}_+ \rangle =
-\frac{\dot{p}}{2K} \tanh \! \frac{p}{K} \, |C_+ \rangle
+
\frac{\dot{p}}{2\sqrt{Kp}} \sqrt{\tanh \! \frac{p}{K}} \, a^{\dagger} |C_- \rangle.
\label{eq-dotCplus}\end{aligned}$$
Substituting Eq. (\[eq-dotCplus\]) into Eq. (\[eq-Hcounter\]), we obtain $$\begin{aligned}
H_{\mathrm{counter}}
\approx
-\frac{i}{2} \left(
\frac{\dot{p}}{2K} \tanh \! \frac{p}{K} \,
|C_+ \rangle \langle C_+|
+
\frac{\dot{p}}{2\sqrt{Kp}} \sqrt{\tanh \! \frac{p}{K}} \,
|C_+ \rangle \langle C_-| a
\right)
+
\mbox{H.c.},
\label{eq-Hcounter2}\end{aligned}$$ where H.c. denotes the Hermitian conjugate.
Using $\displaystyle a^2|C_+ \rangle = \frac{p}{K} |C_+ \rangle$ and $\displaystyle \left( |C_+ \rangle \langle C_-| a \right) |C_+ \rangle
= \sqrt{\frac{p}{K} \tanh \! \frac{p}{K}} \, |C_+ \rangle
= \sqrt{\frac{K}{p} \tanh \! \frac{p}{K}} \, a^2 |C_+ \rangle$, $H_{\mathrm{counter}}$ acting on $|C_+ \rangle$ is approximated as follows: $$\begin{aligned}
H_{\mathrm{counter}}
\approx
-\frac{i}{2} \left(
\frac{\dot{p}}{2p} \tanh \! \frac{p}{K} \, a^2
+
\frac{\dot{p}}{2\sqrt{Kp}} \sqrt{\tanh \! \frac{p}{K}} \,
\sqrt{\frac{K}{p} \tanh \! \frac{p}{K}} \, a^2
\right)
+
\mbox{H.c.}
=
-\frac{i}{2} \frac{\dot{p}}{p} \tanh \! \frac{p}{K} \, a^2
+
\mbox{H.c.}
\label{eq-Hcounter3}\end{aligned}$$ Thus, we obtain Eqs. (11) and (12) in the main text.
To confirm the validity of the above derivation, we performed numerical simulations of a KPO without the output coupling, where the pump amplitude $p(t)$ is increased linearly from 0 to $2K$ at time $10K^{-1}$. The results are summarized in Fig. \[fig-shortcut\] together with the results in the cases without shortcut-to-adiabaticity technique and with the shortcut-to-adiabaticity technique proposed in Ref. 15. The technique proposed in Ref. 15 also uses an imaginary pump amplitude $p'(t)$, but it is defined as $$\begin{aligned}
p'(t)
=\frac{2\dot{\alpha}_0 \sqrt{1-2e^{-2|\alpha_0|^2}}}{1+2\alpha_0}
=
\frac{\dot{p} \sqrt{1-2e^{-2p/K}}}{\sqrt{Kp}+2p},
\label{eq-pi-Puri}\end{aligned}$$ where $\alpha_0 = \sqrt{p/K}$ is used. While the results with the technique in Ref. 15 exhibit oscillations, the results with our technique change monotonically and the final fidelity is almost perfect. These results clearly show the usefulness of our shortcut-to-adiabaticity technique for a KPO.
![Simulation results for shortcut to adiabaticity. (a) Pump amplitudes, $p(t)$ (black long-dashed line), $p'(t)$ in our method (cyan solid line), and $p'(t)$ in Ref. 15 (red short-dashed line). (b) Average photon number in KPO, $\langle a^{\dagger} a \rangle$. (c) Fidelity between the final state and the ideal even cat state with amplitude of $\sqrt{2}$. (d) Magnification of (c) around the final time. In (b)–(d), cyan solid, red short-dashed, and black long-dashed lines correspond to our method, the method proposed in Ref. 15, and the case of no $p'(t)$ (without shortcut-to-adiabaticity technique), respectively.[]{data-label="fig-shortcut"}](shortcut.eps){width="12cm"}
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A shortcut-to-adiabaticity technique for a KPO has been proposed in Ref. . This also uses an imaginary pump amplitude $p'(t)$, but the definition of $p'(t)$ is different from the present one in Eq. (\[eq-pi\]). We found that our definition of $p'(t)$ outperforms the definition proposed in Ref. . See Appendix \[sec-shortcut\] for the details.
| 0 |
---
abstract: 'Shell model calculations are performed for magnetic dipole excitations in $^8\mbox{Be}$ and $^{10}\mbox{Be}$ in which all valence configurations plus $2\hbar\omega$ excitations are allowed (large space). We study both the orbital and spin excitations. The results are compared with the ‘valence space only’ calculations (small space). The cumulative energy weighted sums are calculated and compared for the $J=0^+$ $T$=$0$ to $J=1^+$ $T$=$1$ excitations in $^8\mbox{Be}$ and for $J=0^+$ $T$=$1$ to both $J=1^+$ $T$=$1$ and $J$=$1^+$ $T$=$2$ excitations in $^{10}\mbox{Be}$. We find for the $J=0^+$ $T$=$1$ to $J=1^+$ $T$=$1$ isovector transitions in $^{10}\mbox{Be}$ that the summed strength in the space is less than in the space. We find that the high energy energy-weighted isovector orbital strength is smaller than the low energy strength for transitions in which the isospin is changed, but for $J=0^+$ $T$=$1$ to $J=1^+$ $T$=$1$ in $^{10}\mbox{Be}$ the high energy strength is larger. We find that the low lying orbital strength in $^{10}\mbox{Be}$ is anomalously small, when an attempt is made to correlate it with the $B(E2)$ strength to the lowest $2^+$ states. On the other hand a sum rule of Zheng and Zamick which concerns the total $B(E2)$ strength is reasonably satisfied in both $^8\mbox{Be}$ and $^{10}\mbox{Be}$. The Wigner supermultiplet scheme is a useful guide in analyzing shell model results. In $^{10}Be$ and with a $Q \cdot Q$ interaction the $T=1$ and $T=2$ scissors modes are degenerate, with the latter carrying $\frac{5}{3}$ of the $T=1$ strength.'
---
M. S. Fayache, S. Shelly Sharma\* and L. Zamick
[*Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855*]{}
[\**Departamento de Física, Universidade Estadual de Londrina, Londrina, Parana, 86051-970, Brazil*]{}
The Experimental Situation
==========================
From our perspective, much experimental information is lacking in the nuclei $^8\mbox{Be}$ and $^{10}\mbox{Be}$. For example, no $J=1^+$ states have been identified in $^{10}\mbox{Be}$. The $B(E2)$ from the $2_1^+$ state of $^8\mbox{Be}$ to the $J=0^+$ ground state is not known -this is understandable because of the large decay width to two alpha particles.
The following states and their properties are of interest to us:
[**[(a) $^8\mbox{Be}$]{}**]{}
The $J=2_1^+$ state has an excitation energy of $3.04$ $MeV$. The $J=4_1^+$ state is at $11.4$ $MeV$. This is consistent with an $J(J+1)$ spectrum of a rotational band, but it should be recalled that any spin-independent interaction gives an $J(J+1)$ spectrum in the $p$ shell. The $J=1_1^+$ $T=1$ state, which we discussed extensively in a previous publication [@fay] is at $17.64$ $MeV$ and the $J=1_1^+$ $T=0$ state is at $18.15$ $MeV$.
The $B(M1)$ from the $17.64$ $MeV$ state to the ground state has a strength of $0.15$ $W.u.$ or $B(M1)$$\put(2,8.){\vector(0,-9){9}}$ $=$ $0.27{\mu_N}^2$. The $B(M1)$ of this state to the $2_1^+$ state is $0.12$ $W.u.$ or $B(M1)$$\put(2,8.){\vector(0,-9){9}}$ $=$ $0.21{\mu_N}^2$ [@ajz]. Of course $B(M1)\uparrow$=$3B(M1)\downarrow$
[**[(b) $^{10}\mbox{Be}$]{}**]{}
The $J=2_1^+$ state is at $3.368$ $MeV$ and the $J=2_2^+$ state at $5.960$ $MeV$. We recall that with a spin independent interaction the $2_1^+$ and $2_2^+$ would be degenerate. The experimental spectrum looks more vibrational. However, the values of $B(E2)$$\put(2,0.){\vector(0,9){9}}$ from the $J=0^+$ ground state to the $2_1^+$ state is very strong: $B(E2)$$\put(2,0.){\vector(0,9){9}}$ $=$ $52$ $e^2fm^4$. Raman et. al. deduce from this a deformation parameter $\beta$ $=$ $1.13$ [@ram]. As mentioned above, there are no $J=1^+$ states mentioned in the compilation of Ajzenberg-Selove [@ajz]. Also the $J=4^+$ state has not been found.
The Interactions
================
We have chosen two types of interactions to do the calculations. First we use a short range ‘simplified realistic’ ($x,y$) interaction previously used by Zheng and Zamick [@ann], and then we use a long-range quadrupole-quadrupole interaction. By choosing these two extremes, we make sure that the results we obtain are not too dependent on the specifics of the model.
In more detail, the($x,y$) Hamiltonian is:
$$H=\sum T_i + \sum_{i<j}V(ij)$$
where
$$V=V_c+xV_{so}+yV_t$$
with $c \equiv$central, $s.o. \equiv$spin-orbit, and $t
\equiv$tensor.
For ($x,y$)=(1,1) the matrix elements of this interaction are close to those of realistic G matrices such as Bonn A. We can study the effects the spin-orbit and tensor interactions by varying $x$ and $y$.
Note that we do not add any single-particle energies to the above Hamiltonian. Rather, we let the single-particle energies be implicitly generated by $H$. Hence, if we set $x$=0 i.e. turn off the spin-orbit interaction, we will also be turning off the one-body spin-orbit splitting coming from this interaction.
As a counterpoint, we repeat all the calculations with the $Q \cdot Q$ Hamiltonian
$$H_{Q}=\sum_i \frac{p_i^2}{2m} + \frac{1}{2}m\omega^2r_i^2
-\chi\sum_{i<j}Q \cdot Q$$
Note that we have added the term $\frac{1}{2}m\omega^2r^2$ which is not present for the ($x,y$) interaction. The reason for this is that $Q \cdot Q$ cannot generate any single-particle potential energy splitting whereas the ($x,y$) interaction can.
Whereas the ($x,y$) interaction like all realistic interactions is of short range, the $Q \cdot Q$ interaction is long range. Yet, as we shall see some of the results (but not all) are rather similar for the two interactions. Since the best milieu for the existence of scissors mode excitations (orbital magnetic dipole excitations) are strongly deformed systems, one would expect the $Q \cdot Q$ Hamiltonian to yield strong scissors modes. But is this also true for the realistic interaction ? We will address this question. Another motivation for introducing the $Q \cdot Q$ Hamiltonian is that it is easy to establish a connection via energy weighted sum rule techniques between isovector orbital $B(M1)$’s and isoscalar and isovector $B(E2)$’s.
We shall be performing the calculations, not only in the $0\hbar\omega$ space (small space) but also in a space which allows $2\hbar\omega$ admixtures (large space). For the $Q \cdot Q$ Hamiltonian in the small space the energy matrix is proportional to $\chi$. Hence the energy eigenvalues depend linearly on $\chi$, but the eigenfunctions (and $B(M1)$’s and $B(E2)$’s) are independent of the interaction strength. In a large space calculation there is one more parameter: the energy splitting induced by $\frac{p^2}{2m} + \frac{1}{2}m\omega^2r^2$ i.e. $2\hbar\omega$. Thus the wave function and the corresponding $B(M1)$’s and $B(E2)$’s will also depend on $\chi$.
We have chosen values of $\chi$ appropriate for the large space calculation. We also use these same values in the small space. One can argue that in the small space one should use a renormalized value $\chi'$ which is close to twice $\chi$. However, as mentioned above, the wave function and hence $B(M1)$ and $B(E2)$ will not change, only the energies. By choosing the same $\chi$ in the two spaces it is easier to see what the differences in the two calculations are. The values of $\chi$ are 0.5762 $\frac{MeV}{fm^4}$ for $^8Be$ and 0.3615 $\frac{MeV}{fm^4}$ for $^{10}Be$.
The summed magnetic dipole strength
===================================
In Table I we give the summed magnetic dipole strength ($\sum_i~ B(M1:
0_1^+,~T=1 \rightarrow 1_i^+,~T=1)$ and $\sum_i~ B(M1:
0_1^+,~T=1 \rightarrow 1_i^+,~T=2)$ ) broken up into isoscalar and isovector and spin and orbit and where we use the ($x,y$) interaction with $x=1$, $y=1$. We first discuss the behaviour as a function of the size of the model space. Later we will make a comparison of the behaviour in $^8\mbox{Be}$ and $^{10}\mbox{Be}$. There are striking differences for the two nuclei.
Our small space calculation consists of all configurations of the form $(0s)^4(0p)^4$ for $^8\mbox{Be}$ and $(0s)^4(0p)^6$ for $^{10}\mbox{Be}$. The large space consists of those configurations plus $2\hbar\omega$ excitations. Thus one can either excite two particles to the next major shell or excite one particle through two major shells. We also give results for the summed strength in the [*low-large space*]{} -this is the low energy part of the large space covering an energy range more or less equal to that of the small space. It is easy to identify the low energy sector because there is a fairly wide plateau in the summed strength which separates the low energy rise from the high energy rise.
Usually the large space summed strength is somewhat larger than the small space strength e.g. for the isovector orbital strength in $^8\mbox{Be}$ the values shown in Table $I$ are $0.6701$ $\mu_N^2$ and $0.7283$ $\mu_N^2$ respectively. But there is one glaring exception. For the case of $J^{\pi}=0^+$ $T=1$ $\rightarrow$ $J^{\pi}=1^+$ $T=1$ transitions in $^{10}\mbox{Be}$, the summed isovector spin strength in the large space is $2.08~10^{-2}$ $\mu_N^2$ but in the small space it is $2.34~10^{-2}$ $\mu_N^2$. For the orbital strength it is the other way around but for the physical case ($g_{l\pi}=1$, $g_{l\nu}=0$, $g_{s\pi}=5.586$, $g_{s\nu}=-3.826$) the spin prevails and the summed strength in the $1.952$ $\mu_N^2$ is less than in the $2.09$ $\mu_N^2$.
Thus it is not always true that the net result of higher shell admixtures is to rob strength from the low energy sector and move it to higher energies. In some cases the total strength gets depleted.
We next compare the low energy sum in the large space with the small space sum. In all cases the latter is larger than the low energy sum, thus indicating that there is a quenching of the low energy part due to higher shell admixtures. The hindrance factor $[(low~large)/small]$ is $0.88$ for the isovector orbital in $^8\mbox{Be}$, $0.73$ for the isovector spin in $^8\mbox{Be}$, $0.77$ for the total $M1$ in $^{10}\mbox{Be}$ etc.
Note that the total $M1$’s for $^{10}\mbox{Be}$ are somewhat larger than for $^8\mbox{Be}$. However there is a dramatic drop in the orbital strength in $^{10}\mbox{Be}$ relative to that in $^8\mbox{Be}$. The large space summed orbital strength for $^8\mbox{Be}$ is $0.73$ $\mu_N^2$ whereas for $^{10}\mbox{Be}$ (to $J=1^+$ $T=1$ and $T=2$) the value is ($0.196~+~0.183$)=$0.38$ $\mu_N^2$. In the low energy sector the $^8\mbox{Be}$ value is $0.59$ $\mu_N^2$ whereas for $^{10}\mbox{Be}$ it is $0.10~+~0.13~=~0.23$ $\mu_N^2$, less than half the value for $^8\mbox{Be}$.
From the systematics of orbital transitions in heavy nuclei one concludes that the proper milieu for isovector orbital transitions is strongly deformed nuclei. Can one conclude that $^{10}\mbox{Be}$ is not strongly deformed? The answer, by examining the tables of Raman et. al. [@ram] is no! There is a strong $E2$ connecting the $0_1^+$ and $2_1^+$ states in $^{10}\mbox{Be}$. From this the authors conclude that the deformation parameter $\beta$ is about $1.13$ -quite enormous. Of course $^8\mbox{Be}$ might have an even stronger $E2$ transition -there is no data on this in the Raman paper [@ram], probably because of the rapid decay of the $2_1^+$ state into two alpha particles.
The cumulative energy weighted strength for orbital transitions in $^8\mbox{Be}$ and $^{10}\mbox{Be}$
=====================================================================================================
In this section we present results and figures for the cumulative energy weighted sum of magnetic dipole strength.
We are motivated in so doing by various energy-weighted sum rules that have been developed e.g. by Zheng and Zamick [@zz], Heyde and de Coster [@ibm], Moya de Guerra and Zamick [@dz1], Nojarov [@no], Hamamoto and Nazarewicz [@hz] and Fayache and Zamick [@fay]. We will focus in particular on the orbital strength for which the operator is ($\vec{L_{\pi}}-\vec{L_{\nu}}$)/2. In a previous publication we presented results for the ($x,y$) interaction with $x$=$1$, $y$=$1$ for $^8\mbox{Be}$ [@fay]. In this work the quadrupole-quadrupole interaction results are compared with the ($x,y$) interaction results, and furthermore we extend the calculation to $^{10}\mbox{Be}$. In the latter nucleus one does not have $N=Z$ and this leads to big differences.
Whereas in $^8\mbox{Be}$ there is only one isospin channel for isovector transitions $J=0_1^+$ $T=0$ $\rightarrow$ $J=1^+$ $T=1$, in $^{10}\mbox{Be}$ there are two: $J=0_1^+$ $T=1$ $\rightarrow$ $J=1^+$ $T=1$ and $J=0_1^+$ $T=1$ $\rightarrow$ $J=1^+$ $T=2$. The low lying $J=1^+$ $T=1$ states in $^{10}\mbox{Be}$ are expected to have much smaller excitation energies than the $J=1^+$ $T=1$ states in $^8\mbox{Be}$. This makes it easier to look for such states experimentally.
In Table II we present the results for the summed energy weighted strengths for the ($x,y$) interaction. As a crude orientation it should be noted that simple models e.g. the Nilsson model used by de Guerra and Zamick [@dz1] and a model by Nojarov [@no] would have the ‘large’ result be twice the ‘low large’ result. On the other hand Hamamoto and Nazarewicz [@hz] have argued that the ‘large’ result should be much more than twice the ‘low large’ result. The actual ratios for the ($x,y$) and $Q \cdot Q$ interactions for this calculation (all $0\hbar\omega$ configurations plus $2\hbar\omega$ excitations) are
----------- ------------------------------------------- --------- -------------
($x,y$) $Q \cdot Q$
$^8Be$ $J=0^+$ $T=0$ $\rightarrow$ $J=1^+$ $T=1$ 1.75 1.37
$^{10}Be$ $J=0^+$ $T=1$ $\rightarrow$ $J=1^+$ $T=2$ 2.00 1.52
$^{10}Be$ $J=0^+$ $T=1$ $\rightarrow$ $J=1^+$ $T=1$ 3.22 3.68
$^{10}Be$ (‘$T=1$’ + ‘$T=2$’) 2.52 2.33
----------- ------------------------------------------- --------- -------------
For $^{10}Be$ we should actually compare the theoretical models with the combined result (‘$T=1$’ + ‘$T=2$’).
These results indicate that the simple models are not too bad as a first orientation but there are fluctuations -sometimes the ratio is less than two, sometimes greater. We will discuss these matters in more detail in the context of the figures.
In Figs $1$ and $2$ we show the cumulative energy weighted strength distributions in $^8\mbox{Be}$ for the ($x,y$) interaction and for the Hamiltonian $H_{Q}$ i.e. the quadrupole-quadrupole interaction. The results for the two interactions are quite similar. The outstanding feature is that there are two rises separated by a rather wide plateau. For the ($x,y$) interaction the first rise is to a plateau at about $12$ $\mu_N^2MeV$ followed by a second rise to about $20.8$ $\mu_N^2MeV$. For the $Q \cdot Q$ interaction the first plateau is at $10.25$ $\mu_N^2MeV$ and the second at $14$ $\mu_N^2MeV$. A simple self-consistent Nilsson model was shown to give the second plateau at twice the energy of the first plateau [@zz] [@ibm]. That is to say the high energy rise was equal to the low energy rise. In the more detailed calculations performed here the high energy rise is less than the low energy rise.
We next turn to $^{10}\mbox{Be}$. Here there are two channels: $J=0^+$ $T=1$ $\rightarrow$ $J=1^+$ $T=1$ and $J=0^+$ $T=1$ $\rightarrow$ $J=1^+$ $T=2$. Let us discuss the latter channel first. The behaviour for $T=2$ in $^{10}\mbox{Be}$ is similar to that for $T=1$ in $^8\mbox{Be}$. As shown in Figs $3$ and $4$ for the ($x,y$) and $Q \cdot Q$ interactions respectively, there are two rises separated by a plateau and here the second rise is about twice the first rise for the ($x,y$) interaction. For the $Q \cdot Q$ interaction (with $\chi=0.3615$) the low energy rise is to $1.7$ $\mu_N^2MeV$ and the next rise is to $2.6$ $\mu_N^2MeV$ -only $1.5$ to one.
There is a big difference in the cumulative energy weighted distributions, shown in Figs 5 and 6, for the $J=0^+$ $T=1$ $\rightarrow$ $J=1^+$ $T=1$ channel. For the ($x,y$) interaction the first plateau (at about $2.5$ $\mu_N^2MeV$) is not very flat, but the most outstanding feature in the curve is that the high energy rise is much larger than the low energy rise. The energy weighted sum reaches up to about $8$ $\mu_N^2MeV$. Thus the high energy rise is over three time the low energy rise. For the $Q \cdot Q$ interaction, the first plateau is better defined -it is located at $1$ $\mu_N^2MeV$ and the cumulative sum extends to about $3.8$ $\mu_N^2MeV$.
The Zheng-Zamick Sum Rule
=========================
Energy weighted sum rules for magnetic dipole transitions, be they spin or orbital, are highly model dependent. An energy weighted sum rule for [*isovector orbital*]{} magnetic dipole transitions for the quadrupole-quadrupole interaction $Q \cdot Q$ was developed by Zheng and Zamick [@zz]. This was motivated by the work of the Darmstadt group [@zr] [@rich] showing a linear relationship between summed orbital $B(M1)$ strength and the square of the deformation parameter i.e. $\delta^2$.
The result was
$$\sum_n (E_n-E_0)B(M1)_o=\frac{9\chi}{16\pi}\left\{\sum_i[B(E2,0_1
\rightarrow 2_i)_{isoscalar} - B(E2,0_1 \rightarrow 2_i)_{isovector}]
\right\}~~~~(EWSR)$$
where $B(M1)_o$ is the value for the isovector orbital $M1$ operator ($g_{l\pi}=0.5$ $g_{l\nu}=-0.5$ $g_{s\pi}=0$ $g_{s\nu}=0$) and the operator for the $E2$ transitions is $\sum_{protons}e_pr^2Y_2$ $+$ $\sum_{neutrons}e_nr^2Y_2$ with $e_p=1$, $e_n=1$ for the isoscalar transition, and $e_p=1$, $e_n=-1$ for the isovector transition.
Let us now describe in detail how this sum rule works. The sum rule should work for single-shell as well mixed-shell space.
We first consider the case of $^8\mbox{Be}$. We have the following values in a large space calculation for the $H_{Q}$ interaction corresponding to orbital $M1$ excitations from the $J=0^+$ $T=0$ ground state to all $1^+$ $T=1$ states:
= 1. Energy weighted isovector orbital $M1$ strength: = = $\mu_N^2MeV$\
2. The isoscalar summed strength $B(E2;1,1)$: $237.46$ $e^2fm^4$\
3. The isovector summed strength $B(E2;1,-1)$: $89.611$ $e^2fm^4$\
4. The right hand side ($\frac{9\chi}{16\pi}=0.1032$): $\mu_N^2MeV$.\
We don’t get exact agreement ($14.04$ $\mu_N^2MeV$ vs. $15.25$ $\mu_N^2MeV$) but it is reasonably close. One possible reason for the disagreement is that spurious states have been removed and/or that only $2\hbar\omega$ excitations to the $\Delta N=2$ shell are taken into account [@ibm] [@zr].
There have been other approaches, especially in the context of the Interacting Boson Model [@ibm] which relate the energy weighted orbital magnetic sum to the $B(E2)$ of the lowest $2^+$ state. As a matter of curiosity we shall examine in our calculation what happens if we take only the lowest $2^+$ state in the right hand side of the sum rule ($EWSR$).
For the $2_1^+$ state in $^8\mbox{Be}$ we obtain (in our large space calculation) $B(E2;1,1)=196.76$ and $B(E2;1,-1)=0$ (because the $2_1^+$ state has $T=0$). The right hand side becomes $20.30$ $\mu_N^2MeV$. We get a answer using the lowest $2^+$ state than we do if we use all $2^+$ states in the $0\hbar\omega$ and $2\hbar\omega$ space. The reason for this is that when the lowest $2^+$ state is excluded, the isovector $B(E2)$ is larger than the isoscalar $B(E2)$.
Can we also get agreement for the sum rule in the small space $(0s)^4(0p)^4$ for $^8\mbox{Be}$? Now the numbers are:
= 1. Energy weighted isovector orbital $M1$ strength: = = $\mu_N^2MeV$\
2. The isoscalar summed strength $B(E2;1,1)$: $72.54$ $e^2fm^4$\
3. The isovector summed strength $B(E2;1,-1)$: $10.37$ $e^2fm^4$\
4. The right hand side ($\frac{9\chi}{16\pi}=0.1032$): $\mu_N^2MeV$.\
We get perfect agreement.
We next consider $^{10}\mbox{Be}$. The relevant numbers for the large space calculation are:
= 1. Energy weighted $M1$ strength: = $J=0^+,~T=1$ $\rightarrow$ $J=1^+,~T=1$ = $ 3.811 $ = $\mu_N^2MeV$\
$J=0^+,~T=1$ $\rightarrow$ $J=1^+,~T=2$ $ 2.602 $ $\mu_N^2MeV$\
Left Hand Side $\mu_N^2MeV$\
2. $\sum B(E2;1,1)$ $J=0^+,~T=1$ $\rightarrow$ $J=2^+,~T=1$ $251.4$ $e^2fm^4$\
3.(a) $\sum B(E2;1,-1)$ $J=0^+,~T=1$ $\rightarrow$ $J=2^+,~T=1$ $94.02$ $e^2fm^4$\
(b) $\sum B(E2;1,-1)$ $J=0^+,~T=1$ $\rightarrow$ $J=2^+,~T=2$ $48.78$ $e^2fm^4$\
(c) $\sum B(E2;1,-1)$ Total $142.8$ $e^2fm^4$\
4. Right hand Side ($\frac{9\chi}{16\pi}=0.0647$): $\mu_N^2MeV$\
For $^{10}\mbox{Be}$ we are also curious to see what happens if we use only the lowest $2^+$ state in the right hand side of the sum rule. But we have to be careful! It turns out that there is substantial $B(E2)$ strength to the two lowest $J=2^+$ states. This can be understood from the fact that with a $Q \cdot Q$ interaction in a small space calculation ($(0s)^4(0p)^6$) the two lowest $2^+$ states are exactly degenerate. The states belong to the $[f]=[42]$ representation. The $Q \cdot Q$ interaction fails to remove the degeneracy of these states. Another way of stating this is that the ($\lambda \mu$) values for both states are (22), and the allowed values of the $K$ quantum number in the Nilsson scheme are $K=\mu$, $\mu-2$, etc. Thus the $2^+$ states have $K=0$ and $K=2$.
When we go to the large space calculation with a $Q \cdot Q$ interaction, limiting the excitations to $2\hbar\omega$, the degeneracy is removed but the states are still fairly close together. The calculated values are:
= = $E2(1,1)$ = $E2(1,-1)$\
$2^+_1$ $E=2.08$ $MeV$ 64.94 12.32\
$2^+_2$ $E=2.92$ $MeV$ 93.38 10.11\
Thus, using the calculated values of $B(E2)$ for the lowest two $2^+$ $T=1$ states in $^{10}\mbox{Be}$, we get for the right hand side of the sum rule a value of $8.80$ $\mu_N^2MeV$. Again, as in the case of $^8\mbox{Be}$, this is larger than the value $7.03$ $\mu_N^2MeV$ that is obtained by using all $2^+$ $T=1$ and all $2^+$ $T=2$ states.
The corresponding numbers in small space for $^{10}\mbox{Be}$ are:
= 1. Energy weighted $M1$ strength: = $J=0^+,~T=1$ $\rightarrow$ $J=1^+,T=1$ = $ 0.7597 $ = $\mu_N^2MeV$\
$J=0^+,~T=1$ $\rightarrow$ $J=1^+,~T=2$ $1.266$ $\mu_N^2MeV$\
Left Hand Side $\mu_N^2MeV$\
2. $\sum B(E2;1,1)$ $J=0^+,~T=1$ $\rightarrow$ $J=2^+,T=1$ $68.31$ $e^2fm^4$\
3.(a) $\sum B(E2;1,-1)$ $J=0^+,~T=1$ $\rightarrow$ $J=2^+,~T=1$ $33.80$ $e^2fm^4$\
(b) $\sum B(E2;1,-1)$ $J=0^+,T=1$ $\rightarrow$ $J=2^+,T=2$ $3.203$ $e^2fm^4$\
(c) $\sum B(E2;1,-1)$ Total $37.003$ $e^2fm^4$\
4. Right hand Side ($\frac{9\chi}{16\pi}=0.0647$): $\mu_N^2MeV$\
A discussion of the calculated B(E2) values
===========================================
Although the main thrust of this work is on $B(M1)$ values, we have established a connection with $B(E2)$ for the orbital case. It is therefore appropriate to discuss the calculated $B(E2)$ values -comparing the behaviours in $^8\mbox{Be}$ and $^{10}\mbox{Be}$, and comparing the different interactions that have been used (see tables $III$ and $IV$.)
In making the comparison between $^8\mbox{Be}$ and $^{10}\mbox{Be}$ we should lump together the $B(E2)$’s of the first two $2^+$ states in $^{10}\mbox{Be}$ because with the interactions used here -especially $Q \cdot Q$- these states are nearly degenerate. (However, experimentally the states are well separated $E_{2_1^+}= 3.368$ $MeV$ and $E_{2_2^+}= 5.958$ $MeV$). When this is done we find that the $B(E2)$ values in the two nuclei are comparable.
For the ($x,y$) interaction, the calculated (large space) value of $B(E2)$ to the lowest two $2^+$ states in $^{10}\mbox{Be}$ is $22.90$ $e^2fm^4$, whereas it is $25.97$ $e^2fm^4$ to the lowest $2^+$ state in $^8\mbox{Be}$. With the $Q \cdot Q$ interaction the two values are respectively $46.40$ and $49.16$ $e^2fm^4$. One big difference between the two interactions is the ratio of large to small space values for corresponding $B(E2)$ values. In $^8\mbox{Be}$ the ratio of the large sum to the small sum is $1.98$ for the ($x,y$) interaction whereas it is much larger $3.28$ for the $Q \cdot Q$ interaction. There is much more core polarization with the $Q \cdot Q$ interaction than with the ($x,y$) interaction.
There have been many discussions concerning the correlation of summed orbital $M1$ strength and the $B(E2)$ from the $J=0^+$ ground state to the first $2^+$ state. The latter is an indication of the nuclear deformation. We have noted that the calculated values of $B(E2)$ are about the same in $^8\mbox{Be}$ and $^{10}\mbox{Be}$. Thus we would expect the orbital $M1$ strengths in the two nuclei to be about the same.
There is a certain ‘vagueness’ in what is meant by ‘strength’. It is clear that the experiments thus far sample only low energy strengths up to about $6$ $MeV$ in heavy deformed nuclei [@zr] [@rich]. Also some of the theories involve summed strength per se and others involve the energy weighted strength. Rather than enter into deep philosophical discussions about what is meant by strength, we will give a variety of ratios of strength $\frac{^{10}\mbox{Be}}{^8\mbox{Be}}$ in Table $V$. We see that all the ratios, be they non-energy weighted or energy weighted, be they in small spaces or in large spaces, are substantially less than one. In forming the ratios, we added for the numerator ($^{10}\mbox{Be}$) the $J=0^+$ $T=1$ to $J=1^+$ $T=1$ and $J=0^+$ $T=1$ to $J=1^+$ $T=2$ strengths.
A comparison of the $J=1^+$ $\rightarrow$ $0_1^+$ and $J=1^+$ $\rightarrow$ $2_1^+$ Magnetic Dipole Transitions
===============================================================================================================
Let us assume that the $0_1^+$ and $2_1^+$ states are members of a $K=0$ rotational band and that the $1^+$ states have $K=1$. We can then use the rotational formula of Bohr and Mottelson (Eq. 4-92) in their book [@bm] ($K_1=0$, $K_2=1$) (We use the notation $\left[\begin{array}{lll}J_1 & J_2 & J\\ M_1 & M_2 &
M\end{array}\right]$ for a Clebsch-Gordan coefficient):
$$\langle K_2I_2 || M(\lambda) || K_1=0 I_1 \rangle
=\sqrt{2(2I_1+1)}\left[\begin{array}{lll}I_1 & \lambda & I_2\\ 0 & K_2 &
K_2\end{array}\right]
\langle K_2 | M(\lambda,\nu=K_2)|K_1=0\rangle$$
From this we can easily deduce
$$r=\frac{B(M1)_{J=1^+, K=1 \rightarrow J=2^+, K=0}}{B(M1)_{J=1^+, K=1
\rightarrow J=0^+, K=0}} =\frac{1}{2}$$
Note, however, that the experimental ratio for $^8Be$ from the $J=1^+$ $T=1$ state at 17.64 $MeV$ (see section 1) is $\frac{0.12}{0.15}=0.8$. Bohr and Mottelson later discuss corrections to the above simple formula.
We can obtain a value for the above ratio by forming an intrinsic state and projecting out states of angular momentum $J=0$ and $J=2$. We assume an extreme prolate shape for $^8Be$ and put the four valence nucleons ($N\uparrow$, $N\downarrow$, $P\uparrow$, $P\downarrow$) in the lowest Nilsson orbit with $\Lambda=0$. The asymptotic wave function is $NrY_{10}$.
The $J=0$ wave function is
$$N'\sum_L \left[\begin{array}{lll}1 & 1 & L\\0 & 0 &
0\end{array}\right] \left[\begin{array}{lll}1 & 1 & L\\0 & 0 &
0\end{array}\right] \left[\begin{array}{lll}L & L & 0\\0 & 0 &
0\end{array}\right] [L~L]^0 = 0.74536 [0~0]^0~+~0.66666 [2~2]^0$$
where the notation $[L_\pi~L_\nu]^J$ is used.
The $J=2$ wave function is
$$N''\left[\begin{array}{lll}1 & 1 & 0\\0 & 0 &
0\end{array}\right] \left[\begin{array}{lll}1 & 1 & 2\\0 & 0 &
0\end{array}\right] [0~2]^2~+~ \left[\begin{array}{lll}1 & 1 &
2\\0 & 0 &
0\end{array}\right] \left[\begin{array}{lll}1 & 1 & 0\\0 & 0 &
0\end{array}\right] [2~0]^2$$
$$~+~ \left[\begin{array}{lll}1 & 1& 2\\0 & 0 &
0\end{array}\right] \left[\begin{array}{lll}1 & 1 & 2\\0 & 0 &
0\end{array}\right] \left[\begin{array}{lll}2 & 2 & 2\\0 & 0 &
0\end{array}\right] [2~2]^2$$
i.e.
$$\psi^{J''=2}=0.62361[0~2]^2~+~0.62361[2~0]^2+0.47141[2~2]^2$$
We don’t actually have to specify the $J=1^+$ state in detail to get the ratio. We note that only the component $[2~2]^1$ of the $J=1$ wave function can contribute to the $M1$ transition. The probability of this component factors out in the ratio. With some additional Racah algebra, we can show that $r=\frac{7}{8}=0.875$. This should be compared with the value $r=\frac{1}{2}$ of the simple rotational formula and with the experimental value $r_{exp}=0.8$. We see that we get better agreement by using this projection method.
To complete the story, using results described in the next section, we are able to deduce that for $^8Be$ $B(M1)_{1^+\rightarrow
2_1^+}$=$\frac{7}{4\pi}~\mu_N^2$
Supermultiplet Scheme with a $Q \cdot Q$ interaction
====================================================
A. Supermultiplet Scheme in $^8Be$
----------------------------------
The $Q \cdot Q$ interaction that we have been using fits in nicely with the $L-S$ supermultiplet scheme of Wigner [@wig]. For the $p$ shell the unitary group $U(3)$ is relevant since there are three states: $L=1$, $M=$1, 0 and -1. A very useful reference for this section is the book by Hammermesh [@ham].
If the Hamiltonian were a Casimir operator of $U(3)$ all states of a given special symmetry $[f]=[f_1,~f_2,~f_3]$ would be degenerate. For the case of $1p$ shell a state with a given particle symmetry $[f_1,~f_2,~f_3]$ is identical to a quantum oscillator symmetry state [@Har; @Elliot] $(\lambda, \mu)=(f_1-f_2,f_2-f_3)$. The states $(\lambda, \mu,L)$ are eigenstates of our $Q \cdot Q$ interaction which is a linear combination of the Casimir operator of $SU(3)$ and an $L\cdot L$ interaction. The latter gives rise to a terminating rotational $L(L+1)$ spectrum for states of different $L$ but with the same $[f]$. Amusingly, as has been pointed out by many, one gets identical bands in all $p$ shell nuclei with this model provided the coefficient of $L\cdot L$ is fixed.
Unlike in the $s,~d$ shell, nothing new is added by using the quantum numbers ($\lambda,~ \mu$) instead of \[$f_1,~f_2,~f_3$\] for $1p$ shell states. This is because the number of different $M$ states availbale for particles (3) coincides with the number of possible directions for oscillator quanta $(a^{\dagger}_x,~a^{\dagger}_y,~a^{\dagger}_z)$ and a single creation operator correponds to each particle.
In more detail, the Casimir operator is $\tilde{C_2}=Q\cdot Q~-3\vec{L} \cdot
\vec{L}$. Hence,
$$\begin{aligned}
\langle-\chi Q\cdot Q\rangle_{\lambda~\mu~L} & = &\bar{\chi}\left[-\langle
\tilde{C_2}\rangle_{\lambda~\mu}~+~3L(L+1)\right]\\
& = &
\bar{\chi}\left[-4(\lambda^2+\mu^2+\lambda\mu+3(\lambda+\mu)+3L(L+1)\right]\end{aligned}$$
where $\bar{\chi}=\chi \frac{5b^4}{32\pi}$ with $b$ the harmonic oscillator length parameter ($b^2=\frac{\hbar}{m\omega}$). The magnetic dipole modes in the $L~S~T$ representation are:
$L=1$ $S=0$ $T=0$ = $L=0$ $S=1$ $T=1$ (isovector spin mode)\
$L=1$ $S=0$ $T=1$ (scissors mode) $L=1$ $S=1$ $T=0$\
$L=0$ $S=1$ $T=0$ $L=1$ $S=1$ $T=1$\
With the $Q\cdot Q$ interaction that we have chosen, transitions from the $L=0~S=0$ ground state in $^8Be$ to all of these modes except one will vanish. The only surviving mode is the $L=1~S=0~T=1$ scissors mode.
Let us give briefly the energies and some properties of the states in $^8Be$ ($\bar{\chi}=0.1865$):
\(a) \[f\]=\[4,0\] ($\lambda,\mu$)=(4,0) Ground State Band
$L$ = $S$ = $T$ = $\frac{E}{\bar{\chi}}$ = $E^*(MeV)$\
0 0 0 $-112$ 0\
2 0 0 $-94$ 3.36\
4 0 0 $-52$ 11.19\
\(b) \[f\]=\[3,1\] ($\lambda,\mu$)=(2,1) -contains the scissors mode ($L=1$, $S=0$, $T=1$).
Note that the ($S,T$) combinations (0,1), (1,0) and (1,1) are allowed.
$L$ = $\frac{E}{\bar{\chi}}$ = $E^*(MeV)$\
1 $-58$ 10.07\
2 $-46$ 12.31\
3 $-28$ 15.67\
\(c) \[f\]=\[2,2\] ($\lambda,\mu$)=(0,2) The ($S,T$) combinations (0,0), (0,2), (2,0) and (1,1) are allowed.
$L$ = $\frac{E}{\bar{\chi}}$ = $E^*(MeV)$\
0 $-40$ 13.43\
2 $-22$ 16.78\
\(d) \[f\]=\[2,1,1\] ($\lambda,\mu$)=(1,0) The ($S,T$) combinations (0,1), (1,0), (1,1), (1,2) and (2,1) are allowed.
$L$ = $\frac{E}{\bar{\chi}}$ = $E^*(MeV)$\
1 $-10$ 19.02\
Note that this supermultiplet also has a state with the quantum numbers of the scissors mode $L=1~S=0~T=1$.
Some further comments are in order. The scissors mode state in $^{156}Gd$, as a single band state originally discovered in electron scattering [@bo], was found when finer resolution ($\gamma,\gamma'$) experiments were performed to consist of many states [@berg; @bohle]. This was a beautiful example of intermediate structure. The supermultiplet scheme here affords a concrete example of the origin of the intermediate structure. Our scissors mode state at an energy of $-58~\bar{\chi}$ is degenerate with an $L=1~S=1~T=1$ state. If we introduce spin-dependent interactions the two states will admix and the degeneracy will be removed. We will get intermediate structure.
B. Supermultiplet Scheme in $^{10}Be$
-------------------------------------
We now give the energies and some properties of the states in $^{10}Be$ ($\bar{\chi}=0.1286$): (a) \[f\]=\[4,2\] ($\lambda,\mu$)=(2,2) (includes ground state).
Allowed states:
$L$ = $S$ = $T$ = $\frac{E}{\bar{\chi}}$ = $E^*(MeV)$\
0 0 1 $-96$ 0\
$2_1$ 0 1 $-78$ 2.32\
$2_2$ 0 1 $-78$ 2.32\
3 0 1 $-60$ 4.63\
4 0 1 $-36$ 7.72\
Note that the $2_1^+$ and $2_2^+$ states are degenerate in this scheme. This arises from the fact that, in a rotational picture, the $K$ values that are allowed are $\mu,~\mu-2,~...$. Thus we have a degeneracy of $J=2_1^+$ $K=0$ and $J=2_2^+$ $K=2$. This degeneracy does not correspond to the experimental situation -the $2_1^+$ and $2_2^+$ states are at $3.368~MeV$ and $5.960~MeV$ respectively -well separated.
As a practical matter this degeneracy gives problems for the shell model code $OXBASH$ [@oxbash]. Shell model routines often give wrong answers for transition rates when the states involved are degenerate. To overcome this difficulty, we have introduced weak additional terms in the Hamiltonian to remove the degeneracy e.g. we use a weak one-body spin-orbit interaction $-\xi \vec{l}\cdot\vec{s}$ with $\xi=0.1~MeV$.
\(b) Two degenerate bands \[f\]=\[4,1,1\] ($\lambda,\mu$)=(3,0), \[f\]=\[3,3\] ($\lambda,\mu$)=(0,3)
Allowed states:
$L$ = $S$ = $T$ = $\frac{E}{\bar{\chi}}$ = $E^*(MeV)$\
1 1 1 $-66$ 3.853\
3 1 1 $-36$ 7.716\
We get the low-lying $1^+$ states (one from \[4,1,1\] and one from \[3,3\]). Note however that we have $L=1,~S=1$, hence the states cannot be excited by either the orbital operator or the spin operator.
\(c) Band which contains the scissors mode \[f\]=\[3,2,1\] ($\lambda,\mu$)=(1,1)
$L$ = $S$ = $T$ = $\frac{E}{\bar{\chi}}$ = $E^*(MeV)$\
scissors mode 1 0 1 $-30$ 8.49\
scissors mode 1 0 2 $''$ $''$\
1 1 1 $''$ $''$\
1 0 2 $''$ $''$\
1 2 1 $''$ $''$\
There are also several $L=2$ states with $\frac{E}{\bar{\chi}}=-18$ and $E^*=10.03~MeV$.
Note that the $L=1,~S=0$ scissors modes finally make their appearance. There are two branches -one with isospin $T=1$ and one with isospin $T=2$. These two scissors mode states are in energy in the supermultiplet scheme $^{10}Be$.
\(d) \[f\]=\[2,2,2\] ($\lambda,~\nu$)=(0,0)
$L$ = $\frac{E}{\bar{\chi}}$ = $E^*(MeV)$\
0 $0$ 12.34\
The (S T) values are (0 1), (1 0), (1 2), (2 1), (0 4) and (4 0).
B(M1) Transitions in the $SU(3)$ Scheme
---------------------------------------
For ${ ^8}$Be and $^{10}Be$ the strength for orbital M1 transitions from $J^{\pi}=0^+$ to scissors mode states can be obtained in the $SU(3)$ scheme by observing that in the process the ground state intrinsic state is transformed to the corresponding intrinsic state of $1^+$ state. For example in ${ ^8}$Be, the orbital isovector M1 operator, $\mu=\sum_i l_i~ \tau_z^i=(L^\pi - L^{\nu})$ ,transforms the four nucleon intrinsic state $(4,0)$ into the intrinsic state of $1^+ 1$ state that is (21). The operator $(L^{\pi}_0 - L^{\nu}_0)$ is the generator of the scissors mode. The orbital part of ground state and $1^+ 1$ state of ${ ^8}$Be in $SU(3)$ scheme can be projected out from the corresponding maximum weight intrinsic states $\vert[f](\lambda,\mu)\epsilon,\Lambda ,\rho>$ where $\epsilon=2\lambda+\mu$ and $\Lambda = \rho =\frac {1}{2} \mu$, by using the following projection[@Har], $$\vert[f](\lambda,\mu)K,L,M> =\frac {(2L+1)}{a(\lambda \mu L K)}
\int d \Omega D^{L}_{M,K}(\Omega) R(\Omega)
\vert[f](\lambda,\mu)\epsilon,\Lambda,\rho>.
\label{proj}$$ Eq.(\[proj\]) is a general equation for projecting out orbital part of the wave function for a given L from a $(\lambda, \mu)$ state. In particular for ${ ^8}$Be we have, $$\vert[4](40)0,0,0>=\frac{1}{a(4000)} \int d \Omega
D^{0}_{00}(\Omega) R(\Omega)
\vert[4](40)8,0,0>$$ and $$\vert[31](21)1,1,0>=\frac{3}{a(2111)}\int d \Omega
D^{1}_{01}(\Omega) R(\Omega)
\vert[31](21)5,\frac {1}{2},\frac {1}{2}>.$$ Knowing the \[f\] representation of the $(\lambda, \mu)$ states one can look up the corresponding conjugate charge spin states to get definite JT states.
The orbital magnetic transition strength B(M1) between these states is calculated by a method similar to that outlined in Appendix A of ref.[@reta] and is found to be
$$\begin{array}{c}
B(M1)= \frac {9}{16 \pi} \vert <[31](21)110,S=0,T=1 \vert
(L^{\pi}_0 - L^{\nu}_0)
\vert[4](40)000, S=0, T=0>\vert^2\\
= \frac {2}{\pi}\mu_n^2 =0.637~\mu_n^2
\end{array}$$
In an analogous fashion one can calculate the scissors mode M1 transition strengths for the nucleus $^{10}Be$. We find the following results for SU(3) limit transition strengths in $^{10}Be$:
$$B(M1)(0^+1 \rightarrow 1^+1)=\frac {9}{32 \pi}\mu_n^2
=0.0895\mu_n^2$$ $$B(M1)(0^+1 \rightarrow 1^+2)=\frac {15}{32 \pi}\mu_n^2
=0.1492\mu_n^2 .$$
Realistic Spin-Orbit Interaction and Restoration of $SU(3)$ Symmetry
--------------------------------------------------------------------
As pointed out before, an important role of spin dependent part of interaction is to remove the degeneracies present in the $SU(3)$ limit by mixing up the same final angular momentum states arising due to a given intrinsic state as well as from different intrinsic states. In a realistic interaction, the relative strengths of spin dependent and spin independent part of interaction determine whether the wavefunctions are close to SU(3) scheme or a (j-j) coupling scheme is a better description of the system.
To understand further the part played by spin independent part of the full realistic interaction in the restoration of $SU(3)$ symmetry, we consider a small space calculation with the full spin-orbit part of the $(x,y)$ interaction plus a variable $Q.Q$ interaction. Figure.(7) is a plot of isovector orbital, spin and total strength for M1 transitions from $J=0_1^+ T=0 \rightarrow J=1^+ T=1$ states versus $t$, the parameter multiplying the full Q.Q interaction matrix elements for ${ ^8}$Be. For the spin part we use the operator 9.412$\sum \sigma
t_z$ i.e. we the large isovector factor. In ${ ^8}$Be, with increasing $t $ the orbital isovector strength is seen to approach the SU(3) limit value of $0.637$$\mu_n^2$. The contribution of isovector spin transition, on the other hand to total B(M1) decreases as $t$ becomes large. This is because the $SU(4)$ limit is being approached and in this limit the spin contribution vanishes.
In Fig.(7) with only spin-orbit part of realistic interaction in play, the calculated M1 transition strength for $0^+ 0$ to $1^+ 1$ transitions has a large spin flip contribution and is found tobe as large as 9.7 $\mu_n^2$. In a small space calculation with full realistic interaction$(x,y)$(Table I), a total B(M1) value of 1.0547$\mu_n^2$ is obtained with an isovector orbital transition strength of 0.67$\mu_n^2$ and an isovector spin contribution of 0.38$\mu_n^2$. Of course to get the physical $B(M1)$ we add the spin and orbital and square. The spin $B(M1)$ is a factor of 25 lower here than the $t=0$ value in Fig 7. It still has some effect because of the factor 9.412. We may note that the orbital transition strength arising due to full realistic interaction is very close to the $SU(3)$ limit indicating that the realistic interaction favors a restoration of $SU(3)$ symmetry. The large space realistic interaction calculation inspite of the correlations induced by shell mixings results in a B(M1) value 1.2866$\mu_n^2$ and isovector orbital transition strength of 0.728$\mu_n^2$ indicating that the wavefunctions are still very close to $SU(3)$ wave functions.
In $ ^{10}$Be the situation is more interesting due to splitting of scissors mode strength into two degenerate states in SU(3) limit. Figures (8) and (9) show the orbital and spin part respectively of M1 transition strength for transitions from ground state to $J=1^+ T=1$ states, $ J=1^+ T=2$ states and all $J=1^+$ states. For ground state to $J=1^+ T=1$ transitions the orbital B(M1) is seen to dip to a minimum for $t=0.3$ before it starts increasing so as to approach its SU(3) limit. An opposite trend is observed in the corresponding spin strength that shows some increase, reaches a maximum and then tends to the $SU(4)$ limit of zero. The characteristic behaviour at $t=0.3$ is possibly a manifestation of a shape change at small deformation before the system stabilizes by acquiring a permanent deformation. The M1 transition sums for ground state to $J=1^+ T=2$ states, show a behaviour similar to that observed for ground state to $J=1^+ T=1$ transitions in ${ ^8}$Be.
Magnetic Dipole Transitions To Individual States
================================================
We here present several tables of magnetic dipole transitions from the $J=0^+$ ground states of $^8Be$ and $^{10}Be$ to individual $J=1^+$ states. We use both the ($x,y$) interaction with $x=1,~y=1$ and the $Q
\cdot Q$ interaction.
Concerning the latter, we learned in the previous section that there are many degenerate states in the $0\hbar\omega$ calculation when a $Q
\cdot Q$ interaction is used. Unfortunately, most shell model routines, including the one used here, give erroneous results for transition rates when there are degeneracies. In all our small space ($0\hbar\omega$) calculations using $Q \cdot Q$ we have added a small spin-orbit interaction $-\xi \vec{l} \cdot \vec{s}$ with $\xi=0.1~MeV$. This works but it introduces an artificial complexity in our tables. However, it is easy to see by eye what states would be degenerate if the spin-orbit interaction is removed. Alternatively, one can use the analytic expressions for the energies in the previous section.
The columns in Tables $VI$ through $XIII$ are defined as in Table $I$:
= $g_{l\pi}$ = $g_{l\nu}$ = $g_{s\pi}$ = $g_{s\nu}$\
(a) Isovector Orbital 0.5 -0.5 0 0\
(b) Isovector Spin 0 0 0.5 -0.5\
(c) Physical 1 0 5.586 -3.826\
Calculated Magnetic Dipole Transitions in $^8Be$
------------------------------------------------
In Tables $VI$ and $VII$ we present the details of the $0\hbar\omega$ calculated $B(M1)$ values from the $J=0^+,~T=0$ ground state of $^8Be$ to the $J=1^+,~T=1$ excited states.
For the realistic ($x,y$) interaction, the isovector orbital strength is concentrated in three states at 13.7, 16.6 and 18.0 $MeV$. The sum of the orbital strengths is $\sum B(M1)\uparrow=0.67~\mu_{N}^2$. This is in fair agreement with the experimental value $B(M1)\uparrow=0.81~\mu_{N}^2$ (which is actually deduced from the downward $\gamma$ decay of the 17.64 $MeV$ $J=1^+,~T=1$ state to the ground state). However, in the experiment all the strength is concentrated in state whereas in our calculation we have considerable fragmentation. On the other hand, if we look at the physical transitions, there is much more concentration in one state at 13.73 $MeV$ with $B(M1)\uparrow=0.72~\mu_{N}^2$. We will discuss this more soon.
With the $Q \cdot Q$ interaction, all the orbital strength is concentrated in the (2-fold degenerate) state at $10.1~MeV$ with a summed strength $B(M1)\uparrow=0.64~\mu_{N}^2$, very similar to that of the ($x,y$) interaction. The energy is too low compared with experiment, but we must remember that we did not renormalize the strength $\chi$ to allow for $\Delta N=2$ admixtures. Note that the isovector spin transitions are zero with the $Q \cdot Q$ interaction. This is because we are at the $SU(4)$ limit since $Q \cdot Q$ is a spin-independent interaction.
Note that with the $Q \cdot Q$ interaction the summed orbital strength is $\frac{2}{\pi}~\mu_{N}^2$, confirming the expressions that were derived in the previous section. Coming back to the ($x,y$) interaction, we see that here also the isovector spin transitions are very weak. But note that for the 13.73 $MeV$ state whereas the orbital value of $B(M1)\uparrow$ is 0.2569 $\mu_{N}^2$ and the spin value is 0.0013 $\mu_{N}^2$, the physical value is 0.7155 $\mu_{N}^2$. The reason is that the spin and orbit amplitudes add coherently and that the spin amplitude is multiplied by the factor $9.412$.
For other states there is destructive interference between spin and orbit. For example, for the 16.64 $MeV$ state, the value of $B(M1)\uparrow$ is 0.234 $\mu_{N}^2$ for the orbital case but only 0.065 $\mu_{N}^2$ for the physical case.
Calculated Magnetic Dipole Transitions in $^{10}
Be$
------------------------------------------------
In Tables $VIII$ and $IX$ we present the details of the $0\hbar\omega$ calculated $B(M1)$ values from the $J=0^+~T=1$ ground state of $^{10}Be$ to the $J=1^+~T=1$ and to $J=1^+~T=2$ excited states. We caution the reader that whereas for $^8Be$ we presented the results in units of $\mu_{N}^2$ (Tables $VI$ and $VII$), for $^{10}Be$ we use $10^-2$ $\mu_{N}^2$ as the unit. The reason for this is that the orbital transitions to states in $^{10}Be$ are considerably smaller than those in $^8Be$.
Let us look at the $Q \cdot Q$ interaction (Table $IX$) first. There are several outstanding features which are explained in the previous section on $L~S$ coupling and supermultiplet symmetry.
The first two $J=1^+~T=1$ states are degenerate at $E^*=3.86$ $MeV$. They carry no spin or orbital strength from the from the ground state. The \[f\] symmetries are \[4 1 1\] and \[3 3\]. They have additional quantum numbers $L=1~S=1~T=1$. Since the isovector orbital operator ($\vec{L_{\pi}}$-$\vec{L_{\nu}}$) cannot change both $L$ and $S$ from zero to one, the orbital $B(M1)$ vanishes. A similar argument holds for the isovector spin operator. These lowest two states are therefore not scissors mode states.
Then we have a four fold set of degenerate states with \[3 2 1\] symmetry at about 8.5 $MeV$ excitation which does include the $L=1~S=0~T=1$ scissors mode. We note that for $^{10}Be$, the $T=1$ scissors mode is with the $T=2$ scissors mode also at 8.5 $MeV$ excitation. This again is a prediction of the supermultiplet theory.
The summed isovector orbital strength is $\frac{9}{32\pi}$ $\mu_{N}^2$ from $J=0^+~T=1$ to the $J=1^+~T=1$ states, and it is $\frac{15}{32\pi}$ $\mu_{N}^2$ to the $J=1^+~T=2$ states. We have in effect a ($2T+1$) rule:
$$\frac{(2T+1)_{T=2}}{(2T+1)_{T=1}}=\frac{5}{3}$$
This coincides with the ratio of $T=2$ to $T=1$ strength.
Recalling that the $^8Be$ strength was $\frac{2}{\pi}$ $\mu_{N}^2$, we see that the ratio of total strength $\frac{^{10}Be}{^8Be}$ is $\frac{3}{8}$.
We now come back to Table $VIII$ which shows the same calculational results with the ‘realistic’ ($x,y$) interaction. There are several similarities but also some differences with the $Q \cdot Q$ results. Just as with the $Q \cdot Q$ interaction, the orbital transitions to the lowest two $J=1^+$ $T=1$ states at 6.14 and 7.68 $MeV$ are very weak 0.16 and 0.17 $\times~(10^{-2}\mu_{N}^2)$ respectively. However, the spin transitions, which with $Q \cdot Q$ were also zero, are now sufficiently strong so as to have a visible effect. For example, the physical $B(M1)\uparrow$ to the 7.68 $MeV$ state is calculated to be 1.85 $\mu_{N}^2$. This is certainly measurable.
As with the $Q \cdot Q$ interaction, the scissors mode states with the ($x,y$) interaction are at a much higher energy than the lowest two $1^+$ states 19 $MeV$. Also, the $J=1^+$ $T=1$ and $T=2$ excitations are roughly in the same energy range -the $Q \cdot Q$ interaction has them degenerate. The ratio of $T=2$ to $T=1$ orbital strength is about the same for the ($x,y$) interaction as for $Q \cdot Q$ 1.44 vs. $\frac{5}{3}$.
One major difference is that the energy scale is larger for the ($x,y$) interaction than for $Q \cdot Q$. The lowest and higher energies in Table $VIII$ are 6.14 $MeV$ and 30.96 $MeV$ whereas in Table $IX$ they are 3.85 $MeV$ and 12.35 $MeV$.
In Tables X and XI we present results of large space calculations for $^8Be$ to be compared with the corresponding small space Tables VI and VII. Likewise in Tables XII and XIII we present the large space results for $^{10}Be$ to be compared with tables VIII and IX. We do not show all the states here, only the low energy sector. The excitation energies are in general larger in the large space calculations. The major changes occur when one has nearly degenerate levels sharing some strength. For example, the lowest two $1^+$ states in the large space calculations, which are at 7.38 $MeV$ and 9.62 $MeV$ have almost equal $M1$ strengths 0.64 $\mu_{N}^2$ and 0.79 $\mu_{N}^2$. In the small space, the lowest state has only 0.011 $\mu_{N}^2$ and the second one 1.8 $\mu_{N}^2$.
A sensible attitude is to assume that neither calculation is accurate enough to give the detailed distribution of strength between the two states -only the summed strength for the two states should be compared with experiment.
With the $Q \cdot Q$ interaction in the large space, the degeneracy encountered in the small space calculation is removed. In part, this is due to the fact that we do not include the single-particle terms $\sum_{i=j}Q(i) \cdot Q(j)$. These will induce a single-particle splitting between $1s$ and $0d$ in the $N=2$ shell. But since degeneracies give us trouble in our shell model diagonalizations, we are happy to leave the calculation as is.
With $Q \cdot Q$ the scissors mode strength in $^8Be$ gets pushed up from the small space value of 10.1 $MeV$ to the large space value of 15.5 $MeV$. For $^{10}Be$ the corresponding numbers are 8.5 $MeV$ and 11.3 $MeV$ for the $J=1^+,~T=1$ states and essentially the same for $J=1^+,~T=2$ states. That is, the near degeneracy of the $T=1$ and $T=2$ scissors modes in $^{10}Be$ is maintained in the large space calculation.
Lastly, we reiterate the fact that the shell model calculations here yield not only colective magnetic states but also show intermediate structure. That this structure is a natural occurence is shown by the supermultiplet model, where for a given $[f]_{L=1}$ there are several $S$ and $T$ values possible. It is of course very difficult to get the details of the intermediate structure to come out right, but it is good to be able to explain the origin of this structure.
Additional Comments and Closing Remarks
=======================================
We can gain further insight into the nature of $^8Be$ and $^{10}Be$ by evaluating the quadrupole moments of the $J=2^+$ states. A small space calculation gives the following values:
= $Q \cdot Q~-0.1\vec{l}\cdot\vec{s}$ = ($x,y$) interaction\
$^8Be~~J=2^+,~T=0$ $Q=-8.02~e~fm^2$ $Q=-7.86~e~fm^2$\
$^{10}Be~~J=2_1^+,~T=1$ $Q=-2.52~e~fm^2$ $Q=-7.68~e~fm^2$\
$^{10}Be~~J=2_2^+,~T=1$ $Q=+2.06~e~fm^2$ $Q=+6.91~e~fm^2$\
In the rotational model the quadrupole moment of the $2^+$ of a $K=0$ band is $-\frac{2}{7}Q_0$ where $Q_0$ is the intrinsic quadrupole moment. Thus a negative $Q$ corresponds to a prolate shape and a positive $Q$ to an oblate shape. From the above, $^8Be$ acts as a normal deformed nucleus of the prolate shape.
It has been pointed out by Harvey that in the $SU(3)$ scheme, whenever $\mu$ is less than or equal to $\lambda$ the nucleus becomes oblate [@Har]. For the ground state band in $^8Be$ $\lambda$ is bigger than $\mu$ but for $^{10}Be$ $\lambda$ and $\mu$ are equal. The situation with $^{10}Be$ is somewhat confusing. With the $Q \cdot Q$ interaction, which one might think would favor deformation, the quadrupole moment of the $2_1^+$ state drops to -2.52 $e~fm^2$. Recall that for a perfect vibrator, the value of $Q$ is zero, so it would appear that $^{10}Be$ is headed in that direction. However with the realistic interaction, which contains a large spin-orbit interaction that one might think would oppose deformation, the quadrupole moment of $^{10}Be$ becomes more negative -almost the same as that of $^8Be$. Note also that the calculated values of $Q$ for the $2_1^+$ and $2_2^+$ states are nearly equal but opposite to both interactions.
We have learned many interesting things by considering scissors modes in light nuclei. First of all there is evidence for their existence. This evidence comes strangely from a nucleus whose ground state is unstable -$^8Be$. We learn of the existence from the inverse process i.e. $\gamma$ decay from the $J=1^+,~T=1$ state at $17.64~MeV$ to the ground state [@ajz]. The decay to the $2_1^+$ state, presumably a member of a $K=0$ rotational band, is also observed and this suggests that theoretical studies (and experimental ones as well whenever possible) should be made not only between between $J=1^+~K=1$ and $J=0^+~K=0$ states but also between $J=1^+~K=1$ and $J=2^+~K=0$ states. This will make the picture of scissors modes more complete. In this work we considered but one example and showed that the ratio of $J=1^+$ decay rate to $J=2^+$ vs $J=0^+$ deviates from the simple rotational formula result of 0.5. Further studies along these lines are being planned.
To make the picture even more complete, one can also study the decay of $J=1^+~K=1$ to $J=2^+~K=2$. We were almost forced into such a study by the fact that in the $SU(3)$ scheme there is a two-fold degeneracy of the lowest $J=2^+$ states in $^{10}Be$ [@ham; @Elliot]. Presumably, these two states are admixtures of $2^+~K=0$ and $2^+~K=2$.
The shell model approach used here [@oxbash] enables us to study fragmentation or intermediate structure. We find for example that with an electromagnetic probe there are, besides the strong scissors mode states, some almost states. These have separately substantial orbital contributions and substantial spin contributions to the magnetic dipole excitations but the physical $B(M1)$ is very small because of the destructive interference of the spin and orbital amplitudes. For example, as seen in Table X, in $^8Be$ we calculate that the low lying orbital strength is shared almost equally between two states (at 18.0 $MeV$ and 20.8 $MeV$) -the strengths being 0.22 $\mu_N^2$ and 0.30 $\mu_N^2$ respectively. However, the physical $B(M1)$’s are 0.62 $\mu_N^2$ and 0.12 $\mu_N^2$. Thus one can miss considerable orbital strength into these ‘invisible’ sttaes if one uses only an electromagnetic probe.
Another thing we learn is that although spin excitations are strongly suppressed they cannot be ignored. In the $SU(4)$ limit, the spin matrix elements vanish and there is a clear tendency in our calculations in that direction. However, since the isovector spin operator is multiplied by a factor of 9.412, the spin and orbital contributions tend to be on the same footing.
In the example of the above paragraph in the decay of the (calculated) 18.0 $MeV$ state, the orbital $B(M1)$ is only 0.22 $\mu_N^2$ but the physical one which induces the spin contribution is 0.62 $\mu_N^2$. In $^{10}Be$ the first two $J=1^+$ states are calculated to be excited mainly by the spin operator and the $B(M1)$’s should be substantial 0.5 $\mu_N^2$. Yet in the $U(3)-SU(4)$ limit, these lowest two states \[f\]=\[4,1,1\] and \[3,3\] should not be excited at all either by the spin or by the orbital operators.
We have found the Wigner supermultiplet scheme [@wig] combined with the $SU(3)$ scheme [@Har; @Elliot] a very useful guide to the more complicated shell model calculations that we have performed. There is the added simplicity in the $p$ shell that there is a one-to-one correpondence between a given \[f\] symmetry and the $(\lambda,\mu)$ symmetries. Many interesting properties about scissors modes can be literally read off the pages of the book by Hammermesh [@ham]. For example, there is the fact that the $T=1$ and $T=2$ scissors mode states in $^{10}Be$ are degenerate in energy. This is an exact result with the $Q \cdot Q$ interaction in a $p$ shell calculation. Results very close to this are obtained with a realistic interaction, but we frankly didn’t notice this until we made an $SU(3)$ analysis. Also the non-obvious fact that the lowest two $1^+$ states in $^{10}Be$ are not scissors mode states is made clear by such an analysis.
Also the fact that scissors mode states everywhere, including $^{156}Gd$, have intermediate structure [@berg; @bohle] is made clear by the supermultiplet scheme. For a given $L=1$ state there are often several $S,~T$ combinations which are degenerate. The removal of the degeneracy and the mixing of these states e.g. by a spin-orbit interaction leads to fragmentation and intermediate structure.
By extending the shell model calculations to ‘large space’ i.e. by including $2\hbar\omega$ excitations, we were able to calculate the cumulative energy weighted strength distribution for isovector orbital excitations. The results which are shown in several figures are characterized by a low energy rise followed by a second plateau. The shapes of the distributions were similar for the two contrasting interactions used here -the ‘realistic’ short range interaction and the schematic quadrupole-quadrupole interaction. The results were compared witht the simple Nilsson model [@dz1; @no] which predicts that the energy-weighted sum at high energy (beyond the first plateau) should equal the low energy rise i.e. the ratio $\frac{total}{low~energy}$ should be 2:1. The actual calculated ratios witht the ($x,y$) interaction are 1.75 for $^8Be$ and 2.52 for $^{10}Be$. The corresponding numbers for the $Q \cdot Q$ interaction were 1.37 and 2.33. We see that the Nilsson model is not bad as a first orientation but there are fluctuations. The fact that the above ratios are larger for the ($x,y$) interaction than for the $Q \cdot Q$ interaction may support the idea of Hamamoto and Nazarewicz [@hz] that the symmetry energy will cause the ratio to increase. Our calculations however do not support their claim that the high energy part of the energy weighted orbital strength should than the low energy part -certainly not for a ‘normal’ rotational nucleus like $^8Be$. For $^{10}Be$ the ratio is however somewhat larger than the Nilsson model prediction.
Whether this is due to the atypical properties of $^{10}Be$ mentioned in the text or is a harbinger of what will happen for most other nuclei remains to be seen. From an experimental point of view, it would be helpful to have more data on $^{10}Be$. Not only have no $J=1^+$ states been identified but neither has the $J=4_1^+$ state been seen. The location of this state might help us decide whether $^{10}Be$ is rotational or vibrational.
At any rate, we should examine a larger range of nuclei and look into more detail about the symmetry energy in order to be able to make more definitive statements about the systematics of the cumulative energy weighted distributions. We note that the Zheng-Zamick sum rule [@zz] is able to handle the divergent behaviour between $^8Be$ and $^{10}Be$. This sum rule involves the difference between isoscalar and isovector summed $B(E2)$ strength, whereas corresponding expressions by Heyde and de Coster [@ibm] based on the $I.B.A.$ model [@iach; @diep] as well as empirical analyses [@zr] involve only $B(E2)$ to the lowest $2^+$ states. Even here more sharpening up is in order.
Our initial reason for studying lighter nuclei is that they would give us insight into the behaviour of heavier nuclei, and we could carry out more complete calculations in the low $A$ region. But then we found many results which made light nuclei studies fascinating for their own sake. One rather broad lesson we have learned in the light nucleus study is that there can be considerable change in going from one nucleus to the next, and perhaps in heavier nuclei too much effort has been made to make the nuclei fit into a smooth pattern. For example, we find that whereas in $^8Be$ the lowest $1^+$ state is dominantly a scissors mode state, in $^{10}Be$ the lowest $1^+$ states are not scissors mode states at all -they can only be reached by the spin operator. We should perhaps be looking for more variety of behaviour in heavier nuclei. Lastly the supermultiplet scheme which we found extremely useful was originally thought to be of interest only in light nuclei where the spin-orbit interaction is small relative to the residual interaction. However, it is now being realized that even in heavy nuclei -especially for superdeformed states this scheme may once again be very relevant. This would make our light nuclear studies all the more important.
This work was supported by the Department of Energy Grant No. DE-FG05-86ER-40299. S.S. Sharma would like to thank the Department of Physics at Rutgers University for its hospitality and to acknowledge financial support from $CNP_q$, Brazil. We thank E. Moya de Guerra for familiarizing us with her work in collaboration with J. Retamosa, J.M. Udias and A. Poves. We thank I. Hamamoto for her interest. We also thank N. Sharma for useful comments about matrix diagonalization.
\(a) Small Space $(0s)^4(0p)^6$
\(b) Large Space $(0s)^4(0p)^6$ $+$ all $2\hbar\omega$ excitations
\(c) Low energy part of Large Space (up to the first plateau)
\(d) $g_{l\pi}=0.5$ $g_{l\nu}=-0.5$ $g_{s\pi}=0$ $g_{s\nu}=0$
\(e) $g_{l\pi}=0$ $g_{l\nu}=0$ $g_{s\pi}=0.5$ $g_{s\nu}=-0.5$
\(f) $g_{l\pi}=0.5$ $g_{l\nu}=0.5$ $g_{s\pi}=0$ $g_{s\nu}=0$. The value for isoscalar spin is the same as for isoscalar orbital in the case of $^{10}Be$ $J=0^+,~T=1$ $\rightarrow$ $J=1^+,~T=1$. For $\Delta T=1$ the isoscalar case gives zero.
\(g) $g_{l\pi}=1$ $g_{l\nu}=0$ $g_{s\pi}=5.586$ $g_{s\nu}=-3.826$
[**Figure (1):**]{} The cumulative sum of the energy-weighted isovector orbital $B(M1)$ strength for the $0_1^+,~0~\rightarrow~1^+,1$ transitions in $^8Be$ with the realistic interaction ($x=1,~y=1$).
[**Figure (2):**]{} Same as Figure 1 but with the $Q \cdot Q$ interaction.
[**Figure (3):**]{} The cumulative sum of the energy-weighted isovector orbital $B(M1)$ strength for the $0_1^+,~1~\rightarrow~1^+,2$ transitions in $^{10}Be$ with the realistic interaction ($x=1,~y=1$).
[**Figure (4):**]{} Same as Figure 3 but with the $Q \cdot Q$ interaction.
[**Figure (5):**]{} The cumulative sum of the energy-weighted isovector orbital $B(M1)$ strength for the $0_1^+,~1~\rightarrow~1^+,1$ transitions in $^{10}Be$ with the realistic interaction ($x=1,~y=1$).
[**Figure (6):**]{} Same as Figure 5 but with the $Q \cdot Q$ interaction.
[**Figure (7):**]{} $\sum B(M1)(J=0_1^+ T=0 \rightarrow
J=1^+ T=1$ states) versus $t$ for $^8Be$. The solid line, dashed line and dot-dash line are the total, spin and orbital parts of $\sum B(M1)$ .
[**Figure (8):**]{} The orbital part of $\sum B(M1)$ versus $t$ for $ ^{10}Be$. The solid line, dashed line and dot-dash line are the total, $\sum B(M1)(J=0_1^+ T=1 \rightarrow
J=1^+ T=1$ states) and $\sum B(M1)(J=0_1^+ T=1 \rightarrow J=1^+ T=2$ states) respectively.
[**Figure (9):**]{} Same as in Fig.(8) for spin part of $\sum B(M1)$.
[99]{} M.S. Fayache and L.Zamick, Physics Letters [**B**]{} [**338**]{}, (1994)421. F. Ajzenberg-Selove, Nucl. Phys. [**A 490**]{}, (1988)1-225 S. Raman, C.H. Malarkey, W.T. Milner, C.W. Nesta, J.R. and P.H. Stelson, ATOMIC DATA AND NUCLEAR DATA TABLES, [**36**]{}, (1987)1-96 D.C. Zheng and L. Zamick, Ann. of Phys. [**206**]{}, (1991)106. L. Zamick and D.C. Zheng, Phys. Rev. C [**44**]{}, (1991)2522; C [bf 46]{}, (1992)2106. K. Heyde and C. de Coster, Phys. Rev. C [**44**]{}, (1991)R2262. E. Moya de Guerra and L. Zamick, Phys. Rev. C [**47**]{}, (1993)2604. R. Nojarov, Nuclear Physics [**A 571**]{}, (1994)93. I. Hamamoto and W. Nazarewicz, Phys. Lett [**297 B**]{}, (1992)25. W. Ziegler, C. Rangacharyulu, A. Richter and C. Spieler, Phys. Rev. Lett. [**65**]{}, (1990)2515. A. Richter, Nuclear Physics [**A 507**]{}, (1990)99 A. Bohr and B. Mottelson, Nuclear Structure, Vol. II (Benjamin, New York, 1975) E.P. Wigner, Physical Review [**51**]{},(1937)107 M. Hammermesh, Group Theory and its Applications to Physical Problems, Addison-Wesley, Reading MA 1962. M. Harvey, Advances in Nuclear Physics Vol. I, edited by Michel Baranger and Erich Vogt, Plenum press, 67(1968) J.P Elliot, Proc. Royal Soc. [**A 245**]{}: (1958) 128 and 562. D. Bohle, A. Richter, W. Steffen, A.E.L. Dieprink, N. Lo Iudice, F. Palumbo and O. Scholten, Phys. Lett. [**137 B**]{}, (1984)27. U.E.P. Berg et. al., Phys. Lett. [**B 149**]{}(1984)59 D. Bohle et. al., Nucl. Phys. [**A 458**]{}(1986)205 B. A. Brown, A. Etchegoyen and W. D. M. Rae, The computer code OXBASH, MSU-NSCL report number 524(1992). J. Retamosa, J. M. Udias, A. Poves and E. Moya de Guerra, Nucl. Phys. [**[A511]{}**]{}221(1990) F. Iachello, Nuclear Physics [**A 358**]{}, (1981)890 A.E.L. Dieprink, Prog. in Part. and Nucl. Phys. [**9**]{},(1983)121
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abstract: 'This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions, which are based on coherence results for self-adjunctions (adjunctions where an endofunctor is adjoint to itself), ordinals in $\varepsilon_0$ play a prominent role. The paper ends by considering how the notion of Frobenius algebra induces the collapse of the hierarchy of ordinals in $\varepsilon_0$, and by raising the question of the exact categorial abstraction of the notion of Frobenius algebra.'
author:
- |
and [Zoran Petri' c]{}\
[Mathematical Institute, SANU]{}\
[Knez Mihailova 36, p.f. 367, 11001 Belgrade, Serbia]{}\
[email: {kosta, zpetric}@mi.sanu.ac.yu]{}
title: Ordinals in Frobenius Monads
---
\#1\#2
[\#1.]{} [*\#2*]{}
[*Mathematics Subject Classification* ([*2000*]{}): $\;$03G30, 03E10, 16H05, 16W30, 18C15, 18D10, 55N22]{}
[: Frobenius monad, Frobenius algebra, self-adjunction, bijunction, coherence, split equivalence, transfinite ordinal, separable algebra, cobordism, topological quantum field theory]{}
Introduction
============
The purpose of this paper is to connect two seemingly distant and unrelated topics: Frobenius algebras and ordinals contained in the infinite denumerable ordinal $\varepsilon_0$ (namely, the least ordinal $\xi$ such that ${\omega^\xi=\xi}$). Frobenius algebras play an important role in topology, mathematical physics and algebra (see [@K03] and references therein), while $\varepsilon_0$ is usually deemed interesting only for set-theorists and proof-theorists.
The categorial abstraction of the notion of Frobenius algebra leads to the notion of Frobenius monad (for some more details, see below). The structure of a Frobenius monad is given by a category with an endofunctor that bears both the structure of a monad (or triple) and a comonad, and satisfies moreover additional conditions called Frobenius equations (see the next section).
The notion of Frobenius monad is closely related to a special kind of adjoint situation where two functors (not necessarily distinct) are both left and right adjoint to each other (see [@M65], [@CMS97], [@K02], [@M03], [@S04], [@L06], [@CH09], and further references in these papers). Adjunction is a central notion in category theory, in logic, and perhaps in mathematics in general (see [@ML98] and [@LAW69a]), and the connection of this notion with the notion of Frobenius monad may serve to explain the importance of the latter.
One of the goals of this paper is to show that the notion of adjunction where two functors are both left and right adjoint to each other amounts, in a sense that we will make precise, to the notion of self-adjunction, which we have investigated in [@DP03]. A self-adjunction is an adjoint situation where an endofunctor is both left and right adjoint to itself. So we find a close relationship between Frobenius monads and self-adjunctions. Through this relationship, we can prove coherence results for Frobenius monads, by relying on a coherence result that we have previously established for self-adjunctions. (That self-adjunction arises in the context of Frobenius monads was noted in [@RSW05], Note after Definition 2.9; this is however implicit already in [@LAW69], pp. 151-152, in [@CW87], Theorem 2.4, and in [@K03], Chapter 2.)
These coherence results assert that there is a faithful functor from a freely generated Frobenius monad to manageable model categories, which we will consider in this paper. This faithful functor is here an isomorphism. With our model categories we can easily decide whether a diagram of arrows commutes. In logical terms, this is like proving completeness with respect to a manageable model, which helps us to solve the decision problem. Coherence here is analogous to the isomorphism that exists between the syntactically constructed freely generated monad and the simplicial category (see [@DP08b], Section 3, [@D08], Section 4, and references therein).
The coherence we establish is also the gist of the connection between the notions of Frobenius monad and two-dimensional topological quantum field theory (2TQFT). A 2TQFT may be understood as a functor from the category *2Cob*, whose arrows are cobordisms in dimension 2, to the category $\mbox{\it
Vect}_K$ of finite-dimensional vector spaces over the field $K$. In terms of category theory, a Frobenius algebra is characterized by a monoidal functor from the Frobenius monad freely generated by a single object to $\mbox{\it Vect}_K$, modulo the strictification of $\mbox{\it Vect}_K$ with respect to its monoidal structure given by the tensor product and $K$ (cf. the beginning of Section 7). A Frobenius algebra is the image of the object $1$ of the Frobenius monad. The main result here is that 2TQFTs correspond bijectively, modulo a skeletization of *2Cob*, to commutative Frobenius algebras. This result is stated officially as a result about equivalence of categories (see [@K03], Section 3.3).
An alternative result with the same mathematical content is that the free commutative Frobenius monad is isomorphic to the skeleton of *2Cob*. From that alternative result, the former result follows immediately. This alternative result may be conceived as a coherence result for commutative Frobenius monads.
Our coherence results for Frobenius monads mentioned above are more general. They deal with Frobenius monads in general, and not only commutative ones. Because of that, infinite ordinals contained in $\varepsilon_0$ enter into the picture. They arise naturally in our principal model category, which bears some similarity to *2Cob*. It is a kind of planar version of *2Cob*. Something related to this model category has been described topologically in a 2-categorial context in [@KL01] (Appendix C; see also [@L08]). The infinite ordinal structure of the model category is however mentioned neither in this book, nor in the papers mentioned in the third paragraph, nor in [@K03]. In [@K03] (Section 3.6.20) we find only the vague conclusion that this ordinal structure, with which we want to deal, is “nearly about any possible drawing you can imagine”. This structure is the main novelty we obtain when we reject commutativity and pass to Frobenius monads in general.
This structure could be described by other means than by the ordinals in $\varepsilon_0$. What we need is the commutative monoid with one unary operation freely generated by the empty set of generators (see Section 6). This monoid can be isomorphically represented in the positive integers too, but we believe its isomorphic representation in $\varepsilon_0$, which is quite natural, is worth investigating.
Towards the end of his book [@K03] (Sections 3.6.16-27), J. Kock discusses heuristically a project to describe geometrically the freely generated Frobenius monad, and leaves the matter as a challenge to the reader (Section 3.6.26). In this paper, one can find an answer to this challenge.
To make the hierarchy of ordinals in $\varepsilon_0$ collapse, and pass to something that amounts to *2Cob*, we need not assume commutativity. In the last two sections of this paper, we show how the notion of Frobenius algebra requires that the notion of Frobenius monad be extended with further assumptions, which produce the collapse of the hierarchy. The culprit for this collapse is the symmetry of $\mbox{\it Vect}_K$, without assuming that the Frobenius algebra is commutative (the Frobenius objects in symmetric monoidal categories of [@H04], Section 2, involve such a collapse too). We know that such a collapse must take place, but we do not know what should be its exact extent. In that context, we consider the collapse brought by the assumption of separability in Frobenius algebras, for which the exact categorial abstraction is the notion of separable matrix Frobenius monad in the last section of the paper. We leave however as an open question what is the exact categorial abstraction of the notion of Frobenius algebra.
This paper is a companion to [@DP08b], but, except for some side comments, an acquaintance with that paper is not indispensable. We rely however, as we said above, on the results of [@DP03], and we assume an acquaintance with parts of that earlier paper, though some of the essential matters we need are reviewed in Section 6. We assume also the reader is acquainted with some basic notions of category theory, which may all be found in [@ML98], but, for the sake of notation, we define some of these basic notions below.
The free Frobenius monad
========================
A *Frobenius monad* is a structure made of a category $\cal
A$, an endofunctor $M$ of $\cal A$ (i.e. a functor from $\cal A$ to $\cal A$) and the natural transformations
=$\varepsilon^\Box\,$=$:M\strt I_{\cal
A}$,= $\varepsilon^\Diamond\,$=$:I_{\cal A}\strt M$,\
$\delta^\Box$$:M\strt MM$, $\delta^\Diamond$$:MM\strt M$,
for $I_{\cal A}$ being the identity functor of $\cal A$, such that $\langle{\cal
A},M,\varepsilon^\Diamond,\delta^\Diamond\rangle$ is a monad, $\langle{\cal A},M,\varepsilon^\Box,\delta^\Box\rangle$ is a comonad, and, moreover, for every object $A$ of $\cal A$, the following *Frobenius* equations hold: $$M\delta^\Diamond_A\cirk\delta^\Box_{MA}=\delta^\Diamond_{MA}\cirk
M\delta^\Box_A=\delta^\Box_A\cirk\delta^\Diamond_A.$$ (For easier comparison, we use here, with slight modifications, the notation with the modal superscripts $\Box$ and $\Diamond$, which was introduced in [@DP08b].)
The equations defining the notions of monad and comonad are given below. For the Frobenius equations the reader may consult [@K03] (in particular, Lemma 2.3.19, and [@DP08b], Sections 6-7; as far as we know, and according to [@K06], the first appearance of these equations is in [@CW87]). Lawvere introduced in [@LAW69] (pp. 151-152) the notion of Frobenius monad with the equations $$M\varepsilon^\Box_A\cirk M\delta^\Diamond_A\cirk\delta^\Box_{MA}=
\varepsilon^\Box_{MA}\cirk\delta^\Diamond_{MA}\cirk
M\delta^\Box_A=\delta^\Diamond_A,$$ or, alternatively, the dual equations $$\delta^\Diamond_{MA}\cirk M\delta^\Box_A\cirk
M\varepsilon^\Diamond_A=
M\delta^\Diamond_A\cirk\delta^\Box_{MA}\cirk\varepsilon^\Diamond_{MA}=
\delta^\Box_A,$$ which can replace the Frobenius equations. (In the terminology of [@DP08b], Section 8, a Frobenius monad is a dyad, or codyad, where $\Box$ and $\Diamond$ coincide.)
The category *Frob* of the Frobenius monad freely generated by a single object, denoted by $0$, has as objects the natural numbers ${n\geq 0}$, where $n$ stands for a sequence of $n$ occurrences of $M$; so $Mn$ is ${n\pl 1}$. The arrows of this category are defined syntactically as equivalence classes of *arrow terms*, which are defined inductively as follows. The primitive arrow terms of *Frob* are
$\mj_n\!:n\str n$,\
=$\varepsilon^\Box_n\,$=$:n\pl 1\str
n$,= $\varepsilon^\Diamond_n\,$=$:n\str n\pl 1$,\
$\delta^\Box_n$$:n\pl 1\str n\pl 2$, $\delta^\Diamond_n$$:n\pl 2\str n\pl 1$.
The remaining arrow terms of *Frob* are defined inductively out of these with the clauses:
īf ${f\!:n\str m}$ and ${g\!:m\str k}$ are arrow terms, then so is ${(g\cirk f)\!:n\str k}$;\
if ${f\!:n\str m}$ is an arrow term, then so is ${Mf\!:n\pl
1\str m\pl 1}$.
We take for granted the outermost parentheses of arrow terms, and omit them. (Further omissions of parentheses will be permitted by the associativity of $\cirk$.)
The least equivalence relation, congruent with respect to $\cirk$ and $M$, by which we obtain the arrows of *Frob* is such that, first, we have the *categorial* equations of composition with $\mj$ and associativity of composition $\cirk$, and the *functorial* equations for $M$ (see [@DP08b], Section 2). We have next the *naturality* equations:
($\varepsilon^\Box$ [*nat*]{})=$f\cirk\varepsilon^\Box_n=\varepsilon^\Box_m\cirk
Mf$,($\varepsilon^\Diamond$ [*nat*]{})=$\varepsilon^\Diamond_m\cirk
f=Mf\cirk\varepsilon^\Diamond_n$,\
($\delta^\Box$ [*nat*]{})$MMf\cirk\delta^\Box_n=\delta^\Box_m\cirk
Mf$,($\delta^\Diamond$ [*nat*]{})$\delta^\Diamond_m\cirk
MMf=Mf\cirk\delta^\Diamond_n$,
the *comonad* and *monad* equations:
=$(\delta^\Box)$=$M\delta^\Box_n\!$=$\cirk\delta^\Box_n=
\delta^\Box_{n+1}\cirk\delta^\Box_n$,= $(\delta^\Diamond)$=$\delta^\Diamond_n\cirk
M\delta^\Diamond_n$=$=\delta^\Diamond_n\cirk
\delta^\Diamond_{n+1}$,\
$(\Box\beta)$ $\varepsilon^\Box_{n+1}$$\cirk\delta^\Box_n=\mj_{n+1}$, $(\Diamond\beta)$$\delta^\Diamond_n\cirk
\varepsilon^\Diamond_{n+1}$$=\mj_{n+1}$,\
$(\Box\eta)$ $M\varepsilon^\Box_n$$\cirk\delta^\Box_n=\mj_{n+1}$, $(\Diamond\eta)$$\delta^\Diamond_n\cirk
M\varepsilon^\Diamond_n$$=\mj_{n+1}$,
and, finally, the *Frobenius* equations where $A$ is replaced by $n$. The equations $(\delta^\Box)$ and $(\delta^\Diamond)$ are redundant in this axiomatization (see [@K03], Proposition 2.3.24, and [@DP08b], Section 6; they do not seem however to be redundant when the Frobenius equations are replaced by Lawvere’s equations).
The category *Frob* has a strict monoidal structure. The $\otimes$ of this monoidal structure is addition on objects. We define ${\mj_n\otimes f}$ as $M^n f$, where $M^n$ is a sequence of ${n\geq 0}$ occurrences of $M$, while ${f\otimes\mj_n}$ is defined by increasing the subscripts of $f$ by the natural number $n$. Then for ${f_1\!:n_1\str m_1}$ and ${f_2\!:n_2\str m_2}$ we have $$f_1\otimes f_2=_{df}(f_1\otimes\mj_{m_2})\cirk(\mj_{n_1}\otimes
f_2).$$ The category *Frob* was envisaged as a monoidal category in [@K03] (Section 3.6.16).
The category $\cal M$ of the monad freely generated by a single object $0$ is defined like *Frob* save that we omit the arrow terms $\varepsilon^\Box_n$ and $\delta^\Box_n$, and whatever involves them. By omitting $\varepsilon^\Diamond_n$ and $\delta^\Diamond_n$, we define analogously the comonad freely generated by $0$.
Free adjunctions and monads
===========================
An adjunction is given by two categories $\cal A$ and $\cal B$, a functor $F$ from $\cal B$ to $\cal A$, the *left adjoint*, a functor $G$ from $\cal A$ to $\cal B$, the *right adjoint*, a natural transformation ${\gamma\!:I_{\cal
B}\strt GF}$, the *unit* of the adjunction, and a natural transformation ${\varphi\!:FG\strt I_{\cal A}}$, the *counit* of the adjunction, which satisfy the following *triangular* equations for every object $B$ of $\cal B$ and every object $A$ of $\cal A$: $$\varphi_{FB}\cirk
F\gamma_B=\mj_{FB},\hspace{5em}G\varphi_A\cirk\gamma_{GA}=\mj_{GA}.$$
The adjunction freely generated by a single object $0$ on the $\cal B$ side is defined in syntactical terms analogously to *Frob* (see [@D99], Chapter 4, for a detailed exposition). In this free adjunction, the objects of $\cal B$ are $0$, $GF0$, $GFGF0$, etc., while those of $\cal A$ are $F0$, $FGF0$, $FGFGF0$, etc. This notion of freely generated adjunction is essentially the same as a 2-categorial notion that may be found in [@A74], [@SS86] (cf. also [@KS74]) and [@L08]. If we consider the sub-2-category of the 2-category *Cat* of categories whose only 0-cells are $\cal A$ and $\cal B$, whose 1-cells are made of $F$ and $G$, and whose 2-cells are made of $\varphi$, $\gamma$, $F$ and $G$, we obtain a 2-category isomorphic in the 2-categorial sense to the free category *Ad* of [@A74] (called *Adj* in [@SS86]). This does not depend on the number of generators of our free adjunction, provided it is not zero, and they may be either on the $\cal A$ side or on the $\cal B$ side.
The connection of our notion of free adjunction with the 2-category *Ad* may also be construed as follows. In addition to what we have above, we should consider the adjunction freely generated by a different object on the $\cal A$ side, which altogether gives us four disjoint categories. These four categories are isomorphic respectively to the categories *Hom*$({\cal A},{\cal A})$, *Hom*$({\cal A},{\cal B})$, *Hom*$({\cal B},{\cal B})$ and *Hom*$({\cal B},{\cal
A})$ that may be found in the 2-categorial approach of [@A74] and the other references above. Roughly speaking, one has only to understand our freely generated objects as 1-cells, and add 0-cells, to pass to the 2-categorial approach. In contradistinction to that approach, we restrict ourselves to syntactically constructed free adjunctions within the category *Cat*, and we make explicit the free generators, but the mathematical content is essentially the same. (The mathematical content changes by moving to a new level of categorification with the pseudoadjunctions of [@V92] and [@L00].)
We give a new simple proof of the following result of [@A74] (*Corollaire* 2.8), which connects the category $\cal M$ of the free monad defined at the end of the preceding section with the category $\cal B$ of the adjunction freely generated by $0$ on the $\cal B$ side. This result is interesting for us, because it is at the base of a more complicated result concerning *Frob* that we establish in Section 5.
This isomorphism is proved syntactically by defining first by induction a functor $I$ from $\cal M$ to $\cal B$ for which we have
=$I0=0$,=$I(n\pl
1)=GFIn$,=\
$I\varepsilon^\Diamond_n=\gamma_{In}$,$I\delta^\Diamond_n=G\varphi_{FIn}$,$I\mj_n=\mj_{In}$,\
$I(h_2\cirk h_1)=Ih_2\cirk Ih_1$,$IMh=GFIh$.
We verify that $I$ is indeed a functor by induction on the length of derivation of an equation of $\cal M$.
Next we define by induction a functor $J$ from the category ${\cal
B}+{\cal A}$, which is the disjoint union of the categories $\cal
B$ and $\cal A$ of the free adjunction, to the category $\cal M$. For $J$ we have
=$J0=0$,=$JGFB=JFB=JB\pl
1$,=\
$J\gamma_B=\varepsilon^\Diamond_{JB}$,$J\varphi_A=\delta^\Diamond_{JA-1}$, $J\mj_C=\mj_{JC}$,\
$J(h_2\cirk h_1)=Jh_2\cirk Jh_1$,$JGf=Jf$,$JFg=MJg$.
To verify that $J$ is indeed a functor, which is done by induction on the length of derivation of an equation, we had to define it from ${\cal B}+{\cal A}$, but there is an obvious functor $J_{\cal
B}$ from $\cal B$ to $\cal M$ obtained by restricting $J$.
It is straightforward to verify by induction on the complexity of objects and arrow terms that $I$ and $J_{\cal B}$ are inverse to each other. So the categories $\cal M$ and $\cal B$ are isomorphic.
A more involved, graphical, proof of this proposition may be found in [@D08] (Sections 6-8).
If our free adjunction is generated by a single object on the $\cal A$ side, then we establish the isomorphism of $\cal A$ with the category of the comonad freely generated by a single object (see the end of Section 2).
Bijunctions and self-adjunctions
================================
We call *trijunction* a structure made of the categories $\cal A$ and $\cal B$, the functor $U$ from $\cal A$ to $\cal B$, and the functors $L$ and $R$ from $\cal B$ to $\cal A$, such that $L$ is left adjoint to $U$, with the unit $\gamma^{\cal
B}\!:I_{\cal B}\strt UL$ and counit $\varphi^{\cal A}\!:LU\strt
I_{\cal A}$, and $R$ is right adjoint to $U$, with the unit $\gamma^{\cal A}\!:I_{\cal A}\strt RU$ and counit $\varphi^{\cal
B}\!:UR\strt I_{\cal B}$. This notion plays an important role in [@DP08b].
We call *bijunction* a trijunction where the functors $L$ and $R$ are equal. We write in this context $P$ instead of $L$ and $R$. (Various terms have been used for bijunctions and related notions: in [@M65] one finds *strongly adjoint* pairs of functors, in [@CMS97] *Frobenius functors*, in [@K02] *biadjunction*, which has already been introduced for something else in [@S80], in [@S04] one finds *autonomous category* and *Frobenius pseudomonoid*, and in [@L06] *ambidextrous adjunction*.)
A *self-adjunction* is an adjunction where the categories $\cal A$ and $\cal B$ are equal, and the functors $F$ and $G$, which are now endofunctors, are also equal. We write in this context $\cal S$ for $\cal A$ and $\cal B$, and $F$ for both $F$ and $G$. So the unit and counit of a self-adjunction are respectively ${\gamma\!:I_{\cal S}\strt FF}$ and ${\varphi\!:FF\strt I_{\cal S}}$. Every self-adjunction is a bijunction. (Self-adjunction is not often mentioned in textbooks of category theory—an exception is [@AHS], Chapter 5, Exercise 19G; the notion of self-adjoint functor of [@T99] is a related but different notion.)
The bijunction freely generated by a single object $0$ on the $\cal A$ side is defined in syntactical terms analogously to *Frob* in Section 2. The objects of the category $\cal A$ are here $0$, $PU0$, $PUPU0$, etc., while those of $\cal B$ are $U0$, $UPU0$, $UPUPU0$, etc.
We define analogously the free self-adjunction generated by a single object $0$. An object of the category $\cal S$ of this self-adjunction is of the form $F^n0$, where $F^n$ is a sequence of ${n\geq 0}$ occurrences of $F$. We identify this object with $n$, so that $Fn$ is ${n\pl 1}$. (One can find in [@DP03] a more detailed construction of $\cal S$, which is there called ${\cal L}_c$.) The category $\cal S$ is the disjoint union of the categories $\cal S_A$, whose objects are even, and $\cal S_B$, whose objects are odd.
For $\cal C$ being one of the categories $\cal A$ and $\cal B$ of the penultimate paragraph, and a subscript of one of the categories $\cal S_A$ and $\cal S_B$ of the preceding paragraph, we can prove the following.
We define first by induction the functors $H_{\cal C}$ from $\cal S_C$ to $\cal C$, for $\alpha$ being $\varphi$ or $\gamma$:
=$H_{\cal A}0=0$,=$H_{\cal A}(2n\pl
2)=PH_{\cal B}(2n\pl 1)$,=$H_{\cal B}(2n\pl
1)=UH_{\cal A}2n$,\
$H_{\cal A}\alpha_{2n}=\alpha^{\cal A}_{H_{\cal A}2n}$, $H_{\cal B}\alpha_{2n+1}=\alpha^{\cal B}_{H_{\cal B}(2n+1)}$,\
$H_{\cal C}\mj_n=\mj_{H_{\cal C}n}$,$H_{\cal C}(h_2\cirk
h_1)=H_{\cal C}h_2\cirk H_{\cal C}h_1$,\
$H_{\cal A}Fg=PH_{\cal B}g$,$H_{\cal B}Ff=UH_{\cal A}f$.
Next we define by induction the functors $K_{\cal C}$ from $\cal
C$ to $\cal S_C$:
=$H_{\cal A}0=0$,=$H_{\cal A}(2n\pl
2)=PH_{\cal B}(2n\pl 1)$,=$H_{\cal B}(2n\pl
1)=UH_{\cal A}2n$,
$K_{\cal A}0=0$,$K_{\cal A}PB=K_{\cal B}B\pl 1$,$K_{\cal
B}UA=K_{\cal A}A\pl 1$,\
$K_{\cal C}\alpha^{\cal C}_C=\alpha_{K_{\cal C}C}$,\
$K_{\cal C}\mj_C=\mj_{K_{\cal C}C}$,$K_{\cal C}(h_2\cirk
h_1)=K_{\cal C}h_2\cirk K_{\cal C}h_1$,\
$K_{\cal A}Pg=FK_{\cal B}g$,$K_{\cal B}Uf=FK_{\cal A}f$.
We verify by induction on the length of derivation of an equation that $H_{\cal C}$ and $K_{\cal C}$ are indeed functors. Next we verify by induction on the complexity of objects and arrow terms that $H_{\cal C}$ and $K_{\cal C}$ are inverse to each other. So the categories $\cal C$ and $\cal S_C$ are isomorphic.
Frobenius monads and self-adjunctions
=====================================
We want to prove the following result concerning the category *Frob* of the free Frobenius monad of Section 2 and the category $\cal S_A$ of the free self-adjunction of the preceding section.
We define first by induction a functor $I$ from *Frob* to $\cal S_A$:
=$I0=0$,=$I(n\pl
1)=GFIn$,=
$In=2n$,\
$I\varepsilon^\Box_n=\varphi_{2n}$,$I\delta^\Box_n=F\gamma_{2n+1}$,\
$I\varepsilon^\Diamond_n=\gamma_{2n}$,$I\delta^\Diamond_n=
F\varphi_{2n+1}$,$I\mj_n=\mj_{2n}$,\
$I(h_2\cirk h_1)=Ih_2\cirk Ih_1$,$IMh=FFIh$.
Next we define by induction a functor $J$ from $\cal S$ to *Frob*:
=$I0=0$,=$I(n\pl
1)=GFIn$,=
$J2n=n$,$J(2n\pl 1)=n\pl 1$,\
$J\varphi_{2n}=\varepsilon^\Box_n$,$J\gamma_{2n+1}=\delta^\Box_n$,\
$J\gamma_{2n}=\varepsilon^\Diamond_n$,$J\varphi_{2n+1}=\delta^\Diamond_n$, $J\mj_C\!=\mj_{JC}$,\
$J(h_2\cirk h_1)=Jh_2\cirk Jh_1$,$JFg=Jg$, for $g$ in $\cal
S_B$,$JF\!f\!=MJf$, for $f$ in $\cal S_A$.
We verify by induction on the length of derivation of an equation that $I$ and $J$ are indeed functors. We will not dwell on that verification for $I$, while for $J$ we have to verify first that $$J(h\cirk\varphi_n)=J(\varphi_m\cirk FFh).$$ If $h$ is from $\cal S_A$, then we use the equation ($\varepsilon^\Box$ [*nat*]{}) of Section 2. If $h$ is from $\cal
S_B$, then we proceed by induction on the complexity of $h$, by using the Frobenius equations and the equations $(\delta^\Diamond)$ and ($\delta^\Diamond$ [*nat*]{}) of Section 2. Note that if $h$ is from $\cal S_B$, then $Jh$ can be neither $\varepsilon^\Box_k$ nor $\varepsilon^\Diamond_k$. We proceed analogously for $$J(\gamma_m\cirk h)=J(FFh\cirk \gamma_n).$$ To verify ${Jh_1=Jh_2}$ for ${h_1=h_2}$ a triangular equation, we use the equations $(\Box\beta)$, $(\Box\eta)$, $(\Diamond\beta)$ and $(\Diamond\eta)$ of Section 2.
There is an obvious functor $J_{\cal A}$ from $\cal S_A$ to *Frob* obtained by restricting $J$, and it is straightforward to verify by induction on the complexity of arrow terms that $I$ and $J_{\cal A}$ are inverse to each other. So the categories *Frob* and $\cal S_A$ are isomorphic.$\dashv$
From this proposition and from the Proposition of the preceding section we can conclude that *Frob* is isomorphic to the category $\cal A$ of the bijunction freely generated by a single object on the $\cal A$ side, but the isomorphism we have established in this section is more interesting for us, as it will become clear in the next section.
Coherence for Frobenius monads
==============================
Out of the category $\cal S$ of the free self-adjunction of Section 4, we build a monoid ${\cal S}^\ast$ (which in [@DP03], Section on ${\cal L}_c$ *and* ${\cal L}_\omega$, is called ${\cal L}_c^\ast$, while $\cal S$ is called ${\cal
L}_c$). On the arrows of $\cal S$, we define a total binary operation $\ast$ based on composition in the following manner: for $f\!:m\str n$ and $g\!:k\str l$, $$g\ast f=_{df}\left\{
\begin{array}{ll}
g\cirk F^{k-n}f & {\mbox{\rm if }}n\leq k
\\
F^{n-k}g\cirk f & {\mbox{\rm if }}k\leq n.
\end{array}
\right.$$
Next, let $f\equiv g$ if and only if for some $k$ and $l$ we have $F^kf=F^lg$ in $\cal S$. It is easy to check that $\equiv$ is an equivalence relation on the arrows of $\cal S$, which satisfies moreover $${\mbox{\rm if }}f_1\equiv f_2\; {\mbox{\rm and }} g_1\equiv g_2,
\; {\mbox{\rm then }}g_1\ast f_1\equiv g_2\ast f_2.$$
For every arrow $f$ of $\cal S$, let $[f]$ be $\{g\mid f\equiv
g\}$, and let ${\cal S}^\ast$ be $\{[f]\mid f$ is an arrow of $\cal S\}$. With $$\begin{array}{l}
\mj=_{df}[\mj_0],
\\[.1cm]
[g][f]=_{df}[g\ast f],
\end{array}$$ we can check that ${\cal S}^\ast$ is indeed a monoid, which in [@DP03] is shown to be isomorphic to the monoid ${\cal
L}_\omega$. The monoid ${\cal S}^\ast$ is built of syntactical material, coming from the category $\cal S$, which is presented by generators and equations. The monoid ${\cal L}_\omega$ is just another presentation by generators and equations of ${\cal
S}^\ast$. A reason for introducing it in [@DP03] was to make simpler reduction to normal form, without being encumbered by the sources and targets of arrow terms. We presuppose the reader is acquainted with ${\cal L}_\omega$, but indications about how this monoid is presented will be given below when we deal with composition in *Frobse*. This monoid interests us here only as an auxiliary, leading towards the geometric categories *Frz* and *Frobse*, which we will consider below.
Out of the material of the monoid ${\cal S}^{\ast}$ we return to $\cal S$, by building a category isomorphic to $\cal S$, in the following way. Let ${\cal S}^{\ast t}$ be the category whose objects are the natural numbers, and whose arrows are the triples ${\langle [f], n, m\rangle}$ such that there is an arrow ${g:\!n\str m}$ in $[f]$; the arrow ${\langle [f], n, m\rangle}$ is of *type* $n\str m$ in ${\cal S}^{\ast t}$, which means that its source is $n$ and its target $m$. The composition of ${\langle [f_1], n, m\rangle}$ and ${\langle [f_2],m,k\rangle}$ is defined as ${\langle [f_2][f_1],n,k\rangle}$, and the identity arrow on $n$ is ${\langle [\mj_0],n,n\rangle}$. We can prove the following.
The functor from $\cal S$ to ${\cal S}^{\ast t}$ giving this isomorphism is identity on objects and an arrow $f\!:n\str m$ is mapped to ${\langle [f], n, m\rangle}$. To prove the proposition, we rely on the fact that for every element $[f]$ of ${\cal
S}^\ast$, and every arrow ${g:\!n\str m}$ in $[f]$, the arrow $g$ is the only arrow in $[f]$ of the type ${n\str m}$. To establish this fact we need to establish the following.
An analogous lemma (called the $\cal L$ Cancellation Lemma) was proved in [@DP03], but with a restriction on the type of $f$ and $g$. It was stated there that the restriction can be lifted, and it was suggested how to achieve that. The suggestion envisaged two ways, one of which is rather straightforward (both of these ways are however lengthy, and for lack of space a detailed exposition was omitted.) As a matter of fact, there is a direct way to prove the $\cal S$ Cancellation Lemma along the lines of the ${\cal K}_c$ Cancellation Lemma of [@DP03]. Here are indications for this direct proof (which presuppose an acquaintance with [@DP03]).
[Proof of the $\cal S$ Cancellation Lemma.]{} We proceed as in the proof of the ${\cal K}_c$ Cancellation Lemma of [@DP03] until we reach the case that $f=\varphi_0\cirk f'$ and $g=\varphi_0\cirk g'$. Then we must ensure that $\varphi_0$ in $\varphi_0\cirk f'$ and $\varphi_0$ in $\varphi_0\cirk g'$ are tied in $\delta(\psi(f))$, which is equal to $\delta(\psi(g))$, to circles encompassing the same circular forms. It is always possible to achieve that. We conclude that $\delta(\psi(f'))\cong_{\cal L} \delta(\psi(g'))$. This is because $\delta(\psi(\varphi_0\cirk f'))$ and $\delta(\psi(\varphi_0\cirk
g'))$ are both $\cal L$-equivalent to $\delta(c_1^\alpha)$, for some $\alpha\geq 1$, while $\delta(\psi(f'))$ and $\delta(\psi(g'))$ must both be $\cal L$-equivalent to the same $\delta(b_1^{\beta}c_1^{\gamma})$, for $\beta$ and $\gamma$ lesser than $\alpha$. We conclude that ${f'\equiv g'}$, and since $f'$ and $g'$ are of type $0\str 2$, by the $\cal L$ Cancellation Lemma of [@DP03], we have that $f'=g'$ in $\cal S$. From that we obtain that $f=g$ in $\cal S$.
As $\cal S$, the isomorphic category ${\cal S}^{\ast t}$ is the disjoint union of two categories ${\cal S}_{\cal A}^{\ast t}$ and ${\cal S}_{\cal B}^{\ast t}$, isomorphic respectively to $\cal
S_A$ and $\cal S_B$. If ${\langle [f], n, m\rangle}$ is in ${\cal
S}_{\cal A}^{\ast t}$, then $n$ and $m$ are even, and if it is in ${\cal S}_{\cal B}^{\ast t}$, then they are odd. We will now define a category ${\cal L}^{\cal A}_\omega$, isomorphic to ${\cal
S}_{\cal A}^{\ast t}$. This category, made out of the material of the monoid ${\cal L}_\omega$, interests us only as a stepping stone towards the isomorphic geometric categories *Frz* and *Frobse*, which we will consider in a moment.
The objects of the category ${\cal L}^{\cal A}_\omega$ are again the natural numbers. An arrow of this category will be obtained from an arrow ${\langle [f], n, m\rangle}$ of ${\cal S}_{\cal
A}^{\ast t}$ by replacing the class $[f]$ by the corresponding element $e$ of ${\cal L}_\omega$, and $n$ and $m$ by respectively $n/2$ and $m/2$; the type of ${\langle e,n/2,m/2\rangle}$ in ${\cal L}^{\cal A}_\omega$ is ${n/2\str m/2}$. We divide the numbers in the types by two to obtain types that will correspond to the types in *Frob* (see the preceding section).
In this way, with every element of ${\cal L}_\omega$ we associate in ${\cal L}^{\cal A}_\omega$ a denumerable infinity of types. The generator $a^\alpha_{2n+1}$ of ${\cal L}_\omega$ (see [@DP03], Section on *Normal forms in* ${\cal L}_\omega$) will have as associated types $n\pl l\pl 1\str n\pl l$, for every $l\geq 0$, while the generator $b^\alpha_{2n+1}$ will have $n\pl l\str n\pl
l\pl 1$, and the generator $c^\alpha_{2n+1}$ will have $n\pl l\str
n\pl l$. (This typing is explained by the typing of the friezes below.) The generator $a^\alpha_{2n+2}$ will have as associated types $n\pl l\pl 2\str n\pl l\pl 1$, the generator $b^\alpha_{2n+2}$ will have $n\pl l\pl 1\str n\pl l\pl 2$, and the generator $c^\alpha_{2n+2}$ will have $n\pl l\pl 1\str n\pl l\pl
1$. Multiplication of terms now becomes composition, and takes the types into account. Two typed terms of ${\cal L}_\omega$ stand for the same arrow of the category ${\cal L}^{\cal A}_\omega$ if and only if they are of the same type and equal in ${\cal L}_\omega$. We can then assert the following.
This follows immediately from the isomorphism of *Frob* and $\cal S_A$, proved in the preceding section, the isomorphism of $\cal S_A$ and ${\cal S}_{\cal A}^{\ast t}$, which follows from Proposition 1, and the isomorphism of ${\cal S}_{\cal
A}^{\ast t}$ and ${\cal L}^{\cal A}_\omega$.
From [@DP03] (Section on ${\cal L}_\omega$, ${\cal K}_\omega$ *and friezes*) one can infer that the category ${\cal
L}^{\cal A}_\omega$ is isomorphic to a category *Frz* whose arrows are diagrams called *friezes* with associated types. Roughly speaking, a frieze is a tangle without crossings in whose regions we find circular forms that correspond bijectively to the ordinals contained in the infinite ordinal $\varepsilon_0$. In [@DP03] one can found a proof that ${\cal L}_\omega$ is isomorphic to the monoid of friezes, and from that the isomorphism of the categories ${\cal L}^{\cal A}_\omega$ and *Frz* follows. So, by Proposition 2, the categories *Frob* and *Frz* are isomorphic. By this last isomorphism, the arrows on the left are mapped to the friezes on the right, with the type associated to the friezes being those of the arrows:
(200,60)(-60,0)
(-120,25)[(0,0)\[l\][${\varepsilon^\Box_n\!:n\pl 1\str
n}$]{}]{}
[ (0,10)[(1,0)[200]{}]{} (0,10)[(0,1)[30]{}]{} (0,40)[(1,0)[200]{}]{}]{}
(30,0)[(0,0)\[b\][$1$]{}]{} (90,0)[(0,0)\[b\][$2n\pl 1$]{}]{} (30,45)[(0,0)\[b\][$1$]{}]{} (90,45)[(0,0)\[b\][$2n\pl 1$]{}]{} (120,45)[(0,0)\[b\][$2n\pl 2$]{}]{} (150,45)[(0,0)\[b\][$2n\pl 3$]{}]{}
(30,10)[(0,1)[30]{}]{} (90,10)[(2,1)[60]{}]{}
(105,40)[(30,30)\[b\]]{}
(50,1)[$\ldots$]{} (50,46)[$\ldots$]{} (147,25)[$\ldots$]{}
(200,70)(-60,0)
(-120,25)[(0,0)\[l\][${\delta^\Box_n\!:n\pl 1\str n\pl
2}$]{}]{}
[ (0,10)[(1,0)[200]{}]{} (0,10)[(0,1)[30]{}]{} (0,40)[(1,0)[200]{}]{}]{}
(30,0)[(0,0)\[b\][$1$]{}]{} (90,0)[(0,0)\[b\][$2n\pl 1$]{}]{} (120,0)[(0,0)\[b\][$2n\pl 2$]{}]{} (150,0)[(0,0)\[b\][$2n\pl 3$]{}]{} (180,0)[(0,0)\[b\][$2n\pl 4$]{}]{}
(30,45)[(0,0)\[b\][$1$]{}]{} (90,45)[(0,0)\[b\][$2n\pl 1$]{}]{} (120,45)[(0,0)\[b\][$2n\pl 2$]{}]{}
(30,10)[(0,1)[30]{}]{} (90,10)[(0,1)[30]{}]{} (180,10)[(-2,1)[60]{}]{}
(135,10)[(30,30)\[t\]]{}
(55,25)[$\ldots$]{} (177,25)[$\ldots$]{}
(200,70)(-60,0)
(-120,25)[(0,0)\[l\][${\varepsilon^\Diamond_n\!:n\str
n\pl 1}$]{}]{}
[ (0,10)[(1,0)[200]{}]{} (0,10)[(0,1)[30]{}]{} (0,40)[(1,0)[200]{}]{}]{}
(30,45)[(0,0)\[b\][$1$]{}]{} (90,45)[(0,0)\[b\][$2n\pl 1$]{}]{} (30,0)[(0,0)\[b\][$1$]{}]{} (90,0)[(0,0)\[b\][$2n\pl 1$]{}]{} (120,0)[(0,0)\[b\][$2n\pl 2$]{}]{} (150,0)[(0,0)\[b\][$2n\pl 3$]{}]{}
(30,10)[(0,1)[30]{}]{} (90,40)[(2,-1)[60]{}]{}
(105,10)[(30,30)\[t\]]{}
(50,1)[$\ldots$]{} (50,46)[$\ldots$]{} (147,25)[$\ldots$]{}
(200,70)(-60,0)
(-120,25)[(0,0)\[l\][${\delta^\Diamond_n\!:n\pl 2\str
n\pl 1}$]{}]{}
[ (0,10)[(1,0)[200]{}]{} (0,10)[(0,1)[30]{}]{} (0,40)[(1,0)[200]{}]{}]{}
(30,45)[(0,0)\[b\][$1$]{}]{} (90,45)[(0,0)\[b\][$2n\pl 1$]{}]{} (120,45)[(0,0)\[b\][$2n\pl 2$]{}]{} (150,45)[(0,0)\[b\][$2n\pl 3$]{}]{} (180,45)[(0,0)\[b\][$2n\pl 4$]{}]{}
(30,0)[(0,0)\[b\][$1$]{}]{} (90,0)[(0,0)\[b\][$2n\pl 1$]{}]{} (120,0)[(0,0)\[b\][$2n\pl 2$]{}]{}
(30,10)[(0,1)[30]{}]{} (90,10)[(0,1)[30]{}]{} (180,40)[(-2,-1)[60]{}]{}
(135,40)[(30,30)\[b\]]{}
(55,25)[$\ldots$]{} (177,25)[$\ldots$]{}
When ${n=0}$, the vertical thread connecting 1 at the top with 1 at the bottom does not exist in the first and the third frieze. Note that our friezes are “thin” tangles that may be conceived as the boundaries of the corresponding *thick* tangles of [@KL01].
A *circular form* is a finite collection of nonintersecting circles in the plane factored through homeomorphisms of the plane mapping one collection into another (see the definition of $\cal
L$-equivalence of friezes in [@DP03], Section on *Friezes*). The circular forms obtained by composing friezes are coded by the ordinals contained in $\varepsilon_0$ in the following way. The circular form consisting of no circles is coded by 0. If the circular forms $c_1$, $c_2$ and $c$ are coded by the ordinals $\alpha_1$, $\alpha_2$ and $\alpha$ respectively, then the circular form ${c_1c_2}$ (the disjoint union of $c_1$ and $c_2$) is coded by the natural sum ${\alpha_1\sharp\,\alpha_2}$, and the circular form
(12,7) (5,2) (5,0)[(0,0)\[b\][$c$]{}]{}
($c$ inside a new circle) is coded by $\omega^\alpha$. So a single circle is coded by $\omega^0$, which is equal to 1 (see [@DP03], Section on *Finite multisets, circular forms and ordinals*).
Let $\cal F$ be the commutative monoid with one unary operation freely generated by the empty set of generators. The elements of $\cal F$ may be identified with the hierarchy of finite multisets obtained by starting from the empty multiset as the only urelement, or by finite nonplanar trees with arbitrary finite branching, or by circular forms. A monoid isomorphic to $\cal F$ is the commutative monoid ${\langle\varepsilon_0,\sharp,0,\omega^{-}\rangle}$ where $\sharp$ is binary natural sum, and we have the additional unary operation $\omega^{-}$ (for more details on these matters, see [@DP03]). Note that though the elements of $\varepsilon_0$ greater than or equal to $\omega$ are associated with infinite ordinals, they may be used to code finite objects, such as circular forms. Another monoid isomorphic to $\cal F$ is the commutative monoid ${\langle\textbf{N}^+,\cdot,1,p_{\underline{\;\;}}\,\rangle}$ where $\textbf{N}^+$ is the set of natural numbers greater than 0, the operation $\cdot$ is multiplication, and $p_n$ is the $n$-th prime number (we are indebted for this remark to a suggestion of Marko Stoši' c).
The isomorphism of *Frob* with *Frz* may be understood as a geometrical description of *Frob*. Towards the end of his book [@K03] (Sections 3.6.20 ff), Kock was looking for such a description, but not exactly in the same direction. The category *Frobse*, isomorphic to *Frz*, which we will consider below, gives another alternative approach to the geometrization of *Frob* sought by Kock.
The isomorphism of *Frob* and *Frz* may be understood also as a coherence result, which provides a decision procedure for equality of arrows in *Frob*. This decision procedure involves a syntactical description of friezes given by the monoid ${\cal L}_\omega$ of [@DP03], and a reduction to normal form.
Instead of the category *Frz*, one can use an alternative isomorphic category, which we will call *Frobse*. In the arrows of this category, the regions of friezes stand for equivalence classes of an equivalence relation whose domain is split into a source part and a target part, which are both copies of $\textbf{N}^+$. Such equivalence relations were called *split equivalences* in [@DP03a]. Split equivalences are related to *cospans* in the base category *Set* (see [@ML98], XII.7, and [@RSW05], Example 2.4), but unlike cospans they do not register the common target of the two arrows making the cospan.
The split equivalences we envisage for *Frobse* are *nonintersecting* in the following sense. Let the source and target elements be identified respectively with the positive and negative integers (so 0 does not correspond to any element). For $a,b,c,d\in\textbf{Z}-\{0\}$, we say that ${(a,b)}$ *intersects* ${(c,d)}$ when either $a<c<b<d$ or $c<a<d<b$. An equivalence relation on ${\textbf{Z}-\{0\}}$ is *nonintersecting* when if $a$ and $b$ are in one equivalence class, while $c$ and $d$ are in another equivalence class, then ${(a,b)}$ does not intersect ${(c,d)}$. (This is related to the *nonoverlapping* segments of [@DP03], Section on *Friezes*.)
For example, instead of the frieze on the left-hand side, which is an arrow of *Frz* of the type ${2\pl l\str 1\pl l}$, we have the nonintersecting split equivalence on the right-hand side, which is an arrow of *Frobse* of the same type:
(200,40)
(0,35)[(0,0)\[b\][$1$]{}]{} (20,35)[(0,0)\[b\][$2$]{}]{} (40,35)[(0,0)\[b\][$3$]{}]{} (60,35)[(0,0)\[b\][$4$]{}]{}
(20,0)[(0,0)\[b\][$1$]{}]{} (40,0)[(0,0)\[b\][$2$]{}]{}
(-10,32)[(0,0)\[b\][$1$]{}]{} (10,32)[(0,0)\[b\][$2$]{}]{} (30,32)[(0,0)\[b\][$3$]{}]{} (50,32)[(0,0)\[b\][$4$]{}]{} (70,32)[(0,0)\[b\][$5$]{}]{}
(10,8)[(0,0)\[t\][$1$]{}]{} (30,8)[(0,0)\[t\][$2$]{}]{} (50,8)[(0,0)\[t\][$3$]{}]{}
(20,10)[(-1,1)[20]{}]{} (40,10)[(1,1)[20]{}]{}
(30,30)[(20,20)\[b\]]{}
(150,35)[(0,0)\[b\][$1$]{}]{} (170,35)[(0,0)\[b\][$2$]{}]{} (190,35)[(0,0)\[b\][$3$]{}]{} (210,35)[(0,0)\[b\][$4$]{}]{}
(170,0)[(0,0)\[b\][$1$]{}]{} (190,0)[(0,0)\[b\][$2$]{}]{}
(140,32)[(0,0)\[b\][$1$]{}]{} (160,32)[(0,0)\[b\][$2$]{}]{} (180,32)[(0,0)\[b\][$3$]{}]{} (200,32)[(0,0)\[b\][$4$]{}]{} (220,32)[(0,0)\[b\][$5$]{}]{}
(160,8)[(0,0)\[t\][$1$]{}]{} (180,8)[(0,0)\[t\][$2$]{}]{} (200,8)[(0,0)\[t\][$3$]{}]{}
(160,10)[(-1,1)[20]{}]{} (200,10)[(1,1)[20]{}]{} (180,10)[(0,1)[10]{}]{} (180,20)[(-2,1)[20]{}]{} (180,20)[(2,1)[20]{}]{}
(180,29)
The thick white regions on the left-hand side become thin black equivalence classes on the right-hand side, and the thin black threads on the left-hand side become white regions on the right-hand side. We will not obtain in this way on the right-hand side every nonintersecting split equivalence.
The equivalence classes of those we obtain satisfy some additional conditions. First, they are all finite, and all but finitely many of them are such that they have just two elements—one at the top and one at the bottom. Secondly, they are either *even* or *odd*, depending on whether their members are even or odd; we have only such even and odd equivalence classes. Finally, two classes of the same parity cannot be immediate neighbours in the following sense. The classes $A$ and $B$ are *immediate neighbours* when for every ${a\in A}$ and every ${b\in B}$ and every class $C$ and every ${c_1,c_2\in C}$, if ${(a,b)}$ intersects ${(c_1,c_2)}$, then $C$ is either $A$ or $B$. The nonintersecting split equivalences that satisfy these additional conditions concerning their equivalence classes will be called *maximal* split equivalences.
Note that in maximal split equivalences the odd equivalence classes are completely determined by the even equivalence classes, and vice versa. We cannot however reject either of them because of the ordinals. In the regions of friezes one finds finitely many circular forms that correspond to ordinals in $\varepsilon_0$, and we will assign these ordinals to the equivalence classes of maximal split equivalences.
Maximal split equivalences together with a function assigning ordinals in $\varepsilon_0$ to the equivalence classes, so that all but finitely many have zero as value, will be called *Frobenius* split equivalences. Frobenius split equivalences with types associated to them are the arrows of *Frobse*. For example, to the frieze on the left-hand side we assign the Frobenius split equivalence on the right-hand side:
(280,60)
(0,55)[(0,0)\[b\][$1$]{}]{} (20,55)[(0,0)\[b\][$2$]{}]{} (40,55)[(0,0)\[b\][$3$]{}]{} (60,55)[(0,0)\[b\][$4$]{}]{} (80,55)[(0,0)\[b\][$5$]{}]{} (100,55)[(0,0)\[b\][$6$]{}]{}
(20,0)[(0,0)\[b\][$1$]{}]{} (80,0)[(0,0)\[b\][$2$]{}]{}
(-10,52)[(0,0)\[b\][$1$]{}]{} (10,52)[(0,0)\[b\][$2$]{}]{} (30,52)[(0,0)\[b\][$3$]{}]{} (50,52)[(0,0)\[b\][$4$]{}]{} (70,52)[(0,0)\[b\][$5$]{}]{} (90,52)[(0,0)\[b\][$6$]{}]{} (110,52)[(0,0)\[b\][$7$]{}]{}
(10,8)[(0,0)\[t\][$1$]{}]{} (50,8)[(0,0)\[t\][$2$]{}]{} (90,8)[(0,0)\[t\][$3$]{}]{}
(20,10)[(-1,2)[20]{}]{} (80,10)[(1,2)[20]{}]{}
(50,50)[(20,20)\[b\]]{} (50,50)[(60,40)\[b\]]{}
(33,43) (33,43) (70,20) (-5,25) (1,25) (-8,25) (-8,25)
(180,55)[(0,0)\[b\][$1$]{}]{} (200,55)[(0,0)\[b\][$2$]{}]{} (220,55)[(0,0)\[b\][$3$]{}]{} (240,55)[(0,0)\[b\][$4$]{}]{} (260,55)[(0,0)\[b\][$5$]{}]{} (280,55)[(0,0)\[b\][$6$]{}]{}
(200,0)[(0,0)\[b\][$1$]{}]{} (260,0)[(0,0)\[b\][$2$]{}]{}
(170,52)[(0,0)\[b\][$1$]{}]{} (190,52)[(0,0)\[b\][$2$]{}]{} (210,52)[(0,0)\[b\][$3$]{}]{} (230,52)[(0,0)\[b\][$4$]{}]{} (250,52)[(0,0)\[b\][$5$]{}]{} (270,52)[(0,0)\[b\][$6$]{}]{} (290,52)[(0,0)\[b\][$7$]{}]{}
(190,8)[(0,0)\[t\][$1$]{}]{} (230,8)[(0,0)\[t\][$2$]{}]{} (270,8)[(0,0)\[t\][$3$]{}]{}
(190,10)[(-1,2)[20]{}]{} (270,10)[(1,2)[20]{}]{} (230,10)[(0,1)[10]{}]{} (230,20)[(-4,3)[40]{}]{} (230,20)[(4,3)[40]{}]{} (230,50)[(40,20)\[b\]]{} (230,49)
(179,30)[(0,0)\[r\][$\omega^{\omega^{\omega^0}\!\sharp \omega^0}$]{}]{} (232,23)[(0,0)\[tl\][$\omega^0$]{}]{} (234,38.5)[(0,0)\[t\][$\omega^{\omega^0}$]{}]{} (232,50)[(0,0)\[tl\][$0$]{}]{} (282,30)[(0,0)\[l\][$0$]{}]{}
All the Frobenius split equivalences are generated by composition from the following *generating* Frobenius split equivalences, which are correlated with the elements of the monoid ${\cal
L}_\omega$ mentioned on the left of the following pictures (see [@DP03], Section on *Normal forms in* ${\cal L}_\omega$ *and* ${\cal K}_\omega$), where we omit mentioning that an equivalence class bears $0$; here, ${k\geq 1}$ and ${\alpha,\beta\in\varepsilon_0}$:
(310,40)
(22,20)[(0,0)\[l\][$a^\alpha_k$]{}]{}
(125,20)[(0,0)\[l\][$\ldots$]{}]{} (215,20)[(0,0)\[r\][$\ldots$]{}]{}
(115,32)[(0,0)\[b\][$1$]{}]{} (135,32)[(0,0)\[b\][$k$]{}]{} (155,32)[(0,0)\[b\][$k\pl 1$]{}]{} (175,32)[(0,0)\[b\][$k\pl 2$]{}]{} (195,32)[(0,0)\[b\][$k\pl 3$]{}]{}
(115,5)[(0,0)\[b\][$1$]{}]{} (155,5)[(0,0)\[b\][$k$]{}]{} (195,5)[(0,0)\[b\][$k\pl 1$]{}]{}
(115,10)[(0,1)[20]{}]{} (155,10)[(0,1)[10]{}]{} (155,20)[(-2,1)[20]{}]{} (155,20)[(2,1)[20]{}]{} (195,10)[(0,1)[20]{}]{}
(155,27)[(0,0)\[t\][$\alpha$]{}]{}
(155,29)
(310,40)
(22,20)[(0,0)\[l\][$b^\beta_k$]{}]{}
(125,20)[(0,0)\[l\][$\ldots$]{}]{} (215,20)[(0,0)\[r\][$\ldots$]{}]{}
(115,5)[(0,0)\[b\][$1$]{}]{} (135,5)[(0,0)\[b\][$k$]{}]{} (155,5)[(0,0)\[b\][$k\pl 1$]{}]{} (175,5)[(0,0)\[b\][$k\pl 2$]{}]{} (195,5)[(0,0)\[b\][$k\pl 3$]{}]{}
(115,32)[(0,0)\[b\][$1$]{}]{} (155,32)[(0,0)\[b\][$k$]{}]{} (195,32)[(0,0)\[b\][$k\pl 1$]{}]{}
(115,10)[(0,1)[20]{}]{} (155,30)[(0,-1)[10]{}]{} (155,20)[(-2,-1)[20]{}]{} (155,20)[(2,-1)[20]{}]{} (195,30)[(0,-1)[20]{}]{}
(155,13)[(0,0)\[b\][$\beta$]{}]{}
(155,11)
(310,40)
(22,20)[(0,0)\[l\][$c^\alpha_k$]{}]{}
(125,20)[(0,0)\[l\][$\ldots$]{}]{} (201,20)[(0,0)\[r\][$\ldots$]{}]{}
(115,5)[(0,0)\[b\][$1$]{}]{} (155,5)[(0,0)\[b\][$k$]{}]{} (175,5)[(0,0)\[b\][$k\pl 1$]{}]{}
(115,32)[(0,0)\[b\][$1$]{}]{} (155,32)[(0,0)\[b\][$k$]{}]{} (175,32)[(0,0)\[b\][$k\pl 1$]{}]{}
(115,10)[(0,1)[20]{}]{} (155,30)[(0,-1)[20]{}]{} (175,30)[(0,-1)[20]{}]{}
(154,20)[(0,0)\[r\][$\alpha$]{}]{}
The composition of Frobenius split equivalences is made according to the following reductions, which are correlated with the equations of ${\cal L}_\omega$ on the left of the following pictures, for ${l\leq k}$:
(310,60)
(-13,30)[(0,0)\[l\][$(aa)$]{}]{} (32,31)[(0,0)\[l\][$a^\alpha_k a^\beta_l=a^\beta_l
a^\alpha_{k+2}$]{}]{}
(150,54)[(0,0)\[b\][$l$]{}]{} (170,54)[(0,0)\[b\][$l\pl 2$]{}]{} (190,54)[(0,0)\[b\][$k\pl 2$]{}]{} (210,54)[(0,0)\[b\][$k\pl 4$]{}]{}
(160,5)[(0,0)\[b\][$l$]{}]{} (200,5)[(0,0)\[b\][$k$]{}]{}
(160,10)[(0,1)[20]{}]{} (160,32)[(0,1)[10]{}]{} (160,42)[(-1,1)[10]{}]{} (160,42)[(1,1)[10]{}]{}
(200,10)[(0,1)[10]{}]{} (200,20)[(-1,1)[10]{}]{} (200,20)[(1,1)[10]{}]{} (190,32)[(0,1)[20]{}]{} (200,32)[(0,1)[20]{}]{} (210,32)[(0,1)[20]{}]{}
(160,50) (200,29)
(160.5,48.5)[(0,0)\[t\][$\beta$]{}]{} (200,27)[(0,0)\[t\][$\alpha$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(250,54)[(0,0)\[b\][$l$]{}]{} (270,54)[(0,0)\[b\][$l\pl 2$]{}]{} (290,54)[(0,0)\[b\][$k\pl 2$]{}]{} (310,54)[(0,0)\[b\][$k\pl 4$]{}]{}
(260,5)[(0,0)\[b\][$l$]{}]{} (300,5)[(0,0)\[b\][$k$]{}]{}
(300,10)[(0,1)[20]{}]{} (300,32)[(0,1)[10]{}]{} (300,42)[(-1,1)[10]{}]{} (300,42)[(1,1)[10]{}]{}
(260,10)[(0,1)[10]{}]{} (260,20)[(-1,1)[10]{}]{} (260,20)[(1,1)[10]{}]{} (250,32)[(0,1)[20]{}]{} (260,32)[(0,1)[20]{}]{} (270,32)[(0,1)[20]{}]{}
(300,50) (260,29)
(300,48)[(0,0)\[t\][$\alpha$]{}]{} (260.5,27.5)[(0,0)\[t\][$\beta$]{}]{}
(170,31)[(0,0)\[l\][$\ldots$]{}]{} (280,31)[(0,0)\[l\][$\ldots$]{}]{}
(310,60)
(-13,30)[(0,0)\[l\][$(c2)$]{}]{} (32,31)[(0,0)\[l\][$c^\alpha_k
c^\beta_k=c^{\alpha\sharp\beta}_k$]{}]{}
(180,54)[(0,0)\[b\][$k$]{}]{} (180,5)[(0,0)\[b\][$k$]{}]{} (280,54)[(0,0)\[b\][$k$]{}]{} (280,5)[(0,0)\[b\][$k$]{}]{}
(180,10)[(0,1)[20]{}]{} (180,32)[(0,1)[20]{}]{} (280,10)[(0,1)[42]{}]{}
(182,20)[(0,0)\[l\][$\alpha$]{}]{} (182,42)[(0,0)\[l\][$\beta$]{}]{} (282,31)[(0,0)\[l\][$\alpha\sharp\beta$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(310,60)
(-13,30)[(0,0)\[l\][$(cc)$]{}]{} (10,31)[(0,0)\[l\][[for $l<k,\;\;$]{} $c^\alpha_k
c^\beta_l=c^\beta_l c^\alpha_k$]{}]{}
(160,54)[(0,0)\[b\][$l$]{}]{} (190,54)[(0,0)\[b\][$k$]{}]{} (260,54)[(0,0)\[b\][$l$]{}]{} (290,54)[(0,0)\[b\][$k$]{}]{} (160,5)[(0,0)\[b\][$l$]{}]{} (190,5)[(0,0)\[b\][$k$]{}]{} (260,5)[(0,0)\[b\][$l$]{}]{} (290,5)[(0,0)\[b\][$k$]{}]{}
(160,10)[(0,1)[20]{}]{} (160,32)[(0,1)[20]{}]{} (180,10)[(0,1)[20]{}]{} (180,32)[(0,1)[20]{}]{} (190,10)[(0,1)[20]{}]{} (190,32)[(0,1)[20]{}]{}
(260,10)[(0,1)[20]{}]{} (260,32)[(0,1)[20]{}]{} (280,10)[(0,1)[20]{}]{} (280,32)[(0,1)[20]{}]{} (290,10)[(0,1)[20]{}]{} (290,32)[(0,1)[20]{}]{}
(192,20)[(0,0)\[l\][$\alpha$]{}]{} (158,42)[(0,0)\[r\][$\beta$]{}]{}
(292,42)[(0,0)\[l\][$\alpha$]{}]{} (258,20)[(0,0)\[r\][$\beta$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(164,31)[(0,0)\[l\][$\ldots$]{}]{} (264,31)[(0,0)\[l\][$\ldots$]{}]{}
(310,60)
(-13,30)[(0,0)\[l\][$(ab\:1)$]{}]{} (32,31)[(0,0)\[l\][$a^\alpha_l b^\beta_{k+2}=b^\beta_k
a^\alpha_l$]{}]{}
(150,54)[(0,0)\[b\][$l$]{}]{} (170,54)[(0,0)\[b\][$l\pl 2$]{}]{} (200,54)[(0,0)\[b\][$k\pl 2$]{}]{}
(160,5)[(0,0)\[b\][$l$]{}]{} (190,5)[(0,0)\[b\][$k$]{}]{} (210,5)[(0,0)\[b\][$k\pl 2$]{}]{}
(160,10)[(0,1)[10]{}]{} (160,20)[(-1,1)[10]{}]{} (160,20)[(1,1)[10]{}]{} (150,32)[(0,1)[20]{}]{} (160,32)[(0,1)[20]{}]{} (170,32)[(0,1)[20]{}]{}
(200,52)[(0,-1)[10]{}]{} (200,42)[(-1,-1)[10]{}]{} (200,42)[(1,-1)[10]{}]{} (190,30)[(0,-1)[20]{}]{} (200,30)[(0,-1)[20]{}]{} (210,30)[(0,-1)[20]{}]{}
(160,29) (200,33) (160,27)[(0,0)\[t\][$\alpha$]{}]{} (200,35)[(0,0)\[b\][$\beta$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(250,54)[(0,0)\[b\][$l$]{}]{} (270,54)[(0,0)\[b\][$l\pl 2$]{}]{} (300,54)[(0,0)\[b\][$k\pl 2$]{}]{}
(260,5)[(0,0)\[b\][$l$]{}]{} (290,5)[(0,0)\[b\][$k$]{}]{} (310,5)[(0,0)\[b\][$k\pl 2$]{}]{}
(260,10)[(0,1)[20]{}]{} (260,32)[(0,1)[10]{}]{} (260,42)[(-1,1)[10]{}]{} (260,42)[(1,1)[10]{}]{}
(300,32)[(0,1)[20]{}]{} (300,30)[(0,-1)[10]{}]{} (300,20)[(-1,-1)[10]{}]{} (300,20)[(1,-1)[10]{}]{}
(260,51) (300,11) (260,49)[(0,0)\[t\][$\alpha$]{}]{} (300,13)[(0,0)\[b\][$\beta$]{}]{}
(174,31)[(0,0)\[l\][$\ldots$]{}]{} (275,31)[(0,0)\[l\][$\ldots$]{}]{}
(310,60)
(-13,30)[(0,0)\[l\][$(ab\:3.1)$]{}]{} (32,31)[(0,0)\[l\][$a^\alpha_k b^\beta_{k+1}=c^\beta_k
c^\alpha_{k+1}$]{}]{}
(160,54)[(0,0)\[b\][$k$]{}]{} (180,54)[(0,0)\[b\][$k\pl 1$]{}]{}
(170,5)[(0,0)\[b\][$k$]{}]{} (190,5)[(0,0)\[b\][$k\pl 1$]{}]{}
(170,10)[(0,1)[10]{}]{} (170,20)[(-1,1)[10]{}]{} (170,20)[(1,1)[10]{}]{} (190,10)[(0,1)[20]{}]{} (160,32)[(0,1)[20]{}]{} (180,52)[(0,-1)[10]{}]{} (180,42)[(-1,-1)[10]{}]{} (180,42)[(1,-1)[10]{}]{}
(170,29) (180,33) (170,27)[(0,0)\[t\][$\alpha$]{}]{} (180,35)[(0,0)\[b\][$\beta$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(270,54)[(0,0)\[b\][$k$]{}]{} (290,54)[(0,0)\[b\][$k\pl 1$]{}]{}
(270,5)[(0,0)\[b\][$k$]{}]{} (290,5)[(0,0)\[b\][$k\pl 1$]{}]{}
(270,10)[(0,1)[20]{}]{} (270,32)[(0,1)[20]{}]{}
(290,10)[(0,1)[20]{}]{} (290,32)[(0,1)[20]{}]{}
(269,20)[(0,0)\[r\][$\beta$]{}]{} (291,42)[(0,0)\[l\][$\alpha$]{}]{}
(310,60)
(-13,30)[(0,0)\[l\][$(ab\:3.2)$]{}]{} (32,31)[(0,0)\[l\][$a^\alpha_{k+1} b^\beta_k=c^\alpha_k
c^\beta_{k+1}$]{}]{}
(170,54)[(0,0)\[b\][$k$]{}]{} (190,54)[(0,0)\[b\][$k\pl 1$]{}]{}
(160,5)[(0,0)\[b\][$k$]{}]{} (180,5)[(0,0)\[b\][$k\pl 1$]{}]{}
(180,10)[(0,1)[10]{}]{} (180,20)[(-1,1)[10]{}]{} (180,20)[(1,1)[10]{}]{} (160,10)[(0,1)[20]{}]{} (190,32)[(0,1)[20]{}]{} (170,52)[(0,-1)[10]{}]{} (170,42)[(-1,-1)[10]{}]{} (170,42)[(1,-1)[10]{}]{}
(170,33) (180,29) (180,27)[(0,0)\[t\][$\alpha$]{}]{} (170,35)[(0,0)\[b\][$\beta$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(270,54)[(0,0)\[b\][$k$]{}]{} (290,54)[(0,0)\[b\][$k\pl 1$]{}]{}
(270,5)[(0,0)\[b\][$k$]{}]{} (290,5)[(0,0)\[b\][$k\pl 1$]{}]{}
(270,10)[(0,1)[20]{}]{} (270,32)[(0,1)[20]{}]{}
(290,10)[(0,1)[20]{}]{} (290,32)[(0,1)[20]{}]{}
(269,20)[(0,0)\[r\][$\alpha$]{}]{} (291,42)[(0,0)\[l\][$\beta$]{}]{}
(310,60)
(-13,30)[(0,0)\[l\][$(ab\:3.3)$]{}]{} (32,31)[(0,0)\[l\][$a^\alpha_k
b^\beta_k=c^{\omega^{\alpha\sharp\beta}}_k$]{}]{}
(180,54)[(0,0)\[b\][$k$]{}]{}
(180,5)[(0,0)\[b\][$k$]{}]{}
(180,10)[(0,1)[10]{}]{} (180,20)[(-1,1)[10]{}]{} (180,20)[(1,1)[10]{}]{} (180,52)[(0,-1)[10]{}]{} (180,42)[(-1,-1)[10]{}]{} (180,42)[(1,-1)[10]{}]{}
(180,33) (180,29) (180,27)[(0,0)\[t\][$\alpha$]{}]{} (180,35)[(0,0)\[b\][$\beta$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(280,54)[(0,0)\[b\][$k$]{}]{} (280,5)[(0,0)\[b\][$k$]{}]{}
(280,10)[(0,1)[42]{}]{}
(281,31)[(0,0)\[l\][$\omega^{\alpha\sharp\beta}$]{}]{}
(310,60)
(-13,30)[(0,0)\[l\][$(ac\:1)$]{}]{} (32,31)[(0,0)\[l\][$a^\alpha_k c^\gamma_l=c^\gamma_l
a^\alpha_k$]{}]{}
(160,54)[(0,0)\[b\][$l$]{}]{} (180,54)[(0,0)\[b\][$k$]{}]{} (200,54)[(0,0)\[b\][$k\pl 2$]{}]{}
(160,5)[(0,0)\[b\][$l$]{}]{} (190,5)[(0,0)\[b\][$k$]{}]{}
(160,10)[(0,1)[20]{}]{} (160,32)[(0,1)[20]{}]{} (190,10)[(0,1)[10]{}]{} (190,20)[(-1,1)[10]{}]{} (190,20)[(1,1)[10]{}]{} (180,32)[(0,1)[20]{}]{} (190,32)[(0,1)[20]{}]{} (200,32)[(0,1)[20]{}]{}
(190,29) (190,27)[(0,0)\[t\][$\alpha$]{}]{} (159,42)[(0,0)\[r\][$\gamma$]{}]{}
(164,31)[(0,0)\[l\][$\ldots$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(260,54)[(0,0)\[b\][$l$]{}]{} (280,54)[(0,0)\[b\][$k$]{}]{} (300,54)[(0,0)\[b\][$k\pl 2$]{}]{}
(260,5)[(0,0)\[b\][$l$]{}]{} (290,5)[(0,0)\[b\][$k$]{}]{}
(260,10)[(0,1)[20]{}]{} (260,32)[(0,1)[20]{}]{}
(290,10)[(0,1)[20]{}]{} (290,32)[(0,1)[10]{}]{} (290,42)[(-1,1)[10]{}]{} (290,42)[(1,1)[10]{}]{}
(290,51) (290,49)[(0,0)\[t\][$\alpha$]{}]{} (259,20)[(0,0)\[r\][$\gamma$]{}]{}
(270,31)[(0,0)\[l\][$\ldots$]{}]{}
(310,60)
(-13,30)[(0,0)\[l\][$(ac\:2)$]{}]{} (32,31)[(0,0)\[l\][$a^\alpha_l c^\gamma_{k+2}=c^\gamma_k
a^\alpha_l$]{}]{}
(160,54)[(0,0)\[b\][$l$]{}]{} (180,54)[(0,0)\[b\][$l\pl 2$]{}]{} (200,54)[(0,0)\[b\][$k\pl 2$]{}]{}
(170,5)[(0,0)\[b\][$l$]{}]{} (200,5)[(0,0)\[b\][$k$]{}]{}
(200,10)[(0,1)[20]{}]{} (200,32)[(0,1)[20]{}]{} (170,10)[(0,1)[10]{}]{} (170,20)[(-1,1)[10]{}]{} (170,20)[(1,1)[10]{}]{} (160,32)[(0,1)[20]{}]{} (170,32)[(0,1)[20]{}]{} (180,32)[(0,1)[20]{}]{}
(170,29) (170,27)[(0,0)\[t\][$\alpha$]{}]{} (201,42)[(0,0)\[l\][$\gamma$]{}]{}
(184,31)[(0,0)\[l\][$\ldots$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(260,54)[(0,0)\[b\][$l$]{}]{} (280,54)[(0,0)\[b\][$l\pl 2$]{}]{} (300,54)[(0,0)\[b\][$k\pl 2$]{}]{}
(270,5)[(0,0)\[b\][$l$]{}]{} (300,5)[(0,0)\[b\][$k$]{}]{}
(300,10)[(0,1)[20]{}]{} (300,32)[(0,1)[20]{}]{}
(270,10)[(0,1)[20]{}]{} (270,32)[(0,1)[10]{}]{} (270,42)[(-1,1)[10]{}]{} (270,42)[(1,1)[10]{}]{}
(270,51) (270,49)[(0,0)\[t\][$\alpha$]{}]{} (301,20)[(0,0)\[l\][$\gamma$]{}]{}
(280,31)[(0,0)\[l\][$\ldots$]{}]{}
(310,60)
(-13,30)[(0,0)\[l\][$(ac\:3)$]{}]{} (32,31)[(0,0)\[l\][$a^\alpha_k c^\gamma_{k+1}=
a^{\alpha\sharp\gamma}_k$]{}]{}
(170,54)[(0,0)\[b\][$k$]{}]{} (190,54)[(0,0)\[b\][$k\pl 2$]{}]{}
(180,5)[(0,0)\[b\][$k$]{}]{}
(180,10)[(0,1)[10]{}]{} (180,20)[(-1,1)[10]{}]{} (180,20)[(1,1)[10]{}]{} (170,32)[(0,1)[20]{}]{} (180,32)[(0,1)[20]{}]{} (190,32)[(0,1)[20]{}]{}
(180,29) (180,27)[(0,0)\[t\][$\alpha$]{}]{} (179,42)[(0,0)\[r\][$\gamma$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(270,54)[(0,0)\[b\][$k$]{}]{} (290,54)[(0,0)\[b\][$k\pl 2$]{}]{}
(280,5)[(0,0)\[b\][$k$]{}]{}
(280,10)[(0,1)[22]{}]{} (280,32)[(-1,2)[10]{}]{} (280,32)[(1,2)[10]{}]{}
(280,51) (280,49)[(0,0)\[t\][$\alpha\sharp\gamma$]{}]{}
If we disregard the ordinals, then this is exactly like composition of split equivalences.
There are moreover reductions corresponding to the equations $(bb)$, ${(ab\:2)}$, ${(bc\:1)}$, ${(bc\:2)}$ and ${(bc\:3)}$ of [@DP03] (Section on *Normal forms in* ${\cal L}_\omega$), which are analogous to $(aa)$, ${(ab\:1)}$, ${(ac\:1)}$, ${(ac\:2)}$ and ${(ac\:3)}$. We do not mention here trivial reductions involving $c^0_k$, which is equal to 1. As a limit case, where ${l=k}$, of the reduction corresponding to $(aa)$ we have
(140,60)
(0,54)[(0,0)\[b\][$l$]{}]{} (20.5,54)[(0,0)\[b\][$l\pl 2$]{}]{} (41.5,54)[(0,0)\[b\][$l\pl 4$]{}]{}
(30,5)[(0,0)\[b\][$l$]{}]{}
(30,10)[(0,1)[10]{}]{} (30,20)[(-2,1)[20]{}]{} (30,20)[(1,1)[10]{}]{}
(10,32)[(0,1)[10]{}]{} (10,42)[(-1,1)[10]{}]{} (10,42)[(1,1)[10]{}]{}
(30,32)[(0,1)[20]{}]{} (40,32)[(0,1)[20]{}]{}
(30,29) (10,51) (10,49)[(0,0)\[t\][$\beta$]{}]{} (30,27)[(0,0)\[t\][$\alpha$]{}]{}
(70,31)[(0,0)[$\leadsto$]{}]{}
(99,54)[(0,0)\[b\][$l$]{}]{} (120.5,54)[(0,0)\[b\][$l\pl 2$]{}]{} (140.5,54)[(0,0)\[b\][$l\pl 4$]{}]{}
(110,5)[(0,0)\[b\][$l$]{}]{}
(110,10)[(0,1)[10]{}]{} (110,20)[(-1,1)[10]{}]{} (110,20)[(2,1)[20]{}]{}
(130,32)[(0,1)[10]{}]{} (130,42)[(-1,1)[10]{}]{} (130,42)[(1,1)[10]{}]{}
(100,32)[(0,1)[20]{}]{} (110,32)[(0,1)[20]{}]{}
(110,29) (130,51) (110,27)[(0,0)\[t\][$\beta$]{}]{} (130,49)[(0,0)\[t\][$\alpha$]{}]{}
and analogously in other limit cases. The limit case ${l=k}$ of ${(ab\:1)}$ corresponds to one of the Frobenius equations:
(140,60)
(0,54)[(0,0)\[b\][$l$]{}]{} (32.5,54)[(0,0)\[b\][$l\pl 2$]{}]{}
(10,5)[(0,0)\[b\][$l$]{}]{} (42.5,5)[(0,0)\[b\][$l\pl 2$]{}]{}
(10,10)[(0,1)[10]{}]{} (10,20)[(-1,1)[10]{}]{} (10,20)[(1,1)[10]{}]{}
(32,52)[(0,-1)[10]{}]{} (32,42)[(-1,-1)[10]{}]{} (32,42)[(1,-1)[10]{}]{}
(32,10)[(0,1)[20]{}]{} (42,10)[(0,1)[20]{}]{} (0,32)[(0,1)[20]{}]{} (10,32)[(0,1)[20]{}]{}
(10,29) (32,33) (32,35)[(0,0)\[b\][$\beta$]{}]{} (10,27)[(0,0)\[t\][$\alpha$]{}]{}
(70,31)[(0,0)[$\leadsto$]{}]{}
(110,54)[(0,0)\[b\][$l$]{}]{} (130,54)[(0,0)\[b\][$l\pl 2$]{}]{}
(110,5)[(0,0)\[b\][$l$]{}]{} (130,5)[(0,0)\[b\][$l\pl 2$]{}]{}
(120,30)[(0,-1)[10]{}]{} (120,20)[(-1,-1)[10]{}]{} (120,20)[(1,-1)[10]{}]{}
(120,32)[(0,1)[10]{}]{} (120,42)[(-1,1)[10]{}]{} (120,42)[(1,1)[10]{}]{}
(120,11) (120,51) (120,13)[(0,0)\[b\][$\beta$]{}]{} (120,49)[(0,0)\[t\][$\alpha$]{}]{}
We believe that our Frobenius split equivalences are more handy than the diagrams that may be found in [@KL01] (Appendix C), to which they should be equivalent. They are more handy because the circular forms are coded efficiently by ordinals, while in the diagrams of [@KL01] they make complicated patterns that are defined in all possible ways in terms of the generators. What these diagrams miss essentially is the reduction corresponding to the equation ${(ab\:3.3)}$.
The friezes appropriate for trijunctions (see [@DP08b], Section 8) are such that circular components and circular forms do not arise. Such friezes can be replaced by maximal split equivalences, without ordinals. As we said above, in maximal split equivalences, the odd equivalence classes are completely determined by the even equivalence classes, and vice versa. By rejecting the odd equivalence classes, we obtain the split equivalences that correspond to the categories $S5_{\Box\Diamond}$ and $5S_{\Box\Diamond}$ by the functor $G$; by rejecting the even equivalence classes, we obtain those that come with the functor $G^d$ (see [@DP08b], Sections 6-7). Coherence for trijunction could be proved with respect to nonintersecting split equivalences for which either odd or even equivalence classes are rejected.
Frobenius monads and matrices
=============================
Let *Mat* be the skeleton of the category $\mbox{\it Vect}_K$ of finite-dimensional vector spaces over the field $K$, with linear transformations as arrows. The objects of *Mat* are the natural numbers, which are dimensions of the objects of $\mbox{\it Vect}_K$, and its arrows are matrices. The category *Mat* is strictly monoidal (in it the canonical arrows of its monoidal structure are identity arrows).
In this section we will show how the requirement of having a faithful functor into *Mat* induces a collapse of the ordinals of *Frob*. This means that the usual notion of Frobenius algebra is not exactly caught by the notion of Frobenius monad. There are further categorial equations implicit in the notion of Frobenius algebra, which do not hold in every Frobenius monad. We will describe in this section these equations, and show their necessity. We leave open the question whether they are also sufficient to describe categorially the notion of Frobenius algebra.
There is no faithful monoidal functor from the strictly monoidal category *Frob* into *Mat*. A necessary condition to obtain such a functor would be to extend the definition of *Frob* with some new equations, for whose formulation we need the following abbreviations:
= $(\delta^\Box_n)^0$= $=\mj_{n+1}$,=$(\delta^\Diamond_n)^0$= $=\mj_{n+1}$,\
$(\delta^\Box_n)^{k+1}$ $=\delta^\Box_{n+k}\cirk(\delta^\Box_n)^k$, $(\delta^\Diamond_n)^{k+1}$ $=(\delta^\Diamond_n)^k\cirk\delta^\Diamond_{n+k}$,\
$\Phi^k_n=_{df}\varepsilon^\Box_n\cirk(\delta^\Diamond_n)^k\cirk(\delta^\Box_n)^k\cirk\varepsilon^\Diamond_n$.
Our new equations are then all of the following equations, for ${k,n\geq 0}$:
$(\Phi)$$\Phi^k_n=M^n\Phi^k_0$,
where $M^n$ is a sequence of ${n\geq 0}$ occurrences of $M$. Equations with the same force as $(\Phi)$, which we will also call $(\Phi)$, are, for ${k,n\geq 0}$,
$(\Phi)$= $\Phi^k_n$ = $=M^n\Phi^k_0$,
$\Phi^k_{n+1}$ $=M\Phi^k_n$.
These equations do not hold in *Frob*, as can be seen with the help of the monoid ${\cal L}_\omega$, where the corresponding equations
=$(\Phi)$= $\Phi^k_n$ = $=M^n\Phi^k_0$,
$(\Phi c)$ $c^{\omega^k}_{2n+1}$ $=c^{\omega^k}_1$
do not hold. These equations hold in the monoid ${\cal K}_\omega$ of [@DP03].
Let the category $\mbox{\it Frob}'$ be defined like *Frob* save that we have in addition all the equations $(\Phi)$, and let ${\cal L}_\omega'$ be the monoid defined like ${\cal L}_\omega$ save that we have in addition all the equations $(\Phi c)$. If all the subscripts $n$ that may be found in defining $\Phi^k_n$ are replaced by $A$, while ${n\pl 1}$ and ${n\pl k}$ are replaced respectively by $MA$ and ${M^kA}$, then the equations $(\Phi)$ become
=$(\Phi)$= $\Phi^k_n$ = $=M^n\Phi^k_0$,
$\Phi^k_{M^{\!n}\!A}$ $=M^n\Phi^k_A$or$\Phi^k_{MA}=M\Phi^k_A$,
which we will also call $(\Phi)$, and which are the equations characterizing the class of Frobenius monads in which $\mbox{\it
Frob}'$ is the free one generated with a single object.
In the language of the free self-adjunction of Section 4, let $\kappa^0_{2n+1}$ stand for $\mj_{2n+1}$, and let $\kappa^{k+1}_{2n+1}$ be ${\kappa^k_{2n+1}\cirk\varphi_{2n+1}\cirk\gamma_{2n+1}}$. Consider then the category $\cal S'$ constructed like the category $\cal S$ of the free self-adjunction save that we have in addition for every ${k,n\geq 0}$ the equation $$\varphi_{2n}\cirk
F\kappa^k_{2n+1}\cirk\gamma_{2n}=F^{2n}(\varphi_0\cirk
F\kappa^k_1\cirk\gamma_0),$$ where $F^m$ is a sequence of ${m\geq 0}$ occurrences of $F$. The category $\cal S'$ is related to $\mbox{\it Frob}'$ as the category $\cal S$ is related to *Frob*; this is shown as in Section 5. On the other hand, $\cal S'$ is related to ${\cal
L}_\omega'$ as $\cal S$ is related to ${\cal L}_\omega$; this is shown as in [@DP03] (Section on ${\cal L}_c$ *and* ${\cal
L}_\omega$).
We can infer that $\mbox{\it Frob}'$ is isomorphic to a category whose arrows are the elements of the monoid ${\cal L}_\omega'$ with types associated to them (see the preceding section). This result may be understood as a coherence result, which provides a decision procedure for equality of arrows in $\mbox{\it Frob}'$. The normal form involved in this decision procedure would serve also for the isomorphism with the category $\mbox{\it Frz}'$, which we will consider in a moment. We will deal with this normal form later (see the second paragraph after the proof of Lemma $2m\pl 2$).
One could consider a category $\mbox{\it Frz}'$ analogous to the category *Frz* of the previous section, which would be isomorphic to our category derived from ${\cal L}_\omega'$. We will not describe $\mbox{\it Frz}'$ in detail, but just make a few indications. For the arrows of $\mbox{\it Frz}'$ we would take, instead of friezes, two-manifolds made out of friezes in the following way. The regions of friezes may be chessboard-coloured by making the leftmost region white, and then alternating black and white for subsequent regions. For example, one of the friezes we had above is chessboard-coloured as follows:
(100,60)
(0,55)[(0,0)\[b\][$1$]{}]{} (20,55)[(0,0)\[b\][$2$]{}]{} (40,55)[(0,0)\[b\][$3$]{}]{} (60,55)[(0,0)\[b\][$4$]{}]{} (80,55)[(0,0)\[b\][$5$]{}]{} (100,55)[(0,0)\[b\][$6$]{}]{}
(20,0)[(0,0)\[b\][$1$]{}]{} (80,0)[(0,0)\[b\][$2$]{}]{}
(-10,52)[(0,0)\[b\][$1$]{}]{} (10,52)[(0,0)\[b\][$2$]{}]{} (30,52)[(0,0)\[b\][$3$]{}]{} (50,52)[(0,0)\[b\][$4$]{}]{} (70,52)[(0,0)\[b\][$5$]{}]{} (90,52)[(0,0)\[b\][$6$]{}]{} (110,52)[(0,0)\[b\][$7$]{}]{}
(10,8)[(0,0)\[t\][$1$]{}]{} (50,8)[(0,0)\[t\][$2$]{}]{} (90,8)[(0,0)\[t\][$3$]{}]{}
(40,50)[(1,0)[20]{}]{}
(40,50)[(1,0)[20]{}]{}
(0,50)[(1,0)[20]{}]{} (80,50)[(1,0)[20]{}]{} (20,10)[(-1,2)[20]{}]{} (17,49.2)[(1,0)[3]{}]{} (17,48.4)[(1,0)[3]{}]{} (17,47.6)[(1,0)[3]{}]{} (17,46.8)[(1,0)[3]{}]{} (17,46)[(1,0)[3]{}]{} (80,49.2)[(1,0)[2]{}]{} (80,48.4)[(1,0)[2]{}]{} (80,47.6)[(1,0)[2]{}]{} (80,46.8)[(1,0)[2]{}]{} (80,46)[(1,0)[2]{}]{} (79.8,45.2)[(1,0)[2]{}]{} (74,25)[(1,0)[10]{}]{}
(59.7,30)(0,-.6)[9]{}[(1,0)[13]{}]{} (67.4,24.4)(-.3,-.4)[14]{}[(-1,0)[7]{}]{} (74.5,30)(.75,.4)[16]{}[(-1,0)[10]{}]{} (80,24.4)(.2,-.4)[18]{}[(-1,0)[7]{}]{} (63,10.4)(0,.4)[12]{}[(1,0)[11]{}]{} (20,10)[(1,0)[60]{}]{}
(20,10)(.4,0)[44]{}[(-1,2)[20]{}]{} (37.8,10)[(-1,2)[14.5]{}]{} (38.2,10)[(-1,2)[14]{}]{} (38.6,10)[(-1,2)[13]{}]{} (39,10)[(-1,2)[12.5]{}]{} (39.4,10)[(-1,2)[12]{}]{} (39.8,10)[(-1,2)[11.9]{}]{} (40.2,10)[(-1,2)[11.8]{}]{} (41,10)(.8,0)[2]{}[(-1,2)[11.15]{}]{} (42.6,10)[(-1,2)[10.5]{}]{} (43.4,10)(.8,0)[4]{}[(-1,2)[10.3]{}]{} (37,10)(.4,0)[80]{}[(-1,2)[9.8]{}]{}
(80,10)[(1,2)[20]{}]{} (80,10)(-.4,0)[24]{}[(1,2)[20]{}]{} (58.5,10)(-.4,0)[40]{}[(1,2)[9.8]{}]{} (79,25)(-.4,0)[24]{}[(1,2)[12.5]{}]{}
(50,50)[(20,20)\[b\]]{} (50,50)[(19.5,19.5)\[b\]]{} (50,50)[(19,19)\[b\]]{} (50,50)[(18.5,18.5)\[b\]]{} (50,50)[(18,18)\[b\]]{} (50,50)[(17.5,17.5)\[b\]]{} (50,50)[(17,17)\[b\]]{} (50,50)[(16.5,16.5)\[b\]]{} (50,50)[(16,16)\[b\]]{} (50,50)[(15.5,15.5)\[b\]]{} (50,50)[(15,15)\[b\]]{} (50,50)[(14.5,14.5)\[b\]]{} (50,50)[(14,14)\[b\]]{} (50,50)[(13.5,13.5)\[b\]]{} (50,50)[(13,13)\[b\]]{} (50,50)[(12.5,12.5)\[b\]]{} (50,50)[(12,12)\[b\]]{} (50,50)[(11.5,11.5)\[b\]]{} (50,50)[(11,11)\[b\]]{} (50,50)[(10.5,10.5)\[b\]]{} (50,50)[(10,10)\[b\]]{} (50,50)[(9.5,9.5)\[b\]]{} (50,50)[(9,9)\[b\]]{} (50,50)[(8.5,8.5)\[b\]]{} (50,50)[(8,8)\[b\]]{} (50,50)[(7.5,7.5)\[b\]]{} (50,50)[(7,7)\[b\]]{} (50,50)[(6.5,6.5)\[b\]]{} (50,50)[(6,6)\[b\]]{} (50,50)[(5.5,5.5)\[b\]]{} (50,50)[(5,5)\[b\]]{} (50,50)[(4.5,4.5)\[b\]]{} (50,50)[(4,4)\[b\]]{} (50,50)[(3.5,3.5)\[b\]]{} (50,50)[(3,3)\[b\]]{} (50,50)[(2.5,2.5)\[b\]]{} (50,50)[(2,2)\[b\]]{} (50,50)[(1.5,1.5)\[b\]]{} (50,50)[(1,1)\[b\]]{}
(50,50)[(60,40)\[b\]]{} (50,50)[(61,41)\[b\]]{}
(33,43) (33,43) (33,43) (33,43) (33,43) (33,43) (33,43) (33,43) (33,43) (33,43) (33,43) (33,43) (33,43)
(70,20) (70,20) (70,20) (70,20) (-5,25) (1,25) (1,25) (1,25) (1,25) (-8,25) (-8,25) (-8,25) (-8,25) (-8,25) (-8,25)
(-7,34.6)[(1,0)[4]{}]{} (-7,34.2)[(1,0)[4]{}]{} (-9.8,33.8)[(1,0)[9.6]{}]{} (-9.8,33.4)[(1,0)[9]{}]{} (-11,33)[(1,0)[12]{}]{} (-11,32.6)[(1,0)[11]{}]{} (-12,32.2)[(1,0)[14]{}]{} (-12,31.8)[(1,0)[13]{}]{} (-13,31.4)[(1,0)[15]{}]{} (-13,31)[(1,0)[15]{}]{} (-13.5,30.6)[(1,0)[16]{}]{} (-13.5,30.2)[(1,0)[16]{}]{} (-14,29.8)[(1,0)[5]{}]{} (-14,29.4)[(1,0)[3.5]{}]{} (-14.3,29)[(1,0)[3.5]{}]{} (-14.3,21)[(1,0)[3.5]{}]{} (-14,20.6)[(1,0)[3.5]{}]{} (-14,20.2)[(1,0)[5]{}]{} (-13.5,19.8)[(1,0)[16]{}]{} (-13.5,19.4)[(1,0)[17]{}]{} (-13,19)[(1,0)[15]{}]{} (-13,18.6)[(1,0)[16]{}]{} (-12,18.2)[(1,0)[13]{}]{} (-12,17.8)[(1,0)[14]{}]{} (-11,17.4)[(1,0)[11]{}]{} (-11,17)[(1,0)[12]{}]{} (-9.8,16.6)[(1,0)[9]{}]{} (-9.8,16.2)[(1,0)[9.6]{}]{} (-8,15.8)[(1,0)[6]{}]{} (-8,15.4)[(1,0)[6.5]{}]{} (-4,30.2)[(1,0)[7]{}]{} (-4,29.8)[(1,0)[7.5]{}]{} (-3.5,29.4)[(1,0)[7]{}]{} (-3.5,29)[(1,0)[7.5]{}]{} (-3.2,28.6)[(1,0)[7]{}]{} (-3.2,28.2)[(1,0)[7]{}]{} (-3.2,27.8)[(1,0)[8]{}]{} (-3.2,27.4)[(1,0)[8]{}]{} (-4,20.6)[(1,0)[8]{}]{} (-4,20.2)[(1,0)[8]{}]{} (-3.5,21.4)[(1,0)[8]{}]{} (-3.5,21)[(1,0)[8]{}]{} (-3.2,22.2)[(1,0)[8]{}]{} (-3.2,21.8)[(1,0)[8]{}]{} (-3.2,22.6)[(1,0)[8]{}]{}
Then consider the two-manifolds with boundary made of the compact black regions, which we will call *black friezes*, and on black friezes consider the equivalence relation based on homeomorphisms that preserve all the points on the top and bottom line (this is like the $\cal K$-equivalence of [@DP03], Section on *Friezes*). So the following black frieze would be equivalent to the black frieze above:
(100,60)
(0,55)[(0,0)\[b\][$1$]{}]{} (20,55)[(0,0)\[b\][$2$]{}]{} (40,55)[(0,0)\[b\][$3$]{}]{} (60,55)[(0,0)\[b\][$4$]{}]{} (80,55)[(0,0)\[b\][$5$]{}]{} (100,55)[(0,0)\[b\][$6$]{}]{}
(20,0)[(0,0)\[b\][$1$]{}]{} (80,0)[(0,0)\[b\][$2$]{}]{}
(-10,52)[(0,0)\[b\][$1$]{}]{} (10,52)[(0,0)\[b\][$2$]{}]{} (30,52)[(0,0)\[b\][$3$]{}]{} (50,52)[(0,0)\[b\][$4$]{}]{} (70,52)[(0,0)\[b\][$5$]{}]{} (90,52)[(0,0)\[b\][$6$]{}]{} (110,52)[(0,0)\[b\][$7$]{}]{}
(10,8)[(0,0)\[t\][$1$]{}]{} (50,8)[(0,0)\[t\][$2$]{}]{} (90,8)[(0,0)\[t\][$3$]{}]{}
(40,50)[(1,0)[20]{}]{}
(40,50)[(1,0)[20]{}]{}
(0,50)[(1,0)[20]{}]{} (80,50)[(1,0)[20]{}]{} (20,10)[(-1,2)[20]{}]{} (17,49.2)[(1,0)[3]{}]{} (17,48.4)[(1,0)[3]{}]{} (17,47.6)[(1,0)[3]{}]{} (17,46.8)[(1,0)[3]{}]{} (17,46)[(1,0)[3]{}]{} (80,49.2)[(1,0)[2]{}]{} (80,48.4)[(1,0)[2]{}]{} (80,47.6)[(1,0)[2]{}]{} (80,46.8)[(1,0)[2]{}]{} (80,46)[(1,0)[2]{}]{} (79.8,45.2)[(1,0)[2]{}]{} (74,25)[(1,0)[10]{}]{}
(59.7,30)(0,-.6)[9]{}[(1,0)[13]{}]{} (67.4,24.4)(-.3,-.4)[14]{}[(-1,0)[7]{}]{} (74.5,30)(.75,.4)[16]{}[(-1,0)[10]{}]{} (80,24.4)(.2,-.4)[18]{}[(-1,0)[7]{}]{} (63,10.4)(0,.4)[12]{}[(1,0)[11]{}]{} (20,10)[(1,0)[60]{}]{}
(20,10)(.4,0)[44]{}[(-1,2)[20]{}]{} (37.8,10)[(-1,2)[14.5]{}]{} (38.2,10)[(-1,2)[14]{}]{} (38.6,10)[(-1,2)[13]{}]{} (39,10)[(-1,2)[12.5]{}]{} (39.4,10)[(-1,2)[12]{}]{} (39.8,10)[(-1,2)[11.9]{}]{} (40.2,10)[(-1,2)[11.8]{}]{} (41,10)(.8,0)[2]{}[(-1,2)[11.15]{}]{} (42.6,10)[(-1,2)[10.5]{}]{} (43.4,10)(.8,0)[4]{}[(-1,2)[10.3]{}]{} (37,10)(.4,0)[80]{}[(-1,2)[9.8]{}]{}
(80,10)[(1,2)[20]{}]{} (80,10)(-.4,0)[24]{}[(1,2)[20]{}]{} (58.5,10)(-.4,0)[40]{}[(1,2)[9.8]{}]{} (79,25)(-.4,0)[24]{}[(1,2)[12.5]{}]{}
(50,50)[(20,20)\[b\]]{} (50,50)[(19.5,19.5)\[b\]]{} (50,50)[(19,19)\[b\]]{} (50,50)[(18.5,18.5)\[b\]]{} (50,50)[(18,18)\[b\]]{} (50,50)[(17.5,17.5)\[b\]]{} (50,50)[(17,17)\[b\]]{} (50,50)[(16.5,16.5)\[b\]]{} (50,50)[(16,16)\[b\]]{} (50,50)[(15.5,15.5)\[b\]]{} (50,50)[(15,15)\[b\]]{} (50,50)[(14.5,14.5)\[b\]]{} (50,50)[(14,14)\[b\]]{} (50,50)[(13.5,13.5)\[b\]]{} (50,50)[(13,13)\[b\]]{} (50,50)[(12.5,12.5)\[b\]]{} (50,50)[(12,12)\[b\]]{} (50,50)[(11.5,11.5)\[b\]]{} (50,50)[(11,11)\[b\]]{} (50,50)[(10.5,10.5)\[b\]]{} (50,50)[(10,10)\[b\]]{} (50,50)[(9.5,9.5)\[b\]]{} (50,50)[(9,9)\[b\]]{} (50,50)[(8.5,8.5)\[b\]]{} (50,50)[(8,8)\[b\]]{} (50,50)[(7.5,7.5)\[b\]]{} (50,50)[(7,7)\[b\]]{} (50,50)[(6.5,6.5)\[b\]]{} (50,50)[(6,6)\[b\]]{} (50,50)[(5.5,5.5)\[b\]]{} (50,50)[(5,5)\[b\]]{} (50,50)[(4.5,4.5)\[b\]]{} (50,50)[(4,4)\[b\]]{} (50,50)[(3.5,3.5)\[b\]]{} (50,50)[(3,3)\[b\]]{} (50,50)[(2.5,2.5)\[b\]]{} (50,50)[(2,2)\[b\]]{} (50,50)[(1.5,1.5)\[b\]]{} (50,50)[(1,1)\[b\]]{}
(50,50)[(60,40)\[b\]]{} (50,50)[(61,41)\[b\]]{}
(-5,43) (-5,43) (-5,43) (-5,43) (-5,43) (-5,43) (-5,43) (-5,43) (-5,43) (-5,43) (-5,43) (-5,43) (-5,43)
(70,20) (70,20) (70,20) (70,20) (-5,25) (1,25) (1,25) (1,25) (1,25) (-8,25) (-8,25) (-8,25) (-8,25) (-8,25) (-15,37)
(-7,34.6)[(1,0)[4]{}]{} (-7,34.2)[(1,0)[4]{}]{} (-9.8,33.8)[(1,0)[9.6]{}]{} (-9.8,33.4)[(1,0)[9]{}]{} (-11,33)[(1,0)[12]{}]{} (-11,32.6)[(1,0)[11]{}]{} (-12,32.2)[(1,0)[14]{}]{} (-12,31.8)[(1,0)[13]{}]{} (-13,31.4)[(1,0)[15]{}]{} (-13,31)[(1,0)[15]{}]{} (-13.5,30.6)[(1,0)[16]{}]{} (-13.5,30.2)[(1,0)[16]{}]{} (-14,29.8)[(1,0)[5]{}]{} (-14,29.4)[(1,0)[3.5]{}]{} (-14.3,29)[(1,0)[3.5]{}]{} (-14.3,21)[(1,0)[3.5]{}]{} (-14,20.6)[(1,0)[3.5]{}]{} (-14,20.2)[(1,0)[5]{}]{} (-13.5,19.8)[(1,0)[16]{}]{} (-13.5,19.4)[(1,0)[17]{}]{} (-13,19)[(1,0)[15]{}]{} (-13,18.6)[(1,0)[16]{}]{} (-12,18.2)[(1,0)[13]{}]{} (-12,17.8)[(1,0)[14]{}]{} (-11,17.4)[(1,0)[11]{}]{} (-11,17)[(1,0)[12]{}]{} (-9.8,16.6)[(1,0)[9]{}]{} (-9.8,16.2)[(1,0)[9.6]{}]{} (-8,15.8)[(1,0)[6]{}]{} (-8,15.4)[(1,0)[6.5]{}]{} (-4,30.2)[(1,0)[7]{}]{} (-4,29.8)[(1,0)[7.5]{}]{} (-3.5,29.4)[(1,0)[7]{}]{} (-3.5,29)[(1,0)[7.5]{}]{} (-3.2,28.6)[(1,0)[7]{}]{} (-3.2,28.2)[(1,0)[7]{}]{} (-3.2,27.8)[(1,0)[8]{}]{} (-3.2,27.4)[(1,0)[8]{}]{} (-4,20.6)[(1,0)[8]{}]{} (-4,20.2)[(1,0)[8]{}]{} (-3.5,21.4)[(1,0)[8]{}]{} (-3.5,21)[(1,0)[8]{}]{} (-3.2,22.2)[(1,0)[8]{}]{} (-3.2,21.8)[(1,0)[8]{}]{} (-3.2,22.6)[(1,0)[8]{}]{}
The category $\mbox{\it Frz}'$ is related to the category *2Cob* of [@K03] (Section 1.4), whose arrows are cobordisms of dimension 2. An arrow of $\mbox{\it Frz}'$ may be conceived as a kind of “thin” cobordism.
As we associated the category *Frobse* to *Frz*, so we may look for a category $\mbox{\it Frobse}'$ like *Frobse* to associate to $\mbox{\it Frz}'$. We will previously demonstrate however the necessity of the equations $(\Phi)$ for faithful monoidal functors into *Mat*, and consider the consequences for ordinals of having $(\Phi)$ and related equations.
The necessity of $(\Phi)$ follows from the fact that *Mat* is a symmetric strictly monoidal category, which has a symmetry natural isomorphism $c_{n,m}\!:n\otimes m\str m\otimes n$ for which we have the equation
${(c1)}$$c_{1,m}=c_{m,1}=\mj_m$
(where 1 in the subscripts of $c$ is the unit object of *Mat*). Hence, for every arrow ${f\!:1\str 1}$ of *Mat*, we have $$\mj_m\otimes f=(\mj_m\otimes f)\cirk c_{1,m}=
c_{1,m}\cirk(f\otimes\mj_m)=f\otimes\mj_m.$$ Since for every monoidal functor $G$ from *Frob* to *Mat* we have $G0=1$ (where 0 is the unit object of *Frob*), and since $G\Phi^k_n$ is of the form ${\mj_{n\cdot
p}\otimes f}$ for ${f\!:1\str 1}$, we have $G\Phi^k_n=G\Phi^k_0$. So, from the faithfulness of $G$, the equation $(\Phi)$ follows.
In the reasoning above $c$ can be a braiding natural isomorphism, instead of a symmetry natural isomorphism. We would have the equation ${(c1)}$, and the equation $(\Phi)$ would again be imposed by the faithfulness of $G$. So we could replace *Mat* by a braided strictly monoidal category (cf. [@K03], Section 3.6.27).
We defined above the monoid ${\cal L}_\omega'$ as ${\cal
L}_\omega$ with the equation $(\Phi c)$ added. In ${\cal
L}_\omega'$ the hierarchy of $\varepsilon_0$ collapses to $\omega^\omega$. This means that every element of ${\cal
L}_\omega'$ is definable in terms of $e^\beta_n$, for $e$ being $a$, $b$ or $c$, and ${\beta\in\omega^\omega}$. We can restrict the terms $e^\beta_n$ even further, to those in the following table, without altering the structure of the normal form for ${\cal L}_\omega$ of [@DP03] (Section on *Normal forms in* ${\cal L}_\omega$):
---------------------- ----------- ---------------------------------------
$e$ $n$ $\beta$
\[.5ex\] $c$ 1 $\hspace{0.7em}\beta\in\omega^\omega$
\[.3ex\] $c$ $2m\pl 2$ $\beta\in\omega$
\[.3ex\] $a$ and $b$ $2m\pl 1$ $\beta\in\omega$
\[.3ex\] $a$ and $b$ $2m\pl 2$ $\beta=0$
---------------------- ----------- ---------------------------------------
This is shown as follows.
By Cantor’s Normal Form Theorem (see, for example, [@KM], VII.7, Theorem 2, p. 248, or [@L79], IV.2, Theorem 2.14, p.127), for every ordinal ${\alpha>0}$ in $\varepsilon_0$ there is a unique finite ordinal ${n\geq 1}$ and a unique sequence of ordinals $\alpha_1\geq\ldots\geq\alpha_n$ contained in $\alpha$, i.e. lesser than $\alpha$, such that $\alpha=\omega^{\alpha_1}\sharp\ldots\sharp\,\omega^{\alpha_n}$. So every ordinal in $\varepsilon_0$ can be named by using the operations of the monoid ${\langle\varepsilon_0,\sharp,0,\omega^{-}\rangle}$ mentioned in the previous section.
Let $\beta_0$ be $\omega^0$, which is equal to 1, and let ${\beta_k\!:\varepsilon_0^k\str\varepsilon_0}$, for ${k\geq 1}$, be defined by
= If $k=0$, then $\beta_0'$ = $=\omega^0=1=\beta_0$.
$\beta_k(\alpha_1,\ldots,\alpha_k)$ $=\omega^{\omega^{\alpha_1}\sharp\ldots\sharp\,\omega^{\alpha_k}}$.
By Cantor’s Normal Form Theorem, to name the ordinals in $\varepsilon_0$ we can replace the unary operation $\omega^{-}$ by the operations $\beta_k$ for every ${k\geq 0}$. So the name of every ordinal in $\varepsilon_0$ can be written in terms of $0$, $\sharp$ and $\beta_k$. We proceed by induction on the complexity of such a name to define the map $'$ from $\varepsilon_0$ to $\omega^\omega$:
= If $k=0$, then $\beta_0'$ = $=\omega^0=1=\beta_0$.
$0'$ $=0$,\
$(\alpha_1\sharp\,\alpha_2)'$ $=\alpha_1'\sharp\,\alpha_2'$,\
$\beta_0'$ $=\omega^0=1=\beta_0$,\
$\beta_k(\alpha_1,\ldots,\alpha_k)'$ $=\omega^k\sharp\,\alpha_1'\sharp\ldots\sharp\,\alpha_k'$, for $k\geq 1$.
We can then prove the following lemmata.
We proceed by induction on the size of $\alpha$. If ${\alpha=0}$, then we use the following equation of ${\cal
L}_\omega$:
$c^0_{2m+1}=c^0_1=\mj$.
In the induction step we have
=$c^{\alpha_1\sharp\,\alpha_2}_{2m+1}$ = $=c^{\alpha_1'\sharp\,\alpha_2'}_{2m+1}$, by ${(c2)}$ and the induction hypothesis,\
$c^{\beta_0}_{2m+1}$ $=c^{\beta_0'}_1$, by $(\Phi c)$,\
for $k\geq 1$,\
$c^{\beta_k(\alpha_1,\ldots,\alpha_k)}_{2m+1}=a^0_{2m+1}c^{\omega^{\alpha_1}}_{2m+2}\ldots
c^{\omega^{\alpha_k}}_{2m+2} b^0_{2m+1}$, by ${(ab\:3.3)}$, ${(ac\:3)}$ and ${(c2)}$.
For every $i\in\{1\ldots,k\}$, we have, by the same equations,
=$c^{\alpha_1\sharp\,\alpha_2}_{2m+1}$ = $c^{\omega^{\alpha_i}}_{2m+2}$ $=a^0_{2m+2}c^{\alpha_i}_{2m+3}b^0_{2m+2}$.
Then, by the induction hypothesis and the equations ${(ac\:1)}$, ${(bc\:1)}$ and ${(c2)}$, for $d^0$ being $\mj$, and $d^{n+1}$ being $d^na^0_{2m+2}b^0_{2m+2}$, we obtain
$c^{\beta_k(\alpha_1,\ldots,\alpha_k)}_{2m+1}$ = $=a^0_{2m+1}d^k
b^0_{2m+1}c_1^{\alpha_1'\sharp\ldots\sharp\,\alpha_k'}$\
$=c^{\omega^k}_{2m+1}c^{\alpha_1'\sharp\ldots\sharp\,\alpha_k'}_1$, by ${(ab\:3.3)}$ and ${(ac\:3)}$,\
$=c^{\beta_k(\alpha_1,\ldots,\alpha_k)'}_1$, by $(\Phi c)$ and ${(c2)}$. \` $\dashv$
We have
=$c^{\alpha_1\sharp\,\alpha_2}_{2m+1}$ = $c^{\omega^{\alpha}}_{2m+2}$ $=a^0_{2m+2}c^{\alpha}_{2m+3}b^0_{2m+2}$, by ${(ab\:3.3)}$ and ${(ac\:3)}$,\
$=c^{\alpha'}_1c^1_{2m+2}$, by the preceding lemma, ${(ac\:1)}$ and ${(ab\:3.3)}$.\` $\dashv$
With these two lemmata, we can show that the terms $e^\beta_n$ in the table above are sufficient to define every element of ${\cal
L}_\omega'$ without altering the structure of our normal form. This is clear for the terms $c^\alpha_n$. We also have
=$c^{\alpha_1\sharp\,\alpha_2}_{2m+1}$ =
$a^\alpha_{2m+2}$ $=a^0_{2m+2}c^\alpha_{2m+3}$, by ${(ac\:3)}$,\
$=c^{\alpha'}_1 a^0_{2m+2}$, by Lemma $2m+1$ and ${(ac\:1)}$;\
$a^{\omega^{\alpha_1}\sharp\ldots\sharp\,\omega^{\alpha_n}}_{2m+1}$ $=a^0_{2m+1}c^{\omega^{\alpha_1}}_{2m+2}\ldots
c^{\omega^{\alpha_n}}_{2m+2}$, by ${(ac\:3)}$ and $(c2)$,\
$=c^{\alpha_1'\sharp\ldots\sharp\,\alpha_n'}a^n_{2m+1}$, by Lemma $2m+2$, ${(ac\:1)}$ and ${(c2)}$,
and analogous equations with $a$ replaced by $b$.
Consider terms of ${\cal L}_\omega'$ in the form exactly like the normal form of ${\cal L}_\omega$ in [@DP03] save that all the generators $a^\alpha_i$, $b^\beta_j$ and $c^\gamma_k$ are terms from our table. We say that such terms are in *normal form*. This is the normal form we mentioned previously, which we can use to decide equations in ${\cal L}_\omega'$, and to prove the isomorphism with $\mbox{\it Frz}'$, along the lines of [@DP03].
We can now sketch how the category $\mbox{\it Frobse}'$ analogous to *Frobse* and isomorphic to $\mbox{\it Frz}'$ would look like. Its arrows will be based on Frobenius split equivalences where the function assigning ordinals will follow restrictions in accordance with our table:
\(1) an even class is mapped to an ordinal in $\omega$,
\(2) an odd class containing 1 is mapped to an ordinal in $\omega^\omega$,
\(3) an odd class not containing 1 is mapped to 0.
Even classes correspond to the black regions of the black friezes and odd classes to the white regions; the odd class containing 1 corresponds to the leftmost white region. The ordinals of (1) register the number of white holes in the black regions, and those of (2) the number of black disks and the number of white holes in them.
Composition in $\mbox{\it Frobse}'$ would be defined by reductions based on the equations of ${\cal L}_\omega'$, like those we gave for *Frobse*. Essentially, we would have to change only the reductions corresponding to ${(ab\:3.1)}$, ${(ab\:3.2)}$ and ${(ab\:3.3)}$. We could have instead
(310,60)
(-13,30)[(0,0)\[l\][$(ab\:3.1)$]{}]{} (32,31)[(0,0)\[l\][$a^n_{2m+1} b^0_{2m+2}=c^n_{2m+2}$]{}]{}
(157,54)[(0,0)\[b\][$2m\pl 1$]{}]{} (180,54)[(0,0)\[b\][$2m\pl 2$]{}]{}
(167,5)[(0,0)\[b\][$2m\pl 1$]{}]{} (190,5)[(0,0)\[b\][$2m\pl 2$]{}]{}
(170,10)[(0,1)[10]{}]{} (170,20)[(-1,1)[10]{}]{} (170,20)[(1,1)[10]{}]{} (190,10)[(0,1)[20]{}]{} (160,32)[(0,1)[20]{}]{} (180,52)[(0,-1)[10]{}]{} (180,42)[(-1,-1)[10]{}]{} (180,42)[(1,-1)[10]{}]{}
(170,29) (180,33) (170,27)[(0,0)\[t\][$n$]{}]{} (180,35)[(0,0)\[b\][$0$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(290,54)[(0,0)\[b\][$2m\pl 2$]{}]{}
(290,5)[(0,0)\[b\][$2m\pl 2$]{}]{}
(290,10)[(0,1)[42]{}]{}
(291,32)[(0,0)\[l\][$n$]{}]{}
(310,60)
(32,31)[(0,0)\[l\][$a^0_{2m+2} b^n_{2m+3}=c^n_{2m+2}$]{}]{}
(157,54)[(0,0)\[b\][$2m\pl 2$]{}]{} (180,54)[(0,0)\[b\][$2m\pl 3$]{}]{}
(167,5)[(0,0)\[b\][$2m\pl 2$]{}]{} (190,5)[(0,0)\[b\][$2m\pl 3$]{}]{}
(170,10)[(0,1)[10]{}]{} (170,20)[(-1,1)[10]{}]{} (170,20)[(1,1)[10]{}]{} (190,10)[(0,1)[20]{}]{} (160,32)[(0,1)[20]{}]{} (180,52)[(0,-1)[10]{}]{} (180,42)[(-1,-1)[10]{}]{} (180,42)[(1,-1)[10]{}]{}
(170,29) (180,33) (170,27)[(0,0)\[t\][$0$]{}]{} (180,35)[(0,0)\[b\][$n$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(290,54)[(0,0)\[b\][$2m\pl 2$]{}]{}
(290,5)[(0,0)\[b\][$2m\pl 2$]{}]{}
(290,10)[(0,1)[42]{}]{}
(291,32)[(0,0)\[l\][$n$]{}]{}
(310,60)
(-13,30)[(0,0)\[l\][$(ab\:3.3)$]{}]{} (32,31)[(0,0)\[l\][$a^n_{2m+1}
b^l_{2m+1}=c^{\omega^{n+l}}_1$]{}]{}
(177,54)[(0,0)\[b\][$2m\pl 1$]{}]{}
(177,5)[(0,0)\[b\][$2m\pl 1$]{}]{}
(180,10)[(0,1)[10]{}]{} (180,20)[(-1,1)[10]{}]{} (180,20)[(1,1)[10]{}]{} (180,52)[(0,-1)[10]{}]{} (180,42)[(-1,-1)[10]{}]{} (180,42)[(1,-1)[10]{}]{}
(180,33) (180,29) (180,27)[(0,0)\[t\][$n$]{}]{} (180,35)[(0,0)\[b\][$l$]{}]{}
(230,31)[(0,0)[$\leadsto$]{}]{}
(290,54)[(0,0)\[b\][$1$]{}]{}
(290,5)[(0,0)\[b\][$1$]{}]{}
(290,10)[(0,1)[42]{}]{}
(291,31)[(0,0)\[l\][$\omega^{n+l}$]{}]{}
(310,60)
(32,31)[(0,0)\[l\][$a^0_{2m+2} b^0_{2m+2}=c^1_{2m+2}$]{}]{}
(177,54)[(0,0)\[b\][$2m\pl 2$]{}]{}
(177,5)[(0,0)\[b\][$2m\pl 2$]{}]{}
(180,10)[(0,1)[10]{}]{} (180,20)[(-1,1)[10]{}]{} (180,20)[(1,1)[10]{}]{} (180,52)[(0,-1)[10]{}]{} (180,42)[(-1,-1)[10]{}]{} (180,42)[(1,-1)[10]{}]{}
(180,33) (180,29)
(230,31)[(0,0)[$\leadsto$]{}]{}
(290,54)[(0,0)\[b\][$2m\pl 2$]{}]{}
(290,5)[(0,0)\[b\][$2m\pl 2$]{}]{}
(290,10)[(0,1)[42]{}]{}
(291,31)[(0,0)\[l\][$1$]{}]{}
and analogous reductions for ${(ab\:3.2)}$.
Separable matrix Frobenius monads
=================================
In the preceding section, we saw how symmetry in the category *Mat* induces a collapse of the ordinals in $\varepsilon_0$ of *Frob* into the ordinals in $\omega^\omega$. In all that, we have not considered commutative Frobenius monads, which play a central role in connection with topological quantum field theories. With commutative Frobenius monads, our ordinals are still contained in $\omega^\omega$, as in the preceding section.
Another collapse of ordinals comes with separability (see [@DI71], [@C91] and [@RSW05]). The *separability* equation for Frobenius monads is the equation $$\delta^\Diamond_A\cirk\delta^\Box_A=\mj_{MA}.$$ If we consider extending *Frob* with this equation, we just replace $A$ by $n$. To state the consequence of the corresponding equation ${c^1_{2n+2}=1}$ for ${\cal L}_\omega$, we need some terminology.
Let the ordinal $0$ be of *even height*. If $\alpha_1,\ldots,\alpha_n$ are all of *even (odd) height*, then $\omega^{\alpha_1}\sharp\ldots\sharp\,\omega^{\alpha_n}$ is of *odd (even) height*. If an ordinal in $\varepsilon_0$ is of even or odd height, we say that it has a *homogeneous height*. Not all ordinals in $\varepsilon_0$ have a homogeneous height. The consequence of the separability equation for ${\cal
L}_\omega$ is that every $c^\alpha_n$ is equal to $c^{\alpha'}_n$ for $\alpha'$ an ordinal in $\varepsilon_0$ of homogeneous height; if $n$ is ${2m\pl 2}$, then $\alpha'$ is of even height, and if $n$ is ${2m\pl 1}$, then $\alpha'$ is of odd height.
If we combine the separability equation with the equation $(\Phi)$ of the preceding section, then the ordinals in $\varepsilon_0$ collapse to the ordinals in $\omega$. More precisely, the consequence for ${\cal L}_\omega$ is that we could take as primitive only the terms $e^k_n$, for $e$ being $a$, $b$ or $c$, and ${k\in\omega}$, where only $c^k_1$ may have ${k\geq 0}$; in all other cases, ${k=0}$. In the presence of the separability equation, the equation
$(\Phi^0)$$\Phi^0_{MA}=M\Phi^0_A$
has the same force as the equations $(\Phi)$. According to our definition, $\Phi^0_A$ is ${\varepsilon^\Box_A\cirk\varepsilon^\Diamond_A}$.
We call Frobenius monads that satisfy $(\Phi^0)$ and the separability equation *separable matrix* Frobenius monads. For separable matrix Frobenius monads, we can answer positively the question of sufficiency left open at the beginning of the preceding section. Namely, there is a faithful monoidal functor $F$ from the separable matrix Frobenius monad generated by a single object into the category *Mat*. In fact, something stronger holds: for every natural number ${p\geq 2}$, there is a functor $F$ as above such that ${F(1)=p}$. We will not prove this in detail, but just give some indications.
Our task is to represent in *Mat* an ordered pair made of a maximal split equivalence (see Section 6) and a natural number, which is the ordinal ${k\in\omega}$ tied to $c^k_1$. We may reject the odd equivalence classes from this maximal split equivalence, and then represent the remaining split equivalence in a Brauerian manner (see [@DP03], [@DP03a] and [@DP06]). The natural number $k$ will be mapped to the scalar $p^k$. This is analogous to representing ${\cal K}_c$ in *Mat* (in the section with that name in [@DP03]), but is not exactly the same. In the free self-adjunction ${\cal K}_c$ of the $\cal K$ type (corresponding to Temperley-Lieb algebras), the ordinals in $\varepsilon_0$ of ${\cal L}_\omega$ also collapse to natural numbers, and are not tied to particular regions of the frieze. This is analogous to what we have with separable matrix Frobenius monads, but is not exactly the same. The difference is that for ${\cal K}_c$ all circles are counted, while here we count circles tied to ${\varepsilon^\Box_A\cirk\varepsilon^\Diamond_A}$, which may be moved according to the equation $(\Phi)$ or $(\Phi^0)$, and do not count circles tied to ${\delta^\Diamond_A\cirk\delta^\Box_A}$, according to the separability equation. We will deal with these matters in more detail on another occasion.
Let us sum up matters from the preceding section and the present one. We know that the equation $(\Phi)$ is necessary for the existence of a faithful monoidal functor $F$ into the category *Mat*. We do not know whether $(\Phi)$ is sufficient. If it were, then we could legitimately call Frobenius monads that satisfy $(\Phi)$ *matrix* Frobenius monads. We know on the other hand that $(\Phi)$ together with the separability equation is sufficient for the existence of such an $F$, but we do not know whether the separability equation is necessary, though this necessity does not seem likely. Since ordinals in separable matrix Frobenius monads have collapsed to natural numbers, with these monads we reach the boundary we set ourselves for this paper, where we wanted to investigate the role of bigger ordinals in Frobenius monads.
[. Work on this paper was supported by the Ministry of Science of Serbia (Grant 144013).]{}
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| 0 |
---
abstract: 'During the last years, several algorithmic meta-theorems have appeared (Bodlaender *et al*. \[FOCS 2009\], Fomin *et al*. \[SODA 2010\], Kim *et al*. \[ICALP 2013\]) guaranteeing the [*existence*]{} of linear kernels on sparse graphs for problems satisfying some generic conditions. The drawback of such general results is that it is usually not clear how to derive from them [*constructive*]{} kernels with reasonably low [*explicit*]{} constants. To fill this gap, we recently presented \[STACS 2014\] a framework to obtain explicit linear kernels for some families of problems whose solutions can be certified by a subset of [*vertices*]{}. In this article we enhance our framework to deal with [*packing*]{} problems, that is, problems whose solutions can be certified by collections of [*subgraphs*]{} of the input graph satisfying certain properties. ${\mathcal F}$-<span style="font-variant:small-caps;">Packing</span> is a typical example: for a family ${\mathcal F}$ of connected graphs that we assume to contain at least one planar graph, the task is to decide whether a graph $G$ contains $k$ vertex-disjoint subgraphs such that each of them contains a graph in ${\mathcal F}$ as a minor. We provide explicit linear kernels on sparse graphs for the following two orthogonal generalizations of ${\mathcal F}$-<span style="font-variant:small-caps;">Packing</span>: for an integer $\ell {\geqslant}1$, one aims at finding either minor-models that are pairwise at distance at least $\ell$ in $G$ (<span style="font-variant:small-caps;">$\ell$-${\mathcal}{F}$-Packing</span>), or such that each vertex in $G$ belongs to at most $\ell$ minors-models (<span style="font-variant:small-caps;">${\mathcal}{F}$-Packing with $\ell$-Membership</span>). Finally, we also provide linear kernels for the versions of these problems where one wants to pack [*subgraphs*]{} instead of minors.'
author:
- 'Valentin Garnero[^1]'
- 'Christophe Paul$^{\dagger}$'
- 'Ignasi Sau$^{\dagger}$'
- 'Dimitrios M. Thilikos$^{\dagger}$[^2]'
bibliography:
- 'linearkernels.bib'
title: 'Explicit linear kernels for packing problems[^3]'
---
**Keywords:** Parameterized complexity; linear kernels; packing problems; dynamic programming; protrusion replacement; graph minors.
Introduction {#sec:intro}
============
**Motivation.** A fundamental notion in parameterized complexity (see [@CyganFKLMPPS15] for a recent textbook) is that of *kernelization*, which asks for the existence of polynomial-time preprocessing algorithms producing equivalent instances whose size depends exclusively on the parameter $k$. Finding kernels of size polynomial or linear in $k$ (called *linear kernels*) is one of the major goals of this area. A pioneering work in this direction was the linear kernel of Alber *et al*. [@AFN04] for <span style="font-variant:small-caps;">Dominating Set</span> on planar graphs, generalized by Guo and Niedermeier [@GuNi07] to a family of problems on planar graphs. Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result of Bodlaender *et al*. [@BFL+09] on graphs of bounded genus. It was followed-up by similar results on larger sparse graph classes, such as graphs excluding a minor [@FLST10] or a topological minor [@KLP+12].
The above results guarantee the [*existence*]{} of linear kernels on sparse graph classes for problems satisfying some generic conditions, but it is hard to derive from them [*constructive*]{} kernels with [*explicit*]{} constants. We recently made in [@KviaDP] a significant step toward a fully constructive meta-kernelization theory on sparse graphs with explicit constants. In a nutshell, the main idea is to substitute the algorithmic power of CMSO logic that was used in [@BFL+09; @FLST10; @KLP+12] with that of dynamic programming (DP for short) on graphs of bounded decomposability (i.e., bounded treewidth). We refer the reader to the introduction of [@KviaDP] for more details. Our approach provides a DP framework able to construct linear kernels for families of problems on sparse graphs whose solutions can be certified by a subset of [*vertices*]{} of the input graph, such as $r$-<span style="font-variant:small-caps;">Dominating Set</span> or <span style="font-variant:small-caps;">Planar-$\mathcal{F}$-Deletion</span>.
**Our contribution.** In this article we make one more step in the direction of a fully constructive meta-kernelization theory on sparse graphs, by enhancing the existing framework [@KviaDP] in order to deal with [*packing*]{} problems. These are problems whose solutions can be certified by collections of [*subgraphs*]{} of the input graph satisfying certain properties. We call these problems *packing-certifiable*, as opposed to *vertex-certifiable* ones. For instance, deciding whether a graph $G$ contains at least $k$ vertex-disjoint cycles is a typical packing-certifiable problem. This problem, called <span style="font-variant:small-caps;">Cycle Packing</span>, is [[FPT]{}]{} as it is minor-closed, but it is unlikely to admit polynomial kernels on general graphs [@BodlaenderTY11].
As an illustrative example, for a family of connected graphs ${\mathcal F}$ containing at least one planar graph, we provide a linear kernel on sparse graphs for the <span style="font-variant:small-caps;">${\mathcal F}$-Packing</span> problem[^4]: decide whether a graph $G$ contains at least $k$ vertex-disjoint subgraphs such that each of them contains a graph in ${\mathcal F}$ as a minor, parameterized by $k$. We provide linear kernels as well for the following two [*orthogonal generalizations*]{} of ${\mathcal F}$-<span style="font-variant:small-caps;">Packing</span>: for an integer $\ell {\geqslant}1$, one aims at finding either minor-models that are pairwise at distance at least $\ell$ in $G$ (<span style="font-variant:small-caps;">$\ell$-${\mathcal}{F}$-Packing</span>), or such that each vertex in $G$ belongs to at most $\ell$ minors-models (<span style="font-variant:small-caps;">${\mathcal}{F}$-Packing with $\ell$-Membership</span>). While only the [*existence*]{} of linear kernels for <span style="font-variant:small-caps;">${\mathcal F}$-Packing</span> was known [@BFL+09], to the best of our knowledge no kernels were known for <span style="font-variant:small-caps;">$\ell$-${\mathcal}{F}$-Packing</span> and <span style="font-variant:small-caps;">${\mathcal}{F}$-Packing with $\ell$-Membership</span>, except for <span style="font-variant:small-caps;">$\ell$-${\mathcal}{F}$-Packing</span> when $\mathcal{F}$ consists only of a triangle and the maximum degree is also considered as a parameter [@AtminasKR14]. We would like to note that the kernels for <span style="font-variant:small-caps;">${\mathcal F}$-Packing</span> and for <span style="font-variant:small-caps;">${\mathcal}{F}$-Packing with $\ell$-Membership</span> apply to minor-free graphs, while those for <span style="font-variant:small-caps;">$\ell$-${\mathcal}{F}$-Packing</span> for $\ell {\geqslant}2$ apply to the smaller class of apex-minor-free graphs. We also provide linear kernels for the versions of the above problems where one wants to pack [*subgraphs*]{} instead of minors (as one could expect, the kernels for subgraphs are considerably simpler than those for minors). We call the respective problems <span style="font-variant:small-caps;">$\ell$-${\cal F}$-Subgraph-Packing</span> and <span style="font-variant:small-caps;">${\mathcal}{F}$-Subgraph-Packing with $\ell$-Membership</span>. While the first problem can be seen as a broad generalization of $\ell$-<span style="font-variant:small-caps;">Scattered Set</span> (see for instance [@BFL+09; @KviaDP]), the second one was recently defined by Fernau *et al*. [@FernauLR15], motivated by the problem of discovering overlapping communities (see also [@RomeroL14-WALCOM; @RomeroL14-CSR] for related problems about detecting overlapping communities): the parameter $\ell$ bounds the number of communities that a member of a network can belong to. More precisely, the goal is to find in a graph $G$ at least $k$ subgraphs isomorphic to a member of $\mathcal{F}$ such that every vertex in $V(G)$ belongs to at most $\ell$ subgraphs. This type of overlap was also studied by Fellows *et al*. [@FellowsGKNU11] in the context of graph editing. Fernau *et al*. [@FernauLR15] proved, in particular, that the <span style="font-variant:small-caps;">$\mathcal{F}$-Subgraph-Packing with $\ell$-Membership</span> problem is -hard for all values of $\ell {\geqslant}1$ when ${\mathcal F}= \{F\}$ and $F$ is an arbitrary connected graph with at least three vertices, but polynomial-time solvable for smaller graphs. Note that <span style="font-variant:small-caps;">$\mathcal{F}$-Subgraph-Packing with $\ell$-Membership</span> generalizes the <span style="font-variant:small-caps;">$\mathcal{F}$-Subgraph-Packing</span> problem, which consists in finding in a graph $G$ at least $k$ vertex-disjoint subgraphs isomorphic to a member of $\mathcal{F}$. The smallest kernel for the <span style="font-variant:small-caps;">$\mathcal{F}$-Subgraph-Packing</span> problem [@Moser09] has size $O(k^{r-1})$, where $\mathcal{F} = \{F\}$ and $F$ is an arbitrary graph on $r$ vertices. A list of references of kernels for particular cases of the family $\mathcal{F}$ can be found in [@FernauLR15]. Concerning the kernelization of <span style="font-variant:small-caps;">$\mathcal{F}$-Subgraph-Packing with $\ell$-Membership</span>, Fernau *et al*. [@FernauLR15] provided a kernel on general graphs with $O((r+1)^r k^r)$ vertices, where $r$ is the maximum number of vertices of a graph in $\mathcal{F}$. In this article we improve this result on graphs excluding a fixed graph as a minor, by providing a linear kernel for <span style="font-variant:small-caps;">$\mathcal{F}$-Subgraph-Packing with $\ell$-Membership</span> when $\mathcal{F}$ is any family of (not necessarily planar) connected graphs.
**Our techniques: vertex-certifiable vs. packing-certifiable problems**. It appears that packing-certifiable problems are intrinsically more involved than vertex-certifiable ones. This fact is well-known when speaking about [[FPT]{}]{}-algorithms on graphs of bounded treewidth [@CyganNPPRW11; @LokshtanovMS11], but we need to be more precise with what we mean by being “more involved” in our setting of obtaining kernels via DP on a tree decomposition of the input graph. Loosely speaking, the framework that we presented in [@KviaDP] and that we need to redefine and extend here, can be summarized as follows. First of all, we propose a general definition of a problem *encoding* for the tables of DP when solving parameterized problems on graphs of bounded treewidth. Under this setting, we provide three general conditions guaranteeing that such an encoding can yield a so-called *protrusion replacer*, which in short is a procedure that replaces large “protrusions” (i.e., subgraphs with small treewidth and small boundary) with “equivalent” subgraphs of constant size. Let us be more concrete on these three conditions that such an encoding ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ needs to satisfy in order to obtain an explicit linear kernel for a parameterized problem $\Pi$. The first natural condition is that on a graph $G$ without boundary, the optimal size of the objects satisfying the constraints imposed by ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ coincides with the optimal size of solutions of $\Pi$ in $G$; in that case we say that ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ is a *$\Pi$-encoder*. On the other hand, we need that when performing DP using the encoding ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$, we can use tables such that the maximum difference among all the values that need to be stored is bounded by a function $g$ of the treewidth; in that case we say that ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ is *$g$-confined*. Finally, the third condition requires that ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ is “suitable” for performing DP, in the sense that the tables at a given node of a tree decomposition can be computed using only the information stored in the tables of its children (as it is the case of practically all natural DP algorithms); in that case we say that ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ is *DP-friendly*. These two latter properties exhibit some fundamental differences when dealing with vertex-certifiable or packing-certifiable problems.
Indeed, as discussed in more detail in Section \[sec:generic\], with an encoding ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ we associate a function $f^{{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ that corresponds, roughly speaking, to the maximum size of a partial solution that satisfies the constraints defined by ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$. In order for an encoder to be $g$-confined for some function $g(t)$ of the treewidth $t$, for some vertex-certifiable problems such as $r$-<span style="font-variant:small-caps;">Scattered Set</span> (see [@KviaDP]) we need to “force” the confinement artificially, in the sense that we directly discard the entries in the tables whose associated values differ by more than $g(t)$ from the maximum (or minimum) ones. Fortunately, we can prove that an encoder with this modified function is still DP-friendly. However, this is not the case for packing-certifiable problems such as <span style="font-variant:small-caps;">$\mathcal{F}$-Packing</span>. Intuitively, the difference lies on the fact that in a packing-certifiable problem, a solution of size $k$ can contain arbitrarily many vertices (for instance, if one wants to find $k$ disjoint cycles in an $n$-vertex graph with girth $\Omega(\log n)$) and so it can as well contain arbitrarily many vertices from any subgraph corresponding to a rooted subtree of a tree decomposition of the input graph $G$. This possibility prevents us from being able to prove that an encoder is DP-friendly while still being $g$-confined for some function $g$, as in order to fill in the entries of the tables at a given node, one may need to retrieve information from the tables of other nodes different from its children. To circumvent this problem, we introduce another criterion to discard the entries in the tables of an encoder: we recursively discard the entries of the tables whose associated partial solutions [*induce*]{} partial solutions at some lower node of the rooted tree decomposition that need to be discarded. That is, if an entry of the table needs to be discarded at some node of a tree decomposition, we [*propagate*]{} this information to all the other nodes.
**Organization of the paper.** Some basic preliminaries can be found in Section \[sec:prelim\], including graph minors, parameterized problems, (rooted) tree decompositions, boundaried graphs, the canonical equivalence relation $\equiv_{\Pi,t}$ for a problem $\Pi$ and an integer $t$, FII, protrusions, and protrusion decompositions. The reader not familiar with the background used in previous work on this topic may see [@BFL+09; @FLST10; @KLP+12; @KviaDP]. In Section \[sec:generic\] we introduce the basic definitions of our framework and present an explicit protrusion replacer for packing-certifiable problems. Since many definitions and proofs in this section are quite similar to the ones we presented in [@KviaDP], for better readability we moved the proofs of the results marked with ‘$[\star]$’ to Appendix \[ap:framework\]. Before moving to the details of each particular problem, in Section \[sec:applications\] we summarize the main ingredients that we use in our applications. The next sections are devoted to showing how to apply our methodology to various families of problems. More precisely, we start in Section \[sec: FPack\] with the linear kernel for <span style="font-variant:small-caps;">Connected-Planar-${\mathcal}{F}$-Packing</span>. This problem is illustrative, as it contains most of the technical ingredients of our approach, and will be generalized later in the two orthogonal directions mentioned above. Namely, in Section \[sec: rFPack\] we deal with the variant in which the minor-models are pairwise at distance at least $\ell$, and in Section \[sec: FMemb\] with the version in which each vertex can belong to at most $\ell$ minor-models. In Section \[sec: FSub\] we adapt the machinery developed for packing minors to packing subgraphs, considering both variants of the problem. For the sake of completeness, each of the considered problems will be redefined in the corresponding section. Finally, Section \[sec:conclusions\] concludes the article.
Preliminaries {#sec:prelim}
=============
In our article graphs are undirected, simple, and without loops. We use standard graph-theoretic notation; see for instance [@Die05]. We denote by $d_G(v,w)$ the distance in $G$ between two vertices $v$ and $w$ and by $d_G(W_1,W_2) = \min\{ d_G(w_1,w_2) : w_1\in W_1, w_2\in W_2 \}$ the distance between two sets of vertices $W_1$ and $W_2$ of $G$. Given $S \subseteq V(G)$, we denote by $N(S)$ the set of vertices in $V(G) \setminus S$ having at least one neighbor in $S$.
A parameterized graph problem ${\Pi}$ is called *packing-certifiable* if there exists a language $L^{\Pi}$ (called [*certifying language for $\Pi$*]{}) defined on pairs $(G,{\cal S})$, where $G$ is a graph and ${\cal S}$ is a collection of subgraphs of $G$, such that $(G,k)$ is a [<span style="font-variant:small-caps;">Yes</span>]{}-instance of $\Pi$ if and only if there exists a collection ${\cal S}$ of subgraphs of $G$ with $|{\cal S}| {\geqslant}k$ such that $(G,{\cal S}) \in L^{\Pi}$.
In the above definition, for the sake of generality we do not require the subgraphs in the collection ${\cal S}$ to be pairwise distinct. Also, note that the subclass of packing-certifiable problems where each subgraph in ${\cal S}$ is restricted to consist of a single vertex corresponds to the class of vertex-certifiable problems defined in [@KviaDP].
For a class of graphs [${\mathcal}{G}$]{}, we denote by $\Pi_{\ensuremath{{\mathcal}{G}}\xspace}$ the problem $\Pi$ where the instances are restricted to contain graphs belonging to [${\mathcal}{G}$]{}. With a packing-certifiable problem we can associate in a natural way an optimization function as follows.
\[defi:optimization-function\] Given a packing-certifiable parameterized problem $\Pi$, the *maximization function* $f^{\Pi}: \Gamma^* \rightarrow \mathbb{N}\cup\{-\infty\}$ is defined as $$\begin{aligned}
f^{\Pi}(G)=\
\left\{\begin{array}{lll}
& \max\{|{\cal S}| : (G,{\cal S}) \in L^{\Pi}\} & \mbox{, if there exists such an ${\cal S}$ and}\\
& - \infty & \mbox{, otherwise}.
\end{array}\right.\end{aligned}$$
\[defi:boundaried\] A *boundaried graph* is a graph $G$ with a set $B \subseteq V (G)$ of distinguished vertices and an injective labeling $\lambda_G: B \to \mathbb{N}$. The set $B$ is called the *boundary* of $G$ and it is denoted by $\partial(G)$. The set of labels is denoted by $\Lambda(G) = \{\lambda_G(v) : v \in \partial(G) \}$. We say that a boundaried graph is a *$t$-boundaried graph* if $\Lambda(G) \subseteq \{1, \ldots ,t\}$.
We denote by [${\mathcal}{B}_t$]{}the set of all $t$-boundaried graphs.
\[defi:gluing\] Let $G_1$ and $G_2$ be two boundaried graphs. We denote by $G_1 \oplus G_2$ the graph obtained from $G$ by taking the disjoint union of $G_1$ and $G_2$ and identifying vertices with the same label in the boundaries of $G_1$ and $G_2$. In $G_1 \oplus G_2$ there is an edge between two labeled vertices if there is an edge between them in $G_1$ or in $G_2$.
Given $G = G_1 \oplus G_2$ and $G_2'$, we say that $G' = G_1 \oplus G_2'$ is the graph obtained from $G$ by *replacing* $G_2$ with $G_2'$. The following notion was introduced by Bodlaender *el al*. [@BFL+09].
\[defi:cano\] Let $\Pi$ be a parameterized problem and let $t \in \mathbb{N}$. Given $G_1,G_2 \in {\ensuremath{{\mathcal}{B}_t}\xspace}$, we say that $G_1 \equiv_{\Pi} G_2$ if $\Lambda(G_1) = \Lambda(G_2)$ and there exists a transposition constant ${\ensuremath{\Delta_{\Pi,t}}\xspace}(G_1,G_2) \in \mathbb{Z}$ such that for every $H \in {\ensuremath{{\mathcal}{B}_t}\xspace}$ and every $k \in \mathbb{Z}$, it holds that $(G_1 \oplus H, k) \in \Pi$ if and only if $(G_2 \oplus H, k+{\ensuremath{\Delta_{\Pi,t}}\xspace}(G_1,G_2)) \in \Pi$.
A *tree decomposition* of a graph $G$ is a couple $(T,{\mathcal}{X} = \{ B_x : x \in V(T) \})$, where $T$ is a tree and such that $\bigcup_{x \in V(T)} B_x = V(G)$, for every edge $\{u,v\} \in E (G)$ there exists $x \in V(T)$ such that $u,v \in B_x$, and for every vertex $u \in V(G)$ the set of nodes $\{ x\in V(T) : u \in B_x \}$ induce a subtree of $T$. The vertices of $T$ are referred to as *nodes* and the sets $B_x$ are called bags.
A *rooted tree decomposition* $(T,{\mathcal}{X},r)$ is a tree decomposition with a distinguished node $r$ selected as the *root*. A *nice tree decomposition* $(T,{\mathcal}{X},r)$ (see [@Klo94]) is a rooted tree decomposition where $T$ is binary and for each node $x$ with two children $y,z$ it holds $B_x =B_y =B_z$ and for each node $x$ with one child $y$ it holds $B_x =B_y \cup \{u\}$ or $B_x =B_y \setminus \{ u\}$ for some $u \in V(G)$. The *width* of a tree decomposition is the size of a largest bag minus one. The *treewidth* of a graph, denoted by ${{\mathbf{tw}}}(G)$, is the smallest width of a tree decomposition of $G$. A *treewidth-modulator* of a graph $G$ is a set $X \subseteq V(G)$ such that ${{\mathbf{tw}}}(G-X) {\leqslant}t$, for some fixed constant $t$.
Given a bag $B$ (resp. a node $x$) of a rooted tree decomposition $T$, we denote by $G_B$ (resp. $G_x$), the subgraph induced by the vertices appearing in the subtree of $T$ rooted at the node corresponding to $B$ (resp. the node $x$). We denote by [${\mathcal}{F}_t$]{}the set of all $t$-boundaried graphs that have a rooted tree decomposition of width $t-1$ with all boundary vertices contained in the root-bag. Obviously ${\ensuremath{{\mathcal}{F}_t}\xspace}\subseteq {\ensuremath{{\mathcal}{B}_t}\xspace}$. (Note that graphs can be viewed as 0-boundaried graphs, hence we use a same alphabet $\Gamma$ for describing graphs and boundaried graphs.)
\[defi:prot\] Let $t,\alpha$ be positive integers. A *$t$-protrusion* $Y$ of a graph $G$ is an induced subgraph of $G$ with $|\partial(Y)| {\leqslant}t$ and $ {{\mathbf{tw}}}(Y) {\leqslant}t-1$, where $\partial(Y)$ is the set of vertices of $Y$ having neighbors in $V(G) \setminus V(Y)$. An *$(\alpha,t)$-protrusion decomposition* of a graph $G$ is a partition ${\cal P}=Y_{0}\uplus Y_{1}\uplus \cdots \uplus Y_{\ell}$ of $V(G)$ such that for every $1\leqslant i\leqslant \ell$, $N(Y_{i})\subseteq Y_{0}$, $\max\{\ell, |Y_{0}|\}\leqslant \alpha$, and for every $1\leqslant i\leqslant \ell$, $Y_i\cup N(Y_i)$ is a $t$-protrusion of $G$. When $(G,k)$ is the input of a parameterized problem with parameter $k$, we say that an $(\alpha,t)$-protrusion decomposition of $G$ is *linear* whenever $\alpha =O(k)$.
We say that a rooted tree decomposition of a protrusion $G$ (resp. a boundaried graph $G$) is *boundaried* if the boundary $\partial(G)$ is contained in the root bag. In the following we always consider boundaried nice tree decompositions of width $t-1$, which can be computed in polynomial time for fixed $t$ [@Klo94; @Bod96].
A framework to replace protrusions for packing problems {#sec:generic}
=======================================================
In this section we restate and in many cases modify the definitions given in [@KviaDP] in order to deal with packing-certifiable problems; we will point out the differences. As announced in the introduction, missing proofs can be found in Appendix \[ap:framework\].
**Encoders.** In the following we extend the definition of an encoder given in [@KviaDP Definition 3.2] so that it is able to deal with packing-certifiable problems. The main difference is that now the function ${\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}$ is incorporated in the definition of an encoder, since as discussed in the introduction we need to consider an additional scenario where the entries of the table are discarded (technically, this is modeled by setting those entries to “$-\infty$”) and for this we will have to deal with the partial solutions particular to each problem. In the applications of the next sections, we will call such functions that propagate the entries to be discarded *relevant*. We also need to add a condition about the [*computability*]{} of the function [$f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}$]{}, so that encoders can indeed be used for performing dynamic programming.
\[defi:encod\] An *encoder* is a triple ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}= ({\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace},{\ensuremath{L^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace},{\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace})$ where
- is a function in $2^\mathbb{N} \to 2^{\Upsilon^*}$ that maps a finite subset of integers $ I \subseteq \mathbb{N}$ to a set ${\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(I)$ of strings over some alphabet $\Upsilon$. Each string $R \in {\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(I)$ is called an *encoding*. The *size* of the encoder is the function $s_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}} : \mathbb{N} \to \mathbb{N}$ defined as $s_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}(t) := \max \{|{\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(I)| : I \subseteq \{1,\ldots,t\}\} $, where $|{\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(I)|$ denotes the number of encodings in ${\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(I)$;
- is a computable language which accepts triples $(G,{\cal S},R)\in \Gamma^* \times \Sigma^* \times \Upsilon^*$, where $G$ is a boundaried graph, ${\cal S}$ is a collection of subgraphs of $G$ and $R \in {\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(\Lambda(G))$ is an encoding. If $(G,{\cal S},R)\in {\ensuremath{L^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}$, we say that ${\cal S}$ *satisfies* the encoding $R$ in $G$; and
- is a computable function in $\Gamma^* \times \Upsilon^* \to \mathbb{N}\cup\{ -\infty\}$ that maps a boundaried graph $G$ and an encoding $R\in {\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(\Lambda(G))$ to an integer or to $-\infty$.
As it will become clear with the applications described in the next sections, an encoder is a formalization of the tables used by an algorithm that solves a packing-certifiable problem $\Pi$ by doing DP over a tree decomposition of the input graph. The encodings in ${\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(I)$ correspond to the entries of the DP-tables of graphs with boundary labeled by the set of integers $I$. The language Łidentifies certificates which are partial solutions satisfying the boundary conditions imposed by an encoding.
The following definition differs from [@KviaDP Definition 3.3] as now the function [$f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}$]{}is incorporated in the definition of an encoder [[${\mathcal}{E}_{}$]{}]{}.
\[defi:pi-encod\] Let $\Pi$ be a packing-certifiable problem. An encoder [[${\mathcal}{E}_{}$]{}]{}is a *$\Pi$-encoder* if ${\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(\emptyset)$ is a singleton, denoted by $\{R_\emptyset\}$, such that for any $0$-boundaried graph $G$, ${\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R_\emptyset) = f^\Pi(G)$.
The following definition allows to control the number of possible distinct values assigned to encodings and plays a similar role to FII or *monotonicity* in previous work [@BFL+09; @KLP+12; @FLST10].
\[def:confined\] An encoder ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ is *$g$-confined* if there exists a function $g : \mathbb{N} \to \mathbb{N}$ such that for any $t$-boundaried graph $G$ with $\Lambda(G) = I$ it holds that either $\{R \in {\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(I) : {\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R) \neq - \infty \}= \emptyset\ $ or $\ \max_{R} \{{\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R)\neq - \infty \}\ -\ \min_{R} \{{\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R) \neq - \infty\} \ {\leqslant}\ g(t)$.
For an encoder [[${\mathcal}{E}_{}$]{}]{}and a function $g$, in the next sections we will denote the [*relevant*]{} functions discussed before by ${\ensuremath{\bar f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}_{g}}\xspace}$ to distinguish them from other functions that we will need.
**Equivalence relations and representatives.** We now define some equivalence relations on $t$-boundaried graphs.
\[defi:equiv\] Let [[${\mathcal}{E}_{}$]{}]{}be an encoder, let $G_1,G_2 \in {\ensuremath{{\mathcal}{B}_t}\xspace}$, and let ${\ensuremath{{\mathcal}{G}}\xspace}$ be a class of graphs.
1. $G_1 {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}^*}\xspace} G_2$ if $\Lambda(G_1)=\Lambda(G_2)=: I$ and there exists an integer ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}(G_1,G_2)$ (depending on $G_1, G_2$) such that for any encoding $R \in {\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(I)$ we have ${\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G_1,R) = {\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G_2,R) - {\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace} (G_1,G_2)$.
2. $G_1 {\ensuremath{\sim_{{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace} G_2$ if either $G_1 \notin {\ensuremath{{\mathcal}{G}}\xspace}$ and $G_2 \notin {\ensuremath{{\mathcal}{G}}\xspace}$, or $G_1,G_2 \in {\ensuremath{{\mathcal}{G}}\xspace}$ and, for any $H \in {\ensuremath{{\mathcal}{B}_t}\xspace}$, $H \oplus G_1 \in {\ensuremath{{\mathcal}{G}}\xspace}$ if and only if $H \oplus G_2 \in {\ensuremath{{\mathcal}{G}}\xspace}$.
3. $G_1 {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace}^*}\xspace} G_2$ if $G_1 {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}^*}\xspace} G_2$ and $G_1 {\ensuremath{\sim_{{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace} G_2$.
4. If we restrict the graphs $G_1 , G_2$ to be in [${\mathcal}{F}_t$]{}, then the corresponding equivalence relations, which are a restriction of [[${\mathcal}{E}_{}$]{}]{}and [[${\mathcal}{E}_{}$]{}]{}, are denoted by [[${\mathcal}{E}_{}$]{}]{}and [[${\mathcal}{E}_{}$]{}]{}, respectively.
If for all encodings $R$, ${\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G_1,R) = {\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G_2,R) = -\infty$, then we set ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace} (G_1,G_2):=0$ (note that any fixed integer would satisfy the first condition in Definition \[defi:equiv\]). Following the notation of Bodlaender *et al*. [@BFL+09], the function ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}$ is called the *transposition function* for the equivalence relation [[${\mathcal}{E}_{}$]{}]{}. Note that we can use the restriction of ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}$ to couples of graphs in [${\mathcal}{F}_t$]{}to define the equivalence relation [[${\mathcal}{E}_{}$]{}]{}.
In the following we only consider classes of graphs whose membership can be expressed in Monadic Second Order (MSO) logic. Therefore, we know that the number of equivalence classes of [${\mathcal}{G}$]{}is finite [@Buc60], say at most $r_{{\ensuremath{{\mathcal}{G}}\xspace},t}$, and we can state the following lemma.
$[\star]$ \[lem:nb class\] Let [${\mathcal}{G}$]{}be a class of graphs whose membership is expressible in MSO logic. For any encoder [[${\mathcal}{E}_{}$]{}]{}, any function $g: \mathbb{N} \rightarrow \mathbb{N}$ and any integer $t \in \mathbb{N}$, if [[${\mathcal}{E}_{}$]{}]{}is $g$-confined then the equivalence relation [[${\mathcal}{E}_{}$]{}]{}has at most $ r({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace}):=(g(t)+2)^{s_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}(t)} \cdot 2^t \cdot r_{{\ensuremath{{\mathcal}{G}}\xspace},t} $ equivalence classes. In particular, the equivalence relation [[${\mathcal}{E}_{}$]{}]{}has at most $ r({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace})$ equivalence classes as well.
\[defi:DP-friend\] An equivalence relation [[${\mathcal}{E}_{}$]{}]{}is *DP-friendly* if, for any graph $G \in {\ensuremath{{\mathcal}{B}_t}\xspace}$ with $\partial(G) = A$ and any two boundaried graphs $H$ and $G_B$ with $G= H \oplus G_B$ such that $G_B$ has boundary $B \subseteq V(G)$ with $|B| {\leqslant}t$ and $A\cap V(G_B) \subseteq B$, the following holds. Let $G'\in {\ensuremath{{\mathcal}{B}_t}\xspace}$ with $\partial(G') = A$ be the graph obtained from $G$ by replacing the subgraph $G_B$ with some $G_B' \in {\ensuremath{{\mathcal}{B}_t}\xspace}$ such that $G_B {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace}^*}\xspace} G_B'$. Then $G {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace}^*}\xspace} G'$ and ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}(G,G') = {\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}(G_B, G_B')$.
The following useful fact states that for proving that [[${\mathcal}{E}_{}$]{}]{}is DP-friendly, it suffices to prove that $G {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}^*}\xspace} G'$ instead of $G {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace}^*}\xspace} G'$.
$[\star]$ \[fait:equiv\] Let $G\in {\ensuremath{{\mathcal}{B}_t}\xspace}$ with a separator $B$, let $G_B {\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace} G_B'$, and let $G' \in {\ensuremath{{\mathcal}{B}_t}\xspace}$ as in Definition \[defi:DP-friend\]. If $G {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}^*}\xspace} G'$, then $G {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace}^*}\xspace} G'$.
In order to perform a protrusion replacement that does not modify the behavior of the graph with respect to a problem $\Pi$, we need the relation [[${\mathcal}{E}_{}$]{}]{}to be a refinement of the canonical equivalence relation $\equiv_{\Pi,t}$.
$[\star]$\[lem:refine eq\] Let $\Pi$ be a packing-certifiable parameterized problem defined on a graph class ${\cal G}$, let [[${\mathcal}{E}_{}$]{}]{}be an encoder, let $g: \mathbb{N} \rightarrow \mathbb{N}$, and let $G_1, G_2 \in {\ensuremath{{\mathcal}{B}_t}\xspace}$. If [[${\mathcal}{E}_{}$]{}]{}is a $g$-confined $\Pi$-encoder and [[${\mathcal}{E}_{}$]{}]{}is DP-friendly, then the fact that $G_1 {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace}^*}\xspace} G_2$ implies the following:
- $G_1 \equiv_\Pi G_2$; and
- $\Delta_{\Pi,t}(G_1,G_2) = {\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}(G_1,G_2)$.
In particular, this holds when $G_1, G_2 \in {\ensuremath{{\mathcal}{F}_t}\xspace}$ and $G_1 {\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace} G_2$.
\[defi:progres\] Given an encoder [[${\mathcal}{E}_{}$]{}]{}and an equivalence class $\mathfrak{C} \subseteq {\ensuremath{{\mathcal}{F}_t}\xspace}$ of [[${\mathcal}{E}_{}$]{}]{}, a graph $G \in \mathfrak{C}$ is a *progressive representative* of $\mathfrak{C}$ if for any $G'\in \mathfrak{C}$, it holds that ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}(G,G') {\leqslant}0$.
$[\star]$\[lem:progres size\] Let [${\mathcal}{G}$]{}be a class of graphs whose membership is expressible in MSO logic. For any encoder [[${\mathcal}{E}_{}$]{}]{}, any function $g: \mathbb{N} \rightarrow \mathbb{N}$, and any $t \in \mathbb{N}$, if [[${\mathcal}{E}_{}$]{}]{}is $g$-confined and [[${\mathcal}{E}_{}$]{}]{}is DP-friendly, then any equivalence class of [[${\mathcal}{E}_{}$]{}]{}has a progressive representative of size at most $b({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace}) := 2^{r({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace})+1} \cdot t$, where $r({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace})$ is the function defined in Lemma \[lem:nb class\].
**An explicit protrusion replacement.** The next lemma specifies conditions under which, given an upper bound on the size of the representatives, a generic DP algorithm can provide in linear time an explicit protrusion replacer.
$[\star]$ \[lem:comput repres\] Let [${\mathcal}{G}$]{}be a class of graphs, let [[${\mathcal}{E}_{}$]{}]{}be an encoder, let $g: \mathbb{N} \rightarrow \mathbb{N}$, and let $t \in \mathbb{N}$ such that [[${\mathcal}{E}_{}$]{}]{}is $g$-confined and [[${\mathcal}{E}_{}$]{}]{}is DP-friendly. Assume we are given an upper bound $b {\geqslant}t$ on the size of a smallest progressive representative of any class of [[${\mathcal}{E}_{}$]{}]{}. Given a $t$-protrusion $Y$ inside some graph, we can compute a $t$-protrusion $Y'$ of size at most $b$ such that $Y {\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace} Y'$ and ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}(Y',Y) {\leqslant}0$. Furthermore, such a protrusion can be computed in time $ O(|Y|)$, where the hidden constant depends only on ${{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,b,{\ensuremath{{\mathcal}{G}}\xspace}$, and $t$.
Let us now piece everything together to state the main result of [@KviaDP] that we need to reprove here for packing-certifiable problems. For issues of constructibility, we restrict [${\mathcal}{G}$]{}to be the class of $H$-(topological)-minor-free graphs.
$[\star]$ \[theo:main\] Let [${\mathcal}{G}$]{}be the class of graphs excluding some fixed graph $H$ as a (topological) minor and let $\Pi$ be a parameterized packing-certifiable problem defined on [${\mathcal}{G}$]{}. Let [[${\mathcal}{E}_{}$]{}]{}be an encoder, let $g: \mathbb{N} \rightarrow \mathbb{N}$, and let $t \in \mathbb{N}$ such that [[${\mathcal}{E}_{}$]{}]{}is a $g$-confined $\Pi$-encoder and [[${\mathcal}{E}_{}$]{}]{}is DP-friendly. Given an instance $(G,k)$ of $\Pi$ and a $t$-protrusion $Y$ in $G$, we can compute in time $O(|Y|)$ an equivalent instance $(G-(Y-\partial(Y)) \oplus Y',k')$ where $Y'$ is a $t$-protrusion with $|Y'| {\leqslant}b({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace})$ and $k'{\leqslant}k$ and where $b({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace})$ is the function defined in Lemma \[lem:progres size\].
Such a protrusion replacer can be used to obtain a kernel when, for instance, one is able to provide a protrusion decomposition of the instance.
$[\star]$ \[coro:main\] Let [${\mathcal}{G}$]{}be the class of graphs excluding some fixed graph $H$ as a (topological) minor and let $\Pi$ be a parameterized packing-certifiable problem defined on [${\mathcal}{G}$]{}. Let [[${\mathcal}{E}_{}$]{}]{}be an encoder, let $g: \mathbb{N} \rightarrow \mathbb{N}$, and let $t \in \mathbb{N}$ such that [[${\mathcal}{E}_{}$]{}]{}is a $g$-confined $\Pi$-encoder and [[${\mathcal}{E}_{}$]{}]{}is DP-friendly. Given an instance $(G,k)$ of $\Pi$ and an $(\alpha k,t)$-protrusion decomposition of $G$, we can construct a linear kernel for $\Pi$ of size at most $ (1+b({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace}))\cdot \alpha \cdot k$, where $b({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace})$ is the function defined in Lemma \[lem:progres size\].
Main ideas for the applications {#sec:applications}
===============================
In this section by sketch the main ingredients that we use in our applications for obtaining the linear kernels, before going through the details for each problem in the next sections.
**<span style="font-variant:small-caps;">General methodology</span>.** The next theorem will be fundamental in the applications.
\[theo:prot dec\] Let $c,t$ be two positive integers, let $H$ be an $h$-vertex graph, let $G$ be an $n$-vertex $H$-topological-minor-free graph, and let $k$ be a positive integer. If we are given a set $X \subseteq V(G)$ with $|X| {\leqslant}c \cdot k$ such that ${{\mathbf{tw}}}(G-X) {\leqslant}t$, then we can compute in time $O(n)$ an $((\alpha_{H} \cdot t \cdot c)\cdot k, 2t + h)$-protrusion decomposition of $G$, where $\alpha_{H}$ is a constant depending only on $H$, which is upper-bounded by $40 h^2 2 ^{5 h \log h}$.
A typical application of our framework for obtaining an explicit linear kernel for a packing-certifiable problem $\Pi$ on a graph class [${\mathcal}{G}$]{}is as follows. The first task is to define an encoder ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ and to prove that for some function $g: \mathbb{N} \rightarrow \mathbb{N}$, [[${\mathcal}{E}_{}$]{}]{}is a $g$-confined $\Pi$-encoder and [[${\mathcal}{E}_{}$]{}]{}is DP-friendly. The next ingredient is a polynomial-time algorithm that, given an instance $(G,k)$ of $\Pi$, either reports that $(G,k)$ is a <span style="font-variant:small-caps;">Yes</span>-instance (or a <span style="font-variant:small-caps;">No</span>-instance, depending on the problem), or finds a treewidth-modulator of $G$ with size $O(k)$. The way to obtain this algorithm depends on each particular problem and in our applications we will use a number of existing results in the literature in order to find it. Once we have such a linear treewidth-modulator, we can use Theorem \[theo:prot dec\] to find a linear protrusion decomposition of $G$. Finally, it just remains to apply Corollary \[coro:main\] to obtain an explicit linear kernel for $\Pi$ on ${\ensuremath{{\mathcal}{G}}\xspace}$; see Figure \[fig:scheme\] for a schematic illustration.
Let us provide here some generic intuition about the additional criterion mentioned in the introduction to discard the entries in the tables of an encoder. For an encoder ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}= ({\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace},{\ensuremath{L^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace},{\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace})$ and a function $g: \mathbb{N} \to \mathbb{N}$, we need some notation in order to define the *relevant function* [$\bar f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}_{g}$]{}, which will be an appropriate modification of [$f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}$]{}. Let $G \in {\ensuremath{{\mathcal}{B}_t}\xspace}$ with boundary $A$ and let $R_A$ be an encoding. We (recursively) define $R_A$ to be *irrelevant for [$\bar f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}_{g}$]{}* if there exists a certificate ${\mathcal}{S}$ such that $(G,{\mathcal}{S},R_A)\in {\ensuremath{L^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}$ and $|{\mathcal}{S}|= {\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R_A)$ and a separator $B \subseteq V(G)$ with $|B| {\leqslant}t$ and $B \neq A$, such that ${\mathcal}{S}$ [*induces*]{} an encoding $R_B$ in the graph $G_B \in {\ensuremath{{\mathcal}{B}_t}\xspace}$ with $ {\ensuremath{\bar f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}_{g}}\xspace}(G_B,R_B) = -\infty$. Here, by using the term “induces” we implicitly assume that ${\mathcal}{S}$ defines an encoding $R_B$ in the graph $G_B$; this will be the case in all the encoders used in our applications.
To define [$\bar f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}_{g}$]{}, we will always use the following [*natural*]{} function [$f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}$]{}, which for each problem $\Pi$ is meant to correspond to an extension to boundaried graphs of the maximization function $f^{\Pi}$ of Definition \[defi:optimization-function\]. For a graph $G$ and an encoding $R$, this natural function is defined as ${\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R) = \max \{k : \exists {\mathcal}{S}, |{\mathcal}{S}| {\geqslant}k, (G,{\mathcal}{S},R) \in {\ensuremath{L^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}\}$. Then we define the function ${\ensuremath{\bar f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}_{g}}\xspace}$ as follows:
$$\label{eq: relevant f}
{\ensuremath{\bar f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}_{g}}\xspace}(G,R_A) =\
\left\{\begin{array}{lll}
& -\infty, & \text{if } {\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R_A) + g(t) < \max \{{\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R): R \in {\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(\Lambda(G)) \}, \\
& & \text{or if $R_A$ is irrelevant for {\ensuremath{\bar f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}_{g}}\xspace}.} \\
& {\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R_A), & \mbox{otherwise}.\\
\end{array}\right.$$
That is, we will use the modified encoder $({\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace},{\ensuremath{L^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace},{\ensuremath{\bar f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}_{g}}\xspace})$. We need to guarantee that the above function [$\bar f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}_{g}$]{}is computable, as required[^5] in Definition \[defi:encod\]. Indeed, from the definition it follows that an encoding $R_A$ defined at a node $x$ of a given tree decomposition is irrelevant if and only if $R_A$ can be obtained by combining encodings corresponding to the children of $x$, such that at least one of them is irrelevant. This latter property can be easily computed recursively on a tree decomposition, by performing standard dynamic programming. We will omit this computability issue in the applications, as the same argument sketched here applies to all of them.
In order to obtain the linear treewidth-modulators mentioned before, we will use several results from [@BFL+09; @FLST10; @FLRS10], which in turn use the following two propositions. For an integer $r{\geqslant}2$, let ${\rm \Gamma}_{r}$ be the graph obtained from the $(r\times r)$-grid by triangulating internal faces such that all internal vertices become of degree $6$, all non-corner external vertices are of degree 4, and one corner of degree 2 is made adjacent to all vertices of the external face (the [*corners*]{} are the vertices that in the underlying grid have degree 2). As an example, the graph $\Gamma_6$ is shown in Figure \[fig-gamma-k\].
\[prop:tw-minor\] There is a function $f_m:\mathbb{N}\rightarrow\mathbb{N}$ such that for every $h$-vertex graph $H$ and every positive integer $r$, every $H$-minor-free graph with treewidth at least $f_{m}(h)\cdot r$, contains the $(r\times r)$-grid as a minor.
\[prop:tw-contraction\] There is a function $f_c:\mathbb{N}\rightarrow\mathbb{N}$ such that for every $h$-vertex apex graph $H$ and every positive integer $r$, every $H$-minor-free graph with treewidth at least $f_{c}(h)\cdot r$, contains the graph ${\rm \Gamma}_{r}$ as a contraction.
The current best upper bound [@KaKo12] for the function $f_m$ is $f_m (h) = 2^{ O(h^2 \log h)}$ and, up to date, there is no explicit bound for the function $f_c$. We would like to note that this non-existence of explicit bounds for $f_c$ is an issue that concerns the [*graph class*]{} of $H$-minor-free graphs and it is perfectly compatible with our objective of providing explicit constants for particular [*problems*]{} defined on that graph class, which will depend on the function $f_c$.
Let us now provide a sketch of the main basic ingredients used in each of the applications.
**Packing minors.** Let ${\mathcal}{F}$ be a fixed finite set of graphs. In the <span style="font-variant:small-caps;">${\mathcal}{F}$-Packing</span> problem, we are given a graph $G$ and an integer parameter $k$ and the question is whether $G$ has $k$ vertex-disjoint subgraphs $G_1,\ldots,G_k$, each containing some graph in ${\mathcal}{F}$ as a minor. When all the graphs in ${\mathcal}{F}$ are connected and ${\mathcal}{F}$ contains at least one planar graph, we call the problem <span style="font-variant:small-caps;">Connected-Planar-${\mathcal}{F}$-Packing</span>. The encoder uses the notion of rooted packing introduced by Adler *et al*. [@ADF+11], which we also used in [@KviaDP] for <span style="font-variant:small-caps;">Connected-Planar-${\mathcal}{F}$-Deletion</span>. To obtain the treewidth-modulator, we use the *Erdős-Pósa property* for graph minors [@ErPo65; @RoSe86; @ChChSTOC13]. More precisely, we use that on minor-free graphs, as proved by Fomin *et al*. [@FST11], if $(G,k)$ is a <span style="font-variant:small-caps;">No</span>-instance of <span style="font-variant:small-caps;">Connected-Planar-${\mathcal}{F}$-Packing</span>, then $(G,k')$ is a <span style="font-variant:small-caps;">Yes</span>-instance of <span style="font-variant:small-caps;">Connected-Planar-${\mathcal}{F}$-Deletion</span> for $k' = O(k)$. Finally, we use a result of Fomin *et al*. [@FLST10] that provides a polynomial-time algorithm to find treewidth-modulators for <span style="font-variant:small-caps;">Yes</span>-instances of <span style="font-variant:small-caps;">Connected-Planar-${\mathcal}{F}$-Deletion</span>. The obtained constants involve, in particular, the currently best known constant-factor approximation of treewidth on minor-free graphs.
**Packing scattered minors.** Let ${\mathcal}{F}$ be a fixed finite set of graphs and let $\ell$ be a positive integer. In the $\ell$-${\mathcal}{F}$-Packing problem, we are given a graph $G$ and an integer parameter $k$ and the question is whether $G$ has $k$ subgraphs $G_1,\ldots,G_k$ pairwise at distance at least $\ell$, each containing some graph from ${\mathcal}{F}$ as a minor. The encoder for [<span style="font-variant:small-caps;">$\ell$-$\cal F$-Packing</span>]{}is a combination of the encoder for [<span style="font-variant:small-caps;">$\cal F$-Packing</span>]{}and the one for $\ell$-<span style="font-variant:small-caps;">Scattered Set</span> that we used in [@KviaDP]. For obtaining the treewidth-modulator, unfortunately we cannot proceed as for packing minors, as up to date no linear Erdős-Pósa property for packing scattered planar minors is known; the best bound we are aware of is $O(k\sqrt{k})$, which is [*not*]{} enough to obtain a linear kernel. To circumvent this problem, we use the following trick: we (artificially) formulate [<span style="font-variant:small-caps;">$\ell$-$\cal F$-Packing</span>]{}as a vertex-certifiable problem and prove that it fits the conditions required by the framework of Fomin *et al*. [@FLST10] to produce a treewidth-modulator. (We would like to stress that this formulation of the problem as a vertex-certifiable one is [*not*]{} enough to apply the results of [@KviaDP], as one has to further verify the necessary properties of the encoder are satisfied and it does not seem to be an easy task at all.) Once we have it, we consider the original formulation of the problem to define its encoder. As a drawback of resorting to the general results of [@FLST10] and, due to the fact that [<span style="font-variant:small-caps;">$\ell$-$\cal F$-Packing</span>]{}is contraction-bidimensional, we provide linear kernels for the problem on the (smaller) class of apex-minor-free graphs. **Packing overlapping minors.** Let ${\mathcal}{F}$ be a fixed finite set of graphs and let $\ell$ be a positive integer. In the <span style="font-variant:small-caps;">$\cal F$-Packing with $\ell$-Membership</span> problem, we are given a graph $G$ and an integer parameter $k$ and the question is whether $G$ has $k$ subgraphs $ G_1,\dots,G_k$ such that each subgraph contains some graph from $\cal F$ as a minor, and each vertex of $G$ belongs to at most $\ell$ subgraphs. The encoder is an enhanced version of the one for packing minors, in which we allow a vertex to belong simultaneously to several minor-models. To obtain the treewidth-modulator, the situation is simpler than above, thanks to the fact that a packing of models is in particular a packing of models with $\ell$-membership. This allows us to use the linear Erdős-Pósa property that we described for packing minors and therefore to construct linear kernels on minor-free graphs.
**Packing scattered and overlapping subgraphs.** The definitions of the corresponding problems are similar to the ones above, just by replacing the minor by the subgraph relation. The encoders are simplified versions of those that we defined for packing scattered and overlapping minors, respectively. The idea for obtaining the treewidth-modulator is to apply a simple reduction rule that removes all vertices not belonging to any of the copies of the subgraphs we are looking for. It can be easily proved that if a reduced graph is a <span style="font-variant:small-caps;">No</span>-instance of the problem, then it is a <span style="font-variant:small-caps;">Yes</span>-instance of $\ell'$-<span style="font-variant:small-caps;">Dominating Set</span>, where $\ell'$ is a function of the integer $\ell$ corresponding to the problem and the largest diameter of a subgraph in the given family. We are now in position to use the machinery of [@FLST10] for $\ell'$-<span style="font-variant:small-caps;">Dominating Set</span> and find a linear treewidth-modulator.
Conclusions and further research {#sec:conclusions}
================================
In this article we generalized the framework introduced in [@KviaDP] to deal with packing-certifiable problems. Our main result can be seen as a [*meta-theorem*]{}, in the sense that as far a particular problem satisfies the generic conditions stated in Corollary \[coro:main\], an explicit linear kernel on the corresponding graph class follows. Nevertheless, in order to verify these generic conditions and, in particular, to verify that the equivalence relation associated with an encoder is DP-friendly, the proofs are usually quite technical and one first needs to get familiar with several definitions. We think that it may be possible to simplify the general methodology, thus improving its applicability.
Concerning the explicit bounds derived from our results, one natural direction is to reduce them as much as possible. These bounds depend on a number of intermediate results that we use along the way and improving any of them would result in an improvement on the overall kernel sizes. It is worth insisting here that some of the bounds involve the (currently) [*non-explicit*]{} function $f_c$ defined in Proposition \[prop:tw-contraction\], which depends exclusively on the considered graph class (and not on each particular problem). In order to find explicit bounds for this function $f_c$, we leave as future work using the linear-time deterministic protrusion replacer recently introduced by Fomin *et al*. [@FLM+15], partially inspired from [@KviaDP].
**Acknowledgement.** We would like to thank Archontia C. Giannopoulou for insightful discussions about the Erdős-Pósa property for scattered planar minors.
[^1]: AlGCo project-team, CNRS, LIRMM, Université de Montpellier, Montpellier, France.
[^2]: Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece.
[^3]: [Emails: Valentin Garnero: [[email protected]]{}, Christophe Paul: [[email protected]]{}, Ignasi Sau: [[email protected]]{}, Dimitrios M. Thilikos: [[email protected]]{}]{}.
[^4]: We would like to clarify here that in our original conference submission of [@KviaDP] we claimed, among other results, a linear kernel for <span style="font-variant:small-caps;">${\mathcal F}$-Packing</span> on sparse graphs. Unfortunately, while preparing the camera-ready version, we realized that there was a bug in one of the proofs and we had to remove that result from the paper. It turned out that for fixing that bug, several new ideas and a generalization of the original framework seemed to be necessary; this was the starting point of the results presented in the current article.
[^5]: The fact that the values of the function [$\bar f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}_{g}$]{}can be calculated is important, in particular, in the proof of Lemma \[lem:comput repres\], since we need to be able to compute equivalence classes of the equivalence relation [[${\mathcal}{E}_{}$]{}]{}.
| 0 |
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abstract: 'Serre obtained a sharp bound on how often two irreducible degree $n$ complex characters of a finite group can agree, which tells us how many local factors determine an Artin $L$-function. We consider the more delicate question of finding a sharp bound when these objects are primitive, and answer these questions for $n=2,3$. This provides some insight on refined strong multiplicity one phenomena for automorphic representations of $\operatorname{GL}(n)$. For general $n$, we also answer the character question for the families $\operatorname{PSL}(2,q)$ and $\operatorname{SL}(2,q)$.'
address:
- 'Department of Mathematics, University of Oklahoma, Norman, OK 73019 USA'
- 'Department of Mathematics, Occidental College, Los Angeles, CA 90041 USA'
author:
- Kimball Martin
- Nahid Walji
title: 'Distinguishing finite group characters and refined local-global phenomena'
---
Introduction
============
In this paper, we consider two questions about seemingly different topics:
1. How often can two characters of a finite group agree?
2. How many local Euler factors determine an $L$-function?
The first question is just about characters of finite groups, and the second is a refined local-global principle in number theory. However, it has been observed, notably by Serre, that being able to say something about (1) allows one to say something about (2), which is our primary motivation, though both are natural questions. Our main results about the first question are for comparing primitive characters of degree $\le 3$ and characters of $\operatorname{PSL}(2,q)$ or $\operatorname{SL}(2,q)$. This will yield sharp bounds on how many Euler factors one needs to distinguish primitive 2- or 3-dimensional $L$-functions of Galois representations. We address them in turn.
Distinguishing group characters
-------------------------------
Let $G$ be a finite group, and $\rho, \rho'$ be two complex representations of $G$ with characters $\chi, \chi'$. We will study the quantities $$\delta(\rho, \rho') = \delta(\chi, \chi') =
\frac {| \{ g \in G : \chi(g) \ne \chi'(g) \} |}{|G|}.$$
Specifically, let $\delta_n(G)$ be the minimum of $\delta(\rho, \rho')$ as $\rho, \rho'$ range over pairs of inequivalent irreducible $n$-dimensional representations of $G$, with the convention that $\delta_n(G) = 1$ if there are no such pairs $\rho, \rho'$. Note that $\delta_n(G)$ tells us what fraction of elements of $G$ we must check to distinguish irreducible degree $n$ characters. Put $d_n = \inf_G \{ \delta_n(G) \}$.
An elementary consequence of orthogonality relations is
\[prop1\] We have $d_n \ge \frac 1{2n^2}$.
Buzzard, Edixhoven and Taylor constructed examples to show this bound is sharp when $n$ is a power of 2, which Serre generalized this to arbitrary $n$ (see [@ramakrishnan:motives]).
\[thm:serre\] For any $n$, there exists $G$ such that $\delta_n(G) = \frac 1{2n^2}$, so $d_n = \frac 1{2n^2}$.
In particular, the infimum in $d_n$ is a minimum. We will recall the proof of Proposition \[prop1\] and Serre’s construction in Section \[sec:serre\]. For now, the main points to note are that Serre’s examples must be solvable and the representations are induced.
In this paper, we consider two kinds of refinements of determining $d_n$. The first refinement is about restricting to primitive representations and the second is about restricting to certain families of groups.
Define $\delta^\natural_n(G)$ to be the infimum of $\delta(\rho, \rho')$ where $\rho, \rho'$ range over pairs of inequivalent irreducible primitive $n$-dimensional complex representations of $G$. Let $d^\natural_n = \inf_G \{ \delta_n^\natural(G) \}$. From Serre’s theorem, we get a trivial bound $d^\natural_n \ge d_n = \frac 1{2n^2}$.
Our first result is to determine $d^\natural_n$ for $n \le 3$.
\[thm1\] We have $d_1^\natural = \frac 12$, $d_2^\natural = \frac 14$ and $d_3^\natural = \frac 27$. Furthermore, $\delta^\natural_2(G) = \frac 14$ if and only if $G$ is an extension of $H \times_{C_2} C_{2m}$ where $m \in \mathbb N$ and $H=[48,28]$ or $H=[48,29]$. Also, $\delta^\natural_3(G) = \frac 27$ if and only if $G$ is an extension of $\operatorname{PSL}(2,7)$.
Here $G$ being an extension of $H$ by some $N \lhd G$ means $G/N \simeq H$. The groups $[48,28]$ and $[48,29]$ are the two groups of order 48 which are extensions of $S_4$ by the cyclic group $C_2$ and contain $\operatorname{SL}(2,3)$.
The $n=1$ case is already contained in Proposition \[prop1\] as $d_1 = d_1^\natural$. For $n=2, 3$, these bounds are much better than the trivial bounds $d_2^\natural \ge \frac 18$ and $d_3^\natural \ge \frac 1{18}$ from Proposition \[prop1\]. For $n=2$, related results were previously obtained by the second author in [@walji] and will be discussed below.
Note that while $d_n$ is a strictly decreasing sequence for $n \ge 1$, our result says this is not the case for $d_n^\natural$.
In a slightly different direction, one can look for stronger lower bounds than $\frac 1{2n^2}$ for certain families of groups. We do not begin a serious investigation of this here, but just treat two basic families of finite groups of Lie type which are related to the calculations for $\delta^\natural_2(G)$ and $\delta^\natural_3(G)$.
\[thm2\] We compute $\delta_n(G)$ and $\delta^{\natural}_n(G)$ where $G = \operatorname{PSL}(2,q)$ and $G=\operatorname{SL}(2,q)$; for $n$ not listed explicitly below, $\delta_n(G) = \delta^{\natural}_n(G)=1$.
For $G = \operatorname{SL}(2,q)$ with $q$ arbitrary or for $G = \operatorname{PSL}(2,q)$ with $q$ even, $$\begin{aligned}
\delta_n(G) = \delta^{\natural}_n (G) \
\left\{ \begin{array}{cll}
= \frac{2}{q}& \text{if }n = \frac{q \pm 1}{2} \text{ and $q$ is odd},& \\
\ge \frac{1}{6}& \text{if }n = q - 1,&\\
\end{array} \right.\end{aligned}$$ and $\delta_{q+1} (G) \ge \frac{1}{6}$ whereas $\delta^{\natural}_{q+1} (G) = 1$.
For $G = \operatorname{PSL}(2,q)$ and $q$ odd, $$\begin{aligned}
\delta_n(G) = \delta^{\natural}_n (G) \ \left\{ \begin{array}{cll}
= \frac{2}{q}&\text{if } n = \frac{q - 1}{2} \text{ and $q \equiv 3 \bmod 4$},& \\
= \frac{2}{q}&\text{if } n = \frac{q + 1}{2} \text{ and $q \equiv 1 \bmod 4$},& \\
\ge \frac{1}{6}&\text{if } n = q - 1,&\\
\end{array} \right.\end{aligned}$$ and $\delta_{q+1} (G) \ge \frac{1}{6}$ whereas $\delta^{\natural}_{q+1} (G) = 1$.
We remark that we completely determine $\delta_{q \pm 1}(G)$ for $G=\operatorname{SL}(2,q)$ and $\operatorname{PSL}(2,q)$ in Section \[sec:sl2q\], but the exact formulas are a bit complicated and depend on divisibility conditions of $q \mp 1$. In particular, $\delta_{q \pm 1}(\operatorname{SL}(2,q)) = \frac 16$ if and only if $12 | (q \mp 1)$, and $\delta_{q \pm 1}(\operatorname{PSL}(2,q)) = \frac 16$ if and only if $24 | (q \mp 1)$.
The values for $\operatorname{SL}(2,q)$ immediately give the following bounds.
$d_{(q\pm 1)/2}^\natural \leq \frac{2}{q}$ for $q$ any odd prime power greater than 3.
Note the upper bound in the corollary for $q=7$ is the exact value of $d^\natural_3$.
Even though Theorem \[thm1\] implies $d_n^\natural$ is not a decreasing sequence for $n \ge 1$, this corollary at least suggests that $d_n^\natural \to 0$ as $n \to \infty$.
The proof of Theorem \[thm1\] relies on consideration of various cases according to the possible finite primitive subgroups of $\operatorname{GL}_2({\mathbb C})$ and $\operatorname{GL}_3({\mathbb C})$ which are “minimal lifts”, and about half of these are of the form $\operatorname{PSL}(2,q)$ or $\operatorname{SL}(2,q)$ for $q \in \{ 3, 5, 7, 9 \}$. Thus Theorem \[thm2\] is a generalization of one of the ingredients for Theorem \[thm1\]. However, most of the work involved in the proof of Theorem \[thm1\] is the determination of and reduction to these minimal lifts, as described in Section \[sec:general\].
Distinguishing $L$-functions
----------------------------
Let $F$ be a number field, and consider an $L$-function $L(s)$, which is a meromorphic function of a complex variable $s$ satisfying certain properties, principally having an Euler product $L(s) = \prod L_v(s)$ where $v$ runs over all primes of $F$ for $s$ in some right half-plane. For almost all (all but finitely many) $v$, we should have $L_v(s) = (p_v(q_v^{-s}))^{-1}$ where $q_v$ is the size of the residue field of $F_v$ and $p_v$ a polynomial of a fixed degree $n$, which is the degree of the $L$-function.
Prototypical $L$-functions of degree $n$ are $L$-functions $L(s, \rho) = \prod L(s, \rho_v)$ of $n$-dimensional Galois representations $\rho : \operatorname{Gal}(\bar F/F) \to \operatorname{GL}_n({\mathbb C})$ (or into $\operatorname{GL}_n(\overline{{\mathbb Q}_p})$) and $L$-functions $L(s,\pi) = \prod L(s, \pi_v)$ of automorphic representations $\pi$ of $\operatorname{GL}_n({\mathbb A}_F)$. In fact it is conjectured that all (nice) $L$-functions are automorphic. These $L$-functions are local-global objects, and one can ask how many local factors $L_v(s)$ determine $L(s)$.
First consider the automorphic case: suppose $\pi, \pi'$ are irreducible cuspidal automorphic representations of $\operatorname{GL}_n({\mathbb A}_F)$, $S$ is a set of places of $F$ and we know that $L(s, \pi_v) = L(s, \pi'_v)$ for all $v \not \in S$. Strong multiplicity one says that if $S$ is finite, then $L(s, \pi) = L(s, \pi')$ (in fact, $\pi \simeq \pi'$). Ramakrishnan [@ramakrishnan:motives] conjectured that if $S$ has density $< \frac 1{2n^2}$, then $L(s, \pi) = L(s, \pi')$, and this density bound would be sharp. This is true when $n=1$, and Ramakrishnan also showed it when $n=2$ [@ramakrishnan:SMO].
Recently, in [@walji] the second author showed that when $n=2$ one can in fact obtain stronger bounds under various assumptions, e.g., the density bound $\frac 18$ from [@ramakrishnan:SMO] may be replaced by $\frac 14$ if one restricts to non-dihedral representations (i.e., not induced from quadratic extensions) or by $\frac 29$ if the representations are not twist-equivalent.
Our motivation for this project was to try to understand an analogue of [@walji] for larger $n$. However the analytic tools known for $\operatorname{GL}(2)$ that are used in [@walji] are not known for larger $n$. Moreover, the classification of $\operatorname{GL}(2)$ cuspidal representations into dihedral, tetrahedral, octahedral and icosahedral types has no known nice generalization to $\operatorname{GL}(n)$. So, as a proxy, we consider the case of Galois (specifically Artin) representations. The strong Artin conjecture says that all Artin representations all automorphic, and Langlands’ principle of functoriality says that whatever is true for Galois representations should be true (roughly) for automorphic representations as well.
Consider $\rho, \rho'$ be irreducible $n$-dimensional Artin representations for $F$, i.e., irreducible $n$-dimensional continuous complex representations of the absolute Galois group $\operatorname{Gal}(\bar F/F)$ of $F$. For almost all places $v$ of $F$, we can associate a well-defined Frobenius conjugacy class ${\mathrm{Fr}}_v$ of $\operatorname{Gal}(\bar F/F)$, and $L(s, \rho_v)$ determines the eigenvalues of $\rho({\mathrm{Fr}}_v)$, and thus $\operatorname{tr}\rho({\mathrm{Fr}}_v)$. Let $S$ be a set of places of $F$, and suppose $L(s, \rho_v) = L(s, \rho'_v)$, or even just $\operatorname{tr}\rho({\mathrm{Fr}}_v) = \operatorname{tr}\rho'({\mathrm{Fr}}_v)$, for all $v \not \in S$.
Continuity means that $\rho$ and $\rho'$ factor through a common finite quotient $G = \operatorname{Gal}(K/F)$ of $\operatorname{Gal}(\bar F/F)$, for some finite normal extension $K/F$. View $\rho, \rho'$ as irreducible $n$-dimensional representations of the finite group $G$. The Chebotarev density theorem tells us that if $C$ is a conjugacy class in $G$, then the image of ${\mathrm{Fr}}_v$ in $\operatorname{Gal}(K/F)$ lies in $C$ for a set of primes $v$ of density $\frac{|C|}{|G|}$. This implies that if the density of $S$ is $< \delta_n(G)$ (or $< \delta_n^\natural(G)$ if $\rho, \rho'$ are primitive), then $\rho \simeq \rho'$, i.e., $L(s, \rho) = L(s, \rho')$. Moreover, this bound on the density of $S$ is sharp.
Consequently, Proposition \[prop1\] tells us that if the density of $S$ is $< \frac 1{2n^2}$, then $L(s, \rho) = L(s, \rho')$, and Serre’s result implies this bound is sharp. (See Rajan [@rajan] for an analogous result on $\ell$-adic Galois representations.) In fact, this application to Galois representations was Serre’s motivation, and it also motivated the bound in Ramakrishnan’s conjecture. For us, the Chebotarev density theorem together with Theorem \[thm1\] yields
Let $\rho$, $\rho'$ be irreducible primitive $n$-dimensional Artin representations for $F$. Suppose $\operatorname{tr}\rho({\mathrm{Fr}}_v) = \operatorname{tr}\rho'({\mathrm{Fr}}_v)$ for a set of primes $v$ of $F$ of density $c$.
1. If $n=2$ and $c > \frac 34$, then $\rho \simeq \rho'$.
2. If $n=3$ and $c > \frac 57$, then $\rho \simeq \rho'$.
When $n=2$, if $\rho$ and $\rho'$ are automorphic, i.e., satisfy the strong Artin conjecture, then the above result already follows by [@walji]. When $n=2$, the strong Artin conjecture for $\rho$ is known in many cases—for instance, if $\rho$ has solvable image by Langlands [@langlands] and Tunnell [@tunnell], or if $F={\mathbb Q}$ and $\rho$ is “odd” via Serre’s conjecture by Khare-Wintenberger [@khare-wintenberger]. We remark that the methods of [@walji] are quite different than ours here.
The above corollary suggests the following statement may be true: if $\pi, \pi'$ are cuspidal automorphic representations of $\operatorname{GL}_3({\mathbb A}_F)$ which are not induced from characters and $L(s, \pi_v) = L(s, \pi'_v)$ for a set of primes $v$ of density $> \frac 57$, then $\pi \simeq \pi'$. Since not all cuspidal $\pi, \pi'$ come from Artin representations, the $\frac 57$ bound is not even conjecturally sufficient for general $\pi, \pi'$. However, it seems reasonable to think that coincidences of a large fraction of Euler factors only happen for essentially algebraic reasons, so the density bounds are likely to be the same in both the Artin and automorphic cases.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank a referee for pointing out an error in an earlier version. The first author was partially supported by a Simons Collaboration Grant. The second author was supported by Forschungskredit grant K-71116-01-01 of the University of Zürich and partially supported by grant SNF PP00P2-138906 of the Swiss National Foundation. This work began when the second author visited the first at the University of Oklahoma. The second author would like to thank the first author as well as the mathematics department of the University of Oklahoma for their hospitality.
Notation and Background
=======================
Throughout, $G$, $H$ and $A$ will denote finite groups, and $A$ will be abelian. Denote by $Z(G)$ the center of $G$.
If $G$ and $N$ are groups, by a (group) extension of $G$ by $N$ we mean a group $H$ with a normal subgroup $N$ such that $H/N \simeq G$. The extension is called central or cyclic if $N$ is a central or cyclic subgroup of $H$.
If $G$, $H$, and $Z$ are groups such that $Z \subset Z(G) \cap
Z(H)$, then the central product $G \times_Z H$ of $G$ and $H$ with respect to $Z$ is defined to be direct product $G \times H$ modulo the central subgroup $\{ (z, z) : z \in Z \}$.
If $\chi_1, \chi_2$ are characters of $G$, their inner product is $(\chi_1, \chi_2) = |G|^{-1} \sum_G \chi_1(g)
\overline{\chi_2(g)}$.
We denote a cyclic group of order $m$ by $C_m$.
Finite subgroups of $\operatorname{GL}_n({\mathbb C})$
------------------------------------------------------
Next we recall some definitions and facts about finite subgroups of $\operatorname{GL}_n({\mathbb C})$.
Let $G$ be a finite subgroup of $\operatorname{GL}_n({\mathbb C})$, so one has the standard representation of $G$ on $V = {\mathbb C}^n$. We say $G$ is reducible if there exists a nonzero proper subspace $W \subset V$ which is fixed by $G$.
Suppose $G$ is irreducible. Schur’s lemma implies that $Z(G) \subset Z(\operatorname{GL}_n({\mathbb C}))$. In particular, $Z(G)$ is cyclic. If there exists a nontrivial decomposition $V = W_1 \oplus \cdots \oplus W_k$ such that $G$ acts transitively on the $W_j$, then we say $G$ is imprimitive. In this case, each $W_j$ has the same dimension, and the standard representation is induced from a representation on $W_1$. Otherwise, call $G$ primitive.
Let $A \mapsto \bar A$ denote the quotient map from $\operatorname{GL}_n({\mathbb C})$ to $\operatorname{PGL}_n({\mathbb C})$. Similarly, if $G \subset \operatorname{GL}_n({\mathbb C})$, let $\bar G$ be the image of $G$ under this map. We call the projective image $\bar G$ irreducible or primitive if $G$ is. Finite subgroups of $\operatorname{PGL}_n({\mathbb C})$ have been classified for small $n$, and we can use this to describe the finite subgroups of $\operatorname{GL}_n({\mathbb C})$.
Namely, suppose $G \subset \operatorname{GL}_n({\mathbb C})$ is irreducible. Then $Z(G)$ is a cyclic subgroup of scalar matrices, and $\bar G = G/Z(G)$. Hence the irreducible finite subgroups of $\operatorname{GL}_n({\mathbb C})$, up to isomorphism, are a subset of the set of finite cyclic central extensions of the irreducible subgroups $\bar G$ of $\operatorname{PGL}_n({\mathbb C})$.
Let $H$ be an irreducible subgroup of $\operatorname{PGL}_n({\mathbb C})$. Given one cyclic central extension $G$ of $H$ which embeds (irreducibly) in $\operatorname{GL}_n({\mathbb C})$, note that the central product $G \times_{Z(G)} C_m$ also does for any cyclic group $C_m \supset Z(G)$, and has the same projective image as $G$. (Inside $\operatorname{GL}_n({\mathbb C})$, this central product just corresponds to adjoining more scalar matrices to $G$.) Conversely, if $G \times_{Z(G)} C_m$ is an irreducible subgroup of $\operatorname{GL}_n({\mathbb C})$, so is $G$. We say $G$ is a minimal lift of $H$ to $\operatorname{GL}_n({\mathbb C})$ if $G$ is an irreducible subgroup of $\operatorname{GL}_n({\mathbb C})$ with $\bar G \simeq H$ such that $G$ is not isomorphic to $G_0 \times_{Z(G_0)} C_m$ for any proper subgroup $G_0$ of $G$.
Serre’s construction {#sec:serre}
--------------------
Here we explain the proof of Proposition \[prop1\] and describe Serre’s construction.
Suppose $\chi_1$ and $\chi_2$ are two distinct irreducible degree $n$ characters of a finite group $G$. Let $Y$ be the set of elements of $G$ such that $\chi_1(g) \ne \chi_2(g)$. Then we have $$|G| ( (\chi_1, \chi_1) - (\chi_1, \chi_2) ) =
\sum_Y \chi_1(g) (\overline{\chi_1(g)} - \overline{\chi_2(g)}).$$ Using the bound $|\chi_i(g)| \le n$ for $i=1,2$ and orthogonality relations, we see $$|G| = |G| ( (\chi_1, \chi_1) - (\chi_1, \chi_2) )
\le 2n^2 |Y|.$$ This proves Proposition \[prop1\].
We now recall Serre’s construction proving Theorem \[thm:serre\], which is briefly described in [@ramakrishnan:motives] using observations from [@serre Sec 6.5].
Let $H$ be an irreducible subgroup of $\operatorname{GL}_n({\mathbb C})$, containing $\zeta I$ for each $n$-th root of unity $\zeta$, such that $\bar H$ has order $n^2$. This means that $H$ is of “central type” with cyclic center. Such $H$ exist for all $n$. For instance, one can take $\bar H = A \times A$, where $A$ is an abelian group of order $n$. Some nonabelian examples of such $\bar H$ are given by Iwahori and Matsumoto [@iwahori-matsumoto Sec 5]. Iwahori and Matsumoto conjectured that groups of central type are necessarily solvable and this was proved using the classification of finite simple groups by Howlett and Isaacs [@howlett-isaacs].
Since $|H| = n^3$ and $|Z(H)| = n$, the identity $\sum |\operatorname{tr}h|^2 = |H|$ implies $\operatorname{tr}h = 0$ for each $h \in H \setminus Z(H)$, i.e., the set of $h \in H$ such that $\operatorname{tr}h = 0$ has cardinality $n^3-n = (1 - \frac 1{n^2}) |H|$.
Let $G = H \times \{ \pm 1 \}$ and consider the representations of $G$ given by $\rho = \tau \otimes 1$ and $\rho' = \tau \otimes
\text{sgn}$, where $\tau$ is the standard representation of $H$ and $\text{sgn}$ is the nontrivial character of $\{ \pm 1 \}$. Then $\operatorname{tr}\rho(g) = \operatorname{tr}\rho'(g) = 0$ for $2(n^3-n) = (1 - \frac 1{n^2} ) |G|$ elements of $G$. On the remaining $2n$ elements of $Z(G)$, $\operatorname{tr}\rho$ and $\operatorname{tr}\rho'$ must differ on precisely $n$ elements, giving $G$ with $\delta_n(G) = \frac 1{2n^2}$ as desired.
Finally that $\rho$ and $\rho'$ so constructed are induced for $n > 1$. It suffices to show $\tau$ is induced. Since $\bar H$ is solvable, there is a subgroup of prime index $p$, so there exists a subgroup $K$ of $H$ of index $p$ which contains $Z=Z(H)$. Put $\chi = \operatorname{tr}\tau$. Now $\sum_K |\chi(k)|^2
= \sum_Z |\chi(k)|^2 = |H|$. On the other hand $\sum_K |\chi(k)|^2 \le
\sum_{i=1}^r \sum_K |\psi_i(k)|^2 = r |K|$, where $\chi |_K = \psi_1 + \cdots + \psi_r$ is the decomposition of $\chi|_K$ into irreducible characters of $K$. Thus $r \ge p$ and we must have equality, which means $\tau$ is induced from a $\psi_i$. We note that, more generally, Christina Durfee informed us of a proof that $\rho$, $\rho'$ must be induced if $\delta(\rho, \rho') = \frac 1{2n^2}$.
General Methods {#sec:general}
===============
Central extensions and minimal lifts
------------------------------------
The first step in the proof of Theorem \[thm1\] is the determination of the minimal lifts of irreducible finite subgroups of $\operatorname{PGL}_2({\mathbb C})$ and $\operatorname{PGL}_3({\mathbb C})$. Here we explain our method for this.
Let $G$ be a group and $A$ an additive abelian group, which we view as a $G$-module with trivial action. Then a short exact sequence of groups $$\label{eq:ses}
0 \to A \overset{\iota}{\to} H \overset{\pi}{\to} G \to 1,$$ where $\iota$ and $\pi$ are homomorphisms, such that $\iota(A) \subset Z(H)$ gives a central extension $H$ of $G$ by $A$. Let $M(G,A)$ be the set of such sequences. (Note these sequences are often called central extensions, but for our purpose it makes sense to call the middle term $H$ the central extension.) We say two sequences in $M(G,A)$ are equivalent if there is a map $\phi$ that makes this diagram commute: $$\label{eq:cd}
\raisebox{-0.5\height}{\includegraphics{commd}}
$$ Let ${\tilde}M(G,A)$ be $M(G,A)$ modulo equivalence.
If two sequences in $M(G,A)$ as above are equivalent, then $H \simeq H'$. However the converse is not true. E.g., taking $G\simeq A \simeq C_p$, then $|{\tilde}M(G,A)| = p$ but there are only two isomorphism classes of central extensions of $C_p$ by itself, namely the two abelian groups of order $p^2$.
Let ${\mathrm{Cent}}(G,A)$ be the set of isomorphism classes of central extensions of $G$ by $A$. Then the above discussion shows we have a surjective but not necessarily injective map $\Phi : {\tilde}M(G,A) \to {\mathrm{Cent}}(G,A)$ induced from sending a sequence as in to the isomorphism class of $H$.
Viewing $A$ as a trivial $G$-module, we have a bijection between ${\tilde}M(G,A)$ and $H^2(G,A)$, with the class $0 \in H^2(G,A)$ corresponding to all split sequences in $M(G,A)$. We can use this to help determine minimal lifts of irreducible subgroups of $\operatorname{PGL}_n({\mathbb C})$. We recall $H_1(G,{\mathbb Z})$ is the abelianization of $G$, and $H_2(G,{\mathbb Z})$ is the Schur multiplier of $G$.
\[prop:31\] Let $G$ be an irreducible subgroup of $\operatorname{PGL}_n({\mathbb C})$. Then any minimal lift of $G$ to $\operatorname{GL}_n({\mathbb C})$ is a central extension of $G$ by $C_m$ for some divisor $m$ of the exponent of $H_1(G,{\mathbb Z}) \times H_2(G,{\mathbb Z})$.
Any lift of $G$ to an irreducible subgroup $H \subset \operatorname{GL}_n({\mathbb C})$ corresponds to an element of ${\mathrm{Cent}}(G,A)$ where $A=C_m$ for some $m$, and thus corresponds to at least one element of $H^2(G,A)$. The universal coefficients theorem gives us the exact sequence $$\label{eq:uct}
0 \to \mathrm{Ext}(H_1(G,{\mathbb Z}), A) \to H^2(G, A) \to \mathrm{Hom}(H_2(G,{\mathbb Z}), A) \to 0.$$ Let $m'$ be the gcd of $m$ with the exponent of $H_1(G,{\mathbb Z}) \times H_2(G,{\mathbb Z})$. Recall that $\mathrm{Ext}(\bigoplus {\mathbb Z}/n_i {\mathbb Z}, A) = \bigoplus A/n_i A$, so $\mathrm{Ext}(H_1(G,{\mathbb Z}), C_m) = \mathrm{Ext}(H_1(G,{\mathbb Z}), C_{m'})$. An analogous statement is true for $\mathrm{Hom}(H_2(G,{\mathbb Z}), -)$ so $|H^2(G, C_m)| = |H^2(G, C_{m'})|$.
Assume $m \ne m'$. Consider a sequence as in with $A = C_{m'}$. This gives a sequence $$0 \to C_m \to H \times_{C_{m'}} C_m \to G \to 1$$ in $M(G,C_m)$ by extending $\iota : C_{m'} \to H$ to be the identity on $C_m$. Note if one has an equivalence $\phi$ of two sequences in $M(G, C_m)$ constructed in this way, then commutativity implies $\phi(H) = H$ so restricting the isomorphism $\phi$ on the middle groups to $H$ yields and equivalence of the corresponding sequences in $M(G, C_{m'})$. Hence all elements of ${\tilde}M(G, C_m)$ arise from “central products” of sequences in $M(G, C_{m'})$, and thus no elements of ${\mathrm{Cent}}(G, C_m)$ can be minimal lifts.
When $H_1(G,{\mathbb Z}) \times H_2(G, {\mathbb Z}) \simeq 1$, then $H^2(G,A) = 0$ for any abelian group $A$, which means all central extensions are split, i.e., ${\mathrm{Cent}}(G, A) = \{ G \times A \}$ for any $A$. When $H_1(G,{\mathbb Z}) \times H_2(G, {\mathbb Z}) \simeq {\mathbb Z}/2{\mathbb Z}$, then tells us that $|H^2(G, C_m)|$ has size 1 or 2 according to whether $m$ is odd or even, so there must be a unique nonsplit extension ${\tilde}G \in {\mathrm{Cent}}(G,C_2)$. Then the argument in the proof tells us any cyclic central extension of $G$ is a central product of either $G$ or ${\tilde}G$ with a cyclic group.
However, in general, knowing $H_1(G,{\mathbb Z})$ and $H_2(G, {\mathbb Z})$ is not enough to determine the size of ${\mathrm{Cent}}(G, C_m)$. When $|{\mathrm{Cent}}(G, C_m)| < |H^2(G, C_m)|$, we will sometimes need a way to verify that the central extensions of $G$ by $C_m$ we exhibit exhaust all of ${\mathrm{Cent}}(G, C_m)$. For this, we will use a lower bound on the size of the fibers of $\Phi$, i.e., a lower bound on the number of classes in ${\tilde}M(G, A)$ a given central extension $H \in {\mathrm{Cent}}(G,A)$ appears in.
The central automorphisms of a group $H$ with center $Z$, denoted ${\mathrm{Aut}}_Z(H)$, are the automorphisms $\sigma$ of $H$ which commute with the projection $H \to H/Z$, i.e., satisfy $\sigma(h)h^{-1} \in Z$ for all $h \in H$.
\[prop:32\] Let $A$ be abelian and $H \in {\mathrm{Cent}}(G,A)$ such that $A = Z:= Z(H)$. Then $|\Phi^{-1}(H)| \ge \frac{|{\mathrm{Aut}}(Z)|}{|{\mathrm{Aut}}_Z(H)|}$. Moreover, if $H$ is perfect, then $|\Phi^{-1}(H)| \ge |{\mathrm{Aut}}(Z)|$.
Recall $H$ being perfect means $H$ equals its derived group, i.e., $H_1(H,{\mathbb Z}) = 0$. In particular, non-abelian simple groups are perfect. By , central extensions of perfect groups are simpler to study. In fact a perfect group $H$ possesses a universal central extension by $H_2(H,{\mathbb Z})$.
Consider a commuting diagram of sequences as in with $H'=H$. Suppose $\pi = \pi'$, which forces $\phi \in {\mathrm{Aut}}_Z(H)$ and $\iota'(A) = \ker \pi = \iota(A)$. Fixing $\pi$ and $\iota$, there are $|{\mathrm{Aut}}(Z)|$ choices for $\iota'$, which gives $|{\mathrm{Aut}}(Z)|$ elements of $M(G,A)$. Each different $\iota'$ must induce a different central automorphism $\phi \in {\mathrm{Aut}}_Z(H)$. Thus at most $|{\mathrm{Aut}}_Z(H)|$ of these $|{\mathrm{Aut}}(Z)|$ bottom sequences can lie in the same equivalence class, which proves the first statement.
Adney and Yen [@adney-yen] showed $|{\mathrm{Aut}}_Z(H)| = |\mathrm{Hom}(H,Z)|$ when $H$ has no abelian direct factor. Consequently, ${\mathrm{Aut}}_Z(H) = 1$ when $H$ is perfect.
Reduction to minimal lifts {#sec:reduction}
--------------------------
Let $G$ be a finite group and $\rho_1, \rho_2$ be two inequivalent irreducible representations of $G$ into $\operatorname{GL}_n({\mathbb C})$. Let $N_i = \ker \rho_i$ and $G_i = \rho_i(G)$ for $i = 1, 2$. We want to reduce the problem of finding lower bounds for $\delta(\rho_1,
\rho_2)$ to the case where $G_1$ and $G_2$ are minimal lifts of $\bar G_1$ and $\bar G_2$. Note that $\delta(\rho_1, \rho_2)$ is unchanged if we factor through the common kernel $N_1 \cap N_2$, so we may assume $N_1 \cap N_2 = 1$. Then $N_1 \times N_2$ is a normal subgroup of $G$, $N_1 \simeq \rho_2(N_1) \lhd G_2$ and $N_2 \simeq \rho_1(N_2) \lhd G_1$.
Write $G_i = H_i \times_{Z(H_i)} Z_i$ for $i=1, 2$, where $H_i$ is a minimal lift of $\bar G_i$ to $\operatorname{GL}_n({\mathbb C})$ and $Z_i$ is a cyclic group containing $Z(H_i)$.
For a subgroup $H$ of $\operatorname{GL}_n({\mathbb C})$, let $\alpha_n(H)$ be the minimum of $\frac{ | \{ h \in H : \operatorname{tr}h \ne 0 \} | }{|H|}$ as one ranges over all embeddings (i.e., faithful $n$-dimensional representations) of $H$ in $\operatorname{GL}_n({\mathbb C})$.
\[lem:34\] Let $m = |\rho_1(N_2) \cap Z(G_1)|$. Then $\delta(\rho_1, \rho_2) \ge \frac{m-1}{m} \alpha_n(H_1)$.
Let $K = N_2 \cap \rho_1^{-1}(Z(G_1))$, so $\rho_1(K)$ is a cyclic subgroup of $Z(G_1)$ of order $m$ and $\rho_2(K) = 1$. Fix any $g \in G$. Then as $k$ ranges over $K$, $\operatorname{tr}\rho_1(gk)$ ranges over the values $\zeta \operatorname{tr}\rho(g)$, where $\zeta$ runs through all $m$-th roots of 1 in ${\mathbb C}$, attaining each value equally often. On the other hand, $\operatorname{tr}\rho_2(gk) = \operatorname{tr}\rho_2(g)$ for all $k \in K$. So provided $\operatorname{tr}\rho_1(g) \ne 0$, $\operatorname{tr}\rho_1$ and $\operatorname{tr}\rho_2$ can agree on at most $\frac{1}m |K|$ values on the coset $gK$. Then note that the fraction of elements $g \in G$ for which $\operatorname{tr}\rho_1(g) \ne 0$ is the same as the fraction of elements in $h \in H_1$ for which $\operatorname{tr}h \ne 0$.
We say a subgroup $H_0$ of a group $H$ is $Z(H)$-free if $H_0 \ne 1$ and $H_0 \cap Z(H) = 1$. The above lemma implies that if $G_1$ has no $Z(G_1)$-free normal subgroups, then $\delta(\rho_1, \rho_2) \ge \frac {\alpha_n(H_1)} 2$ or $N_2 = 1$ (as the $K$ in the proof must be nontrivial). This will often allow us to reduce to the case where $N_2 = 1$, and similarly $N_1 = 1$, i.e., $G = G_1 = G_2$, when we can check this property for $G_1$ and $G_2$. The following allows us to simply check it for $H_1$ and $H_2$.
If $H_1$ has no $Z(H_1)$-free normal subgroups, then $G_1$ has no $Z(G_1)$-free normal subgroups.
Suppose $H_1$ has no $Z(H_1)$-free normal subgroups, but that $N$ is a $Z(G_1)$-free normal subgroup of $G_1$. Let $N' = \{ n \in H_1 : (n, z) \in G_1 = H_1 \times_{Z(H_1)} Z_1
\text{ for some } z \in Z_1 \}$. Then $N' \lhd H_1$. If $N' = 1$, then $N \subset Z_1 = Z(G_1)$, contradicting $N$ being $Z(G_1)$-free. Hence $N' \ne 1$ and must contain a nontrivial $a \in Z(H_1)$. But then $(a,z) \in N \cap Z(G_1)$ for some $z \in Z_1$, which also contradicts $N$ being $Z(G_1)$-free.
This will often allow us to reduce to the case where $G = H \times_{Z(H)} A$ for some cyclic group $A \supset Z(H)$, where we can use the following.
\[lem:35\] Let $H$ be a finite group, $A \supset Z(H)$ an abelian group and $G = H \times_{Z(H)} A$. Then $\delta^\natural_n(G) \ge \min
\{ \frac 12 \alpha_n(H), \delta^\natural_n(H) \}$.
We may assume $m = |A| > 1$. Let $\rho_1, \rho_2: G \to \operatorname{GL}_n({\mathbb C})$ be distinct primitive representations of $G$. They pull back to $H \times A$, so for $i=1,2$ we can view $\rho_i = \tau_i \otimes \chi_i$ where $\tau_i: H \to \operatorname{GL}_n({\mathbb C})$ is primitive and $\chi_i: A \to {\mathbb C}^\times$. By a similar argument to the proof of Lemma \[lem:34\], we have that $\delta(\rho_1, \rho_2) \ge \frac{m-1}m \alpha_n(H)$ if $\chi_1 \ne \chi_2$. If $\chi_1 = \chi_2$, it is easy to see $\delta(\rho_1, \rho_2) = \delta(\tau_1, \tau_2)$.
In the simplest situation, this method gives the following.
\[cor:method\] Let $\mathcal H$ be the set of minimal lifts of $\bar G_1$ and $\bar G_2$ to $\operatorname{GL}_n({\mathbb C})$. Suppose that $H$ has no $Z(H)$-free normal subgroups for all $H \in \mathcal H$. Then $$\delta(\rho_1, \rho_2) \ge \min \{ \frac 12 \alpha_n(H), \delta^\natural_n(H) : H \in \mathcal H \}.$$
This corollary will address most but not all cases of our proof of Theorem \[thm1\]. Namely, when $n=3$, it can happen that $\bar G_1$ has a lift $H \simeq \bar G_1$ which is simple, so $H$ is a $Z(H)$-free normal subgroup of itself. So we will need to augment this approach when $H_1$ or $H_2$ is simple.
Primitive degree 2 characters
=============================
In this section we will prove the $n=2$ case of Theorem \[thm1\].
We used the computer package GAP 4 [@gap] for explicit group and character calculations in this section and the next. We use the notation $[n, m]$ for the $m$-th group of order $n$ in the Small Groups Library, which is accessible by the command `SmallGroup(n,m)` in GAP. We can enumerate all (central or not) extensions of $G$ by $N$ in GAP if $|G||N| \le 2000$ and $|G||N| \ne 1024$ as all groups of these orders are in the Small Groups Library. We can also compute homology groups $H_n(G,{\mathbb Z})$ using the HAP package in GAP.
Finite subgroups of $\operatorname{GL}_2({\mathbb C})$
------------------------------------------------------
Recall the classification of finite subgroups of $\operatorname{PGL}_2({\mathbb C}) \simeq \mathrm{SO}_3({\mathbb C})$. Any finite subgroup of $\operatorname{PGL}_2({\mathbb C})$ is of one of the following types:
(A) cyclic
(B) dihedral
(C) tetrahedral ($A_4 \simeq \operatorname{PSL}(2,3)$)
(D) octahedral ($S_4$)
(E) icosahedral ($A_5 \simeq \operatorname{PSL}(2,5) \simeq \operatorname{PSL}(2,4) \simeq \operatorname{SL}(2,4)$)
Now suppose $G$ is a subgroup of $\operatorname{GL}_2({\mathbb C})$ with projective image $\bar G$ in $\operatorname{PGL}_2({\mathbb C})$. If $\bar G$ is cyclic, $G$ is reducible. If $\bar G$ is dihedral, then $G$ is not primitive.
Assume $\bar G$ is primitive. Then we have the following possibilities.
(C) Suppose $\bar G = A_4 \simeq \operatorname{PSL}(2,3)$. Here $H_1(A_4, {\mathbb Z}) \simeq {\mathbb Z}/3{\mathbb Z}$ and $H_2(A_4, {\mathbb Z}) \simeq {\mathbb Z}/2{\mathbb Z}$. There is one nonsplit element of ${\mathrm{Cent}}(A_4, C_2)$, namely $\operatorname{SL}(2,3)$; one nonsplit element of ${\mathrm{Cent}}(A_4, C_3)$, namely \[36, 3\]; and one element of ${\mathrm{Cent}}(A_4, C_6)$ which is not a central product with a smaller extension, namely \[72, 3\]. Of these central extensions (and the trivial extension $A_4$), only $\operatorname{SL}(2,3)$ and $[72, 3]$ have irreducible faithful 2-dimensional representations.
Thus $\operatorname{SL}(2,3)$ and $[72, 3]$ are the only minimal lifts of $A_4$ to $\operatorname{GL}_2({\mathbb C})$. We check that neither $H=\operatorname{SL}(2,3)$ nor $H=[72, 3]$ has $Z(H)$-free normal subgroups. In both cases, we have $\alpha_2(H) = \frac 34$, and $\delta_2^\natural(H) = \frac 23$.
(D) Next suppose $\bar G = S_4$. Note $H_1(S_4, {\mathbb Z}) \simeq H_2(S_4, {\mathbb Z}) \simeq {\mathbb Z}/2{\mathbb Z}$. There are 3 nonsplit central extensions of $S_4$ by $C_2$: \[48, 28\], \[48, 29\], \[48, 30\]. Neither $S_4$ nor \[48, 30\] have faithful irreducible 2-dimensional representations, but both \[48, 28\] and \[48, 29\] do.
Thus $H=[48, 28]$ and $H=[48, 29]$ are the minimal lifts of $S_4$ to $\operatorname{GL}_2({\mathbb C})$. Neither of them have $Z(H)$-free normal subgroups. In both cases we compute $\alpha_2(H) = \frac 58$ and $\delta_2^\natural(H) = \frac 14$.
(E) Last, suppose $\bar G = A_5 = \operatorname{PSL}(2,5)$. This group is perfect and $H_2(A_5, {\mathbb Z}) \simeq {\mathbb Z}/2{\mathbb Z}$, with $\operatorname{SL}(2,5)$ being the nontrivial central extension by $C_2$ (the universal central extension). Note $A_5$ has no irreducible 2-dimensional representations.
Hence there is only one minimal lift of $A_5$ to $\operatorname{GL}_2({\mathbb C})$, $H=\operatorname{SL}(2,5)$. We can check that $\operatorname{SL}(2,5)$ has no $Z(\operatorname{SL}(2,5))$-free normal subgroups, $\alpha_2(\operatorname{SL}(2,5)) = \frac 34$ and $\delta_2^\natural(\operatorname{SL}(2,5)) = \frac 25$ (cf. Theorem \[thm2\]).
Comparing characters
--------------------
Let $\rho_1, \rho_2: G \to \operatorname{GL}_2({\mathbb C})$ be inequivalent primitive representations. By Corollary \[cor:method\], $$\delta(\rho_1, \rho_2) \ge \min \left( \frac 12 \left\{ \frac 34, \frac 58, \frac 34 \right\} \cup
\left\{ \frac 23, \frac 14, \frac 25 \right\} \right) = \frac 14.$$ This shows $d_2^\natural \ge \frac 14$. Furthermore, we can only have $\delta(\rho_1, \rho_2) = \frac 14$ if $\bar G_1$ or $\bar G_2$ is $S_4$, which implies $G_1$ or $G_2$ is of the form $H \times_{C_2} C_{2m}$ for some $m$ with $H=[48, 28]$ or $H=[48, 29]$. Thus we can only have $\delta_2^\natural(G) = \frac 14$ if $G$ is an extension of $H \times_{C_2} C_{2m}$ where $m \in \mathbb N$ and $H=[48, 28]$ or $H=[48, 29]$. Moreover, if $G$ is such an extension $\delta_2^\natural(G)$ equals $\frac 14$ because $\delta_2^\natural(H)$ does.
This completes the proof of Theorem \[thm1\] when $n=2$.
Primitive degree 3 characters
=============================
Here we prove the $n=3$ case of Theorem \[thm1\].
Finite subgroups of $\operatorname{GL}_3({\mathbb C})$
------------------------------------------------------
First we review the classification of finite subgroups $\operatorname{GL}_3({\mathbb C})$. The classification can be found in Blichfeldt [@blichfeldt] or Miller–Blichfeldt–Dickson [@MBD]. We follow the classification system therein. The description involves 3 not-well-known groups, $G_{36} = [36,9]$, $G_{72} = [72, 41]$, and $G_{216} = [216,153]$. Explicit matrix presentations for preimages in $\operatorname{GL}_3({\mathbb C})$ are given in [@me:thesis Sec 8.1].
Any finite subgroup $G$ of $\operatorname{GL}_3({\mathbb C})$ with projective image $\bar G$ is one of the following types, up to conjugacy:
(A) abelian
(B) a nonabelian subgroup of $\operatorname{GL}_1({\mathbb C}) \times \operatorname{GL}_2({\mathbb C})$
(C) a group generated by a diagonal subgroup and ${\begin{pmatrix}}& 1 & \\ && 1 \\ 1 & & \\ {\end{pmatrix}}$
(D) a group generated by a diagonal subgroup, ${\begin{pmatrix}}& 1 & \\ && 1 \\ 1 & & \\ {\end{pmatrix}}$ and a matrix of the form ${\begin{pmatrix}}a && \\ && b \\ &c & {\end{pmatrix}}$
(E) $\bar G \simeq G_{36}$
(F) $\bar G \simeq G_{72}$
(G) $\bar G \simeq G_{216}$
(H) $\bar G \simeq A_5 \simeq \operatorname{PSL}(2,5) \simeq \operatorname{PSL}(2,4) \simeq \operatorname{SL}(2,4)$
(I) $\bar G \simeq A_6 \simeq \operatorname{PSL}(2,9)$
(J) $\bar G \simeq \operatorname{PSL}(2,7)$
Of these types, (A), (B) are reducible, (C), (D) are imprimitive, and the remaining types are primitive. The first 3 primitive groups, (E), (F) and (G), have non-simple projective images, whereas the latter 3, (H), (I) and (J), have simple projective images.
Now we describe the minimal lifts to $\operatorname{GL}_3({\mathbb C})$ of $\bar G$ for cases (E)–(J).
(E) We have $H_1(G_{36}, {\mathbb Z}) \simeq {\mathbb Z}/4{\mathbb Z}$ and $H_2(G_{36}, {\mathbb Z}) \simeq
{\mathbb Z}/3{\mathbb Z}$. The nonsplit extension of $G_{36}$ by $C_2$ is \[72, 19\]. There is one non split extension of $G_{36}$ by $C_4$ which is not a central product, \[144, 51\]. However, $G_{36}$, \[72, 19\] and \[144, 51\] all have no irreducible 3-dimensional representations.
There is 1 nonsplit central extension of $G_{36}$ by $C_3$, \[108, 15\]; there is one by $C_6$ which is not a central product, \[216, 25\]; there is one by $C_{12}$ which is not a central product, \[432, 57\]. All of these groups have faithful irreducible 3-dimensional representations.
Hence any minimal lift of $G_{36}$ to $\operatorname{GL}_3({\mathbb C})$ is $H=[108, 15]$, $H=[216, 25]$ or $H=[432, 57]$. In all of these cases, $H$ has no $Z(H)$-free normal subgroups, $\alpha_3(H) = \frac 79$ and $\delta_3^\natural(H) = \frac 12$.
(F) We have $H_1(G_{72}, {\mathbb Z}) \simeq {\mathbb Z}/2{\mathbb Z}\times {\mathbb Z}/2{\mathbb Z}$ and $H_2(G_{72}, {\mathbb Z}) \simeq {\mathbb Z}/3{\mathbb Z}$. There is a unique nonsplit central extension of $G_{36}$ by $C_2$, \[144, 120\]; a unique central extension of $G$ by $C_3$, \[216, 88\]; and a unique central extension of $G$ by $C_6$ which is not a central product, \[432, 239\]. Of these extensions (including $G_{72}$), only the latter two groups have faithful irreducible 3-dimensional representations.
Thus there are two minimal lifts of $G_{72}$ to $\operatorname{GL}_3({\mathbb C})$, $H=[216, 88]$ and $H=[432, 239]$. In both cases, $H$ has no $Z(H)$-free normal subgroups, $\alpha_3(H) = \frac 89$ and $\delta_3^\natural(H) = \frac 12$.
(G) We have $H_1(G_{216}, {\mathbb Z}) \simeq H_2(G_{216}, {\mathbb Z}) \simeq {\mathbb Z}/3{\mathbb Z}$. There are 4 nonsplit central extensions of $G_{216}$ by $C_3$: \[648, 531\], \[648, 532\], \[648, 533\], and \[648, 534\]. Neither $G_{216}$ nor \[648, 534\] has irreducible faithful 3-dimensional representations.
Thus there are three minimal lifts of $G_{216}$ to $\operatorname{GL}_3({\mathbb C})$, $H=[648, 531]$, $H=[648, 532]$, and $H=[648, 533]$. In all cases $H$ has no $Z(H)$-free normal subgroups, $\alpha_3(H) = \frac {20}{27}$ and $\delta_3^\natural(H) = \frac 49$.
(H) As mentioned in the $n=2$ case, $A_5 \simeq \operatorname{PSL}(2, 5)$ is perfect and we have $H_2(A_5, {\mathbb Z}) \simeq {\mathbb Z}/2{\mathbb Z}$. The nontrivial extension by $C_2$ (the universal central extension) is $\operatorname{SL}(2,5)$, but $\operatorname{SL}(2,5)$ has no faithful irreducible 3-dimensional representations.
Thus the only minimal lift of $A_5$ to $\operatorname{GL}_3({\mathbb C})$ is $A_5$ itself. We have $\alpha_3(A_5) = \frac 23$ and $\delta_3^\natural(A_5) = \delta_3^\natural(\operatorname{PSL}(2,5)) =
\frac 25$ (cf. Theorem \[thm2\]).
(I) The group $A_6$ is also perfect, but (along with $A_7$) exceptional among alternating groups in that $H_2(A_6, {\mathbb Z}) \simeq {\mathbb Z}/6 {\mathbb Z}$. Neither $A_6 \simeq \operatorname{PSL}(2,9)$, nor its double cover $\operatorname{SL}(2,9)$, has irreducible 3-dimensional representations. There is a unique nonsplit central extension of $A_6$ by $C_3$, sometimes called the Valentiner group, which we denote $V_{1080} = [1080, 260]$ and is also a perfect group. It is known (by Valentiner) that $V_{1080}$ has an irreducible faithful 3-dimensional representation.
To complete the determination of minimal lifts of $A_6$ to $\operatorname{GL}_3({\mathbb C})$, we need to determine the central extensions of $A_6$ by $C_6$. Here we cannot (easily) proceed naively as in the other cases of testing all groups of the appropriate order because we do not have a library of all groups of order 2160. We have $|{\tilde}M(A_6, C_6)| = 6$, with one class accounted for by the split extension and one by $\operatorname{SL}(2,9) \times_{C_2} C_6$. Since $V_{1080}$ must correspond to two classes in ${\tilde}M(A_6, C_3)$, $V_{1080} \times_{C_3} C_6$ corresponds to two classes in ${\tilde}M(A_6, C_6)$ by the proof of Proposition \[prop:31\]. Since $A_6$ is perfect, it has a universal central extension by $C_6$, which we denote ${\tilde}A_6$. By Proposition \[prop:32\], ${\tilde}A_6$ accounts for the remaining 2 classes in ${\tilde}M(A_6, C_6)$, and thus we have described all elements of ${\mathrm{Cent}}(A_6, C_6)$. The group ${\tilde}A_6$ is the unique perfect group of order 2160 and can be accessed by the command `PerfectGroup(2160)` in GAP, and we can check that it has no faithful irreducible 3-dimensional representations.
Hence $V_{1080}$ is the unique minimal lift of $A_6$ to $\operatorname{GL}_3({\mathbb C})$. We note $H=V_{1080}$ has no $Z(H)$-free normal subgroups, $\alpha_3(H) = \frac 79$, and $\delta_3^\natural(H) = \frac 25$.
(J) The group $\operatorname{PSL}(2,7)$ is perfect and $H_2(\operatorname{PSL}(2,7), {\mathbb Z}) \simeq {\mathbb Z}/2{\mathbb Z}$. Since $\operatorname{SL}(2,7)$ has no faithful irreducible 3-dimensional representations, any minimal lift of $\operatorname{PSL}(2,7)$ to $\operatorname{GL}_3({\mathbb C})$ is just $H = \operatorname{PSL}(2,7)$. Here $\alpha_3(H) = \frac 23$ and $\delta_3^\natural(H) = \frac 27$ by Theorem \[thm2\].
[L|CCCCCC]{} |G & 3 & 1 & 0 & 2 & 2 & 3\
G\_[36]{} & & 34 & 29 & & &\
G\_[72]{} & 1[72]{} & 78 & 1[9]{} && &\
G\_[216]{} & 1[216]{} & 58 & 7[27]{} & & & 19\
A\_5& 1[60]{} & 14 & 13 & 25 &\
A\_6 & 1[360]{} & 38 & 29 & 25 &\
(2,7) & 1[168]{} & 38 & 13 & & 27 &
Comparing characters
--------------------
Let $G$ be a finite group and $\rho_1, \rho_2: G \to \operatorname{GL}_3({\mathbb C})$ be two inequivalent primitive representations. Let $G_i, N_i, H_i, Z_i$ be as in Section \[sec:reduction\]. As before, we may assume $N_1 \cap N_2 = 1$, so $G$ contains a normal subgroup isomorphic to $N_1 \times N_2$ whose image in $G_1$ is $N_2$ and image in $G_2$ is $N_1$.
\[prop:deg3\] Suppose at least one of $\bar G_1$, $\bar G_2$ is simple. Then $\delta(\rho_1, \rho_2) \ge \frac 27$, with equality only if $\bar G_1 \simeq \bar G_2 \simeq
\operatorname{PSL}(2,7)$.
Say $\bar G_1$ is simple. Then by above, $H_1$ is isomorphic to one of $A_5$, $V_{1080}$ and $\operatorname{PSL}(2,7)$.
Suppose $\bar G_1 \not \simeq \bar G_2$. For $i=1,2$, the fraction of $g\in G$ for which $|\operatorname{tr}\rho_i(g)| = x$ is the same as the fraction of $h \in H_i$ for which $|\operatorname{tr}h| = x$. Calculations show that the proportion of such $g \in G$ (given $x$) depends neither on the minimal lift $H_i$ nor its embedding into $\operatorname{GL}_3({\mathbb C})$, but just on $\bar G_i$. These proportions are given in Table \[tabby\].
If $\bar G_1 \simeq \operatorname{PSL}(2,7)$, we see $\delta(\rho_1, \rho_2) \ge \frac 27$ just from considering elements with absolute character value $\sqrt 2$. Looking at other absolute character values shows this inequality is strict.
If $\bar G_1 \simeq A_5$ or $A_6$ and $\bar G_2$ is not isomorphic to $A_5$ or $A_6$, then considering elements with absolute character value $\frac{1\pm \sqrt 5}2$ shows $\delta(\rho_1, \rho_2) \ge \frac 25$.
So assume $\bar G_1 \simeq A_5$ and $\bar G_2 \simeq A_6$. Then $G_1 = A_5 \times C_m$ and $G_2 \simeq V_{1080} \times_{C_3} C_{3r}$ for some $m, r \in \mathbb N$. Suppose $\delta(\rho_1, \rho_2) < \frac 13$. By Lemma \[lem:34\], $\rho_1(N_2)$ and $\rho_2(N_1)$ are either $Z(G_1)$- and $Z(G_2)$-free normal subgroups of $G_1$ and $G_2$ or trivial. This forces $N_1 = 1$, so $G \simeq G_1$, but it is impossible for a quotient of $G_1$ to be isomorphic to $G_2$. Hence $\delta(\rho_1, \rho_2) \ge \frac 13 > \frac 27$ in this case.
Suppose $\bar G_1 \simeq \bar G_2$.
First suppose $N_1$ or $N_2$ is trivial, say $N_1$. Then $G \simeq G_1$. By Lemma \[lem:35\], we have $\delta_3^\natural(G) \ge \min \{ \frac 13, \delta_3^\natural(H_1) \}$. Thus $\delta_3^\natural(G) = \frac 27$ if and only if $H_1 = \operatorname{PSL}(2,7)$.
So assume $N_1$ and $N_2$ are nontrivial. By Lemma \[lem:34\], we can assume $\rho_1(N_2)$ and $\rho(N_1)$ are $Z(G_1)$- and $Z(G_2)$-free normal subgroups of $G_1$ and $G_2$. This is only possible if $N_1 \simeq N_2 \simeq H_1 \simeq H_2$ is isomorphic to $A_5$ or $\operatorname{PSL}(2,7)$.
Let $N = \rho_1^{-1}(N_2) \lhd G$ and we identify $N = N_1 \times N_2$. Fix $g \in G$. Then for any $n_1 \in N_1$, $\operatorname{tr}\rho_1(g(n_1, 1)) = \operatorname{tr}\rho_1(g)$ but $\operatorname{tr}\rho_2(g(n_1,1)) = \operatorname{tr}\rho_2(g(n_1,1))$. Since $\rho_2( g(N_1 \times 1)) = H_2 \times \{ z \}$ for some $z \in Z_2$, the fraction of elements of $g (N_1 \times 1)$ (and thus of $G$) on which $\operatorname{tr}\rho_1$ and $\operatorname{tr}\rho_2$ can agree is at most the maximal fraction of elements of $H_1$ with a given trace. By Table \[tabby\] this is less than $\frac 12$ for either $\bar G_1 \simeq A_5$ or $\bar G_1 \simeq \operatorname{PSL}(2,7)$.
To complete the proof of Theorem \[thm1\] for $n=3$, it suffices to show $\delta(\rho_1, \rho_2) > \frac 27$ when $\bar G_1$ and $\bar G_2$ are each one of $G_{36}$, $G_{72}$ and $G_{216}$. Using Corollary \[cor:method\], in this situation we see $$\delta(\rho_1, \rho_2) \ge \min \left( \frac 12 \left\{ \frac 79, \frac 89,
\frac {20}{27} \right\} \cup \left\{ \frac 12, \frac 12, \frac 49 \right \} \right) = \frac {10}{27}.$$ This finishes Theorem \[thm1\].
Families $\operatorname{SL}(2,q)$ and $\operatorname{PSL}(2,q)$ {#sec:sl2q}
===============================================================
We consider ${\operatorname{SL}}(2,q)$ and ${\operatorname{PSL}}(2,q)$, for even and odd prime powers $q$. We separate these into three subsections: $\operatorname{SL}(2,q)$, $q$ odd; ${\operatorname{SL}}(2,q) \simeq {\operatorname{PSL}}(2,q) $, $q$ even; and ${\operatorname{PSL}}(2,q)$, $q$ odd. We refer to, and mostly follow the notation of, Fulton–Harris [@fulton-harris] for the representations of these groups.
Choose an element $\Delta \in {\mathbb F}_q^\times - ({\mathbb F}_q^\times )^2$. Denote by ${\mathbb E}:={\mathbb F}_q (\sqrt{\Delta})$ the unique quadratic extension of ${\mathbb F}_q$. We can write the elements of ${\mathbb E}$ as $a + b \delta$, where $\delta := \sqrt \Delta$. The norm map $N: {\mathbb E}^\times \rightarrow {\mathbb F}_q^\times $ is then defined as $N (a + b \delta) = a^2 - b^2 \Delta$. We also denote ${\mathbb E}^1$ to be the kernel of the norm map.
$\operatorname{SL}(2,q)$, for odd $q$
-------------------------------------
The order of $\operatorname{SL}(2,q)$ is $(q + 1)q (q-1)$. We begin by describing the conjugacy classes for $\operatorname{SL}(2,q)$:
(A) $I$.
(B) $-I$.
(C) Conjugacy classes of the form $[c_2(\epsilon,\gamma)]$, where $c_2 (\epsilon, \gamma) = \left( \begin{array}{cc}
\epsilon&\gamma \\
&\epsilon
\end{array} \right),$ where $\epsilon = \pm 1$ and $\gamma = 1 \text{ or } \Delta$. So there are four conjugacy classes, each of size $(q ^2 -1)/2$.\
(D) Conjugacy classes of the form $[c_3(x)]$, where $c_3 (x) = \left( \begin{array}{cc}
x& \\
&x ^{-1}
\end{array} \right)$ with $x \neq \pm 1$. Since the conjugacy classes $c_3 (x)$ and $c_3 (x ^{-1})$ are the same, we have $(q-3)/2$ different conjugacy classes, each of size $q (q + 1)$.\
(E) Conjugacy classes of the form $[c_4(z)]$, where $c_4 (z) = \left( \begin{array}{cc}
x&\Delta y \\
y&x
\end{array} \right)$ where $z = x + \delta y \in {\mathbb E}^1$ and $z \neq \pm 1$. Since $c_4 (z) = c_4 (\bar{z})$ we have $(q-1)/2$ conjugacy classes, each of size $q (q-1)$.\
We give a brief description of the representations that appear in the character table. The first set of representations, denoted $W_\alpha$, are induced from the subgroup $B$ of upper triangular matrices. Given a character $\alpha \in \widehat{{\mathbb F}_q^\times}$, we can extend this to a character of $B$, which we then induce to a $(q + 1)$-dimensional representation $W_\alpha$ of ${\rm SL}_2({\mathbb F}_q)$. If $\alpha^2 \neq 1$, then the induced representation is irreducible. If $\alpha = 1$, then $W_1$ decomposes into its irreducible consituents: the trivial representation $U$ and the Steinberg representation $V$. If $\alpha ^2 = 1$ and $\alpha \ne 1$, then it decomposes into two irreducible constituents denoted $W^+$ and $W^-$.
For the remaining irreducible representations, we consider characters $\alpha$ and $\varphi$ of the diagonal subgroup $A$ and the subgroup $S:=\{c_4 (z) \mid z \in {\mathbb E}^1\}$, respectively, where the characters agree when restricted to $A \cap S$. Then we construct a virtual character $\pi_\varphi := {\rm Ind}^G_A (\alpha) - W_\alpha - {\rm Ind}^G_S (\varphi)$ (note that the virtual character will not depend on the specific choice of $\alpha$).
When $\varphi = \overline{\varphi}$, $\pi_\varphi$ decomposes into two distinct characters. In the case when $\varphi$ is trivial, $\pi_1$ decomposes into the difference between the characters for the Steinberg representation and the trivial representation. If $\varphi$ is the unique (non-trivial) order 2 character of $S$, then $\pi_\varphi$ decomposes into two distinct irreducible characters of equal dimension; we will label the corresponding representations $X^+$ and $X^-$. If $\varphi \neq \overline{\varphi}$, then $\pi_\varphi$ corresponds to an irreducible representation, which we denote as $X_\varphi$. Two irreducibles $X_\varphi$ and $X_{\varphi'}$ are equivalent if and only if $\varphi = \varphi'$ or $\varphi = \overline{\varphi'}$. We note that out of all the irreducible representations, the imprimitive representations are exactly all the $W_\alpha$ (for $\alpha^2 \neq 1$).
We define some notation that will appear in the character table for $\operatorname{SL}(2,q)$. Let $\alpha \in \widehat{{\mathbb F}_q^\times}$ with $\alpha \neq \pm 1$, and $\varphi$ a character of ${\mathbb E}^1$ with $\varphi ^2 \neq 1$. Fix $\tau$ to be the non-trivial element of $\widehat{{\mathbb F}_q^\times / ({\mathbb F}_q^\times)^2 }$, and let $$\begin{aligned}
s^\pm (\epsilon,\gamma) &= \frac{1}{2} (\tau (\epsilon) \pm \tau (\epsilon \gamma) \sqrt{\tau (-1)q}),\\
u^\pm (\epsilon,\gamma) &= \frac{1}{2} \epsilon (-\tau (\epsilon) \pm \tau (\epsilon \gamma) \sqrt{\tau (-1)q}).\end{aligned}$$ Lastly, we define $\psi$ to be the non-trivial element of $\widehat{{\mathbb E}^1 / ({\mathbb E}^1)^2}$. The character table is:
$[I]$ $[-I]$ $[c_2(\epsilon, \gamma)]$ $[c_3(x)]$ $[c_4(z)]$
------------- ----------------- ------------------- ----------------------------------- --------------------------- --------------------------------- ------------------------------------
Size: 1 1 $\frac{q ^2 -1}{2}$ $q (q + 1)$ $q (q-1)$
Rep \#
$U$ 1 1 1 1 1 1
$X^\pm $ 2 $\frac{q-1}{2}$ $\frac{q-1}{2} \cdot \psi (-1)$ $u^\pm (\epsilon,\gamma)$ 0 $- \psi (z)$
$W^\pm$ 2 $\frac{q + 1}{2}$ $\frac{q + 1}{2} \cdot \tau (-1)$ $s^\pm (\epsilon,\gamma)$ $\tau (x)$ 0
$X_\varphi$ $\frac{q-1}{2}$ $q-1$ $(q-1)\varphi (-1)$ $-\varphi (\epsilon)$ 0 $-\varphi (z) - \varphi (z ^{-1})$
$V$ 1 $q$ $q$ 0 1 $-1$
$W_\alpha$ $\frac{q-3}{2}$ $q + 1$ $(q + 1)\alpha (-1)$ $\alpha (\epsilon)$ $\alpha (x) + \alpha (x ^{-1})$ $0$
### [**The pair of representations $X^\pm$:**]{.nodecor} {#the-pair-of-representations-xpm .unnumbered}
The two $(q-1)/2$-dimensional representations $X^+$ and $X^-$ have the same trace character values for exactly all group elements outside of $[c_2 (\epsilon,\gamma)]$, so we have $\delta (X^+, X^-) = {2}/{q}.$
### [**The pair of representations $W^\pm$:**]{.nodecor} {#the-pair-of-representations-wpm .unnumbered}
The two $(q + 1)/2$-dimensional representations $W^+$ and $W^-$ have the same trace character values exactly for all group elements outside of the $[c_2 (\epsilon,\gamma)]$ conjugacy classes. So again we have $\delta (W^+, W^-) = {2}/{q}.$
### [**$(q-1)$-dimensional representations:**]{.nodecor} {#pi-a .unnumbered}
There are $(q-1)/2$ such representations, denoted $X_\varphi$, where $\varphi \in \widehat{{\mathbb E}^1}$, for $\varphi ^2 \neq 1$. Note that $|{\mathbb E}^1| = q + 1$.
In order to determine $\delta (X_\varphi, X_{\varphi'})$, we need to find the number of $z \in {\mathbb E}^1$ for which $\varphi (z) + \varphi (z^{-1}) = \varphi' (z) + \varphi' (z^{-1})$, and whether $\varphi (-1) = \varphi' (-1)$.
We begin with the first equation. Note that ${\rm Im} (\varphi), {\rm Im} (\varphi') \subset \mu_{q+1}$, where $\mu_n$ denotes the $n$th roots of unity. Then $\varphi (z) + \varphi (z^{-1} )$ is of the form $\zeta^a + \zeta^{-a}$, where $\zeta$ is the primitive $(q+1)$th root of unity $e^ {2\pi i / (q+1)}$ and $a$ is a non-negative integer less than $q+1$. Now $\zeta^a + \zeta^{-a} = \zeta^b + \zeta^{-b}$ for some $0 \leq a,b < q+1$ implies that $a = b$ or $(q+1) -b$. So $\varphi (z) + \varphi (z^{-1}) = \varphi' (z) + \varphi' (z^{-1})$ iff $\varphi (z) = \varphi'(z)$ or $\varphi(z) = \varphi' (z^{-1})$.
If $\varphi (z) = \varphi' (z)$, then this is equivalent to $(\varphi') ^{-1} \varphi(z) = 1$, and the number of $z$ for which this holds is $| {\rm ker}\, (\varphi') ^{-1} \varphi|$. The number of $z$ for which $\varphi (z) = \varphi' (z ^{-1})$ is $| {\rm ker}\, \varphi' \varphi |$. Thus the number of $z \in
{\mathbb E}^1$ for which $\varphi (z) + \varphi (z^{-1}) = \varphi' (z) + \varphi'(z ^{-1})$ is $$\begin{aligned}
| {\rm ker}\, (\varphi') ^{-1} \varphi | + |{\rm ker}\, \varphi' \varphi | - |{\rm ker}\, \varphi' \varphi \cap {\rm ker}\, (\varphi')^{-1} \varphi |.\end{aligned}$$
Now ${\mathbb E}^1$ is a cyclic group, so we can fix a generator $g$. The elements of $\widehat{{\mathbb E}^1}$ can then be denoted as $\{\varphi_0, \varphi_1, \varphi _2, \dots, \varphi_{q}\}$, where $\varphi_m$ is defined via $\varphi_m (g) = \zeta^m$. Note that $|{\rm ker}\, \varphi_m |= (m, q+1).$ Define $$\begin{aligned}
M_{k}(m,m') := \frac{(m + m', k) + (m-m', k) - (m+m', m-m',k) - 1 - t_{m,m'}}{2},\end{aligned}$$ where $t_{m,m'}=1$ if both $k$ and $m + m'$ are even, and $0$ otherwise.
Then:
For distinct integers $0 \leq m, m' < q+1$, we have $$|\{[c_4(z)] : \varphi_m (z) + \varphi_m (z ^{-1}) = \varphi_{m'} (z) + \varphi_{m'} (z ^{-1}) \}| = M_{q + 1} (m, m').$$
If $m$ and $m'$ have the same parity, then $\varphi_m (-1) = \varphi_{m'} (-1)$ so $$\begin{aligned}
\label{eq:Xphi-even}
\delta (X_{\varphi_m},X_{\varphi_{m'}})
=\frac{1}{q+1} \left(\frac{q-1}{2} - M_{q+1}(m,m')\right).\end{aligned}$$
If $m$ and $m'$ have different parity, then $$\begin{aligned}
\label{eq:Xphi-odd}
\delta (X_{\varphi_m},X_{\varphi_{m'}})
= \frac{1}{q ^2 -1}\left(\frac{q ^2 +1}{2} - M_{q+1}(m,m')(q - 1)\right).\end{aligned}$$
To determine the minimum possible value of $\delta$ above, we consider the maximum possible size of $M_k (m,m')$.
\[max-lem\] Suppose $k=2^j \ge 8$. Then $$\max M_k(m,m') = 2^{j-2} -1 = \frac k4-1,$$ where $m, m'$ run over distinct classes in ${\mathbb Z}/k{\mathbb Z}\setminus \{ 0, \frac k2 \}$ with $m \not
\equiv \pm m'$.
Suppose $k \in 2 \mathbb N$ is not a power of 2 and let $p$ be the smallest odd prime dividing $k$. Then $$\max M_k(m,m') = \begin{cases}
\frac k4 \left( 1 + \frac 1 p \right) - 1 & k \equiv 0 \pmod 4 \\
\frac{k-2}4 & k \equiv 2 \pmod 4, \\
\end{cases}$$ where $m, m'$ range as before.
In all cases above, the maximum occurs with $m, m'$ of the same parity if and only if $4 | k$.
Let $d = (m+m', k)$ and $d' = (m-m', k)$, so our restrictions on $m, m'$ imply that $d, d'$ are proper divisors of $k$ of the same parity. Note that any pair of such $d, d'$ arise from some $m, m'$ if $d \ne d'$, and the case $d = d' = \frac k2$ does not occur. Then $M_k(m,m') = \frac 12 ( d + d' - (d, d') -1 - t_{m,m'})$, and $m, m'$ have the same parity if and only if $d, d'$ are both even.
The case $k=2^j$ has a maximum with $d = \frac k2$ and $d'= \frac k4$.
Suppose $k=2pk'$ as in the second case. Then note $d + d' - (d, d')$ is maximised when $d = \frac k2$ and $d' = \frac kp$, which is an admissible pair if $k'$ is even. Otherwise, we get a maximum when $d = \frac k2$ and $d' = \frac k{2p}$.
In all cases we have $$\label{eq:max-bound}
\max M_k(m,m') \le \frac k3 - 1,$$ and equality is obtained if and only if $12 | k$ for suitable $m, m'$ of the same parity. This leads to an exact formula for $\delta_{q-1}(\operatorname{SL}(2,q))$ with $q > 3$ odd by combining with and . We do not write down the final expression, but just note the consequence that $\delta_{q-1}(\operatorname{SL}(2,q)) \ge \frac 16$ with equality if and only $12 | (q+1)$.
### [**$(q + 1)$-dimensional representations:**]{.nodecor} {#rho-a .unnumbered}
Consider $W_\alpha, W_{\alpha'}$, where $\alpha, \alpha' \in \widehat{{\mathbb F}_q^\times} - \{\pm 1\}$ and $\alpha \neq \alpha'$. Since $ |{\mathbb F}_q^\times| = q - 1$, we know that ${\rm Im}(\alpha) < \mu_{q - 1}$. So, given a generator $g$ of the cyclic group ${\mathbb F}_q^\times$, we define the elements of $\widehat{{\mathbb F}_q^\times}$ as: $\alpha_m (g) = \zeta^m$, where $\zeta := e ^{2\pi i / (q - 1)}$, and $0 \leq m \leq q-2$.\
Using similar arguments to the $(q-1)$-dimensional case above, we have:
For distinct integers $0 \leq m, m' < q-1$, we have $$\begin{aligned}
&\left| \{[c_3(x)] : \alpha_m (x) + \alpha_m (x ^{-1}) = \alpha_{m'} (x) + \alpha_{m'} (x ^{-1}) \} \right|
= M_{q-1}(m,m').\end{aligned}$$
Given that the value of $\alpha_m (-1)$ is $+1$ if $m$ is even and $-1$ if $m$ is odd, we obtain that if $m$ and $m'$ have the same parity, then $$\begin{aligned}
\delta (W_{\alpha_m},W_{\alpha_{m'}})
=\frac{1}{q-1} \left(\frac{q - 3}{2} - {M_{q-1}(m,m')}{}\right).\end{aligned}$$ Whereas if $m$ and $m'$ have different parity, then $$\begin{aligned}
\delta (W_{\alpha_m},W_{\alpha_{m'}}) = \frac{1}{q ^2 -1}\left(\frac{q ^2 -3}{2} - {M_{q-1}(m,m')}{}(q + 1)\right).\end{aligned}$$
Combining these with Lemma \[max-lem\] for $q > 5$ gives a formula for $\delta_{q+1}(\operatorname{SL}(2,q))$. In particular, gives $\delta_{q+1}(\operatorname{SL}(2,q)) \ge \frac 16$, with equality if and only if $12 | (q-1)$.
$\operatorname{SL}(2,q)$, for even $q$
--------------------------------------
We keep the notation from the previous section. The order of $\operatorname{SL}(2,q)$ is again $q(q + 1)(q-1)$. The conjugacy classes for $\operatorname{SL}(2,q)$, $q$ even, are as follows:
(A) $I$.
(B) $[N]= \left[\left( \begin{array}{cc}
1&1 \\
0&1
\end{array} \right)\right]$. This conjugacy class is of size $q ^2 -1$.
(C) $[c_3 (x)]$, where $c_3 (x) = \left( \begin{array}{cc}
x& \\
&x ^{-1}
\end{array} \right)$, with $x \neq 1$. We note that $[c_3 (x)] = [c_3 (x ^{-1})]$, so there are $(q-2)/2$ such conjugacy classes. Each one is of size $q (q + 1)$.
(D) $[c_4 (z)]$, where $ c_4 (z) = \left( \begin{array}{cc}
x&\Delta y \\
y&x
\end{array} \right)$ for $z = x + \delta y \in {\mathbb E}^1$ with $z \neq 1$. Since $c_4 (z) = c_4 (\bar{z})$, there are $q/2$ such conjugacy classes, each of size $q (q-1)$.\
The representations for $q$ even are constructed similarly to the case of $q$ odd, with a couple of differences: Since, for $q$ even, the subgroup $S$ has odd order, it does not have characters of order two, and so the irreducible representations $X ^\pm $ do not arise. Similarly, the character $\alpha$ cannot be of order two, and so the irreducible representations $W ^\pm$ do not occur. The character table is:
$[I]$ $[N]$ $[c_3(x)]$ $[c_4(z)]$
------------- ----------- --------- ----------- --------------------------------- ------------------------------------
Size: 1 $q ^2 -1$ $q(q + 1)$ $q (q-1)$
Rep \#
$U$ 1 $1$ 1 1 $1$
$X_\varphi$ $q/2$ $q-1$ $-1$ 0 $-\varphi (z) - \varphi (z ^{-1})$
$V$ 1 $q$ 0 1 $-1$
$W_\alpha$ $(q-2)/2$ $q + 1$ 1 $\alpha (x) + \alpha (x ^{-1})$ $0$
### [**Representations of dimension $q-1$:**]{.nodecor} {#representations-of-dimension-q-1 .unnumbered}
The analysis here is similar to that in Section \[pi-a\], which gives us: $$\delta (X_{\varphi_m}, X_{\varphi_{m'}} ) = \frac{1}{q + 1}\left(\frac{q}{2} - M_{q + 1}(m,m')\right).$$
Analogous to Lemma \[max-lem\], we have when $k\ge 3$ is odd, $$\label{eq:max-even}
\max M_{k}(m,m') =
\begin{cases}
\frac 12 \left( \frac kp - 1 \right) & k = p^j \\
\frac 12 \left( \frac k{p_1 p_2}(p_1+p_2-1) - 1 \right) & k = p_1 p_2 k'
\end{cases}$$ where $m, m'$ run over all nonzero classes of ${\mathbb Z}/k{\mathbb Z}$ such that $m \not \equiv \pm m'$ and in the latter case are the two smallest distinct primes dividing $k$. The above two equations give an exact expression for $\delta_{q-1}(\operatorname{SL}(2,q))$, $q \ge 4$. For $k$ odd, note $$\label{eq:max-bound-even}
\max M_{k}(m,m') \le \frac{7k-15}{30},$$ with equality if and only if $15 | k$. Thus $\delta_{q-1}(\operatorname{SL}(2,q)) \ge \frac 4{15}$ with equality if and only if $15 | (q+1)$.
### [**Representations of dimension $q + 1$:**]{.nodecor} {#representations-of-dimension-q-1-1 .unnumbered}
A similar analysis to that in Section \[rho-a\] gives $$\delta (W_{\alpha_m}, W_{\alpha_{m'}} ) = \frac{1}{q-1}\left(\frac{q-2}{2} - M_{q - 1}(m,m') \right).$$ Combining this with gives an exact formula for $\delta_{q+1}(\operatorname{SL}(2,q))$ for $q \ge 8$, and from , we again get $\delta_{q+1}(\operatorname{SL}(2,q)) \ge \frac{4}{15}$ with equality if and only if $15 | (q-1)$.
$\operatorname{PSL}(2,q)$, for odd $q$
--------------------------------------
The order of $\operatorname{PSL}(2,q)$ is $\frac{1}{2}q (q ^2 -1)$ if $q$ is odd. The conjugacy classes are as follows:
(A) $I$.
(B) $[c_2 (\gamma)]$, where $c_2(\gamma) = c_2 (1,\gamma)= \left( \begin{array}{cc}
1&\gamma \\
&1
\end{array} \right)$ for $\gamma \in \{1, \Delta\}$.
(C) $[c_3 (x)]$, $(x \neq \pm 1)$, where $c_3 (x)$ is as in the previous two sections. Since $c_3 (x) = c_3 (-x) = c_3 (1/x) = c_3 (-1/x)$, the number of such conjugacy classes when $q \equiv 3 \pmod 4$ is $(q-3)/4$. In this case, all of the $c_3 (x)$ conjugacy classes have size $q (q + 1)$.
If $q \equiv 1 \pmod 4$, then $-1$ is a square in ${\mathbb F}_q$ and there is a conjugacy class denoted by $c_3 (\sqrt{-1})$ which has size $q (q + 1)/2$; the remaining $c_3 (x)$ conjugacy classes (there are $(q-5)/4$ such classes) have size $q (q + 1)$.
(D) $[c_4 (z)]$, for $z \in {\mathbb E}^1, z \neq \pm 1$, where $c_4 (z)$ is defined as in the previous two sections. Since $c_4 (z) = c_4 (\bar{z}) = c_4 (-z) = c_4 (- \bar{z})$, when $q \equiv 1 \pmod 4$, the number of such conjugacy classes is $(q-1)/4$, and they are all of size $q (q-1)$. When $q \equiv 3 \pmod 4$, we can choose $\Delta$ to be $-1$ (since it is not a square), and so we see that $\delta \in {\mathbb E}^1$. The conjugacy class associated to $c_4 (\delta)$ has size $q (q-1)/2$, whereas the rest of the $c_4 (z)$ conjugacy classes (of which there are $(q-3)/4$ such classes) have size $q (q-1)$.\
The representations of $\operatorname{PSL}(2,q)$ are the representations of $\operatorname{SL}(2,q)$ which are trivial on $-I$; this depends on the congruence class of $q$ modulo 4.
### $q \equiv 1 \pmod 4$
\
For the character table below, the notation is the same as in previous subsections.
$[I]$ $[c_2(\gamma)]$ $[c_3(\sqrt{-1})]$ $[c_3(x)]$ $[c_4(z)]$
------------- ----------------- ------------------- --------------------- ----------------------- --------------------------------- ------------------------------------
Size: 1 $\frac{q ^2 -1}{2}$ $\frac{q (q + 1)}{2}$ $q (q + 1)$ $q (q-1)$
Rep $\#$ 1 2 1 $\frac{q-5}{4}$ $\frac{q-1}{4}$
$U$ 1 1 1 1 1 1
$W^\pm$ 2 $\frac{q + 1}{2}$ $s^\pm (1,\gamma)$ $\tau (\sqrt{-1})$ $\tau (x)$ 0
$X_\varphi$ $\frac{q-1}{4}$ $q-1$ $-1$ 0 0 $-\varphi (z) - \varphi (z ^{-1})$
$V$ 1 $q$ 0 1 1 $-1$
$W_\alpha$ $\frac{q-5}{4}$ $q + 1$ 1 $2\alpha (\sqrt{-1})$ $\alpha (x) + \alpha (x ^{-1})$ $0$
\
### [**Representations $W ^\pm$:**]{.nodecor} {#representations-w-pm .unnumbered}
The trace characters of these $(q + 1)/2$-dimensional representations agree everywhere but for the conjugacy classes $[c_2 (\gamma)]$. This gives us $\delta (W^+,W^-) = {2}/{q}.$
### [**Representations of dimension $q -1$:**]{.nodecor} {#representations-of-dimension-q--1 .unnumbered}
Assume $q \ge 9$. Any two representations $X_\varphi, X_{\varphi'}$ have trace characters that may differ only for the conjugacy classes $[c_4 (z)]$. We may view $\varphi$ as a map into $\mu_{\frac{q+1}2}$ and parameterize the $\varphi$ by $\varphi_m$ for nonzero $m \in {\mathbb Z}/\frac{q+1}2 {\mathbb Z}$ similar to before. Analogously, we obtain $$\delta (X_{\varphi_m}, X_{\varphi_{m'}}) = \frac{1}{q + 1} \left(\frac{q-1}{2} - 2M_{\frac{q + 1}2}(m,m')\right).$$ From , this gives $\delta_{q-1}(\operatorname{PSL}(2,q)) \ge \frac 4{15}$, with equality if and only if $30 | (q+1)$.
### [**Representations of dimension $q + 1$:**]{.nodecor} {#representations-of-dimension-q-1-2 .unnumbered}
Assume $q \ge 13$. The analysis follows in a similar manner to that in previous sections. View $\alpha : {\mathbb F}_q^\times / \{ \pm 1 \} \to \mu_{\frac{q-1}2}$, and we can parametrize such $\alpha$ by $m \in {\mathbb Z}/\frac{q-1}2 {\mathbb Z}$ as before. One difference is that we must consider the case when $x = \sqrt{-1}$. Note that this is the only conjugacy class of the form $[c_3 (x)]$ that has size $q (q + 1)/2$. We find that $\alpha_m (\sqrt{-1}) = \alpha_{m'}(\sqrt{-1})$ if and only if $m, m'$ have the same parity. Overall we get $$\delta (W_{\alpha_m}, W_{\alpha_{m'}}) = \frac{1}{q -1} \left(\frac{q-5}{2} - 2M_{\frac{q-1}2}(m,m')
+ 1 - t_{m,m'} \right).$$ From we get $\delta_{q+1}(\operatorname{PSL}(2,q)) \ge \frac 16$ with equality if and only if $24 | (q-1)$.
### $q \equiv 3 \pmod 4$
$[I]$ $[c_2(\gamma)]$ $[c_3(x)]$ $[c_4(z)]$ $[c_4(\delta)]$
------------- ----------------- ----------------- --------------------- --------------------------------- ------------------------------------ -----------------------
Size: 1 $\frac{q ^2 -1}{2}$ $q (q + 1)$ $q (q-1)$ $\frac{q (q - 1)}{2}$
Rep Number 1 2 $\frac{q-3}{4}$ $\frac{q-3}{4}$ 1
$U$ 1 1 1 1 1 1
$X^\pm$ 2 $\frac{q-1}{2}$ $u^\pm (1,\gamma)$ 0 $-\psi (z)$ $-\psi (\delta)$
$X_\varphi$ $\frac{q-3}{4}$ $q-1$ $-1$ 0 $-\varphi (z) - \varphi (z ^{-1})$ $-2 \varphi (\delta)$
$V$ 1 $q$ 0 1 $-1$ 1
$W_\alpha$ $\frac{q-3}{4}$ $q + 1$ 1 $\alpha (x) + \alpha (x ^{-1})$ $0$ 0
\
where $u^\pm(1,\gamma)$ and $\psi$ are defined as before.\
### [**Representations $X^\pm$:**]{.nodecor} {#representations-xpm .unnumbered}
For $W^\pm$, the characters of the representations $X^\pm$ agree everywhere but for the conjugacy classes $[c_2 (\gamma)]$, so: $\delta (X^+, X^-) = {2}/{q}.$
### [**Representations of dimension $q-1$:**]{.nodecor} {#representations-of-dimension-q-1-3 .unnumbered}
Assume $q \ge 11$. Any two representations $X_\varphi, X_{\varphi'}$ have trace characters that may differ only for the conjugacy classes $[c_4 (z)]$. In the case of the conjugacy class $[c_4 (\delta)]$, we note that $\delta$ has order 2 in ${\mathbb E}^1/\{ \pm 1 \}$. Parametrize the nontrivial maps $\varphi: {\mathbb E}^1/ \{ \pm 1 \} \to \mu_{\frac{q+1}2}$ by $1 \le m \le \frac{q-3}4$ as before. Then $\varphi_m (\delta) = \varphi_{m'}(\delta)$ if and only if $m, m'$ have the same parity. We obtain $$\begin{aligned}
\delta (X_{\varphi_m}, X_{\varphi_{m'}}) = \frac{1}{q + 1} \left(\frac{q-3}{2} - 2M_{\frac{q+1}2}(m,m') + 1 - t_{m,m'} \right).\end{aligned}$$ By , we get $\delta_{q-1}(\operatorname{PSL}(2,q)) \ge \frac 16$ with equality if and only if $24 | (q+1)$.
### [**Representations of dimension $q + 1$:**]{.nodecor} {#representations-of-dimension-q-1-4 .unnumbered}
Assume $q \ge 11$. We obtain $$\delta (W_{\alpha_m}, W_{\alpha_{m'}}) = \frac{1}{q -1} \left(\frac{q-3}{2} - 2M_{\frac{q-1}2}(m,m')\right).$$ By , we get $\delta_{q+1}(\operatorname{PSL}(2,q)) \ge \frac 4{15}$, with equality if and only if $30 | (q-1)$.
| 0 |
---
author:
- |
Tatsuo Azeyanagi$^a$, Masanori Hanada$^{b,c}$, Tomoyoshi Hirata$^{a}$ and Tomomi Ishikawa$^{d}$\
Department of Physics, Kyoto University,\
Kyoto 606-8502, Japan\
Theoretical Physics Laboratory, RIKEN Nishina Center,\
Wako, Saitama 351-0198, Japan\
Department of Particle Physics, Weizmann Institute of Science,\
Rehovot 76100, Israel\
RIKEN BNL Research Center, Brookhaven National Laboratory,\
Upton, New York 11973, USA\
E-mail:
title: 'Phase structure of twisted Eguchi-Kawai model'
---
Introduction
============
The large-$N$ gauge theories provide fruitful features to both phenomenology and string theory. They are simplified in the large-$N$ limit while preserving essential features of QCD [@'t; @Hooft:1973jz]. Additionally, dimensional reductions of ten-dimensional ${\cal N}=1$ super Yang-Mills theory (matrix model) are expected to provide nonperturbative formulations of superstring theory [@Banks:1996vh; @Ishibashi:1996xs; @Dijkgraaf:1997vv; @Maldacena:1997re], and can also be regarded as effective actions of D-branes [@Witten:1995ex]. Furthermore, their twisted reduced versions, which we study in this article, can provide a nonperturbative formulation of the gauge theories on noncommutative spaces (NCYM) [@Aoki:1999vr; @Ambjorn:1999ts]. In order to study the nonperturbative nature of these theories, numerical simulations using lattice regularizations are quite efficient. (Non-lattice simulations are also applicable for the reduced models. See references [@Hanada:2007ti; @Anagnostopoulos:2007fw] for the recent progress.)
In the large-$N$ limit there is an equivalence between the gauge theory and its zero-dimensional reduction, which is known as [*Eguchi-Kawai equivalence*]{} [@Eguchi:1982nm]. Here, we consider the $SU(N)$ gauge theory (YM) on $D$-dimensional periodic lattice with the Wilson’s plaquette action $$S_W=-\beta N\sum_x\sum_{\mu\neq\nu}{\rm Tr}
~U_{\mu}(x)U_{\nu}(x+\hat{\mu})U^\dagger_{\mu}(x+\hat{\nu})U^\dagger_{\nu}(x),$$ where $U_{\mu}(x)\ (\mu=1,..., D)\in SU(N)$ are link variables and $\beta$ is the inverse of the bare ’t Hooft coupling. In the large-$N$ limit the space-time degrees of freedom can be neglected, and then this theory can be equivalent to a model defined on a single hyper-cube, $$S_{EK}=-\beta N\sum_{\mu\neq\nu}{\rm Tr}
~U_{\mu}U_{\nu}U^\dagger_{\mu}U^\dagger_{\nu},$$ which is called the Eguchi-Kawai model (EK model). The equality was shown by observing that the Schwinger-Dyson equations for Wilson loops (loop equations) in both theories are the same. In the EK model the loop equations can naively have open Wilson line terms, which do not exist in the original gauge theory side due to the gauge invariance. Therefore we need to assume that the global $\mathbb{Z}_N^D$ symmetry $$U_\mu\to e^{i\theta_\mu}U_\mu,$$ which eliminates the non-zero expectation value of the open Wilson lines, is not broken spontaneously. However, soon after the discovery of the equivalence, it was found that the $\mathbb{Z}_N^D$ symmetry is actually broken for $D>2$ in the weak coupling region [@Bhanot:1982sh]. Although the naive EK equivalence does not hold, modifications were proposed for this issue. They are quenched Eguchi-Kawai model (QEK model) [@Bhanot:1982sh; @Parisi:1982gp; @Gross:1982at] and twisted Eguchi-Kawal model (TEK model) [@GonzalezArroyo:1982ub]. Historically, most of the works previously done were based on the TEK model because this model is theoretically interesting and numerically more practical (and this model describes the NCYM as mentioned before).
In the TEK model, twisted boundary conditions are imposed and then the $\mathbb{Z}_N^D$ symmetry is ensured in the weak coupling limit. It is not obvious whether the symmetry is broken or not in the intermediate coupling region. There is no guarantee for not violating the symmetry. Numerical simulations in the 1980s, however, suggested that the $\mathbb{Z}_N^D$ symmetry is not broken throughout the whole coupling region. Then we have believed that the TEK model actually describes the large-$N$ limit of the gauge theory.
Recently some indication about the $\mathbb{Z}_N^D$ symmetry breaking was surprisingly reported in several context around the TEK model [@IO03; @Teper:2006sp][@Guralnik:2002ru; @Bietenholz:2006cz]. The most relevant discussion for the present article was done by Teper and Vairinhos in [@Teper:2006sp][^1] . They showed that the $\mathbb{Z}_N^D$ symmetry is really broken in the intermediate coupling region by the Monte-Carlo simulation for the $D=4$ TEK model with the standard twist. Our work in this article is along this line and we mainly concentrate on investigating locations of the symmetry breaking from the weak coupling side in $(\beta, N)$ plane. By the Monte-Carlo simulation we clarify the linear behavior of critical lattice coupling $$\beta_c^L\sim L^2,
\label{EQ:scaling_of_beta_c^L}$$ where $\beta_c^L$ represents critical lattice coupling from the weak coupling side and $L$ is the lattice size we have considered. This result means that [*the continuum limit of the planar gauge theory cannot be described by the TEK model*]{} from the argument of the scaling behavior around the weak coupling limit. This discussion can be also applied to the NCYM case.
This article is organized as follows. In the next section we review the TEK model briefly and fix our setup. In section \[SEC:breaking\] we show the numerical results for the $\mathbb{Z}_N^D$ symmetry breaking of the TEK model and find the scaling behavior (\[EQ:scaling\_of\_beta\_c\^L\]). In section \[SEC:discussions\] we give the validation for the numerical result, and also discuss whether the TEK model has a continuum limit or not.
Twisted Eguchi-Kawai model
==========================
Action and Wilson loop
----------------------
In this study, we treat the $D=4$ case. The TEK model [@GonzalezArroyo:1982ub] is a matrix model defined by the partition function $$Z_{TEK}=\int\prod_{\mu=1}^4 dU_{\mu}\exp(-S_{TEK}),$$ with the action $$S_{TEK}=-\beta N\sum_{\mu\neq\nu}Z_{\mu\nu}{\rm Tr}
~U_\mu U_\nu U_\mu^\dagger U_\nu^\dagger,
\label{EQ:TEK_action}$$ where $U_{\mu}$ and $dU_\mu\ (\mu=1,2,3,4)$ are link variables and Haar measure. The phase factors $Z_{\mu\nu}$ are $$Z_{\mu\nu}=\exp\left(2\pi i n_{\mu\nu}/N\right),\qquad
n_{\mu\nu}=-n_{\nu\mu}\in \mathbb{Z}_N.$$ The Wilson loop operator also contains the phase $Z({\cal C})$ as $$W_{TEK}({\cal C})\equiv Z({\cal C})\langle\hat{W}({\cal C})\rangle,$$ where $\hat{W}({\cal C})$ is the trace of the product of link variables along a contour $\cal C$ and $Z({\cal C})$ is the product of $Z_{\mu\nu}$’s which correspond to the plaquettes in a surface whose boundary is $\cal C$. This model is obtained by dimensional reduction of the Wilson’s lattice gauge theory with the twisted boundary condition. With these definitions, the loop equations in the TEK model take the same form as those in the ordinary lattice gauge theory if the $\mathbb{Z}_N^4$ symmetry, which we discuss in section \[SEC:U(1)symmetry\], is not broken.
Twist prescriptions and classical solutions {#SEC:twist prescription}
-------------------------------------------
In the weak coupling limit, the path-integral is dominated by the configuration which gives the minimum to the action. This configuration $U^{(0)}_\mu=\Gamma_{\mu}$ satisfies the ’t Hooft algebra $$\Gamma_{\mu}\Gamma_{\nu}=Z_{\nu\mu}\Gamma_{\nu}\Gamma_{\mu},$$ and is called “twist-eater”. The most popular twist might be the minimal symmetric twist (standard twist) $$\begin{aligned}
n_{\mu\nu}&=&\left(\begin{array}{cccc}
0 & L & L & L\\
-L & 0 & L & L\\
-L & -L & 0 & L\\
-L & -L & -L & 0
\end{array}\right),\qquad N=L^2.
\label{EQ:minimal_sym_form}\end{aligned}$$ This twist represents $L^4$ lattice. In order to construct the classical solution for this twist, it is convenient to use the $SL(4, \mathbb{Z})$ transformation for the coordinates on $\mathbb{T}^4$ [@vanBaal:1985na]. Using the $SL(4, \mathbb{Z})$ transformation we can always rewrite the $n_{\mu\nu}$ in the skew-diagonal form $$\begin{aligned}
n_{\mu\nu}&\longrightarrow&n_{\mu\nu}'=V^T n_{\mu\nu}V=
\left(\begin{array}{cc|cc}
0 & L & 0 & 0\\
-L & 0 & 0 & 0\\
\hline
0 & 0 & 0 & L\\
0 & 0 & -L & 0
\end{array}\right),
\label{EQ:minimal_skew_form}\end{aligned}$$ where $V$ is a $SL(4, \mathbb{Z})$ transformation matrix. This form makes the construction of the twist-eater easy. Here we define $L\times L$ “shift” matrix $\hat{S}_L$ and “clock” matrix $\hat{C}_L$ by $$\begin{aligned}
\hat{S}_L=\left(
\begin{array}{ccccc}
0 & ~1 & 0 & ~\cdots & ~0 \\
\vdots & ~0 & 1 & ~\ddots & \vdots \\
\vdots & & \ddots & ~\ddots & ~0 \\
0 & & & ~\ddots & ~1 \\
1 & ~0 & \cdots & ~\cdots & ~0
\end{array}
\right), \qquad
\hat{C}_L=\left(
\begin{array}{ccccc}
1 & & & & \multirow{2}{0mm}[-1mm]{\it\huge O} \\
& e^{2\pi i/L} & & & \\
& & e^{2\pi i\cdot 2/L} & & \\
& & & \ddots & \\
\multirow{2}{0mm}[4mm]{\it\huge O} & & & & e^{2\pi i(L-1)/L}
\end{array}
\right),\end{aligned}$$ which satisfy the little ’t Hooft algebra $$\hat{C}_L\hat{S}_L = e^{-2\pi i/L}\hat{S}_L\hat{C}_L.$$ Using these matrices, the twist-eater configuration for the skew-diagonal form (\[EQ:minimal\_skew\_form\]) is easily constructed as $$\begin{aligned}
\begin{array}{ll}
\Gamma_1=\hat{C}_L \otimes \mathbbm{1}_L, \quad &
\Gamma_2=\hat{S}_L \otimes \mathbbm{1}_L, \\
\Gamma_3=\mathbbm{1}_L \otimes \hat{C}_L, &
\Gamma_4=\mathbbm{1}_L \otimes \hat{S}_L.
\end{array}\end{aligned}$$ From (\[EQ:minimal\_skew\_form\]) we can also construct the twist-eater configuration for the minimal symmetric twist (\[EQ:minimal\_sym\_form\]) as $$\begin{aligned}
\begin{array}{ll}
\Gamma_1=\quad\;\hat{C}_L \otimes \mathbbm{1}_L, \quad &
\Gamma_2=\hat{S}_L\hat{C}_L \otimes \hat{C}_L, \\
\Gamma_3=\hat{S}_L\hat{C}_L \otimes \hat{S}_L, &
\Gamma_4=\quad\;\hat{S}_L \otimes \mathbbm{1}_L.
\end{array}\end{aligned}$$ Although these forms are different only by the coordinate transformation, they can give different results except the weak coupling limit as seen in next section.
Another kind of the twist we consider in this article is $$\begin{aligned}
n_{\mu\nu}=
\left(\begin{array}{cc|cc}
0 & mL & 0 & 0 \\
-mL & 0 & 0 & 0 \\
\hline
0 & 0 & 0 & mL\\
0 & 0 & -mL & 0
\end{array}
\right),\qquad N=mL^2
\label{EQ:generic_twist}\end{aligned}$$ with classical solution $$\begin{aligned}
\begin{array}{ll}
\Gamma_1=\hat{C}_L \otimes \mathbbm{1}_L \otimes \mathbbm{1}_m, \quad &
\Gamma_2=\hat{S}_L \otimes \mathbbm{1}_L \otimes \mathbbm{1}_m, \\
\Gamma_3=\mathbbm{1}_L \otimes \hat{C}_L \otimes \mathbbm{1}_m, &
\Gamma_4=\mathbbm{1}_L \otimes \hat{S}_L \otimes \mathbbm{1}_m.
\end{array}
\label{EQ:generic_twist-eater}\end{aligned}$$ While we write the twist using the skew-diagonal form here, we can always rewrite it in the symmetric form by the $SL(4, \mathbb{Z})$ transformation. We call this twist “generic twist” in this article, and the minimal twists (\[EQ:minimal\_sym\_form\]) and (\[EQ:minimal\_skew\_form\]) are particular cases ($m=1$) of the generic twist. As is well known, the TEK model can describe the NCYM theory [@Aoki:1999vr; @Ambjorn:1999ts]. Expanding the matrix model around noncommutative tori background, we can obtain noncommutative $U(m)$ Yang-Mills theory on fuzzy tori. (Note that this interpretation is possible even at finite-$N$.) Because fuzzy torus can be used as a regularization of fuzzy ${\mathbb R}^4$, it is naively possible to give a nonperturbative formulation of the NCYM on fuzzy ${\mathbb R}^4$ by taking a suitable large-$N$ limit in the TEK model. (See appendix \[sec:NCYM\] for details.) However, we will see later it is not the case because of the $\mathbb{Z}_N^4$ symmetry breaking. In the NCYM interpretation the shift and clock matrices can be regarded as matrix realization of a fuzzy torus. From this point of view, twist prescription (\[EQ:generic\_twist\]) provides YM theories on $m$-coincident four-dimensional fuzzy tori.
$\mathbb{Z}_N^4$ symmetry {#SEC:U(1)symmetry}
-------------------------
The $\mathbb{Z}_N^4$ symmetry plays a crucial role in the Eguchi-Kawai equivalence. Generally, the YM theory with a periodic boundary condition has a critical size. If we shrink the volume of the system beyond the critical size, we encounter the center symmetry breaking, which is just the same as the finite temperature system. In the EK model, which is a single hyper-cubic model, the critical size corresponds to $\beta_c\sim 0.19$ in the lattice coupling. In the region less than the $\beta_c$ – the strong coupling region – the center symmetry ${\mathbb Z}_N^4$ is maintained. On the other hand, in the region larger than $\beta_c$ – the weak coupling region – the symmetry is spontaneously broken, and then the EK equivalence does not hold.
The TEK model avoids this problem by imposing the twisted boundary condition on the system instead of the periodic one. In the weak coupling limit the path integral is dominated by the vacuum configuration, which is twist-eater configurations, as we already mentioned. These configurations are invariant under global ${\mathbb Z}_L^4$ transformation $$U_\mu\to e^{i\theta_\mu}U_\mu, \qquad e^{i\theta_\mu}\in {\mathbb Z}_L,$$ which is regarded as the $U(1)^4$ symmetry in the large-$N$ limit. As a result, $W_{TEK}({\cal C})$ is zero if $\cal C$ is an open contour in the weak coupling limit.
A key point is that the solution for this problem is obvious only at the classical level. That is to say, there is no guarantee to maintain the ${\mathbb Z}_L^4$ symmetry if we take into account the quantum fluctuation. Going away from the weak coupling limit, the configurations fluctuate around the twist-eater. The situation can be displayed in the eigenvalue distribution of the link variables. In the weak coupling limit the $N$ eigenvalues distribute regularly and uniformly on the unit circle in the complex plane, and then they are ${\mathbb Z}_L$ symmetric. If we decrease $\beta$, the eigenvalues begin to fluctuate around the location of the twist-eater. If the fluctuation is not too large, the ${\mathbb Z}_L$ symmetric distribution is maintained. However, large fluctuation can make the uniform distribution shrink to a point, which corresponds to $U_\mu=\mathbbm{1}_N$ configuration. In the strong coupling region the distribution is randomly uniform, and then the symmetry is restored.
Although there is no guarantee to maintain the ${\mathbb Z}_N^4$ symmetry in the intermediate coupling region, the 1980s numerical simulations suggested that the symmetry was unbroken. And this caused us to believe that the EK equivalence in the TEK model does hold throughout the whole coupling region.
Limiting procedure {#SEC:limiting_procedure}
------------------
As is well known, the scaling of the YM lattice theory behaves as $\beta\sim\log a^{-1}$ around the weak coupling limit, where $a$ is the lattice spacing, and which is obtained by one-loop perturbative calculation of the renormalization group equation. If we wish to construct the TEK model which corresponds to the YM theory by the EK equivalence, the scaling of the TEK model should obey that of the YM theory. In the TEK model, the lattice size $L$ relates to $N$. (For the twist we consider in this article, the relation is $N=mL^2$.) Then, the YM system with fixed physical size $l=aL$ can be obtained by the scaling $$\beta\sim\log a^{-1}\sim\log N.
\label{EQ:one-loop_scaling}$$ In order to obtain the large-$N$ limit with infinite volume, we should increase $\beta$ slower than the scaling (\[EQ:one-loop\_scaling\]). If it is not the case, the system shrinks to a point.
In the case of the NCYM, the scaling near the weak coupling limit is essentially same as the YM theory, that is, $\beta\sim\log a^{-1}$. (See appendix \[sec:NCYM\].) But if we wish to make the TEK model corresponding to the NCYM, there is a constraint $a^2L=al=fixed$, which means that we take a scheme in which the noncommutative parameter $\theta$ is fixed. Then, both the continuum limit and the infinite volume limit are simultaneously taken (double scaling limit). Regardless of difference of the constraint, the scaling for the NCYM we should take is the same as that of the ordinary YM (\[EQ:one-loop\_scaling\]) by the nature of the logarithm scaling.
$\mathbb{Z}_N^4$ symmetry breaking in the TEK model {#SEC:breaking}
===================================================
As mentioned in the previous section, the $\mathbb{Z}_N^4$ symmetry breaking had not been observed in the older numerical simulation. However, there are several recent reports which indicate the symmetry breaking [@IO03; @Teper:2006sp; @Bietenholz:2006cz]. In [@Teper:2006sp], the symmetry breaking in the $D=4$ $SU(N)$ TEK model was studied in the case of the standard twist up to $N=144=12^2$. The authors of [@Teper:2006sp] performed the Monte-Carlo simulation starting both from a randomized configuration (“hot start”) and from the twist-eater solution (“cold start”). In both cases the $\mathbb{Z}_N^4$ symmetry begins to break at $N\ge 100=10^2$. At $N=144$ the symmetry breaking and restoration patterns they observed are $$\begin{aligned}
\begin{array}{cccccccccccl}
\mathbb{Z}_N^4 & \xrightarrow{\beta_c^H} & \mathbb{Z}_N^3 & \longrightarrow &
\mathbb{Z}_N^2 & \longrightarrow & \mathbb{Z}_N^1 &
\multicolumn{3}{c}{\longrightarrow} & \mathbb{Z}_N^0 \;\; &
\mbox{($N=144$, standard, hot start)},\\
\mathbb{Z}_N^4 & \longleftarrow & \mathbb{Z}_N^3 & \longleftarrow &
\mathbb{Z}_N^2 & \longleftarrow & \mathbb{Z}_N^1 & \longleftarrow &
\mathbb{Z}_N^0 & \xleftarrow{\beta_c^L} & \mathbb{Z}_N^4 \;\; &
\mbox{($N=144$, standard, cold start)},
\end{array}
\label{EQ:breaking_pattern_sym}\end{aligned}$$ where $\beta_c^H$ and $\beta_c^L$ are the first breaking point for the hot start and that for cold start, respectively. Note that although there is recovery of the symmetry for the cold start, the symmetry remains broken for the hot start.
In this section we show the results of the numerical simulation for this symmetry breaking phenomena. In order to argue about the possibility of the continuum and large-$N$ limiting procedure for this model, we mainly focus on the first breaking point for the cold start $\beta_c^L$, which depends on $N$.[^2]
Simulation method
-----------------
In our simulation we use the pseudo-heatbath algorithm. The algorithm is based on [@Fabricius:1984wp], and in each sweep over-relaxation is performed five times after multiplying $SU(2)$ matrices. The number of sweeps is $O(1000)$ for each $\beta$. We scan the symmetry breaking on the resolution of $\Delta\beta=0.005$, and then we always quote the value $\pm 0.0025$ as the error due to the resolution. Note that the breaking points are ambiguous because the breakdown of the $\mathbb{Z}_N^4$ symmetry is a first-order transition. As an order parameter for detecting the $\mathbb{Z}_N^4$ breakdown, we measure the Polyakov lines $$\begin{aligned}
P_\mu\equiv\left\langle\left|\frac{1}{N}{\rm Tr}~U_{\mu}\right|\right\rangle.\end{aligned}$$
Simulation results
------------------
### Minimal symmetric twist {#minimal-symmetric-twist .unnumbered}
First of all we treat the minimal symmetric twist (\[EQ:minimal\_sym\_form\]). This twist is the most standard and is also used in the paper [@Teper:2006sp]. In our study we only investigate the first $\mathbb{Z}_N^4$ symmetry breaking point from weak coupling limit, that is, $\beta_c^L$ for this twist. (For more detailed information about the symmetry breaking phenomena, see [@Teper:2006sp].) The obtained results are in table \[TAB:minimal symmetric twist\] and plotted in figure \[FIG:minimal symmetric twist\]. The symmetry breaking points and patterns ($\mathbb{Z}_N^4\xrightarrow{\beta_c^L}\mathbb{Z}_N^3$ for $N=100$; $\mathbb{Z}_N^4\xrightarrow{\beta_c^L}\mathbb{Z}_N^0$ for $N>100$) are consistent with the results in [@Teper:2006sp] up to $N=144$. In this work we explore the simulation for larger $N$. From figure \[FIG:minimal symmetric twist\] we can find clear linear dependence of $\beta_c^L$ on $N(=L^2)$ for $N\gtrsim169$. The fitted result in linear function using $N\geq169$ data is $$\beta_c^L\sim 0.0011N+0.21.
\label{EQ:bc_L_minimal_sym}$$ A theoretical argument for this linear behavior is discussed in section \[SEC:discussions\].
$N$ $L$ $\beta_c^L$ $N$ $L$ $\beta_c^L$
------- ------ -------------------- ------- ------ --------------------
$100$ $10$ $0.3525\pm 0.0025$ $225$ $15$ $0.4575\pm 0.0025$
$121$ $11$ $0.3625\pm 0.0025$ $256$ $16$ $0.4875\pm 0.0025$
$144$ $12$ $0.3775\pm 0.0025$ $289$ $17$ $0.5275\pm 0.0025$
$169$ $13$ $0.3975\pm 0.0025$ $324$ $18$ $0.5675\pm 0.0025$
$196$ $14$ $0.4225\pm 0.0025$
: Critical lattice coupling from the weak coupling side $\beta_c^L$ for the minimal symmetric twist.[]{data-label="TAB:minimal symmetric twist"}
### Minimal skew-diagonal twist {#minimal-skew-diagonal-twist .unnumbered}
Twists can be always transformed into the skew-diagonal form by $SL(4,\mathbb{Z})$ transformation as we mentioned in section \[SEC:twist prescription\]. As it were, the minimal symmetric twist (\[EQ:minimal\_sym\_form\]) is equivalent to the minimal skew-diagonal twist (\[EQ:minimal\_skew\_form\]) in the weak coupling limit. However, both forms can represent different features by taking into account the quantum fluctuation. Actually, the $\mathbb{Z}_N^4$ symmetry is already broken at $N=25$. This fact enables us to observe the $N$-dependence of the critical points easily. Not only is the symmetry breaking point different from the symmetric form, so is the breaking and restoration pattern. Figure \[FIF:pattern\_N100\_cold-start\] shows the expectation value of the plaquette (top) and the Polyakov lines (besides the top) versus $\beta$ for the cold start at $N=100$. For $N\geq100$ we find the $\mathbb{Z}_N^4$ symmetry breaking and restoration pattern: $$\mathbb{Z}_N^4\leftarrow\mathbb{Z}_N^3\leftarrow\mathbb{Z}_N^2\leftarrow
\mathbb{Z}_N^0\xleftarrow{\beta_c^L}\mathbb{Z}_N^4\quad
\mbox{($N=100$, minimal skew-diagonal, cold start)},$$ which represents a difference from the symmetric form case (\[EQ:breaking\_pattern\_sym\]). The first breaking pattern $\mathbb{Z}_N^4\xrightarrow{\beta_c^L}\mathbb{Z}_N^0$ is, however, the same as that in the symmetric twist. (We note that for $N\le 81$ the first breaking pattern is $\mathbb{Z}_N^4\xrightarrow{\beta_c^L}\mathbb{Z}_N^2$, which resembles the pattern $\mathbb{Z}_N^4\xrightarrow{\beta_c^L}\mathbb{Z}_N^3$ at $N=100$ for the symmetric form [@Teper:2006sp].)
Table \[TAB:minimal skew-diagonal\] shows the first breaking points for the cold start $\beta_c^L$ and for the hot start $\beta_c^H$. These data are plotted in figure \[FIG:m=1B\_low\] for $\beta_c^L$ and figure \[FIG:m=1B\_high\] for $\beta_c^H$. Again, we find clear linear dependence on $N$ for $\beta_c^L$, as we found for the symmetric form. Additionally, we also find clear dependence on $1/N$ for $\beta_c^H$. The fitted results are $$\begin{aligned}
\beta_c^L &\sim& 0.0034N+0.25,\label{EQ:bc_L_minimal_skew}\\
\beta_c^H &\sim& \frac{2.9}{N}+0.18,\label{EQ:bc_H_minimal_skew}\end{aligned}$$ where we used only $N\geq64$ data for $\beta_c^L$, whereas all data are used for $\beta_c^H$. As $N$ is increased the $\beta_c^H$ approaches a point $0.190$, where the phase transition $\mathbb{Z}_N^4\xrightarrow{\beta_c^H}\mathbb{Z}_N^3$ takes place in the original EK model. These results suggest that the quantum fluctuation is so large that the $\mathbb{Z}_N^4$ symmetry is broken in exactly the same region as that in the original EK model. The lines for transitions $\beta_c^L$ and $\beta_c^H$ seem to intersect around the bulk transition point $\beta_c^B\sim0.35$, which corresponds to $N\sim 20$ for the twist considered here. For smaller values than $N\sim 20$, we did not observe a signal of breakdown of the $\mathbb{Z}_N^4$ symmetry.
$N$ $L$ $\beta_c^H$ $\beta_c^L$ $N$ $L$ $\beta_c^H$ $\beta_c^L$
------ ----- -------------------- -------------------- ------- ------ -------------------- --------------------
$9$ $3$ - - $64$ $8$ $0.2225\pm 0.0025$ $0.4625\pm 0.0025$
$16$ $4$ - - $81$ $9$ $0.2125\pm 0.0025$ $0.5175\pm 0.0025$
$25$ $5$ $0.2925\pm 0.0025$ $0.3625\pm 0.0025$ $100$ $10$ $0.2075\pm 0.0025$ $0.5875\pm 0.0025$
$36$ $6$ $0.2575\pm 0.0025$ $0.3925\pm 0.0025$ $121$ $11$ $0.2025\pm 0.0025$ $0.6525\pm 0.0025$
$49$ $7$ $0.2375\pm 0.0025$ $0.4225\pm 0.0025$ $144$ $12$ $0.1975\pm 0.0025$ $0.7325\pm 0.0025$
: Critical lattice coupling from the weak coupling side $\beta_c^L$ and from strong coupling side $\beta_c^H$ for the minimal skew-diagonal twist $(m=1)$.[]{data-label="TAB:minimal skew-diagonal"}
------------------- --
\[FIG:m=1B\_low\]
------------------- --
### Generic skew-diagonal twist {#generic-skew-diagonal-twist .unnumbered}
Here, we show the numerical result of the generic twist (\[EQ:generic\_twist\]). For this twist we use the skew-diagonal form because the $\mathbb{Z}_N^4$ symmetry breaking occurs at smaller $N$ than that in the symmetric form, which makes our investigation much easier.
We measure $\beta_c^L$ for this twist up to $m=4$. Table \[TAB:m=234\] shows the $\beta_c^L$ for $m=2, 3, 4$ and that for $m=1$ is presented in table \[TAB:minimal skew-diagonal\]. These data are plotted in figure \[FIG:bc\_L\_mall\_skew1\]. From this figure we can find that the $\beta_c^L$ for each $L$ are reduced as we increase $m$, and the dependence is linear in $1/m$. The data at $1/m=0$ in this plot are linearly extrapolated values. An interesting point is the behavior for the case $L=5$. While the $\mathbb{Z}_N^4$ symmetry breaking is observed for $m=1, 2$ and $3$, it is not seen for $m=4$ because the $\beta_c^L$ reaches the bulk transition point $\beta_c^B\sim0.35$ by increasing $m$. Figure \[FIG:bc\_L\_mall\_skew2\] represents the same data in figure \[FIG:bc\_L\_mall\_skew1\], but the horizon axis is $L^2$. As we have seen in the $m=1$ case, the data for $L\geq8$ are well fitted by the linear function of $L^2$ for each $m$. From these figures, we find that the data for $L\geq8$ are well fitted globally by a function: $$\beta_c^L\sim0.0034L^2+\frac{0.060}{m}+0.19.
\label{EQ:bc_L_generic_skew}$$
------ ------- ------------------- ------- ------------------- ------- -------------------
$L$ $N$ $\beta_c^L$ $N$ $\beta_c^L$ $N$ $\beta_c^L$
$5$ $50$ $0.3525\pm0.0025$ $75$ $0.3475\pm0.0025$ - -
$6$ $72$ $0.3675\pm0.0025$ $108$ $0.3575\pm0.0025$ $144$ $0.3525\pm0.0025$
$7$ $98$ $0.3925\pm0.0025$ $147$ $0.3875\pm0.0025$ $196$ $0.3825\pm0.0025$
$8$ $128$ $0.4375\pm0.0025$ $192$ $0.4275\pm0.0025$ $256$ $0.4225\pm0.0025$
$9$ $162$ $0.4925\pm0.0025$ $243$ $0.4825\pm0.0025$ $324$ $0.4775\pm0.0025$
$10$ $200$ $0.5575\pm0.0025$ $300$ $0.5475\pm0.0025$ $400$ $0.5425\pm0.0025$
------ ------- ------------------- ------- ------------------- ------- -------------------
: $\beta_c^L$ for the generic skew-diagonal twist ($m=2, 3, 4$). See also Tab. \[TAB:minimal skew-diagonal\] for $m=1$.[]{data-label="TAB:m=234"}
[lr]{}
\[FIG:bc\_L\_mall\_skew1\]
&
Discussions {#SEC:discussions}
===========
In this section we discuss the numerical results obtained in the previous section and the validity of taking the large-$N$ and continuum limit for this model.
Theoretical estimation of the $\mathbb{Z}_N^4$ symmetry breaking point {#SEC:theoretical_estimation}
----------------------------------------------------------------------
In the previous section we showed our numerical results. In particular, we elaborately investigated $\beta_c^L$, the first $\mathbb{Z}_N^4$ breaking point from the cold start. From our investigation, we found the clear linear behavior like (\[EQ:bc\_L\_minimal\_sym\]), (\[EQ:bc\_L\_minimal\_skew\]) and (\[EQ:bc\_L\_generic\_skew\]). These behaviors can be obtained through the following consideration.
### Energy difference between twist-eater $\Gamma_{\mu}$ and identity $\mathbbm{1}_N$ configurations {#energy-difference-between-twist-eater-gamma_mu-and-identity-mathbbm1_n-configurations .unnumbered}
We simply assume that the $\mathbb{Z}_N^4$ breaking is a transition from twist-eater phase $U_{\mu}=\Gamma_{\mu}$ to identity configuration phase $U_{\mu}=\mathbbm{1}_N$. For plainness, we consider $\mathbb{Z}_N^4\xrightarrow{\beta_c^L}\mathbb{Z}_N^0$ type breaking here. Of course we can treat $\mathbb{Z}_N^4\xrightarrow{\beta_c^L}\mathbb{Z}_N^3\xrightarrow{\beta_c^L}
\mathbb{Z}_N^2\xrightarrow{\beta_c^L}\mathbb{Z}_N^1\xrightarrow{\beta_c^L}
\mathbb{Z}_N^0$ (cascade) type breaking at a $\beta_c^L$, but the obtained behavior is not different from the former type. Firstly, we focus on the classical energy difference between these configurations. The energy can be easily calculated from the action (\[EQ:TEK\_action\]) as $$\begin{aligned}
\Delta S
&=&S_{TEK}[U_{\mu}=\mathbbm{1}_N]-S_{TEK}[U_{\mu}=\Gamma_{\mu}]\nonumber\\
&=&\beta N^2\sum_{\mu\neq\nu}
\left\{ 1-\cos\left(\frac{2\pi n_{\mu\nu}}{N}\right)\right\}
\simeq 2\pi^2\beta\sum_{\mu\neq\nu}n_{\mu\nu}^2.
\label{EQ:potential_difference}\end{aligned}$$ For the generic twist, it becomes $$\Delta S=
\begin{cases}
24\pi^2\beta m^2L^2\qquad & \mbox{(symmetric form)},\\
8\pi^2\beta m^2L^2\qquad & \mbox{(skew-diagonal form)}.
\label{EQ:potential_difference_generic}
\end{cases}$$ Note that the symmetric form is roughly three times more stable than the skew-diagonal form if both twists have equal quantum fluctuations. This is the reason that the $\mathbb{Z}_N^4$ symmetry breaking for the skew-diagonal form can occur at quite smaller $N$ than that for the symmetric form, as is observed in our simulation.
### Quantum fluctuations and symmetry breaking {#SSEC:fluctuation .unnumbered}
Going away from the weak coupling limit, the system has quantum fluctuations. Here we naively expect that the $\mathbb{Z}_N^4$ symmetry is broken if the fluctuation around twist-eater configuration exceeds the energy difference $\Delta S$. Because the system describes $O(N^2)$ interacting gluons, it is natural to assume that their quantum fluctuations provide $O(N^2)$ value to the effective action. For the generic twist the quantum fluctuation is $O(m^2L^4)$. Combined with the fact (\[EQ:potential\_difference\_generic\]), we can estimate the critical point $\beta_c^L$ as $$\beta_c^L\sim L^2,\label{EQ:linear_behavior}$$ which is consistent with the numerical results (\[EQ:bc\_L\_minimal\_sym\]), (\[EQ:bc\_L\_minimal\_skew\]) and (\[EQ:bc\_L\_generic\_skew\]). In addition we can explain the difference of the coefficient of $L^2$ in (\[EQ:bc\_L\_minimal\_sym\]), (\[EQ:bc\_L\_minimal\_skew\]) and (\[EQ:bc\_L\_generic\_skew\]) between the symmetric and the skew-diagonal form, which is roughly three times different, by the factor in (\[EQ:potential\_difference\_generic\]).
Although the above crude estimation reproduces the linear $L^2$ behavior of $\beta_c^L$, we cannot explain the dependence on $m$. To catch the behavior completely, we need to make the discussion more sophisticated. However, we do not pursue this issue here because the $m$ dependence can be negligible at the larger $N$.
This argument can be applied for other twist prescriptions like taking the twist phase as $\exp(i\pi(L+1)/L)$, which is usually used for describing noncommutative spaces. (See appendix \[sec:NCYM\].)
Continuum and large-$N$ limit
-----------------------------
We have shown that the linear $L^2$ dependence of the critical point $\beta_c^L$ could be explained by the theoretical discussion in this section. While our simulation is restricted in the small $N$ region, we confirm that the behavior must continue to $N=\infty$ by combining with the discussion. Then the EK equivalence is valid only in the region $\beta >\beta_c^L\sim N$ even in the weak coupling limit and the large-$N$ limit. As we mentioned in the section \[SEC:limiting\_procedure\], both the ordinary YM with fixed physical volume and the NCYM theory with fixed noncommutative parameter have essentially logarithm scaling (\[EQ:one-loop\_scaling\]) near the weak coupling limit. Then, because $\beta_c^L$ grows faster than the logarithm, the EK equivalence does not hold in the continuum limit.
Conclusions
===========
In order to study the nonperturbative nature of the large-$N$ gauge theory by lattice simulations, the large-$N$ reduction is very useful property for saving the computational effort. In this paper, we studied the phase structure of the TEK model, which has been a major way to realize the large-$N$ reduction. Contrary to the naive hope in old days, at least in ordinary twist prescriptions as investigated in this paper, the $\mathbb{Z}_N^4$ symmetry is broken even in the weak coupling region and hence a continuum limit as the planar gauge theory cannot be described by the TEK model. For the NCYM, the situation is the same. We can also consider a lot of variation for the twist prescription and the combination of reduced and non-reduced dimension. For example, in [@Guralnik:2002ru; @Bietenholz:2006cz], four-dimensional model with two commutative and two noncommutative directions was studied using two-dimensional lattice action. However, the $\mathbb{Z}_N^4$ symmetry is broken also in this model, and hence we cannot take a naive continuum limit.
Another way for the reduction is the QEK model, in which the eigenvalues of the link variables are quenched. The QEK model might have no problem in principle, but its computational cost is larger than that of the TEK model. Although the TEK and QEK model are reduced models to a single hyper-cube, recent studies deviate from them. The contemporary method might be the partial reduction [@Narayanan:2003fc]. This work showed that the large-$N$ reduction is valid above some critical physical size $l_c$. This means that for a lattice size $L$ the reduction holds below some lattice coupling $\beta(L)$. In order to take continuum limits we should avoid the bulk transition point $\beta_c^B$, causing the condition $\beta_c^B<\beta(L)$ to be necessary. That is, there is a lower limitation for the lattice size $L_c$ for the continuum reduction. In addition, the twist prescription is also applicable to the partial reduction [@GonzalezArroyo:2005dz]. Due to the twisted boundary condition, the lower limitation $L_c$ can be reduced. Therefore, combination of the twist prescription and the partial reduction would be quite efficient in the current situation.
Note also that NCYM on fuzzy ${\mathbb R}^4$ could be realized by using TEK with quotient conditions [@Ambjorn:1999ts] which give a periodic condition to eigenvalues and hence quantum fluctuation is suppressed. Further study in this direction would be important. The numerical computations in this work were in part carried out at the Yukawa Institute Computer Facility. The authors would like to thank Sinya Aoki, Hikaru Kawai, Jun Nishimura, Masanori Okawa, Yoshiaki Susaki, Hiroshi Suzuki and Shinichiro Yamato for stimulating discussions and comments. M. H. was supported by Special Postdoctoral Researchers Program at RIKEN. T. H. would like to thank the Japan Society for the Promotion of Science for financial support.
Double scaling limit as the noncommutative Yang-Mills theory {#sec:NCYM}
============================================================
The TEK model can be used to formulate gauge theories on noncommutative spaces nonperturbatively [@Aoki:1999vr; @Ambjorn:1999ts; @Connes:1997cr]. In this appendix, we give a review for the construction of the NCYM from the TEK model [@Aoki:1999vr], a discussion for the scaling and some supplemental comments for our analysis.
By taking $U_\mu=e^{iaA_\mu}$, where $a$ corresponds to the lattice spacing, and expanding the action of the TEK model (\[EQ:TEK\_action\]), we have its continuum version $$S_{TEK, continuum}=-\frac{1}{4g^2}\sum_{\mu\neq\nu}
{\rm Tr}\left([A_\mu,A_\nu]-i\theta_{\mu\nu}\right)^2
\label{EQ:continuum_TEK}$$ up to higher order terms in $a$, where $$\theta_{\mu\nu}=\frac{2\pi n_{\mu\nu}}{Na^2},\quad
\frac{1}{2g^2}=a^4 \beta N.$$ Then, by expanding the action around a classical solution of (\[EQ:continuum\_TEK\]) $$A_\mu^{(0)}=\hat{p}_\mu, \qquad
[\hat{p}_\mu,\hat{p}_\nu]=i\theta_{\mu\nu},$$ we obtain the $U(1)$ NCYM on fuzzy ${\mathbb R}^4$ as follows. Let us define the “noncommutative coordinate” $\hat{x}^\mu=\left(\theta^{-1}\right)^{\mu\nu}\hat{p}_\nu$. Then we have $$[\hat{x}^\mu,\hat{x}^\nu]=-i(\theta^{-1})^{\mu\nu}.$$ This commutation relation is the same as that of coordinate on fuzzy ${\mathbb R}^4$ with noncommutativity parameter $\theta$, and hence functions of $\hat{x}$ can be mapped to functions on fuzzy ${\mathbb R}^4$. More precisely, we have the following mapping rule: $$\begin{aligned}
\begin{array}{ccc}
f(\hat{x})=\sum_k\tilde{f}(k)e^{ik\hat{x}} & \leftrightarrow &
f(x)=\sum_k\tilde{f}(k)e^{ikx},\\
f(\hat{x})g(\hat{x}) & \leftrightarrow &
f(x)\star g(x),\\
i[\hat{p}_{\mu},\ \cdot\ ] & \leftrightarrow &
\partial_{\mu},\\
{\rm Tr} & \leftrightarrow &
\frac{\sqrt{\det\theta}}{4\pi^2}\int d^4x,
\end{array}\end{aligned}$$ where $\star$ represents the noncommutative star product, $$f(x)\star g(x)=
f(x)\exp\left(-\frac{i}{2}\overset{\leftarrow}{\partial}_{\mu}
(\theta^{-1})^{\mu\nu}\overset{\rightarrow}{\partial}_{\nu}\right) g(x),$$ and we obtain $U(1)$ NCYM action $$S_{U(1)NC}=-\frac{1}{4g_{NC}^2}\int d^4x ~F_{\mu\nu}\star F_{\mu\nu}$$ with coupling constant $$g_{NC}^2=4\pi^2g^2/\sqrt{\det\theta}.$$ In the same way, by expanding the action (\[EQ:continuum\_TEK\]) around $A_\mu^{(0)}=\hat{p}_\mu\otimes\mathbbm{1}_m$, $U(m)$ NCYM can be obtained. From (\[EQ:generic\_twist-eater\]), it is apparent that the generic twist gives the $U(m)$ NCYM. Intuitively, the vacuum configuration (\[EQ:generic\_twist-eater\]) describes $m$-coincident fuzzy tori and fuzzy ${\mathbb R}^4$ is realized as a tangent space.
In order to keep the noncommutative scale $\theta$ finite, we should take the double scaling limit with $$\begin{aligned}
a^{-1}\sim\sqrt{L}\sim N^{1/4}. \end{aligned}$$ One-loop beta function for $U(m)$ NCYM is given by [@Minwalla:1999px][^3] $$\begin{aligned}
\beta_{\rm 1-loop}(g_{NC})=
-\frac{2}{(4\pi)^2}\frac{11}{3}mg_{NC}^3+O(g_{NC}^5). \end{aligned}$$ Therefore, the ’t Hooft coupling $\beta$ scales as $$\begin{aligned}
\beta \sim \frac{1}{g_{NC}^2} \sim \log N. \end{aligned}$$ Then, the scaling we should take for the NCYM is just the same as that for the ordinary YM, and $\mathbb{Z}_N^D$ symmetry is broken in the scaling limit. Therefore, fuzzy torus crunches to a point and hence the fuzzy ${\mathbb R}^4$ cannot be realized[^4].
Of course, we can also use other twist prescriptions. In order to make the periodicity of the discretized fuzzy torus correct, we usually take the twist as $\exp(i\pi(L+1)/L)$ [@Ambjorn:1999ts]. Regardless of the difference of the twist, the conclusion might not be altered. Here we repeat the discussion in section \[SEC:theoretical\_estimation\]. In this case, the $\mathbb{Z}_N^4$ is likely to break down to ${\mathbb Z}_2^4$. The difference between potentials in twist-eater and ${\mathbb Z}_2^4$-preserving configurations is $$\Delta S\sim\beta N^2\left\{
1-\cos\left(\frac{\pi}{L}\right)\right\}\sim\beta m^2 L^2,$$ which is the same order as (\[EQ:potential\_difference\]). Then the behavior of the critical point $\beta_c^L$ (\[EQ:linear\_behavior\]) is not changed.
[^1]: In [@Guralnik:2002ru; @Bietenholz:2006cz] a similar model with two commutative and two noncommutative dimensions were studied in the context of NCYM. In this case the instability of ${\mathbb Z}_N$ preserving vacuum was observed even in a perturbative calculation. This instability arises due to UV/IR mixing.
[^2]: Strictly speaking, the symmetry preserved in the weak coupling region is not $\mathbb{Z}_N^4$ but $\mathbb{Z}_L^4$. However, $\mathbb{Z}_L^4$ is sufficient for the Eguchi-Kawai equivalence so we do not dare to distinguish them in this article.
[^3]: Renormalizability of the NCYM is a delicate problem. For example, see [@Bichl:2001cq], in which the renormalizability is discussed by a perturbation expansion.
[^4]: Although the fuzzy ${\mathbb R}^4$ cannot be realized, another kind of noncommutative space with finite physical volume may exist. In the case of a four-dimensional model with two commutative and two noncommutative directions, such a limit was found numerically [@Bietenholz:2006cz].
| 0 |
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abstract: 'Spin off events and impacts can eject boulders from an asteroid surface and rubble pile asteroids can accumulate from debris following a collision between large asteroids. These processes produce a population of gravitational bound objects in orbit that can impact an asteroid surface at low velocity and with a distribution of impact angles. We present laboratory experiments of low velocity spherical projectiles into a fine granular medium, sand. We delineate velocity and impact angles giving ricochets, those giving projectiles that roll-out from the impact crater and those that stop within their impact crater. With high speed camera images and fluorescent markers on the projectiles we track spin and projectile trajectories during impact. We find that the projectile only reaches a rolling without slipping condition well after the marble has reached peak penetration depth. The required friction coefficient during the penetration phase of impact is 4-5 times lower than that of the sand suggesting that the sand is fluidized near the projectile surface during penetration. We find that the critical grazing impact critical angle dividing ricochets from roll-outs, increases with increasing impact velocity. The critical angles for ricochet and for roll-out as a function of velocity can be matched by an empirical model during the rebound phase that balances a lift force against gravity. We estimate constraints on projectile radius, velocity and impact angle that would allow projectiles on asteroids to ricochet or roll away from impact, finally coming to rest distant from their initial impact sites.'
address:
- 'Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA'
- 'Dept. of Computer Science, University of Rochester, Rochester, NY, 14627, USA'
- 'Colorado Center for Astrodynamics Research, The University of Colorado Boulder, UCB 431, Boulder, CO 80309-0431, United States'
- 'Department of Mechanical Engineering, University of Rochester, Rochester, NY 14627, USA'
- 'Department of Earth and Environmental Sciences, University of Rochester, Rochester, NY 14627, USA'
- 'Lunar and Planetary Lab, University of Arizona, Tucson, AZ, USA'
- 'Laboratoire Lagrange, Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, C.S. 34229, 06304 Nice Cedex 4, France'
author:
- Esteban Wright
- 'Alice C. Quillen'
- Juliana South
- 'Randal C. Nelson'
- Paul Sánchez
- John Siu
- Hesam Askari
- Miki Nakajima
- 'Stephen R. Schwartz'
bibliography:
- 'ricochet\_v03.bib'
title: Ricochets on Asteroids
---
Introduction
============
Impact crater ejecta curtains, seismicity associated with impacts (e.g. @wright19), and mass loss associated with spin-off events (e.g., @holsapple10 [@hirabayashi15]) are processes that would eject a population of rocks and boulders from an asteroid surface. Eventually, this material would be ejected from the asteroid’s vicinity or it would return to hit the asteroid surface. A rubble pile asteroid can be formed following a disruptive collision of two large bodies, including a phase of re-accumulation from previously disrupted but gravitationally bound material [@michel13; @walsh18; @walsh19]. Late stages of re-accumulation involves low velocity impacts onto the asteroid surface. Impacts with objects in solar orbit have a mean relative velocity of $\sim 5$ km/s [@bottke94]. In comparison, a gravitational bound object would impact the asteroid surface at much lower velocity, less than the escape velocity which is $\sim 20$ cm/s for an 500 m diameter object such as Asteroid 101995 Bennu [@scheeres19]. These would be low velocity impacts into surface regolith or rubble at low gravitational acceleration and would include encounters at low or at grazing incidence angle because the projectiles were previously in orbit. Due to its orbital angular momentum, this would also be true of most material that was thrown off the asteroid during spin-off events. Experiments of low velocity and normal impacts into granular media at low surface gravity from drop towers include those by @goldman08 [@Sunday_2016; @murdoch17] and in aircraft those by @brisset18. These experiments studied normal impacts so they did not study the sensitivity of the projectile deceleration profile and stopping depth to impact angle. However debris that is orbiting an asteroid that later hits the asteroid surface would be unlikely to only have high or nearly normal impact angles. Grazing impacts of spherical projectiles on sand or water are more likely to bounce or ricochet (e.g., @birkhoff44 [@soliman76; @daneshi77; @bai81]). Simulations by @maurel18 show that this is also true for non-spherical, low velocity projectiles, such as a lander, in low surface gravity.
Images of asteroids 101995 Bennu and 162173 Ryugu show large (10–50 m) boulders that look as if they were perched on the surface [@sugita19; @walsh19]. See Figure \[fig:bennu\] for a boulder that might unstable if perturbed by large amplitude vibrations. Possible explanations for boulders on the surface of a rubble pile asteroids include the Brazil nut effect (e.g., @matsumura14), which drives previous buried boulders to the surface, and boulder stranding that occurs during landing of ejecta that is launched by an impact generated pressure pulse [@wright19]. Some of the surface boulders on Bennu are so large they probably instead landed on the surface during accumulation following a disruptive large collision of larger asteroids [@walsh19].
A low velocity boulder projectile with a shallow or grazing impact angle might bounce off the surface, landing distant from the site of first impact and from any depression in the surface resulting from material ejected during its first impact. We consider ricochets of low velocity impactors as a way to account for protruding boulders on Bennu and Ryugu’s surfaces. While Bennu and Ryugu might lack surface regions comprised primarily of fine grained material, other asteroids such as 433 Eros [@veverka01; @cheng02], 25143 Itokawa [@miyamoto07] and the moon exhibit smooth regions covered in regolith. The dynamics of low velocity grazing impacts into granular media is also relevant to interpretation of the surfaces of these bodies and for the design of landers that may be going to them. For a review of granular media in solar system bodies see @hestroffer19.
![A perched boulder is 47 ft (14.3 m) long on Bennu. From <https://www.asteroidmission.org/20190405-shelf/> This image was taken by the PolyCam camera on NASA’s OSIRIS-REx spacecraft on April 5 2019 from a distance of 2.8 km. The field of view is 40.5 m. \[fig:bennu\]](bennu_arrow.png){width="3.5in"}
Deployed from the European Space Agency’s Rosetta spacecraft, the Philae lander’s anchoring harpoons failed to fire when approaching comet 67P/Churyumov-Gerasimenko. The lander rebounded twice from the comet surface prior to coming to rest roughly 1 km away from the intended landing site [@biele15]. The first touchdown was at a relative velocity of 1 m/s (and near the local escape velocity) and about $12^\circ$ from normal. The normal velocity component was damped and the outgoing velocity was about 1/3 of the incoming relative velocity. The lander itself contains a damping element which was depressed during this touchdown. Interpretation of the vertical acceleration profile suggest that the lander hit a granular surface with a compressive strength of order a few kPa [@biele15]. Hyabusa2 is a sample return mission to asteroid Ryugu that contains a 10kg lander called MASCOT (Mobile ASteroid SCOuT; @ho16). The soft sphere rubble pile simulations by @maurel18 showed that simulated low velocity impacts (19 cm/s) of MASCOT onto Ryugu’s surface with lower (closer to grazing) impact angles were more likely to bounce and had higher effective coefficients of restitution than normal impacts. The Philae and MASCOT landers illustrate that the understanding of low velocity surface-lander interactions is critical for missions with lander components and influences lander deployment and sample return strategies. Phenomenological models have been proposed to account for experimental measurements of penetration depth of spherical projectiles impacting granular materials in a gravitational field and at normal incidence (e.g., @uehara03 [@tsimring05; @ambroso05; @katsuragi07; @goldman08; @katsuragi13; @murdoch17]). For normal impacts (e.g., @uehara03 [@tsimring05; @ambroso05; @goldman08; @katsuragi13; @murdoch17]), and recent oblique impact experiments [@bester19], most experimental studies have modeled the granular medium with an empirical force law that includes a hydrodynamic drag term proportional to the square of the velocity, which dominates at higher velocity and so at deeper penetration, and a term which accounts for a depth dependent static resistance force which dominates at lower speeds and shallow penetration depths. However a higher velocity projectile at grazing incidence may not penetrate deeply and would still feel a hydrodynamic-like drag. Horizontal motion in granular media can cause lift [@ding11; @potiguar13]. Thus the empirical models primarily developed for normal impacts need to be modified in order to account for impacts at grazing angles.
Phenomenological models for ricochet in sand or water [@birkhoff44; @johnson75; @daneshi77; @soliman76; @bai81] predict a velocity dependent critical angle for ricochet. However, these early models assume low angles of impact and nearly horizontal and constant velocity during the encounter between projectile and substrate and neglect spin. The lift was based on a hydrodynamic pressure exerted on the submerged projectile surface and may not be consistent with more recent empirical force models or experimental measurements. Models consistent with a broader range of phenomena, including ricochets and oblique impacts, would be helpful for understanding processes taking place in low gravity environments.
Laboratory Experiments of Projectiles Impacting Similar Density Granular Material {#sec:exp}
=================================================================================
We carry out experiments of spherical projectiles into sand. We restrict the density of our projectiles to similar materials and densities as the solids in our substrate material so as to be similar to natural low velocity impacts on asteroids. Prior to this study we had explored non-spherical projectiles (pebbles) into coarser substrates (gravel), however we found that experiments were often not easily reproducible. This was likely due to the complex projectile shapes, their spin and phase of impact with respect to rotation, and irregularities in the substrate. We have reduced the degrees of freedom so as to try to understand simpler systems as a first approximation. The projectiles we discuss here are spheres and the granular substrate is comprised of particles that are much smaller than the projectile.
Our granular substrate is dry playground sand. As irregularities in a fine medium are less likely affect the projectile trajectory, and craters are easier to see in a fine medium, we chose fine sand to facilitate measurement of projectile stopping times, crater sizes, depths and morphology. The sand was passed through a sieve, giving only particles less than 0.5 mm in diameter. Density of our sand is $\rho_s = 1.6$ g/cm$^3$. The porosity of our sand was 0.4. We measured the porosity by filling a volume of sand with water until the voids were filled and taking the ratio of the volumes. Prior to each impact the substrate is raked and then scraped flat. It is not pressed or compacted. The rake consists of a linear row of 2 cm long nails. The separation between each pair of neighboring nails is 1 cm. The rake and a close up view of the sand tray is shown in Figure \[fig:setup\_close\]. The low packing fraction of the sand and shallow penetration regime of our impacts suggests that interstitial air would not have a strong affect the impact dynamics (see @royer11).
We use a small spherical projectile. The glass marble projectile we used for most experiments has a mass of 5.57 g, a diameter of 16.15 mm and a density of 2.5 g cm$^{-3}$. This density is within the scatter of different types of quartz and glasses. A small size in the lab is convenient as a $R_p \sim 1$ cm radius projectile at an impact speed of $v_{impact} = 3$ m/s has dimensionless Froude number $Fr \equiv v_{impact}/\sqrt{g R_p} \sim 10$ that is similar to that of a meter radius projectile at an impact speed similar to the escape velocity $\sim 20 $ cm/s on a small asteroid like Bennu with effective gravity $\sim 10^{-4}$ g. The Froude number is related to the dimensionless number, $\pi_2$, used to described impact crater scaling relations [@housen03]; $\pi_2 = 3.22 g R_p/v^2 = 3.22 Fr^{-2}$.
We desire a way to launch projectiles that minimizes initial spin, allows us to adjust impact velocity and impact angle and gives reproducible craters and outcomes (ricochet or not). After trying a rail gun, we settled on a pendulum launcher because it gave us low projectile spin and more repeatable impact velocities and angles. The pendulum is raised to a set height and then drops due to gravity until it hits a horizontal stop-bar. The location of the bar that stops the pendulum as it swings down sets the projectile impact angle. Our pendulum setup is illustrated in Figure \[fig:pend\]. A photograph of the experimental setup is shown in Figure \[fig:setup\] and a close up view in Figure \[fig:setup\_close\]. An illustration of the sandbox, camera and lighting as viewed from above is shown in Figure \[fig:ric\_lighting\].
Prior to letting the pendulum drop, the marble is held up against a thin rubber washer. A red turkey baster handle is used to apply light suction to the marble to hold it in place. A light tap on the pendulum is enough to break the suction and eject the marble. Because the required tap is light, the marble suffers only a small reduction in velocity during ejection. The marble is ejected with some backspin (angular rotation rate $\sim -30$ rad s$^{-1}$), that is usually well below the size of the horizontal velocity component divided by marble radius ($\sim $ 100 to 300 rad s$^{-1}$). Backspin is larger than expected from the pendulum swing which would be $v_{eject}/L_m \sim 5$ rad s$^{-1}$. $v_{eject}$ is the velocity of the marble when ejected from launcher, and $L_m$ is the radial distance from the pivot to the ejection point. The backspin must be caused by uneven friction or suction when the marble exits its holder. Measurements for the marble spin are discussed in more detail in section \[sec:data\].
Craters in our sandbox were increasingly reproducible after mechanical vibrations were reduced. The pendulum rod is a hollow aluminum pipe, replacing a narrower and heavier steel rod that flexed upon impact. The aluminum rod still flexes some on impact and this might be the cause of the ejected marble’s backspin. The pendulum pivot is clamped to a lab table to prevent bounces and vibration during the impact between pendulum and horizontal stop-bar. We adjusted the angle of the marble holder so that the marble is ejected in the same plane of the pendulum. The connection between marble holder and pendulum rod was shimmed and tightened so that it did not rotate. We added super glue to the the thread at the top of the pendulum rod to prevent it from turning. Prior to each impact experiment, we checked that the sand tray is leveled in directions parallel and perpendicular to the marble launcher using a bubble level.
The pendulum itself is $L=94.8$ cm long, but the radial distance from pivot to marble ejection point is $L_m = 84.3$ cm long. Due to its extended mass distribution, the pendulum is a compound pendulum. We measured the moment of inertia of the pendulum from its period of small oscillations ($T=1.78$ s) and its center of mass radius from the pivot, $R_{cm} = 50.5$ cm using the relation $$\frac{I_{pend}}{M_{pend}} = \frac {T^2}{(2\pi)^2} g R_{cm}$$ where $M_{pend}$ and $I_{pend}$ are the mass and moment of inertia (about the pivot point) of the pendulum. Measurements of the pendulum are listed in Table \[tab:pend\]. The inside dimensions of the sandbox are 87.5 cm long, 11.5 cm wide and 6.3 cm deep.
![Side-view illustrations of our pendulum based marble launcher. The pendulum is dropped from a height $h$, setting the velocity of impact. The pendulum swing is stopped by a stop-bar at an angle $\theta_{sb}$ that sets the angle of impact. The marble is ejected from its holder when the pendulum is stopped and then hits the sand. The red bulb is used to apply a weak vacuum that holds the marble in place until the pendulum hits the stop-bar. The lengths and angles shown here are used to estimate the velocity and angle of the marble impact on the sand. See Table \[tab:pend\] for a list of quantities and Table \[tab:nomen\] for nomenclature. \[fig:pend\] ](pend.png){width="3.4in"}
![Side-view illustrations of our pendulum based marble launcher. The pendulum is dropped from a height $h$, setting the velocity of impact. The pendulum swing is stopped by a stop-bar at an angle $\theta_{sb}$ that sets the angle of impact. The marble is ejected from its holder when the pendulum is stopped and then hits the sand. The red bulb is used to apply a weak vacuum that holds the marble in place until the pendulum hits the stop-bar. The lengths and angles shown here are used to estimate the velocity and angle of the marble impact on the sand. See Table \[tab:pend\] for a list of quantities and Table \[tab:nomen\] for nomenclature. \[fig:pend\] ](pend2.png){width="3.4in"}
![Photograph of the experimental setup. The high speed camera and white light are on the left, whereas the blue LEDs are on the right. The sand box is just above the floor. \[fig:setup\] ](setup_small.jpg){width="3.4in"}
![Close up view of the sand tray and pendulum launcher. The marble in the foreground is 16 mm in diameter. Prior to each experiment we raked the sand flat. The rake we used is shown on the left and has nails that penetrate to a depth of 2 cm. \[fig:setup\_close\]](close_up_small.jpg){width="3.5in"}
![A top view of the pendulum marble launcher showing the lighting and high speed camera viewing position. \[fig:ric\_lighting\] ](ric_lighting.png){width="3.4in"}
[0.95]{} ![We show three different types of impact craters. These are photographs taken from above of the sandbox after three impact experiments. The top photo shows a ricochet, the middle one a roll-out and the bottom one a stop event. Ricochet events have a clear gap between the primary and secondary craters when the marble was above the sand. A roll-out event is one where the marble rolled out of the primary crater, but never lost contact with the sand. The stop event occurs when the marble remains within its primary crater. The marble shown has a diameter of 16.15 mm. Ricochets and roll-outs tend to occur at higher velocity and lower (or grazing) impact angles. []{data-label="fig:craters"}](pend030_crater_small.jpg "fig:"){width="1\linewidth"}
[0.95]{} ![We show three different types of impact craters. These are photographs taken from above of the sandbox after three impact experiments. The top photo shows a ricochet, the middle one a roll-out and the bottom one a stop event. Ricochet events have a clear gap between the primary and secondary craters when the marble was above the sand. A roll-out event is one where the marble rolled out of the primary crater, but never lost contact with the sand. The stop event occurs when the marble remains within its primary crater. The marble shown has a diameter of 16.15 mm. Ricochets and roll-outs tend to occur at higher velocity and lower (or grazing) impact angles. []{data-label="fig:craters"}](pend033_crater_small.jpg "fig:"){width="1\linewidth"}
[0.95]{} ![We show three different types of impact craters. These are photographs taken from above of the sandbox after three impact experiments. The top photo shows a ricochet, the middle one a roll-out and the bottom one a stop event. Ricochet events have a clear gap between the primary and secondary craters when the marble was above the sand. A roll-out event is one where the marble rolled out of the primary crater, but never lost contact with the sand. The stop event occurs when the marble remains within its primary crater. The marble shown has a diameter of 16.15 mm. Ricochets and roll-outs tend to occur at higher velocity and lower (or grazing) impact angles. []{data-label="fig:craters"}](pend034_crater_small.jpg "fig:"){width="1\linewidth"}
[0.95]{} ![We show ejecta curtains during three different impact experiments. These three images are from the same experiments shown in Figure \[fig:craters\]. These images are individual frames taken from the three high speed videos.[]{data-label="fig:ejecta"}](pend030_ejecta.png "fig:"){width="1\linewidth"}
[0.95]{} ![We show ejecta curtains during three different impact experiments. These three images are from the same experiments shown in Figure \[fig:craters\]. These images are individual frames taken from the three high speed videos.[]{data-label="fig:ejecta"}](pend033_ejecta.png "fig:"){width="1\linewidth"}
[0.95]{} ![We show ejecta curtains during three different impact experiments. These three images are from the same experiments shown in Figure \[fig:craters\]. These images are individual frames taken from the three high speed videos.[]{data-label="fig:ejecta"}](pend034_ejecta.png "fig:"){width="1\linewidth"}
Delineating ricochets from roll-outs and stops {#sec:delin}
----------------------------------------------
Photographs of three different impact craters are shown in Figure \[fig:craters\]. These photographs were taken from above the sand tray looking downward and after the impacts. Using the impact crater morphology, we classified an impact as a [*ricochet*]{} if there was a clear gap between a primary crater and a secondary one, as shown in Figure \[fig:crater\_ricochet\]. An impact was classified as a [*roll-out*]{} if the marble rolled out of its crater, as shown in Figure \[fig:crater\_rollout\]. An impact was classified as a [*stop*]{} if the marble rested inside its impact crater after impact, as shown in Figure \[fig:crater\_stop\].
Using different initial pendulum heights and stop-bar positions, setting the impact angle, we classified the outcomes of 120 impacts as ricochet, stop or roll-outs. For each impact, we recorded the initial pendulum drop height $h$ and the angle of the stop-bar $\theta_{sb}$. These two measurements were used to estimate the velocity of impact $v_{impact}$ and the angle of impact $\theta_{impact}$, measured from horizontal so that a grazing impact has a low impact angle. The classifications are plotted with different point types and colors in Figure \[fig:ric\] and as a function of computed $v_{impact}$ and $\theta_{impact}$.
We describe how impact velocity and impact angle are computed from the initial pendulum drop height $h$ and stop-bar angle $\theta_{sb}$. The initial drop height from the mechanism base, $h$, and distance of pendulum top to mechanism base when vertical, $d_base$, as shown in Figure \[fig:pend\], are used to calculate the initial angle of the pendulum $\alpha$, (measured from vertical) with $\cos \alpha = 1 - \frac{h-d_{base}}{L}$ and using the pendulum length $L$. The pendulum’s angular velocity at the moment the pendulum stops at the stop-bar and the projectile is ejected from its holder is $$\dot \theta =
\sqrt{2 \frac{M_{pend} }{I_{pend}} R_{cm} g
(\cos \theta_{sb} - \cos \alpha) }.$$ The radius of the pendulum’s center of mass is $R_{cm}$ (from its pivot) and the angle $\theta_{sb}$ is the angle set by the stop-bar from vertical. The speed of the marble when ejected from its holder is $$v_{eject} = L_m \dot \theta.$$ Here $L_m$ is the radial distance from the pivot to the center of the marble when it is in the launcher. The marble’s horizontal and vertical velocity components are $v_{x,eject} = v_{eject} \cos \theta_{sb} $ and $v_{z,eject} = v_{eject} \sin \theta_{sb} $.
We correct for the distance of projectile free fall before hitting the substrate surface, even though this correction is usually small. After it is ejected, the marble freely falls a distance of $$dz = L + d_{base} - d_s - L_m \cos\theta_{sb}$$ to hit the sand, where $d_s$ is the height of sand surface above the pendulum mechanism base. The estimated velocity of projectile impact with the sand substrate is $$v_{impact} = \sqrt{v_{z,eject}^2 + 2 g dz + v_{x,eject}^2}
\label{eqn:v_i}$$ and the angle of impact (measured from horizontal) is $$\theta_{impact}={\rm arctan}\left( \frac{\sqrt{v_{z,eject}^2 + 2 g dz}}{v_{x,eject}}\right). \label{eqn:theta_i}$$
The velocity and angle of impact along with classifications based on impact crater morphology were used to make Figure \[fig:ric\]. Figure \[fig:ric\] shows that crater morphology and impact behavior depends on both impact angle and velocity. At higher velocities and lower impact angles ricochets are more likely. Below a velocity of about 2 m/s grazing impacts had projectiles that rolled out of their crater rather than bounced off the sand. The dividing line between ricochet and roll-out and that between ricochet and stop was sometimes sensitive to vibrations and wobble in the apparatus. We noticed that the location of the ricochet/roll-out line shifted when we inserted shims into the pivot holder to keep it from vibrating during impact. With vibrations reduced, events were repeatable, with series of three or four trials at the same initial pendulum height and stop-bar position giving the same impact crater morphology and event classification. We tentatively assign a $\pm 5$ error to each point in Figure \[fig:ric\] that is due to variations in stiffness in the mechanical launch mechanism that we have tried to minimize. This value was the largest discrepancy between the estimated and measured impact angles. Estimated and measured impact velocities agree. The discrepancy in impact angle is discussed below.
Figure \[fig:ric\] shows that the dividing line between different outcomes is quite sensitive to the angle of impact. The dividing line trend is opposite to that found by @soliman76 as we see the critical grazing impact angle dividing ricochets from roll-outs increases as a function of impact velocity, rather than decreases. There are some differences between our experiments and theirs that might explain this difference in behavior. Our projectile density is similar to the substrate density (glass marbles into sand), whereas @soliman76 used denser projectiles (steel, aluminum and lead balls into sand). Their projectiles have higher velocity (theirs were up to 180 m/s and ours are below 5 m/s). Both our and their projectiles are spherical. It is useful to define a dimensionless number known as the Froude number $$Fr \equiv \frac{v}{\sqrt{gR_p}}$$ with $R_p$ the projectile radius. Both sets of experiments can be considered at high Froude number where impact velocity gives $Fr > 1$. The Froude number of our impacts are 10–20, whereas those by @soliman76 are 300–600. Our projectiles remain in a shallow penetration regime, where the maximum penetration depth rarely exceeds the projectile diameter. We designed our experiment to minimize initial projectile spin. Unfortunately projectile spin is not discussed by these early works (though see the discussion on bouncing bombs by @johnson98). Models that predict granular flow above a particular stress level [@bagnold54] might account for the higher lift we infer in our lower velocity experiments that give us ricochets at high impact angles, as the medium could be effectively stiffer at lower impact velocity. Alternatively lower velocity and lower projectile density, giving shallower levels of penetration, may be increasing the likelihood of ricochets in our experiments. Since collisions on asteroids are likely to have similar projectile and substrate densities, our experiments suggest that ricochets could be common in the low velocity regime.
![Classified impacts as a function of impact angle and velocity. Impact angle is measured from horizontal so low $\theta_{impact}$ is a grazing impact. Black circles, denoted ‘Stop’ in the legend, are impacts where the marble stayed within its impact crater. Blue triangles, denoted ‘Roo’ in the legend are roll-outs. Red squares (Ric) are ricochets. Green squares (Roo/i) are on the dividing line of ricochet and roll-out. Brown circles (S/Roo) are on the dividing line of stop and roll-out. Some of the ricochets bounced twice and are labelled with a red diamond. Each point represents a single impact trial using the pendulum launcher. \[fig:ric\] ](ric.png){width="3.3in"}
The critical angle for ricochet {#sec:crit}
-------------------------------
A ricochet takes place when the lift force is at least large enough to overcome the gravitational force and this must happen before drag forces reduce the horizontal velocity component velocity to zero [@johnson75; @soliman76; @bai81]. For spherical projectiles, a lift force dependent on the square of the depth times the square of the velocity and a constant downward gravitational acceleration were adopted by @soliman76 to estimate a critical impact angle for ricochet $$\theta_{cr}^2 \sim \frac{1}{10} \frac{\rho_s}{\rho_p} - \frac{4 R_p g}{v_{impact}^2} .
\label{eqn:soliman}$$ At a particular velocity, ricochets occur at impact angles below the critical one. The term on the right is proportional to the inverse of the square of the Froude number and was originally calculated for ricochets on water [@johnson75; @birkhoff44]. The term on the left is only dependent on the substrate and projectile density ratio. At high velocities the critical angle only depends on the density ratio. In the limit of high velocity, equation \[eqn:soliman\] predicts a critical angle of $15^\circ$ for our substrate to projectile density ratio of $\rho_s/\rho_p \sim 0.66$. At the higher velocities in Figure \[fig:ric\], we saw ricochets at impact angles up to $40^\circ$, so this model does not apply very well in the regime of our experiments.
Equation \[eqn:soliman\] predicts that the critical angle is larger at higher velocity. This behavior is seen on water, but ricochets on sand can deviate from this behavior and can scale in the opposite way with critical angle decreasing at higher velocity (see Figure 8 by @soliman76). @bai81 modified equation \[eqn:soliman\] with the addition of a constant pressure term dependent on parameter $K'$ adding to the lift, $$\theta_{cr}^2 \sim \frac{1}{10} \frac{\rho_s}{\rho_p} - \frac{4 R_p g}{v_{impact}^2} + \frac{K'}{v_{impact}^2}.
\label{eqn:bai}$$ This additional term allowed them to account for a decreasing critical angle with increasing impact velocity, which was seen in their experiments.
As we see an increase in impact angle with increasing velocity, the simpler model by @soliman76 might give a line that matches the division seen in our experiments. We found a similar line that does delineate the impact outcomes and it is shown in Figure \[fig:ric\_flip2\] as a dotted orange line. Figure \[fig:ric\_flip2\] shows the same experiments as Figure \[fig:ric\] except we rotated the axes so that $\theta_{cr}(v_{impact})$ is a function of the Froude number or $\bar v_{impact} = v_{impact}/\sqrt{g R_p}$ with $R_p$ the marble radius. The orange dotted line that separates the ricochets from the roll-outs is $$\theta_{cr,ric}^2 = 0.65 - \frac{55}{\bar v^2}
\label{eqn:orange}$$ with $\theta_{cr,ric}$ in radians. A similar line, separating roll-outs from stops is shown as a grey dot-dashed line on Figure \[fig:ric\_flip2\]. $$\theta_{cr,roo}^2 = 0.88 - \frac{55}{\bar v^2} .
\label{eqn:gray}$$
The second term in equation \[eqn:soliman\] is 4 times the square of the Froude number but our orange dotted line requires a number 13 times larger than this. The constant term in Equation \[eqn:soliman\] was predicted to be 1/10th the density ratio. Our substrate to marble density ratio is about 0.64 so the size of the constant term in Equation \[eqn:orange\] is about 10 times higher than expected. The orange dotted line on Figure \[fig:ric\_flip2\] (from equation \[eqn:orange\]) is not consistent with the ricochet model by @soliman76.
The orange dotted line on Figure \[fig:ric\_flip2\] (equation \[eqn:orange\]) represents our first attempt to model the line dividing ricochets from other types of events. This expression will be revisited and improved later on in this paper only after the experimental measurements are discussed.
![The dots are the same events from experiments shown Figure \[fig:ric\]. Here the lower $x$ axis is the impact Froude number or the impact velocity in units of $\sqrt{g R_p}$ where $R_p$ is the marble radius. The top axis is velocity in $m/s$. The left $y$ axis is the grazing impact angle in degrees and that on the right in radians. The orange dotted line shows equation \[eqn:orange\] which is in the form of equation \[eqn:soliman\] (based on that by @soliman76) but with larger coefficients. The gray dot-dashed line is equation \[eqn:gray\] and approximately separates the roll-outs from the stop events. \[fig:ric\_flip2\] ](ric_flip2.png){width="3.3in"}
Trajectories
============
The marbles were painted with a black undercoat. On top of the undercoat they were painted with 18 dots of fluorescent paint to aid in tracking the projectile’s spin. We lit the experiment with bright blue LEDs, causing the paint dots to fluoresce green. The LEDs are CREE XLamp XT-E Royal Blue that peak at 450 nm.[^1] We were careful to use a fluorescent paint that is detected as bright when viewed with our high speed video camera. We used 4 wide angle blue LEDs to light the sandbox, primarily from the right side, as shown in Figures \[fig:pend\] \[fig:ric\_lighting\], and \[fig:setup\]. The 4 blue light sources made this lighting fairly diffuse. We also lit the impact region with a single bright white light from the top left side (see the photograph in Figure \[fig:setup\]). With projectiles moving left to right, this gave a single white reflection on the marble that could be seen from the front of the experiment during most of the impact. The ejecta curtain tended to obscure the right side of the marble during the impact. We used the white reflection to track the marble’s center of mass motion.
We filmed the impacts with a Krontech Chronos 1.4 high speed camera at 3000 frames per second. The marble diameter (16.15 mm) was used to find the pixel scale in the video image frames. The high speed videos were taken from a 45$^\circ$ angle from vertical (see Figure \[fig:setup\]), allowing us to track both horizontal and vertical projectile motions. The impact craters were used to verify that marble trajectories remained in the pendulum’s plane. Videos used to track the projectiles can be found in the supplemental materials.
Data reduction {#sec:data}
--------------
As our high speed camera takes color images, using weighted sums of red, green and blue color channels we could emphasize the white light reflection or remove it and focus on the fluorescent green markers. The white light reflections were used to track the marble center of mass. The fluorescent green markers were used to measure the marble spin.
We use the soft-matter particle tracking software package `trackpy` [@trackpy] to identify and track the fluorescent dots and reflections on the projectile seen in individual video frames. `Trackpy` is a software package for finding blob-like features in video, tracking them through time, linking and analyzing the trajectories. It implements and extends in Python the widely-used Crocker-Grier algorithm for finding single-particle trajectories [@crocker96].
We first tracked the position of the white light reflection on the marble in each video frame. Prior to tracking we used a series of images to construct a median image which was subtracted from each video frame. We adjusted the radius and integrated peak brightness so that the reflection was identified by the tracking software and also so that the number of sand particles tracked is reduced. We eliminated spurious tracks by hand, leaving only the tracked white light reflection. The white light reflection track measured from three high speed videos is shown on top of a sum of images in Figure \[fig:CM\_seq\]. The horizontal axis is the marble’s position in x and the vertical axis is the projected z direction as the camera was positioned 45 above the sand tray’s surface plane.
The positions of the white light reflection plus a constant offset gives us an estimate for the marble center of mass position as a function of time. We adopt a coordinate system with $x$ increasing in the horizontal direction along the direction of the projectile motion and $z$ increasing in the vertical direction. Projectile trajectories remained nearly in the $xz$ plane (pendulum plane). The $x$ and $z$ marble position vectors were interpolated from the arrays of tracked positions so as to be evenly sampled in time. The vertical positions were corrected to take into account the camera viewing angle from horizontal. Because the camera frame was oriented parallel to the horizontal direction of the projectile motion, we did not need to correct the $x$ direction for camera viewing angle. The time vector is computed from frame numbers by dividing by the video frame rate (3000 fps). We estimated the time and position of impact from the first frame showing an ejecta curtain. We median filtered the $x$ and $z$ position arrays using a width of 11 samples which is 3.6 ms at a sampling rate of 3000 Hz. To compute velocities and accelerations, we smoothed the arrays using a Savinsky-Golay filter with widths of 15 and 17 samples respectively. We checked that the white light reflection used to track the center of mass of the marble did not change position on the surface during its motion. Trajectories of the marble center of mass as a function of time are shown in Figure \[fig:pend\_traj\]. The origin of these plots correspond to the time and location of impact.
To measure the marble spin we used the video frames’ green channel, showing the fluorescent markers. We shifted each image using the previously computed marble center of mass positions to put the marble in the center of the image. We then tracked the fluorescent markers again using the `trackpy` software package. Tracks of the fluorescent markers spanning whole videos are shown in Figure \[fig:spintracks\] for three videos. Even though the marble spin varies as a function of time, the tracks in each figure lie on similar arcs. This implies that the spin orientation did not significantly vary throughout the video. This does not imply that the marble’s angular rotation rate remained fixed. The orientation of the arcs are consistent with a horizontal spin axis and the camera orientation angle, $\theta_{cam}$, of 45 with respect to vertical.
To measure the spin angular rotation rate we fit tracks during short intervals of time ($\sim10$ ms). Assuming a spherical surface, each track is described by the spin orientation angle, the angular rotation rate and an initial fluorescent dot position on the marble surface. We constructed a minimization function that is the sum of differences between predicted (via rotation) and observed tracked particle positions. We simultaneously fit for the initial dot positions and the angular rotation rate. Comparisons between tracks and fit tracks for a series of time intervals are shown in Figure \[fig:pend030\_fit\]. This figure shows that our fit angular rotation rates successfully put modeled tracks on top of those ones observed.
The vertical error bars for the angular rotation rates were determined by varying parameters that went into the angular velocity fitting routine such as center of mass position, camera angle, and projectile radius. The largest source of error was identified from the uncertainty in the radius. An uncertainty of 3 pixels in the radius gave a 10% error in the spin values shown in Figure \[fig:pend030\_fit\]. A radius of 33 pixels was used for our projectile spin fitting. Horizontal error bars for the angular rotation rates show the time interval used to measure the spin.
Table \[tab:exp\_list\] lists experiments along with initial pendulum arm settings: height h, the stopping bar angle $\theta_{sb}$, and distance of sand surface to the pendulum base $d_s$. Predicted impact velocities $v_{imp}$ and angles $\theta_{imp}$ were found using equations \[eqn:v\_i\] & \[eqn:theta\_i\] respectively. Measured impact velocities and angles were obtained from marble trajectories just before impact. The predicted and measured impact velocities are consistent. The measured impact angles were about 5 lower than predicted for the ricochet and roll-out events. The discrepancy between the predicted and measured angles are attributable to errors in the pendulum setup. Soft rubber was used on the launcher to better hold the marble. This could lead to a nonuniform suction causing the marble to not separate from the launcher once it hits the stopping bar. The pendulum hitting the stopping bar also caused the post to bend slightly. This was minimized by using a thicker post. The pendulum arm can also bounce when hitting the stopping bar. Lead weights were added to the base of the pendulum arm to reduce shaking during marble launch.
\[fig:rico\_seq\]
\[fig:rollout\_seq\]
\[fig:stop\_seq\]
[0.3]{}
[0.3]{}
[0.3]{}
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Length of pendulum $L$ 94.8 cm
Period of small oscillations $T$ 1.78 s
Moment of inertia divided by mass $\left(\frac{I}{M}\right)_{pend}$ 3972 cm$^2$
Radius of pendulum’s center of mass $R_{cm}$ 50.5 cm
Radius of marble holder from pivot $L_m$ 84.3 cm
Distance of pendulum tip to base $d_{base}$ 1.5 cm
Distance of base to sand surface $d_s$ 9.9 cm
Diameter of marble $2 R_p$ 16.15 mm
Mass of marble $m_p$ 5.57 g
Density of marble $\rho_p$ 2.5 g/cm$^3$
Unit of velocity $\sqrt{g R_p}$ 28.1 cm/s
Density of sand $\rho_s$ 1.6 g/cm$^3$
Sand angle of repose $\theta_s$ 32$^\circ$
Coefficient of friction $\mu$ 0.51
Camera Angle $\theta_{cam}$ 45
------------------------------------- ----------------------------------- --------------
: Quantities[]{data-label="tab:pend"}
Notes: The value for $d_s$ (as shown in Figure \[fig:pend\]) reported here is for all experiments shown in Figure \[fig:ric\]. The coefficient of friction for the sand is computed from its angle of repose $\mu = {\rm atan}(\theta_s)$.
------------------------------------ ---------------------- --
Projectile mass $m_p$
Surface gravitational acceleration $g$
Projectile radius, if spherical $R_p$
Granular substrate mean density $\rho_s$
Projectile density $\rho_p$
Projectile velocity at impact $v_{impact}$
Projectile velocity vector ${\bf v}$
Projectile cross sectional area $A$
Critical impact angle $\theta_{cr}$
Froude number
Horizontal coordinate $x$
Vertical coordinate $z$
Normalized vertical coordinate $\bar z = |z| / R_p$
Depth below surface level
Impact angle $\theta_{impact}$
Drag force $F_d$
Lift force $F_L$
Coefficient of static friction $\mu_s$
Angle of repose $\theta_r$
Stopping time $t_s$
Maximum penetration depth $d_{mp}$
Horizontal velocity component $v_{xmp}$
at maximum depth
Time of maximum height $t_m$
during rebound
Height reached in rebound $z(t_m)$
Drag coefficients $\alpha_x, \beta_x $
Lift coefficient $c_L$
Effective friction coefficient $\mu_{eff}$
Vertical, horizontal acceleration $a_z$, $a_x$
Angular acceleration $\dot{\omega}$
Angle of stop-bar $\theta_{sb}$
Height of tip of pendulum arm $h$
Drop angle of pendulum arm $\alpha$
------------------------------------ ---------------------- --
: Nomenclature[]{data-label="tab:nomen"}
\
For a grazing impact $\theta=0$. The vertical coordinate is positive upward. The horizontal coordinate is positive with the initial direction of projectile motion.
\
The values for the pendulum, sand and marble, $L$, $L_m$, $d_{base}$, $(I/m)_{pend}$, $m_p$, $\rho_p$, and $\rho_{s}$ are the same for all events and given in Table \[tab:pend\]. The predicted impact angle and velocity were predicted from the initial pendulum height and stop-bar position using equations \[eqn:v\_i\] and \[eqn:theta\_i\].
Shapes of Trajectories
----------------------
The trajectories shown in Figure \[fig:pend\_traj\] are measured from three different videos. The panels, from left to right, are for a ricochet, a roll-out, and a stop impact event. The top two panels show the x and z position of the white light reflection on the projectile as functions of time. The origin was chosen to be the location and time of impact. The x position increases for all time for all cratering events. The stop event has a decrease in x for late times that corresponds to the marble unable to escape its own initial impact crater and rolling backwards.
Inspection of the depths as a function of time (second panels from top in Figure \[fig:pend\_traj\]) shows that the marble bounced above the surface (where $z>0$) during the ricochet event. The bottom surface of the marble did rise above the surface level breaking contact with the sand. The bottom of the marble remained below surface level after penetration for the roll-out and stop events. In the stop event (rightmost figure in Figure \[fig:pend\_traj\]) and after maximum penetration depth, the bottom of the marble rose to about $z=-0.5$ cm which places the center of mass of the marble near the surface level. This can be contrasted with the roll-out event where the bottom of the marble rose to a height nearly level with the surface and placing the marble’s center of mass well above the surface level. Likely we can think of a roll-out event as one with depth reached after maximum penetration high enough to put the center of mass above the surface and allowing the marble to roll-out of its crater.
This is consistent with the crater morphology for these events that is shown in Figure \[fig:craters\] and provides confirmation that our tracking software is working. The maximum depth is closer to the point of impact than the crater center giving azimuthally asymmetric shapes which has been seen in other oblique impact treatments (e.g., @soliman76 [@daneshi77; @bai81]).
Prior studies (e.g., @katsuragi13) call the maximum depth reached a penetration depth. The penetration depth is reached when the vertical velocity component changes sign. For our tracked videos we have listed measurements at the time of maximum depth in Table \[tab:max\_pen\].
The trajectories have similar peak acceleration values, with peak vertical acceleration component $a_z \sim 24$ g and peak horizontal acceleration $|a_x|$ slightly less at $|a_x| \sim 22$ g. [@katsuragi07], [@vandermeer17], and [@goldman08] had accelerations that were the same order of magnitude for similar impact velocities. All were normal impact experiments with steel projectiles. [@vandermeer17] used sand as a substrate whereas the other two used glass beads.
As shown in the bottom panels of Figure \[fig:pend\_traj\], the marble’s spin increases while the deceleration is high right after impact. On the bottom panel, the gray lines show $v_x/R_p$ and the red dots show the spin or angular rotation rate. When the two coincide, the marble is rolling without slipping. We see that a rolling without slip condition is not reached until later times when the projectiles are at lower velocities. The marble’s surface was moving with respect to the sand while the ejecta curtains were launched. The ricochet event (that in the left figure in Figure \[fig:pend\_traj\]) did not achieve rolling without slip until the marble fell back into the sand. The roll-out event (the middle figure) rolled without slipping past $t \approx 0.14$ s while the marble continued to roll across the sand.
\
Quantities measured at the moment of maximum penetration from the trajectories plotted in Figure \[fig:pend\_traj\] for the three experiments presented.
![The effective friction coefficient between the projectile and the sand for each experiment presented as a function of time from impact. The friction coefficient was computed using Equation \[mu\_fric\] and accelerations measured from the tracked positions of the projectile and angular accelerations from the fitted spin data. \[fig:friction\] ](friction_coeff.png){width="3.2in"}
Estimating an effective friction coefficient from the spin
----------------------------------------------------------
We estimate an effective friction coefficient between marble and sand using the rate of change of spin during the impact. We assume that the friction force $F_\mu = \mu_{eff} F_N$ on the marble is set by an effective friction coefficient times a normal force. We estimate the normal force from the size of the vertical acceleration $F_N = m_p a_z$. The torque on the marble $\tau = I \dot \omega \approx R_p F_\mu$. For a homogeneous sphere the moment of inertia $I = \frac{2}{5} m_p R_p^2$. Putting these together, we estimate the friction coefficient $$\mu_{eff} \sim \frac{I \dot\omega}{R_p m_p a_z} \sim \frac{2 R_p \dot \omega}{5 a_z} .
\label{mu_fric}$$ Using the rate of change of spin $\dot \omega(t)$ and vertical acceleration $a_z(t)$ from our projectile trajectories we use equation \[mu\_fric\] to estimate the effective friction coefficient as a function of time $\mu_{eff}(t)$. Equation \[mu\_fric\] requires dividing by acceleration. If the acceleration is low, the result is noisy. To mitigate this effect, we only measured the effective friction coefficient during the early and high acceleration phase of impact.
The effective friction coefficient measured near the time of impact for each experiment is shown in Figure \[fig:friction\]. We computed the angular acceleration by passing our angular rotation rates through a Savinsky-Golay filer with width of 25 samples. The high values prior to impact are spurious and due to dividing by low value for the acceleration prior to impact. We only plotted the estimated friction coefficient during the early and high acceleration phase of impact for the same reason.
The coefficient of dynamic friction of glass to glass contact is 0.4. Our measured values peak around 0.1 which is below that expected for sand and marble friction contacts. Equation \[mu\_fric\] can also be approximated as $\mu_{eff} \approx 2 R_p \Delta \omega / (5 \Delta v_z)$, using a change in spin $\Delta \omega$ and a change in the vertical velocity component $\Delta v_z$. Using changes in both quantities during the high acceleration phases of the impacts, we confirm that the estimated friction coefficient is approximately 0.1. This check ensures that our estimate is not affected by how we smoothed the data.
We found that the value of the effective friction coefficient $\mu_{eff} \sim 0.1 $ is remarkably low. The low value for the suggests that sand particles are acting like lubricant, or ball bearings rolling under the marble. Alternatively, contacts on the front size of the marble could be partly cancelling the torque exerted from contacts on the bottom of the marble.
Trends {#sec:trends}
------
Phenomenological models of low velocity impacts into granular media have primarily been developed for normal impacts (e.g., @tsimring05 [@katsuragi07; @goldman08; @katsuragi13; @brzinski13; @murdoch17]). The developed empirical force laws are based upon measurements of impact penetration depth, duration and trajectories as a function of depth or time during the impact (e.g., @katsuragi13). To help pin down the force laws for non-normal impacts we use our trajectories to search for relations between acceleration and other parameters such as velocity and depth.
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-- --
In Figure \[fig:a\_zv2\] we show the projectile horizontal and vertical components of acceleration from three experiments and in units of $g$. The accelerations are plotted as a function of combinations of depth $|\bar z| = |z|/R_p$ normalized by the marble radius and velocity in units of $\sqrt{g R_p} = 28.13$ cm/s. The dimensionless $\bar v = v/\sqrt{gR_p}$ is akin to a Froude number. Each trajectory is labeled with different colors and marker types. Each point is at a different time with positions, velocity and acceleration shown on our trajectory plots (Figures \[fig:pend\_traj\]). The color and marker size depends upon whether the vertical velocity component is positive or negative. The lighter colors and larger markers are for the initial penetration phases where the projectile is moving downward into the granular media. The darker colors and smaller markers show the later part of the trajectories when the projectile is moving upward and the velocities are lower. The colors and point types are shown in the legends with labels ending with ’d’ corresponding to the initial downward penetration phase. Labels ending with ’u’ correspond to the later rebound phase when the projectiles move upward.
In the left panel in Figure \[fig:a\_zv2\] we plot acceleration components as a function of $|\bar z| \bar v^2$. We also show gray lines with a slope of 0.18 and 0.15 in the horizontal and vertical directions respectively. These slopes were found to approximately match both the horizontal and vertical accelerations in the slower and later (upward) rebound phases of the trajectories (and as seen on the lower left side of both panels). In this later phase, the accelerations scale with the square of the velocity, as would be expected from hydrodynamic drag or lift forces. When the projectile is moving upward, the vertical acceleration should be called ‘lift’ rather than drag as both velocity and acceleration are in the upward direction. The $|\bar z| \bar v^2 $ scaling does not match the accelerations during the earlier penetration phases of these trajectories.
The right panel of Figure \[fig:a\_zv2\] shows the horizontal and vertical components of the projectile’s acceleration as a function of $|\bar z| \bar v^2 \sin \theta_{impact}$. The vertical component $a_z$ has a black fit line with a slope of 0.6. It can be seen from both panels that the penetration phase of $a_z$ does not scale as $\bar z \bar v^2$ alone but with the sine of the impact angle.
During the penetration phase a drag-like horizontal force with acceleration $a_x \propto -v^2$ is approximately supported by the top left panel in Figure \[fig:a\_zv2\]. An upward drag-like vertical force dependent on the square of the velocity and sine of the impact angle seems approximately supported by the lower right panel in Figure \[fig:a\_zv2\]. We will use these relations to leverage prior normal impact studies to approximate estimate the maximum penetration depth and horizontal component of velocity at the time of maximum penetration in section \[sec:models\].
We searched for combinations using scaling polynomials of powers of $\bar z$ and components of $\bar {\bf v}$ that would put all the acceleration points on a single curve. If this were possible, such a curve would have allowed us to create an empirical force law that could be integrated to predict projectile trajectories. Unfortunately we failed to find simple combinations of depth and velocity for collapsing our trajectories to a single curve.
Figure \[fig:a\_zv2\] illustrates that a single force law does not fit both penetration and rebound phases of the trajectories. The trends seen here suggest that during the rebound phase, the vertical and horizontal forces scale with depth and velocity, and are similar in their dependence. However the forces during the penetration phase, before the time of maximum depth, must differ in form compared to those in the rebound phase. This might due to compaction of granular medium in front of the projectile, acting like ramp in front of a snow plow (e.g., [@percier11]). Simulations that take into account the response of the granular medium are probably required to match our projectile trajectories.
Phenomenological models {#sec:models}
=======================
Empirical models for Normal Impacts
-----------------------------------
Phenomenological models have been proposed to account for experimental measurements of penetration depth of non-spinning spherical projectiles impacting granular materials in a gravitational field and at normal incidence (e.g., @ambroso05 [@tsimring05; @katsuragi07; @goldman08; @katsuragi13; @altshuler14; @murdoch17]). The equation of motion of the projectile’s vertical position during the impact, $$\frac{d^2 z}{dt^2} = -g + \frac{F_d}{m} \label{eqn:motion}$$ where $z$ is the vertical coordinate with $z=0$ at the point of impact on the granular surface, $m$ is the projectile mass and $g$ is the downward vertical acceleration due to gravity. An empirical form for the vertical force from the granular substrate decelerating the projectile $$F_d = F_z(z) + B(z) v + \alpha v^2 \label{eqn:normal}$$ where $F_z(z)$ is a depth dependent force and called a hydrostatic, frictional or quasi-static resistance force term. The $v^2$ term describes an inertial or hydrodynamic-like drag force (e.g., @allen57 [@tsimring05; @katsuragi07; @goldman08; @pachecovazquez11; @murdoch17]). Sometimes a velocity dependent term, here with coefficient $B(z)$, is included that looks like a low Reynolds number drag term (e.g., @allen57 [@goldman08]).
The force law of equation \[eqn:normal\] is commonly only applied while the projectile decelerates (e.g., @goldman08 [@katsuragi13]). A maximum penetration depth $d_{mp}$, is reached when the vertical velocity first reaches zero. The time at which this happens (after impact) is called a stopping or collision time $t_s$. The equation of motion may not have a fixed point at this depth and at this time, so would give subsequent upward acceleration. However the post penetration phase upward motion is irrelevant when estimating a stopping time and a maximum penetration depth.
Recent normal impacts experiments into granular media using Atwood machines find that past a certain impact velocity, the maximum penetration depth is approximately independent of effective gravity and is approximately proportional to the collision velocity, $d_{mp} \propto v_{impact}$ [@goldman08; @murdoch17]. The experiments by @goldman08 have impact velocities of a few m/s and effective gravity 0.1 to 1 g and those by @murdoch17 have velocities 1 to 40 cm/s and effective gravity 10$^{-2}$ to 1 g. The collision time or duration is approximately independent of impact velocity and the effective gravity [@goldman08; @murdoch17]. These findings build upon prior experiments at 1 g that found similar scaling laws [@debruyn04; @ambroso05; @ciamarra04; @tsimring05; @katsuragi13]. The equations of motion with empirical force law in the form of Equation \[eqn:normal\] can match the trends measured for the maximum penetration depth and collision time, (e.g., @tsimring05 [@goldman08; @katsuragi13]).
![Different phases of an impact. \[fig:phases\]](ricochet.png){width="3.5in"}
Empirical model for ricochet and roll-out lines
-----------------------------------------------
Prior empirical models for ricochet that we introduced in section \[sec:crit\] [@birkhoff44; @johnson75; @daneshi77; @soliman76; @bai81], assume that the horizontal velocity component is nearly constant and that the lift force depends on it with $F_L \propto v_x^2$. The assumed lift is dependent on depth and computed by integrating a hydrostatic pressure applied to the submerged surface of the spherical projectile. This pressure is estimated following Bernoulli’s principle. The velocity dependence of the lift force resembles that of the hydrodynamic drag-like force that has been used to model normal impacts (e.g., @katsuragi07).
Our trajectories and trends in them discussed in section \[sec:trends\] imply that a single empirical model, based on that developed for normal impacts, would not give a good description for both the penetration phase and rebound phases of impacts (see Figure \[fig:phases\] for an illustration of these phases). We attempt to improve upon prior ricochet models in sand by using scaling developed for normal impacts into granular media to estimate a maximum penetration depth. We then use a simple but different model for the post penetration or rebound phase of impact to estimate a criterion for ricochet and roll-out events.
At the moment of maximum penetration, the horizontal velocity is $v_{xmp}$, the depth is $d_{mp}$ and the vertical velocity component $v_z =0$. We first estimate $v_{xmp}$ and $d_{mp}$ from the impact angle $\theta_{impact}$ and the impact velocity $v_{impact}$. We then find the height reached in the rebound phase from a vertical equation of motion that has a lift force that is dependent on the horizontal component of velocity.
To estimate the horizontal component of velocity at the moment of maximum penetration, we assume that the horizontal velocity component during the penetration phase is described with a hydrodynamic-like drag $$\frac{dv_x}{dt} = - \alpha_x v_x^2 . \label{eqn:avx}$$ Here $\alpha_x$ is a drag coefficient that has units of inverse length. For hydrodynamic drag on a sphere of radius $R_p$, the drag force is proportional to the projectile cross sectional area and the drag coefficient depends on the density ratio and projectile radius, $$\alpha_x \approx \frac{\rho_s}{\rho_p} \frac{3}{4 R_p}. \label{eqn:alphax}$$ Equation \[eqn:avx\] has solution $$v_x(t) = \frac{v_{x0}}{v_{x0} \alpha_x t + 1} ,\label{eqn:vxt}$$ where the initial horizontal velocity $v_{x0} = v_{impact} \cos \theta_{impact}$. The horizontal velocity component at the time of maximum penetration $$v_{xmp}
%= \frac{v_{x0}}{v_{x0} \alpha_x t_s + 1}
=
\frac{v_{impact} \cos \theta_{impact}}{v_{impact} \cos \theta_{impact} \alpha_x t_s + 1}, \label{eqn:vxmp}$$ in terms of the stopping time $t_s$. Following experimental studies of normal impacts [@murdoch17], we assume that the stopping time $t_s$ is independent of velocity and effective gravity. We also assume that the stopping time $t_s$ is independent of impact angle, as supported by recent experiments of oblique impacts [@bester19].
Following experimental studies of normal impacts (e.g., @goldman08 [@murdoch17]), we assume that the depth of maximum penetration $d_{mp}$ is proportional to impact velocity. We expect that the depth of maximum penetration $d_{mp}$ would be lower for shallower impact angles, so we assume $$d_{mp} = k_d t_s v_{impact} \sin \theta_{impact}, \label{eqn:dmp}$$ with unit-less parameter $k_d$. We estimate $\alpha_x$ and $k_d$ from measurements of our tracked trajectories. This angular dependence is consistent with penetration depth proportional to the initial z component of velocity, and this is approximately supported by the recent oblique impact experiments by @bester19 (see their Figure 2).
We assume that the lift during the rebound phase is proportional to the square of the horizontal velocity component, as did prior models for ricochet [@soliman76; @bai81]. These ricochet models assumed that the horizontal component of velocity was nearly constant during the impact and that the lift was depth dependent. Our trajectories show that the horizontal component of velocity varies significantly during rebound. Likely the lift during rebound is both depth dependent and time dependent through its sensitivity to the horizontal velocity component.
To roughly characterize a regime for ricochet and roll-out we ignore the depth dependence of the lift but we take into account its time dependence. For the rebound phase we use an equation of motion in the vertical direction $$\frac{dv_z}{dt} = c_L v_x(t)^2 - g \label{eqn:vz_rebound}$$ with lift coefficient $c_L$ that is in units of inverse length. The left term is lift and the right term is the gravitational acceleration. With $z=0$, the bottom edge of the projectile touches the substrate surface and with $z=-2R_p$, it is entirely submerged.
The horizontal component of velocity $v_x(t)$ in the rebound phase follows equation \[eqn:vxt\] but with a drag coefficient $\beta_{x}$ that might differ from that present during the penetration phase (that we called $\alpha_x$). In the rebound phase $$v_x(t) = \frac{v_{xmp}}{v_{xmp} \beta_{x} t + 1} \label{eqn:vx_rebound}$$ where $v_{xmp}$ is the horizontal velocity component at the moment of maximum depth (estimated in equation \[eqn:vxmp\]) and time is measured from the beginning of the rebound phase. Initial conditions are $v_x(0) = v_{xmp}$, $v_z(0) = 0$ and $z(0) = - d_{mp}$. We integrate Equation \[eqn:vz\_rebound\] using Equation \[eqn:vx\_rebound\] for $v_x(t)$ $$\begin{aligned}
v_z(t) &= - \frac{c_L}{\beta_x} \frac{v_{xmp} }{ (v_{xp} \beta_x t + 1)} - gt + \frac{c_L}{\beta_x} v_{xp} \nonumber \\
& = \frac{c_L v_{xmp}^2 t}{ (v_{xmp} \beta_x t + 1)} - gt. \label{eqn:vzt}\end{aligned}$$ The constant of integration is determined by requiring $v_z(0) = 0$ at the moment of maximum depth. The maximum height (or minimum depth) during rebound is subsequently reached when $v_z(t_m)=0$ where $$t_m =\frac{c_L v_{xmp}^2 - g}{v_{xmp} \beta _x}. \label{eqn:tm}$$ We integrate Equation \[eqn:vzt\] to find the height in the rebound phase $$\begin{aligned}
z(t) = - \frac{c_L}{\beta_x^2} \ln (v_{xp} \beta_x t + 1) - \frac{g t^2}{2}+ \frac{c_L}{\beta_x} v_{xp} t - d_{mp}.
\label{eqn:zt}\end{aligned}$$ The height $z(t_m)$ gives a maximum height during the rebound phase.
Let’s examine the time $t_m$ (equation \[eqn:tm\]) which is the time in the rebound phase when height $z(t)$ reaches an extremum. We require $t_m >0$ for the rebound trajectory to rise and not sink. This gives condition $$c_L v_{xmp}^2 >g.$$ If the horizontal velocity component does not significantly vary during the impact then is equivalent to $$\cos^2 \theta_{impact} > \frac{g}{v_{impact}^2 c_L}.$$ In the limit of low impact angle this condition becomes $$\theta_{impact}^2 < 1 - \frac{g}{v_{impact}^2 c_L}$$ which is similar to the expression for the critical angle giving ricochet by @soliman76. A comparison between this equation and equation \[eqn:orange\], the orange line we adjusted to match the ricochet line on Figure \[fig:ric\_flip2\], gives a lift coefficient $c_L \approx 0.02 R_p^{-1}$. Henceforth we allow the horizontal velocity component to decay during the impact. With horizontal drag, a larger lift coefficient would be required for ricochet to occur.
We can use $z(t_m)$ computed using equations \[eqn:vxmp\], \[eqn:dmp\], \[eqn:tm\], and \[eqn:zt\] to estimate the height reached during the rebound phase. These are functions of initial $v_{impact}, \theta_{impact}$, and coefficients $\beta_x, c_L,\alpha_x, t_s, k_p$. The coefficients $\alpha_x, t_s, k_p$ can be estimated from our impact trajectories. The drag and lift coefficients $\beta_x, c_L$ can be adjusted. The result is an estimate of the height $z(t_m)$ as a function of impact velocity $v_{impact}$ and angle $\theta_{impact}$.
If $z(t_m) >0$ then the projectile rises above the level of the substrate and we would classify the event as a ricochet. If $0>z(t_m) > -R_p$ then the center of mass of the projectile rises above the substrate level and the projectile could roll. We assign the condition $z(t_m) = 0$ to be the line dividing ricochet from roll-out and $z(t_m) = -R_p$ to be the line dividing roll-out from stop events. By computing $z(t_m)$ and adjusting $c_L, \beta_x$ to match our experimental event classifications, we find an empirical model for these two dividing lines.
We measured stopping times, $t_s$, maximum penetration depths $d_{mp}$ and horizontal velocity components $v_{xmp}$ at the maximum depth for the three videos that we tracked. These quantities are listed in Table \[tab:max\_pen\]. The three tracked videos have stopping time $t_s \sim 0.01$ s. From the maximum penetration depths and using equation \[eqn:dmp\] we estimate the factor $k_p \sim 0.5$. The horizontal velocity component measured at maximum penetration depth divided by the initial horizontal velocity component is about 0.3 for our three tracked videos. From the horizontal velocity components measured at the maximum depth, impact angles and velocities and using equation \[eqn:vxmp\] we estimate the drag coefficient during penetration phase $\alpha_x \sim 0.4 R_p^{-1}$. This is similar to that expected for ballistic drag (as estimated in equation \[eqn:alphax\]) and is consistent with trends seen in the tracked trajectories, (shown in Figure \[fig:a\_zv2\]).
In Figure \[fig:ric\_flip3\] we show with a color map the rebound height $z(t_m)/R_p$ computed at different values of impact velocity and angle. The colorbar shows the value of $z(t_m)/R_p$. The $x$ axis is a Froude number or impact velocity in units of $\sqrt{g R_p}$. The rebound height in the rebound phase was computed using equations \[eqn:vxmp\], \[eqn:dmp\], \[eqn:tm\], and \[eqn:zt\] and the above estimated values for stopping time, stopping depth parameter $k_p$ and drag coefficient $\alpha_x$. The remaining parameters used are lift coefficient $c_L = 0.15/R_p$ and rebound phase drag coefficient $\beta_x = 0.1/R_p$. On this plot, we also show our experiment impact classifications that were described previously in section \[sec:delin\] and shown in Figures \[fig:ric\] and \[fig:ric\_flip2\]. The upper dashed yellow line shows a $z(t_m) = -R_p$ contour and the lower white dotted line shows a depth $z(t_m) = 0$ contour. These are estimates for the critical angle giving roll-out and that giving ricochet. The model is a pretty good match to the experimental ricochet/roll-out line, but overestimates the critical angle for the roll-out/stop line, particularly at lower velocities. The rolling marble, as seen in the roll-out event trajectory shown in Figure \[fig:pend\_traj\]), stays at a particular equilibrium depth while rolling. A better prediction for the roll-out/stop dividing line might be made by computing the height that lets lower surface of the marble rise above this equilibrium level during the rebound phase.
We find that rebound drag coefficient must be smaller than the penetration phase drag coefficient, $\beta_x < \alpha_x$. Otherwise, the ricochet line on Figure \[fig:ric\_flip3\] does not rise with increasing velocity. A lower value of the rebound drag coefficient is consistent with the shallow slope in $v_x$ seen in Figure \[fig:pend\_traj\] in the rebound phases. There is some degeneracy between rebound phase drag and lift coefficients, $\beta_x$ and $c_L$. This degeneracy is not surprising, since their ratio appears in equation \[eqn:zt\] for the height reached during rebound. Extremely low values of $c_L$ would give rebound phases that are longer than we observed.
As was true for prior stopping depth and time estimates (e.g., [@katsuragi13]), our model does not have an equilibrium fixed point at the maximum height reached during the rebound phase. After this height is reached, the gravitational acceleration in the model would cause the projectile to drop forever. A more complete model could add a hydrostatic-like force term, dominating at low velocity, that allows the projectile to reach a final equilibrium resting condition at a shallow depth. We opted to use a time dependent but depth independent lift force and a constant gravitational acceleration. The result is an acceleration that is approximately linearly dependent on time. It might be possible to derive a similar looking model with depth dependent forces. We attempted to do so with constant but depth dependent hydrostatic and lift terms but had less success with them.
Application to low-g environments
=================================
Using dimensionless numbers and scaling arguments laboratory experiments can be used to predict phenomena in regimes that are difficult to reach experimentally. With that idea in mind we discuss using scaling laws developed for for crater impacts and ejecta curtains [@holsapple93] to apply the results of our laboratory results at 1 g to asteroid surfaces. [@holsapple93] defines three dimensionless parameters that have historically been denoted $\pi_2, \pi_3$ and $\pi_4$. These dimensionless are used to give regimes and scaling relations for the crater efficiency, $\pi_1$ (sometimes called $\pi_V$), which is the ratio of the ejecta mass to the projectile mass. The first of the dependent dimensionless variables is $\pi_2 \equiv ga/U^2$ where $a$ is the radius of the projectile and $U$ is its velocity. This is the same as the inverse of the square root of the Froude number. The $\pi_2$ parameter is defined as the ratio of the lithostatic pressure to the dynamic pressure generated by the impact at a depth of the projectile’s radius. The next dimensionless parameter $\pi_3 = Y/\rho U^2$ is the ratio of the crustal material strength $Y$ to the dynamic pressure of the impact $\rho U^2$. The strength of our sand is low and we can assume that regolith on an asteroid will also be low compared to the dynamic pressure from an impact. This results in a small value for $\pi_3$ and so we can neglect it in our scaling argument. The last parameter, $\pi_4$ is the ratio of the substrate to the projectile density, which in our experiments was 0.64. @holsapple93 ignores this parameter in the scaling relations since its value is confined to be near unity. Since our density ratio is not significantly different than unity we follow him by neglecting this parameter as well. This leaves only a single important dimensionless parameter, $\pi_2$ which is directly dependent on the Froude number.
Are there additional dimensionless parameters that might be important in the granular impact setting that were not considered by [@holsapple93]? The ratio of projectile radius to grain size radius might be important. In our experiments this ratio is large (and equal to about 32). The ratio is large enough that it probably does not affect our experimental results, however planetary surfaces can have both larger and smaller sized particles near the surface.
We adopt the assumption that we can scale our laboratory experiments to a low g asteroid environment by matching the Froude number. Future experimental studies at low g facilities and using granular media of different size distributions could test this assumption.
We estimate the conditions (the velocity and impact angle) that would allow a rock to ricochet on asteroids such as Bennu or Ryugu. The escape velocity from a spherical object of radius $R_a$ can be written in terms of its surface gravity $g_a = GM/R_a^2$ $$v_{esc} = \sqrt{\frac{2GM}{R_a}} = \sqrt{2 g_a R_a}.$$
We can write impact velocity in units of $\sqrt{g_a R_p}$ as $$\bar v^2 = \frac{v_{impact}^2}{{g_a R_p} }= 2 {\frac{R_a}{R_p} } \left (\frac{v_{impact}}{v_{esc}}\right)^2.
\label{eqn:va}$$ We insert this velocity into equation \[eqn:orange\] for the critical angle allowing ricochet, giving us the critical angle as a function of projectile and asteroid radius. These are plotted on Figure \[fig:regime\]. The series of black, red, and orange lines are for impacts at the escape velocity. Each line is labelled with the critical impact angle and points to the right and below the line allow ricochets below this labelled impact angle. The series of blue and green lines are for ricochets at a tenth of the escape velocity. The axes on this plot are log10 of asteroid and projectile diameters in meters.
Figure \[fig:regime\] shows that few meter diameter and smaller boulders on 500 m diameter asteroid such as Bennu, when hitting a region of level granular material at the escape velocity, would be likely to ricochet. Since most impacts would not be normal, this would apply to a large fraction of such objects. At lower velocities ricochets would only be likely for few cm sized objects.
We find that large boulders, such as the 14 m one shown in Figure \[fig:bennu\] would be above the ricochet line and so would not ricochet. However this size boulder is near enough to the ricochet line that it might roll upon impact. If use equation \[eqn:gray\] instead of equation \[eqn:orange\] to make this plot, then a 14 m diameter rock is on the line allowing roll-out to take place at impact angles below $30^\circ$.
If Froude number is relevant for matching grazing impact behavior at 1 g to that on asteroids, then we infer that boulders on Bennu would have rolled or ricocheted upon impact for near escape velocity impacts.
![ Projectile diameters on different asteroids that would ricochet. The $x$ axis is asteroid diameter, the $y$ axis is projectile diameter. The lines show critical angles allow ricochet and are computed using the empirical relation of equation \[eqn:orange\] and \[eqn:va\]. The black, brown, red and orange set of solid lines is for impact velocities at the escape velocity. The black, blue and green dashed set of lines is for impact velocities at 0.1 that of the escape velocity. The angle allowing ricochet for each line is labelled in degrees. Ricochets occur below and to the right of each line. Smaller and faster objects are more likely to ricochet. \[fig:regime\]](regime.png){width="3.5in"}
Summary and Discussion
======================
We have carried out laboratory experiments of glass spherical projectiles (marbles) impacting level sand at a range of impact angles. Impact velocities range from 2–5 m/s and grazing impact angles (measured with 0 corresponding to a grazing impact) range from about 10 to 50$^\circ$. Our projectile material (glass) has density similar to that of the grains in the granular substrate (sand). We use a pendulum projectile launcher to reduce projectile spin.
We use high speed camera images to track projectile motion and spin. The projectiles spin up when they penetrate the sand, however, the friction coefficient required is low, suggesting that the sand particles fluidize near the projectile and effectively lubricate the projectile surface.
We find that projectiles can ricochet or roll-out of their initial impact crater, and that this is likely at higher impact velocities and lower grazing impact angles. This trend is opposite to that found from experiments at higher velocities and higher projectile density into sand that were done by @soliman76.
We delineate lines between ricochet, roll-out and stop events as a function of impact velocity and angle. The dividing lines for these classes of events are empiricaly matched by quadratic relation for the square of the critical impact angle, that is in the same form as that derived by @soliman76, but has larger coefficients. We explore an empirical model for the post maximum penetration (rebound) phase of impact, balancing a lift force that is dependent upon the square of the horizontal velocity component against gravitational acceleration. This model estimates a maximum height reached in the rebound phase of the impact. A condition for ricochet is the projectile center of mass reaching a maximum height rising above the surface. A maximum height just reaching, but not above, the surface gives a condition for projectiles that roll-out of their impact crater. With adjustment of lift and drag coefficients, this empirical model can match our experiment ricochet and roll-out dividing lines.
The projectile trajectories show different scaling in penetration and rebound phases, making it difficult to find simple empirical force laws for the impact dynamics that cover inertial regimes and a lower velocity end phase. Likely a numerical simulation that includes an inertial regime for the granular medium is required to fully understand low velocity or shallow impact dynamics at oblique angles into granular media. Extending resistive force theory (e.g., @ding11) into the inertial and low gravity regimes could be one way to improve empirical models.
We have tried to simplify our experiments by minimizing projectile spin and using spherical projectiles. In future we would like to explore how spin direction and rate affects the impact. Non-spherical projectiles would be harder to track but also interesting to characterize. We would also like to explore the role of irregularities in the substrate, surface level variations and different granular size distribution.
Due to their small size, our projectiles are in a high Froude number regime where $v/\sqrt{g R_p} \sim 8$ to 17. If this dimensionless number governs behavior in low gravity environments then our projectiles match m sized projectiles near the escape velocity on small asteroids such as Bennu. The large range of angles allowing ricochet would then imply that projectiles in this regime would predominantly be found distant from their impact crater. Experiments in effective low surface gravity could be used to better understand low velocity impact phenomena in low g environments and improve upon this rough estimate.
0.3 truein
1 truein **Acknowledgements**
This material is based upon work supported in part by NASA grant 80NSSC17K0771, and National Science Foundation Grant No. PHY-1757062.
We thank Jim Alkins for helpful discussions regarding machining. We are grateful to Tony Dimino for advise, lent equipment, and suggestions that inspired and significantly improved this manuscript.
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[^1]: <https://www.cree.com/led-components/media/documents/XLampXTE.pdf>
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abstract: |
We present high-resolution X-ray observations of , the 89 ms pulsar associated with the Vela supernova remnant. We have acquired two observations separated by one month to search for changes in the pulsar and its environment following an extreme glitch in its rotation frequency. We find a well-resolved nebula with a toroidal morphology remarkably similar to that observed in the Crab Nebula, along with an axial Crab-like jet. Between the two observations, taken $\sim3\times10^5$s and $\sim3\times10^6$s after the glitch, the flux from the pulsar is found to be steady to within $0.75
\%$; the $3 \sigma$ limit on the fractional increase in the pulsar’s X-ray flux is $\simlt 10^{-5}$ of the inferred glitch energy. We use this limit to constrain parameters of glitch models and neutron star structure. We do find a significant increase in the flux of the nebula’s outer arc; if associated with the glitch, the inferred propagation velocity is $\simgt0.7c$, similar to that seen in the brightening of the Crab Nebula wisps.
We propose an explanation for the X-ray structure of the Vela synchrotron nebula based on a model originally developed for the Crab Nebula. In this model, the bright X-ray arcs are the shocked termination of a relativistic equatorial pulsar wind that is contained within the surrounding kidney-bean shaped synchrotron nebula comprising the post-shock, but still relativistic, flow. In a departure from the Crab model, the magnetization parameter $\sigma$ of the Vela pulsar wind is allowed to be of order unity; this is consistent with the simplest MHD transport of magnetic field from the pulsar to the nebula, where $B \leq 4 \times 10^{-4}$ G. The inclination angle of the axis of the equatorial torus with respect to the line of sight is identical to that of the rotation axis of the pulsar as previously measured from the polarization of the radio pulse. The projection of the rotation axis on the sky may also be close to the direction of proper motion of the pulsar if previous radio measurements were confused by orthogonal-mode polarized components. We review effects that may enhance the probability of alignment between the spin axis and space velocity of a pulsar, and speculate that short-period, slowly moving pulsars are just the ones best-suited to producing synchrotron nebulae with such aligned structures. Previous interpretations of the compact Vela nebula as a bow-shock in a very weakly magnetized wind suffered from data of inadequate spatial resolution and less plausible physical assumptions.
author:
- 'D. J. Helfand, E. V. Gotthelf, & J. P. Halpern'
title: The Vela Pulsar and its Synchrotron Nebula
---
Introduction
============
Within two years of the discovery of radio pulses from CP1919+21, magnetized, rotating neutron stars were firmly established as the origin of these remarkable signals. Furthermore, the steady increase in pulse period recorded for all sources provided an explanation for the pulsar power source: rotational kinetic energy. The detection of a decelerating 33 msec pulsar in the Crab Nebula solved the long-standing mystery of what powered this unique nebula: the spin-down rate of the Crab pulsar implied an energy loss rate, $\dot E \sim 5 \times 10^{38}$ erg s$^{-1}$, more than enough to cover the radiation losses observed from radio to gamma ray frequencies. By 1974, the basic model of the electrodynamics of pulsar magnetospheres and their coupling to the surrounding synchrotron-emitting plasma was in place (Rees and Gunn 1974), although a detailed understanding of the processes involved continues to elude us (e.g., Arons 1998).
After a decade of timing observations, it became clear that most pulsars were not defect-free clocks which simply slowed smoothly as rotational energy was transformed into an electromagnetic outflow. Two distinct types of non-monotonic behavior were established: “timing noise” characterized by a stochastic wandering in pulse phase and/or frequency which appeared to afflict most pulsars (Helfand, Taylor, and Backus 1980; Cordes and Helfand 1980; Cordes and Downs 1985; D’Amico et al. 1998), and “glitches”, an apparently instantaneous increase in the pulse frequency (a spin-up) accompanied by a simultaneous change in the spin-down rate; these rare events were found to be most prevalent in young objects (Reichly and Downs 1971; Lyne 1996 and references therein). Thirty years after the first glitch in the Vela pulsar was recorded, a total of 65 events have been seen in 27 different pulsars (Lyne 1996; Wang et al. 2000).
Glitches are a sudden fractional increases in the pulsar spin frequency with $\delta\nu / \nu \approx 10^{-9} \ {\rm to} \ 6
\times 10^{-6}$. No pulsar with a characteristic age of $>10^6$ yr has been observed to glitch more than once, but some young objects experience these events roughly annually. The best studied and most prolific in terms of large glitches is the first object in which a glitch was seen – the Vela pulsar. A dozen events have been recorded over the past three decades and daily monitoring continues. The largest event yet observed occurred in 2000 January ($\delta\nu / \nu =
3.14 \times 10^{-6}$) and provided the stimulus for the observations reported here.
In this paper, we report new results on based on observations acquired with the High Resolution Camera. The data enable us for the first time to distinguish morphological details of the synchrotron nebula surrounding , and reveal a striking picture of bilateral symmetry reminiscent of the loops and jets recently resolved in the Crab Nebula (Weisskopf et al. 2000). We offer an interpretation of the nebula’s structure which requires an MHD wind with a high magnetization parameter (unlike that inferred for the Crab). We also construct a high quality X-ray pulse profile and set tight upper limits on any change in the profile following the glitch, constraining models for the neutron star interior. Finally, we demonstrate an apparent brightening in the Nebula a month after the spin-up event; whether this was stimulated by the glitch or is a phenomena akin to the flickering wisps in the Crab Nebula remains a question for future observations to answer.
In section 2, we describe in some detail the analysis procedures required to extract quantitative information from our HRC data, in part as a cautionary tale for other early users of this instrument. We then go on to explore the morphology of the pulsar’s synchrotron nebula (§3), the soft X-ray pulse profile and limits on changes thereto (§4), and a search for changes in the nebula following the glitch (§5). The Discussion (§6) begins by developing a model which accounts for the nebula’s geometry with respect to the pulsar, as well as its anomalously low value of $L_X / \dot E$; we then go on to derive constraints on glitch models from the temporal changes we see (and don’t see) following the glitch. The final section (§7) summarizes our conclusions and assesses the prospects for future observations.
Observations
============
In response to an IAU Circular announcing a large Vela glitch on 2000 January 16.319 (Dodson et al. 2000), we submitted a Target of Opportunity request to the Observatory (Weisskopf, O’Dell, and van Speybroeck 1996) to observe the pulsar as soon as practical, followed by a second observation roughly one month later in order to search for changes in the pulsar’s flux, pulse profile, and/or surrounding nebula. The observations were carried out on 20 January 2000 and 21 February 2000, $\sim 3.5$ and $\sim 35$ days after the glitch using the imaging High Resolution Camera (HRC-I; Murray et al. 1997). Integration times of $\sim 50$ ksec were achieved in both observations.
The HRC-I detector on-board is sensitive to X-rays over the $0.08-10.0$ keV range, although essentially no energy information on the detected photons is available. Photons are time-tagged with a nominal precision of 15.6 $\mu$s; in this work, their arrival times were corrected to the solar system barycenter using a beta version of [AXBARY]{}. The data were collected during a portion of the orbit which avoided regions of high background contamination from the bright Earth and radiation belt passages; the second observation was, however, found to be partially contaminated by particle activity, most likely of solar wind origin (see below). The pulsar was centered at the on-axis position of the HRC where the point-spread function (PSF) has a minimum half-power diameter (the radius enclosing 50% of total source counts) of $\sim 0\farcs5$, which increases with energy. Images were extracted centered on the pulsar and binned using the native HRC $0\farcs13175 \times 0\farcs13175$ pixel size into $1024 \times 1024$ pixel images ($2.5^{\prime}$ on a side).
We began our analysis using event data calibrated by the initial processing and made available through the public archive. The first observation revealed several problems in the standard data sets and further problems were subsequently found during the analysis of the second observation. These problems affected both the spatial and timing analyses and had significant implications for the proper interpretation of the data. We alerted the HRC hardware and software teams to the instrument and data processing anomalies, and received considerable support in working through the problems. We document here the various artifacts discovered and the steps taken to correct for, or eliminate, them in our final data sets; our goals in doing so are 1) to allow for the replication of our results, and 2) to alert other early HRC users to problems they may encounter. In fact, we found it necessary to reprocess the data from Level 0.5 using custom scripts which incorporated improved filtering and processing tools, making use of several beta versions of software provided by the HRC team.
In the first observation we found evidence for a significant “ghost image” which appeared as a spectacular jet-like feature emanating from the pulsar along the detector $v$-axis. Examination by the instrument team found that the standard processing had failed to screen out all events flagged as instrumental. After filtering with a beta version of [SCREEN\_HRC]{} with the mask parameter set to $32771$, a much truncated jet-like feature was still apparent. In order to isolate any detector-centric artifacts, we obtained our second observation with a roll angle offset by $36^{\circ}$ from the first. As discussed below, we were able to confirm the reality of the residual jet-like feature in the cleaned images.
Independent of any filtering, the initial images showed the pulsar to be broader than the nominal PSF. To separate the pulsar from any proximate nebular emission, we followed the same phase-resolved imaging analysis described in Gotthelf & Wang (2000) for their HRC observation of PSR 0540$-$69, the 50 ms pulsar in the LMC. This separation, however, failed completely. By plotting the arrival times of the pulsar centroid, we observed that the sky coordinates of the pulsar wandered in a sinusoidal fashion with an amplitude of $0\farcs3$ and the periodicity of the programmed telescope dither. This accounted for the pulsar’s non-point-like appearance in the time-integrated image. Discussions with the attitude aspect team, however, showed a high-quality aspect reconstruction for the Vela observations.
Further analysis by the HRC team revealed a systematic problem with one of the three anode preamplifiers which causes the coarse position algorithm to mis-place photon locations depending on the photon input position relative to the HRC tap gaps. This explains the apparent wandering of the pulsar centroid at the dither frequency: a fraction of the detected photons are displaced along the detector coordinates by a fixed amount. Indeed, we were able to ascertain that the apparent diffuse flux was produced by faint echoes of the pulsar itself along the two orthogonal detector axes [^1].
To eliminate the echoes, we initially used the bright pulsar as a fiducial point to re-aspect the photons and thus take out the detector-induced wobble in a statistical sense. Subsequently, the instrument team made available a beta version of a code to identify and correct the mis-placed photons – [HRC\_EVT0\_CORRECT]{} – along with updated degap parameters ($cfu1=1.068$; $cfu2=0.0$; $cfv1=1.045$; $cfv2=0.0$) for use in [HRC\_PROCESS\_EVENTS]{}. This software, together with the new parameters, effectively eliminated the echo problem. The images produced by the two methods are indistinguishable and the pulsar now matches the PSF to within its estimated uncertainty.
In the second observation, we noted additional artifacts in the sky image resembling a rabbit-ear antenna extending $20^{\prime\prime}$ from the pulsar in orthogonal directions along the detector axes with point-like sources at the ends. Temporal analysis of the region showed that the rabbit-ear counts occurred during a number of specific time intervals lasting tens of seconds. Examination of the mission timeline parameters showed that the occurrence of these events always followed the “AOFF\_GAP” times by a few hundred seconds. Furthermore, data drop-outs were found for tens of seconds at the “AOFF\_GAP” times and during the following intervals when the spurious counts were recorded. We wrote an algorithm to generate a new good-time-intervals file which eliminates these intervals based on the “AOFF\_GAP” times.
Two additional detector issues needed consideration when extracting accurate timing information: telemetry saturation and a hardware time-stamp mis-assignment. Although data obtained during the first observation displayed nominal background levels, the second observation was plagued by intervals of telemetry saturation induced by high background levels; such occurrences can seriously affect timing studies by introducing spurious periods in the power spectrum aliased with the full buffer rate of $\sim 4$ ms. We filtered out telemetry-saturated time intervals with the dead-time fraction criteria of ${\rm DFT} > 0.9$. A further complication for precision timing was recently discovered by the HRC hardware team: the time-stamps for each event are mis-assigned to the following event. Based on the [VALID\_EVT\_COUNT]{} count rate of $\sim 500$ cps, the average error in the assigned photon arrival time is 2 ms or a 2% phase error for the Vela pulsar; assuming roughly Poisson fluctuations in the HRC count rate over the observation interval, the maximum error for any photon will be $\le 3$ ms. Thus, with 25 phase bins across the 88 ms pulsar period, few, if any, photons have been misassigned and we have taken no mitigating action to correct this error.
To compare directly the two observations, we reprocessed both data sets starting from the Level 0.5 event files using identical methods and filter/screening/processing criteria, compensating for incorrect keyword values, producing correct GTI extensions, etc. This resulted in a total of 50.3 ks and 45.3 ks integration times for the first and second observation, respectively. Despite all the initial discrepancies and artifacts in the two observations, this reprocessing produced effectively identical images, light curves and count rates. We are thus confident that we have eliminated all currently recognized instrumental artifacts in the final data sets upon which we base the analysis herein.
An Image of the Vela Pulsar
===========================
A global view of the Vela pulsar and its environment as seen by the HRC is presented in Figure 1. The pulsar is embedded in a complex region of previously resolved thermal X-ray emission from the Vela supernova remnant that is present throughout this image and extends far beyond its boundaries. The X-ray jet noted by Markwardt and Ogelman (1995) is essentially overresolved in this image and extends far to the south of the image boundary; it is evident as a faint enhancement in the diffuse emission extending to the southeast and south of the bright pulsar nebula.
The superb spatial resolution of the HRC provides the first look at the structure of the synchrotron nebula in the immediate vicinity of ; Figure 2 shows an image constructed from the two observations which have been centered on the pulsar and summed. The bright point source representing the pulsar has an extent roughly consistent with the local PSF. Apparently emanating from the pulsar, towards the southeast, is a linear, jet-like feature $10^{\prime\prime}$ in length. There is also evidence for a counter jet in the opposite direction. These jets have a position angle of $130^{\circ}$ (measured East of North), and are aligned to within $8^{\circ}\pm5^{\circ}$ degrees with the pulsar’s proper motion vector (Bailes et al. 1990; DeLuca, Mignani, and Caraveo 2000).
Concentric with the pulsar is a diffuse outer arc of emission perpendicular to the jet. This feature is roughly elliptical in shape and subtends an angle of $\sim 150$ degrees as seen from the pulsar. Interior to this arc is an elliptical ring of emission with a curvature very similar to the outer arc. The pulsar, jet, and arcs are embedded in a extended nebula of faint diffuse emission which has been described as ”kidney-bean" shaped (Markwardt and Ögelman 1998). The configuration of the jet feature relative to the nebula is reminiscent of the image of the Crab Nebula (Fig. 3; see Weisskopf et al. 2000).
We determined the count rates by extracting counts from the various regions discussed above (see Figure 4). For each source region we carefully estimated the background. For the diffuse emission, we determined the HRC detector background derived from an annulus $13.2^{\prime\prime}$ wide exterior to the kidney bean emission ($r>52.7^{\prime\prime}$). We extracted counts from the pulsar using a circular aperture 264 in radius and estimated background from the surrounding annulus, $2\farcs64 < r < 3\farcs43$. Table 1 summarizes the properties of the individual components, including their estimated sizes and intensities.
[rcc]{} A) Pulsar (Obs 1)& point-like & $2.012 \pm 0.006$ (Obs 2)& & $1.997 \pm 0.007$ B) Nebula (Obs 1)& $20^{\prime\prime} \times 10^{\prime\prime}$ NE-SW & $2.720 \pm 0.007$ (Obs 2)& & $2.762 \pm 0.008$ C) Arc (Obs 1)& & $0.675 \pm 0.004$ (Obs 2)& & $0.700 \pm 0.004$ D) Jet (Obs 1)& $10^{\prime\prime}$ long SE-NW & $0.037 \pm 0.001$ (Obs 2)& & $0.036 \pm 0.001$
The best current measurements for the Vela pulsar and remnant place it at a distance of only $250\pm30$ pc (Cha, Sembach, and Danks 1999 and references therein); in all that follows we scale by $d = 250 d_{250}$ pc.
Vela X, the $\sim 100^{\prime}$ diameter, flat-spectrum radio component near the center of the Vela remnant (Milne 1968; Bock et al. 1998) is generally regarded as the pulsar’s radio synchrotron nebula. While soft X-rays from this region are detected, they are primarily thermal in nature, and represent emission from the hot plasma which fills the entire remnant (Kahn et al. 1985; Lu and Aschenbach 2000). Even the bright radio filament detected by Bietenholz, Frail and Hankins (1991) shows no corresponding enhancement in our X-ray image, although more constraining limits will be derivable from ACIS observations.
The compact X-ray source near the pulsar was first recognized by Kellogg et al. (1973) as having a harder spectrum. Subsequent observations with increasing angular resolution (Harnden et al. 1985 and references therein; Ögelman, Finley, and Zimmermann 1993; Markwart and Ögelman 1998) localized the compact nebula to a region $\sim 2^{\prime}$ in extent roughly centered on the pulsar. Ögelman et al. (1993) used the nominal PSF of the ROSAT PSPC and an [*ad hoc*]{} model for the surface brightness of the diffuse emission to attempt a deconvolution of the pulsar and its nebula and to obtain spectral fits to the two components. They found that a blackbody effective temperature of 0.15 keV adequately characterized the point source, while the extended emission exhibited a power law spectrum with a photon index of $\sim 2.0$; a column density of $N_H = 1 \times 10^{20}$ cm$^{-2}$ is marginally consistent with both components. Markwardt and Ögelman (1998) subsequently revised the division of the flux between the point source and nebula based on ROSAT HRI observations. Seward et al. (2000) attempted to isolate the pulsar emission temporally and found somewhat lower blackbody temperatures. The RXTE observations of Gurkan et al. (2000) found a similar power law index for the nebula, but a much higher normalization; while this could indicate diffuse synchrotron X-ray emission from a larger area (given their one-degree field of view), it could also result from background modeling problems, since Vela is a weak source for RXTE.
As described above, our HRC image allows us to separate cleanly the pulsar from the nebular emission. We find count rates for the pulsar and the nebula minus the pulsar (within a radius of $50^{\prime\prime}$) completely consistent with those of Markwardt and Ögelman (1998) using the Ögelman et al. (1993) spectral parameters ($kT = 0.15$ keV; $N_H = 1 \times 10^{20}$ cm$^{-2}$), confirming these as a useful characterization of the observed flux. This leads to an unabsorbed, bolometric luminosity for the pulsar blackbody emission of $\approx 1.5 \times 10^{32} d^{2}_{250}$ erg s$^{-1}$. Note, however, that this luminosity is inconsistent with the adopted temperature ($T=1.7
\times 10^6$K) and distance (250 pc) for a uniformly radiating blackbody with a radius of 10 km; even for the minimum neutron star radius consistent with reasonable equations of state ($R\sim7$ km), the derived $L_X$ is too high by a factor of 20. Lowering $T$ to $8.5
\times 10^5$K as advocated by Seward et al. (2000) yields self-consistent values for $L_X, R, T$, and $d$, and raises the intrinsic luminosity by $\sim30\%$. Adjusting the distance, and including such effects as non-grey opacity in the neutron star atmosphere and a non-uniform temperature distribution over the surface will also affect the calculated luminosity. For the purpose of a comparison with glitch models (§6), we adopt $T=1.0 \times 10^6$ K.
The integrated luminosity of the whole nebula ($r<52.7^{\prime\prime}$ minus the pulsar) in the $0.1-10$ keV band is $3.5 \times 10^{32} d^{2}_{250}$ erg s$^{-1}$, corresponding to $4.9
\times 10^{-5}$ of the pulsar’s spin down luminosity. This ratio of $L_{neb}/ \dot E$ is significantly lower than that for any other pulsar, and is a major constraint on models for coupling the pulsar wind to the nebula (see §6).
The X-ray pulse profile
=======================
After many unsuccessful searches, Ögelman et al. (1993) were the first to detect X-ray pulsations from the Vela pulsar using the ROSAT PSPC. Their observations revealed a complex profile, not obviously related to the pulse profiles previously recorded at radio, optical, and gamma-ray wavelengths. The observed pulsed fraction of $4.4 \pm
1.1\%$ was diluted by the inability of the PSPC to resolve the pulsar from the surrounding nebula; using the approximate model described above, the authors estimated a soft X-ray pulsed fraction of 11%. Seward et al. (2000) constructed a higher signal-to-noise profile by combining six ROSAT HRC observations; they estimated a pulsed fraction of 12% divided between a broad component (8%) and two narrow peaks (4%). Strickman et al. (1999) and Gurkan et al. (2000) have recently derived 2-30 keV profiles based on RXTE observations of Vela. Strickman et al. illustrate a trend in which the component separation of the main pulse increases with energy.
We have determined the X-ray pulse period for our two observations and compared them to the radio ephemeris (Don Backer, personal communication). We began by constructing a periodigram around a narrow range of periods centered on the expected period $\pm 0.1$ ms, sampled in increments of $0.05 \times P^2/T$, where $T$ is the observation duration, and $P$ is the test period. For each trial period, we folded photons extracted from a $r = 2\farcs64$ aperture centered on the pulsar position using 25 phase bins and computed the $\chi^2$ of the resultant profile. We find a highly significant signal ($> 8
\sigma$) at $P = 89.32842(5)$ ms at epoch 51563.314043 MJD (TBD) and $P = 89.32876(6)$ ms at epoch 51595.370251 MJD (TBD), completely consistent with the observed radio periods. The uncertainty was estimated according to the method of Leahy (1987). For each fold, we adoped the period derivative determined from the radio ephemeris. Our measurements of the period are consistent (within the errors) with the radio prediction.
In order to compare the pulse profiles for the two observations we used the phase-connected radio ephemeris to fold and align them. We present the sum and difference profiles in Figure 5. This phase alignment is completely consistent with one we computed empirically by cross-correlating the two profiles; this suggests that the clock is stable to a few ms over a month. We can also derive the absolute radio to X-ray phase offset if we assume that the absolute time assignment is accurate (the calibration of this quantity has not yet been finalized). The phase of the radio peak relative to the X-ray profile is indicated in Figure 5.
The $\sim 200,000$ counts, uncontaminated by nebular emission, provides us with the highest signal-to-noise X-ray pulse profile for Vela yet reported (Figure 5). Greater than 99% of the counts from the blackbody component detected by the HRC fall in the $0.1-2.4$ keV ROSAT band. We compute a pulsed fraction by integrating the counts in the light curve above the lowest point and dividing by the total counts within a radius of $r=2.64^{\prime\prime}$ and subtracting the small amount of nebular background in this extraction radius (see Table 1). Our value for the pulsed fraction is $7.1 \pm 1.1\%$; the quoted error is dominated by the Poisson uncertainty in the number of counts recorded in the bin representing the light curve minimum. This value is somewhat lower than those cited above, but has the advantage of utilizing a direct measure of the total point-source contribution with subarcsecond resolution. We measure a separation between the two peaks of the main component of $\delta\phi \sim 0.325$, consistent with a linear extrapolation of the energy-dependence of this quantity reported by Strickman et al. (1999).
The background-corrected pulsar count rates were found to be $2.012
\pm 0.006$ and $1.997 \pm 0.007$ c s$^{-1}$, respectively, for the first and second epochs; the overall count-rate is constant to within 0.75% (the second observation is $1.5\sigma$ fainter than the first). Thus, the $3\sigma$ limit on any increase in the pulsar luminosity in response to energy input from the glitch is $<1.2 \times 10^{30}$ erg s$^{-1}$ or $\Delta T \sim 0.2\%$, 35 days ($3 \times 10^6$ s) after the event[^2]. The lower half of Figure 5 shows the difference between the two observations as a function of pulse phase, where the second dataset has simply been scaled by the ratio of the total integration times; no single bin has a discrepancy exceeding 1.5 sigma. This constancy in both the pulsed luminosity and pulse profile set interesting constraints on the glitch mechanism (see §6.4).
Changes in the Nebula
=====================
While the primary energy release from a glitch must be within (or on the surface of) the neutron star, the response of the star’s magnetosphere could result in the release of energy to the synchrotron nebula, triggering changes in its morphology and/or brightness. Even without the stimulus of a glitch, the optical wisps of the Crab Nebula near the pulsar have been shown to change on timescales of weeks, presumably in response to instabilities in the relativistic wind from the pulsar (Hester et al. 1995). Greiveldinger and Aschenbach (1999) have also reported changes in the X-ray surface brightness of the Crab Nebula on larger scales and somewhat longer timescales. Thus, we have examined our two images of the Vela nebula carefully in a search for surface brightness fluctuations.
We examined the count rate in the kidney-bean region bracketed by the outer-arc and an inner circle with $r = 1\farcs32$ centered on the pulsar. No significant change was observed between the two observations. Similarly, no measurable change in the count rate associated with the jet-like feature was found. Comparisons of other regions defined in Table 1 also showed no change, with the notable exception of the outer arc itself which appeared to increase in brightness by $\sim 5\%$ in the second observation.
To investigate this further, we examined regions congruent with the morphology of the nebula by constructing radial bins which are elliptical in shape; the ratio of the semi-major to semi-minor axes of the ellipse is 1.76, and the elliptical annuli are oriented at a position angle of $50^{\circ}$ (east of north). As Figure 6 shows, the sector of the radial profile encompassing the bright northwestern arc exhibits a $7.8\sigma$ excess between semi-minor axis radii of $13.5^{\prime\prime}$ and $18.0^{\prime\prime}$ in the sense that the source is brighter in the second observation. No other sectors or radii show any significant changes. In Figure 7, we display the azimuthal profile of the whole nebula in the elliptical ring $4.5^{\prime\prime}$ wide centered on these radii. The residuals are positive (the second observation is brighter) throughout the range $250^\circ$ to $30^\circ$; the excess is significant at the $7.6 \sigma$ level and represents a brightening of 5.3%. The excess energy being radiated in the HRC band amounts to $\sim 3 \times 10^{30}$ erg s$^{-1}$. For the assumed geometry discussed in $\S 6.1$, the outer arc lies at a distance of $1.05 \times 10^{17}$ cm from the pulsar, requiring signal propagation at $\simgt 0.7c$ if the impetus for the brightening originated from the pulsar at the time of the glitch.
Discussion
==========
Geometry and Kinematics of the Nebular Structure
------------------------------------------------
Figure 8 shows our proposed model for the Vela X-ray nebula. We assume that the two prominent arc-like features lie along circular rings highlighting shocks in which the energy of an outflowing equatorial wind is dissipated to become the source of synchrotron emission for the compact nebula extending to the boundary of the “bean”. One reason that the arcs are not complete rings might be that the emission is from outflowing particles which Doppler boost their emission in the forward direction.
This is essentially the picture for the similar arcs surrounding the Crab pulsar first suggested by Aschenbach and Brinkmann (1975) and later elaborated by other authors (Arons et al. 1998 and references therein). The main difference is that the dark cavity which contains the unshocked pulsar wind in the Kennel & Coroniti (1984a) model of the Crab is small compared with the volume of the Crab Nebula, while the radius of the Vela nebula (the bean), is barely twice as large as its pulsar wind cavity (see Figure 4). We assume that the two rings straddle the equator symmetrically, and suppose that the deficit of emission exactly in the equatorial plane is related to the fact that this is where the direction of a toroidally wrapped magnetic field changes sign; i.e., the field may vanish there. The semimajor axis of the ring $a = 25.7^{\prime\prime}$, and the ratio $a/b = 1.67$ specifies the angle that the axis of the torus (i.e., the rotation axis of the pulsar) makes with the line of sight, $\zeta = {\rm
cos}^{-1}(b/a) = 53^{\circ}.2$. The angle $\Psi_0 = 130^{\circ}$ is the position angle of the axis of the torus on the plane of the sky, defined according to convention as the angle measured to the east from north. The direction of rotation (the sign of $\Omega$) is arbitrary.
The projected separation of the two rings is measured as $s =
17.7^{\prime\prime}$. The half opening angle of the wind $\theta$ is then given by ${\rm tan}\,\theta\ =\ s/(2a\,{\rm sin}\,\zeta) = 0.43
(\theta = 23.\!^{\circ}3)$, and the radius of the shock is $r_s =
a\,d\,/{\rm cos}\,\theta$, where $d$ is the distance to the pulsar. For $d = 250$ pc we find $r_s = 1.05 \times 10^{17}$ cm.
The rotating vector model of pulsar polarization (Radhakrishnan & Cooke 1969) is commonly used to derive information about the geometry of the pulsar magnetic inclination and viewing angles. The angles $\zeta$ and $\Psi_0$ can in principle be evaluated independently using information derived from polarization measurements of the radio pulse. In particular, the swing in position angle $\Psi (t)$ of linear polarization across the pulse is very sensitive to $\zeta$, the angle between the line of sight and the rotation axis. The angle $\alpha$ between the magnetic axis and the rotation axis is much more difficult to measure unless $\alpha \approx \zeta$ – [*i.e.*]{}, unless the line of sight passes near the center of the polar cap. Accordingly, $\alpha$ is often assumed while $\zeta$ is fitted. For example, Krishnamohan & Downs (1983) assume $\alpha = 60^{\circ}$ in their model for Vela; this is consistent with the value $\alpha = 65^{\circ}$ derived by Romani and Yadigaroglu (1995) from a fit of their geometric gamma-ray emission models to Vela’s pulse profile. When an interpulse is observed, $\alpha$ is often inferred to be $90^{\circ}$ (that is, both polar caps are visible in this case). With these definitions (see Figure (8) or Figure (13) of Krishnamohan & Downs (1983) for the geometry),
$${\rm tan}(\Psi (t) - \Psi_0)\ =\ {{\rm sin}\,\phi (t) \over
{\rm cot}\,\alpha\ {\rm sin}\,\zeta\ -\ {\rm cos}\,\zeta\
{\rm cos}\,\phi (t)}.\eqno(1)$$
Here $\phi(t)$ is the longitude of the emitting region, which increases linearly with time, and $\Psi_0$ is the position angle of the rotation axis of the pulsar projected on the sky as in Figure (8).
In the context of the rotating vector model, $\Psi_0$ is identical to the position angle of polarization $\Psi$ at the peak of the pulse where the magnetic dipole axis crosses the rotation axis ($\phi (t) = 0$ in Equation (1)). Because the emission mechanism is thought to be curvature radiation from particles moving along magnetic field lines, the electric vector is tangent to those field lines, rather than perpendicular to them as is the case with synchrotron radiation. While this measurement is in principle straightforward, in practice it is not routinely accomplished. Observations at two or more frequencies are needed to determine (and to correct for) the interstellar rotation measure, and to demonstrate that the intrinsic polarization is in fact frequency-independent. Another complication is that a pulse is often composed of several identifiable components, some of which can be polarized in the orthogonal mode (a result of propagation effects in the magnetosphere) obscuring the “true” polarization. Furthermore, the pulse itself might not even contain an identifiable core component, being composed instead of emission from random patches within a cone (e.g., Lyne and Manchester 1988; Manchester 1995; Deshpande and Rankin 1999). Accordingly, measurements of $\Psi_0$ are rarely attempted, measurements of $\alpha$ are rarely trusted, and measurements of $\zeta$ are rarely questioned.
In fact, there are several determinations of $\Psi_0$ for the Vela pulsar that are not in particularly good agreement with each other; we review a representative subset here. The original value of Radhakrishnan & Cooke (1969) is $\Psi_0 = 47^{\circ}$ with an uncertainty of $\approx 5^{\circ}$. Hamilton et al. (1977) made measurements over several years, all of which are consistent with $\Psi_0 = 64^{\circ} \pm 1.\!^{\circ}5$. A detailed decomposition into four separate pulse components was performed by Krishnamohan & Downs (1983), in which they concluded that one of the components was polarized in the mode orthogonal to the other three. However, they did not attempt an absolute measurement of the angle $\Psi_0$. Bietenholz et al. (1991) measured $\Psi_0 = 35^{\circ}$ using the VLA (although this measurement may not be directly comparable, in that it represents a mean value weighted by the degree of linear polarization rather than the value at the center of symmetry of the position angle curve). Thus, while the published values of $\Psi_0$ differ by as much as $30^{\circ}$, it appears that none is even close to being aligned with the axis of the nebula, and that all are roughly perpendicular to it.
Interestingly, the model of Krishnamohan & Downs (1983) produces a precise (albeit model-dependent) value of the angle between the rotation axis and the line of sight, $\zeta = 55.\!^{\circ}57 \pm 0.\!^{\circ}15$. This value agrees well with the inclination angle of our postulated equatorial wind torus to the line of sight derived by fitting an ellipse to the shape of the X-ray features: $\zeta = {\rm cos}^{-1}(b/a)
= 53{^\circ}.2$. This striking coincidence gives us courage to pursue the basic physics of the equatorial wind model using the geometry of Figure 8, and even to be so bold as to suggest that [*all*]{} of the radio determinations of $\Psi_0$ for Vela are incorrect by $90^{\circ}$ because of incorrect mode identification ($\it i.e.$, perhaps three out of four of the pulse components are actually polarized in the orthogonal mode). In this case, $\Psi_0 =
130^{\circ}$ (as inferred from the orientation of the X-ray torus), and can be identified with the projected direction of the pulsar rotation axis. Speculations about the true orientation of $\Psi_0$ in pulsars go back to Tademaru (1977), who first discussed the possible alignment between spin axis and proper motion in the context of the radiation rocket hypothesis (Harrison & Tademaru 1975).
Implications of the Proper Motion
---------------------------------
For both the Crab and Vela pulsars, the direction of proper motion (transverse velocity $v_t$) is strikingly close to the projected X-ray symmetry axis of the inferred equatorial wind and polar jet-like structures. The proper motion of the Vela pulsar has been measured with comparable accuracy using both radio interferometry (Bailes et al. 1990) and images from [*HST*]{} (De Luca et al. 2000). The resulting mean value $0.\!^{\prime\prime}056 \pm 0.\!^{\prime\prime}004$ yr$^{-1}$ at a position angle of $302^{\circ} \pm 4^{\circ}$ is within $8^{\circ}$ of the axis of the nebula ($\Psi_0 + 180^{\circ} = 310^{\circ}$). The transverse velocity $v_t = 65$ km s$^{-1}$ at $d = 250$ pc. The proper motion of the Crab pulsar is $0.\!^{\prime\prime}018 \pm
0.\!^{\prime\prime}003$ yr$^{-1}$ at a position angle of $292^{\circ}
\pm 10^{\circ}$ (Caraveo & Mignani 1999), which corresponds to $v_t =
123$ km s$^{-1}$ at $d = 2000$ pc. The axis of the Crab’s toroidal optical and X-ray structure is $299^{\circ}$ (Hester et al. 1995), only $7^{\circ}$ from the direction of proper motion. The probability that two such close alignments will occur by chance when drawn from a pair of uncorrelated distributions is 0.7%. We also note that both Vela and the Crab are rather slow moving compared to typical young pulsars; Lyne & Lorimer (1994) found a mean velocity for young pulsars of between 400 and 500 km s$^{-1}$.
If these relationships are not a coincidence, then they may be understandable in terms of the scenario proposed by Spruit & Phinney (1998) who suggested that the rotation axes and space velocities of pulsars could be connected through the nature of the “kicks” given to neutron stars at birth. Spruit & Phinney argue that the rotation rate of the progenitor stellar core is too slow in the few years before the formation of the neutron star for pulsar spin periods to be explained by simple conservation of angular momentum during core collapse. Instead, it is likely that the same asymmetric kicks (whatever the cause) that are responsible for the space velocities of pulsars, are also the dominant contributors to their initial spin rates. If neutron stars acquire their velocities from a single momentum impulse, then their rotation axes should be perpendicular to their space velocities. If, however, they receive many random, independently located impulses over time, as might result from convection which leads to anisotropic neutrino transport or anisotropic fallback, then their velocities and spins should be uncorrelated in direction. However, if those multiple thrusts are not short in duration relative to the resulting rotation period, it is possible that kicks applied perpendicular to the rotation axis will average out, while those that are along the rotation axis will accumulate. In the latter case, particularly germane for short rotation periods, the space velocity will be preferentially aligned with the rotation axis. This is actually the scenario preferred by Spruit & Phinney, for which they appeal to long duration (several-second) thrusts that could result from the effect of parity violation in neutrino scattering in a magnetic field.
Thus, in the context of the above models, we speculate that the Crab and Vela pulsars have relatively low space velocities because the components of their kicks perpendicular to their rotation axes were averaged out and, as a result, their final space velocities were aligned closely with their spin axes (cf. Lai, Chernoff, and Cordes 2001). We also note that alignment of the spin axis and proper motion is a natural consequence of the Harrison and Tademaru (1975) photon rocket acceleration mechanism. With the recent revision of Lai et al. (2001), an initial spin period as long as 6 ms suffices to account for the measured transverse component of the velocity in the maximum acceleration case. Since pulsars that rotate most rapidly at birth are also the ones most capable of powering synchrotron nebulae, either scenario might argue for a stronger than average correlation of the axes of such nebulae with the proper motion directions of their parent pulsars.
Before the toroidal arcs in the Vela nebula were resolved by [*Chandra*]{}, Markwardt & Ögleman (1998) interpreted the overall shape of the nebula as seen by the [*ROSAT*]{} HRI as being [*determined*]{} by the space velocity of the pulsar. In particular, they noted that the outline of the nebula, which is dominated by the bean shape, resembles a bow shock whose symmetry axis at position angle $295^{\circ}$ is identical to the direction of the radio proper motion ($297^{\circ}$). However, the rather uniform and gently curved outline of this structure could only be reconciled with the sharper, asymmetric curve expected of a bow shock if the space velocity were nearly along the line of sight. This requirement, coupled with the large absolute velocity needed for the pulsar to exceed the speed of sound in the surrounding supernova remnant forced Markwardt & Ögleman to conclude that the velocity vector is less than $22^{\circ}$ from the line of sight. This notion of the compact Vela nebula as a bow shock also led Chevalier (2000) to a considerably different model of its physics. As we shall argue below, the [*Chandra*]{} observations do not support such a bow-shock interpretation, but instead favor a physical model in which the entire structure is a synchrotron nebula similar in physics to the Crab, but with an interesting difference in one of its parameters.
A Physical Model of the Nebula
------------------------------
In the basic Kennel & Coroniti (1984a,b) model of the Crab Nebula, a relativistic pulsar wind terminates in an MHD shock, which produces the nonthermal distribution of particles and post-shock magnetic field that comprise the synchrotron nebula. Although the pulsar wind is assumed to carry the entire spin-down luminosity of the pulsar, specific wind parameters such as the particle velocity and the fraction of the power carried in magnetic fields are not known [*a priori*]{}. Rather, they are inferred by using the results of the shock jump conditions to model the spectrum of the Crab Nebula, and also to match the observed radii of the MHD shock and the outer boundary of the Nebula. It is necessary to adopt outer boundary conditions; a natural one is to require the final velocity of the flow to match the observed expansion velocity of the Nebula, although it is not clear how this outer boundary condition is communicated to the inner MHD shock which is a factor of 20 smaller in radius.
A peculiar result of the Kennel & Coroniti model is that the wind magnetization parameter $\sigma$ is required to be $\approx 0.003$ in the Crab. That is, the fraction of power carried in $B$ field is much less than 1%. Such a small fraction is required in order that sufficient compression occurs in the shock to convert the bulk flow energy into random energy of the particles so that they can radiate the observed synchrotron luminosity. Highly magnetized shocks produce less radiation because there is little energy dissipation and, for the Crab, would supply an insufficient number of X-ray emitting particles. Furthermore, highly magnetized shocks are weak because all of the energy dissipation allowed by the jump conditions is used in making the small increase in $B$ field needed to conserve magnetic flux. The post-shock flow velocity is still relativistic.
The reason that such a small magnetization is difficult to understand is that the pulsar magnetic field energy carried out to the radius of the shock in the simplest MHD wind should be of the same order of magnitude as the spin-down power, as the following argument shows. The total wind energy flux at the shock is
$${I\,\Omega\,\dot\Omega \over 4\pi\,r_s^2}\ =\
\left ( {B_p^2\,R^6\,\Omega^4 \over 6\,c^3}\right )
\ {1 \over 4\pi\,r_s^2}\ =\
{B_p^2\,R^6\,\Omega^4 \over 24\pi\,c^3\,r_s^2}
\eqno(2)$$
while the transported pulsar magnetic field $B_s$ at the location of the shock is
$$B_s \approx {B_p \over 2} \,\left ( {R \over r_{lc}} \right )^3\,
\left ( {r_{lc} \over r_s} \right )\ =\
{B_p\over 2} {R^3\,\Omega^2 \over c^2\,r_s}\eqno(3)$$
where $r_{lc}$ is the radius of the light cylinder defined as $r_{lc} =
c/\Omega$ and $R$ is the neutron star radius at which the magnetic field strength is $B_p$. Therefore, the magnetic energy flux at $r_s$ is
$$c\,{B_s^2 \over 8\pi}\ \simeq\ {B_p^2\,R^6\,\Omega^4 \over 32\pi\,c^3\,r_s^2}\ ,
\eqno(4)$$
comparable to the value in Equation (2). While various solutions to this paradox for the Crab have been proposed, we argue here that the dimensions and spectrum of the Vela synchrotron nebula are in much better accord with $\sigma \sim 1$.
A basic application of the pulsar wind model to the Vela synchrotron nebula was made by de Jager, Harding, & Strickman (1996) in conjunction with their detection of Vela with the Oriented Scintillation Spectrometer Experiment (OSSE) on the [*Compton Gamma-Ray Observatory*]{}. We summarize their conclusions here. de Jager et al. noted that the unpulsed part of the hard X-ray spectrum extends with power-law photon index $\Gamma = 1.73$ up to 0.4 MeV, which implies that the synchrotron nebula radiates $\sim 2 \times 10^{33}(E_{\rm max}/0.4\,{\rm MeV})^{0.27}$ ergs s$^{-1}$, or only $\sim 3 \times 10^{-3}$ of the pulsar spin-down power. The absence of an observed spectral break limits the residence time $\tau_r$ of the electrons in the nebula that radiate in this energy range to less than their synchrotron lifetime $\tau_s = 5.1 \times 10^8/
(\gamma\,B^2)$ s. Since the electrons radiating at the highest observed energy $E_{\rm max}$ have $\gamma = [2\pi\,m\,c\,E_{\rm max}/(h\,e\,B)]^{1/2}$, the upper limit on the nebular magnetic field is $B < 20\ \tau_r^{-2/3}$ G. To make use of this limit, de Jager et al. assumed that the residence time would be $\tau_r \approx r/c_s$ where the nebula outer radius $r \sim 2 \times 10^{17}$ cm, and $c_s = c/\sqrt 3$, the velocity of sound in a relativistic plasma. For these values, $B < 4 \times 10^{-4}$ G. This requirement is compatible with a strongly magnetized pulsar wind for which little change in $B$ occurs across the shock, while the post-shock field continues to decline inversely proportional to the distance from the pulsar. Indeed, equation (3) predicts a pre-shock field of $\sim 1.5 \times 10^{-4}$ G, consistent with the limit derived above; thus, the wind may remain relativistic across the nebula, which extends a factor of 2 in radius beyond the shock.
Such a synchrotron nebula is in approximate pressure balance with the surrounding supernova remnant. Markwart & Ögelman (1997) found by fitting a two-temperature thermal model to the [*ASCA*]{} spectrum of the inner remnant that the thermal pressure is $\approx 8.5 \times
10^{-10}$ erg cm$^{-3}$. This compares well with the pulsar wind pressure at $r$, $\dot E/(4\pi\,r^2\,c) = 5.2 \times 10^{-10}$ erg cm$^{-3}$. The entire compact X-ray nebula, then, is consistent with being powered by a strongly magnetized pulsar wind shock whose still relativistic downstream flow is confined by the Vela SNR. The modest production of synchrotron electrons in such a shock naturally explains the very low value of $L_X/\dot E$ observed.
In summary, the match between the nebular field upper limit derived from the [*OSSE*]{} data and the value found from equation 3 using our measured value of $r_s$, and the fact that the pressure corresponding to this field strength matches the confining thermal pressure of the X-ray gas leads us to conclude that our high-magnetization model of the Vela nebula is both self-consistent and plausible. Thus, in a sense, Vela may be a more natural realization of the Kennel & Coriniti model than is the Crab, for which the model was created.
While this appears to be a quite satisfactory model, certain details are subject to additional constraints. First, since the radiation from the nebula is so inefficient, energetic electrons must be able to escape to much larger distance scales before losing all of their energy to synchrotron radiation. Indeed, the radio luminosity of the 100-arcminute Vela X region, $\sim8 \times 10^{32}$ erg s$^{-1}$, could be one manifestation of the escaping electrons. Second, there is a natural upper energy to the synchrotron spectrum when the electron gyroradius $r_g$ exceeds the radius of the nebula. Since $$r_g\ =\ 1.6 \times 10^{17}\
\left ( {B \over 10^{-4}\ {\rm G}} \right )^{-3/2} \
\left ( {E \over 100\ {\rm MeV}} \right )^{1/2} \ {\rm cm}
\eqno(5)$$ this is not a restrictive limit. Thus, we find the de Jager et al (1996) description of the Vela synchrotron nebula basically in accord with the [*Chandra*]{} observations.
An alternative picture was proposed by Chevalier (2000) based on the bow-shock interpretation of Markwardt & Ögleman (1998), and a simplified version of the Kennel & Coroniti model. If the Vela synchrotron nebula is energized by a shock between a relativistic pulsar wind and the surrounding supernova remnant, then the bow shock travels at the velocity of the pulsar $v_p$, which necessarily exceeds the sound speed in the hot confining medium. In the first place, such a large pulsar velocity is hardly likely, since the thermal sound speed $\sqrt{dP/d\rho}\,\approx\,875$ km s$^{-1}$ in the Vela SNR according to the [*ASCA*]{} spectral analysis of Markwardt & Ögleman (1997). Markwardt & Ögleman (1998) assumed that $v_p
\geq 260$ km s$^{-1}$ is sufficient.
An additional consequence of this scenario, however, is that the residence time of the emitting particles in the nebula is much longer than in the de Jager et al. model, $\tau_r \sim r/v_p$ instead of $\tau_r
\sim \sqrt 3 r/c$. Consequently, Chevalier (2000) was forced to assume $\tau_r \sim 10^3$ yr, requiring an extremely small magnetization parameter, $\sigma < 10^{-4}$, in order that the model’s radiated luminosity not exceed the observed X-ray luminosity. We consider that the physical difficulties of the bow-shock interpretation, in conjunction with the new [*Chandra*]{} evidence that the X-ray morphology is dominated by a pair of arcs resembling similar toroidal structures in the Crab Nebula, strongly disfavor such a model. Chevalier’s revised estimate of $\sigma$ is 0.06 after adopting our assumption of relativistic post-shock flow. Determining whether $\sigma$ is actually of order unity or not will require more detailed modeling
While our model provides a plausible explanation for the torroidal arcs and outer nebula, it says nothing about the other striking feature of the image, the jet and counterjet. The jet is $10^{\prime\prime}$ long, giving it a deprojected length of $4.1\times 10^{16}$ cm. At $v=65$ km s$^{-1}$, it has taken the pulsar $\sim 200$ yr to travel this distance. The synchrotron lifetime of electrons producing 1 keV emission ranges from 5 yr to 40 yr for the fields of $(1-4)\times 10^{-4}$ G discussed above. Thus, the jet is not simply a wake, but must be supplied with particles from the pulsar (as the existence of the counterjet also suggests). The luminosity required is modest: $L_X \sim 5 \times 10^{30}$ erg s$^{-1}$, roughly 1% of the nebular luminosity and $< 10^{-6} \dot E$. While we have no scenario to propose, the existence of such features in both Vela and the Crab suggests understanding their origin may prove useful in modeling particle flow from young pulsars.
Thermal Emission Constraints on the Neutron Star Interior
---------------------------------------------------------
Several models have been advanced to explain the sudden apparent change in the moment of inertia of a glitching neutron star. Originally, starquakes, resulting from the release of strain in the stellar crust induced by the change in the equilibrium ellipticity of the star as it slows, were invoked (Ruderman 1969). But the magnitude and frequency of the Vela glitches could not be explained by this model, and a picture involving the sudden unpinning from the inner crust of superfluid vortices in the core of the star became the dominant paradigm (Anderson and Itoh 1975). Observations of the relaxation of the star back toward its original spin-down rate suggest that $\sim 1\%$ of the star’s mass is involved in the event (Alpar et al. 1988; Ruderman, Zhu, and Chen 1998), implying a total energy release of $\sim 10^{42}$ ergs.
The fate of this energy is unclear and predictions concerning the observable consequences vary widely. The timescale for energy deposited at the base of the crust to diffuse outward, the fraction of the surface area whose temperature will be affected, and the secondary effects, such as the rearrangement of the surface magnetic field which could dump energy into the surrounding synchrotron nebula, are all uncertain by one or more orders of magnitude.
Our stringent upper limit on a change in the X-ray flux from the neutron star within the 35 days following the glitch allows us to begin setting meaningful constraints on the parameters of the neutron star and the glitch. Seward et al. (2000) provide a concise introduction to the published models for the thermal response of the stellar surface to a glitch generated by the sudden unpinning of superfluid vortex lines deep in the star (Van Riper, Epstein, and Miller 1991; Chong and Cheng 1994; Hirano et al. 1997, and Cheng, Li, and Suen 1998), and we need not repeat it here. We follow their approach in deriving parameter limits from the models.
The allowable change in the pulsar’s flux derived in §4 corresponds to a fractional change in the surface temperature of $ <0.2\% $. For timescales of $\sim 30$ days, we are primarily sensitive to stars with small radii ($R<14$ km) which correspond to soft or moderate equations of state. Using Figure 2 of Van Riper et al. (1991), we can set a limit of $E_{glitch} < 10^{42}$ ergs independent of the depth of occurrence within the inner crust. For depths corresponding to local densities $ \rho < 10^{13}$ gm cm$^{-3}$, $E_{glitch} < 3 \times
10^{41}$ ergs, and for shallow events ($\rho \sim 10^{12}$ gm cm$^{-3}$), the glitch energy must be less than $10^{41}$ ergs. For the softest equation of state used by Hirano et al. (1997) corresponding to a $1.4M_{\odot}$ star with a radius of 11 km, glitch depths shallower than $10^{13}$ gm cm$^{-3}$ require an energy deposition of less than $10^{41}$ ergs. With observations of similar sensitivity $\sim300$ and $\sim3000$ days after the event, we could rule out $E_{glitch} \sim 10^{43}$ erg for all equations-of-state and glitch-depth combinations, and require $E_{glitch}<10^{41}$ ergs for soft and moderate equations of state for depths $\rho <10^{13.5}$ gm cm$^{-3}$. Note that on the longest timescales (appropriate for deep glitches in stars with very stiff equations of state), Vela may be an inappropriate target for constraining glitch parameters, since another glitch may well have occurred before the thermal pulse has peaked; indeed, if even 10% of the glitch energy appears as surface thermal emission, the total X-ray luminosity can be powered by events with $<E_{glitch}>\sim2 \times 10^{41}$ ergs.
Summary and Conclusions
=======================
We have presented a high-resolution X-ray image of the Vela pulsar revealing a highly structured surrounding nebula. We interpret the nebula’s morphology in the context of the shocked MHD wind model developed by Kennel and Coroniti (1984a,b) for the Crab Nebula, and find that the Vela nebula allows a large magnetization parameter, possibly of order unity. This picture also provides a natural explanation for the low $L_X/\dot E$ of the Vela nebula. We speculate that the alignments of the symmetry axes of the Crab and Vela nebulae with the proper motion vectors of their respective pulsars should be expected preferentially in rapidly spinning young pulsars with surrounding X-ray synchrotron nebulae if the causal connection between spin and proper motion suggested by Spruit and Phinney (1998) is correct.
Our two observations, centered 3.5 and 35 days after the largest glitch yet recorded from the pulsar, allow us to set significant limits on changes in the pulse profile and stellar luminosity which can be used to constrain glitch model parameters. We find that, for soft and moderate equations of state, the glitch energy must be $<10^{42}$ ergs; an additional observation a year following the event will substantially tighten this constraint. An apparent change in the nebula surface brightness between the two observations may or may not be a consequence of the glitch; the implied velocity of the disturbance, assuming that it originates near the pulsar, is $\sim 0.7
c$, similar to the velocity inferred from changes in the Crab Nebula wisps.
Future observations with and XMM can be used to gain further insight into the structure of the neutron star and its surrounding nebula. An observation $\sim 1.5$ yr after the glitch, now scheduled, will further constrain models for the glitch and the parameters of the neutron star. Additional HRC observations will also be required to decide whether the nebula changes reported here are a consequence of the glitch, or whether they occur routinely in response to instabilities in the pulsar’s relativistic wind, as appears to be the case in the Crab Nebula. Data from the EPIC PN camera on XMM will yield spectral clues helful in understanding the complex pulse profile, while either EPIC or ’s ACIS could be used to search for spectral changes caused by synchrotron energy losses and/or internal shocks in the nebula.
We are grateful to the Chandra Science Center Director, Dr. Harvey Tananbaum, for making this TOO possible. We also wish to acknowledge Dr. Steven Murray and Dr. Michael Juda for many extremely helpful discussions concerning HRC issues, and for kindly making available beta-version software. We thank Drs. Fernando Camilo and George Pavlov for discussion of the pulsar timing issues and assistance in determining the radio pulse phase, and Don Backer for providing the radio ephemeris. This work was funded in part by NASA LTSA grant NAG5-7935 (E.V.G.), and SAO grant DD0-1002X (D.J.H). This is contribution \#692 of the Columbia Astrophysics Laboratory.
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[^1]: This problem does not affect early HRC-I observations obtained prior to an increase in the instrument gain by a factor of two, such as those of the 50 ms pulsar PSR 0540$-$69 (see Gotthelf & Wang 2000).
[^2]: This estimate includes a $\sim 20\%$ correction for the fraction of the bolometric luminosity lying below the HRC band.
| 0 |
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abstract: 'We show that weakly guiding nonlinear waveguides support stable propagation of rotating spatial solitons (azimuthons). We investigate the role of waveguide symmetry on the soliton rotation. We find that azimuthons in circular waveguides always rotate rigidly during propagation and the analytically predicted rotation frequency is in excellent agreement with numerical simulations. On the other hand, azimuthons in square waveguides may experience spatial deformation during propagation. Moreover, we show that there is a critical value for the modulation depth of azimuthons above which solitons just wobble back and forth, and below which they rotate continuously. We explain these dynamics using the concept of energy difference between different orientations of the azimuthon.'
address: |
$^1$Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Stra[ß]{}e 38, 01187 Dresden, Germany\
$^2$State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, 710119 Xi’an, China\
$^3$Institute of Condensed Matter Theory and Solid State Optics, Friedrich Schiller University, Max-Wien-Platz 1, 07743 Jena, Germany\
$^4$Laser Physics Center, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200, Australia
author:
- 'Yiqi Zhang$^{1,2}$, Stefan Skupin$^{1,3}$, Fabian Maucher$^1$, Arpa Galestian Pour$^3$, Keqing Lu$^{2}$, Wieslaw Królikowski$^4$'
title: Azimuthons in weakly nonlinear waveguides of different symmetries
---
[10]{}
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Introduction
============
Spatial solitons are nonlinear localized states that keep their form during propagation due to the balance between diffraction and self-induced nonlinear potential [@Stegeman_science_1999]. Recently, there has been a lot of interest in a generalized type of spatial solitons, the so-called azimuthons. These are azimuthally modulated beams, that exhibit steady angular rotation upon propagation [@desyatnikov_prl_2005]. They can be considered as azimuthally perturbed optical vortices, i.e. beams with singular phase structure [@coullet_oc_1989; @desyatnikov_prl_2005; @lashkin_pra_2008]. Theoretical studies demonstrated both, stable and unstable propagation of azimuthons [@buccoliero_ol_2008; @lopez-aguayo_oe_2006; @buccoliero_prl_2007], and the first experimental observation of optical azimuthons was recently achieved in rubidium vapors [@minovich_oe_2009].
It appears that higher order solitonic structures and optical vortices are generally unstable in typical nonlinear media with local (Kerr-like) response [@kruglov_jmo_1992; @skryabin_pre_1998]. On the other hand, it has been shown that a spatially nonlocal nonlinear response provides stabilization of various complex solitonic structures including vortices [@suter_pra_1993; @bang_pre_2002; @krolikowski_job_2004; @briedis_oe_2005]. Consequently, azimuthons and their dynamics have been studied almost exclusively in the context of spatially nonlocal nonlinear media [@lopez-aguayo_ol_2006; @buccoliero_prl_2007; @stefan_oe_2008; @Fabian:oqe:09]. In spite of the fact that there are various physical settings exhibiting nonlocality such as nematic liquid crystals [@peccianti_ol_2002; @conti_prl_2003; @conti_prl_2004], Bose-Einstein condensates [@lashkin_pra_2008; @nath_pra_2007; @lashkin_pra_2009], plasmas [@litvak_plasmas_1975], thermo-optical materials [@rotschild_prl_2005] etc., experimental realization of such systems is always quite involved. Moreover, from the theoretical point of view, nonlocal media are quite challenging for numerical modeling and analytical treatment.
In this paper we propose a much simpler and experimentally accessible optical system to study the propagation of azimuthons: a weakly nonlinear optical multi-mode waveguide. Here, weakly nonlinear means that the nonlinear induced index change, which is proportional to the intensity of the optical beam, is small compared to the index profile (or trapping potential) of the waveguide. Following [@stefan_pre_2004], we can expect that weakly nonlinear azimuthons are stable in multi-mode waveguides.
The paper is organized as follows: In Sec. \[modelling\], we briefly introduce the general model equation for beam propagation in weakly nonlinear waveguides, and then we discuss in detail the properties of dipole azimuthons in circular and square waveguides in Sec. \[dipoles\], respectively. In Sec. \[higherorder\], higher order azimuthons are investigated, and in Sec. \[conclusion\] we conclude.
Mathematical modeling {#modelling}
=====================
We consider the evolution of a continuous wave (CW) optical beam with amplitude $\mathcal{E} (\xi, \eta, \zeta)$, where $(\xi,\eta)$ and $\zeta$ denote the transverse and longitudinal coordinates, respectively. Then the propagation of this beam in a weakly-guiding waveguide with Kerr nonlinearity in the scalar, slowly varying envelope approximation is described by the following equation [@boyd_book_2008]: $$i \frac{\partial}{\partial \zeta} \mathcal{E} +
\frac{1}{2k_0} \left( \frac{\partial ^2}{\partial \xi^2} + \frac{\partial ^2}{\partial \eta^2} \right) \mathcal{E} +
k_0 \frac{n_2}{n_b} |\mathcal{E}|^2 \mathcal{E} + k_0 \frac{n(\xi,\eta)-n_b}{n_b} \mathcal{E}=0,
\label{physicaleq}$$ where $k_0=2 \pi n_b / \lambda_0$ refers to the carrier central wave number in the medium; $n_2$ is the Kerr nonlinear coefficient, $n(\xi,\eta)$ the linear refractive index distribution, $n_b$ the background index, and $\lim_{\xi,\eta \rightarrow \infty}n(\xi,\eta)=n_b$. We consider a weakly-guiding waveguide, so both the linear and nonlinear induced index change $|n-n_b|$ and $n_2|\mathcal{E}|^2$ are small compared to the mean index $n_b$, and at the same time $n_2|\mathcal{E}|^2 \ll |n-n_b|$, to guarantee a weak nonlinearity.
From the mathematical point of view, Eq. (\[physicaleq\]) is a (2+1)-dimensional nonlinear Schr[ö]{}dinger (NLS) equation with linear potential representing the waveguide profile. For technical convenience, we rescale Eq. (\[physicaleq\]) to dimensionless quantities with $x=\xi/r_0$, $y=\eta/r_0$, $z=\zeta/(2 k_0 r_0^2)$, and $\sigma=\mathrm{sgn}(n_2)$, where $r_0$ represents the transverse spatial extent of the waveguide. Then, the two-dimensional (2D) NLS equation for the scaled wave function $\psi = k_0 r_0 \sqrt{2|n_2|/n_b} \mathcal{E}$ reads $$i\frac{\partial}{\partial z} \psi + \left( \frac{\partial ^2}{\partial x^2}
+ \frac{\partial ^2}{\partial y^2} \right) \psi +
\sigma |\psi|^2 \psi + V \psi=0,
\label{partial}$$ with an “attractive” bounded potential $V=2k_0^2 r_0^2 (n-n_b)/n_b$, given by the spatial profile of the refractive index of the waveguide. Here we will consider propagation in step index waveguides with parameter $V(x,y)$ given as follows $$V(x,y)=
\begin{cases}
V_0 & \mathrm{~where~}
\begin{cases}
\sqrt{x^2+y^2} \leq 1 & \mathrm{~for~circular~waveguide}
\\
|x| \leq 1 ~\&~ |y| \leq 1 & \mathrm{~for~square~waveguide}
\end{cases}
\\
0 & \mathrm{elsewhere}
\end{cases}.$$ $V_0$ is the height of the potential and determines the number of linear waveguide modes.
Considering the specific case of silica-made waveguides, typical values for the parameters are $n_b=1.4$, $|n-n_b| \leq 9 \times 10^{-3}$, $n_2 = 3\times 10^{-16} ~\mathrm{cm}^2/\mathrm{W}$ at a vacuum wavelength of $\lambda_0=790$ nm, and those values do not vary much when we choose any $\lambda_0$ in the range from visible to near-infrared. The laser-induced-damage-threshold (LIDT) for synthetic fused silica is about $10 ~\mathrm{J}/\mathrm{cm}^2$ for nanosecond pulses (according to Eq. (1) and Table 2 in Ref. [@kuzuu_ao_1999]) corresponding to $\sim 10^{10} ~\mathrm{W}/\mathrm{cm}^2$, up to which the model should be valid. For shorter, femtosecond pulses this damage threshold is $1000$ times higher, but since we consider “CW” beams the lower value for nanosecond pulses is relevant. It is justified to neglect temporal effects on beam propagation, because the dispersion length is already of the order of kilometers for pulse durations of a few tens of picoseconds, much longer than propagation distances considered in this work. Throughout this paper, we choose $V_0=1000$, and find that $r_0 \approx 25.0 ~\mu \mathrm{ m}$ through $r_0=\sqrt{\left|n_bV/\left[2k_0^2(n-n_b)\right]\right|}$, which nicely corresponds to the radius of a standard circular multi-mode fiber [@agrawal_book_2009]. In addition, in order to guarantee intensities smaller than the LIDT the condition $|\psi|^2<1/3$ should be fulfilled.
Note that in Eq. (\[partial\]) $\sigma$ is the sign of nonlinearity: $\sigma =1$ ($\sigma=-1$) represents focusing (defocusing) nonlinearity. The main difference between weakly nonlinear waveguides with focusing and defocusing nonlinearity is that defocusing nonlinearity supports higher amplitudes of the wave function, and in general leads to more stable and robust configurations. In this paper, however, we consider the experimentally relevant case of a focusing nonlinearity.
Rotating localized dipoles {#dipoles}
==========================
Dipole azimuthons in circular waveguides
----------------------------------------
Azimuthons are a straightforward generalization of the usual ansatz for stationary solutions [@desyatnikov_prl_2005]. They represent spatially rotating structures and hence involve an additional parameter, the rotation frequency $\omega$ (see also [@skryabin_pre_2002]), so we seek approximate solutions of the form $$\psi (r,\phi,z) = U (r, \phi - \omega z) \exp (i \kappa z),
\label{ansatz}$$ where $r=\sqrt{x^2+y^2}$ and $\phi$ the azimuthal angle in the transverse plane $(x,y)$, $U$ is the stationary profile, $\omega$ the rotation frequency, and $\kappa$ the propagation constant. For $\omega=0$, azimuthons become ordinary (non-rotating) solitons. The simplest family of azimuthons is the one connecting the dipole soliton with the single charged vortex soliton [@lopez-aguayo_ol_2006]. A single charged vortex consists of two equal-amplitude dipole-shaped structures representing real and imaginary part of $U$. If these two components differ in amplitude, the resulting structure forms a “rotating dipole” azimuthon. If one of the components is zero we deal with the dipole soliton, which consists of two out-of-phase humps and does not rotate for symmetry reasons. In a first attempt, let us assume we know the radial shape of the linear vortex mode $F(r)$ which is normalized according to $\pi \int r|F(r)|^2 \mathrm{d}r =1$. Then, using separation of variables, we consider the simplest so-called “rotating dipole” azimuthon with ansatz [@stefan_oe_2008] $$U(r,\phi-\omega z)=AF(r)\left[\cos(\phi-\omega z)+i B \sin(\phi-\omega z) \right],
\label{complexamplitude}$$ where $A$ is an amplitude factor, and $1-B$ the azimuthal modulation depth of the resulting ring-like structure. Because we are operating in the weakly nonlinear regime, using linear waveguide modes as initial conditions for nonlinear (soliton) solutions is a quite good approximation.
After plugging Eq. (\[ansatz\]) into Eq. (\[partial\]), multiplying by $U^*$ and $\partial U^* / \partial \phi$ respectively, and integrating over the transverse coordinates we end up with [@stefan_oe_2008]
$$-\kappa P + \omega L_z + I + N = 0,$$
$$-\kappa L_z + \omega P' + I' + N' = 0.$$
This system relates the propagation constant $\kappa$ and the rotation frequency $\omega$ of the azimuthons to integrals over their stationary amplitude profiles:
$$P = \iint r|U(r)|^2 ~\mathrm{d}r~\mathrm{d}\phi,$$
$$L_z = -i\iint r\frac{\partial U(r)}{\partial \phi} U^*(r) ~\mathrm{d}r~\mathrm{d}\phi,$$
$$I = \iint rU^*(r) \Delta_{\perp} U(r) ~\mathrm{d}r~\mathrm{d}\phi,$$
$$N = \iint r\left[\sigma|U(r)|^2+V \right] |U(r)|^2 ~\mathrm{d}r~\mathrm{d}\phi,$$
$$P' = \iint r\left| \frac{\partial U(r)}{\partial \phi} \right|^2 ~\mathrm{d}r~\mathrm{d}\phi,$$
$$I' = i\iint r\frac{\partial U^*(r)}{\partial \phi} \Delta_{\perp} U(r) ~\mathrm{d}r~\mathrm{d}\phi,$$
$$N' = i\iint r\left[\sigma|U(r)|^2+V \right] \frac{\partial U^*(r)}{\partial \phi} U(r) ~\mathrm{d}r~\mathrm{d}\phi.$$
The first two quantities $(P$ and $L_z)$ have straightforward physical meanings, namely power and angular momentum. The integrals $I$ and $I'$ are related to the diffraction mechanism of the system, whereas $N$ and $N'$ account for waveguide and nonlinearity. Thus we can formally solve for the rotation frequency with these quantities and obtain $$\omega=\frac{P(I'+N')-L_z(I+N)}{L_z^2-PP'}.
\label{angularfrequency}$$
After inserting Eq. (\[complexamplitude\]) into Eq. (\[angularfrequency\]), it turns out that only the nonlinear term contributes to the rotation frequency $\omega$ (see also Eq. (10) in Ref. [@fabian_pra_2010]). In order to give an estimate for the rotation frequency, we use the linear stationary modal profile $F(r)$ of a circular waveguide expressed in terms of Bessel functions of first kind $J_1$ and modified Bessel function of second kind $K_1$: $$F(r) = C \times
\begin{cases}
J_1 ( \sqrt{V_0-\kappa} r) & \mathrm{for}~ 0\leq r\leq1
\\
\dfrac{J_1(\sqrt{V_0-\kappa})}{K_1(\sqrt{\kappa})} \cdot K_1 (\sqrt{\kappa}r) & \mathrm{for}~ r>1
\end{cases},$$ where $C$ is a normalization factor such that $\pi\int|F(r)|^2rdr=1$. Thus, the analytically predicted rotation frequency is $$\omega=\frac{\pi \sigma}{2} \int r |F(r)|^4 \mathrm{d}r \cdot A^2 B.
\label{analyticalangular}$$ From the above equation, one can see that sense of rotation is opposite for focusing and defocusing nonlinearities. Moreover, as $\omega = 2 k_0 r_0^2 \cdot 2 \pi/ \ell$, one can find from Eq. (\[analyticalangular\]) an expression for the physical distance $\ell$ over which the azimuthon in a circular waveguide performs one full rotation: $$\ell= \frac{\displaystyle 8 k_0 r_0^2}{\displaystyle A^2 B \int r |F(r)|^4 \mathrm{d}r}.
\label{physicaldistance}$$ For example, assuming propagation in a silica fiber with the parameters discussed earlier (see Sec. \[modelling\]) and taking $A=0.4$, $B=0.5$, we obtain $\ell \approx 2.4 ~\mathrm{m}$.
Figure \[circularrelations\] shows the dependence of the azimuthon rotation frequency as a function of its amplitude $A$ (left panel) and the modulation parameter $B$ (right panel). Blue solid lines represent the analytical predictions and, in excellent agreement, red dots are obtained from numerical simulations. An exemplary propagation of the dipole azimuthon in circular waveguide rotating with $\omega \approx 0.037$ is illustrated in Fig. \[circularevolution\]. The top iso-surface plot displays 3D rigid and continuous rotation of the azimuthon during propagation. The row below depicts the transverse intensity (left) and phase (right) distribution of the dipole azimuthon during propagation after rotating by $\pi/2$.
![Azimuthon rotation frequency $\omega$ versus amplitude factor $A$ (left panel) and amplitude ratio $B$ (right panel). Blue solid lines show analytical predictions from Eq. (\[analyticalangular\]). Red dots denote results obtained from numerical simulations of Eq. (\[partial\]). The values of $A$ and $B$ are shown next to the lines.[]{data-label="circularrelations"}](relation_ca "fig:"){width="40.00000%"} ![Azimuthon rotation frequency $\omega$ versus amplitude factor $A$ (left panel) and amplitude ratio $B$ (right panel). Blue solid lines show analytical predictions from Eq. (\[analyticalangular\]). Red dots denote results obtained from numerical simulations of Eq. (\[partial\]). The values of $A$ and $B$ are shown next to the lines.[]{data-label="circularrelations"}](relation_cb "fig:"){width="40.00000%"}
![The propagation of a dipole azimuthon with $A=0.4,~B=0.5$ in the circular waveguide. The iso-surface plot on the top clearly displays the spiraling of the azimuthon during propagation. Figures in the row below depict the azimuthon’s transverse intensity distribution at input, and after rotating $\pi/2$, respectively. The white line indicates the waveguide boundaries.[]{data-label="circularevolution"}](cd_prop "fig:"){width="55.00000%"}\
![The propagation of a dipole azimuthon with $A=0.4,~B=0.5$ in the circular waveguide. The iso-surface plot on the top clearly displays the spiraling of the azimuthon during propagation. Figures in the row below depict the azimuthon’s transverse intensity distribution at input, and after rotating $\pi/2$, respectively. The white line indicates the waveguide boundaries.[]{data-label="circularevolution"}](cdipole0 "fig:"){width="35.00000%"} ![The propagation of a dipole azimuthon with $A=0.4,~B=0.5$ in the circular waveguide. The iso-surface plot on the top clearly displays the spiraling of the azimuthon during propagation. Figures in the row below depict the azimuthon’s transverse intensity distribution at input, and after rotating $\pi/2$, respectively. The white line indicates the waveguide boundaries.[]{data-label="circularevolution"}](cdipole85 "fig:"){width="35.00000%"}
Azimuthon-like dipoles in square waveguides
-------------------------------------------
Because of the lack of circular symmetry, azimuthons in the strict sense of Eq. (\[complexamplitude\]) (preservation of shape during rotation with constant frequency) cannot exist in a square waveguide. However, as we will show below, the azimuthon-like behavior is still possible even though the beam propagation is accompanied by variation of the beam transverse intensity distribution. To set the initial condition, we use two linear degenerated orthogonal dipole modes $D_1$, $D_2$ (as in the case of the circular waveguide), which are normalized according to $\iint |D_{1,2}|^2~\mathrm{d}x~\mathrm{d}y=1$, and superpose them as before to form the azimuthon-like object $$U(x,y,z=0)=A\left(D_1+iBD_2\right).
\label{squareansatz}$$ The field Eq. (\[squareansatz\]) is then used as an initial condition to the nonlinear Schr[ö]{}dinger equation Eq. (\[partial\]). However, in contrast to the circular waveguide, the orientation of the dipoles $D_1$ and $D_2$ is important in nonlinear regime. If the two orthogonal dipoles $D_1$ and $D_2$ are oriented along the diagonals of the waveguide cross-section (see first subplot of Fig. \[square\_r\_evolution\]), for a given amplitude $A$ rotation occurs only if the modulation parameter $B$ exceeds a critical value $B=B_{\rm cr}$. Moreover, the rotation is no longer constant as in the case of cylindrical waveguide but fluctuates and hence in what follows we use its average value (termed “average frequency”) $\bar \omega=1/L\int_0^L\omega(z)~\mathrm{d}z$ with $L$ being the propagation distance corresponding to one full $2\pi$ rotation. This threshold value $B_{\rm cr}$ decreases if we choose different initial dipole orientations [^1], and appears to be zero (within our numerical accuracy) for parallel orientation with respect to the waveguide boundaries.
![The propagation of the azimuthon-like dipole, that rotates in the square waveguide. Top row: intensity (left) and phase (right) distribution corresponding to the input and soliton rotation by $\pi/4$, respectively. The white line indicates the waveguide boundaries. The iso-surface plot at the bottom displays the rotation and deformation during propagation. The initial amplitude and modulation parameters are $A=0.4,~B=0.5$.[]{data-label="square_r_evolution"}](sdr0 "fig:"){width="35.00000%"} ![The propagation of the azimuthon-like dipole, that rotates in the square waveguide. Top row: intensity (left) and phase (right) distribution corresponding to the input and soliton rotation by $\pi/4$, respectively. The white line indicates the waveguide boundaries. The iso-surface plot at the bottom displays the rotation and deformation during propagation. The initial amplitude and modulation parameters are $A=0.4,~B=0.5$.[]{data-label="square_r_evolution"}](sdr30 "fig:"){width="35.00000%"}\
![The propagation of the azimuthon-like dipole, that rotates in the square waveguide. Top row: intensity (left) and phase (right) distribution corresponding to the input and soliton rotation by $\pi/4$, respectively. The white line indicates the waveguide boundaries. The iso-surface plot at the bottom displays the rotation and deformation during propagation. The initial amplitude and modulation parameters are $A=0.4,~B=0.5$.[]{data-label="square_r_evolution"}](sdr_prop "fig:"){width="55.00000%"}
Let us focus on the case where the initial dipole-like field structure is oriented along the diagonal of the waveguide cross-section, and from now on the notation $D_1$ and $D_2$ stands for this orientation. Figure \[square\_r\_evolution\] shows an exemplary propagation of the resulting azimuthon-like solution with $B>B_{\rm cr}$, and an average rotation frequency of $\bar \omega \approx 0.0262$. The top row depicts intensity and phase distribution at different propagation distance while the bottom plots illustrate the full 3D evolution of the “soliton”. The rotation (counter-rotating w.r.t. phase) accompanied by beam deformation is clearly visible. For an amplitude ratio $B$ less than the critical value ($B<B_{\rm cr}$) the azimuthon no longer rotates but swings back and forth and hence its average rotation frequency is zero. The propagation of this wobbling dipole with an amplitude ratio $B$ smaller than the critical value is displayed in Fig. \[square\_t\_evolution\]. The right panels show the maximum angle which the dipole attains during propagation. The 3D surface plot in the bottom row clearly illustrates the swinging movement of the “soliton” upon propagation.
![The propagation of a dipole azimuthon that wobbles in the square waveguide. The first row depicts the dipole at input and maximum displacement, respectively. The white line indicates the waveguide boundaries. The iso-surface plot below displays the twist and deformation during propagation. The chosen amplitude factor and ratio are $A=0.4,~B=0.2$.[]{data-label="square_t_evolution"}](sdt0 "fig:"){width="35.00000%"} ![The propagation of a dipole azimuthon that wobbles in the square waveguide. The first row depicts the dipole at input and maximum displacement, respectively. The white line indicates the waveguide boundaries. The iso-surface plot below displays the twist and deformation during propagation. The chosen amplitude factor and ratio are $A=0.4,~B=0.2$.[]{data-label="square_t_evolution"}](sdt40 "fig:"){width="35.00000%"}\
![The propagation of a dipole azimuthon that wobbles in the square waveguide. The first row depicts the dipole at input and maximum displacement, respectively. The white line indicates the waveguide boundaries. The iso-surface plot below displays the twist and deformation during propagation. The chosen amplitude factor and ratio are $A=0.4,~B=0.2$.[]{data-label="square_t_evolution"}](sdt_prop "fig:"){width="55.00000%"}
Due to the azimuthon profile deformation, it is not possible to find an analytical expression of $\bar \omega$ as a function of $A$ and $B$ as in the case of circular waveguide (i.e. Eq. (\[analyticalangular\])). Therefore, we need to resort to numerical simulations. In Fig. \[squarerelations\] we show the numerically determined relation between $\bar \omega$ and the azimuthon parameters $A$ (left panel) and $B$ (right panel). The threshold-like behavior is evident in the right graph of this figure: the region $B<B_{\rm cr}$, in which the soliton wobbles, is depicted by a green line.
![Averaged rotation frequency $\bar \omega$ versus amplitude factor $A$ (left panel) and amplitude ratio $B$ (right panel) in a square waveguide. Red dots are numerical results obtained from Eq. (\[partial\]), blue curves ($B<0.4$) and blue lines ($B>0.4$) are the fitting results to the numerical simulations. The green line represents the superposed dipoles twist during propagation. The values of $A$ and $B$ are shown next to the curves. The black square represents the analytical estimate for $B_{\rm cr}$.[]{data-label="squarerelations"}](relation_sa "fig:"){width="40.00000%"} ![Averaged rotation frequency $\bar \omega$ versus amplitude factor $A$ (left panel) and amplitude ratio $B$ (right panel) in a square waveguide. Red dots are numerical results obtained from Eq. (\[partial\]), blue curves ($B<0.4$) and blue lines ($B>0.4$) are the fitting results to the numerical simulations. The green line represents the superposed dipoles twist during propagation. The values of $A$ and $B$ are shown next to the curves. The black square represents the analytical estimate for $B_{\rm cr}$.[]{data-label="squarerelations"}](relation_sb "fig:"){width="40.00000%"}
Let us have a closer look at the boundary between domains of azimuthon rotation ($B>B_{\rm cr}$) and wobbling ($B<B_{\rm cr}$). We find numerically that the critical value $B_{\rm cr}$, which separates those two domains, depends very weakly on the azimuthon amplitude $A$: we are in weakly nonlinear limit, and may use linear waveguide modes to elucidate nonlinear propagation properties of the azimuthons. The appearance of a threshold value $B_{\rm cr}$, above which the azimuthon rotates, can be explained by considering the Hamiltonian associated with propagation equation Eq. (\[partial\]), $$\mathcal{H}=\iint \left(|\nabla \psi|^2-\frac{1}{2}|\psi|^4-V|\psi|^2 \right)~\mathrm{d}x ~\mathrm{d}y,
\label{hamiltonian}$$ which is a conserved quantity upon propagation. Depending on their orientation, stationary dipoles ($B=0$) have a slightly different value of $\mathcal{H}$ in nonlinear regime. If we take our diagonal (w.r.t. waveguide cross-section) dipoles $D_1$ and $D_2$ from above, we can construct a parallel dipole $D_p$ in the following way: $$D_p=\frac{D_1+D_2}{\displaystyle \sqrt{\iint |D_1+D_2|^2 ~\mathrm{d}x ~\mathrm{d}y}}.$$ The intensity distribution of this dipole $D_p$ is very close to that shown in the snapshot of the rotating azimuthon shown in the right panel in Fig. \[square\_r\_evolution\]. One can show using the definition of the Hamiltonian Eq. (\[hamiltonian\]) that $\mathcal{H}(D_p)>\mathcal{H}(D_1)=\mathcal{H}(D_2)$. In fact, it turns out that those two dipole orientations (parallel and diagonal) correspond to the extremal values of the Hamiltonian. Now, if the azimuthon rotates it has to pass through all possible orientations, i.e., its value of $\mathcal{H}$ has to be larger than $\mathcal{H}(D_p)$. We can use this reasoning to estimate the value of $B_{\rm cr}$: The left panel of Fig. \[angularmomentumcurve\] depicts the dependence of the Hamiltonian of superposed diagonal dipoles as a function of $B$ (red line) for constant power $P$. The blue line represents the value of the Hamiltonian of the parallel dipole $\sim D_p$ at the same power level. As the red curve monotonically increases with modulation parameter $B$, the intersection of the two curves gives an estimate of the critical value of the azimuthon modulation $B_{\rm cr}$, which is shown in the right panel in Fig. \[squarerelations\] by a black square. In other words, in order to rotate from the diagonal to vertical position the azimuthon must overcome some kind of energy barrier represented by the difference of the values of the Hamiltonian in those two basic states. As Fig. \[angularmomentumcurve\] shows this can be achieved by increasing the value of $B$ in the diagonal state to $B_1=0.342$, which is very close to the value found numerically (see right panel in Fig. \[squarerelations\]). Similar phenomena have been identified, for instance, in discrete nonlinear systems where the discreteness introduces the so called Peierls-Nabarro potential which has to be overcome by a soliton to become mobile [@Kivshar_Campbell].
![ Left panel: Hamiltonian of $A(D_1+iBD_2)$ (red curve) as a function of $B$ and Hamiltonian of $D_3$ (blue line) (for the same power $P$). The intersection of the two curves defines $B_{\rm cr}$ (depicted by the black square). Right panel: The angular momenta of the rotating superposed dipoles (corresponding to Fig. \[square\_r\_evolution\], blue curve), and the twisting ones (corresponding to Fig. \[square\_t\_evolution\], red curve). The other curves depict the cases of $B$ slightly above (black) and bellow (green) $B_{\rm cr}$. All curves are obtained for $A=0.4$. []{data-label="angularmomentumcurve"}](hamil "fig:"){width="45.00000%"} ![ Left panel: Hamiltonian of $A(D_1+iBD_2)$ (red curve) as a function of $B$ and Hamiltonian of $D_3$ (blue line) (for the same power $P$). The intersection of the two curves defines $B_{\rm cr}$ (depicted by the black square). Right panel: The angular momenta of the rotating superposed dipoles (corresponding to Fig. \[square\_r\_evolution\], blue curve), and the twisting ones (corresponding to Fig. \[square\_t\_evolution\], red curve). The other curves depict the cases of $B$ slightly above (black) and bellow (green) $B_{\rm cr}$. All curves are obtained for $A=0.4$. []{data-label="angularmomentumcurve"}](sangular "fig:"){width="45.00000%"}
It should be emphasized that the crucial difference between azimuthon dynamics in circular and square waveguide originates from the fact that angular momentum of the beam is conserved in the circular waveguide, whereas it changes considerably in the square waveguide. In the right panel of Fig. \[angularmomentumcurve\] we display the angular momentum of the beam in the square waveguide for four different values of modulation parameter $B$. If $B$ is above the critical value $B_{\rm cr}$, the angular momentum does not change its sign as the blue curve shows. If $B<B_{\rm cr}$, the angular momentum changes periodically its sign in propagation (red curve), which indicates the swinging motion of the soliton. For values $B$ close to the critical value the variation of the angular momentum are largest as shown by the green and black curves. At the same time, the appearance of broad plateaus indicates slow propagation dynamics.
It is worth mentioning that we also investigated other waveguide symmetries, such as hexagonal waveguides, and found the behavior of the dipole azimuthons to be qualitatively similar to that of the square waveguide, i.e., they either rotate or wobble, depending on modulation depth and initial orientation. However, the difference of the values of the Hamiltonian for the two extremal dipoles is now smaller, resulting in a lower threshold value $B_{\rm cr}$. We believe that this threshold behavior in the propagation dynamics of azimuthons is generic for any non-circular-symmetric waveguide structure which supports degenerated dipole modes.
Rotating higher order localized modes {#higherorder}
=====================================
In analogy to dipole-azimuthons discussed so far, it is possible to consider dynamics of higher order azimuthons constructed by using pairs of degenerated higher order waveguide modes. In particular, in a circular waveguide degenerate higher order modes have the form of quadrupoles, hexapoles, octapoles, decapoles, etc. and other modes with even number of lobes (optical necklaces). In a square waveguide, one can identify pairs of degenerate modes in a form of hexapoles, octapoles, decapoles, dodecapoles, etc. (optical matrices). Interestingly, the quadrupoles found in square waveguides are not degenerate and hence the propagation dynamics of their corresponding superposed states in weakly nonlinear regime is dominated by linear mode beating.
![The propagation of a hexapole azimuthon with $A=0.4,~B=0.5$ in the circular waveguide. Left panel shows the original hexapole azimuthon and the corresponding phase; right panel shows the iso-surface plot of the propagation.[]{data-label="cquadr_evolution"}](chexa0 "fig:"){width="45.00000%"} ![The propagation of a hexapole azimuthon with $A=0.4,~B=0.5$ in the circular waveguide. Left panel shows the original hexapole azimuthon and the corresponding phase; right panel shows the iso-surface plot of the propagation.[]{data-label="cquadr_evolution"}](chexa_prop "fig:"){width="50.00000%"}
Fig. \[cquadr\_evolution\] displays an example of hexapole azimuthon in a circular waveguide. The left panel represents the initial intensity and phase distribution of the azimuthon. The 3D surface plot in the right panel depicts stable and smooth rotation of the hexapole azimuthon. As far as the higher order azimuthons of the square waveguide are concerned, they share the dynamical properties of dipole azimuthons discussed before. As an example, Fig. \[hexa\_evolution\] shows the rotating (top row) and swinging (middle row) hexapoles. Due to the beam deformation, the hexapole with $B=0.5$ transforms into a ($2 \times 3$) matrix while rotating. The left two panels in the third row of Fig. \[hexa\_evolution\] display the rotating dodecapole azimuthon and its deformation into a ($3 \times 4$) solitonic matrix; the right panels show rotating icosapole azimuthon and its deformation into a ($4 \times 5$) solitonic matrix.
![The propagation of superposed hexapoles in the square waveguide. The first row shows a rotating azimuthon with $A=0.4,~B=0.5>B_{\rm cr}$ (Media 1), and the second row shows a twisting azimuthon with $A=0.4,~B=0.2<B_{\rm cr}$ (Media 2). The superposed dodecapoles (left two panels), and superposed icosapoles (right two panels) as well as their deformations are shown in the third row; in both cases we choose $A=0.4$ and $B=0.5$.[]{data-label="hexa_evolution"}](shexa_r0 "fig:"){width="35.00000%"} ![The propagation of superposed hexapoles in the square waveguide. The first row shows a rotating azimuthon with $A=0.4,~B=0.5>B_{\rm cr}$ (Media 1), and the second row shows a twisting azimuthon with $A=0.4,~B=0.2<B_{\rm cr}$ (Media 2). The superposed dodecapoles (left two panels), and superposed icosapoles (right two panels) as well as their deformations are shown in the third row; in both cases we choose $A=0.4$ and $B=0.5$.[]{data-label="hexa_evolution"}](shexa_r30 "fig:"){width="35.00000%"}\
![The propagation of superposed hexapoles in the square waveguide. The first row shows a rotating azimuthon with $A=0.4,~B=0.5>B_{\rm cr}$ (Media 1), and the second row shows a twisting azimuthon with $A=0.4,~B=0.2<B_{\rm cr}$ (Media 2). The superposed dodecapoles (left two panels), and superposed icosapoles (right two panels) as well as their deformations are shown in the third row; in both cases we choose $A=0.4$ and $B=0.5$.[]{data-label="hexa_evolution"}](shexa_t0 "fig:"){width="35.00000%"} ![The propagation of superposed hexapoles in the square waveguide. The first row shows a rotating azimuthon with $A=0.4,~B=0.5>B_{\rm cr}$ (Media 1), and the second row shows a twisting azimuthon with $A=0.4,~B=0.2<B_{\rm cr}$ (Media 2). The superposed dodecapoles (left two panels), and superposed icosapoles (right two panels) as well as their deformations are shown in the third row; in both cases we choose $A=0.4$ and $B=0.5$.[]{data-label="hexa_evolution"}](shexa_t30 "fig:"){width="35.00000%"}\
![The propagation of superposed hexapoles in the square waveguide. The first row shows a rotating azimuthon with $A=0.4,~B=0.5>B_{\rm cr}$ (Media 1), and the second row shows a twisting azimuthon with $A=0.4,~B=0.2<B_{\rm cr}$ (Media 2). The superposed dodecapoles (left two panels), and superposed icosapoles (right two panels) as well as their deformations are shown in the third row; in both cases we choose $A=0.4$ and $B=0.5$.[]{data-label="hexa_evolution"}](sdodeca "fig:"){width="35.00000%"} ![The propagation of superposed hexapoles in the square waveguide. The first row shows a rotating azimuthon with $A=0.4,~B=0.5>B_{\rm cr}$ (Media 1), and the second row shows a twisting azimuthon with $A=0.4,~B=0.2<B_{\rm cr}$ (Media 2). The superposed dodecapoles (left two panels), and superposed icosapoles (right two panels) as well as their deformations are shown in the third row; in both cases we choose $A=0.4$ and $B=0.5$.[]{data-label="hexa_evolution"}](sicosa "fig:"){width="35.00000%"}
Conclusion
==========
In conclusion, we demonstrated numerically stable propagation of azimuthons, i.e. localized rotating nonlinear waves in weakly nonlinear waveguides. Depending on the waveguide profile, different nonlinearity induced propagation dynamics can be observed. We showed that azimuthons in circular waveguides rotate continuously. The analytically predicted dependence of rotation frequency $\omega$ as a function of soliton parameters was found to be in excellent agreement with numerical simulations. Further on, we discussed propagation of azimuthon-like structures in square waveguides. We showed that their dynamics critically depends on the initial conditions. In particular, we found a threshold-like behavior in the propagation dynamics, separating rotating and wobbling solutions. We showed that this effect is associated with different values of the Hamiltonian for different azimuthon orientations. As our analysis relies on physical parameters of actual multi-mode waveguides, our findings may open a relatively easy route to experimental observations of stable azimuthons.
Acknowledgement
===============
Numerical simulations were partly performed on the SGI XE Cluster and the Sun Constellation VAYU Cluster of the Australian Partnership for Advanced Computing (APAC). This research was supported by the Australian Research Council.
[^1]: In the linear system, any dipole orientation is possible, and can be constructed from orthogonal basis ($D_1,D_2$) by superposition.
| 0 |
---
abstract: 'The recognition problem for attribute-value grammars(AVGs) was shown to be undecidable by Johnson in 1988. Therefore, the general form of AVGs is of no practical use. In this paper we study a very restricted form of AVG, for which the recognition problem is decidable (though still ${\mathord{\it NP}}$-complete), the R-AVG. We show that the R-AVG formalism captures all of the context free languages and more, and introduce a variation on the so-called [*off-line parsability constraint*]{}, the [*honest parsability constraint*]{}, which lets different types of R-AVG coincide precisely with well-known time complexity classes.'
author:
- 'Leen Torenvliet[^1]'
- 'Marten Trautwein[^2]'
date: |
*University of Amsterdam\
Department of Mathematics and Computer Science\
Plantage Muidergracht 24\
1018 TV Amsterdam*
title: 'A Note on the Complexity of Restricted Attribute-Value Grammars'
---
\[section\] \[thm\][Corollary]{}
\[thm\][Lemma]{}
Introduction
============
Although a universal feature theory does not exist, there is a general understanding of its objects. The objects of feature theories are abstract linguistic objects, e.g., an object “sentence,” an object “masculine third person singular,” an object “verb,” an object “noun phrase.” These abstract objects have properties like “tense,” “number,” “predicate,” “subject.” The values of these properties are either atomic, like “present” and “singular,” or abstract objects, like “verb” and “noun-phrase.” The abstract objects are fully described by their properties and their values. Multiple descriptions for the properties and values of the abstract linguistic objects are presented in the literature. Examples are:
1. Feature graphs, which are labeled rooted directed acyclic graphs $G=(V,A)$, where $F$ is a collection of labels, a sink in the graph represents an atomic value and the labeling function is an injective function $f:V\times A\mapsto F$.
2. Attribute-value matrices, which are matrices in which the entries consist of an attribute and a value or a reentrance symbol. The values are either atomic or attribute-value matrices.
From a computational point of view, all descriptions that are used in practical problems are equivalent. Though there exist some theories with a considerably higher expressive power [@blackburn.ea:93]. For this paper we adopt the feature graph description, which we will define somewhat more formal in the next section. Attribute Value Languages(AVL) [@smolka:92] consist of sets of logical formulas that describe classes of feature graphs, by expressing constraints on the type of paths that can exist within the graphs. To wit: In a sentence like “a man walks” the edges labeled with “person” that leave the nodes labeled “a man” and “walks” should both end in a node labeled “singular.” Such a constraint is called a “path equation” in the attribute-value language.
A rewrite grammar [@chomsky:56] can be enriched with an AVL to construct an Attribute Value Grammar(AVG), which consists of pairs of rewrite-rules and logical formulas. The rewrite rule is applicable to a production (nonterminal) only if the logical formula that expresses the relation between left- and right-hand side of the rule evaluates to true. The recognition problem for attribute-value grammars can be stated as: Given a grammar $G$ and a string $w$ does there exist a derivation in $G$, that respects the constraints given by its AVL, and that ends in $w$. As the intermediate productions correspond to feature graphs this question can also be formulated as a question about the existence of a consistent sequence of feature graphs that results in a feature graph describing $w$. For the rewrite grammar, any formalism in the Chomsky hierarchy (from regular to type $0$) can be chosen. From a computational point of view it is of course most desirable to restrict oneself to a formalism that on the one hand gives enough expressibility to describe a large fragment of the (natural) language, and on the other hand is restrictive enough to preserve feasibility. For a discussion on the linguistic significance of such restrictions, see [@perrault:84].
Johnson [@johnson:88] proved that attribute-value grammars that are as restrictive as being equipped with a rewrite grammar that is regular can already give rise to an undecidable recognition problem. Obviously, to be of any practical use, the rewrite grammar or the attribute-value language must be more restrictive. Johnson proposed to add the [*off-line parsability constraint*]{}, which is respected if the rewrite grammar has no chain- or $\epsilon$-rules. Then, the number of applications in a production is linear and the size of the structure corresponding to the partial productions is polynomial. Hence as by a modification of Smolka’s algorithm [@smolka:92] consistency of intermediate steps can be checked in quadratic time, the complexity of the recognition problem can at most be (nondeterministic) polynomial time. This observation was made in [@lp-95-01], which also has an ${\mathord{\it NP}}$-hardness proof of the recognition problem.
We further investigate the properties of these restricted AVGs (R-AVGs). In the next section, we give some more formal definitions and notations. In Section \[WGC\] we show that the class of languages generated by an R-AVG (R-AVGL) includes the class of context free languages (CFL). It follows that any easily parsable class of languages (like CFL) is a proper subset of R-AVGL, unless ${\mathord{\it P}}={\mathord{\it NP}}$. Likewise, R-AVGL is a proper subset of the class of context sensitive languages, unless ${\mathord{\it NP}}={\mathord{\it PSPACE}}$. In Section \[HPC\] we propose a further refinement on the off-line parsability constraint, which allows R-AVGs that respect this constraint to capture [*precisely*]{} complexity classes like ${\mathord{\it NP}}$ or ${\mathord{\it NEXP}}$. That is, for any language $L$ that has an ${\mathord{\it NP}}$-parser, there exists an R-AVG, say $G$, such that $L=L(G)$. Though our refinement, the [*honest parsability constraint*]{} is probably not a property that can be decided for arbitrary R-AVGs, we show that R-AVGs can be equipped with restricting mechanisms that enforce this property. The techniques that prove Theorem \[ravgl\] and Theorem \[npishpavg\] result from Johnson’s work. Therefore, the proofs of these theorems are deferred to the appendices.
Definitions and Notation
========================
Attribute-Value Grammars
------------------------
The definitions in this section are in the spirit of [@johnson:88 Section 3.2] and [@smolka:92 Sections 3–4]. Consider three sets of pairwise disjoint symbols.
$A$, the finite set of constants, denoted ($a,b,c,\dots$)
$V$, the countable set of variables, denoted ($x,y,z,\ldots$)
$L$, the finite set of attributes, also called features, denoted ($f,g,h,\ldots$)
An [*$f$-edge*]{} from $x$ to $s$ is a triple $(x,f,s)$ such that $x$ is a variable, $f$ is an attribute, and $s$ is a constant or a variable. A [*path*]{}, $p$, is a, possibly empty, sequence of $f$-edges $(x_1,f_1,x_2),(x_2,f_2,x_3),\ldots,(x_{n},f_n,s)$ in which the $x_i$ are variables and $s$ is either a variable or a constant. Often a path is denoted by the sequence of its edges’ attributes, in reversed order, e.g., $p = f_n\ldots f_1$. Let $p$ be a path, $ps$ denotes the path that starts from $s$, where $s$ is a constant only if $p$ is the empty path. If the path is nonempty, $p = f_n\ldots f_1 \;(n geq 1)$, then $s$ is a variable. For paths $ps$ and $qt$ we write $ps\doteq qt$ iff $p$ and $q$ start in $s$ and $t$ respectively and end in the same variable or constant. The expression $ps\doteq qs$ is called a [*path equation*]{}. A [*feature graph*]{} is either a pair $(a,\emptyset)$, or a pair $(x,E)$ where $x$ is the root and $E$ a finite set of $f$-edges such that:
1. if $(y,f,s)$ and $(y,f,t)$ are in $E$, then $s=t$;
2. if $(y,f,s)$ is in $E$, then there is a path from $x$ to $y$ in $E$.
An [*attribute-value language*]{} ${\cal A}(A,V,L)$ consists of sets of logical formulas that describe feature graphs, by expressing constraints on the type of paths that can exist within the graphs.
- The terms of an attribute-value language ${\cal A}(A,V,L)$ are the constants and the variables $s,t\in A\cup V$.
- The formulas of an attribute-value language ${\cal A}(A,V,L)$ are path equations and Boolean combinations of path equations. Thus all formulas are either $ps\doteq qt$, where $ps$ and $qt$ are paths, or $\phi \wedge \psi$, $\phi \vee \psi$, or $\neg \phi$, where $\phi$ and $\psi$ are formulas.
Assume a finite set ${\mathord{\rm Lex}}$ (of lexical forms) and a finite set ${\mathord{\rm Cat}}$ (of categories). ${\mathord{\rm Lex}}$ will play the role of the set of terminals and ${\mathord{\rm Cat}}$ will play the role of the set of nonterminals in the productions.
A [*constituent structure tree*]{} (CST) is a labeled tree in which the internal nodes are labeled with elements of Cat and the leaves are labeled with elements of Lex.
Let $T$ be a constituent structure tree and $F$ be a set of formulas in an attribute-value language ${\cal A}(A,V,L)$. An [*annotated constituent structure tree*]{} is a triple $\mathopen{<}T,F,h\mathclose{>}$, where $h$ is a function that maps internal nodes in $T$ onto variables in $F$.
A [*lexicon*]{} is a finite subset of ${\mathord{\rm Lex}}\times
{\mathord{\rm Cat}}\times {\cal A}(A,\{x_0\},L)$. A set of [*syntactic rules*]{} is a finite subset of $\bigcup_{i\geq 1} {\mathord{\rm Cat}}\times{\mathord{\rm Cat}}^i\times{\cal A}(A,\{x_0,\ldots,x_i\},L)$. An [*attribute-value grammar*]{} is a triple $\mathopen{<}\mathord{\rm lexicon},\mathord{\rm rules},\mathord{\rm start}
\mathclose{>}$, where lexicon is a lexicon, rules is a set of syntactic rules and start is an element of ${\mathord{\rm Cat}}$.
\
1. [**[@balcazar.ea:88 p .150]** ]{} A class ${\cal C}$ of sets is [*recursively presentable*]{} iff there is an effective enumeration $M_1, M_2, \ldots$ of deterministic Turing machines which halt on all their inputs, and such that ${\cal C} = \{
L(M_i) \mid i = 1, 2, \ldots \}$.
2. We say that a class of grammars ${\cal G}$ is [*recursively presentable*]{} iff the class of sets $\{L(G) \mid G \in {\cal G} \}$ is recursively presentable.
Restricted Attribute-Value Grammars
-----------------------------------
The only formulas that are allowed in the attribute-value language of restricted attribute-value grammars (R-AVGs) are path-equations and conjunctions of path-equations (i.e. disjunctions and negations are out). We will denote the attribute-value language of an R-AVG by ${\cal A'}(A,V,L)$ to make the distinction clear. The CST of an R-AVG is produced by a chain- and $\epsilon$-rule free regular grammar. The CST of an R-AVG can be either a left-branching or a right-branching tree, since the grammar contains at most one nonterminal in each rule.
The set of syntactic rules of a restricted attribute-value grammar is a subset of $\bigcup_{i\geq 1, k\leq 1}
{\mathord{\rm Cat}}\times{\mathord{\rm Lex}}^i\times {\mathord{\rm Cat}}^k\times {\cal A'}(A,\{x_0,x_k\},L)$. A [*restricted attribute-value grammar*]{} is a pair $\mathopen{<}\mathord{\rm rules},\mathord{\rm start}
\mathclose{>}$, where rules is a set of syntactic rules and start is an element of ${\mathord{\rm Cat}}$.
An R-AVG $\mathopen{<}\mathord{\rm rules},\mathord{\rm start}
\mathclose{>}$ [*generates*]{} an annotated constituent structure tree $\mathopen{<}T,F,h\mathclose{>}$ iff
1. the root node of $T$ is start, and
2. every internal node of $T$ is licensed by a syntactic rule, and
3. the set $F$ is consistent, i.e., describes a feature graph.
Let $\phi[x/y]$ stand for the formula $\phi$ in which all variable $y$ is substituted for variable $x$. An internal node $v$ of an annotated constituent structure tree is [*licensed*]{} by a syntactic rule $(c_0, l_1,\ldots,l_i,\phi)$ iff
1. the node $v$ is labeled with category $c_0$, $h(v) = n_0$, and
2. all daughters of $v$ are leaves, which are labeled with $l_1 \ldots l_i$, and
3. $\phi[x_0/n_0]$ is in the set $F$.
An internal node $v$ of an annotated constituent structure tree is [*licensed*]{} by a syntactic rule $(c_0, l_1,\ldots,l_i, c_1,\phi)$ iff
1. the node $v$ is labeled with category $c_0$, $h(v) = n_0$, and
2. one of $v$’s daughters is an internal node, $v_1$, which is labeled with category $c_1$, and $h(v_1) = n_1$, and
3. the daughters of $v$ that are leaves are labeled with $l_1 \ldots l_i$, and
4. $\phi[x_0/n_0, x_1/n_1]$ is in the set $F$.
Weak Generative Capacity {#WGC}
========================
In [@lp-95-01], it is shown that the recognition problem for R-AVGs is ${\mathord{\it NP}}$-complete. This seems to indicate that although the mechanism for generating CSTs in R-AVGs is extremely simple, the generative capacity of R-AVGs is different from the generative capacity of e.g., context free languages (CFLs), which have a polynomial time parsing algorithm [@earley:70]. Yet, a priori, there may exist CFLs that do not have an R-AVG.
\[ravgl\] Let $L$ be a context free language. There exists an R-AVG $G$ such that $L=L(G)$.
If $L$ is a context free language, then there exists a context free grammar $G'$ in Greibach normal form such that $L=L(G')$. From this grammar $G'$, we can construct a pushdown store $M$ that accepts exactly the words in $L(G')=L$. Such a pushdown store $M$ is actually a finite state automaton $M'$ with a stack $S$. The finite state automaton $M'$ may be simulated by a chain- and $\epsilon$-rule free regular grammar. Furthermore, we can construct an attribute-value language ${\cal A'}(A,V,L)$ that simulates the stack $S$. Thus it should be clear that there exists an R-AVG $G$ that produces word $w$ iff $w \in L(G')$. Details of this construction are deferred to Appendix \[Greib\].
From this we can draw the conclusion that the class of context free languages is indeed a proper subset of the class of R-AVG languages, unless ${\mathord{\it P}}={\mathord{\it NP}}$.
\[PNP\] Let ${\cal C}$ be a recursively presentable class of grammars such that:
1. $G\in{\cal C}$ can be decided in time polynomial in ${\mathopen|G\mathclose|}$
2. $G\stackrel{*}{\Rightarrow}w$ can be decided in time polynomial in ${\mathopen|G\mathclose|}+{\mathopen|w\mathclose|}$.
If every R-AVG $G$ has a grammar in ${\cal C}$ then ${\mathord{\it P}}={\mathord{\it NP}}$. In fact, for every language $L$ in ${\mathord{\it NP}}$ there is an explicit deterministic polynomial time algorithm.
Let $L$ be a language in ${\mathord{\it NP}}$ and $w\in \{0,1\}^*$. Trautwein [@lp-95-01] provided an R-AVG $G$ and a reduction that maps any formula $F$ onto a string $w_F$ s.t. $G\stackrel{*}{\Rightarrow}w_F$ iff $F\in\mathord{\it SAT}$. It was also shown that any R-AVG has a nondeterministic polynomial time, hence deterministic exponential time, recognition algorithm. Suppose every R-AVG $G$ has a grammar in ${\cal C}$. Then there exists a $G' \in {\cal C}$ with $L(G')=L(G)$. We can decide in polynomial time whether $w_F\in L(G)$ for any $w_F$. So, ${\mathord{\it P}}={\mathord{\it NP}}$.
If every R-AVG $G$ has a grammar in ${\cal C}$, then the algorithm for deciding “$w\in L$?” consists of: use Cook’s reduction to produce a formula $F$ that is satisfiable iff $w\in L$; use Trautwein’s reduction to produce $w_F$ and R-AVG $G$; enumerate grammars in $\cal C$ for the first grammar $G'$ that has a description of length less than $\log\log{\mathopen|w\mathclose|}$ for which $L(G)\cap\{0,1\}^{\leq\log\log{{\mathopen|w\mathclose|}}}=L(G')
\cap\{0,1\}^{\leq\log\log{{\mathopen|w\mathclose|}}}$ accept iff $w\in L(G')$. This gives a polynomial time algorithm that erroneously accepts or rejects $w$ for only a finite number of strings $w$. The theorem now follows from the fact that both ${\mathord{\it P}}$ and ${\mathord{\it NP}}$ are closed under finite variation.
If R-AVGs generate only context free languages then ${\mathord{\it P}}={\mathord{\it NP}}$.
In fact it can be shown directly that R-AVGs also produce non-context free languages.
The context sensitive language $\{a^nb^nc^n\}$ is generated by an R-AVG.
(Sketch) Typically, the R-AVG that generates the language $\{a^nb^nc^n\}$ first generates an amount of $a$’s then an amount of $b$’s and finally an amount of $c$’s. Let us assume that the grammar generates $i$ $a$’s. During the derivation, the feature graph can be used to store the amount of $a$’s that is produced. Once the grammar starts to produce $b$’s , the feature graph will force the grammar to generate exactly $i$ $b$’s and next to generate exactly $i$ $c$’s as well.
The Honest Parsability Constraint and Consequences {#HPC}
==================================================
According to Theorem \[PNP\], it is unlikely that the languages generated by R-AVGs can be limited to those languages with a polynomial time recognition algorithm. Trautwein [@lp-95-01] showed that all R-AVGs have nondeterministic polynomial time algorithms. Is it perhaps the case that any language that has a nondeterministic polynomial time recognition algorithm can be generated by an R-AVG. Does there exist a tight relation between time bounded machines and R-AVGs as e.g., between LBAs and CSLs? The answer is that the off-line parsability constraint that forces the R-AVG to have no chain- or $\epsilon$-rules is just too restrictive to allow such a connection. The following trick to alleviate this problem has been observed earlier in complexity theory. The off-line parsability constraint(OLP) [@johnson:88] relates the amount of “work” done by the grammar to produce a string linearly to the number of terminal symbols produced. It is therefore a sort of honesty constraint that is also demanded of functions that are used in e.g., cryptography. There the deal is, for each polynomial amount of work done to compute the function at least one bit of output must be produced. In such a way, for polynomial time computable functions one can guarantee that the inverse of the function is computable in nondeterministic polynomial time.
As a more liberal constraint on R-AVGs we propose an analogous variation on the OLP
A grammar $G$ satisfies the Honest Parsability Constraint(HPC) iff there exists a polynomial $p$ s.t. for each $w$ in $L(G)$ there exists a derivation with at most $p({\mathopen|w\mathclose|})$ steps.
From Smolka’s algorithm and Trautwein’s observation it trivially follows that any attribute-value grammar that satisfies the HPC (HP-AVG) has an ${\mathord{\it NP}}$ recognition algorithm. The problem with the HPC is of course that it is not a syntactic property of grammars. The question whether a given AVG satisfies the HPC (or the OLP for that matter) may well be undecidable. Nonetheless, we can produce a set of rules that, when added to an attribute-value grammar [*enforces*]{} the HPC. The newly produced language is then a subset of the old produced language with an ${\mathord{\it NP}}$ recognition algorithm. Because of the fact that our addition may simulate any polynomial restriction, we regain the full class of AVG’s that satisfy the HPC. In fact
The class, P-AVGL, of languages produced by the HP-AVGs is recursively presentable.
We will give a detailed construction of such a set of rules in Appendix \[hpavg\]. The existence of such a set of rules and the work of Johnson now gives the following theorem.
\[npishpavg\] For any language $L$ that has an ${\mathord{\it NP}}$ recognition algorithm, there exists a restricted attribute-value grammar $G$ that respects the HPC and such that $L=L(G)$.
(Sketch) Let $M$ be the Turing machine that decides $w\in L$. Use a variation of Johnson’s construction of a Turing machine to create an R-AVG that can produce any string $w$ that is recognized by $M$. Add the set of rules that guarantee that only strings that can be produced with a polynomial number of rules can be produced by the grammar.
Veer out the HPC
================
Instead of creating a counter of logarithmic size as we do in Appendix \[hpavg\], it is quite straightforward to construct a counter of linear size (or exponential size if there is enough time). In fact, for well-behaved functions, the construction of a counter gives a method to enforce any desired time bound constraint on the recognition problem for attribute-value grammars. For instance, for nondeterministic exponential time we could define the Linear Dishonest Parsability Constraint (LDP) (allowing a linear exponential number of steps) which would give.
The class of languages generated by R-AVGs obeying the LDP condition is exactly ${\mathord{\it NE}}$.
Acknowledgements {#acknowledgements .unnumbered}
================
We are indebted to E. Aarts and W.C. Rounds for their valuable suggestions on an early presentation of this work.
[BDG88]{} J. Balcázar, J. Díaz, and J. Gabarró. . Springer-Verlag, New York, 1988.
P. Blackburn and E. Spaan. A modal perspective on the computational complexity of attribute value grammar. , 2(2):129–169, 1993.
N. Chomsky. Three models for the description of language. , 2(3):113–124, 1956.
J. Earley. An efficient context-free parsing algorithm. , 13(2):94–102, February 1970.
J. Hopcroft and J. Ullman. . Addison Wesley, Reading, MA, 1979.
M. Johnson. , volume 16 of [*CSLI Lecture Notes*]{}. CSLI, Stanford, 1988.
C. Perrault. On the mathematical properties of linguistic theories. , 10(3–4):165–176, 1984.
G. Smolka. Feature-constraint logics for unification grammars. , 12(1):51–87, 1992.
T. Sudkamp. . Addison Wesley, Reading, MA, 1988.
M. Trautwein. Assessing complexity results in feature theories. ILLC Research Report and Technical Notes Series LP-95-01, University of Amsterdam, Amsterdam, 1995. Submitted to the CMP-LG archive.
Simulating a Context Free Grammar in GNF {#Greib}
========================================
A context free grammar (CFG) is a quadruple $\langle N, \Sigma, P, S \rangle$, where $N$ is a set of nonterminals, $\Sigma$ is a set of terminals, $P$ is a set of productions, and $S \in N$ is the start nonterminal. A CFG is in Greibach normalform (GNF) if, and only if, the productions are of one of the following forms, where $a\in\Sigma, A\in N, A_1\ldots A_n \in N\setminus\{S\}$ and $\epsilon$ the empty string (c.f., [@hopcroft.ea:79], [@sudkamp:88]): $$\begin{aligned}
A &\rightarrow &a\,A_1\ldots A_n \\
A &\rightarrow &a \\
S &\rightarrow &\epsilon\end{aligned}$$
Given a GNF $G = \langle N, \Sigma, P, S \rangle$, we can construct a restricted attribute-value grammar (R-AVG) $G'$ that simulates grammar $G$. R-AVG $G'$ consists of the same set of nonterminals and terminals as GNF $G$. The productions of R-AVG $G'$ are described by Table \[RulesSim\]. The only two attributes of R-AVG $G'$ are and . R-AVG $G'$ contains $|N| + 1$ atomic values, one atomic value for each nonterminal and the special atomic value \$. The R-AVG $G'$ uses the feature graph to encode a push-down stack, similar to the encoding of a list. The stack will be used to store the nonterminals that still have to be rewritten.
The three syntactic abbreviations below are used to clarify the simulation. We use represent a stack by a Greek letter, or a string of symbols; the top of the stack is the leftmost symbol of the string. Let $x_0$ encode a stack $\gamma$, then the formulas in the abbreviation ${\mbox{\sc push}}(A_0\ldots A_n)$ express that $x_1$ encodes a stack $A_0\ldots A_n\gamma$. Likewise, the formulas in the abbreviation ${\mbox{\sc pop}}(A)$ express that $x_0$ encodes a stack $A\gamma$, and $X_1$ encodes the stack $\gamma$. The abbreviation expresses that $x_0$ encodes an empty stack. $$\begin{aligned}
{\mbox{\sc push}}(A_0\ldots A_n) &\mbox{stands for}
&{\mbox{\sc top}}(x_1) \doteq A_0 \wedge \\
&&{\mbox{\sc top rest}}(x_1) \doteq A_1 \wedge \\
&&{\centering \vdots}\\
&&{\mbox{\sc top rest}}^n(x_1) \doteq A_n \wedge\\
&&{\mbox{\sc rest}}^{n+1}(x_1) \doteq x_0 \\
{\mbox{\sc pop}}(A) &\mbox{stands for}
&{\mbox{\sc top}}(x_0) \doteq A \wedge \\
&&{\mbox{\sc rest}}(x_0) \doteq x_1 \\
{\mbox{\sc empty-stack}} &\mbox{stands for} &x_0 \doteq \$\end{aligned}$$
We have to prove that GNF $G$ and its simulation by R-AVG $G'$ generate (almost) the same language. Obviously, R-AVG $G'$ cannot generate the empty string. However, for all non-empty strings the following theorem holds.
Start nonterminal $S$ of GNF $G$ derives string $\alpha$ ($\alpha \in \Sigma^+$) if, and only if, start nonterminal $S$ of R-AVG $G'$ derives string $\alpha$ with the empty stack.
There are two cases to consider. First, $S$ derives string $\alpha$ in one step. Second, $S$ derives string $\alpha$ in more than one step. The lemma below is needed in the proof of the second case.
Case I
: Let start nonterminal $S$ derive string $\alpha$ in one step. GNF $G$ contains a production $S \rightarrow \alpha$ iff R-AVG $G'$ contains a production $S \rightarrow \alpha$ with the equation . So, $S$ derives $\alpha$ in a derivation of GNF $G$ iff $S$ derives $\alpha$ with an empty stack in the derivation of R-AVG $G'$.
Case II
: Initial nonterminal $S$ of GNF $G$ derives string $\alpha=\beta\beta'$ in more than one step iff there is a left-most derivation $S \stackrel{*}{\Rightarrow} \beta A
\Rightarrow
\beta\beta'$. GNF $G$ contains production $A \rightarrow \beta'$ iff R-AVG $G'$ contains production $A \rightarrow \beta'$ with the equation . By the next lemma: $S \stackrel{*}{\Rightarrow}
\beta A$ iff $S \stackrel{*}{\Rightarrow} \beta A$ with the empty stack. Hence $S$ derives $\alpha$ for GNF $G$ iff $S$ derives $\alpha$ with empty stack for R-AVG $G'$.
Start nonterminal $S$ derives $\alpha A\gamma$ ($\alpha \in \Sigma^+,
A\gamma \in (N \setminus \{S\})^+$) in a left-most derivation of GNF $G$ if, and only if, nonterminal $S$ derives $\alpha A$ with stack $\gamma\$$ (\$ is the bottom-of-stack symbol) in the derivation of R-AVG $G'$.
The lemma is proven by induction on the length of the derivation.
Basis
: If $S$ derives $\alpha A \gamma$ in one step, then GNF $G$ contains production $S \rightarrow \alpha A \gamma$ and R-AVG $G'$ contains production $S \rightarrow \alpha A$ with stack $\gamma\$$. If $S$ derives $\alpha A$ with stack $\gamma\$$ in one step, then R-AVG $G'$ contains production $S \rightarrow \alpha A$ with stack $\gamma\$$ and GNF $G$ contains production $S \rightarrow \alpha A \gamma$.
Induction
: The induction hypotheses states that $S \stackrel{n}{\Rightarrow} \alpha A
\gamma$ in GNF $G$ iff $S \stackrel{n}{\Rightarrow} \alpha A$ with stack $\gamma\$$ in R-AVG $G'$. Next, we distinguish three cases.
1. GNF $G$ contains a production $A \rightarrow a A_1A_2\ldots A_n$. Hence there is a left-most derivation $S \stackrel{n+1}{\Rightarrow}
\alpha a A_1A_2 \ldots A_n\gamma$. GNF $G$ contains the production $A \rightarrow
a A_1A_2 \ldots A_n$ iff R-AVG $G'$ contains a production $A \rightarrow a A_1$ with equation ($A_2\ldots A_n$). Since the induction hypotheses states that there is a derivation $S \stackrel{n}{\Rightarrow} \alpha A$ with stack $\gamma\$$, there is a derivation $S \stackrel{n+1}{\Rightarrow} \alpha a A_1$ with stack $A_2 \ldots
A_n\gamma\$$.
2. GNF $G$ contains a production $A \rightarrow a$ and $\gamma = B'\gamma'$. Hence there is a left-most derivation $S \stackrel{n+1}{\Rightarrow}
\alpha a B' \gamma'$. GNF $G$ contains the production $A \rightarrow a$ iff R-AVG $G'$ contains productions $A \rightarrow a B$ with equation (B), for all $B \in N \setminus \{S\}$. Hence by the induction hypotheses, there is a derivation $S \stackrel{n+1}{\Rightarrow} \alpha a B'$ with stack $\gamma'\$$.
3. GNF $G$ contains a production $A \rightarrow a$ and $\gamma = \epsilon$. Then there is a left-most derivation $S \stackrel{n+1}{\Rightarrow}
\alpha a$. GNF $G$ contains the production $A \rightarrow a$ iff R-AVG $G'$ contains production $A \rightarrow a$ with equation . Hence by the induction hypotheses, there is a derivation $S \stackrel{n+1}{\Rightarrow} \alpha a$ with stack $\$$.
Because every context free language is generated by some GNF $G$, every context free language is generated by some R-AVG $G'$.
Constructing an Honestly Parsable Attribute-Value Grammar {#hpavg}
=========================================================
In this section we show how to add a binary counter to an attribute-value grammar (AVG). This counter enforces the Honest-Parsability Constraint (HPC) upon the AVG. To keep this section legible we sometimes use the attribute-value matrices (AVMs) as descriptions. In Section \[create\], we show how to create a counter for the AVG. In Section \[avg2hpc\] we show how to extend the syntactic rules and the lexicon of the AVG.
Arithmetic by AVGs
------------------
We start with a little bit of arithmetic.
#### Natural numbers.
The AVMs below encode natural numbers in binary notation. The sequences of attributes and in these AVMs encode natural numbers, from least- to most-significant bit. The attribute has value 1 (or 0) if, and only if, it has a sister attribute (or ).
1. The AVMs [${\left[\begin{array}{ll} {\mbox{\sc v}} &0\\ {\mbox{\sc 0}} &{\mbox{\it +}} \end{array}\right]}$]{} and [${\left[\begin{array}{ll} {\mbox{\sc v}} &1\\ {\mbox{\sc 1}} &{\mbox{\it +}} \end{array}\right]}$]{} encode the natural numbers zero and one.
2. The AVMs [${\left[\begin{array}{ll} {\mbox{\sc v}} &0\\ {\mbox{\sc 0}} &[F] \end{array}\right]}$]{} and [${\left[\begin{array}{ll} {\mbox{\sc v}} &1\\ {\mbox{\sc 1}} &[F] \end{array}\right]}$]{} encode natural numbers iff the AVM $[F]$ encodes a natural number.
#### Syntactic rules that tests two numbers for equality.
Assume a nonterminal $A$ with some AVM [${\left[\begin{array}{ll} {\mbox{\sc n}} &[F]\\ {\mbox{\sc m}} &[H] \end{array}\right]}$]{}, where $[F]$ and $[H]$ encode natural number $x$ and $y$, respectively. We present one syntactic rule that derives from this nonterminal $A$ a nonterminal $B$ with AVM [${\left[\begin{array}{ll} {\mbox{\sc n}} &[F]\\ {\mbox{\sc m}} &[H] \end{array}\right]}$]{} if $x = y$.
Clearly, this simple test takes one step. A more sophisticated test, which also tests for inequality, would compare $[F]$ and $[G]$ bit-by-bit. Such a test would take $O(\min(\log(x),\log(y)))$ $=O(\min({\mathopen|[F]\mathclose|},{\mathopen|[H]\mathclose|}))$ derivation steps.
#### Syntactic rules that multiply by two.
Assume a nonterminal $A$ with some AVM ${\left[\begin{array}{ll} {\mbox{\sc n}} &[F] \end{array}\right]}$, where $[F]$ encodes natural number $x$. We present one syntactic rule that derives from this nonterminal $A$ a nonterminal $B$ with the AVM ${\left[\begin{array}{ll} {\mbox{\sc n}} &[H] \end{array}\right]}$, where $[H]$ encodes natural number $2x$.
The number in $[H]$ equals two times in $[F]$ if, and only if, the least-significant bit of in $[H]$ is 0, and the remaining bits form the same sequence as the number in $[F]$. Multiplication by two takes one derivation step.
#### Syntactic rules that increments by one.
Assume a nonterminal $A$ with some AVM ${\left[\begin{array}{ll} {\mbox{\sc n}} &[F] \end{array}\right]}$, where $[F]$ encodes natural number $x$. We present five syntactic rules that derive from this nonterminal $A$ a nonterminal $C$ with AVM ${\left[\begin{array}{ll} {\mbox{\sc n}} &[H] \end{array}\right]}$, where $[H]$ encodes natural number $x+1$.
The increment of requires two additional pointers in the AVM of $A$: attribute points to the next bit that has to be incremented; attribute points to the most-significant bit of the (intermediate) result. These additional pointers are hidden from the AVMs of the nonterminals $A$ and $C$.
The five rules from Table \[Tinc\] increment by one. Nonterminal $A$ rewrites, in one or more steps, to nonterminal $C$, potentially through a number of nonterminals $B$.
The first and fourth rule of Table \[Tinc\] state that adding one to a zero bit sets this bit to one and ends the increment. The second and third rule state that adding one to a one bit sets this bit to zero and the increment continues. The fifth rule states that adding one to the most-significant bit sets this bit to zero and yields a new most-significant one bit. We claim that $A \stackrel{*}{\Rightarrow} C$ takes $O(\log(x)) = O({\mathopen|[F]\mathclose|})$ derivation steps.
Rules, similar to the ones above, can be given that decrement the attribute by one. We only have to take a little extra care that the number 0 cannot be decremented.
#### Syntactic rules that sum two numbers.
In this section we use the previous test and increment rules (indicated by =). Assume a nonterminal $A$ with some AVM [${\left[\begin{array}{ll} {\mbox{\sc n}} &[F]\\ {\mbox{\sc m}} &[H] \end{array}\right]}$]{}, where $[F]$ and $[H]$ encode natural number $x$ and $y$, respectively. We present syntactic rules (Table \[Tcovsum\]–\[Tstopsum\]) that derive from this nonterminal $A$ a nonterminal $C$ with AVM [${\left[\begin{array}{ll} {\mbox{\sc n}} &[F']\\ {\mbox{\sc m}} &[H] \end{array}\right]}$]{}, where $[F']$ encodes the natural number $x + y$.
The increment of by is similar to the increment by one. Here, three additional pointers are required: the attributes and point to the bits in and respectively that have to be summed next; attribute points to the most-significant bit of the (intermediate) result. In the addition two states are distinguished. In the one state, the carry bit is zero, indicated by nonterminal $A'$. In the other state, the carry bit is one, indicated by nonterminal $B$. We claim that $A \stackrel{*}{\Rightarrow} C$ takes $O(\max(\log(x), \log(y))) = O(\max({\mathopen|[F]\mathclose|},{\mathopen|[H]\mathclose|}))$ derivation steps.
#### Syntactic rules that sum a sequence of numbers.
In this section we use the previous summation rules (indicated by =). Assume a nonterminal $A$ with some AVM ${\left[\begin{array}{ll} {\mbox{\sc l}} &[F'] \end{array}\right]}$, where $[F']$ encodes a list of numbers. To wit $$[F'] \;=\;
\mbox{\footnotesize
${\left[\begin{array}{ll} {\mbox{\sc f}} &[G_1] \\ {\mbox{\sc r}} &{\left[\begin{array}{ll} {\mbox{\sc f}} &[G_2] \\
{\mbox{\sc r}} &\ldots{\left[\begin{array}{ll} {\mbox{\sc f}} &[G_n] \\ {\mbox{\sc r}} &{\mbox{\it +}} \end{array}\right]} \end{array}\right]} \end{array}\right]}$ }$$ where $[G_i]$ encodes natural number $x_i$. We present syntactic rules (Table \[Tlist\]) that derive from this nonterminal $A$ a nonterminal $B$ with AVM [${\left[\begin{array}{ll} {\mbox{\sc suml}} &[F]\\ {\mbox{\sc l}} &[F'] \end{array}\right]}$]{}, where $[F]$ encodes the natural number $\Sigma_i x_i$.
The summation requires an additional pointer in the AVM $[F']$: attribute points to the next element in the list that has to be summed. We claim that $A \stackrel{*}{\Rightarrow} B$ takes $O(\Sigma_i\,\log(x_i)) = O({\mathopen|[F']\mathclose|})$ derivation steps.
Creating a counter of logarithmic size\[create\]
------------------------------------------------
Create an AVM of the following form: [$${\left[\begin{array}{ll} {\mbox{\sc counter}} &{\left[\begin{array}{ll}
{\mbox{\sc size}} &{\left[\begin{array}{ll} {\mbox{\sc 1}}\cup{\mbox{\sc 0}} &\ldots[{\mbox{\sc 1}}\;+] \end{array}\right]} \\
{\mbox{\sc n}} &{\left[\begin{array}{ll} {\mbox{\sc v}} &1 \cup 0\\ {\mbox{\sc 1}}\cup{\mbox{\sc 0}}
&{\left[\begin{array}{ll} {\mbox{\sc v}} &1 \cup 0\\ \ldots &[{\mbox{\sc 1}}\;+] \end{array}\right]} \end{array}\right]} \\
{\mbox{\sc m}} &{\left[\begin{array}{ll} {\mbox{\sc v}} &1 \cup 0\\ {\mbox{\sc 1}}\cup{\mbox{\sc 0}}
&{\left[\begin{array}{ll} {\mbox{\sc v}} &1 \cup 0\\ \ldots &[{\mbox{\sc 1}}\;+] \end{array}\right]} \end{array}\right]} \\
{\mbox{\sc poly}} &{\left[\begin{array}{ll} {\mbox{\sc 1}}\cup{\mbox{\sc 0}} &\ldots[{\mbox{\sc 1}}\;+] \end{array}\right]} \end{array}\right]} \end{array}\right]}$$ ]{}
Attribute is used to distinguish the AVMs that encodes the counter from those in the original attribute-value grammar. We will neglect the attribute in the remainder of this section, because it is not essential here. The attributes , , and encode natural numbers. The attribute records the size of the string that will be generated. The attribute records the maximum number of derivation steps that is allowed for a string of size . The attributes and are auxiliary numbers.
The construction of the counter starts with an initiation-step. The further construction of the counter consists of cycles of two phases. Each cycle starts in nonterminal $A$.
#### Initiation step and first phase.
The initiation-step sets the numbers and to 0, and the numbers and to 1. In the first phase of each cycle, the numbers and are incremented by 1.
#### The second phase of the cycle.
In this phase the numbers and are compared. If is twice , then ($i$) number is extended by $k$ bits, ($ii$) number is doubled, and ($iii$) number is set to 0. If is less than twice , nothing happens.
The left rule of the second phase doubles the number in the second and the third equation. The test “Is equal to 2?” therefore reduces to one (the first) equation. The fourth equation extend the number with $k$ bits. The fifth and sixth equations set the number to 0.
The right rule is always applicable. If the right rule is used where the left rule was applicable, then the number will never be equal to $2{\mbox{\sc m}}$ in the rest of the derivation. Thus will not be extended any more.
We claim that the left rule appears $\log(n)$ times and the right rule $O(n)$ times in a derivation for input of size $n$. Obviously, the number is $O(2^{k\log{i}}) = O(i^k)$ when the number is $i$.
From AVG to HP-AVG\[avg2hpc\]
-----------------------------
In this section we show how to transform an AVG into an AVG that satisfies the HPC (HP-AVG). Since all computation steps of the HP-AVG only require a linear amount of derivation steps, total derivations of HP-AVGs have polynomial length.
We can divide the attributes of the HP-AVG into two groups. The attributes that encode the counters, and the attributes of the original AVG. The former will be embedded under the attribute , the latter under the attribute . In the sequel, we mean by $\phi|{\mbox{\sc grammar}}$ the formula $\phi$ embedded under the attribute , i.e., the formula obtained from $\phi$ by substituting the variables $x_i$ by ${\mbox{\sc grammar}}(x_i)$.
The HP-AVG is obtained from the AVG in three steps: change the start nonterminal, the lexicon and the syntactic rules. First, the HP-AVG contains the rules of the previous section, which construct the counter. The nonterminal $S$ from Table \[T1st\] is the start nonterminal of the HP-AVG. For the nonterminal $A$ the start nonterminal of the AVG is taken. Nonterminal $B$ from Table \[T2nd\] is a fresh nonterminal, not occurring in the AVG.
Second, the HP-AVG contains an extension of the lexicon of the AVG. The entries of the lexicon are extended in the following way. The size of the lexical form is set to one, and the amount of derivation steps is zero. Thus, if $(w,X,\phi)$ is the lexicon of the AVG, then $(w,X,\psi)$ is the lexicon of the HP-AVG, where $$\begin{aligned}
\psi &= &\phi|{\mbox{\sc grammar}} \\
&\wedge &{\mbox{\sc v size counter}}(x_0) \doteq 1 \\
&\wedge &{\mbox{\sc 1 size counter}}(x_0) \doteq + \\
&\wedge &{\mbox{\sc poly counter}}(x_0) \doteq +\end{aligned}$$
Third, the HP-AVG contains extensions of the syntactic rules of the AVG. The syntactic rules are extended in the following way. The numbers and of the daughter nonterminals are collected in the lists and . Both lists are summed. The number of the mother nonterminal is equal to the sum of ’s, and the number of the mother nonterminal is one more than the sum of ’s. Thus, if $(X_0,X_1,\ldots,X_n, \phi)$ is a syntactic rule of the AVG, then $(X_0,X_1,\ldots,X_n, \psi)$ is a syntactic rule of the HP-AVG, where $$\begin{aligned}
\psi &= &\phi|{\mbox{\sc grammar}} \\
&\wedge &{\mbox{\sc sums counter}}(x_0) = \Sigma\,{\mbox{\sc slist counter}}(x_0) \\
&\wedge &{\mbox{\sc size counter}}(x_0) = {\mbox{\sc sums counter}}(x_0) \\
&\wedge &{\mbox{\sc sump counter}}(x_0) = \Sigma\,{\mbox{\sc plist counter}}(x_0) \\
&\wedge &{\mbox{\sc sump counter}}(x_0) = y \\
&\wedge &{\mbox{\sc poly counter}}(x_0) = y+1 \\
&\wedge &{\mbox{\sc f r$^i$ slist counter}}(x_0) \doteq {\mbox{\sc size counter}}(x_i)
\;\; (0 \leq i < n) \\
&\wedge &{\mbox{\sc r$^n$ slist counter}}(x_0) \doteq + \\
&\wedge &{\mbox{\sc f r$^i$ plist counter}}(x_0) \doteq {\mbox{\sc poly counter}}(x_i)
\;\; (0 \leq i < n) \\
&\wedge &{\mbox{\sc r$^n$ plist counter}}(x_0) \doteq +\end{aligned}$$
Now, a derivation for the HP-AVG starts with a nondeterministic construction of a counter with value $n$ and a counter with value $O(n^k)$. Then, the derivation of the original AVG is simulated, such that ($i$) the mother nonterminal produces a string of size $n$ if, and only if the daughter nonterminals together produce a string of size $n$, and ($ii$) the mother nonterminal makes $n^k+1$ derivation steps if, and only if the daughter nonterminals together make $n^k$ derivation steps.
[^1]: The author was supported in part by HC&M grant ERB4050PL93-0516.
[^2]: The author was supported by the Foundation for language, speech and logic (TSL), which is funded by the Netherlands organization for scientific research (NWO)
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author:
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Wei Xiao$^\dagger$, Hao Helen Zhang$^\ddagger$, and Wenbin Lu$^\dagger$\
\
\
bibliography:
- 'OTR.bib'
title: Robust regression for optimal individualized treatment rules
---
> [*Abstract:*]{} Because different patients may response quite differently to the same drug or treatment, there is increasing interest in discovering individualized treatment rule. In particular, people are eager to find the optimal individualized treatment rules, which if followed by the whole patient population would lead to the “best” outcome. In this paper, we propose new estimators based on robust regression with general loss functions to estimate the optimal individualized treatment rules. The new estimators possess the following nice properties: first, they are robust against skewed, heterogeneous, heavy-tailed errors or outliers; second, they are robust against misspecification of the baseline function; third, under certain situations, the new estimator coupled with pinball loss approximately maximizes the outcome’s conditional quantile instead of conditional mean, which leads to a different optimal individualized treatment rule comparing with traditional Q- and A-learning. Consistency and asymptotic normality of the proposed estimators are established. Their empirical performance is demonstrated via extensive simulation studies and an analysis of an AIDS data.
>
> [*Key words and phrases:*]{} Optimal individualized treatment rules; Personalized medicine; Quantile regression; Robust regression.
Introduction
============
Given the same drug or treatment, different patients may respond quite differently. Factors causing individual variability in drug response are multi-fold and complex. This has raised increasing interests of individualized medicine, where customized medicine or treatment is recommended to each individual according to his/her characteristics, including genetic, physiological, demographic, environmental, and other clinical information. The rule that applied in personalized medicine to match each patient with a target treatment is called individualized treatment rule (ITR), and our goal is to find the “optimal” one, which if followed by the whole patient population would lead to the “best” outcome. For many complex diseases such as cancer and AIDS, the optimal individualized treatment rule or regime is a dynamical treatment process, involving a sequence of treatment decisions made at different time points throughout the disease evolving course.
Q-learning [@watkins1992q; @murphy2005generalization] and A-learning [@murphy2003optimal; @robins2004optimal] are two main approaches for finding optimal dynamic individualized treatment rules based on clinical trials or observational data. Q-learning is based on posing a regression model to estimate the conditional expectation of the outcome at each time point, and then applying a backward recursive procedure to fit the model. A-learning, on the other hand, only requires modeling the contrast function of the treatments at each time point, is therefore more flexible and robust to a model misspecification. See [@schulte2014q] for a complete review and comparison of these two methods under various scenarios, in terms of the parameter estimation accuracy and the estimation of expected outcomes. Q- and A-learning have good performance when model is correctly specified but are sensitive to model misspecification. To overcome this shortcoming, several “direct” methods have been proposed, which maximize value functions directly instead of modeling the conditional mean. See [@ZhaoYingQi2012OWL; @Zhang2013Robust] for example.
All existing methods for optimal individualized treatment rule estimation, including Q-learning and A-learning, belong to mean regression as they estimate the optimal estimator by maximizing expected outcomes. In the case of single decision point, Q-learning is equivalent to the least-squares regression. Least-squares estimates are optimal if the errors are i.i.d. normal random variables. However, skewed, heavy-tailed, heteroscedastic errors or outliers of the response are frequently encountered. In such situations, the efficiency of the least square estimates is impaired. One extreme example is that when the response takes i.i.d. Cauchy errors, neither Q-learning nor A-learning can consistently estimate the optimal ITR. For example, in AIDS Clinical Trials Group Protocol 175 (ACTG175) data [@hammer1996trial], HIV-infected subjects were randomized to four regimes with equal probabilities, and our objective is to find the optimal ITR for each patient based on their age, weight, race, gender and some other baseline measurements. The response CD4 count of the data follows a skewed, heteroscedastic errors, which weakens the efficiency of classical Q- and A-learning. A method to estimate optimal ITR which is robust against skewed, heavy-tailed, heteroscedastic errors or outliers is highly valuable. One possible solution is to construct the optimal decision rule based on the conditional median or quantiles of response given covariates than based on average effects.
In the following, we present a simple example where a quantile-based decision rule is more preferable than a mean-based decision rules. We use higher value of response $Y$ to indicate more favorable outcomes. Figure \[fig:plot1\] plots the conditional density of $Y$ under two treatments, $A$ and $B$, given a binary covariate $X$ which takes the value of male and female. Under the comparison based on conditional means, $A$ and $B$ are exactly equivalent. However, conditional quantiles provide us more insight. For the male group, the conditional distribution of response given treatment $B$ is a log-normal and skewed to the right. Therefore, treatment $B$ is less favorable when either 50% or 25% conditional quantile are considered. For the female group, the conditional distribution of response given treatment $A$ is a standard normal while a Cauchy distribution given treatment $B$. Therefore, if we make a comparison based on $25\%$ conditional quantile, treatment $A$ is more favorable.
![The distribution functions of the response $Y$, in a randomized clinical trial with two treatments, $A$ and $B$, for male (two panels on the left) and female (two panels on the right). The solid lines with triangle symbol, dashed line, and dotted lines are the conditional mean, $50\%$ quantile, and $25\%$ quantile functions of $Y$ given the gender and the treatment, respectively.[]{data-label="fig:plot1"}](plot1.eps){width="6in"}
In this paper, we propose a general framework for optimal individualized treatment rule estimation based on robust regression, including quantile regression and the regression based on Huber’s loss and $\epsilon$-insensitive loss. The proposed methodology has the following desired features. First, the new decision rule obtained by maximizing the conditional quantile, which is suitable for skewed, heavy-tailed errors or outliers. Second, the proposed estimator requires only modeling the contrast function between two treatments, and is therefore robust against misspecification of the baseline function. This property is shared by A-learning. Third, empirical results from our comprehensive numerical study suggest favorable performance of the new robust regression estimator.
The rest of the paper is organized as follows. In Section 2, we first review the classical Q- and A- learning methods. Then we propose the new procedure and method and discuss its connection with existing methods. In Section 3, we study and prove the asymptotic properties of the proposed method, including consistency and asymptotic normality. In Section 4, a comprehensive numerical study is conducted to assess finite sample performance of the new procedure. In Section 5, we apply the method to ACTG175 data. Concluding remarks are given in Section 6. Throughout the paper, we use upper case letters to denote random variables and lower case letters to denote their values.
New Optimal Treatment Estimation Framework: Robust Regression
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Basic Notations and Assumptions
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For simplicity, we consider a single stage randomized clinical trial with two treatments. For each patient, the observed data is $({{\bm X}},A,Y)$, where ${{\bm X}}\in\mathcal{X}={\mathrm{I \! R} \mathit{^{p}}}$ denotes the baseline covariates, $A\in\mathcal{A}=\{0,1\}$ denotes the treatment assigned to the patient, and $Y$ is the real-valued response, which is coded so that higher values indicate more favorable clinical outcomes. An ITR $g$ is a function mapping from $\mathcal{X}$ to $\mathcal{A}$.
We first review the potential outcome framework [@neyman1923applications; @rubin1974estimating; @rubin1986comment]. The potential outcome $Y^*(a)$ is the outcome for an arbitrary individual has s/he received treatment $a$. In actuality, at most one of the potential outcomes can be observed for any individual. The optimal ITR under mean regression, which maximizes the expected outcome, is $g^{\mathrm{opt}}_\mu={\mathrm{argmax}}_{g\in\mathcal{G}}{\mbox{E}}[Y^*\{g({{\bm X}})\}]$. Define the propensity score $\pi({{\bm X}})\triangleq P(A=1|{{\bm X}})$. Following [@rubin1974estimating] and [@rubin1986comment], we can compute the expectation of the potential outcome under the following two key assumptions.
- **Stable Unit Treatment Value Assumption (SUTVA):** a patient’s observed outcome is the same as the potential outcome for the treatment that s/he actually received. Based on [@rubin1986comment], the SUTVA assumption implies that the value of the potential outcome for a subject does not depend on what treatments other subject receive. Specifically, we can write the SUTVA assumption as $$Y_i=Y_i^*(1)A_i+Y_i^*(0)(1-A_i),\;i=1,\ldots,n.$$ This is also referred as consistency assumption.
- **Strong Ignorability Assumption:** the treatment assignment $A$ for an individual is independent of the potential outcomes conditional on the covariates ${{\bm X}}$, i.e., $A\bot \{Y^*(a)\}_{a\in\mathcal{A}}|{{\bm X}}$. For a randomized clinical trial, this assumption is satisfied automatically. For an observational study, as clinicians make decisions based only on all past available information, this assumption essentially assumes no unmeasured confounders.
For consistent estimation of the optimal treatment rule, we also need to assume
- **Positivity Assumption:** $0<\pi({{\bm x}})<1$, $\forall {{\bm x}}\in\mathcal{X}$.
Existing Learning Methods: Q-learning and A-learning
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Define the Q-function $Q({{\bm x}},a)\triangleq{\mbox{E}}(Y|{{\bm x}},a)$. Under assumptions (C1)-(C2), one can show that $g^{\mathrm{opt}}_\mu({{\bm X}})={\mathrm{argmax}}_{a\in\mathcal{A}}Q({{\bm x}},a)={\mathrm{argmax}}_{a\in\mathcal{A}}{\mbox{E}}(Y|{{\bm X}},A=a)$. This suggests that, in order to find $g^{\mathrm{opt}}_\mu$, we only need to estimate the conditional expectation of $Y$ given $({{\bm X}},A)$. This result serves as the foundation of Q- and A-learning framework. We further define the value function $V_\mu(g)={\mbox{E}}_{{{\bm X}}}[Q\{{{\bm X}},g({{\bm X}})\}]$ which is simply the marginal mean outcome under the ITR $g$, and $g^{\mathrm{opt}}_\mu={\mathrm{argmax}}_gV_\mu(g)$.
Define the $\tau$-th conditional quantile of $Y$ given $({{\bm X}},A)$ as $Q_{\tau}({{\bm X}},A)\triangleq\inf\{y: F_{Y|{{\bm X}},A}(y)\geq\tau\}$. Then we define the value function based on the $\tau$-th conditional quantile as $V_{\tau-q}(g)={\mbox{E}}_{{{\bm X}}}[Q_{\tau}\{{{\bm X}},g({{\bm X}})\}]$, which is an analog to the definition of $V_\mu(g)$. The optimal ITR which maximizes the $\tau$-th conditional quantile is then defined as $$g^{\mathrm{opt}}_{\tau}({{\bm x}})=\underset{a\in\mathcal{A}}{\mathrm{argmax}}Q_{\tau}({{\bm x}},a),\;\tau\in[0,1],$$ and $g^{\mathrm{opt}}_\tau={\mathrm{argmax}}_gV_{\tau-q}(g)$.
Consider the general model ${\mbox{E}}(Y|{{\bm X}},A)=h_0({{\bm X}})+AC_0({{\bm X}})$, where $h_0({{\bm X}})$ represents the baseline effect, and $C_0({{\bm X}})$ denotes the contrast effect as $$C_0({{\bm X}})={\mbox{E}}(Y|{{\bm X}},A=1)-{\mbox{E}}(Y|{{\bm X}},A=0).$$ Therefore, $g^{\mathrm{opt}}_\mu({{\bm X}})={\mathrm{1}}\{C_0({{\bm X}})>0$}. In Q-learning, a parametric model is often employed as a working model, $${\mbox{E}}(Y|{{\bm X}},A)=h({{\bm X}}; {{\mbox{\boldmath $\gamma$}}})+AC({{\bm X}};{{\mbox{\boldmath $\beta$}}}),$$ where $h({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$ and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ are posited parametric models for $h_0({{\bm X}})$ and $C_0({{\bm X}})$ respectively. Commonly a linear model is assumed for simplicity and interpretability, i.e., $h({{\bm X}};{{\mbox{\boldmath $\gamma$}}})={{\mbox{\boldmath $\gamma$}}}{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}$ and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})={{\mbox{\boldmath $\beta$}}}{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}$, where $\tilde{{{\bm X}}}=({{\mathbf 1}},{{\bm X}}{^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$. Given the observation $\{(Y_i,{{\bm X}}_i,A_i);\;i=1,\ldots,n\}$, the Q-learning procedure estimates the parameters $({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})$ by minimizing the squared error loss $$L_{1n}({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})=\frac{1}{n}\sum_{i=1}^{n}\left\{Y_i-h({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})-A_iC({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})\right\}^2.$$ Denote the optimized point as $(\hat{{{\mbox{\boldmath $\beta$}}}}^Q,\hat{{{\mbox{\boldmath $\gamma$}}}}^Q)$. The estimated optimal ITR based on Q-learning is then $\hat{g}^{Q}({{\bm x}})\triangleq{\mathrm{1}}\{C({{\bm x}};\hat{{{\mbox{\boldmath $\beta$}}}}^Q)>0\}$, which is a consistent estimator of $g^{\mathrm{opt}}_\mu({{\bm x}})$ if both $h({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$ and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ are correctly specified.
A-learning is a semiparametric improvement of Q-learning by modeling only the contrast function $C_0({{\bm X}})$ rather than the full Q-function. This is reasonable based on the observation that the optimal ITR $g^{\mathrm{opt}}_\mu$ only depends on $C_0({{\bm X}})$. By positing $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ for the contrast function, in A-learning, one can estimate coefficients ${{\mbox{\boldmath $\beta$}}}$ by solving the following estimating equation $$\sum_{i=1}^{n}\lambda({{\bm X}}_i)\left\{A_i-\pi({{\bm X}}_i)\right\}\left\{Y_i-A_iC({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})-h({{\bm X}}_i)\right\}=0,
\label{eq:eeofAlearn}$$ where $\lambda({{\bm X}}_i)$ and $h({{\bm X}}_i)$ are arbitrary functions, and $\lambda({{\bm X}}_i)$ has the same dimension as ${{\mbox{\boldmath $\beta$}}}$. Denote the solution to by $\hat{{{\mbox{\boldmath $\beta$}}}}^{A}$. If $\mathrm{var}(Y|X)$ is constant and $C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})$ is correctly specified, the optimal choices of $\lambda(\cdot)$ and $h(\cdot)$ are $\lambda({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})=\partial/\partial {{\mbox{\boldmath $\beta$}}}C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})$ and $h({{\bm X}}_i)=h_0({{\bm X}}_i)$ [@robins2004optimal]. In practice, one may pose models, say $\pi({{\bm X}}_i;{{\mbox{\boldmath $\phi$}}})$ and $h({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})$ for $\pi({{\bm X}}_i)$ and $h({{\bm X}}_i)$ respectively, and take $\lambda({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})=\partial/\partial {{\mbox{\boldmath $\beta$}}}C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})$. Under randomized designs, the propensity score $\pi({{\bm X}}_i)$ is known. Otherwise, a logistic model can be proposed. Under the assumption that $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ is correctly specified, the double robustness property of A-learning states that as long as one of $\pi({{\bm X}};{{\mbox{\boldmath $\phi$}}})$ and $h({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$ is correctly specified, $\hat{g}^{A}({{\bm x}})\triangleq{\mathrm{1}}\{C({{\bm x}};\hat{{{\mbox{\boldmath $\beta$}}}}^A)>0\}$ is consistent estimator of $g^{\mathrm{opt}}_\mu({{\bm x}})$.
Recently, [-@lu2011variable] propose a variant of A-learning by a loss-based learning framework. Rewrite $$\begin{aligned}
{\mbox{E}}(Y|{{\bm X}},A) =& h_0({{\bm X}})+AC_0({{\bm X}})\\
=& \varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}C_0({{\bm X}}),\end{aligned}$$ where $\varphi_0({{\bm X}})=h_0({{\bm X}})+\pi({{\bm X}})C_0({{\bm X}})$. Based on the expression above, [@lu2011variable] propose to estimate $({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})$ by minimizing the following loss function $$L_{2n}({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})=\frac{1}{n}\sum_{i=1}^{n}\left[Y_i-\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})-\{A_i-\pi({{\bm X}}_i)\}C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})\right]^2,
\label{eq:A-loss}$$ where $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$, $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ are proposed models for $\varphi_0({{\bm X}})$ and $C_0({{\bm X}})$ respectively. Denote the minimizer of as $(\hat{{{\mbox{\boldmath $\beta$}}}}^A_{LS},\hat{{{\mbox{\boldmath $\gamma$}}}}^{A}_{LS})$. [@lu2011variable] show that $\hat{g}^{A}_{LS}({{\bm x}})\triangleq{\mathrm{1}}\{C({{\bm x}};\hat{{{\mbox{\boldmath $\beta$}}}}^A_{LS})>0\}$ is a consistent estimator of $g^{\mathrm{opt}}_\mu({{\bm x}})$ when the propensity score $\pi({{\bm X}})$ is known or can be consistently estimated from the data, and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ is correctly specified. We refer to this method as least square A-learning (lsA-learning).
One main advantage of the lsA-learning, compared to the classical A-learning, is its square loss, making the procedure easy to be coupled with penalized regression to achieve variable selection in high dimensional data. Specifically, [@lu2011variable] propose to identify important nonzero coefficients in ${{\mbox{\boldmath $\beta$}}}$ by applying an adaptive LASSO penalty to . Under some regularity conditions, both the selection consistency and asymptotic normality of the estimator are established in [@lu2011variable]. The downside of lsA-learning is that one direction of the double robustness property of the classical A-learning is lost, i.e., when $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$ is correctly specified, ${{\mbox{\boldmath $\beta$}}}$ may still not be consistent if the propensity score $\pi({{\bm X}})$ is not consistently estimated. Finally, it can be shown that lsA-learning and Q-learning are equivalent when $\pi({{\bm X}})$ is constant and both $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$ and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ take the linear form (with the space of $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ included in the space of $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$). Similar properties hold for A-learning and Q-learning [@schulte2014q].
New Proposal: Robust Regression
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Skewed, heavy-tailed, heteroscedastic errors or outliers of the response $Y$ are frequently encountered in clinical trials. It is well known that ordinary least square estimation fails to produce a reliable estimator in such situations. The immediate consequence is the efficiency loss in the estimators produced by Q-, A-, and lsA-learning. This motivates us to adopt robust regression techniques in optimal treatment regime estimation.
We consider the following additive model, $$Y_i=\varphi_0({{\bm X}}_i)+\{A_i-\pi({{\bm X}}_i)\}C({{\bm X}}_i;\beta_0)+\epsilon_i,\; i=1,\ldots,n,
\label{eq:model_nointeraction}$$ where $\varphi_0({{\bm X}})$ is the baseline function, $C({{\bm X}};\beta_0)$ is the contrast function, $\pi({{\bm X}})$ is the propensity score, and $\epsilon$ is the error term which satisfies the conditional independence assumption $\epsilon\perp A|{{\bm X}}$. We point out that the error term defined in can be very general. For example, we could take $\epsilon=\sum_{j=1}^{K}\sigma_j({{\bm X}})e_j$ for any $K\geq 1$ that allows the error distribution to change with ${{\bm X}}$, used to model heterogeneous errors, where $\sigma_j({{\bm X}})$ are arbitrary positive functions and $e_{j}\perp (A,{{\bm X}})$ for all $j=1,\ldots,K$. Throughout the paper, we assume $\{(Y_i,{{\bm X}}_i,A_i,\epsilon_i),i=1,\ldots,n\}$ are i.i.d random samples of the population.
We propose to estimate $({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}})$ by minimizing $$L_{3n}({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}})=\frac{1}{n}\sum_{i=1}^{n}M\left[Y_i-\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})-\{A_i-\pi({{\bm X}}_i)\}C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})\right],
\label{eq:A-general-loss}$$ where ${{\mbox{\boldmath $\gamma$}}}\in\Gamma$, ${{\mbox{\boldmath $\beta$}}}\in\mathcal{B}$ and $M:{\mathrm{I \! R} \mathit{^{\rightarrow}}} [0,\infty)$ is a convex function with minimum achieved at 0. Denote the minimizer of as $(\hat{{{\mbox{\boldmath $\beta$}}}}^R_{M},\hat{{{\mbox{\boldmath $\gamma$}}}}^{R}_{M})$, and the estimated ITR is then $\hat{g}^{R}_M({{\bm x}})\triangleq{\mathrm{1}}\{C({{\bm x}};\hat{{{\mbox{\boldmath $\beta$}}}}^R_M)>0\}$. In the following, we refer the robust regression with loss function $M(x)$ as RR(M)-learning. In this article, we consider the following three types of loss functions, i.e., the pinball loss $$M(x)=\rho_\tau(x)\triangleq
\begin{cases}
(\tau-1)x, &\text{if } x<0\\
\tau x, &\text{if } x\geq0
\end{cases}$$ where $0<\tau<1$, the Huber loss $$M(x)=H_\alpha(x)\triangleq
\begin{cases}
0.5x^2, &\text{if } |x|<\alpha\\
\alpha|x|-0.5\alpha^2, &\text{if } |x|\geq\alpha
\end{cases}$$ for some $\alpha>0$, and the $\epsilon$-insensitive loss $$M(x)=J_\epsilon(x)\triangleq\max(0, |x|-\epsilon)$$ for some $\epsilon>0$. The pinball loss are frequently applied for quantile regression [@koenker2005quantile], and the Huber losses and the $\epsilon$-insensitive are robust against heavy tailed errors or outliers. A dramatic difference of pinball loss, Huber loss and $\epsilon$-insensitive loss, compared with the square loss, is that they penalize large deviances linearly instead of quadratically. This property makes them more robust when dealing with responses with non-normal type of errors.
Asymptotic Properties {#section:asymptotic}
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Consistency of Robust Regression: Pinball Loss
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Under the conditional independence assumption $\epsilon\perp A|{{\bm X}}$, we have $$\begin{aligned}
Q({{\bm X}},A)=& \varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}C({{\bm X}};\beta_0)+\mu_\epsilon({{\bm X}});\\
Q_\tau({{\bm X}},A)=& \varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}C({{\bm X}};\beta_0)+F^{-1}_{\epsilon}({{\bm X}};\tau).\end{aligned}$$ where $\mu_{\epsilon}({{\bm X}})$ and $F^{-1}_{\epsilon}({{\bm X}};\tau)$ denote the mean and the $\tau$-th quantile of $\epsilon$ conditional on ${{\bm X}}$ respectively. Therefore, in this situation, we have $g^{\mathrm{opt}}_\mu=g^{\mathrm{opt}}_{\tau}={\mathrm{1}}\{C({{\bm X}};{{\mbox{\boldmath $\beta$}}}_0)>0\}$. In other words, the underlying ITR which maximize the population mean and $\tau$-th quantile are equivalent. For a good ITR $\hat{g}={\mathrm{1}}\{C({{\bm X}};\hat{{{\mbox{\boldmath $\beta$}}}})>0\}$, it is reasonable to require $\hat{{{\mbox{\boldmath $\beta$}}}}$ to be a consistent estimator of ${{\mbox{\boldmath $\beta$}}}_0$. This consistency result is first shown for the robust regression with pinball loss, which is given in Theorem 1. We allocate all the proofs into the Appendix A.
Under regularity conditions (A1)-(A8) in the Appendix A, if the contrast function in is correctly specified and $\pi({{\bm x}})$ is known, then $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)}\inprob{{\mbox{\boldmath $\beta$}}}_0$ for all $\tau\in(0,1)$, where $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)}$ is the solution of when $M(x)=\rho_{\tau}(x)$.
**Remarks:**
1. Theorem 1 doesn’t assume the finiteness of $E(Y)$. Therefore it can be applied to the cases when $\epsilon_i$ follows a Cauchy distribution.
2. After fitting the model, the Assumption (A2), $\epsilon\perp A|{{\bm X}}$, can be verified by applying conditional independence test with $\hat{r}(\hat{{{\mbox{\boldmath $\beta$}}}},\hat{{{\mbox{\boldmath $\gamma$}}}})$ and $A$ given ${{\bm X}}$, where $\hat{r}(\hat{{{\mbox{\boldmath $\beta$}}}},\hat{{{\mbox{\boldmath $\gamma$}}}})$ is the estimated residual and $\hat{r}(\hat{{{\mbox{\boldmath $\beta$}}}},\hat{{{\mbox{\boldmath $\gamma$}}}})=Y-\varphi({{\bm X}};\hat{{{\mbox{\boldmath $\gamma$}}}})-\{A-\pi({{\bm X}})\}C({{\bm X}};\hat{{{\mbox{\boldmath $\beta$}}}})$. See [@lawrance1976conditional; @su2007consistent; @song2007testing; @huang2010testing; @zhang2012kernel] for more discussion of conditional independence hypothesis tests. In particular, we demonstrate the usefulness of the test by applying the Kernel-based conditional independence test (KCI-test, [@zhang2012kernel]) in Section 5. KCI-test doesn’t assume functional forms among variables and thus suits our need.
When the conditional independence assumption ($\epsilon\perp A|{{\bm X}}$) does not hold, $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)}$ may no longer be a consistent estimator of ${{\mbox{\boldmath $\beta$}}}_0$. This is intuitively reasonable as $\epsilon$ contains extra information with respect to $A$. In fact, a general result which can be derived in this case is that, $(\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)},\hat{{{\mbox{\boldmath $\gamma$}}}}^{R}_{\rho(\tau)})$ minimizes a weighed mean-square error loss function with specification error [@angrist2006quantile; @lee2013interpretation].
Instead of assuming response $Y$ takes an additive error term $\epsilon$ as in , we assume the conditional quantile function $Q_\tau({{\bm X}},A)=\varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}C({{\bm X}};{{\mbox{\boldmath $\beta$}}}_0(\tau))$, where we redundantly represent the baseline function and contrast function as $\varphi_0(\cdot)$ and $C(\cdot)$ respectively. Notice that we use ${{\mbox{\boldmath $\beta$}}}_0(\tau)$ instead of ${{\mbox{\boldmath $\beta$}}}_0$ to emphasize that the true ${{\mbox{\boldmath $\beta$}}}$ may vary with respect to $\tau$. The proposed model is $\hat{Q}({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})=\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})+\{A-\pi({{\bm X}})\}C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ with $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ correctly specified. Define $$\label{eq:population_beta_tau}
\left({{\mbox{\boldmath $\beta$}}}(\tau),{{\mbox{\boldmath $\gamma$}}}(\tau)\right)=\underset{{{\mbox{\boldmath $\beta$}}}\in\mathcal{B},{{\mbox{\boldmath $\gamma$}}}\in\Gamma}{\mathrm{argmin}}{\mbox{E}}\left[\rho_{\tau}\{Y-\hat{Q}({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})\}
-\rho_{\tau}\{Y-\hat{Q}({{\mbox{\boldmath $\beta$}}}',{{\mbox{\boldmath $\gamma$}}}')\}\right]$$ where $({{\mbox{\boldmath $\beta$}}}',{{\mbox{\boldmath $\gamma$}}}')$ is any fixed point in $\mathcal{B}\times\Gamma$. Define the QR specification error as $\Delta_{\tau}({{\bm X}},A;{{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})\triangleq\hat{Q}({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})-Q_\tau({{\bm X}},A)$. Define the quantile-specific residual as $\epsilon_{\tau}\triangleq Y-Q_\tau({{\bm X}},A)$ with conditional density function $f_{\epsilon_{\tau}}(\cdot|{{\bm X}},A)$. Then we have the following approximation theorem. The proof of the theorem follows Theorem 1 of [@angrist2006quantile], and is omitted for brevity.
Suppose that (i) the conditional density $f_{Y}(y|{{\bm X}},A)$ exists a.s.; (ii)${\mbox{E}}[Q_{\tau}({{\bm X}},A)]$ and ${\mbox{E}}[\Delta^2_{\tau}({{\bm X}},A;{{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})]$ are finite; (iii) $\left({{\mbox{\boldmath $\beta$}}}(\tau),{{\mbox{\boldmath $\gamma$}}}(\tau)\right)$ uniquely solves . Then $$\left({{\mbox{\boldmath $\beta$}}}(\tau),{{\mbox{\boldmath $\gamma$}}}(\tau)\right)=\underset{{{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}}}{\mathrm{argmin}}{\mbox{E}}[w_{\tau}({{\bm X}},A;{{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})\Delta^2_{\tau}({{\bm X}},A;{{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})]$$ where $$w_{\tau}({{\bm X}},A;{{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})=\int_{0}^{1}(1-u)f_{\epsilon_{\tau}}(u\Delta_{\tau}({{\bm X}},A;{{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})|{{\bm X}},A)du.$$ \[thm:approximation\_quantile\]
**Remarks:**
1. Theorem 2 shows that $\hat{Q}\left({{\mbox{\boldmath $\beta$}}}(\tau),{{\mbox{\boldmath $\gamma$}}}(\tau)\right)$ is a weighted least square approximation to $Q_{\tau}({{\bm X}},A)$. In other word, $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}}(\tau))+\{A-\pi({{\bm X}})\}C({{\bm X}};{{\mbox{\boldmath $\beta$}}}(\tau))$ is close to $\varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}C({{\bm X}};{{\mbox{\boldmath $\beta$}}}_0(\tau))$. So even though it is not true that ${{\mbox{\boldmath $\beta$}}}(\tau)={{\mbox{\boldmath $\beta$}}}_0(\tau)$ holds exactly, the difference between them is small in general . This coupled with the fact that $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)}\inprob{{\mbox{\boldmath $\beta$}}}(\tau)$ (proved in Theorem \[thm:asymptotic\_normality\_pinball\]), leads to the conclusion that approximately ITR $\hat{g}^{R}_{\rho(\tau)}({{\bm x}})$ $(\triangleq{\mathrm{1}}\{C({{\bm x}};\hat{{{\mbox{\boldmath $\beta$}}}}^R_{\rho(\tau)})>0\})$ maximizes the $\tau$-th conditional quantile. This observation is justified numerically in Section 4.2.
2. When there exists ${{\mbox{\boldmath $\gamma$}}}_0\in\Gamma$ such that $\varphi_0({{\bm X}})\equiv\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}}_0)$, then we have ${{\mbox{\boldmath $\beta$}}}(\tau)={{\mbox{\boldmath $\beta$}}}_0(\tau)$.
Consistency of Robust Regression: Other Losses
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Under model and the assumption $\epsilon\perp A|{{\bm X}}$, similar consistency results can be established for Huber loss and the $\epsilon$-insensitive loss, as stated in Theorem \[thm:consistency\_huber\].
\[thm:consistency\_huber\] Under regularity conditions (A1)-(A8), if the contrast function in is correctly specified and $\pi({{\bm x}})$ is known, then we have
(a) $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{H(\alpha)}\inprob{{\mbox{\boldmath $\beta$}}}_0$ for all $\alpha>0$, where $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{H(\alpha)}$ is the solution of when $M(x)=H_{\alpha}(x)$;
(b) $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{J(\epsilon)}\inprob{{\mbox{\boldmath $\beta$}}}_0$ for all $\epsilon>0$, where $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{J(\epsilon)}$ is the solution of when $M(x)=J_{\epsilon}(x)$.
Asymptotic Normality: Pinball Loss
----------------------------------
Without loss of generality, in this section we assume both the $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$ and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ take the linear form: $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})=\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\gamma$}}}$ and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})=\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\beta$}}}$, where $\tilde{{{\bm X}}}=(1,{{\bm X}}{^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$. Denote $\hat{{{\mbox{\boldmath $\beta$}}}}(\tau)=\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)}$ and $\hat{{{\mbox{\boldmath $\gamma$}}}}(\tau)=\hat{{{\mbox{\boldmath $\gamma$}}}}^{R}_{\rho(\tau)}$. Denote ${{\bm W}}=(\{A-\pi({{\bm X}})\}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}},\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$, ${{\mbox{\boldmath $\theta$}}}(\tau)=({{\mbox{\boldmath $\beta$}}}(\tau){^{\mbox{\tiny {\sf T}}}},{{\mbox{\boldmath $\gamma$}}}(\tau){^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$, $\hat{{{\mbox{\boldmath $\theta$}}}}(\tau)=(\hat{{{\mbox{\boldmath $\beta$}}}}(\tau){^{\mbox{\tiny {\sf T}}}},\hat{{{\mbox{\boldmath $\gamma$}}}}(\tau){^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$ and $J(\tau)\triangleq{\mbox{E}}\left[f_Y({{\bm W}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\theta$}}}(\tau)|{{\bm X}},A){{\bm W}}{{\bm W}}{^{\mbox{\tiny {\sf T}}}}\right]$. Under the following regularity conditions, which is the same as the assumptions assumed in [@angrist2006quantile] and [@lee2013interpretation], we have the asymptotic normality of $\hat{{{\mbox{\boldmath $\theta$}}}}(\tau)$, which is given in Theorem \[thm:asymptotic\_normality\_pinball\].
- $\{(Y_i,{{\bm X}}_i,A_i,\epsilon_i),i=1,\ldots,n\}$ are i.i.d random variables;
- the conditional density $f_Y(y|{{\bm X}}={{\bm x}},A=a))$ exists, and is bounded and uniformly continuous in y, uniformly in ${{\bm x}}$ over the support of ${{\bm X}}$;
- $J(\tau)$ is positive definite for all $\tau\in(0,1)$, where ${{\mbox{\boldmath $\theta$}}}(\tau)$ is uniquely defined in ;
- ${\mbox{E}}\|{{\bm X}}\|^{2+\epsilon}$ for some $\epsilon>0$.
\[thm:asymptotic\_normality\_pinball\] If regularity conditions (B1)-(B4) are hold, we have
1. (**Uniform Consistency**) $\sup_{\tau}\|\hat{{{\mbox{\boldmath $\theta$}}}}(\tau)-{{\mbox{\boldmath $\theta$}}}(\tau)\|=o_p(1)$;
2. (**Asymptotic Normality**) $J(\cdot)\sqrt{n}(\hat{{{\mbox{\boldmath $\theta$}}}}(\cdot)-{{\mbox{\boldmath $\theta$}}}(\cdot))$ converge in distribution to a zero mean Gaussian process with covariance function $\Sigma(\tau,\tau')$ defined as $$\Sigma(\tau,\tau')={\mbox{E}}\left[\left(\tau-{\mathrm{1}}\{Y<{{\bm W}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\theta$}}}(\tau)\}\right)
\left(\tau'-{\mathrm{1}}\{Y<{{\bm W}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\theta$}}}(\tau)\}\right){{\bm W}}{{\bm W}}{^{\mbox{\tiny {\sf T}}}}\right].$$
The proof is given in [@angrist2006quantile], and the asymptotic covariance matrix of $\hat{{{\mbox{\boldmath $\theta$}}}}(\tau)$ can be estimated by either a bootstrap procedure [@hahn1997bayesian] or a nonparametric kernel method [@angrist2006quantile]. We adopt the parametric bootstrap approach to estimate the asymptotic covariance matrix in Section 5. Under model the result of Theorem \[thm:asymptotic\_normality\_pinball\] can be further simplified, which is given in Theorem 5.
Under the condition of Theorem 4, if further we assume $Y=\varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\beta$}}}_0+\epsilon$, and $\epsilon\perp A|{{\bm X}}$, then
1. $\sup_{\tau}\|\hat{{{\mbox{\boldmath $\beta$}}}}(\tau)-{{\mbox{\boldmath $\beta$}}}_0\|=o_p(1)$;
2. $\sqrt{n}(\hat{{{\mbox{\boldmath $\beta$}}}}(\tau)-{{\mbox{\boldmath $\beta$}}}_0)\indist N({{\mathbf 0}}, J_{11}^{-1}(\tau)\Sigma_{11}(\tau,\tau)J_{11}^{-1}(\tau))$, where $$\begin{aligned}
J_{11}(\tau)=& {\mbox{E}}\left[f_{\epsilon}\left(\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\gamma$}}}(\tau)-\varphi_0({{\bm X}})|{{\bm X}}\right)
\pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\right],\\
\Sigma_{11}(\tau,\tau)=& {\mbox{E}}\left\{\left[\tau-{\mathrm{1}}\{\epsilon<\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\gamma$}}}(\tau)-\varphi_0({{\bm X}})\}\right]^2
\pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\right\}.
\end{aligned}$$ Furthermore, we have $\Sigma_{11}(\tau,\tau)\leq\left(\tau^2+|1-2\tau|\right){\mbox{E}}\left[\pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\right]$.
Comparing the asymptotic normality of $\hat{{{\mbox{\boldmath $\beta$}}}}(\tau)$ with $\hat{{{\mbox{\boldmath $\beta$}}}}^A_{LS}$ yields interesting insights. Assuming that ${\mbox{E}}(Y|{{\bm X}},A)=\varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\beta$}}}_0$ and $({{\mbox{\boldmath $\beta$}}}_0,{{\mbox{\boldmath $\gamma$}}}^*)={\mathrm{argmin}}_{({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})}{\mbox{E}}[Y-\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})-\{A-\pi({{\bm X}})\}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\beta$}}}]^2$, the asymptotic normality property of $\hat{{{\mbox{\boldmath $\beta$}}}}^{A}_{LS}$ can then be established, which is summarized in Theorem \[thm:lsA\]. Its proof has been omitted, and readers are referred to [@lu2011variable].
\[thm:lsA\] Under the regularity condition of A1-A4 of [@lu2011variable], $$\sqrt{n}(\hat{{{\mbox{\boldmath $\beta$}}}}^{A}_{LS}-{{\mbox{\boldmath $\beta$}}}_0)\indist N(0,U_{11}^{-1}\Omega_{11}U_{11}^{-1}),$$ where $U_{11}={\mbox{E}}\left[\pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\right]$ and $$\Omega_{11}={\mbox{E}}\left[\left\{\varphi_0({{\bm X}})-\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}}^*)+\epsilon\right\}^2
\pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\right]$$
**Remarks:**
1. When the family of functions $\{\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}}),{{\mbox{\boldmath $\gamma$}}}\in\Gamma\}$ cannot well approximate the unknown baseline function $\varphi_0({{\bm X}})$, the $\Omega_{11}$ term in the asymptotic variance of $\hat{{{\mbox{\boldmath $\beta$}}}}^{A}_{LS}$ may explode, which makes $\hat{{{\mbox{\boldmath $\beta$}}}}^{A}_{LS}$ less efficient than $\hat{{{\mbox{\boldmath $\beta$}}}}(\tau)$.
2. When $Y=\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\gamma_0+\{A-\pi({{\bm X}})\}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\beta$}}}_0+\epsilon$, $\epsilon\perp(A,{{\bm X}})$, $\pi({{\bm X}})\equiv0.5$ and $\epsilon\sim N(0,\sigma^2)$, the asymptotic variance of $\hat{{{\mbox{\boldmath $\beta$}}}}(\tau=0.5)$ is $2\pi\sigma^2{\mbox{E}}(\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}})^{-1}$, which is strictly larger than $4\sigma^2{\mbox{E}}(\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}})^{-1}$ (the asymptotic variance of $\hat{{{\mbox{\boldmath $\beta$}}}}^{A}_{LS}$).
Numerical Results: Simulation Studies {#section:simulation}
=====================================
To demonstrate finite sample performance of the proposed robust regression methods for optimal treatment rule estimation, we conduct two simulation studies: the errors independent with treatments, and the errors interactive with treatments, respectively.
Simulation Study I: error terms independent with treatment
----------------------------------------------------------
We consider the following two models with p=3,
- Model I: $$Y_i=1+(X_{i1}-X_{i2})(X_{i1}+X_{i3})+\{A_i-\pi({{\bm X}}_{i})\}{{\mbox{\boldmath $\beta$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i+\sigma({{\bm X}}_{i})\epsilon_i,$$ where ${{\bm X}}_{i}=(X_{i1},X_{i2},X_{i3}){^{\mbox{\tiny {\sf T}}}}$ are multivariate normal with mean 0, variance 1, and $\mathrm{Corr}(X_{ij},X_{ik})=0.5^{|j-k|}$, $\tilde{{{\bm X}}}_i=(1,{{\bm X}}_i{^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$ and ${{\mbox{\boldmath $\beta$}}}_0=(0,1,-1,1){^{\mbox{\tiny {\sf T}}}}$.
- Model II: $$Y_i={{\mbox{\boldmath $\gamma$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i+\{A_i-\pi({{\bm X}}_{i})\}{{\mbox{\boldmath $\beta$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i+\sigma({{\bm X}}_{i})\epsilon_i,$$ where ${{\mbox{\boldmath $\gamma$}}}_0{^{\mbox{\tiny {\sf T}}}}=(0.5,4,1,-3)$, and ${{\bm X}}_{i}$, $\tilde{{{\bm X}}}_{i}$ and ${{\mbox{\boldmath $\beta$}}}_0$ are the same as Model I.
We take linear forms for both the baseline and the contrast functions, where $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})={{\mbox{\boldmath $\gamma$}}}{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}$ and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})={{\mbox{\boldmath $\beta$}}}{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}$. We assume the propensity scores $\pi(\cdot)$ are known, and we study both the constant case $(\pi({{\bm X}}_i)=0.5)$ and the non-constant case $(\pi({{\bm X}}_i)=\mathrm{logit}({{\bm X}}_{i1}-{{\bm X}}_{i2}))$. In addition, We consider two different $\sigma({{\bm X}}_{i})$ functions, i.e., the homogeneous case with $\sigma({{\bm X}}_{i})=1$, and the heterogenous case with $\sigma({{\bm X}}_{i})=0.5+(X_{i1}-X_{i2})^2$. The simulation results under constant and non-constant propensity scores are similar. Thus, for brevity, we only report the constant case and allocate the result of non-constant case to the Appendix B. The results of Model I and II with constant propensity score are given in Table \[table:modelI\_constant\] and \[table:modelII\_constant\] respectively.
[@ll ccc ccc ccc]{}\
& & & &\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} n & method & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} 100 & LS & 1.32 (0.040) & 80.7 & 1.06 & 2.36 (0.081) & 75.7 & 1.57 & & 58.4 & 3.75\
& P(0.5) & 1.44 (0.042) & 80.1 & 1.13 & 1.73 (0.051) & 78.0 & 1.31 & 2.69 (0.077) & 75.2 & 1.63\
& P(0.25) & 1.90 (0.057) & 78.3 & 1.34 & 1.63 (0.051) & 79.0 & 1.29 & 5.29 (0.168) & 70.4 & 2.25\
& Huber & 1.15 (0.034) & 81.9 & 0.93 & 1.45 (0.044) & 79.9 & 1.13 & 2.61 (0.072) & 74.9 & 1.66\
200 & LS & 0.68 (0.021) & 85.6 & 0.59 & 1.10 (0.033) & 82.0 & 0.91 & & 58.7 & 3.70\
& P(0.5) & 0.73 (0.021) & 85.3 & 0.62 & 0.78 (0.021) & 84.1 & 0.70 & 1.23 (0.037) & 81.3 & 0.99\
& P(0.25) & 0.92 (0.028) & 84.0 & 0.75 & 0.70 (0.023) & 86.0 & 0.59 & 2.48 (0.079) & 75.7 & 1.64\
& Huber & 0.58 (0.017) & 86.8 & 0.50 & 0.66 (0.018) & 85.5 & 0.58 & 1.24 (0.035) & 80.8 & 1.03\
400 & LS & 0.33 (0.009) & 90.3 & 0.26 & 0.56 (0.016) & 87.1 & 0.46 & & 59.2 & 3.61\
& P(0.5) & 0.35 (0.010) & 90.0 & 0.29 & 0.37 (0.010) & 89.0 & 0.34 & 0.56 (0.016) & 87.1 & 0.48\
& P(0.25) & 0.43 (0.013) & 89.1 & 0.34 & 0.33 (0.010) & 90.7 & 0.25 & 1.16 (0.037) & 82.9 & 0.86\
& Huber & 0.28 (0.008) & 91.1 & 0.22 & 0.31 (0.009) & 90.2 & 0.27 & 0.58 (0.017) & 86.7 & 0.49\
800 & LS & 0.17 (0.005) & 93.2 & 0.13 & 0.26 (0.008) & 90.9 & 0.23 & & 59.4 & 3.59\
& P(0.5) & 0.17 (0.005) & 93.1 & 0.13 & 0.19 (0.005) & 92.1 & 0.17 & 0.29 (0.009) & 90.7 & 0.24\
& P(0.25) & 0.22 (0.007) & 92.4 & 0.16 & 0.18 (0.006) & 93.6 & 0.12 & 0.59 (0.019) & 87.3 & 0.48\
& Huber & 0.14 (0.004) & 93.8 & 0.11 & 0.16 (0.005) & 93.1 & 0.14 & 0.29 (0.008) & 90.5 & 0.25\
\
& & & &\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} n & method & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} 100 & LS & 3.24 (0.110) & 74.7 & 1.70 & 8.98 (0.561) & 68.6 & 2.44 & & 56.2 & 4.05\
& P(0.5) & 1.70 (0.060) & 80.5 & 1.08 & 1.80 (0.064) & 80.1 & 1.08 & 3.45 (0.124) & 75.1 & 1.69\
& P(0.25) & 2.50 (0.085) & 77.4 & 1.42 & 2.51 (0.079) & 76.8 & 1.46 & 9.13 (0.341) & 67.2 & 2.66\
& Huber & 1.70 (0.057) & 80.4 & 1.10 & 1.87 (0.063) & 79.2 & 1.16 & 4.27 (0.155) & 72.8 & 1.93\
200 & LS & 1.54 (0.050) & 80.6 & 1.06 & 4.71 (0.244) & 73.4 & 1.85 & & 55.2 & 4.17\
& P(0.5) & 0.78 (0.028) & 86.7 & 0.53 & 0.90 (0.032) & 85.3 & 0.63 & 1.49 (0.052) & 81.9 & 0.95\
& P(0.25) & 1.16 (0.039) & 83.5 & 0.81 & 1.23 (0.039) & 82.0 & 0.91 & 3.95 (0.150) & 73.2 & 1.90\
& Huber & 0.77 (0.025) & 86.4 & 0.55 & 0.94 (0.032) & 84.5 & 0.69 & 1.94 (0.071) & 79.3 & 1.19\
400 & LS & 0.80 (0.026) & 86.0 & 0.58 & 2.69 (0.136) & 77.8 & 1.34 & & 54.7 & 4.26\
& P(0.5) & 0.39 (0.013) & 90.5 & 0.27 & 0.44 (0.017) & 89.6 & 0.32 & 0.71 (0.024) & 86.9 & 0.50\
& P(0.25) & 0.56 (0.019) & 88.8 & 0.37 & 0.66 (0.020) & 86.9 & 0.50 & 1.70 (0.055) & 79.6 & 1.17\
& Huber & 0.38 (0.012) & 90.4 & 0.27 & 0.48 (0.017) & 88.8 & 0.36 & 0.91 (0.029) & 84.9 & 0.65\
800 & LS & 0.41 (0.013) & 89.9 & 0.29 & 1.35 (0.150) & 83.1 & 0.82 & & 56.5 & 4.00\
& P(0.5) & 0.18 (0.006) & 93.6 & 0.12 & 0.20 (0.007) & 92.6 & 0.16 & 0.36 (0.013) & 91.0 & 0.25\
& P(0.25) & 0.28 (0.009) & 92.2 & 0.18 & 0.31 (0.010) & 90.8 & 0.24 & 0.89 (0.031) & 85.8 & 0.60\
& Huber & 0.19 (0.006) & 93.3 & 0.13 & 0.22 (0.007) & 92.1 & 0.18 & 0.47 (0.017) & 89.2 & 0.34\
[@ll ccc ccc ccc]{}\
& & & &\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} n & method & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} 100 & LS & 0.24 (0.006) & 91.1 & 0.21 & 1.23 (0.061) & 82.4 & 0.87 & & 58.6 & 3.73\
& P(0.5) & 0.36 (0.010) & 89.0 & 0.32 & 0.39 (0.012) & 88.8 & 0.34 & 0.80 (0.024) & 84.2 & 0.69\
& P(0.25) & 0.45 (0.012) & 87.8 & 0.40 & 0.13 (0.004) & 93.4 & 0.12 & 2.37 (0.083) & 76.0 & 1.49\
& Huber & 0.25 (0.007) & 90.8 & 0.22 & 0.31 (0.010) & 90.3 & 0.26 & 0.99 (0.029) & 82.4 & 0.84\
200 & LS & 0.11 (0.003) & 93.7 & 0.10 & 0.52 (0.018) & 87.3 & 0.45 & & 58.7 & 3.69\
& P(0.5) & 0.17 (0.005) & 92.4 & 0.16 & 0.17 (0.005) & 92.4 & 0.15 & 0.32 (0.009) & 89.5 & 0.30\
& P(0.25) & 0.20 (0.005) & 91.8 & 0.18 & 0.06 (0.002) & 95.6 & 0.05 & 1.03 (0.033) & 82.1 & 0.88\
& Huber & 0.12 (0.003) & 93.6 & 0.11 & 0.13 (0.003) & 93.5 & 0.12 & 0.43 (0.013) & 87.9 & 0.40\
400 & LS & 0.05 (0.001) & 95.7 & 0.05 & 0.26 (0.008) & 90.7 & 0.23 & & 59.4 & 3.60\
& P(0.5) & 0.09 (0.002) & 94.5 & 0.08 & 0.09 (0.002) & 94.5 & 0.08 & 0.15 (0.004) & 92.8 & 0.14\
& P(0.25) & 0.10 (0.002) & 94.2 & 0.09 & 0.03 (0.001) & 96.9 & 0.02 & 0.44 (0.012) & 87.9 & 0.39\
& Huber & 0.06 (0.001) & 95.5 & 0.05 & 0.06 (0.002) & 95.4 & 0.06 & 0.21 (0.006) & 91.6 & 0.19\
800 & LS & 0.03 (0.001) & 96.9 & 0.03 & 0.13 (0.004) & 93.5 & 0.11 & & 59.4 & 3.58\
& P(0.5) & 0.04 (0.001) & 96.1 & 0.04 & 0.04 (0.001) & 96.2 & 0.04 & 0.07 (0.002) & 95.1 & 0.06\
& P(0.25) & 0.05 (0.001) & 95.8 & 0.05 & 0.01 (0.000) & 97.9 & 0.01 & 0.20 (0.005) & 91.5 & 0.19\
& Huber & 0.03 (0.001) & 96.8 & 0.03 & 0.03 (0.001) & 96.8 & 0.03 & 0.10 (0.002) & 94.2 & 0.09\
\
& & & &\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} n & method & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} 100 & LS & 1.97 (0.072) & 79.8 & 1.13 & 7.75 (0.514) & 70.4 & 2.22 & & 56.4 & 4.02\
& P(0.5) & 0.84 (0.029) & 86.1 & 0.55 & 1.21 (0.045) & 84.3 & 0.74 & 1.82 (0.071) & 80.5 & 1.07\
& P(0.25) & 1.37 (0.049) & 82.1 & 0.90 & 1.56 (0.051) & 80.5 & 1.04 & 6.20 (0.261) & 69.8 & 2.25\
& Huber & 0.84 (0.031) & 85.9 & 0.57 & 1.33 (0.046) & 82.8 & 0.85 & 2.69 (0.106) & 77.0 & 1.42\
200 & LS & 0.99 (0.035) & 84.7 & 0.66 & 4.16 (0.237) & 75.2 & 1.62 & & 55.1 & 4.19\
& P(0.5) & 0.41 (0.014) & 90.2 & 0.28 & 0.58 (0.024) & 89.4 & 0.37 & 0.79 (0.030) & 86.7 & 0.52\
& P(0.25) & 0.64 (0.021) & 87.4 & 0.45 & 0.74 (0.024) & 86.1 & 0.54 & 2.48 (0.096) & 76.9 & 1.40\
& Huber & 0.39 (0.013) & 90.3 & 0.27 & 0.69 (0.027) & 87.7 & 0.45 & 1.17 (0.044) & 83.4 & 0.78\
400 & LS & 0.51 (0.018) & 89.0 & 0.35 & 2.48 (0.133) & 79.3 & 1.20 & & 54.7 & 4.25\
& P(0.5) & 0.20 (0.007) & 93.2 & 0.14 & 0.29 (0.011) & 92.6 & 0.17 & 0.32 (0.011) & 91.2 & 0.22\
& P(0.25) & 0.30 (0.009) & 91.3 & 0.22 & 0.39 (0.012) & 89.9 & 0.28 & 0.99 (0.030) & 83.0 & 0.78\
& Huber & 0.20 (0.007) & 93.2 & 0.14 & 0.34 (0.012) & 91.4 & 0.22 & 0.53 (0.016) & 88.4 & 0.37\
800 & LS & 0.25 (0.008) & 92.2 & 0.17 & 1.25 (0.159) & 84.2 & 0.73 & & 56.4 & 4.00\
& P(0.5) & 0.10 (0.004) & 95.3 & 0.07 & 0.14 (0.006) & 94.7 & 0.09 & 0.16 (0.006) & 93.9 & 0.11\
& P(0.25) & 0.14 (0.005) & 94.0 & 0.10 & 0.18 (0.006) & 92.9 & 0.14 & 0.49 (0.015) & 88.0 & 0.39\
& Huber & 0.09 (0.004) & 95.3 & 0.06 & 0.17 (0.006) & 93.9 & 0.11 & 0.26 (0.009) & 91.8 & 0.19\
Comparison is made among four methods. They are: lsA-learning, robust regression with $\rho_{0.5}$ (RR($\rho_{0.5}$)), robust regression with $\rho_{0.25}$ (RR($\rho_{0.25}$)), and robust regression with Huber loss (RR(H)). The error terms $\epsilon_i$ are taken as standard i.i.d. normal, log-normal or Cauchy distribution, and independent with both $A$ and ${{\bm X}}$. It is easy to check that the conditional independence assumption $\epsilon\perp A|{{\bm X}}$ is satisfied, and $g^{\mathrm{opt}}_{\mu}=g^{\mathrm{opt}}_{\tau}={\mathrm{1}}\{{{\mbox{\boldmath $\beta$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i>0\}$. We consider four different sample sizes 100, 200, 400 and 800. To evaluate the performance of each method, we compare three groups of criteria: (1) the mean squared error $\|\hat{{{\mbox{\boldmath $\beta$}}}}-{{\mbox{\boldmath $\beta$}}}_{0}\|^2_2$ (mse), which measures the distance between estimated parameters and the true parameter ${{\mbox{\boldmath $\beta$}}}_0$; (2) the percentage of making correct decisions (PCD), which are calculated based on a validation set with 10000 observations. Specifically, we take the formula $100*\left(1-\sum_{i=1}^{N_T}|{\mathrm{1}}\{\hat{{{\mbox{\boldmath $\beta$}}}}{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i>0\}-{\mathrm{1}}\{{{\mbox{\boldmath $\beta$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i>0\}|/N_T\right)$ with $N_T=10000$; (3) the differences of $V_\mu(g)$ and $V_{0.5-q}(g)$ between the optimal ITR and the estimated ITR, where $\delta_{\mu}=V_{\mu}(g_{\mu}^{\mathrm{opt}})-V_{\mu}(\hat{g})$ and $\delta_{\tau}=V_{\tau-q}(g_{\mu}^{\mathrm{opt}})-V_{\tau-q}(\hat{g})$, $\forall\tau\in(0,1)$. $V_{\mu}(g)$ and $V_{\tau-q}(g)$ (defined in Section 2.1) are estimated from the validation set as well, and they evaluate the overall performance of an ITR $g$, where the former one focuses on the response’s mean and the latter one focuses on the response’s conditional $\tau$-th quantile. Under our setting, $\delta_{\mu}=\delta_{0.5}$ when they both exists. Thus, only $\delta_{0.5}$ is reported. For each scenario, we take 1000 replications. All numbers in the tables are based on the sample average of all replications. We further report the standard errors of mse to evaluate the variability of the corresponding statistics.
When the propensity score is constant, lsA-learning is equivalent to both Q- and A-learning under our setting. If we compare the performance of the methods under homogeneous and heterogeneous errors, the first thing we find is that lsA-learning works much worse under the heterogeneous errors, while all other methods are generally less affected by the heterogeneity of the errors. When the baseline function is misspecified as in Model I, under the homogeneous normal errors, RR(H) works slightly better than lsA-learning, while $\mathrm{RR}(\rho_{0.25})$ works the worst. However, the difference in general is small. For the homogeneous log-normal errors, again RR(H) works the best, while $\mathrm{RR}(\rho_{0.5})$ and $\mathrm{RR}(\rho_{0.25})$ have similar performance, and lsA-learning works the worst. Under the homogeneous Cauchy errors, $\mathrm{RR}(\rho_{0.5})$ works the best and RR(H) has a close performance. The lsA-learning is no longer consistent, and its mse explodes. The actual numbers are too large and thus leave as blank in Table \[table:modelI\_constant\] and \[table:modelII\_constant\]. Furthermore, with the Cauchy errors, the PCD of lsA-learning are less than 60% under all scenarios, while other methods’ PCD can be as high as 90%. When baseline function is correctly specified as in Model II, under homogeneous normal errors, lsA-learning performs the best. However, in this case RR(H) also has a very close performance, and thus makes no difference from a practical point of view to choose between these two methods. The results of Model II under other cases draw similar conclusion as Model I. To sum up, the overall conclusion is that, under the conditional independence assumption, the proposed robust regression method RR(M) is more efficient than Q-, A- and lsA-learning in the circumstances when observations have skewed, heterogeneous or heavy-tailed errors. On the other hand, when the error terms indeed follows i.i.d. normal distribution, the loss of efficiency of RR(M) is not significant. This is especially true when Huber loss is applied.
Simulation Study II: error terms interactive with treatment
-----------------------------------------------------------
We consider the following model with p=2, $$Y_i=1 + 0.5\sin[\pi(X_{i1}-X_{i2})]+
0.25(1+X_{i1}+2X_{i2})^2+(A_i-\pi({{\bm X}}_{i})){{\mbox{\boldmath $\theta$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i+\sigma({{\bm X}}_{i},A_i)\epsilon_i,$$ where ${{\bm X}}_{i}=(X_{i1},X_{i2}){^{\mbox{\tiny {\sf T}}}}$, $\tilde{{{\bm X}}}_i=(1,{{\bm X}}_i{^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$, $\sigma({{\bm X}}_{i},A_i)=1+A_i d_0 X_{i1}^2$, ${{\mbox{\boldmath $\theta$}}}_0{^{\mbox{\tiny {\sf T}}}}=(0.5,2,-1)$ and $X_{ik}$ are i.i.d. Uniform\[-1,1\].
Similar as Section 4.1, we take linear forms for both the baseline and the contrast functions, where $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})={{\mbox{\boldmath $\gamma$}}}{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}$, $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})={{\mbox{\boldmath $\beta$}}}{^{\mbox{\tiny {\sf T}}}}{{\bm W}}$ and ${{\bm W}}=(\tilde{{{\bm X}}},X_{1}^2,X_{2}^2,X_{1}X_{2})$. $d_0=5$, 10 or 15. The error terms $\epsilon_i$ follows i.i.d. N(0,1) or Gamma(1,1)-1 distribution. The propensity scores $\pi(\cdot)$ are known, and we consider both the constant case $\pi({{\bm X}}_i)=0.5$ and the non-constant case $\pi({{\bm X}}_i)=\mathrm{logit}({{\bm X}}_{i1}-{{\bm X}}_{i2})$. We report only the result of the constant case (Table \[table:interacted\_constant\_ps\]), and allocate the non-constant case to Appendix B.
---------------------------------------------------------- ------- ----- ----------------- ----------------- ------------------ ----------------- ----------------- ------------------ ----------------- ----------------- ------------------ ----------------- ----------------- ------------------
(r)[4-6]{} (lr)[7-9]{} (lr)[10-12]{} (lr)[13-15]{} Error $d_0$ n $ \delta_{\mu}$ $ \delta_{0.5}$ $ \delta_{0.25}$ $ \delta_{\mu}$ $ \delta_{0.5}$ $ \delta_{0.25}$ $ \delta_{\mu}$ $ \delta_{0.5}$ $ \delta_{0.25}$ $ \delta_{\mu}$ $ \delta_{0.5}$ $ \delta_{0.25}$
Normal 5 100 0.16 0.16 0.31 0.16 0.16 0.27 0.25 0.25 0.17 0.14 0.14 0.26
200 0.09 0.09 0.24 0.10 0.10 0.19 0.18 0.18 0.09 0.08 0.08 0.19
400 0.05 0.05 0.18 0.07 0.07 0.12 0.15 0.15 0.05 0.05 0.05 0.13
800 0.02 0.02 0.14 0.05 0.05 0.09 0.14 0.14 0.04 0.03 0.03 0.09
10 100 0.28 0.28 0.92 0.22 0.22 0.81 0.39 0.39 0.40 0.21 0.21 0.82
200 0.19 0.19 0.85 0.15 0.15 0.71 0.33 0.33 0.28 0.13 0.13 0.72
400 0.12 0.12 0.79 0.10 0.10 0.60 0.30 0.30 0.23 0.09 0.09 0.63
800 0.06 0.06 0.73 0.07 0.07 0.50 0.27 0.27 0.22 0.06 0.06 0.54
15 100 0.35 0.35 1.55 0.25 0.25 1.40 0.47 0.47 0.62 0.26 0.26 1.43
200 0.27 0.27 1.48 0.18 0.18 1.31 0.45 0.45 0.45 0.18 0.18 1.34
400 0.19 0.19 1.47 0.13 0.13 1.17 0.44 0.44 0.37 0.12 0.12 1.23
800 0.12 0.12 1.39 0.09 0.09 1.03 0.41 0.41 0.35 0.08 0.08 1.07
Gamma 5 100 0.15 0.18 0.31 0.15 0.11 0.16 0.22 0.12 0.09 0.12 0.09 0.15
200 0.09 0.12 0.26 0.10 0.06 0.10 0.18 0.08 0.05 0.08 0.05 0.09
400 0.05 0.07 0.21 0.08 0.03 0.07 0.16 0.06 0.04 0.06 0.02 0.07
800 0.02 0.04 0.17 0.07 0.03 0.06 0.15 0.06 0.03 0.05 0.02 0.07
10 100 0.26 0.33 0.90 0.22 0.16 0.54 0.39 0.13 0.27 0.22 0.14 0.50
200 0.19 0.29 0.88 0.17 0.08 0.44 0.37 0.10 0.22 0.17 0.07 0.41
400 0.12 0.24 0.87 0.13 0.04 0.39 0.35 0.08 0.19 0.14 0.03 0.36
800 0.06 0.17 0.78 0.12 0.03 0.37 0.33 0.07 0.19 0.13 0.02 0.35
15 100 0.36 0.57 1.52 0.30 0.31 0.98 0.53 0.19 0.40 0.32 0.28 0.89
200 0.28 0.53 1.51 0.22 0.19 0.81 0.55 0.16 0.29 0.24 0.16 0.71
400 0.19 0.47 1.50 0.17 0.13 0.73 0.57 0.15 0.26 0.21 0.11 0.63
800 0.11 0.43 1.50 0.15 0.11 0.71 0.58 0.15 0.24 0.18 0.09 0.62
---------------------------------------------------------- ------- ----- ----------------- ----------------- ------------------ ----------------- ----------------- ------------------ ----------------- ----------------- ------------------ ----------------- ----------------- ------------------
: Summary results with constant propensity scores when errors interacted with treatment. Least square stands for lsA-learning. Pinball(0.5) stands for robust regression with pinball loss and parameter $\tau=0.5$. Pinball(0.25) stands for robust regression with pinball loss and parameter $\tau=0.25$. Huber stands for robust regression with Huber loss, where parameter $\alpha$ is tuned automatically with R function rlm.[]{data-label="table:interacted_constant_ps"}
We compare the performance of four methods: lsA-learning, robust regression with $\rho_{0.5}$ ($\mathrm{RR}(\rho_{0.5})$), robust regression with $\rho_{0.25}$ ($\mathrm{RR}(\rho_{0.25})$) and robust regression with Huber loss ($\mathrm{RR}(H)$). We consider four different sample sizes 100, 200, 400 and 800. For each scenario, we again simulate 1000 replications. When error terms are interactive with treatment, the true ${{\mbox{\boldmath $\beta$}}}_0$ associated with $g_{\mu}^{\mathrm{opt}}$ and $g_{\tau}^{\mathrm{opt}}$ are different. Specifically, under our model, ${{\mbox{\boldmath $\beta$}}}_0=({{\mbox{\boldmath $\theta$}}}_0{^{\mbox{\tiny {\sf T}}}},0,0,0){^{\mbox{\tiny {\sf T}}}}$ for $g_{\mu}^{\mathrm{opt}}$, ${{\mbox{\boldmath $\beta$}}}_0=({{\mbox{\boldmath $\theta$}}}_0{^{\mbox{\tiny {\sf T}}}},d_0 F^{-1}_{\epsilon}(0.5),0,0){^{\mbox{\tiny {\sf T}}}}$ for $g_{0.5}^{\mathrm{opt}}$ and ${{\mbox{\boldmath $\beta$}}}_0=({{\mbox{\boldmath $\theta$}}}_0{^{\mbox{\tiny {\sf T}}}},d_0 F^{-1}_{\epsilon}(0.25),0,0){^{\mbox{\tiny {\sf T}}}}$ for $g_{0.25}^{\mathrm{opt}}$. Thus, the two criteria, mse and PCD used in simulation study I, are no longer meaningful. So we evaluate the performance of methods in this simulation study based on value differences $\delta_{\mu}$, $\delta_{0.5}$ and $\delta_{0.25}$.
Based on Theorem \[thm:lsA\], we can prove that $\hat{g}^{A}_{LS}({{\bm x}})$ is consistent which converges to $g^{\mathrm{opt}}_{\mu}$ as sample size goes to infinity. This is shown in Table \[table:interacted\_constant\_ps\] such that the $\delta_{\mu}$ column for the lsA-learning method converges to 0 as sample size increases. We also know under Normal error terms, $\delta_{0.5}=\delta_{\mu}$. Thus, the $\delta_{0.5}$ column for the lsA-learning method also converges to 0. However, all other columns in Table \[table:interacted\_constant\_ps\] converge to a positive constant instead of 0 as sample size goes to infinity.
Another observation we discover from Table \[table:interacted\_constant\_ps\] is $\mathrm{RR}(H)$ and $\mathrm{RR}(\rho_{0.5})$ perform similarly. One additional observation we have is even though lsA-learning outperform all other methods in $\delta_{\mu}$ when sample size is large. It may be worse than $\mathrm{RR}(\rho_{0.5})$ and $\mathrm{RR}(H)$ when sample size is small. This is due to the fact that lsA-learning is inefficient under the heteroscedastic or skewed errors. The last observation we have is overall lsA-learning, $\mathrm{RR}(\rho_{0.5})$ and $\mathrm{RR}(\rho_{0.25})$ perform best at the columns $\delta_{\mu}$, $\delta_{0.5}$ and $\delta_{0.25}$ accordingly. The reason is given in the Remark under Theorem \[thm:approximation\_quantile\], which shows that $\hat{g}^{R}_{\rho(\tau)}$ $(\triangleq{\mathrm{1}}\{C({{\bm x}};\hat{{{\mbox{\boldmath $\beta$}}}}^R_{\rho(\tau)})>0\})$ in general approximates the unknown optimal ITR $g^{\mathrm{opt}}_{\tau}$ even when the conditional independence assumption $\epsilon\perp A|{{\bm X}}$ does not hold.
Application to AIDS study {#section:aids}
=========================
We illustrate the proposed robust regression method to data from AIDS Clinical Trials Group Protocol 175 (ACTG175), which has been previously studied by various authors [@leon2003semiparametric; @tsiatis2008covariate; @zhang2008improving; @lu2011variable]. In the study, 2139 HIV-infected subjects were randomized to four different treatment groups in equal proportions, and the treatment groups are zidovudine (ZDV) monotherapy, ZDV + didanosine (ddI), ZDV + zalcitabine, and ddI monotherapy. Following [@lu2011variable], we choose CD4 count $(\mathrm{cells/mm}^3)$ at $20\pm5$ weeks post-baseline as the primary continuous outcome $Y$, and include five continuous covariates and seven binary covariates as our covariates. They are: 1. age (years), 2. weight (kg), 3. karnof=Karnofsky score (scale of 0-100), 4. cd40=CD4 count $(\mathrm{cells/mm}^3)$ at baseline, 5. cd80=CD8 count $(\mathrm{cells/mm}^3)$ at baseline, 6. hemophilia=hemophilia (0=no, 1=yes), 7. homosexuality=homosexual activity (0=no, 1=yes), 8. drugs=history of intravenous drug use (0=no, 1=yes), 9. race (0=white, 1=non-white), 10. gender (0=female, 1=male), 11. str2= antiretroviral history (0=naive, 1=experienced), and 12. sympton=symptomatic status (0=asymptomatic, 1=symptomatic). For brevity, we only compare the treatment ZDV + didanosine (ddI) $(A=1)$ and ZDV + zalcitabine $(A=0)$, and restrict our samples to subjects receiving these two treatments. Thus, the propensity scores $\pi({{\bm X}}_i)\equiv 0.5$ in our restricted samples as the patients are assigned into one of two treatments with equal probability.
In our analysis, we assume linear models for both the baseline and the contrast functions. For interpretability, we keep the response $Y$ (the CD4 count) at its original scale, which is also consistent with the way clinicians think about the outcome in practice [@tsiatis2008covariate]. We plot the scatter plot of response Y against age. It shows some skewness and heterogeneity. With some preliminary analysis (fitting full model with lsA-learning and RR(M)), we find that only covariates age, homosexuality and race may possibly interact with the treatment. So in our final model, only these three covariates are included in the contrast function, while at the same time we still keep all twelve covariates in the baseline function. The estimated coefficients associated with their corresponding standard errors and p-values are given in Table \[table:aids\], where standard errors are estimated with 1000 bootstrap samples (parametric bootstrap) and p-values are calculated with normal approximation. Only coefficients included in the contrast function are shown.
---------------------------------------------------------- -------- ------- ----------- -------- ------- ----------- -------- ------- ----------- -------- ------- -----------
(r)[2-4]{} (lr)[5-7]{} (l)[8-10]{} (l)[11-13]{} Variable Est. SE PV Est. SE PV Est. SE PV Est. SE PV
intercept -42.61 32.93 0.196 -33.45 37.32 0.370 -35.77 39.17 0.361 -42.76 31.40 0.173
age 3.13 0.85 **0.000** 2.62 0.97 **0.007** 2.46 1.06 **0.020** 2.80 0.79 **0.000**
homosexuality -40.66 16.73 **0.015** -33.18 17.68 0.061 -35.38 18.28 0.053 -27.33 15.19 0.072
race -25.70 17.69 0.146 -33.56 18.12 0.064 -34.21 18.32 0.062 -25.29 16.08 0.116
---------------------------------------------------------- -------- ------- ----------- -------- ------- ----------- -------- ------- ----------- -------- ------- -----------
: Analysis results for AIDS data. Est. stands for estimate; SE stands for standard error; PV stands for p-value. All p-values which are significant at level 0.1 are highlighted.[]{data-label="table:aids"}
From Tables \[table:aids\], we make the following observations. First, lsA-learning (equivalent to Q- and A-learning with this model setting) and robust regression with pinball loss and Huber loss all have estimates with the exact same signs. Second, the estimated coefficients are distinguishable across different methods. Third, the covairiate homosexuality is significant under lsA-learning, but it is not significant under robust regression with either pinball losses or Huber loss, when the significant level $\alpha$ is set to 0.05.
We could further estimate the values $(V_{\mu}(\hat{g}))$ associated with each method by either the inverse probability weighted estimator (IPWE) [@robin2000marginal] or the augmented inverse probability weighted estimator (AIPWE) [@robins1994estimation], where $$\begin{aligned}
\hat{V}^{\mathrm{IPWE}}_{\mu}(\hat{g})=&\frac{\sum_{i=1}^n{\mathrm{1}}{\{A_i=\hat{g}({{\bm X}}_i)\}}Y_i/p(A_i|{{\bm X}}_i)}
{\sum_{i=1}^n{\mathrm{1}}{\{A_i=\hat{g}({{\bm X}}_i)\}}/p(A_i|{{\bm X}}_i)},\\
\hat{V}^{\mathrm{AIPWE}}_{\mu}(\hat{g})=&\frac{1}{n}\sum_{i=1}^{n}\hat{{\mbox{E}}}(Y_i|{{\bm X}}_i,\hat{g}({{\bm X}}_i))
+\frac{1}{n}\sum_{i=1}^{n}\frac{{\mathrm{1}}{\{A_i=\hat{g}({{\bm X}}_i)\}}}{p(A_i|{{\bm X}}_i)}\left[Y_i-\hat{{\mbox{E}}}(Y_i|{{\bm X}}_i,A_i)\right],\end{aligned}$$ $\hat{{\mbox{E}}}(Y_i|{{\bm X}}_i,A_i))=\varphi({{\bm X}}_i;\hat{{{\mbox{\boldmath $\gamma$}}}})+\left\{A_i-p(A_i|{{\bm X}}_i)\right\}C({{\bm X}}_i;\hat{{{\mbox{\boldmath $\beta$}}}})$, and $p(A_i|{{\bm X}}_i)\equiv0.5$. Both $\hat{V}^{\mathrm{IPWE}}_{\mu}(\hat{g})$ and $\hat{V}^{\mathrm{AIPWE}}_{\mu}(\hat{g})$ are consistent estimator of value $V_{\mu}(\hat{g})$, and their asymptotic covariance matrix can also be consistently estimated from the data [@zhang2012robust; @mckeague2014estimation]. The estimates of $(V_{\mu}(\hat{g}))$ and their corresponding 95% confidence interval of four methods based on both IPWE and AIPWE are given in Table \[table:aids\_value\].
Estimator method Value SE CI
----------- --------------- -------- ------ ------------------
IPWE Least Square 405.05 6.72 (391.88, 418.22)
Pinball(0.5) 406.77 6.71 (393.63, 419.92)
Pinball(0.25) 406.07 6.73 (392.87, 419.26)
Huber 407.03 6.71 (393.87, 420.18)
AIPWE Least Square 404.39 6.12 (392.40, 416.38)
Pinball(0.5) 405.93 6.13 (393.92, 417.94)
Pinball(0.25) 403.60 6.62 (390.62, 416.58)
Huber 406.00 6.15 (393.95, 418.04)
: Result of estimated values and their corresponding 95% confidence interval for four methods based on IPWE and AIPWE.SE stands for standard error. CI stands for 95% confidence interval.[]{data-label="table:aids_value"}
From Table \[table:aids\_value\], robust regression with $\rho_{0.5}$ and Huber loss perform slightly better than lsA-learning, while robust regression with $\rho_{0.25}$ performs worse than lsA-learning when the values $(V_{\mu}(\hat{g}))$ is estimated based on AIPWE. We conduct KCI-test to check the conditional independence assumption $\epsilon\perp A|{{\bm X}}$. For $\mathrm{RR}(\rho(0.5))$, $\mathrm{RR}(\rho(0.25))$ and RR(H), their p-values associated with KCI-test are 0.060, 0.002 and 0.083 respectively. The conditional independence assumption holds at the significance level of 0.05 for $\mathrm{RR}(\rho(0.5))$ and RR(H), so the estimated ITR can be thought to maximize $V_{\mu}(g)$. On the other hand, this assumption doesn’t hold for $\mathrm{RR}(\rho(0.25))$, and its estimated ITR doesn’t maximize $V_{\mu}(g)$, instead it approximately maximizes $V_{0.25-q}(g)$. This partly explains the relatively bad performance of RR($\rho_{0.25}$) in Table \[table:aids\_value\]. Again, as $\mathrm{RR}(\rho(0.5))$ and RR(H) are more robust against heterogeneous, right skewed errors comparing with the least square method, they slightly outperform lsA-learning in term of $V_{\mu}(g)$.
Discussion
==========
In this article, we propose a new general loss based robust regression framework for estimating the optimal individualized treatment rules. This new method has the desired property to be robust against skewed, heterogeneous, heavy-tailed errors and outliers. And similar as A-learning, it produces consistent estimates of the optimal ITR even when the baseline function is misspecified. However, the consistency of the proposed method does require the key conditional independence assumption $\epsilon\perp A|{{\bm X}}$, which is somewhat stronger than the condition needed for the consistency of Q- and A-learning $({\mbox{E}}(\epsilon|{{\bm X}},A)=0)$. So there are situations when the classical Q- and A-learning are more appropriate to apply. Furthermore, we also point out in the article that when pinball loss $\rho_{\tau}$ is chosen and the assumption $\epsilon\perp A|{{\bm X}}$ doesn’t hold, the estimated ITR approximately maximize the conditional $\tau$-th quantile and thus maximize $V_{\tau-q}(g)$. From a practice point of view, there are situations when maximizing $V_{\tau-q}(g)$ is a much more reasonable approach comparing with maximizing $V_{\mu}(g)$, especially when the conditional distribution of response $Y$ is highly skewed to one side.
In practice, there are cases when multiple treatment groups need to be compared simultaneously. For brevity, we have limited our discussion to two treatment groups. However, the proposed method can be readily extended to multiple cases by just replacing equation with the following more complex form, $$L_{3n}({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}})=\frac{1}{n}\sum_{i=1}^{n}M\left[Y_i-\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})-\sum_{k=1}^{K-1}(I(A_i=k)-\pi_k({{\bm X}}_i))C_k({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}}_k)\right],
\label{eq:A-general-loss2}$$ where $\mathcal{A}=\{1,\ldots,K\}$, $K$-th treatment is the baseline treatment, $\pi_k({{\bm X}}_i)=\Pr(A_i=k|{{\bm X}}_i)$ and $C_k({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}}_k)$ denotes the contrast function comparing $k$-th treatment and the baseline treatment. All Theorems can be easily extended to this multiple treatments setting as well.
When the dimension of prognostic variables is high, regularized regression is needed in order to produce parsimonious yet interpretable individualized treatment rules. Essentially this is a variable selection problem in the context of M-estimator, which has been previously studied in [-@wu2009variable; -@li2011nonconcave], etc. This is an interesting topic that needs further investigation. Another interesting direction is to extend the current method to the multi-stage setting, where sequential decisions are made along the time line.
Appendix A: Proof of Asymptotic Properties {#appendix-a-proof-of-asymptotic-properties .unnumbered}
==========================================
We consider the following additive model, $$Y_i=\varphi_0({{\bm X}}_i)+\{A_i-\pi({{\bm X}}_i)\}C({{\bm X}}_i;\beta_0)+\epsilon_i,\; i=1,\ldots,n,$$ where $\varphi_0({{\bm X}})$ is the baseline function, $C({{\bm X}};\beta_0)$ is the contrast function, $\pi({{\bm X}})$ is the propensity score, and $\epsilon$ is the error term. We estimate $({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}})$ by minimizing $$L_{3n}({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}})=\frac{1}{n}\sum_{i=1}^{n}M\left[Y_i-\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})-\{A_i-\pi({{\bm X}}_i)\}C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})\right],
\label{eq:A-general-loss}$$ where ${{\mbox{\boldmath $\gamma$}}}\in\Gamma$, ${{\mbox{\boldmath $\beta$}}}\in\mathcal{B}$ and $M:{\mathrm{I \! R} \mathit{^{\rightarrow}}} [0,\infty)$ is a convex function with minimum achieved at 0. We consider the following three types of loss functions, i.e., the pinball loss $$M(x)=\rho_\tau(x)\triangleq
\begin{cases}
(\tau-1)x, &\text{if } x<0\\
\tau x, &\text{if } x\geq0
\end{cases}$$ where $0<\tau<1$, the Huber loss $$M(x)=H_\alpha(x)\triangleq
\begin{cases}
0.5x^2, &\text{if } |x|<\alpha\\
\alpha|x|-0.5\alpha^2, &\text{if } |x|\geq\alpha
\end{cases}$$ for some $\alpha>0$, and the $\epsilon$-insensitive loss $$M(x)=J_\epsilon(x)\triangleq\max(0, |x|-\epsilon)$$ for some $\epsilon>0$. Define $\Delta C({{\bm x}};{{\mbox{\boldmath $\beta$}}})=C({{\bm x}};{{\mbox{\boldmath $\beta$}}})-C({{\bm x}};{{\mbox{\boldmath $\beta$}}}_0)$. Assume ${{\mbox{\boldmath $\gamma$}}}\in\Gamma$, ${{\mbox{\boldmath $\beta$}}}\in\mathcal{B}$ and ${{\mbox{\boldmath $\gamma$}}}'$ is any arbitrary fix point in $\Gamma$.
**Regularity conditions A:**
- $\{(Y_i,{{\bm X}}_i,A_i,\epsilon_i),i=1,\ldots,n\}$ are i.i.d random variables.
- $\epsilon_i\perp A_i|{{\bm X}}_i$ $\forall i=1,\ldots,n$.
- ${\mbox{E}}|\Delta C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})|<\infty$ $\forall{{\mbox{\boldmath $\beta$}}}\in\mathcal{B}$.
- $\Pr\{{{\bm x}}\in\mathcal{X}:\;\Delta C({{\bm x}};{{\mbox{\boldmath $\beta$}}})\neq 0\}>0$ for all ${{\mbox{\boldmath $\beta$}}}\neq{{\mbox{\boldmath $\beta$}}}_0$.
- ${\mbox{E}}|\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})|<\infty$ $\forall{{\mbox{\boldmath $\gamma$}}}\in\Gamma$.
- $G_2({{\mbox{\boldmath $\gamma$}}})$ has unique minimizer ${{\mbox{\boldmath $\gamma$}}}^*$, where $G_2({{\mbox{\boldmath $\gamma$}}})$ is the pointwise limit of $L_{3n}({{\mbox{\boldmath $\beta$}}}_0,{{\mbox{\boldmath $\gamma$}}})-L_{3n}({{\mbox{\boldmath $\beta$}}}_0,{{\mbox{\boldmath $\gamma$}}}')$ in probability.
- $L_{3n}({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})$ is strictly convex with respect to $({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})$.
- $\epsilon|{{\bm X}}={{\bm x}}$ has nonzero density on $\mathbb{R}$ for almost all ${{\bm x}}\in\mathcal{X}$.
$\left|\rho_\tau(x-y)-\rho_\tau(x)\right|\leq |y|$, for all $\tau\in(0,1)$.
$$\begin{aligned}
\left|\rho_\tau(x-y)-\rho_\tau(x)\right| &= \left|\tau\left\{(x-y)_{+}-x_{+}\right\}+(1-\tau)\left\{(x-y)_{-}-x_{-}\right\}\right|\\
&\leq|(x-y)_{+}-x_{+}|+|(x-y)_{-}-x_{-}|=|y|
\end{aligned}$$
$$\begin{aligned}
\rho_\tau(x-y)-\rho_\tau(x)=&-\tau y{\mathrm{1}}\{x\geq 0\}+(1-\tau)y{\mathrm{1}}\{x< 0\}+(y-x){\mathrm{1}}\{x\geq 0\}{\mathrm{1}}\{y>x\}\\
&+(x-y){\mathrm{1}}\{x< 0\}{\mathrm{1}}\{y< x\},
\end{aligned}$$
for all $\tau\in(0,1)$.
Denote $D=\rho_\tau(x-y)-\rho_\tau(x)$.
1. If $x\geq0$, $y\leq0$ $\Rightarrow$ $D=-\tau y$;
2. If $x\geq0$, $y>0$, $|x|\geq|y|$ $\Rightarrow$ $D=-\tau y$;
3. If $x\geq0$, $y>0$, $|x|<|y|$ $\Rightarrow$ $D=-\tau y+(y-x)$;
4. If $x<0$, $y\geq0$ $\Rightarrow$ $D=(1-\tau)y$;
5. If $x<0$, $y<0$, $|x|\geq|y|$ $\Rightarrow$ $D=(1-\tau)y$;
6. If $x<0$, $y<0$, $|x|<|y|$ $\Rightarrow$ $D=(1-\tau)y+(x-y)$;
Combining the above 6 cases, Lemma 2 is proved.
**Proof of Theorem 1.**
Recall that the loss function defined in takes the form $$L_{3n}({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}})=\frac{1}{n}\sum_{i=1}^{n}\rho_\tau\left[\varphi_0({{\bm X}}_i)-\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})+\epsilon_i
-(A_i-\pi({{\bm X}}_i))\Delta C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})\right].$$ By definition, $$\begin{aligned}
(\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)},\hat{{{\mbox{\boldmath $\gamma$}}}}^{R}_{\rho(\tau)}) =& {\mathrm{argmin}}_{({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})}L_{3n}({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})-L_{3n}({{\mbox{\boldmath $\beta$}}}_0,{{\mbox{\boldmath $\gamma$}}}')\\
=& {\mathrm{argmin}}_{({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})}\left[L_{3n}({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})-L_{3n}({{\mbox{\boldmath $\beta$}}}_0,{{\mbox{\boldmath $\gamma$}}})\right]+
\left[L_{3n}({{\mbox{\boldmath $\beta$}}}_0,{{\mbox{\boldmath $\gamma$}}})-L_{3n}({{\mbox{\boldmath $\beta$}}}_0,{{\mbox{\boldmath $\gamma$}}}')\right],
\end{aligned}$$ Define $$\begin{aligned}
S_{1n}({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}}) =& L_{3n}({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}})-L_{3n}({{\mbox{\boldmath $\beta$}}}_0, {{\mbox{\boldmath $\gamma$}}})=1/n\sum_{i=1}^n d_{1i};\\
S_{2n}({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}}) =& L_{3n}({{\mbox{\boldmath $\beta$}}}_0, {{\mbox{\boldmath $\gamma$}}})-L_{3n}({{\mbox{\boldmath $\beta$}}}_0, {{\mbox{\boldmath $\gamma$}}}')=1/n\sum_{i=1}^n d_{2i}
\end{aligned}$$ where $$\begin{aligned}
d_{1i} =& \rho_\tau\left[\varphi_0({{\bm X}}_i)-\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})+\epsilon_i
-(A_i-\pi({{\bm X}}_i))\Delta C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})\right]-\rho_\tau\left[\varphi_0({{\bm X}}_i)-\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})+\epsilon_i\right],\\
d_{2i} =& \rho_\tau\left[\varphi_0({{\bm X}}_i)-\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})+\epsilon_i\right]-\rho_\tau\left[\varphi_0({{\bm X}}_i)-\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}}')+\epsilon_i\right].
\end{aligned}$$ By Lemma 1, A3 and A5, ${\mbox{E}}|d_{1i}|\leq{\mbox{E}}|(A_i-\pi({{\bm X}}_i))\Delta C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})|\leq{\mbox{E}}|\Delta C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})|<\infty$ and ${\mbox{E}}|d_{2i}|\leq{\mbox{E}}|\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})-\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}}')|\leq{\mbox{E}}|\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})|+{\mbox{E}}|\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}}')|
<\infty$. Then, by Law of Large Number, $\forall\;{{\mbox{\boldmath $\beta$}}}\in\mathcal{B}$, ${{\mbox{\boldmath $\gamma$}}}\in\Gamma$, we have $S_{1n}({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}})\inprob G_1({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})\triangleq{\mbox{E}}(D)$, and $S_{2n}({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}})\inprob G_2({{\mbox{\boldmath $\gamma$}}})$, where $$\begin{aligned}
D=&\rho_\tau\left[\varphi_0({{\bm X}})-\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})+\epsilon
-\{A-\pi({{\bm X}})\}\Delta C({{\bm X}};{{\mbox{\boldmath $\beta$}}})\right]-\rho_\tau\left[\varphi_0({{\bm X}})-\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})+\epsilon\right].
\end{aligned}$$ Below we show that a) $({{\mbox{\boldmath $\beta$}}}_0,{{\mbox{\boldmath $\gamma$}}}^*)$ is the minimizer of $G_1({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})+G_2({{\mbox{\boldmath $\gamma$}}})$, b) $({{\mbox{\boldmath $\beta$}}}_0,{{\mbox{\boldmath $\gamma$}}}^*)$ is the unique minimizer. The consistency then follows from the argmax continuous mapping theorem under Assumption (A7).
Denote $K_1=\varphi_0({{\bm X}})-\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})+\epsilon$, $K_2=\{A-\pi({{\bm X}})\}\Delta C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$. By Lemma 2, $$\begin{aligned}
D =& -\tau K_2{\mathrm{1}}\{K_1\geq0\}+(1-\tau) K_2{\mathrm{1}}\{K_1<0\}+(K_2-K_1){\mathrm{1}}\{K_1\geq0\}{\mathrm{1}}\{K_2>K_1\}\\
&+(K_1-K_2){\mathrm{1}}\{K_1<0\}{\mathrm{1}}\{K_2<K_1\}.
\end{aligned}$$ Since $\epsilon\perp A|{{\bm X}}$ and $\Pr(A|{{\bm X}})=\pi({{\bm X}})$, applying double expectation rule with ${{\bm X}}$, we have ${\mbox{E}}[-\tau K_2{\mathrm{1}}\{K_1\geq0\}]={\mbox{E}}[(1-\tau) K_2{\mathrm{1}}\{K_1<0\}]=0$. Thus, $$G_1({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})={\mbox{E}}[(K_2-K_1){\mathrm{1}}\{K_1\geq0\}{\mathrm{1}}\{K_2>K_1\}]+{\mbox{E}}[(K_1-K_2){\mathrm{1}}\{K_1<0\}{\mathrm{1}}\{K_2<K_1\}].
\label{eq:Gfunction}$$ It is easy to check $G_1({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})\geq0$ and achieves minimal value 0 at point $({{\mbox{\boldmath $\beta$}}}_0,{{\mbox{\boldmath $\gamma$}}})$ for all ${{\mbox{\boldmath $\gamma$}}}\in\Gamma$. In addition, by A6, we know $G_2({{\mbox{\boldmath $\gamma$}}})$ has unique minimizer ${{\mbox{\boldmath $\gamma$}}}^*$. Combining the above two facts, a) is proved.
Combining A4, A8 and , we could prove $G_1({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})>0$ for all ${{\mbox{\boldmath $\beta$}}}\neq{{\mbox{\boldmath $\beta$}}}_0$ and ${{\mbox{\boldmath $\gamma$}}}\in\Gamma$. So b) holds.
**Proof of Theorem 3.**
\(a) When $M(x)=H_{\alpha}(x)$, the proof follows similar steps as Theorem 1. The only difference is that $G_1({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})$ takes a different expression now and we need to redo the proof of 1) $G_1({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})>0$ $\forall{{\mbox{\boldmath $\beta$}}}\neq{{\mbox{\boldmath $\beta$}}}_0$, ${{\mbox{\boldmath $\gamma$}}}\in\Gamma$, and 2) $G_1({{\mbox{\boldmath $\beta$}}}_0,{{\mbox{\boldmath $\gamma$}}})=0$ $\forall{{\mbox{\boldmath $\gamma$}}}\in\Gamma$. By definition, $G_1({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})\triangleq{\mbox{E}}(D)$, where $$\begin{aligned}
D=H_\alpha\left[\varphi_0({{\bm X}})-\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})+\epsilon
-\{A-\pi({{\bm X}})\}\Delta C({{\bm X}};{{\mbox{\boldmath $\beta$}}})\right]-H_\alpha\left[\varphi_0({{\bm X}})-\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})+\epsilon\right].
\end{aligned}$$ Then, 2) holds immediately. Denote $K_1=\varphi_0({{\bm X}})-\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})+\epsilon$, $K_2=\{A-\pi({{\bm X}})\}\Delta C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$. We have the following four cases:
1. If $K_1>\alpha$ then $H_\alpha(K_1-K_2)\geq\alpha(K_1-K_2)-0.5\alpha^2$. Thus, $D\geq-\alpha K_2$;
2. If $K_1<-\alpha$ then $H_\alpha(K_1-K_2)\geq\alpha(K_2-K_1)-0.5\alpha^2$. Thus, $D\geq\alpha K_2$;
3. If $K_1\in[-\alpha,\alpha]$ and $K_1-K_2\in[-\alpha,\alpha]$ then $D=1/2(K_1-K_2)^2-1/2K_1^2=-K_1K_2+1/2K_2^2$;
4. If $K_1\in[-\alpha,\alpha]$ and $K_1-K_2\not\in[-\alpha,\alpha]$ then $H_\alpha(K_1-K_2)\geq1/2(K_1-K_2)^2-\left\{1/2(\alpha+|K_2|)^2-\left[\alpha(\alpha+|K_2|)-1/2\alpha^2\right]\right\}=1/2(K_1-K_2)^2-1/2K_1^2$. Thus, $D\geq1/2(K_1-K_2)^2-1/2K_1^2-1/2K_2^2=-K_1K_2$.
Combining the above four equalities and inequalities, $$\begin{aligned}
G_1({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})\geq& {\mbox{E}}[-\alpha K_2{\mathrm{1}}\{K_1>\alpha\}] + {\mbox{E}}[\alpha K_2{\mathrm{1}}\{K_1<-\alpha\}] +
{\mbox{E}}[-K_1K_2{\mathrm{1}}\{K_1\in[-\alpha,\alpha]\}]\\
&+ {\mbox{E}}\left[1/2K_2^2{\mathrm{1}}\left(\{K_1\in[-\alpha,\alpha]\}\cup\{K_1-K_2\in[-\alpha,\alpha]\}\right)\right]
\end{aligned}$$ Since $\epsilon\perp A|{{\bm X}}$ and $\Pr(A|{{\bm X}})=\pi({{\bm X}})$, applying double expectation rule with ${{\bm X}}$, we have ${\mbox{E}}[-\alpha K_2{\mathrm{1}}\{K_1>\alpha\}]={\mbox{E}}[\alpha K_2{\mathrm{1}}\{K_1<-\alpha\}]={\mbox{E}}[-K_1K_2{\mathrm{1}}\{K_1\in[-\alpha,\alpha]\}]=0$. Thus, $$G_1({{\mbox{\boldmath $\beta$}}};{{\mbox{\boldmath $\gamma$}}})\geq{\mbox{E}}\left[1/2K_2^2{\mathrm{1}}\left(\{K_1\in[-\alpha,\alpha]\}\cup\{K_1-K_2\in[-\alpha,\alpha]\}\right)\right].
\label{eq:Gfunction_thm2}$$ Combining , A4 and A8, we can check that 1) holds. Thus, part (a) is proved.
\(b) When $M(x)=J_{\epsilon}(x)$, similarly $D=J_\epsilon\left(K_1-K_2\right)-J_\epsilon\left(K_1\right)$. Notice that we have the following three cases:
1. If $K_1>\epsilon$ then $D\geq -K_2$;
2. If $K_1<-\epsilon$ then $D\geq K_2$;
3. If $K_1\in[-\epsilon,\epsilon]$ then $D\geq 0$;
The rest of the proof follows similar steps as part (a).
**Proof of Theorem 5.**
From Theorem 1, ${{\mbox{\boldmath $\beta$}}}_{\tau}={{\mbox{\boldmath $\beta$}}}_0$. Plugging this into Theorem 4 and applying double expectation rules, we have $$J(\tau)={\mbox{E}}\left[f_{\epsilon}\left(\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\gamma$}}}(\tau)-\varphi_0({{\bm X}})|{{\bm X}}\right)
\left(\begin{array}{cc}
\pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}& {{\mathbf 0}}\\
{{\mathbf 0}}& \tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\end{array}
\right)\right]$$ and $$\Sigma(\tau,\tau)={\mbox{E}}\left\{\left[\tau-{\mathrm{1}}\left\{\epsilon<\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\gamma$}}}(\tau)-\varphi_0({{\bm X}})\right\}\right]^2
\left(\begin{array}{cc}
\pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}& {{\mathbf 0}}\\
{{\mathbf 0}}& \tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\end{array}
\right)\right\}.$$ Thus, $\sqrt{n}(\hat{{{\mbox{\boldmath $\beta$}}}}(\tau)-{{\mbox{\boldmath $\beta$}}}_0)\indist N({{\mathbf 0}}, J_{11}^{-1}(\tau)\Sigma_{11}(\tau,\tau)J_{11}^{-1}(\tau))$, where $J_{11}^{-1}(\tau)$ and $\Sigma_{11}(\tau,\tau)$ are defined as in Theorem 5. Conditional on ${{\bm X}}$, ${\mathrm{1}}\left\{\epsilon<\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\gamma$}}}(\tau)-\varphi_0({{\bm X}})\right\}$ is a binomial random variable with $p=\Pr\left(\epsilon<\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\gamma$}}}(\tau)-\varphi_0({{\bm X}})\right)$. Then, ${\mbox{E}}\left\{\left[\tau-{\mathrm{1}}\{\epsilon<\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\gamma$}}}(\tau)-\varphi_0({{\bm X}})\}\right]^2|{{\bm X}}\right\}=(p-\tau)^2+p(1-p)\leq \tau^2+|1-2\tau|$. Thus, $\Sigma_{11}(\tau,\tau)\leq\left(\tau^2+|1-2\tau|\right){\mbox{E}}\left[\pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\right]$.
Appendix B: Additional Simulation Results {#appendix-b-additional-simulation-results .unnumbered}
=========================================
We conducted additional simulations with non-constant propensity scores. Specifically, we considered the following examples.
Examples with error terms independent with treatment {#examples-with-error-terms-independent-with-treatment .unnumbered}
----------------------------------------------------
We consider the following two models with p=3,
- Model I: $$Y_i=1+(X_{i1}-X_{i2})(X_{i1}+X_{i3})+\{A_i-\pi({{\bm X}}_{i})\}{{\mbox{\boldmath $\beta$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i+\sigma({{\bm X}}_{i})\epsilon_i,$$ where ${{\bm X}}_{i}=(X_{i1},X_{i2},X_{i3}){^{\mbox{\tiny {\sf T}}}}$ are multivariate normal with mean 0, variance 1, and $\mathrm{Corr}(X_{ij},X_{ik})=0.5^{|j-k|}$, $\tilde{{{\bm X}}}_i=(1,{{\bm X}}_i{^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$ and ${{\mbox{\boldmath $\beta$}}}_0=(0,1,-1,1){^{\mbox{\tiny {\sf T}}}}$.
- Model II: $$Y_i={{\mbox{\boldmath $\gamma$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i+\{A_i-\pi({{\bm X}}_{i})\}{{\mbox{\boldmath $\beta$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i+\sigma({{\bm X}}_{i})\epsilon_i,$$ where ${{\mbox{\boldmath $\gamma$}}}_0{^{\mbox{\tiny {\sf T}}}}=(0.5,4,1,-3)$, and ${{\bm X}}_{i}$, $\tilde{{{\bm X}}}_{i}$ and ${{\mbox{\boldmath $\beta$}}}_0$ are the same as Model I.
We take linear forms for both the baseline and the contrast functions, where $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})={{\mbox{\boldmath $\gamma$}}}{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}$ and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})={{\mbox{\boldmath $\beta$}}}{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}$. We assume the propensity scores $\pi(\cdot)$ are known, and we study the non-constant case $(\pi({{\bm X}}_i)=\mathrm{logit}({{\bm X}}_{i1}-{{\bm X}}_{i2}))$ here. In addition, We consider two different $\sigma({{\bm X}}_{i})$ functions, i.e., the homogeneous case with $\sigma({{\bm X}}_{i})=1$, and the heterogenous case with $\sigma({{\bm X}}_{i})=0.5+(X_{i1}-X_{i2})^2$. The simulation results are given in Table \[table:modelI\_nonconstant\] and Table \[table:modelII\_nonconstant\].
[@ll ccc ccc ccc]{}\
& & & &\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} n & method & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} 100 & LS & 1.70 (0.061) & 81.9 & 0.91 & 2.90 (0.114) & 77.6 & 1.34 & & 59.3 & 3.61\
& P(0.5) & 1.90 (0.069) & 80.1 & 1.09 & 2.13 (0.073) & 78.3 & 1.25 & 3.54 (0.128) & 75.7 & 1.57\
& P(0.25) & 2.35 (0.080) & 78.2 & 1.33 & 1.95 (0.076) & 80.4 & 1.08 & 8.45 (0.431) & 69.8 & 2.28\
& Huber & 1.51 (0.053) & 82.1 & 0.89 & 1.77 (0.065) & 80.6 & 1.02 & 3.67 (0.127) & 75.4 & 1.60\
200 & LS & 0.77 (0.026) & 86.8 & 0.50 & 1.35 (0.045) & 82.2 & 0.91 & & 59.2 & 3.63\
& P(0.5) & 0.88 (0.028) & 85.5 & 0.60 & 1.00 (0.029) & 83.0 & 0.79 & 1.54 (0.050) & 81.1 & 1.00\
& P(0.25) & 1.06 (0.035) & 84.5 & 0.68 & 0.83 (0.027) & 85.9 & 0.59 & 3.61 (0.143) & 74.7 & 1.70\
& Huber & 0.68 (0.022) & 87.3 & 0.46 & 0.81 (0.025) & 85.2 & 0.62 & 1.58 (0.052) & 80.7 & 1.03\
400 & LS & 0.39 (0.012) & 90.2 & 0.28 & 0.65 (0.020) & 86.9 & 0.48 & & 58.0 & 3.79\
& P(0.5) & 0.43 (0.013) & 89.3 & 0.32 & 0.47 (0.014) & 88.4 & 0.38 & 0.73 (0.022) & 86.5 & 0.51\
& P(0.25) & 0.53 (0.016) & 88.5 & 0.38 & 0.41 (0.013) & 90.5 & 0.27 & 1.50 (0.049) & 81.7 & 0.96\
& Huber & 0.34 (0.010) & 90.6 & 0.25 & 0.39 (0.012) & 89.6 & 0.30 & 0.72 (0.022) & 86.3 & 0.53\
800 & LS & 0.18 (0.006) & 93.3 & 0.13 & 0.32 (0.010) & 90.2 & 0.27 & & 58.3 & 3.75\
& P(0.5) & 0.21 (0.007) & 92.7 & 0.15 & 0.24 (0.007) & 91.5 & 0.20 & 0.36 (0.011) & 90.3 & 0.27\
& P(0.25) & 0.28 (0.009) & 92.4 & 0.17 & 0.21 (0.007) & 93.4 & 0.13 & 0.78 (0.026) & 86.9 & 0.50\
& Huber & 0.16 (0.005) & 93.7 & 0.11 & 0.19 (0.006) & 92.6 & 0.15 & 0.37 (0.010) & 89.9 & 0.28\
\
& & & &\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} n & method & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} 100 & LS & 2.84 (0.111) & 78.2 & 1.33 & 9.96 (0.773) & 72.0 & 2.06 & & 55.2 & 4.18\
& P(0.5) & 2.01 (0.082) & 80.6 & 1.09 & 2.18 (0.080) & 79.2 & 1.21 & 4.18 (0.189) & 74.1 & 1.81\
& P(0.25) & 2.91 (0.110) & 76.7 & 1.52 & 3.22 (0.105) & 74.2 & 1.76 & 10.62 (0.475) & 65.3 & 2.87\
& Huber & 1.90 (0.074) & 80.9 & 1.06 & 2.38 (0.090) & 78.1 & 1.32 & 5.06 (0.230) & 71.9 & 2.04\
200 & LS & 1.46 (0.053) & 83.1 & 0.83 & 4.47 (0.371) & 76.8 & 1.51 & & 56.3 & 4.04\
& P(0.5) & 0.92 (0.033) & 86.4 & 0.55 & 0.98 (0.035) & 85.3 & 0.64 & 1.69 (0.065) & 81.5 & 0.98\
& P(0.25) & 1.35 (0.049) & 83.3 & 0.81 & 1.47 (0.049) & 81.6 & 0.97 & 4.73 (0.241) & 71.9 & 2.05\
& Huber & 0.86 (0.030) & 86.6 & 0.52 & 1.02 (0.036) & 84.7 & 0.68 & 2.11 (0.079) & 79.3 & 1.18\
400 & LS & 0.74 (0.029) & 87.4 & 0.47 & 2.65 (0.402) & 81.4 & 1.04 & & 56.2 & 4.06\
& P(0.5) & 0.45 (0.016) & 90.2 & 0.29 & 0.44 (0.017) & 89.5 & 0.34 & 0.79 (0.029) & 87.2 & 0.49\
& P(0.25) & 0.66 (0.025) & 88.3 & 0.41 & 0.70 (0.023) & 86.9 & 0.50 & 2.12 (0.091) & 79.5 & 1.19\
& Huber & 0.43 (0.016) & 90.2 & 0.28 & 0.48 (0.018) & 89.0 & 0.36 & 1.01 (0.036) & 85.0 & 0.65\
800 & LS & 0.36 (0.013) & 90.8 & 0.25 & 1.09 (0.066) & 85.0 & 0.69 & & 56.3 & 4.02\
& P(0.5) & 0.21 (0.008) & 93.2 & 0.14 & 0.24 (0.009) & 92.3 & 0.19 & 0.39 (0.014) & 90.5 & 0.27\
& P(0.25) & 0.33 (0.013) & 91.7 & 0.21 & 0.36 (0.012) & 90.8 & 0.25 & 1.01 (0.034) & 84.9 & 0.65\
& Huber & 0.20 (0.008) & 93.2 & 0.14 & 0.25 (0.009) & 92.1 & 0.19 & 0.49 (0.016) & 89.1 & 0.34\
[@ll ccc ccc ccc]{}\
& & & &\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} n & method & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} 100 & LS & 0.36 (0.011) & 89.8 & 0.29 & 1.65 (0.085) & 80.8 & 1.06 & & 58.7 & 3.69\
& P(0.5) & 0.57 (0.017) & 86.9 & 0.46 & 0.61 (0.026) & 86.4 & 0.55 & 1.31 (0.045) & 81.7 & 0.93\
& P(0.25) & 0.65 (0.020) & 86.2 & 0.52 & 0.22 (0.008) & 91.7 & 0.20 & 4.67 (0.312) & 74.7 & 1.64\
& Huber & 0.38 (0.012) & 89.5 & 0.30 & 0.45 (0.018) & 88.3 & 0.40 & 1.70 (0.060) & 79.5 & 1.14\
200 & LS & 0.16 (0.004) & 92.9 & 0.14 & 0.74 (0.030) & 85.6 & 0.61 & & 59.1 & 3.64\
& P(0.5) & 0.25 (0.007) & 91.2 & 0.21 & 0.26 (0.008) & 90.7 & 0.24 & 0.52 (0.017) & 87.8 & 0.41\
& P(0.25) & 0.30 (0.008) & 90.3 & 0.26 & 0.09 (0.003) & 94.8 & 0.08 & 1.69 (0.074) & 81.3 & 0.92\
& Huber & 0.17 (0.005) & 92.8 & 0.14 & 0.19 (0.006) & 92.2 & 0.17 & 0.70 (0.022) & 86.2 & 0.53\
400 & LS & 0.08 (0.002) & 95.1 & 0.06 & 0.36 (0.013) & 89.7 & 0.30 & & 58.0 & 3.79\
& P(0.5) & 0.12 (0.003) & 93.8 & 0.10 & 0.12 (0.003) & 93.8 & 0.10 & 0.22 (0.006) & 91.6 & 0.19\
& P(0.25) & 0.14 (0.004) & 93.3 & 0.12 & 0.04 (0.001) & 96.5 & 0.03 & 0.63 (0.021) & 86.5 & 0.49\
& Huber & 0.08 (0.002) & 95.0 & 0.07 & 0.09 (0.002) & 94.8 & 0.07 & 0.30 (0.009) & 90.3 & 0.26\
800 & LS & 0.04 (0.001) & 96.5 & 0.03 & 0.18 (0.006) & 92.3 & 0.16 & & 58.2 & 3.76\
& P(0.5) & 0.06 (0.002) & 95.6 & 0.05 & 0.06 (0.002) & 95.6 & 0.05 & 0.10 (0.003) & 94.4 & 0.09\
& P(0.25) & 0.07 (0.002) & 95.3 & 0.06 & 0.02 (0.001) & 97.5 & 0.02 & 0.29 (0.009) & 90.6 & 0.23\
& Huber & 0.04 (0.001) & 96.4 & 0.03 & 0.04 (0.001) & 96.3 & 0.04 & 0.14 (0.004) & 93.2 & 0.12\
\
& & & &\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} n & method & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$ & mse & PCD & $\delta_{0.5}$\
(r)[3-5]{} (lr)[6-8]{} (l)[9-11]{} 100 & LS & 1.45 (0.059) & 82.9 & 0.85 & 8.53 (0.784) & 72.4 & 2.01 & & 54.9 & 4.22\
& P(0.5) & 0.94 (0.034) & 85.6 & 0.61 & 1.29 (0.058) & 83.3 & 0.86 & 2.27 (0.132) & 78.9 & 1.24\
& P(0.25) & 1.46 (0.051) & 81.5 & 0.96 & 1.78 (0.071) & 78.2 & 1.30 & 7.88 (0.422) & 68.1 & 2.46\
& Huber & 0.89 (0.034) & 86.1 & 0.57 & 1.46 (0.067) & 81.7 & 0.99 & 3.28 (0.157) & 75.1 & 1.65\
200 & LS & 0.84 (0.035) & 86.6 & 0.53 & 3.85 (0.358) & 77.6 & 1.43 & & 55.9 & 4.09\
& P(0.5) & 0.44 (0.016) & 90.0 & 0.29 & 0.60 (0.024) & 89.0 & 0.39 & 0.87 (0.034) & 86.3 & 0.56\
& P(0.25) & 0.69 (0.025) & 87.0 & 0.49 & 0.75 (0.024) & 85.5 & 0.59 & 3.08 (0.179) & 75.3 & 1.58\
& Huber & 0.43 (0.016) & 90.3 & 0.28 & 0.66 (0.025) & 87.7 & 0.47 & 1.32 (0.050) & 82.4 & 0.87\
400 & LS & 0.44 (0.020) & 90.3 & 0.28 & 2.34 (0.393) & 82.4 & 0.95 & & 55.9 & 4.09\
& P(0.5) & 0.23 (0.009) & 92.9 & 0.16 & 0.28 (0.011) & 92.5 & 0.19 & 0.39 (0.015) & 90.8 & 0.26\
& P(0.25) & 0.33 (0.011) & 91.0 & 0.23 & 0.36 (0.012) & 90.1 & 0.27 & 1.25 (0.048) & 82.8 & 0.82\
& Huber & 0.22 (0.008) & 93.1 & 0.15 & 0.31 (0.012) & 91.7 & 0.21 & 0.60 (0.022) & 88.0 & 0.43\
800 & LS & 0.23 (0.009) & 93.0 & 0.15 & 0.90 (0.057) & 86.2 & 0.60 & & 56.3 & 4.03\
& P(0.5) & 0.11 (0.004) & 95.0 & 0.07 & 0.14 (0.005) & 94.8 & 0.09 & 0.18 (0.006) & 93.6 & 0.12\
& P(0.25) & 0.17 (0.006) & 93.7 & 0.12 & 0.18 (0.006) & 93.0 & 0.14 & 0.59 (0.017) & 87.3 & 0.44\
& Huber & 0.10 (0.004) & 95.1 & 0.07 & 0.15 (0.006) & 94.2 & 0.11 & 0.29 (0.010) & 91.4 & 0.21\
We firstly notice that lsA-learning works much worse under the heterogeneous errors, while all other methods are generally less affected by the heterogeneity of the errors. When the baseline function is misspecified as in Model I, under the homogeneous normal errors, RR(H) works slightly better than lsA-learning, while $\mathrm{RR}(\rho_{0.25})$ works the worst. The difference in general is small. For the homogeneous log-normal errors, again RR(H) works the best, while $\mathrm{RR}(\rho_{0.5})$ and $\mathrm{RR}(\rho_{0.25})$ work slightly worse. Here lsA-learning has the worst performance. Under the homogeneous Cauchy errors, the lsA-learning is no longer consistent and work the worst. Both $\mathrm{RR}(\rho_{0.5})$ and RR(H) have good performance under the homogeneous Cauchy errors. When baseline function is correctly specified as in Model II, under homogeneous normal errors, lsA-learning performs the best. However, in this case RR(H) also has a very close performance. Under homogeneous log-normal errors, $\mathrm{RR}(\rho_{0.25})$ work the best and lsA-learning work the worst. Under homogeneous Cauchy errors, $\mathrm{RR}(\rho_{0.5})$ has the best performance and RR(H) has a close performance. lsA-learning is again not consistent.
Examples with error terms interacted with treatment {#examples-with-error-terms-interacted-with-treatment .unnumbered}
---------------------------------------------------
We consider the following model with p=2, $$Y_i=1 + 0.5\sin[\pi(X_{i1}-X_{i2})]+
0.25(1+X_{i1}+2X_{i2})^2+(A_i-\pi({{\bm X}}_{i})){{\mbox{\boldmath $\theta$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i+\sigma({{\bm X}}_{i},A_i)\epsilon_i,$$ where ${{\bm X}}_{i}=(X_{i1},X_{i2}){^{\mbox{\tiny {\sf T}}}}$, $\tilde{{{\bm X}}}_i=(1,{{\bm X}}_i{^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$, $\sigma({{\bm X}}_{i},A_i)=1+A_i d_0 X_{i1}^2$, ${{\mbox{\boldmath $\theta$}}}_0{^{\mbox{\tiny {\sf T}}}}=(0.5,2,-1)$ and $X_{ik}$ are i.i.d. Uniform\[-1,1\]. We take linear forms for both the baseline and the contrast functions, where $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})={{\mbox{\boldmath $\gamma$}}}{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}$, $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})={{\mbox{\boldmath $\beta$}}}{^{\mbox{\tiny {\sf T}}}}{{\bm W}}$ and ${{\bm W}}=(\tilde{{{\bm X}}},X_{1}^2,X_{2}^2,X_{1}X_{2})$. $d_0=5$, 10 or 15. The error terms $\epsilon_i$ follows i.i.d. N(0,1) or Gamma(1,1)-1 distribution. The propensity scores $\pi(\cdot)$ are known, and we consider the non-constant case ($\pi({{\bm X}}_i)=\mathrm{logit}({{\bm X}}_{i1}-{{\bm X}}_{i2})$) here. The simulation results are given in Table \[table:interacted\_nonconstant\_ps\].
---------------------------------------------------------- ------- ----- ----------------- ----------------- ------------------ ----------------- ----------------- ------------------ ----------------- ----------------- ------------------ ----------------- ----------------- ------------------
(r)[4-6]{} (lr)[7-9]{} (lr)[10-12]{} (lr)[13-15]{} Error $d_0$ n $ \delta_{\mu}$ $ \delta_{0.5}$ $ \delta_{0.25}$ $ \delta_{\mu}$ $ \delta_{0.5}$ $ \delta_{0.25}$ $ \delta_{\mu}$ $ \delta_{0.5}$ $ \delta_{0.25}$ $ \delta_{\mu}$ $ \delta_{0.5}$ $ \delta_{0.25}$
Normal 5 100 0.19 0.19 0.36 0.20 0.20 0.34 0.30 0.30 0.23 0.17 0.17 0.33
200 0.11 0.11 0.28 0.13 0.13 0.25 0.21 0.21 0.12 0.11 0.11 0.23
400 0.06 0.06 0.21 0.08 0.08 0.17 0.17 0.17 0.06 0.06 0.06 0.16
800 0.03 0.03 0.16 0.06 0.06 0.11 0.15 0.15 0.04 0.04 0.04 0.11
10 100 0.29 0.29 0.93 0.24 0.24 0.88 0.44 0.44 0.50 0.24 0.24 0.88
200 0.21 0.21 0.92 0.18 0.18 0.84 0.37 0.37 0.34 0.17 0.17 0.83
400 0.13 0.13 0.87 0.14 0.14 0.75 0.32 0.32 0.25 0.12 0.12 0.75
800 0.08 0.08 0.80 0.11 0.11 0.64 0.28 0.28 0.21 0.08 0.08 0.64
15 100 0.35 0.35 1.58 0.27 0.27 1.51 0.53 0.53 0.72 0.26 0.26 1.51
200 0.29 0.29 1.56 0.21 0.21 1.47 0.50 0.50 0.54 0.20 0.20 1.47
400 0.21 0.21 1.58 0.17 0.17 1.37 0.48 0.48 0.39 0.15 0.15 1.38
800 0.14 0.14 1.52 0.14 0.14 1.26 0.45 0.45 0.31 0.12 0.12 1.27
Gamma 5 100 0.18 0.21 0.34 0.20 0.17 0.24 0.28 0.18 0.14 0.18 0.15 0.21
200 0.10 0.14 0.29 0.13 0.10 0.15 0.21 0.11 0.07 0.11 0.07 0.13
400 0.06 0.09 0.23 0.10 0.05 0.10 0.18 0.07 0.04 0.07 0.03 0.08
800 0.03 0.06 0.19 0.08 0.03 0.06 0.16 0.06 0.03 0.06 0.02 0.07
10 100 0.27 0.34 0.90 0.28 0.25 0.67 0.46 0.21 0.33 0.28 0.22 0.62
200 0.20 0.32 0.94 0.21 0.16 0.57 0.43 0.14 0.24 0.21 0.13 0.49
400 0.13 0.27 0.92 0.16 0.09 0.46 0.38 0.10 0.18 0.15 0.06 0.39
800 0.08 0.21 0.85 0.13 0.05 0.40 0.35 0.07 0.16 0.13 0.03 0.35
15 100 0.34 0.55 1.49 0.33 0.37 1.09 0.59 0.25 0.46 0.33 0.33 0.99
200 0.27 0.54 1.57 0.26 0.29 1.00 0.60 0.19 0.31 0.27 0.23 0.85
400 0.19 0.50 1.56 0.20 0.21 0.88 0.61 0.15 0.21 0.22 0.14 0.70
800 0.12 0.47 1.58 0.17 0.14 0.76 0.62 0.15 0.18 0.19 0.09 0.63
---------------------------------------------------------- ------- ----- ----------------- ----------------- ------------------ ----------------- ----------------- ------------------ ----------------- ----------------- ------------------ ----------------- ----------------- ------------------
: Summary results with non-constant propensity scores when errors interacted with treatment. Least square stands for lsA-learning. Pinball(0.5) stands for robust regression with pinball loss and parameter $\tau=0.5$. Pinball(0.25) stands for robust regression with pinball loss and parameter $\tau=0.25$. Huber stands for robust regression with Huber loss, where parameter $\alpha$ is tuned automatically with R function rlm.[]{data-label="table:interacted_nonconstant_ps"}
Based on Theorem 6 of the main paper, $\delta_{\mu}$ column for the lsA-learning method in Table \[table:interacted\_nonconstant\_ps\] converges to 0 as sample size increases. Under Normal error terms, we have $\delta_{0.5}=\delta_{\mu}$. Thus, the $\delta_{0.5}$ column for the lsA-learning method under Normal error also converges to 0. All other columns in Table \[table:interacted\_nonconstant\_ps\] converge to a positive constant instead of 0 as sample size goes to infinity. $\mathrm{RR}(H)$ and $\mathrm{RR}(\rho_{0.5})$ perform similarly in Table \[table:interacted\_nonconstant\_ps\]. We also find even though lsA-learning outperform all other methods in $\delta_{\mu}$ when sample size is large. It may be worse than $\mathrm{RR}(\rho_{0.5})$ and $\mathrm{RR}(H)$ when sample size is small due to the fact that lsA-learning is inefficient under the heteroscedastic or skewed errors. Last, we find that lsA-learning, $\mathrm{RR}(\rho_{0.5})$ and $\mathrm{RR}(\rho_{0.25})$ perform best at the columns $\delta_{\mu}$, $\delta_{0.5}$ and $\delta_{0.25}$ accordingly. The reason is given in the Remark under Theorem 2 of the main paper.
| 0 |
---
abstract: |
We develop a novel, general framework for the asymptotic reduction of the bias of $M$-estimators from unbiased estimating functions. The framework relies on additive, empirical adjustments to the estimating functions that depend only on the first two derivatives of the contributions to the estimating functions. The new estimation method has markedly broader applicability than previous bias-reduction methods by applying to models that are either partially-specified or that have a likelihood that is intractable or expensive to compute, and a surrogate objective is employed. The method also offers itself to easy, general implementations for arbitrary models by using automatic differentiation. This is in contrast to other popular bias-reduction methods that require either resampling or evaluation of expectations of products of log-likelihood derivatives. If $M$-estimation is by the maximization of an objective function, then, reduced-bias $M$-estimation can be achieved by maximizing an appropriately penalized objective. That penalized objective relates closely to information criteria based on the Kullback-Leibler divergence, establishing, for the first time, a strong link between reduction of estimation bias and model selection. The reduced-bias $M$-estimators are found to have the same asymptotic distribution, and, hence, the same asymptotic efficiency properties as the original $M$-estimators, and we discuss inference and model selection with reduced-bias $M$-estimates. The properties of reduced-bias $M$-estimation are illustrated in well-used, important modelling settings of varying complexity.\
[Keywords: *automatic differentiation*; *composite likelihood*; *model selection*; *penalized likelihood*; *quasi likelihood*; *quasi Newton-Raphson*]{}
author:
- |
Ioannis Kosmidis\
`[email protected]`\
Department of Statistics, University of Warwick\
Gibbet Hill Road, Coventry, CV4 7AL, UK\
and\
The Alan Turing Institute\
British Library, 96 Euston Road, London, NW1 2DB, UK\
and\
Nicola Lunardon\
`[email protected]`\
Department of Economics, Management and Statistics, University of Milano-Bicocca\
Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy
bibliography:
- 'robustbr.bib'
title: 'Empirical bias-reducing adjustments to estimating functions'
---
=1
Introduction
============
Reduction of estimation bias in statistical modelling is a task that has attracted immense research activity since the early days of the statistical literature. This ongoing activity resulted in an abundance of general bias-reduction methods of wide applicability. As is noted in @kosmidis:2014, the majority of those methods start from an estimator $\hat{{}\theta}$ and, directly or indirectly, attempt to produce an estimator $\tilde{{}\theta}$ of an unknown parameter ${{}\theta}$, which approximates the solution of the equation $$\label{bias_equation}
\hat{{}\theta} - \tilde{{}\theta} = {{}B}_G(\bar{{}\theta}) \, ,$$ with respect to $\tilde{{}\theta}$. In the above equation, $G$ is the typically unknown, joint distribution function of the process that generated the data, ${{}B}_G({{}\theta}) = \expect_G(\hat{{}\theta} - {{}\theta})$ is the bias function, and $\bar{{}\theta}$ is the value that $\hat{{}\theta}$ is assumed to converge to in probability as information about ${{}\theta}$ increases, typically with the volume of the data. The need for approximating the solution of (\[bias\_equation\]) arises because, on one hand, $G$ is either unknown or the expectation with respect to $G$ is not available in closed form, and, on the other hand, the value of $\bar{{}\theta}$ is unknown.
Table \[br\_characteristics\] classifies prominent bias-reduction methods according to various criteria relating to their applicability and operation. Given the size of the literature on bias-reduction methods, we only cite key works that defined or greatly impacted the area.
Bias-reduction methods like the adjusted scores functions approach in @firth:1993, indirect inference in @gourieroux:1993, and iterated bootstrap in @kuk:1995 and @guerrier+dupuis-lozeron+etal:2019, assume that the model can be fully and correctly specified, in the sense that $G$ results from the assumed model for specific parameter values. That assumption allows to either have access to log-likelihood derivatives and expectations of products of those or to simulate from the model. In contrast, bias-reduction methods, like asymptotic bias correction [@efron:1975], bootstrap [@efron+tibshirani:1993; @hall+martin:1988], and jackknife [@quenouille:1956; @efron:1982] can also apply to at least partially-specified models. In this way, they provide the means for improving estimation in involved modelling settings, where researchers have, historically, resorted to surrogate inference functions in an attempt to either limit the number of hard-to-justify modelling assumptions or because the full likelihood function is impractical or cumbersome to compute or construct; see, for example, @wedderburn:1974 for quasi likelihoods, @liang+zeger:1986 for generalized estimating equations, and @lindsay:1988 and @varin+reid+firth:2011 for composite likelihood methods.
Another criterion for classifying bias-reduction methods comes from @kosmidis:2014. Therein, bias-reduction methods are classified according to whether they operate in an explicit manner by estimating ${{}B}_G(\bar{{}\theta})$ from the data and subtracting that from $\hat{{}\theta}$, or in an implicit manner by replacing ${{}B}_G(\bar{{}\theta})$ with $\hat{{{}B}}_G(\tilde{{}\theta})$ in (\[bias\_equation\]) for some estimator $\hat{{{}B}}_G$ of the bias function, and solving the resulting implicit equation.
Bias-reduction methods can also be classified according to whether the necessary approximation of the bias term in (\[bias\_equation\]) is performed analytically or through simulation. The vanilla implementations of asymptotic bias correction and adjusted score functions approximate ${{}B}_G({{}\theta})$ with a function ${{}b}({{}\theta})$ such that ${{}B}_G({{}\theta}) = {{}b}({{}\theta}) + O(n^{-3/2})$, where $n$ is a measure of how the information about ${{}\theta}$ accumulates. On the other hand, jackknife, bootstrap, iterated bootstrap, and indirect inference, generally, approximate the bias by simulating samples from the assumed model or an estimator of $G$, like the empirical distribution function. As a result, and depending on how demanding the computation of $\hat{{}\theta}$ is, simulation-based methods are typically more computationally intensive than analytical methods. Also, implicit, simulation-based methods require special care and ad-hoc considerations when approximating the solution of (\[bias\_equation\]), because the simulation-based estimator of ${{}B}_G({{}\theta})$ is not always differentiable with respect to ${{}\theta}$.
The requirement of differentiation of the log-likelihood or surrogate functions for some of the bias-reduction methods in Table \[br\_characteristics\] is also another area where considerable analytical effort has been devoted to (see, for example, @kosmidis+firth:2009 for multivariate generalized nonlinear models, and @grun+kosmidis+zeileis:2012 for Beta regression models). Nevertheless, differentiation is nowadays a task requiring increasingly less analytical effort because of the availability of comprehensive automatic differentiation routines [@griewank+walther:2008] in popular computing environments; such routines can be found, for example, in the FowardDiff Julia package [@revels+lubin+papamarkou:2016], and the CppAD package for C++ [@bell:2019] that enabled the development of software like the TMB package [@kristensen+nielsen+berg+etal:2016] for R [@rproject] which is a generic framework for fitting and inference from complex random effects models.
The vanilla versions of asymptotic bias correction and the adjusted score functions in @efron:1975 and @firth:1993, respectively, require the computation of expectations of products of log-likelihood derivatives under the model. Those expectations are intractable or expensive to compute for models with intractable or cumbersome likelihoods, and can be hard to derive even for relatively simple models [see, for example @grun+kosmidis+zeileis:2012 for the required expectations in Beta regression models].
----------------------------------------- --------------------- -------------------- ---------- ------------- ----------------- -------------------- --
Method Model specification Bias approximation Type Expectation Differentiation Original estimator
Asymptotic bias correction partial analytical Explicit yes yes yes
Adjusted score functions full analytical Implicit yes yes no
Bootstrap partial simulation-based Explicit no no yes
Jackknife partial simulation-based Explicit no no yes
Iterated bootstrap / Indirect inference full simulation-based Implicit no no yes
RB$M$-estimation partial analytical Implicit no yes no
----------------------------------------- --------------------- -------------------- ---------- ------------- ----------------- -------------------- --
\[br\_characteristics\]
Finally, except of the adjusted scores approach in @firth:1993, all the bias-reduction methods reviewed in Table \[br\_characteristics\] require the original estimator $\hat{{}\theta}$ and they cannot operate without it. For this reason, they directly inherit any of the instabilities that $\hat{{}\theta}$ may have. For example, in multinomial logistic regression, there is always a positive probability of data separation [@albert+anderson:1984] that results in infinite maximum likelihood estimates. Then, asymptotic bias correction, bootstrap, iterated bootstrap, and jackknife cannot be applied. The direct dependence on $\hat{{}\theta}$ may be more consequential for naive implementations of the latter three methods because they are simulation-based; even if data separation did not occur for the original sample, there is always positive probability that it will occur for at least one of the simulated samples. There is no easy way of knowing this before carrying out the simulation and such cases can only be handled in an ad-hoc way.
The current work develops a novel method for the reduction of the asymptotic bias of $M$-estimators from general, unbiased estimating functions. We call the new estimation method reduced-bias $M$-estimation, or RB$M$-estimation in short. Like the adjusted scores approach in @firth:1993, the new method relies on additive adjustments to the unbiased estimating functions that are bounded in probability, and results in estimators with bias of lower asymptotic order than the original $M$-estimators. The key difference is that the empirical adjustments introduced here depend only on the first two derivatives of the contributions to the estimating functions, and they require neither the computation of cumbersome expectations nor the, potentially expensive, calculation of $M$-estimates from simulated samples. Specifically, and as noted in the last row of Table \[br\_characteristics\], RB$M$-estimation i) applies to models that are at least partially-specified; ii) uses an analytical approximation to the bias function that relies only on derivatives of the contributions to the estimating functions; iii) does not depend on the original estimator; and iv) does not require the computation of any expectations. By relying only on derivatives of the contributions to the estimating functions, the new method is typically easier to implement for arbitrary models than other popular bias-reduction methods that require either resampling or the evaluation of moments of products of log-likelihood derivatives. In fact, automatic differentiation can be used to develop generic implementations of the new method whose only required input is an implementation of the contributions to the estimating functions. The GEEBRA Julia package (<https://github.com/ikosmidis/GEEBRA.jl>) by the Authors is a proof-of-concept of such an implementation.
If the estimating functions are the components of the gradient of an objective function, as is the case in maximum likelihood and maximum composite likelihood estimation, then, we show that bias reduction can always be achieved by the maximization of an appropriately penalized version of the objective. This is in contrast to the method in @firth:1993 which does not always have a penalized likelihood interpretation; see, for example, @kosmidis+firth:2009, who derive a necessary and sufficient condition for the existence of bias-reducing penalized likelihood for generalized linear models. Moreover, it is shown that the bias-reducing penalized objective closely relates to information criteria for model selection based on the Kullback-Leibler divergence. The functions of the parameters and the data that are used for bias reduction and model selection differ only by a known scalar constant. These observations, establish, for the first time, a strong link between reduction of bias in estimation and model selection. It is also shown that the RB$M$-estimators have the same asymptotic distribution, hence the same asymptotic efficiency properties, as the original $M$-estimators, and we discuss inference and model selection with RB$M$-estimates.
Section \[modelling\_setting\] introduces notation and sets up the general modelling setting we consider and the assumptions underpinning the theoretical developments. Section \[sec:theory\] and Section \[empirical\_adjustments\] are devoted on the derivation of the bias-reducing, empirical adjustments to the estimating functions and the specification of their components. Section \[sec:inference\] shows that the asymptotic distribution of the RB$M$ estimator is the same as that of the original $M$-estimator, and introduces Wald-type and generalized score approximate pivots that can be used for the construction of inferences. Section \[sec:penalties\] introduces and discusses bias-reducing penalized objectives and the links to model selection. A quasi Newton-Raphson iteration that has the BR$M$-estimates as its fixed point is introduced in Section \[sec:implementation\], along with discussion on its ingredients and on general implementations using automatic differentiation. The finite-sample properties of RB$M$-estimation are illustrated in well-used, important modelling settings of increasing complexity including, the estimation of the ratio of two means with minimal distributional assumptions (Section \[empirical\_adjustments\]), generalized linear models (Section \[sec:glms\]) and quasi likelihoods (Section \[sec:quasi\]), and composite likelihood methods for the estimation of gaussian max-stable processes (Section \[sec:maxstable\]), and for the estimation of multivariate probit models with correlated random effects for longitudinal ordinal responses (Section \[autoregressive\_probit\]). Section \[sec:discussion\] provides a discussion on the developments and possible extensions of the developments in this work, and introduces two alternative RB$M$-estimators.
Modelling setting and assumptions {#modelling_setting}
=================================
Estimating functions {#EF}
--------------------
Suppose that we observe the values ${{}y}_1, \ldots, {{}y}_k$ of a sequence of random vectors ${{}Y}_1, \ldots, {{}Y}_k$ with ${{}y}_i = (y_{i1}, \ldots, y_{ic_i})^\top \in \mathcal{Y} \subset
\Re^c$, possibly with a sequence of covariate vectors ${{}x}_1, \ldots, {{}x}_k$, with ${{}x}_i = (x_{i1}, \ldots, x_{iq_i})^\top \in \mathcal{X} \subset
\Re^{q_i}$. Any two distinct random vectors ${{}Y}_i$ and ${{}Y}_j$ may share random scalar components. We denote the distinct scalar random variables by $Z_1, \ldots, Z_m$, and by $G(z_1, \ldots, z_m)$ their typically unknown, underlying joint distribution function. In general $m \le \sum_{i = 1}^k c_i$, with equality only if ${{}Y}_1, \ldots, {{}Y}_k$ have distinct components. Let ${{}Y} = ({{}Y}_1^\top, \ldots, {{}Y}_k^\top)^\top$, and denote by $X$ the set of ${{}x}_1, \ldots, {{}x}_k$.
On of the common aims in statistical modelling is to estimate at least a sub-vector of an unknown parameter vector ${{}\theta} \in \Theta \subset \Re^p$ using data ${{}y}_1, \ldots, {{}y}_k$ and ${{}x}_1, \ldots, {{}x}_k$. This is most commonly achieved through a vector of $p$ estimating functions $\sum_{i = 1}^k{{}\psi}^i({{}\theta}) = (\sum_{i =
1}^k\psi_1^i({{}\theta}), \ldots, \sum_{i = 1}^k\psi_{p}^i({{}\theta}))^\top$, where ${{}\psi}^i({{}\theta}) = {{}\psi}({{}\theta}, {{}Y}_i, {{}x}_i)$ and $\psi^i_r({{}\theta}) = \psi_r({{}\theta}, {{}Y}_i, {{}x}_i)$, $(r = 1,\ldots,p)$. In particular, ${{}\theta}$ is estimated by the $M$-estimator $\hat{{}\theta}$ [@vaart:1998 Chapter 5], which results by the solution of the system of estimating equations $$\label{estimating_eqautions}
\sum_{i = 1}^k{{}\psi}^i({{}\theta}) = {{}0}_p \, ,$$ with respect to ${{}\theta}$, where ${{}0}_p$ is a $p$-vector of zeros. Prominent examples of estimation methods that fall within the above framework are estimation via quasi likelihoods [@wedderburn:1974] and generalized estimating equations [@liang+zeger:1986]. @stefanski+boos:2002 provide an accessible overview of estimating functions and demonstrate their generality and ability to tackle challenging estimation problems with fewer assumptions than likelihood-based estimation requires.
One way to derive estimating equations is through a, typically stronger, modelling assumption that ${{}Y}_i$ has a distribution function $F_i({{}y}_i | {{}x}_i, {{}\theta})$. The estimator $\hat{{}\theta}$ can then be taken to be the maximizer of the objective function $$\label{objective_function}
l({{}\theta}) = \sum_{i =1}^k \log f_i({{}y}_i | {{}x}_i, {{}\theta}) \, ,$$ where $f_i({{}y}_i | {{}x}_i, {{}\theta})$ is the mixed joint density corresponding to $F_i({{}y}_i | {{}x}_i, {{}\theta})$. The word “mixed” is used here to allow for the fact that some of the components of ${{}y}_i$ may be continuous and some others may be discrete. If the objective function (\[objective\_function\]) is used, then the estimating functions in (\[estimating\_eqautions\]) have ${{}\psi}^i({{}\theta}) = \nabla \log f_i({{}y}_i | {{}x}_i,
{{}\theta})$, assuming that the gradient exists in $\Theta$. Prominent examples of estimation methods that involve an objective function of the form (\[objective\_function\]) are maximum likelihood and maximum composite likelihood [see, for example, @lindsay:1988; @varin+reid+firth:2011].
Assumptions
-----------
The assumptions we employ for the theoretical development in this work are listed below.
1. \[consistency\] Consistency: $$\hat{{}\theta} \stackrel{p}{\longrightarrow} \bar{{{}\theta}} \, ,$$ where $\bar{{{}\theta}}$ is such that $\expect_G ({{}\psi}^i) = {{}0}_p$ for all $i \in \{1, \ldots, k\}$, with ${{}\psi}^i={{}\psi}^i(\bar{{}\theta})$ and $\expect_G(\cdot)$ denoting expectation with respect to the unknown joint distribution function $G$. In particular, we assume that $\hat{{}\theta} - \bar{{{}\theta}} = O_p(n^{-1/2})$, where $n \equiv n(k, m, q)$ is a measure of information about ${{}\theta}$.
2. \[smoothness\] Smoothness: The derivatives of $\psi_r^i({{}\theta})$ exist up to the $5$-th order. In particular, $$l_{R_a}({{}\theta}) = \sum_{i=1}^k \frac{\partial^{a-1}
\psi_{r_1}^i({{}\theta})}{\partial {{}\theta}^{r_2} \cdots \partial
{{}\theta}^{r_a}} \, ,$$ exist for any set $R_a = \{r_1, \ldots, r_a\}$, with $r_j \in \{1, \ldots, p\}$ and $a \in \{1, \ldots, 5\}$, under the convention that $l_r({{}\theta}) = \sum_{i = 1}^k \psi_r^i({{}\theta})$ and that the components of ${}\theta$ are identified by superscripts.
3. \[derivative\_orders\] Asymptotic orders of centred estimating function derivatives: $$H_{R_a} = l_{R_a} - \mu_{R_a} = O_p(n ^{1/2}) \, ,$$ where $\mu_{R_a} = \expect_G(l_{R_a})$, $l_{R_a} = l_{R_a}(\bar{{}\theta})$ and $\mu_{R_a} = \mu_{R_a}(\bar{{}\theta})$ exist for $a \in \{1, \ldots, 5\}$.
4. \[moment\_orders\] Asymptotic orders of joint central moments of estimating functions and their derivatives: $$\nu_{R_{a_1}, S_{a_2}, \ldots, T_{a_b}} = \left\{
\begin{array}{ll} O(n^{(b - 1)/2})\,, & \text{if $b$ is odd} \\
O(n^{b/2})\,, & \text{if $b$ is even}
\end{array} \right. \, ,$$ where $\nu_{R_{a_1}, S_{a_2}, \ldots, T_{a_b}} = \expect_G
(H_{R_{a_1}} H_{S_{a_2}} \cdots H_{T_{a_b}})$ are joint central moments of estimating functions and their derivatives, with $R_{a_1}, S_{a_2}, \ldots, T_{a_b}$ being subsets of $a_1, a_2,
\ldots, a_b > 0$ integers, respectively.
5. \[invertibility\] The matrix with elements $\mu_{rs}$ $(r,s = 1,\ldots,p)$ is invertible.
Assumption \[consistency\] is a working assumption that we make about the unbiasedness of the estimating functions and the consistency of the $M$-estimators. Consistency can sometimes be shown to hold under weak assumptions about $G$ and the asymptotic unbiasedness of the estimating functions; see, for example, @vaart:1998 [Section 5.2] and @huber+ronchetti:2009 [Section 6.2], for theorems on the consistency of $M$-estimators. We assume that there is an index $n$, which is typically, but not necessarily, the number of observations, that measures the rate the information about the parameter ${{}\theta}$ accumulates, and that the difference $\hat{{}\theta} - \bar{{}\theta}$ is of order $n^{-1/2}$ in probability.
Assumption \[smoothness\] allows taking derivatives of the estimating functions across the parameter space when constructing the stochastic Taylor expansions required for the derivation of the empirical bias-reducing adjustments to the estimating functions in Section \[sec:theory\]. Such an assumption covers many well-used estimating functions, like the ones arising in quasi-likelihood estimation, estimation using generalized estimating equations, and maximum likelihood and maximum composite likelihood estimation for a wide range of models. The smoothness assumption does not cover, though, settings where the estimating function or one of its first few derivatives are non-differentiable at particular points in the parameter space. Examples of this kind are the estimating functions for quantile regression and robust regression with Huber loss; see @koenker:2005 and @huber+ronchetti:2009 for textbook-length expositions of topics in quantile and robust regression, respectively.
Assumptions \[derivative\_orders\] and \[moment\_orders\] ensure the existence of the expectations, under the underlying process $G$, of products of estimating functions and their derivatives, and that $\sqrt{n}$-asymptotic arguments are valid. Assumption \[invertibility\] is a technical assumption to ensure that the expectation of the jacobian of the estimating function is invertible, when inverting the stochastic Taylor expansions in Section \[sec:theory\], and is typically assumed for estimation using maximum likelihood and estimating equations [see, for example, @boos+stefanski:2013 Section 7.7].
Adjusted estimating equations for bias reduction {#sec:theory}
================================================
Asymptotic bias {#asymptotic_bias}
---------------
Under the assumptions of Section \[assumptions\] it can be shown that the bias of $\hat{{}\theta}$ is $$\label{bias}
\expect_G(\hat{{}\theta} - \bar{{}\theta}) = O(n^{-1}) \, .$$ This provides some reassurance that, as the information about the parameter ${{}\theta}$ grows, the bias from $\bar{{}\theta}$ will converge to ${{}0}_p$. Nevertheless, the finite sample bias of $\hat{{}\theta}$ is typically not zero. Under the same assumptions, it is also possible to write down (\[bias\]) in the more explicit form $\expect_G(\hat{{}\theta} - \bar{{}\theta}) = {{}b}(\bar{{}\theta}) + O(n^{-3/2})$, where ${{}b}(\bar{{}\theta})$ depends on joint moments of estimating functions and their derivatives under $G$. It is tempting, then, to replace $\hat{{}\theta}$ with a new estimator $\hat{{}\theta} - {{}b}(\hat{{}\theta})$, on the basis that, under \[consistency\], ${{}b}(\hat{{}\theta}) \stackrel{p}{\longrightarrow} {{}b}(\bar{{}\theta})$, and hoping that the new estimator will have better bias properties.
The estimator $\hat{{}\theta} - {{}b}(\hat{{}\theta})$ has been shown to, indeed, have better bias properties when estimation is by maximum likelihood and the model is correctly specified [see, Section 10 @efron:1975 for a proof]. By the model being correctly specified we mean that the unknown joint distribution function $G(z_1, \ldots, z_m)$ is assumed to be a particular member of the family of distributions specified fully by $F_i({{}y}_i | {{}x}_i, {{}\theta})$ when forming (\[objective\_function\]). In that particular case, it is also possible to evaluate ${{}b}({{}\theta})$ and, hence, compute $\hat{{}\theta} - {{}b}(\hat{{}\theta})$ in light of data, because the expectations involved in the joint null moments are with respect to the modelling assumption. This is also the basis of more refined bias reduction methods, like the adjusted score function approach that has been derived in @firth:1993 and explored further in @kosmidis+firth:2009.
In the more general setting of Section \[modelling\_setting\], where the model can be only partially specified, naive evaluation of ${{}b}({{}\theta})$ using $F_i({{}y}_i | {{}x}_i, {{}\theta})$ not only does not lead to reduction of bias, in general, but it can also inflate the bias; see, for example, @lunardon+scharfstein:2017 [Section 2.1] who illustrate the impact of using an incorrect bias function in the estimation of log-odds in longitudinal settings. Furthermore, even in the case that the researcher is comfortable to assume that $F_i({{}y}_i | {{}x}_i, {{}\theta})$ is correctly specified, the applicability of standard bias-reduction methods is hampered whenever $F_i({{}y}_i | {{}x}_i, {{}\theta})$ is impossible or impractical to compute in closed form. In such cases, estimation and inference is based on pseudo likelihoods or estimating functions; an example of this kind is within the framework of max-stable processes for which the evaluation of the joint density becomes quickly infeasible when the number of site locations increases [@davison+gholamrezaee:2011].
In the remainder of this section, we show that the gap between improved estimation methods and applied modelling using general $M$-estimators can be bridged either through simple empirical bias-reducing adjustments to the estimating functions (\[estimating\_eqautions\]) that do not involve expectations, or empirical bias-reducing penalties to the objective function (\[objective\_function\]).
Family of bias-reducing adjustments to estimating functions
-----------------------------------------------------------
Suppose that the assumptions in Section \[assumptions\] hold and consider the estimator $\tilde{{}\theta}$ that results from the solution of the adjusted estimating equations $$\label{adjusted_estimating_equations}
\sum_{i=1}^k {{}\psi}^i({{}\theta}) + {{}A}({{}\theta}) = {{}0}_p \, ,$$ where both ${{}A}({{}\theta}) = {{}A}({{}\theta}, {{}Y}, X)$ and its derivatives with respect to ${{}\theta}$ are $O_p(1)$ as $n$ grows.
Using assumptions \[consistency\]-\[derivative\_orders\] and index notation, with the indices taking values in the set $\{1,\ldots,p\}$, a calculation similar to that in @mccullagh:2018 [Section 7.3] can be used to show that the expansion of ${{}0}_p = \sum_{i=1}^k {{}\psi}^i(\tilde{{}\theta}) + {{}A}(\tilde{{}\theta})$ about $\bar{{}\theta}$ results in a stochastic Taylor expansion for $\tilde{{}\theta} - \bar{{}\theta}$ of the form $$\begin{aligned}
\label{taylor}
\tilde{\theta}^{r} - \bar{\theta}^{r} & = H^{r} + H^{a} H_{a}^{r} + \frac{1}{2} H^{a} H^{b} \mu_{ab}^{r} + A^{r} +\\ \notag
&+ H^{a} H_{a}^{b} H_{b}^{r} + \frac{1}{2} H^{a} H^{b} H_{c}^{r} \mu_{ab}^{c} + \frac{1}{2} H^{a} H^{b} H_{b}^{c} \mu_{ac}^{r}+ \\ \notag
& + \frac{1}{2} H^{a} H^{b} H_{a}^{c} \mu_{cb}^{r} + \frac{1}{4} H^{a} H^{b} H^{c} \mu_{bc}^{d} \mu_{ad}^{r} + \frac{1}{4} H^{a} H^{b} H^{c} \mu_{ab}^{d} \mu_{dc}^{r} + \\ \notag
& + \frac{1}{2} H^{a} H^{b} H_{ab}^{r} + \frac{1}{6} H^{a} H^{b} H^{c} \mu_{abc}^{r} +\\ \notag
& + A^{a} H_{a}^{r} + \frac{1}{2} A^{a} H^{b} \mu_{ba}^{r} + \frac{1}{2} A^{a} H^{b} \mu_{ab}^{r} + A_{a}^{r} H^{a} + O_p(n^{-2}) \, ,\end{aligned}$$ where ${H^{r}_{r_1\ldots r_a}} = - {\mu^{rs}_{}} {H^{}_{sr_1\cdots r_a}}$, ${\mu^{r}_{r_1\cdots r_a}} = - {\mu^{rs}_{}} {\mu^{}_{sr_1\cdots r_a}}$, and ${A^{r}_{r_1\cdots r_a}} = - {\mu^{rs}_{}} {A^{}_{sr_1\cdots r_a}}$, with $\mu^{rs}$ denoting the matrix inverse of $\mu_{rs}$ (assumption \[invertibility\]) and $A_{r_1\cdots r_a} = \partial^{a-1} A_{r_1}({{}\theta}) / \partial
\theta^{r_2} \cdots \partial \theta^{r_a}$.
Taking expectations with respect to the underlying distribution $G$ on both sides of (\[taylor\]), assumption \[moment\_orders\] gives that the bias of $\tilde{{{}\theta}}$ is $$\expect_G{(\tilde{\theta}^{r} - \bar{\theta}^{r})} = -{\mu^{ra}_{}}
\expect_G (A_a) + \frac{1}{2} {\mu^{ra}_{}} {\mu^{bc}_{}}
\left({\nu^{}_{ab,c}} - {\mu^{de}_{}} {\nu^{}_{c,e}}
{\mu^{}_{abd}} \right) + O(n^{-3/2})\, ,$$ where all terms in the right-hand side are understood as being evaluated at $\bar{{}\theta}$.
The above expansion for the bias implies that use of any adjustment ${{}A}$ with $$\label{adjustment}
\expect_G(A_r) = \frac{1}{2} {\mu^{ab}_{}} \left( 2 {\nu^{}_{ra,b}} - {\mu^{cd}_{}} {\nu^{}_{b,d}} {\mu^{}_{rac}} \right) + O(n^{-1/2}) \, ,$$ in (\[adjusted\_estimating\_equations\]), will result in estimators with bias $\expect_G{(\tilde{\theta}^{r} - \bar{\theta}^{r})} = O(n^{-3/2})$, which is smaller, asymptotically, than the bias of $\hat{{}\theta}$ in (\[bias\]). Hence, expression (\[adjustment\]) defines a family of bias-reducing adjustments to estimating functions.
A clear candidate for ${{}A}$ has as $r$th component the first term in the right hand side of (\[adjustment\]). If we assume that the model $F_i({{}y}_i | {{}x}_i, {{}\theta})$ fully specifies $G$ and the estimation method is maximum likelihood, then the Bartlett relations $\mu_{cd} + \nu_{c, d} = 0$ hold [see, for example, @pace+salvan:1997 Section 9.2], and the adjustment (\[adjustment\]) becomes ${\mu^{ab}_{}} \left( 2 {\nu^{}_{ra,b}} + {\mu^{}_{rab}}
\right)/2$. The latter expression coincides with the bias-reducing score adjustment derived in @firth:1993.
For the more general setting of Section \[modelling\_setting\], however, the underlying distribution $G$ is at most only partially specified through $F_i({{}y}_i | {{}x}_i, {{}\theta})$, and the expectations involved in the right-hand side of (\[adjustment\]) cannot be computed. It turns out that, there is a family of empirical adjustments that can always be implemented and delivers bias reduction of $M$-estimators in that more general framework.
Empirical bias-reducing adjustments {#empirical_adjustments}
===================================
Derivation
----------
Consider data-dependent quantities ${l^{}_{r|s}}({{}\theta})$ and ${l^{}_{rs|t}}({{}\theta})$ such that $\expect_G({l^{}_{r|s}}) = {\nu^{}_{r, s}}$ and $\expect_G({l^{}_{rs|t}}) = {\nu^{}_{rs, t}}$, with ${H^{}_{r | s}} = {l^{}_{r | s}} - {\nu^{}_{r, s}} = O_p(n^{1/2})$ and ${H^{}_{rs | t}} = {l^{}_{rs | t}} - {\nu^{}_{rs, t}} =
O_p(n^{1/2})$. Then, a simple calculation under the assumptions of Section \[assumptions\] gives that equation (\[adjustment\]) is satisfied by an empirical adjustment of the form $$\label{general_empirical_adjustment}
A_r({{}\theta}) = \frac{1}{2} {l^{ab}_{}}({{}\theta}) \left\{ 2 {l^{}_{ra|b}}({{}\theta}) - {l^{cd}_{}}({{}\theta}) {l^{}_{b|d}}({{}\theta}) {l^{}_{rac}}({{}\theta})\right\} \, ,$$ where ${l^{st}_{}}({{}\theta})$ is the matrix inverse of ${l^{}_{ts}}({{}\theta})$. The matrix form of expression (\[general\_empirical\_adjustment\]) sets the $r$th element of the vector of empirical bias-reducing adjustments to $$\label{general_empirical_adjustment_matrix}
A_r({{}\theta}) = - \trace\left\{{{}j}({{}\theta})^{-1} {{}d}_r({{}\theta}) \right\} -
\frac{1}{2}\trace\left[ {{}j}({{}\theta})^{-1} {{}e}({{}\theta}) \left\{{{}j}({{}\theta})^{-1}\right\}^\top {{}u}_r({{}\theta}) \right] \, ,$$ where ${{}u}_r({{}\theta})=\sum_{i=1}^k \nabla \nabla^\top \psi_r^i({{}\theta})$, and ${{}j}({{}\theta})$ is the matrix with $s$th row $-\sum_{i=1}^k \nabla \psi_s^i({{}\theta})$, assumed to be invertible but not necessarily symmetric. The matrix ${{}j}({{}\theta})$ coincides with the negative hessian matrix $\nabla \nabla^\top l({{}\theta})$ when estimation is through the maximisation of an objective function like (\[objective\_function\]), and is the observed information matrix when estimation is through maximum likelihood. The $p \times p$ matrices ${{}e}({{}\theta})$ and ${{}d}_r({{}\theta})$ correspond to the quantities ${l^{}_{r|s}}({{}\theta})$ and ${l^{}_{rs|t}}({{}\theta})$, respectively, and need to be exactly defined in order to use adjustment (\[general\_empirical\_adjustment\]) in estimation problems.
Specification of p and p under independence {#independence}
-------------------------------------------
If the assumption that ${{}Y}_1, \ldots, {{}Y}_k$ are independent under $G$ is plausible, then $$\nu_{r, s} = \sum_{i = 1}^k E_G( \psi_r^i \psi_s^i) + \sum_{i \ne j} E_G( \psi_r^i) E_G(\psi_s^j) \, .$$ Assumption \[consistency\] on the unbiasedness of the estimating function implies that the second term in the right-hand side of the above expression is zero. Hence, $l_{r|s} = \sum_{i = 1}^k \psi_r^i \psi_s^i$ is such that $\expect_G(l_{r|s}) = \nu_{r, s}$ and, under assumption \[derivative\_orders\], ${H^{}_{r | s}} = {l^{}_{r | s}} - {\nu^{}_{r, s}} = O_p(n^{1/2})$.
In matrix notation, ${{}e}({{}\theta})$ can be taken to be the $p \times p$ matrix ${{}\Psi}({{}\theta})^\top {{}\Psi}({{}\theta})$, where ${{}\Psi}({{}\theta})$ is the $n \times p$ matrix with $(i, s)$th element $\psi_s^i({{}\theta})$. In other words, the $(s, t)$th element of ${{}e}({{}\theta})$ is $$\label{e_mat}
[{{}e}({{}\theta})]_{st} = \sum_{i = 1}^k \psi_s^i({{}\theta}) \psi_t^i({{}\theta}) \,.$$ A similar argument gives that ${{}d}_r({{}\theta})$ can be taken to be the $p \times p$ matrix $\tilde{{{}\Psi}}_r({{}\theta})^\top {{}\Psi}({{}\theta})$ where $\tilde{{{}\Psi}}_r({{}\theta})$ is the $n \times p$ matrix with $(i, s)$th element $\partial \psi_r^i / \partial \theta^s$. In other words, the $(s, t)$th element of ${{}d}_r({{}\theta})$ is $$\label{d_mat}
[{{}d}_r({{}\theta})]_{st} = \sum_{i = 1}^k \left\{ \frac{\partial}{\partial \theta^s} \psi_r^i({{}\theta}) \right\} \psi_t^i({{}\theta}) \, .$$
The above expressions can directly be used in settings that involve observations of $k$ independent random vectors with dependent components. Examples of such settings are the generalized estimating equations in @liang+zeger:1986 for estimating marginal regression parameters for correlated responses, and the composite likelihood approach in @varin+czado:2010 for estimating multivariate probit models with random effects to account for serial dependence in longitudinal settings with binary and ordinal outcomes. Section \[autoregressive\_probit\] considers a case study using the latter models.
Reduced-bias $M$-estimation for partially-specified models
----------------------------------------------------------
Under the assumptions of Section \[assumptions\] and the assumptions about $G$ required for the specification of ${{}e}({{}\theta})$ and ${{}d}_r({{}\theta})$ in Section \[independence\], the solution of the adjusted estimating equations $\sum_{i=1}^k{{}\psi}^i({{}\theta}) + {{}A}({{}\theta}) =
0_p$, with ${{}A}({{}\theta})$ as in (\[general\_empirical\_adjustment\]), will result in RB$M$-estimators that have $O(n^{-3/2})$ bias. This is a major advantage over past bias-reduction methods [see, e.g. @cordeiro+mccullagh:1991; @firth:1993] whose applicability is limited to cases where $\sum_{i=1}^k{{}\psi}^i({{}\theta})$ is the gradient of the log-likelihood function of a correctly-specified model.
The example below demonstrates the benefits of using the empirical bias-reducing adjustment in the estimation of a ratio of means from realizations of independent pairs of random variables without any further assumptions on the joint distributions of the pairs.
\[ratio\_estimation\]
[**Ratio of two means**]{} Consider a setting where independent pairs of random variables $(X_1, Y_1), \ldots, (X_n, Y_n)$ are observed, and suppose that interest is in the ratio of the mean of $Y_i$ to the mean of $X_i$, that is $\theta = \mu_Y / \mu_X$, with $\mu_X = \expect_G(X_i)$ and $\mu_Y = \expect_G(Y_i) \ne 0$ $(i = 1, \ldots, n)$.
Assuming that sampling is from an infinite population, one way of estimating $\theta$ without any further assumptions about the joint distribution of $(X_i, Y_i)$ is to set up an unbiased estimating equation of the form (\[estimating\_eqautions\]), with $\psi^i(\theta) = Y_i - \theta X_i$. Then, the $M$-estimator is $$\label{M_ratio}
\hat\theta = \arg \mathop{\rm solve}_{\theta \in \Re} \left\{\sum_{i = 1}^n \psi^i(\theta) = 0 \right\} = \frac{s_Y}{s_X} \, ,$$ where $s_X = \sum_{i = 1}^n X_i$ and $s_Y = \sum_{i = 1}^n Y_i$. The estimator $\hat\theta$ is generally biased, as can be shown, for example, by an application of the Jensen inequality assuming that $X_i$ is independent of $Y_i$, and efforts have been made in reducing its bias; see, for example, @durbin:1959, for an early work in that direction using the jackknife.
For the estimating function for the ratio of means, $j(\theta) = s_X$, $u(\theta) = 0$, and, using (\[e\_mat\]) and (\[d\_mat\]), $e(\theta) = s_{YY} + \theta^2 s_{XX} - 2\theta s_{XY}$ and $d(\theta) = -s_{XY} + \theta s_{XX}$, where $s_{XX} = \sum_{i = 1}^n X_i^2$, $s_{YY} = \sum_{i = 1}^n Y_i^2$ and $s_{XY} = \sum_{i = 1}^n X_i Y_i$. So, the empirical bias-reducing adjustment in (\[general\_empirical\_adjustment\]) is $s_{XY}/s_{X} - \theta s_{XX}/s_{X}$, and the solution of the adjusted estimating equations results in the RB$M$-estimator $$\label{BR_ratio}
\tilde\theta = \arg \mathop{\rm solve}_{\theta \in \Re} \left\{\sum_{i = 1}^n \psi^i(\theta) + A(\theta) = 0 \right\} = \frac{s_Y + \frac{s_{XY}}{s_{X}}}{s_X + \frac{s_{XX}}{s_{X}}} \, .$$ In this case, the empirical bias-reducing adjustment has the side-effect of producing an estimator that is more robust to small values of $s_X$ than the standard $M$-estimator is. In particular, as $s_X$ becomes smaller in absolute value, $\hat\theta$ diverges, while $\tilde{\theta}$ converges to $s_{XY}/s_{XX}$, which is the slope of the regression line through the origin of $y$ on $x$. As a result, when $\mu_X$ is small in absolute value, $\tilde\theta$ has not only smaller bias, as granted by the developments in the current paper, but also smaller variance than $\hat\theta$, and, hence, smaller mean squared error.
To illustrate the performance of the RB$M$-estimator $\tilde\theta$ of the ration $\theta$ we assume that $(X_1, Y_1), \ldots, (X_n, Y_n)$ are independent random vectors from a bivariate distribution constructed through a gaussian copula with correlation $0.5$, to have an exponential marginal with rate $1/2$ for $X_i$ and a normal marginal for $Y_i$ with mean $10$ and variance $1$. For each $n \in \{10, 20, 40, 80, 160, 320\}$, we simulate $N_n = 250 \times 2^{16} / n$ samples. Calibrating the simulation size in this way guarantees a fixed simulation error for the simulation-based estimate of the bias $\expect_G(\hat\theta - \theta)$ of the $M$-estimator $\hat\theta$. For each sample we estimate $\theta$ using the $M$-estimator $\hat\theta$, the RB$M$-estimator $\tilde\theta$, and the jackknife estimator $$\label{Ja_ratio}
\theta^* = n \hat\theta - \frac{n - 1}{n} \sum_{i = 1}^n \hat\theta_{(-i)} \, ,$$ where $\hat\theta_{(-i)} = \sum_{j \ne i} Y_i/\sum_{j \ne i}
X_i$. Note here that use of the bias-reducing adjusted score function in @firth:1993 requires expectations of products of log-likelihood derivatives, which require fully specifying the bivariate distribution for $(X_i, Y_i)$. Despite the simplicity of the current setting, even if one could confidently specify that distribution, the required expectations involve non-trivial analytic calculations and may not even be available in closed-form.
-- ---------------- ------- ------- ------ ------ ------ ------
$\hat\theta$ 0.53 0.25 0.12 0.06 0.03 0.01
$\theta^*$ -0.07 -0.01 0.00 0.00 0.00 0.00
$\tilde\theta$ 0.17 0.05 0.01 0.00 0.00 0.00
$\hat\theta$ 3.76 1.47 0.65 0.31 0.15 0.07
$\theta^*$ 3.04 1.29 0.61 0.30 0.15 0.07
$\tilde\theta$ 3.12 1.31 0.61 0.30 0.15 0.07
$\hat\theta$ 1.37 0.91 0.62 0.44 0.30 0.21
$\theta^*$ 1.31 0.88 0.61 0.43 0.30 0.21
$\tilde\theta$ 1.29 0.88 0.61 0.43 0.30 0.21
$\hat\theta$ 0.46 0.47 0.48 0.48 0.49 0.49
$\theta^*$ 0.60 0.56 0.54 0.53 0.52 0.51
$\tilde\theta$ 0.54 0.54 0.53 0.52 0.52 0.51
-- ---------------- ------- ------- ------ ------ ------ ------
: Simulation-based estimates of the bias (B; $\expect_G\left(\theta_n - \theta\right)$), mean squared error (MSE; $\expect_G\left\{(\theta_n - \theta)^2\right\}$), mean absolute error (MAE; $\expect_G\left(\left|\theta_n - \theta\right|\right)$), and probability of underestimation (PU; $P_G(\theta_n < \theta)$) for the estimation of a ratio of means in the setting of Example \[ratio\_estimation\]. The estimator $\theta_n$ is either the $M$-estimator $\hat\theta$ in (\[M\_ratio\]), the Jackknife estimator $\theta^*$ in (\[Ja\_ratio\]) or the RB$M$-estimator $\tilde\theta$ in (\[BR\_ratio\]). The simulation-based estimates are computed from $N_n = 250 \times 2^{16} / n$ simulated samples with $n \in \{10, 20, 40, 80, 160, 320\}$ independent pairs each from a bivariate distribution constructed through a gaussian copula with correlation $0.5$, to have an exponential marginal with rate $1/2$ for $X_i$ and a normal marginal for $Y_i$ with mean $10$ and variance $1$ ($\theta = 5$). Figures are reported in 2 decimal places, and a figure of $0.00$ indicates an estimated bias between $-0.005$ and $0.005$. The simulation error for the estimates of the bias is between $0.001$ and $0.002$.
\[ratio\_simulation\]
Table \[ratio\_simulation\] shows the simulation-based estimates of the bias, mean-squared error, mean absolute deviation, and probability of underestimation for $\hat\theta$, $\tilde\theta$, and $\theta^*$. As with the jackknife, use of the empirical bias-reducing adjustment results in a marked reduction of the bias. The estimated slopes of the regression lines of the logarithm of the absolute value of the estimated biases for $\hat\theta$, $\theta^*$, and $\tilde\theta$ on the values of $n$ are $-1.046$, $-1.445$, and $-1.610$, respectively, which are in close agreement to the theoretical slopes of $-1$, $-3/2$, $-3/2$.
Reduction of the bias in this setting leads also in marked reduction in mean squared error and mean absolute deviation, with $\tilde\theta$ and $\theta^*$ performing similarly, and significantly better than $\hat\theta$. The RB$M$-estimator $\tilde{\theta}$ also has probability of underestimation closer to $0.5$, hence it is closer to being median unbiased than is $\theta^*$.
Asymptotic distribution of reduced-bias $M$-estimators and inference {#sec:inference}
====================================================================
Under the assumptions of Section \[assumptions\], the argument in @stefanski+boos:2002 [Section 2] applied to the first term of the stochastic Taylor expansion in (\[taylor\]) with ${{}A}({{}\theta}) = {{}0}_p$ gives that the $M$-estimator $\hat{{}\theta}$ is such that $${{}Q}(\bar{{}\theta})^{1/2} (\hat{{}\theta} - \bar{{}\theta}) \stackrel{d}{\longrightarrow} N_p({{}0}_p, {{}I}_p)$$ as $n$ increases, where ${{}Q}({{}\theta}) = {{{}V}({{}\theta})}^{-1}$ with ${{}V}({{}\theta}) = {{}B}({{}\theta})^{-1} {{}M}({{}\theta}) \{{{}B}({{}\theta})^{-1}\}^\top$. In the latter expression, ${{}M}({{}\theta}) = \expect_G[ \sum_{i=1}^k \sum_j^k {{}\psi^i}({{}\theta})\{ {{}\psi}^j({{}\theta})\}^\top ]$, and ${{}B}({{}\theta})$ is a $p \times p$ matrix with $r$th row $-\sum_{i=1}^k \expect_G\{\nabla \psi_r^i({{}\theta}) \}$. The notation ${{}D} \stackrel{d}{\longrightarrow} N_p({{}0}_p, {{}I}_p)$ indicates that the random $p$-vector ${{}D}$ converges in distribution to a multivariate Normal with mean ${}0_p$ and variance-covariance matrix the $p \times p$ identity matrix ${{}I}_p$.
From the stochastic Taylor expansion (\[taylor\]), it becomes directly apparent that $\tilde{{{}\theta}} - \bar{{{}\theta}}$ and $\hat{{{}\theta}} - \bar{{{}\theta}}$ have exactly the same $O_p(n^{-1/2})$ term in their expansions because ${{}A}({{}\theta}) = O_p(1)$. Hence, the RB$M$-estimator $\tilde{{}\theta}$ is also such that $$\label{asymptotic_normality}
{{}Q}(\bar{{}\theta})^{1/2} (\tilde{{}\theta} - \bar{{}\theta}) \stackrel{d}{\longrightarrow} N_p({{}0}_p, {{}I}_p) \, .$$
An implication of (\[asymptotic\_normality\]) is that $\hat{{{}V}}({{}\theta}) = {{}j}({{}\theta})^{-1} {{}e}({{}\theta}) \left\{{{}j}({{}\theta})^{-1}\right\}^\top$ evaluated at ${{}\theta}=\tilde{{}\theta}$ is a consistent estimator of the variance-covariance matrix of $\tilde{{}\theta}$. This is exactly as in the case where $\hat{{{}V}}(\hat{{}\theta})$ is used as an estimator of the variance-covariance matrix of $\hat{{}\theta}$ in the framework of $M$-estimation [see, for example, @stefanski+boos:2002 Section 2]. It is important to note that the expression for $\hat{{{}V}}({{}\theta})$ appears unaltered in the second term of the right-hand-side of expression (\[general\_empirical\_adjustment\_matrix\]) for the empirical bias-reducing adjustment. As a result, the value of ${{}V}(\tilde{{}\theta})$ and, hence, that of estimated standard errors for the parameters, is, in general, readily available at the last step of the iterative process that is used to solve the adjusted estimating equations. Section \[sec:implementation\], provides more details on the implementation of solvers for the bias-reducing adjusted estimating equations.
In addition, if the model is correctly specified, and $l({{}\theta})$ in (\[objective\_function\]) is the log-likelihood, then the second Bartlett identity gives ${{}M}({{}\theta}) = {{}B}({{}\theta})$, which implies that ${{}Q}({{}\theta})$ is the expected information matrix. As a result, the RB$M$-estimator is asymptotically efficient, exactly as the maximum likelihood estimator and the reduced-bias estimator in @firth:1993 are.
-------------------------------------- ------ ------ ------ ------ ------ ------
$\var_G(\hat\theta)$ 3.49 1.41 0.64 0.30 0.15 0.07
$\expect_G\{\hat{V}(\hat\theta)\}$ 2.55 1.20 0.59 0.29 0.14 0.07
$\var_G(\tilde\theta)$ 3.09 1.31 0.61 0.30 0.15 0.07
$\expect_G\{\hat{V}(\tilde\theta)\}$ 2.16 1.10 0.56 0.28 0.14 0.07
-------------------------------------- ------ ------ ------ ------ ------ ------
: Simulation-based estimates of the variances of the $M$-estimator $\hat\theta$ and the RB$M$-estimator $\tilde\theta$ ($\var_G(\hat\theta)$ and $\var_G(\tilde\theta)$, respectively) and of the mean of $\hat{V}(\hat\theta)$ and $\hat{V}(\tilde\theta)$ ($\expect_G\{\hat{V}(\hat\theta)\}$ and $\expect_G\{\hat{V}(\tilde\theta)\}$, respectively), from the simulation study of Example \[ratio\_estimation\]. See, also Table \[ratio\_simulation\].
\[ratio\_inference\]
[**Ratio of two means (continued)**]{} Table \[ratio\_inference\] shows the estimates of the actual variances of $\hat\theta$ and $\tilde\theta$ for $n \in \{10, 20, 40, 80, 160, 320\}$ and the estimate of the mean of $\hat{V}(\hat\theta)$ and $\hat{V}(\tilde\theta)$, respectively. As expected by the above discussion, the large sample approximations to the variance of the estimator converge to the actual variances as the sample size increases.
Another implication of (\[asymptotic\_normality\]) is that asymptotically valid inferential procedures, like hypothesis tests and confidence regions for the model parameters, can be constructed based on the Wald-type and generalized score approximate pivots of the form $$\begin{aligned}
\label{pivots}
W_{(e)}({{}\theta}) & = (\tilde{{}\theta} - {{}\theta})^\top \left\{\hat{{{}V}}(\tilde{{}\theta}) \right\}^{-1}(\tilde{{}\theta} - {{}\theta}) \, , \\ \notag
W_{(s)}({{}\theta}) & = \left\{\sum_{i = 1}^k{{}\psi}^i({{}\theta}) + {{}A}({{}\theta})\right\}^\top \left\{{{}e}(\tilde{{}\theta})\right\}^{-1} \left\{\sum_{i = 1}^k{{}\psi}^i({{}\theta}) + {{}A}({{}\theta})\right\}\,,\end{aligned}$$ respectively, which, asymptotically, have a $\chi^2_p$ distribution. These pivots are direct extensions of the Wald-type and generalized score pivots, respectively, that are typically used in $M$-estimation [see, @boos:1992 for discussion about generalized score tests in $M$-estimation].
Empirical bias-reducing penalties to objective functions {#sec:penalties}
========================================================
Bias-reducing penalized objectives {#sec:penalty}
----------------------------------
When estimation is through the maximisation of (\[objective\_function\]), ${{}j}({{}\theta})$ is a symmetric matrix. Then, an extra condition that determines how $l_{r|s}$ behaves under differentiation guarantees that the bias-reducing adjustment (\[general\_empirical\_adjustment\]) always corresponds to an additive penalty to the objective function used for estimation. In particular, if $l_{r|s}$ admits a chain rule under differentiation, that is $$\label{diff_condition}
\frac{\partial}{\partial {\theta^{a}_{}}} {l^{}_{r|s}} = {l^{}_{ar | s}} + {l^{}_{r | as}} \, ,$$ then bias reduction through empirical bias-reducing adjustments is formally equivalent to the maximization of a penalized objective function of the form $$\label{penalized_likelihood}
l({{}\theta}) - \frac{1}{2}\trace \left\{{{}j}({{}\theta})^{-1} {{}e}({{}\theta}) \right\} \, ,$$ assuming that the maximum exists. This result greatly facilitates implementation of bias reduction for a much wider class of models and estimation methods than other bias-reduction methods via adjusted score functions [see, for example @firth:1993], where focus is on cases where $l({{}\theta})$ is the log-likelihood function of a correctly-specified model, and a bias-reducing penalized log-likelihood does not always exist.
The matrix ${{}e}({{}\theta})$ that has been derived under the assumption of independence in Section \[independence\] satisfies condition (\[diff\_condition\]).
Bias reduction and model selection based on Kullback-Leibler divergence {#sec:model_selection}
-----------------------------------------------------------------------
Suppose that $l({{}\theta})$ is the log-likelihood function based on an assumed parametric model $F$. @takeuchi:1976 showed that $$\label{tic}
- 2 l(\hat{{}\theta}) + 2 \trace \left\{{{}j}(\hat{{}\theta})^{-1} {{}e}(\hat{{}\theta}) \right\}$$ is an estimator of the expected Kullabck-Leibler divergence of the underlying process $G$ to the assumed model $F$, where $\hat{{}\theta}$ is the maximum likelihood estimator. Expression (\[tic\]) is widely known as the Takeuchi information criterion (TIC), and, in contrast to the Akaike Information Criterion (AIC; @akaike:1974), is robust against deviations from the assumption that the model used is correct. @claeskens:2008 [Section 2.5] provide a more thorough discussion on TIC and its formal relationship to AIC.
Model selection from a set of parametric models then proceeds by computing $\hat{{}\theta}$ for each model and selecting the model with the smallest TIC value (\[tic\]), or equivalently, the model with the largest value for $$\label{tic2}
l(\hat{{}\theta}) - \trace \left\{{{}j}(\hat{{}\theta})^{-1} {{}e}(\hat{{}\theta}) \right\} \, .$$ A direct comparison of expressions (\[tic2\]) and (\[penalized\_likelihood\]) reveals a previously unnoticed close connection between bias reduction in maximum likelihood estimation and model selection. Specifically, both bias reduction and TIC model selection rely on exactly the same penalty $\trace \left\{{{}j}({{}\theta})^{-1} {{}e}({{}\theta})
\right\}$, but differ in the strength of penalization; bias reduction is achieved by using half that penalty, while valid model selection requires stronger penalization by using one times the penalty.
As discussed in Section \[sec:inference\], the RB$M$-estimator $\tilde{{}\theta}$ has the same asymptotic distribution as $\hat{{}\theta}$. Then, the derivation of TIC [see, for example, @claeskens:2008 Section 2.3] works also with the RB$M$-estimator in place of the maximum likelihood estimator. As a result, TIC and, under more assumptions, AIC at the RB$M$-estimates are asymptotically equivalent to their typical versions at the maximum likelihood estimates. The same holds for reduced-bias estimators of @firth:1993. In other words, TIC model selection can proceed by selecting the model with the largest value of $$\label{tic3}
l(\tilde{{}\theta}) - \trace \left\{{{}j}(\tilde{{}\theta})^{-1} {{}e}(\tilde{{}\theta}) \right\} \, ,$$ and AIC model selection using the largest value of $l(\tilde{{}\theta}) - p$. The quantity in (\[tic3\]) is readily available once (\[penalized\_likelihood\]) has been maximized to obtain the RB$M$-estimates; the only requirement for model selection is to adjust, from $1/2$ to $1$, the factor of the value of $\trace \left\{{{}j}({{}\theta})^{-1} {{}e}({{}\theta})
\right\}$ after maximization.
@varin+vidoni:2005 developed a model selection procedure when the objective $l({{}\theta})$ is a composite likelihood [see, @varin+reid+firth:2011 for a review of composite likelihood methods]. The composite likelihood information criterion (CLIC) derived in [@varin+vidoni:2005] has the same functional form as TIC in (\[tic\]). So, the link between model selection and bias reduction exists also when $l({{}\theta})$ is the logarithm of a composite likelihood.
Implementation {#sec:implementation}
==============
Apart from special cases, like the estimation of the ratio of two means in Example \[ratio\_estimation\], and as is the case for general estimating functions, the solution of the bias-reducing adjusted estimating equations (\[adjusted\_estimating\_equations\]) is, typically, not available in closed form, and iterative procedures are used to approximate that solution.
A general iterative procedure of this kind results from a modification of the Newton-Raphson iteration that in the $u$th iteration updates the current estimate ${{}\theta}^{(u)}$ to a new value ${{}\theta}^{(u + 1)}$ as $$\label{quasiNR}
{{}\theta}^{(u + 1)} := {{}\theta}^{(u)} + a_u \left\{{{}j}\left({{}\theta}^{(u)}\right)\right\}^{-1} \left\{ \sum_{i=1}^k {{}\psi}^i\left({{}\theta}^{(u)}\right) + {{}A}\left({{}\theta}^{(u)}\right) \right\}\, ,$$ where $a_u$ is a deterministic sequence of positive constants that can be used to implement various schemes to further control the step size, like step-halving. Iteration (\[quasiNR\]) defines a quasi Newton-Raphson procedure with the correct fixed point, rather than a full Newton-Raphson iteration that would have $a_u = 1$ and the matrix of derivatives of $\sum_{i=1}^k {{}\psi}^i({{}\theta}) + {{}A}({{}\theta})$ in the place of ${{}j}({{}\theta})$. The $M$-estimates from the solution of $\sum_{i=1}^k {{}\psi}^i({{}\theta}) = {{}0}_p$ are obvious starting values for the quasi Newton-Raphson procedure. Candidate stopping criteria include $|{{}\theta}^{(u + 1)} - {{}\theta}^{(u)}|/a_u < \epsilon$ and $||\sum_{i=1}^k {{}\psi}^i\left({{}\theta}^{(u)}\right) + {{}A}\left({{}\theta}^{(u)}\right)||_1 < \epsilon$, for some $\epsilon > 0$, where $||\cdot||_1$ is the L1 norm.
Typically, quasi Newton-Raphson will have first-order convergence to the solution of the adjusted estimating equations, compared to the second-order convergence that full Newton-Raphson has. The advantage of using quasi Newton-Raphson instead of full Newton-Raphson is that all quantities required to implement (\[quasiNR\]) are readily available once an implementation of the empirical bias-reducing adjustments is done.
The fact that the empirical bias-reducing adjustment in (\[general\_empirical\_adjustment\]) depends only on derivatives of estimating functions, enables general implementations by deriving the derivatives $\partial \psi_r^i({{}\theta}) / \partial \theta_s$ and $\partial^2 \psi_r^i({{}\theta}) / \partial \theta_s \partial
\theta_t$ $(r, s, t = 1, \ldots, p)$, either analytically or by using automatic differentiation techniques [@griewank+walther:2008]. Those derivatives can be combined together to produce ${{}u}_r({{}\theta})$, ${{}j}({{}\theta})$, ${{}e}({{}\theta})$ and ${{}d}_r({{}\theta})$, and, then, matrix multiplication and a numerical routine for matrix inversion can be used for an easy, general implementation of (\[general\_empirical\_adjustment\_matrix\]).
For implementations using automatic differentiation, in particular, the only required input is an appropriate implementation of the contributions ${{}\psi}^i({{}\theta})$ to the estimating functions. The automatic differentiation routines will, then, produce implementations of the required first and second derivatives of the contributions. The GEEBRA Julia package (<https://github.com/ikosmidis/GEEBRA.jl>) provides a proof-of-concept of such an implementation.
When estimation is through the maximization of an objective function, the RB$M$-estimates can be computed using general numerical optimization procedures for the maximization of the penalized objective function (\[penalized\_likelihood\]), like those provided by the `optim` function in R or the Optim Julia package, that can operate by numerically approximating the gradient of (\[penalized\_likelihood\]). In such cases, bias reduction can be performed using only routines for matrix multiplication and inversion, and the contributions to the estimating function and their first derivatives for the implementation of ${{}j}({{}\theta})$ and ${{}e}({{}\theta})$. Those derivatives can again be obtained using an automatic differentiation library.
The discussion in Section \[sec:penalty\] implies that using the empirical bias-reducing adjustments in maximum likelihood and maximum composite likelihood estimation is readily available for all models for which there are implementations of TIC and CLIC, respectively [see, e.g., @padoan+bevilacqua:2015 for estimation of random fields based on composite likelihoods].
Generalized linear models {#sec:glms}
=========================
Bias-reducing penalized likelihood
----------------------------------
Consider a sequence of $n$ vectors $(y_1, {{}x}_1^\top)^\top, \ldots, (y_n, {{}x}_n^\top)^\top$, where $y_i \in \mathcal{Y} \subset \Re$, and ${{}x}_i \in \mathcal{X} \subset \Re^p$. Suppose that $y_1, \ldots ,y_n$ are realizations of random variables $Y_1, \ldots, Y_n$ which, conditionally on covariate vectors ${{}x}_1, \ldots, {{}x}_n$, are assumed to be independent and distributed according to a generalized linear model. The $i$th log-likelihood contribution is $$\label{loglikelihood_glm}
\log f_i(y_i | {{}x}_i, {{}\beta}, \phi) = \frac{m_i}{\phi} \left\{ y_i \theta_i - \kappa(\theta_i) - c_1(y_i) \right\} - \frac{1}{2} a\left(-\frac{m_i}{\phi}\right) \, ,$$ for sufficiently smooth functions $\kappa(\cdot)$, $c_1(\cdot)$, and $a(\cdot)$, where the conditional mean and variance of $Y_i$ are associated to ${}\beta$ and ${{}x}_i$ as $\mu_i = d \kappa(\theta_i) / d\theta_i = h(\eta_i)$ and $\var(Y_i | {{}x}_i) = \phi v(\mu_i) / m_i$, respectively, with $\eta_i = {{}x}_i^\top {{}\beta}$, and a sufficiently smooth function $h(\cdot)$ $(i = 1, \ldots, n)$. The parameter $\phi$ is a known or unknown dispersion parameter, $m_1, \ldots, m_n$ are known observation weights, and $v(\mu_i) = d^2 \kappa(\theta_i) / d \theta_i^2$ is the variance function. A few prominent generalized linear models are binomial logistic regression models, Poisson log-linear models, gamma regression models, and normal linear regression models.
Bias reduction of the maximum likelihood estimator $\hat{{}\beta}$ and $\hat\phi$ for fully-specified generalized linear models has been studied extensively. Landmark studies in this direction include: @cordeiro+mccullagh:1991, who derive a closed-form estimator of the bias of $\hat{{}\beta}$ and $\hat\phi$ and subtract that from the estimates; @kosmidis+firth:2009, who show that iterating @cordeiro+mccullagh:1991’s reweighted least squares reduced-bias estimator results in the solution of the bias-reducing adjusted score equations in @firth:1993, and derive a necessary and sufficient condition for the existence of a bias-reducing penalized likelihood for generalized linear models; and @kosmidis+kennepagui+sartori:2019, who derive the connections between mean and median bias reduction for generalized linear models and propose a unifying implementation through a quasi Fisher scoring procedure.
In contrast to the bias-reduction methods proposed for generalized linear models in @kosmidis+firth:2009, the empirical bias-reducing adjustment to the score function always corresponds to a penalty to the log-likelihood function about ${{}\beta}$ and $\phi$. According to Section \[sec:penalty\], if $\phi$ is unknown, the only ingredients required in the penalty are the observed information matrix about ${{}\beta}$ and $\phi$, ${{}j}^{(G)}({{}\beta}, \phi)$, and the sum of the outer products of the gradient of (\[loglikelihood\_glm\]) across observations, ${{}e}^{(G)}({{}\beta}, \phi)$. The bias-reducing penalized log-likelihood is, then $$\sum_{i = 1}^n \log f_i(y_i | {{}x}_i, {{}\beta}, \phi) -
\frac{1}{2} \trace \left[ \left\{ {{}j}^{(G)}({{}\beta},
\phi)\right\}^{-1} {{}e}^{(G)}({{}\beta}, \phi) \right] .$$ The closed-form expressions for ${{}j}^{(G)}({{}\beta}, \phi)$ and ${{}e}^{(G)}({{}\beta}, \phi)$ are given in Appendix \[glm\_appendix\]. If $\phi$ is fixed, as is, for example, for the binomial and Poisson distributions, then the penalty involves only the $({{}\beta}, {{}\beta})$ blocks of ${{}j}^{(G)}({{}\beta}, \phi)$ and ${{}e}^{(G)}({{}\beta}, \phi)$. Some algebra, after plugging the expressions in Appendix \[glm\_appendix\] in (\[penalized\_likelihood\]), shows that the RB$M$-estimator of ${{}\beta}$ results as the maximizer of the bias-reducing penalized log-likelihood $$\label{pen_loglik_glm}
\sum_{i = 1}^n m_i \left\{ y_i \theta_i - \kappa(\theta_i) - \frac{1}{2} s_i \frac{d_i}{v_i}(y_i - \mu_i) \right\} \, ,$$ where $d_i = d h(\eta_i) / d\eta_i$ and $s_i$ is the $i$th diagonal element of the matrix ${{}X}({{}X}^\top {{}Q} {{}X})^{-1} {{}X}^\top \tilde{{{}W}}$ with ${{}X}$ the $n \times p$ matrix with rows ${{}x}_1, \ldots, {{}x}_n$, and with the diagonal matrices $\tilde{{{}W}}$ and ${{}Q}$ as defined in Appendix \[glm\_appendix\].
-- -- ----------- -- -------- -------- -------- --------
Parameter
$\beta_1$ -0.047 -0.018 -0.011 -0.008
$\beta_4$ 0.019 0.007 0.004 0.005
$\beta_5$ 0.058 0.027 0.016 0.007
$\beta_1$ -0.001 0.003 0.002 -0.002
$\beta_4$ -0.001 -0.001 -0.001 0.002
$\beta_5$ 0.004 -0.001 0.001 0.000
$\beta_1$ -0.013 -0.000 0.000 -0.002
$\beta_4$ 0.002 -0.001 -0.001 0.002
$\beta_5$ 0.020 0.005 0.002 0.001
$\beta_1$ 0.140 0.064 0.033 0.017
$\beta_4$ 0.133 0.062 0.034 0.016
$\beta_5$ 0.059 0.024 0.011 0.006
$\beta_1$ 0.121 0.060 0.032 0.016
$\beta_4$ 0.118 0.059 0.033 0.016
$\beta_5$ 0.046 0.021 0.011 0.005
$\beta_1$ 0.124 0.060 0.032 0.016
$\beta_4$ 0.120 0.059 0.033 0.016
$\beta_5$ 0.050 0.022 0.012 0.005
-- -- ----------- -- -------- -------- -------- --------
: Bias and mean squared error (MSE) of the maximum likelihood (ML) estimator, RB$M$-estimator, and the adjusted scores (AS) estimator in @firth:1993 of $\beta_1$, $\beta_4$ and $\beta_5$ in the probit regression model with $\mu_i = \Phi(\beta_1 + \beta_4 x_{i4} + \beta_4 x_{i5})$ $(i = 1, \ldots, n)$, $n \in \{75, 150, 300, 600\}$. The figures are based on $10\, 000$ samples simulated from the model with $\mu_i = \Phi(-0.5 + 0.5 x_{i4} + 0.5 x_{i5})$. For each value of $n$, the covariate values $x_{i4}$ and $x_{i5}$ $(i = 1, \ldots, n)$ are generated independently and independent of each other from a Bernoulli distribution with probability $3/4$, and an exponential distribution with rate $1$, respectively, and are held fixed across samples. No boundary estimates were encountered, though the figures should be interpreted as simulation-based estimates of the bias and mean-squared error conditional on all estimates being finite.
\[glm\_bias\_mse\]
Probit regression
-----------------
The performance of the RB$M$-estimator is compared here to the performance of the maximum likelihood estimator and the reduced-bias estimator of @firth:1993 in a probit regression model with $\mu_i = \Phi(\beta_1 + \sum_{t = 2}^5 \beta_t x_{it})$ $(i = 1, \ldots, n)$, where $\Phi(\cdot)$ is the cumulative distribution function of a standard Normal distribution. The covariate values $x_{i2}, x_{i3}, x_{i4}, x_{i5}$ $(i = 1, \ldots, n)$ are generated independently and independent of each other from a standard normal distribution, Bernoulli distributions with probabilities $1/4$ and $3/4$, and an exponential distribution with rate $1$, respectively. Note that the necessary and sufficient condition of @kosmidis+firth:2009 [Theorem 1] is not satisfied for this model, and hence, there is no bias-reducing penalized log-likelihood that corresponds to the adjusted score functions of @firth:1993. Nevertheless, the bias-reducing penalized log-likelihood (\[pen\_loglik\_glm\]) is well-defined.
There are $16$ possible nested probit regression models with an intercept $\beta_1$, depending on which of $\beta_2, \ldots, \beta_5$ are zero or non-zero. For $n \in \{75, 150, 300, 600\}$, we simulate $n$ covariate values, as detailed in the previous paragraph, and conditional on those we simulate $10\, 000$ samples of $n$ response values from a probit regression model with ${{}\beta} = (-0.5, 0, 0, 0.5, 0.5)^\top$. For each sample, we estimate all $16$ possible models using maximum likelihood, the adjusted score functions approach in @firth:1993, and maximum bias-reducing penalized likelihood. Maximum likelihood estimates are computed using the `glm` function in R [@rproject], and the @firth:1993 adjusted scores estimates are computed using the `brglm_fit` method from the `brglm2` R package [@brglm2]. The RB$M$-estimates resulting from the numerical maximization of are computed using the `nlm` R function (see the scripts for probit regression supplied in the Supporting Materials).
There were $7$ samples for which separation occurred for at least one of the 16 models, and no separated samples were observed under the data generating model that involves only the intercept, $x_{i4}$ and $x_{i5}$. The detection of separated data sets was done prior to fitting each model using the linear programming algorithms in @konis:2007, as implemented in the `detect_separation` method of the `brglm2` R package [@brglm2]. In those cases, maximum likelihood and maximum penalized likelihood result in estimates on the boundary of the parameter space. In contrast, as has also been observed in the 2007 PhD thesis by Ioannis Kosmidis [@kosmidis:2007], the adjusted score approach of @firth:1993 always resulted in finite estimates.
Table \[glm\_bias\_mse\] shows the simulation-based estimates of the bias and mean squared error of the three estimators for the true model that has $\beta_2 = \beta_3 = 0$, conditionally on the maximum likelihood estimate not being on the boundary of the parameter space. The conditioning is carried out by ignoring the separated samples when taking averages. As is immediately apparent, both the adjusted score and maximum penalized likelihood approaches to bias reduction result in estimators with smaller conditional bias and mean squared errors than the maximum likelihood estimator. The mean squared error of the RB$M$-estimator tends to be slightly larger than that of the adjusted score estimator, but the differences diminish fast as the sample size increases.
Figure \[model\_selection\] shows the selection proportion among the 16 models based on AIC and TIC for each of the three estimators. As expected by the discussion in Section \[sec:model\_selection\], the probability of selecting the model with $\beta_2 = \beta_3 = 0$ increases with the sample size for both information criteria and for all estimation methods. There are only small discrepancies on the selection proportions between estimation methods, which tend to disappear as the sample size increases. Finally, it is worth noting that AIC model selection tends to be more confident on what the true model is than TIC, illustrating less variability in the selected proportions. In the current study, for the larger samples sizes this results in selecting the correct model more often than TIC does. However, in smaller samples sizes, AIC selects the wrong model with $\beta_2 = \beta_3 = \beta_4 = 0$ notably more times than the correct model.
Quasi likelihoods {#sec:quasi}
=================
Preamble
--------
Compared to generalized linear models, much less work has been carried out for reducing the bias of quasi-likelihood estimators. At the time of writing this paper, apart from bias reduction based on resampling schemes, like the bootstrap and the jackknife [see, for example @wu:1986 Section 9], we were unable to find an analytical approach for the reduction of the bias for quasi-likelihood estimators. @heyde:1997 [Section 4.4] notes that an adjusted score approach to bias reduction is possible for quasi likelihoods, but provides no further development in that direction. Also, @paul+zhang:2014 develop a bias-reduction method for generalized estimating equations though their development seems to focus on cases where the distribution of the longitudinal outcomes is correctly specified [see, also @lunardon+scharfstein:2017 who show that @paul+zhang:2014 development is not delivering bias reduction under mispecification of the working intra-subject covariance matrix].
Consider again a sequence of $n$ vectors $(y_1, {{}x}_1^\top)^\top, \ldots, (y_n, {{}x}_n^\top)^\top$, where $y_i \in \mathcal{Y} \subset \Re$, and ${{}x}_i \in \mathcal{X} \subset \Re^p$. Suppose that $y_1, \ldots ,y_n$ are realizations of random variables $Y_1, \ldots, Y_n$ which are assumed independent conditionally on covariate vectors ${{}x}_1, \ldots, {{}x}_n$, and that interest is in the estimation of a parameter ${{}\beta}$ in the assumed mean relation $\mu_i = \expect(Y_i | {{}x}_i) = h(\eta_i)$ with $\eta_i = {{}x}_i^\top {{}\beta}$, for some known function $h(\cdot)$ $(i = 1, \ldots, n)$. Interest is also in allowing for the variance to vary with the mean as $\var(Y_i | {{}x}_i) = \phi v(\mu_i) / m_i$, where $\phi$ is a dispersion parameter, $m_1, \ldots, m_n$ are known observation weights, and $v(\mu_i)$ is a variance function.
Under the assumption that the assumed relationship between the mean and the regression parameters ${{}\beta}$ is correctly specified, a consistent estimator $\hat{{}\beta}$ for ${{}\beta}$ can be obtained by solving the quasi-likelihood equations $\sum_{i= 1}^n \psi^i_s({{}\beta}, \phi) = 0$ $(s = 1, \ldots, p)$ with $$\label{quasilikelihood}
\psi^i_s({{}\beta}, \phi) = \frac{m_i d_i}{\phi v_i} (Y_i - \mu_i) x_{is} \,,$$ where $d_i = \partial \mu_i / \partial \eta_i$ and $v_i = v(\mu_i)$ [see, @wedderburn:1974; @mccullagh:1983 for introduction and study of quasi-likelihood methods]. @mccullagh:1983 [Section 4] has shown that the quasi-likelihood estimator generalizes the Gauss-Markov optimality of least squares estimators by having the minimum asymptotic variance amongst all estimators resulting from unbiased estimating equations that are linear in $y_i$.
By the discussion in Section \[sec:inference\], if the expression for $\var(Y_i | {{}x}_i)$ is also correctly specified, then the variance-covariance matrix for $\hat{{}\beta}$ can be estimated as $\hat\phi \left\{{{}X}^\top {{}W}(\hat{{{}\beta}})^{-1} {{}X}\right\}^{-1}$, for some estimator $\hat\phi$ of the dispersion parameter, where ${{}W}({{}\beta})$ is a diagonal matrix with $i$th diagonal element $m_i d_i^2/v_i$ . Otherwise, the variance-covariance matrix for $\hat{{}\beta}$ can be estimated by $\hat{{{}V}}(\hat{{}\beta}, \hat\phi)$ as defined in Section \[sec:inference\] or variants of it; see, for example, @firth:1993b [Section 2] for a concise discussion on the estimation of the variance-covariance matrix of $\hat{{}\beta}$.
@mccullagh+nelder:1989 recommend to estimate $\phi = m_i \var(Y_i | {{}x}_i) / v(\mu_i)$ using the moment estimator $$\label{dispersion}
\hat\phi_R = \frac{1}{R} \sum_{i = 1}^n \frac{m_i}{v(\hat\mu_i)} (Y_i - \hat\mu_i)^2 \, ,$$ with $\hat\mu_i = h(\hat{{}\beta}^\top {{}x}_i)$ and $R = n - p$ to account for the estimation of the $p$-vector of regression parameters ${{}\beta}$. The estimator $\hat\phi_n$ is consistent under the same assumptions that $\hat\phi_{n - p}$ is consistent, but $\hat\phi_{n - p}$ is considered to be less biased than $\hat\phi_n$.
It is well-known that quasi-likelihood methods about ${{}\beta}$ are closely related to maximum likelihood methods for generalized linear models. If the distribution of $Y_i | {{}x}_i$ is fully-specified to be an exponential family with mean $\mu_i$ and variance $\phi v_i /m_i$, then it is straightforward to see that (\[quasilikelihood\]) is the partial derivative of the $i$th log-likelihood contribution (\[loglikelihood\_glm\]) with respect to $\beta_s$. So, the discussion and results below also cover estimation of the parameters of generalized linear models with unknown dispersion estimated by $\hat\phi_R$.
Bias reduction in the estimation of p and $\phi$
------------------------------------------------
There exist two immediate possibilities for the use of the empirical bias-reducing adjustments (\[general\_empirical\_adjustment\_matrix\]) in quasi-likelihood estimation. The first is to adjust the quasi-likelihood equations for ${{}\beta}$, and after solving those, to use $\hat\phi_{n-p}$ in (\[dispersion\]) for the estimation of $\phi$ with $\hat\mu_i$ replaced by $\tilde\mu_i = h({{}x}_i^\top\tilde{{}\beta})$, where $\tilde{{}\beta}$ is the RB$M$-estimator.
The second possibility is to adjust the quasi-likelihood estimating functions for ${{}\beta}$ and the estimating function $\sum_{i = 1}^n\psi_{p + 1}^i({{}\beta}, \phi)$ with $$\label{phi_ee}
\psi_{p + 1}^i({{}\beta}, \phi) = \frac{m_i}{v(\mu_i)} (Y_i - \mu_i)^2 - \frac{R}{n} \phi \, .$$ Under the assumption that $\expect(Y_i | {{}x}_i)$ and $\var(Y_i | {{}x}_i)$ are correctly specified, the estimating function $\sum_{i = 1}^n\psi_{p + 1}^i({{}\beta}, \phi)$ has expectation $(n - R)\phi$. So, assumption \[consistency\] on the unbiasedness of the estimating equations is satisfied for $R = n$. In other words, the estimating equations that result in $\hat\phi_n$ in (\[dispersion\]), rather than those for $\hat\phi_{n - p}$, are the ones that should be adjusted to produce the RB$M$-estimator of $\phi$.
According to Section \[sec:implementation\], the necessary quantities for the implementation of the empirical bias-reducing adjustment to the quasi-likelihood equations are the first and second derivatives of (\[quasilikelihood\]) with respect to ${{}\beta}$. If reduced-bias estimation of $\phi$ is also required, then the first and second derivatives of (\[quasilikelihood\]) with respect to $\phi$ and those of $m_i (Y_i - \mu_i)^2/v(\mu_i) - \phi$ with respect to ${{}\beta}$ and $\phi$ are also required. The closed-form expressions for ${{}j}^{(Q)}({{}\beta}, \phi)$, ${{}e}^{(Q)}({{}\beta}, \phi)$, ${{}d}_r^{(Q)}({{}\beta}, \phi)$ and $u_r^{(Q)}({{}\beta}, \phi)$ $(r = 1, \ldots, p + 1)$ are given in Appendix \[quasi\_appendix\], and are ready to be used for the implementation of the quasi Newton-Raphson iteration in Section \[sec:implementation\] for bias reduction in the estimation of both ${{}\beta}$ and $\phi$.
For bias reduction in the estimation of only ${{}\beta}$, the empirical bias-reducing adjustment is composed using only the $({{}\beta}, {{}\beta})$ blocks of ${{}j}^{(Q)}({{}\beta}, {{}\phi})$, ${{}e}^{(Q)}({{}\beta}, \phi)$, ${{}d}_r^{(Q)}({{}\beta}, \phi)$ and ${{}u}_r^{(Q)}({{}\beta}, \phi)$ $(r = 1, \ldots, p)$ in Appendix \[quasi\_appendix\]. Then, a direct computation shows that the empirical bias-reducing adjustment is inversely proportional to $\phi$, exactly as (\[quasilikelihood\]) is. Hence, the value of the dispersion parameter is not needed for getting the RB$M$-estimate of ${{}\beta}$ with the empirical bias-reducing adjustment in quasi likelihoods and generalized linear models. In fact, similarly to (\[pen\_loglik\_glm\]), the RB$M$-estimates of ${{}\beta}$ can be obtained by maximizing the bias-reducing penalized quasi log-likelihood $$\sum_{i = 1}^n m_i \left\{ \int_{y_i}^{\mu_i} \frac{y_i - t}{v(t)} d
t - \frac{1}{2} s_i \frac{d_i}{v_i}(y_i - \mu_i) \right\} \, ,$$ with respect to ${{}\beta}$ which does not involve the dispersion parameter $\phi$. In contrast, @kosmidis+kennepagui+sartori:2019 show that the bias-reduction method in @firth:1993 requires the joint estimation of ${{}\beta}$ and $\phi$ in generalized linear models with an unknown dispersion parameter.
Over-dispersion in count responses {#sec:over_simu}
----------------------------------
In order to illustrate the impact of the empirical bias-reducing adjustment in quasi-likelihood estimation we assume that the responses $Y_1, \ldots, Y_n$ are generated independently from a negative binomial regression model with mean $\mu_i = \expect(Y_i | {{}x}_i) = \exp\{\beta_0 + \beta_1 x_{i1}+
\beta2 x_{i2}\}$ and variance $\var(Y_i | {{}x}_i) = \phi \mu_i$ $(i = 1, \ldots, n)$. This model is discussed in @mccullagh+nelder:1989 [Section 6.2.3] as a way to model count responses with dispersion that is greater than what is implied by a Poisson log-linear model. We generate two sets $\{x_{11}, \ldots, x_{n_01}\}$ and $\{x_{12}, \ldots, x_{n_02}\}$ of covariate values, independently and independent of each other from a Bernoulli distribution with probability $0.5$ and an exponential distribution with rate $2$, respectively. Conditionally on those covariate values, we simulate $N_0 = 4000$ samples of $n_0 = 20$ response values at $(\beta_0, \beta_1, \beta_2, \phi)^\top = (2, 1, -1, 6)$. Such a parameter setting implies that the conditional variances of the observed counts are 6 times greater than what would have been prescribed by a Poisson log-linear model. We also simulate $N_r= 2^r N_0$ samples of $n_r = 2^r n_0$ response values, after replicating each of the $n_0$ covariate settings $2^r$ times for $r \in \{1, \ldots, 7 \}$. This tuning of the simulation and sample sizes with $r$ guarantees that the simulation standard error when estimating the scaled bias ($n_r$ times the bias) of an estimator is $O(\sqrt{n_0} / \sqrt{N_0})$ and, hence, asymptotically bounded for any $r$. According to the theory in Section \[sec:theory\], the scaled bias for the quasi-likelihood estimator is $O(1)$ and for the reduced-bias estimator is $O(n^{-1/2})$. As a result, the bias of the reduced-bias estimator should converge to zero as $r$ increases, while that of the quasi-likelihood estimator should stabilize at a value that is not necessarily zero. Figure \[negbin\], shows the estimated scaled biases for various estimators for $\beta_0, \beta_1, \beta_2, \phi$ along with approximate $99\%$ Wald-type intervals for the scaled biases. The estimators for ${{}\beta}$ examined are the maximum likelihood estimator, the quasi-likelihood estimator, and the RB$M$-estimators in the estimation of only ${{}\beta}$, and of both ${{}\beta}$ and $\phi$. The quasi-likelihood and RB$M$-estimators of ${{}\beta}$ are examined not only for the correctly-specified variance function $V(\mu) = \mu$, but also for the incorrectly-specified variance function $V(\mu) = \mu^2$. As shown in the bottom right of Figure \[negbin\], there were very few cases where iteration (\[quasiNR\]), as we implemented it, did not converge (see the scripts for quasi likelihoods and negative binomial regression supplied in the Supporting Materials). These have been excluded from the computation of the scaled biases. The scaled biases for the estimators of $\phi$ are only provided for the correctly-specified model.
As expected from theory, the scaled biases of the maximum likelihood and quasi-likelihood estimators of ${{}\beta}$ converge to non-zero values. In contrast, the scaled biases of both versions of the RB$M$-estimator converge to zero, demonstrating that the empirical bias-reducing adjustment delivers estimators with bias of smaller order. When the variance function is correctly specified, the empirical bias-reducing adjustment delivers an estimator of $\phi$ with second-order bias. In contrast, the bias of $\hat\phi_{n - p}$ is of the same asymptotic order to that of the maximum likelihood estimator, and, hence, $\hat\phi_{n - p}$ does not have bias of smaller asymptotic order for arbitrary quasi likelihoods.
Figure \[negbin\_variance\] shows the variances of the estimators examined in Figure \[negbin\]. For correctly-specified variance function, the variances of the RB$M$-estimators are almost the same to the variances of the quasi-likelihood estimators for the sample sizes considered. Also, the variance of the RB$M$-estimator of $\phi$ is almost the same to that of $\hat\phi_{n - p}$. When the variance function is misspecified, the RB$M$-estimators do considerably better in terms of finite sample variance than the quasi-likelihood estimator for $n_0 = 20$. As expected from Section \[sec:inference\], the differences in the variances of all estimators vanish as the sample size increases.
Gaussian max-stable processes {#sec:maxstable}
=============================
Extreme climate events are becoming more frequent all over the world, with strong impact on the built environment. As a result, their statistical modelling and prediction has attracted a lot of research. Vanilla likelihood and Bayesian approaches for spatial extreme processes face challenges, because the direct generalization of the classical multivariate extreme value distributions to the spatial case is a max-stable process for which the evaluation of the likelihood becomes increasingly more intractable as the number of site locations increases [@davison+gholamrezaee:2011].
Several works have proposed the use of computationally appealing surrogates to the likelihood, like composite likelihoods, which are formed by specifying marginal or conditional densities for subsets of site locations [see @padoan+ribadet+sisson:2010; @genton+ma+sang:2011; @davison+gholamrezaee:2011; @huser+davison:2013 among others]. Nevertheless, standard bias-reduction methods, like the one in @firth:1993 and @kuk:1995 in Table \[br\_characteristics\], are either infeasible or computationally expensive because the calculation of the bias function involves either integrals with respect to the true underlying joint density or requires resampling and refitting.
An example of such a surrogate to the likelihood function is the pairwise likelihood introduced in @padoan+ribadet+sisson:2010 under a block maxima approach to the modelling of extremes. Suppose that $y_1({}s),\ldots,y_k({}s)$ with ${}s \in \{{}s_1,\ldots,{}s_L\}$, ${}s_j \in \Re^2$, are $k$ independent observations at $L$ site locations. The pairwise log-likelihood formed from the collection of $L(L-1)/2$ distinct pairs of locations is $$\label{pwl_mev}
l({{}\theta})=\sum_{i=1}^k\sum_{l>m} \log f(y_i({}s_l),y_i({}s_{m}) | {{}\theta}) \,,$$ where $$\begin{aligned}
\label{BivDensMev}
f(y_i({}s_l),y_i({}s_{m})|{{}\theta}) = \exp\left\{-\frac{\Phi(w_{lm})}{y_i({}s_l)}-\frac{\Phi(v_{lm})}{y_i({}s_{m})}\right\} \left[ \frac{v_{lm}\phi(w_{lm})}{a^2_{lm}y^2_i({}s_l)y_i({}s_{m})} + \frac{w_{lm}\phi(v_{lm})}{a^2_{lm}y^2_i({}s_{m})y_i({}s_l)} \right. & \\ \notag
+ \left. \left\{ \frac{\Phi(w_{lm})}{y^2_i({}s_l)} + \frac{\phi(w_{lm})}{a_{lm}y^2_i({}s_l)} - \frac{\phi(v_{lm})}{a_{lm}y_i({}s_l)y_i({}s_{m})} \right\}
\left\{ \frac{\Phi(v_{lm})}{y^2_i({}s_{m})} + \frac{\phi(v_{lm})}{a_{lm}y^2_i({}s_{m})} - \frac{\phi(w_{lm})}{a_{lm}y_i({}s_l)y_i({}s_{m})} \right\} \right],\end{aligned}$$ is the joint density of $Y_i({}s_l)$ and $Y_i({}s_{m})$ $(l, m = 1, \ldots, L; l \ne m)$, with $\Phi(\cdot)$ and $\phi(\cdot)$ the distribution and density function of the standard normal distribution, respectively. In the above expression, $a_{lm} \equiv a_{lm}({{}\theta}) = \{({}s_l-{}s_m)^\top{{}\Sigma}^{-1}({{}\theta})({}s_l-{}s_m)\}^{1/2}$, $w_{lm}\equiv w_{lm}({{}\theta}) = a_{lm}/2+\log\{y_i({}s_l)/y_i({}s_{m})\}/a_{lm}$, and $v_{lm} = a_{lm}-w_{lm}$. The parameter vector is ${{}\theta}=(\sigma^2_{1}, \sigma^2_{2}, \sigma^2_{12})^\top$, which collects the distinct elements of the $2\times 2$ covariance matrix ${{}\Sigma}({{}\theta})$ that governs the spatial dependence, and has diagonal elements $\sigma^2_1$ and $\sigma^2_2$, with $\sigma_{12}^2$ in the off-diagonals.
The maximizer of is the maximum pairwise likelihood estimator $\hat{{}\theta}$, while the RB$M$-estimator $\tilde{{}\theta}$ results by maximizing the penalized version of $\l({}\theta)$ in (\[penalized\_likelihood\]). Closed-form expressions for ${{}j}({{}\theta})$ and ${{}e}({{}\theta})$ are given in Section S2 of the Supporting Materials.
Simulations are run by generating independent observations $y_1({}s),\ldots,y_k({}s)$ from a Gaussian max-stable process observed at $L=50$ site locations. The locations are generated uniformly on a $[0, 40] \times [0, 40]$ region. We consider sample sizes $k \in \{10, 20, 40, 80, 160\}$ with corresponding number of simulations equal to $4000 k$, and true parameter settings ${{}\theta} = (2000, 3000, 1500)^\top$ and ${{}\theta} = (20, 30, 15)^\top$, imposing strong and weak spatial dependence, respectively. These parameter values correspond to ${{}\Sigma}_4$ and ${{}\Sigma}_5$ in Table 1 of @padoan+ribadet+sisson:2010. In Figure \[MEV\_plots\] (top panel), we show the simulation-based estimates of $\log |\expect_G\{\hat{{}\theta}-\bar{{}\theta}\}|$ and $\log |\expect_G\{\tilde{{}\theta}-\bar{{}\theta}\}|$ as functions of $\log n$, where $n = k$. These curves have roughly slopes $-1$, and between $-3/2$ and $-2$, respectively as expected by the asymptotic theory in Section \[sec:theory\], demonstrating the reduction of the bias that $\tilde{{}\theta}$ delivers. Furthermore, $\tilde{{}\theta}$ appears to have smaller finite-sample mean squared error, and hence smaller variance than the maximum pairwise likelihood estimator $\hat{{}\theta}$; see bottom panel in Figure \[MEV\_plots\]. The simulation results provide evidence for the superiority of the RB$M$-estimator. The differences between $\hat{{}\theta}$ and $\tilde{{}\theta}$ will tend to be significant in the small to moderate samples that are typically observed in settings involving block-maxima.
[![Results of the simulation study for Gaussian max-stable process: right and left panels refer to $\Sigma_4$ and $\Sigma_5$, respectively. Summaries for $\hat{{}\theta}$ and $\tilde{{}\theta}$ are depicted in black and gray, respectively. The top panels show simulation-based estimates of the logarithm of the absolute bias of $\hat{{}\theta}$ and $\tilde{{}\theta}$ (solid). The dashed line is a line with slope $-1$ corresponding to the theoretical rate of the bias of $\hat{{}\theta}$. The shadowed region is defined by lines with slopes $-3/2$ and $-2$, which correspond to the theoretical rate of the bias of $\tilde{{}\theta}$. The bottom panels show the simulation-based estimates of the root mean squared error of $\hat{{}\theta}$ and $\tilde{{}\theta}$.[]{data-label="MEV_plots"}](Figures/RATE_Sigma4 "fig:")]{} [![Results of the simulation study for Gaussian max-stable process: right and left panels refer to $\Sigma_4$ and $\Sigma_5$, respectively. Summaries for $\hat{{}\theta}$ and $\tilde{{}\theta}$ are depicted in black and gray, respectively. The top panels show simulation-based estimates of the logarithm of the absolute bias of $\hat{{}\theta}$ and $\tilde{{}\theta}$ (solid). The dashed line is a line with slope $-1$ corresponding to the theoretical rate of the bias of $\hat{{}\theta}$. The shadowed region is defined by lines with slopes $-3/2$ and $-2$, which correspond to the theoretical rate of the bias of $\tilde{{}\theta}$. The bottom panels show the simulation-based estimates of the root mean squared error of $\hat{{}\theta}$ and $\tilde{{}\theta}$.[]{data-label="MEV_plots"}](Figures/RATE_Sigma5 "fig:")]{} [![Results of the simulation study for Gaussian max-stable process: right and left panels refer to $\Sigma_4$ and $\Sigma_5$, respectively. Summaries for $\hat{{}\theta}$ and $\tilde{{}\theta}$ are depicted in black and gray, respectively. The top panels show simulation-based estimates of the logarithm of the absolute bias of $\hat{{}\theta}$ and $\tilde{{}\theta}$ (solid). The dashed line is a line with slope $-1$ corresponding to the theoretical rate of the bias of $\hat{{}\theta}$. The shadowed region is defined by lines with slopes $-3/2$ and $-2$, which correspond to the theoretical rate of the bias of $\tilde{{}\theta}$. The bottom panels show the simulation-based estimates of the root mean squared error of $\hat{{}\theta}$ and $\tilde{{}\theta}$.[]{data-label="MEV_plots"}](Figures/VAR_Sigma4 "fig:")]{} [![Results of the simulation study for Gaussian max-stable process: right and left panels refer to $\Sigma_4$ and $\Sigma_5$, respectively. Summaries for $\hat{{}\theta}$ and $\tilde{{}\theta}$ are depicted in black and gray, respectively. The top panels show simulation-based estimates of the logarithm of the absolute bias of $\hat{{}\theta}$ and $\tilde{{}\theta}$ (solid). The dashed line is a line with slope $-1$ corresponding to the theoretical rate of the bias of $\hat{{}\theta}$. The shadowed region is defined by lines with slopes $-3/2$ and $-2$, which correspond to the theoretical rate of the bias of $\tilde{{}\theta}$. The bottom panels show the simulation-based estimates of the root mean squared error of $\hat{{}\theta}$ and $\tilde{{}\theta}$.[]{data-label="MEV_plots"}](Figures/VAR_Sigma5 "fig:")]{}
Migraine severity study {#autoregressive_probit}
=======================
We consider the data from a longitudinal study on the determinants of migraine severity, as has been analyzed in @varin+czado:2010. The study involved $n = 133$ Canadian patients that were asked to rate headache pain levels on an ordinal scale and record them at four pre-specified daily occasions. Potential explanatory variables, such as subject specific information and meteorological data, were collected with the pain ratings.
Suitable regression models for longitudinal ordinal responses that account for subject-specific variability are probit models with random effects. Let $Y_{ij}$ be the pain rating by the $i$th individual at time $t_{ij}$, and let ${{}x}_{ij}$ be the $r$-dimensional vector containing explanatory variables with $x_{i1} = 1$ $(i = 1, \ldots, n; j = 1, \ldots, n_i)$. Suppose that $Y_{ij}$ takes values from $\{1, \ldots, h\}$ and that it results from the discretization of a continuous random variable $Z_{ij}$ according to the rule $Y_{ij} = y_{ij} \iff \alpha_{y_{ij}-1} < Z_{ij} \leq
\alpha_{y_{ij}}$, where ${{}\alpha}=(\alpha_0,\alpha_1,\ldots,\alpha_h)^\top$ is a vector of cut-points with $\alpha_0<\alpha_1 \ldots <\alpha_h$, $\alpha_0 = -\infty$, $\alpha_1 = 0$, and $\alpha_h = \infty$, where $Z_{ij} = {{}x}^\top_{ij}{{}\beta} + U_i + \epsilon_{ij}$. The $r$-dimensional vector ${{}\beta}$ collects the regression parameters, $U_i$ are random effects, and $\epsilon_{ij}$ are random errors. Under normality assumptions on both the random effects and errors, the joint distribution of $(Z_{i1}, \ldots, Z_{in_i})$ is multivariate normal. The computation of the log-likelihood function about ${{}\alpha}$, ${{}\beta}$ and the parameters of the distribution of the random effects and errors is typically impractical because it requires the approximation of $n$ integrals, with the $i$th integral being of dimension $n_i$.
@varin+czado:2010 have overcome those computational challenges by resorting to a pairwise likelihood whose specification requires the joint distribution for the pairs $(Y_{ij}, Y_{ik})$ $(j, k = 1, \ldots, n_i; j\neq k)$ and, in turn, the numerical approximation of bivariate integrals only.
Under the assumption that $U_i \sim N(0, \sigma^2)$ and $\epsilon_{ij} \sim N(0, 1)$, with $\cov(U_i, U_j)=0$, $\cov(\epsilon_{ik}, \epsilon_{il}) = 0$, and $\cov(U_{i}, \epsilon_{ik}) = 0$, the pairwise log-likelihood about ${{}\theta}=({{}\alpha}^\top, {{}\beta}^\top, {{}\omega}^\top,
\sigma^2)^\top$ that is introduced in @varin+czado:2010 is $$\label{pllik_varin_czado}
l({{}\theta}) = \sum_{i=1}^n\sum_{j>k}^{n_i}\log P(Y_{ij}=y_{ij}, Y_{ik}=y_{ik} ; {{}\theta}) \mathbb{I}_{[-q,q]}(t_{ij} - t_{ik}),$$ where $\mathbb{I}_{[-q,q]}(a)$ takes value $1$ if $a \in [-q, q]$ and $0$, otherwise, and $$\label{int_varin_czado}
P(Y_{ij}=y_{ij}, Y_{ik}=y_{ik}; {{}\theta})=\int^{\tilde\alpha_{y_{ij}}}_{\tilde\alpha_{y_{ij}-1}}\int^{\tilde\alpha_{y_{ik}}}_{\tilde\alpha_{y_{ik}-1}}\phi_2(u,v; \rho_{ijk})du dv \, ,$$ with $\tilde\alpha_{y_{ij}} = (\alpha_{y_{ij}} - {{}x}^\top_{ij}
{{}\beta})(1 + \sigma^2)^{-1/2}$ and $\phi_2(\cdot; \rho_{ijk})$ denoting the bivariate normal density function with zero means and correlation $\rho_{ijk} = (1 + \sigma^2)^{-1}\omega_i^{|t_{ij}-t_{ik}|}$, allowing for subject heterogeneity in the correlation functions. Computational speed and statistical efficiency gains [see, for example @joe+lee:2009; @bevilacqua+gaetan+mateu+etal:2012] can be achieved by discarding too distant pairs through a suitable choice of $q$.
@varin+czado:2010 analyzed the migraine data by using and derived the associated inferential tools. We repeat their analysis using the bias-reducing penalized version of and compare the results; the derivation and closed-form expressions for the matrices ${}e({}\theta)$ and ${}j({}\theta)$ in (\[penalized\_likelihood\]) are given in Section S3 of the Supporting Materials.
The five explanatory variables we consider are university degree status, analgesics intake, change of atmospheric pressure between consecutive days, humidity, and windchill. A brief description of those variables and the response is in Table \[table:migraine\]. We set a base model that has the university degree status and analgesics intake as explanatory variables, and use the CLIC at the maximum pairwise likelihood and the RB$M$-estimates to select the best out of the $8$ possible models that result by including or not the possible combinations of the remaining $3$ explanatory variables. As in @varin+czado:2010, we use $q = 12$ and set the autocorrelation structure in all models to be $\rho_{ijk} = (1 + \sigma^2)^{-1}\omega_u^{|t_{ij}-t_{ik}|}$, with $u = 0$ if the $i$th patient does not make use of analgesics, and $u = 1$, otherwise.
We define the bias-reducing penalized pairwise log-likelihood for ${{}\xi}={{}\xi({}\theta)} = ({{}\gamma}^\top, {{}\beta}^\top,
{{}\tau}^\top, \lambda)^\top$, where $\tau_u = \log\{\omega_u/(1 - \omega_u)\}$ $(u = 0, 1)$, $\lambda = \log\sigma^2$, $\gamma_0=0$, and $\gamma_m = \log(\alpha_{m+1}-\alpha_{m})$ $(m = 1, \ldots, h-1)$. In this way, all parameters take values on the real line avoiding numerical issues during maximization. By the equivariance properties of the maximum pairwise likelihood estimator, $\hat{{}\xi}={}\xi(\hat{{}\theta})$, where $\hat{{}\theta}$ is the maximizer of . Note here that back-transforming the RB$M$-estimators ${{}\gamma}$, ${{}\tau}$ and $\lambda$ does not result in RB$M$ estimators for ${{}\alpha}$, ${{}\omega}$, $\sigma^2$ because the bias-reducing penalty is not parametrization invariant under non-linear transformations of the parameters. Nevertheless, the RB$M$-estimators of the parameters of interest ${{}\beta}$ will have improved bias in either parameterization.
Table \[table:migraine clic\] reports the CLIC information criteria for models estimated using and its penalized version. Table \[table:migraine clic\] also reports the CLIC weights, defined as $w_k = \exp(-\Delta_k)/\sum_{i=1}^8 \exp(-\Delta_i)$, $\Delta_k=(\text{CLIC}_k-\min_c \text{CLIC}_c)/2$. According to the CLIC weights for models estimated by maximum pairwise likelihood, the best and second best models are the base model with change of atmospheric pressure, and the base model, respectively. We arrive at the same conclusion by using the CLIC weights at the RB$M$-estimates, but there is stronger evidence in favour of the base model with the CLIC weight increasing from $0.28$ to $0.38$.
------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------
Variable Categories
\[.5ex\] Headache pain rating (response) no/mild/moderate/painful/severe/intense
University degree yes/no
Analgesics intake yes/no
Change of atmospheric pressure high to low/low to high/unchanged
Humidity $<$60%/60%-80%/$>$80%
Windchill $(-50^\circ\text{C},-10^\circ\text{C})$/$(-10^\circ\text{C},0^\circ\text{C})$/$(0^\circ\text{C},10^\circ\text{C})$/$(10^\circ\text{C},30^\circ\text{C})$
------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------
: Variables in the migraine data. The response variable was recorded on an ordinal scale with six categories whose full length description may be found in @varin+czado:2010 [Table 1]. The correspondence between the levels of the variable referring to the change of atmospheric pressure and the actual change in pressure are from high ($>$1013 hPa) to low pressure ($\leq$ 1013 hPa), from low to high pressure, and unchanged level of pressure (either from low to low or from high to high).
\[table:migraine\]
-- -------- ---------- ----------- --------- ------- --------- -------
Change Humidity Windchill CLIC $w_k$ CLIC $w_k$
- - - 5919.17 0.28 5940.74 0.38
\* - - 5918.26 0.44 5940.40 0.45
- \* - 5921.11 0.11 5943.80 0.08
- - \* 5925.00 0.02 5949.91 0.00
\* \* - 5920.60 0.14 5943.87 0.08
\* - \* 5924.45 0.02 5950.01 0.00
- \* \* 5927.40 0.00 5953.45 0.00
\* \* \* 5927.09 0.01 5953.80 0.00
-- -------- ---------- ----------- --------- ------- --------- -------
: CLIC statistics and CLIC weights for the models considered for the migraine data set, with $q=12$. CLIC for pairwise likelihood refers to , whereas CLIC for the bias-reducing penalized pairwise likelihood refers to once multiplied by $-2$. The symbol \* indicates that a variable is included in the model and $w_k$ refers to the CLIC weight of each model. CLIC values for pairwise likelihood displayed here differ slightly from the ones in @varin+czado:2010 [Table 4] because in our analysis the penalty term in the CLIC has been evaluated using the actual hessian of rather than the approximation used in @varin+czado:2010 [equation 3.1].
\[table:migraine clic\]
Estimates from the best two models according to CLIC weights are displayed in Table \[table:migraine est\]. While the maximum pairwise likelihood estimator is equivariant, the corresponding estimated standard errors are not, so that hypotheses tests based on Wald-type statistics for the main effects refer to the ${{}\xi}$ parametrization. However, the differences between the estimated standard errors for ${{}\beta}$ in the two parametrizations are small and the conclusions are the same as the ones stated in @varin+czado:2010. Furthermore, the difference between the maximum pairwise likelihood and RB$M$-estimates of ${{}\beta}$ are small, indicating that estimation bias has not been a major concern in that case study. In particular, analgesics intake is related to higher pain level ratings, the university degree acts in the opposite direction, and the decrease of atmospheric pressure is associated with worse pain ratings. Overall, the estimated standard errors for the RB$M$-estimator $\tilde{{}\xi}$ are only slightly smaller than those for $\hat{{}\xi}$.
---------------------- ---------- ------- ---------- ------- -- ---------- ------- ---------- -------
$\gamma_2$ $-$0.530 0.088 $-$0.529 0.088 $-$0.534 0.088 $-$0.533 0.088
$\gamma_3$ $-$0.602 0.069 $-$0.601 0.069 $-$0.602 0.069 $-$0.602 0.069
$\gamma_4$ $-$0.430 0.061 $-$0.429 0.061 $-$0.428 0.060 $-$0.427 0.061
$\gamma_5$ $-$0.330 0.092 $-$0.330 0.092 $-$0.329 0.092 $-$0.329 0.093
Intercept $-$0.474 0.230 $-$0.521 0.230 $-$0.461 0.230 $-$0.510 0.231
University $-$0.523 0.156 $-$0.524 0.158 $-$0.537 0.156 $-$0.537 0.158
Analgesics 0.558 0.221 0.560 0.223 0.544 0.220 0.546 0.222
Change - unchanged 0.031 0.046 0.033 0.046
Change - high to low 0.164 0.051 0.164 0.052
$\tau_0$ 0.886 0.322 0.891 0.329 0.915 0.306 0.921 0.312
$\tau_1$ 1.253 0.110 1.252 0.110 1.250 0.106 1.249 0.106
$\tau_1-\tau_0$ 0.368 0.347 0.361 0.353 0.335 0.334 0.327 0.340
$\lambda$ $-$0.567 0.196 $-$0.566 0.198 $-$0.541 0.189 $-$0.541 0.191
---------------------- ---------- ------- ---------- ------- -- ---------- ------- ---------- -------
: Estimates and estimated standard errors (s.e.) from maximum pairwise likelihood ($\hat{{}\xi}$) and from maximum bias-reducing penalized pairwise likelihood ($\tilde{{}\xi}$) for the best two models, according to the CLIC values in Table \[table:migraine clic\]. Columns 1-2 and 5-6 refer to base models, whereas 3-4 and 7-8 to the models including the change of atmospheric pressure between consecutive days. The baseline categories for university degree status and analgesics intake is “no”, and for the variable change of atmospheric pressure between consecutive days is “low to high”.
\[table:migraine est\]
Discussion and further work {#sec:discussion}
===========================
We have developed a novel, general framework for the asymptotic reduction of the bias of general $M$-estimators from sufficiently smooth, unbiased estimating functions. Bias reduction is achieved by the adjustment of the estimating functions by quantities that are bounded in probability, and depend only on the first and second derivatives of contributions to the estimating functions. The RB$M$-estimates can be computed using the general quasi Newton-Raphson iteration in Section \[sec:implementation\] that again requires only the contributions to the estimating functions and the first two derivatives of those.
The resulting estimators have the same asymptotic distribution, and, hence, they are asymptotically as efficient as the original, unadjusted $M$-estimators. As detailed in Section \[sec:inference\], uncertainty quantification can be carried out using the empirical estimate $\hat{{{}V}}({{}\theta})$ of the variance-covariance matrix of that asymptotic distribution. The expression for $\hat{{{}V}}({{}\theta})$ appears in expression (\[general\_empirical\_adjustment\_matrix\]) for the empirical adjustment, and, hence, is readily available at the last iteration of the quasi Newton-Raphson fitting procedure. Inferences can be constructed using the Wald-type and generalized score pivots in expression (\[pivots\]).
An alternative estimator with $o(n^{-1})$ bias in general $M$-estimation problems under the assumptions of Section \[assumptions\] is $$\label{bias_correction}
{{}\theta}^\dagger = \hat{{}\theta} + {{}j}(\hat{{}\theta})^{-1} {{}A}(\hat{{}\theta}) \, ,$$ where the $r$th component of ${{}A}({{}\theta})$ is as in expression (\[general\_empirical\_adjustment\_matrix\]) for the empirical bias-reducing adjustments. According to the classification of bias-reduction methods in @kosmidis:2014, the estimator (\[bias\_correction\]) defines an explicit bias-reduction method. The value of ${{}\theta}^\dagger$ in (\[bias\_correction\]) can be computed with a single step of the quasi Newton-Raphson procedure of Section \[sec:implementation\] with $a_1 = 1$, starting at the $M$-estimator $\hat{{}\theta}$. The choice between using ${{}\theta}^\dagger$ and the fully iterated version $\tilde{{}\theta}$ that has been introduced earlier in this paper is application-dependent. For example, for the estimation of a ratio of two means in Example \[ratio\_estimation\], $\theta^\dagger = \hat\theta (1 - s_{XX} / s_{X}^2) + s_{XY}/ s_X^2$, which, like $\hat\theta = s_Y/s_X$ and unlike the fully-iterated $\tilde\theta$, is not robust to small values of $s_X$. Furthermore, if $M$-estimation is through the maximization of an objective function, then $\tilde{{}\theta}$ is the maximizer of the penalized objective (\[penalized\_likelihood\]). As discussed in Section \[sec:implementation\], the maximization of (\[penalized\_likelihood\]) can be performed using a general numerical optimization routine that operates by numerically approximating the gradient of (\[penalized\_likelihood\]) and requires only the estimating function contributions and their first derivatives. In contrast, by its definition, ${{}\theta}^\dagger$ requires also the second derivatives of the estimating function contributions.
A derivation similar to that in Section \[derivation\] and Section \[sec:penalty\] can be used to show that another estimator with $o(n^{-1})$ bias results by the maximization of the bias-reducing penalized objective $$\label{penalized_likelihood2}
l({{}\theta}) + \frac{1}{2} \log \det \left\{{{}j}({{}\theta}) \right\} - \frac{1}{2} \log \det\left\{ {{}e}({{}\theta}) \right\} \, .$$ The resulting estimator has, again, the same asymptotic distribution to the original $M$-estimators (see Section \[sec:inference\]), however, it is, typically, less attractive than $\tilde{{}\theta}$ from a computational point of view, because of the need to compute the logarithms of two determinants.
We also showed that the bias-reducing penalized objective (\[penalized\_likelihood\]) closely relates to model selection procedures using the Kullback-Leibler divergence. Such model selection procedures include the use of the TIC for maximum likelihood estimation, and the CLIC for maximum composite likelihood estimation. The functions of the parameters and the data that are used for bias reduction and model selection differ only by a known scalar constant. These results establish, for the first time, a close relation between bias reduction in estimation and model selection.
Furthermore, TIC and CLIC are still consistent information criteria when evaluated at the RB$M$-estimates, and, hence, their value is readily available once the bias-reducing penalized likelihood has been maximized. The only difference with the standard versions of TIC and CLIC is that when evaluated at the RB$M$-estimates these criteria are only asymptotically invariant to non-linear transformations of the parameters. The same justification we provided in Section \[sec:model\_selection\] for the use of information criteria at the reduced-bias estimates can be used to justify the use of information criteria at estimates arising from the additive adjustment of estimating functions by alternative $O_p(1)$ quantities, like the median reduced-bias estimates discussed in @kenne+salvan+sartori:2017 and @kosmidis+kennepagui+sartori:2019, and the mean reduced-bias estimators in @firth:1993.
If there is only $k = 1$ multivariate observation from the underlying process, application-dependent conditions need to be used for the appropriate definition of ${{}e}({{}\theta})$ in the penalized objective function in (\[penalized\_likelihood\]) or of ${{}e}({{}\theta})$ and ${{}d}_r({{}\theta})$ in the empirical bias-reducing adjustment (\[general\_empirical\_adjustment\_matrix\]). For example, in the context of time series and spatial data that do not seriously depart from the condition of stationarity, one can consider window sub-sampling for the definition of ${{}e}({{}\theta})$ and ${{}d}_r({{}\theta})$ [see, for example @carlstein:1986; @heagerty+lumley:2000 for definitions and guidance on the choice of the window size].
@lunardon:2018 showed that bias reduction in maximum likelihood estimation using the adjustments in @firth:1993 can be particularly effective for inference about a low-dimensional parameter of interest in the presence of high-dimensional nuisance parameters, while providing, at the same time, improved estimates of the nuisance parameters. Current research investigates the performance of the reduced-bias estimator from empirically adjusted estimating functions for general $M$-estimation in stratified settings, extending the arguments and optimality results in @lunardon:2018 when maximum composite likelihood and other $M$-estimators are employed.
Supporting materials
====================
The supporting materials include computer code to fully reproduce all numerical results and figures in the paper. The organization of the computer code is detailed in Section S1 of the supporting materials document. Section S2 and Section S3 of that document derive the mathematical expressions that are used in Section \[sec:maxstable\] and Section \[autoregressive\_probit\], respectively.
Acknowledgements
================
Ioannis Kosmidis is supported by The Alan Turing Institute under the EPSRC grant EP/N510129/1.
Appendix
========
For notational simplicity, the dependence of the various quantities below on ${{}\beta}$ and/or $\phi$ is suppressed.
Expressions for the bias-reducing penalty for generalized linear models {#glm_appendix}
-----------------------------------------------------------------------
$${{}j}^{(G)} = \left[
\begin{array}{cc}
{{}j}_{{{}\beta}{{}\beta}} & {{}j}_{{{}\beta}\phi} \\
{{}j}_{{{}\beta}\phi}^\top & {{}j}_{\phi\phi}^{(G)} \\
\end{array}
\right] \quad \text{and} \quad
{{}e}^{(G)} = \left[
\begin{array}{cc}
{{}e}_{{{}\beta}{{}\beta}} & {{}e}_{{{}\beta}\phi}^{(\text{G})} \\
\left\{{{}e}_{{{}\beta}\phi}^{(\text{G})}\right\}^\top & {{}e}_{\phi\phi}^{(G)} \\
\end{array}
\right] \, ,$$
where $$\begin{aligned}
{{}j}_{{{}\beta}{{}\beta}} & = \frac{1}{\phi} {{}X}^\top {{}Q} {{}X} \, , &
{{}j}_{\phi\phi}^{(G)} & = \frac{1}{\phi^3} {{}1}_n^\top ({{}R} - {{}A}') {{}1}_n + \frac{1}{2\phi^4} {{}1}_n^\top {{}A}'' {{}1}_n \, , &
{{}j}_{{{}\beta}\phi} & = \frac{1}{\phi^2} {{}X}^\top \tilde{{{}W}} {{}1}_n \, , \\
{{}e}_{{{}\beta}{{}\beta}} & = \frac{1}{\phi^2} {{}X}^\top \tilde{{{}W}}^2 {{}X} \, , &
{{}e}_{\phi\phi}^{(G)} & = \frac{1}{4 \phi^4} {{}1}_n^\top ({{}R} - {{}A}')^2 {{}1}_n \, , &
{{}e}_{{{}\beta}\phi}^{(G)} & = \frac{1}{2\phi^3} {{}X}^\top \tilde{{{}W}} ({{}R} - {{}A}') {{}1}_n \, .\end{aligned}$$ In the above expressions, ${{}1}_n$ is an $n$-vector of ones. The $n \times n$ diagonal matrices ${{}Q}$, $\tilde{{{}W}}$, ${{}R}$, ${{}A}'$, ${{}A}''$ have $i$th diagonal element $q_i = b_i d_i - b_i' (y_i - \mu_i)$, $\tilde{g}_i = b_i (y_i - \mu_i)$, $r_i = -2 m_i (y_i \theta_i - b(\theta_i) - c_1(y_i))$ (deviance residual), $a_i' = m_i a'(-m_1/\phi)$, $a_i'' = m_i^2 a''(-m_1/\phi)$, respectively, where $b_i = m_i d_i / v_i$, $b_i' = m_i (d_i'/v_i - d_i^2 v_i'/v_i^2)$, $d_i' = d^2 \mu_i / d\eta_i^2$, $v_i' = d v_i /d \mu_i$ $(i = 1, \ldots, n)$, and $a'(u) = d a(u) / d u$, $a''(u) = d^2 a(u) / d u^2$.
Expressions for the empirical bias-reducing adjustment for quasi likelihoods {#quasi_appendix}
----------------------------------------------------------------------------
$${{}j} = \left[
\begin{array}{cc}
{{}j}_{{{}\beta}{{}\beta}} & {{}j}_{{{}\beta}\phi} \\
{{}j}_{\phi{{}\beta}}^{(\text{Q})} & {{}j}_{\phi\phi}^{(Q)} \\
\end{array}
\right] \quad \text{and} \quad
{{}e} = \left[
\begin{array}{cc}
{{}e}_{{{}\beta}{{}\beta}} & {{}e}_{{{}\beta}\phi}^{(\text{Q})} \\
\left\{{{}e}_{{{}\beta}\phi}^{(\text{Q})}\right\}^\top & {{}e}_{\phi\phi}^{(Q)} \\
\end{array}
\right] \, ,$$
where $$\begin{aligned}
{{}j}_{\phi\phi}^{(Q)} & = n \, , &
{{}j}_{\phi{{}\beta}}^{(Q)} & = {{}1}_n^\top {{}F} {{}X} \, , \\
{{}e}_{\phi\phi}^{(Q)} & = {{}1}_n^\top ({{}K} - \phi {{}I}_n)^2 {{}1}_n \, , &
{{}e}_{{{}\beta}\phi}^{(Q)} & = \frac{1}{\phi} {{}X}^\top \tilde{{{}W}} ({{}K} - \phi {{}I}_n) {{}1}_n \, .\end{aligned}$$ In the above expressions, ${{}I}_n$ is the $n \times n$ identity matrix. The $n \times n$ diagonal matrices ${{}F}$ and ${{}K}$ have $i$th diagonal element $f_i = 2 c_i d_i (y_i - \mu_i) - c_i' (y_i - \mu_i)^2$ and $k_i = c_i (y_i - \mu_i)^2$, respectively, with $c_i = m_i / v_i$ and $c_i' = - m_i v_i' d_i / v_i^2$ $(i = 1, \ldots, n)$.
In addition, $${{}u}_r({{}\beta}, \phi) = \left[
\begin{array}{cc}
{{}u}_{r, {{}\beta}{{}\beta}} & {{}u}_{r, {{}\beta}\phi} \\
{{}u}_{r, {{}\beta}\phi}^\top & {{}u}_{r, \phi\phi} \\
\end{array}
\right] \quad \text{and} \quad
{{}u}_{p+1}({{}\beta}, \phi) = \left[
\begin{array}{cc}
{{}u}_{p+1, {{}\beta}{{}\beta}} & {{}u}_{p+1, {{}\beta}\phi} \\
{{}u}_{p+1, {{}\beta}\phi}^\top & {{}u}_{p+1, \phi\phi} \\
\end{array}
\right] \quad \quad (r = 1, \ldots, p) \, ,$$ where $$\begin{aligned}
{{}u}_{r, {{}\beta}{{}\beta}} & = -\frac{1}{\phi} {{}X}^\top {{}Q}' {{}T}_r {{}X} \, &
{{}u}_{r, \phi\phi} & = \frac{2}{\phi^3} {{}1}_n^\top {{}T}_r \tilde{{{}W}} {{}1}_n \, , &
{{}u}_{r, {{}\beta}\phi} & = \frac{1}{\phi^2} {{}X}^\top {{}Q} {{}T}_r {{}1}_n \, , \\
{{}u}_{p + 1, {{}\beta}{{}\beta}} & = {{}X}^\top {{}S} {{}X} \,, &
{{}u}_{p + 1, \phi\phi} & = 0 \, , &
{{}u}_{p + 1, {{}\beta}\phi} & = {{}0}_p \, . \\ \end{aligned}$$ The $n \times n$ diagonal matrices ${{}Q}'$, ${{}S}$, and ${{}T}_r$ have $i$th diagonal element $q_i' = 2 b_i' d_i + b_i d_i' - b_i'' (y_i - \mu_i)$, $s_i = c_i'' (y_i - \mu_i)^2 - 4 c_i' d_i (y_i - \mu_i) - 2 c_i d_i'
(y_i - \mu_i) + 2 c_i d_i^2$, and $x_{ir}$, respectively $(i = 1, \ldots, n)$, with $c_i'' = - m_i \left\{ v_i'' d_i^2 / v_i^2 - v_i' d_i'/v_i^2 - 2 d_i^2
(v_i')^2 / v_i^3 \right\}$ and $b_i'' = m_i (d_i'' v_i - 3 d_i d_i' v_i' - d_i^3 v_i'')/v_i^2 + m_i
d_i^3 (v_i')^2/v_i^3$.
Finally, $${{}d}_r({{}\beta}, \phi) = \left[
\begin{array}{cc}
{{}d}_{r, {{}\beta}{{}\beta}} & {{}d}_{r, {{}\beta}\phi} \\
{{}d}_{r, \phi{{}\beta}} & {{}d}_{r, \phi\phi} \\
\end{array}
\right] \quad \text{and} \quad
{{}d}_{p+1}({{}\beta}, \phi) = \left[
\begin{array}{cc}
{{}d}_{p+1, {{}\beta}{{}\beta}} & {{}d}_{p+1, {{}\beta}\phi} \\
{{}d}_{p+1, \phi{{}\beta}}^\top & {{}d}_{p+1, \phi\phi} \\
\end{array}
\right] \quad \quad (r = 1, \ldots, p) \, ,$$ where $$\begin{aligned}
{{}d}_{r, {{}\beta}{{}\beta}} & = \frac{1}{\phi^2} {{}X}^\top {{}Q} {{}T}_r \tilde{{{}W}} {{}X} \, ,&
{{}d}_{r, \phi\phi} & = \frac{1}{\phi^2} {{}1}_n^\top \tilde{{{}W}} {{}T}_r ({{}K} - \phi {{}I}_n) {{}1}_n \, , \\
{{}d}_{r, {{}\beta}\phi} & = \frac{1}{\phi} {{}X}^\top {{}Q} {{}T}_r ({{}K} - \phi {{}I}_n) {{}1}_n \, , &
{{}d}_{r, \phi{{}\beta}} & = \frac{1}{\phi^3} {{}1}_n ^\top \tilde{{{}W}}^2 {{}T}_r {{}X} \, , \\
{{}d}_{p + 1, {{}\beta}{{}\beta}} & = \frac{1}{\phi} {{}X}^\top {{}F} \tilde{{{}W}} {{}X} \,, &
{{}d}_{p + 1, \phi\phi} & = {{}1}_n^\top ({{}K} - \phi {{}I}_n) {{}1}_n \, , \\
{{}d}_{p + 1, {{}\beta}\phi} & = {{}X}^\top {{}F} ( {{}K} - \phi {{}I}_n) {{}1}_n\, , &
{{}d}_{p + 1, \phi{{}\beta}} & = \frac{1}{\phi} {{}1}_n^\top \tilde{{{}W}} {{}X} \, . \end{aligned}$$
| 0 |
---
abstract: 'We calculate the gauge invariant cumulants (and moments) associated with the Zak phase in the Rice-Mele model. We reconstruct the underlying probability distribution by maximizing the information entropy and applying the moments as constraints. When the Wannier functions are localized within one unit cell, the probability distribution so obtained corresponds to that of the Wannier function. We show that in the fully dimerized limit the magnitude of the moments are all equal. In this limit, if the on-site interaction is decreased towards zero, the distribution shifts towards the midpoint of the unit cell, but the overall shape of the distribution remains the same. Away from this limit, if alternate hoppings are finite, and the on-site interaction is decreased, the distribution also shifts towards the midpoint of the unit cell, but it does this by changing shape, by becoming asymmetric around the maximum, as well as by shifting. We also follow the probability distribution of the polarization in cycles around the topologically non-trivial point of the model. The distribution moves across to the next unit cell, its shape distorting considerably in the process. If the radius of the cycle is large, the shift of the distribution is accompanied by large variations in the maximum.'
author:
- 'M. Yahyavi and B. Hetényi'
title: 'Reconstruction of the polarization distribution of the Rice-Mele model'
---
\[sec:intro\]
Introduction
============
One way to derive the Berry phase [@Berry84; @Pancharatnam56; @Xiao10] is to form a product of scalar products between quantum states at different points of the space of external parameters (Bargmann invariant [@Bargmann64]) and to take the continuous limit along a cyclic curve. An extension [@Souza00; @Hetenyi14] of this derivation, keeping higher order terms, leads to gauge invariant cumulants (GIC) associated with the Berry phase. One is lead to ask two questions. The GICs give information of the distribution of what physical quantity? Can one reconstruct the probability distribution from the GICs?
The answer to the first question depends on the physical context in which the Berry phase is defined. In a crystalline solid the Berry phase (or Zak phase [@Zak89], in this context) corresponds to the macroscopic polarization. Zak showed [@Zak89] that the phase itself corresponds to the expectation value of the position over a Wannier function. For the higher order GICs Souza, Wilkens and Martin [@Souza00] showed that they only correspond to the cumulants of the distribution of the position associated with Wannier functions, if the Wannier functions themselves are localized within the unit cell (non-overlapping among different unit cells). Indeed, in the construction of tight-binding based lattice models, one starts with a continuum description, and assumes a localized basis of non-overlapping Wannier functions (see for example, Ref. [@Essler05]). In practice, however, constructing such a localized basis is not trivial [@Marzari97].
The distribution of the polarization gauges the extent to which the system is localized in the full configuration space, a criterion [@Kohn64] which distinguishes an insulator from a conductor. The second GIC was shown [@Kudinov91; @Souza00] to be proportional to the integrated frequency dependent conductivity (sum rule). A gauge dependent definition of the spread (similar to the second GIC) was used to define the maximally localized Wannier function [@Marzari97]. Also, the second cumulant was proposed [@Resta99] to distinguish conductors from insulators. In Ref. [@Hetenyi14] the simplest system with a Berry phase, an isolated spin-$\frac{1}{2}$ particle in a magnetic field, was considered, and it was shown that (based on calculating the first four cumulants) the moments of this underlying distribution are all equal.
The Zak phase was measured in Ref. [@Atala13] in an optical lattice setup which corresponds to the experimental realization of the Su-Schrieffer-Heeger (SSH) model [@Su79] and its extension the Rice-Mele (RM) model [@Rice82]. The RM model is a lattice model with an alternating on-site potential, and hoppings with alternating strengths, depending on whether a given bond is odd or even. An interesting characteristic [@Vanderbilt93; @Xiao10] of the RM model is its topological behavior which manifests when an adiabatic cycle in the parameter space of the Hamiltonian encircles the point ($\Delta=0$, $J=J'=1$). Due to the fact that the polarization as a function of the parameters of the Hamiltonian is not single valued, the polarization in such a process changes by a “polarization quantum.” A recent related study [@Nakajima15] realized quantized adiabatic charge pumping [@Thouless83], also in the RM model.
In this paper we calculate the leading GICs associated with the Zak phase for the RM model. Based on the GICs (or associated moments, GIMs) we approximately reconstruct the distribution associated with the polarization. The RM model is a lattice model, which implies that the underlying Wannier functions are non-overlapping among different unit cells, and that the GICs correspond to the distribution associated with the Wannier function. Hence, our reconstructed probabilities correspond to the squared modulus of the Wannier function. We show that in the fully dimerized limit the GIMs should all have the same magnitude, and that the sign of odd GIMs switch sign with respect to the direction of the polarization. We also focus on the line of the parameter plane where the polarization shows a line of discontinuity (see Fig. \[fig:mom1\], lower panel, left inset). We also present two model calculations in which the evolution of the probability distribution is followed around the topologically nontrivial point of the RM model. As expected, the distribution migrates to the next unit cell, although its shape varies considerably during the cycle.
Reconstructing a probability distribution from knowledge of a finite set of moments is an ill-posed mathematical problem which already has a long history [@Smoluchowski17], although there has been a renewed interest in the last decades [@John07; @DeSouza10]. The scientific applications are also quite broad; image processing [@Sluzek05], calculating magnetic moments [@Berkov00], or molecular electronic structure [@Bandyopadhyay05]. In our study, we opt for a reconstruction based on maximizing the entropy [@Bandyopadhyay05; @Collins77; @Mead84] of the underlying probability distribution.
This paper is organized as follows. In the next section we introduce the GICs associated with the Zak phase. We then discuss their connection to the distribution associated with the Wannier functions. In section \[sec:resp\] we discuss the connection of the cumulants to response functions, after which the reconstruction procedure is presented. In section \[sec:SSH\] the Su-Schrieffer-Heeger and Rice-Mele models are introduced. Subsequently, the behavior of the moments for the fully dimerized limit is studied. Section \[sec:results\] contains our results and analysis before concluding our work.
Gauge invariant cumulants associated with the Zak phase {#sec:cml}
=======================================================
Consider a one-dimensional system whose Hamiltonian which is periodic in $L$. We take Bloch functions parametrized by the crystal momentum, $\Psi_0(K)$ on a grid of $M$ points $K_I = 2\pi I /(M L) - \pi/L$, with $I=0,...,M-1$. The Zak phase can be derived from a product of the form $$\label{eqn:phi_dscrt}
\phi_{Zak} = \mbox{Im} \ln \prod_{I=0}^{M-1} \langle \Psi_0(K_I)|\Psi_0(K_{I+1})\rangle,$$ by taking the continuous limit ($M\rightarrow \infty$). The product in Eq. (\[eqn:phi\_dscrt\]) is known as the Bargmann invariant [@Bargmann64]. We will derive the Zak phase, as well as the associated gauge invariant cumulants (GIC). We start by equating the product in Eq. (\[eqn:phi\_dscrt\]) to a cumulant expansion, $$\label{eqn:cum_exp}
\left[\prod_{I=0}^{M-1} \langle
\Psi_0(K_I)|\Psi_0(K_{I+1})\rangle\right]^{\Delta K} =
\exp\left( \sum_{n=1}^\infty \frac{(i \Delta K)^n}{n!}\tilde{C}_n\right),$$ with $\Delta K = 2 \pi/M$. We now expand both sides and equate like powers of $\Delta K$ term-by-term, mindful of the fact that the left-hand side includes a product over $I$. For example, the first-order term will be $$\tilde{C}_1 = i \sum_{I=0}^{M-1}\Delta K
\gamma_1(K_I)$$ the second will be $$\tilde{C}_2 = - \sum_{I=0}^{M-1}\Delta K [\gamma_2(K_I) - \gamma_1(K_I)^2]$$ with $\gamma_i(K) = \langle \Psi_0(K) | \partial^i_K |
\Psi_0(K)\rangle$. Straightforward algebra and taking the continuous limit ($\Delta K \rightarrow 0$, $M\rightarrow \infty$) gives up to the fourth order term, $$\begin{aligned}
\label{eqn:cmlnts}
C_1 &=& i \frac{L}{2\pi} \int_{-\frac{\pi}{L}}^{\frac{\pi}{L}} d K \gamma_1 \\ \nonumber
C_2 &=& -\frac{L}{2\pi} \int_{-\frac{\pi}{L}}^{\frac{\pi}{L}} d K [\gamma_2 - \gamma_1^2] \\ \nonumber
C_3 &=& -i \frac{L}{2\pi} \int_{-\frac{\pi}{L}}^{\frac{\pi}{L}} d K [\gamma_3 -3 \gamma_2 \gamma_1+ 2\gamma_1^3] \\ \nonumber
C_4 &=& \frac{L}{2\pi} \int_{-\frac{\pi}{L}}^{\frac{\pi}{L}} d K [\gamma_4 -3 \gamma_2^2 -4\gamma_3\gamma_1 + 12 \gamma_1^2\gamma_2 -6\gamma_1^4]\end{aligned}$$ The quantities $C_n$ in Eq. (\[eqn:cmlnts\]) are the GICs associated with the Zak phase (the Zak phase itself being equal to $C_1$). The difference between $\tilde{C}_i$ and $C_i$ is the multiplicative factor $L/2\pi$, which is also how the phase is defined by Zak [@Zak89]. This assures that the first moment corresponds to the average position associated with square modulus of the Wannier function (Eq. (10) in Ref. [@Zak89]). When the underlying probability distribution is well defined the associated moments can be defined based on the cumulants. Following this standard procedure we also define a set of moments. For the first four moments the expressions are $$\begin{aligned}
\label{eqn:muC}
\mu_C^{(1)} &=& C_1 \\ \nonumber
\mu_C^{(2)} &=& C_2 + C_1^2 \\ \nonumber
\mu_C^{(3)} &=& C_3 + 3 C_2 C_1 + C_1^3 \\ \nonumber
\mu_C^{(4)} &=& C_4 + 4 C_3 C_1 + 3 C_2^2 + 6 C_2 C_1^2 + C_1^4.\end{aligned}$$ As discussed below, when the Wannier functions of a particular model are localized within the unit cell, these moments correspond to the moments of the polarization, alternatively, to the distribution of the Wannier functions themselves.
We remark that in general, the Berry phase is a physically well-defined observable, which is thought not to correspond to an operator acting on the Hilbert space. The Zak phase, however, is known to correspond to the total position, and is the basic quantity in expressing the polarization in the modern theory [@King-Smith93; @Resta94; @Resta98].
Connection to the distribution of Wannier centers
=================================================
Cumulants of the type described in the previous section appear in the theory of polarization [@Souza00]. In this section we connect the cumulants to the distribution of Wannier centers. We consider a typical term contributing to cumulant $C_M$, which can be written in the form $$\label{eqn:C_m}
C_{M,\alpha} = \frac{L}{2 \pi}
\int_{-\frac{\pi}{L}}^{\frac{\pi}{L}} dK \prod_{i=1}^d \langle u_{nK} | \partial^{m_i}_K| u_{nK} \rangle,$$ where $\sum_{i=1}^d m_i = M$ and where we have used the periodic Bloch functions $u_{nK}(x)$ as a basis. The periodic Bloch functions can be written in terms of Wannier functions, $$u_{nK}(x) = \sum_{p=-\infty}^\infty \exp(iK(p L - x)) a_n(x - p L),$$ where $a_n(x)$ denote the Wannier functions. With this definition it holds that $$\frac{L}{2\pi} \int_{-\pi/L}^{\pi/L} d K \int_0^L d x |u_{nK}(x)|^2 =
\int_{-\infty}^{\infty} d x |a_n(x)| = 1.$$
We can rewrite a scalar product appearing in Eq. (\[eqn:C\_m\]) as $$\begin{aligned}
\nonumber
\langle u_{nK} | \partial^{m}_K| u_{nK} \rangle = \sum_{ \Delta p= - \infty}^\infty \exp(-iK \Delta p L )
\int_{-\infty}^\infty dx \\ a_n^*(x - \Delta p L) (-i x)^m a_n(x). \hspace{2cm}
\label{eqn:scalarprod}\end{aligned}$$ Substituting Eq. (\[eqn:scalarprod\]) $C_{M,\alpha}$ and integrating in $K$ results in $$\begin{aligned}
C_{M.\alpha} =
\sum_{\Delta p_1 = -\infty}^\infty
...
\sum_{\Delta p_d = -\infty}^\infty \delta[\Delta P , 0] \hspace{1cm}\\ \hspace{.5cm}
\prod_{j=1}^d
\left \{ \int_{-\infty}^\infty dx_j (-i x_j)^{m_j}
a_n^*(x_j - \Delta p_j L) a_n(x_j) \right \}
, \nonumber\end{aligned}$$ where $\Delta P = \sum_{j=1}^d \Delta p_j$ and $\delta[\Delta P , 0]$ is a Kronecker delta.
We note that if the Wannier functions are localized in one unit cell, then the summation in the scalar product of Eq. (\[eqn:scalarprod\]) will be restricted to the term $\Delta p = 0$. In this case, the cumulants $C_M$ will correspond to those of the Wannier centers.
Relation to response functions {#sec:resp}
==============================
The second GIC associated with the polarization gives a sum rule for the frequency-dependent conductivity. This was shown for a finite system by Kudinov [@Kudinov91], and the derivation was extended to periodic systems by Souza, Wilkens, and Martin [@Souza00], by replacing the ordinary matrix elements of the total position operator by their counterparts valid in the crystalline case. Their result is $$C_2 = \frac{\hbar}{\pi q_e^2 n_0} \int \frac{d \omega}{\omega} \bar{\sigma}(\omega),$$ where $q_e$ denotes the charge, $n_0$ the density, and $\bar{\sigma}(\omega)=(V/8\pi^3)\int d{\bf k} \sigma^{\bf k}(\omega)$.
For an insulating (gapped) system one can show that the second cumulant provides an upper bound for the dielectric susceptibility, $\chi$. This was shown by Baeriswyl [@Baeriswyl00] for an open system. This derivation is also easily extended to periodic systems by the appropriate replacement of the total position matrix elements, resulting in, $$\label{eqn:chi}
\chi \leq \frac{2q_e}{V \Delta_g} C_2.$$ In this equation $\Delta_g$ denotes the gap, $V$ denotes the volume of the system.
For higher order cumulants, the derivation of relations such as Eq. ($\ref{eqn:chi}$) are not possible. However, in the classical limit, the cumulants correspond exactly to the response functions of their respective order ($C_2$ gives $\chi$, $C_3$ gives the first non-linear response function, etc.).
Reconstruction of the probability distribution
==============================================
If the Wannier functions can be assumed to be localized within a unit cell, the moments calculated based on the GICs correspond to the actual moments associated with the Wannier orbitals. If all the moments are known, the full probability distribution can be reconstructed. However, in practice, usually only a finite number of cumulants are available. In this case the cumulants can be used as constraints to improve the form of the probability distribution. The first and second cumulants give the average and the variance, and if only these two are available, the best guess for the probability distribution is a Gaussian. Higher order cumulants refine this guess. The third cumulant (skewness) provides information about the asymmetry of the distribution around the mean, while the fourth order one, (kurtosis) represents how sharp the maximum of the distribution is approached from either side.
Below we calculate the GICs of the Rice-Mele model, which is a lattice model (in other words, the Wannier functions are completely localized on particular sites), and approximately reconstruct the probability distribution of the polarization. Our reconstruction is based [@Bandyopadhyay05; @Collins77; @Mead84] on maximizing the information entropy under the constraints provided by the moments calculated. The expression for the entropy we use is $$\label{eqn:SPx}
S[P(x)] = - \int d x P(x) \ln P(x),$$ minimized as a functional of $P(x)$ under the constraints $$\mu_P^{(k)} = \int d x P(x) x^k,$$ as well as the constraint that $P(x)$ is normalized. The functional minimization of Eq. (\[eqn:SPx\]) under the constraints results in the functional differential equation $$\label{eqn:Sopt}
\frac{\delta}{\delta P(x)} \left[ S[P(x)]
- \sum_k A_k (\mu_P^{(k)} - \mu_C^{(k)})
\right] = 0,$$ where $\mu_C^{(k)}$ are the moments obtained from the cumulants of the Berry phase (see Eq. (\[eqn:muC\])), and $A_k$ are Lagrange multipliers. The solution of Eq. (\[eqn:Sopt\]) is $$P(x) = C\exp\left( - \sum_k A_k x^k\right),$$ where $C$ is the normalization constant. We determine the constants $A_k$ by numerically minimizing the quantity $$\label{eqn:chisquared}
\chi^2 = \sum_k (\mu_P^{(k)}-\mu_C^{(k)})^2,$$ as a function of $A_k$. As our initial guess in all cases studied below, we take the Gaussian distribution defined by the first two cumulants obtained for the particular case. The minimization procedure we applied is the simulated annealing technique [@Kirkpatrick83]. Below our reconstructions are based on calculating the first six GIMs in all cases.
Su-Schrieffer-Heeger and Rice-Mele models {#sec:SSH}
=========================================
The SSH model was first introduced [@Su79] to understand the properties of one-dimensional polyacetylene. The RM model is an extension of the SSH model, it includes an additional term, consisting of an alternating on-site potential, added in order to extend the SSH model to diatomic polymers. In recent decades it has been studied extensively due to the wealth of interesting physical phenomena it displays: topological soliton excitation, fractional charge, and non-trivial edge states[@Takayama80; @Su80; @Jackiw76; @Heeger88; @Ruostekoski02; @Li14]. It was also realized as a system of cold atoms trapped in an optical lattice in one dimension recently [@Atala13]. The Berry phase in the RM model was studied by Vanderbilt and King-Smith [@Vanderbilt93]. In that study the point of the model in parameter space of the model which is metallic (and which is responsible for the topologically nontrivial behavior) was encircled in parameter space. This leads to the increase of $C_1$ (the Berry phase, or the polarization) by one polarization quantum, consistent with the quantization of charge transport [@Thouless83; @King-Smith93].
The hopping part of the SSH Hamiltonian reads: $$\hat{H}_{SSH} = -J \sum_{i=1}^{N/2} c_{i,A}^\dagger c_{i,B}
-J' \sum_{i=1}^{N/2} c_{i,B}^\dagger c_{i+1,A} + \mbox{H.c.},$$ where $N$ denotes the number of sites, the on-site potential has the form $$\hat{H}_{\Delta} = -\Delta \sum_{i=1}^{N/2} c_{i,A}^\dagger c_{i,A}
+ \Delta \sum_{i=1}^{N/2} c_{i,B}^\dagger c_{i,B}.$$ The model is shown schematically in Fig. \[fig:model\]. This figure shows the one-dimensional lattice, including sublattices, the alternating hoppings, and the on-site potential. The unit cell is indicated in shaded yellow. Also shown is the continuous variable $x$, which runs from $-\infty$ to $\infty$, and will serve as the axis for the reconstructed probability distributions of the polarization calculated below.
The hoppings can also be expressed in terms of the average hopping $t$ and the deviation $\delta$ as $$\label{eqn:tdelta}
J = \frac{t}{2} + \frac{\delta}{2}, J' = \frac{t}{2} - \frac{\delta}{2}.$$ The total Hamiltonian we consider is $$\hat{H} = \hat{H}_{SSH} + \hat{H}_{\Delta}.$$ The parameters $J$ and $J'$ are hopping parameters corresponding to hopping along alternating bonds. We take the lattice constant to be unity (the unit cell is two lattice constants). The parameter $\Delta$ denotes the on-site potential, whose sign alternates from site to site. This model is metallic for $J=J'$ and $\Delta=0$ but is insulating for all other values of the parameters. In reciprocal space this Hamiltonian becomes $$\hat{H} = \sum_k \left(
\begin{array}{cc} \Delta & -\rho_k \\
-\rho_k^* & -\Delta,
\end{array}
\right)$$ where $$\rho_k = J e^{ik} + J' e^{-ik}.$$ At a particular value of $k$ we can write the eigenstate for the lower band as $$\left(
\begin{array}{c} \alpha_k \\
\beta_k
\end{array}
\right) =
\left(
\begin{array}{c} \sin\left(\frac{\theta_k}{2}\right) \\
e^{-i \phi_k}\cos\left(\frac{\theta_k}{2}\right)
\end{array}
\right),$$ where $$\begin{aligned}
\theta_k &=& \arctan \left( \frac{|\rho_k|}{\Delta}\right) \\
\phi_k &=& \arctan \left( \frac{(J-J')\sin(k)}{(J+J')\cos(k)}\right). \nonumber\end{aligned}$$ The cumulants can now be written in terms of the eigenstates. For example, $$C_1 = \frac{i}{\pi} \int_{-\pi/2}^{\pi/2} dk ( \alpha_k^* \partial_k \alpha_k + \beta_k^*
\partial_k \beta_k ),$$ and the other cumulants can be constructed accordingly (note that the unit cell is $L=2$).
Fully dimerized limit {#sec:FDL}
=====================
Here we show that in the fully dimerized limit the GIMs should all have the same magnitude. In Ref. [@Hetenyi14] we pointed out that the Berry phase can be related to an observable $\hat{O}$ fixed by requiring that $$\label{eqn:cnd}
\partial_K H(K) = i [H(K),\hat{O}].$$ This definition does not uniquely fix the operator $\hat{O}$. For example, for the magnetic field example the matrix $\sigma_z/2$ or $(\sigma_z +I)/2$ both satisfy Eq. (\[eqn:cnd\]). This arbitrariness causes a shift in the first cumulant. However, only the operator $(\sigma_z+I)/2$ will give a distribution in which all moments are equal, since this matrix has the form $$(\sigma_z+I)/2 = \left(
\begin{array}{cc} 1 & 0 \\
0 & 0
\end{array}
\right),$$ and is equal to itself when raised to any power.
In the case of the RM model we first write the Hamiltonian with the parameter $K$ explicitly as $$\begin{aligned}
\hat{H}(K) &= -J \exp(i K) \sum_{j=1}^{L/2} c_{j,A}^\dagger c_{j,B} + \mbox{H.c.}\\
& \nonumber
-J' \exp(i K) \sum_{j=1}^{L/2} c_{j,B}^\dagger c_{j+1,A} + \mbox{H.c.} + \hat{H}_{\Delta}. \nonumber\end{aligned}$$ The operator $\partial_K \hat{H}(K)$ is the current, $$\begin{aligned}
\partial_K \hat{H}(K) &= -i J \exp(i K) \sum_{j=1}^{L/2} c_{j,A}^\dagger c_{j,B} + \mbox{H.c.}\\
& \nonumber
-i J' \exp(i K) \sum_{j=1}^{L/2} c_{j,B}^\dagger c_{j+1,A} + \mbox{H.c.}. \nonumber\end{aligned}$$ We now write a form for the operator $\hat{O}$ as $$\hat{O} = \sum_{j=1}^{L/2} x_j c_{j,A}^\dagger c_{j,A} +
y_j c_{j,B}^\dagger c_{j,B}.$$ Evaluating the commutator gives $$\begin{aligned}
i [\hat{H}(K),\hat{O}] = i \sum_{j=1}^{L/2}(y_j - x_j) J \exp(iK) c_{j,A}^\dagger c_{j,B} + \mbox{H.c.} \nonumber \\
i \sum_{j=1}^{L/2}(x_{j+1} - y_j) J' \exp(iK) c_{j+1,A}^\dagger c_{j,B} + \mbox{H.c.} \hspace{1cm}\end{aligned}$$ For the case $J'=0$ we can chose $x_j=0$ and $y_j=1$, so that $i[\hat{H}(K),\hat{O}]$ corresponds to the current. This is not the only choice, but with this choice the operator $\hat{O}$ when written in $k$-space corresponds to $$\hat{O} = \sum_k ( c^\dagger_{k,A} c^\dagger_{k,B} )
\left(
\begin{array}{cc} 0 & 0 \\
0 & 1
\end{array}
\right)
\left(
\begin{array}{c} c_{k,A} \\
c_{k,B}
\end{array}
\right),$$ which gives equal moments. Clearly, the choice for the spatial coefficients $x_j$ and $y_j$ are due to the fact that in this case the system consists of a set of independent dipoles. When $J$ is taken to zero, and $J'$ kept finite, then the appropriate choice to fix $\hat{O}$ is $x_j=0$ and $y_j=-1$. If instead the sign of $\Delta$ is changed $\hat{O}$ is again defined by the $x_j=0$ and $y_j=-1$. These results are clearly due to the reversal of the direction of the dipole moment within the unit cell. The results presented in Fig. \[fig:mom01\] corroborate our derivation.
Results and Analysis {#sec:results}
====================
We first look at the system with $J'=0$. In this case, the band structure of the system is simply two flat lines in the Brillouin zone. The system can be thought of as a simple two-state system. We calculated the first four GICs, from which we obtained the corresponding GIMs. The results are shown in the uppermost panel of Fig. \[fig:mom01\]. The moments as a function of $\Delta/J$ all fall on the same curve in this case. If the hopping parameters $J$ and $J'$ are switched (not shown), the sign of the odd moments changes, the even moments remain the same. These results are in accordance with section \[sec:FDL\].
Fig. \[fig:mom01\] also shows the cumulants for different ratios; $J'/J=0.3, 0.5, 0.7$. The deviation of the cumulants from one another is more pronounced, and increases with an increase of $J'/J$. However, the moments become equal for any $J'/J$ when $\Delta \rightarrow \pm \infty $. In this case also, the system becomes an independent array of two state systems. The band energies in all these cases vary continuously with $k$ across the Brillouin zone.
The results for the case $J'/J=1$ are also shown separately in Fig. \[fig:mom1\], as well as the limits $J' \rightarrow_\pm J$. For finite $\Delta/J$ the odd cumulants are zero, indicating an even probability distribution. The ratio of the second and fourth cumulants rule out a Gaussian. As $\Delta/J \rightarrow 0$ a discontinuity in the slope of the band develops. In this case, the cumulants $C_2$ and $C_4$ diverge. The lower panel in this figure shows what happens when $J'$ is close to $J$ (bigger or smaller) but the two are not quite equal ($J' = J +
\epsilon$, $\epsilon$ a small number). We see that in this case the first moment is one or minus one, depending on the sign of $\epsilon$, and zero is not approached as $\epsilon \rightarrow 0$ from either side. The left inset in the lower panel shows the behavior of the first moment on the $\Delta-\delta$ plane, indicating a discontinuity along the line $\Delta<0,
\delta = 0$ (the well-known result of Vanderbilt and King-Smith [@Xiao10; @Vanderbilt93]). The moments and cumulants we find are consistent with the behavior shown in the left inset of the lower panel of the figure.
In Fig. \[fig:prob\] we show examples of reconstructed probability distributions for $J'/J=0.0, 0.3, 0.5, 0.7$, in each case for several values of $\Delta/J$. $\chi^2$ (defined in Eq. (\[eqn:chisquared\])) is tabulated in the appendix (Table \[tab:chi2\]). The most localized example ($J'/J =
0$ and $\Delta/J=-2$) shows a sharp peak around $x=1$; as $\Delta$ decreases the curves shift to the left and spread out, but their shape is always very similar (for smaller values of $\Delta/J$ this is emphasized in the inset). The maximum of the probability distribution is always between zero and one. These curves are all cases for which all the moments are equal. As the alternate hoppings ($J'$) are turned on, the shifting occurs in a qualitatively different manner. Initially ($\Delta/J=-2$ in all cases) the curves are centered very near $x=1$. $\Delta/J=-2$ is for most cases well in the region where the moments are equal. As $\Delta$ decreases, the distributions shift, but they do this by becoming asymmetric about their mean, with the density increasing on the side left of the maxima of the distributions. The shape of the distributions changes considerably. This is clearly due to the fact that in these latter cases the moments vary as $\Delta$ is varied, and they are not all equal. The maxima for the cases for which $J'/J \neq 0$ shift much less as $\Delta/J$ is varied. When $\Delta/J$ changes sign (results not shown), the polarization becomes centered around $x=0$ end of the unit cell and the probability distributions are reflections of the ones shown in Fig. \[fig:prob\] across $x = 1/2$.
The probability distributions for the case $J'=J=1$ are also shown separately in Fig. \[fig:prob1\] (with $\chi^2$ tabulated in Table \[tab:chi2\]), as well as the case $J'$ close to $J$. All of the $J'=J$ distributions are symmetric around the origin. As $\Delta/J \rightarrow 0$ the distribution broadens, and it is clear that a conducting phase is approached [@Resta99]. If $\Delta<0$ then the polarizations are localized near $x = \pm1$ depending on whether $J'$ is smaller or larger than $J$. This is consistent with Fig. \[fig:mom1\].
In Figs. \[fig:CircleR1\] and \[fig:CircleR02\] we show the evolution of the reconstructed probability distributions along two cyclic paths which encircle the topologically non-trivial point of the RM model, one with radius unity, the other with radius $0.2$ in the $\Delta/t$, $\delta/t$ plane. In these calculations the parametrization was different from the previous ones, here $t$ was set to unity, rather than $J$ (see Eq. (\[eqn:tdelta\])). For the points $A^*$, $B$, ... in Figs. \[fig:CircleR1\] and \[fig:CircleR02\] the values of $\Delta/J$ and $J'/J$ are shown in Table \[tab:chi2\_circles\]. The upper panel in both figures show the evolution of the different GIMs(GICs). The even moments are single-valued, the odd ones are not. This follows from gauge invariance properties of the cumulants (Eq. (\[eqn:cmlnts\])). The first cumulant is only gauge invariant modulo $2 \pi$ times an integer [@Xiao10; @Vanderbilt93], the others do not change at all due to a gauge transformation. The odd GIMs depend on combinations of the GICs which involve odd combinations of the cumulants, therefore they are not multivalued in general. In both sets of figures (\[fig:CircleR1\] and \[fig:CircleR02\]) the points $A^*$ are not exactly on the $\phi = -\pi/2$ axis, but instead we numerically realize the limit $\phi = \lim_{\delta \Phi
\rightarrow 0^+} (-\pi/2 + \delta \Phi)$. In the actual calculation we took $\delta \Phi = 2 \pi/1000$. Also, the point $\phi = -\pi/2$ or $\phi =
3\pi/2$ is excluded from the curves shown in the upper panels of the two figures.
The example with radius unity (Fig. \[fig:CircleR1\]) remains mostly in the fully dimerized limit, as can be seen in the upper panel of the figure. The odd moments and even moments are always equal. Except for a small region near $\phi/\pi = 0.5$ the absolute values of the moments are equal. The lower panel shows the evolution of the probability distribution along the path. Starting from a relatively sharp distribution localized near $x=1$, the maximum moves to the left. Before reaching half the unit cell, the distribution spreads. After passing through the midpoint the system, where the maximum is the smallest, the distribution begins to localize again until $x=0$. From there this tendency is repeated. Indeed, the distribution ends up at $x=-1$ at the end of the process: the Wannier function “walked” to an equivalent position in the next unit cell. For the case of the smaller radius (0.2, Fig. \[fig:CircleR1\]) the initial distribution is broader, and as the cycle is traversed, the maximum of the distribution oscillates with a smaller amplitude, but the “walking” to a new equivalent position still occurs.
In both Figs. \[fig:CircleR1\] and \[fig:CircleR02\] it is clear that the odd moments do not correspond to single-valued functions. The values of the odd moments depend on whether we approach the original point from which the cycle begins ($\delta = 0, \Delta<0$) from the left or the right. At the same time, the probability distributions for some cases with $\delta = 0, \Delta<0$ are shown in Fig. \[fig:prob1\]; they are centered around zero and they spread as $\Delta/J$ is decreased. This suggests the limiting cases from either direction give different results from the result for fixing the Hamiltonian parameters such that $\delta = 0,\Delta<0$.
Conclusion
==========
We studied the gauge invariant cumulants associated with the Zak phase. We have shown that for localized Wannier functions they correspond to the cumulants of the Wannier centers. They are also related to the dielectric response functions of a given system. We calculated the cumulants for the Rice-Mele model. In the limit of isolated dimers, all the moments (extracted from the gauge invariant cumulants) are equal. This can be justified for this case by constructing the operator which corresponds to the Berry phase explicitly. Deviations from this behavior come about when the hopping parameters are both finite. For a system with equal hopping parameters the odd cumulants vanish. We have also reconstructed the full probability distribution of the polarization based on the gauge invariant cumulants and have studied how they evolve as functions of different parameters of the Hamiltonian. In particular we calculated the evolution of the distribution around the topologically non-trivial point of the model. We anticipate that detailed experimental measurements can also provide a probability distribution of the polarization for comparison with our predictions.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors acknowledge financial support from the Turkish agency for basic research (TÜBITAK, grant no. 113F334). We also thank L. G. M. de Souza for helpful discussions on the topic of probability reconstruction from moments.
Appendix {#appendix .unnumbered}
========
In Table \[tab:chi2\] values of the negative base ten logarithm of $\chi^2$ rounded down to the first digit (defined in Eq. (\[eqn:chisquared\])) is shown for the reconstructed probabilities in Figs. \[fig:prob\] and \[fig:prob1\]. In all cases $\chi^2$ decreased at least eight orders of magnitude from its initial value during the simulated annealing calculation.
In Table \[tab:chi2\_circles\] the values of the parameters according to the parametrization used in Figs. \[fig:mom01\]-\[fig:prob1\] are shown. Also shown are values of the negative base ten logarithm of $\chi^2$ rounded down to the first digit for probability distributions corresponding to the points in Figs. \[fig:CircleR1\] and \[fig:CircleR02\].
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![(Color online.) Schematic representation of the Rice-Mele model. $\Delta$ represents the on-site potential, $A$ and $B$ refer to the different sublattices. $J$ and $J'$ are the alternating hoppings. The unit cell is indicated in yellow. The $x$ label corresponds to localization within the unit cell ($-1<x<1$). The variable $x$ is continuous, below, in our subsequent calculations, the probability distribution will be shown as a function of $x$. The unit of $x$ is the lattice constant.[]{data-label="fig:model"}](./Model6-eps-converted-to.pdf){width="\linewidth"}
![Moments for $J'/J=0, 0.3, 0.5, 0.7$ as a function of $\Delta/J$. In these calculations $J=1$. For $J'/J=0$ the curves are identical. The insets show the corresponding cumulants.[]{data-label="fig:mom01"}](./MOM0-eps-converted-to.pdf "fig:"){width="\linewidth"} ![Moments for $J'/J=0, 0.3, 0.5, 0.7$ as a function of $\Delta/J$. In these calculations $J=1$. For $J'/J=0$ the curves are identical. The insets show the corresponding cumulants.[]{data-label="fig:mom01"}](./MOM03-eps-converted-to.pdf "fig:"){width="\linewidth"} ![Moments for $J'/J=0, 0.3, 0.5, 0.7$ as a function of $\Delta/J$. In these calculations $J=1$. For $J'/J=0$ the curves are identical. The insets show the corresponding cumulants.[]{data-label="fig:mom01"}](./MOM05-eps-converted-to.pdf "fig:"){width="\linewidth"} ![Moments for $J'/J=0, 0.3, 0.5, 0.7$ as a function of $\Delta/J$. In these calculations $J=1$. For $J'/J=0$ the curves are identical. The insets show the corresponding cumulants.[]{data-label="fig:mom01"}](./MOM07-eps-converted-to.pdf "fig:"){width="\linewidth"}
![(Color online.) Upper panel: Moments for $J'/J=1$ as a function of $\Delta/J$. In these calculations $J=1$. The inset shows the corresponding cumulants. In the limit $\Delta/J \rightarrow 0$ (the topological point of the model) the even cumulants diverge, while the odd cumulants are always zero for this case. Lower panel: first two moments and cumulants (right inset) for $J=1,J'=J \pm 0.006$. Left inset shows the first moment on the $\Delta-\delta$ plane, indicating the singular behavior along the line $\Delta < 0, \delta = 0$.[]{data-label="fig:mom1"}](./MOM1-eps-converted-to.pdf "fig:"){width="\linewidth"} ![(Color online.) Upper panel: Moments for $J'/J=1$ as a function of $\Delta/J$. In these calculations $J=1$. The inset shows the corresponding cumulants. In the limit $\Delta/J \rightarrow 0$ (the topological point of the model) the even cumulants diverge, while the odd cumulants are always zero for this case. Lower panel: first two moments and cumulants (right inset) for $J=1,J'=J \pm 0.006$. Left inset shows the first moment on the $\Delta-\delta$ plane, indicating the singular behavior along the line $\Delta < 0, \delta = 0$.[]{data-label="fig:mom1"}](./CM001F4-eps-converted-to.pdf "fig:"){width="\linewidth"}
![(Color online.) Normalized probability distribution of the polarization for different parameters of the Rice-Mele Hamiltonian. In these calculations $J=1$. The unit of length in these figures is the lattice constant. Different values of $\Delta/J$ are shown for $J'/J=0.0,
0.3, 0.5, 0.7$. In the topmost panel ($J'/J=0$) the inset shows the distribution for the cases $\Delta/J = 0.0,-0.1,-0.2,-0.3,-0.4,-0.5$)[]{data-label="fig:prob"}](./Jp0-eps-converted-to.pdf "fig:"){width="0.9\linewidth"} ![(Color online.) Normalized probability distribution of the polarization for different parameters of the Rice-Mele Hamiltonian. In these calculations $J=1$. The unit of length in these figures is the lattice constant. Different values of $\Delta/J$ are shown for $J'/J=0.0,
0.3, 0.5, 0.7$. In the topmost panel ($J'/J=0$) the inset shows the distribution for the cases $\Delta/J = 0.0,-0.1,-0.2,-0.3,-0.4,-0.5$)[]{data-label="fig:prob"}](./Jp03-eps-converted-to.pdf "fig:"){width="0.9\linewidth"} ![(Color online.) Normalized probability distribution of the polarization for different parameters of the Rice-Mele Hamiltonian. In these calculations $J=1$. The unit of length in these figures is the lattice constant. Different values of $\Delta/J$ are shown for $J'/J=0.0,
0.3, 0.5, 0.7$. In the topmost panel ($J'/J=0$) the inset shows the distribution for the cases $\Delta/J = 0.0,-0.1,-0.2,-0.3,-0.4,-0.5$)[]{data-label="fig:prob"}](./Jp05-eps-converted-to.pdf "fig:"){width="0.9\linewidth"} ![(Color online.) Normalized probability distribution of the polarization for different parameters of the Rice-Mele Hamiltonian. In these calculations $J=1$. The unit of length in these figures is the lattice constant. Different values of $\Delta/J$ are shown for $J'/J=0.0,
0.3, 0.5, 0.7$. In the topmost panel ($J'/J=0$) the inset shows the distribution for the cases $\Delta/J = 0.0,-0.1,-0.2,-0.3,-0.4,-0.5$)[]{data-label="fig:prob"}](./Jp07-eps-converted-to.pdf "fig:"){width="0.9\linewidth"}
![(Color online.) Normalized probability distribution of the polarization for cases $J'=J$ and $J' = J \pm \epsilon$ ($\epsilon =
0.006$). In these calculations $J=1$. The unit of length is the lattice constant. Different values of $\Delta/J$ are shown. Upper panel(lower panel): $\Delta /J < 0$ ($\Delta /J > 0$).[]{data-label="fig:prob1"}](./Pr3-eps-converted-to.pdf){width="\linewidth"}
{width="\linewidth"} {width="\linewidth"}
{width="\linewidth"} {width="\linewidth"}
Fig. \[fig:prob\] panel 1 $\Delta/J$ $-\log_{10}(\chi^2)$ $\Delta/J$ $-\log_{10}(\chi^2)$
--------------------------- ------------ ---------------------- ------------ ----------------------
$J'/J = 0$ $-2$ $8$ $-1$ $7$
$J'/J = 0$ $-0.5$ $8$ $-0.4$ $6$
$J'/J = 0$ $-0.3$ $7$ $-0.2$ $6$
$J'/J = 0$ $-0.1$ $7$ $0$ $6$
Fig. \[fig:prob\] panel 2 $\Delta/J$ $-\log_{10}(\chi^2)$ $\Delta/J$ $-\log_{10}(\chi^2)$
$J'/J = 0.3$ $-2$ $6$ $-1$ $5$
$J'/J = 0.3$ $-0.5$ $6$ $-0.4$ $4$
$J'/J = 0.3$ $-0.3$ $5$ $-0.2$ $4$
$J'/J = 0.3$ $-0.1$ $5$ $0$ $4$
Fig. \[fig:prob\] panel 3 $\Delta/J$ $-\log_{10}(\chi^2)$ $\Delta/J$ $-\log_{10}(\chi^2)$
$J'/J = 0.5$ $-2$ $5$ $-1$ $4$
$J'/J = 0.5$ $-0.5$ $5$ $-0.4$ $4$
$J'/J = 0.5$ $-0.3$ $5$ $-0.2$ $4$
$J'/J = 0.5$ $-0.1$ $5$ $0$ $4$
Fig. \[fig:prob\] panel 4 $\Delta/J$ $-\log_{10}(\chi^2)$ $\Delta/J$ $-\log_{10}(\chi^2)$
$J'/J = 0.7$ $-2$ $5$ $-1$ $4$
$J'/J = 0.7$ $-0.5$ $5$ $-0.4$ $4$
$J'/J = 0.7$ $-0.3$ $4$ $-0.2$ $4$
$J'/J = 0.7$ $-0.1$ $4$ $0$ $4$
Fig. \[fig:prob1\] $\Delta/J$ $-\log_{10}(\chi^2)$ $\Delta/J$ $-\log_{10}(\chi^2)$
$J'/J = 1.0$ $-1$ $8$ $-0.6$ $7$
$J'/J = 1.0$ $-0.9$ $8$ $-0.5$ $7$
$J'/J = 1.0$ $-0.8$ $7$ $-0.4$ $7$
$J'/J = 1.0$ $-0.7$ $7$
: $-\log_{10}\chi^2$ rounded to the first digit shown for the reconstructed probabilities in Figs. \[fig:prob\] and \[fig:prob1\]. []{data-label="tab:chi2"}
------- ------------ ------------------ ---------------------- ------------ ---------- ----------------------
$\Delta/J$ $J'/J$ $-\log_{10}(\chi^2)$ $\Delta/J$ $J'/J$ $-\log_{10}(\chi^2)$
$A^*$ $-1.9875$ $0.9875$ $8$ $-.3995$ $.9975$ $6$
$B$ $-1.154$ $.333$ $7$ $-.3149$ $.8181$ $6$
$C$ $-.5359$ $.072$ $7$ $-.1705$ $.7047$ $6$
$D$ $0$ $0$ $6$ $0$ $.6666$ $6$
$E$ $.5359$ $.072$ $8$ $.1705$ $.7047$ $6$
$F$ $1.154$ $.333$ $7$ $.3149$ $.8181$ $6$
$G$ $2$ $1$ $7$ $.4$ $1$ $6$
$H$ $3.4641$ $3$ $6$ $.3849$ $1.2222$ $6$
$I$ $7.4641$ $13.928$ $8$ $.2419$ $1.4189$ $6$
$J$ $0$ $\infty$ $(1/0)$ $7$ $0$ $1.5$ $6$
$K$ $-7.4641$ $13.928$ $7$ $-.2419$ $1.4189$ $6$
$L$ $-3.4641$ $3$ $6$ $-.3849$ $1.2222$ $6$
------- ------------ ------------------ ---------------------- ------------ ---------- ----------------------
: Values of the parameters according to the parametrization used in Figs. \[fig:mom01\]-\[fig:prob1\] are shown. Also shown are values of $-\log_{10}\chi^2$ rounded to the first digit for probability distributions corresponding to the points in Figs. \[fig:CircleR1\] and \[fig:CircleR02\]. []{data-label="tab:chi2_circles"}
| 0 |
---
abstract: 'For the explosion mechanism of Type Ia supernovae (SNe Ia), different scenarios have been suggested. In these, the propagation of the burning front through the exploding white dwarf star proceeds in different modes, and consequently imprints of the explosion model on the nucleosynthetic yields can be expected. The nucleosynthetic characteristics of various explosion mechanisms is explored based on three two-dimensional explosion simulations representing extreme cases: a pure turbulent deflagration, a delayed detonation following an approximately spherical ignition of the initial deflagration, and a delayed detonation arising from a highly asymmetric deflagration ignition. Apart from this initial condition, the deflagration stage is treated in a parameter-free approach. The detonation is initiated when the turbulent burning enters the distributed burning regime. This occurs at densities around $10^{7}$ g cm$^{-3}$ – relatively low as compared to existing nucleosynthesis studies for one-dimensional spherically symmetric models. The burning in these multidimensional models is different from that in one-dimensional simulations as the detonation wave propagates both into unburned material in the high density region near the center of a white dwarf and into the low density region near the surface. Thus, the resulting yield is a mixture of different explosive burning products, from carbon-burning products at low densities to complete silicon-burning products at the highest densities, as well as electron-capture products synthesized at the deflagration stage. Detailed calculations of the nucleosynthesis in all three models are presented. In contrast to the deflagration model, the delayed detonations produce a characteristic layered structure and the yields largely satisfy constraints from Galactic chemical evolution. In the asymmetric delayed detonation model, the region filled with electron capture species (e.g., $^{58}$Ni, $^{54}$Fe) is within a shell, showing a large off-set, above the bulk of $^{56}$Ni distribution, while species produced by the detonation are distributed more spherically.'
author:
- |
K. Maeda, F.K. Röpke, M. Fink, W. Hillebrandt,\
C. Travaglio, F.-K. Thielemann
title: |
Nucleosynthesis in Two-dimensional Delayed Detonation Models\
of Type Ia supernova Explosions
---
INTRODUCTION
============
There is a consensus that Type Ia supernovae (SNe Ia) are the outcome of a thermonuclear explosion of a carbon-oxygen white dwarf (WD) (e.g., Wheeler et al. 1995; Nomoto et al. 1997; Branch 1998). For the progenitor, the Chandrasekhar-mass ($M_{\rm Ch}$) WD model has been favored for a majority of SNe Ia (e.g., Höflich & Khokhlov 1996; Nugent et al. 1997; Fink et al. 2007; Mazzali et al. 2007).
The mass of the WD could reach $M_{\rm Ch}$ by several evolutionary paths, either by a mass-transfer from a binary giant/main-sequence companion (single degenerate scenario; e.g., Whelan & Iben 1973; Nomoto 1982) or as a result of merging with a binary degenerate WD companion (double degenerate scenario; e.g., Iben & Tutukov 1984; Webbink 1984). As the WD has accreted a sufficient amount of material, the central density of the WD increases and the heating rate by the carbon fusion exceeds the cooling rate by neutrino emission. The evolution is then followed by a simmering phase with convective carbon burning, lasting for about a century. At the end of this simmering phase, the temperature rises to the point where convection can no longer efficiently transport away the energy produced by the carbon burning. This is a stage where the burning becomes dynamical, initiating a thermonuclear flame that propagates outward and disrupts the WD, i.e., a supernova explosion (Nomoto et al. 1984; Woosley & Weaver 1986).
Once the thermonuclear flame is ignited, there are two possible modes of the propagation: subsonic deflagration and supersonic detonation. A prompt ignition of the detonation flame is disfavored because the resulting nucleosynthesis yield conflicts with Galactic chemical evolution (Arnett 1969) and fails to produce the strong intermediate-mass element features observed in SNe Ia. Thus, the explosion should start with a subsonic deflagration flame. The deflagration stage may last until the end of the explosion (the deflagration model; Nomoto et al. 1984), while it is also possible that the deflagration flame turns into the detonation wave \[the delayed detonation model, or the deflagration-detonation transition (DDT) model; Khokhlov 1991; Yamaoka et al. 1992; Woosley & Weaver 1994; Iwamoto et al. 1999\].
The supernova explosion phase has been investigated by “classical” one-dimensional spherically symmetric models; the classical deflagration W7 model of Nomoto et al. (1984) has successfully explained the basic contribution of SNe Ia to Galactic chemical evolution, as well as basic features of observed spectra and light curves of individual SNe Ia of a normal class (Branch et al. 1985). Some improvement in these observational aspects has been obtained by introducing a delayed detonation (e.g., Höflich & Khokhlov 1996; Iwamoto et al. 1999).
These models, however, treat the propagation speed of the deflagration flame as a parameter. Moreover, the deflagration flame is hydrodynamically unstable and non-sphericity is thus actually essential (e.g., Niemeyer et al. 1996). Recent investigations of the explosion models have intensively addressed these issues (e.g., Reinecke et al. 2002; Gamezo et al. 2003; Röpke & Hillebrandt 2005; Bravo & García-Senz 2006; Röpke et al. 2006b; Schmidt & Niemeyer 2006). With high resolution multi-dimensional hydrodynamic simulations, coupled with an appropriate sub-grid model to capture turbulence effects on unresolved scales, recent studies provide essentially “parameter-free” simulations for the initial deflagration stage, where only the structure of the pre-supernova WD and the distribution of the initial deflagration ignition sparks set up the initial conditions.
The multi-dimensional simulations have been performed with different initial conditions to see if SNe Ia in general can be explained in a framework of a pure deflagration explosion, as is summarized by Röpke et al. (2007b). Detailed nucleosynthesis calculations have been performed for some deflagration models (Travaglio et al. 2004a, 2005; Kozma et al. 2005, Röpke et al. 2006a). These studies indicate that the pure deflagration explosion can explain a part of SNe Ia, up to relatively weak and faint SNe Ia in the normal population. However, it has also been shown that a subsequent detonation phase is probably necessary to account for typical normal SNe Ia and brighter ones (Gamezo et al. 2005; Röpke et al. 2007b).
Delayed detonation models have also been investigated with multi-dimensional simulations (Gamezo et al. 2005; Plewa 2007; Golombek & Niemeyer 2005; Röpke & Niemeyer 2007c; Bravo & García-Senz 2008). However, compared to the deflagration models as summarized above, the investigation of multi-dimensional delayed detonation models is still at the very initial stage. The radiation transfer based on the multi-dimensional models has been examined by Kasen et al. (2009). Detailed nucleosynthesis studies have rarely been done (but see Bravo & García-Senz 2008). In this paper, we present results from detailed nucleosynthesis calculations, based on two-dimensional delayed detonation models and, for comparison, a pure deflagration model.
The two delayed detonation models presented here can be regarded as extreme cases – not necessarily with respect to their $^{56}$Ni production and brightness, but with respect to symmetries/asymmetries in the explosion phase. While in one model, the deflagration was ignited in an approximately spherical configuration at the center of the WD, the other model features an off-center ignition and propagation of the initial deflagration flame similar to the three-dimensional simulations of Röpke et al. (2007d). It has been suggested that the convection in the simmering phase may be dominated by a dipolar mode, and there is a good possibility that the deflagration is initiated in an off-center way (e.g., Woosley et al. 2004, Kuhlen et al. 2006). In the pure-deflagration model, such an explosion cannot account for normal SNe Ia, because the small burning surface area should result in inefficient production of $^{56}$Ni (see, e.g., Röpke et al. 2007d). However, this can be possibly overcome in the delayed detonation scenario, as the detonation can potentially produce a large amount of $^{56}$Ni.
The paper is organized as follows. In §2, we present methods and models, in §3, we present our results. Discussion of these results from a view point of the chemical evolution is given in §4. The paper is closed in §5 with concluding remarks.
[ccccccc]{} $M_{\rm burn}$ & 1.21 & 0.61 & 1.01 & 1.25 & 0.522 & 0.393\
$E_{\rm nuc}$ & 1.78 & 0.91 & 1.46 & 1.80 & 0.767 & 0.522\
$E_{\rm K}$ & 1.28 & 0.41 & 0.96 & 1.30 & &\
$E_{\rm K}$ (hyd) & 1.30 & 0.41 & 0.93 & 1.27 & &
Methods and Models
==================
Explosion Models
----------------
In this paper, we concentrate on two-dimensional models. Although some results might be affected by the imposed symmetry (e.g., Travaglio et al. 2004a; Röpke et al. 2007b), we believe that examining 2D models is a natural step forward. Moreover, it illustrates how nucleosynthesis in the different models proceeds, and highlights differences with respect to one-dimensional models.
In particular, we focus on three models in this paper as follows:
- [**C-DEF:** ]{} A globally spherically symmetric pure-deflagration explosion in which the deflagration was ignited within the *c3*-shape boundary, as was done by Reinecke et al. (1999a). About $2\times10^{-2}\,M_{\odot}$ were initially incinerated to trigger the deflagration. This model is similar to one presented already in Travaglio et al. (2004a).
- [**C-DDT:** ]{} A delayed detonation model which follows the 2D spherical deflagration C-DEF model. A prescription for the DDT is given below.
- [**O-DDT:** ]{} A delayed detonation model which follows an extremely off-center deflagration. The deflagration is ignited by 29 bubbles distributed within an opening angle with respect to the $z$-axis of 45 degrees. The outermost bubble was placed at a distance of $\sim 180$ km from the center. About $1\times 10^{-5}\,M_{\odot}$ were initially incinerated to trigger the deflagration. The DDT is treated in the same way as in the C-DDT model.
The exploding WD had a central density of $2.9 \times 10^{9}\,\mathrm{g}\,
\mathrm{cm}^{-3}$. In the hydrodynamic explosion simulations, burning was treated in a simplified way. Only five species (carbon, oxygen, alpha-particles, a representative for iron-peak elements, and a representative for intermediate mass elements) were followed. The deflagration and detonation fronts were modeled with the level-set approach (Reinecke et al. 1999b; Golombek & Niemeyer 2005; Roepke & Niemeyer 2007c). After passage of the zero-level set representing the combustion waves, the material was converted from the carbon/oxygen fuel mixture to an approximate nuclear ash composition. At high densities, NSE is reached and the ash was modeled as a temperature- and density-dependent mixture of iron-peak elements and alpha-particles. In the NSE region, electron captures neutronizing the ashes were followed. In the hydrodynamic simulations, $512 \times 512$ cells are used for C-DEF and C-DDT models, and $1024 \times 512$ cells are used for O-DDT model.
In the delayed detonation models, we assumed the deflagration-to-detonation transition to take place once the flame enters the distributed burning regime (e.g., Röpke & Niemeyer 2007c) at a fuel density of $\le 1 \times 10^{7}\,\mathrm{g}\,
\mathrm{cm}^{-3}$ (hereafter $\rho_{\rm DDT}$). We emphasize that in contrast to the classical 1D models, the DDT is not exclusively parameterized by $\rho_{\rm DDT}$, but the requirement on reaching the distributed burning implies also a turbulence criterion (see Röpke 2007a for an evaluation in three-dimensional models). An estimate for entering the distributed regime is the equality of the laminar flame width and the Gibson scale – the scale at which the turbulent velocity fluctuations equal the laminar flame speed. When the Gibson scale becomes smaller than the laminar flame width, turbulence affects (and ultimately destroys) the laminar flame structure. This is a prerequisite for DDT, however, it is not a sufficient criterion (e.g. Woosley 2007; Woosley et al. 2009). The microphysics of DDT is not yet fully known, and therefore we apply the necessary condition of entering the distributed regime here only. We emphasize that the hydrodynamic models are still in a preliminary stage and used here only in order to demonstrate the nucleosynthesis associated with them. For more robust predictions of the $^{56}$Ni production and the implied brightness of the events, more elaborate hydrodynamical simulations should be used (F.K. Röpke et al., in prep.); the models presented here are understood as a case study demonstrating typical nucleosynthesis for the different explosion processes.
Results on some synthetic observables (e.g., light curves) derived for similar two-dimensional models can be found in Kasen et al. (2009), and details on hydrodynamic calculations will be presented elsewhere (F.K. Röpke et al., in prep.). Note that the present models cover extreme cases in the sequence of “classical” delayed detonation models where the DDT takes place before the deflagration wave reaches the WD surface (e.g., Kasen et al. 2009). For the situation in which only a few initial bubbles are distributed within a small solid angle with a large off-set, the “gravitationally confined detonation” model (e.g. Jordan et al. 2008; Meakin et al. 2009) has been suggested as an alternative explosion mechanism. This would be a more extreme case than our O-DDT model.
Nucleosynthesis
---------------
We apply the tracer particle method to the calculations of nucleosynthesis. The essence is to follow thermal histories of Lagrangian particles, which are passively advected in hydrodynamic simulations, and then to employ detailed nuclear reaction network calculations to each particle separately. The method was first applied to core-collapse supernovae (Nagataki et al. 1997; Maeda et al. 2002; Maeda & Nomoto 2003), and has become popular in the field thanks to its simplicity and its applicability to multi-dimensional problems (e.g., Travaglio et al. 2004b). Travaglio et al. (2004a) applied the method to multi-dimensional, purely deflagration explosion models of SNe Ia.
In the setup for our hydrodynamic simulations, $80^{2}$ tracer particles are distributed uniformly in mass coordinate, such that each particle represents the same mass of $\sim 2.2 \times 10^{-4} M_{\odot} (=M_{\rm wd}/6400)$. The particles are advected passively, following the velocity field at each time step of the Eulerian hydrodynamic simulations. The thermal history each particle experiences is recorded. The number of the tracer particles is sufficient to accurately follow the nucleosynthesis. Seitenzahl et al. (in preparation) find that in two-dimensional SN Ia simulations with $80^{2}$ tracer particles all isotopes with abundances higher than $\sim 10^{-5}$ are reproduced with an accuracy of better than 5% (except for $^{20}$Ne).
The nuclear postprocessing calculations are then performed for each particle separately. To this end, we recalculate the temperature from the recorded internal energy, rather than directly using the value obtained by the hydrodynamic simulations (see Travaglio et al. 2004a). In deriving the temperature, the electron fraction ($Y_{\rm e}$) is assumed to be 0.5, which introduces some errors when the electron captures are very active. When $T_{9} = T/10^9$ K $ > 6$, we follow the abundance evolution by applying the Nuclear Statistical Equilibrium (NSE) abundance (i.e., the abundance specified by $\rho$, $T$, and $Y_{\rm e}$) rather than fully solving the reaction network, as the NSE is reached in this regime of high density and temperature. In order to correctly follow the evolution of $Y_{\rm e}$, weak interactions are computed along with the NSE abundance.
Throughout this paper, the initial C+O WD composition is assumed as follows: X($^{12}$C) $= 0.475$, X($^{16}$O) $=0.5$, and X($^{22}$Ne) $= 0.025$ in mass fractions. This is consistent with the compositions used in the W7 model, roughly corresponding to the initially solar CNO composition: Metallicity is represented by $^{22}$Ne, assuming that the CNO cycle in the H-burning has converted all heavy elements to $^{14}$N, and then it is reprocessed to $^{22}$Ne by $^{14}$N($\alpha, \gamma$)$^{18}$F(${\rm e}^{+}, \nu_{\rm e}$) $^{18}$O($\alpha, \gamma$)$^{22}$Ne in the He-burning.
The reaction network (Thielemann et al. 1996) includes 384 isotopes up to $^{98}$Mo. The electron capture rates, which strongly affect nucleosynthesis at the beginning of the deflagration stage, are taken from Langanke & Martinez-Pinedo (2000) and Martinez-Pinedo et al. (2000). Detailed comparisons between the new rates and those of Fuller et al. (1982, 1985) are presented by Brachwitz et al. (2000) for the spherical 1D deflagration model W7 (see also Thielemann et al. 2004).
Results
=======
Characteristic Burning Regimes
------------------------------
Before presenting the result of our calculations, we summarize the basics of explosive burning taking place in the deflagration and detonation stages. Typical burning products can be characterized by the maximum temperatures ($T_{\rm max}$) and densities ($\rho_{\rm max}$) attained by the material under consideration, after the passage of the thermonuclear flame \[see, e.g., Arnett (1996), Thielemann et al. (1998), and references therein, for a review of explosive nucleosynthesis\].
The explosive burning in thermonuclear supernovae proceeds in different regimes, mainly characterized by the temperatures reached in the nuclear ashes (Thielemann et al. 1986). In deflagrations, this directly translates into characteristic fuel densities ahead of the flame. At the first stage of the deflagration, the density is higher than $10^{8}$ g cm$^{-3}$. Temperatures rise to $T_{{\rm max}, 9} \equiv
T_{\rm max}/10^{9}$K $\gsim 6$. At this temperature, NSE applies, and thus the final composition is determined by freezeout processes (represented by $T_{\rm max}$ and $\rho_{\rm max}$) and the efficiency of electron capture reactions. Because of the high density, electron capture reactions are important. Dominant species in this “complete silicon burning with electron capture region” are stable $^{56}$Fe, $^{54}$Fe, $^{58}$Ni, and radioactive $^{56}$Ni (which decays into $^{56}$Fe). In the case of even stronger electron capture reactions, the main products are $^{50}$Ti, $^{54}$Cr, and $^{58}$Fe (e.g., Thielemann et al. 2004).
Following the expansion of the WD, the deflagration proceeds progressively at the lower density. At $T_{{\rm max}, 9}
\sim 5 - 6$ and $\rho_{\rm max} \sim 5 \times 10^{7} - 10^{8}$ g cm$^{-3}$, NSE still applies. The electron captures are no longer important, and thus the initial $Y_{\rm e}$ is virtually preserved. The dominant species are $^{56}$Ni and $^{58}$Ni.
In contrast to spherically-symmetric delayed detonation models, the detonation in our two-dimensional setups is triggered at certain spots at the flame front, but at other locations, deflagration burning can still proceed for some while. It is characterized by successively lower temperature and density. At $T_{{\rm max}, 9} \sim 3 - 5$, oxygen burning or incomplete silicon burning is a result, leaving mainly intermediate-mass-elements (IME) such as $^{28}$Si and $^{32}$S. At $T_{{\rm max}, 9} \sim 2 - 3$, carbon burning or neon burning is a result, characterized by abundant $^{16}$O and $^{24}$Mg in consumption of $^{12}$C. At even lower temperatures ($T_{{\rm max}, 9} \lsim 2$), virtually no major thermonuclear reactions take place.
The detonation wave can also be characterized by the same tendency, although the temperature reached in the ashes is not a unique function of the fuel density anymore but also depends on the shock strength, the effect of which needs further investigation in hydrodynamic simulations. In the delayed detonation framework, the detonation is ignited after significant expansion of the WD, and thus the electron captures are almost always unimportant (§3.4). If there is still a high density region left after the deflagration, the detonation wave can convert the material chiefly into $^{56}$Ni by complete silicon burning. As the density decreases, the detonation nucleosynthesis is characterized by oxygen burning, carbon burning, and eventually burning ceases. Compared to the corresponding deflagration case at a similar burning stage, these stages are usually encountered at lower fuel densities, as the detonation compresses the burning material.
Structure of the ejecta
-----------------------
Table 1 shows the global features of our calculations. In the C-DEF model, more than half of the WD material is left unburned. The final kinetic energy, $\sim 4 \times 10^{50}$ erg, is significantly lower than that inferred for normal SNe Ia. In the C-DDT model, $\sim 0.5 M_{\odot}$ of the C/O material is incinerated by the detonation wave, after the deflagration already has burned $\sim 0.5 M_{\odot}$. The incinerated mass and nuclear energy release in the deflagration stage of the O-DDT model are smaller than the C-DDT model. The detonation wave, however, burns a larger amount of material and produces a larger amount of nuclear energy in the O-DDT model than in the C-DDT model. The final incinerated mass and the kinetic energy in the O-DDT model are comparable to those in the W7 model. Hereafter, we discuss how these results can be understood in terms of the flame propagation and the DDT. Note that the nuclear energy release in the hydrodynimic simulations (with a simplified treatment for the nuclear reactions) and that in the detailed reaction network calculations are consistent within 5%.
Figures 1 – 3 show the distribution of selected species at 10 seconds after the ignition of the deflagration flame. At this time, burning has ceased and the SN material is already almost in a homologous expansion. The C-DEF model (Fig. 1) shows large-scale mixing, and thus it does not possess a clearly layered structure as predicted in exactly spherical 1D models (e.g., W7). A large amount of unburned carbon and oxygen are left, being mixed down to the central region.
In contrast, the detonation wave produces a more layered structure, since this supersonically propagating wave is unaffected by hydrodynamic instabilities (at least on the large scales resolved in the models). In the C-DDT model, the deflagration burns out much of the center of the WD before the detonation is triggered. Therefore, the detonation mainly burns material towards the stellar surface, but it also propagates down the fuel funnels between the fingers of the deflagration ash (Fig. 1). In the C-DDT model, the whole WD, including the central region, experiences the strong expansion in the vigorous deflagration stage. Therefore, the unburned material left after the deflagration is processed in the detonation stage mainly by carbon and oxygen burning, producing O and IME, but virtually no Fe-peak elements (Fig. 2). In this particular model, Fe-peak elements are thus produced mostly in the initial deflagration stage.
In the O-DDT model the burning proceeds in an aspherical manner. Figures 4 and 5 show the temporal evolution in the O-DDT model. The deflagration frame, ignited off-center, floats outward, and spreads laterally (Röpke et al. 2007d). It creates a large blob of ashes in the upper hemisphere of the WD. Due to the lateral expansion in the outer layers of the star, the neutron-rich Fe-peak elements produced by electron captures end up in a characteristic off-center shell-like layer (Fig. 3). The detonation triggers on top of this blob at $\sim$ 1 second. It cannot cross ash regions (Maier & Niemeyer 2006) and has to burn around the ash blob in order to reach the central parts of the WD. Consequently, it initially propagates only outward (at $\sim$ 1.1 second).
At about 1.5 sec after the deflagration ignition, the detonation wave has burned around the ash blob and propagates inward. Since the off-center deflagration did not release much energy, the central parts of the WD are still dense and contain mostly unburned material. This material is converted to Fe-peak (predominantly $^{56}$Ni) by the detonation wave.
Characteristic thermal properties
---------------------------------
To understand the characteristic structure of the ejecta described in §3.2, we examine details of the thermal history of the tracer particles. Figure 6 shows the maximum temperature ($T_{\rm max}$) and density ($\rho_{\rm max}$) for each tracer particle (§3.1) after flame passage. For comparison, the same values for the classical W7 model are also shown. In Figure 7, average behaviors are plotted, which were obtained by taking the mass-average as a function of $T_{\rm max}$.
Figures 6 and 7 show that the distribution of ($T_{\rm max}$, $\rho_{\rm max}$) is similar for all the models at $T_{{\rm max}, 9} \gsim 3$ (corresponding to $\rho_{\rm max}
\sim \rho_{\rm DDT}$). This reflects the fact that the local properties of the deflagration flame are basically independent from the geometry. Interestingly, despite the different prescription of the deflagration flame propagation and different expansion time scale (see e.g., Travaglio et al. 2004a), we note a similarity with the outcome of the W7 in this respect.
At $T_{{\rm max}, 9} \lsim 3$, the models differ. Note that the behavior of W7 in this temperature regime shown in Figures 6 and 7 is not real; the deflagration is turned off at $T_{{\rm max}, 9} \sim 2$, where the flame is already near the surface in the W7 model. Our deflagration model extends to lower $T_{\rm max}$ and $\rho_{\rm max}$. In the two delayed detonation models, this temperature range corresponds to the detonation phase, and thus the behavior is expected to deviate from the deflagration models, and indeed it does. The two detonation models show a similar $T_{\rm max} - \rho_{\rm max}$ distribution; the compression and the maximum temperature are roughly linear functions of the unburned density.
Figure 6 shows a significant dispersion in the $T_{\rm max} - \rho_{\rm max}$ distribution around the mean value, reaching nearly one order of magnitude in $\rho_{\rm max}$ for given $T_{\rm max}$, at $T_{{\rm max}, 9} \lsim 5$. This is a unique behavior in the multi-D models, potentially leading to a variety of burning products otherwise not expected in strictly spherically symmetric models like W7.
The different hydrodynamic properties lead to different amounts of material as a function of $T_{\rm max}$ (or $\rho_{\rm max}$), and this is the main reason why different nucleosynthesis features appear. Figure 8 shows the mass of the material as a function of $T_{\rm max}$. It is seen that the deflagration is efficient in W7, and thus the W7 model has a much larger amount of the mass with $T_{{\rm max}, 9} \gsim 6$ than the 2D models. At $T_{{\rm max}, 9} \lsim 3 - 4$, there appears a peak in the C-DDT model; this is due to the detonation wave burning the fuel material left behind the deflagration stage. The O-DDT model shows a characteristic distribution; two peaks at $T_{{\rm max}, 9} \sim 3 - 4$ and at $T_{{\rm max}, 9} \sim 5 - 6$. The former is the detonation wave propagating outward, as also seen in the C-DDT model. The latter one at higher $T_{\rm max}$ is due to the detonation wave propagating inward, near the central region where the fuel still has high densities. The gap between the two peaks corresponds to the (off-center) shell structure of the deflagration products (i.e., neutron-rich Fe-peak elements; Fig. 3).
Deflagration and Electron Captures
----------------------------------
In the densest region in the deflagration stage, electron capture reactions proceed to synthesize neutron-rich Fe-peak elements under NSE conditions. Figure 9 shows the electron fraction ($Y_{\rm e}$) as a function of $\rho_{\rm max}$, for the O-DDT model. The same figure for the other 2D models (not shown) is similar to the O-DDT model. Figure 9 shows that the electron capture becomes important at $\rho_{\rm max}
\gsim 5 \times 10^{8}$ g cm$^{-3}$, and $Y_{\rm e}$ as low as $0.463$ is realized in the highest density region. Thus, the main product of the highest density region is stable $^{56}$Fe ($Y_{\rm e} \sim 0.464$), followed by $^{54}$Fe ($Y_{\rm e} \sim 0.481$) and $^{58}$Ni ($Y_{\rm e} \sim 0.483$) in the lower density region. Since $Y_{\rm e} > 0.46$, $^{50}$Ti ($Y_{\rm e} \sim 0.440$), $^{54}$Cr ($Y_{\rm e} \sim 0.444$) and $^{58}$Fe ($Y_{\rm e} \sim 0.448$) are not produced abundantly in the present models. The behavior in $Y_{\rm e}$ and the resulting electron capture reactions are largely consistent with the 2D deflagration model presented in Travaglio et al. (2004a). The electron captures, however, are less efficient in the 2D models than in the 3D deflagration model (Travaglio et al. 2004a) and in the classical 1D models (Thielemann et al. 2004; see also §3.6). This is likely due to the weaker deflagration in the present models than in the others (1D and 3D), although an uncertainty is involved in the treatment of the temperature evolution in the NSE regime (§2.2).
Figure 10 shows the comparison between $Y_{\rm e}$ before and after the DDT. In the C-DDT model, $Y_{\rm e}$ is not affected by the detonation wave as is consistent with previous 1D models. On the other hand, in the O-DDT model the detonation slightly affects $Y_{\rm e}$ especially of the material which was hardly processed by the deflagration ($\sim Y_{\rm e} \gsim 0.496$). The change in $Y_{\rm e}$ introduced by the detonation is at most $\Delta Y_{\rm e} \sim 0.02$, much smaller than the change in the deflagration stage at $\gsim 10^{8} - 10^{9}$g cm$^{-3}$. This is a result of electron capture reactions at the detonation wave propagating near the WD center, since the central region is still at high density (e.g., Meakin et al. 2009). The change is not as large as that in the GCD models of Jordan et al. (2008) and Meakin et al. (2009), which resulted in $\Delta Y_{\rm e} \sim 0.05$. This is because the distribution of the initial bubbles in our simulations is not as extreme as theirs (see §2.1) and the WD pre-expansion before the onset of the detonation was stronger in our model.
Distribution of Nucleosynthesis Products
----------------------------------------
Figure 11 shows the radial abundance distribution, for which the material is angle-averaged within each velocity bin. Note that our models are followed up to $\sim 10$ seconds, thus the velocity is proportional to the radial distance, and the distribution in velocity space corresponds in good approximation to that in spatial space. The O-DDT model has a very aspherical ejecta structure. Therefore, the angle-dependent radial abundance distribution in this model is given in Figure 12.
In the C-DEF model, different burning regions are macroscopically well-mixed. As a consequence, it forms a broad region in which the electron capture products ($^{56}$Fe, $^{54}$Fe, and $^{58}$Ni), complete-silicon burning products ($^{56}$Ni), intermediate mass elements ($^{32}$S, $^{28}$Si, $^{24}$Mg), and unburned elements ($^{16}$O and $^{12}$C) coexist (when averaging in the zenith angle). This region is surrounded by unburned material at radii not reached by the flame. An enhancement of C and O near the center due to the large-scale deflagration mixing is visible. These are typical of 2D deflagration models; this smoothing effect tends to be suppressed in 3D deflagration models (Röpke et al. 2007b).
In the C-DDT model, the inner region below $\sim 8,000$ km s$^{-1}$ has a similar structure to that in the deflagration model. The difference is seen in the innermost region, where unburned C and O are burned into intermediate mass elements by the detonation wave. In this model, C is almost completely burned, but there still remains O with a mass fraction of $\sim 0.1$, since the detonation in this model leads to carbon or oxygen burning.
Surrounding the deflagration region is the oxygen burning layer, where $^{28}$Si, $^{32}$S, and $^{24}$Mg are the main products. This is surrounded by a mixture of carbon burning products and unburned material. Note that the carbon burning region and the unburned region are microscopically separated. As shown in Figure 2, the unburned region (with almost initial C+O composition) is not fully homogeneously distributed, and there are “fingers” of carbon burning regions (where oxygen and IMEs are produced in consumption of carbon).
\
\
The radial composition distribution in the O-DDT model follows that of the C-DDT model. A striking difference, however, is that the region of deflagration ashes is not in the center but confined within $V \sim 5,000 - 10,000$ km s$^{-1}$. The innermost region is converted to $^{56}$Ni, by the complete silicon burning taking place at the inward detonation wave.
[ccccccc]{} C & 4.99E-02 & 3.64E-01 & 1.21E-02 & 3.49E-03 & 4.17E-01 & 4.86E-01\
N & 1.11E-06 & 1.54E-06 & 1.15E-06 & 2.80E-07 & 2.55E-06 & 1.14E-05\
O & 1.40E-01 & 4.27E-01 & 3.76E-01 & 1.51E-01 & 4.61E-01 & 5.21E-01\
F & 7.19E-10 & 1.04E-09 & 1.54E-09 & 2.70E-10 & 5.40E-10 & 3.56E-10\
Ne & 4.28E-03 & 2.51E-02 & 1.13E-02 & 4.13E-03 & 2.49E-02 & 2.82E-02\
Na & 6.59E-05 & 1.59E-04 & 2.07E-04 & 5.87E-05 & 7.09E-05 & 6.15E-05\
Mg & 1.63E-02 & 9.92E-03 & 5.90E-02 & 2.16E-02 & 4.81E-03 & 2.27E-03\
Al & 1.00E-03 & 7.23E-04 & 3.22E-03 & 1.17E-03 & 3.45E-04 & 1.99E-04\
Si & 1.67E-01 & 5.19E-02 & 3.38E-01 & 2.87E-01 & 3.39E-02 & 1.38E-02\
P & 3.86E-04 & 1.91E-04 & 1.36E-03 & 5.91E-04 & 1.34E-04 & 8.34E-05\
S & 7.97E-02 & 1.98E-02 & 1.19E-01 & 1.27E-01 & 1.33E-02 & 6.05E-03\
Cl & 1.38E-04 & 4.15E-05 & 3.69E-04 & 2.20E-04 & 5.81E-05 & 7.83E-05\
Ar & 1.31E-02 & 2.94E-03 & 1.62E-02 & 2.20E-02 & 2.41E-03 & 1.61E-03\
K & 6.45E-05 & 1.82E-05 & 1.71E-04 & 1.31E-04 & 3.25E-05 & 8.11E-05\
Ca & 9.76E-03 & 2.05E-03 & 6.91E-03 & 1.70E-02 & 1.72E-03 & 1.60E-03\
Sc & 1.33E-07 & 2.55E-08 & 2.38E-07 & 2.29E-07 & 1.59E-07 & 1.59E-05\
Ti & 3.94E-04 & 9.97E-05 & 2.08E-04 & 8.72E-04 & 1.53E-04 & 1.98E-03\
V & 4.04E-05 & 2.15E-05 & 2.95E-05 & 6.52E-05 & 1.29E-04 & 3.01E-03\
Cr & 5.28E-03 & 3.57E-03 & 3.52E-03 & 1.04E-02 & 4.41E-03 & 2.07E-02\
Mn & 6.72E-03 & 7.35E-03 & 5.95E-03 & 7.35E-03 & 1.83E-02 & 4.42E-02\
Fe & 7.61E-01 & 3.89E-01 & 3.60E-01 & 6.51E-01 & 3.36E-01 & 2.06E-01\
Co & 8.29E-04 & 8.20E-04 & 6.70E-04 & 6.36E-04 & 1.31E-03 & 3.22E-03\
Ni & 1.19E-01 & 8.42E-02 & 7.50E-02 & 8.05E-02 & 6.88E-02 & 4.30E-02\
Cu & 2.55E-06 & 8.45E-07 & 7.82E-07 & 2.13E-06 & 1.38E-06 & 1.26E-05\
Zn & 3.80E-05 & 1.01E-05 & 8.82E-06 & 3.50E-05 & 1.08E-06 & 5.48E-06\
Total Yields
------------
Figure 13 shows the nucleosynthesis yield for each model, integrated over the whole ejecta. The ratios of the masses of elements to those of the W7 model are plotted in Figure 14. Tables 2 and 3 list the masses of the elements and isotopes (after radioactive decays) while major long-lived radioactive isotopes are given in Table 4. For the C-DDT and O-DDT models, the values at the time of the first DDT ($\sim 1.15$ sec for C-DDT and $\sim 1.05$ sec for O-DDT) are also shown in Tables 2 and 3.
For comparison, we present the yields of the W7 model, calculated by ourselves using the thermal history of the original model. Details are different from Iwamoto et al. (1999), because of the updated electron capture rates (see Brachwitz et al. 2000).
The result for the C-DEF model is consistent with the similar 2D model in the previous study (Travaglio et al. 2004a). First of all, the mass of $^{56}$Ni is only $\sim 0.25 M_{\odot}$, smaller than in the W7 model ($\sim 0.64 M_{\odot}$). A large amount of C and O are left unburned. The final C/O ratio is larger than in the W7 model, because of the weak C-burning in our model resulting in almost the original, unburned C/O ratio near the surface (Fig. 11). IMEs are underproduced as compared to the W7 model. Note, however, that generally full three-dimensional models alleviate this problem (e.g., Travaglio et al. 2004a; Röpke et al. 2007b).
In the C-DDT model, the density of the WD is already low when the DDT takes place, and the temperature of material after the passage of the detonation wave does not reach $\sim 5 \times 10^{9}$ K (Fig. 8). The detonation therefore produces at most IMEs; Fe-peak elements are produced in the initial deflagration stage, and the mass of $^{56}$Ni is almost the same as in the deflagration model. Consequently, the abundance pattern of the Fe-peak elements (including $^{56}$Ni) is very similar to that of the C-DEF model (Figs. 13 and 14). IMEs are produced much more abundantly than in the W7 model. Because of the carbon burning, the C/O ratio is much smaller than for W7.
Somewhat surprisingly, the O-DDT model predicts the abundance pattern similar to W7 (Fig. 13), with most of elements consistent with W7 within a factor of two (Fig. 14). The mass of $^{56}$Ni ($\sim 0.54 M_{\odot}$) is comparable with the W7 model ($\sim 0.64 M_{\odot}$) (but see §4). IMEs tend to be overproduced by a factor of $\sim 2$, because of the carbon and oxygen burning in the detonation phase. Carbon is almost entirely consumed by the carbon burning.
A comparison between the W7 model and our 2D models shows that very strong electron capture reactions are not efficient in the 2D models (§3.4). $^{50}$Ti, $^{54}$Cr, and $^{58}$Fe, produced at the highest density resulting in $Y_{\rm e} < 0.46$, are not abundantly produced in the 2D models.
Radioactive Isotopes
--------------------
The amount of some radioactive isotopes in our detonation models is quite different from W7 and even with canonical, spherical delayed detonation models. Comparison between our results (Tab. 4) and Table 4 of Iwamoto et al. (1999) shows that radioactive isotopes lighter than Fe-peak tend to be more abundant, with the tendency of larger amounts of these isotopes for smaller density at the detonation. On the other hand, the amount of some Fe-peak radioactive isotopes can change dramatically from the classical 1D delayed detonation model, due to updated electron capture rates in our calculations (§2.2).
[ccccccc]{} $^{12}$C & 4.99E-02 & 3.64E-01 & 1.21E-02 & 3.49E-03 & 4.17E-01 & 4.86E-01\
$^{13}$C & 9.57E-07 & 2.29E-06 & 2.80E-06 & 8.75E-07 & 1.16E-06 & 1.35E-06\
$^{14}$N & 1.11E-06 & 1.54E-06 & 1.15E-06 & 2.79E-07 & 2.55E-06 & 1.14E-05\
$^{15}$N & 1.71E-09 & 1.65E-09 & 3.32E-09 & 7.06E-10 & 2.79E-09 & 9.71E-09\
$^{16}$O & 1.40E-01 & 4.27E-01 & 3.76E-01 & 1.51E-01 & 4.62E-01 & 5.21E-01\
$^{17}$O & 3.59E-08 & 4.51E-08 & 7.61E-08 & 1.46E-08 & 7.60E-08 & 3.96E-07\
$^{18}$O & 1.13E-09 & 1.80E-09 & 2.42E-09 & 5.24E-10 & 2.04E-09 & 2.57E-08\
$^{19}$F & 7.19E-10 & 1.04E-09 & 1.54E-09 & 2.70E-10 & 5.40E-10 & 3.56E-10\
$^{20}$Ne & 1.69E-03 & 6.13E-03 & 1.10E-02 & 4.05E-03 & 3.06E-03 & 2.65E-03\
$^{21}$Ne & 1.06E-05 & 1.66E-05 & 2.36E-05 & 4.35E-06 & 7.40E-06 & 4.09E-06\
$^{22}$Ne & 2.57E-03 & 1.90E-02 & 2.82E-04 & 7.62E-05 & 2.18E-02 & 2.55E-02\
$^{23}$Na & 6.59E-05 & 1.59E-04 & 2.07E-04 & 5.87E-05 & 7.09E-05 & 6.15E-05\
$^{24}$Mg & 1.62E-02 & 9.55E-03 & 5.84E-02 & 2.14E-02 & 4.61E-03 & 2.14E-03\
$^{25}$Mg & 6.12E-05 & 1.72E-04 & 2.95E-04 & 8.90E-05 & 9.50E-05 & 6.84E-05\
$^{26}$Mg & 3.79E-05 & 1.93E-04 & 3.49E-04 & 1.13E-04 & 1.12E-04 & 5.94E-05\
$^{27}$Al & 1.00E-03 & 7.23E-04 & 3.22E-03 & 1.17E-03 & 3.45E-04 & 1.99E-04\
$^{28}$Si & 1.64E-01 & 5.02E-02 & 3.27E-01 & 2.83E-01 & 3.31E-02 & 1.34E-02\
$^{29}$Si & 9.72E-04 & 6.52E-04 & 3.64E-03 & 1.47E-03 & 3.49E-04 & 1.94E-04\
$^{30}$Si & 2.04E-03 & 9.88E-04 & 7.45E-03 & 2.83E-03 & 4.88E-04 & 1.87E-04\
$^{31}$P & 3.86E-04 & 1.91E-04 & 1.36E-03 & 5.91E-04 & 1.34E-04 & 8.34E-05\
$^{32}$S & 7.74E-02 & 1.87E-02 & 1.07E-01 & 1.21E-01 & 1.27E-02 & 5.76E-03\
$^{33}$S & 3.62E-04 & 1.01E-04 & 1.04E-03 & 5.18E-04 & 7.71E-05 & 6.36E-05\
$^{34}$S & 1.96E-03 & 9.80E-04 & 1.11E-02 & 5.65E-03 & 5.50E-04 & 2.23E-04\
$^{36}$S & 4.10E-07 & 1.49E-07 & 1.30E-06 & 4.78E-07 & 6.19E-08 & 1.97E-08\
$^{35}$Cl & 1.10E-04 & 3.41E-05 & 2.90E-04 & 1.70E-04 & 4.96E-05 & 5.87E-05\
$^{37}$Cl & 2.81E-05 & 7.45E-06 & 7.87E-05 & 5.03E-05 & 8.54E-06 & 1.96E-05\
$^{36}$Ar & 1.21E-02 & 2.54E-03 & 1.21E-02 & 1.93E-02 & 2.19E-03 & 1.49E-03\
$^{38}$Ar & 9.46E-04 & 3.98E-04 & 4.02E-03 & 2.67E-03 & 2.15E-04 & 1.20E-04\
$^{40}$Ar & 6.10E-09 & 1.32E-09 & 1.46E-08 & 5.49E-09 & 7.76E-10 & 8.48E-10\
$^{39}$K & 6.01E-05 & 1.71E-05 & 1.60E-04 & 1.22E-04 & 3.07E-05 & 6.43E-05\
$^{40}$K & 4.34E-08 & 7.17E-09 & 7.75E-08 & 3.88E-08 & 1.71E-08 & 1.95E-07\
$^{41}$K & 4.36E-06 & 1.10E-06 & 1.18E-05 & 8.74E-06 & 1.79E-06 & 1.66E-05\
$^{40}$Ca & 9.73E-03 & 2.04E-03 & 6.82E-03 & 1.69E-02 & 1.71E-03 & 1.52E-03\
$^{42}$Ca & 2.38E-05 & 8.38E-06 & 8.71E-05 & 6.15E-05 & 5.73E-06 & 7.91E-06\
$^{43}$Ca & 7.27E-08 & 1.84E-08 & 1.31E-07 & 1.61E-07 & 5.11E-08 & 3.38E-06\
$^{44}$Ca & 8.40E-06 & 1.94E-06 & 3.31E-06 & 1.59E-05 & 5.66E-06 & 6.22E-05\
$^{46}$Ca & 3.65E-11 & 2.45E-12 & 5.80E-11 & 2.30E-11 & 1.80E-12 & 4.32E-10\
$^{48}$Ca & 2.82E-13 & 1.94E-17 & 4.27E-16 & 1.75E-16 & 2.44E-16 & 3.70E-12\
$^{45}$Sc & 1.33E-07 & 2.55E-08 & 2.38E-07 & 2.29E-07 & 1.59E-07 & 1.59E-05\
$^{46}$Ti & 1.16E-05 & 3.57E-06 & 3.34E-05 & 2.52E-05 & 3.32E-06 & 5.78E-05\
$^{47}$Ti & 5.15E-07 & 1.36E-07 & 7.98E-07 & 1.08E-06 & 8.80E-07 & 8.15E-05\
$^{48}$Ti & 3.65E-04 & 9.23E-05 & 1.65E-04 & 8.16E-04 & 1.38E-04 & 1.34E-03\
$^{49}$Ti & 1.48E-05 & 3.68E-06 & 8.52E-06 & 2.95E-05 & 1.12E-05 & 4.97E-04\
$^{50}$Ti & 1.85E-06 & 9.98E-10 & 3.77E-09 & 1.34E-08 & 2.85E-08 & 7.73E-07\
$^{50}$V & 3.50E-09 & 3.82E-09 & 7.13E-09 & 4.88E-09 & 6.24E-08 & 4.99E-06\
$^{51}$V & 4.04E-05 & 2.15E-05 & 2.95E-05 & 6.52E-05 & 1.28E-04 & 3.01E-03\
$^{50}$Cr & 2.61E-04 & 1.56E-04 & 3.15E-04 & 3.84E-04 & 2.03E-04 & 2.42E-03\
$^{52}$Cr & 4.32E-03 & 2.70E-03 & 2.60E-03 & 8.91E-03 & 2.92E-03 & 9.14E-03\
$^{53}$Cr & 6.60E-04 & 7.12E-04 & 6.03E-04 & 1.08E-03 & 1.28E-03 & 9.12E-03\
$^{54}$Cr & 3.48E-05 & 6.85E-08 & 1.82E-07 & 6.68E-07 & 1.17E-06 & 1.07E-05\
$^{55}$Mn & 6.72E-03 & 7.35E-03 & 5.95E-03 & 7.34E-03 & 1.83E-02 & 4.42E-02\
$^{54}$Fe & 7.77E-02 & 8.15E-02 & 7.01E-02 & 7.40E-02 & 5.98E-02 & 6.55E-02\
$^{56}$Fe & 6.57E-01 & 2.95E-01 & 2.78E-01 & 5.59E-01 & 2.57E-01 & 1.11E-01\
$^{57}$Fe & 2.56E-02 & 1.26E-02 & 1.23E-02 & 1.80E-02 & 1.93E-02 & 2.91E-02\
$^{58}$Fe & 2.29E-04 & 1.36E-05 & 1.27E-05 & 1.57E-05 & 6.42E-05 & 3.63E-04\
$^{59}$Co & 8.29E-04 & 8.20E-04 & 6.70E-04 & 6.36E-04 & 1.31E-03 & 3.22E-03\
$^{58}$Ni & 1.06E-01 & 7.24E-02 & 6.63E-02 & 7.03E-02 & 6.19E-02 & 3.90E-02\
$^{60}$Ni & 9.75E-03 & 1.11E-02 & 8.05E-03 & 8.35E-03 & 6.77E-03 & 3.68E-03\
$^{61}$Ni & 2.73E-04 & 7.20E-05 & 6.95E-05 & 2.05E-04 & 4.92E-05 & 2.30E-04\
$^{62}$Ni & 2.52E-03 & 6.35E-04 & 5.85E-04 & 1.61E-03 & 7.41E-05 & 1.07E-04\
$^{64}$Ni & 2.20E-07 & 6.32E-11 & 1.52E-10 & 1.32E-09 & 7.15E-09 & 3.61E-07\
$^{63}$Cu & 1.77E-06 & 6.49E-07 & 5.99E-07 & 1.34E-06 & 1.36E-06 & 1.21E-05\
$^{65}$Cu & 7.78E-07 & 1.96E-07 & 1.82E-07 & 7.85E-07 & 1.94E-08 & 5.56E-07\
$^{64}$Zn & 1.43E-05 & 3.74E-06 & 3.23E-06 & 1.38E-05 & 9.80E-07 & 5.11E-06\
$^{66}$Zn & 2.37E-05 & 6.39E-06 & 5.58E-06 & 2.13E-05 & 1.02E-07 & 3.36E-07\
$^{67}$Zn & 4.94E-11 & 2.29E-11 & 2.28E-11 & 8.14E-11 & 2.23E-10 & 2.32E-08\
$^{68}$Zn & 9.59E-09 & 2.71E-09 & 2.44E-09 & 9.55E-09 & 2.36E-10 & 6.69E-09\
$^{70}$Zn & 2.36E-14 & 2.60E-19 & 2.89E-18 & 5.53E-17 & 5.78E-15 & 2.93E-11\
Discussion: Implications for Chemical Evolution
===============================================
In this paper, we present our first results for nucleosynthesis in 2D delayed detonation models. The models investigated in this paper represent extreme cases[^1]; the deflagration is initiated either at the center (C-DDT) or by off-center bubbles confined in the narrow angle with respect to the $z$-axis (O-DDT). The important variation, we have not examined in this paper, is the deflagration bubbles distributed more or less spherically, but at a distance from the center (e.g., Kasen et al. 2009). In this case, we expect that the propagation of the burning will produce an ejecta structure that falls in between the O-DDT and the C-DDT models. Some material in the central region is left unexpanded in the deflagration stage, and the detonation can burn the material there to the Fe-peak elements. Thus, we expect that nucleosynthesis features are similar to the O-DDT model, except for the strong angle-dependence seen in the O-DDT model. Depending on the initial placement of the deflagration bubbles, the deflagration ashes may stay more confined to the center than in the O-DDT model.
Also, we should note that the treatment of the DDT in the hydrodynamic calculation is still preliminary (§2.1). According to simulations with several different prescriptions, especially with different values for $\rho_{\rm DDT}$, we believe that the prescription used in this paper results in a relatively weak detonation (see also Gamezo et al. 2005; Bravo & García-Senz 2008). Therefore, although the models presented here are extreme with respect to the ejecta structure, they do not necessarily reflect the range of $^{56}$Ni production possible within the framework of the delayed detonation model. The results of an extended survey of delayed detonation models will be presented elsewhere (F. Röpke et al., in prep). Despite these caveats, the general features found in this paper are not expected to be sensitive to such details.
[ccccc]{} $^{22}$Na & 2.01E-08 & 1.01E-07 & 1.46E-07 & 5.40E-08\
$^{26}$Al & 5.18E-07 & 1.69E-06 & 2.47E-06 & 8.77E-07\
$^{36}$Cl & 2.08E-06 & 4.74E-07 & 5.22E-06 & 2.06E-06\
$^{39}$Ar & 6.79E-09 & 1.53E-09 & 1.69E-08 & 7.75E-09\
$^{40}$K & 4.34E-08 & 7.17E-09 & 7.75E-08 & 3.90E-08\
$^{41}$Ca & 4.35E-06 & 1.10E-06 & 1.18E-05 & 8.85E-06\
$^{44}$Ti & 8.37E-06 & 1.93E-06 & 3.21E-06 & 1.59E-05\
$^{48}$V & 4.32E-08 & 1.68E-08 & 9.76E-08 & 1.09E-07\
$^{49}$V & 1.05E-07 & 1.00E-07 & 3.07E-07 & 2.69E-07\
$^{53}$Mn & 1.64E-04 & 4.93E-04 & 3.38E-04 & 2.25E-04\
$^{55}$Fe & 1.79E-03 & 4.17E-03 & 2.89E-03 & 1.93E-03\
$^{60}$Fe & 3.33E-09 & 9.86E-15 & 2.29E-13 & 6.93E-12\
$^{55}$Co & 4.89E-03 & 3.18E-03 & 3.07E-03 & 5.40E-03\
$^{56}$Co & 1.21E-04 & 1.18E-04 & 1.06E-04 & 1.04E-04\
$^{57}$Co & 9.52E-04 & 1.94E-03 & 1.40E-03 & 9.37E-04\
$^{60}$Co & 4.32E-08 & 5.30E-10 & 1.19E-09 & 3.30E-09\
$^{56}$Ni & 6.40E-01 & 2.45E-01 & 2.46E-01 & 5.40E-01\
$^{57}$Ni & 2.46E-02 & 1.06E-02 & 1.09E-02 & 1.71E-02\
$^{59}$Ni & 4.66E-04 & 7.24E-04 & 5.78E-04 & 4.22E-04\
$^{63}$Ni & 4.82E-08 & 4.28E-11 & 1.85E-10 & 1.22E-09\
The nucleosynthesis features in the W7 model have been shown to be roughly consistent with the Galactic chemical evolution (e.g., Iwamoto et al. 1999; Goswami & Prantzos 2000). One problem is that the ratio $^{58}$Ni/$^{56}$Fe is too large in the W7 to compensate the over-solar production of $^{58}$Ni in core-collapse SNe. The problem is basically solved by introducing a delayed detonation. Affecting primarily lower density material, it produces a large amount of $^{56}$Ni but virtually no $^{58}$Ni due to the inefficiency of electron captures here. We see that the ratio $^{58}$Ni/$^{56}$Fe is slightly decreased in the O-DDT model compared to the W7, marginally satisfying the constraint from the chemical evolution \[($^{58}$Ni/$^{56}$Fe)/($^{58}$Ni/$^{56}$Fe)$_{\odot}$ $\sim 3$\]. The situation should become better if the detonation burns more material to NSE than in the present prescription. We should also note one limitation in the present analysis, i.e., the treatment of the temperature in the NSE regime. The temperature is extracted assuming $Y_{\rm e} = 0.5$, which should introduce some errors when the electron capture reactions are active. This could to some extent affect the nucleosynthesis in the initial deflagration phase (§3.4), and therefore the ratio $^{58}$Ni/$^{56}$Fe.
The intermediate mass elements can also be used to constrain SN Ia models. Assuming a typical fraction of $\sim 20$% for the frequency of SNe Ia as compared to core-collapse SNe, then (Si/Fe)/(Si/Fe)$_{\odot}$ $\lsim 0.5$ is required to compensate the over-solar production in core-collapse SNe. The O-DDT model has the ratio (Si/Fe)/(Si/Fe)$_{\odot} \sim 1$; thus, the model can explain at most a half of SNe Ia. The C-DDT is not at all favored, since (Si/Fe)/(Si/Fe)$_{\odot} \sim 2$, i.e., it produces too much IMEs relative to Fe. The deflagration models (W7 and C-DEF) do not have the problem in the Si/Fe ratio \[(Si/Fe)/(Si/Fe)$_{]\odot} < 0.5$\]. In the O-DDT model, all the elements heavier than N are consistent with W7 within a factor of two.
In reality, the solar abundances are the superposition of contributions from SNe at various metallicities, and we expect that the “average” SNe Ia to occur at a metallicity less than solar. We indeed expect some improvement if we consider a lower metallicity (see e.g., Travaglio et al. 2005): a decreased metallicity should result in a smaller $^{58}$Ni/$^{56}$Fe ratio in the delayed detonation models. Also, the smaller metallicity is expected to lead to the smaller amount of $^{54}$Fe. Mg and Al are also affected: If the metallicity is 10% of the solar value, then the amount of $^{24}$Mg could increase by $\sim 50$%, and that of $^{27}$Al could decrease by a factor of a few. These changes do not conflict with the Galactic chemical evolution, since the ratios (Mg, Al)/Fe are much smaller than the solar values in the O-DDT models.
Summarizing, the hypothesis that about half of SNe Ia are represented by the extreme O-DDT model, is not rejected by the chemical evolution arguments. Note that this should also apply to globally symmetric, but off-center, delayed detonation models to some extent, as these models should share the basic feature that the central region is burned to Fe-peak and $^{56}$Ni by the detonation (§4). Therefore, multi-D delayed detonation models can potentially account for a majority of SNe Ia.
Concluding Remarks
==================
We have presented results of the detailed nucleosynthesis calculations for 2D delayed detonation models, focusing on an extremely off-center model (O-DDT model). Features are different from classical 1D models. Unlike a globally symmetric delayed detonation model following the central ignition of the deflagration flame (C-DDT model), the detonation wave proceeds both in the high density region near the center of a white dwarf and in the low density region near the surface. Thus, the resulting yield is a mixture of different explosive burning products, from carbon-burning at low densities to complete silicon-burning at the highest densities, as well as the electron-capture products from the deflagration stage.
The evolution of the deflagration flame can be different in 2D and in 3D simulations (e.g., Röpke et al. 2007b). We believe that the global feature found in this study, i.e., the detonation wave propagating into the innermost region, would not be changed substantially in 3D simulations. However, some details would be affected: For example, 3D simulations tend to result in stronger deflagration than in 2D simulations for similar initial conditions (e.g., Travaglio et al. 2004a). As the stronger deflagration is followed by the weaker detonation in our DDT models, this would result in a shift of the overall energetic and the nucleosynthesis production as compared to the 2D models. On the other hand, although in our 2D simulations the detonation wave has to burn around the deflagration ash blob before reaching to the center, the deflagration ash blob may have holes, depending on the ignition geometry through which the detonation wave can directly penetrate into the center in 3D simulations. Thus, future 3D simulations are important to obtain the robust model predictions as a function of the model input (e.g., distribution of the deflagration ignition sparks).
The yield of the O-DDT model largely satisfies constraints from the Galactic chemical evolution despite the low DDT density as compared to 1D delayed detonation models. This is a result of qualitatively different behavior of the detonation propagation in 1D and multi-D models; the outward propagation in former (which is also the case in the C-DDT model), while the inward propagation in the latter.
The O-DDT model could thus account for a fraction of SNe Ia (especially bright ones). As less-extreme (more spherical) models are expected to share the common properties in the integrated yield, the multi-D delayed detonation model could potentially account for a main population of SNe Ia.
The delayed detonation models also provide a characteristic layered structure, unlike multi-dimensional, especially 2D, deflagration models [^2]. In the O-DDT model, the region filled by electron capture species (e.g., $^{58}$Ni, $^{54}$Fe) shows a large off-set, and the region is within a shell above the bulk of the $^{56}$Ni distribution near the center. These can be directly tested by observations.
We note that the distribution of the nucleosynthsis products in our 2D DDT models is somewhat different from the result of Bravo & García-Senz (2008). Their 3D DDT models lack a clear abundance stratification, and they are characterized by large mass fractions of Fe-peak elements near the surface regions. In contrast, the 3D DDT models of Röpke & Niemeyer (2007c), do produce a layered structure with the surface dominated by IMEs – similar to our present 2D models (§3.5). It seems that (1) the deflagration is stronger in Bravo & García-Sent (2008) than in our 2D models and in 3D models of Röpke & Niemeyer (2007c), and (2) the detonation wave is mainly propagating inward in Bravo & García-Senz (2008) although it is propagating both inward (producing Fe-peak elements) and outward (producing IME) in our simulations. As a result, the amount of unburned material is smaller in our models. The cause of the different flame propagation is not clear, but likely due to different treatment of thermonuclear flames. The overall abundance distribution of the 3D DDT models of Gamezo et al. (2005), on the other hand, is similar to our 2D models, producing the stratified configuration. In their simulations, they initiated the detonation from the center at relatively high DDT density, and the detonation wave propagates outward, producing the layered composition structure as the temperature drops following the detonation propagation. This is, indeed, quite different from our models, in which the detonation is initiated at relatively low DDT density but propagates inward to the high density central region.
The observational consequences from similar models have been discussed by Kasen et al. (2009) for the early photospheric phase (see also Hillebrandt et al. 2007; Sim et al. 2007). Here, we summarize some additional, expected observational characteristics, especially in the late-time nebular phase (taken after a few hundred days). We emphasize that the late-time spectroscopy is currently the most effective way to hunt for the signature of the DDT model in the innermost region. See also Maeda et al. (2010) who discussed the following points in details.
- [**Abundance Stratification and carbon near the surface:** ]{} Overall, the stratified composition structure obtained in our 2D models is consistent with the result of the “abundance tomography” (Stehle et al. 2005; Mazzali et al. 2008). Spectrum synthesis models as compared to the observed early-phase photospheric spectra show that the mass fraction of carbon at $\sim 10,000 - 14,000$ km s$^{-1}$ should be smaller than $0.01$ (e.g., Branch et al. 2003; Thomas et al. 2007; Tanaka et al. 2008). This constraint is satisfied by our DDT models at $10,000$ km s$^{-1}$, although it is marginal at $14,000$ km s$^{-1}$. The latter could, however, be improved by changing the DDT criterion such that the transition on average proceeds at higher densities (e.g., Iwamoto et al. 1999). Another interesting observational target is the inhomogeneous distribution of the unburned C+O pockets near the surface both in C-DDT and O-DDT models (§3.5), although the total amount of such unburned material is small in the DDT models. Detectable carbon absorption lines may appear only when the observed line-of-sight intersects a large number of such unburned pockets, which may explain the low frequency of SNe Ia showing the C absorption lines detected (e.g., Tanaka et al. 2006).
- [**\[O I\] $\lambda\lambda$6300,6363:** ]{} The main problem in the pure deflagration model is existence of unburned carbon and oxygen mixed down to the central region. This should produce a strong \[O I\] $\lambda\lambda$6300, 6363 doublet in late-time spectra, although no such signature has been detected in the observations (Kozma et al. 2005). This tension is alleviated in 3D deflagration models, with the mass fraction of unburned elements going down to 10 % (Röpke et al. 2007b) or even to smaller values (e.g., Travaglio et al. 2004a). The DDT model does not have this problem, as unburned material near the center is burned in the subsequent detonation. A detailed study of late-time nebular spectra should provide us with information on this issue.
- [**Line shifts:** ]{} Profiles of nebular emission lines can be used to effectively trace the distribution of the burning products, as is proven to be efficient for core-collapse SNe (Maeda et al. 2002; Mazzali et al. 2005; Maeda et al. 2008; Modjaz et al. 2008; Taubenberger et al. 2009). The O-DDT model predicts that (1) the distribution of stable Fe-peak elements is off-set following the asymmetric deflagration flame propagation, while the detonation products (e.g., a large fraction of $^{56}$Ni) are distributed more or less in a spherical manner (Figs. 5 and 12). Recently, Maeda et al. (2010) found that the expected variation of the line wavelength is seen in nebular spectra of SNe Ia, indicating that the above configuration can be relatively common in SNe Ia.
- [**Line profiles:** ]{} Detailed profiles of the nebular emission lines can be used to infer the distribution of emitting ions. It has been suggested to use Near-Infrared (NIR) spectroscopy to hunt for the geometry, as NIR \[Fe II\] lines are not severely blended (e.g., Höflich et al. 2004; Motohara et al. 2006). This can also be done using several IR lines like \[Co III\] 11.88$\micron$ (Gerardy et al. 2007). The best data to date have been obtained for SN 2003hv, showing a hole in the distribution of $^{56}$Ni[^3]. Because the mixing is introduced by the deflagration, such a hole is difficult to understand in the present models (and most of SN Ia explosion models; but see Meakin et al. 2009). This issue remains unresolved, and need further study preferentially in 3D simulation.
We would like to thank Ken’ichi Nomoto for kindly providing the thermal history of the W7 model. This research has been supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. The work of K.M. is also supported through the Grant-in-Aid for Young Scientists (20840007) of Japanese Society for Promotion of Science (JSPS). The work of F.K.R. is supported through the Emmy Noether Program of the German Research Foundation (DFG; RO 3676/1-1) and by the Cluster of Excellence “Origin and Structure of the Universe” (EXC 153). The work by F.-K.T. is supported by the Swiss National Science Foundation(SNF) and the Alexander von Humholdt Foundation. The calculations have been performed on IBM Power5 system at Rechenzentrum Garching (RZG) of the Max-Planck Society.
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[^1]: Note that the O-DDT model is still less extreme than the GCD model (Jordan et al. 2008; Meakin et al. 2009) in the distribution of the initial bubbles.
[^2]: Note that the 3D deflagration models can also potentially produce the layered structure (Röpke et al. 2007b).
[^3]: Note, however, that the central wavelengths of the emission lines are shifted with respective to the explosion center, and this “line shift” can be well explained by the off-set DDT scenario (Maeda et al. 2010).
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[MRI-P-030701]{}\
[IC/2003/53]{}\
hep-ph/0307117\
[**Bhabha Scattering with Radiated Gravitons at Linear Colliders**]{}\
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$^a$ The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
E-mail: [[email protected]]{}
$^b$Harish-Chandra Research Institute, Chhatnag Road, Allahabad 211019, India.
E-mail: [[email protected], [email protected]]{}
$^c$Department of Physics, Indian Institute of Technology, Kanpur 208016, India.
E-mail: [[email protected]]{}
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> We study the process $e^+ e^- \to e^+ e^- \not{\!\!E}$ at a high-energy $e^+ e^-$ collider, where the missing energy arises from the radiation of Kaluza-Klein gravitons in a model with large extra dimensions. It is shown that at a high-energy linear collider, this process can not only confirm the signature of such theories but can also sometimes be comparable in effectiveness to the commonly discussed channel $e^+ e^- \to \gamma \not{\!\!E}$, especially for a large number of extra dimensions and with polarized beams. We also suggest some ways of distinguishing the signals of a graviton tower from other types of new physics signals by combining data on our suggested channel with those on the photon-graviton channel.
1 cm
PACS No. 04.50.+h, 11.10.Kk, 11.25.Mj, 12.60.-i, 13.88.+e
Introduction
============
It is a matter of common belief in the high-energy physics community that there must be physics beyond the Standard Model (SM) around the TeV scale. This belief is founded in the fact that some of the parameters of the SM can be made stable against quantum corrections only by invoking new physics at the TeV scale. With this in mind, an enormous international effort is being poured into the construction of the 14-TeV LHC at CERN, which is expected to run by 2008 and come up with signals for new physics. However, as the LHC is a hadronic machine, it is unlikely to yield signals which are unequivocal as to the underlying theory. It is therefore widely deemed necessary to build a high-energy $e^+ e^-$ collider to confirm the nature and detailed properties of this new physics.
A high-energy $e^+ e^-$ collider operating in the range 500 GeV to a few TeV must be of the linac type to avoid large synchrotron radiation losses. Such a machine would require a very high luminosity in order to be competitive with other accelerators. This is, in fact, an essential feature of the design[@LC] of proposed linear colliders. The relatively clean environment of an $e^+ e^-$ collider allows the complete reconstruction of individual events, making it possible to carry out precision tests of new physics. Moreover, one has the facility of polarizing the beams, which can play a significant role in background elimination. It is also possible to run such a machine in the $e^- e^-$, $e\gamma$ and $\gamma\gamma$ modes. In this paper we consider the possibility of using a high energy $e^+ e^-$ collider to detect signatures of low-scale quantum gravity in a model with large extra dimensions. In particular, we focus on Bhabha scattering with radiated gravitons.
One of the most exciting theoretical developments of recent years has been the idea that there could be one or more extra spatial dimensions and the observable Universe could be confined to a four-dimensional hypersurface in a higher-dimensional ‘bulk’ spacetime[@Akama]. Such ideas, which can be motivated by superstring theories, give rise to elegant solutions[@ADD] to the well-known gauge hierarchy problem of high energy physics, which is just the instability against quantum corrections mentioned above. What is even more interesting, perhaps, is the suggestion[@GRW; @HLZ] that there could be observable signals of quantum gravity at current and future accelerator experiments, and this possibility has spawned a vast and increasing body of work over the past five years. This relatively new set of ideas, commonly dubbed ‘Brane World Phenomenology’, bases itself on two main principles: the concept of hidden compact dimensions and the string-theoretic idea of $D_p$-branes. The simplest brane-world scenario is the so-called Arkani-Hamed—Dimopoulos—Dvali (ADD) model[@ADD], in which there are $d$ extra spatial dimensions, compactified on a $d$-torus of radius $R_c$ each way. Together with the four canonical Minkowski dimensions, this constitutes the ‘bulk’ spacetime. In this scenario the radius $R_c$ of the extra dimensions can be as large as a quarter of a millimeter[@LCP]. However, the SM fields are confined to a four-dimensional slice of spacetime, with thickness not more than $10^{-17}$ cm, which is called the ‘brane’. If the ADD model is embedded in a string-theoretic framework, the ‘brane’ is, in fact a $D_3$-brane, i.e. a 3+1 dimensional hypersurface on which the ends of open strings are confined[@Polchinksi]. However, it is not absolutely essential to embed the model in a string theory, and the word ‘brane’ or ‘wall’ is then used simply to denote the hypersurface (or thin slice) where the SM fields are confined. A crucial feature of this model is that gravity, which is a property of spacetime itself, is free to propagate in the bulk. As a result
- Planck’s constant in the bulk $M_S$, identified with the ‘string scale’, and fixed by the string tension $\alpha'$, is related to Planck’s constant on the brane $M_P$ ($\simeq~1.2\times 10^{19}$ GeV) by $$(M_S)^{2+d} = (4\pi)^{d/2} ~\Gamma(d/2) ~M_P^2 ~(R_c)^{-d}$$
which means that for $R_c \sim 0.2$ mm, it is possible to have $M_S$ as low as a TeV for $d \geq 2$. (The normalization of reference [@HLZ] has been adopted here). This solves the gauge hierarchy problem simply by bringing down the scale of new physics (i.e. strong gravity in this case) to about a TeV and thereby providing a natural cut-off to the SM, since the string scale $M_S$ now controls graviton-induced processes on the brane.
- There are a huge number of massive Kaluza-Klein excitations of the (bulk) graviton field on the brane, with masses $m_n = n/R_c$, and these collectively produce gravitational excitations of electroweak strength, which may be observable at current experiments and those planned in the near future[@Kubyshin].
It is only fair to mention that a major drawback of the ADD model is that it creates a new hierarchy between the ‘string scale’ $M_s \sim 1$ TeV and the size of the extra dimensions $R_c^{-1} \sim
1$ $\mu$eV. In fact, the huge size of the extra dimensions (compared to the Planck length) is not stable under quantum corrections, which tend to shrink it down until $M_s \sim R_c^{-1} \sim M_P \sim
10^{19}$ GeV, at which stage the original hierarchy problem is reinstated. Nevertheless, there are several variants of the ADD model which address this problem in various ways, and some of these ideas may not be far from the truth. From a phenomenological point of view, it is, therefore, reasonable to postpone addressing the stability issue, and proceed to study the minimal ADD model and its consequences for experiment.
The experimental consequences of ADD gravity have been mainly studied in the context of of ($a$) real graviton emission and ($b$) virtual graviton exchange. In the former case the final state gravitons in the ADD model are ‘invisible’, escaping the detector because of their feeble individual interactions ($\sim M_P^{-1}$) with matter. The final state, involving missing energy due to gravitons, will be built up by making an incoherent sum over the tower of graviton modes. In the case of virtual gravitons, the final state is built up by making a coherent sum. In either case, it may be shown[@GRW; @HLZ] that, after summing, the Planck mass $M_P$ cancels out of the cross-section, leaving an interaction of near-electroweak strength.
At an $e^+ e^-$ collider, the most-frequently discussed[@GRW; @MPP; @Wilson] signal for ADD gravitons is the process $e^+ e^- \to \gamma G_n$, where $G_n$ is the $n$th Kaluza-Klein excitation of the graviton field. This process leads to single photon events with missing energy and momentum, and is expected to be among the earliest signals of ADD gravity at an $e^+ e^-$ collider. In addition, the process $e^+ e^- \to \mu^+ \mu^- G_n$ has also been studied[@Eboli]. In this work, we have studied the process $$e^+ e^- \to e^+ e^- G_n$$ which is simply Bhabha scattering with a radiated graviton. The final state would contain an electron-positron pair with substantial missing energy. The reasons for taking up a study of this process are as follows.
- At an $e^+ e^-$ collider, final states with an $e^+ e^-$ pair will be one of the first things to be analyzed, since this is needed for beam calibration[@LC]. Among such states, it should be a simple matter to select events with large missing energy as well.
- Unlike $e^+ e^- \to \mu^+ \mu^- G_n$, this process does not suffer from large $s$-channel suppression, especially if one considers final state electrons (positrons) in relatively forward (backward) directions.
- Taken in conjunction with the process $e^+ e^- \to \gamma G_n$, this process could help to distinguish signals of the ADD scenario from other physics models beyond the SM. This is elaborated in the subsequent discussions.
- In some ways, as we shall see, this process has signatures more distinct from other new physics effects than $e^+ e^- \to \gamma G_n$.
In the following section we describe the salient features of our calculation of the signal process. Backgrounds and some strategies for their elimination are discussed in section 3. Section 4 contains the results of our numerical analysis. In section 5 we discuss how this process can be used to pinpoint the model if signatures of the suggested type are indeed observed. We summarize and conclude in the final section.
The Process $e^+ e^- \to e^+ e^- G_n$
======================================
As gravity couples to the energy-momentum tensor, each component of the tower of ADD gravitons couples to all SM fields, as well as to each SM interaction vertex. The relevant Feynman rules can be found, for example, in Refs.[@GRW; @HLZ]. Using these, it can be shown that the process $e^+ e^- \to e^+ e^- G_n$ is driven, at tree-level, by 28 Feynman diagrams. Some representative diagrams are shown in Fig. 1, which are obtained by ‘dressing’ an $s$-channel process $e^+ e^- \to e^+ e^-$ with graviton radiation. Fig. 1($a$) represents a graviton emission from one of the external electron legs; there will be 4 such diagrams with the graviton emitted from [*each*]{} leg in turn. Fig. 1($b$) represents a graviton emission from the gauge boson propagator. Fig. 1($c$) represents a graviton emission from one vertex; there will be another such with graviton emission from the other vertex. Thus Fig. 1 encompasses 14 diagrams, 7 each with $Z$ and photon exchanges. In addition, there will be another set of 14 diagrams with the $\gamma,Z$ exchange in the $t$-channel, making 28 diagrams in all. This latter set is absent in the case of $e^+ e^- \to \mu^+ \mu^- G_n$.
=1.3 pt
(90,90)(0,0) (0,20)(30,40) (30,40)(0,60) (30,40)(60,40)[4]{}[3]{} (60,40)(90,20) (90,60)(60,40) (15,50)(45,70)[3]{}[5]{} (45,70)(15,50)[-3]{}[5]{} (0,15)\[l\][$e^-$]{} (0,70)\[l\][$e^+$]{} (45,25)\[c\][$\gamma ,Z$]{} (100,15)\[r\][$e^+$]{} (100,70)\[r\][$e^-$]{} (59,75)\[r\][$G$]{} (45,8)\[c\][$(a)$]{}
(90,90)(0,0) (0,20)(30,40) (30,40)(0,60) (30,40)(60,40)[4]{}[3]{} (60,40)(90,20) (90,60)(60,40) (45,40)(45,70)[3]{}[5]{} (45,70)(45,40)[-3]{}[5]{} (0,15)\[l\][$e^-$]{} (0,70)\[l\][$e^+$]{} (45,25)\[c\][$\gamma ,Z$]{} (100,15)\[r\][$e^+$]{} (100,70)\[r\][$e^-$]{} (45,80)\[c\][$G$]{} (45,8)\[c\][$(b)$]{}
(90,90)(0,0) (0,20)(30,40) (30,40)(0,60) (30,40)(60,40)[4]{}[3]{} (60,40)(90,20) (90,60)(60,40) (60,40)(90,40)[3]{}[5]{} (90,40)(60,40)[-3]{}[5]{} (0,15)\[l\][$e^-$]{} (0,70)\[l\][$e^+$]{} (45,25)\[c\][$\gamma ,Z$]{} (100,15)\[r\][$e^+$]{} (100,70)\[r\][$e^-$]{} (105,40)\[r\][$G$]{} (45,8)\[c\][$(c)$]{}
Figure 1. [*Representative Feynman diagrams for the process $e^+ e^- \to e^+ e^- G_n$.*]{}
Considering the process $$e^-(k_1,\lambda_1) ~e^+(k_2,\lambda_2) \to G_n(p_1) ~e^+(p_2) ~e^-(p_3)$$ with a single Kaluza-Klein mode $G_n(p_1)$ in the final state, we obtain the unpolarized cross-section $$\sigma(m_n) = \frac{1}{2s} \int
\frac{d^3\vec{p_1}}{(2\pi)^3 ~2E_1}
~\frac{d^3\vec{p_2}}{(2\pi)^3 ~2E_2}
~\frac{d^3\vec{p_3}}{(2\pi)^3 ~2E_3}
~\delta^4(k_1 + k_2 - p_1 - p_2 - p_3)
~~\frac{1}{4}\sum_{\lambda_1, \lambda_2} |M_n(\lambda_1, \lambda_2)|^2
\label{cross}$$ where the final state helicities are summed over. This cross-section is a function of the mass $m_n$ of the graviton excitation.
The squared matrix element $|M_n(\lambda_1, \lambda_2)|^2$ for the process can be written as the sum $$|M_n(\lambda_1, \lambda_2)|^2 =
\left|\sum_{i = 1}^{28} M_n^{(i)}(\lambda_1, \lambda_2)\right|^2$$ where $|M_n^{(i)}(\lambda_1, \lambda_2)|^2$ represents the squared helicity amplitude arising from the $i$th diagram. In our calculation, we make use of the helicity amplitude technique to write down amplitudes for all these 28 diagrams and evaluate the individual terms in the above sum using the subroutine HELAS[@Hagiwara].
Recalling that the observed final state, namely, $e^+ e^-$ plus missing energy, consists of an incoherent sum over the tower of graviton modes, we get $$\begin{aligned}
\sigma(e^+ e^- \to e^+ e^- + \not{\!\!E})
& = & \sum_n \sigma(m_n) \nonumber \\
& \simeq & \int_0^{\sqrt{s}} dm ~\rho(m) ~\sigma(m)\end{aligned}$$ approximating the discrete (but closely-spaced) tower of states by a continuum. The density of states is given by[@HLZ] $$\rho(m) = \frac{2~R_c^d ~m^{d-1}}{(4\pi)^{d/2} ~\Gamma(d/2)} \ ,$$ and the integration is cut off at the kinematic limit $\sqrt{s}$.
As the calculation, even using HELAS, is long and cumbersome, some checks on the numerical results are called for. The most useful check is provided by the Ward identities arising from general coordinate invariance, which constitute an essential feature of any theory involving gravity. We can write the amplitude for the emission of any graviton in the form $$M_n(\lambda_1, \lambda_2)
= T^{\mu\nu}(\lambda_1, \lambda_2) ~\epsilon^{(n)*}_{\mu\nu}(p_1)$$ where $\epsilon^{(n)}_{\mu\nu}(p_1)$ is the polarization tensor for the $n$th (massive) graviton mode. The tensor $T^{\mu\nu}(\lambda_1, \lambda_2)$ is [*the same for every mode*]{}, including the massless mode $\epsilon^{(0)}_{\mu\nu}(p_1)$, which is the usual graviton of four-dimensional Einstein gravity. This must now satisfy the Ward identities $$p_1^\mu T_{\mu\nu}(\lambda_1,\lambda_2)
= p_1^\nu T_{\mu\nu}(\lambda_1,\lambda_2) = 0 \ ,$$ where we note that $$\begin{aligned}
T^{\mu\nu}(\lambda_1, \lambda_2) = \sum_{i = 1}^{28}
T_i^{\mu\nu}(\lambda_1, \lambda_2)\end{aligned}$$ with $i$ indicating the $i$th diagram, as above. The consistency check therefore requires a perfect cancellation, for each choice of $\lambda_1$ and $\lambda_2$, between all such terms[^1], which is highly sensitive to errors in signs and factors. We have also checked that if only the $s$-channel results are taken our numerical results are in close agreement with those of Ref.[@Eboli], and this, of course, implies that the $t$-channel contributions can be easily checked using crossing symmetry.
In our numerical analysis of this problem, we have also considered diagrams mediated by gravitons in place of $\gamma,Z$. Such diagrams, in fact, are of the same order in perturbation theory after summation over the graviton propagators, but it turns out that the contributions are very small — typically less than 1% for experimentally-allowed values of $M_S$ — due to the smallness of the effective graviton coupling after summation.
It is also worth mentioning that contributions due to graviscalars in the ADD model can also be neglected. It can be shown[@HLZ] that diagrams with graviscalars coupling to external fermion legs and vertices undergo large cancellations and the final contribution is proportional to the electron mass. This leaves only the diagrams with graviscalars emitted from the massive $Z$-propagator, which is again suppressed at high energies by a factor of $M_Z^2/s$ compared to the gravitensor contributions, the latter being jacked up by derivative couplings.
Background Elimination
======================
The actual signal for the process considered in the previous section consists of a hard electron and a hard positron, with substantial missing energy. The SM backgrounds arise from all processes of the form $e^+ e^-
\to e^+ e^- \nu \bar{\nu}$, where the neutrinos can be of any flavor. At the lowest order, there are 96 Feynman diagrams which give rise to this final state. Some of the principal sub-processes are listed below. $$\begin{aligned}
e^+ e^- & \to & e^\pm W^\mp \nu_e^{\!\!\!\!\!\!(-)}
\to e^+ e^- \nu_e \bar{\nu}_e \nonumber \\
e^+ e^- & \to & e^+ e^- Z \to e^+ e^- (\nu \bar{\nu}) \nonumber \\
e^+ e^- & \to & \nu_e \bar{\nu}_e Z \to \nu_e \bar{\nu}_e (e^+ e^-) \nonumber \\
e^+ e^- & \to & W^+ W^- \to (e^+ \nu_e) (e^- \bar{\nu}_e) \nonumber \\
e^+ e^- & \to & ZZ \to (e^+ e^-) (\nu \bar{\nu}) \nonumber \\
e^+ e^- & \to & \gamma^*Z \to (e^+ e^-) (\nu \bar{\nu}) \end{aligned}$$ Of course, for every process with real $W/Z$, there will also be many diagrams with off-shell particles. Taken together, all these diagrams constitute a formidable background to the suggested signal. A judiciously chosen set of kinematic cuts, however, enable us to reduce these backgrounds enormously. For example, diagrams with the neutrino pair (missing energy) arising from a real $Z$ boson can be easily removed by putting a cut on the $\nu\bar{\nu}$ (missing) invariant mass. A similar cut on the $e^+ e^-$ pairs would seem obvious but it is not advisable, since it would remove a large part of the [*signal*]{} arising from radiative return to the $Z$-pole through graviton emission. This, however, is not a serious problem, as the background contribution due to $e^+ e^- \to \nu_e \bar{\nu}_e Z \to
\nu_e \bar{\nu}_e (e^+ e^-)$ is not very large. On the other hand, the background due to the process $e^+ e^- \to e^\pm W^\mp
~\nu_e^{\!\!\!\!\!\!(-)}$ is more problematic, as most cuts tending to reduce this also affect the signal adversely.
In principle, an additional source of background can be the ‘two-photon’ process $e^+ e^- \to e^+ e^- e^+ e^-$ where one electron-positron pair escapes detection by being emitted close to the beam pipe. Here one has to remember that such forward electrons, being rather hard (with energy on the order of $100$ GeV or above), should be detectable in the end caps of the electromagnetic calorimeter, although their energy resolution will be rather poor[@LC]. Thus it should be possible to impose a veto on hard electrons up to within a few degrees of the beam pipe, whereby the above background can be virtually eliminated. Events with the forward electrons even closer to the beam pipe[@Chen] are removed via a cut on missing $p_T$.
Taking all these considerations into account, we focus on an $e^+ e^-$ collider operating at a center-of-mass energy of 500 GeV (1 TeV). The initial set of kinematic cuts used in our analysis are as follows:
- The final state electron (positron) should be at least $10^0$ away from the beam pipe. This tames the collinear singularities arising from $t$-channel photon exchange. At the same time, it also ensures that any background effects from beamstrahlung are mostly eliminated[@Godbole].
- The electron (positron) should have a transverse momentum $p_T^e > 10$ GeV.
- We demand a missing transverse momentum $p_T^{\rm miss} > 15 (25)$ GeV for $\sqrt{s} =$ 500 GeV (1 TeV). As mentioned above, this also helps in reducing the two-photon background.
- The tracks due to electron and positron must be well-separated, with $\Delta R > 0.2$, where $R = \sqrt{\Delta\eta^2 + \Delta\phi^2}$, in terms of the pseudorapidity $\eta$ and the azimuthal angle $\phi$.
- The opening angle between the electron and positron tracks is required to be limited by $5^0 < \theta_{e^+e^-} < 175^0$. This ensures elimination of possible cosmic ray backgrounds, and also ensures sufficient missing energy.
- The missing invariant mass $M_{\rm miss}$ should satisfy the cut $|M_{\rm miss} - M_Z| > 10$ GeV. This important cut eliminates many SM backgrounds in which two final state neutrinos arise from a $Z$-decay.
= 9.0 cm=8.0cm =9.0 cm=8.0cm
Incorporating all these cuts into a Monte Carlo event generator, we obtain the distribution in missing invariant mass shown in Figure 2. It is clear from the figure that the missing invariant mass of the signal is much harder than that of the background. This is easy to understand if we realize that the missing mass is a measure of the energy of the emitted graviton. Now it is well-known that the higher the energy, the more strongly the graviton is coupled to matter and consequently the higher the production cross-section. Moreover, a higher energy leads to a higher density of states, and this further enhances the cross-section. In Figure 2, we have also shown the value of the signal when only $s$-channel diagrams are considered, which would be the case if we were to study processes like $e^+ e^- \to \mu^+ \mu^- G_n$ or $e^+ e^- \to \tau^+ \tau^- G_n$. Comparison with the signal of interest makes it obvious that there is a substantial contribution to the Bhabha scattering signal from the $t$-channel, which makes it more viable as a signal of ADD gravity than the other two. In fact, the emission of a high-energy graviton (corresponding to large missing invariant mass) reduces the energy of the underlying Bhabha scattering process, causing a large enhancement due to the $Z$-pole. This is the reason why the signal for large $M_{\rm miss}$ is completely dominated by the $s$-channel contribution. At the same time, when the graviton is relatively soft, the $s$-channel diagrams are strongly suppressed, but the $t$-channel contributions are not; as a result the signal is dominated by the $t$-channel contribution for low $M_{\rm miss}$. We can therefore obtain a clear separation of signal from background by imposing a cut $$M_{\rm miss} > 350 ~(450) ~{\rm GeV~~for}~\sqrt{s} = 500 ~{\rm GeV}~( 1 ~{\rm T
eV})$$ These choices are optimal, after taking into account the fact that the signal falls rapidly with increasing $M_s$ and $d$.
An important strategy for background reduction is the use of the beam polarization facility at a linear collider. If the electron (positron) beam has a right (left) polarization efficiency ${\cal P}_e$ (${\cal P}_p$), the cross-section formula corresponding to equation \[cross\] is obtained by the replacement $$\begin{aligned}
\sum_{\lambda_1, \lambda_2} |M_n(\lambda_1, \lambda_2)|^2
& \longrightarrow &
~~~(1 + {\cal P}_e)(1 - {\cal P}_p) |M_n(+,+)|^2
+ (1 + {\cal P}_e)(1 + {\cal P}_p) |M_n(+,-)|^2
\nonumber \\
&& +~(1 - {\cal P}_e)(1 - {\cal P}_p) |M_n(-,+)|^2
+ (1 - {\cal P}_e)(1 + {\cal P}_p) |M_n(-,-)|^2
\nonumber \\\end{aligned}$$ Typical values for the polarization efficiencies[@LC] used in this analysis are ${\cal P}_e = 0.8$ and ${\cal P}_p = 0.6$, which tend to favor the first two terms on the right side of the above expression. The use of polarized beams leads to a drastic reduction in the SM background, chiefly because it causes suppression of the $W$-induced diagrams. The numerical effects are presented in the next section.
Results and Discussions
=======================
After numerical evaluation of signal and background, we find it convenient to present our results, using the significance $S/\sqrt{B}$, where $S (B) =
{\cal L} \sigma_{S(B)}$, for given values of $d$ and $M_S$, Here $\sigma_{S(B)}$ denotes the cross-section for the signal (background), while ${\cal L}$ is the integrated luminosity. Our numerical results are presented assuming ${\cal L} = 500$ fb$^{-1}$; however, it is a simple matter to scale the significance values for different values of luminosity.
In Figure 3($a$) we show curves showing the variation of the significance with the string scale $M_S$, for $d = 2, 3, 4 ,5$ and $6$ respectively, for $\sqrt{s} = 500$ GeV. Figure 3($b$) shows a similar plot for $\sqrt{s} = 1$ TeV. In both cases, we have considered unpolarized electron and positron beams.
= 9.0 cm=8.0cm =9.0 cm=8.0cm
As has been already mentioned, the use of beam polarization improves changes of detecting the signal quite dramatically. This is illustrated in Figure 4, which is similar to Figure 3, except that the polarization efficiencies have been taken to be ${\cal P}_e = 0.8$ and ${\cal P}_p = 0.6$ for the electron and positron respectively.
= 9.0 cm=8.0cm =9.0 cm=8.0cm
From the above graphs, it is straightforward to find out the maximum value of $M_s$ that can be probed at the linear collider for any given value of $d$, both for polarized and unpolarized beams. In Table 1, we present the values of $M_s$ that one can probe at 99% confidence level.
$$\begin{array}{|c|r|c|c|c|c|c|c|}
\hline
\sqrt{s}
& d = & 2 & 3 & 4 & 5 & 6 \\
\hline\hline
0.5
& {\rm polarized} & 4.53 & 2.79 & 2.01 & 1.60 & 1.35 \\
& {\rm unpolarized} & 3.16 & 2.09 & 1.59 & 1.30 & 1.12 \\
\hline\hline
1.0
& {\rm polarized} & 7.40 & 4.63 & 3.41 & 2.76 & 2.36 \\
& {\rm unpolarized} & 5.36 & 3.57 & 2.74 & 2.28 & 2.36 \\
\hline
\end{array}$$
Table 1. [ *99% C.L. discovery limits on the string scale $M_S$ for an ADD scenario using (graviton) radiative Bhabha scattering, for different numbers of extra dimensions. An integrated luminosity of $500~fb^{-1}$ has been assumed. All numbers are in TeV.*]{}
If one compares these with the corresponding reach of the process[@GRW; @MPP; @Wilson] $e^+ e^- \to \gamma + \not{\!\!E}$, one will notice that search limits are approximately of the same order. However, the comparison should not be made too literally, because of several reasons. First, the polarization efficiencies in our case, while matching with those of [@Wilson], are different from those of [@GRW], while the center-of-mass energy at which reference [@Wilson] has calculated the effects is slightly different from ours. In addition, one needs to match the event selection criteria more carefully for a full-fledged comparison. Finally, it should be borne in mind that in our analysis we have adopted a normalization of the string scale $M_s$, which, though widely used, is not necessarily uniform in the literature. However, in spite of such non-uniformities, Figures 3 and 4, together with Table 1, can perhaps be taken as faithful indications of the fact that the predictions on radiative Bhabha scattering are comparable to those on the photon-graviton channel, so far as the limits of the probe on the string scale in ADD models are concerned.
Pinning Down the Model
======================
A few comments are in order on how to distinguish graviton signals involving missing energy from similar ones arising from other kinds of new physics. For example, let us consider the well-known process $e^+ e^- \to \gamma G_n$, leading to a single-photon-plus-missing-energy signal. Such signals can be obtained in several other models. A model with an extra $Z'$ boson could lead to a process $e^+ e^- \to \gamma Z'$ with the $Z'$ decaying to neutrinos. Similarly, in supersymmetric models, a process like $e^+ e^- \to \gamma \widetilde{\chi}^0_1
\widetilde{\chi}^0_1$ (where $\widetilde{\chi}^0_1$ is the lightest neutralino) would also lead to a single photon signal. Although the $Z'$ signal can be differentiated by looking for a resonant peak in the missing invariant mass, the supersymmetric signal is more difficult to disentangle.
The situation for the process considered in this paper is somewhat more encouraging. It is true that both types of new physics considered in the preceding paragraph can produce the same signal. Processes of the form $e^+ e^- \to e^+ e^- Z'$, with the $Z'$ decaying to neutrinos are one possibility. In supersymmetry, too, it is possible to pair-produce either sleptons or charginos, followed by decays to electrons (positrons) and invisible particles. The $Z'$ signal can again be differentiated by a peak in the missing invariant mass. The supersymmetric signal, in this case, always arises from production of a pair of real sparticles, which should emerge in opposite hemispheres and, if light enough, will be highly boosted. As a result, the observed electron and positron should lie, most of the time, in opposite hemispheres. For the graviton signal, however, there is no such correlation; in fact the distribution in the opening angle between $e^+$ and $e^-$ turns out to be practically flat. This distinction clearly does not work very well when the produced sparticles are heavy.
= 9.0 cm=8.0cm =9.0 cm=8.0cm
A more elegant way of distinguishing these signals from those other forms of new physics is to compare the cross-sections arising from [*both*]{} the processes $e^+ e^- \to \gamma G_n$ and $e^+ e^- \to e^+ e^- G_n$, both of which are determined by the two parameters $d$ and $M_S$, and hence, will have some correlation. In Figure 5, we have plotted the cross-sections $\sigma(\gamma G_n)$ versus $\sigma( e^+ e^- G_n)$ for different values of $d$. In view of the discussion in the previous section, we would like to note that here the rates for $e^+ e^- \to \gamma G_n$ have been computed with the same normalization of the string scale as that used for the Bhabha scattering process. Each curve corresponds to variation of $M_S$ over the range allowed by current experimental constraints. Once experimental data are available from a high energy $e^+ e^-$ machine, we should be able to pinpoint a small region (ideally a point) on the graph(s) in Figure 5. The position of this would immediately direct us to the value of $d$; comparison of the cross-sections would now yield a measurement of the value of $M_S$. The identification of the model parameters should be unambiguous since the curves for different values of $d$ do not intersect except at the origin. In any case, the area in the vicinity of the origin is limited by the search limits indicated in Table 1 since it corresponds to very high values of $M_S$. The shaded region corresponds to $M_S = \sqrt{s}$ beyond which the theoretical calculations are unreliable. It should, of course be remembered that further information on the fundamental parameters of theory can be extracted from various kinematic distributions which are sensitive to these parameters.
Summary and Conclusions
=======================
In the paper we have considered the process $e^+ e^- \to e^+ e^- G_n$, which is essentially Bhabha scattering with radiated gravitons in the ADD model. After identifying suitable event selection criteria, we find that this process can act as an effective probe of large extra dimensions at a high-energy $e^{+} e^{-}$ collider, especially with a center-of-mass energy of order 1 TeV and with polarized beams. The string scale $M_S$ that can be probed in this channel is found to be comparable to that which is accessible through the alternative process $e^+ e^- \to \gamma G_n$. It is also shown that a study of the (graviton) radiative Bhabha scattering process may provide some further handle on the essential characteristics of ADD-like theories. And finally, taking a cue from the wisdom that, in coming to any conclusion on new physics possibilities, it is always advantageous to have more than one type of data, we have demonstrated how our predictions can be combined with those on the photon-graviton channel to obtain rather trustworthy revelations on models with large extra dimensions.
**Acknowledgments**
[The authors acknowledge useful discussions with D.Choudhury, R.M.Godbole, U.Mahanta, and S.K.Rai. This work was initiated as part of the activity of the Indian Linear Collider Working Group (Project No. SP/S2/K-01/2000-II of the Department of Science and Technology, Government of India). SD thanks the SERC, Department of Science and Technology, Government of India for partial support. The work of BM was partially supported by the Board of Research in Nuclear Sciences (BRNS), Government of India. SR thanks the Harish-Chandra Research Institute for hospitality while this paper was being written. ]{}
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[^1]: In fact the $s$ and $t$-channel diagrams form two independent gauge-invariant sets, so that the actual cancellation takes place between 14 diagrams at a time.
| 0 |
---
abstract: 'We consider a -based transmission scheme, where data is embedded into the imaginary part of the nonlinear discrete spectrum. Inspired by probabilistic amplitude shaping, we propose a scheme as a means to increase the data rate of the system. We exploit the fact that for an -based transmission scheme, the pulses in the time domain are of unequal duration by transmitting them with a dynamic symbol interval and find a capacity-achieving distribution. The scheme shapes the information symbols according to the capacity-achieving distribution and transmits them together with the parity symbols at the output of a encoder, suitably modulated, via time-sharing. We furthermore derive an achievable rate for the proposed scheme. We verify our results with simulations of the discrete-time model as well as with simulations.'
author:
- |
Andreas Buchberger, Alexandre Graell i Amat, \
Vahid Aref, and Laurent Schmalen, [^1][^2][^3][^4]
title: |
Probabilistic Eigenvalue Shaping for\
Nonlinear Fourier Transform Transmission
---
Discrete spectrum, nonlinear Fourier transform (NFT), probabilistic shaping, soliton communication.
Introduction {#sec:introduction}
============
propagation in optical fibers is severely impaired by nonlinear effects that should be either compensated or utilized for the design of the communication system. The [@yousefi2014nftI-III] provides a method to transform a signal from the time domain into a nonlinear frequency domain (spectrum), where the channel acts as a multiplicative filter on the signal. The nonlinear spectrum consists of a continuous and a discrete part. Both parts can be used to transmit information, either separately or jointly, and several schemes have been presented in theory and practice [@yousefi2014nftI-III; @dong2015; @aref2015ecoc; @aref2016ecoc; @geisler2016ecoc; @hari2016]. However, very little is known so far about the of the received signal in the nonlinear spectral domain when it is contaminated by channel noise.
In [@shevchenko2018tcom], a simplified communication system modulating only the imaginary part of the eigenvalues in the discrete nonlinear spectrum was presented. For this scheme, an approximation for the conditional of the channel can be obtained in closed form. In general, for a given channel, the capacity-achieving distribution is not known and is often different from the conventional distribution with equispaced signal points and uniform signaling. Hence, some form of shaping is required [@forney1984]. Two popular methods of shaping are probabilistic shaping and geometric shaping. In geometric shaping, the capacity-achieving distribution is mimicked by optimizing the position of the constellation points for equiprobable signaling [@sun1993_geometric_shaping] whereas probabilistic shaping uses uniformly spaced constellation points and approximates the capacity-achieving distribution by assigning different probabilities to different constellation points [@forney1984].
The main drawback of probabilistic shaping is its practical implementation. An abundance of probabilistic shaping schemes have been presented, most suffering from high decoding complexity, low flexibility in adapting the spectral efficiency, or error propagation. For a literature review on probabilistic shaping, we refer the reader to [@bocherer2015_bw_efficient_rate_matched_ldpc Section II].
Recently, a new scheme called has been proposed in [@bocherer2015_bw_efficient_rate_matched_ldpc]. Compared to other shaping schemes, yields high flexibility and close-to-capacity performance over a wide range of spectral efficiencies for the channel while still allowing bit-metric decoding. Although originally introduced for the channel, can be applied to other channels with a symmetric capacity-achieving input distribution assuming a sufficiently high spectral efficiency.
In this paper, we consider a similar -based transmission scheme to the one presented in [@shevchenko2018tcom], where data is embedded into the imaginary part of the nonlinear discrete spectrum. As a means to increase the data rate, we demonstrate that the concept of can be adapted to this -based transmission system. In particular, we propose a scheme, enabling similar low complexity and bit-metric decoding as . We take advantage of the dependence of the pulse length on the data for the -based transmission system and transmit each pulse as soon as the previous one has been transmitted rather than with a fixed interval as in [@shevchenko2018tcom], yielding increased data rate. Accordingly, we find the capacity-achieving input distribution, maximizing the time-scaled . For ease of notation, we refer to the maximized as capacity noting that it is in fact the constrained capacity of a system transmitting first-order solitons. The scheme then shapes the information symbols according to the capacity-achieving distribution by a . The information symbols are also encoded by a encoder and the parity symbols at the output of the encoder are suitably modulated. The resulting sequence of modulated symbols and the sequence at the output of the are transmitted via time-sharing. We further derive an achievable rate for such a scheme. We demonstrate via discrete-time Monte-Carlo and simulations, that performs at around $\SI{2}{\decibel}$ from capacity using off-the-shelf codes. The proposed scheme yields a significant improvement of up to twice the data rate compared to an unshaped system as in [@shevchenko2018tcom].
It is important to note that although first-order solitons do not outperform conventional coherent systems due to their spectrally inefficient pulse shape compared to a Nyquist pulse shape, they have some other advantages. For instance, the first-order soliton transmission does not require compensation or as dispersion and nonlinearity are balanced and hence compensated. This work attempts to approach the limits of current NFT-based systems. To improve the spectral efficiency further, one should use higher-order solitons as well as the continuous part of the nonlinear spectrum together [@aref2016ecoc]. However, the channel equalization will not be as easy as the one for the first-order solitons and the channel model is not yet fully known.
The remainder of the paper is organized as follows. In [Section ]{}, we describe pulse propagation in an optical fiber and the -based transmission scheme. In [Section ]{}, we optimize the input distribution and in [Section ]{}, we introduce and describe the proposed scheme and derive an achievable rate. In [Section ]{}, we present numerical results for , both from Monte-Carlo simulation and simulation, and in [Section ]{} we draw some conclusions.
Notation: The following notation is used throughout the paper. ${\mathfrak{R}\{\cdot\}}$ and ${\mathfrak{I}\{\cdot\}}$ denote the real and the imaginary part of a complex number, respectively, and $\jmath=\sqrt{-1}$ denotes the imaginary unit. Vectors are typeset in bold, e.g., $\bm{x}$, are capitalized, e.g., $X$, and hence vectors of are capitalized bold, e.g., $\bm{X}$. The of an $X$ is written as $p_X(x)$ and its expectation as ${\mathbb{E}_{X}\!{\mathopen{}\mathclose\bgroup\originalleft}\{{x}{\aftergroup\egroup\originalright}\}
}$. The conditional of $Y$ given $X$ is denoted as $p_{Y|X}(y|x)$. The of an $X$ is denoted by $P_X(x)$. The transpose of a vector or matrix is given as $(\cdot )^\mathsf{T}$. A set is denoted by a capitalized Greek letter, e.g., $\Lambda$, and its cardinality by $|\Lambda|$. We write $\log_a(\cdot)$ for the logarithm of base $a$ and $\ln(\cdot)$ for the natural logarithm.
Nonlinear Fourier Transform-based Transmission System {#sec:system}
=====================================================
Pulse Propagation and the Nonlinear Fourier Transform {#subsec:NFT}
-----------------------------------------------------
Pulse propagation in optical fibers is governed by a partial differential equation, the stochastic , $$\begin{gathered}
\jmath\frac{\partial u(\tau,\ell)}{\partial \ell} + \jmath\frac{\alpha}{2}u(\tau,\ell) - \frac{\beta_2}{2}\frac{\partial^2 u(\tau,\ell)}{\partial \tau^2} + \gamma u(\tau,\ell) |u(\tau,\ell)|^2 \\= n(\tau,\ell ),\nonumber$$ where $u(\tau,\ell)$ denotes the envelope of the electrical field as a function of the position $\ell$ along the fiber and time $\tau$, $\alpha$ the attenuation, $\beta_2$ the second order dispersion, $\gamma$ the nonlinearity parameter, and $n(\tau, \ell)$ is a white Gaussian process in time and in space with spectral density $\sigma_0^2$. The spectral density depends on the system and for distributed Raman amplification is given as $\sigma_0^2 = \alpha K_T h \nu_0$, where $K_T$ is the temperature-dependent phonon occupancy factor, and $h\nu_0$ is the average photon energy [@shevchenko2018tcom]. A general closed-form solution of the stochastic does not exist. In some special cases, e.g., for noisefree and lossless fibers, special solutions like, e.g., solitons, exist. Furthermore, we consider the in normalized form in the focusing regime, i.e., $\beta_2 < 0$, under the assumption of ideal distributed Raman amplification, i.e., $\alpha = 0$, $$\begin{aligned}
\jmath\frac{\partial q(t,z)}{\partial z} + \frac{\partial^2 q(t,z)}{\partial t^2} + 2 q(t,z) |q(t,z)|^2 = 0,
\label{eqn:nlse_losless_noisefree}\end{aligned}$$ where $t=\tau / \sqrt{|\beta_2| L/2}$, $z=\ell / L$, $q = u\sqrt{\gamma L}/\sqrt{2}$, and $L$ is the length of the fiber. In this case, the is an integrable partial differential equation for which a pair of operators, called Lax pair, can be found. The eigenvalues of such an operator remain invariant during noiseless propagation and the Lax pair can be used to solve the partial differential equation. Solutions of [()]{} can be uniquely represented in terms of its eigenvalues via the so-called . For a given position $z$, the of a signal $q(t)$ (we drop the position $z$ for simplicity of presentation) with support on the time interval $t\in [t_1,t_2]$, is calculated by solving the partial differential equation $$\begin{aligned}
\frac{\partial \bm{v}(t,\lambda)}{\partial t} = \begin{pmatrix}
-\jmath\lambda & q(t)\\
- q(t)^*& \jmath\lambda
\end{pmatrix}\bm{v}(t,\lambda),\label{eqn:nft_dgl}\end{aligned}$$ where $\bm{v}(t,\lambda)=\begin{pmatrix} v_1(t,\lambda) & v_2(t,\lambda) \end{pmatrix}$ is the eigenvector of the auxiliary operator, with boundary conditions $$\begin{aligned}
\bm{v}^{(1)}(t,\lambda) &\rightarrow {\begin{pmatrix} 0 & 1 \end{pmatrix}^\mathsf{T}} {\operatorname{e}^{\jmath\lambda t}}, &\text{as } t &\rightarrow t_2\\
\bm{v}^{(2)}(t,\lambda) &\rightarrow {\begin{pmatrix} 1 & 0 \end{pmatrix}^\mathsf{T}} {\operatorname{e}^{-\jmath\lambda t}}, &\text{as } t &\rightarrow t_1,\end{aligned}$$ and $\lambda$ is the spectral component. Solving [()]{} gives rise to the continuous and discrete nonlinear spectrum $$\begin{aligned}
\hat{q}(\lambda ) &= \frac{b(\lambda)}{a(\lambda)}, \lambda\in{\mathbb{R}}& \tilde{q}(\lambda_i) &= \frac{b(\lambda_i)}{\mathrm{d}a(\lambda) / \mathrm{d}\lambda|_{\lambda=\lambda_i}}, \lambda_i\in{\mathbb{C}}^{+},\end{aligned}$$ respectively, where $a(\lambda ) = \lim_{t\rightarrow t_2} v_1^{(2)}(t,\lambda) {\operatorname{e}^{\jmath\lambda t}}$, $b(\lambda ) = \lim_{t\rightarrow t_2} v_2^{(2)}(t,\lambda) {\operatorname{e}^{-\jmath\lambda t}}$, and $\lambda_i$ are the zeros of $a(\lambda)$, $\lambda_i\in{\mathbb{C}}^{+}$, a finite set of isolated complex zeros, referred to as eigenvalues. Hence, the represents the signal in the nonlinear spectral domain, where the influence of the channel on the signal is a multiplicative filter.
As a counterpart to the that transforms a signal from the time domain to the nonlinear spectral domain, the transforms a signal from the nonlinear spectral domain to the time domain. For an in-depth mathematical description of the , we refer the interested reader to [@yousefi2014nftI-III].
Soliton Transmission
--------------------
![Block diagram of the -based system.[]{data-label="fig:system:block_diagram_nft"}](block_diagram_nft)
As in [@shevchenko2018tcom], we embed information in the imaginary part of the discrete spectrum, also referred to as eigenvalues. Hence, the input of the channel is an $X\in\Lambda=\{\lambda_1,\ldots,\lambda_M\}$, where $\Lambda$ is the set of eigenvalues, $\lambda_i$ is the $i$th eigenvalue, and $M$ is the order of the modulation. The eigenvalues $\{\lambda_i\}$ are assumed to be ordered in ascending order by their imaginary parts. Furthermore, the output of the channel is an $Y\in\Psi$, where $\Psi = \{y\in\mathbb{C}: {\mathfrak{R}\{y\}}=0, {\mathfrak{I}\{y\}}\geq 0\}$. A block diagram is depicted in [Fig.]{}. The information embedded in a single eigenvalue $\lambda\in\Lambda$ is transformed to a time-domain signal $q(t,0)$ via the where the transmitter is located at position $z=0$ along the fiber. At position $z=1$, the receiver calculates the discrete spectrum $\psi\in\Psi$ from the received signal $q(t,1)$ via the . The time-domain signal corresponds to first order solitons, i.e., $$\begin{aligned}
q(t,0) &= 2{\mathfrak{I}\{\lambda\}} {\operatorname{sech}{\mathopen{}\mathclose\bgroup\originalleft}(2{\mathfrak{I}\{\lambda\}} t{\aftergroup\egroup\originalright})}.\end{aligned}$$ For the to be valid, the signal must have finite support, i.e., before transmitting the next pulse, the previous one must have returned to zero. As the pulses in general have infinite tails, we truncate them when they fall below a threshold close to zero. We define the pulse over the smallest support containing a fraction $(1-\delta ) $ of the energy of the pulse and hence, we can formally define the pulse width as follows.
The pulse width of $\lambda$ is defined as the smallest support containing a fraction $(1-\delta ) $ of the energy of the pulse, $$\begin{aligned}
T(\lambda, \delta ) &\triangleq \frac{1}{2{\mathfrak{I}\{\lambda\}}} \ln{\mathopen{}\mathclose\bgroup\originalleft}(\frac{2}{\delta} -1{\aftergroup\egroup\originalright}),\end{aligned}$$ where $0 < \delta < 1$.
The value of the cutoff parameter $\delta$ must be chosen in a way such that soliton-soliton interactions are negligible. For longer transmission distances, $\delta$ decreases, i.e., the pulses must be spaced further apart. Furthermore, the condition $$\begin{aligned}
{\operatorname{e}^{-2{\mathfrak{I}\{\lambda\}} \Delta(\lambda, \delta )}}
&= {\operatorname{e}^{-\ln{\mathopen{}\mathclose\bgroup\originalleft}(\frac{2}{\delta} -1{\aftergroup\egroup\originalright})}} \ll 1
\label{eqn:guard_time_condition}\end{aligned}$$ must be fulfilled [@shevchenko2018tcom].
At this point, it is important to comment on the memorylessness of the system emanating from the absence of soliton-soliton interactions. A pulse train of well-separated first order solitons was investigated in [@shevchenko2018tcom] for launch powers of $\SI{-1.5}{\dBm}$ and $\SI{1.45}{\dBm}$ and transmission over $\SI{500}{\kilo\meter}$ and $\SI{2000}{\kilo\meter}$. It was shown via simulations that the correlation between the symbols at the receiver is essentially zero, concluding that the channel is indeed memoryless in the transmission range of $\SI{500}{\kilo\meter}$ to $\SI{2000}{\kilo\meter}$ and transmit power range of $\SI{-1.5}{\dBm}$ to $\SI{1.45}{\dBm}$ for which the model [()]{} is applicable. While this approach is not a rigorous proof, the results indicate that memorylessness is a valid assumption. Although the transmission scheme is different in [@shevchenko2018tcom], the underlying condition that any two pulses need to be sufficiently separated is the same. Hence, we can treat the -based transmission system in this work as a memoryless channel.
In a practical system, we assume distributed Raman amplification and noise with received power spectral density $\sigma^2$ to compensate for the lossy fiber and be able to use the to relate the input and the output. The conditional of such a system has been derived via a perturbative approach and the Fokker-Planck equation method [@derevyanko2005] and is used to design a communication system in [@shevchenko2018tcom]. It is given by
$$\begin{aligned}
p_{Y|X}(\psi|\lambda) &= \frac{2}{\sigma^2}\sqrt{\frac{{\mathfrak{I}\{\psi\}}}{{\mathfrak{I}\{\lambda\}}}}{\operatorname{e}^{-2\frac{{\mathfrak{I}\{\lambda\}}+{\mathfrak{I}\{\psi\}}}{\sigma^2}}}{\operatorname{I}_{1}{\mathopen{}\mathclose\bgroup\originalleft}(\frac{4\sqrt{{\mathfrak{I}\{\lambda\}}{\mathfrak{I}\{\psi\}}}}{\sigma^2}{\aftergroup\egroup\originalright})},
\label{eqn:pdf_yx_sac}\end{aligned}$$
where $\psi$ is the received symbol as in [Fig.]{}, and ${\operatorname{I}_{1}{\mathopen{}\mathclose\bgroup\originalleft}(\cdot{\aftergroup\egroup\originalright})}$ is the modified Bessel function of the first kind of order one. The power spectral density of the received noise $\sigma^2$ is normalized and relates to real world units as $\sigma^2 = \gamma\sqrt{L^3}\sigma_0^2 / {\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{2|\beta_2|}{\aftergroup\egroup\originalright})$. The is defined as $\mathrm{SNR}\triangleq 4{\mathbb{E}_{X}\!{\mathopen{}\mathclose\bgroup\originalleft}\{{{\mathfrak{I}\{\lambda\}}}{\aftergroup\egroup\originalright}\}
}/\sigma^2$. It is important to note that the model [()]{} assumes the noise intensity to be small such that it can be treated as a perturbation to the soliton. Hence, the model is only applicable if the signal energy is not the same order as that of the noise. Furthermore, for very high signal powers, [()]{} is no longer valid either since the impact of the inelastic scattering effects (i.e., stimulated Raman or Brillouin scattering) is not considered within the 1st-order perturbation approach. For a detailed derivation of the model, we refer the reader to [@derevyanko2005].
In [@shevchenko2018tcom], the shortest possible symbol interval is defined by the pulse duration of $\lambda_1$, i.e., the longest pulse. However, this tends to be inefficient since especially for short pulses, the guard interval between two consecutive pulses is longer than necessary and thereby limits the data rate. Here, we exploit the effect of varying pulse lengths and transmit each pulse as soon as the previous one has returned to zero. This concept is depicted in [Fig.]{}, where pulse sequences with fixed and varying symbol interval are compared. The figure clearly shows the advantage of a varying pulse interval and also demonstrates the aforementioned inefficiencies. The data rate of a system with varying symbol intervals depends on the distribution of the data. Thus, we define the average symbol interval as follows.
![Comparison of a pulse sequence with static symbol intervals and dynamic symbol intervals.[]{data-label="fig:system:plot_comparison_spacing"}](plot_comparison_spacing)
\[def:avg\_duration\] The average symbol interval is $$\begin{aligned}
\bar{T}(X) &\triangleq \sum_{k=1}^{M} p_X (\lambda_k) {T{\mathopen{}\mathclose\bgroup\originalleft}(\lambda_k{\aftergroup\egroup\originalright})} = {\mathbb{E}_{X}\!{\mathopen{}\mathclose\bgroup\originalleft}\{{{T{\mathopen{}\mathclose\bgroup\originalleft}(\lambda{\aftergroup\egroup\originalright})}}{\aftergroup\egroup\originalright}\}
}.\end{aligned}$$
In [@shevchenko2018tcom], only eigenvalues with an imaginary part larger than zero are used. We extend this by allowing ${\mathfrak{I}\{\lambda\}}=0$. In the time domain, this results in a pulse with amplitude zero, i.e., we do not transmit anything. We define its corresponding duration as the same as the duration of the shortest pulse, ${T{\mathopen{}\mathclose\bgroup\originalleft}(\lambda=0{\aftergroup\egroup\originalright})} \triangleq {T{\mathopen{}\mathclose\bgroup\originalleft}(\lambda_M{\aftergroup\egroup\originalright})}$.
As any practical system can handle only a maximum peak power and a maximum bandwidth, we enforce a peak power constraint which relates to a maximum eigenvalue constraint. Especially in systems with lumped amplification and , such a constraint is required as eigenvalues fluctuate depending on their amplitude, which decreases the performance [@zafrulla2003].
We note that the varying symbol interval introduces additional challenges on detection. In particular, an erroneously detected symbol may lead to error propagation, insertion errors (detection of symbols when none was transmitted), deletion errors (not detecting a transmitted symbol), or the loss of synchronization. To calculate the capacity, however, we neglect these effects. Hence, the results can be seen as an upper bound on the performance.
Capacity Achieving Distribution {#sec:optimum_input_distribution}
===============================
From [Fig.]{}, it is intuitive that pulses with short duration should be transmitted more frequently than pulses with long duration. However, shorter pulses are more perturbed by noise than longer pulses. Hence, the optimal input distribution to the channel as described by the conditional [()]{} is not the conventional uniform distribution. The channel capacity is obtained by maximizing the , $$\begin{aligned}
{\mathbb{I}(X;Y)} &\triangleq {\mathbb{E}_{X,Y}\!{\mathopen{}\mathclose\bgroup\originalleft}\{{\log_2{\mathopen{}\mathclose\bgroup\originalleft}({\frac{p_{Y|X}(Y|X)}{\sum_{\tilde{\lambda}\in\Lambda}p_{Y|X}(Y|\tilde{\lambda}) p_X(\tilde{\lambda})}}{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}\}
}\end{aligned}$$ over all possible input distributions $p_X(\lambda )$. Here, due to the variable transmission duration, we need to consider the under a variable cost constraint $\bar{T}(\cdot)$ [@verdu1990], $$\begin{aligned}
{\mathsf{I}(X;Y)} &\triangleq \frac{{\mathbb{I}(X;Y)}}{\bar{T}(X)}.
\label{eqn:mutual_information_time_scaled}\end{aligned}$$ To emphasize that the cost of a symbol is its corresponding pulse duration, we refer to the in the form of [()]{} as time-scaled . We can therefore define the capacity as $$\begin{aligned}
{\mathsf{C}}&\triangleq \sup_{p_X(\lambda)} {\mathsf{I}(X;Y)}
\label{eqn:def_cstar}\end{aligned}$$ where we set the supremum to zero if the set of distributions therein is empty. The capacity-achieving distribution, denoted by $p_X^*(\lambda )$, is in the set for which the supremum is non-zero.
As the ${\mathbb{I}(X;Y)}$ is concave in $p_X(\lambda )$ and $\bar{T}(X)$ is linear in $p_X(\lambda )$ and positive, the time-scaled ${\mathsf{I}(X;Y)}$ is quasiconcave [@stancu-minasian_frac_programming Table 2.5.2]. We can solve [()]{} and obtain the corresponding capacity-achieving distribution numerically.
![Optimal distribution for different .[]{data-label="fig:optDist:example_opt_dist"}](optimal_distribution_combined_sufigures "fig:") -0.5cm
Exemplary results of the capacity-achieving distribution are shown in [Fig.]{}. We note that the lowest and highest amplitudes are always used with equal and high probability. For low , only these are used, i.e., is optimal. Furthermore, the capacity-achieving distribution is discrete and is of exponential-like shape with the exception of a point mass at zero as it can be seen in [Fig.]{}.
Note that ${\mathsf{C}}$ assumes memorylessness, which does not necessarily hold due to the variable symbol interval. Hence, ${\mathsf{C}}$ is, in fact, the constraint capacity under the assumption of a memoryless channel and the constraint of transmitting only first-order solitons. However, for notational simplicity, we refer to it simply as capacity with its corresponding capacity-achieving distribution.
In the case of a noiseless channel, it is possible to derive a closed form solution to [()]{} under the assumption of a finite discretization.
\[lemma:noiseless\] Let $\lambda_1, \lambda_2, \ldots, \lambda_M$ be $M \geq 2$ eigenvalues with $0 \leq {\mathfrak{I}\{\lambda_1\}} < {\mathfrak{I}\{\lambda_2\}} < \ldots < {\mathfrak{I}\{\lambda_M\}}$ and let ${T{\mathopen{}\mathclose\bgroup\originalleft}(\lambda_k{\aftergroup\egroup\originalright})}$ be the time of transmitting a pulse with eigenvalue $\lambda_k$. Let $r$ be the unique real positive root of the polynomial $\sum_{k=1}^Mx^{-T(\lambda_k)} - 1$. Then, in the noiseless case, the capacity is obtained as $${\mathsf{C}}= \log_2(r)$$ and the capacity-achieving distribution is given by $$\begin{aligned}
P_X^\diamond(\lambda_k) &= {\operatorname{e}^{-\ln(r){T{\mathopen{}\mathclose\bgroup\originalleft}(\lambda_k{\aftergroup\egroup\originalright})}}}, & k&=1,\ldots,M.
\label{eqn:opt_dist_noiseless}\end{aligned}$$
Suppose that the $k$-th eigenvalue is transmitted with probability $P_k$. For any fixed average symbol interval $\bar{T}(X) = \sum_kP_kT(\lambda_k)$, where ${T{\mathopen{}\mathclose\bgroup\originalleft}(\lambda_M{\aftergroup\egroup\originalright})} \leq \bar{T}(X) \leq {T{\mathopen{}\mathclose\bgroup\originalleft}(\lambda_1{\aftergroup\egroup\originalright})}$, we are interested in the distribution that maximizes the entropy while leading to the average symbol duration $\bar{T}(X)$. It is known that this distribution takes the form [@cover_inf_theory Ch. 12] $$\begin{aligned}
P_k &= \frac{{\operatorname{e}^{-\theta {T{\mathopen{}\mathclose\bgroup\originalleft}(\lambda_k{\aftergroup\egroup\originalright})}}}}{\xi(\theta)}
\label{eqn:maximizer}\end{aligned}$$ where $\xi(\theta) = \sum_i{\operatorname{e}^{-\theta T(\lambda_i)}}$ ensures that $\sum_k P_k = 1$ and $\theta$ has to be selected such that $\sum_kP_kT(\lambda_k) = \bar{T}$. In the noiseless case, the MI is given by ${\mathbb{I}(X;Y)} = {\mathbb{H}(X)}$. The entropy ${\mathbb{H}(X)}$ then is $$\begin{aligned}
{\mathbb{H}(X)} =: {\mathbb{H}(\theta)} &= -\sum_{k=1}^M P_k \log_2{\mathopen{}\mathclose\bgroup\originalleft}(\frac{{\operatorname{e}^{-\theta {T{\mathopen{}\mathclose\bgroup\originalleft}(\lambda_k{\aftergroup\egroup\originalright})}}}}{\xi(\theta)}{\aftergroup\egroup\originalright})\\
&= \frac{1}{{\ln(2)}}\sum_{k=1}^M P_k (\theta {T{\mathopen{}\mathclose\bgroup\originalleft}(\lambda_k{\aftergroup\egroup\originalright})} + \ln(\xi(\theta)))\\
&= \frac{\theta \bar{T}(X)}{{\ln(2)}} + \log_2(\xi(\theta)).\end{aligned}$$ The time-scaled MI hence takes the form $$\begin{aligned}
{\mathsf{I}(X;Y)} &= \frac{\theta}{\ln(2)} + \frac{\log_2(\xi(\theta))}{\bar{T}(X)} \\
&= \frac{\theta}{\ln(2)} + \frac{\log_2(\xi(\theta))}{\sum_kP_kT(\lambda_k)} \\
&= \frac{\theta}{\ln(2)} + \frac{\log_2(\sum_k{\operatorname{e}^{-\theta T(\lambda_k)}})\sum_k{\operatorname{e}^{-\theta T(\lambda_k)}}}{\sum_k{\operatorname{e}^{-\theta T(\lambda_k)}}T(\lambda_k)}\,. \end{aligned}$$ In order to maximize ${\mathsf{I}(X;Y)}$, we find the optimal parameter $\theta$ by setting $\xi(\theta) = 1$. This can be seen by setting the derivative of ${\mathsf{I}(X;Y)}$ to zero, with $$\begin{gathered}
\frac{\partial}{\partial\theta} {\mathsf{I}(X;Y)} = \\
\log_2{\mathopen{}\mathclose\bgroup\originalleft}(\sum_k{\operatorname{e}^{-\theta T(\lambda_k)}}{\aftergroup\egroup\originalright}){\mathopen{}\mathclose\bgroup\originalleft}(\frac{\sum_k{\operatorname{e}^{-\theta T(\lambda_k)}}}{\sum_kT(\lambda_k){\operatorname{e}^{-\theta T(\lambda_k)}}}{\aftergroup\egroup\originalright})^2\mathop{\mathrm{var}}(T(\lambda))\, ,\end{gathered}$$ where $\mathop{\mathrm{var}}(T(\lambda))$ denotes the variance of $T(\lambda_k)$ for the given $\theta$. By assumption, as all $T(\lambda_k)$ are different, the middle part of this expression is strictly positive and $\mathop{\mathrm{var}}(T(\lambda)) > 0$. Hence, it is easy to see that this derivative can only be zero if $\sum_k{\operatorname{e}^{-\theta T(\lambda_k)}} = 1$. The optimal $\theta$ is hence found by setting $\xi(\hat{\theta}) = 1$. Consider the polynomial $$f(x) = \sum_{k=1}^Mx^{-T(\lambda_k)} - 1.$$ As this polynomial is monotonically decreasing for positive $x$, with $\lim_{x\to 0^+}f(x) = +\infty$ and $\lim_{x\to+\infty}f(x)=-1$, $f(x)$ has exactly one positive real root. Let $r$ be the unique positive real root of $f(x)$. Then $\hat{\theta} = \ln(r)$. Inserting $\hat{\theta}$ into ${\mathsf{I}(X;Y)}$ and proves the lemma.
![Time-scaled of the optimal distribution for linearly spaced constellations with $M$ points (colored with markers solid), and of a system as in [@shevchenko2018tcom] (with markers dotted). As a reference, the capacity ${\mathsf{C}}$ is plotted as well (black solid without markers). For the cutoff parameter, $\delta=0.005$ was used.[]{data-label="fig:shaping:mi_comparison"}](mi_comparison)
We clearly see that [()]{} is of exponential shape with an additional point mass at zero. Furthermore, we note that the shape of the distribution is mostly caused by the variable pulse duration. The noise then determines the optimal location and optimal number of constellation points.
For a transmission system, the is an upper bound on the achievable rate. In [Fig.]{} we evaluate the time-scaled for various input distributions for a cutoff parameter $\delta=0.005$. The capacity is depicted with a black solid line. To reduce the complexity of implementation, we constrain the constellation $\Lambda$ to $M$ linearly spaced points from $\lambda_1=0$ to $\lambda_M$, i.e., $$\begin{aligned}
\lambda_i &= (i-1)\frac{\lambda_M}{M-1} \text{ for } i=1,\ldots,M,\end{aligned}$$ and plot the corresponding time-scaled in colored solid lines with markers. We note that the time-scaled is very close to the capacity curve until it saturates. Increasing the modulation order $M$ shows significant increase in the time-scaled . For comparison purposes, we also plot the time-scaled for a system with fixed symbol duration and conventional uniform distribution on a linearly spaced constellation as in [@shevchenko2018tcom]. We observe that the rate saturates at very low values and that increasing the modulation order $M$ shows only slight improvement.
![Block diagram of the scheme.[]{data-label="fig:shaping:system_block_diagram_pes"}](block_diagram_pes)
Probabilistic Eigenvalue Shaping {#sec:shaping_time_share}
================================
In the previous section, we observed a significant gap between the time-scaled of the system in [@shevchenko2018tcom] and the capacity. This gap is referred to as shaping gap. In order to close it, we propose a system as shown in [Fig.]{}, inspired by [@bocherer2015_bw_efficient_rate_matched_ldpc].
In the scheme, the sequence of uniformly distributed data bits is mapped to a sequence of positive amplitudes distributed half Gaussian by a . The binary image of this sequence is encoded by a systematic code, resulting in uniformly distributed parity bits, which are then used to map the sequence of half Gaussian distributed symbols to a stream of Gaussian distributed symbols.
As the capacity-achieving distribution $p_X^*(\lambda)$ is not symmetric, cannot be directly applied here. However, in order to keep the benefits of , we wish to apply the before the . We describe in the following with reference to [Fig.]{}. The binary data sequence $\bm{u}$ of length $k_\mathsf{s}$ bits is mapped by the to a sequence of eigenvalues $\bm{\lambda}\in\Lambda^{n_\mathsf{s}}$ of length $n_\mathsf{s}$ distributed according to $p_X^*(\lambda)$. The can be used for that purpose [@schulte2016_ccdm]. It is asymptotically optimal as its rate $R_\mathsf{s}$ approaches the entropy of the desired channel input $X$, $$\begin{aligned}
R_\mathsf{s} &=\frac{k_\mathsf{s}}{n_\mathsf{s}} \rightarrow {\mathbb{H}(X)} \text{ as } n_\mathsf{s} \rightarrow\infty.\end{aligned}$$ For large block sizes, the gap between $R_\mathsf{s}$ and ${\mathbb{H}(X)}$ is sufficiently small and can be neglected. Note that some of the possible eigenvalues may occur with probability zero.
We consider the modulation order $M$ to be a power of two such that we can define its binary image. The binary image of $\bm{\lambda}$, $\bm{bi}(\bm{\lambda} )$, is then encoded by a systematic encoder with information block length $k_\mathsf{c}$, code length $n_\mathsf{c}$, and rate $R_\mathsf{c}=\frac{k_\mathsf{c}}{n_\mathsf{c}}$. The code is denoted by $\mathcal{C}$, with $|\mathcal{C}|=2^{k_\mathsf{c}}$. The parity bits at the output of the encoder are mapped to a sequence of eigenvalues $\bm{\lambda}_\mathsf{par}\in \Lambda_\mathsf{par}$ with modulation order $M_\mathsf{par}=|\Lambda_\mathsf{par}|$ and $\Lambda_\mathsf{par}\subseteq\Lambda$ by the block $\bm{s}(\cdot )$ in [Fig.]{} such that they are uniformly distributed.
Assuming that a high code rate $R_\mathsf{c}$ is used, we accept a small penalty with respect to the optimal channel input distribution and transmit $\bm{\lambda}$ and $\bm{\lambda}_\mathsf{par}$ via time-sharing. The major difference of compared to is the fact that the channel input distribution is not the optimal distribution due to the time-sharing with the sequence $\bm{\lambda}_\mathsf{par}$. Consequently, this causes a performance degradation. However, is highly flexible as the spectral efficiency can be adapted by the and the code rate $R_\mathsf{c}$, and a single code can be used. Note that every eigenvalue is protected by the code as is performed after the and decoding and demapping can be performed independently. Thus, shares these advantages with .
We wish for a high code rate $R_\mathsf{c}$ to keep the performance degradation due to the time-sharing low. More precisely, we wish to maximize the number of symbols distributed according to $p_X^*(\lambda)$. The ratio between information symbols and coded symbols, denoted by $R_\mathsf{ts}$, is an indication for the expected performance degradation, $$\begin{aligned}
R_\mathsf{ts} &= \frac{\frac{n_\mathsf{c} R_\mathsf{c}}{\log_2(M)}}{\frac{n_\mathsf{c}R_\mathsf{c}}{\log_2(M)} + \frac{n_\mathsf{c}(1-R_\mathsf{c})}{\log_2(M_\mathsf{par})}}\nonumber\\
&= \frac{R_\mathsf{c}\log_2(M_\mathsf{par})}{\log_2(M)(1-R_\mathsf{c})+R_\mathsf{c}\log_2(M_\mathsf{par})}.
\label{eqn:shaping:R_ts}\end{aligned}$$
Parity symbols
--------------
The parity symbols at the output of the code encoder are uniformly distributed. In [Fig.]{}, we observed that with uniform signaling, i.e., $\Lambda_\mathsf{par}=\{\lambda_1, \lambda_M\}$ and $M_\mathsf{par}=2$, is optimal for low as it achieves capacity and performs reasonably well for high . However, we note from [Fig.]{} that for a higher order modulation, even with uniform signaling, higher rates are possible. Hence, here we consider a scenario where $M_\mathsf{par}>2$. We further increase the rate by only using a subset of $\Lambda$ and by picking the eigenvalues such that they are not uniformly spaced.
Consider the information symbol alphabet $\Lambda=\{\lambda_1,\ldots,\lambda_8\}$ with $M=8$. For the $\Lambda_\mathsf{par}$, we could pick $\Lambda_\mathsf{par}=\{\lambda_1, \lambda_6, \lambda_7,\lambda_8\}$ with $p_X(\lambda)=\{0.25, 0.25, 0.25, 0.25\}$ and $M_\mathsf{par}=4$.
To find the function $\bm{s}(\cdot )$ that maps the parity symbols onto $\lambda\in\Lambda_\mathsf{par}$, we use a greedy algorithm as described in [Algorithm]{}. It starts with , i.e., $\Lambda_\mathsf{par}=\{\lambda_1, \lambda_M\}$. For each of the remaining symbols $\lambda\in\Lambda\setminus\Lambda_\mathsf{par}$, it calculates the time-scaled of $\lambda\cup\Lambda_\mathsf{par}$, finds the symbol $\lambda$ for which the time-scaled of $\lambda\in\Lambda\setminus\Lambda_\mathsf{par}$ is maximized, and adds it to $\Lambda_\mathsf{par}$. All symbols with a greater or equal imaginary part than $\lambda$ are removed, i.e., the eigenvalues $\{\lambda'\in\Lambda : {\mathfrak{I}\{\lambda'\}}\geq{\mathfrak{I}\{\lambda\}}\}$ are removed. This process is repeated until there are no symbols left. We then choose the set of symbols that gives the highest time-scaled as $\Lambda_\mathsf{par}$. We note that this procedure does not guarantee an optimal solution. However, for $M=\{4,8\}$ an exhaustive search gives the same result as that of [Algorithm]{}.
Constellation $\Lambda$ Constellation $\Lambda_\mathsf{par}$ $\Lambda_\mathrm{placed} = \{\lambda_1, \lambda_M\}$ $\Lambda_\mathsf{par} = \{\lambda_1, \lambda_M\}$
$\Lambda_\mathrm{not\,placed} = \Lambda \setminus \Lambda_\mathsf{par}$ Calculate ${\mathsf{I}(\Lambda_\mathrm{placed} \cup \lambda_i)}$ $\lambda_\mathrm{max} := \arg\max {\mathsf{I}(\cdot)}$ $\Lambda_\mathrm{placed} = \Lambda_\mathrm{placed} \cup \lambda_\mathrm{max}$ $\Lambda_\mathsf{par} = \Lambda_\mathrm{placed}$ $\Lambda_\mathrm{not\,placed} = \Lambda_\mathrm{not\,placed}\setminus\{\lambda:\lambda\in\Lambda_\mathrm{not\,placed}, {\mathfrak{I}\{\lambda\}}\geq{\mathfrak{I}\{\lambda_\mathrm{max}\}}\}$
$\Lambda_\mathsf{par}$
In [Fig.]{}, we show $\Lambda_\mathsf{par}$ for different modulation orders and . For $M=4$, we note that for low gives the best result. Increasing the results in a third level being added. The same behavior is observed for $M=8$. Compared to $M=4$, the third level is introduced at a slightly lower . This results from the fact that for $M=8$, different constellation points are available. For $M=16$, we note that again a third level appears when increasing the . When further increasing it, this third level moves to an eigenvalue with larger imaginary part and consequently a fourth level at an eigenvalue with lower imaginary part appears. This behavior can be observed repeatedly. To map the binary parity bits to the constellation points, we require $M_\mathsf{par}$ to be a power of two. As this is not always the case (see [Fig.]{}), we pick the largest power of two that is smaller or equal than the number of constellation points given by [Algorithm]{}.
Achievable Rate of Probabilistic Eigenvalue Shaping {#sec:achievable_rate}
---------------------------------------------------
To characterize the performance of , we derive the achievable rate of , denoted by $\mathsf{R}_\mathsf{ps}$. We assume that the channel is memoryless and that the decoder performs bit-metric decoding.
The achievable rate of is $$\begin{aligned}
\mathsf{R}_\mathsf{ps} &= R_\mathsf{ts}{\mathopen{}\mathclose\bgroup\originalleft}({\mathbb{H}(X)} - \sum\limits_{i=1}^{m}{\mathbb{H}(X^{\mathsf{B}}_i|Y^{\mathsf{B}}_i)} {\aftergroup\egroup\originalright}) \nonumber\\
&\quad + {\mathopen{}\mathclose\bgroup\originalleft}(1-R_\mathsf{ts}{\aftergroup\egroup\originalright}){\mathopen{}\mathclose\bgroup\originalleft}(m_\mathsf{par} - \sum\limits_{i=1}^{m_\mathsf{par}}{\mathbb{H}(X^{\mathsf{B}}_{\mathsf{par},i}|Y^{\mathsf{B}}_{\mathsf{par},i})}{\aftergroup\egroup\originalright}).\label{eqn:air}\end{aligned}$$
The achievable rate for has been derived in [@bocherer2017_achievable_rate_ps]. For a system employing time-sharing, the resulting achievable rate is the average of the achievable rate of the two transmission schemes.
![Achievable rates for different code rates with $\Lambda_\mathsf{par}$ according to [Algorithm]{} for a cutoff parameter $\delta=0.005$.[]{data-label="fig:shaping:achievable_rate_air"}](air_time_share)
In [Fig.]{}, we plot the capacity and the achievable rate [()]{} for different code rates $R_\mathsf{c}=\{$$1/4$, $1/3$, $2/5$, $1/2$, $3/5$, $2/3$, $3/4$, $4/5$, $5/6$, $8/9$, $9/10\}$ and modulation orders for a cutoff parameter $\delta=0.005$. $\Lambda_\mathsf{par}$ and hence $M_\mathsf{par}$ are chosen according to the results of [Algorithm]{}. For each modulation order, we notice that the curves cross at a certain . For below this point, the lowest code rate (corresponding to the highest curve) gives the best performance whereas for above this point, the highest code rate (corresponding to the highest curve) gives the best performance. We note the influence of time-sharing, which results in a gap between the achievable rate and capacity. The gap increases for lower code rates $R_\mathsf{c}$ as the channel input distribution deviates more from the optimal one.
Numerical Evaluation {#sec:performance}
====================
In this section, we evaluate the performance of the scheme via discrete-time Monte-Carlo and simulations. For the mapping $\bm{bi}(\cdot)$ (see [Fig.]{}), we use Gray labeling. Also, for the , we use the binary codes of the standard with code length $n_\mathsf{c}=64800$ and code rates $R_\mathsf{c}=\{$$1/4$, $1/3$, $2/5$, $1/2$, $3/5$, $2/3$, $3/4$, $4/5$, $5/6$, $8/9$, $9/10\}$. For the parity symbols, we use the constellation arising from [Algorithm]{}, depicted in [Fig.]{}.
Detection {#sec:detection}
---------
For the simulation, we simulate a continuous signal and hence, we require a detector. We use the following method to deal with the variable pulse durations: We set a threshold $\theta$ sufficiently higher than the noise. Once the magnitude of the signal rises above $\theta$, we save the time as $t_\mathsf{start}$ and when the magnitude of the signal falls below $\theta$, we save the time as $t_\mathsf{end}$. We then extend the interval bounded by $t_\mathsf{start}$ and $t_\mathsf{end}$, i.e., $\tilde{t}_\mathsf{start} = t_\mathsf{start} - \delta_t$ and $\tilde{t}_\mathsf{end} = t_\mathsf{end} + \delta_t$. Calculating the over the interval $[\tilde{t}_\mathsf{start}, \tilde{t}_\mathsf{end}]$ using the spectral method [@yousefi2014nftI-III Part II, Section IV] and only considering the imaginary part of the discrete eigenvalue gives the received symbol $y$. This approach requires that the is sufficiently high. As the model has the same requirement due to the perturbation approach, this requirement is fulfilled.
It may happen that due to noise, a received pulse never rises above the threshold $\theta$. In this case, the shortest duration is assumed (i.e., the duration of the pulse with amplitude zero). This scenario can be avoided by choosing the threshold sufficiently lower than the lowest amplitude. Furthermore, due to the shape of the capacity-achieving distribution, lower amplitudes are less likely, hence preventing this scenario.
To find the best threshold, we tested the performance for different values of $\theta$ and found that the performance of a threshold at $75 \%$ of the lowest non-zero amplitude of the constellation works best. We observed that small deviations of the threshold do not affect the performance significantly whereas setting the threshold too high (missing symbols with low amplitude) or to low (detecting a symbol where there is none) leads to performance degradation. Furthermore, we assume synchronization sequences spread sufficiently far apart in order not to impact the rate. We assume synchronization to be ideal such that it is guaranteed that error propagation is limited.
Numerical Results
-----------------
We perform Monte-Carlo simulations of the discrete-time model [()]{} and show the results in [Fig.]{}, where we plot the transmission rate at a of $10^{-5}$ for $M=4,8$ and $16$. The highest transmission rate for each modulation order corresponds to the highest code rate $R_\mathsf{c}$. We notice that the gap to capacity for $M=4$ is smaller than for $M=8$ and $M=16$. If we consider $\Delta M = M - M_\mathsf{par}$, i.e., the difference of the modulation order of $\Lambda$ and $\Lambda_\mathsf{par}$, we note that for a low $M$, $\Delta M$ is low was well. For example, for $M=4$, $\Delta M \leq 2$. Hence, the rate loss due to time-sharing is small. For $M=16$, the gap to capacity is smaller than for $M=8$. Considering the relevant range, we note that $\Delta M$ is smaller for $M=16$ than for $M=8$ and thus explaining the smaller rate loss.
We also simulated the transmission over a fiber using simulations transmitting a train of solitons. We consider a with parameters as in [Table]{} and two different amplification schemes, distributed Raman amplification and lumped amplification using . For both schemes, the peak power constraint is chosen such that the effect of the can be neglected, i.e., $\lambda_\mathsf{max}=2\jmath$. We employ the detection schemes as described in and choose the cutoff-parameter $\delta=0.005$, i.e., $99.5\,\%$ of the energy is contained in the pulse, for which the condition [()]{} is fulfilled. This then leads to a similar cutoff parameter as in [@shevchenko2018tcom]. For each modulation order $M=\{4,8,16\}$, we determine the furthest distance over which we achieve a of less than $10^{-5}$ and consider the rate gain compared to an unshaped system as in [@shevchenko2018tcom]. This results for $M=\{4,8,16\}$ in transmission over $\SI{3200}{\kilo\meter}$, $\SI{3040}{\kilo\meter}$, and $\SI{2960}{\kilo\meter}$ at a rate gain of $\SI{20}{\percent}$, $\SI{26}{\percent}$, and $\SI{95}{\percent}$, respectively. The results do not differ for distributed and lumped amplification as this is ensured by the peak power constraint.
Conclusion {#sec:conclusion}
==========
In this paper, we presented a probabilistic shaping scheme for an -based transmission system embedding information in the imaginary part of the discrete spectrum. It shapes the information symbols according to the capacity-achieving distribution and transmits them via time-sharing together with the uniformly distributed, suitably modulated parity symbols. We exploited the fact that the pulses of the signal in the time domain are of unequal length to improve the data rate compared to [@shevchenko2018tcom]. We used the time-scaled and derived the capacity-achieving distribution in closed form for the noiseless case and numerically in the general case. We showed that significantly improves the performance of an -based transmission scheme, and can almost double the data rate. As a possible extension of our work, the continuous spectrum can be used to increase the spectral efficiency [@aref2016ecoc].
![Performance of time sharing with parity symbols according to [Algorithm]{}. The rate points correspond to a performance at $\text{BER}=10^{-5}$. The highest transmission rate for each modulation order corresponds to the highest code rate $R_\mathsf{c}$.[]{data-label="fig:performance:mi_monte_carlo_perf_geometric"}](mi_monte_carlo_perf)
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank the anonymous reviewers for their feedback and comments which helped to improve this paper significantly. Especially, we would like to acknowledge one of the reviewers for proposing an elegant way to prove [Lemma ]{}, which is included in this paper.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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[^1]: This work was funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 676448.
[^2]: A. Buchberger is with the Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, SE-412 96, Sweden and Nokia Bell Labs, Lorenzstr. 10, 70435 Stuttgart, Germany, e-mail: [email protected].
[^3]: A. Graell i Amat is with the Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, SE-412 96, Sweden, e-mail: [email protected].
[^4]: V. Aref and L. Schmalen are with Nokia Bell Labs, Lorenzstr. 10, 70435 Stuttgart, Germany, e-mail: {firstname.lastname}@nokia-bell-labs.com.
| 0 |
---
abstract: 'The sensitivity of a low-noise superconducting transition edge sensor (TES) is determined by the thermal conductance of the support structure that connects the active elements of the device to the heat bath. Low-noise devices require conductances in the range 0.1 to 10pWK$^{-1}$, and so have to rely on diffusive phonon scattering in long, narrow, amorphous SiN$_{\rm x}$ legs. We show that it is possible to manufacture and operate TESs having short, ballistic low-dimensional legs (cross section 500 $\times$ 200nm) that contain multi-element phononic interferometers and ring resonators. These legs transport heat in effectively just 5 elastic modes at the TES’s operating temperature ($<$ 150mK), which is close to the quantised limit of 4. The phononic filters then reduce the thermal flux further by frequency-domain filtering. For example, a micromachined 3-element ring resonator reduced the flux to 19 % of a straight-legged ballistic device operating at the quantised limit, and 38 % of a straight-legged diffusive reference device. This work opens the way to manufacturing TESs where performance is determined entirely by filtered, few-mode, ballistic thermal transport in short, low-heat capacity legs, free from the artifacts of two level systems.'
author:
- 'E.A. Williams'
- 'S. Withington'
- 'C.N. Thomas'
- 'D.J. Goldie'
- 'D. Osman'
bibliography:
- 'References.bib'
title: Superconducting transition edge sensors with phononic thermal isolation
---
Introduction
============
There is considerable interest in developing superconducting Transition Edge Sensors (TESs)[@IrwinChapter] for astronomy and space science. For ground-based photometric measurements at long wavelengths (3 mm - 300$\mu$m), Noise Equivalent Powers (NEPs) of 10$^{-17}$WHz$^{-1/2}$ are required [@posada2015fabrication; @westbrook2016development; @AlMnAdvancedACTPol; @BICEP2KECKSPIDER]; for space-based measurements and Earth Observation at long wavelengths, NEPs of 10$^{-18}$WHz$^{-1/2}$ are necessary; for space-based measurements with cooled-aperture telescopes at FIR wavelengths, such as SPICA (200 - 30$\mu$m) [@SafariSPICA; @SafariNewImproved; @SPICANewFramework; @goldie2016performance; @goldie2012ultra], NEPs of 10$^{-19}$WHz$^{-1/2}$ and better are the necessary target. Time and energy resolved photon counting TESs are being developed for the x-ray space telescope Athena (0.2 - 12keV) [@gottardi2016development; @den2014requirements; @akamatsu2016tes; @TESMicrocalorimeterATHENA], and for general utilitarian applications at optical wavelengths (1550 - 400nm) [@cabrera1998SinglePhoton; @portesi2008OpticalPhoton; @eisaman2011invited; @SPDNatPhot2009; @QKDNoiseFree].
State-of-the-art TESs have many favourable characteristics, but they also have a number of shortcomings. To achieve low-noise operation, a low thermal conductance ($G =$ 0.1 - 10pWK$^{-1}$) is needed between the active elements of the device and the heat bath. TESs are usually fabricated on SiN$_{\rm x}$ membranes, and thin ($H =$ 200nm - 1$\mu$m), narrow ($W =$ 1 - 10$\mu$m), long ($L =$100 - 700$\mu$m) legs patterned into the membrane, using Deep Reactive Ion Etching (DRIE), to achieve the necessary thermal isolation. The lower the target NEP, the lower the thermal conductance required, and this leads to quite extreme geometries. In the case of ultra-low-noise imaging arrays, long legs ($L =$ 500$\mu$m - 1mm) prevent tight optical packing, and inefficient optical coupling schemes must be used to minimise the effects of the large pixel-to-pixel spacing. In addition, SiN$_{\rm x}$ is a highly disordered dielectric and contains an abundance of Two Level Systems (TLSs) [@anderson1972anomalous; @phillips1972tunneling; @zink2004specific]. TLSs result in a specific heat that is many hundreds of times higher than the Debye value and, when combined with low thermal conductance, this leads to devices that are too slow for some applications. Also, phonon trapping in long, narrow legs causes localised transport, which results in wide variations (at least $\pm$ 15 %) in the performance of even notionally identical devices on the same wafer.
In a previous paper we demonstrated that it is possible to manufacture SiN$_{\rm x}$ TESs having tiny ballistic support legs ($H =$ 200nm, $W < 1\,\mu $m, $L = $ 1 - 4$\mu$m) [@DJBallistic]. The thermal conductance and thermal fluctuation noise in these devices was found to be fully predicted by heat transport calculations based solely on the dispersion curves of elastic modes calculated using the bulk elastic constants of the material. Moreover, the uniformity in performance was high as a consequence of having eliminated resonant phonon scattering in the disorder of the material.
At low temperatures ($\leq$ 150mK), heat is transported in low-dimensional dielectric bars through a small number of elastic modes. In our ballistic devices [@DJBallistic], approximately 6-7 modes were excited, which is close to the quantised limit of 4: one compressional, one torsional, and in-plane and out-of-plane flexure. In a subsequent series of experiments [@ThermalAttenuation2017], we measured the thermal elastic attenuation length of these modes to be 20$\mu$m, and so our short-legged TESs were operating well within the ballistic limit. It can be shown, and was found in practice, that the ballistic, few mode limit corresponds to an NEP of approximately 10$^{-18}$WHz$^{-1/2}$. This NEP cannot be reduced further by increasing the length, because there is no scattering, or reducing the cross section, because we have already reached the quantised limit. The question arises as to whether it is possible to incorporate micromachined phononic filters into the low-dimensional legs of low-noise TESs in order to reduce the NEP below the ballistic quantised limit.
The incorporation of phononic filters would have a number of benefits: First, it should be possible to manufacture low-$G$ devices having legs that are significantly shorter than their long-legged diffusive counterparts. Second, the reduction in $G$ would be brought about by a phase coherent scattering process, which is likely to have a beneficial effect on the thermal fluctuation noise in the legs, as compared with that generated by a dissipative diffusive process. Third, we would like to manufacture devices using crystalline Si membranes [@ThermalSilicon; @SiliconCMB], as this would significantly reduce the heat capacity of the device, but the dispersion curves of Si are very similar to those of SiN$_{\rm x}$, and the elastic attenuation length considerably larger due to the low density of TLSs. Therefore phononic filters are needed if crystalline Si-membrane devices, which would have exceedingly long phonon mean free paths, are to be produced having NEPs of better than 10$^{-18}$WHz$^{-1/2}$.
The objectives of the exploratory work described here were as follows: (i) to determine whether TESs having low-dimensional phononic filters can be manufactured at all; (ii) to develop and compare manufacturing techniques using optical lithography (OL) and electron beam lithography (EBL); (iii) to investigate whether TESs with phononic filters behave in a conventional way; (iv) to determine whether thermal conductance can in practice be reduced significantly below the few-mode quantised limit; and (v) to investigate uniformity in performance between notionally identical devices. The experimental work was based solely on SiN$_{\rm x}$ membranes, but the results give direct information about the likely behaviour of phononic devices based on crystalline Si membranes.
Theory
======
Elastic waves and ballistic thermal power {#subsec:Theory_Elastic}
-----------------------------------------
The thermal flux through a uniform, low-dimensional, ballistic, dielectric bar can be calculated directly from the dispersion curves of the discrete elastic modes [@DJBallistic]. Here we summarise the calculation because it is central to the subject matter of the paper, and because the ballistic limit will be used later for normalising experimental data.
The classical elastic wave equation is $$\rho\omega^2 u_i +C_{ijkl}\frac{\partial^2 u_k}{\partial x_j \partial x_l} = 0,
\label{eq:ElasticWave}$$ where $u_i$ is the displacement field in Cartesian direction $i$, $C_{ijkl}$ the fourth-rank stiffness tensor, $\rho$ the mass density, $\omega$ the angular frequency, and the Einstein summation notation has been assumed. Equation (\[eq:ElasticWave\]) can be solved by adopting a general basis for the displacement field, $$\label{eq:BasisExp}
u_i = a_{ir}\psi_{ir},$$ where $a_{ir}$ is the $r$’th expansion coefficient of the $i$-directed displacement and $\psi_{ir}$ is the associated basis function. Equation (\[eq:BasisExp\]) may be substituted into Eq. (\[eq:ElasticWave\]), and the resulting algebraic equations solved numerically to give the dispersion curves of the propagating modes. Although a variety of basis functions, such as Gaussian-Hermite polynomials, could be used for this purpose, we have found power-series expansions to be particularly effective [@nishiguchi1997acoustic].
In the case of a homogeneous, isotropic, insulating dielectric such as SiN$_{\rm x}$, the stiffness tensor simplifies, and the modal calculation requires only the mass density, $\rho$, Young’s modulus, $E$, and Poisson’s ratio, $\nu_p$, of the material, in addition to the height, $H$, and width, $W$, of the bar. Averaging over any specific microstructure in favour of the bulk elastic properties is appropriate given the long dominant phonon wavelengths ($>$1$\mu$m) at low temperatures.
Figure \[fig:Modes\_Plot\](a) shows the dispersion profiles of the low-order modes of the experimentally considered geometry $H=$200 nm, $W=$ 500nm, with $\rho =$ 3.14gcm$^{-3}$, $E=$ 280GPa and $\nu_p =$0.28 [@vlassak1992new; @walmsley2007poisson; @MaterialsCRC]. With the exception of the four lowest-order modes, all modes have a cut-off frequency that increases as the cross-sectional area is reduced. Each mode can be assigned to one of four 2-dimensional displacement symmetries: compressional, in-plane and out-of-plane flexural, and torsional. The lowest order mode in each group is a principal mode with no cut-off. Propagation of these four principal modes at all frequencies imposes a fundamental lower limit on the power transmitted ballistically along a straight bar, even as its dimensions are reduced such that the higher order modes carry negligible power. This is often called the ‘quantised limit’.
cd ![\[fig:Modes\_Plot\] (a) Frequency, $\nu$, against wavenumber, $q$, of elastic modes with cut-off frequencies up to 25 GHz for a SiN$_{\rm x}$ bar having cross section 500 $\times$ 200nm. Longitudinal, in-plane transverse, out-of-plane transverse and torsional symmetries are shown in blue, green, red and magenta respectively, with the principal modes dashed. (b) Total net power spectral density (PSD) for a bar with termination temperatures of 68mK and 135mK, with the ordinate matching that of (a). (c) Mode number $i$, against cut off frequency $\nu_{i}$, showing quadratic dependence.](Figure1.pdf "fig:"){width="3.37in"}
In the context of TESs, each leg consists of a SiN$_{\rm x}$ bar terminating at the central island with a temperature taken to be the superconducting transition temperature $T_C$, and at the surrounding silicon wafer, held at the bath temperature $T_B$. The net thermal power transmitted from the island to the heat bath in the ballistic limit is therefore obtained by summing over the power carried by each mode, giving $$P_{bal} = \displaystyle \sum_i \int_{\nu_i}^\infty B(\nu,T_C) - B(\nu,T_B) \, \mathrm{d}\nu,
\label{eq:Pbal}$$ where $\nu_i$ is the cut-off frequency of the $i$th mode and $$B(\nu,T) = \frac{h\nu}{\mathrm{e}^{h\nu/k_BT}-1}$$ is the single-mode Power Spectral Density (PSD). Previously, we have demonstrated a strong agreement between the net thermal power given by Eq. (\[eq:Pbal\]) and measurements on TESs with leg lengths less than 4$\mu$m, and a range of widths [@DJBallistic]. This work confirmed that the net power can be calculated from first principles, through the bulk elastic constants, without free parameters, independently of the precise stoichiometry of the SiN$_{\rm x}$.
Figure \[fig:Modes\_Plot\](b) shows the total net PSD, $B(\nu,T_C) - B(\nu,T_B)$, summed over all simulated modes, for $T_C = 135$mK and $T_B = 68$mK. Sharp discontinuities are evident where individual modes cut on, corresponding to the intercepts of the dispersion curves with the ordinate of Fig. \[fig:Modes\_Plot\](a). At these experimentally representative temperatures, the PSD rolls off such that modes with $\nu_i \gtrsim 25$GHz carry negligible power. Figure \[fig:Modes\_Plot\](c) shows mode number $i$ against cut-on frequency $\nu_i$, indicating the number of propagating modes as a function of frequency. Above the four principal modes, the number of modes increases quadratically with frequency.
In numerical work, it is convenient to normalise calculated powers to the power carried by a single principal mode, $P_{qua}$, which defines an effective number of propagating modes: $$N_{eff} = \frac{P_{bal}}{P_{qua}} = \frac{\displaystyle \sum_i \int_{\nu_i}^\infty B(\nu,T_C) - B(\nu,T_B) \, \mathrm{d}\nu}{\int_{0}^\infty B(\nu,T_C) - B(\nu,T_B) \, \mathrm{d}\nu}.$$ $P_{qua}$ is the net power that would be carried by a single elastic mode. In the limit of narrow legs operating at low temperatures, the effective number of modes approaches $N_{eff}=4$. The experimentally measured power $P$ may be substituted for $P_{bal}$, as is done in Section \[sec:Results\_Discussion\], to calculate the actual effective number of modes propagating in test structures, $N_{eff}=P/4P_{qua}$, for which the ballistic case is an upper limit.
In experimental work, it is convenient to normalise the measured power $P$ flowing from the TES island to the heat bath to the straight-leg ballistic limit, $$\epsilon = \frac{P}{4N_{eff}P_{qua}}=\frac{P}{4P_{bal}},
\label{eq:Epsilon}$$ where the factor of 4 arises because each TES has 4 legs. In the case of a phononic thermal filter, $\epsilon$ quantifies the level of power attenuation achieved relative to the multi-mode ballistic case.
Phononic interferometers
------------------------
The central question of this paper is whether it is possible to achieve a significant reduction in thermal flux by introducing phononic filters into low-dimensional dielectric bars. In work on TESs, it is common practice to describe the heat flux in the legs by the equation $$P = K \left( T_{C}^{n} - T_{B}^{n} \right),
\label{eq:ThermalFlux}$$ where $K$ is a parameter that determines the overall magnitude of the flux, and $n$ is a parameter that describes the functional dependence on temperature. For truly ballistic transport in a single-mode structure $n=2$, whereas for ballistic transport in a highly multimode structure $n=4$ [@withington2011low]. In general, for diffusive transport in a few-mode structure, $n$ is intermediate between these two values. It follows from Eq. (\[eq:ThermalFlux\]) that the differential thermal conductance is given by $$G = n K T_{C}^{(n-1)}.
\label{eq:ThermalCond}$$ Both $K$ and $n$ change when a phononic filter is introduced, and therefore the flux and thermal conductance can in principle change in different ways. In what follows, we shall measure $K$ and $n$ directly for a variety of filters.
At first sight, it seems as if a suitable phononic filter might comprise alternating sections of narrow and wide bars, but simulations indicate that is difficult to achieve large acoustic impedance ratios, and the effect on power transmission is relatively small. More troublesome is the fact that the dominant phonon wavelengths at 100 mK are of order 2 $\mu$m, or shorter, and therefore optical lithography cannot be used easily to define steps that are highly abrupt on a scale size of $\lambda / 4$, diminishing the effectiveness of the filter.
An alternative approach is to make the legs wider and introduce periodic patterns of holes, thereby creating a truly phononic lattice [@PhononWaveBandgap; @ComprehensiveTopology; @HeatGuidingFocusing]. Such phononic crystals have been employed, for example, as support structures for micro-mechanical resonators, to reduce coupling loss due to elastic wave propagation to the substrate [@mohammadi2009high; @hsu2011reducing; @feng2014phononic]. Although this approach produces good filter characteristics, the number of transmission channels available, prior to the filter characteristic being applied, is high. Another way of thinking about this same problem is that the phononic lattice comprises a large number of low-dimensional links, each of which transports at least 4 modes. Thus the filter characteristic must compensate for the large increase in the number of underlying modes simply to break even.
We have taken a different approach based on few-mode elastic interferometers and ring resonators: An interferometer is formed by dividing a leg into two paths, one of which is longer than the other. Simulations based on multimode travelling wave calculations [@DJThesis] indicate that flux reductions of 25-75 % are possible, depending on the number of interferometers used in series. We have measured the thermal elastic attenuation length in SiN$_{\rm x}$ to be 20$\mu$m, and given that this is much larger than a typical wavelength, individual filters behave in a phase coherent way. Large series arrays of interferometers, however, become comparable to the attenuation length, and so operate in the diffusive to ballistic transition. To some extent, absorption isolates the effects of one interferometer on another. In other words, locally the structure behaves as a phase-coherent few-element filter, but globally, the structure conducts diffusively and behaves phase incoherently.
For the purposes of this paper, we define interferometers to be two-path elements that divide and recombine the travelling waves gradually. Another option is a ring resonator design, in which case one benefits from the modal scattering that takes place at the junctions, as well as the interferometric effects of the ring.
Effective thermal response time
-------------------------------
In the results that follow we report direct measurements of thermal fluxes in multi-stage interferometers and ring resonators patterned into the low-dimensional legs of SiN$_{\rm x}$ TESs. Furthermore, as an additional indicator of reduction in thermal differential conductance, we also measure the time constants of phononic devices.
![\[fig:Circuits\] (a): Thévenin equivalent representation of the TES bias circuit, where the TES is shown as a variable resistor $R(T,I)$. $R_L$ is the internal resistance of the voltage source, $V$. $R_L$ is the sum of a 1.45m$\Omega$ bias resistor and a $\approx 1$m$\Omega$ stray resistance. $L$ corresponds to the input inductance of the SQUID and any stray wiring inductance. $V_{TES}$ is the voltage across the TES. (b): Thermal circuit showing the TES with heat capacity $C$ coupled to the heat bath via thermal conductance $G$. $T$ and $T_B$ are the temperatures of the TES and the heat bath respectively . $P_J$ represents the Joule power dissipated in the TES, and $P_B$ the thermal power flowing to the heat bath.](Figure2.pdf){width="3.37in"}
The effective thermal time constant $\tau_{eff}$ of a TES may be determined from its response to a small step in bias voltage. The induced current response may be derived from the coupled differential equations that describe the TES electrical and thermal circuits. Figure \[fig:Circuits\](a) shows a Thévenin-equivalent representation of a bias circuit connected to a TES having a current and temperature dependent resistance $R(T,I)$, where the current is read out using an inductively coupled SQUID circuit. $R_L$ is the sum of the bias and stray resistances, and $L$ represents the input inductance of the SQUID and any additional stray inductance due to wiring. Figure \[fig:Circuits\](b) shows a representation of the thermal circuit, where to a first approximation, the TES has a single heat capacity, $C$, coupled to the heat bath via support legs with thermal conductance, $G$. The differential electrical and thermal equations are then $$L \frac{\mathrm{d}I}{\mathrm{d}t} = V-IR_L-IR(T,I),$$ and $$C \frac{\mathrm{d}T}{\mathrm{d}t} = -P_B+P_J,$$ respectively. $T$ is the temperature of the central island, $P_B$ is the power flow to the heat bath, and $P_J= I^{2} R(T,I)$ is the Joule power dissipated in the TES bilayer. Notice that we distinguish between $T_{C}$, which is the critical temperature of the bilayer defined by some point on the superconducting transition, and $T$, which is the temperature of the bilayer as the instantaneous operating point moves up and down the transition.
It is standard practice in TES physics, to expand non-linear terms such as $P_B$, $P_J$ and $R(T,I)$ to first order in the small-signal limit around the steady state operating point $T_0$, $I_0$ and $R_0$, giving [@IrwinChapter] $$\frac{\mathrm{d}}{\mathrm{d}t}
\begin{pmatrix}
\delta I \\
\delta T
\end{pmatrix}
=-
\begin{pmatrix}
\frac{1}{\tau_{el}} & \frac{\alpha P_{J0}}{T_0I_0L} \\[0.7em]
-\frac{I_0R_0(2+\beta)}{C} & \frac{1}{\tau_I}
\end{pmatrix}
\begin{pmatrix}
\delta I \\
\delta T
\end{pmatrix}
+
\begin{pmatrix}
\frac{\delta V}{L} \\[0.5em]
0
\end{pmatrix},
\label{eq:Diff_Matrix}$$ where $\delta I = I-I_0$, $\delta T = T-T_0$, $P_{J0} = I_0^2R_0$, and $\delta V$ represents a small change in the applied bias voltage. The resistance-temperature and resistance-current sensitivities are given by $\alpha = (\partial \ln R/\partial \ln T)_I$ and $\beta = (\partial \ln R/\partial \ln I)_T$ respectively. The time constants $\tau_{el} = L/(R_L+R_0(1+\beta))$ and $\tau_I = \tau/(1-P_{J0}\alpha/(GT_0))$ represent electrical and thermal time constants. The natural thermal time constant in the absence of electrothermal feedback, $\alpha = 0$, and strictly $\beta = 0$, is given by $\tau = C/G$.
Adapting the approach of Lindeman[@LindemanThesis; @IrwinChapter], Eq. (\[eq:Diff\_Matrix\]) may be solved for the specific case of a small step in bias voltage, $\delta V$ at $t=0$, subsequently maintained over the course of a measurement, giving $$\delta I = \frac{\delta V}{L}\frac{1}{\lambda_+-\lambda_-}(A_-\mathrm{e}^{-\lambda_-t}-A_+\mathrm{e}^{-\lambda_+t}+B),
\label{eq:RtMod}$$ where $\lambda_\pm = \tau_\pm^{-1}$ are eigenvalues of the matrix in Eq. \[eq:Diff\_Matrix\], $A_\pm = 1-\lambda_\pm^{-1}\tau_I^{-1}$ and $B=\tau_I^{-1}(\lambda_-^{-1}-\lambda_+^{-1})$. For low inductance, $\tau_+ \ll \tau_-$, such that $$\begin{aligned}
\tau_+ & \to \tau_{el} \\
\tau_- & \to \tau_{eff},\end{aligned}$$ where $\tau_{eff}$ is the effective thermal time constant, given by $$\tau_{eff} = \tau\frac{1+\beta+R_L/R_0}{1+\beta+R_L/R_0+(1-R_L/R_0)P_{J0}\alpha/(GT_0)}$$ $$\label{eq:tauEff_Approx}
\approx \frac{\tau}{1+\frac{\alpha}{n}(1- T_B^n / T_0^n )}.$$ Equation \[eq:tauEff\_Approx\] is a simplified form following the assumptions that $\beta$ and $R_{L}/R_{0}$ are small such that $R(T,I)\approx R(T)$, and the device is driven from a near-perfect voltage source [@irwin1998thermal; @IrwinThesis]. The empirical expression $P_{J0} = K(T_0^n-T_B^n)$ was used here, Eq. (\[eq:ThermalFlux\]), which is standard in the TES community.
The time constant $\tau_{eff}$ governs the rate at which the current stabilises after a voltage step has been applied, through the dominant exponential term in Eq. \[eq:RtMod\]. This response time is significantly shortened from its natural value $\tau = C/G$ due to negative electrothermal feedback when the TES is voltage-biased in its transition, where $\alpha \gg 0$. Since $\tau_{eff}$ is approximately proportional to $C/G$, a TES with reduced $G$ is expected to have a larger $\tau_{eff}$, which can be tested experimentally by fitting Eq. \[eq:RtMod\] to $\delta I(t)$. Measurements of $\tau_{eff}$ therefore provide an independent, relative measure of the differential thermal conductances of devices, as distinct from the thermal fluxes, assuming of course that the heat capacities of the devices are the same.
Experiment {#sec:Experiment}
==========
Transition Edge Sensors having a variety of patterned phononic legs were fabricated on 200nm thick, low-stress, amorphous SiN$_{\rm x}$ membranes. Every TES had an identical $80 \times 80$$\mu$m MoAu bilayer with 3 gold bars deposited on the upper surface, giving transition temperatures $T_C = 135 \pm 4$mK. Phononic structures were classified according to the number of filters in series per leg, m, and the filter style. Distinct filter styles were termed either ‘interferometers’ (mI), with pointed elliptical loops connected by collinear microbridges, or ‘ring resonators’ (mR), with typically angled connections intersecting circular rings: Fig. \[fig:TES\_Images\]. The primary difference between the two styles lies in the way in which power is divided and recombined upon entering and leaving a filter section: see later. A number of straight-leg control devices (mIC) were also fabricated, with lengths equal to the direct end-to-end lengths of the interferometers mI.
In previous work we have always used optical lithography (OL) and reactive ion etching (RIE) to pattern the the SiN$_{\rm x}$, followed by deep reactive ion etching (DRIE) to release the membrane from its supporting Si substrate[@glowacka2012fabrication]. Through this method we have been able to fabricate narrow legs, down to $W=$ 700nm, with a high degree of reliability and reproducibility. This method was also used to fabricate our previous few-mode ballistic devices, and we have successfully produced prototype interferometers using OL. The devices reported in this paper, however, used EBL and RIE to pattern the membranes. This required the development of direct-write EBL processing to pattern the legs and define the sputtered Nb bias leads. These new techniques then had to be combined with conventional OL to fabricate the main body of the TES. Using this hybrid method, we have been able to fabricate interferometers and ring resonators having leg cross sections of only 300 $\times$ 200nm, which ensures that only 4 elastic modes are excited in each bar of a structure for temperatures below 100mK.
![\[fig:TES\_Images\] (a) Optical microscopy of a typical ultra-low-noise TES with four long diffusive SiN$_{\rm x}$ support legs and a far-infrared absorber. Insets (b)-(g) show representative phononic legs of each type tested. These are labelled mI for an interferometer style, or mR for a ring resonator style, where m is the number of filters in series. Inset (h) shows a triple interferometer with Nb wiring on the SiN$_{\rm x}$. The cross section of each bar was 500 $\times$ 200nm for all devices tested. Note the difference between the interferometers and ring resonators. (g) Scanning Electron Microscopy image of a 1R phononic leg, viewed obliquely.](Figure3.pdf){width="3.37in"}
Figure \[fig:TES\_Images\](a) shows one of our traditional ultra-low-noise devices having long, straight legs ($W =$ 1.5$\mu$m, $L =$ 640$\mu$m). The MoAu bilayer, and bars, can be seen as a small gold-coloured square with lateral bars, and the $\beta$-phase Ta FIR absorber as a large gold-rimmed square. Around the outside of Fig. \[fig:TES\_Images\](a), (b - g), we show a number of the phononic filters fabricated. These comprised single, double and triple interferometers and ring resonators, and all of the features had cross sections of 500 $\times$ 200nm. Table \[tab:TES\_Props\] lists the devices tested, with the path difference $\Delta L$ engineered between the arms of the filters in each phononic leg. Figure \[fig:TES\_Images\](d) shows the Nb wiring, for bias and readout, on one of the 3-element interferometers, with an alignment tolerance of 50nm. Nb is significantly less stiff than SiN$_{\rm x}$ and therefore does not influence the elastic modes of the structure even though its thickness is comparable with that of the SiN$_{\rm x}$. The superconducting wiring also contributes negligible electronic heat conduction because the quasiparticle density is exceedingly small at low temperatures. Figure \[fig:TES\_Images\](h) shows a Scanning Electron Micrograph of a 1R phononic leg, viewed obliquely.
Figure \[fig:TES\_Images\] illustrates that it is possible to fabricate few-mode multi-element interferometers, with 500nm wide features, outstanding definition, and well-aligned Nb wiring. It is remarkable that these tiny patterned legs are perfectly able to support the main body of the TES, and can be fabricated with high yield, which was due in part to our ability to control film stresses in the main body of the device. As will be seen later, it is also notable that these devices performed perfectly well as TESs, with no evidence of anomalous behaviour, such as weak links or additional stray resistance where the Nb leads meandered over the arms of the interferometers. As will be seen later, the thermal properties of these tiny structures were fully consistent with few-mode elastic behaviour even though they were supporting the relatively large central island of the TES. This occurs because the bulk elastic constants are relatively insensitive to static strain, and furthermore the dispersion relationships are insensitive to the bulk elastic constants.
Device $\Delta L$ ($\mu m$) $n$ $K$ (pW/K$^n$) $T_C$ (mK) $\epsilon$ $G$ (pW/K) $N_{eff}$ $\tau_{eff}$ (ms)
--------- ---------------------- ------ ---------------- ------------ ------------ ------------ ----------- -------------------
1IC$_a$ - 2.70 22.2 141.1 0.66 2.16 3.39 0.54
1IC$_b$ - 2.53 15.4 134.5 0.62 1.81 3.13 0.64
2IC$_a$ - 2.43 14.2 137.4 0.69 2.01 3.47 0.53
2IC$_b$ - 2.37 12.3 133.3 0.67 1.86 3.36 0.64
3IC$_a$ - 2.28 7.5 138.3 0.48 1.36 2.42 0.85
1I$_a$ 1 2.74 20.1 141.4 0.55 1.84 2.83 0.60
1I$_b$ 1.5 2.59 13.7 131.7 0.49 1.41 2.44 0.78
1R$_a$ 0 2.69 15.2 137.2 0.45 1.42 2.30 1.20
2I$_a$ 1, 1.5 2.62 11.8 127.8 0.40 1.10 1.98 0.90
2I$_b$ 1, 1.75 2.67 13.3 134.5 0.41 1.25 2.07 0.73
2R$_a$ 1, 1.75 2.60 8.1 127.4 0.28 0.78 1.39 1.27
3I$_a$ 1, 1.5, 2 2.47 6.5 136.7 0.29 0.86 1.48 1.04
3I$_b$ 1, 1.25, 1.75 2.48 6.7 133.2 0.30 0.85 1.49 1.52
3R$_a$ 1, 1.25, 1.75 2.44 4.0 128.5 0.19 0.51 0.94 1.88
3R$_b$ 1, 1.3, 1.6 2.51 4.4 129.3 0.19 0.51 0.92 1.28
Each TES was voltage-biased with a low impedance source ($\approx$ 1.5m$\Omega$) and read out using a two-stage SQUID amplifier as a low-noise current-to-voltage converter. The TES and SQUID chips were mounted in an optically blackened light-tight box and cooled to a base temperature of 68mK in an adiabatic demagnetisation refrigerator (ADR). The bath temperature of the TES chip was taken to be that of the copper housing, held constant to within 200$\mu$K by means of the residual current in the ADR magnet. Current and voltage offsets and stray resistances were identified and compensated for in data processing. The TES current response to a step in voltage was obtained by biasing the TES in its transition and superimposing a square wave on the bias input, with small amplitude compared to the voltage width of the transition. Current response was averaged over multiple leading-edge voltage steps.
Results and Discussion {#sec:Results_Discussion}
======================
A TES voltage-biased within its transition self-regulates its temperature due to negative electrothermal feedback. In the steady state, $dT / dt= 0$, the net power flow from the island to the heat bath is equal to the Joule power dissipated in the bilayer. The power flow is therefore given by $P_{B} = P_{J} = IV_{TES}$, allowing $P_{B}$ to be obtained from a series of $I$-$V_{TES}$ curves taken over a range of bath temperatures. If the electrothermal feedback is strong, the TES temperature is essentially constant within the transition at $T_C$, allowing $K$ and $n$ to be found.
Figure \[fig:PT\_Inset\_PV\](a) shows thermal power against the voltage across the TES, $V_{TES}$, for device 3R$_b$, for a set of bath temperatures, $T_{B}$. The topmost curve corresponds to the lowest bath temperature used, $T_{B0}$. The power is essentially constant across the voltage range for which the bilayer is in its transition, indicating the presence of strong electrothermal feedback. This plot is representative of all devices tested, and demonstrates that the presence of the phononic filters in the legs (4 for each device), and associated Nb wiring layer, does not introduce artifacts into the operation of the device.
The power averaged over the transition region is shown in Fig. \[fig:PT\_Inset\_PV\](b) as a function of $T_B$ for devices 3IC (green), 3I$_b$ (red), and 3R$_b$ (magenta). Figure \[fig:PT\_Inset\_PV\](b) displays a clear reduction in the power transmitted through the triple phononic filters relative to the straight leg control, with further improved attenuation for the ring resonator 3R$_b$ over the interferometer 3I$_b$. This behaviour is reproduced almost identically in devices 3I$_a$ and 3R$_a$, For all of the devices tested, Eq. (\[eq:ThermalFlux\]) was fitted to data of this kind under the assumption that the temperature of the TES was maintained constant at nearly $T_C$, which is true for sufficiently sharp transitions. $K$, $n$ and $T_C$ were free parameters in the fitting process, with $T_C$ corresponding to the intercept of the curve on the $P_{B}=0$ axis.
![\[fig:PT\_Inset\_PV\] (a) Power against voltage across the TES, $V_{TES}$, for device 3R$_b$, for a subset of $T_B$ from $T_{B0}=67.5$mK to 125mK from top to bottom. (b) Net power flow from the TES island to the heat bath, $P_{B}$, against bath temperature, $T_B$, for devices 3IC (green, solid), 3I$_b$ (red, dashed) and 3R$_b$ (magenta, compound dashed). Points show measured data, and lines indicate model fits according to Eq. (\[eq:ThermalFlux\]). ](Figure4.pdf){width="3.37in"}
For each device, the measured power at the lowest bath temperature, $T_{B0} = 68 \pm 1$mK, was used to calculate the normalised power per leg, $\epsilon$, according to Eq. \[eq:Epsilon\]. In this way, the measured fluxes were normalised to the theoretical ballistic power for a device with straight legs with termination temperatures $T_{B0}$ and $T_C$. The total thermal conductance $G$ of the support structure was determined from Eq. \[eq:ThermalCond\]. Table \[tab:TES\_Props\] lists measured values of $n$, $K$, $T_C$, $\epsilon$, $G$, and $N_{eff}$.
![\[fig:Epsilon\_L\]Net power flow through each leg, $\epsilon$, against leg length, $L$, for the lowest bath temperature measured, $T_{B0}$, normalised to the theoretical ballistic power for a straight leg, given $T_{B0}$ and $T_C$. Straight leg control, interferometer and ring resonator structures are shown in green, red with border and magenta with central dot respectively. Circles, triangles and diamonds represent legs with one, two and three filters, with control devices matching their corresponding phononic designs. Filled markers are plotted with respect to the direct end-to-end leg length. For phononic legs, open markers show the same $\epsilon$ against the equivalent length of a straight leg that would give the same thermal conductance in a purely diffusive model. The blue line shows $\epsilon_{r}$ for diffusive phonon transport according to Eq. (\[eq:Epsilon\_L\_Diff\]) with an acoustic attenuation length of 20$\mu$m.](Figure5.pdf){width="3.37in"}
Figure \[fig:Epsilon\_L\] shows the normalised flux $\epsilon$ against leg length $L$ for all of the devices tested. Also shown for comparison (solid blue line) is an analytical model for heat transport in the diffusive to ballistic regime: $$\epsilon_{r} = (1+L/L_a)^{-1}.
\label{eq:Epsilon_L_Diff}$$ In previous work[@ThermalAttenuation2017], we determined the acoustic attenuation length, $L_a$, to be 20$\mu$m in SiN$_{\rm x}$ at low temperatures. This was achieved by fitting Eq. (\[eq:Epsilon\_L\_Diff\]) to data from a set of straight leg devices having lengths, 1 - 490$\mu$m, which span the diffusive to ballistic transition. An $\epsilon$ of less than unity indicates that a filter has a transmission factor lower than the ballistic case, and an $\epsilon$ of less than $\epsilon_{r}$ indicates that a filter has a transmission factor lower than its straight-legged counterpart, where some diffusive scattering is present.
In order to compare the flux of a phononic filter with a straight-legged device, it is necessary to assign an equivalent length to the filter, and this can be done in a variety of ways. In Fig. \[fig:Epsilon\_L\], filled markers show the normalised flux as a function of the overall end-to-end length of each phononic leg, equal to the length of the corresponding straight reference legs spanning the same gap. It could be argued, however, that the actual length of the path travelled should be used. For a purely diffusive process, where $G \propto 1/L$, the greater path length of a curved leg would reduce $G$ relative to a straight leg device with the same end-to-end length irrespective of any coherent destructive process. This should be taken into account, but it is still not clear which path along a multistage filter should be used.
For a fully diffusive process, it is possible to define an equivalent length, $L_{eq}$, based on the notion of thermal conductances in parallel. In a single interferometer for example, $L_{eq} = L_{l1}+(1/L_{a1}+1/L_{a2})^{-1}+L_{l2}$, where $L_{l1}$ and $L_{l2}$ correspond to the straight linking sections and $L_{a1}$ and $L_{a2}$ to the lengths of the different paths around the filter. $L_{eq}$ is therefore a single equivalent length giving the same $G$ as a chain of series and parallel conductances representing a phononic structure, for $G \propto 1/L$. This constitutes a more appropriate definition of length in Eq. \[eq:Epsilon\_L\_Diff\] for phononic structures, because it does not mistakenly imply that a reduction in $\epsilon$ due to the longer path length of the interferometer is necessarily due to coherent interference. The open markers in Fig. \[fig:Epsilon\_L\] show $\epsilon$ against $L_{eq}$ for all phononic devices. All of the open markers are to the left of the solid markers because the parallel arms reduce the effective length.
The phononic legs show a clear reduction in transmitted power relative both to their corresponding straight leg control devices and the diffusive attenuation expected from Eq. \[eq:Epsilon\_L\_Diff\]. This presents strong evidence that micromachined phononic filters can be used to reduce thermal flux. Moreover, the reductions achieved are comparable with those predicted previously[@DJThesis]. A maximum flux reduction to 19 % of the ballistic limit is achieved for the 3R devices, corresponding to 38 % of the flux in fully diffusive devices. From Eq. \[eq:Epsilon\_L\_Diff\], a leg length of 87$\mu$m would be necessary to achieve this attenuation in the absence of the phononic filter, a more than threefold increase from the 24$\mu$m end-to-end length actually used. The expected monotonic decrease in $\epsilon$ with number of filters per leg is also observed within both the interferometer and ring resonator groups [@DJThesis].
Within pairs of devices of the same type, the greatest difference in $\epsilon$ for different filter path lengths, $\Delta L$, is observed between 1I$_a$ and 1I$_b$, with the larger $\Delta L$ giving the lower transmission. For devices of type 2I, 3I and 3R, variations in $\Delta L \leq 0.25$$\mu$m of the second and third filter stages show negligible effect on $\epsilon$. This insensitivity is reasonable since a change in differential path length, $\Delta L$, shifts the fringe of the filter, in frequency space, relative to the wide band blackbody spectrum, changing the transmitted flux very little. In the next phase of the work, we will carry out detailed simulations of precise designs in order to understand the degree to which modelling can be used to predict and optimise behaviour.
Figure \[fig:Epsilon\_L\] shows that ring resonators perform significantly better than their interferometer counterparts, for single, double and triple designs, including the case where $\Delta L$ is the same for the two types. The origin of this improvement is likely to be due to the way in which the principle modes scatter at the junctions. An elastic wave reaching an interferometric filter may maintain its symmetry as it divides between the two arms, whereas in a ring resonator, the incoming wave encounters a perpendicular bar and mode conversion takes place; for example from a torsional wave to two out of phase flexures. Similarly, ring resonators allow waves to propagate multiple times around the ring, possibly contributing to Fabry-Pérot-like enhanced attenuation.
At the lowest bath temperatures, $T_B\approx68$ mK, used in this experiment, the effective number of modes transporting heat in a purely ballistic leg is approximately 5, which is very close to the quantised limit of 4. For $\epsilon_{r} = 0.5$, from Eq. \[eq:Epsilon\_L\_Diff\] for $L=L_{eq}=20.7$ in the case of the 3-element ring resonator, the calculated effective number of modes is then 2.5. However, Table \[tab:TES\_Props\] shows that for both 3R devices, the effective number of modes is 0.92, from the measured power. Thus the phononic structures have significantly reduced the effective number of modes through frequency-domain filtering.
Figure \[fig:G\_L\] plots the differential conductance $G$ against leg length. Indeed the small-signal behaviour of a TES depends on $G$ rather than on the absolute value of flux. The trends in conductance are essentially the same as those in flux, with minor differences due to the effect on $G$ of variations in $T_C$ between devices. The 2- and 3- stage ring resonators significantly reduce the conductance below the ballistic value of about 2.2 pWK$^{-1}$. In the case of phononic filters, Table \[tab:TES\_Props\] shows that the reduction in $\epsilon$ and $G$ is associated with a reduction in $K$, with $n$ staying almost constant. In the ballistic case, we find $n \approx 2.5$,[@DJBallistic] which is slightly above the single-mode value $n=2$. In the case of phononic filters, it seems that $n \approx 2.5$ also. This is very different to the case of long, narrow diffusive legs where $n$ takes on values of unity and below, which we have always regarded as being an indicator of the effects of TLS loss in the disordered SiN$_{\rm x}$ [@ThermalAttenuation2017].
The values of $G$ achieved with few-mode ballistic and phononic legs are already highly suitable for many applications, but in particular it should be noted that if we were to use a 3-stage ring resonator with $T_{B} =$ 50mK and $T_{C} = $ 100mK, then $G =$ 0.3pWK$^{-1}$, and we would be close to the requirement $G= 0.2$pWK$^{-1}$ for the ultra-low-noise TESs needed for SPICA. Now, however, the legs would only be of order 25$\mu$m long, rather than the 600-700$\mu$m long legs currently used.
Of particular note is the remarkable consistency in $\epsilon$ and $G$ between different devices of similar design. In fact some of the points on Fig. \[fig:Epsilon\_L\] and Fig. \[fig:G\_L\] are difficult to distinguish. In conventional TESs having narrow, straight legs, hundreds of microns long, different research teams see conductance variations of $\pm15\%$ or higher between notionally identical devices, even from the same wafer. This variation is attributed to phonon localisation, where elastic waves are reflected by impedance discontinuities due to disorder in the dielectric, creating resonant cells that exaggerate variations in elastic properties[@ThermalAttenuation2017]. The reproducibility seen in Fig. \[fig:Epsilon\_L\] strongly suggests that phononic filters are capable of producing highly uniform arrays, eliminating the troublesome effects of localisation seen in conventional devices.
![\[fig:G\_L\] Conductance, $G$, against leg length, $L$, for the lowest bath temperature measured, $T_{B0}$, Straight leg control, interferometer and ring resonator structures are shown in green, red with border and magenta with central dot respectively. Circles, triangles and diamonds represent legs with one, two and three filters, with control devices matching their corresponding phononic designs. Filled markers are plotted with respect to the direct end-to-end leg length. For phononic legs, open markers show the same $\epsilon$ against the equivalent length of a straight leg that would give the same thermal conductance in a purely diffusive model.](Figure6.pdf){width="3.37in"}
As an independent indicator of the reduction in $G$, we measured the effective thermal time constants of the phononic TESs. Because all of the TESs were identical, apart from the different leg designs, we would expect the time constants to follow $G$ in the appropriate way. Figure \[fig:Rt\_Fit\] shows the change in current, $\delta I$, in response to a small step in bias voltage at $t=0$ for device 3R$_b$ at 90.3mV, corresponding to a point on the transition where the resistance of the bilayer was 28 % of its normal value. The first dip on the leading edge is due to the electrical response of the TES, and its bias circuit, whereas the slowly rising trailing edge is due to the electrothermal relaxation.
![\[fig:Rt\_Fit\] TES current response, $\delta I$, to a small step increase in voltage at time $t=0$, in arbitrary units for device 3R$_b$ at 90.3mV, 28% of the TES normal resistance. The red line shows $\delta I$ according to the model given by Eq. \[eq:RtMod\]: $\tau_{eff} = 1.12$ms, $\tau_{el} = 2.16$$\mu$s, $\tau_{I} = -0.48$ms.](Figure7.pdf){width="3.37in"}
$\delta I(t)$ given by Eq. (\[eq:RtMod\]) was fitted to the measured data with a scaling pre-factor to give $\alpha = 551$ and $\beta = 0.85$, corresponding to $\tau_{eff} = 1.12$ms, $\tau_{el} = 2.16$$\mu$s and $\tau_{I} = -0.48$ms, for this particular bias point. The steady state values of $I_0$, $R_0$ and $P_{J0}$ were taken from $I$-$V_{TES}$ measurements, $T_0=T_C$ was assumed, $L$ was derived from impedance measurements with the bilayer in its fully superconducting state, and $C=41.8$fJ$K^{-1}$ was calculated using the volumes and specific heats of the various materials used. We found that although the values of $\alpha$ obtained scale with the value of $C$ assumed, the fitted values of $\tau_{eff}$ obtained do not change; in other words, there is a linear degeneracy between the $\alpha$ and $C$, but the same value of $\tau_{eff}$ always results. Figure \[fig:Rt\_Fit\] is typical of the data taken, and the fit is in good agreement with the model despite using only a single heat capacity $C$.
![\[fig:TauEff\_G\] Effective thermal time constant, $\tau_{eff}$, against thermal conductance $G$ for all devices. As in Fig. \[fig:Epsilon\_L\], straight leg control, interferometer and ring resonator structures are shown in green, red with border and magenta with central dot respectively. Circles, triangles and diamonds represent legs with one, two and three filters, with control devices matching their corresponding phononic designs. Error bars correspond to standard error in the mean of multiple measurements of $\tau_{eff}$ at different bias points. The blue line shows the fitted model $\tau_{eff}\propto1/G$.](Figure8.pdf){width="3.37in"}
Figure \[fig:TauEff\_G\] plots $\tau_{eff}$ against $G$ for all of the devices tested. The error bars correspond to the standard error in the plotted mean of multiple measurements of $\tau_{eff}$ for different bias points across the transition, where applicable for each device. $\tau_{eff}$ is expected to be approximately inversely proportional to $G$ from Eq. (\[eq:tauEff\_Approx\]), and so the simple model $\tau_{eff}\propto 1/
\/G$ is shown as a blue line on Fig. \[fig:TauEff\_G\]. The general agreement offers additional evidence that the phononic filters reduced the differential conductance of the legs in the expected way. There are various reasons why, however, this proportionality may not be exact. The heat capacity will increase slightly as bias voltage is reduced through the transition as the MoAu bilayer becomes increasingly superconducting. Additionally, unknown sources of heat capacity may exist, for example due to residual SiO$_2$ used as an etch stop in fabrication. The steady state values $I_0$, $R_0$ and $P_{J0}$ vary with bias point, although the impact on $\tau_{eff}$ between measurements on the same device is typically small. Notwithstanding these considerations, Fig. \[fig:TauEff\_G\] shows an overall decrease in $\tau_{eff}$ with increasing $G$, which implies that the interferometrically reduced values of $G$ derived through the measured values of $K$ and $n$ are true differential conductances.
Conclusions
===========
We have successfully manufactured a range of superconducting transition edge sensors having few mode, phononic thermal isolation in the legs. By using electron beam lithography we were able to pattern interferometers and ring resonators into legs having cross-sectional dimensions of only 500 $\times$ 200nm. At temperatures of around 100mK each leg effectively transports heat in just 5 elastic modes, which is close to the quantised limit of 4. The phononic filters then reduced the thermal flux and conductance further. Nb bias leads were patterned on the filters to an alignment tolerance of better than 50nm. The manufacturing process proved to be highly reliable, giving robust devices with high dimensional definition.
Significant reductions in thermal flux and thermal conductance were recorded, with the ring resonators giving the highest rejection ratios. No artifacts were seen in behaviour, making the devices suitable for many applications. The device-to-device variation in thermal conductance of notionally identical devices was exceedingly small, and well below the $\pm$ 15 % frequently seen in conventional long-legged designs. It should also be noted that the temperature exponent $n$ stayed at its near ballistic value of 2.5, in contrast to the case of long narrow legs, where diffusive transport due to two level systems reduces the exponent to typically 1, and below.
A key advantage of phononic filters is that it is possible to approach the very lowest $G$s seen with long ($>$ 700$\mu$m) diffusive legs, but using significantly shorter structures. The attenuation length of the low-order modes in SiN$_{\rm x}$ is 20$\mu$m, and the ring resonators are typically 5$\mu$m in diameter, and therefore by placing a large number of ring resonators in series, dividing a diffusive leg into phase-coherent phononic cells, one would expect to be able to realise thermal conductances significantly smaller than anything achieved to date. Also, one would expect to be able to create ultra-low-noise TESs using crystalline ballistic Si membranes, which would have many advantages.
The next stage in our work will be to carry out a range of scattered travelling wave simulations in diffusive structures in an attempt to identify optimised filters with even higher levels of attenuation. We have already developed a modelling technique for patterned phononic structures operating in the ballistic to diffusive regime, and this will be reported in an upcoming paper.
The authors are grateful to Science and Technology Facilities Council for funding this work. Emily Williams is grateful for a PhD studentship from the NanoDTC, Cambridge, EP/L015978/1.
| 0 |
---
abstract: 'Transverse dunes appear in regions of mainly unidirectional wind and high sand availability. A dune model is extended to two dimensional calculation of the shear stress. It is applied to simulate dynamics and morphology of transverse dunes which seem to reach translational invariance and do not stop growing. Hence, simulations of two dimensional dune fields have been performed. Characteristic laws were found for the time evolution of transverse dunes. Bagnold’s law of the dune velocity is modified and reproduced. The interaction between transverse dunes led to interesting results which conclude that small dunes can pass through bigger ones.'
author:
- 'Veit Schwämmle$^{(1)}$ and Hans J. Herrmann$^{(1)}$'
bibliography:
- 'dune.bib'
title: Modelling transverse dunes
---
Introduction
============
Transverse dunes have been investigated with constant interest recently. Field measurements have been carried out in order to obtain more knowledge about airflow and sediment transport over transverse dunes [@Wilson72; @Lancaster82; @Lancaster83; @Mulligan88; @Burkinshaw93; @McKenna2000]. Several numerical models have been proposed to model dune formation [@Wippermann86; @zeman-jensen:88; @Fisher88; @Stam97; @NishimoriXX; @boxel-arens-van_dijk:99; @van_dijk-arens-boxel:99; @herrmann-sauermann:2000; @MomijiWarren2000; @SauermannKroy2001; @KroySauermann2002]. But still many questions are not answered and some efforts are needed to better understand dune morphology and dynamics or even to anticipate dune formation. The simulations in this article will try to answer some questions and justify the importance of further field measurements.
First some general aspects will be introduced. A model for dunes in three dimensions will show why translational invariance appears in transverse dune formation. The time evolution and the dune velocity in a two dimensional model with constant sand influx will be presented. The time evolution of a model of two dimensional dunes with periodic boundary conditions will give some more insight into dune field dynamics. A final discussion of the statement that dunes behave like solitons will close this article.
General aspects of transverse dunes {#sec:trans_intro}
===================================
About 40% of all terrestrial sand seas are covered by transverse dunes. They are mostly located in sand seas where sand is available. Thus in larger sand seas one mainly finds ensembles of many transverse dunes which interact with respect to their dynamics. The crest to crest spacing ranges from a few meters to over 3 km [@Breed79]. They are common in the Northern Hemisphere, for example in China, and along coasts. On Mars transverse dunes dominate the sand seas.
![An aerial photo (STS047–153A–263) of transverse dunes in Lop Nur, China. Image courtesy of Earth Sciences and Image Analysis Laboratory, NASA Johnson Space Center (http://eol.jsc.nasa.gov)[]{data-label="fig:transv_photo"}](Figure1.eps){width="80.00000%"}
An example of a field of transverse dunes in China is shown in Figure \[fig:transv\_photo\]. Normally transverse dunes have more irregular patterns and even smaller hierarchies of smaller transverse dunes can be found in a field of big transverse dunes. Strong winds coming essentially from the same direction are the main environment where this type of dune can be found. proposed that in highly mobile environments cross winds distort the dune shape.
![Two dimensional cut through a single transverse dune profile $h(x)$ (solid line) and the sand flux $q(x)$ (dashed line) over it. The wind is coming from the left.[]{data-label="fig:transv_single"}](Figure2.eps){width="80.00000%"}
In Figure \[fig:transv\_single\] a dune of a height of about $27$ m is depicted which is part of a two dimensional calculation of our model. This dune is situated between other similar dunes which together result in a simulation of a dune field of a length of four kilometers. The dashed curve shows the sand flux which is set to zero at the slip face where the shear stress is not strong enough to entrain sand. This occurs in the region of flow separation, defining the boundary between quasi-laminar flow and the turbulent layer of the eddy after the brink. The brink separates windward side and slip face. In this case the brink and the crest do not coincide which means in this case that this dune has not yet reached a stationary state.
The dune model {#sec:model}
==============
The model described here can be seen as a minimal model including the main processes of dune morphology. As predecessor of the model described here the work of revealed interesting new insights into dynamics and formation. The model is now extended to a two dimensional shear stress calculation (longitudinal and lateral direction), a full sand bed and different boundary conditions. Nevertheless the wind is restricted to be constant and unidirectional in time. In this article the model is used to simulate transverse dune fields. In every iteration the horizontal shear stress ${\boldsymbol{\rm \tau}}$ of the wind, the saltation flux ${\boldsymbol{\rm q}}$ and the flux due to avalanches are calculated. The time scale of these processes is much smaller than the time scale of changes in the dune surface so that they are treated to be instantaneous. In the calculation of the surface evolution a time step of 3–5 hours is used. In the following the different steps at every iteration are explained.
#### The air shear stress $\tau$ at the ground:
The shear stress perturbation over a single dune or over a dune field is calculated using the algorithm of . The $\tau_x$-component points in wind direction and the $\tau_y$-component denotes the lateral direction. The calculation is made in Fourier space, where $k_x$ and $k_y$ denote the wave numbers, $$\begin{aligned}
\label{eq:tau_x}
\hat\tau_x(k_x,k_y) =
\frac{h(k_x,k_y) k_x^2}{|k|}
\frac{2}{U^2(l)} \cdot \nonumber \\
\left( 1 +
\frac{2 \ln L|k_x| + 4 \gamma + 1 +
i \, \text{sign}(k_x) \pi}{\ln l/z_0} \right),\end{aligned}$$ and $$\label{eq:tau_y}
\hat\tau_y(k_x,k_y) = \frac{h(k_x,k_y) k_x k_y}{|k|}
\frac{2}{U^2(l)},$$ where $|k| = \sqrt{k_x^2 + k_y^2}$ and $\gamma=0.577216$ (Euler’s constant). $U(l)$ is the normalized velocity of the undisturbed logarithmic profile at the height of the inner region $l$ [@SauermannPhD2001] defined in . The roughness length $z_0$ is set to $0.0025$ m and the so called characteristic length $L$ to $10$ m.
Equations (\[eq:tau\_x\]) and (\[eq:tau\_y\]) are calculated in Fourier space and have to be multiplied with the logarithmic velocity profile in real space in order to obtain the total shear stress. The surface is assumed to be instantaneously rigid and the effect of sediment transport is incorporated in the roughness length $z_0$. For slices in wind direction the separation streamlines in the lee zone of the dunes are fitted by a polynomial of third order going through the point of the brink $x_0$. The length of the separation streamlines is determined by allowing a maximum slope of $14^0$ [@SauermannPhD2001]. Where the separation streamlines cross the surface at the reatachment point $x_1$, a new stream line is calculated as third polynomial attaching $x_0$ and $x_1$. The separation bubble guarantees a smooth surface and the shear stress in the area of the separation bubble is set equal to zero. Problems can arrive due to numerical fluctuations in the value of the slope of the brink where the separation bubble begins and its influence on the calculation of a separation streamline for each slice. To get rid of this numerical error the surface is Fourier-filtered by cutting the small frequencies. Figure \[fig:transv\_sinus\] depicts the separation bubble and an interesting similarity of the separation bubble profile to a sine function.
![The transverse dune profile $h(x)$, its separation bubble $s(x)$ and a sine function. The separation bubble ensures a smooth surface. The wind is coming from the left.[]{data-label="fig:transv_sinus"}](Figure3.eps){width="80.00000%"}
#### The saltation flux $q$:
The time to reach the steady state of sand flux over a new surface is several orders of magnitude smaller than the time scale of the surface evolution. Hence, the steady state is assumed to be reached instantaneously. The length scale of the model is too large to include sand ripples. Nevertheless the kinetics and the characteristic length scale of saltation influence the calculation by breaking the scale invariance of dunes and by determining the minimal size of a barchan dune [@SauermannKroy2001]. A calculation of the saltation transport by the well known flux relations [@Bagnold41; @Lettau78; @Sorensen91] would restrict the model to saturated sand flux which is not the case for example at the foot of the windward side of a barchan dune due to little sand supply or at the end of the separation bubble in the interdune region between transverse dunes due to the vanishing shear stress in the separation bubble. The sand density $\rho(x,y)$ and the grain velocity ${\boldsymbol{\rm u}}(x,y)$ are integrated in vertical direction and calculated from mass and momentum conservation, respectively. We simplified the closed model of [@SauermannKroy2001] by neglecting the time dependent terms and the convective term of the grain velocity ${\boldsymbol{\rm u}}(x,y)$. The expansion to two dimensions yields Equations (\[eq:3d\_rho\]) and (\[eq:3d\_u\]) where $\rho$ and ${\boldsymbol{\rm u}}$ are determined from the before obtained shear stress and the gradient of the actual surface, $$\label{eq:3d_rho}
\text{div} \, (\rho \, {\boldsymbol{\rm u}})
= \frac{1}{T_s} \rho \left( 1 - \frac{\rho}{\rho_s} \right) \;
\begin{cases}
\Theta(h) & \rho < \rho_s\\
1 & \rho \ge \rho_s
\end{cases}
,$$ with $$\label{eq:rho_s_tau}
\rho_s = \frac{2 \alpha}{g} \left( |{\boldsymbol{\rm \tau}}| - \tau_t \right) \quad \quad
T_s = \frac{2 \alpha | {\boldsymbol{\rm u}}|}{g} \, \frac{\tau_t}{\gamma (|{\boldsymbol{\rm \tau}}| - \tau_t)}.$$ and $$\label{eq:3d_u}
\frac{3}{4} \, C_d \frac{\rho_{\text{air}}}{\rho_{\text{quartz}}} d^{-1} \, ({\boldsymbol{\rm {v_{\text{eff}}}}} - {\boldsymbol{\rm u}})|{\boldsymbol{\rm {v_{\text{eff}}}}} - {\boldsymbol{\rm u}}|
- \frac{g}{2 \alpha} \frac{{\boldsymbol{\rm u}}}{|{\boldsymbol{\rm u}}|}
- g \, {\boldsymbol{\rm \nabla}} \, h
= 0,$$ where $v_{\text{eff}}$ is the velocity of the grains in the saturated state,\
$${\boldsymbol{\rm v}}_{\text{eff}} = \frac{2 {\boldsymbol{\rm u}}_*}{\kappa |{\boldsymbol{\rm u}}_*|} \cdot \nonumber$$ $$\label{eq:3d:v_eff_of_tau_g0}
\left( \sqrt{\frac{z_1}{z_m} u_*^2 +
\left( 1 - \frac{z_1}{z_m} \right)
\, u_{*t}^2}
+ \left(
\ln{\frac{z_1}{z_0}} -2 \right) \, \frac{u_{*t}}{\kappa}
\right),$$ and $$\label{eq:ustar_s}
u_* = \sqrt{\tau / {\rho_{\text{air}}}}$$ The constants and model parameters have been taken from [@SauermannKroy2001] and are summarized here: $g=9.81\,$m$\,$s$^{-2}$,$\kappa=0.4$, ${\rho_{\text{air}}}=1.225\,$kg$\,$m$^{-3}$, ${\rho_{\text{quartz}}}=2650\,$kg$\,$m$^{-3}$, $z_m=0.04\,$m, $z_0=2.5~10^{-5}\,$m, $D=d=250\,\mu$m, $C_d=3$, $u_{*t}=0.28\,$m$\,$s$^{-1}$, $\gamma=0.4$,$\alpha=0.35$ and $z_1=0.005\,$m. The sand density and the sand velocity define the sand flux over a surface element ${\boldsymbol{\rm q}}(x,y)={\boldsymbol{\rm u}}(x,y) \rho(x,y)$.
#### Avalanches:
Surfaces with slopes which exceed the maximal stable angle of a sand surface, the called [*angle of repose*]{} $\Theta \approx 34^o$, produce avalanches which slide down in the direction of the steepest descent. The unstable surface relaxes to a somewhat smaller angle. For the study of dune formation two global properties are of interest. These are the sand transport downhill due to gravity and the maintenance of the angle of repose. To determine the new surface after the relaxation by avalanches the model proposed by is used, $$\label{eq:dhdt_a}
\frac{\partial h}{\partial t} = -
C_a R \left( \left| \nabla h \right| - \tan \Theta \right)$$ $$\label{eq:drho_adt}
\frac{\partial R}{\partial t}
+ \nabla \left(R {\boldsymbol{\rm u}}_a \right) =
C_a R \left( \left| \nabla h \right| - \tan \Theta \right),$$ where $h$ denotes the height of the sand bed, $R$ the height of the moving layer, $C_a$ is a model parameter and the velocity ${\boldsymbol{\rm u}}_a$ of the sand grains in the moving layer is obtained by, $$\label{eq:3d:ua}
{\boldsymbol{\rm u}}_a = - u_a \frac{\nabla h}{\tan \Theta},$$ where $u_a$ is the velocity at the angle of repose. Like in the calculation of the sand flux the steady state of the avalanche model is assumed to be reached instantaneously. In the dune model a certain amount of sand is transported over the brink to the slip face and in every iteration the sand grains are relaxed over the slip face by this avalanche model determining the steady state.
#### The time evolution of the surface
The calculation of the sand flux over a not stationary dune surface leads to changes by erosion and deposition of sand grains. The change of the surface profile can be expressed using the conservation of mass, $$\partial_t \rho + \nabla \Phi = 0 \, ,
\label{eq:time_evol_h}$$ where $\rho$ is the sand density and $\Phi$ the sand flux per time unit and area. Both $\rho$ and $\Phi$ are now integrated over the vertical coordinate assuming that the dune has a constant density of $\rho_{\text{sand}}$, $$\label{eq:height_rho}
h = \frac{1}{{\rho_{\text{sand}}}} \int \rho dz , \quad \quad
{\boldsymbol{\rm q}} = \int {\boldsymbol{\rm \Phi}} dz.$$ Thus Equation (\[eq:time\_evol\_h\]) can be rewritten, as $$\label{eq:masscons_h}
\frac{\partial h}{\partial t} = - \frac{1}{{\rho_{\text{sand}}}} \nabla{{\boldsymbol{\rm q}}}.$$ Finally, it is noted that Equation (\[eq:masscons\_h\]) is the only remaining time dependent equation and thus defines the characteristic time scale of the model which is normally between 3–5 hours for every iteration.
#### The initial surface and boundary conditions:
As initial surface we take a plain sand bed of arbitrary sand height over the solid ground. The surface can additionally be disturbed by small Gaussian hills. An initial surface has to be smooth (at least under consideration of the separation bubble) and have slopes not larger than the angle of repose. The boundary conditions influence the surface height $h$ with its separation bubble, the sand flux $q$ and the height $R$ of the moving layer of the avalanche model. The boundary in both directions $x$ and $y$ with respect to the direction of the incoming wind are open or periodic. At open boundary in $x$–direction an additional parameter controls the sand influx $q_{in}$ into the simulated dune field. It is set constant along the lateral direction at $x=0$.
All calculations presented in the following sections are made with the conditions of a completely filled sand bed and unidirectional wind. All simulations model dune fields instead of single dunes.
The model of three dimensional dunes and translational invariance {#sec:trans_2dim}
=================================================================
The main aim in this section is to justify why models of two dimensional dunes can be used in the following sections. The advantage to omit the lateral dimension makes it possible to look at larger dune fields within a still tolerable cost of computional time.
A plain initial surface would lead to no change of the height profile. This is because the system needs at least one small fluctuation to begin dune growth. Therefore as initial surface a large number of low Gaussian hills is introduced.
![Surface after $1.49$ years. The shear velocity is $u_*=0.45 $ m s$^{-1}$, the boundary conditions are periodic in wind direction and open in lateral direction. Some slip faces can be seen. One unit corresponds to the length of $2$ m.[]{data-label="fig:transv_2d2"}](Figure4BW.eps){width="80.00000%"}
The simulation models dune dynamics for a dune field of a length and width of $400$ m and $200$ m, respectively. The boundary conditions are periodic in wind direction and open in lateral direction and the shear velocity is $u_*=0.45 $ m s$^{-1}$. First the Gaussian hills lead to a growth at their corresponding positions on the dune field. After some time they build a slip face which extends its size in lateral direction.
We assume that transverse dunes try to reach a state of translational invariance. For an illustration of this dynamics see Figure \[fig:transv\_2d2\]. The slip faces become wider until they reach the lateral boundary and extend over the entire width of the simulated fields. Thus the slip face traps all the sand going over the brink. The trapped sand relaxes there through avalanching and maintains the angle of repose constant. Shear stresses of the wind field transport more sand over the brink than that which is transported down at the slip face by avalanches. Hence, transverse dunes are growing wherever there is a part of their lee zone in which saltation transport can be neglected (Section \[sec:trans\_const\]). When dunes grow the length of their separation bubble increases. But for a dune field with a limited length due to the periodic boundary condition and to a certain number of dunes which increase their mutual distance there is a state where the number of dunes must decrease by one. This leads to a displacement of the dune with the lowest height which looses its sand to the next dune situated upwind. In this state the system breaks the symmetry of translational invariance and a part of the slip face disappears. There sand is transported to the following dune by saltation. When the dune has vanished once again the system approaches translational invariance. Hence, the effect of converging dunes perturbs the steady growth of transverse dunes in a field with a periodic boundary.
![Surface of a transverse dune field with periodic boundary conditions in both directions $1.19$ years after initiation. The shear velocity is $u_*=0.4 $ m s$^{-1}$. Height units are in meters, length and width units in two meters.[]{data-label="fig:transv_2d4"}](Figure5.eps){width="80.00000%"}
To get more information how a system of transverse dunes shows translational invariance a simulation of a dune fiel with periodic boundary in both horizontal directions is made. The field extends over $400$ m in length and width. As initial surface also small Gaussian hills are used. Figure \[fig:transv\_2d4\] shows the height profile after 5,000 iterations which corresponds to a time of $1.19$ years. The results of this simulation yield the same conclusions as the simulation with open lateral boundary. Also each merging of two dunes leads to a breaking of the symmetry. Figure \[fig:transv\_2d6\] shows the system at a state close to translational invariance. The final state of the calculation is reached when only one dune is left. A simulation with periodic boundaries in both directions shows no state where the system has a structure or oscillations in lateral direction.
![Surface of a transverse dune field with periodic boundary conditions in both directions $9.5$ years after initiation. The system reached a state close to translational invariance. The shear velocity is $u_*=0.4$ m s$^{-1}$. One unit corresponds to the length of $2$ m.[]{data-label="fig:transv_2d6"}](Figure6BW.eps){width="80.00000%"}
The conclusion from these calculations for three dimensional dunes would be that an open system of transverse sand dunes reaches translational invariance under ideal unidirectional wind. In the precedent simulations the periodic boundary inhibited the system to break the invariance. A calculation with open boundary conditions in both horizontal directions which is closer to real dune fields would give more information about this assumption but is rather complicated due to the fact that the dunes move out of the simulated area. More insight is given in the following section. Assuming translational invariance simulations of fields of two dimensional dunes are much more effective. They consume much less computional time and give the opportunity to simulate larger dune fields. In the following two sections we consider two situations of two dimensional dunes, a model with constant sand influx and a model with periodic boundary. Both models lead to new interesting conclusions.
The model of two dimensional dunes with constant sand influx {#sec:trans_const}
============================================================
The free parameters for this simulation are the sand influx $q_{in}$ and the shear velocity $u_*$. As initial surface we choose a plain ground filled with sand because the sand influx differs at least a little bit from the saturation flux on the dune field. Thus dune formation is initiated at the beginning of the dune field, i.e. where wind comes in.
Time evolution {#subsec:time_evol_2dc}
--------------
The height profile of a dune field with a length of 4 km is presented at different times. The sand influx is set $q_{in}=0.017$ kg m$^{1}$s$^{-1}$ and the shear velocity $u_*=0.5$ m s$^{-1}$. A sand influx $q_{in}$ which is not equal to the sand flux of saltation transport over a plain surface lowers or raises the inlet of the dune field constantly. This initiates a small oscillatory structure which begins to move in wind direction (Figure \[fig:transv\_evo1\]) and generates more and more small dunes.
![Surface of a two dimensional simulation with constant sand influx after $4.56$ years in the left figure and $45.58$ years on the right. The shear velocity is $u_*=0.5$ m s$^{-1}$ and sand influx is $q_{in}=0.017$ kg m$^{-1}$s$^{-1}$. On the left the first dune does not have a slip face. A final stationary surface is not reached.[]{data-label="fig:transv_evo3"}](Figure7.eps){width="80.00000%"}
![Surface of a two dimensional simulation with constant sand influx after $4.56$ years in the left figure and $45.58$ years on the right. The shear velocity is $u_*=0.5$ m s$^{-1}$ and sand influx is $q_{in}=0.017$ kg m$^{-1}$s$^{-1}$. On the left the first dune does not have a slip face. A final stationary surface is not reached.[]{data-label="fig:transv_evo3"}](Figure8.eps){width="80.00000%"}
The initiating dunes at the inlet of the dune field have increasing size in length and height. The increasing difference between the starting points and the first crest leads to the creation of bigger dunes. With increasing size the dunes have a lower velocity $v_{dune}$ (Section \[subsec:dunevel\]). Hence, dune spacing, the distance between adjacent crests, increases with time and no dune collides or converges with another. So real dune fields can maintain a structure of translational invariance without the effect of the breaking of symmetry which was found in Section \[sec:trans\_2dim\]. Dune fields where the sand influx stays rather constant in time can have a more regular structure than dune fields where the sand influx varies strongly with respect to time.
The slip face of the first dune is missing or is short due to the short evolution time (Figure \[fig:transv\_evo3\]). An observation of this absence for example at a coast where the sea provides a certain amount of sand supply was not found in the literature. The simulation depicted here and any other simulation of dune fields with lengths of 1 km to 4 km do not show a system that reaches a stationary state. According to older dune fields and less climate changes produce larger transverse dunes. The smaller slope at the brink found in this modeling agrees also with qualitative observations. In the model a dune height of $100$ m is reached in roughly $50$ years. Estimates have predicted some $10,000$ years to develop a $100$ m dune. Probably this large difference can be explained by a smaller average wind velocity, changes of wind direction over longer periods and changes in sand supply and climate acting on real dunes. The so called memory (the time to build a dune beginning with a plain sand bed) is related to the ratio of height of the dune and annual rate of sand flux $H/Q_{ann}$ [@Cooke93]. The memory can vary by about four orders of magnitude in time.
The results of the numerical calculations with different sand influxes $q_{in}$ at the same shear velocities lead to the conclusion that there is a direct dependency between influx and height growth of the first dune (Figure \[fig:transv\_scalhq\]). The nearer the sand influx gets to the saturation flux of saltation transport the slower increases the height of the first dune. Hence, dune fields where the sand influx varies strongly in time around the saturation flux of saltation initiate first dunes with different heights. So there can be smaller dunes moving faster into bigger ones and the symmetry breaking explained in the Section \[sec:trans\_2dim\] will occur.
![Evolution of the height of the dunes beginning at the position where the wind comes in. The shear velocity is $u_* = 0.4$ ms$^{-1}$. The height increases with the square root of time.[]{data-label="fig:transv_scalhq"}](Figure9.eps){width="80.00000%"}
In the following some relations found for the time evolution of transverse dunes are presented. Figure \[fig:transv\_scalhq\] indicates also that height versus time increases with a power law as was also found in the model of , $$h(t) \propto \sqrt{a \cdot t},$$ where $a$ is a parameter which is dependent on shear velocity and sand influx. $a$ is a measure for the growth rate. The growth rate seems to be smaller for a larger distance from the beginning of the dune field. There the dunes contain longer slip faces because of their higher age. We observe from Figure \[fig:transv\_scalhq\] that the growth rate should converge to a constant value for very large distances. The same relation is found for the spacing $d_{ij}$ of the dunes $i$ and $j$ (Figure \[fig:transv\_scalsc\]), $$d_{ij}(t) \propto \sqrt{b \cdot t},$$ where $b$ denotes a parameter which measures the spacing rate. This spacing rate approaches the same value for all dunes far away from the influx region. The values fitting well to these rates are approximately $b=8.53 \cdot 10^{-5}$ m$^2$s$^{-1}$ and $b=1.56 \cdot 10^{-4}$ m$^2$s$^{-1}$ for a shear velocity $u_*=0.4$ m s$^{-1}$ and $u_*=0.5$ m s$^{-1}$, respectively.
![Comparing the evolution of the spacing between dunes for different shear velocities. The spacing increases with the square root of time.[]{data-label="fig:transv_scalsc"}](Figure10.eps){width="80.00000%"}
Figure \[fig:transv\_spach\] shows the spacing height relationship which seems to be linear. This agrees with the data of measurements of transverse dunes in the Namib Sand Sea in Namibia by . For different shear stresses the slope stays constant whereas the axis intercept of $h$ increases with higher shear velocities. Hence, the dunes in a transverse dune field are located closer to each other for higher shear velocities.
![Relationship between spacing and height of the dunes for the shear velocities $u_* = 0.4$ ms$^{-1}$ and $u_* = 0.5$ ms$^{-1}$. The dependency seems to be linear.[]{data-label="fig:transv_spach"}](Figure11.eps){width="80.00000%"}
Dune velocity {#subsec:dunevel}
-------------
The validity of Bagnold’s law, $$v_{dune} = \frac{\Phi_{dune}}{h},
\label{eq:transv_vdune}$$ where $\Phi_{dune}$ is the bulk flux of sand blown over the brink has been shown by observations of real dunes. According to a better fit is given by using instead of the height $h$ the characteristic length $l$, i.e. the length of the envelope comprising the height profile and the separation bubble. Bagnold’s law already fits quite well. Nevertheless, the Equation (\[eq:transv\_vdune\]) is generalized. The length of the envelope can be expressed as a function of the height. The function is developed into a Taylor series and orders higher than the linear order are neglected. This finally yields, $$v_{dune} = \frac{\Phi_{dune}}{h+C},
\label{eq:transv_vdune2}$$ where $C$ denotes a constant. In Figure \[fig:transv\_scalv2\] the dune velocities with respect to the height and their fits to Equation (\[eq:transv\_vdune2\]) are compared for the shear velocities $u_*=0.4$ m s$^{-1}$ and $u_*=0.5$ m s$^{-1}$. The observed bulk fluxes are $\Phi_{dune}=454.5$ m$^2$s$^{-1}$ and $\Phi_{dune}=833.3$ m$^2$s$^{-1}$ with the corresponding constants $C=0.45$ m and $C=1.08$ m, respectively. The values are smaller than the bulk fluxes of isolated 2-dimensional dunes calculated by on plain ground without sand. Thus the velocities are also smaller than those observed for isolated transverse dunes on plain ground without sand. found less speed-up of the wind velocity over continuous sand dunes than over isolated transverse dunes. This agrees with the results in this model.
![Dune velocity $v_{dune}$ versus height $h$ for the shear stresses $u_*=0.4$ m s$^{-1}$ and $u_*=0.5$ m s$^{-1}$. The velocity decreases proportional to the reciprocal height.[]{data-label="fig:transv_scalv2"}](Figure12.eps){width="80.00000%"}
The model of two dimensional dunes with periodic boundary {#sec:trans_periodic}
=========================================================
The simulations in this section have a periodic boundary condition in wind direction. So the parameter of sand influx that was used additionally before is no more available. Sand influx is set equal to the outflux. The avalanches flow down even if the slip face is divided by the boundary. Also the separation bubble follows the periodic boundary conditions. The calculations are made with dune fields of a length of two kilometers. The ground is completely filled up with sand. The initial surface consists of small Gaussian hills which disturb a plain surface.
Time evolution {#time-evolution}
--------------
The situation is similar to the simulations of three dimensional transverse dune fields. In the beginning many dunes of different size grow in the system but after some time the dunes approach same heights. As in the three dimensional case all the dunes keep growing so that the number of dunes has to decrease. A more detailed description of the process of merging dunes is given in the following section. The periodic boundary forces a decrease of the number of dunes and this process makes the evolution rather complex. Figures \[fig:transv\_1dp1\], \[fig:transv\_1dp2\] and \[fig:transv\_1dp3\] show the height profile of a dune field of a length of two kilometers at three time steps. Figure \[fig:transv\_1dp1\] depicts a surface with dunes of different sizes which seem to interact strongly with each other. The regular state of the system in Figure \[fig:transv\_1dp2\] is disturbed in Figure \[fig:transv\_1dp3\]. The number of dunes decreases quite regularly versus time (Figure \[fig:transv\_nu\_d\]). This process is very slow and the dunes grow much slower than in the model with open boundary.
![Surface of a dune field with a length of two kilometers. The shear velocity is $u_*=0.45$ m s$^{-1}$ after $31.7$ years, the boundary conditions are periodic. The structure is quite irregular.[]{data-label="fig:transv_1dp1"}](Figure13.eps){width="80.00000%"}
![Surface of a dune field with a length of two kilometers. The shear velocity is $u_*=0.45 $m s$^{-1}$ after $95.1$ years, the boundary conditions are periodic. There are three dunes left with similar heights.[]{data-label="fig:transv_1dp2"}](Figure14.eps){width="80.00000%"}
![Surface of a dune field with a length of two kilometers. The shear velocity is $u_*=0.45$ m s$^{-1}$ after $190.2$ years, the boundary conditions are periodic. The number of dunes will decrease by one after coalescence of two dunes.[]{data-label="fig:transv_1dp3"}](Figure15.eps){width="80.00000%"}
![The number of dunes decreases quite regularly in time.[]{data-label="fig:transv_nu_d"}](Figure16.eps){width="80.00000%"}
Do transversal dunes behave like solitons? {#sec:trans_soliton}
------------------------------------------
proposed that barchan dunes behave like solitons. Solitons are self-stabilizing wave packs which do not change their shape during their propagation not even after collision with other waves. They are found in non-linear systems like for example in shallow water waves. Observations of barchan dunes seem to give evidence that small dunes can migrate over bigger ones without being absorbed completely. A closer examination of the merging of two or more dunes from calculations of a two dimensional dune field with periodic boundary let to some interesting observations. In this case the dune field cannot break its translational scale invariance to reach a faster colliding of two adjacent dunes. Thus the supposition would be that a smaller dune due to its higher velocity collides with the next bigger one in wind direction without passing over it. That is not the case. As an example see the Figure \[fig:transv\_sol1\]. A small dune climbs up the windward side of the following bigger dune. As it reaches the same height as the following one it seems to hand over the state of being the smaller dune. The dune that was bigger before then wanders towards the following dune of the dune field. This process can be observed several times. The fact that the volume of the smaller dune decreases after every time it passes a dune let us conclude that there will be a final dune which will absorb the small one. Hence, these dunes do not behave exactly like solitons because of their loss of volume. The suggestion of that dunes of different hierarchies (for example dunes and mega dunes) could exist with each other without interactions is not valid in our case. Nevertheless the fact that small dunes can migrate over others leads to the conclusion that different hierarchies of dune sizes can coexist in a field of transverse dunes. The loss of volume demonstrates that there is some interaction between different hierarchies of dune sizes.
Finally Figure \[fig:transv\_sol1\] shows a part of the same simulation where a very small dune finally swallowed the bigger one and the number of dunes decreases by one.
![Left: Wandering of a small dune over a big one. Right: Coalescence of small dune in a big one. These results are part of a simulation of a transverse dune field with a length of 2km. The shear velocity is $u_*=0.5 $m s$^{-1}$ and boundary conditions are periodic.[]{data-label="fig:transv_sol1"}](Figure17BW.eps){width="80.00000%"}
Conclusions {#sec:t_concl}
===========
It was shown how transverse dune fields can develop with respect to time. None of the simulations gave evidence that a final stage would be reached, neither the model with periodic nor the model with open boundaries. More knowledge about the still not very well understood trapping efficiency in the lee side of the dunes and the inclusion of this effect as done by could lead to stable final states. The use of translational invariance, shown in the three dimensional model, made it possible to restrict to the model of two dimensional dunes. All simulations also showed that the evolution of a slip face leads to a continuous growth of transversal dunes, and no final state will be reached. In the model of two dimensional dunes with constant influx the height of the dunes increases proportional with respect to square root of time. The same power law is found for the crest-to-crest spacing in the dune field. The difference of influx and saturation flux seems to play a crucial role in dune size. The same relation between dune velocity and height is found for barchan dunes. Two dimensional modeling with periodic boundary shows that different hierarchies of dunes with different sizes can exist but they interact with each other.
| 0 |
---
abstract: 'A well known construction of B. Dubrovin and K. Saito endows the parameter space of a universal unfolding of a simple singularity with a Frobenius manifold structure. In our paper we present a generalization of this construction for the singularities of types $A$ and $D$, that gives a solution of the open WDVV equations. For the $A$-singularity the resulting solution describes the intersection numbers on the moduli space of $r$-spin disks, introduced recently in a work of the second author, E. Clader and R. Tessler. In the second part of the paper we describe the space of homogeneous polynomial solutions of the open WDVV equations associated to the Frobenius manifolds of finite irreducible Coxeter groups.'
address:
- 'A. Basalaev:Faculty of Mathematics, National Research University Higher School of Economics, Usacheva str., 6, 119048 Moscow, Russian Federation, and Skolkovo Institute of Science and Technology, Nobelya str., 3, 121205 Moscow, Russian Federation'
- 'A. Buryak:School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom'
author:
- Alexey Basalaev
- Alexandr Buryak
title: Open Saito theory for $A$ and $D$ singularities
---
Introduction
============
Frobenius manifolds, introduced by B. Dubrovin in the early 90s, gave a geometric approach to study solutions of the [*WDVV equations*]{} $$\begin{gathered}
\label{eq:WDVV equations}
\frac{\d^3 F}{\d t^\alpha\d t^\beta\d t^\mu}\eta^{\mu\nu}\frac{\d^3 F}{\d t^\nu\d t^\gamma\d t^\delta}=\frac{\d^3 F}{\d t^\delta\d t^\beta\d t^\mu}\eta^{\mu\nu}\frac{\d^3 F}{\d t^\nu\d t^\gamma\d t^\alpha},\quad 1\le\alpha,\beta,\gamma,\delta\le N,\end{gathered}$$ where $F = F(t^1,\dots,t^N)$ is an analytic function defined on some open subset $M\subset{\mathbb C}^N$, $\eta=(\eta_{\alpha\beta})$ is an $N\times N$ symmetric non-degenerate matrix with complex coefficients, $(\eta^{\alpha\beta}):=\eta^{-1}$ and we use the convention of sum over repeated Greek indices. The WDVV equations appear in many areas of mathematics, including singularity theory and curve counting theories in algebraic geometry. In Gromov–Witten theory the WDVV equations describe the structure of primary Gromov–Witten invariants in genus $0$ and naturally come from a certain relation in the cohomology of the moduli space of stable curves.
Suppose that a function $F$ satisfies the WDVV equations together with the additional assumption $$\begin{gathered}
\label{eq:unit condition for F}
\frac{\d^3 F}{\d t^1\d t^\alpha\d t^\beta}=\eta_{\alpha\beta}.\end{gathered}$$ The function $F$ defines a commutative product $\circ$ on each tangent space $T_p M$ by $$\frac{{{\partial}}}{{{\partial}}t^\alpha} \circ \frac{{{\partial}}}{{{\partial}}t^\beta}:=\frac{{{\partial}}^3F}{{{\partial}}t^\alpha {{\partial}}t^\beta {{\partial}}t^\gamma} \eta^{\gamma \delta} \frac{{{\partial}}}{{{\partial}}t^\delta}, \quad 1 \le \alpha,\beta \le N.$$ One can immediately see that the WDVV equations are equivalent to the associativity of this product and property means that the vector field $\frac{\d}{\d t^1}$ is the unit. One can go in the opposite direction and consider a manifold with a commutative, associative algebra structure and a symmetric, non-degenerate bilinear form on each tangent space. Under certain conditions such a manifold in special local coordinates, called the [*flat coordinates*]{}, can be described by a solution $F$ of the WDVV equations, satisfying property . The conditions, needed for the existence of a function $F$, were systematically studied by B. Dubrovin [@Dub96; @Dub99], who called manifolds, satisfying these conditions, [*Frobenius manifolds*]{}. The function $F$ is then called a [*Frobenius manifold potential*]{}. The bilinear form is traditionally called a [*metric*]{}.
In his fundamental works [@Sai82; @Sai83] K. Saito constructed a flat metric on the parameter space of a universal unfolding of any simple singularity. B. Dubrovin [@Dub98] then proved that together with a certain commutative, associative algebra structure on each tangent space this metric defines a Frobenius manifold structure on the parameter space of the universal unfolding. These Frobenius manifolds are often called the [*Saito Frobenius manifolds*]{}. Remarkably, the same Frobenius manifolds appear in the study of the geometry of the moduli spaces of algebraic curves with certain additional structures, the so-called Fan–Jarvis–Ruan–Witten (FJRW) theory [@FJR13]. This is one of the manifestations of mirror symmetry.
In the same way, as the WDVV equations appeared in Gromov–Witten theory, another system of non-linear PDEs, called the *open WDVV equations*, appeared more recently in open Gromov–Witten theory [@HS12 Theorem 2.7] (see also [@PST14; @BCT18; @BCT19]). Let $F = F(t^1,\dots,t^N)$ be a solution of the WDVV equations , satisfying condition . The open WDVV equations associated to $F$ are the following PDEs for a function $F^o = F^o(t^1,\ldots,t^N,s)$, depending on an additional variable $s$: $$\begin{aligned}
\frac{\d^3F}{\d t^\alpha\d t^\beta\d t^\mu}\eta^{\mu\nu}\frac{\d^2F^o}{\d t^\nu\d t^\gamma}+\frac{\d^2F^o}{\d t^\alpha\d t^\beta}\frac{\d^2F^o}{\d s\d t^\gamma}=&\frac{\d^3F}{\d t^\gamma\d t^\beta\d t^\mu}\eta^{\mu\nu}\frac{\d^2F^o}{\d t^\nu\d t^\alpha}+\frac{\d^2F^o}{\d t^\gamma\d t^\beta}\frac{\d^2F^o}{\d s\d t^\alpha},&&1\le\alpha,\beta,\gamma\le N,\label{eq:open WDVV,1}\\
\frac{\d^3F}{\d t^\alpha\d t^\beta\d t^\mu}\eta^{\mu\nu}\frac{\d^2F^o}{\d t^\nu\d s}+\frac{\d^2F^o}{\d t^\alpha\d t^\beta}\frac{\d^2F^o}{\d s^2}=&\frac{\d^2F^o}{\d s\d t^\beta}\frac{\d^2F^o}{\d s\d t^\alpha},&&1\le\alpha,\beta\le N.\label{eq:open WDVV,2}\end{aligned}$$ Solutions of equations , , relevant in open Gromov-Witten theory and also in the works [@PST14; @BCT18; @BCT19], satisfy the additional condition $$\begin{gathered}
\label{eq:unit condition for Fo}
\frac{\d^2 F^o}{\d t^1\d t^\alpha}=0,\qquad \frac{\d^2 F^o}{\d t^1\d s}=1.\end{gathered}$$
The solutions of the open WDVV equations from the works [@BCT18; @BCT19] are associated to the Saito Frobenius manifold of the $A$-singularity and they were constructed using ideas of FJRW theory. So it is natural to ask whether the Dubrovin–Saito construction of the Frobenius manifolds corresponding to simple singularities admits a generalization, that produces solutions of the open WDVV equations. In our paper we present such a generalization for the singularities of types $A$ and $D$. For the $A$-singularity our construction gives a polynomial solution that coincides with the one from [@BCT18; @BCT19]. For the $D$-singularity our solution has a simple pole along the variable $s$.
Additionally, in both $A$- and $D$-cases our solution of the open WDVV equations has the following remarkable feature. The Saito Frobenius manifold of a simple singularity has two natural coordinate systems. The first one is given by the parameters of a universal unfolding of a simple singularity. The second coordinate system is given by the flat coordinates of the metric. We show that for the singularities $A$ and $D$ the transition functions between these two coordinate systems coincide with the coefficients of powers of the variable $s$ in the expansion of our solution of the open WDVV equations.
The Saito Frobenius manifolds of simple singularities together with their certain submanifolds form a class of Frobenius manifolds, that is, via a construction of B. Dubrovin [@Dub98], in a natural bijection with the class of finite irreducible Coxeter groups (see also [@Zub94]). This class of Frobenius manifolds plays a fundamental role in the theory of Frobenius manifolds, because of the following result of C. Hertling, conjectured by B. Dubrovin [@Dub98]. Recall that a Frobenius manifold potential $F$ is called homogeneous, if there exists a vector field $E$ of the form $$\begin{gathered}
\label{eq:Euler field for Frobenius}
E=\sum_{\alpha=1}^N\underbrace{(q_\alpha t^\alpha+r^\alpha)}_{=:E^\alpha}\frac{\d}{\d t^\alpha},\quad q_\alpha,r^\alpha\in{\mathbb C},\quad q_1=1,\end{gathered}$$ such that $$\begin{gathered}
E(F)=E^\alpha\frac{\d F}{\d t^\alpha}=(3-\delta)F+\frac{1}{2}A_{\alpha\beta}t^\alpha t^\beta+B_\alpha t^\alpha+C,\quad\text{for some $\delta,A_{\alpha\beta},B_\alpha,C\in{\mathbb C}$}.\end{gathered}$$ The number $\delta$ is called the [*conformal dimension*]{} and the vector field $E$ is called the [*Euler vector field*]{}. C. Hertling proved that any generically semisimple Frobenius manifold (see Section \[subsection:flat F-manifolds\] for definition), whose potential is polynomial $F \in {{\mathbb{C}}}[t^1,\dots,t^N]$ and homogeneous with the Euler vector field of the form $E=q_\alpha t^\alpha\frac{\d}{\d t^\alpha}$, where $q_\alpha>0$, can be expressed as the product of the Frobenius manifolds corresponding to finite irreducible Coxeter groups [@Hert02 Theorem 5.25].
In the second part of the paper we study the space of polynomial solutions of the open WDVV equations associated to the Frobenius manifolds of finite irreducible Coxeter groups. Note that all solutions of the open WDVV equations, considered in the works [@HS12; @PST14; @BCT18; @BCT19], are associated to a homogeneous Frobenius potential $F$ and, moreover, the function $F^o$ satisfies the homogeneity condition $$\begin{gathered}
\label{eq:homogeneity for Fo}
E^\alpha\frac{\d F^o}{\d t^\alpha}+\frac{1-\delta}{2}s\frac{\d F^o}{\d s}=\frac{3-\delta}{2}F^o+D_\alpha t^\alpha+{\widetilde{D}}s+E,\quad\text{for some $D_\alpha,{\widetilde{D}},E\in{\mathbb C}$}.\end{gathered}$$ We see that the degree of the variable $s$ is determined by the conformal dimension of the Frobenius manifold. We will call a solution of the open WDVV equations homogeneous, if it satisfies condition .
In our paper we describe the space of homogeneous polynomial solutions of the open WDVV equations associated to the Frobenius manifolds of [*all*]{} finite irreducible Coxeter groups. In particular, this space is non-empty only for the Coxeter groups $A_N$, $B_N$ and $I_2(k)$.
Our approach to study solutions of the open WDVV equations is based on the following crucial observation of P. Rossi. Let $F = F(t^1,\ldots,t^N)$ be a Frobenius manifold potential and $F^o=F^o(t^1,\ldots,t^N,s)$ be a solution of the open WDVV equations satisfying . Then the $(N+1)$-tuple of functions $\left(\eta^{1\mu}\frac{\d F}{\d t^\mu},\ldots,\eta^{N\mu}\frac{\d F}{\d t^\mu},F^o\right)$ forms a [*vector potential*]{} of a [*flat F-manifold*]{}. This allows us to use the theory of flat F-manifolds to study solutions of the open WDVV equations.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to thank Claus Hertling for useful discussions.
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 797635. The first named author was supported by RSF Grant No. 19-71-00086.
Flat F-manifolds and Frobenius manifolds {#section:definitions}
========================================
In this section we recall the definitions and the main properties of flat F-manifolds and Frobenius manifolds. We also explain how solutions of the open WDVV equations correspond to flat F-manifolds of special type.
Flat F-manifolds {#subsection:flat F-manifolds}
----------------
The notion of a flat F-manifold was introduced in [@Man05].
A flat F-manifold $(M,\nabla,\circ)$ is the datum of a complex analytic manifold $M$, an analytic connection $\nabla$ in the tangent bundle $T M$, an algebra structure $(T_p M,\circ)$ with unit $e$ on each tangent space analytically depending on the point $p\in M$ such that the one-parameter family of connections $\nabla+z\circ$ is flat and torsionless for any $z\in{\mathbb C}$, and $\nabla e=0$.
For a flat F-manifold $(M,\nabla,\circ)$ consider flat coordinates $t^\alpha$, $1\le\alpha\le N$, $N=\dim M$, for the connection $\nabla$ such that $e = \frac{\d}{\d t^1}$. Then locally there exist analytic functions $F^\alpha(t^1,\ldots,t^N)$, $1\leq\alpha\leq N$, such that the second derivatives $$\begin{gathered}
\label{eq:structure constants of flat F-man}
c^\alpha_{\beta\gamma}=\frac{\d^2 F^\alpha}{\d t^\beta \d t^\gamma}\end{gathered}$$ give the structure constants for the multiplication $\circ$, $$\begin{gathered}
\frac{\d}{\d t^\beta}\circ\frac{\d}{\d t^\gamma}=c^\alpha_{\beta\gamma}\frac{\d}{\d t^\alpha}.\end{gathered}$$ From the associativity of the multiplication and the fact that the vector field $\frac{\d}{\d t^1}$ is the unit it follows that $$\begin{aligned}
\frac{\d^2 F^\alpha}{\d t^1\d t^\beta} &= \delta^\alpha_\beta, && 1\leq \alpha,\beta\leq N,\label{eq:axiom1 of flat F-man}\\
\frac{\d^2 F^\alpha}{\d t^\beta \d t^\mu} \frac{\d^2 F^\mu}{\d t^\gamma \d t^\delta} &= \frac{\d^2 F^\alpha}{\d t^\gamma \d t^\mu} \frac{\d^2 F^\mu}{\d t^\beta \d t^\delta}, && 1\leq \alpha,\beta,\gamma,\delta\leq N.\label{eq:axiom2 of flat F-man}\end{aligned}$$ The $N$-tuple of functions $(F^1,\ldots,F^N)$ is called the [*vector potential*]{} of our flat F-manifold.
Conversely, if $M$ is an open subset of ${\mathbb C}^N$ and $F^1,\ldots,F^N\in{\mathcal{O}}(M)$ are functions satisfying equations and , then these functions define a flat F-manifold $(M,\nabla,\circ)$ with the connection $\nabla$, given by $\nabla_{\frac{\d}{\d t^\alpha}}\frac{\d}{\d t^\beta}=0$, and the multiplication $\circ$, given by the structure constants .
\[remark:algebraic construction of multiplication\] Let $M\subset{\mathbb C}^N$ be an open subset in the Zariski topology. The tangent spaces $T_p M$ can be endowed with an algebra structure, algebraically depending on the point $p\in M$, using the following construction. Denote by ${\mathcal{O}}^{\mathrm{alg}}$ the sheaf of algebraic functions on $M$. Let $R$ be an ${\mathcal{O}}^{\mathrm{alg}}(M)$-algebra, which is free as an ${\mathcal{O}}^{\mathrm{alg}}(M)$-module with a basis $\phi_1,\ldots,\phi_N\in R$. Denote by $v_1,\ldots,v_N$ the standard coordinates on ${\mathbb C}^N$ and by ${\mathcal{T}}_M^{\mathrm{alg}}$ the algebraic tangent sheaf of $M$. Define an isomorphism of ${\mathcal{O}}^{\mathrm{alg}}(M)$-modules $\Psi\colon{\mathcal{T}}_M^{\mathrm{alg}}(M)\to R$ by $\Psi(\frac{\d}{\d v_i}):=\phi_i$. Thus, the sheaf ${\mathcal{T}}_M^{\mathrm{alg}}$ becomes a sheaf of ${\mathcal{O}}^{\mathrm{alg}}$-algebras, that endows the tangent spaces $T_p M$ with an algebra structure algebraically depending on the point $p\in M$.
Consider an analytic manifold $M$ with an algebra structure $(T_p M,\circ)$ on each tangent space analytically depending on the point $p\in M$. We see that a connection $\nabla$, endowing our manifold $M$ with a flat F-manifold structure, can be completely described by a choice of coordinates $t^1,\ldots,t^N$ on $M$ such that the structure constants $c^\alpha_{\beta\gamma}$ of multiplication in these coordinates satisfy the integrability condition $$\frac{\d c^\alpha_{\beta\gamma}}{\d t^\sigma}=\frac{\d c^\alpha_{\beta\sigma}}{\d t^\gamma}$$ together with the condition $c^\alpha_{1,\beta}=\delta^\alpha_\beta$. In this paper we will construct flat $F$-manifolds exactly by presenting flat coordinates as above.
A flat F-manifold $(M,\nabla,\circ)$ is called [*conformal*]{}, if it is equipped with a vector field $E$, called the [*Euler vector field*]{}, such that $\nabla\nabla E=0$, $[e,E]=e$ and ${\mathcal{L}}_E(\circ)=\circ$. This means that in the flat coordinates the Euler vector field $E$ has the form $$E=\underbrace{(q^\alpha_\beta t^\beta+r^\alpha)}_{=:E^\alpha}\frac{\d}{\d t^\alpha},\quad q^\alpha_\beta,r^\alpha\in{\mathbb C},\quad q^\alpha_1=\delta^\alpha_1,$$ and the vector potential $(F^1,\ldots,F^N)$ satisfies the condition $$\begin{gathered}
\label{eq:homogeneity for F-man}
E^\mu\frac{\d F^\alpha}{\d t^\mu}=q^\alpha_\beta F^\beta+F^\alpha+A^\alpha_\beta t^\beta+B^\alpha,\quad\text{for some $A^\alpha_\beta,B^\alpha\in{\mathbb C}$}.\end{gathered}$$
A point $p\in M$ of an $N$-dimensional flat F-manifold $(M,\nabla,\circ)$ is called *semisimple* if $T_pM$ has a basis of idempotents $\pi_1,\dots,\pi_N$ satisfying $\pi_k \circ \pi_l = \delta_{k,l} \pi_k$. Moreover, locally around such a point one can choose coordinates $u^i$ such that $\frac{\d}{\d u^k}\circ\frac{\d}{\d u^l}=\delta_{k,l}\frac{\d}{\d u^k}$. These coordinates are called the [*canonical coordinates*]{}. In particular, this means that the semisimplicity is an open property. The flat F-manifold $(M,\nabla,\circ)$ is called semisimple, if a generic point of $M$ is semisimple.
Frobenius manifolds
-------------------
For a complex analytic manifold $M$ we denote by ${\mathcal{T}}_M$ the analytic tangent sheaf of $M$.
A flat F-manifold $(M,\nabla,\circ)$ is called a Frobenius manifold if the tangent spaces $T_pM$ are equipped with a symmetric non-degenerate bilinear form $\eta$ analytically depending on the point $p\in M$ such that $\nabla \eta = 0$ and for any $X,Y,Z \in \T_M$ the following condition is satisfied: $$\eta(X\circ Y, Z) = \eta(X,Y\circ Z).$$ The connection $\nabla$ is then the Levi-Civita connection associated to the form $\eta$. A Frobenius manifold will be denoted by the triple $(M,\eta,\circ)$. The form $\eta$ is traditionally called a metric.
Let $(M,\eta,\circ)$ be a Frobenius manifold and consider the flat coordinates $t^1,\ldots,t^N$ of the metric $\eta$ and the vector potential $(F^1,\ldots,F^N)$. Then locally there exists an analytic function $F$ such that $F^\alpha = \eta^{\alpha\beta}\frac{\d F}{\d t^\beta}$ and $\frac{\d^3 F}{\d t^1\d t^\alpha\d t^\beta}=\eta_{\alpha\beta}$, where $(\eta_{\alpha\beta})$ is the matrix of the form $\eta$ in the coordinates $t^1,\ldots,t^N$. The function $F$ satisfies the WDVV equations and is called the Frobenius manifold potential.
A Frobenius manifold $(M,\eta,\circ)$ is called conformal if the corresponding flat F-manifold is conformal and the metric $\eta$ satisfies the condition $${\mathcal{L}}_E\eta=(2-\delta)\eta,\quad\text{for some $\delta\in{\mathbb C}$},$$ where ${\mathcal{L}}_E$ denotes the Lie derivative. The number $\delta$ is then called the conformal dimension of the Frobenius manifold. The Frobenius manifold potential $F$ satisfies then the condition $$E(F) = (3-\delta) F + \frac{1}{2}A_{\alpha\beta}t^\alpha t^\beta+B_\alpha t^\alpha+C,\quad\text{for some $A_{\alpha\beta},B_\alpha,C\in{\mathbb C}$}.$$ In the theory of Frobenius manifolds it is typically assumed that one can choose flat coordinates such that the matrix $\left(\frac{\d E^\alpha}{\d t^\beta}\right)$ is diagonal and so the Euler vector field has form .
The papers [@Dub96; @Dub99] contain a systematic study of the theory of Frobenius manifolds.
Extensions of flat F-manifolds and the open WDVV equations {#subsection:flat F-manifolds and open WDVV}
----------------------------------------------------------
Consider a flat F-manifold structure, given by a vector potential $(F^1,\ldots,F^{N+1})$ on an open subset ${M\times U\in{\mathbb C}^{N+1}}$, where $M$ and $U$ are open subsets of ${\mathbb C}^N$ and ${\mathbb C}$, respectively. Suppose that the functions $F^1,\ldots,F^N$ don’t depend on the variable $t^{N+1}$, varying in $U$. Then the functions $F^1,\ldots,F^N$ satisfy equations and, thus, define a flat F-manifold structure on $M$. In this case we call the flat F-manifold structure on $M\times U$ an [*extension*]{} of the flat F-manifold structure on $M$.
Consider the flat F-manifold associated to a Frobenius manifold, given by a potential $F(t^1,\ldots,t^N)\in{\mathcal{O}}(M)$ and a metric $\eta$, $F^\alpha=\eta^{\alpha\mu}\frac{\d F}{\d t^\mu}$, $1\le\alpha\le N$. It is easy to check that a function $F^o(t^1,\ldots,t^N,s)\in{\mathcal{O}}(M\times U)$ satisfies equations , and if and only if the $(N+1)$-tuple $(F^1,\ldots,F^N,F^o)$ is a vector potential of a flat F-manifold. Here we identify $s=t^{N+1}$. This defines a correspondence between solutions of the open WDVV equations, satisfying property , and flat F-manifolds, extending the Frobenius manifold given. This observation belongs to Paolo Rossi.
Saito Frobenius manifolds
=========================
In this section we recall the Dubrovin–Saito construction of a Frobenius manifold structure on the parameter space of a universal unfolding of a simple singularity.
Let us first recall the list of polynomials defining simple singularities: $$\begin{aligned}
f_{A_N}(x,y)=&\frac{x^{N+1}}{N+1} + y^2,&& N\ge 1,\\
f_{D_N}(x,y)=&\frac{x^{N-1}}{N-1} + x y^2,&& N\ge 4,\\
f_{E_6}(x,y)=&x^4 + y^3,\\
f_{E_7}(x,y)=&x^3y + y^3,\\
f_{E_8}(x,y)=&x^5 + y^3.\end{aligned}$$ The associated [*local algebra*]{} is defined by $${\mathcal{A}}_W:={\mathbb C}[x,y]\left/\left(\frac{\d f_W}{\d x},\frac{\d f_W}{\d y}\right)\right.,$$ where $W$ is one of the types $A_N$, $D_N$ or $E_N$. Because $x=y=0$ is an isolated critical point of $f_W$, the local algebra ${\mathcal{A}}_W$ turns out to be a finite-dimensional vector space. Denote $\dim{\mathcal{A}}_W=:N$. A [*universal unfolding*]{} of the singularity of $f_W$ is a function $\Lambda_W\colon {{\mathbb{C}}}^{2}\times{{\mathbb{C}}}^{N} \to {{\mathbb{C}}}$ of the form $$\begin{gathered}
\Lambda_W(x,y,v_1,\dots,v_N)=f_W+\sum_{i=1}^N v_i\phi_i,\quad\phi_i\in{\mathbb C}[x,y],\end{gathered}$$ such that the classes of polynomials $\phi_1,\ldots,\phi_N$ form a basis of the local algebra ${\mathcal{A}}_W$. Explicitly, universal unfoldings of the ADE singularities are given by $$\begin{aligned}
\Lambda_{A_N} &= \frac{x^{N+1}}{N+1} + y^2+ \sum_{k=1}^N v_k x^{k-1},\\
\Lambda_{D_N} &= \frac{x^{N-1}}{N-1} + x y^2 + \sum _{k=1}^{N-1} v_k x^{k-1} + v_{N}y,\\
\Lambda_{E_6} &= x^4 + y^3 + v_1 + v_2 x + v_3 y + v_4 x^2 + v_5 x y + v_6 x^2 y,\\
\Lambda_{E_7} &= x^3y + y^3 + v_1 + v_2 x + v_3 y + v_4 x^2 + v_5 x y + v_6 x^3 + v_7 x^4,\\
\Lambda_{E_8} &= x^5 + y^3 + v_1 + v_2 x + v_3 y + v_4 x^2 + v_5 x y + v_6 x^3 + v_7 x^2y + v_8 x^3y.\end{aligned}$$
Consider the quotient ring $${{\widehat{\mathcal{A}}}}_W:={{\mathbb{C}}}[x,y,v_1,\dots,v_N]/\left({{\partial}}_x\Lambda_W,{{\partial}}_y\Lambda_W\right).$$ As a ${\mathbb C}[v_1,\ldots,v_N]$-module, the space ${{\widehat{\mathcal{A}}}}_W$ has dimension $N$ with a basis given by the classes $[\phi_1],\ldots,[\phi_N]\in{{\widehat{\mathcal{A}}}}_W$ of the polynomials $\phi_1,\ldots,\phi_N$. Identifying the ${\mathbb C}[v_1,\ldots,v_N]$-modules ${\mathcal{T}}_{{\mathbb C}^N}^{\mathrm{alg}}({\mathbb C}^N)$ and ${{\widehat{\mathcal{A}}}}_W$ via the isomorphism $\Psi_W$ defined by $$\begin{gathered}
\Psi_W\left(\frac{{{\partial}}}{{{\partial}}v_k}\right) := \left[\phi_k\right], \quad 1 \le k \le N,\end{gathered}$$ by Remark \[remark:algebraic construction of multiplication\], we endow the tangent spaces $T_p {\mathbb C}^N$ with a multiplication. The structure constants of it are polynomials in the coordinates $v_1,\ldots,v_N$.
A flat metric can be introduced as follows. It is easy to see that there exist unique positive rational numbers $q_x,q_y,q_1,\ldots,q_N$ such that $$q_x x \frac{{{\partial}}\Lambda_W}{{{\partial}}x} + q_y y \frac{{{\partial}}\Lambda_W}{{{\partial}}y} + \sum_{k=1}^N q_k v_k \frac{{{\partial}}\Lambda_W}{\d v_k}=\Lambda_W.$$ There is a unique index $1\le l\le N$ such that the number $q_l$ is minimal. For the singularities $A_N$ and $E_N$ we have $l=N$ and in the $D_N$-case we have $l=N-1$. Denote by $(c_v)_{i,j}^k$ the structure constants of multiplication on ${\mathbb C}^N$ in the coordinates $v_1,\ldots,v_N$. Define a bilinear form $\eta_W$ on ${\mathbb C}^N$ by $$\eta_W\left(\frac{\d}{\d v_i},\frac{\d}{\d v_j}\right):=(c_v)^l_{i,j}.$$ This bilinear form together with the multiplication, introduced above, define a Frobenius manifold structure on ${\mathbb C}^N$, often called the Saito Frobenius manifold. It is semisimple and conformal with an Euler vector field $E_W$ given by $$E_W=\sum_{k=1}^N q_k v_k \frac{{{\partial}}}{\d v_k}.$$ The conformal dimension is $\delta=1-q_l$.
The coordinates $v_1,\dots,v_N$ are not flat whenever $N \ge 3$.
There exist unique [*global*]{} flat coordinates $t^i(v_1,\ldots,v_N)$ on ${\mathbb C}^N$ such that $$t^i(v_1,\ldots,v_N)=v_i+O(v_*^2).$$ They satisfy the quasi-homogeneity condition $$\begin{gathered}
\label{eq:homogeneity of flat coordinates}
E_W(t^i(v_1,\ldots,v_N))=q_i t^i(v_1,\ldots,v_N),\end{gathered}$$ and, hence, the Euler vector field in the flat coordinates $t^i$ is given by $$E_W=\sum_{i=1}^N q_i t^i\frac{\d}{\d t^i}.$$ The Frobenius manifold structure in the flat coordinates $t^i$ is described by a polynomial potential $F_W(t^1,\ldots,t^N)\in{\mathbb C}[t^1,\ldots,t^N]$, which we fix by requiring that it doesn’t contain monomials of degree less than $3$. Then the polynomial $F_W$ satisfies the condition $$\begin{gathered}
\label{eq:homogeneity of potential for Coxeter}
E_W(F_W)=(3-\delta)F_W.\end{gathered}$$
Explicit formulas for the flat coordinates of the Saito Frobenius manifolds of simple singularities are given in [@NY98]. For the $A_N$-case the formula is $$t^\gamma(v_1,\ldots,v_N) = \hspace{-0.2cm}\sum_{\substack{\alpha_1,\ldots,\alpha_N\ge 0\\ \sum (N+2-i)\alpha_i = N+2-\gamma}}\hspace{-0.2cm}\frac{1}{N+1-\gamma}\prod_{k=0}^{\sum\alpha_i-1}\left(N+1-\gamma-k(N+1)\right)\frac{\prod v_i^{\alpha_i}}{\prod \alpha_i!}, \quad 1\le\gamma \le N.$$ For the $D_N$-case the formula is $$\begin{aligned}
&t^\gamma(v_1,\ldots,v_N) = \hspace{-0.2cm}\sum_{\substack{\alpha_1,\ldots,\alpha_{N-1}\ge 0\\ \sum (N-i)\alpha_i = N-\gamma}}\hspace{-0.1cm} \left(-\frac{1}{2}\right)^{\sum\alpha_i-1} \prod_{k=0}^{\sum\alpha_i-2} \left(2\gamma-1 + 2k(N-1) \right) \frac{\prod v_i^{\alpha_i}}{\prod \alpha_i!}, \quad 1\le\gamma \le N-1,\label{eq:Dn flat coordinates}\\
&t^N(v_1,\ldots,v_N)=v_N.\notag\end{aligned}$$ Note that in this case the coordinates $t^1,\ldots,t^{N-1}$ depend only on $v_1,\ldots,v_{N-1}$.
Extended $r$-spin theory and the open WDVV equations for the $A$-singularity {#section:extended r-spin}
============================================================================
Here we explain how the Saito Frobenius manifold of the $A$-singularity together with a certain solution of the open WDVV equations appear in the intersection theory on the moduli spaces of algebraic curves.
Let $r=N+1$. For integers $0\le\alpha_1,\ldots,\alpha_n\le r-1$, using the geometry of algebraic curves with an $r$-spin structure, one can construct a cohomology class $$W^r_{0,n}(\alpha_1,\ldots,\alpha_n)\in H^{2d}({\overline{\mathcal M}}_{0,n},\mbQ),\quad d=\frac{\sum\alpha_i-(r-2)}{r},$$ called [*Witten’s class*]{}, on the moduli space ${\overline{\mathcal M}}_{0,n}$ of stable curves of genus $0$ with $n$ marked points (see e.g. [@PPZ16]). Here we assume that the class $W^r_{0,n}(\alpha_1,\ldots,\alpha_n)$ is equal to zero, if $d$ is not an integer or $d<0$. The class $W^r_{0,n}(\alpha_1,\ldots,\alpha_n)$ vanishes, if one of the $\alpha_i$’s is equal to $r-1$. Consider the generating series $$F_{{\text{$r$-spin}}}(t^1,\ldots,t^{r-1}):=\sum_{n\ge 3}\frac{1}{n!}\sum_{0\le\alpha_1,\ldots,\alpha_n\le r-2}\left(\int_{{\overline{\mathcal M}}_{0,n}}W^r_{0,n}(\alpha_1,\ldots,\alpha_n)\right)\prod_{i=1}^n t^{\alpha_i+1}.$$ The functions $F_{A_N}$ and $F_{{\text{$r$-spin}}}$ are related by [@JKV01a] $$F_{A_N}(t^1,\ldots,t^N)=(-r)^{-3}F_{{\text{$r$-spin}}}((-r)t^1,\ldots,(-r)t^N).$$ This is one of the simplest cases of mirror symmetry. Denote $${\left<}\tau_{\alpha_1}\ldots\tau_{\alpha_n}{\right>}_{A_N}:=\left.\frac{\d^n F_{A_N}}{\d t^{\alpha_1}\ldots\d t^{\alpha_n}}\right|_{t^*=0},\quad 1\le\alpha_1,\ldots,\alpha_n\le N.$$ We have (see e.g. [@LVX17 page 4]) $$\begin{gathered}
\label{eq:3-point and 4-point for AN}
{\left<}\tau_{\alpha_1}\tau_{\alpha_2}\tau_{\alpha_3}{\right>}_{A_N}=\delta_{\alpha_1+\alpha_2+\alpha_3,N+2},\qquad{\left<}\tau_{\alpha_1}\tau_{\alpha_2}\tau_{\alpha_3}\tau_{\alpha_4}{\right>}_{A_N}=-\min(\alpha_i-1,N+1-\alpha_i).\end{gathered}$$ These formulas will be used later.
In [@JKV01b] the authors noticed that the construction of Witten’s class $W^r_{0,n}(\alpha_1,\ldots,\alpha_n)$ can be extended to the case when $\alpha_1=-1$ and $0\le\alpha_2,\ldots,\alpha_n\le r-1$. In [@BCT19] the authors considered the generating series $$F^{\mathrm{ext}}_{{\text{$r$-spin}}}(t^1,\ldots,t^r):=\sum_{n\ge 2}\frac{1}{n!}\sum_{0\le\alpha_1,\ldots,\alpha_n\le r-1}\left(\int_{{\overline{\mathcal M}}_{0,n+1}}W^r_{0,n+1}(-1,\alpha_1,\ldots,\alpha_n)\right)\prod_{i=1}^n t^{\alpha_i+1}$$ and proved that it satisfies the open WDVV equations, associated to the potential $F_{{\text{$r$-spin}}}$, together with property . Here one should identify $t^r=s$. It occurs that after a simple transformation the function $F^{\mathrm{ext}}_{{\text{$r$-spin}}}$ also gives the generating series of intersection numbers on the moduli space of $r$-spin disks, introduced in [@BCT18 Theorem 1.3].
Introduce a function $F^o_{A_N}(t^1,\ldots,t^N,s)$ by $$F^o_{A_N}(t^1,\ldots,t^N,s):=(-r)^{-2}F^{\mathrm{ext}}_{{\text{$r$-spin}}}((-r)t^1,\ldots,(-r)t^N,(-r)s).$$ Clearly, it satisfies the open WDVV equations, associated to $F_{A_N}$, together with condition . In [@BCT18 Proposition 5.1] the authors found an explicit formula for the coefficients of $F^{\mathrm{ext}}_{{\text{$r$-spin}}}$ that gives $$\begin{gathered}
{\left<}\tau_{\alpha_1}\ldots\tau_{\alpha_n}\sigma^k{\right>}^o_{A_N}=
\begin{cases}
(n+k-2)!,&\text{if $\sum_{i=1}^n(N+2-\alpha_i)+k=N+2$},\\
0,&\text{otherwise},
\end{cases}\end{gathered}$$ where we use the notation $${\left<}\tau_{\alpha_1}\ldots\tau_{\alpha_n}\sigma^k{\right>}^o_{A_N}:=\left.\frac{\d^{n+k} F^o_{A_N}}{\d t^{\alpha_1}\ldots\d t^{\alpha_n}\d s^k}\right|_{t^*=s=0}.$$ In [@Bur18] the author proved that the coefficients of the function $F^o_{A_N}$ are related to the expression of the coordinates $v_k$ in the terms of the flat coordinates $t^1,\ldots,t^N$ on the Saito Frobenius manifold of the singularity $A_N$ by $$\begin{gathered}
\label{eq:FoAN and the flat coordinates}
\frac{\d F^o_{A_N}}{\d s}=\frac{s^{N+1}}{N+1}+\sum_{k=1}^Ns^{k-1}v_k(t^1,\ldots,t^N).\end{gathered}$$
Generalized Dubrovin–Saito construction for the singularities $A$ and $D$ {#section: Mwext explicit}
=========================================================================
In this section, for the singularites of types $A$ and $D$ we present a generalization of the Dubrovin–Saito construction that produces a flat F-manifold that extends the Saito Frobenius manifold and, therefore, gives a solution of the open WDVV equations. In the $A_N$-case this solution coincides with the function $F^o_{A_N}$. In both $A$- and $D$-cases, the coefficients of powers of the variable $s$ in this solution coincide with the transition functions between two coordinate systems on the Saito Frobenius manifold.
$A_N$-case
----------
Consider the space $M^{\mathrm{ext}}_{A_N}:={\mathbb C}^{N+1}$ with coordinates $v_1,\ldots,v_{N+1}$. Consider the quotient ring $${{\widehat{\mathcal{A}}}}^{{\mathrm{ext}}}_{A_N}:={{\mathbb{C}}}[x,y,w,v_1,\ldots,v_{N+1}] \Big/ \left(w - {{\partial}}_x\Lambda_{A_N}, {{\partial}}_y \Lambda_{A_N}, wx - v_{N+1} w \right).$$ As a ${\mathbb C}[v_1,\ldots,v_{N+1}]$-module, the space ${{\widehat{\mathcal{A}}}}^{{\mathrm{ext}}}_{A_N}$ is free of dimension $N+1$, and the elements $[1],[x],\ldots,[x^{N-1}],[w]$ form a basis. To show that any other element $[x^i y^j w^k]$ can be expressed in terms of them, first note that, obviously, $[y]=0$ and $[w x]=v_{N+1}[w]$. We also see that $$\begin{gathered}
\label{eq:w2 for AN}
[w^2]=\left[w\left(x^N+\sum_{k=2}^{N}(k-1)v_k x^{k-2}\right)\right]=\left(v_{N+1}^N+\sum_{k=2}^N(k-1)v_k v_{N+1}^{k-2}\right)[w].\end{gathered}$$ Using the relation $[x^N]=[w]-\sum_{k=2}^N(k-1)v_k[x^{k-2}]$, we can express any element $[x^p]$ with $p\ge N$ in terms of the elements $[1],[x],\ldots,[x^{N-1}],[w]$.
Identifying the ${\mathbb C}[v_1,\ldots,v_{N+1}]$-modules ${\mathcal{T}}^{\mathrm{alg}}_{M^{\mathrm{ext}}_{A_N}}(M^{\mathrm{ext}}_{A_N})$ and ${{\widehat{\mathcal{A}}}}^{\mathrm{ext}}_{A_N}$ via the isomorphism $\Psi^{\mathrm{ext}}_{A_N}$ defined by $$\begin{gathered}
\Psi_W^{\mathrm{ext}}\left(\frac{{{\partial}}}{{{\partial}}v_k}\right) := [x^{k-1}], \quad 1 \le k \le N,\qquad \Psi_W^{\mathrm{ext}}\left(\frac{{{\partial}}}{{{\partial}}v_{N+1}} \right) := [w],\end{gathered}$$ by Remark \[remark:algebraic construction of multiplication\], we endow the tangent spaces $T_pM_{A_N}^{\mathrm{ext}}$ with a multiplication and, clearly, the structure constants of it are polynomials in the coordinates $v_1,\ldots,v_{N+1}$.
Consider the flat coordinates $t^\alpha=t^\alpha(v_1,\dots,v_N)$, $1 \le \alpha \le N$, the potential $F_{A_N}(t^1,\ldots,t^N)$ of the Frobenius manifold of the singularity $A_N$ and the function $F^o_{A_N}$, described in Section \[section:extended r-spin\].
\
1. The coordinates $t^1(v_1,\ldots,v_N),\ldots,t^N(v_1,\ldots,v_N)$ and $t^{N+1}:=v_{N+1}$ together with the multiplicative structure on $M^{\mathrm{ext}}_{A_N}$, constructed above, define a flat F-manifold structure on $M^{\mathrm{ext}}_{A_N}$.\
2. The vector potential of this flat F-manifold is given by $\left(\eta_{A_N}^{1\alpha} \frac{\d F_{A_N}}{\d t^\alpha}, \ldots, \eta_{A_N}^{N\alpha} \frac{\d F_{A_N}}{\d t^\alpha}, F^o_{A_N}\right)$, where we identify $s=t^{N+1}$.
We denote by $(c^{\mathrm{ext}})^\alpha_{\beta\gamma}$ the structure constants of multiplication in the coordinates $t^1,\ldots,t^{N+1}$ and by $(c^{\mathrm{ext}}_v)^\alpha_{\beta\gamma}$ the structure constants of multiplication in the coordinates $v_1,\ldots,v_{N+1}$.
In order to prove the theorem, we have to check the following equations: $$\begin{aligned}
(c^{\mathrm{ext}})^\alpha_{\beta\gamma}=&\sum_{\mu=1}^N\eta_{A_N}^{\alpha\mu} \frac{\d^3 F_{A_N}}{\d t^\mu\d t^\beta\d t^\gamma},&& 1\le\alpha\le N,&& 1\le\beta,\gamma\le N+1,\label{eq:first property for AN}\\
(c^{\mathrm{ext}})^{N+1}_{\alpha\beta}=&\frac{\d^2 F^o_{A_N}}{\d t^\alpha\d t^\beta},&& 1\le\alpha,\beta\le N+1.&&\label{eq:second property for AN}\end{aligned}$$ Since the subspace ${\mathbb C}[v_1,\ldots,v_{N+1}]{\left<}[w]{\right>}$ is an ideal in the ring ${{\widehat{\mathcal{A}}}}^{{\mathrm{ext}}}_{A_N}$ and the quotient by this ideal coincides with the ring ${{\widehat{\mathcal{A}}}}_{A_N}$, we have $$(c^{\mathrm{ext}}_v)^a_{b,c}=
\begin{cases}
(c_v)^a_{b,c},&\text{if $1\le a,b,c\le N$},\\
0,&\text{if $1\le a\le N$ and one of the indices $b,c$ is equal to $N+1$}.
\end{cases}$$ This implies equation and it remains to prove .
Suppose $1\le\alpha\le N$ and $\beta=N+1$. Since $[xw]=v_{N+1}[w]$, we have $(c^{\mathrm{ext}}_v)^{N+1}_{k,N+1}=v_{N+1}^{k-1}$ for $1\le k\le N$, and, therefore, $$(c^{\mathrm{ext}})^{N+1}_{\alpha,N+1}=\sum_{k=1}^N\frac{\d v_k}{\d t^\alpha}(c^{\mathrm{ext}}_v)^{N+1}_{k,N+1}=\sum_{k=1}^N\frac{\d v_k}{\d t^\alpha}v_{N+1}^{k-1}\stackrel{\text{eq.\eqref{eq:FoAN and the flat coordinates}}}{=}\frac{\d^2 F^o_{A_N}}{\d t^\alpha\d t^{N+1}}.$$
Suppose $\alpha=\beta=N+1$. Then we compute $$(c^{\mathrm{ext}})^{N+1}_{N+1,N+1}=(c^{\mathrm{ext}}_v)^{N+1}_{N+1,N+1}\stackrel{\text{eq.\eqref{eq:w2 for AN}}}{=}v_{N+1}^N+\sum_{k=2}^N(k-1)v_k v_{N+1}^{k-2}\stackrel{\text{eq.\eqref{eq:FoAN and the flat coordinates}}}{=}\frac{\d^2 F^o_{A_N}}{\d (t^{N+1})^2}.$$
Finally, if $1\le\alpha,\beta\le N$, then from the associativity of the algebra ${{\widehat{\mathcal{A}}}}^{{\mathrm{ext}}}_{A_N}$ we get $$(c^{{\mathrm{ext}}})^{N+1}_{\alpha\beta}(c^{\mathrm{ext}})^{N+1}_{N+1,N+1}=(c^{\mathrm{ext}})^{N+1}_{\alpha,N+1}(c^{\mathrm{ext}})^{N+1}_{\beta,N+1}-\sum_{\mu=1}^N c^\mu_{\alpha\beta}(c^{\mathrm{ext}})^{N+1}_{\mu,N+1}.$$ Since the function $F^o_{A_N}$ satisfies and, as we have just proved, $(c^{\mathrm{ext}})^{N+1}_{\gamma,N+1}=\frac{\d^2 F^o_{A_N}}{\d t^\gamma\d t^{N+1}}$ for $1\le\gamma\le N+1$, we obtain $(c^{\mathrm{ext}})^{N+1}_{\alpha\beta}=\frac{\d^2 F^o_{A_N}}{\d t^\alpha\d t^\beta}$.
$D_N$-case
----------
Consider the space $M^{\mathrm{ext}}_{D_N}:={\mathbb C}^N\times{\mathbb C}^*$ with coordinates $v_1,\ldots,v_{N+1}$. Consider the quotient ring $$\begin{gathered}
{{\widehat{\mathcal{A}}}}^{{\mathrm{ext}}}_{D_N} := {{\mathbb{C}}}[x,y,w,v_1,\dots,v_N,v_{N+1},v_{N+1}^{-1}]\Big/ \left(w - v_{N+1} {{\partial}}_x\Lambda_{D_N},{{\partial}}_y \Lambda_{D_N},2wx - v_{N+1}^2w\right).\end{gathered}$$ As a ${{\mathbb{C}}}[v_1,\dots,v_N,v_{N+1},v_{N+1}^{-1}]$-module, the space ${{\widehat{\mathcal{A}}}}^{{\mathrm{ext}}}_{D_N}$ is free of dimension $N+1$ with a basis $[1],[x],\ldots,[x^{N-2}],[y],[w]$. To show that any other element $[x^i y^j w^k]$ can be expressed in terms of them, first note that $$\begin{gathered}
\label{eq:xy for DN}
[xy]=-\frac{v_N}{2}[1],\qquad[w x]=\frac{v_{N+1}^{2}}{2}[w],\qquad [wy]=-\frac{v_N}{v_{N+1}^2}[w],\end{gathered}$$ where the last equation follows from the first two. Similarly to the $A_N$-case, we have $$\begin{gathered}
\label{eq:w2 for DN}
[w^2]=\left(\frac{v_{N+1}^{2N-3}}{2^{N-2}}+\frac{v_N^2}{v_{N+1}^3}+\sum_{k=2}^{N-1}(k-1)v_k \frac{v_{N+1}^{2k-3}}{2^{k-2}}\right)[w].\end{gathered}$$ Using that $[w-v_{N+1} {{\partial}}_x\Lambda_{D_N}]=0$, we obtain $$\begin{gathered}
\label{eq:y2 for DN}
[y^2]=\frac{1}{v_{N+1}}[w]-[x^{N-2}]-\sum_{k=2}^{N-1}(k-1)v_k[x^{k-2}].\end{gathered}$$ Multiplying this equation by $[x]$, we get the relation $$\begin{gathered}
\label{eq:xN-1 for DN}
[x^{N-1}]=\frac{v_{N+1}}{2}[w]+\frac{v_N}{2}[y]-\sum_{k=1}^{N-1}(k-1)v_k[x^{k-1}],\end{gathered}$$ that allows to express any element $[x^p]$ with $p\ge N-1$ in terms of the elements $[1],[x],\ldots,[x^{N-2}]$, $[y]$, $[w]$.
Identifying the ${{\mathbb{C}}}[v_1,\dots,v_N,v_{N+1},v_{N+1}^{-1}]$-modules ${\mathcal{T}}^{\mathrm{alg}}_{M^{\mathrm{ext}}_{D_N}}(M^{\mathrm{ext}}_{D_N})$ and ${{\widehat{\mathcal{A}}}}^{\mathrm{ext}}_{D_N}$ via the isomorphism $\Psi^{\mathrm{ext}}_{D_N}$ defined by $$\begin{gathered}
\Psi_W^{\mathrm{ext}}\left(\frac{{{\partial}}}{{{\partial}}v_k}\right) := [x^{k-1}], \quad 1 \le k \le N-1,\qquad \Psi_W^{\mathrm{ext}}\left(\frac{{{\partial}}}{{{\partial}}v_N} \right) := [y],\qquad \Psi_W^{\mathrm{ext}}\left(\frac{{{\partial}}}{{{\partial}}v_{N+1}} \right) := [w],\end{gathered}$$ we endow the tangent spaces $T_pM_{D_N}^{\mathrm{ext}}$ with a multiplication and, clearly, the structure constants of it belong to the ring ${\mathbb C}[v_1,\ldots,v_N,v_{N+1},v_{N+1}^{-1}]$.
Consider the flat coordinates $t^\alpha=t^\alpha(v_1,\dots,v_N)$, $1 \le \alpha \le N$, and the potential $F_{D_N}(t^1,\ldots,t^N)$ of the Frobenius manifold of the singularity $D_N$. Let $t^{N+1}:=v_{N+1}$ and define a function $F^o_{D_N}(t^1,\ldots,t^{N+1})$ by $$\begin{gathered}
F^o_{D_N}:=\left.\left(\sum_{k=1}^{N-1} \frac{v_k v_{N+1}^{2 k-1}}{2^{k-1}(2 k-1)} + \frac{v_{N+1}^{2 N-1}}{2^{N-2}(2 N-1) (2N-2)} + \frac{v_N^2}{2 v_{N+1}}\right)\right|_{v_i=v_i(t^*)}.\end{gathered}$$
\
1. The coordinates $t^1(v_1,\ldots,v_N),\ldots,t^N(v_1,\ldots,v_N)$ and $t^{N+1}=v_{N+1}$ together with the multiplicative structure on $M^{\mathrm{ext}}_{D_N}$, constructed above, define a flat F-manifold structure on $M^{\mathrm{ext}}_{D_N}$.\
2. The vector potential of this flat F-manifold is given by $\left(\eta_{D_N}^{1\alpha} \frac{\d F_{D_N}}{\d t^\alpha}, \ldots, \eta_{D_N}^{N\alpha} \frac{\d F_{D_N}}{\d t^\alpha}, F^o_{D_N}\right)$.
We denote by $(c^{\mathrm{ext}})^\alpha_{\beta\gamma}$ the structure constants of multiplication in the coordinates $t^1,\ldots,t^{N+1}$ and by $(c^{\mathrm{ext}}_v)^\alpha_{\beta\gamma}$ the structure constants of multiplication in the coordinates $v_1,\ldots,v_{N+1}$.
In order to prove the theorem, we have to check the following equations: $$\begin{aligned}
(c^{\mathrm{ext}})^\alpha_{\beta\gamma}=&\sum_{\mu=1}^N\eta_{D_N}^{\alpha\mu} \frac{\d^3 F_{D_N}}{\d t^\mu\d t^\beta\d t^\gamma},&& 1\le\alpha\le N,&& 1\le\beta,\gamma\le N+1,\label{eq:first property for DN}\\
(c^{\mathrm{ext}})^{N+1}_{\alpha\beta}=&\frac{\d^2 F^o_{D_N}}{\d t^\alpha\d t^\beta},&&1\le\alpha,\beta\le N+1.&&\label{eq:second property for DN}\end{aligned}$$ Since the subspace ${\mathbb C}[v_1,\ldots,v_N,v_{N+1},v_{N+1}^{-1}]{\left<}[w]{\right>}$ is an ideal in the ring ${{\widehat{\mathcal{A}}}}^{{\mathrm{ext}}}_{D_N}$ and the quotient by this ideal coincides with the ring ${{\widehat{\mathcal{A}}}}_{D_N}$, we have $$(c^{\mathrm{ext}}_v)^a_{b,c}=
\begin{cases}
(c_v)^a_{b,c},&\text{if $1\le a,b,c\le N$},\\
0,&\text{if $1\le a\le N$ and one of the indices $b,c$ is equal to $N+1$}.
\end{cases}$$ This implies equation and it remains to prove .
We have two substantially different cases: the case $\alpha\in\{N,N+1\}$ or $\beta\in\{N,N+1\}$ and the case $\alpha,\beta\in\{1,2,\dots,N-1\}$.
[*Case $\alpha\in\{N,N+1\}$ or $\beta\in\{N,N+1\}$*]{}. If $\alpha,\beta\in\{N,N+1\}$, then $(c^{\mathrm{ext}})^{N+1}_{\alpha,\beta}=(c^{\mathrm{ext}}_v)^{N+1}_{\alpha,\beta}$ and $\frac{\d^2 F^o_{D_N}}{\d t^\alpha\d t^\beta}=\frac{\d^2 F^o_{D_N}}{\d v_\alpha\d v_\beta}$. The equation $(c^{\mathrm{ext}}_v)^{N+1}_{\alpha,\beta}=\frac{\d^2 F^o_{D_N}}{\d v_\alpha\d v_\beta}$ immediately follows from formulas , and .
If $1\le\alpha\le N-1$ and $\beta=N+1$, then $$\begin{gathered}
(c^{\mathrm{ext}})^{N+1}_{\alpha,N+1}=\sum_{k=1}^{N-1}\frac{\d v_k}{\d t^\alpha}(c^{\mathrm{ext}}_v)^{N+1}_{k,N+1}\stackrel{\text{eq.\eqref{eq:xy for DN}}}{=}\sum_{k=1}^{N-1}\frac{\d v_k}{\d t^\alpha}\frac{v_{N+1}^{2k-2}}{2^{k-1}}=\frac{\d}{\d t^\alpha}\frac{\d F^o_{D_N}}{\d v_{N+1}}=\frac{\d^2 F^o_{D_N}}{\d t^\alpha\d t^{N+1}}.\end{gathered}$$
If $1\le\alpha\le N-1$ and $\beta=N$, then $$\begin{gathered}
(c^{\mathrm{ext}})^{N+1}_{\alpha,N}=\sum_{k=1}^{N-1}\frac{\d v_k}{\d t^\alpha}(c^{\mathrm{ext}}_v)^{N+1}_{k,N}\stackrel{\text{eq.\eqref{eq:xy for DN}}}{=}0=\frac{\d}{\d t^\alpha}\frac{\d F^o_{D_N}}{\d v_N}=\frac{\d^2 F^o_{D_N}}{\d t^\alpha\d t^N}.\end{gathered}$$
[*Case $\alpha,\beta\in\{1,2,\ldots,N-1\}$*]{}. We have to check that $$\begin{gathered}
(c^{\mathrm{ext}})^{N+1}_{\alpha\beta}=\frac{\d^2 F^o_{D_N}}{\d t^\alpha\d t^\beta} \Leftrightarrow \sum_{1\le a,b\le N-1}\frac{\d v_a}{\d t^\alpha}\frac{\d v_b}{\d t^\beta}(c^{\mathrm{ext}}_v)^{N+1}_{a,b}=\sum_{k=1}^{N-1}\frac{\d^2 v_k}{\d t^\alpha\d t^\beta}\frac{v_{N+1}^{2k-1}}{2^{k-1}(2k-1)},\end{gathered}$$ that is equivalent to the equation $$\begin{gathered}
\label{eq:equation for cab for DN}
(c^{\mathrm{ext}}_v)^{N+1}_{a,b}=-\sum_{k=1}^{N-1}\frac{\d v_k}{\d t^\gamma}\frac{\d^2 t^\gamma}{\d v_a\d v_b}\frac{v_{N+1}^{2k-1}}{2^{k-1}(2k-1)},\quad 1\le a,b\le N-1.\end{gathered}$$
Let us compute the structure constants $(c^{\mathrm{ext}}_v)^{N+1}_{a,b}$. Introduce polynomials $\omega_k\in\mbQ[v_1,\ldots,v_{N-1}]$, $k \ge 0$, by $$\begin{gathered}
\omega_k := \sum_{\substack{\alpha_1,\dots,\alpha_{N-1}\ge 0\\ \sum (N-i)\alpha_i = k}} s_1^{\alpha_1}\cdots s_{N-1}^{\alpha_{N-1}} \frac{(\sum \alpha_i )!}{\prod \alpha_i!},\quad\text{where $s_i:=(1-i)v_i$ for $1\le i\le N-1$.}\end{gathered}$$ The first few functions $\omega_k$ are $$\omega_0 = 1, \quad \omega_1 = s_{N-1}, \quad \omega_2 = s_{N-2} + s_{N-1}^2, \quad \omega_3 = s_{N-3} + 2 s_{N-2}s_{N-1} + s_{N-1}^3.$$ The functions $\omega_k$ satisfy the recursion relation $$\begin{gathered}
\label{eq:recursion for omegak}
\omega_{k+1}=\sum_{i=1}^{N-1}s_{N-i}\omega_{k+1-i},\quad k\ge 0,\end{gathered}$$ where we adopt the convention $\omega_j:=0$ for $j<0$.
\[lemma: Dn xp product\] For $1\le a,b\le N-1$ we have $$\begin{gathered}
\label{eq:cabN+1 in Dn}
c_{a,b}^{N+1} = \sum_{k=0}^{a+b-N-1} \omega_k \frac{v_{N+1}^{2(a+b)-2N-1-2k}}{2^{a+b-N-k}}.\end{gathered}$$
Equation is equivalent to the following formula: $$\label{eq:Dn cab classes}
{\mathrm{Coef}}_{[w]}[x^{p-2}] = \sum_{k=0}^{p-N-1} \omega_k \frac{v_{N+1}^{2p-2N-1-2k}}{2^{p-N-k}},\quad 2\le p\le 2N-2,$$ where ${\mathrm{Coef}}_{[w]}[x^{p-2}]$ denotes the coefficient of $[w]$ in the expression for $[x^{p-2}]$ in terms of the basis elements $[1],[x],\ldots,[x^{N-2}],[y],[w]$. For $p\le N$ formula is obvious because both sides of it are equal to zero.
Suppose $p\ge N+1$. Multiplying both sides of equation by $[x^{p-N-1}]$, we get the relation $${\mathrm{Coef}}_{[w]}[x^{p-2}]=\frac{v_{N+1}^{2p-2N-1}}{2^{p-N}}+\sum_{k=1}^{N-1}s_k{\mathrm{Coef}}_{[w]}[x^{p+k-N-2}],$$ that allows to compute the coefficients ${\mathrm{Coef}}_{[w]}[x^{p-2}]$ recursively. Then, using also relation , formula can be easily proved by induction.
Using the lemma, we see that equation can be equivalently written as $$\begin{aligned}
&\sum_{k=0}^{a+b-N-1} \omega_k \frac{v_{N+1}^{2(a+b)-2N-1-2k}}{2^{a+b-N-k}}=-\sum_{k=1}^{N-1}\frac{\d v_k}{\d t^\gamma}\frac{\d^2 t^\gamma}{\d v_a\d v_b}\frac{v_{N+1}^{2k-1}}{2^{k-1}(2k-1)}\Leftrightarrow\\
\Leftrightarrow&\sum_{k=1}^{N-1} \omega_{a+b-N-k} \frac{v_{N+1}^{2k-1}}{2^k}=-\sum_{k=1}^{N-1}\frac{\d v_k}{\d t^\gamma}\frac{\d^2 t^\gamma}{\d v_a\d v_b}\frac{v_{N+1}^{2k-1}}{2^{k-1}(2k-1)}.\end{aligned}$$ So we have to prove that $$\begin{gathered}
\label{eq:identity for DN}
\frac{\d v_k}{\d t^\gamma}\frac{\d^2 t^\gamma}{\d v_a\d v_b}=-\frac{2k-1}{2}\omega_{a+b-N-k}\Leftrightarrow \frac{\d^2 t^\gamma}{\d v_a\d v_b}=-\sum_{k=1}^{N-1}\frac{2k-1}{2}\omega_{a+b-N-k}\frac{\d t^\gamma}{\d v_k}.\end{gathered}$$ Recall that $t^\gamma(v_1,\ldots,v_{N-1})$ is a quasi-homogeneous polynomial of degree $N-\gamma$, if we put $\deg v_a=N-a$. This implies that both sides of the last equation in are zero if $a+b\le N$. Let us assume now that $a+b\ge N+1$. The last equation in is equivalent to $$\frac{\d^2 t^\gamma}{\d v_a\d v_b}-\sum_{i=1}^{a-1}s_{N-i}\frac{\d^2 t^\gamma}{\d v_{a-i}\d v_b}=-\sum_{k=1}^{N-1}\frac{2k-1}{2}\frac{\d t^\gamma}{\d v_k}\left(\omega_{a+b-N-k}-\sum_{i=1}^{a-1}s_{N-i}\omega_{a-i+b-N-k}\right).$$ Note that for $i\ge a$ we have $a-i+b-N-k<0$ and, therefore, by , the expression in the brackets is equal to zero unless $k=a+b-N$. So we come to the following equivalent identity: $$\begin{gathered}
\label{eq:main identity for DN}
\frac{\d^2 t^\gamma}{\d v_a\d v_b}-\sum_{i=1}^{a-1}s_{N-i}\frac{\d^2 t^\gamma}{\d v_{a-i}\d v_b}=-\frac{2(a+b-N)-1}{2}\frac{\d t^\gamma}{\d v_{a+b-N}},\quad
\begin{array}{@{}c@{}} 1\le a,b,\gamma\le N-1, \\ a+b\ge N+1. \end{array}\end{gathered}$$
Note that both sides of are quasi-homogeneous polynomials of degree $a+b-\gamma-N$. Differentiating both sides by $\frac{\d^{\sum\alpha_i}}{\d v_1^{\alpha_1}\ldots \d v_{N-1}^{\alpha_{N-1}}}$, putting $v_j=0$ and using formula , wee see that equation is equivalent to the following family of identities: $$\begin{aligned}
&-\frac{1}{2}A\left(2\gamma-1+2(N-1)\sum\alpha_i\right)+A\sum_{i=1}^{a-1}(N-i-1)\alpha_{N-i}=-A\frac{2(a+b-N)-1}{2}\Leftrightarrow\notag\\
\Leftrightarrow & A\left(-\gamma-(N-1)\sum\alpha_i+\sum_{i=1}^{a-1}(N-i-1)\alpha_{N-i}+a+b-N\right)=0,\label{eq:final identity for DN}\end{aligned}$$ that should be true for any tuple $\alpha_1,\ldots,\alpha_{N-1}\ge 0$ such that $$\begin{gathered}
\label{eq:degree condition for alpha}
\sum_{i=1}^{N-1} (N-i)\alpha_i=a+b-\gamma-N,\end{gathered}$$ and where $A=\left(-\frac{1}{2}\right)^{\sum\alpha_i} \prod_{k=0}^{\sum\alpha_i-1} \left(2\gamma-1 + 2k(N-1)\right)$. Condition implies that $\alpha_i=0$, if $i\le N-a$. Therefore, the summation $\sum_{i=1}^{a-1}$ in can be replaced by the summation $\sum_{i=1}^{N-1}$ and, using , we immediately see that the expression in the brackets in vanishes. This completes the proof of the theorem.
Taking into account the discussion about the relation between solutions of the open WDVV equations and flat F-manifolds from Section \[subsection:flat F-manifolds and open WDVV\], we get the following result.
\
1. The function $F^o_{D_N}$ satisfies the open WDVV equations together with condition and the quasi-homogeneity property $$\sum_{\alpha=1}^N q_\alpha t^\alpha\frac{\d F^o_{D_N}}{\d t^\alpha}+\frac{1-\delta}{2}s\frac{\d F^o_{D_N}}{\d s}=\frac{3-\delta}{2}F^o_{D_N}.$$ 2. We have $$\begin{gathered}
v_k(t^1,\ldots,t^N) =
\begin{cases}
2^{k-1}(2k-1){\mathrm{Coef}}_{s^{2k-1}}F^o_{D_N},&\text{if $1\le k\le N-1$},\\
\sqrt{2{\mathrm{Coef}}_{s^{-1}}F^o_{D_N}},&\text{if $k=N$}.
\end{cases}\end{gathered}$$
Here are the Frobenius manifold potentials for the singularities $D_4$ and $D_5$ together with the constructed solutions of the open WDVV equations[^1]. $$\begin{aligned}
F_{D_4} &= \frac{1}{2} t_1^2 t_3 + \frac{1}{2} t_1 t_2^2-\frac{1}{2} t_1 t_4^2 - \frac{1}{4} t_2 t_3 t_4^2 -\frac{1}{12} t_2^3 t_3 +\frac{1}{24} t_2^2 t_3^3 -\frac{1}{24} t_3^3t_4^2 + \frac{t_3^7}{3360},\\
F^o_{D_4} &= \frac{s^7}{168}+\frac{t_3 s^5}{20} +\left(\frac{t_3^2}{8}+\frac{t_2}{6}\right) s^3+\left(\frac{t_3^3}{12}+\frac{t_2 t_3}{2}+t_1\right) s+\frac{t_4^2}{2 s},\\
F_{D_5} &= \frac{1}{2} t_1^2 t_4 + t_1 t_2 t_3 - \frac{1}{2} t_1 t_5^2 - \frac{1}{6} t_2 t_3^3 - \frac{1}{4} t_2^2 t_3 t_4 + \frac{1}{24} t_2^2 t_4^3 + \frac{1}{6}t_2^3 - \frac{1}{48} t_3^3 t_4^3 + \frac{1}{16} t_3^4 t_4 + \frac{1}{8} t_2 t_3^2 t_4^2+ \\
& \quad + \frac{1}{32256}t_4^9 - \frac{1}{4} t_2 t_4 t_5^2 - \frac{1}{8} t_3 t_4^2 t_5^2 - \frac{1}{64} t_4^4 t_5^2 - \frac{1}{8} t_3^2 t_5^2 + \frac{1}{160} t_3^2 t_4^5,\\
F^o_{D_5} &= \frac{s^9}{576}+\frac{t_4 s^7}{56} +\left(\frac{t_4^2}{16}+\frac{t_3}{20}\right) s^5+\left(\frac{t_4^3}{12}+\frac{t_3 t_4}{4}+\frac{t_2}{6}\right) s^3+\left(\frac{t_4^4}{32}+\frac{t_3 t_4^2}{4} +\frac{t_2 t_4}{2}+\frac{t_3^2}{4}+t_1\right) s+\frac{t_5^2}{2 s}.\end{aligned}$$
Polynomial solutions of the open WDVV equations for finite irreducible Coxeter groups {#sec: polynomial solutions}
=====================================================================================
In this section we first recall a description of the Frobenius manifolds corresponding to finite irreducible Coxeter groups, and then describe the space of homogeneous polynomial solutions of the associated open WDVV equations.
Frobenius manifolds of finite irreducible Coxeter groups
--------------------------------------------------------
[*Finite Coxeter groups*]{} are finite groups of linear transformations of a real $N$-dimensional vector space $V$, generated by reflections. The complete list of finite irreducible Coxeter groups is given by (the dimension of the space $V$ equals the subscript in the name of the group) $$\begin{aligned}
& A_N, N\ge 1 && D_N, N \ge 4, && E_6, && E_7, && E_8, &&\label{eq: ADE coxeters}\\
& B_N, N \ge 2, && F_4, && H_3, && H_4, && I_2(k), k\ge 3, \label{eq: nonADE coxeters}\end{aligned}$$ with the exceptional isomorphisms $A_2\cong I_2(3)$ and $B_2\cong I_2(4)$. By a construction of B. Dubrovin [@Dub98], for such a group $W$ the complexified space of orbits $M_W := (V \otimes {\mathbb C})/W\cong{\mathbb C}^N$ carries a Frobenius manifold structure. For the Coxeter groups $A_N$, $D_N$ and $E_N$ the corresponding Frobenius manifolds coincide with the Saito Frobenius manifolds of simple singularities. By a result of J.-B. Zuber [@Zub94], the Frobenius manifold potentials corresponding to the remaining irreducible Coxeter groups can be explicitly described by $$\begin{aligned}
F_{B_N}(t^1,\ldots,t^N)=&F_{A_{2N-1}}(t^1,0,t^2,0,\ldots,t^{N-1},0,t^N),\label{eq:formula for non-ADE}\\
F_{I_2(k)}(t^1,t^2)=&F_{A_{k-1}}(t^1,0,\ldots,0,t^2),\notag\\
F_{F_4}(t^1,t^2,t^3,t^4)=&F_{E_6}(t^1,0,t^2,t^3,0,t^4),\notag\\
F_{H_4}(t^1,t^2,t^3,t^4)=&F_{E_8}(t^1,0,t^2,0,0,t^3,0,t^4),\notag\\
F_{H_3}(t^1,t^2,t^3)=&F_{D_6}(t^1,0,t^2,0,t^3,\sqrt{-1}t^2).\notag\end{aligned}$$ All the Frobenius manifolds corresponding to finite irreducible Coxeter groups are semisimple.
Euler vector field
------------------
We see that for any finite irreducible Coxeter group $W$, acting on an $N$-dimensional real vector space $V$, the associated Frobenius manifold is described by the polynomial potential $F_W(t^1,\ldots,t^N)$ satisfying the quasi-homogeneity condition $$\sum_{\alpha=1}^N q_\alpha t^\alpha\frac{\d F_W}{\d t^\alpha}=(3-\delta)F_W,\quad q_\alpha>0.$$ The numbers $q_1,\ldots,q_N$ have the following interpretation. Consider the symmetric algebra $S(V\otimes{\mathbb C})$. The subring $S(V\otimes{\mathbb C})^W$ of $W$-invariant polynomials is generated by $N$ algebraically independent homogeneous polynomials, whose degrees $d_1,\ldots,d_N\ge 2$ are uniquely determined by the Coxeter group. The maximal degree $h$ is called the [*Coxeter number*]{} of $W$. Then we have $$q_\alpha=\frac{d_\alpha}{h},\qquad \delta=1-\frac{2}{h}.$$ Note that then in the homogeneity condition for solutions of the open WDVV equations the degree of the extra variable $s$ becomes $$\frac{1-\delta}{2}=\frac{1}{h}.$$
Homogeneous polynomial solutions of the open WDVV equations
-----------------------------------------------------------
In this section we describe the space of homogeneous polynomial solutions of the open WDVV equations associated to the Frobenius manifolds of finite irreducible Coxeter groups. It occurs that for the Coxeter groups different from $A_N$, $B_N$ and $I_2(k)$ there are no such solutions. We prove it in Section \[subsubsection:nonABI\]. For the groups $A_N$, $B_N$ and $I_2(k)$ all solutions can be obtained from the function $F^o_{A_N}$, as is explained in Section \[subsubsection:ABI\].
Consider an irreducible Coxeter group $W$, the potential $F_W$ and a homogeneous polynomial solution $F^o$ of the open WDVV equations, satisfying . Note that equations - involve only the second partial derivatives of $F^o$ and that adding constant and linear terms in the variables $t^1,\ldots,t^N$ and $s$ to $F^o$ just changes the constants $D_\alpha$, ${\widetilde{D}}$ and $E$ in condition . If we remove constant and linear terms in the variables $t^1,\ldots,t^N$ and $s$ from the function $F^o$, then it will satisfy the condition $$\begin{gathered}
\label{eq:homogeneity for open Coxeter}
\sum_{\alpha=1}^N q_\alpha t^\alpha\frac{\d F^o}{\d t^\alpha}+\frac{1-\delta}{2}s\frac{\d F^o}{\d s}=\frac{3-\delta}{2}F^o.\end{gathered}$$
### Irreducible Coxeter groups different from $A_N$, $B_N$ and $I_2(k)$ {#subsubsection:nonABI}
\[theorem: coxeters\] Let $W$ be a finite irreducible Coxeter group different from $A_N$, $B_N$ and $I_2(k)$. Consider the corresponding Frobenius manifold potential $F_W$. Then there are no homogeneous polynomial solutions $F^o$ of the associated open WDVV equations satisfying property .
As we already explained above, we can assume that $F^o$ doesn’t contain constant and linear terms in the variables $t^1,\ldots,t^N$ and $s$ and satisfies condition .
Let $W$ be one of the groups $D_N$, $E_6$, $E_7$ or $E_8$. Let us rewrite equations in the coordinates $v_1,\ldots,v_N$ and $s$: $$\begin{gathered}
(c_v)^\mu_{\alpha\beta}\frac{\d^2 F^o}{\d v_\mu\d s}+\frac{\d^2 F^o}{\d v_\alpha\d v_\beta}
\frac{\d^2 F^o}{\d s^2}+\frac{\d t^{{\widetilde{\alpha}}}}{\d v_\alpha}\frac{\d t^{{\widetilde{\beta}}}}{\d v_\beta}\frac{\d^2 v_\mu}{\d t^{{\widetilde{\alpha}}}\d t^{{\widetilde{\beta}}}}\frac{\d F^o}{\d v_\mu}\frac{\d^2 F^o}{\d s^2}=\frac{\d^2 F^o}{\d v_\alpha\d s}\frac{\d^2 F^o}{\d v_\beta\d s},\quad 1\le\alpha,\beta\le N,\end{gathered}$$ where $(c_v)^\mu_{\alpha\beta}$ denotes the structure constants of multiplication in the coordinates $v_\mu$. Clearly, $\left.\frac{\d F^o}{\d v_\mu}\right|_{v_*=s=0}=0$. Since $\delta\ge 0$, we have $\frac{3-\delta}{2}>2\cdot\frac{1-\delta}{2}$. This implies that $\left.\frac{\d^2 F^o}{\d s^2}\right|_{v_*=s=0}=0$. Therefore, $$\begin{gathered}
\left.\frac{\d}{\d v_\gamma}\left((c_v)^\mu_{\alpha\beta}\frac{\d^2 F^o}{\d v_\mu\d s}\right)\right|_{v_*=s=0}+\left.\frac{\d}{\d v_\gamma}\left(\frac{\d^2 F^o}{\d v_\alpha\d v_\beta}\frac{\d^2 F^o}{\d s^2}\right)\right|_{v_*=s=0}=\left.\frac{\d}{\d v_\gamma}\left(\frac{\d^2 F^o}{\d v_\alpha\d s}\frac{\d^2 F^o}{\d v_\beta\d s}\right)\right|_{v_*=s=0},\end{gathered}$$ for any indices $1\le\alpha,\beta,\gamma\le N$. We will prove that this equation can’t be true by finding indices $2\le\alpha,\beta,\gamma \le N$ such that $$\begin{aligned}
&\left.(c_v)^\mu_{\alpha\beta}\right|_{v_*=0}=0,&&\frac{{{\partial}}(c_v)^\mu_{\alpha\beta}}{{{\partial}}v_\gamma}=A\delta^{\mu,1},\quad A\in{\mathbb C}^*,\label{eq:first condition for nonexistence}\\
&\frac{{{\partial}}^2 F^o}{{{\partial}}v_\alpha {{\partial}}v_\beta}=0,&&\frac{{{\partial}}}{{{\partial}}v_\gamma} \left(\frac{{{\partial}}^2 F^o}{{{\partial}}v_\alpha {{\partial}}s}\frac{{{\partial}}^2 F^o}{{{\partial}}v_\beta {{\partial}}s}\right) = 0.\label{eq:second condition for nonexistence}\end{aligned}$$
[*Case $W = D_N$, $N \ge 4$.*]{} We have $\delta=\frac{N-2}{N-1}$, $\frac{1-\delta}{2}=\frac{1}{2(N-1)}$ and $q_k=\begin{cases}\frac{N-k}{N-1},&\text{if $1\le k\le N-1$},\\\frac{N}{2(N-1)},&\text{if $k=N$}.\end{cases}$ Let us choose $\alpha=2$ and $\beta=\gamma=N$. From $\frac{{{\partial}}\Lambda_{D_N}}{{{\partial}}y} = 2xy + v_N$ we see that $\frac{{{\partial}}}{{{\partial}}v_2} \circ \frac{{{\partial}}}{{{\partial}}v_N} = - \frac{1}{2}v_N \frac{{{\partial}}}{{{\partial}}v_1}$, that implies the properties in line . We have $$\begin{aligned}
&q_2 + q_N = \frac{N-2}{N-1} + \frac{N}{2(N-1)} = \frac{3N-4}{2(N-1)} > \frac{2N-1}{2(N-1)} = \frac{3-\delta}{2}&& \hspace{-0.3cm}\Rightarrow &&\hspace{-0.3cm} \frac{{{\partial}}^2 F^o}{{{\partial}}v_2 {{\partial}}v_N} = \frac{{{\partial}}^3 F^o}{{{\partial}}v_2 {{\partial}}v_N{{\partial}}s} = 0,\\
&2 q_N + \frac{1-\delta}{2} = \frac{2N}{2(N-1)} + \frac{1}{2(N-1)} = \frac{2N+1}{2(N-1)} > \frac{3-\delta}{2} && \hspace{-0.3cm}\Rightarrow &&\hspace{-0.3cm} \frac{{{\partial}}^3 F^o}{{{\partial}}v_N^2 {{\partial}}s} = 0,\end{aligned}$$ that gives the properties in line . So the theorem is proved for the case $W=D_N$.
[*Case $W = E_6$.*]{} We have $\delta=\frac{5}{6}$, $\frac{1-\delta}{2}=\frac{1}{12}$ and $(q_1,\ldots,q_6)=\left(1,\frac{3}{4},\frac{2}{3},\frac{1}{2},\frac{5}{12},\frac{1}{6}\right)$. Let us choose $\alpha=\beta=\gamma=3$. From $\frac{{{\partial}}\Lambda_{E_6}}{{{\partial}}y} = 3y^2 + v_3 + v_5 x + v_6 x^2$ we see that $\frac{{{\partial}}}{{{\partial}}v_3} \circ \frac{{{\partial}}}{{{\partial}}v_3} = - \frac{1}{3} v_3 \frac{{{\partial}}}{{{\partial}}v_1}-\frac{1}{3}v_5\frac{\d}{\d v_2}-\frac{1}{3}v_6\frac{\d}{\d v_4}$, that implies the properties in line . We have $2q_3 = \frac{4}{3} > \frac{13}{12}=\frac{3-\delta}{2}$, implying $\frac{{{\partial}}^2 F^o}{{{\partial}}v_3^2}=\frac{{{\partial}}^3 F^o}{{{\partial}}v_3^2{{\partial}}s} = 0$, that gives the properties in line and proves the theorem for $W=E_6$.
[*Case $W = E_7$.*]{} We have $\delta=\frac{8}{9}$, $\frac{1-\delta}{2}=\frac{1}{18}$ and $(q_1,\ldots,q_7)=\left(1,\frac{7}{9},\frac{2}{3},\frac{5}{9},\frac{4}{9},\frac{1}{3},\frac{1}{9}\right)$. Choose $\alpha=3$, $\beta=4$ and $\gamma=2$. From $\frac{{{\partial}}\Lambda_{E_7}}{{{\partial}}x} = 3 x^2y + v_2 + 2 v_4 x + v_5 y + 3 v_6 x^2 + 4 v_7 x^3$ we see that $\frac{{{\partial}}}{{{\partial}}v_3} \circ \frac{{{\partial}}}{{{\partial}}v_4} = - \frac{1}{3} v_2 \frac{{{\partial}}}{{{\partial}}v_1} -\frac{2}{3}v_4\frac{\d}{\d v_2}-\frac{1}{3}v_5\frac{\d}{\d v_3}-v_6\frac{\d}{\d v_4}-\frac{4}{3}v_7\frac{\d}{\d v_6}$, that implies the properties in line . We have $$\begin{aligned}
&q_3 + q_4 = \frac{11}{9} > \frac{19}{18} = \frac{3-\delta}{2} && \Rightarrow && \frac{{{\partial}}^2 F^o}{{{\partial}}v_3 {{\partial}}v_4} = 0,\\
&q_2 + q_3 + \frac{1-\delta}{2} = \frac{3}{2} > \frac{3-\delta}{2} && \Rightarrow && \frac{{{\partial}}^3 F^o}{{{\partial}}v_2 {{\partial}}v_3 {{\partial}}s} = 0,\\
&q_2 + q_4 + \frac{1-\delta}{2} = \frac{25}{18} > \frac{3-\delta}{2} && \Rightarrow && \frac{{{\partial}}^3 F^o}{{{\partial}}v_2 {{\partial}}v_4 {{\partial}}s} = 0,\end{aligned}$$ that implies the properties in line . This proves the theorem for $W=E_7$.
[*Case $W = E_8$.*]{} We have $\delta=\frac{14}{15}$, $\frac{1-\delta}{2}=\frac{1}{30}$ and $(q_1,\ldots,q_8)=\left(1,\frac{4}{5},\frac{2}{3},\frac{3}{5},\frac{7}{15},\frac{2}{5},\frac{4}{15},\frac{1}{15}\right)$. Choose $\alpha=\beta=\gamma=3$. From $\frac{{{\partial}}\Lambda_{E_8}}{{{\partial}}y} = 3 y^2 + v_3 + v_5 x + v_7 x^2 + v_8 x^3$ we see that $\frac{{{\partial}}}{{{\partial}}v_3} \circ \frac{{{\partial}}}{{{\partial}}v_3} = - \frac{1}{3} v_3 \frac{{{\partial}}}{{{\partial}}v_1} -\frac{1}{3}v_5\frac{\d}{\d v_2}-\frac{1}{3}v_7\frac{\d}{\d v_4}-\frac{1}{3}v_8\frac{\d}{\d v_6}$, that implies the properties in line . We have $2q_3 = \frac{4}{3} > \frac{31}{30} = \frac{3-\delta}{2}$, implying $\frac{{{\partial}}^2 F^o}{{{\partial}}v_3^2}=\frac{{{\partial}}^3 F^o}{{{\partial}}v_3^2\d s} =0$, that completes the proof of the theorem for $W=E_8$.
For the groups $H_3$, $H_4$ and $F_4$ we are going to use the explicit formulas for the corresponding Frobenius potentials from the paper [@Zub94]. Note that these potentials are related to the ones, given by , by certain rescallings $F_W(t^1,\ldots,t^N)\mapsto F_W(\lambda_1 t^1,\ldots,\lambda_N t^N)$, $\lambda_i\in{\mathbb C}^*$, but this doesn’t affect our proof.
For the groups $F_4$ and $H_4$ the corresponding potentials, computed in [@Zub94], are $$\begin{aligned}
F_{F_4}=&\frac{t_4^{13}}{185328}+\frac{t_3^2 t_4^7}{252}+\frac{t_2^2 t_4^5}{60}+\frac{t_2 t_3^2 t_4^3}{6}+\frac{t_3^4 t_4}{12}+\frac{t_2^3 t_4}{6}+\frac{t_1^2 t_4}{2}+t_1 t_2 t_3,\\
F_{H_4}=&\frac{t_4^{31}}{245764125000}+\frac{t_3^2 t_4^{19}}{1539000}+\frac{t_3^3 t_4^{13}}{10800}+\frac{t_2^2 t_4^{11}}{4950}+\frac{t_2 t_3^2 t_4^9}{360}+\frac{t_3^4 t_4^7}{120} +\frac{t_2^2 t_3 t_4^5}{20}+\frac{t_2 t_3^3 t_4^3}{6}+\frac{t_3^5 t_4}{20}\\
&+\frac{t_2^3 t_4}{6}+\frac{t_1^2 t_4}{2} +t_1 t_2 t_3.\end{aligned}$$ Note that the equation $$\begin{gathered}
\left.\frac{\d}{\d t_2}\left(c^\mu_{2,2}\frac{\d^2 F^o}{\d t_\mu\d s}\right)\right|_{t_*=s=0}+\left.\frac{\d}{\d t_2}\left(\frac{\d^2 F^o}{\d t_2\d t_2}\frac{\d^2 F^o}{\d s^2}\right)\right|_{t_*=s=0}=\left.\frac{\d}{\d t_2}\left(\frac{\d^2 F^o}{\d t_2\d s}\frac{\d^2 F^o}{\d t_2\d s}\right)\right|_{t_*=s=0},\end{gathered}$$ where $c^\gamma_{\alpha\beta}$ are the structure constants of multiplication in the coordinates $t_\mu$, can’t be true, because $\left.c^\mu_{2,2}\right|_{t_*=0}=0$, $\frac{{{\partial}}c^\mu_{2,2}}{{{\partial}}t_2}=\delta^{\mu,1}$ and $\frac{{{\partial}}^2 F^o}{{{\partial}}t_2 {{\partial}}t_2}=0$, that follows from .
For the group $H_3$ the Frobenius manifold, computed in [@Zub94], is $$F_{H_3} = \frac{1}{2} t_1^2 t_3 + \frac{1}{2} t_1 t_2^2 +\frac{1}{20} t_2^2 t_3^5 + \frac{1}{6} t_2^3 t_3^2 + \frac{t_3^{11}}{3960}.$$ The general form of a polynomial function $F^o_{H_3}(t_1,t_2,t_3,s)$ satisfying and is $$F_{H_3}^o = s t_1 + c_9 s t_2 t_3^2 + c_8 s^3 t_2 t_3 + c_7 s^5 t_2 + c_6 s t_3^5 + c_5 s^3 t_3^4 + c_4 s^5 t_3^3 + c_3 s^7 t_3^2 + c_2 s^9 t_3 + c_1 s^{11},\quad c_k\in{\mathbb C}.$$ Suppose that it satisfies equation with $\alpha = 3$, $\beta = 2$. A direct computation shows that, applying the derivative $\frac{\d^2}{\d t_2^2}$ to both sides of it, we get $2$ on the left-hand side and $0$ on the right-hand side. This contradition proves the theorem for the case of the group $H_3$.
### Coxeter groups $A_N$, $B_N$ and $I_2(k)$ {#subsubsection:ABI}
Define $$\begin{aligned}
F^o_{B_N}(t^1,\ldots,t^N,s):=&F^o_{A_{2N-1}}(t^1,0,t^2,0,\ldots,t^{N-1},0,t^N,s),&& N\ge 2,\\
F^o_{I_2(k)}(t^1,t^2,s):=&F^o_{A_{k-1}}(t^1,0,\ldots,0,t^2,s),&& k\ge 3.\end{aligned}$$ Let $F^{o,-}_{I_2(k)}:=2t^1 s-F^o_{I_2(k)}$ and denote $F^{o,+}_{I_2(k)}:=F^o_{I_2(k)}$.
Note that if a function $F^o(t^1,\ldots,t^N,s)$ satisfies the open WDVV equations, then the function $\lambda^{-1}F^o(t^1,\ldots,t^N,\lambda s)$ also satisfies them for any $\lambda\ne 0$. Moreover, if $F^o|_{s=0}=0$, then the substitution $\left.\left(\lambda^{-1}F^o(t^1,\ldots,t^N,\lambda s)\right)\right|_{\lambda=0}$ is well defined and is a solution of the open WDVV equations.
\[theorem: An uniqueness\] Let $W$ be one of the groups $A_N$, $B_N$ or $I_2(k)$. Then all solutions $F^o$ of the open WDVV equations satisfying and are given by the family $$F^o=
\left\{
\begin{aligned}
&\lambda^{-1}F^o_{A_N}(t^1,\ldots,t^N,\lambda s),&&\lambda\in{\mathbb C}^*,&&\text{if $W=A_N$, $N\ge 2$},\\
&\lambda^{-1}F^o_{A_1}(t^1,\lambda s),&&\lambda\in{\mathbb C},&&\text{if $W=A_1$},\\
&\lambda^{-1}F^o_{B_N}(t^1,\ldots,t^N,\lambda s),&&\lambda\in{\mathbb C},&&\text{if $W=B_N$, $N\ge 2$},\\
&\lambda^{-1}F^o_{I_2(k)}(t^1,t^2,\lambda s),&&\lambda\in{\mathbb C}^*,&&\text{if $W=I_2(k)$, $k$ is odd},\\
&\lambda^{-1}F^{o,\pm}_{I_2(k)}(t^1,t^2,\lambda s),&&\lambda\in{\mathbb C},&&\text{if $W=I_2(k)$, $k$ is even}.
\end{aligned}\right.$$
[*Case $W=A_N$*]{}. We have $q_\alpha=\frac{N+2-\alpha}{N+1}$ and $\delta=\frac{N-1}{N+1}$. The case $N=1$ is obvious. Suppose that $N\ge 2$ and $F^o$ is a solution of the open WDVV equations, satisfying and . For an $n$-tuple ${{\overline{\alpha}}}=(\alpha_1,\ldots,\alpha_n)$, $1\le\alpha_i\le N$, denote $${\left<}\tau_{{\overline{\alpha}}}\sigma^k{\right>}^o={\left<}\tau_{\alpha_1}\ldots\tau_{\alpha_n}\sigma^k{\right>}^o:=\left.\frac{\d^{n+k}F^o}{\d t^\alpha_1\ldots\d t^{\alpha_n}\d s^k}\right|_{t^*=s=0}.$$ This number is non-zero only if $k=k({{\overline{\alpha}}}):=N+2-\sum_{i=1}^n(N+2-\alpha_i)$.
Note that $$\begin{gathered}
\label{eq:c2betagamma}
c^\gamma_{2,\beta}=
\begin{cases}
0,&\text{if $\gamma>\beta+1$},\\
1,&\text{if $\gamma=\beta+1$},\\
O(t^*),&\text{if $\gamma\le\beta$},
\end{cases}\end{gathered}$$ that follows from . Setting $t^*=0$ in equation with $\alpha=2$ and $2\le\beta\le N$, we get $$\begin{gathered}
{\left<}\tau_\alpha\sigma^\alpha{\right>}^o=(\alpha-1)!\left({\left<}\tau_2\sigma^2{\right>}^o\right)^{\alpha-1},\quad 2\le\alpha\le N,\qquad{\left<}\tau_2\tau_N{\right>}^o{\left<}\sigma^{N+2}{\right>}^o=N!\left({\left<}\tau_2\sigma^2{\right>}^o\right)^N.\end{gathered}$$ Differentiating equation with $\alpha=2$ and $\beta=N$ by $\frac{\d}{\d t^2}$ and setting $t^*=s=0$, we get $-1+{\left<}\tau_2\tau_N{\right>}^o{\left<}\tau_2\sigma^2{\right>}^o=0$, where we use formula for the numbers ${\left<}\tau_{\alpha_1}\tau_{\alpha_2}\tau_{\alpha_3}\tau_{\alpha_4}{\right>}_{A_N}$. We see that ${\left<}\tau_2\sigma^2{\right>}^o\ne 0$ and $${\left<}\sigma^{N+2}{\right>}^o=N!\left({\left<}\tau_2\sigma^2{\right>}^o\right)^{N+1}\ne 0.$$
After the rescaling $F^o(t^1,\ldots,t^N,s)\mapsto\lambda^{-1}F^o(t^1,\ldots,t^N,\lambda s)$ with an appropriate constant $\lambda\ne 0$ we get ${\left<}\tau_2\sigma^2{\right>}^o=1$ and, therefore, $$\begin{gathered}
{\left<}\tau_\alpha\sigma^\alpha{\right>}^o=(\alpha-1)!={\left<}\tau_\alpha\sigma^\alpha{\right>}^o_{A_N},\quad 1\le\alpha\le N,\qquad{\left<}\sigma^{N+2}{\right>}^o=N!={\left<}\sigma^{N+2}{\right>}^o_{A_N}.\end{gathered}$$
Consider now an $n$-tuple ${{\overline{\alpha}}}=(\alpha_1,\ldots,\alpha_n)$, $1\le\alpha_i\le N$, with $n\ge 2$ and $k({{\overline{\alpha}}})\ge 0$. Differentiating equation with $\alpha=\alpha_1$ and $\beta=\alpha_2$ by $\frac{\d^{n-2}}{\d t^{\alpha_3}\ldots\d t^{\alpha_n}}$ and setting $t^*=0$, we get the recursion $$\begin{gathered}
\label{eq:recursion for AN}
\frac{{\left<}\tau_{{{\overline{\alpha}}}}\sigma^{k({{\overline{\alpha}}})}{\right>}^o}{k({{\overline{\alpha}}})!}=\sum_{\substack{I\sqcup J=\{1,\ldots,n\}\\1\in I,\,2\in J}}\frac{{\left<}\tau_{{{\overline{\alpha}}}_I}\sigma^{k({{\overline{\alpha}}}_I)}{\right>}^o{\left<}\tau_{{{\overline{\alpha}}}_J}\sigma^{k({{\overline{\alpha}}}_J)}{\right>}^o}{(k({{\overline{\alpha}}}_I)-1)!(k({{\overline{\alpha}}}_J)-1)!}-\sum_{\substack{I\sqcup J=\{1,\ldots,n\}\\1,2\in I,\,J\ne\emptyset}}\frac{{\left<}\tau_{{{\overline{\alpha}}}_I}\sigma^{k({{\overline{\alpha}}}_I)}{\right>}^o{\left<}\tau_{{{\overline{\alpha}}}_J}\sigma^{k({{\overline{\alpha}}}_J)}{\right>}^o}{k({{\overline{\alpha}}}_I)!(k({{\overline{\alpha}}}_J)-2)!},\end{gathered}$$ where for a subset $I=\{i_1,\ldots,i_{|I|}\}\subset\{1,\ldots,n\}$, $i_1<\ldots<i_{|I|}$, we denote ${{\overline{\alpha}}}_I:=(\alpha_{i_1},\ldots,\alpha_{i_{|I|}})$. The correlators ${\left<}\cdot{\right>}_{A_N}$ don’t appear in this recursion because for any subset $I\subset\{1,\ldots,n\}$ and an index $1\le\mu\le N$ we have $$\begin{gathered}
\sum_{i\in I}(N+2-\alpha_i)+(N+2-\mu)\le\sum_{i=1}^n(N+2-\alpha_i)+(N+2-\mu)=\\
=2N+4-k({{\overline{\alpha}}})-\mu<2N+4\end{gathered}$$ and, therefore, ${\left<}\tau_{{{\overline{\alpha}}}_I}\tau_\mu{\right>}_{A_N}=0$. The recursion determines all the numbers ${\left<}\tau_{{{\overline{\alpha}}}}\sigma^{k({{\overline{\alpha}}})}{\right>}^o$ starting from the numbers ${\left<}\sigma^{N+2}{\right>}^o$ and ${\left<}\tau_\alpha\sigma^\alpha{\right>}^o$. So we conclude that $F^o=F^o_{A_N}$.
[*Case $W=B_N$*]{}. We have $q_\alpha=\frac{N+1-\alpha}{N}$ and $\delta=\frac{N-1}{N}$. The function $F^o_{B_N}$ satisfies the open WDVV equations together with equations and , because, as one can easily check using the quasi-homogeneity of the function $F^o_{A_{2N-1}}$, the correlator ${\left<}\tau_{\alpha_1}\ldots\tau_{\alpha_n}\tau_\mu{\right>}_{A_{2N-1}}$ vanishes, if all the $\alpha_i$’s are odd and $\mu$ is even.
Suppose that $F^o$ is a solution of the open WDVV equations, satisfying and . Since $F_{B_2}=F_{I_2(4)}$, we will consider the $B_2$-case together with the cases $W=I_2(k)$ later. So we assume that $N\ge 3$. Note that a correlator ${\left<}\tau_{\alpha_1}\ldots\tau_{\alpha_n}\sigma^k{\right>}^o$ vanishes unless $\sum_{i=1}^N(N+1-\alpha_i)+\frac{k}{2}=N+\frac{1}{2}$. Setting $t^*=0$ in equation with $\alpha=2$, we get the relations $$\begin{aligned}
{\left<}\tau_\alpha\sigma^{2\alpha-1}{\right>}^o=&\frac{(2\alpha-2)!}{2^{\alpha-1}}\left({\left<}\tau_2\sigma^3{\right>}^o\right)^{\alpha-1},\quad 2\le\alpha\le N,\\
{\left<}\tau_2\tau_N\sigma{\right>}^o{\left<}\sigma^{2N+1}{\right>}^o=&\frac{(2N-1)!}{2^N}\left({\left<}\tau_2\sigma^3{\right>}^o\right)^N.\end{aligned}$$ Differentiating equation with $\alpha=2$ and $\beta=N-1$ by $\frac{\d}{\d t^2}$ and setting $t^*=s=0$, we get ${\left<}\tau_2^2\tau_{N-1}\tau_N{\right>}_{B_N}+{\left<}\tau_2\tau_N\sigma{\right>}^o=0$. Since, by , ${\left<}\tau_2^2\tau_{N-1}\tau_N{\right>}_{B_N}=-1$, we conclude that ${\left<}\tau_2\tau_N\sigma{\right>}^o=1$ and $${\left<}\sigma^{2N+1}{\right>}^o=\frac{(2N-1)!}{2^N}\left({\left<}\tau_2\sigma^3{\right>}^o\right)^N.$$
Suppose that ${\left<}\tau_2\sigma^3{\right>}^o\ne 0$, then ${\left<}\sigma^{2N+1}{\right>}^o\ne 0$. After the rescaling $F^o(t^1,\ldots,t^N,s)\mapsto\lambda^{-1}F^o(t^1,\ldots,t^N,\lambda s)$ with an appropriate constant $\lambda\ne 0$ we get ${\left<}\tau_\alpha\sigma^{2\alpha-1}{\right>}^o={\left<}\tau_\alpha\sigma^{2\alpha-1}{\right>}^o_{B_N}$ and ${\left<}\sigma^{2N+1}{\right>}^o={\left<}\sigma^{2N+1}{\right>}^o_{B_N}$. In the same way, as in the $A_N$-case, there is a recursion similar to , that reconstructs all the correlators ${\left<}\tau_{\alpha_1}\ldots\tau_{\alpha_n}\sigma^k{\right>}^o$ with $n\ge 2$. Therefore, $F^o=F^o_{B_N}$.
Suppose that ${\left<}\tau_2\sigma^3{\right>}^o=0$, then ${\left<}\sigma^{2N+1}{\right>}^o=0$. Consider the decomposition $$F^o=\sum_{k=0}^N P_k(t^1,\ldots,t^N)s^{2k+1},\quad P_k\in{\mathbb C}[t^1,\ldots,t^N].$$ Consider an index $l$ such that $P_l\ne 0$ and $P_{>l}=0$. Since $l<N$, the polynomial $P_l$ can’t be a constant. Suppose $l>0$, then equation implies that $$\frac{\d P_l}{\d t^\alpha}\frac{\d P_l}{\d t^\beta}=\frac{2l}{2l+1}P_l\frac{\d^2 P_l}{\d t^\alpha\d t^\beta},\quad 1\le\alpha,\beta\le N.$$ The space of solutions of the differential equation $(f')^2=\frac{2l}{2l+1}ff''$ for a function $f=f(x)$ is formed by the family $f=C_1(x+C_2)^{-2l}$, $C_1,C_2\in{\mathbb C}^*$, together with the constant solution $f=C$, $C\in{\mathbb C}$. Since $P_l$ is a non-constant polynomial, we come to a contradition. Therefore, $l=0$.
In this case system is equivalent to the system $$c^\gamma_{\alpha\beta}\frac{\d P_0}{\d t^\gamma}=\frac{\d P_0}{\d t^\alpha}\frac{\d P_0}{\d t^\beta},\quad 1\le\alpha,\beta\le N.$$ For $\alpha=2$ we get the relations $$\frac{\d P_0}{\d t^{\beta+1}}+\sum_{1\le\gamma\le\beta}c^\gamma_{2,\beta}\frac{\d P_0}{\d t^\gamma}=\frac{\d P_0}{\d t^2}\frac{\d P_0}{\d t^\beta},\quad 2\le\beta\le N-1,$$ that recursively determine all the derivatives $\frac{\d P_0}{\d t^\beta}$ starting from the derivatives $\frac{\d P_0}{\d t^2}=t^N$ and $\frac{\d P_0}{\d t^1}=1$. This completely determines the polynomial $P_0$. We conclude that $F^o=\left.\left(\lambda^{-1}F^o_{B_N}(t^1,\ldots,t^N,\lambda s)\right)\right|_{\lambda=0}$.
[*Case $W=I_2(k)$*]{}. We have $q_1=1$, $q_2=\frac{2}{k}$, $\delta=\frac{k-2}{k}$ and $F_{I_2(k)}=\frac{(t^1)^2t^2}{2}+\alpha_k\frac{(t^2)^{k+1}}{(k+1)!}$, $\alpha_k\ne 0$. The function $F^o_{I_2(k)}$ satisfies the open WDVV equations together with equations and , because ${\left<}\tau_{\alpha_1}\ldots\tau_{\alpha_n}\tau_\mu{\right>}_{A_{k-1}}=0$, if $\alpha_i\in\{1,k-1\}$ and $\mu\notin\{1,k-1\}$ [@Zub94 Section 1].
Note that if a function $F^o$ satisfies property , then all the open WDVV equations are automatically satisfied except equation with $\alpha=\beta=2$.
Suppose $k=2l+1$, $l\ge 1$. A polynomial $F^o(t^1,t^2,s)$, satisfying and , has the form $$F^o=t^1 s+\sum_{i=0}^{l+1}\beta_i\frac{s^{2l+2-2i}}{(2l+2-2i)!}\frac{(t^2)^i}{i!},\quad\beta_i\in{\mathbb C}.$$ Suppose that the open WDVV equations are satisfied. Equation with $\alpha=\beta=2$ is equivalent to $$\begin{aligned}
&\frac{\d^2 F_{I_2(2l+1)}}{\d(t^2)^3}+\frac{\d^2 F^o}{\d(t^2)^2}\frac{\d^2 F^o}{\d s^2}-\left(\frac{\d^2 F^o}{\d t^2\d s}\right)^2=0\Leftrightarrow\\
\Leftrightarrow & \alpha_{2l+1}\frac{(t^2)^{2l-1}}{(2l-1)!}+\left(\sum_{i=2}^{l+1}\beta_i\frac{s^{2l+2-2i}}{(2l+2-2i)!}\frac{(t^2)^{i-2}}{(i-2)!}\right)\left(\sum_{i=0}^l\beta_i\frac{s^{2l-2i}}{(2l-2i)!}\frac{(t^2)^i}{i!}\right)\\
&-\left(\sum_{i=1}^l\beta_i\frac{s^{2l+1-2i}}{(2l+1-2i)!}\frac{(t^2)^{i-1}}{(i-1)!}\right)^2=0.\end{aligned}$$ The expression on the left-hand side of the last equation has the form $\sum_{i=0}^{2l-1}(t^2)^{2l-1-i}s^{2i}P_i(\beta_0,\ldots,\beta_{l+1})$, where $$\begin{gathered}
P_0=\frac{\alpha_{2l+1}}{(2l-1)!}+\frac{\beta_{l+1}\beta_l}{(l-1)!l!},\qquad P_i=\frac{\beta_{l+1}\beta_{l-i}}{(l-1)!(l-i)!(2i)!}+Q_i(\beta_{l-i+1},\ldots,\beta_{l+1}),\quad 1\le i\le l,\end{gathered}$$ and $Q_i$ are polynomials in $\beta_{l-i+1},\ldots,\beta_{l+1}$. We see that $\beta_{l+1}\ne 0$ and the equations $P_i=0$, $0\le i\le l$, determine the coefficients $\beta_0,\ldots,\beta_l$ in terms of the coefficient $\beta_{l+1}$. Thus, $F^o=\lambda^{-1}F^o_{I_2(2l+1)}(t^1,t^2,\lambda s)$ for some $\lambda\ne 0$.
Suppose $k=2l$, $l\ge 2$, and a polynomial $F^o$ satisfies the open WDVV equations together with equations and . Then $F^o$ has the form $$F^o=t^1 s+\sum_{i=0}^l\beta_i\frac{s^{2l+1-2i}}{(2l+1-2i)!}\frac{(t^2)^i}{i!},\quad\beta_i\in{\mathbb C},$$ and equation with $\alpha=\beta=2$ is equivalent to $$\begin{aligned}
& \alpha_{2l}\frac{(t^2)^{2l-2}}{(2l-2)!}+\left(\sum_{i=2}^l\beta_i\frac{s^{2l+1-2i}}{(2l+1-2i)!}\frac{(t^2)^{i-2}}{(i-2)!}\right)\left(\sum_{i=0}^{l-1}\beta_i\frac{s^{2l-1-2i}}{(2l-1-2i)!}\frac{(t^2)^i}{i!}\right)\\
&-\left(\sum_{i=1}^l\beta_i\frac{s^{2l-2i}}{(2l-2i)!}\frac{(t^2)^{i-1}}{(i-1)!}\right)^2=0.\end{aligned}$$ The expression on the left-hand side has the form $\sum_{i=0}^{2l-2}(t^2)^{2l-2-i}s^{2i}P_i(\beta_0,\ldots,\beta_l)$, where $$\begin{aligned}
P_0=&\frac{\alpha_{2l}}{(2l-2)!}-\frac{\beta_l^2}{((l-1)!)^2},\\
P_i=&\gamma_i\beta_l\beta_{l-i}+Q_i(\beta_{l-i+1},\ldots,\beta_l),\quad \gamma_i=\frac{2l(i-1)}{(l-1)!(2i)!(l-i)!},\quad 1\le i\le l,\end{aligned}$$ and $Q_i$ are polynomials in $\beta_{l-i+1},\ldots,\beta_l$. We see that the equation $P_0=0$ determines $\beta_l$ up to a sign and then the equations $P_i=0$, $2\le i\le l$, determine the coefficients $\beta_0,\ldots,\beta_{l-2}$ in terms of $\beta_{l-1}$. Thus, $F^o=\lambda^{-1}F^{o,\pm}_{I_2(2l)}(t^1,t^2,\lambda s)$ for some $\lambda$.
[BCT18]{}
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[^1]: We follow the convention of B. Dubrovin and use variables with lower indices for the flat coordinates in particular examples.
| 0 |
---
author:
- Walter Dehnen
title: 'A Hierarchical ${\cal O}(N)$ Force Calculation Algorithm'
---
Introduction {#sec:intro}
============
In $N$-body simulations of stellar dynamics (or any other dynamics incorporating long-range forces), the computation, at every time step, of the gravitational forces between $N$ mutually interacting bodies dominates the operational effort. In many situations, such as studies of collisionless stellar systems, the error of these simulations is dominated by the noise in the distribution of the bodies, whose number $N$ is just a numerical parameter. Therefore, instead of computing the forces exactly by direct summation over all pairs of bodies, one may use approximate but much faster methods, allowing substantially larger $N$ and hence significantly reduced noise.
In stellar dynamics, one of the most commonly used approximate methods is the Barnes & Hut tree code [@BH] and its clones. With these methods, one first sorts the bodies into a hierarchical tree of cubic cells and pre-computes multipole moments of each cell. Next, the force at any body’s position and generated by the contents of some cell is calculated by a multipole expansion if the cell is well-separated from the body; otherwise the forces generated by the cell’s child nodes are taken. This technique reduces the number of interactions per body to ${\cal O}(\log N)$ and hence results in an overall complexity of ${\cal O}(N\log N)$.
Another technique used frequently, for instance in molecular dynamics, is Greengard & Rokhlin’s [@FMMa; @FMMa1] fast multipole method (FMM) and its variants. Traditionally, these methods first sort the bodies into a hierarchy of nested grids, pre-compute multipole moments of each cell, and then compute the forces between grid cells by a multipole expansion, usually in spherical harmonics. That is, cells are not only sources but also sinks, which formally reduces the complexity to ${\cal O}(N)$, although the author is not aware of an empirical demonstration in three dimensions[^1].
Currently, no useful implementation of FMM for application in stellar dynamics exists. In fact, it has been shown [@CM] that, for this purpose, the FMM in its traditional form cannot compete with the tree code. The reason, presumably, is two-fold: first, because stellar systems are very inhomogeneous, non-adaptive methods are less useful; second FMM codes are traditionally designed for high accuracy, whereas in collisionless stellar dynamics a relative force error of few $10^{-3}$ is often sufficient.
Here, we describe in detail a new code, designed for application in the low-accuracy regime, that combines the tree code and FMM whereby taking the better of each. In order to be fully adaptive, we use a hierarchical tree of cubic cells. The force is calculated employing [*mutual*]{} cell-cell interactions, in which both cells are source and sink simultaneously. Whether a given cell-cell interaction can be executed or must be split, is decided using an improved multipole-acceptance criterion (MAC). The new code is a further development of the code presented in [@Da], which in turn may be considered an improvement of a code given in [@WS]. It is substantially faster than the tree code and empirically shows a complexity of ${\cal O}(N)$ or even less.
The paper is organized as follows. In [§\[sec:grav\]]{}, the numerical concepts are presented; [§\[sec:alg\]]{} describes the algorithm; in [§\[sec:assess\]]{} the force errors are empirically assessed for some typical stellar dynamical test cases; [§\[sec:perform\]]{} presents empirical measurements of the complexity and performance, also in comparison to other methods; finally, [§\[sec:summ\]]{} sums up and concludes.
Approximating Gravity {#sec:grav}
=====================
The goal is to approximately compute the gravitational potential $\Phi$ and its derivative, the acceleration, at all body positions ${{\mbox{\boldmath$x$}}}_i$ and generated by all other $N-1$ bodies $$\label{potential}
\Phi({{\mbox{\boldmath$x$}}}_i) =-\sum_{j\neq
i}\mu_j\,g({{\mbox{\boldmath$x$}}}_i-{{\mbox{\boldmath$x$}}}_j),$$ where $\mu_i$ is the weight of the $i$th body.
Consider two cells A and B, see Figure \[fig:cells\], with centers of mass ${{\mbox{\boldmath$z$}}}_{{\rm A}}$ and ${{\mbox{\boldmath$z$}}}_{{\rm B}}$, respectively. The Greens function describing the mutual interaction between a body at position ${{\mbox{\boldmath$x$}}}$ in cell A and a body at position ${{\mbox{\boldmath$y$}}}$ in cell B may be Taylor expanded about the separation ${{\mbox{\boldmath$R$}}}\equiv{{\mbox{\boldmath$z$}}}_{{\rm A}}-{{\mbox{\boldmath$z$}}}_{{\rm B}}$: $$\label{Taylor}
g({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$y$}}}) = \sum_{n=0}^p {1\over n!}\,({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$y$}}}-{{\mbox{\boldmath$R$}}})^{(n)}
\odot{\mbox{\boldmath$\nabla$}}^{(n)}g({{\mbox{\boldmath$R$}}})\;+\;{\cal R}_p(g),$$ where $p$ is the order of the expansion, while ${\cal R}_p$ denotes the Taylor series remainder (see also Appendix A). Here, we follow Warren & Salmon [@WS] by using the notational shorthand in which ${{\mbox{\boldmath$x$}}}^{(n)}$ indicates the $n$-fold outer product of the vector ${{\mbox{\boldmath$x$}}}$ with itself, while $\odot$ denotes a tensor inner product. When inserting [(\[Taylor\])]{} into [(\[potential\])]{}, whereby restricting the sum over $j$ to all bodies within cell B, we obtain for the potential at every position ${{\mbox{\boldmath$x$}}}$ in cell A and generated by all bodies in cell B [@WS] $$\begin{aligned}
\label{local-expn}
\Phi_{{{\rm B}}\to{{\rm A}}}({{\mbox{\boldmath$x$}}}) &=&-\sum_{m=0}^p {1\over m!}\,({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$z$}}}_{{\rm A}})^{(m)}
\odot{{\mbox{\boldmath$\sf C$}}^{m,p}}_{{{\rm B}}\to{{\rm A}}}
\;+\;{\cal R}_p(\Phi_{{{\rm B}}\to{{\rm A}}})
\\[1ex] \label{coeffs}
{{\mbox{\boldmath$\sf C$}}^{m,p}}_{{{\rm B}}\to{{\rm A}}} &=& \sum_{n=0}^{p-m} {(-1)^n\over n!}\;{\mbox{\boldmath$\nabla$}}^{(n+m)}
g({{\mbox{\boldmath$R$}}}) \odot {{\mbox{\boldmath$\sf M$}}^{n}}_{{\rm B}},
\\[1ex] \label{multipoles}
{{\mbox{\boldmath$\sf M$}}^{n}}_{{\rm B}}&=& \sum_{{\mbox{\boldmath$\scriptstyley$}}_i\in{{\rm B}}} \mu_i\,({{\mbox{\boldmath$y$}}}_i-{{\mbox{\boldmath$z$}}}_{{\rm B}})^{(n)}.\end{aligned}$$ The summation over $m$ in [equation [(\[local-expn\])]{}]{} represents the evaluation of gravity, represented by the field tensors ${{\mbox{\boldmath$\sf C$}}^{m,p}}_{{{\rm B}}\to{{\rm A}}}$, at the evaluation point ${{\mbox{\boldmath$x$}}}$ within the sink cell A. The computation of the field tensors via the summation over $n$ in [equation [(\[coeffs\])]{}]{} represents the interaction between sink cell A and source cell B, represented by its multipole moments ${{\mbox{\boldmath$\sf M$}}^{n}}_{{\rm B}}$.
The symmetry between and of the Taylor expansion [(\[Taylor\])]{} has the important consequence that, if [equations [(\[local-expn\])]{}]{} and [(\[coeffs\])]{} are used to compute ${\mbox{\boldmath$\nabla$}}\Phi_{{{\rm B}}\to{{\rm A}}}$ [*and*]{} ${\mbox{\boldmath$\nabla$}}\Phi_{{{\rm A}}\to{{\rm B}}}$, Newton’s third law is satisfied by construction. For instance, the sum over all forces of $N$ bodies vanishes within floating point accuracy.
Note that the highest-order multipole moments, ${{\mbox{\boldmath$\sf M$}}^{n=p}}$, contribute only to the coefficients ${{\mbox{\boldmath$\sf C$}}^{0,p}}$, and hence affect only $\Phi$ but not ${\mbox{\boldmath$\nabla$}}\Phi$. Since, in stellar dynamics the acceleration rather than the potential is to be computed, one may well ignore ${{\mbox{\boldmath$\sf M$}}^{p}}$, reducing CPU-time and memory requirements.
The formulae used in the standard tree code can be obtained by setting ${{\mbox{\boldmath$x$}}}={{\mbox{\boldmath$z$}}}_{{\rm A}}$, corresponding to body sinks. In this case, potential and acceleration are approximated by $-{{\mbox{\boldmath$\sf C$}}^{0,p}}_{{{\rm B}}\to{{\rm A}}}$ and ${{\mbox{\boldmath$\sf C$}}^{1,p}}_{{{\rm B}}\to{{\rm A}}}$, respectively.
Gravity Between Well-Separated Cells {#sec:hard}
------------------------------------
In our implementation, we stick to a third order expansion ($p=3)$, whereby ignoring octopoles ${{\mbox{\boldmath$\sf M$}}^{3}}_{{\rm B}}$ (see remark above). The dipole ${{\mbox{\boldmath$\sf M$}}^{1}}_{{\rm B}}$ vanishes by construction and the Taylor-series coefficients for spherical Greens functions read (with Einstein’s sum convention) $$\label{hard-coeffs}
\begin{array}{lcl}
{{\sf C}^{0}}_{{{\rm B}}\to{{\rm A}}} &=& {{\sf M}^{}}_{{\rm B}}\big[D^0
+ {{\textstyle{1\over2}}}{\tilde{\sf M}{}^{2}}_{{{\rm B}}ii}\,D^1
+ {{\textstyle{1\over2}}}R_iR_j{\tilde{\sf M}{}^{2}}_{{{\rm B}}ij}D^2
\big], \\[1ex]
{{\sf C}^{1}}_{{{\rm B}}\to{{\rm A}},\,i} &=& {{\sf M}^{}}_{{\rm B}}\big[R_i\big(D^1
+ {{\textstyle{1\over2}}}{\tilde{\sf M}{}^{2}}_{{{\rm B}}jj}D^2
+ {{\textstyle{1\over2}}}R_jR_k{\tilde{\sf M}{}^{2}}_{{{\rm B}}jk}D^3\big)
+ R_j{\tilde{\sf M}{}^{2}}_{{{\rm B}}ij}D^2
\big], \\[1ex]
{{\sf C}^{2}}_{{{\rm B}}\to{{\rm A}},\,ij} &=& {{\sf M}^{}}_{{\rm B}}\big[
\delta_{ij}\,D^1+ R_iR_j\,D^2
\big], \\[1ex]
{{\sf C}^{3}}_{{{\rm B}}\to{{\rm A}},\,ijk} &=& {{\sf M}^{}}_{{\rm B}}\big[\big(
\delta_{ij}R_k+\delta_{jk}R_i
+\delta_{ki}R_j\big)D^2
+R_iR_jR_kD^3\big],
\end{array}$$ where ${{\sf M}^{}}_{{\rm B}}\equiv{{\sf M}^{0}}_{{\rm B}}$ is the mass of cell B and ${\tilde{\sf M}{}^{2}}_{{\rm B}}\equiv{{\sf M}^{2}}_{{\rm B}}/{{\sf M}^{0}}_{{\rm B}}$ its [*specific*]{} quadrupole moment, while $$\label{Dn}
D^m\equiv \left.
\left({1\over r}{\partial\over\partial r}\right)^m
g(r) \right|_{r=|{\mbox{\boldmath$\scriptstyleR$}}|}.$$ In practice, we calculate the coefficients ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{m}}_{{{\rm B}}\to{{\rm A}}}
\equiv{{\sf M}^{}}_{{\rm A}}{{\mbox{\boldmath$\sf C$}}^{m}}_{{{\rm B}}\to{{\rm A}}}$, because these obey the symmetry relations ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{2}}_{{{\rm B}}\to{{\rm A}}}= {\tilde{{\mbox{\boldmath$\sf C$}}}{}^{2}}_{{{\rm A}}\to{{\rm B}}}$ and ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{3}}_{{{\rm B}}\to{{\rm A}}}=-{\tilde{{\mbox{\boldmath$\sf C$}}}{}^{3}}_{{{\rm A}}\to{{\rm B}}}$ (for $p=3$), which arise from the mutuality of gravity. Since we consider always both directions of any interaction, exploiting these relations substantially reduces the operational effort.
Accumulating Taylor Coefficients {#sec:accum}
--------------------------------
After, for each cell, the coefficients of all its interactions have been accumulated as ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{m}}_{{\rm A}}= \sum_{{\rm B}}{\tilde{{\mbox{\boldmath$\sf C$}}}{}^{m}}_{{{\rm B}}\to{{\rm A}}}$, where the sum includes all interaction partners B of cell A, we transform back to ${{\mbox{\boldmath$\sf C$}}^{m}}_{{\rm A}}={\tilde{{\mbox{\boldmath$\sf C$}}}{}^{m}}_{{\rm A}}/{{\sf M}^{}}_{{\rm A}}$. Next, for each body, the Taylor series of all relevant cells (those that contain the body), have to be accumulated by first translating to a common expansion center and then adding coefficients.
In contrast to expansions in spherical harmonics, the translation of the Cartesian expansion [(\[local-expn\])]{} to a different center ${{\mbox{\boldmath$z$}}}$ is straightforward. Let ${{\mbox{\boldmath$\sf C$}}^{m,p}}_0$ be the coefficients for expansion center ${{\mbox{\boldmath$z$}}}_0$, then the coefficients for expansion center ${{\mbox{\boldmath$z$}}}_1$ are $$\label{translate}
{{\mbox{\boldmath$\sf C$}}^{m,p}}_1 = \sum_{n=0}^{p-m} {1\over n!}\,({{\mbox{\boldmath$z$}}}_0-{{\mbox{\boldmath$z$}}}_1)^{(n)} \odot
{{\mbox{\boldmath$\sf C$}}^{m+n,p}}_0.$$
The Multipole Acceptance Criterion (MAC) {#sec:mac}
----------------------------------------
For the expansion [(\[local-expn\])]{} to converge, we must have $|{{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$y$}}}-{{\mbox{\boldmath$R$}}}|
<|{{\mbox{\boldmath$R$}}}|$ for all body-body interactions ‘caught’ by a single cell-cell (or cell-body) interaction (see Appendix A). In order to ensure this, we first obtain, for each node, an upper limit $r_{\rm max}$ for the distance of any body within the node from the center of mass (bodies naturally have $r_{\rm max}=0$). We take $r_{\rm max}$ to be either the distance from the cell’s center of mass to its most distant corner [@SW], or [@Benz] $$\max_{{\rm child\;nodes}\;i}\{r_{i,\rm max}+|{{\mbox{\boldmath$z$}}}-{{\mbox{\boldmath$z$}}}_i|\},$$ whatever is smaller. Then $|{{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$y$}}}-{{\mbox{\boldmath$R$}}}|<\theta|{{\mbox{\boldmath$R$}}}|\;\forall\;{{\mbox{\boldmath$x$}}}\in{{\rm A}}\;{\rm
and}\;{{\mbox{\boldmath$y$}}}\in{{\rm B}}$, i.e. the nodes are [*well-separated*]{}, if $$\label{well-separated}
|{{\mbox{\boldmath$R$}}}| > r_{\rm A,crit} + r_{\rm B,crit}
\quad{\rm with}\quad
r_{\rm crit} = r_{\rm max} / \theta.$$ The [*tolerance parameter*]{} $\theta$ controls the accuracy of the approximation: for Newtonian forces, the error made in $d$ dimensions by the $p$th order expansion of the form [(\[local-expn\])]{} is ([equation [(\[error:acc\])]{}]{}) $$\begin{aligned}
\label{error:rel}
|{\cal R}_p({\mbox{\boldmath$\nabla$}}\Phi_{{{\rm B}}\to{{\rm A}}})| &\le&
{(p+1)\theta^p\over(1-\theta)^2}\;{{{\sf M}^{}}_{{\rm B}}\over R^2}
\\ \label{error:abs} &\propto&
{\theta^{p+2}\over(1-\theta)^2}\;r_{\rm B,max}^{d-2}
\propto
{\theta^{p+2}\over(1-\theta)^2}\;{{{\sf M}^{}}_{{\rm B}}}^{(d-2)/d}\end{aligned}$$ where ${{\sf M}^{}}_{{\rm B}}\propto r_{\rm B,max}^d$ has been assumed. Since ${{\sf M}^{}}_{{\rm B}}/R^2$ is, to lowest order, the acceleration due to the interaction, [equation [(\[error:rel\])]{}]{} tells us that with constant $\theta$, which is standard tree-code practice, the [*relative*]{} error introduced by every single interaction is approximately constant. [Equation [(\[error:abs\])]{}]{}, however, shows that for the most important case of $d=3$ this practice results in larger [*absolute*]{} errors for interactions with bigger and hence on average more massive cells. This implies that these interactions will dominate the [*total*]{} error of any body’s acceleration. It is, therfore, expedient to balance the absolute errors of the individual interactions, which can be approximately achieved by using a mass-dependent tolerance parameter as follows $$\label{open:mass}
{\theta^{p+2}\over(1-\theta)^2} =
{\theta_{\rm min}^{p+2}\over(1-\theta_{\rm min})^2}
\left({{{\sf M}^{}}\over{{\sf M}^{}}_{\rm tot}}\right)^{(2-d)/d}$$ where ${{\sf M}^{}}_{\rm tot}$ is the mass of the root cell, while $\theta_{\rm min}=
\theta({{\sf M}^{}}_{\rm tot})$ is the new tolerance parameter. If $r_{\rm A,max}\sim
r_{\rm B,max}$ and ${{\sf M}^{}}_{{\rm A}}\sim{{\sf M}^{}}_{{\rm B}}$, this method results in approximately constant absolute acceleration errors. Note that [equation [(\[open:mass\])]{}]{} results in a very weak decrease of $\theta$ with increasing mass: $\theta\propto{{\sf M}^{-1/15}}$ for $d=3$, $p=3$ and $\theta\ll1$.
Instead of [equation [(\[error:abs\])]{}]{}, one can obtain a stricter error limit incorporating the first $p+1$ multipole moments (see Appendix A), which may be turned into an MAC [@WS]. However, our choice [(\[open:mass\])]{} is (i) much simpler and (ii) overcomes already the main disadvantage of $\theta=\,$const, the variations of the absolute individual errors.
Direct Summation {#sec:sum}
----------------
For small $N$ the exact force computation via direct summation is not only more accurate than approximate methods but also more efficient. Therefore, we replace the approximate technique by direct summation whenever the latter results in higher accuracy at the same efficiency, see Appendix B for more details.
If an interaction is executed by direct summation, the Taylor coefficients of the interacting cell(s) are not affected, but the coefficients ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{0}}$ and ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{1}}$ of the bodies within the cell(s) are accumulated.
The Algorithm {#sec:alg}
=============
Tree Building and Preparation {#sec:build}
-----------------------------
In the first stage, a hierarchical tree of cubic cells is build, as described in [@BH], albeit cells containing $s$ or less bodies are not divided. Next, the tree is linked such that every cell holds the number of cell children[^2], a pointer to its first cell child, as well as the number of body children, of all body descendants, and a pointer to the first body child. Since cell as well as body children are contiguous in memory (to arrange this we actually use tiny copies of the bodies, called ‘souls’, that only hold a pointer to their body and its mass, position, acceleration and potential) these data allow fast and easy access to all cell and body children of any given cell and to all bodies contained in it.
Next, the masses ${{\sf M}^{}}$, centers of mass ${{\mbox{\boldmath$z$}}}$, radii $r_{\rm max}$ and $r_{\rm
crit}$, and specific quadrupole moments ${\tilde{\sf M}{}^{2}}$ of each cell are computed in a recursive way from the properties of the child nodes.
A new Generic Tree-Walk Algorithm {#sec:walk}
---------------------------------
One important new feature of our code is the mutual treatment of all interactions: both interacting nodes are source and sink simultaneously. The standard tree-walk, as for instance implemented by the generic algorithm given in [@WS], as well as the the usual FMM coefficient accumulation algorithm, e.g. in [@FMMb], contain an inherent asymmetry between sinks and sources, and thus cannot be used for our purposes. Instead, our algorithm approximates the forces in two steps: an [*interaction phase*]{}, incorporating [equation [(\[coeffs\])]{}]{}, and an [*evaluation phase*]{}, incorporating [equation [(\[local-expn\])]{}]{}.
### The Interaction Phase {#sec:iact}
Because of the mutuality of the interactions, we cannot accumulate the Taylor coefficients ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{n}}$ ‘on the walk’, but each node must accumulate the coefficients of all its interactions in its own private memory. Cells need storage for ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{0}}$ to ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{3}}$, while bodies only need to accumulate ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{0}}$ and ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{1}}$, i.e. potential and acceleration. The accumulation of these coefficients is done by the following algorithm with the root cell for arguments $A$ and $B$.
try to perform the interaction between[node]{}s $A$ and $B$;(it cannot be performed) [**if**]{} ($A=B$) [**for**]{}(all pairs $\{a,b\}$ of child[node]{}s of $A$) [Interact]{}($a,b$); [**else if**]{} ($r_{\rm max}(A) > r_{\rm max}(B)$) [**for**]{}(all child[node]{}s $a$ of $A$) [Interact]{}($a,B$); [**else**]{} [**for**]{}(all child[node]{}s $b$ of $B$) [Interact]{}($A,b$);
Thus, if an interaction cannot be executed, using the formulae of the last section – see Appendix B for details – it is split. In case of a mutual interaction, the bigger node is divided and up to eight new mutual interactions are created, while a self interaction of a cell results in up to 36 new interactions between its child nodes. In practice, we use a non-recursive code incorporating a stack of interactions.
### The Evaluation Phase {#sec:eval}
Finally, the Taylor coefficients relevant for each body are accumulated and the expansion is evaluated at every body’s position. After transforming ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{}}$ to ${{\mbox{\boldmath$\sf C$}}^{}}$ for every cell and body, this is done by the following recursive algorithm, which is initially called with the root cell and an empty Taylor series as arguments.
$T_A$ =[Taylor series]{} due to the ${{\mbox{\boldmath$\sf C$}}^{n}}$ of[cell]{} $A$; translate center of $T_0$ to center of mass of $A$; $T_A$ += $T_0$; (all[body]{} children of $A$) { evaluate $T_A$ at[body]{}’s position; add to[body]{}’s potential and acceleration; } [**for**]{}(all[cell]{} children $C$ of $A$) [Evaluate Gravity]{}($C$, $T_A$);
Thus, the coefficients of the Taylor-series that is eventually evaluated at some body’s position have been added up from all hierarchies of the tree and hence account for all interactions of all cells that contain the body.
Error Assessment {#sec:assess}
================
Two types of force errors are involved in collisionless $N$-body simulations of stellar dynamics. One is the unavoidable error between the smooth force field of the underlying stellar system modeled and the forces [*estimated*]{} from the positions of $N$ bodies (which are sampled from this stellar system). This [ *estimation error*]{} can be reduced by increasing the number $N$ of bodies in conjunction with a careful [*softening*]{}: the Newtonian Greens function $g({{\mbox{\boldmath$x$}}})=G/|{{\mbox{\boldmath$x$}}}|$ is replaced with a non-singular function that approaches the Newtonian form for $|{{\mbox{\boldmath$x$}}}|$ larger than the softening length $\epsilon$. But at fixed $N$, it cannot be decreased below a certain optimum value [@Merr; @Db].
The other type of error is introduced by an approximate rather than exact computation of these estimated forces. While this [*approximation error*]{} can be reduced to (almost) any size (at the price of increasing computational effort), it is sufficient to reduce it well below the level of the estimation error. We will now first assess the approximation error alone and then consider the total error.
Accuracy of the Approximation {#sec:accur}
-----------------------------
We estimate the accuracy of the approximated forces for various choices of the parameters controlling the algorithm and for two typical astrophysical situations: a spherical Hernquist [@Hern] model galaxy, which has density and force per unit mass $$\label{hernquist}
\rho({{\mbox{\boldmath$x$}}}) = {{\sf M}_{\rm tot} \over 2\pi}
{r_0\over |{{\mbox{\boldmath$x$}}}| \big(r_0+|{{\mbox{\boldmath$x$}}}|\big)^3},\qquad
{{\mbox{\boldmath$F$}}}({{\mbox{\boldmath$x$}}}) = - {{{\mbox{\boldmath$x$}}}\over|{{\mbox{\boldmath$x$}}}|}
{ G{\sf M}_{\rm tot}\over\big(r_0+|{{\mbox{\boldmath$x$}}}|\big)^2},$$ and a group of five such galaxies at different positions and with various scale radii $r_0$. In either case, we sample $N=10^4$, $10^5$, and $10^6$ bodies. We use standard Plummer softening, where $g(r)=G/\sqrt{r^2+\epsilon^2}$, with softening lengths $\epsilon$ chosen such as to minimize the estimation error [@Db]. In order to single out the approximation error, we compare with the forces obtained by a computation via direct summation (in double precision; in case of $N=10^6$ for the first $10^5$ bodies only). As measure for the relative error, we compute for each body [@CM] $$\label{error}
\varepsilon \equiv |a_{\rm approx} - a_{\rm direct}| / a_{\rm direct},$$ where $a$ denotes the magnitude of the acceleration. [Figure \[fig:error\]]{} plots the mean relative error, $\overline\varepsilon$, and that at the 99th percentile, $\varepsilon_{99\%}$, versus the CPU time needed by an ordinary PC (Pentium III/933MHz/Linux/compiler: gcc version 2.95.2) for both constant and mass-dependent tolerance parameter. This figure allows several interesting observations.
1. For the same $\theta=\,$const and stellar system, the errors decrease with increasing $N$. This is because, at constant [*relative*]{} force error per individual interaction (as is the case for $\theta=\,$const), the total error of some body’s force scales with the inverse square root of the number of individual interactions contributing, which increases with $N$.
2. At the same operational effort, as measured by the CPU time, the mass-dependent tolerance parameter employing [equation [(\[open:mass\])]{}]{} results in smaller errors than $\theta=\,$const, for $N>10^4$. This advantage becomes more pronounced for larger $N$, because, with $\theta=\,$const, the [ *absolute*]{} force errors of individual interactions contributing to some body’s force vary stronger with increasing $N$ (due to the larger range of cell masses), such that balancing them becomes more beneficial.
3. Finally, a relative error of $\varepsilon_{99\%}$ of a few per cent or $\overline\varepsilon$ of a few $10^{-3}$ at $N=10^5$, which is commonly accepted to be sufficient in astrophysical contexts [@CM], requires a tolerance parameter $\theta=\,$const$\,\simeq0.65$ or $\theta_{\rm
min}\simeq0.5$.
The Total Force Error {#sec:error}
---------------------
The important question here is for which choices of $\theta$ is the approximation error negligible compared to the estimation error? To answer this question, we have performed some experiments using samples of $N=10^4$, $10^5$ and $10^6$ bodies drawn from a Hernquist model and computed the mean averaged squared error (MASE) of the force (per unit mass) [@AT]: $$\label{mase}
{\rm MASE}({{\mbox{\boldmath$F$}}}) = \left\{{1\over{\sf M_{\rm tot}}} \sum_i \mu_i
\big(\hat{{{\mbox{\boldmath$F$}}}}_i - {{\mbox{\boldmath$F$}}}({{\mbox{\boldmath$x$}}}_i)\big)^2 \right\}.$$ Here, $\hat{{{\mbox{\boldmath$F$}}}}_i$ and ${{\mbox{\boldmath$F$}}}({{\mbox{\boldmath$x$}}})$ are, respectively, the approximately estimated force for the $i$th body and the true force field of the stellar system. The curly brackets denote the ensemble average over many possible random realizations of the same underlying stellar density by $N$ bodies. We used $10^7/N$ ensembles and computed the MASE$({{\mbox{\boldmath$F$}}})$[^3] for various values of $\theta$, but always at optimum $\epsilon$ [@Db]. As one might already have guessed from the behavior of the approximation error in [Figure \[fig:error\]]{}, the relative increase of the MASE$({{\mbox{\boldmath$F$}}})$ is negligible: even for $\theta=0.9$, the approximation error contributes less than one per cent, in agreement with the findings of [@AT]. Based on this result, one may advocate the usage of tolerance parameters larger than $\theta_{\rm
min}\simeq0.5$. However, the distribution of approximation errors is not normal, and the rms error, which is essentially measured by (the square root of) the MASE$({{\mbox{\boldmath$F$}}})$, may well underestimate the danger of using large $\theta$. We therefore cannot recommend using $\theta_{\rm min}{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}}0.7$.
Performance Tests {#sec:perform}
=================
Scaling with $N$ {#sec:scaling}
----------------
We measured the CPU time consumption for both constant $\theta=0.65$ and $\theta=\theta({{\sf M}^{}})$ with $\theta_{\rm min}=0.5$, using $s=6$ in either case. [Figure \[fig:N\]]{} plots the consumed time (averaged over many experiments) per body versus $N$ plotted on a logarithmic scale. For the case of $\theta=\theta({{\sf M}^{}})$, [Table \[tab:N\]]{} gives the number of cells as well as the number of individual body-body (B-B), cell-body (C-B) and cell-cell (C-C) interactions, where the latter two are split into those done via a Taylor expansion and direct summation (subscripts ‘app’ and ‘dir’), respectively.
[@rrrrrrrr]{} $N$ & $N_{\rm cells}$ & B-B & C-B$_{\rm app}$ & C-B$_{\rm dir}$ & C-C$_{\rm app}$ & C-C$_{\rm dir}$ & total 1000 & 370 & 210 & 3746 & 2049 & 4057 & 1600 & 11662 3000 & 1090 & 472 & 16870 & 5184 & 30101 & 4472 & 57099 10000 & 3558 & 753 & 61873 & 11914 & 163554 & 11887 & 249981 30000 & 10607 & 494 & 170634 & 24984 & 572322 & 20102 & 790474 100000 & 35282 & 112 & 429035 & 73350 & 1954508 & 53466 & 2516146 300000 & 106065 & 66 & 1041836 & 205821 & 5457445 & 137138 & 6859114 1000000 & 353342 & 1 & 2918326 & 621802 & 16105065 & 393311 & 19023391 3000000 & 1060650 & 2 & 7771584 & 1689115 & 42974890 & 1047627 & 53645379
The tree code requires ${\cal O}(N\log N)$ operations, corresponding to a rising straight line in [Fig. \[fig:N\]]{}. For our code, however, there is a turn-over at $N\sim10^4$, above which the CPU time per body approaches a constant[^4] (for $\theta=\,$const) or even decreases with $N$ (for $\theta=\theta({{\sf M}^{}})$), i.e. the total number of operations becomes ${\cal O}(N)$ or less, which is also evident from the number of interactions in [Table \[tab:N\]]{}.
In order to understand these scalings, let us first consider the simpler case of $\theta=\,$const and a homogeneous distribution of bodies. Then, eight-folding $N$ is equivalent to arranging eight copies of the old root cell into the octants of the new root cell [@BH], and the total number of interactions rises from $N_I$ to $8N_I+N_+$, where $N_+$ interactions are needed for the mutual forces between these octants. In terms of a differential equation, this gives $$\label{ode}
{{\rm d} N_I\over{\rm d} N}
\simeq {N_I\over N} {\Delta\ln N_I\over\Delta\ln N}
\approx {N_I\over N} + {N_+\over N8\ln 8},$$ where the first term on the right-hand side accounts for the [ *intra-domain*]{} and the second for the [*inter-domain*]{} interactions. [Equation [(\[ode\])]{}]{} has solution $$\label{sol}
N_I \simeq c_0 N + {N\over8\ln8} \int {N_+\over N^2}\;{\rm d} N.$$ In the tree code, every body requires a constant number of additional interactions, i.e. $N_+\propto N$, and the second term in [(\[sol\])]{} becomes $\propto N\ln N$ dominating $N_I(N)$. However, in the new code, $N_+$ grows sub-linear for large $N$, since a constant number of cell-cell interactions accounts for most new interactions of all bodies. In this case, the second term on the right-hand side of [equation [(\[sol\])]{}]{} also grows sub-linear with $N$ and $N_I$ will eventually be dominated by the first term. That is, in contrast to the tree code, the inter-domain interactions are neglible at large $N$ when compared to the intra-domain interactions. The transition value of $N$ will depend on the tolerance parameter $\theta$ and the distribution of bodies.
Another way of estimating the scaling of the computational costs with $N$ is similar to the FMM approach [@FMMa; @FMMb]: On each level $l$ of the tree there are $\sim8^l$ cells of mass ${{\sf M}^{}}\sim8^{-l}$, i.e. $n({{\sf M}^{}})\propto{\sf
M}^{-2}$. Each cell has $\propto\theta^{-3}$ interactions, and thus $$\label{scaling}
N_I \propto \int { d{{\sf M}^{}}\over{\sf M}^{2}\theta^3}.$$ Hence, for $\theta=\,$const, $N_I\propto1/{\sf M}_{\rm min}\propto N$, while a shallower scaling results if $\theta({{\sf M}^{}})$ increases towards smaller masses. Empirically we find for $N>30000$ that the CPU time used by the force computation alone (without tree building) is very well fit by the power-law $\propto N^{0.929\pm0.001}$.
Comparison with Other Methods used in Stellar Dynamics {#sec:comp}
------------------------------------------------------
For various computational techniques used in stellar dynamics, [Figure \[fig:var\]]{} plots $N$ versus the CPU time consumption normalized by $N$, both on logarithmic scales. An ordinary direct method running on a general-purpose computer is slower than our code for any $N{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}}100$. The [GRAPE-5]{} system [@GRAPE] obtains a $\sim100$ times higher performance by wiring elementary gravity into special-purpose hardware and, for $N{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}10^4$, is faster than any other method.
The new code presented here is the fastest method running solely on general-purpose computers and the only one faster than ${\cal O}(N\log N)$. In particular, it out-performs the popular tree code by a factor of 10 and more. At $30000{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}N{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}3\times10^7$, the only technique that requires less CPU time is a combination of the tree code with a [GRAPE-5]{} board [@GRAPE; @G-tree]. Here, the speed-up due to the usage of special-purpose hardware does not quite reach that for direct methods, because of tree-building and other overheads that cannot be done on the [GRAPE]{} board.
Comparison with Fast Multipole Methods {#sec:fmm}
--------------------------------------
Because our code relies on cell-cell interactions, it may be considered a variant of FMM, introduced by Greengard & Rokhlin [@FMMa; @FMMa1]. However, it differs in several ways from most implementations of FMM. (i) the expansions are centered on the cells’ centers of mass instead of the geometrical centers; (ii) a Cartesian Taylor series is used instead of an expansion in spherical harmonics; (iii) the interaction partners are determined by a multipole acceptance criterion rather than by their mutual grid position; (iv) the mutuality of interactions is fully exploited; (v) the expansion order $p$ is fixed and the accuracy controlled by the parameter $\theta$.
[@rrrlrrl]{} & & & & & & 20000 &13.3 & 233 &$7.9\cdot10^{-4}$ & 0.97 & 136 &$3.7\cdot10^{-4}$ 50000 &27.7 & 1483 &$5.2\cdot10^{-4}$ & 2.64 & 924 &$3.3\cdot10^{-4}$ 200000 &158 & 24330 &$8.4\cdot10^{-4}$ &10.77 & 14694 &$3.4\cdot10^{-4}$ 500000 &268 &138380 &$7.0\cdot10^{-4}$ &29.42 & 91134 &$3.7\cdot10^{-4}$ 1000000 &655 &563900 &$7.1\cdot10^{-4}$ &58.34 &366218 &$3.5\cdot10^{-4}$
To assess the effect of these differences, we compared our code directly with the 3D adaptive FMM code by Cheng, Greengard & Rokhlin [@FMMb]. We performed a test identical to one reported by these authors ($N$ bodies randomly distributed in a cube) on an identical computer (a Sun UltraSPARC with 167MHz) using the same error measure (eq. (57) of [@FMMb]): $$E = \left[\sum_i(\Phi_i - \tilde\Phi_i)^2 \Big/ \sum_i \Phi_i^2\right]^{1/2}$$ where $\Phi$ and $\tilde\Phi$ are the potential computed by direct summation and the approximate method, respectively (unfortunately, Cheng et al. do not give the more relevant error of the accelerations). [Table \[tab:compare\]]{} gives, in the last three columns, the CPU time ($T_{\rm approx}$) in seconds and error $E$ for our code with $\theta=1$ and $s=6$ as well as the time needed for direct summation in 64bit precision ($T_{\rm direct}$); columns 2-4 report the data from Table I of [@FMMb]. On average our code is faster by more than a factor of ten and twice as accurate (even though we compromised the approximation of the potential by omitting the octopole contributions).
What causes this enormous difference? Since for the direct summation the timings are much more similar, we can rule out differences in hardware, compiler, etc. as cause. An important clue is the fact that our code cannot compete with the Cheng et al. FMM in the regime $E{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}10^{-6}$. While our code tries to reach this goal by decreasing $\theta$, FMM obtains it by increasing $p$. In general, the accuracy as well as the performance are controlled by both the order $p$ and the choice of interaction partners, parameterized in our code by $\theta$. Hence, maximal efficiency at given accuracy is obtained at a unique choice of $(p,\theta)$. Apparently, low orders $p$ are optimal for low accuracies, while high accuracies are most efficiently obtained with high orders, instead of an increasing number of interactions (decreasing $\theta$).
Thus, traditional FMM is less useful in the low-accuracy regime, such as needed in stellar dynamics, in agreement with earlier findings [@CM], and our code may be called a variant of FMM optimized for low-accuracy. Clearly, however, a code for which $p$ and $\theta$ can be adapted simultaneously would be superior to both.
Discussion and Summary {#sec:summ}
======================
Our code for the approximate computation of mutual long-range forces between $N$ bodies extends the traditional Barnes & Hut [@BH] tree code by including cell-cell interactions, similar to fast multipole methods (FMM). However, unlike most implementations of FMM, our code is optimized for comparably low-accuracy, which is sufficient in stellar dynamical applications ([§\[sec:fmm\]]{}).
As a unique feature, our code exploits the mutual character of gravity: both nodes of any interaction are sink and source simultaneously. This results in exact conservation of Newton’s third law and substantially reduces the computational effort, but requires a novel tree-walking algorithm ([§\[sec:walk\]]{}) which preserves the natural symmetry of each interaction. Note that the generic algorithm given in [§\[sec:walk\]]{} is not restricted to long-range force approximations, but can be used for any task that incorporates mutuality, for instance neighbor and collision-partner searching.
Complexity {#sec:scal}
----------
The new code requires ${\cal O}(N)$ or less operations ([Fig. \[fig:N\]]{}) for the approximate computation of the forces of $N$ mutually gravitating bodies. For stellar dynamicists, this is the first competitive code better than ${\cal
O}(N\log N)$. A complexity of ${\cal O}(N)$ was expected for methods based on cell sinks (implying cell-cell interactions) [@FMMa; @FMMb; @WS], but, to my knowledge, hardly ever shown empirically in three dimensions.
Our code obtains a complexity of less than ${\cal O}(N)$ by employing a mass-dependent tolerance parameter $\theta$. The traditional $\theta=\,$constant results in equal [*relative*]{} errors of each interaction, such that the total error of any body’s force is dominated by the interactions with the most massive cells. By slightly increasing the tolerance parameter for less massive cells, we obtain (approximately) equal [*absolute*]{} errors, resulting in a lower total force error than the traditional method for the same number of interactions. Thus, at the same error, we require less interactions. The additional interactions, arising when increasing $N$, occur at ever less massive cells and hence at ever larger tolerance parameters. This causes the computational costs to rise sub-linear with $N{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}}10^4$ (depending on the accuracy requirements).
Performance {#sec:per}
-----------
We have shown that on general-purpose computers our code out-performs any competitor code commonly used in the field of stellar dynamics. A recent adaptive 3D implementation of FMM [@FMMb] is also out-performed by a factor of ten ([§\[sec:fmm\]]{}), which is related to the fact that traditional FMM codes appear to be good only in the high-accuracy regime. The code presented here was optimized for low accuracies, but by increasing the expansion order $p$, one can easily obtain a version suitable for high accuracies, and it remains to be seen how it would perform compared to traditional FMM.
Currently, the only faster method appears to be a [GRAPE]{}-supported tree code [@G-tree], which uses the special-purpose [GRAPE]{} hardware [@GRAPE]. Unfortunately, unlike the tree code, our code cannot be combined with the current [GRAPE]{} hardware. There are, however, no conceptual obstacles against hard-wiring [equations [(\[hard-coeffs\])]{}]{} into special-purpose hardware, which should yield a speed-up comparable to that of tree to [GRAPE]{} tree, i.e. a factor $\sim50$.
Publication of the Code {#sec:pub}
-----------------------
Our code is written in [C++]{}, also includes a purely two-dimensional version (not described here), and is linkable to [C]{} and [FORTRAN]{} programs. The code will be is publicly available from the author upon request.
A full $N$-body code based on this force algorithm is available under the [ NEMO]{} [@nemo] package (http://bima.astro.umd.edu/nemo). Here, we give an error estimate for the Taylor series approximation of gravity. Using the integral form of the Taylor series remainder, we find for the remainder in [equation [(\[Taylor\])]{}]{}, with ${\mbox{\boldmath$\Delta$}}\equiv({{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$z$}}}_{{\rm A}})-({{\mbox{\boldmath$y$}}}-{{\mbox{\boldmath$z$}}}_{{\rm B}})=
{{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$y$}}}-{{\mbox{\boldmath$R$}}}$, $$\begin{aligned}
\label{remain:pot}
{\cal R}_p(g) &=& {{\mbox{\boldmath$\Delta$}}^{(p+1)}\over p!}\,\odot
\int_0^1 dt\,(1-t)^p\,{\mbox{\boldmath$\nabla$}}^{(p+1)} g({{\mbox{\boldmath$R$}}}+{\mbox{\boldmath$\Delta$}} t)
\\[1ex] \label{remain:acc}
{\mbox{\boldmath$\nabla$}}{\cal R}_p(g) &=& {{\mbox{\boldmath$\Delta$}}^{(p)}\over(p-1)!}\,\odot
\int_0^1 dt\,(1-t)^{p-1}\,{\mbox{\boldmath$\nabla$}}^{(p+1)} g({{\mbox{\boldmath$R$}}}+{\mbox{\boldmath$\Delta$}} t),\end{aligned}$$ For Newtonian gravity, $$\begin{aligned}
\left\| {1\over p!}\int_0^1dt\,(1-t)^p\,
{\mbox{\boldmath$\nabla$}}^{(p+1)}g({{\mbox{\boldmath$R$}}}+{\mbox{\boldmath$\Delta$}} t) \right\|
&\le& {1\over(R-|{\mbox{\boldmath$\Delta$}}|)\,R^{p+1}}, \\[1ex]
\left\| {1\over (p-1)!}\int_0^1dt\,(1-t)^{p-1}\,
{\mbox{\boldmath$\nabla$}}^{(p+1)}g({{\mbox{\boldmath$R$}}}+{\mbox{\boldmath$\Delta$}} t) \right\|
&\le& {(p+1)R - p|{\mbox{\boldmath$\Delta$}}| \over (R-|{\mbox{\boldmath$\Delta$}}|)^2\,R^{p+1}}.\end{aligned}$$ For the summation over the source cell, we find $$\Big\| \sum_{{\mbox{\boldmath$\scriptstyley$}}_i\in{{\rm B}}} \mu_i {\mbox{\boldmath$\Delta$}}^{(p)} \Big\|
\le \sum_{k=0}^p {p\choose k} r_{\rm A,max}^k\,\big\|{{\mbox{\boldmath$\sf M$}}^{(p-k)}}_{{\rm B}}\big\|
\le (r_{\rm A,max}+r_{\rm B,max})^p {{\sf M}^{}}_{{\rm B}}$$ with ${{\sf M}^{}}_{{\rm B}}$ the mass of cell B. With $\theta>(r_{\rm A,max}+r_{\rm
B,max})/R$, we finally get $$\begin{aligned}
\label{error:pot}
|{\cal R}_p(\Phi_{{{\rm B}}\to{{\rm A}}})| &\le&
{\theta^{p+1}\over1-\theta} {{{\sf M}^{}}_{{\rm B}}\over R} \\ \label{error:acc}
|{\cal R}_p({\mbox{\boldmath$\nabla$}}\Phi_{{{\rm B}}\to{{\rm A}}})| &\le&
{(p+1)\theta^p\over(1-\theta)^2} {{{\sf M}^{}}_{{\rm B}}\over R^2}.\end{aligned}$$
Here, we give the interaction details for [Algorithm 1]{} in [§\[sec:iact\]]{}. Mutual body-body interactions are done by elementary gravity, while body self-interactions are ignored. Mutual interactions between nodes containing $N_1$ and $N_2$ bodies are treated as follows.
1. If $N_1 N_2 < N_{\rm nn}^{\rm pre}$, execute the interaction by direct summation; otherwise,
2. if the interaction is well-separated, execute it using [equations [(\[hard-coeffs\])]{}]{}; otherwise,
3. if $N_1 N_2 < N_{\rm nn}^{\rm post}$, execute the interaction by direct summation; otherwise,
4. the interaction cannot be executed, but must be split.
Here, $N_{\rm nn}^{\rm pre}$ and $N_{\rm nn}^{\rm post}$ have different values depending whether it is a cell-body (cb) or cell-cell (cc) interaction (see below). Cell self-interactions are done slightly differently:
1. If $N_1 < N_{\rm cs}$, execute the interaction by direct summation; otherwise,
2. the interaction cannot be executed, but must be split.
The numbers $N_{\rm cb}^{\rm pre}$, $N_{\rm cb}^{\rm post}$, $N_{\rm cc}^{\rm
pre}$, $N_{\rm cc}^{\rm post}$, and $N_{\rm cs}$ determine the usage of direct summation. After some experiments, I found the following values to result in most efficient code at given accuracy (but this certainly depends on the implementation details). $$\begin{array}{lcrlcrlcr}
N_{\rm cb}^{\rm pre} &=& 3,&
N_{\rm cc}^{\rm pre} &=& 0,& \\
N_{\rm cb}^{\rm post} &=&128,&
N_{\rm cc}^{\rm post} &=& 16,&
N_{\rm cs} &=& 64.
\end{array}$$ Note that cell-body interactions with as many as 128 bodies in the cell hardly ever occur, since the interaction algorithm favors interactions between roughly equally sized nodes. Thus, cell-body interactions will almost always be executed.
I thank J. Makino from Tokoy University for helpful discussions and R. Ibata and C. Pichon from Strasbourg Observatory for hardware support.
J.E. Barnes and P. Hut, A hierarchical ${\cal O}(N\log N)$ force calculation algorithm, [*Nature*]{} [**324**]{}, 446 (1986) L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, [*J. Comput. Phys.*]{} [**73**]{}, 325 (1987) L. Greengard and V. Rokhlin, A new version of the fast multipole method for the Laplace equation in three dimensions, [*Acta Numerica*]{} [**6**]{}, 229 (1997) H. Cheng. L. Greengard, and V. Rokhlin, A fast adaptive multipole algorithm in three dimensions, [*J. Comput. Phys.*]{} [ **155**]{}, 468 (1999) R. Capuzzo-Colcetta and P. Miocchi, A comparison between the fast multipole algorithm and the tree code to evaluate gravitational forces in 3D, [*J. Comput. Phys.*]{} [**143**]{}, 29 (1998) W. Dehnen, A very fast and momentum-conserving tree code, [*Astrophys. J.*]{} [**536**]{}, L39 (2000) M.S. Warren, and J.K. Salmon, A portable parallel particle program, [*Comput. Phys. Comm.*]{} [**87**]{}, 266 (1995) J.K. Salmon, and M.S. Warren, Skeletons from the Treecode Closet, [*J. Comput. Phys.*]{} [**111**]{}, 136 (1994) W. Benz, R.L. Bowers, A.Q.W. Cameron, and W.H. Press, Dynamic mass exchange in doubly degenerate binaries I. 0.9 and 1.2 $M_\odot$ stars, [*Astrophys. J.*]{} [**348**]{}, 647 (1990) D. Merritt, Optimal Smoothing for N-body Codes, [*Astron. J.*]{} [**111**]{}, 2462 (1996) W. Dehnen, Towards optimal softening in three-dimensional N-body codes - I. minimizing the force error [*Mon. Not. Royal Astron. Soc.*]{} [**324**]{}, 273 (2001), L. Hernquist, An analytic model for spherical galaxies and bulges, [*Astrophys. J.*]{} [**356**]{}, 359 (1990) E. Athanassoula, E. Fady, J.C. Lambert, and A. Bosma, Optimal softening for force calculations in collisionless N-body simulations, [*Mon. Not. Royal Astron. Soc.*]{} [**314**]{}, 475 (2000) A. Kawai, T. Fukushige, J. Makino, and M. Taiji, GRAPE-5: a special purpose computer for N-body simulations, [*Publ. Astron. Soc. Japan*]{} [**52**]{}, 659 (2000) J. Makino, A tree code with special purpose processor, [ *Publ. Astron. Soc. Japan*]{} [**43**]{}, 621 (1991) P.J. Teuben, The Stellar Dynamics Toolbox NEMO, in Astronomical Data Analysis Software and Systems IV, ed. R. Shaw, H.E. Payne and J.J.E. Hayes, PASP Conf Series [**77**]{}, 398 (1995)
[^1]: One of the most recent members of the FMM family, presented by Cheng, Greengard & Rokhlin [@FMMb], does still not show clear ${\cal O}(N)$ behavior at $N=10^6$ (see, e.g., table I of their paper).
[^2]: Hereafter ‘child’ means a direct sub-node of a cell, while ‘descendant’ refers to any node contained within a cell, including the children, the grand-children and so on.
[^3]: The force field of the Hernquist model has a central singularity, causing a 100% force error at $r=0$, which cannot be resolved by $N$-body methods. In our experiments, we have therefore restricted the summation in [equation [(\[mase\])]{}]{} to $|{{\mbox{\boldmath$x$}}}_i|>\epsilon/2$.
[^4]: The slow rise can be entirely attributed to the tree-building, which is an ${\cal O}(N\log N)$ process.
| 0 |
---
abstract: 'Multiple bases are presented for the conclusion that potentials are fundamental in electrodynamics, with electric and magnetic fields as quantities auxiliary to the scalar and vector potentials – opposite to the conventional ordering. One foundation for the concept of basic potentials and auxiliary fields consists of examples where two sets of gauge-related fields are such that one is physical and the other is erroneous, with the information for the proper choice supplied by the potentials. A major consequence is that a change of gauge is not a unitary transformation in quantum mechanics; a principle heretofore unchallenged. The primacy of potentials over fields leads to the concept of a hierarchy of physical quantities, where potentials and energies are primary, while fields and forces are secondary. Secondary quantities provide less information than do primary quantities. Some criteria by which strong laser fields are judged are based on secondary quantities, making it possible to arrive at inappropriate conclusions. This is exemplified by several field-related misconceptions as diverse as the behavior of charged particles in very low frequency propagating fields, and the fundamental problem of pair production at very high intensities. In each case, an approach based on potentials gives appropriate results, free of ambiguities. The examples encompass classical and quantum phenomena, in relativistic and nonrelativistic conditions. This is a major extension of the quantum-only Aharonov-Bohm effect, both in supporting the primacy of potentials over fields, and also in showing how field-based conceptions can lead to errors in basic applications.'
author:
- 'H. R. Reiss'
title: Ascendancy of potentials over fields in electrodynamics
---
Introduction
============
For most of the history of exploring electromagnetic phenomena, it had been believed that knowledge of the electric and magnetic fields in a physical problem is sufficient to define the problem. The scalar and vector potentials, whose spatial and temporal derivatives yield the fields, had been regarded as auxiliary quantities that are useful but not essential. This conclusion was apparently reinforced by the fact that the set of potentials to represent the fields is not unique. Subject to modest restrictions, there exist transformations (called gauge transformations) to other sets of potentials that produce the same fields.
This seemingly straightforward situation was upset by the Aharonov-Bohm effect [@es; @ab]. The simplest realization of this phenomenon is that an electron beam passing outside a solenoid containing a magnetic field will be deflected, even though there is no field outside the solenoid. There is, however, a potential outside the solenoid that suffices to explain the deflection. The effect remained controversial until it was verified experimentally [@tonomura]. This has the basic consequence that potentials are more fundamental than fields. The Aharonov-Bohm effect is founded on a single explicitly quantum-mechanical phenomenon, and commentary about its significance has been in terms of quantum mechanics [@furry; @vaidman].
The concept explored here is different from that of the Aharonov-Bohm effect, and much more consequential. It is shown that a change of gauge can introduce a violation of basic symmetries, even when the usual constraints on allowable gauge transformations have been satisfied. Furthermore, these symmetry violations can occur in classical physics as well as in quantum mechanics with external electromagnetic fields. The consequences of these results are profound. There exist contrasting sets of potentials that yield exactly the same fields, but where one set is consonant with physical requirements but another is not. This proves directly that the selection of the proper set of potentials is the decisive matter, since the predicted fields are the same in both cases. A corollary is that gauge transformations are not unitary transformations. This contradicts the field-based assumption that gauge transformations must preserve the values of measurable quantities. The assumption of unitarity (often implicit) underlies some of the influential articles that have been published on the subject of gauge choice.
Examples employed here to demonstrate the primacy of potentials – a charged particle in interaction with a constant electric field, and a bound electron subjected to a plane-wave field – represent basic physics problems, unlike the narrow specificity of the Aharonov-Bohm effect. An important feature revealed by these examples is that, although the physical consequences of static or quasistatic-electric (QSE) fields are quite similar to plane-wave effects at the low field intensities that exist in the usual atomic, molecular and optical (AMO) physics processes, at the high field intensities now achievable with laser fields, they can be profoundly different. These differences have yet to be fully appreciated in the AMO literature, leading to misconceptions that persist nearly forty years after the first laboratory observation [@ati] of explicit intense-field effects.
A concept introduced here is that of a hierarchy of physical quantities. Since potentials are primary and fields are secondary, it follows that energies are primary and forces are secondary. This ranking resolves the long-standing mystery about why the Schrödinger equation cannot be written directly in terms of electric and magnetic fields, even though the fields were conventionally assumed to be basic physical quantities. All attempts to express the Schrödinger equation directly in terms of fields have resulted in nonlocality [@mandelstam; @dewitt; @belinfante; @levy; @rohrlich; @priou]. This apparent anomaly is one of the enduring puzzles of quantum mechanics. A hierarchy of physical quantities also serves to clarify the current confused situation in the strong-laser community, where field-based intensity measures are employed that are inconsistent with energy-based criteria. One important example of this misdirection is the introduction of the concept of the critical electric field [@sauter; @schwinger] into the discussion of strong laser effects, despite the fact that lasers produce transverse fields and the critical field has well-defined meaning only for longitudinal fields. The basic differences between transverse fields and longitudinal fields are also exhibited in the macroscopic world in terms of the properties of extremely-low-frequency radio waves.
The range of applicability of the concepts examined here is very large, since it encompasses classical electromagnetism, and also relativistic and nonrelativistic quantum mechanics in which the electromagnetic field is regarded as an external classical field.
The limitation to external classical fields is significant, since it places the present work outside the scope of a Yang-Mills theory [@yangmills]. Quantum electrodynamics (QED) is a Yang-Mills theory, but standard QED does not incorporate strong-field theory. Strong-field theories contain an apparent intensity-dependent mass shift, discovered independently by Sengupta [@sengupta] and by the present author [@hrdiss; @hr62]. This mass shift can be explained in terms of a demand for covariance in external fields [@hrup]. The mass shift is a fundamental phenomenon in strong-field physics [@hrje], but it does not exist in the context of the quantized fields of QED [@hrdiss; @hr62; @jehr].
Section II below discusses two basic examples that exhibit pairs of potential choices that describe exactly the same fields, but where one set of potentials is physically acceptable and the other is not. One example is the simplest possible case: the classical interaction of a charged particle with a constant electric field. Of the two possibilities for gauge choice, one contradicts Noether’s Theorem [@noether]. There is no such problem with the alternative gauge. The next example is the interaction of a charged particle with a plane-wave field, such as the field of a laser. In this case, the key factor is that the symmetry principle in question – preservation of the propagation property of a plane-wave field – is not often mentioned, even in the context of very strong fields where this symmetry is crucial [@hrup]. The demand for the preservation of the propagation property imposes a strong limitation on possible gauge transformations. In the presence of a simultaneous scalar interaction, like a Coulomb binding potential, only the radiation gauge is possible [@hrgge]. This limitation to a unique gauge exists in both classical and quantum domains. An important aspect of this problem is that the widely-used dipole approximation in the description of laser-caused effects suppresses this symmetry, thus masking the errors that follow from ignoring this basic property. Both the constant-electric-field and the propagating-field examples admit of only one possible gauge. This lack of gauge-equivalent alternatives is extremely important, since both situations represent commonplace physical environments. This is in contrast to the specialized Aharonov-Bohm effect.
An immediate consequence of the demonstrated fact that some nominally valid gauge transformations can have unphysical consequences is that a gauge transformation is not a unitary transformation. This is discussed in Section III, where it is shown to be related to the construction of exact transition amplitudes.
The impossibility of writing the Schrödinger equation directly in terms of electric and magnetic fields is discussed in Section IV. This is further evidence of the basic nature of potentials, and it also supports the notion of a hierarchy of physical phenomena. Quantum mechanics can be constructed from classical mechanics when expressed in terms of system functions like the Lagrangian or Hamiltonian, whereas a Newtonian form of classical mechanics has such an extension only by extrapolation to a desired result. System functions are related to energies, whereas Newtonian physics involves forces, and forces are directly connected with electric and magnetic fields, as shown by the Lorentz force expression.
The ambiguities inherent in the view that the $\mathbf{E}^{2}-\mathbf{B}^{2}$ and $\mathbf{E}\cdot\mathbf{B}$ Lorentz invariants reliably characterize the electrodynamic environment is another topic examined in Section IV. (The Lorentz invariants, as are all electromagnetic quantities throughout this paper, are stated in Gaussian units.) This concept has an important failure when both invariants are zero, since it associates propagating plane-wave fields with the completely different constant crossed fields. The commonly-held assumption that constant crossed fields are a zero-frequency limit of plane-wave fields (see, for example, Refs. [@nikrit66; @ritus85]), is shown to be untenable.
Another topic in Section IV is the apparent dominance of the electric component of the Lorentz force expression at low frequencies, a field-related conception that draws attention away from the rising importance of the magnetic component of a propagating field as the frequency declines [@hr75; @hr82]. Inappropriate emphasis on the electric field has caused conceptual errors even in relativistic phenomena, as discussed in Section IV in the context of vacuum pair production. A potentials-related approach obviates this electric-field-dominance hazard. The concept of the *critical field* is often mentioned in connection with strong-laser interactions [@arb; @ssb]. The critical field refers to that value of electric field at which spontaneous pair production from the vacuum becomes significant. It has been applied to laser fields in terms of the electric component of a plane-wave field. This is devoid of meaning for laser beams in vacuum because pure electric fields and plane-wave fields are disjoint concepts, as is evident from Section II. The conservation conditions applicable to critical-field considerations cannot be satisfied by a laser. Even were the electric component of a laser field equal to the critical field, pair production cannot occur because pair production from the vacuum by a laser pulse cannot occur unless there is a counter-propagating field to provide the necessary conservation of momentum [@hrdiss; @hr62; @hr71; @burke]. The fact that photons convey momentum is incompatible with the concept of a critical electric field for laser-induced processes.
Section IV concludes with the practical problem of communicating with submerged submarines. This has been done under circumstances that emphasize how different plane-wave fields are from QSE fields
Section V explores the notion of a hierarchy of physical quantities. Potentials are directly related to energies, so they are identified as primary quantities. Fields are derived from potentials, so they are secondary. Forces are determined by fields and so forces are also secondary. The hierarchy concept is related to classical mechanics in that Newtonian physics is couched in terms of forces, and so it is secondary to versions of classical mechanics based on energy-based system functions like the Lagrangian and Hamiltonian. Mechanics formulated with system functions infer Newtonian mechanics, but the converse is not true.
Symmetry violation
==================
The two examples presented have an important qualitative difference. The first example – a constant electric field – is so elementary that the proper choice of potentials is obvious, and there is no motivation to explore the properties of the symmetry-violating alternative potentials. The next example is quite different in that the improper choice of potentials is very attractive to a laser-physics community that is accustomed to the dipole approximation. The requisite propagation property never appears within the dipole approximation, and its violation is thereby invisible.
A preliminary step is to introduce the units and conventions employed in this article, and to add some general remarks about terminology.
Units and conventions
---------------------
Gaussian units are employed for all electromagnetic quantities. The expressions for the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ in terms of the scalar potential $\phi$ and the 3-vector potential $\mathbf{A}$ are$$\mathbf{E}=-\mathbf{\nabla}\phi\mathbf{-}\frac{1}{c}\partial_{t}\mathbf{A},\qquad\mathbf{B}=\mathbf{\nabla\times A.} \label{a}$$ A gauge transformation generated by the scalar function $\Lambda$ is$$\widetilde{\phi}=\phi+\frac{1}{c}\partial_{t}\Lambda,\qquad\widetilde
{\mathbf{A}}=\mathbf{A-\nabla}\Lambda, \label{b}$$ where $\Lambda$ must satisfy the homogeneous wave equation$$\left( \frac{1}{c^{2}}\partial_{t}^{2}-\mathbf{\nabla}^{2}\right)
\Lambda=\partial^{\mu}\partial_{\mu}\Lambda=0. \label{c}$$ Relativistic quantities are expressed with the time-favoring Minkowski metric, with the signature $\left( +---\right) $, where the scalar product of two 4-vectors $a^{\mu}$ and $b^{\mu}$ is$$a\cdot b=a^{\mu}b_{\mu}=a^{0}b^{0}-\mathbf{a\cdot b}. \label{e}$$ The 4-vector potential $A^{\mu}$ incorporates the scalar and 3-vector potentials as$$A^{\mu}:\left( \phi,\mathbf{A}\right) . \label{g}$$ In 4-vector notation, the two gauge transformation expressions in Eq. (\[b\]) become the single expression$$\widetilde{A}^{\mu}=A^{\mu}+\partial^{\mu}\Lambda. \label{h}$$ Both the initial and gauge-transformed 4-vector potentials must satisfy the Lorenz condition$$\partial^{\mu}A_{\mu}=0,\qquad\partial^{\mu}\widetilde{A}_{\mu}=0. \label{j}$$ The propagation 4-vector $k^{\mu}$ consists of the propagation 3-vector $\mathbf{k}$ as the space part, and the amplitude $\left\vert \mathbf{k}\right\vert =\omega/c$ as the time component:$$k^{\mu}:\left( \omega/c,\mathbf{k}\right) . \label{k}$$ The 4-vector $k^{\mu}$ defines the light cone and, according to the rule (\[e\]), it is self-orthogonal:$$k^{\mu}k_{\mu}=\left( \omega/c\right) ^{2}-\mathbf{k}^{2}=0, \label{l}$$ which is an important possibility in this non-Euclidean space.
The concept of transversality refers to the property of plane-wave fields expressed in a relativistic context as *covariant transversality*$$k^{\mu}A_{\mu}=0, \label{m}$$ in terms of the 4-potential $A^{\mu}$. In many textbooks on classical electromagnetic phenomena, transversality is defined as *geometrical transversality*$$\mathbf{k\cdot E}=0\text{ and }\mathbf{k\cdot B}=0, \label{n}$$ in terms of the electric and magnetic fields. It can be shown that covariant transversality infers geometrical transversality.
Terminology
-----------
Despite the conclusion in this paper that potentials are more basic than fields, it is not possible to avoid the use of the term field in a generic sense. For example, one important conclusion reached herein is that vector and scalar potentials provide more information than do electric and magnetic fields in the description of the effects of laser fields. In the preceding sentence, the term laser field is used generically to identify the radiation created by a laser, despite the particular result that potentials are the better approach in the description of that radiation. A similar problem arises when it is concluded that the dipole approximation amounts to the replacement of the transverse field of a laser by the more elementary longitudinal field. In each of the phrases demarcated by quotation marks, the word field is used in a generic sense to identify an electromagnetic phenomenon.
Constant electric field
-----------------------
The problem of a particle of mass $m$ and charge $q$ immersed in a constant electric field of magnitude $E_{0}$ is inherently one-dimensional. For present purposes, nothing is gained by going to three spatial dimensions. The problem is clearly one in which energy is conserved. By Noether’s Theorem [@noether], the Lagrangian must be independent of time $t$, so that the connection between the electric field and potentials given in Eq. (\[a\]) must depend only on the scalar potential $\phi.$ Equation (\[a\]) can then be integrated to give the potentials$$\phi=-xE_{0},\qquad A=0, \label{o}$$ since an additive constant of integration has no physical meaning. The potentials descriptive of this problem are unique, and given by Eq. (\[o\]).
The Lagrangian function is the difference of the kinetic energy $T$ and the potential energy $U$:$$\begin{aligned}
L & =T-U\label{p}\\
& =\frac{1}{2}m\overset{.}{x}^{2}+qxE_{0}. \label{p1}$$ The Lagrangian equation of motion is$$\frac{d}{dt}\frac{\partial L}{\partial\overset{.}{x}}-\frac{\partial
L}{\partial x}=m\overset{..}{x}-qE_{0}=0, \label{q}$$ which is just the elementary Newtonian equation$$m\overset{..}{x}=qE_{0}. \label{r}$$ The simplest initial conditions for this problem – initial position and velocity set to zero – lead to the solution$$x=\frac{qE_{0}}{2m}t^{2}. \label{s}$$ From Eqs. (\[p\]) and (\[p1\]), it follows that$$T=\frac{1}{2m}\left( qE_{0}t\right) ^{2},\qquad U=-\frac{1}{2m}\left(
qE_{0}t\right) ^{2},\qquad T+U=0. \label{t}$$ The anticipated conservation of energy holds true.
Despite the uniqueness of the potentials of Eq. (\[o\]), there exists an apparently proper gauge transformation generated by the function$$\Lambda=ctxE_{0}. \label{u}$$ The gauge-transformed potentials are$$\widetilde{\phi}=0,\qquad\widetilde{A}=-ctE_{0}, \label{v}$$ and the Lagrangian function is [@hrjmo]$$\widetilde{L}=\frac{1}{2}m\overset{.}{x}^{2}-qtE_{0}\overset{.}{x}. \label{w}$$ The kinetic energy is unaltered ($\widetilde{T}=T$), but the new potential energy is$$\widetilde{U}=qtE_{0}\overset{.}{x}, \label{x}$$ which is explicitly time-dependent. The new equation of motion is$$\frac{d}{dt}\left( \frac{\partial\widetilde{L}}{\partial\overset{.}{x}}\right) -\frac{\partial\widetilde{L}}{\partial x}=m\overset{..}{x}-qE_{0}=0,
\label{y}$$ which is identical to that found in the original gauge, so that the solution is the same as Eq. (\[s\]). However, the altered gauge has introduced a fundamental change. The gauge-transformed potential energy is evaluated as$$\widetilde{U}=\frac{1}{m}\left( qE_{0}t\right) ^{2}, \label{z}$$ so that $$\widetilde{T}+\widetilde{U}=\frac{3}{2m}\left( qE_{0}t\right) ^{2}.
\label{aa}$$ The total energy is not conserved, as was presaged by the explicit time dependence of the gauge-transformed Lagrangian (\[w\]).
How did this happen? One constraint placed on gauge transformations (see, for example, the classic text by Jackson [@jackson]) is that the generating function must be a scalar function that satisfies the homogeneous wave equation, as in Eq. (\[c\]). This is satisfied by the function (\[u\]). The only other condition is the Lorenz condition (\[j\]), which is satisfied by the potentials before and after transformation. However, there is no condition that guarantees preservation of symmetries inherent in the physical problem. It is not enough to employ appropriate fields; it is necessary to employ the appropriate potentials to ensure that all aspects of the physical problem are rendered properly.
This writer is unaware of any instance where inappropriate potentials have been accepted and employed in this exceedingly simple problem. The same cannot be said for the next example.
Plane-wave field
----------------
Laser fields are of central importance in contemporary physics, and laser fields are plane-wave fields. A plane-wave field is the only electromagnetic phenomenon that has the ability to propagate indefinitely in vacuum without the continued presence of sources. In the typical laboratory experiments with lasers, the practical consequence of this ability to propagate without need for sources means that all fields that arrive at a target can only be a superposition of plane-wave fields. Any contamination introduced by optical elements like mirrors or gratings can persist for only a few wavelengths away from such elements. On the scale of a typical laboratory optical table, this is negligible.
Plane-wave fields propagate at the speed of light in vacuum; they are fundamentally relativistic. The 1905 principle of Einstein is basic: the speed of light is the same in all inertial frames of reference [@einstein]. The mathematical statement of this principle is that any description of a plane-wave field can depend on the spacetime coordinate $x^{\mu}$ only as a scalar product with the propagation 4-vector $k^{\mu}$. The consequence of this projection of the spacetime 4-vector onto the light cone is that any change of gauge must be such as to be confined to the light cone. That is, with the definition$$\varphi\equiv k^{\mu}x_{\mu}, \label{ab}$$ the field 4-vector must be such that $$A_{pw}^{\mu}=A_{pw}^{\mu}\left( \varphi\right) , \label{ac}$$ where the subscript *pw* stand for *plane-wave*. When the gauge transformation of Eq. (\[h\]) is applied, the gauge-altered 4-vector potential is confined by the condition (\[ac\]) to the form [@hrgge]$$\widetilde{A}^{\mu}=A^{\mu}+k^{\mu}\Lambda^{\prime}, \label{ad}$$ where the gauge-change generating function can itself depend on $x^{\mu}$ only in the form of $\varphi,$ and $$\Lambda^{\prime}=\frac{d}{d\varphi}\Lambda\left( \varphi\right) . \label{ae}$$ As is evident from Eq. (\[l\]), transversality is maintained by the gauge transformation (\[ad\]).
A further limitation arises if an electron is subjected to a scalar binding potential in addition to the vector potential associated with the laser field. A relativistic Hamiltonian function for a charged particle in a plane-wave field contains a term of the form$$\left( i\hslash\partial^{\mu}-\frac{q}{c}A^{\mu}\right) \left(
i\hslash\partial_{\mu}-\frac{q}{c}A_{\mu}\right) . \label{ae1}$$ This occurs in the classical case, in the Klein-Gordon equation of quantum mechanics, and in the second-order Dirac equation of quantum mechanics [@feynmangm; @schweber]. The expansion of the expression in Eq. (\[ae1\]) contains the squared time part$$\left( i\hslash\partial_{t}-\frac{q}{c}A^{0}\right) ^{2}. \label{ae2}$$ If $A^{0}$ contains contributions from both a scalar potential and the time part of the plane-wave 4-vector potential, then executing the square in Eq. (\[ae2\]) would give a term containing the product of these two scalar potentials that is not physical; it does not occur in the reduction of relativistic equations of motion to their nonrelativistic counterparts [@hrgge]. This applies specifically to applications in AMO physics. That is, it must be true that [@hr79; @hrgge]$$A_{pw}^{0}=\phi_{pw}=0. \label{af}$$ This means that gauge freedom vanishes. Only the *radiation gauge* (also known as *Coulomb gauge*) is possible. This is the gauge in which scalar binding influences are described by scalar potentials $\phi$ and laser fields are described by 3-vector potentials $\mathbf{A}$.
Consider the gauge transformation generated by the function [@hr79]$$\Lambda=-A^{\mu}\left( \varphi\right) x_{\mu}. \label{ag}$$ This leads to the transformed gauge$$\widetilde{A}^{\mu}=-k^{\mu}x^{\nu}\left( \frac{d}{d\varphi}A_{\nu}\right) ,
\label{ah}$$ which was introduced in Ref. [@hr79] in an attempt to base the Keldysh approximation [@keldysh] on plane-wave fields rather than on quasistatic electric fields. The transformed 4-potential can also be written as [@hr79]$$\widetilde{A}^{\mu}=-\frac{k^{\mu}}{\omega/c}\mathbf{r\cdot E}\left(
\varphi\right) , \label{ai}$$ thus suggesting a relativistic generalization of the nonrelativistic *length gauge* used by Keldysh and widely employed within the AMO community. The problem with the $\widetilde{A}^{\mu}$ of Eq. (\[ah\]) or (\[ai\]) is that it violates the symmetry (\[ac\]) required of a propagating field. Nevertheless, this $\widetilde{A}^{\mu}$ satisfies the Lorenz condition (\[j\]) and the transversality condition (\[m\]); and the generating function of Eq. (\[ag\]) satisfies the homogeneous wave equation of Eq. (\[c\]) [@hr79]. That is, all the usual requirements for a gauge transformation are met even though the transformed 4-vector potential $\widetilde{A}^{\mu}$ of Eq. (\[ah\]) or (\[ai\]) violates the symmetry required of a propagating field like a laser field.
This violation of a basic requirement for a laser field has unphysical and hence unacceptable consequences. The most obvious is that the covariant statement of the all-important [@hrup] ponderomotive energy $U_{p}$ produces a null result since$$\widetilde{U}_{p}\sim\widetilde{A}^{\mu}\widetilde{A}_{\mu}=0 \label{aj}$$ as a consequence of the self-orthogonality of the propagation 4-vector $k^{\mu}$. The resemblance of Eq. (\[ai\]) to the length-gauge representation of a quasistatic electric field suggests a tunneling model for the relativistic case [@vspopov; @heidelberg], which is inappropriate for strong laser fields. Tunneling can occur only through interference between scalar potentials, and a strong laser field is inherently vector, not scalar.
The basic defect of the potentials (\[ah\]) or (\[ai\]) is violation of the Einstein condition of the constancy of the speed of light in all Lorentz frames, despite the validity of the gauge transformation leading to those potentials. The importance of the physical situation in which this occurs is robust evidence of the significance of the proper choice of potentials, since the electric and magnetic fields attained from the unacceptable potentials (\[ah\]) or (\[ai\]) are exactly the same as those that follow from potentials that satisfy properly the condition (\[ac\]).
Gauge transformations and unitarity
===================================
Unitary transformations in quantum physics preserve the values of physical observables. It was shown above that not all gauge transformations produce physically acceptable results. Therefore, gauge transformations are not unitary transformations. This conclusion is supported by the basic structure of transition amplitudes.
Transition amplitudes without resort to perturbation theory are best expressed by S matrices. These are of two (equivalent) types. The *direct-time* or *post* amplitude is$$\left( S-1\right) _{fi}=-\frac{i}{\hslash}\int_{-\infty}^{\infty}dt\left(
\Phi_{f},H_{I}\Psi_{i}\right) , \label{aq}$$ and the *time-reversed* or *prior* amplitude is$$\left( S-1\right) _{fi}=-\frac{i}{\hslash}\int_{-\infty}^{\infty}dt\left(
\Psi_{f},H_{I}\Phi_{i}\right) . \label{ar}$$ The indices $f$ and $i$ label the final and initial states. The $\Phi$ states are non-interacting states and the $\Psi$ states are fully interacting states satisfying, respectively, the Schrödinger equations$$\begin{aligned}
i\hslash\partial_{t}\Phi & =H_{0}\Phi,\label{as}\\
i\hslash\partial_{t}\Psi & =\left( H_{0}+H_{I}\right) \Psi, \label{at}$$ where $H_{I}$ is the interaction Hamiltonian.
In a gauge transformation, the matrix elements within the time integrations in Eqs. (\[aq\]) and (\[ar\]) transform as$$\begin{aligned}
\left( \Phi_{f},H_{I}\Psi_{i}\right) & \rightarrow\left( \Phi
_{f},\widetilde{H}_{I}\widetilde{\Psi}_{i}\right) ,\label{au}\\
\left( \Psi_{f},H_{I}\Phi_{i}\right) & \rightarrow\left( \widetilde{\Psi
}_{f},\widetilde{H}_{I}\Phi_{i}\right) . \label{av}$$ Because the noninteracting states are unaltered in a gauge transformation, there is no necessary ** equivalence between the two sides of the expressions in Eqs. (\[au\]) and (\[av\]).
Those authors that endorse the favored status of the length gauge [@yang; @kobesmirl; @beckerss; @lss; @jbauer] solve this problem by attaching a unitary operator to all states, including non-interacting states: $\widetilde{\Phi}=U\Phi$, $\widetilde{\Psi}=U\Psi$. All $U$ and $U^{-1}$ operators exactly cancel in the matrix element, and the transition amplitude is unchanged. This is what leads to the property gauge-invariant formalism sometimes ascribed to the length gauge. However, this procedure amounts to an identity or to a change of *quantum picture*, but not to a gauge transformation.
Fundamental contrasts in the applicability of fields and potentials
===================================================================
The first example to be presented is the very basic one of the impossibility of expressing the Schrödinger equation directly in terms of electric and magnetic fields, which should be possible if fields are truly more fundamental than potentials. Other direct examples of difficulties posed by the assumption of the fundamental importance of fields are shown, many of them long employed unnoticed within the strong-field community.
Schrödinger equation
--------------------
The Schrödinger equation$$i\hslash\partial_{t}\Psi\left( t\right) =H\Psi\left( t\right) , \label{ak}$$ when viewed as a statement in a Hilbert space (that is, without selecting a representation such as the configuration representation or the momentum representation) states that the effect of rotating a state vector $\Psi\left(
t\right) $ by the operator $H$ within the Hilbert space produces the same effect as differentiating the vector with respect to time (multiplied by $i\hslash$). Time $t$ is an external parameter upon which the state vectors depend, which accounts for why Eq. (\[ak\]) specifies $t$ as a label independent of the Hilbert space. A unitary transformation preserves this equivalence. This can be stated as$$i\hslash\partial_{t}-\widetilde{H}=U\left( i\hslash\partial_{t}-H\right)
U^{-1}. \label{al}$$ Since Eq. (\[al\]) can be written as$$\widetilde{H}=UHU^{-1}+U\left( i\hslash\partial_{t}U^{-1}\right) ,
\label{am}$$ this shows explicitly that the Hamiltonian operator does not transform unitarily if there is any time dependence in $U$.
An important gauge transformation is that introduced by Göppert-Mayer [@gm], widely employed in the AMO community. This transformation is given by$$U_{GM}=\exp\left( \frac{ie}{\hslash c}\mathbf{r\cdot A}\left( t\right)
\right) , \label{an}$$ which depends explicitly on time when $\mathbf{A}\left( t\right) $ describes a laser field within the dipole approximation, meaning that Eq. (\[am\]) is consequential.
The fact that, in general,$$\widetilde{H}\neq UHU^{-1}, \label{ao}$$ is the explanation for the curious result to be found in many papers (for example, Refs. [@yang; @kobesmirl; @beckerss; @lss; @jbauer]) that the $\mathbf{r\cdot E}$ potential is a preferred potential. If any other potential is employed in solving the Schrödinger equation, then the claim is made that a transformation factor must be employed even on a non-interacting state. There is a logical contradiction inherent in the requirement that a non-interacting state must incorporate a factor that depends on an interaction, but the list of published papers that accept this premise is much longer than the salient examples cited here. The underlying problem is the assumption that a gauge transformation transforms the Hamiltonian unitarily. That problem exists in all of the references just cited, although it is usually submerged in complicated manipulations. It is especially clear in Ref. [@jbauer], where it is specified that all operators $O$ transform under a gauge transformation according to the unitary-transformation rule$$\widetilde{O}=UOU^{-1},\qquad U^{-1}=U^{\dag}. \label{ap}$$ That specification is applied to $H$, in violation of the condition (\[am\]), and to the interaction Hamiltonian $H_{I}$, with no explanation for how it is possible to gauge-transform from the length-gauge interaction to any other gauge in view of the absence of operators in the scalar potential $\mathbf{r\cdot E}$. In the scheme proposed in Refs. [@yang; @kobesmirl; @beckerss; @lss; @jbauer], if the problem is initially formulated in the context of the $\mathbf{r\cdot E}$ potential, it is never possible to transform to any other gauge. This explains the use of the phrase gauge-invariant formulation with respect to $\mathbf{r\cdot E}$ to be found in some published works.
Locality and nonlocality
------------------------
Fields are derived from potentials by the calculus process of differentiation, as exhibited in Eq. (\[a\]). Differentiation is carried out at a point in spacetime. It is *local*. If potentials are to be expressed from fields, that requires integration, consisting of information from a range of spacetime values; it is *nonlocal*. The fact that the Schrödinger equation requires the local information from potentials, and cannot be described by fields without inferring nonlocality in spacetime, is direct evidence that potentials are more fundamental than fields.
Ambiguity in the electromagnetic field tensors
----------------------------------------------
The basic field tensor of electrodynamics is defined as$$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}. \label{aj1}$$ It is important to note that this expression is in terms of the derivatives of potentials rather than the potentials themselves. Thus it is not surprising that the Lorentz invariant found from the inner product of $F^{\mu\nu}$ with itself yields an expression in terms of fields:$$F^{\mu\nu}F_{\mu\nu}=2\left( \mathbf{B}^{2}-\mathbf{E}^{2}\right) .
\label{aj2}$$ A dual tensor can be defined as$$G^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}, \label{aj3}$$ where $\epsilon^{\mu\nu\rho\lambda}$ is the completely asymmetric fourth-rank tensor. (The conventions of Jackson [@jackson] are being employed.) The inner product of the basic and dual tensors gives a second Lorentz invariant:$$G^{\mu\nu}F_{\mu\nu}=-4\mathbf{B\cdot E}, \label{aj4}$$ also in terms of fields. The two Lorentz invariants$$\mathbf{E}^{2}-\mathbf{B}^{2},\quad\mathbf{E\cdot B} \label{aj5}$$ are said to characterize the electrodynamic environment.
An important special case is that of transverse, propagating fields, where$$\mathbf{E}^{2}-\mathbf{B}^{2}=0,\quad\mathbf{E\cdot B}=0. \label{aj6}$$ The properties (\[aj6\]) lead to radiation fields as sometimes being called null fields. (The terms radiation field, propagating field, transverse field, plane-wave field, are here used interchangeably.) Radiation fields propagate at the speed of light in vacuum, and they have the unique character that, after initial formation, they propagate indefinitely in vacuum without the presence of sources.
However, the invariants (\[aj6\]) are not unique to radiation fields; they apply also to constant crossed fields. That is, it is always possible to generate static electric and magnetic fields of equal magnitude that are perpendicular to each other, and will thus possess zero values for both of the Lorentz invariants of the electromagnetic field. Constant crossed fields do not propagate, and they cannot exist without the presence of sources. They are unrelated to radiation fields despite sharing the same values of the Lorentz invariants. Most importantly, constant crossed fields cannot be considered as the zero-frequency limit of radiation fields, as they are sometimes described [@nikrit66; @ritus85]. All radiation fields propagate at the speed of light for all frequencies, no matter how low. There is no possible zero-frequency static limit [@hrtun].
There is no ambiguity when radiation fields and constant crossed fields are expressed in terms of their potentials. Radiation-field potentials possess the periodicity inherent in trigonometric dependence on the $\varphi$ of Eq. (\[ab\]). This is unrelated to the $\phi=-\mathbf{r\cdot E}_{0}$ potential of (\[o\]) for a constant electric field $\mathbf{E}_{0}$, and $\mathbf{A}=-\left( \mathbf{r\times B}_{0}\right) /2$ for a constant magnetic field $\mathbf{B}_{0}$, both of which require source terms for their existence.
Lorentz force
-------------
The force exerted on a particle of charge $q$ moving with velocity $\mathbf{v}$ in a field with electric and magnetic components $\mathbf{E}$ and $\mathbf{B}$ is given by the Lorentz force expression$$\mathbf{F}=q\left( \mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right) .
\label{aj7}$$ In a plane-wave field, the electric and magnetic fields are of equal magnitude: $\left\vert \mathbf{E}\right\vert =\left\vert \mathbf{B}\right\vert
$. Thus, under conditions where $\left\vert \mathbf{v}\right\vert /c\ll1$, the magnetic component of the force is minor as compared to the electric component. The implication is that, as the field frequency declines, the motion-related magnetic component reduces to an adiabatic limit. This concept of adiabaticity justifies the complete neglect of the magnetic field that is a key element of the dipole approximation, applied within the AMO community in the form:$$\mathbf{E}=\mathbf{E}\left( t\right) ,\quad\mathbf{B}=0. \label{aj8}$$ The adiabaticity line of reasoning suggests a so-called adiabatic limit where the field frequency declines to zero, and the plane wave field behaves as a constant crossed field that satisfies the plane-wave condition (\[aj6\]).
The entire line of reasoning that involves the concepts of adiabaticity, adiabatic limit, and a zero-frequency limit for plane-wave fields is field-based and erroneous [@hr101; @hrtun]. When the problem is treated in terms of potentials, it becomes clear that $v/c$ approaches unity for very strong fields even when the frequency is very low, and the magnetic force becomes equivalent to the electric force for very strong fields.
Critical field
--------------
The critical field in electrodynamics is related to the spontaneous breakdown of the vacuum into electron-positron pairs. The critical field is defined as the electric field strength at which the $\pm mc^{2}$ limits for the rest energies of electron and positron in a particle-hole picture are tilted by the electric field to allow tunneling between positive and negative energy states when the spatial limits of the tunnel are separated by an electron Compton wavelength. This type of pair production is called Sauter-Schwinger pair production [@sauter; @schwinger]. It is fundamentally different from Breit-Wheeler pair production [@bw; @hrdiss; @hr62] to be discussed below.
The reason for the fundamental difference is that Sauter-Schwinger pair production is a phenomenon due to electric fields and Breit-Wheeler is due to plane-wave fields. The distinction between pure electric fields and plane-wave fields could hardly be more clear, since both types of fields have unique gauge choices that are contrasting.
The critical field is often mentioned as a goal of strong-field laser facilities, which is basically a *non-sequitur*. The critical field applies only to electric fields and lasers produce plane-wave fields. Even if a laser field were sufficiently intense that its electric component had the magnitude of the QSE critical field, pair production from the vacuum cannot occur because the photons of the laser field convey momentum. A counter-propagating plane-wave field is necessary to satisfy momentum conservation as well as energy conservation in the production of pairs from the vacuum. This is then the two-fields Breit-Wheeler process, which is unrelated to the single-field Sauter-Schwinger process.
Pair production from the vacuum
-------------------------------
For many years the stated ultimate goal of large-laser programs was to achieve a laser intensity such that the electric component of the laser is equal to the critical field discussed in the preceding subsection This magnitude of electric field corresponds to an intensity of about $4.6\times10^{29}W/cm^{2}$.
The problem is that the Schwinger limit applies *only* to electric fields. It does not apply to laser fields [@hrspie], as explained above. The only way to produce momentum balance with a laser field while still producing pairs from the vacuum is to have the laser beam collide with oppositely-directed photons, as proposed in Ref. [@hr71]. This was predicted to be done on a practical basis at a linear accelerator facility such as that at SLAC (Stanford Linear Accelerator Center), with a laser intensity of only slightly greater than $10^{18}W/cm^{2}$. An important note is that the prediction of Ref. [@hr71] was for the use of the then-important ruby laser at a different pulse length and a different energy of the energetic electron beam used to produce the countervailing photon field. The predicted threshold of 25 photons from the laser field in Ref. [@hr71] is altered to 5 photons for the parameters of the experiment that was actually done at SLAC in 1997 [@burke]. The theoretical prediction of an effective threshold intensity of about $10^{18}W/cm^{2}$ is maintained because of the essential independence of the laser frequency that was remarked in Ref. [@hr71].
(One *caveat* about the experiment is that it was reported as a high-order perturbative result, whereas it is readily shown to be at an intensity beyond the radius of convergence of perturbation theory [@epjd]. A second problem is that it was described as light-by-light scattering, which is a different process altogether. Feynman diagrams of these processes have electron and positron as emergent particles in a pair production process, while light-by-light scattering has emergent photons.)
The striking difference between the $4.6\times10^{29}W/cm^{2}$ required to attain a laser electric component equal to the critical field and the actual $1.3\times10^{18}W/cm^{2}$ required for the SLAC experiment is evidence of a misplaced focus on the electric field required for vacuum pair production. The required intensity of the laser field depends on the properties of the counter-propagating field, but it is never as large as the Sauter-Schwinger critical field.
Low frequency limit of a plane-wave field
-----------------------------------------
It is conventional to view low-frequency laser-induced phenomena from the standpoint of the Lorentz force expression of Eq. (\[aj7\]). In a plane-wave field, the electric and magnetic fields are of equal magnitude: $\left\vert
\mathbf{E}\right\vert =\left\vert \mathbf{B}\right\vert $. Thus, under conditions where $\left\vert \mathbf{v}\right\vert /c\ll1$, the magnetic component of the force is minor as compared to the electric component. The implication is that, as the field frequency declines, the motion-related magnetic component reduces to an adiabatic limit. This concept of adiabaticity appears to justify the complete neglect of the magnetic field that is a key element of the dipole approximation, applied within the AMO community in the form given in Eq. (\[aj8\]). Adiabaticity suggests a so-called adiabatic limit, where the field frequency declines to zero, and the plane wave field behaves as a constant crossed field that satisfies the plane-wave condition (\[aj6\]).
The entire line of reasoning that involves the concepts of adiabaticity, adiabatic limit, and a zero-frequency limit for plane-wave fields is field-based and erroneous [@hr101; @hrtun]. When the problem is treated in terms of potentials, it becomes clear that $v/c$ approaches unity for very strong fields. The magnetic force becomes equivalent to the electric force for very strong fields.
Analysis from a potentials standpoint is in stark contrast to a fields-based approach.
The ponderomotive potential energy of a charged particle in a plane-wave field is a fundamental property of the particle [@hrup], and it becomes divergent as the field frequency approaches zero. The immediate consequence is that the limit $\omega\rightarrow0$ causes the dipole approximation to fail [@hr101; @hrtun], and corresponds to an extremely relativistic environment [@hrup]. This contradicts maximally the field-based conclusions.
The ponderomotive energy $U_{p}$ is given by$$U_{p}=\frac{q^{2}}{2mc^{2}}\left\langle \left\vert A^{\mu}A_{\mu}\right\vert
\right\rangle , \label{aj9}$$ where the angle brackets denote a cycle average, and the absolute value needs to be indicated because the 4-vector potential $A^{u}$ is a spacelike 4-vector and its square is thus negative with the metric being employed. The ponderomotive energy is based on potentials. When expressed in terms of field intensity $I$, $U_{p}$ behaves as$$U_{p}\sim I/\omega^{2}, \label{aj10}$$ which explains the relativistic property of the charged particle as the frequency approaches zero.
Extremely-low-frequency radio waves
-----------------------------------
A central issue in this article is that transverse fields and longitudinal fields are fundamentally different, even when they seem to have some properties in common. An effective example is the matter of communicating with submerged submarines with extremely low frequency (ELF) radio waves. The U. S. Navy operated such a system [@wikisanguine], designed to communicate with submarines submerged at depths of the order of $100m$ over an operational range of about half of the Earth’s surface. The point of using extremely low frequencies is the large skin depth in a conducting medium (seawater) that can be achieved. What is most remarkable about the system is that the frequency of $76Hz$ that was used has a wavelength of about $4\times10^{6}m$, which is approximately $0.62$ times the radius of the Earth. The system could convey an intelligible signal over about half of the Earth’s surface. Considering the length of the submarine (about $100m$) to be the size of the receiving antenna, this means that the wavelength to antenna-length ratio is about $4\times10^{4}$. That is, the received radio wave is constant over the entire length of the receiving antenna to a very high degree of accuracy. A nearly-constant electric field cannot be detected at a distance of half of our planet away from its source; a nearly-constant electric field cannot penetrate $100m$ through a conducting medium; and so a nearly-constant electric field cannot convey an intelligible signal in the presence of all of these extremely effective barriers. A longitudinal field is fundamentally different from a transverse field.
The proposed local constant-field approximation (LCFA) for relativistic laser effects [@ritus85; @heidrmp] is based on the presumed similarity of low-frequency laser fields to constant crossed fields. This was shown earlier in this manuscript to be a meaningless association. The application involved in the communication with submerged submarines is a practical demonstration that the LCFA is not a valid concept.
Hierarchy of physical effects
=============================
In basic calculus, if a function $f\left( x\right) $ is known, then so is $df/dx$; differentiation is local. If the derivative is all that is known, the process of integration to find $f\left( x\right) $ requires the knowledge of $df/dx$ over a range of values; integration is nonlocal. In electromagnetism, potentials are the analog of $f\left( x\right) $ and fields correspond to $df/dx$. The example of the Schrödinger equation shows that knowledge of potentials is primary and fields are secondary. This ordering can be extended to other physical quantities. For example, the Lorentz force expression in Eq. (\[aj7\]) relates forces directly to fields, meaning that forces are secondary. A further implication is that Newtonian mechanics, expressed in terms of forces, is secondary to Lagrangian and Hamiltonian mechanics that are expressed in terms of energies; that is, in terms of potentials. It is no accident that mechanics textbooks show that formalisms based on system functions like the Lagrangian or Hamiltonian infer the Newtonian formalism, but to go in the reverse direction requires an extrapolation of concepts.
This ordering, or hierarchy, of physical quantities has consequences in problems that go beyond simple electric-only or magnetic-only fields. In laser-related problems where both electric and magnetic fields are involved, it is possible to arrive at invalid inferences if only secondary influences are regarded as controlling. An example is the common practice of viewing electric fields as being the dominant quantities in long-wavelength circumstances where the dipole approximation appears to be valid. This disguises the fact that extremely low frequencies can lead to relativistic behavior, where electric fields supply inadequate information [@hr101; @hrtun], and inappropriate concepts such as adiabaticity exist.
A cogent example is the critical field. A widely used strong-field QED parameter is simply the ratio of the electric field to the critical field. This has meaning only for static-electric or for QSE fields. For strong-field laser applications, relevant intensity parameters are all ratios of energies. See, for example, the section entitled Measures of intensity in Ref. [@hrrev]. (The early, but still useful review article by Eberly [@eberlyreview] is also instructive in this regard.) In particular, the free-electron intensity parameter $z_{f}=2U_{p}/mc^{2}$ is known to measure the coupling between an electron and a nonperturbatively intense plane-wave field [@hrdiss; @hr62; @hr62b; @hrup], replacing the fine-structure constant of perturbative electrodynamics.
A special insight arises when the $z_{f}$ parameter is related to the fine-structure constant $\alpha$ [@hrup]:$$z_{f}=\alpha\rho\left( 2\lambda\lambdabar_{C}^{2}\right) , \label{aj11}$$ where $\rho$ is the number of photons per unit volume, and $2\lambda
\lambdabar_{C}^{2}$ is approximately the volume of a cylinder of radius equal to the electron Compton wavelength $\lambdabar_{C}$ and a length given by the wavelength $\lambda$ of the laser field. That is, it appears that all of the photons within the cylinder participate in the coupling between the electron and the laser field. The electron Compton wavelength is the usual interaction distance for a free electron, but the wavelength can be a macroscopic quantity. The strong-field physics that arises from the application of the Volkov solution to problems involving the interaction of very intense radiation fields with matter [@sengupta; @hrdiss; @hr62], thus appears to be the bridge connecting quantum electrodynamics with the classical electrodynamics of Maxwell.
Other examples of the hazards of excessive dependence on secondary quantities have been mentioned above in the context of the assumed dominance of the electric component of the Lorentz force at low frequencies, and the inference from the Lorentz invariants that constant crossed fields are related to propagating fields.
Failure of perturbation theory
==============================
Perturbation theory has been the cornerstone of QED since the Nobel-prize winning work of Feynman, Schwinger, and Tomonaga. The radius of convergence of perturbation theory was examined in depth a long time ago [@hrdiss]. The motivation for the study was the demonstration by Dyson [@dyson] that, despite its remarkable quantitative success, standard covariant QED has a zero radius of convergence for a perturbation expansion. The question about whether a strong-field theory based on the Volkov solution of the Dirac equation [@volkov] could be convergent was the motivation for Ref. [@hrdiss]. The answer was affirmative, but with intensity-dependent singularities in the complex coupling-constant plane that limited the convergence. (This explains the $z_{f}$ terminology, since $z_{f}$ was allowed to be complex, and the quantity $z$ is often used for complex numbers.) Physically, convergence fails whenever the field intensity is high enough to cause a channel closing. That is, if a process requires a certain threshold energy to proceed in a field-free process as measured by some photon number $n_{0}$, that threshold energy will be increased due to the need for a free electron to possess the ponderomotive energy $U_{p}$ in the field. When $U_{p}$ is large enough to cause $n_{0}$ to index up to $n_{0}+1$, that is called a channel-closing, and it marks a sufficient condition for the failure of perturbation theory. The upper limit on perturbation theory is therefore set by$$U_{p}<\hslash\omega,\text{ \ or \ }I<4\omega^{3}, \label{aj12}$$ where the last expression is in atomic units, using the equivalence $U_{p}=I/4\omega^{2}$ in atomic units, and $I$ is the laser intensity.
Although the original convergence investigation was done for free electrons, the same limit was found for atomic ionization [@hr80].
The relevance of using an index based on the ratio of the electric field to the critical field as an *ad hoc* limit on perturbation theory for laser effects is strongly questioned here. Such a basic matter as the applicability of perturbation methods to the treatment of the effects of radiation fields is governed by primary quantities like the energies $U_{p}$ and $mc^{2}$, and not by secondary quantities like the magnitude of the electric field.
Overview
========
The Aharonov-Bohm effect introduced a major change in electrodynamics because it showed that potentials are indeed more fundamental than fields. However, although the effect relates to a quantum phenomenon, it has had little effect on the way in which quantum mechanics is employed. The ascendancy of potentials over fields as described above is much more consequential, especially for strong-field phenomena. A simple summarizing statement is that electromagnetic scalar and vector potentials convey more physical information than the electric and magnetic fields derivable from them.
Of special importance is the fact that two cases have been identified where there can be only one acceptable set of potentials, and these two cases are of very wide practical scope. One case is the constant electric field, which is exactly or approximately applicable to a wide variety of AMO and condensed-matter applications. The unique acceptable potential for static electric fields is the length-gauge, or $\mathbf{r\cdot E}$ potential, and this is already widely employed for constant or slowly varying electric-field applications.
The other case with a unique allowable gauge is the propagating-field case, which is of profound importance in laser-based experiments. Such experiments constitute an increasingly large proportion of AMO activities. An essential reminder is that only propagating fields can persist in the absence of sources, so that virtually all laser-based experiments occur in a superposition of propagating-field components. The radiation gauge (or Coulomb gauge) is the only gauge employable without risking the creation of hidden errors due to improper gauge choice. It is unfortunate that, in strong-field laser applications, the widespread use of the dipole approximation introduces precisely that hazard of hidden errors that affects both practical calculations and qualitative insights into the behavior of physical systems subjected to laser fields.
It has been shown that gauge transformations are not, in general, unitary. This has never previously been reported, and it can lead to further errors in addition to the important example explored in Section III above about the putative universality of the length gauge.
The concept of primary and secondary physical quantities has been introduced, with over-dependence on a secondary quantity like the electric field having the capability of leading to important misconceptions.
The criterion of a critical field for longitudinal fields has no relevance to the transverse field of laser-induced processes.
The appearance of mixed quantum and classical quantities in ascertaining the limits on perturbative methods in the application of strong-field theories identifies strong-field physics as the bridge connecting quantum electrodynamics and the classical electromagnetism of Maxwell.
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| 0 |
---
author:
- 'Tomáš Ježo,'
- Paolo Nason
title: 'On the Treatment of Resonances in Next-to-Leading Order Calculations Matched to a Parton Shower.'
---
The problem
===========
At present, several methods exist for the computation of next-to-leading order (NLO) corrections in the Standard Model. When strong and/or electromagnetic interactions are present, these calculations must deal properly with collinear and soft divergences, that must cancel when infrared insensitive (IR-safe) observables are computed. The so-called subtraction methods are generally used in order to deal with this problem. In essence, they work as follows. A generic NLO cross section can be written symbolically as $${\mathrm{d}}\sigma = {\mathrm{d}}\Phi_B (B (\Phi_B) + V (\Phi_B)) + {\mathrm{d}}\Phi_R R
(\Phi_R), \label{eq:basicsigma}$$ where $\Phi_B$ stands for the Born phase space and $\Phi_R$ is the real emission phase space. $B (\Phi_B)$, $V(\Phi_B)$ and $R (\Phi_R)$ represent the Born, Virtual and Real cross section respectively. The real emission process corresponds to the Born process in association with an extra parton.[^1] For the purpose of this example we assume that we do not have hadrons in the initial state, i.e. we consider processes like $Z$ decays into hadrons. The expression in eq. (\[eq:basicsigma\]), in order to make sense, must be evaluated with some form of regularization for the soft and collinear singularities. Assuming that we are using dimensional regularization, the phase space $\Phi_B$ and $\Phi_R$ are evaluated in $d = 4 - 2 \epsilon$ dimensions. The value of an observable $\mathcal{O}$ is then given by $$\langle \mathcal{O} \rangle = \int \mathcal{O} {\mathrm{d}}\sigma = \int {\mathrm{d}}\Phi_B (B (\Phi_B) + V (\Phi_B)) \mathcal{O} (\Phi_B) + \int {\mathrm{d}}\Phi_R
R (\Phi_R) \mathcal{O} (\Phi_R) . \label{eq:observableO}$$ We can think of our observable $\mathcal{O}$ as the cross section in a given histogram bin of some kinematic distribution. Again, in eq. (\[eq:observableO\]) we assume that we have a $d$-dimensional definition for our observable, with the appropriate 4-dimensional limit. If the observable is IR-safe, soft and collinear singularities will cancel in eq. (\[eq:observableO\]), yielding a finite result.
Subtraction method
------------------
In the subtraction method, one introduces a parametrization of the real phase space of the form $\Phi_R = \Phi_R (\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})$, with $${\mathrm{d}}\Phi_R = {\mathrm{d}}\Phi_B {\mathrm{d}}\Phi_{{\ensuremath{\operatorname{rad}}}},$$ where $\Phi_{{\ensuremath{\operatorname{rad}}}}$ has $d - 1$ dimensions, and parametrizes the emission of the extra parton. The parametrization must have a smooth behaviour in the soft and collinear limit. Thus, in the limit of soft emission, the kinematics of all but the soft parton described by $\Phi_R(\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})$ must match the $\Phi_B$ kinematics. In the collinear limit, the kinematics of the system obtained by replacing the two collinear partons in $\Phi_R (\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})$ with a single parton with the appropriate flavour, having momentum equal to the sum of the momenta of the collinear partons, must match the $\Phi_B$ kinematics. One also introduces a simplified approximation to the real cross section, $R_s$, that coincides with $R$ in the soft and collinear singular limits. Eq. (\[eq:observableO\]) is rewritten as $$\begin{aligned}
\langle \mathcal{O} \rangle & = & \int {\mathrm{d}}\Phi_B \left[ B (\Phi_B) + V
(\Phi_B) + \int {\mathrm{d}}\Phi_{{\ensuremath{\operatorname{rad}}}} R_s (\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})
\right] \mathcal{O} (\Phi_B) \\
& + & \int {\mathrm{d}}\Phi_B {\mathrm{d}}\Phi_{{\ensuremath{\operatorname{rad}}}} [R (\Phi_R (\Phi_B,
\Phi_{{\ensuremath{\operatorname{rad}}}})) \mathcal{O} (\Phi_R (\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})) - R_s
(\Phi_R (\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})) \mathcal{O} (\Phi_B)] . \nonumber
\label{eq:subtrmeth} \end{aligned}$$ eq. (\[eq:subtrmeth\]) is clearly identical to eq. (\[eq:observableO\]). It has however the nice property that $1 / \epsilon$ divergences in the square bracket of the first term on the right-hand side of the equation (arising in the virtual term and in the ${\mathrm{d}}\Phi_{{\ensuremath{\operatorname{rad}}}}$ integration of $R_s$) cancel among each other. Furthermore, collinear and soft divergences cancel under the integral sign in the square bracket of the second term. The second term can thus be evaluated in 4 dimensions with numerical methods. The square bracket in the first term can be evaluated analytically once and for all. One usually defines $$V_{{\ensuremath{\operatorname{sv}}}} (\Phi_B) = \lim_{\epsilon \rightarrow 0} \left[ V (\Phi_B) +
\int {\mathrm{d}}\Phi_{{\ensuremath{\operatorname{rad}}}} R_s (\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}}) \right],$$ and the first term becomes the 4-dimensional expression $$\int {\mathrm{d}}\Phi_B [B (\Phi_B) + V_{{\ensuremath{\operatorname{sv}}}} (\Phi_B)] \mathcal{O}
(\Phi_B),$$ that can be computed numerically.
The development of the subtraction method started since the very early QCD computations, already appearing in the bud in the calculation of the Drell-Yan process of ref. [@Altarelli:1979ub]. A more systematic use of it was made in ref. [[@Ellis:1980wv]]{}, in the context of $e^+ e^-$ annihilation into hadrons. In ref. [[@Kunszt:1989km]]{} the calculation of ref. [[@Ellis:1980wv]]{} was implemented as a parton level generator, such that any given observable could be computed with it without any dedicated analytic work, and was in fact used to compute a number of commonly used IR-safe observables for QCD studies at LEP. Subsequently, the subtraction method implemented in parton level generators was applied also for processes initiated by hadrons [[@Mele:1990bq]]{}, and it became common practice to compute the $R_s$ term by using the collinear and the soft approximations in $d$ dimensions (see for example [[@Mangano:1991jk]]{}).
More recently, fully general formulations of the subtraction method have appeared. The procedure of ref. [[@Catani:1996vz]]{}, known as the CS method, uses local subtraction terms for the $R_s$ cross section. The formulation given in ref. [[@Frixione:1995ms]]{}, known as the FKS method, is instead based upon the more traditional phase space parametrizations used in refs. [[@Ellis:1980wv]]{} and [[@Mele:1990bq]]{}.
The subtraction method and resonances
-------------------------------------
When resonances are present, in the zero width limit, the cross section factorizes into the product of production and decay terms. In these cases, a standard subtraction method can be applied independently to the production and decay processes. In fact this was done in refs. [[@Melnikov:2011qx]]{} and [[@Campbell:2012uf]]{} for top production and decay. Problems do arise, however, if finite width effects are fully included, so that also interference among radiation produced in production and decays, or among radiation produced in the decay of different resonances, is included.
On the one hand, the presence of a finite width regulates the singularity associated with the resonance peak, so that, strictly speaking, a subtraction method will formally lead to finite and consistent results. On the other hand, taking the zero width limit, a standard subtraction method approach will lead to divergent results. In order to illustrate this problem, we consider the example of $t$-channel single top production and decay. One Born amplitude for this process is illustrated in fig. \[fig:ST\].
![Single top $t$-channel production. \[fig:ST\]](ST){width="6cm"}
The final state is composed by a $b$ and $d$ quark, a muon neutrino and an anti-muon. We assume for the sake of illustration a massless $b$ quark. The system comprising the final state $b$ quark, the neutrino and the anti-muon have an invariant mass close to the top mass, becoming identical to the top mass in the zero width limit. Consider now the real contribution obtained by adding gluon radiation to the final state. As illustrated above, in generic subtraction methods, the subtraction counterterms are obtained by factorizing the real phase space in terms of a Born phase space times a radiation phase space. The subtraction term for the collinear singularity corresponding to the final state gluon being collinear to the final $b$ uses a Born phase space where the collinear $b g$ pair is merged into a single $b$. The problem with the resonance is better illustrated in the CS subtraction framework, where the kinematics of the subtraction term is built as follows. Calling $k_{\oplus}$ the 4-momentum of the incoming $b$, and $k_b$, $k_g$ the 4-momenta of the final $b$ and $g$ partons, one defines the momentum of the $b$ quark in the underlying Born configuration as $$\bar{k}_b = k_b + k_g - k_{\oplus} \frac{(k_b + k_g)^2}{2 (k_b + k_g) \cdot
k_{\oplus}},$$ in such a way that $\bar{k}_b^2 = 0$. Furthermore, the incoming $b$ quark momentum is redefined as $$\bar{k}_{\oplus} = k_{\oplus} - k_{\oplus} \frac{(k_b + k_g)^2}{2 (k_b +
k_g) \cdot k_{\oplus}},$$ so that the total 4-momentum is conserved. We see that in this way the 4-momentum of the top quark has been altered, near the collinear limit, by an amount $$k_{\oplus} \frac{(k_b + k_g)^2}{2 (k_b + k_g) \cdot k_{\oplus}} \approx
\frac{m_{b g}^2}{E_{b g}} .$$ Since the CS procedure does not impose that the top 4-momentum is the same in the real and subtraction terms[^2] it will turn out that the top virtualities will differ there by an amount of order $m_{b g}^2 / E_{b g}$. The collinear singularity in the real and subtraction terms will thus match only if $$m_{b g}^2 \ll \Gamma_t E_{b g}, \label{eq:collimgamma}$$ where $\Gamma_t$ is the top width. It is easy to see that this is also true in other subtraction methods. For example, in the one used in the , the momentum of the collinear counterterm is built by setting the 3-momentum of the $b$ quark parallel to the sum of the 3-momenta of the $b$ and $g$ particles in the partonic CM frame. Furthermore, the momentum of the $d \bar{{\mu}} \nu$ system is boosted along the direction of the merged $b$ quark in order to conserve 3-momentum, and the absolute value of the $b$ quark momentum is chosen in such a way that the final state CM energy is conserved. This procedure is designed to conserve the mass of the final state system, and the mass of the system that recoils with respect to the splitting partons, i.e. the $\bar{{\mu}} \nu d$ system, while the mass of the top resonance is not conserved.
We thus expect that the collinear singularities present in the real and subtraction terms will be exposed in the narrow width limit, spoiling the convergence of the subtraction method. In fact, double logs of the resonance width will arise in different regions of the real cross section, yielding to a failure of convergence in the limit $\Gamma \rightarrow 0$. It is also clear that, in order to overcome this problem, one must devise a subtraction method such that the resonance mass is the same in the real and subtraction terms when approaching the resonance peak even when the resonance is off-shell by an amount greater than its width.
NLO+PS and resonances
---------------------
If we plan to use an NLO calculation with an interface to a shower generator (NLO+PS from now on), further problems arise due to the resonance treatment.
In the method [@Frixione:2002ik], one should consider the recoil scheme used by the Shower Monte Carlo to build radiation from a decaying resonance and construct the MC counterterms accordingly.
In the method [@Nason:2004rx; @Frixione:2007vw; @Alioli:2010xd], one first computes the inclusive cross section for the production of an event with a given underlying Born configuration. Radiation is then generated according to a Sudakov form factor with the following form: $$\Delta (p_T^2) = \exp \left[ - \int \frac{R (\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})}{B
(\Phi_B)} \theta (k_T (\Phi_{{\ensuremath{\operatorname{rad}}}}) - p_T) {\mathrm{d}}\Phi_{{\ensuremath{\operatorname{rad}}}}
\right] .$$ The mapping of the real phase space into a product of an underlying Born times a radiation phase space is the same used in the NLO subtraction procedure. In general, it will not preserve resonance masses, so that in the $R / B$ ratio, unless the condition (\[eq:collimgamma\]) is met, the numerator and the denominator will not be on the resonance peak at the same time. In case when $R$ is on peak and $B$ is not, this will yield large ratios that badly violate the collinear approximation.
A further problem arises when interfacing the NLO+PS calculation to a Shower generator, in order to generate the next-to-hardest radiation. Shower Monte Carlo’s should be instructed to preserve the mass of the resonances. Thus, radiation should have a resonance assignment. This is generally not available in processes that include interference among radiation generated by different resonances, or by a resonance and the production process itself.
The method
==========
In order to solve the problems mentioned above, we should separate all contributions to the cross section into terms with definite resonance structure. Each term individually should have resonance peaks only in a single, well defined, resonance cascade chain. The mapping into an underlying Born configuration should be defined for these terms in such a way that the resonance masses are preserved. Thus, when looking for parton pairs that can give rise to a collinear singularity, one should only consider pairs arising from the same resonance decay, or directly from the production process.
In the framework, a subtraction method that preserves the resonance masses is already implemented, but it is presently available only for calculations performed in the zero width approximation. In these cases, only one resonance decay chain is possible, and the real emission contributions are separated according to the resonance that originates the radiation. The method is discussed in detail in ref. [[@Campbell:2014kua]]{}. In essence, with this method, the subtraction procedure for initial state radiation is the same one used in ref. [[@Frixione:2007vw]]{} (the FNO paper from now on). For final state radiation arising from the production process the subtraction procedure is also the same one discussed in Section 5.2 of FNO. This procedure is such that the mapping of the real to the underlying Born configuration does not change the four momentum of the final state. In case of radiation from the decay of a resonance, the subtraction procedure is essentially the same, except that it is applied in the resonance frame, and thus does not alter the resonance four momentum and the momenta of all particles that do not have the resonance as an ancestor.
In the general context when finite width effects are to be included, more than one resonance cascade chain (from now on “resonance history”) may be present, and interference between amplitudes with different resonance histories must also be included. We thus need to perform a separation of the cross section into a sum of contributions, each one of them dominating only for a single resonance history. For each of these contributions we should apply the resonance aware subtraction method of ref. [[@Campbell:2014kua]]{}.
In the following we will describe in great detail the procedure adopted for the separation of the cross section contributions into terms with a definite resonance structure. We will discuss the procedure for the terms that have the Born kinematics (i.e. the Born, Virtual and Collinear Remnant terms), and for real terms. For the latter, subtraction terms having the Born kinematics are also present. We will require that in the collinear and soft limits the separation of the contributions associated with given resonance histories in the real term smoothly matches the corresponding separation in the Born kinematics.
The Born resonance histories
----------------------------
We need to single out contributions from the Born term corresponding to several different resonance histories of the final state. Each resonance history corresponds to a [[**[tree graph]{}**]{}]{}, where the [[**[leaves]{}**]{}]{} of the tree are the final state particle, and the [[**[intermediate nodes]{}**]{}]{} are the resonances. In our case, we include in the tree also the two initial state particles, before the root node.
The root of the tree does not correspond necessarily to any real resonance. For uniformity of treatment, we will however associate to the root a fictitious resonance, and we will refer to it as the “production resonance”.
For each given initial and final flavour configurations, we have several possible resonance histories. We will denote with $F_b$ the initial and final flavour structure of the Born process, irrespective of the internal nodes of the resonance history. We will instead denote with $f_b$ the flavour structure including the resonances decay cascade. We will also refer to it as the [[**[resonance history]{}**]{}]{}. Summarizing, we will refer to $F_b$ as the [[**[bare flavour structure]{}**]{}]{} of the process, and to $f_b$ as the [[**[full]{}**]{}]{} [[**flavour**]{}]{} [[**structure**]{}]{}, or simply as the [[**flavour structure**]{}]{}.
The Born contributions will be labeled as $B_{F_b}$. Thus, $B_{F_b}$ is the square of the amplitude for the production of the final state $F_b$, including all possible resonance histories allowed for the process. We separate the Born contribution in the following way: $$B_{F_b} = \sum_{f_b \in T (F_b)} B_{f_b}, \hspace{2em} B_{f_b} = \Pi_{f_b}
B_{F_b},$$ where $T (F)$ is the set of all trees having the same bare flavour structure $F$. The factors $\Pi_{f_b}$ have the property $$\sum_{f_b \in T (F_b)} \Pi_{f_b} = 1 . \label{eq:pifbsum}$$ Furthermore, they must be such that $\Pi_{f_b} B_{F_b}$ must have resonance peaks compatible with the resonance history of $f_b$. One possible definition for the $\Pi_{f_b}$ is the following. With each resonance $i$ in the resonance history, we associate the factor $$\frac{M_i^4}{(s_i - M_i^2)^2 + \Gamma_i^2 M_i^2},$$ and define $$P^{f_b} = \prod_{i \in {\ensuremath{\operatorname{Nd}}} (f_b) } \frac{M_i^4}{(s_i - M_i^2)^2 +
\Gamma_i^2 M_i^2},$$ where $s_i$, $M_i$ and $\Gamma_i$ are respectively the invariant mass of the decay product system, the mass of the resonance and its width. By ${\ensuremath{\operatorname{Nd}}} (f_b)$ we denote the set of all nodes of the resonance tree for $f_b$ (excluding the root). We then define $$\Pi_{f_b} = \frac{P^{f_b}}{\sum_{f_b' \in T (F_b (f_b))} P^{f_b'}},$$ where we have introduced the notation $F_b (f_b)$ to denote the bare flavour structure associated to a given full flavour structure $f_b$. This definition clearly satisfies the property (\[eq:pifbsum\]). Thus $B_{f_b}$ exhibits resonance peaks only in correspondence with resonances in its own resonance history. In fact the $P^{f'_b}$ factors for all alternative resonance histories in the denominator of $\Pi_{f_b}$ cancel the resonance peaks due to alternative resonance histories in $B_{F_b}$. Only the peaks compatible with the $f_b$ resonance structure, that have a corresponding enhancement factor in the numerator, will remain.
It is worth pointing out that our definition of the $\Pi$ factor is certainly not unique. In particular, there is an alternative possibility that is easily implemented if one has access to the individual sub-amplitudes contributing to the total amplitude characterized by $F_b$: $$ B_{F_b} = \Bigl|^{} \sum_i \mathcal{A}_i \Bigr|^2 .$$ The structure of each sub-amplitude represents in this case a resonance history, so that we can create a correspondence $i \leftrightarrow f_b$, and define $$P^{f_b} = | \mathcal{A}_{f_b} |^2 .$$ This possibility may prove convenient with current numerical matrix elements programs, where the numerical calculation of the individual amplitude is a necessary step for the computation of the full matrix element. Since this procedure is gauge dependent, care should be taken in the choice of an appropriate gauge.
### Implementation of the Born resonance histories in the [[****]{}]{}
The internal implementation of the Born flavour structure can be inherited from the present Born level structure in the , starting with the extension of ref. [[@Campbell:2014kua]]{} for the inclusion of narrow width resonances. In this implementation, the full flavour structure of a Born term is represented by two arrays, and , where the index labels the particular Born full flavour structure $f_b$. The index labels the external leg and the internal resonances, with 1 and 2 representing the incoming legs, and the (integer) value of the array represents the corresponding flavour code (that coincides with the PDG code, except for gluons, that are labeled 0). The integer array represents the resonance pointers, so that the whole resonance structure can be reconstructed. For example, for the case of the full flavour structure corresponding to the process $g g \rightarrow (t \rightarrow (W^+ \rightarrow
e^+ \nu_e) b) ( \bar{t} \rightarrow (W^- \rightarrow {\mu}^-
\bar{\nu}_{{\mu}}) \bar{b} )$, we have
.
We see that the resonance pointer list contains zero for particles generated at the production stage (in `POWHEG` we represent the fictitious production stage resonance as having index 0), while for particles produced in resonance decays the corresponding entry is the position of mother resonance in the list.
At variance with the V2 implementation, in the present case we must be prepared to assume that not all Born flavour configurations have the same resonance history and the same number of resonances, so we must admit Born flavour lists of different length. We thus introduce an array , carrying the length of the flavour list for the Born $f_b$ labeled by the index. The entries of this array are set in the user process subroutine.
The `POWHEG BOX` integration program (the integrator [[@Nason:2007vt]]{}) was updated in order to deal with the resonance histories. Since several resonance histories may be present, the integrator was also updated to be able to deal with a discrete (summation) variable. It now computes a multidimensional integral in a unit hypercube and the summation over a discrete index. The discrete index is used to label each resonance history. The phase space generator examines the value of this discrete index, identifies the corresponding resonance history, and chooses automatically a phase space parametrization that performs importance sampling over the resonance regions, generating the resonance virtualities with an appropriate Breit-Wigner distribution.
The real resonance histories and singular regions
-------------------------------------------------
In the case of real graphs, we have more resonance histories, because we have one more final state particle that can belong to resonances. In analogy with the Born case, we introduce $F_r$ and $f_r$ as before, labeling the bare and full flavour structure for a real graph. We will now introduce a label $\alpha_r$, that labels a singular region[^3] compatible (in a sense that we will specify in the following) with a given $f_r$ $$\alpha_r \in {\ensuremath{\operatorname{Sr}}} (f_r) .$$ Also here we will use the notation $f_r (\alpha_r)$ and $F_r (\alpha_r)$ to denote the full and bare flavour structure associated with a given singular region.
We only consider singular regions that are compatible with the given resonance history in the following sense: [*the particles that become collinear should be siblings, i.e. should arise directly from the decay of the same resonance or from the root (if they are directly produced in the hard reaction)*]{}.
We now perform the separation of the real cross section with a given bare flavour structure into singular region contributions: $$R_{\alpha_r} = \frac{P^{f_r (\alpha_r)} d^{- 1} (\alpha_r)}{\sum_{f_r'
{}\in T (F_r (\alpha_r))} P^{f'_r} \sum_{\alpha'_r \in {\ensuremath{\operatorname{Sr}}}
(f'_r)} d^{- 1} (\alpha_r')} R_{F_r (\alpha_r)}, \label{eq:realpartition}$$ where $F_r (\alpha_r)$ stands for the bare flavour structure associated with $\alpha_r$. We require that the real weights $P^{f_r (\alpha_r)}$ are compatible with the Born weights, in the sense that, in the soft or collinear limit, the $P^{f_r (\alpha_r)}$ must approach smoothly a $P^{f_b}$ factor of the corresponding underlying Born. This is certainly the case if they are defined as in the Born case.
We notice that in the standard `POWHEG` scheme, the real contribution to a given region is enhanced if the collinear pair has a smaller transverse momentum than all other possible collinear pairings. In the present scheme, the relative transverse momentum of the pair is no longer the only element that decides about the partition of singular regions. As an example, consider three final state partons $i$, $j$ and $k$. The cross section is parted among the $i,j$ and $i,k$ singular regions, depending upon how small are the relative transverse momenta in the two cases, and how far from the resonance peaks are the resonances containing respectively the $i,j$ and $i,k$ partons.
The $d^{- 1}$ factors used in the have the form $$\begin{aligned}
\label{eq:didef}
d_i & = & [E_i^2 (1 - \cos^2 \theta_i)]^b, \\ \label{eq:dipmdef}
d_i^{\pm} & = & [E_i^2 2 (1 \pm \cos \theta_i)]^b, \\ \label{eq:dijdef}
d_{i j} & = & \left[ \frac{E_i^2 E^2_j}{(E_i + E_j)^2} (1 - \cos \theta_{i
j}) \right]^b, \end{aligned}$$ where $b$ is a positive constant parameter. Eq. (\[eq:didef\]) is used for the collinear region characterized by parton $i$, with energy $E_i$ and angle $\theta_i$ (relative to the beam) in the partonic CM, becoming collinear to either incoming hadrons. Eq. (\[eq:dipmdef\]) is again for initial state collinear regions, but distinguishes among the two collinear directions. Eq. (\[eq:dijdef\]) is used for the region characterised by final state partons $i$ and $j$ becoming collinear. They are commonly evaluated in the partonic rest frame. In the present case, however, in case of final state singularities associated with the decay products of a resonance, it seems more natural to compute them in the resonance rest frame. They thus become dependent upon the full flavour structure $f_r$ of the real contribution. It is however important for the following developments that the $d_{i j}$ factors do not depend upon the resonance structure in the collinear limit. This is in fact the case with our definition, since $$\lim_{i j} d_{i j} = \lim_{{\ensuremath{\operatorname{coll}}}} \left[ \frac{E_i E_j}{(E_i +
E_j)^2} k_i \cdot k_j \right]^b \Longrightarrow [z (1 - z) k_i \cdot k_j]^b$$ where $\lim_{i j}$ denotes the limit for particles $i$ and $j$ becoming collinear and $z$ is the energy fraction. The last expression is obviously Lorentz invariant in the collinear limit. Thanks to this property, it will turn out that the sum of all $R_{\alpha_r}$ associated with the same underlying Born full flavour structure factorizes in the collinear limit[^4] $$\lim_{i j} \sum_{\alpha_r {}{}}^{f_b (\alpha_r) = f_b}
R_{\alpha_r} \propto B_{f_b} \times P_{i j} (z) .$$ This follows from the fact that all (and only) the $\alpha_r$ associated with particles $i, j$ becoming collinear dominate in this limit, and, being all equal, they simplify out in the numerator and denominator of eq. (\[eq:realpartition\]). We emphasize, however, that the $d_{ij}$ terms are not frame independent in the soft limit. This is quite clear from eq. (\[eq:dijdef\]), that in the $E_i\to 0$ limit becomes $$\label{eq:dijsoftlim}
d_{ij}\approx \left[\frac{E_i}{E_j} k_i\cdot k_j \right]^b,$$ that is clearly frame dependent.
As in the Born case, the scheme discussed here is not the only alternative for the partition of the singular regions and of the resonance structure. Using weights equal to the square of individual sub-amplitudes is still a valid alternative, as long as one computes the amplitudes in a physical gauge, in such a way that squared amplitudes also retain the full collinear singularity structure. In this case one does not need to introduce the $d_{i j}$ factors, since the squared amplitudes already have the appropriate singular behaviour in the collinear limit. In order to further pursue this alternative, issues related to the lack of gauge invariance of the individual amplitudes squared should be addressed. In the present work we did not investigate this alternative any further, since we prefer to assume that in general the individual amplitude for the process may not be available.
Example: electroweak $u \bar{u} \rightarrow u \bar{d} \bar{u} d$
----------------------------------------------------------------
We illustrate the separation of the resonance structures in the process $u \bar{u}
\rightarrow u \bar{d} \bar{u} d$, considering only electroweak interactions. In order to simplify the discussion, we will (wrongfully!) assume that only the diagrams illustrated in fig. \[fig:WWfeynman\] contribute to it. We remark that this process is chosen only for illustration purposes. We are aware of the fact that it has no physical relevance and that we are omitting other relevant resonance histories.
![\[fig:WWfeynman\] Feynman diagrams for $u \bar{u} \rightarrow u \bar{d} \bar{u} d$.](WW-feynman)
There is only one $F_b$, corresponding to the bare flavour structure $u
\bar{u} \rightarrow u \bar{d} \bar{u} d$. We have two $f_b$, represented in fig. \[fig:WWborntrees\], corresponding respectively to $u \bar{u}
\rightarrow (W^+ \rightarrow u \bar{d}) (W^- \rightarrow \bar{u} d)$ and $u
\bar{u} \rightarrow (Z \rightarrow u \bar{u}) (Z \rightarrow d \bar{d})$.
![Trees for $u \bar{u} \rightarrow u \bar{d} \bar{u} d$. \[fig:WWborntrees\]](WW-trees)
The $P$ factors for the two configurations are $$\begin{aligned}
P^1_b & = & \frac{M_W^4}{(s_{34} - M_W^2)^2 + \Gamma_W^2 M_W^2} \times
\frac{M_W^4}{(s_{56} - M_W^2)^2 + \Gamma_W^2 M_W^2}\,,\\
P^2_b & = & \frac{M_Z^4}{(s_{35} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2} \times
\frac{M_Z^4}{(s_{46} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2}\, .\end{aligned}$$ Notice that we have assigned the values 1 and 2 to the $f_b$ index of the two flavour configurations depicted in the figure. Particles are labeled by an integer, starting from the lower incoming line, and going through all other particles clockwise. Summarizing, we have two (full) flavour structures for the given bare flavour structure $u \bar{u} \rightarrow u \bar{d} \bar{u} d$. The corresponding Born contributions will be given by $$B_1 = \frac{P^1_b B}{D_b}, \hspace{2em} B_2 = \frac{P^2_b B}{D_b},$$ with $$D_b = P_b^1 + P_b^2 .$$ Notice that $B$ is the full Born contribution, given by the square of the sum of the graphs in fig. \[fig:WWfeynman\]. However, $B_1$ will be dominated by the square of the first graph, and $B_2$ by the second.
The number of real graphs is already quite large, and we do not show the corresponding figures. They are obtained by adding one final state gluon to the Born flavour configuration, and by replacing one of the initial lines with a gluon, adding a corresponding quark of opposite flavour to the final state. Here we focus upon the bare flavour configuration $u \bar{u}
\rightarrow u \bar{d} \bar{u} d g$. The corresponding full flavour configuration trees are depicted in fig. \[fig:WWrealtrees\].
![Trees for $u \bar{u} \rightarrow u \bar{d} \bar{u} d g$.\[fig:WWrealtrees\]](WW-trees-real)
We will now label the gluon as $7$, and keep the same labels used in the Born case for all other particles. The $P$ factors are now $$\begin{aligned}
P^1_r & = & \frac{M_W^4}{(s_{34} - M_W^2)^2 + \Gamma_W^2 M_W^2} \times
\frac{M_W^4}{(s_{56} - M_W^2)^2 + \Gamma_W^2 M_W^2}\,,\\
P^2_r & = & \frac{M_W^4}{(s_{347} - M_W^2)^2 + \Gamma_W^2 M_W^2} \times
\frac{M_W^4}{(s_{56} - M_W^2)^2 + \Gamma_W^2 M_W^2}\,,\\
P^3_r & = & \frac{M_W^4}{(s_{34} - M_W^2)^2 + \Gamma_W^2 M_W^2} \times
\frac{M_W^4}{(s_{567} - M_W^2)^2 + \Gamma_W^2 M_W^2}\,,\\
P^4_r & = & \frac{M_Z^4}{(s_{35} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2} \times
\frac{M_Z^4}{(s_{46} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2} \,,\\
P^5_r & = & \frac{M_Z^4}{(s_{357} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2} \times
\frac{M_Z^4}{(s_{46} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2}\,,\\
P^6_r & = & \frac{M_Z^4}{(s_{35} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2} \times
\frac{M_Z^4}{(s_{467}^{} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2}\,.\end{aligned}$$ The singular regions $\alpha_r$ are displayed in tab. \[tab:alr-regions-example\].
$\alpha_r$ $f_r$ emitter $d^{- 1} (\alpha_r)$
------------ ------- --------- ----------------------
1 1 0 $d^{- 1}_7$
2 2 3 $d^{- 1}_{37,2}$
3 2 4 $d^{- 1}_{47,2}$
4 3 5 $d^{- 1}_{57,3}$
5 3 6 $d^{- 1}_{67,3}$
6 4 0 $d^{- 1}_7$
7 5 3 $d^{- 1}_{37,5}$
8 5 4 $d^{- 1}_{57,5}$
9 6 5 $d^{- 1}_{47,6}$
10 6 6 $d^{- 1}_{67,6}$
: \[tab:alr-regions-example\]
Notice that the final state radiation $d$ factors carry in the subscript the position of the two partons that become collinear, and, after a comma, an index specifying the resonance history. We are in fact assuming that the $d$ factors are computed in the frame of the resonance that owns the two collinear partons. Notice also that the standard (non resonance aware) `POWHEG` implementation would have found 5 regions, one for the initial state radiation, and 4 for final state radiation, corresponding to a gluon being emitted by each final state parton.
It is interesting to see how the singular part of the cross section is shared among the various resonance histories. We consider as an example the gluon emission from particle 3, carrying the $d^{- 1}_{37}$ singularity. In the standard `POWHEG` formulation this region corresponds to a single $\alpha_r$. On the other hand, in our resonance aware extension, that singularity is shared by the $\alpha_r$ number 2 and 7. The one of the two that is more enhanced by resonant propagators (i.e. by its $P$ factor) will dominate over the other. We have $$\begin{aligned}
D_r & = & P^1_r d^{- 1}_7 + P^2_r (d^{- 1}_{37 ,2} + d^{- 1}_{47 ,2}) +
P^3_r (d^{- 1}_{57 ,3} + d^{- 1}_{67 ,3}) \nonumber\\
& + & P^4_r d^{- 1}_7 + P^5_r (d^{- 1}_{37 ,5} + d^{- 1}_{57 ,5}) + P^6_r
(d^{- 1}_{47 ,6} + d^{- 1}_{67 ,6}), \\
R_2 & = & \frac{P^2_r d^{- 1}_{37 ,2} }{D_r} R, \\
R_7 & = & \frac{P^5_r d^{- 1}_{37 ,5} }{D_r} R. \end{aligned}$$ Notice that near the 3,7 collinear singularity, we have $$\begin{aligned}
R_2 & = & \frac{P^2_r d^{- 1}_{37 ,2} }{D_r} R \; \cong \; \frac{P^2_r d^{-
1}_{37 ,2}}{P^2_r d^{- 1}_{37 ,2} + P^5_r d^{- 1}_{37 ,5}} \, R \; \cong
\; \frac{P^2_r}{P^2_r + P^5_r} R, \\
R_7 & = & \frac{P^5_r d^{- 1}_{37 ,5} }{D_r} R \; \cong \; \frac{P^5_r d^{-
1}_{37 ,5}}{P^2_r d^{- 1}_{37 ,2} + P^5_r d^{- 1}_{37 ,5}} R \; \cong \;
\frac{P^5_r}{P^2_r + P^5_r} R, \end{aligned}$$ where the last equality follows from our requirement that the $d$ factors are Lorentz invariant in the collinear limit. We thus see that in the collinear limit the collinear contribution is distributed among the 2 and 5 resonance histories, favouring the one that is nearer the resonance peaks.
The underlying Born corresponding to the real flavour configurations $f_r \in
\{ 1, 2, 3 \}$ is the Born flavour configuration $f_b = 1$. In the limit of vanishing momentum of the additional radiation in leg 7, i.e. when the momenta on the legs 1–6 in the real diagrams can be mapped to a given set of momenta of its underlying Born diagram, all the $P_r$ factors of the real flavour configuration reduce to the $P_b$ factors of the corresponding underlying Born contributions. In our case: $$P_r^1 \rightarrow P^1_b, \hspace{1em} P_r^2 \rightarrow P^1_b, \hspace{1em}
P_r^3 \rightarrow P^1_b ; \hspace{2em} P_r^4 \rightarrow P^2_b, \hspace{1em}
P_r^5 \rightarrow P^2_b, \hspace{1em} P_r^6 \rightarrow P^2_b .$$ This implies that in the same limit: $$D_r = P^1_b (d^{- 1}_7 + d^{- 1}_{37 ,2} + d^{- 1}_{47 ,2} + d^{- 1}_{57
,3} + d^{- 1}_{67 ,3}) + P^2_b (d^{- 1}_7 + d^{- 1}_{37 ,5} + d^{- 1}_{47
,5} + d^{- 1}_{57 ,6} + d^{- 1}_{67 ,6}),$$ so that, in the soft limit, for example $$R_2 \; \cong \; \frac{P^1_b d^{- 1}_{37 ,2} }{P^1_b (d^{- 1}_7 + d^{-
1}_{37 ,2} + d^{- 1}_{47 ,2} + d^{- 1}_{57 ,3} + d^{- 1}_{67 ,3}) +
P^2_b (d^{- 1}_7 + d^{- 1}_{37 ,5} + d^{- 1}_{47 ,5} + d^{- 1}_{57 ,6} +
d^{- 1}_{67 ,6})},$$ and a similar relation holds for all other $\alpha_r$ contributions. As shown in eq. (\[eq:dijsoftlim\]) we cannot drop the resonance history dependence in the $d_{ij}$ factors. Thus, unlike the case of collinear singularities, in the soft limit a full factorization of the $P$ and $d^{- 1}$ factors does not hold in general. We will see in the following sections that this fact leads to a minor complication in the evaluation of the soft contribution.
Soft-collinear contributions {#sec:softcoll}
============================
In the implementation of narrow resonance decays, the soft collinear contributions (to be added to the virtual one) are computed assuming that no interference terms arise between the different resonances (or between a resonance and the direct production). In the finite width case we are considering now, this restriction has to be removed, because such interference terms do arise. Furthermore, the soft-collinear contributions depend upon the adopted subtraction procedure, and we are now departing from the default one used in the . We thus need to discuss in detail and compute the soft-collinear contributions in the present framework.
As specified previously, each singular region $\alpha_r$ is associated with a single full flavour structure $f_r (\alpha_r)$. The treatment of initial state singularities remain the same as in the standard case, since no resonance decays are involved in ISR (initial state radiation). We thus focus upon FSR (final state radiation). Thus, from now on, the singular region $\alpha_r$ corresponds to final state particles $i$ and $j$ becoming collinear. Furthermore, the soft singularity is associated with particle $i$ becoming soft. This is the case when $i$ is a gluon and $j$ is a quark. If also $j$ is a gluon, in the $R_{\alpha_r}$ contribution is multiplied by a factor of the form $2 h (E_j) / (h (E_i) + h(E_j))$, where $h$ is typically a power. This factor damps the soft singularity when particle $j$ becomes soft. Since the cross section is symmetric in the exchange of the two gluons, this procedure leads to the correct result.
Given the singular region, selects a phase space mapping from the Born phase space $\Phi_B$ and the three radiation variables $\xi_i$, $y_{i j}$ and $\phi$ to the full real emission phase space. In the $\alpha_r$ region, partons $i$ and $j$ will arise from the same resonance $k_{{\ensuremath{\operatorname{res}}}}$. The phase space mapping will thus be chosen in such a way that only the momenta of the decay product of the resonance will be affected. For example, in case of partons $i$ and $j$ corresponding to a gluon and a $b$ quark arising from top decay, the phase space mapping will build the radiation phase space starting from the momenta of the top decay products (i.e. the $b$ and the $W^+$), maintaining fixed all remaining momenta together with the top four-momentum. The mapping procedure will correspond to the prescription described in the FNO paper (ref. [[@Frixione:2007vw]]{}), applied to the top decay product in the rest frame of the top. It is the same procedure that is applied in ref. [[@Campbell:2014kua]]{} for the case of $t
\bar{t}$ production and decay in the factorized approach.
Following the notation of the FNO paper, we thus write the phase space as $$\begin{aligned}
{\mathrm{d}}\Phi_{n + 1} & = & (2 \pi)^d \delta^d \hspace{-0.17em}
\hspace{-0.17em} \left( k_{\oplus} + k_{\ominus} - \sum_{l = 1}^{n + 1} k_l
\right) \hspace{-0.17em} \left[ \prod_{l = 1}^{n + 1} {\mathrm{d}}\Phi_l
\right], \nonumber\\
{\mathrm{d}}\Phi_l & = & \frac{{\mathrm{d}}^{d - 1} k_l}{2 k_l^0 (2 \pi)^{d - 1}} .
\label{eq:ddimphasespace}\end{aligned}$$ where $d = 4 - 2 \epsilon$ is the dimensionality of spacetime. Furthermore, we introduce the parametrization $${\mathrm{d}}\Phi_i = \frac{{\mathrm{d}}^{d - 1} k_i}{2 k_i^0 (2 \pi)^{d - 1}} =
\frac{(k_{{\ensuremath{\operatorname{res}}}}^2)^{1 - \epsilon}}{(4 \pi)^{3 - 2 \epsilon}}
\hspace{0.17em} \xi^{1 - 2 \epsilon} {\mathrm{d}}\xi \hspace{0.17em} {\mathrm{d}}\Omega^{3 - 2 \epsilon},$$ $$$$ where $\xi$ is defined as $$\xi = \frac{2 k^0_i}{\sqrt{k_{{\ensuremath{\operatorname{res}}}}^2}} \hspace{0.17em},$$ computed in the rest frame of resonance $k_{{\ensuremath{\operatorname{res}}}}$. In the soft limit, the phase space becomes $${\mathrm{d}}\Phi_{n + 1} \Rightarrow {\mathrm{d}}\Phi_B \frac{(k_{{\ensuremath{\operatorname{res}}}}^2)^{1 -
\epsilon}}{(4 \pi)^{3 - 2 \epsilon}} \hspace{0.17em} \xi^{1 - 2 \epsilon}
{\mathrm{d}}\xi \hspace{0.17em} {\mathrm{d}}\Omega^{3 - 2 \epsilon}$$ where, following the notation of FNO, the underlying Born kinematics is expressed in terms of the barred variables, and ${\mathrm{d}}\Phi_B$ is the underlying Born phase space.
The $\alpha_r$ contribution to the cross section can be written as $$\int R_{\alpha_r} {\mathrm{d}}\Phi_{n + 1} .$$ We now introduce the expansion $$\xi^{- 1 - 2 \epsilon} = - \frac{1}{2 \epsilon} \delta (\xi) + \xi^{- 1 - 2
\epsilon}_+,$$ where $$\xi^{- 1 - 2 \epsilon}_+ = \left( \frac{1}{\xi} \right)_+ - 2 \epsilon
\left( \frac{\log \xi}{\xi} \right)_+ + \ldots,$$ i.e. is defined as a distribution with a vanishing integral between 0 and 1. We get $$\int R_{\alpha_r} {\mathrm{d}}\Phi_{n + 1} = I_{s, \alpha_r} + I_{+, \alpha_r},$$ with $$\begin{aligned}
I_{s, \alpha_r} & = & - \frac{1}{2 \epsilon} \int {\mathrm{d}}\Phi_B
\frac{(k_{{\ensuremath{\operatorname{res}}}}^2)^{1 - \epsilon}}{(4 \pi)^{3 - 2 \epsilon}} {\mathrm{d}}\Omega^{3 - 2 \epsilon} \lim_{\xi \rightarrow 0} [\xi^2 R_{\alpha_r}], \\
I_{+, \alpha_r} & = & \int {\mathrm{d}}\Phi_{n + 1} \frac{\xi^{- 1 - 2
\epsilon}_+}{\xi^{- 1 - 2 \epsilon}_{}} R_{\alpha_r}, \end{aligned}$$ where the meaning of the second equation is simply to replace $\xi^{- 1 - 2
\epsilon}$ with $\xi^{- 1 - 2 \epsilon}_+$ in the real cross section integral, since the corresponding $\delta (\xi)$ contribution has been subtracted out.
Soft terms
----------
We now discuss explicitly the computation of the soft term $I_{s, \alpha_r}$. In the standard treatment, by summing over all singular regions one recovers the full $R$, that can be approximated in the soft limit by the eikonal formula. We cannot follow this procedure now, since it requires that the soft limit is taken in the same frame for all $\alpha_r$, which is not our case. In order to deal with this complication, we employ the following trick. We use the identity $$\int_0^{\infty} {\mathrm{d}}\xi \xi^{- 1 - 2 \epsilon} e^{- \xi} = \Gamma (- 2
\epsilon) = \frac{\Gamma (1 - 2 \epsilon)}{- 2 \epsilon},$$ to rewrite $I_{s, \alpha_r}$ as $$\begin{aligned}
I_{s, \alpha_r} & = & \frac{1}{\Gamma (1 - 2 \epsilon)} \int {\mathrm{d}}\Phi_B
\int_0^{\infty} {\mathrm{d}}\xi \xi^{- 1 - 2 \epsilon} e^{- \xi}
\frac{(k_{{\ensuremath{\operatorname{res}}}}^2)^{1 - \epsilon}}{(4 \pi)^{3 - 2 \epsilon}} {\mathrm{d}}\Omega^{3 - 2 \epsilon} \lim_{\xi \rightarrow 0} [\xi^2 R_{\alpha_r}]
\nonumber\\
& = & \frac{1}{\Gamma (1 - 2 \epsilon)} \int {\mathrm{d}}\Phi_B \int {\mathrm{d}}\Phi_i e^{- \xi} \tilde{R}_{\alpha_r} \nonumber\\
& = & \frac{1}{\Gamma (1 - 2 \epsilon)} \int {\mathrm{d}}\Phi_B \int {\mathrm{d}}\Phi_i e^{- \frac{2 k_i \cdot k_{{\ensuremath{\operatorname{res}}}}}{k_{{\ensuremath{\operatorname{res}}}}^2}}
\tilde{R}_{\alpha_r} \,, \label{eq:isoftres}\end{aligned}$$ where with $\tilde{R}_{\alpha_r}$ we denote the Taylor expansion of $R_{\alpha_r}$ in the soft limit for $k_{_i}$: $$\tilde{R}_{\alpha_r} = \frac{1}{\xi^2} \lim_{\xi \rightarrow 0} [\xi^2
R_{\alpha_r}] .$$ We notice that $\tilde{R}_{\alpha_r}$ is obviously independent from the frame used to define $\xi$. It is in fact obtained from $R_{\alpha_r}$ by linearizing it in the $k_i$ momentum.
In formula (\[eq:isoftres\]) the frame dependence of the soft contribution is all contained in the exponential, the rest of the expression being fully Lorentz invariant. In order to perform the integral we proceed as follows. We rewrite (\[eq:isoftres\]) as $$\begin{aligned}
I_{s, \alpha_r} & = & I_{s, \alpha_r}^{(1)} {}+ I^{(2)}_{s, \alpha_r}
\nonumber\\
I_{s, \alpha_r}^{(1)} & = & \frac{1}{\Gamma (1 - 2 \epsilon)} \int {\mathrm{d}}\Phi_{{\mathrm{B}}} \int {\mathrm{d}}\Phi_i \tilde{R}_{\alpha_r} \left\{ \exp \left[
- \frac{2 k_i \cdot k_{{\ensuremath{\operatorname{res}}}}}{k_{{\ensuremath{\operatorname{res}}}}^2} \right] - \exp \left[
- \frac{2 k_i \cdot m}{m^2} \right] \right\} \nonumber\\
I^{(2)}_{s, \alpha_r} & = & \frac{1}{\Gamma (1 - 2 \epsilon)} \int {\mathrm{d}}\Phi_{{\mathrm{B}}} \int {\mathrm{d}}\Phi_i \tilde{R}_{\alpha_r} \exp \left[ - \frac{2
k_i \cdot m}{m^2} \right], \label{eq:softsplit}\end{aligned}$$ where $m$ is an arbitrary timelike momentum. For definiteness, we choose $m =
q$, the total four-momentum of the final state particles. The $I_{s,
\alpha_r}^{(1)}$ term in eq. (\[eq:softsplit\]) is infrared finite. The $I^{(2)}_{s, \alpha_r}$ term can now be integrated in any frame we like. We then just pick a common frame for all $\alpha_r$ that have the same underlying Born bare flavour structure $F_b$, and sum over all of them. We get $$\sum_{F_b (\alpha_r) = F_b} I^{(2)}_{s, \alpha_r} = \int {\mathrm{d}}\Phi_{{\mathrm{B}}}
\frac{1}{\Gamma (1 - 2 \epsilon)} \int {\mathrm{d}}\Phi_i \exp \left[ - \frac{2
k_i \cdot m}{m^2} \right] \sum_{F_b (\alpha_r) = F_b} \tilde{R}_{\alpha_r} .
\label{eq:softint0}$$ In the sum of $\tilde{R}_{\alpha_r}$, all dependencies upon the partition of the resonance regions and upon the $d^{- 1}$ factors cancel out, yielding the full $\tilde{R}$ for a soft gluon emission with an underlying Born flavour configuration equal to $F_b$. In fact, the bare flavour structure of the $\alpha_r$ such that $F_b (\alpha_r) = F_b$ consist of the same flavour assignment $F_b$ plus one (soft) gluon. Thus, from eq. (\[eq:realpartition\]), it follows that in the sum in eq. (\[eq:softint0\]) all resonance history and $d^{- 1}$ factors simplify. We thus have $$\sum_{F_b (\alpha_r) = F_b} \tilde{R}_{\alpha_r} = 4 \pi \alpha_S
{\mu}_R^{2 \epsilon} \left[ \sum_{l m} B_{l m}^{(F_b)} \frac{k_l \cdot
k_m}{(k_l \cdot k_i ) (k_m \cdot k_i)} - B^{(F_b)} \sum_l \frac{k_l^2}{(k_l
\cdot k_i)^2} C_l \right] .$$ Performing the integration in the rest frame of $m$, we can use again the replacement $$\xi^{- 1 - 2 \epsilon} e^{- \xi} \Rightarrow \frac{\Gamma (1 - 2
\epsilon)}{- 2 \epsilon} \delta (\xi),$$ that leads to the standard calculation of the soft contributions, regardless of their resonance assignment. The corresponding result is reported in Appendix A.1 of the paper (ref. [[@Alioli:2010xd]]{}).
At this stage, we can split the Born terms again in terms of their full flavour structures, and thus compute each contribution using the appropriate (importance sampled according to the resonance structure) phase space.
The treatment of the $I_{s, \alpha_r}^{(1)}$ term of eq. (\[eq:softsplit\]) requires some care, since although the soft singularity is no longer there, it has still a collinear singularity corresponding to the $\alpha_r$ region. We evaluate the $I_{s, \alpha_r}^{(1)}$ integral in the CM rest frame. Other choices are possible, but there is no reason to make more complex choices, since the resonance virtualities in $\tilde{R}_{\alpha_r}$ do not depend upon the soft momentum $k_i$, and thus no particular importance sampling is needed in the $k_i$ integration. We thus write $$\begin{aligned}
I^{(1)}_{s, \alpha_r} & = & \int {\mathrm{d}}\Phi_{{\mathrm{B}}} \frac{1}{\Gamma (1 - 2
\epsilon)} \int \frac{s^{1 - \epsilon}}{(4 \pi)^{3 - 2 \epsilon}}
\hspace{0.17em} \xi^{1 - 2 \epsilon} (1 - y^2)^{- \epsilon} {\mathrm{d}}\xi
\hspace{0.17em} {\mathrm{d}}y \hspace{0.17em} d \Omega^{2 - 2 \epsilon}
\nonumber\\
& \times & \tilde{R}_{\alpha_r} \left\{ \exp \left[ - \frac{2 k_i \cdot
k_{{\ensuremath{\operatorname{res}}}}}{k_{{\ensuremath{\operatorname{res}}}}^2} \right] - \exp [- \xi] \right\}, \end{aligned}$$ where $$y = \cos \theta_{i j},$$ and $j$ is the emitter associated with the $\alpha_r$ region. By expanding $$\begin{aligned}
(1 - y^2)^{- \epsilon} & = & (1 + y)^{- \epsilon} (1 - y) (1 - y)^{- 1 -
\epsilon} \nonumber\\
& = & (1 + y)^{- \epsilon} (1 - y) \left[ - \frac{2^{-
\epsilon}}{\epsilon} \delta (1 - y) + \left( \frac{1}{1 - y} \right)_+ +
\mathcal{O} (\epsilon) \right] \nonumber\\
& = & (1 - y) \left[ - \frac{2^{- 2 \epsilon}}{\epsilon} \delta (1 - y) +
\left( \frac{1}{1 - y} \right)_+ + \mathcal{O} (\epsilon) \right],
\label{eq:ydistrib}\end{aligned}$$ we can write $$I_{s, \alpha_r}^{(1)} = I^{(1)}_{s +, \alpha_r} + I_{s \delta,
\alpha_r}^{(1)},$$ where $$\begin{aligned}
I^{(1)}_{s +, \alpha_r} & = & \int {\mathrm{d}}\Phi_{{\mathrm{B}}} \int \frac{s^{}
\xi}{(4 \pi)^3} \left( \frac{1}{1 - y} \right)_+ \times (1 - y)
\tilde{R}_{\alpha_r} \left\{ e^{- \frac{2 k_i \cdot
k_{{\ensuremath{\operatorname{res}}}}}{k_{{\ensuremath{\operatorname{res}}}}^2} } - e^{- \xi} \right\} {\mathrm{d}}\xi
\hspace{0.17em} {\mathrm{d}}y \hspace{0.17em} {\mathrm{d}}\phi, \phantom{aaaa} \\
I_{s \delta, \alpha_r}^{(1)} & = & - \int {\mathrm{d}}\Phi_{{\mathrm{B}}}
\frac{1}{\Gamma (1 - 2 \epsilon)} \int \frac{s^{1 - \epsilon}}{(4 \pi)^{3 -
2 \epsilon}} \hspace{0.17em} \xi^{1 - 2 \epsilon} \frac{2^{- 2
\epsilon}}{\epsilon} \delta (1 - y) {\mathrm{d}}\xi \hspace{0.17em} {\mathrm{d}}y
\hspace{0.17em} {\mathrm{d}}\Omega^{2 - 2 \epsilon} \nonumber\\
& \times & \lim_{y \rightarrow 1} [(1 - y) \tilde{R}_{\alpha_r}] \left\{
e^{- \frac{2 k_i \cdot k_{{\ensuremath{\operatorname{res}}}}}{k_{{\ensuremath{\operatorname{res}}}}^2}} - e^{- \xi}
\right\} . \end{aligned}$$ The $I^{(1)}_{s +, \alpha_r}$ term has to to be computed numerically. It has no analogue in the previous implementation.
We now work through the $I_{s \delta, \alpha_r}^{(1)}$. We have $$\lim_{y \rightarrow 1} [(1 - y) \tilde{R}_{\alpha_r}] = \frac{32 \pi
\alpha_s {\mu}^{2 \epsilon} C_{j (f_b)}}{s \xi^2} B_{f_b (\alpha_r)},
\label{eq:collapprox}$$ where $C_j$ is the Casimir invariant associated with the particle that underwent the splitting for the region $\alpha_r$. Observe that in deriving the identity (\[eq:collapprox\]) [[[*we have assumed that in the collinear limit the $d^{- 1}$ factors associated with a given pair of final state particles all coincide, irrespective of the resonance structure*]{}]{}]{}, as we have remarked earlier. Using $${\mathrm{d}}\Omega^{2 - 2 \epsilon} = \frac{2 \pi^{1 - \epsilon}}{\Gamma (1 -
\epsilon)}$$ we get $$\begin{aligned}
I_{s \delta, \alpha_r}^{(1)} & = & - \int {\mathrm{d}}\Phi_{{\mathrm{B}}}
\frac{1}{\Gamma (1 - 2 \epsilon)} \int \frac{s^{1 - \epsilon}}{(4 \pi)^{3 -
2 \epsilon}} \hspace{0.17em} \xi^{1 - 2 \epsilon} \frac{2^{- 2
\epsilon}}{\epsilon} \delta (1 - y) {\mathrm{d}}\xi \hspace{0.17em} {\mathrm{d}}y
\hspace{0.17em} {\mathrm{d}}\Omega^{2 - 2 \epsilon} \nonumber\\
& \times & \frac{32 \pi \alpha_s {\mu}^{2 \epsilon} C_{j (f_b)}}{s
\xi^2} B_{f_b (\alpha_r)} \left\{ \exp \left[ - \frac{2 k \cdot
k_{{\ensuremath{\operatorname{res}}}}}{k_{{\ensuremath{\operatorname{res}}}}^2} \right] - \exp \left[ - \frac{2 k \cdot
m}{m^2} \right] \right\} \nonumber\\
& = & - \frac{(4 \pi)^{\epsilon}}{\Gamma (1 - 2 \epsilon) \Gamma (1 -
\epsilon)} \left( \frac{{\mu}^2}{s} \right)^{\epsilon} \frac{\alpha_s
C_{j (f_b)}}{\pi} \int {\mathrm{d}}\Phi_{{\mathrm{B}}} B_{f_b (\alpha_r)}
\frac{1}{\epsilon} \nonumber\\
& \times & \int {\mathrm{d}}\xi \xi^{- 1 - 2 \epsilon} \left\{ \exp \left[ - \xi
\frac{\sqrt{s} \bar{k}_j \cdot k_{{\ensuremath{\operatorname{res}}}}}{\bar{k}_j^0
k_{{\ensuremath{\operatorname{res}}}}^2} \right] - \exp [- \xi] \right\} \,,\end{aligned}$$ where we have written, in the collinear limit $$k_i = \frac{k_i^0}{\bar{k}^0_j} \bar{k}_j = \frac{\xi \sqrt{s}}{2
\bar{k}^0_j} \bar{k}_j,$$ and $\bar{k}_j$ is the momentum of the emitter in the soft limit, i.e. at the underlying Born level. Performing the $\xi$ integration (from 0 to $\infty$), we get $$\begin{aligned}
I_{s \delta, \alpha_r}^{(1)} & = & - \frac{(4 \pi)^{\epsilon}}{\Gamma (1 - 2
\epsilon) \Gamma (1 - \epsilon)} \left( \frac{{\mu}^2}{s}
\right)^{\epsilon} \frac{\alpha_s C_{j (f_b)}}{\pi} \int {\mathrm{d}}\Phi_{{\mathrm{B}}} B_{f_b (\alpha_r)} \frac{1}{\epsilon} \nonumber\\
& \times & \left[ \left( \frac{\sqrt{s} \bar{k}_j \cdot
k_{{\ensuremath{\operatorname{res}}}}}{\bar{k}_j^0 k_{{\ensuremath{\operatorname{res}}}}^2} \right)^{2 \epsilon} \Gamma (-
2 \epsilon) - \Gamma (- 2 \epsilon) \right] \nonumber\\
& = & \frac{\mathcal{N}}{2 \epsilon^2} \left( \frac{Q^2}{s}
\right)^{\epsilon} \frac{\alpha_s C_{j (f_b)}}{\pi} \int {\mathrm{d}}\Phi_{{\mathrm{B}}} B_{f_b (\alpha_r)} \left[ \left( \frac{\sqrt{s} \bar{k}_j
\cdot k_{{\ensuremath{\operatorname{res}}}}}{\bar{k}_j^0 k_{{\ensuremath{\operatorname{res}}}}^2} \right)^{2 \epsilon} - 1
\right] \nonumber\\
& = & \frac{\mathcal{N}}{2 \epsilon^2} \frac{\alpha_s C_{j (f_b)}}{\pi}
\int {\mathrm{d}}\Phi_{{\mathrm{B}}} B_{f_b (\alpha_r)} \left[ \left( \frac{Q \bar{k}_j
\cdot k_{{\ensuremath{\operatorname{res}}}}}{\bar{k}_j^0 k_{{\ensuremath{\operatorname{res}}}}^2} \right)^{2 \epsilon} -
\left( \frac{Q^2}{s} \right)^{\epsilon} \right] \nonumber\\
& = & \mathcal{N} \frac{\alpha_s C_{j (f_b)}}{\pi} \int {\mathrm{d}}\Phi_{{\mathrm{B}}} B_{f_b (\alpha_r)} \nonumber\\
& \times & \left[ \frac{1}{\epsilon} \log
\frac{\sqrt{s} \bar{k}_j \cdot k_{{\ensuremath{\operatorname{res}}}}}{\bar{k}_j^0
k_{{\ensuremath{\operatorname{res}}}}^2} + \left( \log^2 \frac{Q \bar{k}_j \cdot
k_{{\ensuremath{\operatorname{res}}}}}{\bar{k}_j^0 k_{{\ensuremath{\operatorname{res}}}}^2} - \log^2 \frac{Q}{\sqrt{s}}
\right) \right] \,,\end{aligned}$$ where we have introduced the common normalization factor $$\mathcal{N} = \frac{(4 \pi)^{\epsilon}}{\Gamma (1 - \epsilon)} \left(
\frac{{\mu}^2}{Q^2} \right)^{\epsilon} . \label{eq:normfact}$$ The normalization factor in (\[eq:normfact\]) should be the same one adopted in the computation of the virtual term, as defined in the FNO and papers. If this is the case, the $1 / \epsilon$ singularities in the virtual and soft virtual contributions cancel exactly, and one needs only to retain the finite terms.
Collinear terms
---------------
We now turn to the collinear integral $$\begin{aligned}
I_{+, \alpha_r} & = & \int {\mathrm{d}}\Phi_{n + 1} \frac{\xi^{- 1 - 2
\epsilon}_+}{\xi^{- 1 - 2 \epsilon}_{}} R_{\alpha_r} . \end{aligned}$$ We are considering the region where $i, j$ become collinear, and where, as discussed previously, $k_j$ does not lead to a soft singularity. This is the case for $g q$ pairs $i$ should correspond to $g$. In the $g g$ case, the $d^{- 1}_{i j}$ factors are supplemented with an energy damping factor $$\frac{2 h (E_{j_{}})}{h (E_{i_{}}) + h (E_{j_{}})},$$ where, as in ref. [[@Frixione:2007vw]]{}, $h$ is typically defined to be a simple power law. Since there is no soft singularity in $k_j$, in the following we can assume that we have a lower cutoff on the $k_j$ energy, that can be smoothly removed at the end of the calculation, so that in our manipulation we can always assume that $k_j$ is not vanishingly small. We now write the $I_{+, \alpha_r}$ term as $$\begin{aligned}
I_{+, \alpha_r} & = & \int (2 \pi)^d \delta^d \hspace{-0.17em}
\hspace{-0.17em} \left( k_{\oplus} + k_{\ominus} - \sum_{l = 1}^{n + 1} k_l
\right) \hspace{-0.17em} \left[ \prod_{l \neq j, i} {\mathrm{d}}\Phi_l \right]
{\mathrm{d}}\Phi_j {\mathrm{d}}\Phi_i \frac{\xi^{- 1 - 2 \epsilon}_+}{\xi^{- 1 - 2
\epsilon}_{}} R_{\alpha_r}, \end{aligned}$$ where we remind that $\xi$ is evaluated in the frame of the resonance to which $i$ and $j$ belong. We introduce for $\Phi_i$ the phase space (always defined in the rest frame of the resonance) $${\mathrm{d}}\Phi_i = \frac{(k^0_i)^{1 - 2 \epsilon}}{2 (2 \pi)^{3 - 2 \epsilon}}
{\mathrm{d}}k^0_i (1 - y^2)^{- \epsilon} \hspace{0.17em} {\mathrm{d}}y \hspace{0.17em}
{\mathrm{d}}\Omega_i^{2 - 2 \epsilon},$$ where the angular integration is done with respect to the $j$ direction, i.e. $y = 1 - \cos \theta_{i j}$. We separate out the collinear divergent term by using eq. (\[eq:ydistrib\]), that yields $$\begin{aligned}
I_{+, \alpha_r} & = & I_{+ \delta, \alpha_r} + I_{+ +, \alpha_r},
\nonumber\\
I_{+ +, \alpha_r} & = & \int {\mathrm{d}}\Phi_{n + 1} (1 - y) \left( \frac{1}{1 -
y} \right)_+ \xi \left( \frac{1}{\xi} \right)_+ R_{\alpha_r},
\label{eq:Iplusplus}\end{aligned}$$ and $$\begin{aligned}
I_{+ \delta, \alpha_r} & = & \int (2 \pi)^d \delta^d \hspace{-0.17em}
\hspace{-0.17em} \left( k_{\oplus} + k_{\ominus} - \sum_{l = 1}^{n + 1} k_l
\right) \hspace{-0.17em} \left[ \prod_{l \neq j, i} {\mathrm{d}}\Phi_l \right]
\nonumber\\
& \times & {\mathrm{d}}\Phi_j \left[ - \frac{2^{- 2 \epsilon}}{\epsilon} \delta
(1 - y) \right] \frac{(k^0_i)^{1 - 2 \epsilon}}{2 (2 \pi)^{3 - 2 \epsilon}}
{\mathrm{d}}k^0_i {\mathrm{d}}y {\mathrm{d}}\Omega_i^{2 - 2 \epsilon} \frac{\xi^{- 1 - 2
\epsilon}_+}{\xi^{- 1 - 2 \epsilon}_{}} \lim_{y \rightarrow 1} [(1 - y)
R_{\alpha_r}] . \phantom{aaaa}\end{aligned}$$ Eq. (\[eq:Iplusplus\]) should be interpreted as following: take the $\int
{\mathrm{d}}\Phi_{n + 1} R_{\alpha_r}$ expression, work it out in $y$ and $\xi$ variables, and replace $1 / (1 - y)$ and $1 / \xi$ by the corresponding $+$ distributions.
We now write the $\Phi_j$ integral in terms of angular and radial variables, and introduce a variable $k^0 = k_j^0 + k_i^0$ $$\begin{aligned}
I_{+ \delta, \alpha_r} & = & \int (2 \pi)^d \delta^d \hspace{-0.17em}
\hspace{-0.17em} \left( k_{\oplus} + k_{\ominus} - \sum_{l = 1}^{n + 1} k_l
\right) \hspace{-0.17em} \left[ \prod_{l \neq j, k} {\mathrm{d}}\Phi_l \right]
\nonumber\\
& \times & \frac{(k^0_j)^{1 - 2 \epsilon}}{2 (2 \pi)^{3 - 2 \epsilon}}
{\mathrm{d}}k^0_j {\mathrm{d}}\Omega_j^{3 - 2 \epsilon} {\mathrm{d}}k^0 \delta (k^0 - k_i^0
- k_j^0) \nonumber\\
& \times & \left[ - \frac{2^{- 2 \epsilon}}{\epsilon} \delta (1 - y)
\right] \frac{(k^0_i)^{1 - 2 \epsilon}}{2 (2 \pi)^{3 - 2 \epsilon}} {\mathrm{d}}k^0_i {\mathrm{d}}y {\mathrm{d}}\Omega_i^{2 - 2 \epsilon} \frac{\xi^{- 1 - 2
\epsilon}_+}{\xi^{- 1 - 2 \epsilon}_{}} \lim_{y \rightarrow 1} [(1 - y)
R_{\alpha_r}] . \phantom{aaaa}\end{aligned}$$ Because of the $\delta (1 - y)$ factor we have now that $k_j$ and $k_i$ are proportional, and thus $\Omega_j$ represents their common direction. Defining $$z = 1 - \frac{k_i^0}{E} {}{},$$ and defining $$k = k_i + k_j,$$ performing the ${\mathrm{d}}k^0_j$ integration using the energy $\delta$ function we get $k^0_j = z k^0$, and trading $k^0_i$ for $z$ we get $$\begin{aligned}
I_{+ \delta, \alpha_r} & = & \int (2 \pi)^d \delta^d \hspace{-0.17em}
\hspace{-0.17em} \left( k_{\oplus} + k_{\ominus} - \sum_{l = 1}^{n + 1} k_l
\right) \hspace{-0.17em} \left[ \prod_{l \neq j, k} {\mathrm{d}}\Phi_l \right]
\nonumber\\
& \times & \frac{(k^0)^{1 - 2 \epsilon}}{2 (2 \pi)^{3 - 2 \epsilon}}
{\mathrm{d}}k^0 {\mathrm{d}}\Omega_j^{3 - 2 \epsilon} \nonumber\\
& \times & \left[ - \frac{2^{- 2 \epsilon}}{\epsilon} \right]
\frac{(k^0)^{2 - 2 \epsilon}}{2 (2 \pi)^{3 - 2 \epsilon}} z^{1 - 2 \epsilon}
(1 - z)^{1 - 2 \epsilon} {\mathrm{d}}z {\mathrm{d}}\Omega_i^{2 - 2 \epsilon}
\frac{\xi^{- 1 - 2 \epsilon}_+}{\xi^{- 1 - 2 \epsilon}_{}} \lim_{y
\rightarrow 1} [(1 - y) R_{\alpha_r}] . \phantom{aaaa}\end{aligned}$$ We notice that the expression on the second line corresponds to the phase space of the parton into which $i$ and $j$ have merged, i.e. the $\bar{k}_j$ phase space. The whole expression thus becomes $$\begin{aligned}
&& I_{+ \delta, \alpha_r} = \int {\mathrm{d}}\Phi_B \left[ - \frac{2^{- 2
\epsilon}}{\epsilon} \right] \frac{(k^0)^{2 - 2 \epsilon}}{2 (2 \pi)^{3 - 2
\epsilon}} z^{1 - 2 \epsilon} (1 - z)^{1 - 2 \epsilon} {\mathrm{d}}z \left[
\frac{2 \pi^{1 - \epsilon}}{\Gamma (1 - \epsilon)} \right] \frac{\xi^{- 1 -
2 \epsilon}_+}{\xi^{- 1 - 2 \epsilon}_{}} \lim_{y \rightarrow 1} [(1 - y)
R_{\alpha_r}] \nonumber \\
&& = - \frac{1}{\epsilon} \frac{(4 \pi)^{\epsilon}}{\Gamma (1 - \epsilon)}
\frac{2^{- 2 \epsilon}}{8 \pi^2} \int {\mathrm{d}}\Phi_B (\bar{k}^0_j)^{2 - 2
\epsilon} {\mathrm{d}}z z^{1 - 2 \epsilon} (1 - z)^{1 - 2 \epsilon} \frac{\xi^{-
1 - 2 \epsilon}_+}{\xi^{- 1 - 2 \epsilon}_{}} \lim_{y \rightarrow 1} [(1 -
y) R_{\alpha_r}]\,,\end{aligned}$$ where we have replaced $k^0$ with $\bar{k}^0_j$, that is the energy of the underlying Born emitter. We have $$\xi = \xi_{\max} (1 - z), \hspace{2em} \xi_{\max} = \frac{2
\bar{k}^0_j}{\sqrt{k_{{\ensuremath{\operatorname{res}}}}^2}},$$ and $$\xi^{- 1 - 2 \epsilon}_+ = \xi^{- 1 - 2 \epsilon} + \frac{1}{2 \epsilon}
\delta (\xi) = \xi_{\max}^{- 1 - 2 \epsilon} \times \left[ (1 - z)^{- 1 - 2
\epsilon} + \frac{\xi_{\max}^{2 \epsilon}}{2 \epsilon} \delta (1 - z)
\right],$$ so that $$\begin{aligned}
I_{+ \delta, \alpha_r} & = & - \frac{1}{\epsilon} \frac{(4
\pi)^{\epsilon}}{\Gamma (1 - \epsilon)} \frac{2^{- 2 \epsilon}}{8 \pi^2}
\int {\mathrm{d}}\Phi_B (\bar{k}^{_{} 0}_j)^{2 - 2 \epsilon} {\mathrm{d}}z z^{1 - 2
\epsilon} \nonumber\\
& \times & \left[ (1 - z)^{- 1 - 2 \epsilon} + \frac{\xi_{\max}^{2
\epsilon}}{2 \epsilon} \delta (1 - z) \right] \times \lim_{y \rightarrow 1}
[(1 - z)^2 (1 - y) R_{\alpha_r}] . \end{aligned}$$ The Altarelli-Parisi approximation in $4 - 2 \epsilon$ dimension yields $$\begin{aligned}
\lim_{y \rightarrow 1} [(1 - y) R_{\alpha_r}] & = & \frac{8 \pi \alpha_s
{\mu}^{2 \epsilon}}{2 (\bar{k}^{_{} 0}_j)^2 z (1 - z)} P_{\alpha_r} (z)
B_{f_b (\alpha_r)}, \end{aligned}$$ where with $P_{\alpha_r} (z)$ we mean $$\begin{aligned}
P_{g \rightarrow g g} (z) & = & C_A \left( \frac{z}{1 - z} + \frac{1 -
z}{z} + z (1 - z) \right) \frac{2 h (z)}{h (z) + h (1 - z)}, \\
P_{g \rightarrow q \bar{q}} (z) & = & T_F \frac{(1 - z)^2 + z^2 -
\epsilon}{1 - \epsilon}, \\
P_{q \rightarrow q g} (z) & = & C_F \left( \frac{1 + z^2}{1 - z} - \epsilon
(1 - z) \right) . \end{aligned}$$ We thus get $$\begin{aligned}
I_{+ \delta, \alpha_r} & = & - \frac{\mathcal{N}}{\epsilon}
\frac{\alpha_s}{2 \pi^{}} \int {\mathrm{d}}\Phi_B B_{f_b (\alpha_r)} \left(
\frac{Q}{2 \bar{k}^{_{} 0}_j} \right)^{2 \epsilon} \int {\mathrm{d}}z z^{- 2
\epsilon} \nonumber\\
& \times & \left[ (1 - z)^{- 1 - 2 \epsilon} + \frac{\xi_{\max}^{2
\epsilon}}{2 \epsilon} \delta (1 - z) \right] \times (1 - z) P_{\alpha_r}
(z) . \end{aligned}$$ Let us begin by evaluating the integrals $$\begin{aligned}
\int_0^1 {\mathrm{d}}z z^{- 2 \epsilon} (1 - z)^{- 2 \epsilon} \times P_{g
\rightarrow g g} (z) & = & \int_0^1 {\mathrm{d}}z z^{- 2 \epsilon} (1 - z)^{- 2
\epsilon} \times \frac{P_{g \rightarrow g g} (z) + P_{g \rightarrow g g} (1
- z)}{2} \nonumber\\
& = & C_A \int_0^1 {\mathrm{d}}z z^{- 2 \epsilon} (1 - z)^{- 2 \epsilon} \left(
\frac{z}{1 - z} + \frac{1 - z}{z} + z (1 - z) \right) \nonumber\\
& = & C_A \int_0^1 {\mathrm{d}}z z^{- 2 \epsilon} (1 - z)^{- 2 \epsilon} \left(
\frac{2 z}{1 - z} + z (1 - z) \right) \nonumber\\
& = & C_A \int_0^1 {\mathrm{d}}z z^{- 2 \epsilon} (1 - z)^{- 2 \epsilon} \left(
\frac{2}{1 - z} - 2 + z (1 - z) \right) \nonumber\\
& = & C_A \left[ \int_0^1 {\mathrm{d}}z (1 - z)^{- 2 \epsilon} \frac{2}{1 - z}
- 4 \epsilon \int_0^1 {\mathrm{d}}z \frac{\log (z)}{1 - z} \right. \nonumber\\
& + & \left. \int_0^1 {\mathrm{d}}z (- 2 + z (1 - z)) (1 - 2 \epsilon \log [z (1
- z)]) \right] \nonumber\\
& = & C_A \left[ \frac{- 2}{2 \epsilon} + \epsilon \frac{2 \pi^2}{3} -
\epsilon \frac{67}{9} - \frac{11}{6} \right] \nonumber\\
& = & \frac{- 2 C_A}{2 \epsilon} - \epsilon \left( \frac{67}{9} - \frac{2
\pi^2}{3} \right) C_A - \frac{11 C_A}{6}, \end{aligned}$$ where in the first two steps we have used twice the $z \rightarrow 1 - z$ symmetry of the integral. For the $q\to qg$ case we have $$\begin{aligned}
&& \int_0^1 {\mathrm{d}}z z^{- 2 \epsilon} (1 - z)^{- 2 \epsilon} \times P_{q
\rightarrow q g} (z) \nonumber \\
& = & \int_0^1 {\mathrm{d}}z z^{- 2 \epsilon} (1 - z)^{- 2
\epsilon} \times C_F \left( \frac{1 + z^2}{1 - z} - \epsilon (1 - z) \right)
\nonumber\\
& = & C_F \left[ \int_0^1 {\mathrm{d}}z z^{- 2 \epsilon} (1 - z)^{- 2 \epsilon}
\times \left( \frac{2}{1 - z} - (1 + z) - \epsilon (1 - z) \right) \right]
\nonumber\\
& = & C_F \left[ \int_0^1 {\mathrm{d}}z (1 - z)^{- 2 \epsilon} \frac{2}{1 - z}
- 4 \epsilon \int_0^1 {\mathrm{d}}z \frac{\log (z)}{1 - z} \right. \nonumber\\
&&\; + \; \left. \int_0^1 {\mathrm{d}}z (- (1 + z) - \epsilon (1 - z)) (1 - 2
\epsilon \log [z (1 - z)]) \right] \nonumber\\
& = & C_F \left[ \frac{- 2}{2 \epsilon} + \epsilon \frac{2 \pi^2}{3} -
\frac{3}{2} - \epsilon \frac{13}{2} \right]
\; = \; \frac{- 2 C_F}{2 \epsilon} - \epsilon \left[ \frac{13}{2} - \frac{2
\pi^2}{3} \right] C_F - \frac{3 C_F}{2} . \end{aligned}$$ Finally, for the $g\to q\bar{q}$ case: $$\begin{aligned}
\int_0^1 {\mathrm{d}}z z^{- 2 \epsilon} (1 - z)^{- 2 \epsilon} \times P_{g
\rightarrow q \bar{q}} (z) & = & \int_0^1 {\mathrm{d}}z z^{- 2 \epsilon} (1 -
z)^{- 2 \epsilon} \times T_F \frac{(1 - z)^2 + z^2 - \epsilon}{1 -
\epsilon} \nonumber\\
& = & \frac{2 T_F }{3} {}+ \epsilon \frac{23 T_F }{9} \,.\end{aligned}$$ We now define, as usual $$\begin{array}{lllllll}
\gamma_g & = & \frac{11 C_A - 4 T_F n_F}{6}\;, & & \gamma_g' & = & \left(
\frac{67}{9} - \frac{2 \pi^2}{3} \right) C_A - \frac{23}{9} T_F n_F \;,\\
\gamma_q & = & \frac{3}{2} C_F\;, & & \gamma'_q & = & \left( \frac{13}{2} -
\frac{2 \pi^2}{3} \right) C_F \;,
\end{array}$$ and find $$\sum_{\alpha_r \in \alpha_r (f_b)} I_{+ \delta, \alpha_r} = \mathcal{N}
\frac{\alpha_s}{2 \pi^{}} \int {\mathrm{d}}\Phi_B B_{f_b} \left( \frac{Q}{2
\bar{k}^{_{} 0}_j} \right)^{2 \epsilon} \left[ \frac{1 - \xi_{\max}^{2
\epsilon}}{\epsilon^2} C_{j (f_b)} + \frac{\gamma_{j (f_b)}}{\epsilon} +
\gamma'_{j (f_b)} \right],$$ where by $j (f_b)$ we mean the flavour of the $j^{{\ensuremath{\operatorname{th}}}}$ parton in the $f_b$ flavour structure. We get $$\hspace{-0.4cm}
\sum_{\alpha_r \in \alpha_r (f_b)} I_{+ \delta, \alpha_r} = \mathcal{N}
\frac{\alpha_s}{2 \pi^{}} \int {\mathrm{d}}\Phi_B B_{f_b} \left(
\frac{Q^2}{k^2_{{\ensuremath{\operatorname{res}}}}} \right)^{\epsilon} \xi_{\max}^{- 2 \epsilon}
\left[ \frac{1 - \xi_{\max}^{2 \epsilon}}{\epsilon^2} C_{j (f_b)} +
\frac{\gamma_{j (f_b)}}{\epsilon} + \gamma'_{j (f_b)} \right],$$ where we have used for $\xi_{\max}$ the covariant expression $$\xi_{\max} = \frac{2 \bar{k}_j \cdot k_{{\ensuremath{\operatorname{res}}}}}{k_{{\ensuremath{\operatorname{res}}}}^2} .$$ We now expand it as $$\begin{aligned}
&& \hspace{-1cm} \sum_{\alpha_r \in \alpha_r (f_b)} I_{+ \delta, \alpha_r} \; = \; \mathcal{N}
\frac{\alpha_s}{2 \pi^{}} \int {\mathrm{d}}\Phi_B B_{f_b} \left(
\frac{Q^2}{k^2_{{\ensuremath{\operatorname{res}}}}} \right)^{\epsilon} \left[ \frac{- 2 \log
\xi_{\max} + 2 \epsilon \log^2 \xi_{\max}}{\epsilon} C_{j (f_b)} \right.
\nonumber\\
&& \phantom{aaaaaaaaa}
\; + \; \left. \frac{\gamma_{j (f_b)}}{\epsilon} (1 - 2 \epsilon \log
\xi_{\max}) + \gamma'_{j (f_b)} \right] \nonumber\\
&& \; = \; \mathcal{N} \frac{\alpha_s}{2 \pi^{}} \int {\mathrm{d}}\Phi_B B_{f_b}
\left[ \frac{- 2 \log \xi_{\max} + 2 \epsilon \log^2 \xi_{\max} - 2 \epsilon
\log \xi_{\max} \log (Q^2 / k_{{\ensuremath{\operatorname{res}}}}^2)}{\epsilon} C_{j (f_b)} \right.
\nonumber\\
&&\quad + \; \left. \frac{\gamma_{j (f_b)}}{\epsilon} + \gamma_{j (f_b)} \log
\frac{Q^2}{k_{{\ensuremath{\operatorname{res}}}}^2} - 2 \gamma_{j (f_b)} \log \xi_{\max} +
\gamma'_{j (f_b)} \right] \nonumber\\
&& \; = \; \mathcal{N} \frac{\alpha_s}{2 \pi^{}} \int {\mathrm{d}}\Phi_B B_{f_b}
\left[ \frac{- 2 \log \xi_{\max}}{\epsilon} C_{j (f_b)} {}+
\frac{\gamma_{j (f_b)}}{\epsilon} \right. \nonumber\\
&& \quad + \; \left. 2 \log \xi_{\max} \left( \log \xi_{\max} - \log
\frac{Q^2}{k_{{\ensuremath{\operatorname{res}}}}^2} \right) C_{j (f_b)} \right. \nonumber \\
&& \quad + \left. \left( \log
\frac{Q^2}{k_{{\ensuremath{\operatorname{res}}}}^2} - 2 \log \xi_{\max} \right) \gamma_{j (f_b)} +
\gamma'_{j (f_b)} \right]\,.\end{aligned}$$ We now combine this term with the $I_{s \delta, \alpha_r}^{(1)}$ integral: $$\begin{aligned}
I_{A, \alpha_r} & = & I_{+ \delta, \alpha_r} + I_{s \delta, \alpha_r}^{(1)}
\nonumber\\
& = & \mathcal{N} \frac{\alpha_s}{2 \pi^{}} \int {\mathrm{d}}\Phi_B B_{f_b}
\left[ \frac{2}{\epsilon} \log \frac{\sqrt{s}}{2 \bar{k}^0_j} C_{j (f_b)} +
\frac{\gamma_{j (f_b)}}{\epsilon} \right. \nonumber\\
& + & 2 \left( \log \frac{\sqrt{s}}{2 \bar{k}^0_j} {}+ \log
\xi_{\max} \right) \left( \log \frac{\sqrt{s}}{2 \bar{k}^0_j } {}+
\log \xi_{\max} + \log \frac{Q^2}{s} \right) C_{j (f_b)}\\
& + & \left. 2 \log \xi_{\max} \left( \log \xi_{\max} - \log
\frac{Q^2}{k_{{\ensuremath{\operatorname{res}}}}^2} \right) C_{j (f_b)} + \left( \log
\frac{Q^2}{k_{{\ensuremath{\operatorname{res}}}}^2} - 2 \log \xi_{\max} \right) \gamma_{j (f_b)} +
\gamma'_{j (f_b)} \right], \nonumber\end{aligned}$$ where $f_b$ stands for $f_b (\alpha_r)$, and $j$ is the emitter for the region $\alpha_r$. Notice also that now $\bar{k}^0_j$ represents the energy of the emitter in our common reference frame, while earlier, with the same symbol we denoted its energy in the resonance frame. Notice also that $\xi_{\max}$ is now frame dependent. We will denote as $I_A^{(0)}$ the finite part of $I_A$.
Summary
-------
We now summarize the real and soft-collinear terms that need to be included in the calculation:
Real integral: $$\begin{aligned}
I_{+ +, \alpha_r} & = & \int {\mathrm{d}}\Phi_{n + 1} (1 - y) \left( \frac{1}{1
- y} \right)_+ \xi \left( \frac{1}{\xi} \right)_+ R_{\alpha_r},
\label{eq:Iplusplusfinal}
\end{aligned}$$
Collinear terms: $$\begin{gathered}
\hspace{-1cm} I_{A, \alpha_r}^{(0)} = \frac{\alpha_s}{2 \pi^{}} \int {\mathrm{d}}\Phi_B
B_{f_b} \left[ 2 \left( \log \frac{\sqrt{s}}{2 \bar{k}^0_j} {}+ \log
\xi_{\max} \right) \left( \log \frac{\sqrt{s}}{2 \bar{k}^0_j } {}+
\log \xi_{\max} + \log \frac{Q^2}{s} \right) C_{j (f_b)} \right.\\
\hspace{-1cm} + \left. 2 \log \xi_{\max} \left( \log \xi_{\max} - \log
\frac{Q^2}{k_{{\ensuremath{\operatorname{res}}}}^2} \right) C_{j (f_b)} + \left( \log
\frac{Q^2}{k_{{\ensuremath{\operatorname{res}}}}^2} - 2 \log \xi_{\max} \right) \gamma_{j (f_b)}
+ \gamma'_{j (f_b)} \right]_{},
\end{gathered}$$ where $j$ is the emitter and $k_{{\ensuremath{\operatorname{res}}}}$ is the momentum of the resonance that contains the emitter for the region $\alpha_r$. With $\bar{k}^0_j$ we denote the energy of the emitter in our common reference frame, that is the CM frame of the final state. On the other hand, $\xi_{\max}$ is computed in the resonance frame.
Soft terms remain the same as in the standard treatment, and are reported in Appendix A.1 of the paper (ref. [[@Alioli:2010xd]]{}).
Soft mismatch: $$\begin{aligned}
I^{(1)}_{s +, \alpha_r} & = & \int {\mathrm{d}}\Phi_{{\mathrm{B}}} \int_0^{\infty}
{\mathrm{d}}\xi \int_{- 1}^1 {\mathrm{d}}y \int_0^{2 \pi} {\mathrm{d}}\phi \frac{s^{}
\xi}{(4 \pi)^3} \times \Biggl\{ \tilde{R}_{\alpha_r} \left[ e^{- \frac{2
k_i \cdot k_{{\ensuremath{\operatorname{res}}}}}{k_{{\ensuremath{\operatorname{res}}}}^2} } - e^{- \xi} \right]
_{\alpha_r} \nonumber\\
& - & {}{}\frac{32 \pi \alpha_s C_{j (f_b)}}{s
\xi^2} B_{f_b (\alpha_r)} \frac{\left[
e^{- \frac{2 \bar{k}^{}_j \cdot k_{{\ensuremath{\operatorname{res}}}}}{k_{{\ensuremath{\operatorname{res}}}}^2} \frac{k^0_i}{\bar{k}^0_j} } - e^{- \xi}
\right]_{\alpha_r} }{1 - \cos \theta} \Biggr\},
\end{aligned}$$ to be summed over all the $\alpha_r$ with the emitter belonging to a resonance.
The integration phase space is defined in the partonic CM frame, with the third axis pointing along the direction of the emitter $j$.
Collinear terms related to initial state radiation remain the same as in the standard treatment.
In tab. \[tab:newstuffinbox\], we list the terms that needed to be newly implemented in the new, resonance aware version of the that we are presenting here.
Soft $\log \Gamma$ terms
------------------------
In the procedure that we have illustrated, collinear singular regions arise only among partons produced in the decay of the same resonance. This property arises because, in the separation of the singular regions, we restrict ourselves to singular structures that are compatible with the resonance history. While this feature guarantees a smooth cancellation of the collinear logarithms in the subtraction procedure, we cannot expect a corresponding cancellation of all soft, non collinear logarithms. There are in fact two sources of soft radiation with a lower or upper cut off of the order of the resonance virtualities:
Soft radiation arising from the interference of soft emissions from coloured partons belonging to different resonances. These terms have an [[[*upper*]{}]{}]{} cut-off of the order of the resonance width.
Soft emission involving amplitudes with radiation arising from the resonances internal lines. These terms have a [[[*lower*]{}]{}]{} cut off of the order of the resonance width.
We thus expect that in our procedure $\log \Gamma$ terms will arise in the integration of the real cross section. The virtual corrections will also have corresponding $\log \Gamma$ terms, that cancel the real ones when summed together.
In this section we discuss the structure of these soft terms. As we will see, it is possible, in principle, to remove them from the integration of the real cross section, and include them in the soft term, in such a way that their cancellation takes place in the soft-virtual contribution. However, we have not attempted to implement this in the . In view of the relatively large size of the resonance widths in the typical processes that we consider, it is unlikely that they may cause problems in practical NLO calculations. Furthermore, as far as NLO+PS implementations are concerned, these terms are in fact properly treated in our resonance-aware framework, and do not require any further action.
We now discuss the structure of the soft logarithms in the presence of narrow resonances. For simplicity, we assume that all the resonances have comparable widths of order $\Gamma_0$. We consider two regions:
Region $a$: is characterized by soft emissions with energy $\omega$ larger than $\Gamma_0$. In this region $\Gamma_0$ plays the role of an infrared cutoff. The dominant region of integration has $\log \omega$ uniformly distributed between $\log \Gamma_0$ (lower cut-off) and the log of some hard scale in the process (high cut-off), typically of the order of the mass of the resonances.
Region $b$: is characterized by soft emissions with energy $\omega$ less than $\Gamma_0$. The lower limit in this region is regulated in the usual technical ways (like dimensional regularization). Its upper cut-off is $\Gamma_0$.
In region $a$, since $\Gamma_0$ acts as an infrared cutoff, the emissions from resonances internal lines near their mass shell should also be considered as soft. In fig. \[fig:softlogG\]
![\[fig:softlogG\] Insertion of a soft gluon in an internal resonance propagator.](softlogG){width="48.00000%"}
we illustrate the insertion of a soft emission in an internal resonance line. The product of the resonance propagators will be given by $$\begin{gathered}
\frac{1}{p^2 - M^2 + i \Gamma M} \times \frac{1}{(p - k)^2 - M^2 + i \Gamma
M} \\
= \frac{1}{2 p \cdot k} \left[ \frac{1}{(p - k)^2 - M^2 + i \Gamma M} -
\frac{1}{p^2 - M^2 + i \Gamma M} \right] .\end{gathered}$$ Under the assumption that $\omega = k^0 > \Gamma$, the two denominators cannot be near their mass shell at the same time. When the first term in the square bracket is near its mass shell, the process corresponds to the resonance radiating during production. In fact, in this case the $p$ momentum is far from the mass shell by a scale of order $k^0$, while the $p-k$ momentum is near the mass shell by a scale of order $\Gamma$. In coordinate space, this means that the line carrying momentum $p$ has a length of order $1/k^0$, much shorter than the length of order $1/\Gamma$ of the $p-k$ line. Conversely, if the second term is on-shell, radiation is taking place during decay. When squaring the amplitude, interference between these two terms is suppressed, since the two propagators cannot be on-shell at the same time, and the integration is effectively cut off by at a scale of order $\Gamma$, leaving no phase space for soft logarithms to build up. For the same reason, interference from emissions arising at production with emissions from resonance decay, as well as from emissions arising from the decay of different resonances, do not yield soft logarithms, since they also lead to propagators off the resonance peaks in the interfering amplitudes.
Reasoning in terms of radiation and decay times, by assuming $\omega >
\Gamma_0$ we are assuming that radiation time is shorter than the resonances lifetimes. Thus, soft radiation in production cannot interfere with radiation in decays, since they happen at different times, and for the same reason radiation from different resonances cannot interfere. So, as far as soft singularities are concerned, the process can be though of as the product of independent production and decay processes, each one of them with resonances appearing only as initial or final state particles, but not as internal lines. For all these independent components, soft emissions is given by the usual eikonal formula applied only to initial and final state particles, that in this case can also be unstable resonances.
The structure of the soft singularity in region $b$ is determined by the initial and final state particles after the decay of all resonances. The resonances are considered as off-shell particles, as far as soft emissions are concerned, and interference terms from emissions arising from different resonances are not suppressed by small Breit-Wigner weights, since the emission energy is below the resonance widths. In terms of time, this is the case when the time for soft radiation is longer than the resonance widths, so that only particles that live longer than the resonances can contribute.
The form of the soft subtraction term in the $b$ region is the same one that we have adopted in the present method. However, that in our present treatment we are considering unrestricted emissions, while the $b$ region is defined to involve soft energies below $\Gamma_0$. When considering a given underlying Born resonance history, we will thus have the following cases:
In the emissions from pairs of coloured massless partons belonging to the same resonance, the terms in regions $a$ and $b$ will combine, yielding an unrestricted soft energy integral, and no $\log \Gamma$ terms.
In the emission from pairs of coloured massless partons belonging to different resonances, only the terms in the $b$ region will be present. These contributions will be cut-off at energies above $\Gamma_0$, since for larger energies they will push one of the two resonances out of its mass shell, thus damping the cross section. They will thus lead to $\log \Gamma$ contributions to the cross section.
In any emission from an internal resonance leg, the emission energy will have a lower infrared cutoff $\Gamma_0$, and will yield other $\log
\Gamma$ terms.
It is conceivable that our method may be modified, by adding further soft subtraction terms to the real cross section and corresponding integrated soft terms to the soft-virtual cross section, in such a way that the $\log \Gamma$ terms cancel within the soft-virtual contribution. This procedure may make the NLO calculation more convergent in the zero width limit. However, it would have no effect in the generation of radiation according to the method. In , the cancellation of the $\log \Gamma$ terms takes place numerically in the calculation of the $\tilde{B}$ function, between the real and the soft-virtual integral. In the generation of radiation, the cross section is unitarized by construction, so that no further $\log \Gamma$ terms arise in inclusive quantities. We thus did not attempt to implement such an improvement in the present work.
Code organization
=================
The implementation of the subtraction scheme described in the present paper has required an extensive rewriting of several parts of the framework. While we postpone writing a full documentation for the new code, we will describe in the present section the structures that are used to describe the various components of the cross section in terms of flavour and resonance histories.
The flavour structures used to implement our subtraction scheme are organized as follows. The process specific code provides the flavour structure in terms of arrays carrying the flavour of the particles involved in the process, including intermediate resonances. We have the arrays
\
`flst_born(1:flst_bornlength(iborn),iborn), iborn=1...flst_nborn;`\
`flst_bornres(1:flst_bornlength(iborn),iborn), iborn=1...flst_nborn;`\
\
`flst_real(1:flst_reallength(ireal),ireal), ireal=1...flst_nreal;`\
`flst_realres(1:flst_reallength(ireal),ireal), ireal=1...flst_nreal.`
These arrays are set in the user process routines. The following arrays are also set by the user process routines:
`flst_bornresgroup(1:flst_nborn);`\
`flst_nbornresgroup.`
These have the purpose of grouping together the Born full flavour configurations that have similar resonance structure so that they can be integrated together. Thus, the value of (an integer from 1 to ) labels the resonance structure group of the born full flavour structure . This is needed because the groups together flavour structures with similar resonance histories when performing the integration, since these configuration can be integrated with the same importance sampling.
In the user process arrays, each flavour structure can appear only once, following the traditional approach of FNO. In the case of resonances, this leads to a non-trivial subtlety that we describe now. Consider the flavour structure for the subprocess $q \bar{q} \rightarrow e^+
e^- e^+ e^-$. According to the traditional approach of the there is only one flavour structure associated with this process, i.e. only a single ordering of the final state electron and positron will appear. When resonance structures are considered, we realize that we have two ways of pairing the electrons–anti-electrons to build an intermediate $Z$ boson. These two pairings are fully equivalent up to permutations of the final state particles, and thus only one resonance assignment will appear for them. We should not forget, however, that the contribution with a given resonance assignment carries a resonance projection factor. Assigning the ordering 1 to 8 to the $q \bar{q} \rightarrow Z Z \rightarrow e^+ e^- e^+ e^-$, assuming that the $Z$ in position 3 decays into the $e^{{}+} e^-$ pair in position 5-6, and assuming that we have only these two resonance histories, we will have a factor of the form $$\frac{P (5, 6 ; 7, 8)}{P (5, 6 ; 7, 8) + P (5, 8 ; 7, 6)},$$ where, for example, $$P (5, 6 ; 7, 8) = \frac{M_Z^4}{(s_{56} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2} \times
\frac{M_Z^4}{(s_{78} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2} .$$ It is clear now that we should supply a factor of 2 for this graph, since by assigning the resonances we break part of the exchange symmetry for final state identical particles. Thus, the user process should provide the symmetry factor appropriate for the given final state irrespective of resonance assignment. The machinery will take care of supplying the appropriate factors arising from the resonance assignment specification. Thus, the user process list is built from the list of flavour structures for all distinct final states (where distinct means that final state differing by permutations are not allowed). For each final state, the list will be expanded by assigning all possible resonance histories. But, again, full flavour structure differing only by a permutation of the resonance histories will not be allowed. The checks explicitly that no full flavour structures equivalent up to a permutation will appear.
Notice that the factor of two will lead to a total resonance factor of $$\frac{2 P (5, 6 ; 7, 8)}{P (5, 6 ; 7, 8) + P (5, 8 ; 7, 6)},$$ that is not 1. However, by symmetry, an analysis of generated events that does not distinguish among identical final state particles will lead to the same results as if we included both weights
$$\frac{P (5, 6 ; 7, 8)}{P (5, 6 ; 7, 8) + P (5, 8 ; 7, 6)} + \frac{P (5, 8 ;
7, 6)}{P (5, 6 ; 7, 8) + P (5, 8 ; 7, 6)} = 1 .$$
Given the real, the finds all singular regions associated with the real graphs, and builds the corresponding arrays
\
`flst_alr(1:flst_alrlength(alr),alr), alr=1...flst_nalr;`\
`flst_alrres(1:flst_alrlength(alr),alr), alr=1...flst_nalr.`
It furthermore fills the arrays with the emitter of the given singular region, and the array with the multiplicity of the singular region. A multiplicity factor can arise if we have identical partons in the final state. For example, if we have several gluons and a quark in the final state, there will be regions associated with each gluon being collinear to the quark, and the program will find as many regions of this type as there are gluon. It will recognize that all these regions are equivalent, and it will emit a single region with a multiplicity factor equal to the number of equivalent regions.
If there are real contributions that do not have any singular region, they are collected into the “regular” arrays
\
`flst_regular(1:flst_regularlength(ireg),ireg), ire=1...flst_nregular;`\
`flst_regularres(1:flst_regularlength(ireg),ireg), ireg=1...flst_nregular.`
An array is provided also in this case.
The task of finding out the singular and regular contributions is carried out by the subroutine , in the file . In the traditional implementation, at this stage, for each a list of competing singular regions was found. These were needed since each contribution is obtained by multiplying the corresponding real graph by the ratio of the $d^{- 1}_{\alpha_r}$ factor divided by the sum of all the $d^{-
1}_{\alpha_r}$ associated with the other competing singular regions. In the current implementation, we also need to list together the competing resonance histories that lead to the same final state, in order to compute the corresponding resonance projection factor. So, for either the alr, the regular or the Born terms, we have an array of pointers
`flst_XXXnumrhptrs(1:flst_nXXX), flst_XXXrhptrs(maxreshists,flst_nXXX),`
where stands for either of , or . The integer stores the number of resonance histories associated with the full flavour structure. The integers , in case is either or , are indices in the arrays
\
`flst_allrhlength(maxreshists),`\
`flst_allrh(nlegreal,maxreshists), `\
`flst_allrhres(nlegreal,maxreshists),`
that represent the full flavour structure of the competing resonance histories in the real graphs, and in case is , are indices in the arrays
\
`flst_allbornrhlength(maxreshists),`\
`flst_allbornrh(nlegborn,maxreshists), `\
`flst_allbornrhres(nlegborn,maxreshists),`
representing the full flavour structure of the competing resonance histories for the Born graphs. We stress that we cannot use and in place of and , because in the latter also configurations that differ by a permutation of the intermediate resonances are included. The arrays described above are filled by the subroutine in the file, that also computes the multiplicity factor associated with alternative resonance histories that differ only by a permutation of the resonances from the contribution being considered. These arrays are required for the computation of the weights needed to project out a given resonance history contribution from the real and Born amplitudes. Besides these, in the case of the contributions, we also need to know the singular regions associated with a given resonance history. This information is contained in the arrays
\
`flst_allrhnumreg(maxreshists),`\
`flst_allrhreg(1:2,maxregions,maxreshists).`
For each entry of the arrays, gives the number of singular regions, and gives the indices of the two particles becoming collinear in the corresponding singular region.
The example of single top, $t$-channel production
=================================================
We consider the Born level process $b q \rightarrow b e^+ \nu_e q'$. In the following we will label for conciseness as $q$ and $q'$ all light quarks or antiquarks (excluding the $b$) with the appropriate flavour structure that can appear in the process, and we imply also the presence of the corresponding processes with exchanged initial state particles (i.e. $q b$ in this case). This process is dominated by the single top production process $b q \rightarrow (t \rightarrow b (W^+ \rightarrow e^+ \nu_e)) q'$, such that the top quark is not produced at rest in the partonic centre-of-mass. Therefore this process is relevant for testing our formalism. In fact, the standard momentum mapping leading to the underlying Born configuration, in case of collinear radiation from the $b$ quark arising from top decay, would conserve the incoming partons 4-momentum by adjusting the 3-momenta of the $b$ and the $W^+ q'$ systems with appropriate boosts. This procedure would preserve the mass of the $W^+ q'$ system, but not the mass of the top.
At the Born level, it is enough to consider a single resonance history, namely $b q \rightarrow (t \rightarrow b (W^+ \rightarrow e^+ \nu_e)) q'$. Alternatively, one may consider two different resonance histories: $$\begin{aligned}
b q & \rightarrow & (t \rightarrow b (W^+ \rightarrow e^+ \nu_e)) q', \\
b q & \rightarrow & b (W^+ \rightarrow e^+ \nu_e) q' .
\label{eq:secondhist}\end{aligned}$$ The second one is actually not needed, since treating the $b W^+$ system as a resonance (i.e. preserving its mass in the underling Born mapping) rather than preserving the mass of the $W^+ q'$ system, as the would do for the resonance history of eq. (\[eq:secondhist\]), does not lead to any inaccuracy. We did however include this resonance assignment as an option, and used it to test that our setup works also in the case when more than one resonance history is present at the Born level.
We are considering the following real processes: $b q\to e^+ \nu_e q' g$ and $b g \to e^+ \nu_e q q'$. We do not consider processes of the form $q g \to b e^+ \nu_e q' \bar{b}$, that include also $s$-channel contributions. This is adequate for the purposes of the present paper, where we would like to present and validate a method, rather then provide a realistic simulation of single top production.
We will now list the resonance histories for the real contributions corresponding to the choice of a single resonance history at the Born level: $$\begin{aligned}
b q & \rightarrow & g (t \rightarrow b (W^+ \rightarrow e^+ \nu_e)) q', \\
b q & \rightarrow & (t \rightarrow b g (W^+ \rightarrow e^+ \nu_e)) q', \\
b g & \rightarrow & (t \rightarrow b g (W^+ \rightarrow e^+ \nu_e)) q q', \\
b g & \rightarrow & b (Z / \gamma \rightarrow (W^+ \rightarrow e^+ \nu_e)
(W^- \rightarrow q q')) \label{eq:regular-bsing}\\
b g & \rightarrow & (t \rightarrow b g (W^+ \rightarrow e^+ \nu_e)) (W^-
\rightarrow q q') \end{aligned}$$ Notice that the last two processes are really regular ones, since for them no collinear singularity can arise by pairing particles belonging to the same resonance.
The Born (together with the colour correlated Born) and real matrix elements for the process were easily generated using the interface described in ref. [[@Campbell:2012am]]{}. The virtual contribution was extracted by hand from code generated using `MadGraph5_aMC@NLO` [@Alwall:2014hca].
Test at the NLO level
---------------------
We first tested our method (that we refer to as `POWHEG-BOX-RES`) by comparing its NLO level results with a (traditional) `POWHEG-BOX-V2` implementation of the same process. More specifically, we implemented the $bq\to b e^+ \nu+e q'$ process in the `POWHEG-BOX-V2` framework, without the inclusion of any resonance information, and using exactly the same matrix elements used with the new method. The comparison was carried out by using exactly the same choice of scales and parton density functions.
We observe that the process (\[eq:regular-bsing\]) has initial state collinear singularities, due to a contribution arising from an initial state $g \rightarrow b \bar{b}$ splitting followed by the subprocess $b \bar{b} \rightarrow (Z / \gamma \rightarrow (W^+
\rightarrow e^+ \nu_e) (W^- \rightarrow q q'))$. This process is not subtracted in our procedure, since we do not have a corresponding underlying Born contribution. In order to avoid the associated collinear divergence, and only in the framework of the NLO tests that we are discussing in this section, we supplied our cross section with a damping factor of the form $$\label{eq:damping}
\frac{p_{t,b}^2}{p_{t,b}^2 + 20}\;,$$ that damps low transverse momentum $b$ emissions. It is applied to all terms of the cross section, as a function of the corresponding $b$ quark kinematics. We stress that this damping factor is not used when doing phenomenological calculations. It is needed here only to guarantee that the NLO cross sections computed with the traditional `POWHEG BOX` implementation is finite and agrees formally with the one computed with the new method.
In order to perform the NLO test we did not need the virtual contributions, since they are the same in the two approaches. In the “traditional” implementation the Born phase space was generated with importance sampling on the dominant decay chain $b q \rightarrow (t \rightarrow b (W^+ \rightarrow e^+ \nu_e))
q'$. It was found that the new implementation yielded better convergence, speeding up the calculation by about a factor of 2 or more. Very high statistics runs of both implementations were performed, and full numerical agreement was found for both the total cross section and the differential distributions.
Results and comparisons at the full shower level
------------------------------------------------
As mentioned earlier, the process we are considering is singular when final state $b$ quarks have very small transverse momenta. Thus, event generation requires a generation cut or a Born suppression factor [@Alioli:2010qp]. We adopt the latter method in the present work, using a suppression factor of the form $$\frac{p_{t,b}^2}{p_{t,b}^2 + 20}\;.$$ As a result, less and less events are generated as the $b$ transverse momentum becomes small, but the event weight is increased correspondingly, thus yielding a potentially divergent cross section for observables that do not suppress the contribution of small transverse momentum $b$ quarks.
The shower was performed using `Pythia8` version 81.85 [@Sjostrand:2014zea; @Sjostrand:2007gs; @Sjostrand:2006za]. In all cases, `Pythia8` was run with its default initialization, supplemented with the following calls:
pythia.readString("SpaceShower:pTmaxMatch = 1");
pythia.readString("TimeShower:pTmaxMatch = 1");
pythia.readString("SpaceShower:QEDshowerByQ = off"); // From quarks.
pythia.readString("SpaceShower:QEDshowerByL = off"); // From Leptons.
pythia.readString("TimeShower:QEDshowerByQ = off"); // From quarks.
pythia.readString("TimeShower:QEDshowerByL = off"); // From Leptons.
pythia.readString("PartonLevel:MPI = off");
The first two calls cause `Pythia8` to veto emissions harder than `scalup`, if arising at production level, and to allow unrestricted emissions from resonance decays. Furthermore, $b$ hadron decays were switched off with calls of the following kind
pythia.readString("521:mayDecay = off");
pythia.readString("-521:mayDecay = off");
for all $b$ flavoured mesons and baryons.
In order to test our generator, we generated four samples of one million of events each, and compared the relative output. The samples are obtained as follows:
- `NORES` Sample. This is obtained using the traditional `POWHEG-BOX-V2` implementation for the process. The events are fed to `Pythia8`, with the setting listed earlier. `Pythia8` is required to veto radiations at scales harder than the value of the Les Houches variable `scalup`, set equal to the transverse momentum of the `POWHEG` generated radiation for each Les Houches event.
- `RES-HR` Sample. This sample is obtained using the `POWHEG-BOX-RES` implementation of the process. The Les Houches events include the hardest radiation generated by `POWHEG` (the `HR` in `RES-HR` stands for “hardest radiation”). Since `POWHEG` is generating the hardest radiation, besides vetoing radiation in production with the usual `scalup` mechanism, we also forbid any `Pythia8` radiation from top decays harder than `scalup`. We do this by explicitly examining the showered events. If a radiation generated by `Pythia8` in top decay has a transverse momentum greater than `scalup`, the program discards it, and runs `Pythia8` again on the same Les Houches partonic event. This procedure is repeated indefinitely, thus explicitly vetoing any event with radiation harder than `scalup`.
- `RES-AR` Sample. This sample is obtained using the `POWHEG-BOX-RES` implementation of the process. However, rather than keeping only the hardest radiation, we kept both the hardest radiation in top decay and the hardest radiation in production (the `AR` in `RES-AR` stands for “all radiation”) . These radiations are composed into a single event, using the usual `POWHEG` mapping mechanism. In this case, besides the normal `scalup` veto for radiation [*in production*]{}, we must forbid `Pythia8` radiation in top decays if harder than the `POWHEG` generated one [*in top decay*]{}. There are thus two different veto scales, one for production (i.e. the `scalup` value) and one for decay. There are no provisions in the Les Houches Interface for User Processes to store the radiation scale for decaying resonances. The program thus computes explicitly the transverse momentum of radiation in top decay at the Les Houches level. If the showered event contains shower generated radiation in top decay harder than this computed scale, the shower is discarded, and a new shower is generated on the same Les Houches event, repeating the procedure indefinitely until the event passes the required condition.
The `RES-AR` implementation is fully analogous to the `allrad` procedure illustrated in ref. [@Campbell:2014kua]. The method (and the software) for vetoing `Pythia8` radiation is also borrowed from that reference.
- `ST-tch` Sample. This sample is generated using the `ST-tch` `POWHEG` generator of ref. [@Alioli:2009je]. Radiation in decay is not included in this generator, and thus we let `Pythia8` shower the event according to its default setup, vetoing events with radiation in production harder than `scalup`, and with no veto in top decays. In order to match more closely what we include in `RES-HR`, we deleted in the `ST-tch` the real processes initiated by a light (i.e. not $b$) quark and a gluon.
Phenomenological analysis
-------------------------
We have considered the LHC 8 TeV configuration for our phenomenological runs. We have used throughout the MSTW2008 set [@Martin:2009iq] at NLO order. Other PDF sets, like those of refs. [@Lai:2010vv; @Ball:2010de], can be used as well, but we are not interested in a PDF comparison in the present study. We only consider the $b\, \mu^+ \nu_\mu$ final state (i.e. not the conjugate one).
We set the top mass to $m_t=172.5$ GeV. For this value of the mass and PDF choice the computed top width, including NLO strong corrections, is $1.3306$ GeV. We use the same NLO value of the width also in the `ST-tch` generator. In this generator it only affects the top line-shape, since the cross section is determined by the top cross section multiplied by a user supplied branching fraction. On the other hand, in our generator the width must be computed with the same Standard Model parameters that are used in the matrix elements, since the cross section will be proportional to a partial width (depending upon the couplings that are used in the matrix elements) divided by the total width that appears in the denominator of the top propagator.
Since we will compare generators that do not include top resonance information, our analysis will be performed (unless explicitly stated otherwise) without using “Monte Carlo truth” information as far as the top particle is concerned, thus relying solely upon a particle level reconstruction of the top kinematics. We thus define the following objects:
- The lepton. This is the hardest $\mu^+$ in the event.
- The neutrino. This is the hardest $\nu_\mu$ in the event.
- The $W^+$. This is the system formed by the lepton and the neutrino.
- The $b$ hadron. This is the hardest hadron with a $b$ quark content (not a $\bar{b}$!).
- The $b$-jet. Jets are reconstructed using the anti-$k_t$ algorithm [@Cacciari:2008gp], as implemented in the `FastJet` package [@Cacciari:2011ma], with $R=0.5$. The $b$-jet is the jet that contains the $b$ hadron.
- The top quark. This is defined as the system comprising the $W$ and the $b$-jet. Only $b$-jets with a $p_t$ of at least 25 GeV and $|\eta|<4.5$ are accepted for this purpose. In case such a $b$-jet is not found, no top is reconstructed.
[RES-AR]{} and [ST-tch]{} comparison
------------------------------------
We begin by comparing the `RES-AR` and the `ST-tch` results. For this purpose (and only in this case) we have deleted from the `RES-AR` generator the real amplitudes with the final non-$b$ light partons in a colour singlet. This excludes in particular contributions with $tW^-$ associated production, leading to a more meaningful comparison and to a better agreement on the total cross sections, since these contributions are not included in the `ST-tch` generator. In fig. \[fig:Top-pt-RES-AR-ST\]
![\[fig:Top-pt-RES-AR-ST\] Transverse momentum distribution of the top quark, obtained with the [RES-AR]{} and the [ST-tch]{} generators.](plots/pt-top-AR-ST "fig:"){width="48.00000%"} ![\[fig:Top-pt-RES-AR-ST\] Transverse momentum distribution of the top quark, obtained with the [RES-AR]{} and the [ST-tch]{} generators.](plots/y-top-AR-ST "fig:"){width="48.00000%"}
we plot the transverse momentum and the rapidity distributions of the reconstructed top. In these plots no requirement is made on the mass of the reconstructed top. As one can see, reasonable agreement is found with these distributions. In fig. \[fig:Top-mass-RES-AR-ST\]
![\[fig:Top-mass-RES-AR-ST\] Invariant mass of the top quark, obtained with the [RES-AR]{} and the [ST-tch]{} generators, at the reconstructed level and at the MC-truth level.](plots/mass-top-AR-ST "fig:"){width="48.00000%"} ![\[fig:Top-mass-RES-AR-ST\] Invariant mass of the top quark, obtained with the [RES-AR]{} and the [ST-tch]{} generators, at the reconstructed level and at the MC-truth level.](plots/mass-mc-top-AR-ST "fig:"){width="48.00000%"}
we show the mass peak, both for the reconstructed top and for the top particle in the Monte Carlo record (more specifically, we pick the last top in the Monte Carlo event record). It is apparent that the line-shape of the reconstructed top are not in complete agreement. Assuming that this is due to differences in the structure of the $b$-jet, we plot in fig. \[fig:bjet-RES-AR-ST\]
![\[fig:bjet-RES-AR-ST\] Mass and profile of the $b$ jet, obtained with the [RES-AR]{} and the [ST-tch]{} generators.](plots/bjet-mass-AR-ST "fig:"){width="48.00000%"} ![\[fig:bjet-RES-AR-ST\] Mass and profile of the $b$ jet, obtained with the [RES-AR]{} and the [ST-tch]{} generators.](plots/bprofile-AR-ST "fig:"){width="48.00000%"}
the $b$-jet mass and profile, for $b$-jets with transverse momentum above $15$ GeV and $|\eta|<4.5$. The jet profile is defined as follows $$P_{b\operatorname{-jet}}(\Delta_{\mathrm R})= N \int {\mathrm d}\sigma\;\frac{\sum_j p_{T,j} \times \delta(\Delta_{\mathrm R}^{(j,b\operatorname{-jet})}-\Delta_{\mathrm R})}{p_{T,b\operatorname{-jet}}},$$ where $N$ is chosen in such a way that $$\int_0^{0.5} P_{b\operatorname{-jet}}(\Delta_{\mathrm R}) {\mathrm{d}}\Delta_{\mathrm R} = 1\,,$$ where $0.5$ is the $\Delta_{\mathrm R}$ value that defines the $b$-jet. Thus, for $\Delta_{\mathrm R}<0.5$, $P_{b\operatorname{-jet}}(\Delta_{\mathrm R})$ is the fractional distribution of the transverse momentum in the jet. In fig. \[fig:bjet-pt-RES-AR-ST\]
![\[fig:bjet-pt-RES-AR-ST\] Transverse momentum of the $b$ jet, obtained with the [RES-AR]{} and the [ST-tch]{} generators.](plots/bjet-pt-AR-ST){width="48.00000%"}
we show the transverse momentum of the $b$-jet.
We see that these plots show consistently that the $b$-jet is harder and more massive in the `RES-AR` case than in the `ST-tch` one. In particular, the jet profile plot shows that there are more partons sharing the jet momentum in the region with $\Delta_{\rm R}$ near 0.1 in the `RES-AR` case. In the `ST-tch` case more momentum is concentrated at very small $\Delta_R$ (presumably due to the $b$ meson), and there are more partons at larger values of $\Delta_R$, also outside of the jet cone. We interpret these fact as consistent with the reconstructed top mass peak being slightly shifted to the right in the `RES-AR` case.
In order to quantify the shift in the mass extraction that one would get using one or the other Monte Carlo, we define an observable $M_{\rm trec}$, equal to the average value of the reconstructed top mass in a window of $\pm 15$ GeV around $m_t$, in order to mimic the typical experimental resolution on the reconstructed top mass. We get $M_{\rm trec}=170.54(2)$ GeV for the `RES-AR`, and $M_{\rm trec}=169.59(1)$ GeV for the `ST-tch` generator. Thus, extracting the top mass with the `ST-tch` generator we would get a value 1 GeV larger than if we used the `RES-AR` one.
As a further comment on our findings, we remind the reader that, as far as the reconstructed top line-shape is concerned, the `RES-AR` and `ST-tch` generators differ mainly in the way that radiation from the $b$ quark is treated. In the `RES-AR` generator the hardest radiation from the $b$ quark is always handled by `POWHEG`, with `Pythia8` handling the remaining radiation. In the `ST-tch` generator, on the other hand, `POWHEG` generates no radiation from the decaying top. Thus, all radiation from the $b$ quark is handled by `Pythia8`. We must therefore ascribe the differences that we find to the different treatment of radiation in `POWHEG` and `Pythia8`. This issue was also discussed in ref. [@Campbell:2014kua]. In view of its impact on the top mass determination, this topic deserves a more detailed phenomenological study, that goes beyond the scope of the present work.
[RES-AR]{} and [RES-HR]{} comparison
------------------------------------
We now compare the `RES-AR` and `RES-HR` generators. As mentioned earlier, the two generators differ in the way that the Les Houches record is formed after the stage of generation of radiation. In the former, the hardest radiation is kept for both production and top decay independently. So, events with up to two more partons with respect to the Born kinematics are stored in the Les Houches record, and are passed to `Pythia8` for showering. The shower in production is limited to the hardness of the radiation in production, while the shower in top decay is limited by the hardness of radiation in top decay. In the latter, only the hardest radiation of all is kept. Shower radiation, whether from production or from decay, is limited by the scale of the hardest radiation in `POWHEG`.
In fig. \[fig:Top-pt-RES-AR-RES-HR\]
![\[fig:Top-pt-RES-AR-RES-HR\] Transverse momentum distribution of the top quark, obtained with the [RES-AR]{} and the [RES-HR]{} generators.](plots/pt-top-AR-HR "fig:"){width="48.00000%"} ![\[fig:Top-pt-RES-AR-RES-HR\] Transverse momentum distribution of the top quark, obtained with the [RES-AR]{} and the [RES-HR]{} generators.](plots/y-top-AR-HR "fig:"){width="48.00000%"}
we plot the transverse momentum and the rapidity distributions of the reconstructed top. In fig. \[fig:Top-mass-RES-AR-RES-HR\]
![\[fig:Top-mass-RES-AR-RES-HR\] Invariant mass of the top quark, obtained with the [RES-AR]{} and the [RES-HR]{} generators, at the reconstructed level and at the MC-truth level.](plots/mass-top-AR-HR "fig:"){width="48.00000%"} ![\[fig:Top-mass-RES-AR-RES-HR\] Invariant mass of the top quark, obtained with the [RES-AR]{} and the [RES-HR]{} generators, at the reconstructed level and at the MC-truth level.](plots/mass-mc-top-AR-HR "fig:"){width="48.00000%"}
we show the mass peak, both for the reconstructed top and for the top particle in the Monte Carlo record. In fig. \[fig:bjet-RES-AR-RES-HR\]
![\[fig:bjet-RES-AR-RES-HR\] Mass and profile of the $b$-jet, obtained with the [RES-AR]{} and the [RES-HR]{} generators.](plots/bjet-mass-AR-HR "fig:"){width="48.00000%"} ![\[fig:bjet-RES-AR-RES-HR\] Mass and profile of the $b$-jet, obtained with the [RES-AR]{} and the [RES-HR]{} generators.](plots/bprofile-AR-HR "fig:"){width="48.00000%"}
we plot the mass and profile of the $b$-jet, while in fig. \[fig:bjet-pt-RES-AR-RES-HR\]
![\[fig:bjet-pt-RES-AR-RES-HR\] Transverse momentum of the $b$ jet, obtained with the [RES-AR]{} and the [RES-HR]{} generators.](plots/bjet-pt-AR-HR){width="48.00000%"}
we plot its transverse momentum.
We find, as in the case of the `RES-AR` and `ST-tch` comparison, differences in the top lineshape. In this case, however, they are less pronounced. The comparison of observables related to the $b$-jet also follow a similar pattern. They are qualitatively similar to the previous case, but less pronounced, in particular for the case of the $b$-jet transverse momentum distribution. We interpret these findings as being due to the fact that in the `RES-HR` generator the $b$-jet hardest radiation is in part controlled by `POWHEG` and in part by `Pythia8`. We remark that also in the `RES-HR` case `POWHEG` should correct the hardest radiation from the $b$ quark to yield an NLO accurate result, at least for sufficiently hard radiation.
It is again interesting to quantify the shift in the mass extraction that one would get using one or the other Monte Carlo. Computing, as before, our $M_{\rm trec}$ observable, we get $M_{\rm trec}=170.06(3)$ GeV for the `RES-HR` generator, and $M_{\rm trec}=170.55(2)$ GeV for the `RES-AR`. The small difference in the `RES-AR` result with respect to the one given in the previous subsection was due to the fact that there a set of real contributions was left out, as explained earlier.
[NORES]{} and [RES-HR]{} comparison
-----------------------------------
We now compare the `NORES` and `RES-HR` generators. The purpose of this comparison is to see if and how a generator that is not aware of resonance structures can exhibit visible distortions. In this case, MC resonance truth information is not available for the `NORES` generator. It is possible however to try to guess the resonance information from the structure of the event, as we will detail later.
We begin by showing results with the `NORES` contribution without performing any resonance assignment. In fig. \[fig:Top-pt-NORES-RES-HR\]
![\[fig:Top-pt-NORES-RES-HR\] Transverse momentum distribution of the top quark, obtained with the [NORES]{} and the [RES-HR]{} generators.](plots/pt-top-HR-NR "fig:"){width="48.00000%"} ![\[fig:Top-pt-NORES-RES-HR\] Transverse momentum distribution of the top quark, obtained with the [NORES]{} and the [RES-HR]{} generators.](plots/y-top-HR-NR "fig:"){width="48.00000%"}
we plot the transverse momentum and the rapidity distributions of the reconstructed top. In fig. \[fig:Top-mass-NORES-RES-HR\]
![\[fig:Top-mass-NORES-RES-HR\] Invariant mass of the top quark, obtained with the [NORES]{} and the [RES-HR]{} generators, at the reconstructed level.](plots/mass-top-HR-NR){width="48.00000%"}
we show the mass peak for the reconstructed top. In fig. \[fig:bjet-NORES-RES-HR\]
![\[fig:bjet-NORES-RES-HR\] Mass and profile of the $b$ jet, obtained with the [NORES]{} and the [RES-HR]{} generators.](plots/bjet-mass-HR-NR "fig:"){width="48.00000%"} ![\[fig:bjet-NORES-RES-HR\] Mass and profile of the $b$ jet, obtained with the [NORES]{} and the [RES-HR]{} generators.](plots/bprofile-HR-NR "fig:"){width="48.00000%"}
we plot the $b$-jet mass and profile. In fig. \[fig:bjet-pt-NORES-RES-HR\]
![\[fig:bjet-pt-NORES-RES-HR\] Transverse momentum of the $b$ jet, obtained with the [NORES]{} and the [RES-HR]{} generators.](plots/bjet-pt-HR-NR){width="48.00000%"}
we show the $b$-jet transverse momentum.
We see marked distortions in the mass peak, in the $b$-jet mass and profile, and in the transverse momentum distribution. In this case we get $M_{\rm trec}=170.54(2)$ GeV for the `NORES` and $M_{\rm trec}=170.06(3)$ GeV for the `RES-HR` generator. We remark that in this case no MC-truth was available for the top quark mass in the `NORES` case, and thus the corresponding plot is missing.
Finally, we performed a guess resonance assignment on the `NORES` output record, in the following way:
- The $b\, \mu^+\nu_\mu$ system is assigned to the top in all events.
- If the radiated parton is a gluon, and the $g\, b$ system has the colour of a quark, then we compute the transverse momentum of the gluon relative to the beam axis, $k_{T,{\rm isr}}$, relative to the final state light quark, $k_{T,{\rm fsr}}$, and relative to the $b$ quark in the $g\, b\, \mu^+\nu_\mu$ frame, $k_{T,b}$. Furthermore we compute the quantities $$\begin{aligned}
f_1 &=& \frac{1}{(s_{b\, \mu^+ \nu_\mu}-m_t^2)^2+(\Gamma_t m_t)^2}\,, \\
f_2 &=& \frac{1}{(s_{g\, b\, \mu^+ \nu_\mu}-m_t^2)^2+(\Gamma_t m_t)^2}\,.
\end{aligned}$$ The gluon is assigned or not assigned to the top resonance with probabilities proportional to $$\frac{f_2}{k_{T,b}^2}\quad {\rm and} \quad f_1\times \left(\frac{1}{k_{T,{\rm isr}}^2}
+\frac{1}{k_{T,{\rm fsr}}^2}\right) \,.$$
- If the radiated parton is not a gluon, it is not assigned to the top.
We label as `NORES-i` ([i]{} for “improved”) the corresponding generator, and show its comparison with the `RES-HR` output in figs. \[fig:Top-pt-NORES-i-RES-HR\] through \[fig:bjet-pt-NORES-i-RES-HR\].
![\[fig:Top-pt-NORES-i-RES-HR\] Transverse momentum distribution of the top quark, obtained with the [NORES-i]{} and the [RES-HR]{} generators.](plots/pt-top-HR-NRi "fig:"){width="48.00000%"} ![\[fig:Top-pt-NORES-i-RES-HR\] Transverse momentum distribution of the top quark, obtained with the [NORES-i]{} and the [RES-HR]{} generators.](plots/y-top-HR-NRi "fig:"){width="48.00000%"}
![\[fig:Top-mass-NORES-i-RES-HR\] Invariant mass of the top quark, obtained with the [NORES-i]{} and the [RES-HR]{} generators, both at the reconstructed and at the MC-truth level.](plots/mass-top-HR-NRi "fig:"){width="48.00000%"} ![\[fig:Top-mass-NORES-i-RES-HR\] Invariant mass of the top quark, obtained with the [NORES-i]{} and the [RES-HR]{} generators, both at the reconstructed and at the MC-truth level.](plots/mass-mc-top-HR-NRi "fig:"){width="48.00000%"}
![\[fig:bjet-NORES-i-RES-HR\] Mass and profile of the $b$ jet, obtained with the [NORES-i]{} and the [RES-HR]{} generators.](plots/bjet-mass-HR-NRi "fig:"){width="48.00000%"} ![\[fig:bjet-NORES-i-RES-HR\] Mass and profile of the $b$ jet, obtained with the [NORES-i]{} and the [RES-HR]{} generators.](plots/bprofile-HR-NRi "fig:"){width="48.00000%"}
![\[fig:bjet-pt-NORES-i-RES-HR\] Transverse momentum of the $b$ jet, obtained with the [NORES-i]{} and the [RES-HR]{} generators.](plots/bjet-pt-HR-NRi){width="48.00000%"}
We see that now the differences are much less pronounced, and do not seem to affect significantly the determination of the top mass. In fact, the average value of our $M_{\rm trec}$ observable is now $M_{\rm trec}=170.07(2)$ GeV for the `NORES-i` and $M_{\rm trec}=170.06(3)$ GeV for the `RES-HR` generator. Mild distortions are observed for the remaining distributions. On the other hand, we see that the MC-truth top line-shape seems to exhibit unphysical features, presumably due to the way that resonance assignment was performed.
Conclusions
===========
In this work we have presented a formalism for dealing with intermediate resonances in NLO+PS generators built within the `POWHEG` framework. We have formulated a subtraction method such that no double logarithms of the resonance’s width arise separately in the integrated real and in the soft-virtual term of the NLO calculation. Single logarithms of the widths do however arise in the soft-virtual term (in fact in the virtual contribution) and in the integrated real cross section, and cancel only when adding them up. Thus, in the framework of a `POWHEG` generator, these single log terms cancel in the $\tilde{B}$ function, so that both double and single logarithms of the resonance widths are absent there. In `POWHEG`, the generation of radiation is unitarized by construction. Therefore, all soft divergences (including also those that are cut-off by the resonance width) are properly regulated there by Sudakov form factors and/or by finite width effects.
Our formalism is fully general, and has been implemented in a general way in a modified version of the `POWHEG-BOX-V2`. The framework is such that in order to implement a specific process, one must supply the Born, including spin and colour correlated amplitudes, the virtual and the real amplitudes. The framework takes care of everything else: it finds the possible resonance histories and singular regions, it builds a Born phase space consistent with the resonance histories, and it constructs the subtraction terms, the soft-virtual contributions and the mismatch terms described in sec. \[sec:softcoll\]. It then performs the various stages of the `POWHEG` event generation.
In the present work we have considered, as a test case, the process $p p\to \mu^+ \nu_\mu j_b j$, that is dominated by single-top $t$-channel production, in the 5-flavours scheme. Within this framework, we have examined the output of our generator, with particular attention to observables that can have an impact on the top mass measurements.
We have compared two variants of our generators. In the first one we use the traditional `POWHEG` method, dubbed `RES-HR` in this paper, retaining only the hardest radiation, feeding the corresponding partonic event to a shower Monte Carlo, and vetoing any shower-generated radiation harder than the `POWHEG` one. In the second one, dubbed `RES-AR`, the hardest radiations in production and in resonance decays are both kept and combined in the final partonic event. In this case, the veto scales on the hardness of the Shower radiation in production and in decays are different, being set to the corresponding scales of radiation in `POWHEG`. We also compare our generator to the previous single-top, t-channel generator, the `ST-tch` process of ref. [@Alioli:2009je] in the `POWHEG-BOX-V2`. We can briefly summarize our findings as follows. We find differences in the reconstructed top, mostly due to the structure of the $b$-jets, and we ascribe these differences to the fact that the hardest radiation in the $b$-jet is fully determined by the Shower Monte Carlo in the `ST-tch` generator, it is in part determined by `POWHEG` and in part by the Shower Monte Carlo in the `RES-HR` generator, and it is fully determined by `POWHEG` in the `RES-AR` generator.
We have also considered the output of a generator using the same, full matrix elements for the $p p\to \mu^+ \nu_\mu j_b j$ process that we have used in our new generator, implemented however in the traditional `POWHEG-BOX-V2` framework, that is to say without resonance aware formalism. In this context we have again considered two alternative options. In the first one we pass the events to the Shower Monte Carlo without any resonance information. In the second one we reconstruct a most probable resonance history of the event based upon kinematics. The aim of such test is to search for distortions in the generated radiation due to the lack of proper treatment of resonance decays. We have found that if no resonance information is passed to the shower important differences are in fact observed. On the other hand, if we make an educated guess of the event resonance history, and pass it to the shower, smaller differences are present at the reconstructed level, although the top line-shape at the MC-truth level exhibits unphysical features.
We remark that one comparison is still missing in the present work: the comparison to a generator using the on-shell approximation (i.e. a single top generator analogous to the `ttb_NLO_dec` $t\bar{t}$ generator of ref. [@Campbell:2014kua]), in order to check if off-shell and non-resonant contributions at the radiation level are relevant for an accurate simulation of the production of a top resonance.
The relevant code for the present work is available in the `POWHEG BOX` repository at the url <http://powhegbox.mib.infn.it/POWHEG-BOX-RES-beta>. It has to be considered as very preliminary at the present stage, since only one relatively simple process has been implemented with it so far.
Acknowledgments {#acknowledgments .unnumbered}
===============
We wish to thank Kirill Melnikov, Roberto Tenchini and Giulia Zanderighi for useful discussions. We also wish to thank Carlo Oleari for critically examining parts of the paper.
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[^1]: For simplicity we discuss the QCD case. All what we do is straightforwardly extended to the electrodynamics case.
[^2]: It instead imposes that the incoming $b$ momentum minus the momenta of the final $b$ and of the radiated gluon $g$ in the real term equals to the $b$ incoming momentum minus the final $b$ momentum in the subtraction term, and all other momenta remain the same.
[^3]: We assume throughout that the reader is familiar with the notation introduced of the FNO paper [@Frixione:2007vw]. A singular region corresponds to a configuration where two final state particle become collinear, or a final state particle becomes collinear to an initial particle.
[^4]: Notice that with the notation $f_b
(\alpha_r) = f_b$ means: all $\alpha_r$ that leads to a full underlying Born flavour equal to $f_b$. We thus use $f_b$ both as a function name and as a variable, since in the present context this cannot generate confusion.
| 0 |
---
abstract: 'We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree ($\Delta$) of the underlying graph and the number of colours or spins ($q$) in determining whether the dynamics mixes rapidly or not. We find a lower bound $L$ on the number of colours such that Glauber dynamics is rapidly mixing if at least $L$ colours are used. We give a closely-matching upper bound $U$ on the number of colours such that with probability that tends to 1, the Glauber dynamics mixes slowly on random $\Delta$-regular graphs when at most $U$ colours are used. We show that our bounds can be improved if we restrict attention to certain types of graphs of maximum degree $\Delta$, e.g. toroidal grids for $\Delta=4$.'
author:
- |
Magnus Bordewich[^1]\
School of Engineering and Computing Sciences\
Durham University\
Durham, DH1 3LE, UK\
[email protected]\
- |
Catherine Greenhill[^2]\
School of Mathematics and Statistics\
The University of New South Wales\
Sydney NSW 2052, Australia\
[email protected]\
- |
Viresh Patel[^3]\
School of Mathematical Sciences\
Queen Mary, University of London\
London, E1 4NS\
[email protected]
date:
title: Mixing of the Glauber dynamics for the ferromagnetic Potts model
---
Introduction
============
The Potts model was introduced in 1952 [@pot52] as a generalisation of the Ising model of magnetism. The Potts model has been extensively studied not only in statistical physics, but also in computer science, mathematics and further afield. In physics the main interest is in studying phase transitions and modelling the evolution of non-equilibrium particle systems; see [@wu82] for a survey. In computer science, the Potts model is a test-bed for approximation algorithms and techniques. It has also been heavily studied in the areas of discrete mathematics and graph theory, through an equivalence to the Tutte polynomial of a graph [@wel93], and thereby links to the chromatic polynomial and many other graph invariants. The Potts model and its extensions have also appeared many times in the social sciences, for example in modelling financial markets [@voi01] and voter interaction in social networks [@cho07], and in biology [@gla92].
[**Potts Model.**]{} In graph-theoretic language, the Potts model assigns a weight to each possible colouring of a graph (not necessarily proper), and we are interested in sampling from the distribution induced by the weights. The main obstacle to sampling is that the appropriate normalisation factor, the sum of the weights of all colourings, is hard to compute. To be precise: for a graph $G=(V,E)$, a *(spin) configuration* $\sigma$ is a function which assigns to each vertex $i$ a colour $\sigma_i \in \{1, \ldots, q \}$ (also called states or spins). The probability of finding the system in a given configuration $\sigma$ is given by the Gibbs distribution: $$\pi(\sigma) = Z^{-1} e^{\beta \sum_{(i,j)\in E}
J \delta(\sigma_i, \sigma_j)},$$ where $\delta(\sigma_i, \sigma_j)$ is the Kronecker-$\delta$ (taking value 1 $\sigma_i = \sigma_j$, and taking value 0 otherwise); $\beta = (kT)^{-1}>0$ is the inverse temperature (here $k$ is Boltzman’s constant and $T$ is temperature); and $Z=Z(G,\beta,J,q)$, is the *partition function* i.e. the appropriate normalisation factor to make this a probability distribution. The strength of the interaction between neighbouring vertices is given by the coupling constant $J$. If $J>0$ then the bias is towards having many edges with like colours at the endpoints; this is the ferromagnetic region. If $J<0$ then the bias is towards few edges with like colours at the endpoints: this is the anti-ferromagnetic region.
Our results concern only the ferromagnetic region, where $J > 0$, although we discuss some background on the antiferromagnetic region below. We regard $e^{\beta J}$ as a single parameter $\lambda \geq 0$, which we will call the *activity*; thus $\lambda>1$ gives the ferromagnetic region and $\lambda<1$ gives the antiferromagnetic region. Setting $\mu(\sigma)$ to be the number of monochromatic edges in a configuration $\sigma$ (that is, $\mu(\sigma)=\sum_{(i,j)\in E} \delta(\sigma_i, \sigma_j)$), we obtain the formula $$Z(G,\lambda,q)=\sum_{\sigma\in [q]^V} \lambda^{\mu(\sigma)}.$$
[**Computing the partition function.**]{}When $q=1$ the evaluation of the partition function is trivial. It is also trivial when $q=2, \lambda=0$, which is the antiferromagnetic Ising model at zero temperature: here the partition function counts the number of proper 2-colourings of $G$. In all other cases it is \#P-hard to compute the partition function exactly, and thus there can be no efficient algorithm (running in time polynomial in the size of the underlying graph) assuming P$\neq$NP. (Note that the related Tutte polynomial has three additional points on the real plane at which it can be efficiently evaluated [@jae90], but these do not correspond to the ferromagnetic Potts model at physically meaningful points, *i.e.* where $q\geq 1$ and $ \lambda\geq 0$.) As a result of the hardness of exact evaluation, attention has been focused on approximation algorithms. The specific question is: for what classes of graphs and what ranges of $q$ and $\lambda$ is there a fully polynomial randomised approximation scheme (FPRAS) for computing the partition function?
In the anti-ferromagnetic case, $\lambda<1$, there can be no FPRAS for the partition function unless NP=RP, except when $q=1$ (for all $\lambda$) and when $q=2$ and $\lambda=0$ [@gol08]. For the ferromagnetic region, $\lambda>1$, there is only known to be an FPRAS when $q=2$ (the Ising model) for general graphs at any temperature [@jer93]. There is also an FPRAS for the entire ferromagnetic region (no restriction on $q$) if we restrict the underlying graphs to the class of dense graphs (those having minimum degree $\Omega(n)$ [@alo95], or having edge connectivity at least $\Omega(\log n)$ [@kar99]). In terms of approximation complexity, approximating the partition function of the ferromagnetic Potts model is equivalent to \#BIS, which is the problem of approximating the number of independent sets in a bipartite graph [@gol12]. This puts it in an interesting class of approximation problems, namely, those which are \#BIS-equivalent: no such problem is known to be hard, but none have been shown to exhibit an FPRAS [@dye03b].
[**Glauber dynamics.**]{}A standard approach to approximating the partition function is to simulate *Glauber dynamics*. In [Glauber dynamics]{} the following process is iterated (starting from any given configuration): a random vertex updates its colour by selecting a colour according to the local Gibbs distribution induced by the current colourings of its neighbours. (This will be formalised in the next subsection.) The distribution on configurations obtained after $t$ steps of Glauber dynamics converges to an equilibrium given by the global Gibbs distribution on the whole graph, as $t$ goes to infinity. The approximation is achieved by simulating the Glauber dynamics for long enough to generate a sample that is distributed with very nearly the equilibrium distribution. This process is Markov chain Monte Carlo sampling (MCMC) [@jer96]. The close link between sampling and approximate counting means that if Glauber dynamics gets sufficiently close to equilibrium in polynomial time (in the size of the graph) then there is an FPRAS for the partition function. In this case the dynamics is said to mix rapidly.
In the ferromagnetic case, physicists’ understanding of phase transitions indicate that at sufficiently high temperature (all other things being equal) Glauber dynamics will mix rapidly, whereas at sufficiently low temperature Glauber dynamics will mix slowly [@mar99]. The intuitive explanation is as follows: at high temperature in the ferromagnetic region, $\beta$ is small and so $\lambda$ is close to 1; thus all configurations are weighted roughly equally and the Glauber dynamics walks freely over the state space without getting ‘stuck’. At low temperatures in the ferromagnetic region, $\beta$ is large and so $\lambda$ is also large; thus configurations consisting of predominantly one colour are far more heavily weighted than configurations with a balance of colours, so the Glauber dynamics will become trapped in configurations of the former type. However, determining the exact range of temperature in which Glauber dynamics mixes rapidly is, in general, open.
In the [anti-ferromagnetic]{} case, where it is known that there can be no FPRAS in general, the MCMC technique has still yielded many results approximating the partition function for restricted classes of graph, notably bounded-degree graphs. In the zero temperature limit of the anti-ferromagnetic Potts model only proper vertex colourings have non-zero weight. Thus approximating the partition function is equivalent to approximately counting [proper]{} $q$-colourings of the underlying graph. Jerrum [@jer95] first showed that provided the number of colours is more than twice the maximum degree of the graph then the Glauber dynamics will mix rapidly, also proved independently in the physics community by Salas and Sokal [@sal97]. This result has been followed by numerous refinements gradually reducing the ratio of colours to degree required for rapid mixing: see [@fri07] for a recent survey. In this paper we shall investigate the interplay of the maximum degree $\Delta$ of the graph $G$ and the number of colours $q$ in determining whether the convergence of Glauber dynamics for the *ferromagnetic* Potts model is fast (rapid mixing) or slow.
Definitions
-----------
Throughout we shall be concerned with discrete-time, reversible, ergodic Markov chains with finite state space $\Omega$. Let $\mathcal{M}$ be such a Markov chain with transition matrix $P$ and (unique) stationary distribution $\pi$. For $\varepsilon > 0$ and $x \in \Omega$, we define $$\tau_x(\mathcal{M}, \varepsilon) = \min \{ t: \, \|P^t(x, \cdot) - \pi(\cdot)\|_{TV} \leq \varepsilon \},$$ where $\| \cdot \|_{TV}$ denotes total variation distance between two distributions: that is, $$\| \phi - \phi' \|_{TV} := \frac12 \sum_{x \in \Omega}|\phi(x) - \phi'(x)|.$$ for any two probability distributions $\phi$, $\phi'$ on $\Omega$. We define $\tau(\mathcal{M},\varepsilon)= \max_x \tau_x(\mathcal{M}, \varepsilon)$.
Let $G=(V,E)$ be a graph with $n:=|V|$, and let $[q]=\{ 1, \ldots, q \}$ be a set of colours (or spins). We write $\Omega = [q]^V$ for the set of configurations of $G$ (*i.e.* not-necessarily proper $q$-colourings). Fix a constant $\lambda > 1$, which is called the *activity*. The *Gibbs distribution* $\pi = \pi(G, \lambda, q)$ on $\Omega$ is given by $$\pi(\sigma) \propto \lambda^{ {\mu}(\sigma)}$$ for all $\sigma\in\Omega$, where ${\mu}(\sigma)$ denotes the number of monochromatic edges of $G$ in the configuration $\sigma$. More precisely, $\pi(\sigma) = \lambda^{ {\mu}(\sigma)}/Z$, where $Z$ is the partition function $$Z = Z(G, \lambda, q) = \sum_{\sigma \in \Omega} \lambda^{{\mu}(\sigma)}.$$
The *Glauber dynamics* is a very simple Markov chain on $\Omega$, with stationary distribution given by the Gibbs distribution. Given a configuration $X\in\Omega$, a vertex $v\in V$, and a colour $c \in [q]$, let $n(X,v,c)$ denote the number of neighbours of $v$ with colour $c$ in $X$. Define the probability distribution $\phi_X^v$ on $[q]$ by $$\phi_X^v(c) \propto \lambda^{n(X,v,c)}.$$ The transition procedure of the Glauber dynamics from current state $X_t\in\Omega$ is as follows:
- choose a vertex ${\bm{v}}$ of $G$ uniformly at random;
- given that ${\bm{v}}=v$ (here ${\bm{v}}$ is random and $v$ is fixed), choose a colour $c \in [q]$ according to the distribution $\phi = \phi_{X_t}^v$;
- for each $u \in V$ let $X_{t+1}(u) =
\begin{cases}
X_t(u) &\text{if } u \not= v, \\
c &\text{if } u = v.
\end{cases}$
Then $X_{t+1}$ is the new state. We write $\mathcal{M}_{\mathrm{GD}} = \mathcal{M}_{\mathrm{GD}}(G,\lambda,q)$ for the Glauber dynamics as described above.
We say that $\mathcal{M}_{\mathrm{GD}}$ *mixes rapidly* if $\tau(\mathcal{M},\varepsilon)$ is polynomial in $\log{|\Omega|}$, that is, polynomial in $n$. If $\tau(\mathcal{M},\varepsilon)$ is exponential in $n$, then we say that $\mathcal{M}_{\mathrm{GD}}$ *mixes slowly*.
Results
-------
Our main results are stated below. In order to keep the presentation simple at this stage, we sometimes postpone giving the explicit relationships amongst constants and mixing times until later, but in each case, we direct the reader to where a more detailed statement can be found.
In Theorem \[th:main1\] we present our first, and simplest, bound on the number of colours, as a function of $\lambda$ and $\Delta$, that guarantees rapid mixing of Glauber dynamics. Although Theorem \[th:main1\] follows from a standard coupling argument, for completeness we prove it here, as we will need this result later to establish our improved bounds.
\[th:main1\] Let $\Delta,q\geq 2$ be integers and take $\lambda>1$ such that $q\geq \Delta\lambda^\Delta + 1$. Then the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes rapidly for the class of graphs of maximum degree $\Delta$.
Theorem \[th:main1\] will be proved in Section \[se:GD\]; see Proposition \[le:vertex\] for a more detailed statement.
In Theorem \[th:main2\] we improve the exponent of $\lambda$ in the bound, but at the expense of a larger constant. We also show that the exponent achieved is close to the best possible, by proving a corresponding slow-mixing bound for almost all regular graphs of degree $\Delta$.
\[th:main2\] Fix an integer $\Delta \geq 2$. For any $\eta \in (0,1)$ there are constants $c_1$ and $c_2$ (depending on $\eta$ and $\Delta$), such that for any integer $q \geq 2$ and any $\lambda>1$
1. if $q>c_1\lambda^{\Delta-1+\eta}$ then the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes rapidly for the class of connected graphs of maximum degree $\Delta$;
2. if $q<c_2\lambda^{\Delta-1-\tfrac{1}{\Delta-1}-\eta}$ then the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes slowly for almost all regular graphs of degree $\Delta \geq 3$.
Theorem \[th:main2\] is proved at the end of the paper: a more detailed statement of Theorem \[th:main2\](i) can be found in Theorem \[th:glaubercompare\], while a more detailed statement of Theorem \[th:main2\](ii) can be found in Theorem \[th:torpid-random\].
Theorem \[th:main2\](ii) is proved using a conductance argument. It turns out that conductance for the Glauber dynamics is related to the expansion properties of the underlying graph, and so we prove that almost all $\Delta$-regular graphs have the relevant property. This argument alone gives a worse bound than that in Theorem \[th:main2\](ii), but combined with the solution of an interesting extremal problem (proved in Section \[se:extremal\]), which we believe may be of independent interest, we are able to obtain the required improvement.
Theorem \[th:main2\](i) is proved by first using a coupling argument to prove a rapid-mixing result for block dynamics (a more general form of dynamics than Glauber dynamics) and then using a Markov chain comparison argument to obtain rapid mixing for Glauber dynamics. In proving Theorem \[th:main2\](i), we derive a general combinatorial condition on graphs that guarantees rapid mixing of Glauber dynamics (Theorem \[se:grid\] combined with Corollary \[co:comparison\]). This condition can be used to improve the bounds of Theorem \[th:main2\](i) for graph classes of maximum degree $\Delta$ with “low expansion”. We illustrate this in Theorem \[th:main3\] below with the example of the toroidal grid.
\[th:main3\] For any $\eta\in (0,1)$ there are constants $c_3$, $c_4$ and $c_5$ (depending on $\eta$), such that for any positive integer $q$ and any $\lambda>1$
1. if $q>c_3\lambda^{3+\eta}$ then the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes rapidly for the class of connected graphs of maximum degree 4;
2. if $q>c_4\lambda^{2+\eta}$ then the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes rapidly for the toroidal grid;
3. if $q<c_5\lambda^{\tfrac{8}{3}-\eta}$ then the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes slowly for almost all regular graphs of degree $4$.
In particular, for sufficiently large $\lambda$ there is a positive integer $q$ such that the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes rapidly for the toroidal grid, but slowly for almost all regular graphs of degree $4$.
The purpose of Theorem \[th:main3\] is illustrative and it is proved at the end of the paper. Theorem \[th:main3\](i) and (iii) are immediate consequence of Theorem \[th:main2\] (by substituting $\Delta=4$), while Theorem \[th:main3\](ii) is a useful illustration of our general technique applied to the grid. A more detailed statement of Theorem \[th:main3\](ii) is given as part of Theorem \[th:glaubergrid\].
Section \[se:rapid\] contains our results on rapid mixing of Glauber dynamics. Section \[se:extremal\] is devoted to an extremal problem whose solution allows us to obtain improved bounds for our slow-mixing results in Section \[se:tormix\].
Comparison with related results and phase transitions
-----------------------------------------------------
We write $o(1)$ for an expression that tends to $0$ as $q \to \infty$. (The most interesting setting for our results is when $q$ is large.) We now restate our results in terms of the inverse temperature $\beta$, under the assumption that $J=1$, so that $\lambda = e^\beta$.
The results of Theorems \[th:main1\], \[th:main2\](i), \[th:main2\](ii), \[th:main3\](ii) say respectively:
- if $\beta \leq \frac{1 + o(1))}{\Delta}\, \log q$ then the Glauber dynamics of the $q$-state Potts model mixes rapidly on graphs of maximum degree $\Delta$;
- if $\beta \leq \frac{1 + o(1)}{\Delta-1}\, \log q$ then the Glauber dynamics of the $q$-state Potts mixes rapidly on graphs of maximum degree $\Delta$;
- if $\beta > \frac{1 + o(1)}{\Delta - 1 - \nfrac{1}{\Delta-1}}\, \log q $ then the Glauber dynamics of the $q$-state Potts model mixes slowly for almost all regular graphs of degree $\Delta \geq 3$;
- if $\beta < \frac{1+o(1)}{2}\, \log q$ then the Glauber dynamics of the $q$-state Potts model mixes rapidly for toroidal grids.
There is some overlap between Theorem \[th:main1\] and a result of Hayes [@hayes Proposition 14] for $q=2$, which was generalised to arbitrary $q$ by Ullrich [@ull Corollary 2.14]. Ullrich showed that when the inverse temperature $\beta$ satisfies $\beta \leq 2c/\Delta$ for some $0<c<1$, then the Glauber dynamics is rapidly mixing on graphs of maximum degree $\Delta$. Hence our result (a) holds for a wider range of $\beta$ when $q$ is large. (For small values of $q$, Theorem \[th:main1\] does not apply but [@ull Corollary 2.14] is valid).
As we have mentioned, there is often a link between certain phase transitions and the critical inverse temperature of associated dynamics (i.e. an inverse temperature below which the dynamics mix rapidly and above which they mix slowly). We will not define what we mean by phase transitions here but mention only that, for Glauber dynamics of the $q$-state Potts model on a random $\Delta$-regular graph, the relevant phase transition is the transition from unique to non-unique Gibbs measure on the infinite $\Delta$-regular tree. H[ä]{}ggstr[ö]{}m [@Hag] showed that this phase transition occurs at an inverse temperature $\beta_0 = \log B$, where $B$ is the unique value for which the polynomial $$(q-1)x^{\Delta} + (2 - B - q)x^{\Delta - 1} +Bx - 1$$ has a double root in $(0,1)$. While there is no general closed form formula for $\beta_0$, we show in the appendix that $\beta_0 = \frac{\log q}{\Delta - 1} + O(1)$. Thus $\beta_0$ approximately matches the rapid mixing bound of (b).
We note that, in a recent related work, Galanis *et al.* [@GSVY] give a very detailed picture of the phase transitions of the ferromagnetic Potts model on the infinite $\Delta$-regular tree. Using this analysis they also show that show that the Swendsen-Wang process (a MCMC process different to Glauber dynamics) mixes slowly at a specific phase transition point on almost all random regular graphs of degree $\Delta$.
As mentioned earlier, result (d) is only illustrative since sharper bounds for the grid are known. It is known that for the infinite 2-dimensional grid, the phase transition occurs at $q=(\lambda-1)^2$ [@wu82] (i.e. $\beta = \log(1 + q^\frac{1}{2})$) and that rapid mixing occurs for finite grids when $\beta$ is below this threshold; see [@mar99] and Theorem 2.10 of [@ull]. It is conjectured that the Glauber dynamics mixes slowly when $\beta$ is above this threshold; see Remark 2.11 of [@ull]). Borgs, Chayes and Tetali [@BCT] proved that for $q$ sufficiently large and for $\beta > \frac{\log\left(q\right)}{2}+O(q^{-1/2})$, the heat bath Glauber dynamics is slowly mixing on sufficiently large toroidal grids (with a mixing time exponential in $\beta$ and in $L$, the side length of the grid). This improved on the earlier result [@bor99].
Mixing time upper bounds {#se:rapid}
========================
Our goal in this section is to give good lower bounds on the number of colours needed for the Glauber dynamics to mix rapidly. We begin by describing the notions of coupling and path coupling, which are very useful tools in proving upper bounds on mixing times for Markov chains. In Section \[se:GD\], we apply path coupling directly to the Glauber dynamics of bounded-degree graphs to obtain our first lower bound on the number of colours needed for rapid mixing. In Section \[se:BD\], we consider block dynamics, a more general type of dynamics that can be used to sample from the Gibbs distribution. We give a general lower bound on the number of colours needed for rapid mixing of block dynamics (Theorem \[le:block\]). We illustrate how to apply Theorem \[le:block\] to bounded-degree graphs in Section \[se:grid\]. In Section \[se:comparison\], we relate the mixing times of Glauber dynamics to that of the block dynamics and show how this gives various improvements to the bounds obtained in Section \[se:GD\]. This enables us, in Theorems \[th:glaubercompare\] and \[th:glaubergrid\], to prove what is needed for Theorem \[th:main2\] part (i), and Theorem \[th:main3\] parts (i) and (ii). Note that the final proofs of Theorems \[th:main2\] and \[th:main3\] are left until we have all the pieces, at the end of Section \[se:tormix\].
Coupling {#se:coupling}
--------
The notion of coupling (more specifically path coupling [@BubleyDyer]) lies at the heart of our proofs of upper bounds for mixing times. We give the basic setup in this section.
Let $\mathcal{M} = (X_t)$ be a Markov chain with transition matrix $P$. A *coupling* for $\mathcal{M}$ is a stochastic process $(A_t,B_t)$ on $\Omega \times \Omega$ such that each of $(A_t)$ and $(B_t)$, considered independently, is a faithful copy of $(X_t)$. Since all our processes are time-homogeneous, a coupling is determined by its transition matrix: given elements $(a,b)$ and $(a',b')$ of $\Omega \times \Omega$, let $P'((a,b),(a',b'))$ be the probability that $(A_{t+1},B_{t+1}) = (a',b')$ given that $(A_t,B_t) = (a,b)$. Since $(A_t,B_t)$ is a coupling, for each fixed $(a,b) \in \Omega \times \Omega$, we have $$\begin{aligned}
\sum_{b' \in \Omega}P'((a,b),(a',b')) &= P(a,a') \quad \text{ for all } a' \in \Omega; \\
\sum_{a' \in \Omega}P'((a,b),(a',b')) &= P(b,b') \quad \text{ for all } b' \in \Omega .\end{aligned}$$ Under *path coupling*, the coupling is only defined on a subset $\Lambda $ of $ \Omega \times \Omega$. This restricted coupling is then extended to a coupling on the whole of $\Omega \times \Omega$ along paths in the state space $\Omega$. In our setting, we have $\Omega = [q]^V$, where $V$ is the vertex set of some fixed graph. For $\sigma, \sigma' \in \Omega$, we write $d(\sigma, \sigma')$ for the number of vertices on which $\sigma$ and $\sigma'$ differ in colour (that is, the Hamming distance). Define $\Lambda \subseteq \Omega \times \Omega$ by $$\Lambda = \{(\sigma,\sigma'): d(\sigma,\sigma')=1 \}.$$ The key property of $\Lambda$ required for the path coupling method is that for any $\sigma,\sigma'\in\Omega$, by recolouring the $d(\sigma,\sigma')$ disagreeing vertices one by one in an arbitrary order, we obtain a path of length $d(\sigma,\sigma')$ from $\sigma$ to $\sigma'$, with consecutive elements of the path corresponding to an element of $\Lambda$.
\[See [@DG98] for example\] \[le:coup\] Let $\Omega=[q]^V$ and $\Lambda$ be as above, with $n:=|V|$, and let $\mathcal{M}$ be some Markov chain on $\Omega$. Suppose that we can define a coupling $(A,B)\mapsto (A',B')$ for $\mathcal{M}$ on $\Lambda$ such that for some constant $\beta < 1$ and all $(A,B)\in \Lambda$ we have $${\mathbb{E}}(d(A',B')\mid (A,B)) \leq \beta .$$ Then by path coupling we may conclude that $$\tau(\mathcal{M},\varepsilon) \leq \frac{\log(n\, \varepsilon^{-1})}{1-\beta}.$$
Glauber dynamics {#se:GD}
----------------
Our goal in this subsection is to prove Theorem \[th:main1\]. In the subsections that follow, we shall see how we can improve Proposition \[le:vertex\] in some special cases, but in Section \[se:tormix\], we shall see that the bound given below is close to best possible, at least in terms of the exponent of $\lambda$.
We actually prove the following proposition, which immediately implies Theorem \[th:main1\] but also provides a bound on the mixing time. The proof is a standard coupling calculation.
\[le:vertex\] Let $G$ be a graph with maximum degree $\Delta$, and fix an activity $\lambda>1$. Suppose that $q$ is an integer which satisfies $ q \geq \Delta\lambda^\Delta + 1$. Recall that $\mathcal{M}_{\mathrm{GD}} =
\mathcal{M}_{\mathrm{GD}}(G, \lambda, q)$ denotes the Glauber dynamics for the $q$-state Potts model on $G$ at activity $\lambda$. Then $$\tau(\mathcal{M}_{\mathrm{GD}},\varepsilon) \leq
(\Delta+1)\, n\log(n\, \varepsilon^{-1}).$$
Fix $(A,B)\in\Lambda$ and let $u$ be the (unique) vertex which is coloured differently by $A$ and $B$. We define a coupling $(A,B)\mapsto (A',B')$ as follows: let ${\bm{v}}$ be a uniformly random vertex of $G$, and given that ${\bm{v}}=v$, obtain $A'$ (respectively, $B'$) by updating the colour of the vertex $v$ in $A$ (respectively, $B$) according to the distributions $\phi_A := \phi_A^v$ (respectively, $\phi_B := \phi_B^v$). The joint distribution on $(\phi_A, \phi_B)$ is chosen so as to maximise the probability that $A'(v) = B'(v)$. Call this maximised probability $p=p(v,A,B)$. It is not hard to see that $$1-p = \frac{1}{2}\sum_{c \in [q]} |\phi_A(c) - \phi_B(c)| = \|\phi_A - \phi_B \|_{TV}.$$ Observe that $p(v,A,B) = 1$ if $v=u$ or if $v$ is not a neighbour of $u$ (because in both cases, $A$ and $B$ assign the same colours to the neighbours of $v$ and so $\phi_A$ and $\phi_B$ are the same distribution).
Now assume that $v$ is a neighbour of $u$, so that $\phi_A$ and $\phi_B$ are different distributions. Without loss of generality, we may assume that $A(u)=1$ and $B(u)=2$. Let $a_i:=n(A,v,i)$, that is, $a_i$ is the number of neighbours of $v$ coloured $i$ by $A$. Similarly, let $b_i := n(B,v,i)$. Note that $b_1 = a_1 - 1$, $b_2 = a_2+1$ and $b_i = a_i$ for $i = 3, \ldots, q$. Define $$Z_A = \sum_{i=1}^q \lambda^{a_i} \:\:\:\:\: \text{and} \:\:\:\:\:
Z_B = \sum_{i=1}^q \lambda^{b_i} \,=\, Z_A + (1- \lambda^{-1})(\lambda^{a_2+1} - \lambda^{a_1}),$$ and assume without loss of generality that $Z_B \leq Z_A$. It is easy to see that $\phi_A(i) \leq \phi_B(i)$ for $i=2,\ldots, q$ and hence $\phi_A(1) \geq \phi_B(1)$. Thus $$\|\phi_A - \phi_B\|_{TV} = \max_{R \subseteq [q]}|\phi_A(R) - \phi_B(R)| = |\phi_A(1) - \phi_B(1)| =
\frac{\lambda^{a_1}}{Z_A} - \frac{\lambda^{b_1}}{Z_B}.$$ Given ${\bm{a}} = (a_1, \ldots, a_q) \in [\Delta]^q$, define $f({\bm{a}},\lambda,q) = \frac{\lambda^{a_1}}{Z_A} - \frac{\lambda^{b_1}}{Z_B}$, and let $g(\lambda,q)$ be the maximum of $f$ over all ${\bm{a}} \in [\Delta]^q$ subject to $a_1 + \cdots + a_q = \Delta$.
Observe that $$\begin{aligned}
{\mathbb{E}}\left(d(A',B')-1\big| (A,B)\right)
&= (-1){\mathbb{P}}({\bm{v}}=u) + \sum_{v \in N(u)}{\mathbb{P}}({\bm{v}}=v)(1-p(v,A,B)) \\
&\leq -\frac{1}{n} + \frac{\Delta}{n} g(\lambda,q).\end{aligned}$$ We give an easy upper bound for $g(\lambda, q)$ as follows. First, for all ${\bm{a}}\in [\Delta]^q$ we have $$f({\bm{a}}, \lambda, q) \leq \frac{\lambda^{a_1}}{Z_A}.$$ The right hand side of the above is increasing in all directions of the form $e_1 - e_i$, where $e_1, \ldots, e_q$ is the standard basis for ${\mathbb{R}}^q$. Therefore the right hand side is maximised when ${\bm{a}} = (\Delta,0,\ldots, 0)$ giving $$g(\lambda,q) \leq \frac{\lambda^\Delta}{\lambda^\Delta + q-1} \leq
\frac{1}{\Delta+1},$$ using the lower bound on $q$ to obtain the final inequality. Therefore. $$\begin{aligned}
{\mathbb{E}}\left(d(A',B')\big| (A,B)\right) \leq
1+ \frac{1}{n}\left(-1 + \frac{\Delta}{\Delta+ 1}\right)
&= 1- \frac{1}{(\Delta+1)n}.\end{aligned}$$ Applying Lemma \[le:coup\] completes the proof.
Block dynamics {#se:BD}
--------------
In this section we begin the analysis of *block dynamics* in which, at each step, the colours of several vertices (or a *block* of vertices) are updated. We first present the framework and show general results on block dynamics. In the next subsection we discuss suitable choices of blocks and, in Theorem \[pr:3items\], show rapid mixing of block dynamics for certain block systems.
As before, let $G=(V,E)$ be a graph, fix $\lambda >1$ and let $\Omega = [q]^V$, where $[q] = \{1, \ldots, q\}$. Let $\mathscr{S} = \{ S_1,\ldots, S_R\}$ be a collection of subsets of $V$ such that $\cup_{S\in\mathscr{S}} S = V$. Each element of $\mathscr{S}$ is called a *block*, and we call $\mathscr{S}$ a *block system* for $G$. Fix a probability distribution $\psi$ on $\mathscr{S}$. We define a Markov chain $\mathcal{M}_{\mathrm{BD}}=\mathcal{M}_{\mathrm{BD}}^{\mathscr{S}, \psi}(G, \lambda, q)$ with state space $\Omega$, which we call the $(\mathscr{S}, \psi)$-*block dynamics*. We ensure that the new chain also has the Gibbs distribution as its stationary distribution. First we need some more notation.
Given $S\in \mathscr{S}$, for $c\in [q]^{S}$ and $X\in\Omega$ we let $X^{(S,c)}\in \Omega$ be the configuration defined by $$X^{(S,c)}(u) = \begin{cases} X(u) & \text{ if $u\not\in S$,}\\
c(u) & \text{ if $u\in S$.}\end{cases}$$ Let $\mu_{X,S}(c)$ denote the number of monochromatic edges in $X^{(S,c)}$ which are incident with at least one vertex of $S$. Finally, define the distribution $\phi_{X,S}$ on $[q]^{S}$ by $$\phi_{X,S}(c) \propto \lambda^{\mu_{X,S}(c)},
\quad \text{ that is, } \,\,\,
\phi_{X,S}(c) = \frac{\lambda^{\mu_{X,S}(c)}}{Z_{X,S}}$$ where $$Z_{X,S} = \sum_{c \in [q]^{S}} \lambda^{\mu_{X,S}(c)}.$$ The transition procedure of the $(\mathscr{S},\psi)$-block dynamics can now be described. From current state $X_t\in\Omega$, obtain the new state $X_{t+1}\in \Omega$ as follows:
- choose a random ${\bm{S}} \in \mathscr{S}$ according to the distribution $\psi$;
- given that ${\bm{S}}=S$, choose a configuration $c \in [q]^{S}$ for $S$ from the distribution $\phi_{X_t,S}$;
- let $X_{t+1} = {X_t}^{(S,c)}$.
The stationary distribution of this chain is the Gibbs distribution on $\Omega$.
Theorem \[le:block\] below gives a sufficient condition on the number of colours for the $(\mathscr{S},\psi)$-block dynamics to be rapidly mixing. The result is stated in terms of three parameters which we now define.
For $S \subseteq V$, write $\partial S$ for the set of vertices in $V \setminus S$ that have a neighbour in $S$. Write $s:= \max_{S \in \mathscr{S}}|S|$ for the size of the largest block in $\mathscr{S}$. Let ${\bm{S}} \in\mathscr{S}$ be a random block chosen according to the distribution $\psi$. Given $v\in V$, define $$\psi(v) = {\mathbb{P}}(v \in {\bm{S}}), \quad
\psi_{\partial}(v) = {\mathbb{P}}(v \in \partial {\bm{S}}).$$ Our first parameter $\partial^+$ is $$\label{partial-plus}
\partial^+ = \partial^+(\mathscr{S}) =
\max_{S\in\mathscr{S}} |\partial S|^{\min\{ |S|,\, |\partial S|\}}.$$ Let $\psi_{\min} := \min_{v \in V} \psi(v)$ and define our second parameter $\Psi$ by $$\label{Psi}
\Psi = \Psi(\mathscr{S}, \psi) = \max_{v \in V} \frac{\psi_\partial(v)}
{\psi(v)}.$$ These first two parameters are in some sense less important than the third parameter since they are essentially used as crude estimates for quantities that we do not aim to control too precisely.
For the third parameter we require some terminology. Given $A \subseteq V$ and $X\in \Omega$, write $X|_A$ for the configuration $X$ restricted to $A$. Consider a configuration $c\in [q]^S$. A colour used by $c$ is called *free* with respect to $X,S$ if it does not appear in $X|_{\partial S}$. Write $f(X,S,c)$ for the number of free colours in $c$ with respect to $X,S$. For our third parameter, we first define for each positive integer $f$ $$\mu^+_{X,S,f} =
\max \left\{
\frac{\mu_{X,S}(c)}{|S| - f} : c\in [q]^S,\,\, f(X,S,c)=f\right\},$$ where the maximum over an empty set is defined to be zero. We set $$\label{mu-plus}
\mu^+ = \mu^+(\mathscr{S}) = \max_{S\in\mathscr{S}}\, \max_{X\in \Omega} \,
\max_{f=0,\ldots, |S|-1}\,
\mu^+_{X,S,f}.$$ Although the definition of $\mu^+$ gives an a priori dependency on $q$, in all our applications on bounded-degree graphs we can bound $\mu^+$ independently of $q$ (see Proposition \[pr:boundd\]). Hence we suppress this dependence in our notation.
Let us sketch a very informal argument to show that block dynamics mixes rapidly roughly when $q \geq \lambda^{\mu^+}$; this will be formalised in the statement and proof of Theorem \[le:block\]. Fix $X \in \Omega$ and $S \in \mathscr{S}$, where $|S|$ is typically thought of as a small number and $q$ a large number. We are interested in estimating the quantity $q^{|S|} / Z_{X,S}$, which, in the distribution $\phi_{X,S}$, is approximately the probability of choosing a *free* configuration for $S$. A free configuration is one in which each vertex in $S$ receives a distinct free colour, so that $S$ is coloured with $|S|$ free colours in total. If this probability is close to $1$ for all choices of $X,S$ then, intuitively at least, one expects the block dynamics to mix rapidly.
To show $q^{|S|} / Z_{X,S}$ is close to $1$, we must show that the contribution of non-free configurations to $Z_{X,S}$ is relatively small (compared to $q^{|S|}$). Consider the contribution from configurations with a fixed number $f \leq |S|-1$ of free colours. There are approximately $q^f$ such configurations $c$, each contributing $\lambda^{\mu_{X,S}(c)} \leq \lambda^{(|S|-f)\mu^+_{X,S,f}}$ to $Z_{X,S}$, giving a total contribution of at most $q^f \lambda^{(|S|-f)\mu^+_{X,S,f}}$. Comparing to $q^{|S|}$ gives $$q^{|S|}/q^f \lambda^{(|S|-f)\mu^+_{X,S,f}} = [q \lambda^{-\mu^+_{X,S,f}}]^{|S|-f} \geq q \lambda^{-\mu^+_{X,S}}.$$ This last expression is at least $1$ provided $q > \lambda^{\mu^+_{X,S,f}}$, and this inequality holds for all choices of $X,S,f$ if $q > \lambda^{\mu^+}$. From these crude calculations we expect rapid mixing of block dynamics roughly when $q > \lambda^{\mu^+}$.
The following theorem formalises the argument above, giving a sufficient condition on the number of colours for $(\mathscr{S},\psi)$-block dynamics to be rapidly mixing.
\[le:block\] Let $G=(V,E)$ be a connected graph and let $\mathscr{S}$ be a block system for $G$ such that $V\not\in\mathscr{S}$. Let $\psi$ be a distribution on $\mathscr{S}$ and fix $\lambda > 1$. If $$q \geq (2s)^{s+1}\, \partial^+\, \Psi\, \lambda^{\mu^+}$$ *(*where parameters $s$, $\partial^+$, $\Psi$ and $\mu^+$ are as defined above*)* then the $(\mathscr{S},\psi)$-block dynamics $\mathcal{M}_{\mathrm{BD}} =
\mathcal{M}_{\mathrm{BD}}^{\mathscr{S},\psi}(G, \lambda, q)$ satisfies $$\tau(\mathcal{M}_{\mathrm{BD}},\varepsilon) \leq 2\psi_{\min}^{-1}\log(n \varepsilon^{-1}).$$
We remark that for the bound $q \geq (2s)^{s+1}\, \partial^+\, \Psi\, \lambda^{\mu^+}$ in Theorem \[le:block\], we expect the constant multiplicative factor $(2s)^{s+1} \partial^+ \Psi$ can be improved; however we have not attempted to do this in order to keep our treatment simple.
We define a coupling $(A,B)\mapsto (A',B')$ for $\mathcal{M}_{\mathrm{BD}}$ on $\Lambda$ as follows. Given $(A,B)\in\Lambda$, let $u=u(A,B)$ be the (unique) vertex which is coloured differently by $A$ and $B$. We choose a random ${\bm{S}} \in \mathscr{S}$ using the distribution $\psi$, and given that ${\bm{S}}=S$, we obtain $A'$ (respectively, $B'$) by updating the colouring of $S$ in $A$ (respectively, $B$) according to the distribution $\phi_A := \phi_{A,S}$ (respectively, $\phi_B := \phi_{B,S}$); this will give a coupling since $A$ and $B$ are updated using the transition procedure of $\mathcal{M}_{\mathrm{BD}}$. We choose the joint distribution on $(\phi_A, \phi_B)$ so as to maximise the probability that $A'|_S = B'|_S$. Call this maximised probability $p(S,A,B)$. Observe that $p(S,A,B) = 1$ if $u \not\in \partial S$ (because $A$ and $B$ assign the same colours to $\partial S$, so $\phi_A$ and $\phi_B$ are the same distribution). For the case that $u \in \partial S$, we uniformly bound $p(S,A,B)$ by setting $$p := \min_{(A,B)\in\Lambda}\, \min_{S\in \mathscr{S} : u \in \partial S}
\, p(S,A,B).$$ (Let $p=1$ if, for all $S\in\mathscr{S}$, $u\not\in\partial S$.) Now for all $S\in\mathscr{S}$ with $u\in\partial S$ we have $$\begin{aligned}
\label{eq:1}
p(S,A,B) = \sum_{c \in [q]^S} \min(\phi_A(c), \phi_B(c)) &\geq \sum_{c \in [q]^S} \frac{1}{\max(Z_{A,S}, Z_{B,S})} \nonumber \\
&= \frac{q^{|S|}}{\max(Z_{A,S}, Z_{B,S})}.\end{aligned}$$ We claim that $$\label{fred}
\frac{q^{|S|}}{Z_{X,S}} \geq 1 - \frac{1}{2s\Psi}$$ for all $X\in\Omega$ and $S\in\mathscr{S}$. If (\[fred\]) holds then substituting into (\[eq:1\]) gives $$p\geq 1 - \frac{1}{2s\Psi},$$ which in turn implies that $$\begin{aligned}
{\mathbb{E}}\left(d(A',B') - 1\big| (A,B)\right)
&= -{\mathbb{P}}(u \in {\bm{S}}) + \sum_{S \in \mathscr{S}:\, u \in \partial S}{\mathbb{P}}({\bm{S}}=S)|S|(1-p(S,A,B)) \\
&\leq
\, {} - \psi(u) + s\, \psi_{\partial}(u)\, (1-p) \\
&= -\psi(u) \left( 1 - \frac{s\psi_{\partial}(u)}{\psi(u)}(1-p) \right)\\
&\leq -\psi_{\min} \left( 1 - s \Psi\, (1-p) \right) \\
&\leq -\frac{\psi_{\min}}{2}.\end{aligned}$$ The theorem follows from this, by Lemma \[le:coup\]. So it remains to establish (\[fred\]).
Fix $X\in\Omega$ and $S\in\mathscr{S}$. For any configuration $c$, write $Q(c)$ for the set of colours used by $c$. Given a configuration $c\in [q]^S$, the colour classes of $c$ define a partition $P$ of $S$ into (unordered) nonempty parts. (Here, we think of a partition $P$ of $S$ as a set of nonempty parts $\{ P_1, \ldots, P_t\}$ where $P_i \subseteq S$ are disjoint and $\cup_{A \in P}A=S$.) Let $F\subseteq P$ be the set of colour classes corresponding to colours which are free with respect to $X,S$ (in the given configuration $c$).
Conversely, we can start from a partition $P$ of $S$ and a subset $F$ of $P$. Given a set of $|P|$ colours, we can form a configuration of $S$ by assigning a distinct colour to each part of $P$ such that the colour assigned to $A\in P$ belongs to $[q]\setminus Q(X|_{\partial S})$ if and only if $A\in F$. Any configuration which can be formed in this way is called a $(P,F)$-*configuration* of $S$. (Such a configuration is uniquely determined by $(P,F)$ and the map $P\to [q]$ which performs the assignment of colours.)
Let $n(S,P,F)$ be the number of $(P,F)$-configurations of $S$. By definition of $\mu^+$ we have $$Z_{X,S} = \sum_{c\in [q]^S} \lambda^{\mu_{X,S}(c)}
\leq q^{|S|} +
\sum_{(P,F) : |F|\neq |S|} \, n(S,P,F)\, \lambda^{(|S|-|F|)\mu^+}.$$ The first term corresponds to $P=F$ with $|P|=|S|$, arising from a configuration $c\in [q]^S$ in which every vertex in $S$ receives a distinct free colour. (These were called “free configurations” in the sketch proof.) We use $q^{|S|}$ as an upper bound for the number of such configurations. For all other values of $(S,P,F)$ we have the following crude bound: $$n(S,P,F) \leq q_1^{\min\{q_1, |P|-|F|\}}\, (q-q_1)^{|F|}
\leq |\partial S|^{\min\{ |S|, |\partial S|\}} \, q^{|F|}
\leq \partial^+\, q^{|F|},$$ where $q_1 = |Q(X|_{\partial S})|$ and we recall that all parts must be coloured differently. Substituting gives $$Z_{X,S} \leq q^{|S|} + \sum_{(P,F):\, |F|\neq |S|} \,
\partial^+\, q^{|F|}\, \lambda^{(|S|-|F|)\mu^+} .$$ Now applying the bound on $q$ from the theorem statement gives $$\begin{aligned}
\frac{Z_{X,S}}{q^{|S|}} &\leq 1 + \sum_{(P,F) : |F|\neq |S|} \,
\partial^+\, q^{|F|-|S|} \lambda^{(|S|-|F|)\mu^+}\notag\\
&\leq 1 + \sum_{(P,F) : |F|\neq |S|} \partial^+\,
((2s)^{s+1}\, \partial^+\, \Psi\,
\, \lambda^{\mu^+})^{|F|-|S|}\, \lambda^{(|S|-|F|)\mu^+}\notag\\
& \leq 1 + \sum_{(P,F):\, |F| \not= |S|} \,
((2s)^{s+1}\, \Psi )^{|F|-|S|}.
\label{interrupted}\end{aligned}$$ The number of terms in the above sum is at most $(2|S|)^{|S|}$, since there are at most $|S|^{|S|}$ choices of the partition $P$ and at most $2^{|P|} \leq 2^{|S|}$ choices of $F$.
Next, note that $$\Psi = \max_{v \in V} \frac{\psi_\partial(v)}
{\psi(v)}
\geq {\mathbb{E}}_\rho\left(\frac{\psi_\partial(v)}{\psi(v)}\right)
= \sum_{v\in V} \rho(v)\, \frac{\psi_\partial(v)}{\psi(v)}$$ for any probability distribution $\rho$ on $V$. In particular, we can take $\rho(v) = \psi(v)/N$, where $$N = \sum_{v\in V}\psi(v) = \sum_{S\in\mathscr{S}}\, \psi(S)\, |S|
\leq s.$$ With this choice of $\rho$, we obtain the bound $$\begin{aligned}
\Psi &\geq N^{-1}\, \sum_{v\in V} \psi_\partial(v)
= N^{-1}\, \sum_{S\in S}\, \psi(S)\, |\partial S|
\geq s^{-1}\end{aligned}$$ since $\partial S$ is nonempty for all $S\in\mathscr{S}$, as $G$ is connected and $V\not\in\mathscr{S}$. It follows that $(2s)^{s+1}\, \Psi > 1$, and combining this with (\[interrupted\]) gives $$\begin{aligned}
\frac{Z_{X,S}}{q^{|S|}}
& \leq 1 + \frac{1}{2s\Psi}.\end{aligned}$$ Inverting this and using the identity $(1+y)^{-1}\geq 1-y$ establishes (\[fred\]), completing the proof.
Block dynamics for specific examples {#se:grid}
------------------------------------
In this subsection we illustrate how one can use Theorem \[le:block\] to obtain rapid mixing results for block dynamics on graphs of bounded degree. In the next subsection, we shall see how these results for block dynamics can be translated into rapid mixing results for Glauber dynamics.
In order to build some intuition, we begin by investigating the range of possible values of the parameter $\mu^+$. We will need the following notation: given $T\subseteq T' \subset V$, we write ${\mathrm{vol}}(T,T')$ for the set of edges of $G$ that are contained in $T'$ and have at least one endvertex in $T$.
\[pr:boundd\] Let $G=(V,E)$ be a graph of maximum degree $\Delta$ and let $\mathscr{S}$ be any block system for $G$. Then $$\mu^+ = \mu^+(\mathscr{S}) \leq \Delta.$$ If in addition $G$ is regular then $$\frac{\Delta}{2}\leq \mu^+(\mathscr{S}) \leq \Delta.$$
First fix $X\in\Omega$ and $S\in \mathscr{S}$. Given a configuration $c\in [q]^{S}$, let $P$ be the partition of $S$ defined by the nonempty colour classes of $c$. Define $F \subseteq P$ to be the set of colour classes of $c$ which correspond to a colour which does not appear on $X|_{\partial S}$. Let $$A_F = \bigcup_{A \in F} \, A$$ and $$A'_F = \bigcup_{A \in F\, :\, |A|\geq 2} \, A.$$ Since $G$ has maximum degree $\Delta$, a trivial upper bound on $\mu_{X,S}(c)$ is $\Delta|S|$. But note that if a monochromatic edge $e$ is incident to a vertex in $A_F$, then $e$ must have both endpoints in the same part $A$ of $F$. Thus edges incident to vertices in $A_F \setminus A'_F$ do not contribute to $\mu_{X,S}(c)$ and monochromatic edges incident to vertices in $A'_F$ are double counted in the trivial bound. Hence $$\begin{aligned}
\mu_{X,S}(c)
& \leq \Delta(|S|-|A_F|)+\frac{\Delta}{2}\,|A'_F| \\
& = \Delta |S| - \Delta\left(|A_F|-\frac{|A'_F|}{2}\right) \\
&\leq \Delta (|S| - |F|).\end{aligned}$$ Hence the upper bound holds, by definition of $\mu^+$.
Next, suppose that $G$ is $\Delta$-regular with $X \in \Omega$ and $S\in\mathscr{S}$. Consider any configuration $c\in [q]^S$ which assigns a single colour to all of $S$, and where this is the only colour used in $X|_{\partial S}$. Then $$\mu^+ \geq \frac{\mu_{X,S}(c)}{|S|-|F|} =
\frac{|{\mathrm{vol}}(S, S \cup \partial S)|}{|S|} \geq \frac{\Delta}{2},$$ where the last inequality follows because $G$ is regular of degree $\Delta$.
Next we show how to improve the upper bound on $\mu^+$ given in Proposition \[pr:boundd\] by choosing our block system more carefully.
Let $k\geq 2$ be an integer and let $G=(V,E)$ be a graph with $n$ vertices and with maximum degree $\Delta$. Let $$\mathscr{S} = \{ S_v : v\in V\}$$ where for all $v\in V$ the set $S_v\subseteq V$ satisfies $ v\in S_v,\,\,\, |S_v|=k$ and $G[S_v]$ is connected. Then $\mathscr{S}$ is called a $k$-*block system* for $G$. Let $\psi$ be the uniform distribution over $\mathscr{S}$. To apply Theorem \[le:block\] to the $(\mathscr{S},\psi)$-block dynamics we will calculate upper bounds on the parameters $\partial^+$, $\Psi$ and $\mu^+$.
Clearly $|\partial S|\leq \Delta k$ and $\min\{ k,|\partial S|\}\leq k$ for all $S\in\mathscr{S}$. Hence $$\label{partial-plus-k}
\partial^+ \leq (\Delta k)^k.$$ To compute $\Psi$, observe first that $\psi(v) \geq 1/n$ for all $v \in V$ as there are $n$ blocks and each vertex belongs to at least one block. Next, observe that $\psi_\partial(v) \leq \frac{\Delta^k}{n}$: indeed if $v\in\partial S_u$ for some $u \in V$ then $u$ is at distance at most $k$ from $v$ and since and there are at most $\Delta^k$ vertices (excluding $v$) at distance at most $k$ from $v$ in $G$, there are at most $\Delta^k$ out of $n$ blocks containing $u$ in their boundary. Therefore $$\label{psi-k}
\Psi = \max_{v\in V} \, \frac{\psi_\partial(v)}{\psi(v)} \leq \Delta^k.$$ In order to calculate an upper bound on $\mu^+$ we first prove a preliminary result. For $T\subseteq T'\subset V$, recall the notation ${\mathrm{vol}}(T,T')$ introduced above Proposition \[pr:boundd\], and note that ${\mathrm{vol}}(T,T)$ is just the set of edges inside $T$.
For any two sets $A,B$, we write $\delta_{A,B}$ for the indicator function that $A=B$, that is $\delta_{A,B} =1$ if $A=B$ and $\delta_{A,B} =0$ otherwise.
\[pr:vol\] Let $H=(V,E)$ be a connected graph and let $U \subseteq V$. Then $$|{\mathrm{vol}}(U,V)| \geq |U| - \delta_{U,V}.$$
It is sufficient to prove the statement for $H$ a tree. The statement is clear if $U=V$. Now suppose that $U\neq V$ and consider the components $C_1, \ldots, C_r$ of $H[U]$. Then ${\mathrm{vol}}(C_i, V)$ has at least $|C_i|$ edges and is disjoint from ${\mathrm{vol}}(C_j,V)$ for all $j \not= i$. Thus $$|{\mathrm{vol}}(U,V)| = \sum_{i=1}^r |{\mathrm{vol}}(C_i,V)| \geq \sum_{i=1}^r|C_i| = |U|.$$
Next we give an upper bound on the parameter $\mu^+$ for $k$-block systems. For $k\geq 2$ this bound is a slight improvement on the upper bound given in Proposition \[pr:boundd\].
\[mu-bound-k\] Let $G=(V,E)$ be a connected graph with $n$ vertices and maximum degree $\Delta$. Fix an integer $k\in \{ 2,\ldots, n-1\}$ and let $\mathscr{S}$ be any $k$-block system for $G$. Then $$\mu^+ = \mu^+(\mathscr{S}) \leq \Delta - 1 + \nfrac{1}{k}.$$
Fix $X\in\Omega = [q]^V$ and $v\in V$. Given a configuration $c\in [q]^{S_v}$, let $P$ be the partition of $S_v$ defined by the nonempty colour classes of $c$. Define $F \subseteq P$ to be the set of colour classes of $c$ which correspond to a colour which does not appear on $X|_{\partial S_v}$.
Let $$A_F = \bigcup_{A \in F}\, A, \qquad
A_{\overline{F}} = \bigcup_{A \not\in F}\, A$$ and define $a_F = |A_F|$ and $a_{\overline{F}} = |A_{\overline{F}}|$. Writing $\mu_{X,v}=\mu_{X,S_v}$ for ease of notation, we have $$\begin{aligned}
\mu_{X,v}(c) &\leq \left( \sum_{A\in F} |{\mathrm{vol}}(A,A) | \right) +
\left(\sum_{A \not \in F} | {\mathrm{vol}}(A,\, A \cup \partial S_v) |\right) \nonumber \\
&\leq \left(\sum_{A \in F} |{\mathrm{vol}}(A,A) |\right) +
|{\mathrm{vol}}(A_{\overline{F}}, A_{\overline{F}} \cup \partial S_v)| \label{eq:d1}. \end{aligned}$$ Observe that $$|{\mathrm{vol}}( A_{\overline{F}}, A_{\overline{F}}\cup \partial S_v ) |
\leq \Delta a_{\overline{F}} - |{\mathrm{vol}}(A_{\overline{F}},S_v) |
\leq (\Delta - 1) a_{\overline{F}} + \delta_{F,\emptyset}, \label{eq:d2}$$ where the last inequality follows by Proposition \[pr:vol\] and noting that $\delta_{A_{\overline{F}},S_v} = \delta_{F, \emptyset}$.
Next we claim that for $A\in P$ we have $$\label{clear}
|{\mathrm{vol}}(A,A)| \leq (|A|-1)(\Delta - 1).$$ To ease notation, write $a=|A|$. If $a=1,2$ then (\[clear\]) clearly holds (noting that $\Delta \geq 2$ since $G$ is connected). Next, (\[clear\]) holds for $\Delta = 2$ since we have $|{\mathrm{vol}}(A,A)| \leq a - 1$, where the “$-1$” appears because there is at least one edge leaving $A$ (since $G$ is connected). If $a=3$ and $\Delta \geq 3$ then $|{\mathrm{vol}}(A,A)| \leq 3$ and $(a-1)(\Delta - 1) \geq 4$, so (\[clear\]) holds. For $a \geq 4$ and $\Delta \geq 3$, we note that $|{\mathrm{vol}}(A,A)| \leq \Delta a /2$ and check that $\Delta a /2 \leq (a-1)(\Delta-1)$ holds in this case. This proves the claim, establishing (\[clear\]).
Therefore $$\sum_{A \in F} |{\mathrm{vol}}(A,A)| \leq
\sum_{A \in F}(|A| - 1)(\Delta - 1) = (a_F - |F|)(\Delta - 1).
\label{eq:d3}$$ Combining (\[eq:d1\]), (\[eq:d2\]) , and (\[eq:d3\]), we have $$\begin{aligned}
\mu_{X,v}(c) &\leq (a_F - |F|)(\Delta - 1) +
(\Delta - 1)a_{\overline{F}} + \delta_{F,\emptyset} \\
&= (\Delta - 1)(k - |F|) + \delta_{F,\emptyset}.\end{aligned}$$ Assuming that $|F|\neq k$, dividing by $k - |F|$ gives the ratio $\Delta-1$ if $F\neq\emptyset$ and gives $\Delta-1+ k^{-1}$ if $F=\emptyset$. This completes the proof.
Substituting (\[partial-plus-k\]), (\[psi-k\]) and the result of Lemma \[mu-bound-k\] into Theorem \[le:block\] gives the following, noting that $\psi_{\min}\geq \nfrac{1}{n}$.
\[pr:3items\] Let $G=(V,E)$ be a connected graph with $n$ vertices and maximum degree $\Delta$. Fix an integer $k\in \{ 2,\ldots, n-1\}$ and let $\mathscr{S}$ be a $k$-block system for $G$. Let $\psi$ be the uniform distribution on $\mathscr{S}$. Fix $\lambda > 1$. If $$q \geq 2^{k+1}\, \Delta^{2k}\, k^{2k+1}\, \lambda^{\Delta - 1 + k^{-1}}$$ then $\tau(\mathcal{M}_{\mathrm{BD}}, \varepsilon) \leq 2n\log(n \varepsilon^{-1})$.
To further illustrate the use of Theorem \[le:block\] we apply it to the grid. Although our results are not as sharp as those discussed in [@ull], using the structure of the grid we are able to prove an upper bound on $\mu^+$ which is close to the lower bound given in Proposition \[pr:boundd\]. (See Lemma \[mu-bound-grid\] below.)
For convenience, rather than considering the $L \times L$ grid, we consider the toroidal $L$-grid $G=(V,E)$, where $V = ({\mathbb{Z}}/L {\mathbb{Z}})^2$, and $(a,b)(c,d) \in E$ if and only if, in ${\mathbb{Z}}/L{\mathbb{Z}}$, $$\text{either} \quad (a-c = \pm 1 \, \text{ and } \, b-d=0)
\quad \text{or} \quad (b-d = \pm 1 \, \text{ and } \, a-c=0).$$ Note that the toroidal $L$-grid has $n:=L^2$ vertices. The arguments below can be adapted to higher dimensions and to graphs with different grid topologies provided that the graph is locally a grid.
Let $\mathscr{S}$ be the set of all $r\times r$ subgrids of $G$, where $r\leq L-2$. Then $\mathscr{S}$ is a $r^2$-block system. Let $\psi$ be the uniform distribution on $\mathscr{S}$. To apply Theorem \[le:block\] we must calculate upper bounds on the parameters.
Firstly, note that $$\label{partial-plus-grid}
\partial^+ = (4r)^{4r}$$ since $|\partial S| = 4r$ for all $S\in\mathscr{S}$. Next, for $v\in V$ we have $\psi(v) = r^2/L^2$ and $\psi_{\partial}(v) = 4r/L^2$, and so $$\label{psi-grid}
\Psi = \frac{4}{r}.$$ In order to obtain a tighter bound on $\mu^+$ we need more information about expansion properties of the grid. If $U$, $W$ are disjoint sets of vertices, we write $E(U,W)$ for the set of edges with one endvertex in $U$ and one endvertex in $W$.
\[expansion\] Let $G=(V,E)$ be an $L\times L$ grid and let $S \subseteq V$ be the vertices of an $r\times r$ subgrid. If $T \subseteq S$ and $|T| = t'$ then $|{\mathrm{vol}}(T, T)| \leq 2t' - 2 \sqrt{t'}$ and $|{\mathrm{vol}}(T,S \cup \partial S)| \geq 2t' + 2\sqrt{t'}$.
For $T \subseteq S$, we define $\overline{T} = (S \cup \partial S) \setminus T$. First, we claim that $$\label{claim}
\text{ if } \,\, |E(T,\overline{T})| \leq 4t \,\,\,
\text{ then } \,\,\, |T| \leq t^2.$$ To prove the claim, let us choose $T$ such that $|T|$ is maximised subject to $|E(T, \overline{T})| \leq 4t$. We may assume that $G[T]$ is connected or else we can translate components to connect $G[T]$ without increasing $|E(T, \overline{T})|$. Furthermore, we may assume that $T$ is convex (that is, $T$ is a rectangular subgrid) because if $T$ has any “missing corners” (that is, a vertex outside $T$ with at least two neighbours in $T$) then we can add the missing vertex without increasing $|E(T,\overline{T})|$. It is also easy to verify that amongst the rectangles with $|E(T,\overline{T})| = 4t$, the square (with $t^2$ vertices) has the largest area. This completes the proof of the claim.
Now suppose that $|T|=t'$. Using the contrapositive of (\[claim\]), we have $$2\, |{\mathrm{vol}}(T, T)| = 4|T| - |E(T, \overline{T})| \leq 4t' - 4\sqrt{t'},$$ and dividing by two establishes the first statement. The second statement follows since $$|{\mathrm{vol}}(T,S \cup \partial S)| = 4|T| - |{\mathrm{vol}}(T,T)|.$$
For the toroidal grid, we may now give an upper bound for the parameter $\mu^+$ which is close to the lower bound proved in Proposition \[pr:boundd\].
\[mu-bound-grid\] Let $G$ be the toroidal $L \times L$-grid, and let $\mathscr{S}$ be the $r^2$-block system consisting of all $r\times r$ subgrids of $G$. Then $$\mu^+ \leq 2 + \nfrac{2}{r}.$$
For $v \in V$, let $S_v \in \mathscr{S}$ denote the $r \times r$ subgrid in which $v$ is at the “top left” corner. Suppose that $X\in\Omega$ and $v\in V$. For a given $c\in [q]^{S_v}$, let $P$ be the corresponding partition of $S_v$ given by the colour classes of $c$. As usual, let $F \subseteq P$ be the set of colour classes corresponding to colours which do not appear on $X|_{\partial S_v}$.
Recall the notation $A_F$, $A_{\overline{F}}$, $a_F$ and $a_{\overline{F}}$ introduced in Lemma \[mu-bound-k\]. As in (\[eq:d1\]) we write $\mu_{X,v}$ for $\mu_{X,S_v}$, and find that $$\begin{aligned}
\mu_{X,v}(c)
&\leq \left(\sum_{A \in F} |{\mathrm{vol}}(A,A) |\right)
+ |{\mathrm{vol}}(A_{\overline{F}}, A_{\overline{F}} \cup \partial S_v)|. \end{aligned}$$ Using Lemma \[claim\], we have $$\sum_{A \in F} |{\mathrm{vol}}(A,A) | \, \leq \,
\sum_{A\in F}2(|A| - \sqrt{|A|}) \, = \,
2a_F - \sum_{A\in F}2\sqrt{|A|} \, \leq \, 2a_F - 2|F|.$$ In order to bound $|{\mathrm{vol}}(A_{\overline{F}}, A_{\overline{F}} \cup \partial S_v)|$, observe first that ${\mathrm{vol}}(S_v,S_v \cup \partial S_v)$ is the disjoint union of ${\mathrm{vol}}(A_{\overline{F}}, A_{\overline{F}} \cup \partial S_v)$ and ${\mathrm{vol}}(A_F, S_v \cup \partial S_v)$. Thus $$\begin{aligned}
|{\mathrm{vol}}(A_{\overline{F}}, A_{\overline{F}} \cup \partial S_v)|
&= |{\mathrm{vol}}(S_v,S_v \cup \partial S_v)| - |{\mathrm{vol}}(A_F, S_v \cup \partial S_v)| \\
&= 2 r^2 + 2r - |{\mathrm{vol}}(A_F, S_v \cup \partial S_v)| \\
&\leq 2 r^2 + 2r - 2a_F - 2\sqrt{a_F} \hspace{2cm} \text{by Lemma~\ref{expansion}} \\
&\leq 2 r^2 + 2r - 2a_F - 2\sqrt{|F|}.\end{aligned}$$ Combining the three inequalities above, we have $$\begin{aligned}
\mu_{X,v}(c) \leq 2(r^2 - |F|) + 2(r - \sqrt{|F|})
&= (r^2 - |F|) \left( 2 + \frac{2}{r + \sqrt{|F|}} \right) \\
&\leq (r^2 - |F|)\left(2 + \nfrac{2}{r}\right). \end{aligned}$$ For all $F$ with $|F|\neq r^2$, dividing by $r^2-|F|$ gives the value $2+ \nfrac{2}{r}$, completing the proof.
Substituting (\[partial-plus-grid\]), (\[psi-grid\]) and the result of Lemma \[mu-bound-grid\] into Theorem \[le:block\] gives the following, noting that $\psi_{\min} = r^2/L^2$.
\[th:grid\] Let $G$ be the toroidal $L \times L$-grid (with $n=L^2$ vertices) and let $\mathscr{S}$ be the $r^2$-block system consisting of the set of $r\times r$ subgrids of $G$, for some $r\leq L-2$. Given $\lambda > 1$, if $$q \geq 2^{r^2 + 8r+3} \, r^{2r^2 + 4r +1}\, \lambda^{2 + \frac{2}{r}}$$ then for $\mathcal{M}_{\mathrm{BD}} = \mathcal{M}_{\mathrm{BD}}^{\mathscr{S}}(G, \lambda, q)$, we have $\tau(\mathcal{M}_{\mathrm{BD}}, \varepsilon) \leq 2n\log(n \varepsilon^{-1})/r^2$.
Glauber dynamics via Markov chain comparison {#se:comparison}
--------------------------------------------
The mixing time of two Markov chains on the same state space can be compared using comparison techniques, building on the work of Diaconis and Saloff-Coste [@DS]. We now describe the machinery needed to compare the mixing times of the Glauber dynamics and the block dynamics.
Suppose that $\mathcal{M}$ is a reversible, ergodic Markov chain on state space $\Omega$ with transition matrix $P$ and stationary distribution $\pi$. Let $\mathcal{M}'$ be another reversible, ergodic Markov chain on $\Omega$ with transition matrix $P'$ and the same stationary distribution.
We say a transition $(x,y)$ of $\mathcal{M}$ (respectively, $\mathcal{M}'$) is positive if $P(x,y)>0$ (respectively, $P'(x,y)>0$); here we allow the possibility that $x=y$. For every positive transition $(x,y)$ of $\mathcal{M}'$, let $\mathcal{P}_{x,y}$ be the set of paths $\gamma = (x=x_0, \ldots, x_k=y)$ such that all the $x_i$ are distinct and each $(x_i,x_{i+1})$ is a positive transition of $\mathcal{M}$. Let $\mathcal{P} = \cup \mathcal{P}_{x,y}$, where the union is taken over all positive transitions $(x,y)$ of $\mathcal{M}'$ with $x\neq y$.
We write $| \gamma |$ to denote the length of the path $\gamma$ so that, for example, $|\gamma|=k$ for $\gamma = (x_0, \ldots, x_k)$.
An $(\mathcal{M},\mathcal{M}')$-flow is a function $f$ from $\mathcal{P}$ to the interval $[0,1]$ such that for every positive transition $(x,y)$ of $\mathcal{M}'$ with $x\neq y$, we have $$\sum_{\gamma \in \mathcal{P}_{x,y}} f(\gamma) = \pi(x)P'(x,y).$$ For a positive transition $(z,w)$ of $\mathcal{M}$, the congestion of $(z,w)$ is defined to be $$A_{z,w}(f) = \frac{1}{\pi(z)P(z,w)} \sum_{\gamma \in \mathcal{P}:\, (z,w) \in \gamma } |\gamma| f(\gamma).$$ The congestion of the flow is defined to be $A(f) = \max A_{z,w}(f)$, where the maximum is taken over all positive transitions $(z,w)$ of $\mathcal{M}$ with $z\neq w$.
The essence of the comparison technique of Diaconis and Saloff-Coste [@DS] is that the the eigenvalues of $\mathcal{M}$ and $\mathcal{M}'$ can be related using the parameter $A(f)$. Randall and Tetali [@RT Theorem 1] used this result to compare the mixing times of two reversible ergodic Markov chains with the same stationary distribution, under the assumption that the second-largest eigenvalue (of the corresponding transition matrices) is larger in absolute value than the smallest eigenvalue. (See the discussion above Theorem 1 of [@RT].) For convenience, we will use the following theorem, which is obtained from [@DGJM Theorem 10] by specialising to Markov chains with no negative eigenvalues.
*[@DGJM Theorem 10]*\[th:comparison\] Suppose that $\mathcal{M}$ is a reversible ergodic Markov chain with transition matrix $P$ and stationary distribution $\pi$ and that $\mathcal{M}'$ is another reversible ergodic Markov chain with the same stationary distribution. Suppose that $f$ is an $(\mathcal{M},\mathcal{M}')$-flow. If $\mathcal{M}$ has no negative eigenvalues then for any $0 < \delta < \frac{1}{2}$, we have $$\tau_x(\mathcal{M}, \varepsilon) \leq
A(f) \left( \frac{ \tau(\mathcal{M}', \delta) }{ \log(1/ 2\delta)} + 1 \right) \,
\log \frac{1}{\varepsilon \pi(x)}.$$
Now we apply the above theorem to compare the mixing time of the Glauber dynamics and the block dynamics. Write $\tau(\mathcal{M'}) = \tau(\mathcal{M'},\nfrac{1}{2e})$.
\[le:comparison\] Let $G=(V,E)$ be an $n$-vertex graph of maximum degree $\Delta$. Given $\lambda > 1$, a positive integer $q$, a block system $\mathscr{S}$ for $G$ with maximum block size $s$, and $\psi$ a probability distribution on $\mathscr{S}$, write $\mathcal{M} = \mathcal{M}_{\mathrm{GD}}(G, \lambda, q)$ and $\mathcal{M}' = \mathcal{M}_{\mathrm{BD}}^{\mathscr{S}, \psi}(G, \lambda,q)$. Then for all $\varepsilon >0$ we have $$\tau(\mathcal{M}, \varepsilon) \leq 2s\, q^{s+1}\, \lambda^{\Delta (s+1)} \,
\tau(\mathcal{M}')\, n\,
\left(n\log{ (q \lambda^{\Delta /2})} + \log( \varepsilon^{-1})\right).$$
As before, let $P$ and $P'$ be the transition matrices of $\mathcal{M}$ and $\mathcal{M}'$ respectively. We note at the outset that both $\mathcal{M}$ and $\mathcal{M}'$ have the Gibbs distribution $\pi$ as their stationary distribution. It is proved in [@DGU Section 2.1] that the Glauber dynamics $\mathcal{M}$ has no negative eigenvalues, so we may apply Theorem \[th:comparison\].
We construct an $(\mathcal{M},\mathcal{M}')$-flow and analyse its congestion. Recall that a transition in $\mathcal{M}'$ is obtained by starting at some $X \in \Omega = [q]^V$, selecting $S \in \mathscr{S}$ at random using the distribution $\psi$ and then updating the configuration of $S$ to some configuration $c \in [q]^S$ chosen randomly using the distribution $\phi=\phi_{X,S}$. The resulting configuration is denoted by $X^{(S,c)}$. Let $h(X,S,c):=\psi(S)\phi_{X,S}(c)$ be the probability that this pair $(S,c)$ is chosen. In particular, if $(X,Y)$ is a transition of $\mathcal{M}'$ then $$P'(X,Y) = \sum_{(S,c):\, Y=c \cup X^{(S,c)}}\, h(X,S,c),$$ Fix an ordering of the vertices of $G$. For each $X \in \Omega$, $S \in \mathscr{S}$, and a configuration $c \in [q]^S$ of $S$, we define the path $\gamma(X,S,c)$ from $X$ to $X^{(S,c)}$ as follows: starting from $X$, consider each vertex $v \in \{u \in S: X(u) \not= c(u) \}$, one at a time and in increasing vertex order, and change the colour of $v$ from $X(v)$ to $c(v)$. Thus $\gamma(X,S,c)$ is a path in $\Omega$ from $X$ to $X^{(S,c)}$ using positive transitions of $\mathcal{M}$.
We define an $(\mathcal{M}, \mathcal{M}')$-flow $f$ by setting $f(\gamma(X,S,c))=\pi(X)h(X,S,c)$ for all $(X,S,c)$ and $f(\gamma) = 0$ for all other paths $\gamma$. To verify that this is indeed an $(\mathcal{M},\mathcal{M}')$-flow, given a positive transition $(X,Y)$ of $\mathcal{M}'$ with $X\neq Y$, we have $$\sum_{\gamma \in \mathcal{P}_{X,Y}}f(\gamma) =
\sum_{\gamma = \gamma(X,S,c):\, Y= X^{(S,c)} }\, f(\gamma)
= \sum_{(S,c):\, Y = X^{(S,c)}}
\pi(X)\, h(X,S,c) = \pi(X)\, P'(X,Y).$$
Next we bound the congestion of this flow. Let $(Z,W)$ be a positive transition of $\mathcal{M}$ with $Z \not= W$. Then the configurations $Z$ and $W$ differ on only one vertex, say $v$. The path $\gamma(X,S,c)$ uses the transition $(Z,W)$ only if $v \in S$ and the configurations $X$ and $Z$ differ on a subset of $S$. Thus we have $$\begin{aligned}
A_{Z,W}(f)
&= \frac{1}{\pi(Z)P(Z,W)} \, \sum_{\gamma \in \mathcal{P}:\, (Z,W) \in \gamma }
\,\, |\gamma| \, f(\gamma) \\
&\leq \frac{1}{\pi(Z)P(Z,W)} \, \sum_{S:\, v\in S}\,\,
\sum_{X: \,
X|_{\overline{S}} = Z|_{\overline{S}} }\,\, \sum_{c\in [q]^S}\,
|S| \cdot f(\gamma(X,S,c)) \\
&\leq \frac{s}{\pi(Z)P(Z,W)} \, \sum_{S:\, v\in S}\,\, \sum_{X: \,
X|_{\overline{S}} = Z|_{\overline{S}} } \,\, \sum_{c\in [q]^S} \,
\pi(X)\, h(X,S,c) \\
&\leq \frac{s}{P(Z,W)} \, \sum_{S: \, v \in S}\, \, \sum_{X:
X|_{\overline{S}} = Z|_{\overline{S}} }\, \frac{\pi(X)}{\pi(Z)} \, \psi(S).\end{aligned}$$ If $X$ and $Z$ differ on at most $s$ vertices, and hence on at most $\Delta s$ edges, then $$\frac{\pi(X)}{\pi(Z)} \leq \lambda^{\Delta s}.$$ Also, for any positive transition $(Z,W)$ of $\mathcal{M}$ we have $$P(Z,W)^{-1} \leq q \lambda^{\Delta}\, n.$$ Substituting these upper bounds gives $$A_{Z,W}(f) \leq \, s q \lambda^\Delta n \sum_{S: \, v \in S}\,\psi(S) \,
\sum_{X:\, X|_{\overline{S}} = Z|_{\overline{S}} } \,
\lambda^{\Delta s} \,
\leq \, sq \lambda^{\Delta} q^s \lambda^{\Delta s} \psi(v) n \,
\leq \, s q^{s+1} \lambda^{\Delta ( s + 1)} n,$$ since $\psi(v)\leq 1$. We conclude that $A(f) \leq s q^{s+1} \lambda^{\Delta (s+1)} n$.
Now apply Theorem \[th:comparison\] with $\delta = 1/(2e)$. For all $Z \in \Omega$, we have the crude bound $$\pi(Z) \geq (q^n\, \lambda^{m})^{-1} \geq (q^n\, \lambda^{\Delta n/2})^{-1},$$ which leads to $$\begin{aligned}
\tau(\mathcal{M}, \varepsilon) &\leq
sq^{s+1} \lambda^{\Delta (s+1)}n
\left( \tau(\mathcal{M}') + 1 \right) \,
\log{ (q^n \lambda^{\Delta n /2} \varepsilon^{-1})} \\
&\leq 2s\, q^{s+1} \lambda^{\Delta (s+1)}\, n\,
\tau(\mathcal{M}')\,
\left(n\log(q \lambda^{\Delta /2}) + \log(\varepsilon^{-1})\right), \end{aligned}$$ as claimed.
We would expect that the mixing time for Glauber dynamics should decrease as $q$ increases, but the bound given in Lemma \[le:comparison\] becomes worse for larger values of $q$. However, by combining Lemma \[le:comparison\] with Proposition \[le:vertex\], we can avoid this problem.
\[co:comparison\] Let $G=(V,E)$ be an $n$-vertex graph of maximum degree $\Delta$. Given $\lambda > 1$, a positive integer $q$, a block system $\mathscr{S}$ for $G$ with maximum block size $s$, and $\psi$ a probability distribution on $\mathscr{S}$, write $\mathcal{M} = \mathcal{M}_{\mathrm{GD}}(G, \lambda, q)$ and $\mathcal{M}' = \mathcal{M}_{\mathrm{BD}}^{\mathscr{S}, \psi}(G, \lambda,q)$. Then for $\varepsilon > 0$ we have $$\tau(\mathcal{M}, \varepsilon) \leq \begin{cases}
2s (\Delta\lambda^{2\Delta})^{s+1}\, \tau(\mathcal{M}')\, n \left(
n\log(\Delta\lambda^{3\Delta/2}) + \log(\varepsilon^{-1})\right)
& \text{ if $q < \Delta \lambda^\Delta + 1$,}\\
(\Delta+1) n\log(n\varepsilon^{-1}) & \text{ if $q \geq \Delta\lambda^\Delta + 1$}.
\end{cases}$$
If $q < \Delta \lambda^{\Delta} + 1$ then the corollary holds by Lemma \[le:comparison\], while if $q \geq \Delta \lambda^{\Delta} + 1$ then the corollary holds by Proposition \[le:vertex\].
We complete this section by applying the previous corollary to the block dynamics results obtained in the previous subsection to obtain rapid mixing results for Glauber dynamics.
Let $G=(V,E)$ be an $n$-vertex connected graph with maximum degree $\Delta$, and fix $\lambda > 1$. For every positive integer $k \leq n$, if $q \geq 2^{k+1}\Delta^{2k} k^{2k+1} \lambda^{\Delta - 1 + k^{-1}}$ then for $\mathcal{M}_{\mathrm{GD}} = \mathcal{M}_{\mathrm{GD}}(G, \lambda, q)$, we have $$\tau(\mathcal{M}_{\mathrm{GD}}, \varepsilon) \leq
4k\, (\Delta\, \lambda^{2\Delta})^{k+1}\,
n^2\log(2en)\,\left(n\log{ (\Delta \lambda^{3\Delta/2})} + \log(\varepsilon^{-1})
\right).$$ \[th:glaubercompare\]
Take an arbitrary $k$-block system $\mathscr{S}$ for $G$, and let $\psi$ be the uniform distribution on $\mathscr{S}$. Theorem \[pr:3items\] provides a bound on the mixing time of the block dynamics with respect to $\mathscr{S}$. Then apply Corollary \[co:comparison\] to this bound.
Here any $k$-block system $\mathscr{S}$ may be used (recall the definition after the proof of Proposition \[pr:boundd\]). For any connected graph $G=(V,E)$, one can easily obtain a $k$-block system $\mathscr{S} = \{S_v: v \in V\}$ by taking $S_v$ to be the first $k$ vertices in any breadth-first search starting at $v$.
\[th:glaubergrid\] Let $G=(V,E)$ be the toroidal $L \times L$-grid (with $n=L^2$ vertices), and fix $\lambda > 1$. For every positive integer $r \leq L-2$, if $q \geq 2^{r^2+8r+3}\, r^{2r^2 + 4r+1}\, \lambda^{2 + \frac{2}{r}}$ then for $\mathcal{M}_{\mathrm{GD}} = \mathcal{M}_{\mathrm{GD}}(G, \lambda, q)$, we have $$\tau(\mathcal{M}_{\mathrm{GD}}, \varepsilon) \leq
4 \, (4 \lambda^{8})^{r^2+1}\, n^2\, \log(2en)\, \left(n\log(4 \lambda^6) +
\log( \varepsilon^{-1}) \right).$$
We apply Corollary \[co:comparison\] to the mixing time of the block dynamics in Theorem \[th:grid\]. (Recall that the block system used is the set of $r \times r$ subgrids.)
An extremal problem {#se:extremal}
===================
In this section, we investigate how large the partition function of a bounded-degree graph can be. We require this result in the next section, where we give bounds on the number of colours below which Glauber dynamics mixes slowly, although the result may be of independent interest.
In this section, we allow graphs to have multiple edges, but not loops. For fixed numbers $n$ the number of vertices, $m$ the number of edges, $\Delta$ the maximum degree, $\lambda \geq 1$ the activity, and $q$ the number of colours, we define $$Z((n,m, \Delta), \lambda, q) = \max_{G} Z(G, \lambda, q),$$ where the maximum is over all graphs $G$ with $n$ vertices, $m$ edges, and maximum degree $\Delta$.
We now describe the class of graphs that will turn out to be extremal for the above parameter. Fix positive integers $n$, $m$, and $\Delta$ such that $\Delta$ divides $m$ and $m \leq \Delta n /2$. Let $H(n,m,\Delta)=(V,E)$, where $V$ is a set of $n$ vertices and $E$ is obtained by taking any set of $m/\Delta$ independent edges on $V$ and replacing each edge with $\Delta$ multi-edges. Thus $H(n,m,\Delta)$ has $m$ edges and maximum degree $\Delta$.
The main result of this section is the following.
\[thm:ext\] If $G$ is an $n$-vertex graph with $m$ edges and maximum degree $\Delta$, and $q \in {\mathbb{N}}$ and $\lambda \geq 1$ are given, then $$Z(G, \lambda, q) \leq \left( 1 + q^{-1}(\lambda^\Delta-1) \right)^{\lceil m/\Delta\rceil} q^n.$$ In particular, if $ \Delta$ divides $m$, we have equality above for $G = H(n,m,\Delta)$.
This will immediately give us the following corollary.
\[cor:ext\] Let $n,m,\Delta \in {\mathbb{N}}$ be fixed. Given a number of colours $q$, and activity $\lambda \geq 1$, we have $$Z((n,m,\Delta), \lambda, q) \leq \left(1 + q^{-1}(\lambda^\Delta - 1) \right)^{\lceil m/\Delta\rceil} q^n.$$
We begin by giving a brief outline of the proof. Given an $n$-vertex multigraph $G=(V,E)$, and a uniformly random configuration $\sigma$ of $V$ (i.e. $\sigma$ is a uniformly random element of $[q]^V$), let $X$ be the number of monochromatic edges of $G$ in $\sigma$. Observe that $Z(G, \lambda, q) = {\mathbb{E}}( \lambda^X )q^n$. We proceed by decomposing the edges of $G$ into $\Delta$ forests with $\lceil m/\Delta\rceil$ or $\lfloor m/\Delta\rfloor$ edges each. Then we establish that the number of monochromatic edges in a forest with $m'$ edges is distributed as $X \sim {\mathrm{Bin}}(m', q^{-1})$. This allows us to obtain a bound on ${\mathbb{E}}(\lambda^X)$ and hence prove Theorem \[thm:ext\].
\[le:forest\] Let $G=(V,E)$ be a multigraph with $n$ vertices, $m$ edges, and maximum degree $\Delta$. We can find $\Delta$ spanning forests $F_1,\ldots, F_\Delta$ on the vertex set $V$ such that each $F_i$ has $\lceil m/\Delta\rceil$ or $\lfloor m/\Delta\rfloor$ edges and the edges of $F_1,\ldots, F_\Delta$ form a partition of $E$.
Recall that the *size* of a graph is the number of edges in the graph. We begin by disregarding the condition that the forests should have almost equal size, and decompose (the edge set of) $G$ into (the edge sets) of $\Delta$ spanning forests, as follows. (This follows from [@NW], but for completeness we give a brief proof.) Let $G_1:=G$. Iteratively define $F_i$ to be a spanning forest of $G_i$ of maximum size, and let $G_{i+1}$ be obtained from $G_i$ by deleting the edges of $F_i$. By removing the edges of $F_i$ from $G_i$, we reduce the degree of every non-isolated vertex in $G_i$ by at least one, and so, in particular, we reduce the maximum degree of $G_i$ by at least one. Thus $G_r$ is the empty graph for some $r \leq \Delta$, giving a decomposition of (the edge set of) $G$ into (the edge sets of) $\Delta$ spanning forests, $F_1, \ldots, F_\Delta$ (some of which may have no edges).
We denote the size of $F_i$ by $|F_i|$. Observe that if $|F_i| > |F_j|+1$ then $F_i$ has fewer components than $F_j$ (since all the forests are spanning), so $F_i$ has at least one edge that connects two components of $F_j$. Removing this edge from $F_i$ and adding it to $F_j$ keeps both $F_i$ and $F_j$ acyclic, but reduces the imbalance in their sizes. Iteratively applying this operation to any pair of forests whose sizes differ by at least two eventually results in all forests having size $\lceil m/\Delta\rceil$ or $\lfloor m/\Delta\rfloor$.
\[le:bin\] Let $F=(V,E)$ be a forest and let $\sigma$ be a uniformly random configuration of $V$ (i.e. $\sigma$ is a uniformly random element of $[q]^V$). Let $X$ be the number of monochromatic edges of $F$. Then $X \sim {\mathrm{Bin}}(m, q^{-1})$, where $m$ is the number of edges in $F$.
It is sufficient to consider the case when $F$ is a tree. For if not, then we can consider the components of $F$ independently, and use the fact that the sum of $t$ independent binomial random variables of the form ${\mathrm{Bin}}(m_j,p)$ is a binomial random variable ${\mathrm{Bin}}(m_1+\cdots +m_t,p)$.
Now assume that $F$ is a tree, and root $F$ at a vertex $v_0$. Let $v_0, \ldots, v_{n-1}$ be any ordering of the vertices in $V$ such that for every $i$, the parent of $v_i$ is a member of $\{v_1, \ldots, v_{i-1}\}$. We generate a uniformly random configuration of $V$ by colouring each vertex with a uniformly random colour from $[q]$, independently, in the specified order. Each vertex has probability $1/q$ of being given the same colour as its parent, independently of all previous choices, and hence each edge has probability $1/q$ of being monochromatic, independently of all previous choices. Therefore the total number of monochromatic edges satisfies $X \sim {\mathrm{Bin}}(m, q^{-1})$.
We will also need the following result, which follows from a generalization of H[" o]{}lder’s inequality.
\[le:dom\] Let $(X_1, \ldots, X_d)$ be a random, ${\mathbb{R}}^d$-valued vector, and suppose there exists a random variable $X$ such that $X_i \sim X$ for all $i = 1, \ldots, d$. Then for all $\lambda > 0$ we have $${\mathbb{E}}(\lambda^{X_1 + \cdots + X_d}) \leq {\mathbb{E}}(\lambda^{dX}).$$
Let $Z_j = \lambda^{X_j}$ and $p_j = d$ for $j=1,\ldots, d$. Then the result follows from the generalised H[" o]{}lder’s inequality, which states that $${\mathbb{E}}\left(\prod_{j=1}^d |Z_j|\right) \leq \prod_{j=1}^d \left({\mathbb{E}}|Z_j|^{p_j}\right)^{1/p_j}$$ for any random variables $Z_1,\ldots, Z_d$ and any $p_j\geq 1$ such that $\sum_{j=1}^d 1/p_j = 1$. (See for example [@Finner].)
We are now ready to prove Theorem \[thm:ext\].
By Lemma \[le:forest\], we can decompose the edges of $G$ into $\Delta$ spanning forests $F_1, \ldots, F_\Delta$, such that $m_i$, the number of edges in $F_i$, is either $\lceil m/\Delta\rceil$ or $\lfloor m/\Delta\rfloor$.
Let $\sigma$ be a uniformly random configuration of $V$ (i.e. $\sigma$ is a uniformly random element of $[q]^V$, and let $X_i$ be the number of monochromatic edges of $F_i$ in the configuration $\sigma$. We know by Lemma \[le:bin\] that $X_i \sim {\mathrm{Bin}}(m_i ,q^{-1})$. Then ${\mu}(\sigma)$, the number of monochromatic edges of $G$ in $\sigma$, is given by $ {\mu}(\sigma) = X_1 + \cdots + X_\Delta$ and $$\begin{aligned}
Z(G, \lambda, q) = q^n \, {\mathbb{E}}(\lambda^{{\mu}(\sigma)}) = q^n\, {\mathbb{E}}(\lambda^{X_1 + \cdots + X_\Delta}) .\end{aligned}$$
For each $i=1, \ldots, \Delta$, choose $Y_i \sim{\mathrm{Bin}}(\lceil m/\Delta\rceil ,q^{-1})$ such that ${\mathbb{P}}(Y_i \geq X_i) =1$. Then using the above and Lemma \[le:dom\], we have $$\begin{aligned}
Z(G, \lambda, q) \leq q^n\, {\mathbb{E}}(\lambda^{X_1 + \cdots + X_\Delta}) \leq q^n\, {\mathbb{E}}(\lambda^{Y_1 + \cdots + Y_\Delta}) &\leq q^n\, {\mathbb{E}}(\lambda^{\Delta Y_1}) \\ &= q^n\, (1 + q^{-1}(\lambda^\Delta-1))^{\lceil m/\Delta \rceil}.\end{aligned}$$ The last equality holds because $Y_1 \sim {\mathrm{Bin}}(\lceil m/\Delta\rceil,q^{-1})$, so $${\mathbb{E}}(\lambda^{\Delta Y_1})
= \sum_{i=0}^{\lceil m/\Delta \rceil} \binom{\lceil m/\Delta \rceil}{i}\, q^{-i}(1-q^{-1})^{\lceil m/\Delta\rceil -i} \, \lambda^{\Delta i} = (1 + q^{-1}(\lambda^\Delta-1))^{\lceil m/\Delta \rceil}.$$
Finally, it is easy to check that $Z(H(n,m,\Delta), \lambda, q) = q^n\, (1 + q^{-1}(\lambda^\Delta-1))^{m/\Delta }$ when $\Delta$ divides $m$.
Slow mixing {#se:tormix}
===========
We have seen in Section \[se:GD\] that for general graphs with maximum degree $\Delta$, the Glauber dynamics mixes rapidly if $q \geq \Delta \lambda^\Delta + 1$. Some improvements on this were given in Section \[se:comparison\]. In this section, we shall see that these general bounds cannot be improved by much (in terms of the exponent of $\lambda$). We give a bound on the number of colours below which Glauber dynamics almost surely mixes slowly for a uniformly random $\Delta$-regular graph.
The technical tool used for most slow-mixing proofs is conductance [@JS89]. We now introduce the necessary definitions: for convenience, we follow the treatment given in [@DGJM]. Again, $\mathcal{M}$ is a Markov chain with state space $\Omega$, transition matrix $P$ and stationary distribution $\pi$. For $A,B \subseteq \Omega$, define $$Q_{\mathcal{M}}(A,B) = \sum_{x \in A,\, y \in B} \pi(x)P(x,y).$$ We define $$\Phi_{\mathcal{M}}(A) = \frac{Q_{\mathcal{M}}(A, \overline{A})}{\pi(A)\pi(\overline{A})},$$ where $\overline{A}:= \Omega \setminus A$. Finally, we define the *conductance* of $\mathcal{M}$ as $$\Phi_{\mathcal{M}} := \min_{A \subseteq \Omega} \Phi_{\mathcal{M}}(A).$$ We drop the subscript when the Markov chain is clear from the context. Recall that $\tau(\mathcal{M}) = \tau(\mathcal{M},\nfrac{1}{2e})$. Conductance gives a lower bound for the mixing time of a Markov chain via the following result.
*[@DGJM Theorem 17]* \[th:con\] Let $\mathcal{M}$ be an ergodic Markov chain with transition matrix $P$, stationary distribution $\pi$ and conductance $\Phi$. Then $$\tau(\mathcal{M}) \geq \frac{e-1}{2e\, \Phi_{\mathcal{M}}}.$$
Suppose now that $G=(V,E)$ is an $n$-vertex graph, $\lambda \geq 1$ is given, and $q$ is a number of colours. By Theorem \[th:con\], in order to show that $\mathcal{M} = \mathcal{M}_{\mathrm{GD}}(G, \lambda, q)$ mixes slowly, it is sufficient to show that its conductance $\Phi_{\mathcal{M}}$ is exponentially small in $n$.
We will need some more definitions. For $i\in [q]$ and $\sigma \in \Omega$, define $$\sigma_i = |\{v \in V: \sigma(v) = i\}|.$$ Next, define the $r$-shell and $r$-ball around a colour $i$ as follows: $$S_r(i) = \{ \sigma: \sigma_i = n-r \},\qquad
B_r(i) = \{ \sigma: \sigma_i \geq n-r \}.$$ We see that $B_r(i)$ is the set of configurations at distance at most $r$ from the all-$i$ configuration, and $S_r(i)$ is the set of configurations at distance exactly $r$ from the all-$i$ configuration. To simplify notation, we write $B_r = B_r(1)$ and $S_r=S_r(1)$ for the $r$-ball and $r$-shell around colour 1.
For an $n$-vertex graph $G=(V,E)$ and $r$ is a positive integer satisfying $r \leq n/2$, we define $$\alpha_r(G) = \frac{1}{r} \,\max_{\stackrel{S \subseteq V}{|S|=r}}\, e_G(S),$$ where $e_G(S)$ is the number of edges of $G$ inside $S$. This quantity is low when the edge-expansion of $r$-vertex subgraphs of $G$ is high. We now establish a uniform bound on the conductance of $\mathcal{M}_{\mathrm{GD}}(G,\lambda,q)$ which holds when $\alpha_r(G)$ and $q$ are sufficiently small.
\[lem:conductance\] Let $\lambda \geq 1$ and let $\Delta\geq 2$ be an integer. Fix $\kappa\in \big(1,\nfrac{\Delta}{2}\big]$ and let $\beta\in (0,1)$. Suppose that $n\geq \beta^{-1}(2 + \Delta\log_2\lambda)$ is an integer and let $r=\lfloor \beta n\rfloor$. Let $G$ be a $\Delta$-regular, $n$-vertex graph such that $\alpha_r(G)\leq \kappa$. Finally, suppose that $q\geq 2$ is an integer which satisfies $$\label{q-assumption}
q-1
\leq \frac{\beta^2}{256\, e^2}\,\,
\lambda^{\Delta- \kappa - \frac{\kappa^2}{\Delta - \kappa}}.$$ Then the conductance of the Markov chain $\mathcal{M}=\mathcal{M}_{\mathrm{GD}}(G,\lambda,q)$ is bounded by $$\label{eq:conductance}
\Phi_{\mathcal{M}}
\leq \frac{2}{\sqrt{2\pi r}}\, 2^{-r}.$$
We bound $\Phi_{\mathcal{M}}$ by estimating $\Phi_{\mathcal{M}}(B_r)$. Let $P$ be the transition matrix for $\mathcal{M}$ and let $\pi$ be the stationary distribution of $\mathcal{M}$ (that is, the Gibbs distribution). We have $$\begin{aligned}
\Phi_{\mathcal{M}} \leq \Phi_{\mathcal{M}}(B_r) = \frac{\sum_{x \in B_r,\,
y \in \overline{B_r}} \pi(x)P(x,y)}{\pi(B_r)\pi(\overline{B_r})}
&= \frac{\sum_{x \in S_r,\, y \in \overline{B_r}} \pi(x)P(x,y)}{\pi(B_r)\pi(\overline{B_r})} \\
&\leq \frac{\pi(S_r)}{\pi(B_r) \pi(\overline{B_r})} \\
& \leq \frac{2\, \pi(S_r)}{\pi(B_r)},\end{aligned}$$ where the last inequality follows because $\pi(\overline{B_r}) \geq \frac{1}{2}$ (assuming that $q \geq 2$).
Let $Z=Z(G,\lambda,q)$ be the partition function and write $m=\Delta n /2$ for the number of edges in $G$. Now $\pi(B_r) \geq Z^{-1} \lambda^m$ since the all-$1$ configuration belongs to $B_r$. Next we obtain a lower bound on $\pi(S_r)$.
Suppose that $A\subseteq V$ with $|A|=r$. Writing $E(A)$ for the set of edges of $G$ inside $A$, we know that $|E(A)| \leq \alpha_r(G)r\leq \kappa r$. Observe that $|E(A,\overline{A})| = \Delta r - 2|E(A)|$ because $\Delta r$ counts each edge in $E(A)$ twice. Hence $$\begin{aligned}
|E(\overline{A})| = m - |E(A,\overline{A})| - |E(A)|
&= m - (\Delta r - 2|E(A)|) - |E(A)| \\
&= m - \Delta r +|E(A)|\\
& \leq m - (\Delta - \kappa)r. \end{aligned}$$ Therefore $$\begin{aligned}
\pi(S_r)
= Z^{-1} \sum_{\sigma \in S_r} \lambda^{{\mu}(\sigma)}
&= Z^{-1} \sum_{A \subseteq V : |A|=r} \lambda^{|E(\overline{A})|} \cdot Z(G[A],\lambda,q-1) \\
&\leq Z^{-1} \sum_{A \subseteq V : |A|=r} \lambda^{m - (\Delta - \kappa)r} \cdot Z(G[A],\lambda,q-1) \\
&\leq Z^{-1} \binom{n}{r} \lambda^{m - (\Delta - \kappa)r} \cdot
Z((r,\lceil \kappa r \rceil, \Delta) ,\lambda,q-1).\end{aligned}$$ The final inequality uses the fact that when $\lambda\geq 1$, the partition function is nondecreasing under the addition of edges. Combining these bounds shows that $$\label{conductanceZ} \Phi_{\mathcal{M}}
\leq 2\binom{n}{r} \lambda^{- (\Delta - \kappa)r} \cdot
Z((r,\lceil\kappa r \rceil,\Delta) ,\lambda,q-1).$$ Using Corollary \[cor:ext\], we have $$\begin{aligned}
Z((r,\lceil \kappa r\rceil,\Delta) ,\lambda,q-1)
&\leq (1 + (q-1)^{-1}\lambda^\Delta)^{\lceil \kappa r/\Delta \rceil}(q-1)^r \\
&\leq (2(q-1)^{-1}\lambda^\Delta)^{\lceil \kappa r/\Delta \rceil}(q-1)^r \\
&\leq 2\lambda^{\Delta}(2(q-1)^{-1}\lambda^\Delta)^{ \kappa r/\Delta }(q-1)^r \\
&\leq \left( 4 \lambda^{\kappa} (q-1)^{\frac{\Delta - \kappa}{\Delta}} \right) ^r.\end{aligned}$$ Here the second inequality uses the fact that $q-1\leq \lambda^\Delta$ (which follows from (\[q-assumption\])), and the final inequality follows since $\kappa/\Delta \leq \nfrac{1}{2}$ as well as the fact that $2^r \geq 2\lambda^{\Delta}$ (by our choice of sufficiently large $n$). Substituting this into (\[conductanceZ\]) and applying the well-known inequality $$\binom{n}{r} \leq \frac{n^r}{r!} \leq
\frac{1}{\sqrt{2\pi r}} \left(\frac{en}{r}\right)^r$$ gives $$\Phi_{\mathcal{M}} \leq
\frac{2}{\sqrt{2\pi r}}\, \left( \frac{4en}{r}\, \lambda^{-\Delta - 2\kappa}\, (q-1)^{(\Delta-\kappa)/\Delta}
\right)^{r}.$$ Now raising both sides of (\[q-assumption\]) to the power $(\Delta-\kappa)/\Delta$ and rearranging shows that $$\frac{4en}{r}\, \lambda^{- (\Delta - 2\kappa)}
(q-1)^{\frac{\Delta - \kappa}{\Delta}} \leq \frac{\beta n}{4r}\leq \frac12.$$ Therefore $\Phi_{\mathcal{M}} \leq \frac{2}{\sqrt{2\pi r}} \, 2^{-r}$, as claimed.
Let $\mathcal{G}_{n,\Delta}$ denote the uniform probability space of all $\Delta$-regular graphs on the vertex set $[n]=\{ 1,2,\ldots, n\}$, restricting to $n$ even if $\Delta$ is odd. That is, “$G\in\mathcal{G}_{n,\Delta}$” means that $G$ is a uniformly chosen $\Delta$-regular graph on the vertex set $[n]$. In a sequence of probability spaces indexed by $n$, an event holds *asymptotically almost surely* (a.a.s.) if the probability that the event holds tends to 1 as $n\to\infty$.
Next, given $\kappa$ we show how to choose $r$ in order to ensure that with high probability, a random $\Delta$-regular graph $G$ satisfies $\alpha_r(G)\leq \kappa$.
\[le:expander\] Fix $\Delta \geq 3$ and let $\kappa \in \big(1,\nfrac{\Delta}{2}\big]$. Let $$\label{eq:expan}
\beta = \nfrac{1}{2}\, e^{-\big(1 + \frac{2}{\kappa-1}\big)}\,
\bigg( \frac{\Delta}{2\kappa} \bigg)^{ -\big( 1 + \frac{1}{\kappa-1}\big) },$$ and for each positive integer $n\geq \beta^{-1}$, define $r=r(n) = \lfloor \beta n\rfloor$, which is a positive integer. Let $G\in\mathcal{G}_{n,\Delta}$. Then a.a.s. $\alpha_r(G)\leq \kappa$.
We use the configuration model of Bollob[á]{}s [@bollobas] to construct random regular graphs. In this model, to construct a random $\Delta$-regular graph on $n$ vertices, we take $n$ sets (called *buckets*) each containing $\Delta$ labelled objects called *points*. Then we take a random partition $P$ of the $\Delta n$ points into $\Delta n/2$ *pairs*, where each pair is a set of two distinct points. We call $P$ a *pairing*. By replacing each bucket by a vertex and replacing each pair by an edge between the two corresponding vertices, we obtain a multigraph $G(P)$, which may have loops and multiple edges. If $G(P)$ is simple then it is $\Delta$-regular. It has been shown [@bollobas] that a random pairing is simple with probability tending to $\exp{(- \frac{\Delta^2-1}{4})}$ as $n \to \infty$.
Let $m(2a)$ denote the number of pairings of $2a$ points. It is well known that $$m(2a) = \frac{(2a)!}{a!\, 2^a}.$$ Write $[x]_a = x(x-1)\cdots (x-a+1)$ to denote the falling factorial. Now let $\mathcal{P}_{n,\Delta}$ denote the uniform probability space on the set of pairings with $n$ buckets, each containing $\Delta$ points. Let $B$ be a fixed set of $r$ buckets. Given a positive integer $s$, let $m_B(r,s)$ be the number of pairings in $\mathcal{P}_{n,\Delta}$ in which at least $s$ pairs are contained in $B$. We can obtain an overcount of $m_B(r,s)$ in the following way. We first select $s$ pairs within $B$, in $$\frac{ [\Delta r]_{2s}}{s!2^s}$$ ways. Then we pair up the remaining $\Delta n-2s$ points in $m(\Delta n-2s)$ ways. Hence $$m_B(r,s) \leq \frac{[\Delta r]_{2s}}{s! 2^s}\,
\frac{(\Delta n - 2s)!}{(\Delta n/2 - s)! 2^{\Delta n/2-s}}
= \frac{(\Delta r)! (\Delta n - 2s)!}
{ 2^{\Delta n/2} s! (\Delta r - 2s)! (\Delta n/2 - s)! }.$$ Therefore the probability $p(r,s)$ that a random pairing in $\mathcal{P}_{n,\Delta}$ has at least $s$ pairs within $B$ is $$p(r,s)= \frac{m_B(r,s)}{m(\Delta n)} \leq
\binom{\Delta n/2}{s} \frac{ [\Delta r]_{2s}}
{ [\Delta n]_{2s} }
\leq \binom{\Delta n/2}{s} \bigg( \frac{r}{n} \bigg)^{2s}.$$
Let $X(r,s)$ be the random variable which counts the number of sets of $r$ buckets which contain at least $s$ pairs of $P$, for $P\in\mathcal{P}_{n,\Delta}$. Using the inequality $\binom{a}{b} \leq (ea/b)^b$, we have $${\mathbb{E}}(X(r,s)) = \binom{n}{r} p(r,s)
\leq \binom{n}{r}\binom{\Delta n/2}{s} \bigg( \frac{r}{n} \bigg)^{2s}
\leq \left(\frac{en}{r}\right)^r \, \left(\frac{\Delta e r^2}{2sn}
\right)^s.$$ Now fix $s = \lceil\kappa r\rceil$ where, recall, $r =\lfloor\beta n\rfloor$. By definition of $\beta$ we have $\Delta e r < 2\kappa n$, and hence $${\mathbb{E}}(X(r,\lceil \kappa r\rceil))
\leq \bigg( \frac{ne}{r} \,
\bigg( \frac{\Delta e r }{2 \kappa n}\bigg)^{\kappa} \bigg)^{r}\
\leq ((2\kappa)^{-\kappa} \,e^{\kappa+1} \,
\Delta^{\kappa} \,\beta^{\kappa-1})^{r}.$$
When (\[eq:expan\]) holds, we see that $$(2\kappa)^{-\kappa}\, e^{\kappa+1} \, \Delta^\kappa\, \beta^{\kappa-1}
\leq 2^{-(\kappa-1)}$$ and this upper bound is a constant in $(0,1)$ which is independent of $n$. Since $r\geq \beta n-1$ it follows that ${\mathbb{E}}(X(r,\lceil \kappa r\rceil)) = o(1)$, and we conclude that $${\mathbb{E}}(X(r,\lceil \kappa r\rceil) \mid G(P) \text{ is simple}) \leq
\frac{{\mathbb{E}}(X(r,\lceil \kappa r\rceil))}{{\mathbb{P}}(G(P) \text{ is simple})} =o(1).$$ This shows that when (\[eq:expan\]) holds, a.a.s. $G\in\mathcal{G}_{n, \Delta}$ has the property that all subsets of vertices of size $r$ have fewer than $\kappa r$ edges.
Now we can easily show that when $q$ is sufficiently small and $n$ is sufficiently large, the mixing time of the Glauber dynamics is slow for almost all $\Delta$-regular graphs.
\[th:torpid-random\] Fix $\Delta\geq 3$ and let $\kappa \in \big(1, \nfrac{\Delta}{2} \big]$. Suppose that $\beta$ is defined by (\[eq:expan\]) and let $q\geq 2$ be an integer which satisfies (\[q-assumption\]). Let $G\in\mathcal{G}_{n,\Delta}$. Then a.a.s.the Glauber dynamics $\mathcal{M} = \mathcal{M}_{\mathrm{GD}}(G, \lambda, q)$ satisfies $$\tau(\mathcal{M}) \geq 2^{\beta n-4}.$$
For each positive integer $n\geq \beta^{-1}( 2 + \Delta \log_2 \lambda)$, let $r = r(n) = \lfloor \beta n\rfloor$, which is a positive integer. By Lemma \[le:expander\] we know that a.a.s. $G\in\mathcal{G}_{n,\Delta}$ satisfies $\alpha_r(G)\leq \kappa$. Hence a.a.s. the conductance of the corresponding Glauber dynamics $\mathcal{M}_{\mathrm{GD}}(G,\lambda,q)$ is bounded above by $$\frac{2}{\sqrt{2\pi r}}\, 2^{-r}$$ by Lemma \[lem:conductance\]. Applying Theorem \[th:con\] completes the proof.
We conclude this section by proving Theorem \[th:main2\] and Theorem \[th:main3\].
\(i) Given $\eta\in (0,1)$, let $k=\lceil \eta^{-1}\rceil$ and define $c_1 = k2^{k+1}(\Delta k)^{2k}$. If $q > c_1\lambda^{\Delta-1+\eta}$ then $q > c_1 \lambda^{\Delta - 1 + 1/k}$, by choice of $k$. Then the conclusion follows from Theorem \[th:glaubercompare\].
For (ii), given $\eta \in (0,1)$ define $\kappa = 1+\eta/5$. Since $\Delta\geq 3$ we have $\kappa\in \big(1,\nfrac{6}{5}\big)\subseteq (1,\nfrac{\Delta}{2}]$. Define $$c_2 = \nfrac{1}{1024} e^{-4(1 + \frac{1}{\kappa - 1})}\, \left(
\frac{\Delta}{2\kappa}\right)^{-2(1 + \frac{1}{\kappa-1})}.$$ By our choice of $\kappa$ and since $\Delta\geq 3$, we have $$\begin{aligned}
\kappa + \frac{\kappa^2}{\Delta-\kappa}
& \leq 1 + \frac{1}{\Delta-1} + \eta.\end{aligned}$$ Therefore, if $$q-1 \leq c_2 \lambda^{\Delta -1 -\frac{1}{\Delta -1} - \eta}$$ then (\[q-assumption\]) holds, and the result follows by applying Theorem \[th:torpid-random\].
The first and third statement follow from substituting $\Delta=4$ into Theorem \[th:main2\] (i) and (ii), respectively. (So $c_3$ is obtained by substituting $\Delta=4$ in $c_1$, and $c_5$ is obtained from $c_2$ similarly.)
For (ii), let $k = \lceil 2\eta^{-1}\rceil$ and define $c_4 = (8k-1)\, 2^{k^2+8k}\, k^{2k^2 + 4k}$. If $q > c_4 \lambda^{2+\eta}$ then $q > c_4 \lambda^{2 + 2/k}$, by definition of $k$. Then Theorem \[th:glaubergrid\] applies, completing the proof.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We are grateful to Ostap Hryniv and Gregory Markowsky for leading us to the generalised H[" o]{}lder’s inequality (and to [@Finner]) for Lemma \[le:dom\]. We are also grateful to Mario Ullrich for providing feedback on an earlier draft of this paper. We would also like to thank the referees for their helpful comments.
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Appendix {#appendix .unnumbered}
========
Suppose that $q,\, \Delta\geq 3$ are integers and that $B$ is a real number. We prove that the polynomial $$f(x) := (q-1)x^{\Delta} + (2- q -B)x^{\Delta -1} + Bx -1$$ has a double root in $(0,1)$ only if $0 < B = \Theta(q^{\frac{1}{\Delta - 1}})$ i.e. $\log B = \frac{\log q}{\Delta - 1} + O(1)$. Here all asymptotic notation is with respect to $q \to \infty$.
First we note some properties of $f$. Observe that $f''(x) = c_1 x^{\Delta -2} + c_2 x^{\Delta - 3}$ for some constants $c_1, c_2$. Thus $f''(x)$ has at most one root in $(0,1)$. This implies that $f'(x)$ has at most one turning point in $(0,1)$ and hence at most two roots in $(0,1)$. Thus $f(x)$ has at most two turning points in $(0,1)$. This together with the fact that $f(0) = -1$ and $f(1) = 0$ implies that if $f$ has a double root in $(0,1)$, it must be the case that $f(x) \leq 0$ for all $x \in [0,1]$. (To see this, consider the graph of $f$ with the constraints deduced above.)
We show that (i) if $0<B = \omega(q^{\frac{1}{\Delta -1}})$ and $q$ is sufficiently large, then $f(x)>0$ for some $x \in (0,1)$; (ii) if $B\leq 0$ then $f(x) < 0$ for all $x \in (0,1)$; and (iii) if $0 < B = o(q^{\frac{1}{\Delta -1}})$ and $q$ is sufficiently large, then $f(x) < 0$ for all $x \in (0,1)$. Thus in all three cases there is no double root of $f$ in $(0,1)$; the only possibility remaining is that $0<B = \Theta(q^{\frac{1}{\Delta - 1}})$.
Splitting the terms in $f$, we have: $$f(x) = (q-1)x^{\Delta} - (q-2)x^{\Delta -1} - Bx^{\Delta -1} + Bx - 1.$$ First suppose that $0 < B = \omega(q^{\frac{1}{\Delta -1}})$. Then $f(q^{- \frac{1}{\Delta -1}})$ is dominated by the fourth term above, which is positive. Hence $f(q^{- \frac{1}{\Delta -1}})>0$ for $q$ sufficiently large, proving (i).
For (ii) and (iii), first observe that for all $x \in (0,1)$, we have $$f(x) = (x-1) \left( (q-1)x^{\Delta -1} + 1 + \sum_{i=1}^{\Delta -2}(1-B)x^i \right).$$ If $B\leq 0$ then for all $x\in (0,1)$, the second factor on the right hand side is positive and the first factor is negative, establishing (ii).
For the remainder of the proof, suppose that $0 \leq B = o(q^{\frac{1}{\Delta-1}})$. Using the above identity and the fact that $B$ is positive, for all $x\in (0,1)$ we obtain $$\begin{aligned}
f(x)
&\leq (x-1)\left( (q-1)x^{\Delta -1} + 1 + \sum_{i=1}^{\Delta -2}(-B)x^i \right) \\
&\leq (x-1)\left((q-1)x^{\Delta -1} + 1 - \Delta B x \right).\end{aligned}$$ If $x \in (0, q^{-\frac{1}{\Delta-1}}]$ then $\Delta Bx = o(1)$, so $f(x)<0$ (for all sufficiently large $q$). If $x \in [q^{- \frac{1}
{\Delta -1}},1)$ then it is easy to check that $\Delta Bx =
o((q-1)x^{\Delta -1})$, so $f(x)<0$ (for all sufficiently large $q$). Combining these two statements shows that (iii) holds, completing the proof.
[^1]: Supported by EPSRC grant EP/G066604/1
[^2]: Research supported by the Australian Research Council and performed while the second author was on sabbatical at the University of Durham.
[^3]: Research performed while the third author was at the University of Durham. Supported by EPSRC Grant EP/G066604/1
| 0 |
---
abstract: 'An outstanding question in X-ray single particle imaging experiments has been the feasibility of imaging sub 10-nm-sized biomolecules under realistic experimental conditions where very few photons are expected to be measured in a single snapshot and instrument background may be significant relative to particle scattering. While analyses of simulated data have shown that the determination of an average image should be feasible using Bayesian methods such as the EMC algorithm, this has yet to be demonstrated using experimental data containing realistic non-isotropic instrument background, sample variability and other experimental factors. In this work, we show that the orientation and phase retrieval steps work at photon counts diluted to the signal levels one expects from smaller molecules or with weaker pulses, using data from experimental measurements of 60-nm PR772 viruses. Even when the signal is reduced to a fraction as little as 1/256, the virus electron density determined using *ab initio* phasing is of almost the same quality as the high-signal data. However, we are still limited by the total number of patterns collected, which may soon be mitigated by the advent of high repetition-rate sources like the European XFEL and LCLS-II.'
address: |
Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761, Hamburg, Germany\
Center for Free-Electron Laser Science, Deutsches Elektronen Synchrotron DESY, Notkestra[ß]{}e 85, 22607 Hamburg, Germany\
Linac Coherent Light Source, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA, 94025, USA\
Biosciences Division, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA, 94025, USA\
Stanford PULSE Institute, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA, 94025, USA\
Department of Molecular Biology and Genetics, Koc University, Rumelifeneri yolu, Sariyer, Istanbul, 34450 Turkey\
Biodesign Center for Immunotherapy, Vaccines, and Virotherapy, Biodesign Institute at Arizona State University, Tempe 85288, USA\
Biodesign Center for Applied Structural Discovery, Biodesign Institute at Arizona State University, Tempe 85287, USA\
Arizona State University, School of Life Sciences (SOLS), Tempe, Arizona 85287, USA\
Department of Physics, Arizona State University, Tempe, AZ 85287, USA\
Department of Physics, Universit[ä]{}t Hamburg, Luruper Chaussee 149, Hamburg, Germany\
The Hamburg Center for Ultrafast Imaging, Universit[ä]{}t Hamburg, Luruper Chaussee 149, Hamburg, Germany\
Currently with the ARC Centre of Excellence for Advanced Molecular Imaging, School of Physics, The University of Melbourne, Parkville, VIC 3010, Australia
author:
- 'Kartik Ayyer, Andrew J. Morgan, Andrew A. Aquila, Hasan DeMirci, Brenda G. Hogue, Richard A. Kirian, P. Lourdu Xavier, Chun Hong Yoon, Henry N. Chapman, and Anton Barty'
bibliography:
- 'refs.bib'
title: 'Low-signal limit of X-ray single particle diffractive imaging'
---
\[sec:intro\]Introduction
=========================
The potential of X-ray free electron lasers (XFELs) to image biomolecular structures at room temperature without the need for crystallisation has been one of the goals driving their development. For many years, theoretical studies backed by simulated data have suggested that near-atomic resolution of isolated non-crystalline proteins should be possible with currently available XFEL sources [@Neutze:2000; @Ayyer:2016]. To date, published results have focused on large or symmetric particles such as viruses in the 60-500nm size range where the higher signal levels from larger particles is ideal for methods development [@Loh:2010; @Kassemeyer:2013; @Ekeberg:2015; @Aquila:2018]. Results from the single particle imaging initiative at the Linac Coherent Light Source (LCLS) [@Aquila:2015] have been in a similar size range [@Munke:2016; @Reddy:2017].
Imaging individual proteins has so far proven more elusive due to the lower signal-to-background from smaller sized particles and a lower than expected rate of single particle diffraction pattern acquisition [@Aquila:2018]. While theoretical studies indicate that molecular imaging should be achievable using Bayesian algorithms such as the EMC algorithm [@Loh:2009] for near-perfect data simulated assuming currently available XFEL parameters [@Ayyer:2016], this has yet to be demonstrated using experimental data containing realistic instrument background, sample variability and other experimental factors.
This paper addresses the question of whether these above-mentioned experimental effects pose a fundamental roadblock to diffraction-pattern alignment and phasing algorithms in the low signal limit. We achieve this using experimental rather than simulated data. The approach taken is to start with experimentally measured data and progressively reduce the photon count to levels similar to those expected from smaller particles such as individual proteins. This process also mimics data that would be recorded from the same size particles using weaker X-ray pulses such as will soon be available with a high repetition rate from the LCLS-II upgrade.
We start from data collected by the SPI initiative from PR772 viruses [@Reddy:2017] to 8.5-nm resolution. Weak data was generated by keeping only a small, random fraction of photons from each experimental snapshot. These reduced data, or ‘diluted’, patterns contain just a smattering of photons which often look like pure noise to the eye. In addition to diffraction from the virus particles, each diffraction pattern contains instrument background caused by a range of experimental sources. Any structure in the instrument background does not depend on particle orientation, thus after orientation determination this background appears as a spherically symmetric function incoherently added to the 3D Fourier intensities of the object. To account for this background, we develop a modified iterative phasing algorithm which isolates and retrieves this background while reconstructing the electron density, and also show that phase retrieval is robust to statistical noise.
The paper is set out as follows. The reconstruction pipeline and the results of its application to the full data set are described in Section \[sec:recon\_steps\], and a set of metrics including the Fourier Shell Correlation (FSC) and Phase Retrieval Transfer Function (PRTF) for quantifying reconstruction resolution and fidelity are defined in Section \[sec:metrics\]. The experimental data sets are then subsampled by randomly selecting a fraction of photons in every frame, followed by orientation and phasing of the sparsified photon counts in Section \[sec:reduction\]. The quality of the electron densities obtained using the subsampled data sets is evaluated and compared using the metrics of reconstruction quality defined in Section \[sec:metrics\].
We find that the reconstruction quality persists for a significant reduction of data quantity: even when the signal is reduced by as much as 1/256, quality metrics show the virus electron density determined using *ab initio* phasing is of almost the same quality as the high signal data. This suggests that given sufficient number of single particle diffraction patterns from sub-10 nm biomolecules with current XFEL parameters (assuming a proportionate reduction in instrument background), or from 60-nm viruses with a pulse 256 times weaker, one can obtain reliable 3D electron densities with the methods presented here. In order to obtain higher resolution, many more patterns will be required to achieve sufficient statistics. This may soon be within reach with advancements in sample delivery methods as well as with high-repetition-rate XFEL sources such as the European XFEL and LCLS-II.
\[sec:experiment\]Experiment description
========================================
Diffraction snapshots of aerosolized PR772 viruses were collected at the Linac Coherent Light Source (LCLS) as described in [@Reddy:2017]. Briefly, diffraction patterns were recorded on a pnCCD detector in the AMO instrument at the LCLS [@Ferguson:2015] at a photon energy of with the detector placed downstream from the X-ray-sample interaction point, giving a resolution of at the center-edge of the detector and maximum resolution of in the corner of the detector. This data set is available for download from the Coherent X-ray Imaging database [@Maia:2012] as CXIDB 58.
The data set consists of frames with an average signal level of photons/frame. For a virus, the speckles were around 100 pixels wide. The pixels were therefore binned by a factor of 4 in both dimensions after photon conversion to reduce computational costs. Excluding bad pixels and the central speckle, where the detetor was often saturated, there were photons/frame on average. There were on average 22.2 photons/speckle at the detector corner.
Diffraction patterns were recorded at a repetition rate of 120 Hz, however only a small fraction of the X-ray pulses interacted with an object. These so-called “hits” included not only interactions with PR772 virus particles but also with water droplets, multi-particle clusters, and patterns with detector artifacts. Such spurious patterns need to be excluded from analysis. In [@Reddy:2017], Reddy et al describes the classification of the single particle patterns using various machine learning methods, with the data for this study based on the classification by manifold embedding [@Yoon:2011] to obtain a data set consisting of single virus diffraction patterns.
\[sec:recon\_steps\]Reconstruction procedure
============================================
The PR772 virus electron density was reconstructed in a two-step process, illustrated in Fig. \[fig:pipeline\] and detailed below. First, the orientations of a set of noisy diffraction patterns of mostly identical objects in random orientations with variable incident fluence were determined to produce a 3D intensity volume using the EMC algorithm [@Loh:2009]. The three dimensional diffraction volume was then phased using a background-aware phase retrieval algorithm to arrive at the real-space electron density using a combination of the Difference Map [@Elser:2003] and Error Reduction [@Fienup:1978] algorithms.
![Reconstruction of the virus electron density from measured diffraction snapshots is a two step process. First, the orientations of a set of noisy diffraction patterns of mostly identical objects in random orientations with variable incident fluence (top left) are determined to produce a 3D intensity volume (top right). The three dimensional diffraction volume is then phased using a background-aware phase retrieval algorithm to arrive at the real space electron density (bottom). The electron density is shown as both an isosurface plot and a slice through the center of the object.[]{data-label="fig:pipeline"}](SPI_Pipeline.pdf){width="\columnwidth"}
\[sec:merging\_steps\]Alignment: Determining the 3D reciprocal space intensity distribution
-------------------------------------------------------------------------------------------
Orientation determination, alignment and scaling of the diffraction patterns into a 3D diffraction volume was performed using the *Dragonfly* software [@Ayyer:2016]. Data was provided to *Dragonfly* in photon counts since the pnCCD detector used in this experiment could resolve individual photons. A Poisson noise model was therefore used in *Dragonfly*. Both the orientation as well as a relative scale factor was estimated for each pattern to account for incident fluence fluctuations and variations in impact parameter of the virus relative to the beam. The predicted intensities on the detector for a given orientation were multiplied by this scale factor before calculating the probability distribution over orientations (PDOs). These scale factors were updated every iteration using the current estimate for the PDO for each pattern. In order to avoid convergence issues due to the high signal per pattern, the PDO was raised to the power of the deterministic annealing parameter, $\beta$. This parameter was increased from 0.001 by a factor of $\sqrt{2}$ every 10 iterations. The detailed procedure used for this reconstruction is described in Appendix A.
\[sec:phasing\_steps\]Phasing: Iterative phase retrieval with background estimation
-----------------------------------------------------------------------------------
The three dimensional diffraction volume from *Dragonfly* was phased to arrive at the real space electron density using a background-aware iterative projection phase retrieval algorithm as described in Algorithm \[alg:phase\]. The update rule for this algorithm consists of a modulus projection defined to incorporate a spherically symmetric background intensity which is incoherently added to the diffraction signal (“Background aware") in addition to a support constraint on the electron density consisting of a fixed number of voxels rather than a static mask (“Voxel number support").
The iterate $\Psi$ is comprised of both the real space density ${\rho(\mathbf{x})}$ and background ${B(\mathbf{q})}$ $$\Psi = \left\{\rho(\mathbf{x}), B(\mathbf{q})\right\}$$ In practice this consists of two 3D volumes, one for the real-space electron density and the other for the square root of the background intensity. The calculated intensity is the sum of the intensity from the particle plus the background, $$I_\text{calc}[\Psi](\mathbf{q}) = \left|\mathcal{F}[\rho](\mathbf{q})\right|^2 + B^2(\mathbf{q})
\label{eq:icalc}$$ where $\mathcal{F}[\rho]$ is the discrete Fourier transform of the electron density $\rho$. The modulus projection rescales both terms by the ratio to the measured Fourier magnitude, $$P_M[\Psi] = \left\{\mathcal{F}^{-1}\left[\sqrt{\frac{I_\text{meas}(\mathbf{q})}{I_\text{calc}(\mathbf{q})}} \mathcal{F}[\rho](\mathbf{q})\right], \sqrt{\frac{I_\text{meas}(\mathbf{q})}{I_\text{calc}(\mathbf{q})}}B(\mathbf{q})\right\}
\label{eq:pmod}$$ where $I_\text{meas}(\mathbf{q})$ is the measured intensity.
The support projection imposes two different constraints on the two halves of the iterate, $\rho$ and $B$. A constant $N$ is chosen at the beginning representing the number of voxels inside the particle for which the density is allowed to be non-zero. In this case we chose $N=2000$. The modulus-squared electron density values are sorted and the highest $N$ are left unchanged while the rest are set to zero. The background intensities, $B(\mathbf{q})$, are replaced by the spherically symmetric version i.e. the intensities in each radial bin are replaced by their average. The derivation that both these operations are projections is given in Appendix B. Further details regarding masking and alignment of reconstructions from different random starting models are discussed in Appendix C.
$P_M(P_S(x))$
$\beta = 0.7$ $f_M(x) = (1-1/\beta)P_M(x) + (1+1/\beta)x$ $f_S(x) = (1+1/\beta)P_S(x) + (1-1/\beta)x$ $x + \beta\left[P_M(f_S(x)) - P_S(f_M(x)) \right]$
$\Psi_i \gets$ Uniform Random $\Psi_i \gets$ ER($\Psi_i$) (100 times) $\Psi_i \gets$ DM($\Psi_i$) (200 times) $\Psi_i \gets$ ER($\Psi_i$) (100 times) Align all $\Psi_i$ Calculate Phase Retrieval Transfer Function (PRTF) Average over all aligned $\Psi_i$
\[sec:phased\_all\]Reconstruction from the full data set
--------------------------------------------------------
The results of applying the above two-step reconstruction method to all patterns are shown in Fig. \[fig:pipeline\]. The 3D intensity shows strong icosahedral symmetry even though this constraint was not enforced during the reconstruction. The resolution corresponding to the edge of the spherical volume of intensities is . After iterative phasing, the electron density shown in the bottom row was obtained. The contour plot shows an icosahedron with bulges at each vertex while a slice through the object centre shows the presence of a double-walled shell with a slight reduction in density just inside the outer shell, consistent with other treatments of the data [@Kurta:2017; @Rose:2018].
\[sec:metrics\]Quantifying reconstruction quality
=================================================
A set of quantitative metrics are required in order to compare reconstructions and assess overall reconstruction quality, for reconstructions of both the full and diluted data sets. We used two metrics established in the literature, which we define in this section for clarity, and applied them to the reconstruction performed with the full data set described above.
\[sec:fsc\]“Gold-standard” cross correlations
---------------------------------------------
The first of these metrics, inspired by cryo-electron microscopy, involves a slight change in the analysis pipeline itself. The ‘gold-standard‘ Fourier shell correlation from CryoEM [@Henderson:2012] calls for the separation of the dataset into two equal halves. Each half is analyzed independently, the final volumes rotationally aligned, and the relative agreement is calculated as a function of resolution using the Fourier Shell Correlation (FSC) metric: $$\mathrm{FSC}(q) = \operatorname{Re}\left[\frac{\sum\limits_{|\mathbf{q}_i| = q} F_1(\mathbf{q}_i) F_2^*(\mathbf{q}_i)}
{\sqrt{\sum\limits_{|\mathbf{q}_i| = q} |F_1(\mathbf{q_i})|^2}\sqrt{\sum\limits_{|\mathbf{q}_i| = q} |F_2(\mathbf{q_i})|^2}}\right]$$ where $F(\mathbf{q}) = \mathcal{F}[\rho](\mathbf{q})$. In practice, the FSC is calculated in $q$ bins which are shells of a certain thickness.
A similar correlation can also be calculated between the two half-dataset intensities. In order to increase the sensitivity of the correlation, the mean is subtracted in each resolution shell before calculating the cross-correlation i.e. a Pearson correlation coefficient is calculated in each shell independently. $$\mathrm{CC}_{1/2}(q) = \frac%
{\sum\limits_{|\mathbf{q}_i| = q} \left(\mathrm{I}_1 - \overline{\mathrm{I}_1}\right) \left(\mathrm{I}_2 - \overline{\mathrm{I}_2}\right)}%
{\sqrt{\sum\limits_{|\mathbf{q}_i| = q} \left(\mathrm{I}_1 - \overline{\mathrm{I}_1}\right)^2}%
\sqrt{\sum\limits_{|\mathbf{q}_i| = q} \left(\mathrm{I}_2 - \overline{\mathrm{I}_2}\right)^2}}$$ where $\mathrm{I}_k$ is shorthand for $\mathrm{I}_k(\mathbf{q}_i)$ and $\overline{\mathrm{I}_k}$ is the mean intensity in the resolution shell $\overline{\mathrm{I}_k(q)}$. The increased sensitivity due to subtracting the mean is most apparent when there is spherically symmetric background in the intensity reconstruction, as is the case here.
\[sec:prtf\]Phase retrieval transfer function (PRTF)
----------------------------------------------------
The other metric is the phase retrieval transfer function (PRTF) [@Shapiro:2005]. This metric measures the reliability of iterative phasing by (in effect) averaging complex values over may instances of the phasing process.
The first step in the calculation of this metric is to reconstruct a large number of independent density volumes from different random starting guesses. At any given reciprocal-space voxel, $\mathbf{q}$, the argument of the complex Fourier transform of the density (the phase) can be slightly different in each random start. The value of the PRTF at that voxel is the complex sum of the unit complex numbers whose argument is the phase, $\phi$: $$\mathrm{PRTF}(\mathbf{q}) = \frac{1}{N}\left|\sum_{n=1}^N e^{i\phi_n}\right|$$ where there are $N$ independent density volumes. By convention, the azimuthal average of the PRTF is reported as a function of the radial coordinate $\left|\mathbf{q}\right|$. As described in Sec. \[sec:phasing\_steps\], the different reconstructions must be aligned in real-space before calculating the average. A shift in real space is equivalent to a phase ramp which will significantly lower the PRTF. An uncorrected central inversion will negate the phase, leading to a similar reduction [@Marchesini:2006].
One weakness of the PRTF is that it can be unjustifiably high if the support volume is chosen to be too small. As an extreme case, if the support consists of only one voxel, the PRTF (after alignment) will be unity everywhere even though the reconstruction is very poor. One should therefore have a slightly larger support mask which includes some voxels with low density. In the reconstructions performed here, the support volume (2000 voxels) is significantly larger than the nominal volume of a regular icosahedron with a size corresponding to the fringe spacing (which would be 1497 voxels).
We calculate the PRTF from 400 independent reconstructions. This number is important because it needs to be large enough for the PRTF to converge and the voxels with irreproducible phases to average down. Consider for example the case where the phases are completely random, in which case the sum is a 2D random walk in the complex plane with a fixed step size which has an average distance from the origin of $\sqrt{N}$ after $N$ steps. Thus, the expected lower bound on the PRTF if $N$ reconstructions are averaged is $1/\sqrt{N}$, which is 0.05 for the case of 400 the case here. In keeping with convention, the threshold value to determine the reproducible resolution is considered to be $1/e = 0.37$.
![Reconstruction metrics for the full data set as a function of $q$. Top: Fourier Shell Correlation (FSC) plot with the dashed line showing the half-bit threshold. Middle: Intensity CC$_{1/2}$ plot with the dashed line showing the 0.5 cutoff. Bottom: Phase Retrieval Transfer Function (PRTF) plot with the customary $1/e$ cutoff. Error bars represent the standard deviation across 10 random starts.[]{data-label="fig:full-metrics"}](metrics_full.pdf){width="\columnwidth"}
\[sec:full\_recon\]Metrics applied to full data reconstruction
--------------------------------------------------------------
We applied the metrics defined above to the reconstructed intensity and electron density calculated using the procedure described in Sec. \[sec:recon\_steps\]. For the FSC and CC$_{1/2}$ calculations, frames were split into and odd and even halves containing the 1st, 3rd, 5th... and 2nd, 4th, 6th... patterns respectively. This procedure of splitting is chosen in order for both halves to be similarly affected by slowly varying drifts in the experiment. It is also sufficiently random because the “hits" themselves are a random subset of all the patterns collected.
The FSC and CC$_{1/2}$ plots are shown in Fig. \[fig:full-metrics\]. The crystallographic definition of $q$ is used with the full-period resolution, $d = 1/q$. Each of the metrics gives a slightly different estimate of the resolution of the reconstruction. from the half-bit FSC criterion standard common in cryo-electron microscopy [@VanHeel:2005], the resolution is , while using the CC$_{1/2} = 0.5$ cutoff, the intensities are reproducibly reconstructed to a resolution of . The purely phasing metric, PRTF, suggests that the resolution is for both the even and odd data sets. The oscillations apparent in the PRTF plot, which manifest from fringe intensities in the data, further reveal how resolution determined by the PRTF metric can be dramatically affected by whether or not values in one of the local minima happen to lie above or below the 0.37 threshold value. That the resolution estimates differ is not surprising given that different quantities are being measured, and suggests that one should be cautious when reporting a single resolution number. The difference between values further suggests being very conservative with the precision to which resolution is quoted in publication: the mean resolution estimated above is with a standard deviation of , in which case quoting resolution to three significant figures is certainly not appropriate. One should further be careful comparing resolution between publications to make sure that the same values are being compared.
\[sec:reduction\]Results
========================
We now turn our attention to the effect of reducing the amount of data on reconstruction quality using the analysis pipeline described in Section \[sec:recon\_steps\]. Data quantity is reduced in one of two ways. Diffraction patterns can be made weaker to simulate the effect of imaging smaller particles or the effect of a lower intensity X-ray beam. This has two effects: firstly orientation determination is expected to become harder as there is less information in each pattern from which to determine the orientation, and secondly the signal-to-noise ratio of the reconstructed 3D intensities is reduced making phase retrieval more challenging. Alternatively, the number of diffraction patterns can be reduced to simulate the effect or a smaller data set consisting of fewer diffraction patterns of the same signal strength. Computationally reducing the data in this way avoids confounding factors from working with different data sets collected at different times under potentially different experimental conditions.
\[sec:red\_patterns\]Reducing diffraction pattern intensity
-----------------------------------------------------------
To simulate measurement of weaker diffraction patterns we computationally reduced the number of photons in each image to produce diffraction patterns with fewer photons drawn from the same experimental data sets. Reducing the number of photons in each diffraction pattern was done by applying a Bernoulli process to each photon with a certain probability to keep or discard the photon. These selection fractions, $p$, were reduced from $2^{-1}$ to $2^{-10}$ in steps of powers of two. Due to the Poisson nature of the photon counting statistics, this simulates the effect of a factor $p$ weaker incident pulse. The effect of applying this process to a particular diffraction pattern is shown in Figure \[fig:dilution\]. The average number of photons per frame after photon dilution is shown in Table \[table:dilution\], from which it can be seen that photon counts per frame decreases from nearly 35,000 photons per frame at full strength to only 33 photons per frame when diluted to 1/1024 strength.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Four versions of the same diffraction pattern showing the reduction of photons/pattern by a given selection probability, $p$. In each case, the color scale maximizes at 4 photons per pixel. (a) Original pattern (b) $p=1/4$ (c) $p=1/16$ (d) $p=1/256$[]{data-label="fig:dilution"}](frame_4906_1.png "fig:"){width="0.5\columnwidth"} ![Four versions of the same diffraction pattern showing the reduction of photons/pattern by a given selection probability, $p$. In each case, the color scale maximizes at 4 photons per pixel. (a) Original pattern (b) $p=1/4$ (c) $p=1/16$ (d) $p=1/256$[]{data-label="fig:dilution"}](frame_4906_4.png "fig:"){width="0.5\columnwidth"}
(a) (b)
![Four versions of the same diffraction pattern showing the reduction of photons/pattern by a given selection probability, $p$. In each case, the color scale maximizes at 4 photons per pixel. (a) Original pattern (b) $p=1/4$ (c) $p=1/16$ (d) $p=1/256$[]{data-label="fig:dilution"}](frame_4906_16.png "fig:"){width="0.5\columnwidth"} ![Four versions of the same diffraction pattern showing the reduction of photons/pattern by a given selection probability, $p$. In each case, the color scale maximizes at 4 photons per pixel. (a) Original pattern (b) $p=1/4$ (c) $p=1/16$ (d) $p=1/256$[]{data-label="fig:dilution"}](frame_4906_256.png "fig:"){width="0.5\columnwidth"}
(c) (d)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Reconstruction of the 3D intensity from weakened data was performed in the same manner as previously described for all data sets using identical *Dragonfly* reconstruction parameters for all data sets except for the schedule of the deterministic annealing parameter $\beta$. A low value of $\beta$ was not necessary when the signal level was low since this parameter acts to solve convergence issues for very high signals by broadening the PDOs. Appendix A contains details of the parameters for each subset. The 3D intensities from *Dragonfly* were phased with identical parameters in every case to generate electron densities. Each reduced data set was split into two halves and independently reconstructed in order to calculate the “gold-standard" FSC and CC$_{1/2}$, and this whole process was repeated 10 times to obtain error bars on the metrics.
The results of reducing signal strength are summarized in Fig. \[fig:split\_metrics\]. In Fig. \[fig:split\_metrics\](a) we plot one metric, CC$_{1/2}$, as a function of $q$ for both the full data set and a selection fraction of $p=2^{-8}=1/256$. Fig. \[fig:split\_metrics\](a) shows that the reconstruction from the reduced data shows a slightly decreased quality metric compared to the full data set.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Dependency of reconstruction metrics on selection fraction. (a) Plots of CC$_{1/2}$ vs $q$ for the full data set and for a selection fraction of $p=2^{-8}=1/256$. Error bars represent the standard deviation across 10 different random half datasets and the dashed line represents the CC$_{1/2}=0.5$ cutoff. To represent dependence on selection fraction, we plot the metric in grayscale versus both selection fraction and $q$ in panels (b)-(d) with the color representing the metric value. (b) CC$_{1/2}$, the green dashed line shows the $q$ for the CC$_{1/2} = 0.5$ cutoff; (c) PRTF, dashed line shows the typical PRTF$=1/e$ cutoff, and (d) FSC, where the metric never went below the standard half-bit criterion. Each plot is the average of 10 random subsets.[]{data-label="fig:split_metrics"}](cc_split_256_and_1.pdf "fig:"){width="0.45\columnwidth"} ![Dependency of reconstruction metrics on selection fraction. (a) Plots of CC$_{1/2}$ vs $q$ for the full data set and for a selection fraction of $p=2^{-8}=1/256$. Error bars represent the standard deviation across 10 different random half datasets and the dashed line represents the CC$_{1/2}=0.5$ cutoff. To represent dependence on selection fraction, we plot the metric in grayscale versus both selection fraction and $q$ in panels (b)-(d) with the color representing the metric value. (b) CC$_{1/2}$, the green dashed line shows the $q$ for the CC$_{1/2} = 0.5$ cutoff; (c) PRTF, dashed line shows the typical PRTF$=1/e$ cutoff, and (d) FSC, where the metric never went below the standard half-bit criterion. Each plot is the average of 10 random subsets.[]{data-label="fig:split_metrics"}](cc_split_combined.pdf "fig:"){width="0.5\columnwidth"}
(a) (b)
![Dependency of reconstruction metrics on selection fraction. (a) Plots of CC$_{1/2}$ vs $q$ for the full data set and for a selection fraction of $p=2^{-8}=1/256$. Error bars represent the standard deviation across 10 different random half datasets and the dashed line represents the CC$_{1/2}=0.5$ cutoff. To represent dependence on selection fraction, we plot the metric in grayscale versus both selection fraction and $q$ in panels (b)-(d) with the color representing the metric value. (b) CC$_{1/2}$, the green dashed line shows the $q$ for the CC$_{1/2} = 0.5$ cutoff; (c) PRTF, dashed line shows the typical PRTF$=1/e$ cutoff, and (d) FSC, where the metric never went below the standard half-bit criterion. Each plot is the average of 10 random subsets.[]{data-label="fig:split_metrics"}](prtf_combined.pdf "fig:"){width="0.5\columnwidth"} ![Dependency of reconstruction metrics on selection fraction. (a) Plots of CC$_{1/2}$ vs $q$ for the full data set and for a selection fraction of $p=2^{-8}=1/256$. Error bars represent the standard deviation across 10 different random half datasets and the dashed line represents the CC$_{1/2}=0.5$ cutoff. To represent dependence on selection fraction, we plot the metric in grayscale versus both selection fraction and $q$ in panels (b)-(d) with the color representing the metric value. (b) CC$_{1/2}$, the green dashed line shows the $q$ for the CC$_{1/2} = 0.5$ cutoff; (c) PRTF, dashed line shows the typical PRTF$=1/e$ cutoff, and (d) FSC, where the metric never went below the standard half-bit criterion. Each plot is the average of 10 random subsets.[]{data-label="fig:split_metrics"}](FSC_combined.pdf "fig:"){width="0.5\columnwidth"}
(c) (d)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In order to summarise the results as a function of resolution for many different photon dilution levels, in Figs. \[fig:split\_metrics\](b)–\[fig:split\_metrics\](d) we plot each metric in grayscale versus both selection fraction and $q$, where color represents the metric value. The green dashed line in Fig. \[fig:split\_metrics\](b) marks the somewhat arbitrarily chosen CC$_{1/2} = 0.5$ cutoff, and shows how the resolution of the intensity reconstruction becomes progressively worse as $p$ is reduced. One cause of this reduction is just the graininess of the reconstruction due to insufficient total signal. Similarly the green line in Fig. \[fig:split\_metrics\](c) represents the the typical PRTF$=1/e$ cutoff. The step decrease in resolution shown by the PRTF in Fig. \[fig:split\_metrics\](c) occurs when the overall PRTF decreases to the point where the next local minima falls below cutoff threshold, Fig. \[fig:full-metrics\]. The resolution estimated by each metric is tabulated in Table \[table:dilution\].
From the metrics alone one immediately notices that the electron densities do not suffer from such a drastic falloff in resolution at very low signal. In effect, the support constraint during phasing restores the smoothness of the speckles even when the total number of photons per 3D speckle (Shannon voxel) is low, partially negating the effect of insufficient total signal. For the highest photon dilution ($p=1/1024$), the average signal level used to determine the orientations is just 33.9 photons/frame.
[l r c c c c]{} Fraction & ph/fr & Frames & CC$_{1/2}$ & PRTF & FSC\
\
$1$ & & & 9.02 & 10.19 & 8.75\
$1/2$ & & & 9.16 & 9.16 & 8.75\
$1/4$ & & & 9.33 & 9.16 & 8.75\
$1/8$ & & & 9.50 & 9.33 & 8.75\
$1/16$ & & & 9.69 & 9.33 & 8.75\
$1/32$ & & & 9.69 & 9.33 & 8.75\
$1/64$ & & & 11.2 & 9.50 & 8.75\
$1/128$ & & & 11.2 & 9.50 & 8.75\
$1/256$ & & & 11.4 & 10.9 & 8.75\
$1/512$ & & & 11.7 & 10.9 & 8.75\
$1/1024$ & & & 20.1 & 11.2 & 8.75\
\[table:dilution\]
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(a) Histogram of reconstructed electron densities for three different selection fractions. The voxels with low densities are present because the support is slightly larger than the particle. At higher photon counts, one can see a separation between the higher densities in the core of the virus compared to the capsid shell. This distinction disappears at the very low signal levels corresponding to $p=1/1024$. (b) Slices through representative electron densities with the same selection fractions. One can see the gradual disappearance of the double-shell structure with reducing fraction.[]{data-label="fig:dens_hist"}](dens_hist.pdf "fig:"){width="3in"}
(a)
![(a) Histogram of reconstructed electron densities for three different selection fractions. The voxels with low densities are present because the support is slightly larger than the particle. At higher photon counts, one can see a separation between the higher densities in the core of the virus compared to the capsid shell. This distinction disappears at the very low signal levels corresponding to $p=1/1024$. (b) Slices through representative electron densities with the same selection fractions. One can see the gradual disappearance of the double-shell structure with reducing fraction.[]{data-label="fig:dens_hist"}](dens_slices.pdf "fig:"){width="3in"}
(b)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
We also studied the effect of reducing data on the histogram of electron density values retrieved in real space. Figure \[fig:dens\_hist\] shows the histogram of electron densities inside the support mask for three different selection fractions. The plots are averaged over the 20 phasing runs for each fraction (10 random subsets and two halves per subset). The histograms clearly show the degradation in quality as signals are reduced, with the average reconstructed particle tending towards a uniform icosahedral blob with no internal structure. Additionally, the presence of the low density voxels is reassurance that the support was not too tight and the calculated PRTF not artificially high. For selection fractions above $1/32$, the histograms and densities were nearly identical, and are hence not shown for clarity. The difference in electron density histograms suggests that differences in the real space electron density may not be entirely reflected in all of the reconstruction metrics, and that metric cutoff values used to assess resolution may on their own paint a partial picture of reconstruction quality.
\[sec:red\_num\_patterns\]Reducing number of patterns
-----------------------------------------------------
An alternative method of reducing the total number of measured photons is be to select a random subset of full intensity diffraction patterns. By this method one approaches the limit of a few bright patterns.
From the total number of , 10 random subsets were generated with 8192, 4096, 2048, 1024 and 512 patterns respectively. Each of these subsets was split into two halves (the even and odd patterns) and independently reconstructed. The CC$_{1/2}$ plots for the intensity reconstructions for each of the subsets is shown in Fig. \[fig:cc\_nframes\]. Using this approach the metrics remain largely unaffected provided more than 2048 patterns in total are used (1024 in each half data set), indicating that the reconstruction was very stable and supports the hypothesis that there was more than enough data for this resolution. However, with 1024 frames (512 frames in each half), the reconstruction failed 4 out of the 20 times. What happens in this case is that if the number of patterns is reduced too much, they do not fill the 3D reciprocal space volume, leading to artifacts in orientation determination. Since a unique assignment of orientation for just 512 patterns would be insufficient to fully populate reciprocal space, the reconstruction only succeeds due to the PDOs being broad when $\beta$ is low. Even so, there are times when the 3D intensity collapses into a single, or a few planes: orientation determination effectively fails and all frames are assigned to one or a few orientations. Fortunately, this failure mode is easy to identify and exclude from averaging. The failed reconstructions have been retained in this work for the sake of completeness. Other algorithms which use additional constraints on the intensity, from a restricted real-space support, or from additional point-group symmetries, may have better performance in this limit of a few very bright patterns.
![Intensity CC$_{1/2}$ vs $q$ plots as a function of number of frames in the data set. Like in Fig. \[fig:split\_metrics\], each column represents a plot or a different number of frames.[]{data-label="fig:cc_nframes"}](cc_nframes_combined.pdf){width="0.9\columnwidth"}
\[sec:discussion\]Discussion
============================
By sub-sampling the experimental data from PR772 viruses measured in [@Reddy:2017], we show that the reconstruction quality is essentially same as from the full data set with as few as 135 relevant photons/pattern, corresponding to 0.087 photons/speckle at the detector corner. This approaches the limits of prior work using simulated data [@Neutze:2000; @Loh:2009; @Ayyer:2016] or proof-of-principle experiments under highly controlled conditions not realistic for single particle imaging [@Philipp:2012; @AyyerP:2015]. By way of contrast, the results here are based on data derived from experimental measurements on PR772 viruses incorporating particle variability and instrument background, demonstrating that the signal required for X-ray single particle imaging under realistic conditions is much lower than previously demonstrated especially in terms of the number of scattered photons required per frame.
From this numerical experiment we conclude that current SPI algorithms should be capable of processing experimental single particle diffraction patterns when the photon flux in the X-ray focus is 256 times smaller than currently available at LCLS for particles of the same size as PR772. Furthermore, algorithms appear to be more robust for the case of many weak hits than a small number of very strong hits. The extension of this method to smaller particles is not so direct. In order for this analysis to also hold for the case where the particle volume is reduced by the same factor, one requires that the parasitic scatter is also proportionately reduced. At higher photon energies, significantly lower background has already been achieved [@Munke:2016] than present in this data set. Thus, one strategy for the future direction of the field may be to move to hard X-ray instruments where one has reduced scattering cross section (factor 20 lower for 7 keV vs 1.6 keV, as was the case here) but possibly much lower background.
From this analysis we also conclude that analysis algorithms on their own are not the current limiting factor for SPI imaging. Low background data collection has already been demonstrated in the data set of [@Munke:2016] to 6Å resolution. Unfortunately there were insufficient hits from the entire beamtime for a reconstruction to be feasible. The work here suggests that signal levels may have been adequate had sufficient single-particle diffraction patterns been collected. This points to the need to further develop methods for introducing single particles into the X-ray focus in sufficient density to make sufficient measurements at high resolution. Indeed, this could currently be one of the main factors limiting further progress in SPI imaging. Another key conclusion is that further work is needed in the area of single particle diffraction pattern classification to achieve similar noise tolerance as orientation determination, for which the efficacy of machine learning techniques in the limit of low signal still needs to be explored. This result bodes well for the prospects of single particle flash X-ray imaging to near-atomic resolution at high repetition rate XFELs like the European XFEL and LCLS-II and may help guide future XFEL and instrument design.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(a) Virtual powder sum of all patterns, shown with a logarithmic color scale. (b) Detector mask used in orientation determination and intensity reconstruction. The ‘black’ pixels were ignored completely. The ‘ochre’ pixels were used to calculate the average intensity in 3D but not to calculate the orientations. The ‘white’ pixels were used for both orientation and average intensity calculations.[]{data-label="fig:powder_mask"}](powder.pdf "fig:"){width="0.5\columnwidth"} ![(a) Virtual powder sum of all patterns, shown with a logarithmic color scale. (b) Detector mask used in orientation determination and intensity reconstruction. The ‘black’ pixels were ignored completely. The ‘ochre’ pixels were used to calculate the average intensity in 3D but not to calculate the orientations. The ‘white’ pixels were used for both orientation and average intensity calculations.[]{data-label="fig:powder_mask"}](mask_pnccd_back_260_257.pdf "fig:"){width="0.43\columnwidth"}
(a) (b)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\[sec:app\_dragonfly\]Appendix A: Intensity reconstruction details {#secapp_dragonflyappendix-a-intensity-reconstruction-details .unnumbered}
==================================================================
This appendix gives the detailed steps applied to reconstruct the intensity volume from the full dataset with frames shown in Sec. \[sec:full\_recon\]. A similar procedure was used for the reduced data set reconstructions whose results are described in Sec. \[sec:reduction\]. All intensities were reconstructed using Version 1.0.4 of the *Dragonfly* software. The virtual powder sum from all the patterns is shown in Fig. \[fig:powder\_mask\](a). Figure \[fig:powder\_mask\](b) shows the mask used when reconstructing the intensities. The innermost pixels inside the central speckle were not used to determine the orientations because of saturation. Some other regions were completely excluded from either orientation determination or to calculate the average 3D intensity.
``` {language="ini"}
[parameters]
detd = 586
lambda = 7.75
detsize = 260 257
pixsize = 0.3
stoprad = 40
ewald_rad = 650.
polarization = x
[make_detector]
in_mask_file = aux/mask_pnccd_back_260_257.byt
out_detector_file = data/det_pnccd_back.dat
[emc]
in_photons_list = amo86615_PR772.txt
in_detector_file = make_detector:::out_detector_file
num_div = 10
output_folder = data/
log_file = EMC.log
need_scaling = 1
beta = 0.001
beta\_schedule = 1.41421356 10
```
First, the photon converted patterns were downloaded as HDF5 files from the CXIDB. Each file contains patterns from a single experimental run. The photons were first converted to the sparse `.emc` format using the script `h5toemc.py`. The configuration file used for this reconstruction is shown in Fig. \[fig:dragonfly\_config\]. The file specified by `in_mask_file` is provided along with the *Dragonfly* source code and is shown in Fig. \[fig:powder\_mask\](b). The `make_detector.py` utility was used to generate the detector file detailing which voxel was sampled by every pixel. The `ewald_rad` parameter sets the $q$-space size of a voxel which is defined to be `1/lambda/ewald_rad`. `amo86615_PR772.txt` is a text file containing the names of the converted `emc` files from every run. 100 iterations of the EMC algorithm were performed starting from a random starting model (uniform random numbers at each voxel).
For all the cases where the data set was split into two halves, the `selection` option was added in the `[emc]` section and set to `odd_only` and `even_only` for the two halves respectively. Since the intensity reconstruction is invariant to an overall rotation, the two half-data set volumes were rotationally aligned with each other using the `compare` utility in *Dragonfly*. This program maximizes the overall CC$_{1/2}$ between the two models within a radius range and also calculates the value of CC$_{1/2}$ as a function of $q$ (as shown in Fig. \[fig:split\_metrics\](a)).
\[sec:app\_proj\]Appendix B: $P_M$ and $P_S$ are projections {#secapp_projappendix-b-p_m-and-p_s-are-projections .unnumbered}
============================================================
Equation \[eq:pmod\] for $P_M$ describes the rescaling of both the background and signal Fourier magnitudes by the square root of the ratio of measured to calculated intensities. The Fourier space modulus constraint requires that the calculated intensity defined in Eq. \[eq:icalc\] equals the measured modulus $\sqrt{I_\text{meas}}$. $I_\text{calc}$ has three components at each voxel, namely the real and imaginary parts of the Fourier transform of the electron density and the background, which is allowed to vary independently. The constraint set, therefore, represents the surface of a sphere with radius equalling the measured modulus. The projection of a general point, $\{\operatorname{Re}(\mathcal{F}[\rho]), \operatorname{Im}(\mathcal{F}[\rho]), B\}$ to this sphere is just a rescaling of this 3-vector by the ratio of the magnitudes.
The support projection applies different operations to the two halves of the iterate. For the electron density $\rho(\mathbf{x})$, the “voxel number" constraint states that at most $N$ voxels have non-zero density. The projection to this constraint set under a Euclidean metric is just to let these $N$ voxels be the ones with the highest absolute value. Note, however, that unlike the conventional fixed support constraint, this “voxel number” constraint on $\rho$ is non-convex. For the background volume, $B(\mathbf{q})$, the constraint requires that the background be spherically symmetric. Stated another way, the voxels within the same radial bin should have the same value. The projection to this set is to replace the background magnitude by its azimuthally averaged value.
\[sec:app\_masking\]Appendix C: Iterative phasing details {#secapp_maskingappendix-c-iterative-phasing-details .unnumbered}
=========================================================
This appendix contains some additional implementation details about the phase retrieval procedure described in Section \[sec:recon\_steps\]. The code used to perform the reconstructions in this work can be found here: `https://github.com/andyofmelbourne/3D-Phasing`. The configuration file used is described in Fig. \[fig:phasing\_config\].
[0.43]{}
``` {language="ini"}
[input]
script = 'make_input.py'
fname = ''
dtype = float
shape = 125, 125, 125
padd_to_pow2 = True
inner_mask = 6
outer_mask = 57
outer_outer_mask = 64
subtract_percentile = None
mask_edges = True
spherical_support = None
[geom]
energy = 2.56348259328e-16
detector_distance = 586.0e-3
voxel_size = 901.538461538e-6
```
[0.43]{}
script = phase.py
repeats = 400
iters = 100ERA 200DM 200ERA
[phasing_parameters]
voxel_number = 2000
support = None
background = True
hardware = cpu
dtype = double
[output]
path = ''
!$\,$!
As in the intensity reconstructions, the central speckle intensities were not found to be trustworthy and were masked out up to a radius of 6 voxels from the center. This means that during the modulus projection $P_M$, these voxels were left unmodified. In addition to this central region, a 7-voxel thick shell at the edge of the sphere of reconstructed intensities was also masked out in order to avoid ringing artifacts due to truncating half a speckle.
As mentioned in Section \[sec:phasing\_steps\], the reconstruction from the different random starting guesses need to be aligned with respect to each other before averaging and calculating the PRTF. This is done in three steps, first by translating the volumes such that the center of mass of each of them is at the origin. Second, since the objects are assumed to be complex-valued in general, a global phase is removed by subtracting the mean phase over all voxels. Finally, in order to remove a central inversion uncertainty, one solution (for convenience, the first) is taken as the reference For each of the other solutions, the error with respect to the reference for both the original and the center-inverted version is calculated and the one with lower error is retained.
Funding {#funding .unnumbered}
=======
US Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (OBES), under contract DE-AC02-76SF00515; U.S. National Science Foundation (NSF) Science and Technology Center BioXFEL Award 1231306; Australian Research Council Centre of Excellence in Advanced Molecular Imaging (AMI); European Research Council, “Frontiers in Attosecond X-ray Science: Imaging and Spectroscopy (AXSIS)”, ERC-2013-SyG 609920 (2014-2018); The Human Frontier Science Program (RGP0010/2017); Fellowship from the Joachim Herz Stiftung; Cluster of Excellence “The Hamburg Center for Ultrafast Imaging” of the Deutsche Forschungsgemeinschaft (DFG) - EXC 1074 - project ID 194651731; Helmholtz Association through project-oriented funds.
Acknowledgments {#acknowledgments .unnumbered}
===============
We wish to thank the members of the Single Particle Imaging initiative at LCLS who provided valuable feedback regarding this work, such as Ivan Vartanyants, John Spence and Max Rose.
Disclosures {#disclosures .unnumbered}
===========
The authors declare no conflict of interest.
| 0 |
---
abstract: 'The large branching ratios for pure annihilation $\bar{B}_s^0$ $\to$ $\pi^+ \pi^-$ and $\bar{B}_d^0$ $\to$ $K^+ K^-$ decays reported by CDF and LHCb collaborations recently and the so-called ${\pi}K$ and ${\pi}{\pi}$ puzzles indicate that spectator scattering and annihilation contributions are important to the penguin-dominated, color-suppressed tree dominated, and pure annihilation nonleptonic $B$ decays. Combining the available experimental data for $B_{u,d}$ ${\to}$ $\pi \pi$, ${\pi}K$ and $K \bar{K}$ decays, we do a global fit on the spectator scattering and annihilation parameters $X_H({\rho}_H$, ${\phi}_H)$, $X_A^i({\rho}_A^{i},{\phi}_A^{i})$ and $X_A^f({\rho}_A^{f},{\phi}_A^{f})$, which are used to parameterize the endpoint singularity in amplitudes of spectator scattering, nonfactorizable and factorizable annihilation topologies within the QCD factorization framework, in three scenarios for different purpose. Numerically, in scenario II, we get $({\rho}_A^{i},{\phi}_A^{i}[^{\circ}])=(2.88^{+1.52}_{-1.30},-103^{+33}_{-40})$ and $({\rho}_A^{f},{\phi}_A^{f}[^{\circ}])=(1.21^{+0.22}_{-0.25},-40^{+12}_{-8})$ at the $68\%$ confidence level, which are mainly demanded by resolving ${\pi}K$ puzzle and confirm the presupposition that $X_A^i\neq X_A^f$. In addition, correspondingly, the $B$-meson wave function parameter $\lambda_B$ is also fitted to be $0.18^{+0.11}_{-0.08}\, MeV$, which plays an important role for resolving both ${\pi}K$ and $\pi\pi$ puzzles. With the fitted parameters, the QCDF results for observables of $B_{u,d}$ $\to$ $\pi \pi$, $\pi K$ and $K \bar{K}$ decays are in good agreement with experimental measurements. Much more experimental and theoretical efforts are expected to understand the underlying QCD dynamics of spectator scattering and annihilation contributions.'
author:
- Qin Chang
- Junfeng Sun
- Yueling Yang
- Xiaonan Li
title: |
Spectator Scattering and Annihilation Contributions\
as a Solution to the ${\pi}K$ and ${\pi}{\pi}$ Puzzles\
within QCD Factorization Approach
---
Introduction {#sec01}
============
Charmless hadronic $B$-meson decays provide a fertile ground for testing the Standard Model (SM) and exploring the source of $CP$ violation, which attract much attention in the past years. Thanks to the fruitful accomplishment of BABAR and Belle, the constraints on the sides and interior angles of the unitarity triangle significantly reduce the allowed ranges of some of the CKM elements, and many rare $B$ decays are well measured. With the successful running of LHC and the advent of Belle II at SuperKEKB, heavy flavour physics has entered a new exciting era and more $B$ decay modes will be measured precisely soon.
Recently, the evidence of pure annihilation decays $\bar{B}_{s}^{0}$ ${\to}$ ${\pi}^{+}{\pi}^{-}$ and $\bar{B}_{d}^{0}$ ${\to}$ $K^{+}K^{-}$ are firstly reported by CDF Collaboration [@CDFanni], and soon confirmed by LHCb Collaboration [@LHCbanni]. The Heavy Flavor Averaging Group (HFAG) presents their branching ratios [@HFAG] $${\cal B}(\bar{B}_{s}^{0}{\to}{\pi}^{+}{\pi}^{-})
=(0.73{\pm}0.14){\times}10^{-6}
\label{HFAGpipi},$$ $${\cal B}(\bar{B}_{d}^{0}{\to}K^{+}K^{-})
=(0.12{\pm}0.05){\times}10^{-6}
\label{HFAGKK}.$$ Such results, if confirmed, imply unexpectedly large annihilation contributions in $B$ decays and significant flavour symmetry breaking effects between the annihilation amplitudes of $B_{u,d}$ and $B_{s}$ decays, which attract much attention recently, for instance Refs. [@zhu1; @zhu2; @chang1; @xiao1].
Theoretically, as noticed already in Refs. [@pqcd; @relaRef; @du1; @Beneke2], even though the annihilation contributions are formally $\Lambda_{QCD}/m_b$ power suppressed, they are very important and indispensable for charmless $B$ decays. By introducing the parton transverse momentum and the Sudakov factor to regulate the endpoint divergence, there is a large complex annihilation contribution within the perturbative QCD (pQCD) approach [@pqcd; @relaRef]. The latest renewed pQCD estimations ${\cal B}(\bar{B}_s^0 \to \pi^+ \pi^-)$ $=$ $(5.10^{+1.96+0.25+1.05+0.29}_{-1.68-0.19-0.83-0.20}) \times 10^{-7}$ and ${\cal B}(\bar{B}_d^0 \to K^+ K^-)$ $=$ $(1.56^{+0.44+0.23+0.22+0.13}_{-0.42-0.22-0.19-0.09}) \times 10^{-7}$ [@xiao1] give an appropriate account of the CDF and LHCb measurements within uncertainties. In the QCD factorization (QCDF) framework [@Beneke1], the endpoint divergence in annihilation amplitudes is usually parameterized by $X_{A}(\rho_A,\phi_A)$ (see Eq.(\[XA\])). The parameters $\rho_A$ $\sim$ $1$ and $\phi_A$ $\sim$ $-55^{\circ}$ (scenario S4) [@Beneke2] are adopted conservatively in evaluating the amplitudes of $B$ $\to$ $PP$ decays, which lead to the predictions ${\cal B}(\bar{B}_s^0 \to \pi^+ \pi^-)$ $=$ $(0.26^{+0.00+0.10}_{-0.00-0.09}) \times 10^{-6}$ and ${\cal B}(\bar{B}_d^0 \to K^+ K^-)$ $=$ $(0.10^{+0.03+0.03}_{-0.02-0.03}) \times 10^{-6}$ [@Cheng2]. It is obvious that the QCDF prediction of ${\cal B}(\bar{B}_d^0 \to K^+ K^-)$ agrees well with the data Eq.(\[HFAGKK\]), but the one of ${\cal B}(\bar{B}_s^0 \to \pi^+ \pi^-)$ is much smaller than the present experimental measurement Eq.(\[HFAGpipi\]). This discrepancy kindles the passions of restudy on annihilation contributions [@zhu1; @zhu2; @chang1].
At present, there are two major issues among the well-concerning focus on the annihilation contributions within the QCDF framework, one is whether $X_A(\rho_A,\phi_A)$ is universal for $B$ decays, and the other is what its value should be. As to the first issue, there is no an imperative reason for the annihilation parameters $\rho_A$ and $\phi_A$ to be the same for different $B_{u,d,s}$ decays, even for different annihilation topologies, although they were usually taken to be universal in the previous numerical calculation for simplicity [@du1; @Beneke2]. Phenomenologically, it is almost impossible to account for all of the well-measured two-body charmless $B$ decays with the universal values of $\rho_A$ and $\phi_A$ based on the QCDF approach [@zhu2; @chang1; @Beneke2; @Cheng2]. In addition, the pQCD study on $B$ meson decays also indicate that the annihilation parameters $\rho_A$ and $\phi_A$ should be process-dependent. In fact, in the practical QCDF application to the $B$ ${\to}$ $PP$, $PV$ decays (where $P$ and $V$ denote the light pseudoscalar and vector $SU(3)$ meson nonet, respectively), the non-universal values of annihilation phase $\phi_A$ with respect to PP and PV final states are favored (scenario S4) [@Beneke2]; the process-dependent values of $\rho_A$ and $\phi_A$ are given based on an educated guess [@Cheng2; @Cheng1] or the comparison with the updated measurements [@chang1]; the flavour-dependent values of $\rho_A$ and $\phi_A$ are suggested recently in the nonfactorizable annihilation contributions [@zhu2]. In principle the value of $\rho_A$ and $\phi_A$ should differ from each other for different topologies with different flavours, but we hope that the QCDF approach can accommodate and predict much more hadronic $B$ decays with less input parameters. So much attention in phenomenological analysis on the weak annihilation $B$ decays is devoted to what the appropriate values of the parameters $\rho_A$ and $\phi_A$ should be. This is the second issue. In principle, a large value of $\rho_A$ is unexpected by the power counting rules and the self-consistency validation within the QCDF framework. The original proposal is that $\rho_A$ ${\leq}$ $1$ and an arbitrary strong interaction phase $\phi_A$ are universal for all decay processes, and that a fine-tuning of the phase $\phi_A$ is required to be reconciled with experimental data when $\rho_A$ is significantly larger than 1 [@Beneke2]. The recent study on the annihilation contributions show that $\rho_A$ $>$ $2$ and ${\vert}\phi_A{\vert}$ $\geq$ $30^{\circ}$ are acceptable, even necessary, to reproduce the data for some two-body nonleptonic $B_{u,d,s}$ decay modes [@zhu2; @chang1]. In this paper, we will perform a fitting on the parameters $\rho_A$ and $\phi_A$ by considering $B$ ${\to}$ ${\pi}{\pi}$, ${\pi}K$ and $K \bar{K}$ decay modes, on one hand, to investigate the strength of annihilation contribution, on the other hand, to study their effects on the anomalies in $B$ physics, such as the well-known ${\pi}K$ and ${\pi}{\pi}$ puzzles.
The so-called ${\pi}K$ puzzle is reflected by the difference between the direct $CP$ asymmetries for $B^{-}$ ${\to}$ $K^{-}\pi^{0}$ and $\bar{B}^{0}$ ${\to}$ $K^{-}\pi^{+}$ decays. With the up-to-date HFAG results [@HFAG], we get $$\Delta A \equiv A_{CP}(B^{-} {\to} K^{-} {\pi}^{0})
- A_{CP}(\bar{B}^{0} \to K^{-} \pi^{+})
= (12.2 \pm 2.2) \%
\label{acppi},$$ which differs from zero by about $5.5\sigma$. However, the direct $CP$ asymmetries of $A_{CP}(B^{-} \to K^{-} \pi^{0})$ and $A_{CP}(\bar{B}^{0} \to K^{-} \pi^{+})$ are expected to be approximately equal with the isospin symmetry in the SM, numerically for instance $\Delta A \sim 0.5 \%$ in the S4 scenario of QCDF [@Beneke2].
The so-called ${\pi}{\pi}$ puzzle is reflected by the following two ratios of the $CP$-averaged branching fractions [@pipipuz]: $$R_{+-}^{\pi \pi}
\equiv 2 \Big[
\frac{ {\cal B}(B^{-} \to \pi^{-} \pi^0) }
{ {\cal B}(\bar{B}^{0} \to \pi^{+} \pi^{-}) }
\Big]
\frac{ \tau_{B^0} }{ \tau_{B^+} },
\qquad
R_{00}^{\pi \pi}
\equiv 2 \Big[
\frac{ {\cal B}( \bar{B}^{0} \to \pi^0 \pi^0) }
{ {\cal B}( \bar{B}^{0} \to \pi^{+} \pi^{-}) }\Big]
\label{pipipuzzle}.$$ It is generally expected that branching ratio ${\cal B}(\bar{B}^{0} \to \pi^+ \pi^-) \gtrsim
{\cal B}(B^{-} \to \pi^{-} \pi^0)$ and ${\cal B}(\bar{B}^{0} \to \pi^+ \pi^-) \gg
{\cal B}(\bar{B}^{0} \to \pi^0 \pi^0)$ within the SM. To date, the agreement of $R_{+-}^{\pi \pi}$ between the S4 scenario QCDF $R_{+-}^{\pi \pi}(\text{QCDF})$ $=$ $1.83$ [@Beneke2] and the refined experimental data $R_{+-}^{\pi \pi}(\text{Exp.})$ $=$ $1.99 \pm 0.15$ [@HFAG] can be achieved consistently within experimental error, while the discrepancy in $R_{00}^{\pi \pi}$ between the S4 scenario QCDF $R_{00}^{\pi \pi}(\text{QCDF})$ $=$ $0.27$ (where theoretical uncertainties are unenclosed) [@Beneke2] and the progressive experimental data $R_{+-}^{\pi \pi}(\text{Exp.})$ $=$ $1.99 \pm 0.15$ [@HFAG] is unexpectedly large.
It is claimed [@pipipuz] that the so-called ${\pi}{\pi}$ puzzle could be accommodated by the nonfactorizable contributions in SM. It is argued [@Cheng1; @pipipuz] that to solve the so-called ${\pi}K$ puzzle, a large complex color-suppressed tree amplitude $C^{\prime}$ or a large complex electroweak penguin contribution $P_{\rm EW}^{\prime}$ or a combination of them are essential. An enhanced complex $P_{\rm EW}^{\prime}$ with a nontrivial strong phase can be obtained from new physics effects [@pipipuz]. To get a large complex $C^{\prime}$, one can resort to spectator scattering and final state interactions [@Cheng1; @Cheng2]. Recently, the annihilation amplitudes with large parameters $\rho_A$ is suggested to conciliate the recent measurements Eq.(\[HFAGpipi\]) and Eq.(\[HFAGKK\]), so surprisingly, the ${\pi}K$ puzzle is also resolved simultaneously [@zhu2]. Theoretically, the power corrections, such as spectator scattering at the twist-3 order and annihilation amplitudes, are important to account for the large branching ratios and $CP$ asymmetries of penguin-dominated and/or color-suppressed tree-dominated $B$ decays. So, before claiming a new physics signal, it is essential to examine whether power corrections could retrieve “problematic” deviations from the SM expectations. Interestingly, our study show that with appropriate parameters, the annihilation and spectator scattering contributions could provide some possible solutions to the $\pi K$ and $\pi \pi$ puzzles.
Our paper is organized as following. In section \[sec02\], we give a brief overview of the hard spectator and annihilation calculations and recent studies within QCDF. In section \[sec03\], focusing on $\pi K$ and $\pi \pi$ puzzles, the effects of spectator scattering and annihilation contributions on $B$ $\to$ $\pi \pi$, $\pi K$ and $K \bar{K}$ decays are studied in detail in blue[three]{} scenarios. In each scenario, a fitting on relevant parameters are performed. Our conclusions are summarized in section \[sec04\]. Appendix \[app01\] recapitulates the building blocks of annihilation and spectator scattering amplitudes. The input parameters and our fitting approach are given in Appendix \[app02\] and \[app03\], respectively.
Brief Review of Spectator Scattering and Annihilation Amplitudes within QCDF {#sec02}
============================================================================
The effective Hamiltonian for nonleptonic $B$ weak decays is [@Buchalla:1996vs] $$\begin{aligned}
{\cal H}_{\rm eff} &=&
\frac{G_F}{\sqrt{2}}
\sum\limits_{p,q}
V_{pb} V_{pq}^{\ast}
\Big\{
\sum\limits_{i=1}^{10}
C_i O_i + C_{7 \gamma} O_{7 \gamma}
+ C_{8g} O_{8g} \Big\}
+ {\rm h.c.}
\label{eq:eff},
\end{aligned}$$ where $V_{pb} V_{pq}^{\ast}$ ($p$ $=$ $u$, $c$ and $q$ $=$ $d$, $s$) is the product of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements; $C_{i}$ is the Wilson coefficient corresponding to the local four-quark operator $O_i$; $O_{7 \gamma}$ and $O_{8g}$ are the electromagnetic and chromomagnetic dipole operators.
\
With the effective Hamiltonian Eq.(\[eq:eff\]), the QCDF method has been fully developed and extensively employed to calculate the hadronic two-body $B$ decays, for example, see [@du1; @Beneke1; @Beneke2; @Cheng2]. The spectator scattering and annihilation amplitudes (see Fig.\[diag\]) are expressed as the convolution of scattering functions with the light-cone wave functions of the participating mesons [@Beneke1; @Beneke2]. The explicit expressions for the basic building blocks of spectator scattering and annihilation amplitudes have been given by Ref. [@Beneke2], which are also listed in the appendix \[app01\] for convenience. With the asymptotic light-cone distribution amplitudes, the building blocks for annihilation amplitudes of Eq.(\[ai1\]-\[af3\]) could be simplified as [@Beneke2] $$\begin{aligned}
A_1^i & \simeq & A_2^i \simeq
2 \pi \alpha_s \Big[ 9\,\Big( X_A - 4 + \frac{\pi^2}{3} \Big)
+ r_\chi^{M_1} r_\chi^{M_2} X_A^2 \Big]
\label{xai12}, \\
A_3^i & \simeq &
6 \pi \alpha_s \big(r_\chi^{M_1} - r_\chi^{M_2} \big)
\Big( X_A^2 - 2 X_A + \frac{\pi^2}{3} \Big)
\label{xai3}, \\
A_3^f & \simeq &
6 \pi \alpha_s ( r_\chi^{M_1} + r_\chi^{M_2} )
(2 X_A^2 - X_A)
\label{xaf3},
\end{aligned}$$ where the superscripts $i$ (or $f$) refers to gluon emission from the initial (or final) state quarks, respectively (see Fig.\[diag\]). For the $\pi \pi$, $\pi K$ and $K \bar{K}$ final-state, $A_3^i$ is numerically negligible due to $r_\chi^{M_1}$ $\simeq$ $r_\chi^{M_2}$. The model-dependent parameter $X_A$ is used to estimate the endpoint contributions, and expressed as $$\int_0^1 \frac{dx}{x} \to
X_A = (1+ \rho_A e^{i\phi_A})
\ln \frac{m_B}{\Lambda_h}
\label{XA},$$ where $\Lambda_h$ $=$ $0.5$ GeV. For spectator scattering contributions, the calculation of twist-3 distribution amplitudes also suffers from endpoint divergence, which is usually dealt with the same manner as Eq.(\[XA\]) and labelled by $X_H$ [@Beneke2]. Moreover, a quantity $\lambda_B$ is used to parameterize our ignorance about $B$-meson distribution amplitude \[see Eq.(\[hardblock\])\] through [@Beneke2] $$\int_0^1 \frac{ d \xi }{\xi} \Phi_B(\xi)
\ \equiv \ \frac{m_B}{\lambda_B}
\label{lamdef}.$$
The QCDF approach itself cannot give information or/and constraint on the phenomenological parameters of $X_A$, $X_H$ and ${\lambda_B}$. These parameters should be determined from experimental data. To conform with measurements of nonleptonic $B$ ${\to}$ $PP$ decays, we will adopt a similar method used in Ref.[@zhu2] to deal with the contributions from weak annihilation and spectator scattering. Focusing on the flavor dependence, without consideration of theoretical uncertainties, annihilation contributions are reevaluated in detail [@zhu2] to explain the ${\pi}K$ puzzle and the recent measurements on pure annihilation decays $\bar{B}_{s}^{0}$ ${\to}$ ${\pi}^{+}{\pi}^{-}$ and $\bar{B}_{d}^{0}$ ${\to}$ $K^{+}K^{-}$ \[see Eq.(\[HFAGpipi\],\[HFAGKK\])\]. The authors of Ref. [@zhu2] find that the flavour symmetry breaking effects should be carefully considered for $B_{u,d,s}$ decays, and suggest that the parameters of $\rho_A$ and $\phi_A$ in nonfactorizable annihilation topologies $A^{i}_{k}$ \[see Eq.(\[xai12\],\[xai3\])\] should be different from those in factorizable annihilation topologies $A^{f}_{k}$ \[see Eq.(\[xaf3\])\]. (1) For factorizable annihilation topologies, i.e., the gluon emission from the final states Fig.\[diag\](c,d), the flavor symmetry breaking effects are embodied in the decay constants, because the asymptotic light-cone distribution amplitudes of final states are the same. In addition, all decay constants have been factorized outside from the hadronic matrix elements of factorizable annihilation topologies. So $A^{f}_k$ is independent of the initial state, and is the same for $B_{u,d,s}$ annihilation decays to two light pseudoscalar mesons, that is to say, $\rho^f_A$ and $\phi^f_A$ should be universal for $B_{u,d,s}$ $\to$ $PP$ decays. (2) For nonfactorizable annihilation topologies, i.e., the gluon emission from the initial $B$ meson Fig.\[diag\](a,b), besides the factorized decay constants and the same asymptotic light-cone distribution amplitudes, $B$ meson wave functions $\Phi_{B}(\xi)$ are involved in the convolution integrals of hadronic matrix elements. Hence, $A^{i}_k$ should depend on the initial state and be different for $B_{u,d}$ from $B_{s}$ meosn due to flavor symmetry breaking effects, i.e., parameters of $\rho^i_A$ and $\phi^i_A$ should be non-universal for $B_s$ and $B_{u,d}$ meson decays, and be different from parameters of $\rho^f_A$ and $\phi^f_A$ for $A^{f}_k$. In fact, the symmetry breaking effects have been considered in pervious QCDF study on two-body hadronic $B$ decays [@Cheng1; @Cheng2; @Cheng3; @Beneke2; @chang1], but with parameters of $\rho^f_A$ $=$ $\rho^i_A$ and $\phi^f_A$ $=$ $\phi^i_A$. So, it is essential to systematically reevaluate factorizable and nonfactorizable annihilation contributions and preform a global fit on the annihilation parameters with the current available experimental data. In this paper, we will pay much attention to $B_{u,d}$ ${\to}$ $KK$, ${\pi}K$, ${\pi}{\pi}$ decays and the aforementioned ${\pi}K$, ${\pi}{\pi}$ puzzles with a distinction between ($\rho^f_A$, $\phi^f_A$) and ($\rho^i_A$, $\phi^i_A$), i.e., $X_A^i$ $\neq$ $X_A^f$.
As aforesaid [@Cheng1; @pipipuz], the nonfactorizable spectator scattering amplitudes contribute to a large complex $C^{\prime}$, which is important to resolve the ${\pi}K$, ${\pi}{\pi}$ puzzles. From the building block Eq.(\[hardblock\]), it can be easily seen that $B$ meson wave functions $\Phi_{B}(\xi)$ appear in the spectator scattering amplitudes. Therefore, the symmetry breaking effects should also be considered for the quantity $X_H$ that is introduced to parameterize the endpoint singularity in the twist-3 level spectator scattering corrections. Similar to $X_A^i$, the quantity $X_H$ is related to the topologies that gluon emit from the initial $B$ meson. So, for simplicity, the approximation $X_H$ $=$ $X_A^i$ is assumed in our coming numerical evaluation (scenarios I and II, see the next section for detail). Of course, this approximation is neither based on solid ground or from some underlying principle, and should be carefully studied and deserve much research. In fact, our coming phenomenological study (scenarios III) shows that the approximation $X_H$ $=$ $X_A^i$ is allowable with the up-to-date measurement on $B_{u,d}$ ${\to}$ $KK$, ${\pi}K$, ${\pi}{\pi}$ decays. In addition, it can be seen from Eq.(\[hardblock\]) that the spectator scattering corrections depend strongly on the inverse moment parameter ${\lambda}_{B}$ given in Eq.(\[lamdef\]). Recently, the value of ${\lambda}_{B}$ is an increasing concern of theoretical and experimental physicists [@Beneke5; @Beneke4; @Braun; @BaBarBA1; @BaBarBA2; @lambda]. A scrutiny of parameter ${\lambda}_{B}$ becomes imperative. In this paper, we will give some information on ${\lambda}_{B}$ required by present experimental data of $B_{u,d}$ ${\to}$ $K \bar{K}$, ${\pi}K$, ${\pi}{\pi}$ decays.
numerical analysis and discussions {#sec03}
==================================
With the conventions in Ref. [@Beneke2], the decay amplitudes for $B_{u,d}$ ${\to}$ $\pi K$, $K \bar{K}$, $\pi \pi $ decays within the QCDF framework can be written as $$\begin{aligned}
%%%%%%%%%%%%%%%%% B- -> pi- k0
{\cal A}_{ B^- \to \pi^- \bar{K}^0 }
&=&
\sum\limits_{p=u,c} V_{pb}V_{ps}^{\ast}
A_{ \pi K }
\Big\{ \alpha_{4}^{p} - \frac{1}{2} \alpha_{4,{\rm EW}}^{p}
+ {\delta}_{pu} \beta_{2} + \beta_{3}^{p} +
\beta_{3,{\rm EW}}^{p} \Big\}
\label{bm2pimkz}, \\
%%%%%%%%%%%%%%%%% B- -> piz k-
\sqrt{2} {\cal A}_{ B^- \to \pi^0 K^- }
&=&
\sum\limits_{p=u,c} V_{pb}V_{ps}^{\ast}
\Big\{ A_{ \pi K} \Big[ \delta_{pu} ( \alpha_1 + \beta_2 )
+ \alpha_4^p + \alpha_{4,{\rm EW}}^p + \beta_3^p
+ \beta_{3,{\rm EW}}^p \Big]
\nonumber \\
& & + A_{ K \pi } \Big[ \delta_{pu} \alpha_2
+ \frac{3}{2} \alpha_{3,{\rm EW}}^p \Big] \Big\}
\label{amp2}, \\
%%%%%%%%%%%%%%%% B0 -> pi+ k-
{\cal A}_{ \bar{B}^0 \to \pi^+ K^- }
&=& \sum\limits_{p=u,c} V_{pb}V_{ps}^{\ast}
A_{ \pi K} \Big\{ \delta_{pu} \alpha_1
+ \alpha_4^p + \alpha_{4,{\rm EW}}^p
+ \beta_3^p - \frac{1}{2} \beta_{3,{\rm EW}}^p
\Big\}
\label{amp3}, \\
%%%%%%%%%%%%%%%% B0 -> pi0 k0
\sqrt{2} {\cal A}_{ \bar{B}^0 \to \pi^0 \bar{K}^0 }
&=&
\sum\limits_{p=u,c} V_{pb}V_{ps}^{\ast}
\Big\{ A_{ \pi K} \Big[ - \alpha_4^p
+ \frac{1}{2} \alpha_{4,{\rm EW}}^p - \beta_3^p
+ \frac{1}{2} \beta_{3,{\rm EW}}^p \Big]
\nonumber \\
& & + A_{ K \pi } \Big[ \delta_{pu} \alpha_2
+ \frac{3}{2} \alpha_{3,{\rm EW}}^p \Big] \Big\}
\label{b02pi0k0}, \\
%%%%%%%%%%%%%%%% B- -> k- k0
{\cal A}_{ B^- \to K^0 \bar{K}^0 }
&=& \sum\limits_{p=u,c} V_{pb}V_{pd}^{\ast}
A_{ K K} \Big\{ \alpha_4^p
- \frac{1}{2} \alpha_{4,{\rm EW}}^p
+ \delta_{pu} \beta_2 + \beta_{3}^{p}
+ \beta_{3,{\rm EW}}^p \Big\}
\label{bm2kk}, \\
%%%%%%%%%%%%%%%% B0 -> k+ k-
{\cal A}_{ \bar{B}^0 \to K^- K^+}
&=& \sum\limits_{p=u,c} V_{pb}V_{pd}^{\ast}
\Big\{ B_{ \bar{K} K} \Big[ \delta_{pu} b_1 + b_4^p
+ b_{4,{\rm EW}}^p \Big] + B_{ K \bar{K} } \Big[
b_4^p - \frac{1}{2} b_{4,{\rm EW}}^p \Big] \Big\}
\label{amp4}, \\
%%%%%%%%%%%%%%%% B0 -> k0 k0
{\cal A}_{ \bar{B}^0 \to \bar{K}^0 K^0}
&=& \sum\limits_{p=u,c} V_{pb}V_{pd}^{\ast}
\Big\{ A_{ \bar{K} K} \Big[ \alpha_4^p
- \frac{1}{2} \alpha_{4,{\rm EW}}^p + \beta_3^p
+ \beta_4^p - \frac{1}{2} \beta_{3,{\rm EW}}^p
- \frac{1}{2} \beta_{4,{\rm EW}}^p \Big]
\nonumber \\ & &
+ B_{ K \bar{K} } \Big[
b_4^p - \frac{1}{2} b_{4,{\rm EW}}^p \Big] \Big\}
\label{b02kzkz}, \\
%%%%%%%%%%%%%%%% B- -> pi- pi0
\sqrt{2} {\cal A}_{ B^- \to \pi^- \pi^0 }
&=& \sum\limits_{p=u,c} V_{pb}V_{pd}^{\ast}
A_{ \pi \pi }
\Big\{ \delta_{pu} ( \alpha_1 + \alpha_2 )
+ \frac{3}{2} ( \alpha_{3,{\rm EW}}^p
+ \alpha_{4,{\rm EW}}^p) \Big\}
\label{amp5}, \\
%%%%%%%%%%%%%%%% B0 -> pi+ pi-
{\cal A}_{ \bar{B}^0 \to \pi^+ \pi^- }
&=& \sum\limits_{p=u,c} V_{pb}V_{pd}^{\ast}
A_{ \pi \pi }
\Big\{ \delta_{pu} ( \alpha_1 + \beta_1 )
+ \alpha_{4}^p + \alpha_{4,{\rm EW}}^p
+ \beta_3^p + 2 \beta_4^p
\nonumber \\ & &
- \frac{1}{2} \beta_{3,{\rm EW}}^p
+ \frac{1}{2} \beta_{4,{\rm EW}}^p \Big\}
\label{amp6}, \\
%%%%%%%%%%%%%%%% B0 -> pi0 pi0
-{\cal A}_{ \bar{B}^0 \to \pi^0 \pi^0 }
&=& \sum\limits_{p=u,c} V_{pb}V_{pd}^{\ast}
A_{ \pi \pi } \Big\{ \delta_{pu} ( \alpha_2 - \beta_1 )
- \alpha_{4}^p + \frac{3}{2}\alpha_{3,{\rm EW}}^p
+ \frac{1}{2} \alpha_{4,{\rm EW}}^p
\nonumber \\ & &
- \beta_3^p -2 \beta_4^p
+ \frac{1}{2} \beta_{3,{\rm EW}}^p
- \frac{1}{2} \beta_{4,{\rm EW}}^p) \Big\}
\label{amp7}.
\end{aligned}$$
For the sake for convenient discussion, we reiterate the expressions of the annihilation coefficients [@Beneke2], $$\begin{aligned}
{\beta}_{i}^{p} &=& b_{i}^{p} B_{M_{1}M_{2}}/A_{M_{1}M_{2}}
\label{betai}, \\
b_{1} &=&
\frac{C_{F}}{N_{c}^{2}}\, C_{1} A_{1}^{i},
\quad \quad \quad
b_{2} =
\frac{C_{F}}{N_{c}^{2}}\, C_{2} A_{1}^{i}
\label{b12}, \\
b_{3}^{p} &=&
\frac{C_{F}}{N_{c}^{2}}\,
\Big[ C_{3} A_{1}^{i} + C_{5}( A_{3}^{i} + A_{3}^{f} )
+N_{c} C_{6} A_{3}^{f} \Big]
\label{b3}, \\
b_{4}^{p} &=&
\frac{C_{F}}{N_{c}^{2}}\,
\Big[ C_{4} A_{1}^{i} + C_6 A_2^i \Big]
\label{b4}, \\
b_{3,\rm EW}^p &=&
\frac{C_{F}}{N_{c}^{2}}\,
\Big[ C_9 A_1^i + C_7 ( A_3^i + A_3^f )
+ N_c C_8 A_3^f \Big]
\label{b3ew}, \\
b_{4,\rm EW}^p &=&
\frac{C_{F}}{N_{c}^{2}}\,
\Big[ C_{10} A_1^i + C_8 A_2^i \Big]
\label{b4ew}.
\end{aligned}$$
Numerically, coefficients of $b_{3,\rm EW}^p$ and $b_{4,\rm EW}^p$ are negligible compared with the other effective coefficients due to the small electroweak Wilson coefficients, and so their effects would be not discussed in this paper.
In order to illustrate the contributions of annihilation and spectator scattering, we explore three parameter scenarios in which certain parameters are changed freely.
- Scenario I: $B_{u,d}$ ${\to}$ $\pi K$ and $K \bar{K}$ decays, including the $\pi K$ puzzle and pure annihilation decay $B_{d}$ $\to$ $K^- K^+$, are studied in detail. Combining the latest experimental data on the $CP$-averaged branching ratios, direct and mixing-induced $CP$-asymmetries, total 14 observables (see Table.\[pikbr\], \[pikdcp\], \[pikmcp\]) for seven $B_{u,d}$ ${\to}$ $\pi K$, $K \bar{K}$ decay modes \[see Eq.(\[bm2pimkz\]—\[b02kzkz\])\], the fit on four parameters ($\rho^f_A$, $\phi^f_A$) and ($\rho^i_A$, $\phi^i_A$) is performed with the fixed value $\lambda_B$ $=$ 0.2 GeV and the approximation ($\rho_H$, $\phi_H$) = ($\rho^i_A$, $\phi^i_A$), where ($\rho^f_A$, $\phi^f_A$), ($\rho^i_A$, $\phi^i_A$) and ($\rho_H$, $\phi_H$) are assumed to be universal for factorizable annihilation amplitudes, nonfactorizable annihilation amplitudes and spectator scattering corrections, respectively.
- Scenario II: $B_{u,d}$ ${\to}$ $\pi K$, $K \bar{K}$ and $\pi \pi$ decays, including $\pi \pi$ puzzle, are studied. Combining the latest experimental data on the $CP$-averaged branching ratios, direct and mixing-induced $CP$-asymmetries, total 21 observables (see Table.\[pikbr\], \[pikdcp\], \[pikmcp\]) for ten $B_{u,d}$ ${\to}$ $\pi K$, $K \bar{K}$, $\pi \pi$ decay modes \[see Eq.(\[bm2pimkz\]—\[amp7\])\], the fit on five parameters ($\rho^f_A$, $\phi^f_A$), ($\rho^i_A$, $\phi^i_A$) and $\lambda_B$ is performed with the approximation ($\rho_H$, $\phi_H$) = ($\rho^i_A$, $\phi^i_A$).
- Scenario III: As a general scenario, to clarify the relative strength among ($\rho^f_A$, $\phi^f_A$), ($\rho^i_A$, $\phi^i_A$) and ($\rho_H$, $\phi_H$), and check whether the approximation ($\rho_H$, $\phi_H$) = ($\rho^i_A$, $\phi^i_A$) is allowed or not, a fit on such six free parameters is performed.
Other input parameters used in our evaluation are summarized in Appendix \[app02\]. Our fit approach is illustrated in detail in Appendix \[app03\].
Scenario I {#sec0301}
----------
Comparing Eq.(\[amp2\]) with Eq.(\[amp3\]), it can be clearly seen that $\sqrt{2} {\cal A}_{ B^- \to \pi^0 K^-}$ $\simeq$ ${\cal A}_{ \bar{B}^0 \to \pi^+ K^-}$ if $\delta_{pu} \alpha_2$ $+$ $\frac{3}{2} \alpha_{3,{\rm EW}}^p$ is negligible compared with $\delta_{pu} \alpha_1$ $+$ $\alpha_4^p$. Hence it is expected $\Delta A$ $\simeq$ 0 in SM, which significantly disagrees with the current experimental data in Eq.(\[acppi\]), this is the so-called $\pi K$ puzzle. To resolve the $\pi K$ puzzle, one possible solution is that there is a large complex contributions from $\delta_{pu} \alpha_2$ $+$ $\frac{3}{2} \alpha_{3,{\rm EW}}^p$. Many proposals have been offered, such as the enhancement of color-suppressed tree amplitude $\alpha_2$ in Ref.[@Cheng1], significant new physics corrections to the electroweak penguin coefficient $\alpha_{3,{\rm EW}}^p$ in Ref.[@pipipuz], and so on. Indeed, it has been shown [@Beneke2] that the coefficients $\alpha_2$ and $\alpha_{3,{\rm EW}}^p$ are seriously affected by spectator scattering corrections within QCDF framework. Consequently, the nonfactorizable spectator scattering parameters $X_H$ or ($\rho_H$, $\phi_H$) will have great influence on the observable $\Delta A$. Furthermore, a scrutiny of difference between Eq.(\[amp2\]) and Eq.(\[amp3\]), another possible resolution to the $\pi K$ puzzle might be provided by annihilation contributions, such as coefficient $\beta_2$, as suggested in Ref.[@zhu2]. If so, then $\Delta A$ will depend strongly on the nonfactorizable annihilation parameters ($\rho_A^i$, $\phi_A^i$) because $\beta_2$ is proportional to $A_1^i$ in Eq.(\[b12\]). Additionally, it can be seen from Eq.(\[amp2\]) and Eq.(\[amp3\]) that annihilation coefficient $\beta_3^p$ contributes to amplitudes both ${\cal A}_{ B^- \to \pi^0 K^-}$ and ${\cal A}_{ \bar{B}^0 \to \pi^+ K^-}$. If $\beta_3^p$ could offer a large strong phase, then its effect should contribute to the direct $CP$ asymmetries $A_{CP}(B^- \to \pi^0 K^-)$ and $A_{CP}(\bar{B}^0 \to \pi^+ K^-)$ rather than $\Delta A$. Due to the fact that the lion’s share of $\beta_3^p$ comes from $N_{c} C_6 A_3^f$ in Eq.(\[b3\]), the direct $CP$ asymmetries $A_{CP}(B^- \to \pi^0 K^-)$ and $A_{CP}(\bar{B}^0 \to \pi^+ K^-)$ should vary greatly with the factorizable annihilation parameters $X_A^f$, while $\Delta A$ should be insensitive to variation of parameters ($\rho_A^f$, $\phi_A^f$). The above analysis and speculations are confirmed by Fig.\[cpanni\].
From Eq.(\[amp4\]), it is seen that the amplitude ${\cal A}_{ \bar{B}^0 \to K^- K^+}$ depends heavily on coefficients $\beta_1$ and $\beta_4^p$, which are closely associated with the nonfactorizable annihilation parameter $X_A^i$ only. The factorizable annihilation contributions vanish due to the isospin symmetry, which is consistent with the pQCD calculation [@xiao1]. The large branching ratio Eq.(\[HFAGKK\]) would appeal for large nonfactorizable annihilation parameter $X_A^i$ or $\rho_A^i$. The dependence of branching ratio ${\cal B}(\bar{B}^0 \to K^- K^+)$ on the parameters ($\rho_A^i$, $\phi_A^i$) is displayed in Fig.\[branni\].
![The dependence of branching ratio ${\cal B}(\bar{B}^0 \to K^- K^+)$ on nonfactorizable annihilation parameters ($\rho_A^i$, $\phi_A^{i}$). The notes are the same as Fig.\[cpanni\].[]{data-label="branni"}](kkphi.pdf){width="40.00000%"}
$\rho_H$ $=$ $\rho_A^i$ $\phi_H$ $=$ $\phi_A^i\,[^{\circ}]$ $\rho_A^f$ $\phi_A^f[^{\circ}]$
-------- ------------------------- ------------------------------------- ------------------------ ----------------------
Part A $2.82^{+2.73}_{-1.15}$ $-108^{+44}_{-50}$ $1.07^{+0.30}_{-0.20}$ $-40^{+10}_{-11}$
Part B $2.86^{+2.68}_{-1.20}$ $-108^{+42}_{-51}$ $2.72^{+0.30}_{-0.22}$ $166^{+3}_{-4}$
: Numerical results of annihilation parameters in scenario I.[]{data-label="pikfit"}
Exp. [@HFAG] scenario I scenario II S4 [@Beneke2]
--------------------------------- ------------------------- ----------------------------------- ----------------------------------- ---------------
$B^- \to \pi^- \bar{K}^0$ $23.79 \pm 0.75$ $20.53^{+1.52+4.28}_{-0.65-3.87}$ $21.54^{+1.60+4.40}_{-0.68-3.99}$ $20.3$
$B^- \to \pi^0 K^-$ $12.94^{+0.52}_{-0.51}$ $11.29^{+0.88+2.14}_{-0.45-1.96}$ $11.78^{+0.92+2.20}_{-0.47-2.01}$ $11.7$
$\bar{B}^0 \to \pi^+ K^-$ $19.57^{+0.53}_{-0.52}$ $17.54^{+1.34+3.61}_{-0.65-3.27}$ $18.51^{+1.41+3.73}_{-0.67-3.38}$ $18.4$
$\bar{B}^0 \to \pi^0 \bar{K}^0$ $9.93 \pm 0.49$ $8.05^{+0.60+1.84}_{-0.27-1.65}$ $8.60^{+0.65+1.90}_{-0.29-1.72}$ $8.0$
$B^- \to K^- K^0$ $1.19 \pm 0.18$ $1.45^{+0.13+0.32}_{-0.09-0.29}$ $1.51^{+0.13+0.32}_{-0.09-0.29}$ $1.46$
$\bar{B}^0 \to K^- K^+$ $0.12 \pm 0.05$ $0.13^{+0.01+0.02}_{-0.01-0.02}$ $0.15^{+0.02+0.02}_{-0.01-0.02}$ $0.07$
$\bar{B}^0 \to K^0 \bar{K}^0$ $1.21 \pm 0.16$ $1.22^{+0.11+0.27}_{-0.08-0.24}$ $1.32^{+0.12+0.27}_{-0.08-0.25}$ $1.58$
$B^- \to \pi^- \pi^0$ $5.48^{+0.35}_{-0.34}$ $5.20^{+0.64+1.11}_{-0.47-1.00}$ $5.59^{+0.68+1.15}_{-0.51-1.04}$ $5.1$
$\bar{B}^0 \to \pi^+ \pi^-$ $5.10 \pm 0.19$ $5.88^{+0.66+1.66}_{-0.49-1.45}$ $5.74^{+0.64+1.63}_{-0.47-1.42}$ $5.2$
$\bar{B}^0 \to \pi^0 \pi^0$ $1.91^{+0.22}_{-0.23}$ $1.67^{+0.22+0.25}_{-0.19-0.23}$ $2.13^{+0.29+0.32}_{-0.24-0.29}$ $0.7$
$R_{+-}^{\pi \pi}$ $1.99 \pm 0.15$ $1.64^{+0.06+0.13}_{-0.06-0.11}$ $1.80^{+0.07+0.17}_{-0.07-0.13}$ $1.82$
$R_{00}^{\pi \pi}$ $0.75 \pm 0.09$ $0.57^{+0.06+0.16}_{-0.06-0.12}$ $0.74^{+0.08+0.22}_{-0.08-0.17}$ $0.27$
: The CP-averaged branching ratios (in units of $10^{-6}$) of $B$ ${\to}$ $\pi K$, $K \bar{K}$, $\pi \pi$ decays. For the Part A results of scenario I and II, the first and second theoretical uncertainties are caused by the CKM and other input parameters, respectively.[]{data-label="pikbr"}
Exp. [@HFAG] scenario I scenario II S4 [@Beneke2]
--------------------------------- ---------------- ----------------------------------- ----------------------------------- ---------------
$B^- \to \pi^- \bar{K}^0$ $-1.5 \pm 1.9$ $-0.05^{+0.00+0.13}_{-0.00-0.15}$ $-0.17^{+0.01+0.14}_{-0.01-0.15}$ $0.3$
$B^- \to \pi^0 K^-$ $4.0 \pm 2.1$ $3.2^{+0.2+0.6}_{-0.2-0.6}$ $2.5^{+0.1+0.6}_{-0.1-0.6}$ $-3.6$
$\bar{B}^0 \to \pi^+ K^-$ $-8.2 \pm 0.6$ $-7.7^{+0.4+0.9}_{-0.4-0.9}$ $-9.1^{+0.4+0.9}_{-0.5-0.9}$ $-4.1$
$\bar{B}^0 \to \pi^0 \bar{K}^0$ $-1 \pm 10$ $-10.3^{+0.6+0.9}_{-0.6-1.0}$ $-10.6^{+0.6+0.9}_{-0.6-0.9}$ $0.8$
$\Delta A$ $12.2 \pm 2.2$ $10.9^{+0.6+0.9}_{-0.5-0.8}$ $11.6^{+0.6+0.9}_{-0.6-0.8}$ $0.5$
$B^- \to K^- K^0$ $3.9 \pm 14.1$ $-0.6^{+0.0+3.2}_{-0.0-2.9}$ $2.0^{+0.1+3.4}_{-0.1-3.0}$ $-4.3$
$\bar{B}^0 \to K^0 \bar{K}^0$ $-6 \pm 26$ $-17^{+1+2}_{-1-2}$ $-16^{+1+2}_{-1-2}$ $-11.5$
$B^- \to \pi^- \pi^0$ $2.6 \pm 3.9$ $-1.1^{+0.1+0.1}_{-0.1-0.1}$ $-1.2^{+0.1+0.1}_{-0.1-0.1}$ $-0.02$
$\bar{B}^0 \to \pi^+ \pi^-$ $29 \pm 5$ $19^{+1+4}_{-1-4}$ $24^{+2+5}_{-2-4}$ $10.3$
$\bar{B}^0 \to \pi^0 \pi^0$ $43 \pm 24$ $46^{+3+6}_{+3-6}$ $38^{+2+6}_{-2-6}$ $-19.0$
: The direct CP asymmetries (in units of $10^{-2}$) of $B$ ${\to}$ $\pi K$, $K \bar{K}$, $\pi \pi$ decays. The notes on uncertainties are the same as Table\[pikbr\].[]{data-label="pikdcp"}
Exp. [@HFAG] scenario I scenario II
--------------------------------- --------------- ----------------------- ----------------------- --
$\bar{B}^0 \to \pi^0 \bar{K}^0$ $57 \pm 17$ $78^{+3+1}_{-3-1}$ $79^{+3+1}_{-3-1}$
$\bar{B}^0 \to K^- K^+$ — $-86^{+6+0}_{-5-0}$ $-86^{+6+0}_{-5-0}$
$\bar{B}^0 \to K^0 \bar{K}^0$ $-108 \pm 49$ $-10^{+1+0}_{-1-0}$ $-11^{+1+0}_{-1-0}$
$\bar{B}^0 \to \pi^+ \pi^-$ $-65 \pm 6$ $-59^{+11+2}_{-10-3}$ $-60^{+10+2}_{-10-2}$
$\bar{B}^0 \to \pi^0 \pi^0$ — $77^{+6+1}_{-8-2}$ $77^{+7+1}_{-9-2}$
: The mixing-induced $CP$ asymmetries (in units of $10^{-2}$) of $B$ ${\to}$ $\pi K$, $K \bar{K}$, $\pi \pi$ decays. The notes on uncertainties are the same as Table\[pikbr\].[]{data-label="pikmcp"}
To get more information on annihilation and spectator scattering, we perform a fit on the parameters $X_H$ $=$ $X_A^i$ and $X_A^f$, considering the constraints of the $CP$-averaged branching ratios, direct and mixing-induced $CP$-asymmetries, from $B$ ${\to}$ $\pi K$, $K \bar{K}$ decays. The experimental data are summarized in the second column of Tables \[pikbr\]-\[pikmcp\]. Our fitting results are shown by Fig.\[ParaSpacI\], and the corresponding numerical results are listed in Table \[pikfit\]-\[pikmcp\].
It is found that two possible solutions entitled Part A and B in Table \[pikfit\], correspond to almost the same $(\rho_A^i, \phi_A^i)$ $\approx$ $(2.8,-108^{\circ})$. The large errors on parameter $(\rho_A^i, \phi_A^i)$ are mainly caused by the current loose experimental constraints on $CP$ asymmetries measurements for $B$ ${\to}$ ${\pi}K$, $K \bar{K}$ decays. In principle, the pure annihilation $\bar{B}^0 \to K^- K^+$ decays whose amplitudes depend predominantly on $(\rho_A^i, \phi_A^i)$, besides the decays constants, should give rigorous constraint on $X_A^i$. It’s a pity that the available measurement accuracy on its branching ratio is too poor to efficiently confine $(\rho_A^i, \phi_A^i)$ to some tiny spaces. The large $(\rho_A^i,\phi_A^i)$ mean large $X_A^i$ and $X_H$, i.e., there must exist large nonfactorizable annihilation and spectator scattering contributions to accommodate the current measurements. Our fit results on parameter $\rho_A^i$ provide a robust evidence to the educated guesswrok about $\rho_{Ad}^i$ $=$ 2.5 in Ref.[@zhu2]. In fact, the strong phase $\phi_A^i$ educed from measurements of branching ratios for $B^0$ $\to$ $K \bar{K}$ decays in Ref.[@zhu2] can have either positive or negative values with the magnitudes of $\gtrsim$ $100^{\circ}$ (see Fig.5 of Ref.[@zhu2]), where the positive value $\phi_A^i$ $=$ $+100^{\circ}$ used in Ref.[@zhu2] will be excluded by our fit with much more experimental data on $B$ ${\to}$ $\pi K$, $K \bar{K}$ decays. The large value of $\phi_A^i$, corresponding to a large imaginary part of the enhanced complex corrections, also lends some support to the pQCD claim that the annihilation amplitudes can provide a large strong phase [@pqcd].
There are two possible solutions for the factorizable annihilation parameters, namely, Part A $(\rho_A^f,\phi_A^f)$ $\approx$ $(1.1,-40^{\circ})$ and Part B $(\rho_A^f,\phi_A^f)$ $\approx$ $(2.7,166^{\circ})$. From Fig.\[ParaSpacI\], it can be seen that there is no overlap between the regions of $(\rho_A^f,\phi_A^f)$ and $(\rho_A^i,\phi_A^i)$ at the 95% confidence level, which indicates that it might be wrong to treat $(\rho_A^f,\phi_A^f)$ $=$ $(\rho_A^i,\phi_A^i)$ $=$ $(\rho_A,\phi_A)$ as universal parameters for nonfactorizable and factorizable annihilation topologies in pervious studies. Our fit results certify the suggestion of Ref.[@zhu1; @zhu2] that different annihilation topologies should be parameterized by different annihilation parameters, i.e., $(\rho_A^f,\phi_A^f)$ $\neq$ $(\rho_A^i,\phi_A^i)$. Compared with the results of $(\rho_A^i,\phi_A^i)$, the errors on parameter $(\rho_A^f, \phi_A^f)$ are relatively small (see Table \[pikfit\]), because the available measurements on branching ratios for $B$ ${\to}$ ${\pi}K$ decays are highly precise. The conjecture about $(\rho_A^f, \phi_A^f)$ in [@zhu2] is somewhat alike to our fit results of Part A.
The value of term $(2X_A^f-X_A^f)$ in Eq.(\[af3\]) is about $(27.2-i26.2)$ with parameters for Part A and $(28.9-i25.5)$ for Part B, that is to say, these two solutions, Part A and B, will present similar factorizable annihilation contributions. Nevertheless, a small value of $\rho_A^f$ is more easily accepted by the QCDF approach [@Beneke2]. So with the best fit parameters of Part A in Table \[pikfit\], we present our evaluations on branching ratios, direct and mixing-induced $CP$ asymmetries for $B_{u,d}$ ${\to}$ $\pi K$, $K \bar{K}$, $\pi \pi$ decays in the “scenario I” column of Table \[pikbr\], \[pikdcp\] and \[pikmcp\], respectively. For comparison, the results of scenario S4 QCDF [@Beneke2] are also collected in the “S4” column. It is easily found that all theoretical results are in good agreement with experimental data within errors. Especially, the difference $\Delta A$, which $\sim$ 0.5% in scenario S4 QCDF, is enhanced to the experimental level $\sim$ 11%. It is interesting that although $B$ $\to$ $\pi \pi$ decays are not considered in the “scenario I” fit, all predictions on these decays, including the ratios $R_{+-}^{\pi\pi}$ and $R_{00}^{\pi\pi}$, are also in good consistence with the experimental measurements within errors, which implies that the $\pi K$ and $\pi \pi$ puzzles could be resolved by annihilation and spectator corrections, at the same time, without violating the agreement of other observables. The reason will be excavated in Scenario II.
Scenario II {#sec0302}
-----------
From Eq.(\[amp5\]), it is obviously found that the amplitude of $B^-\to\pi^-\pi^0$ decay is independent of annihilation contributions, and dominated by $\alpha_1$ $+$ $\alpha_2$. Moreover, comparing Eq.(\[amp6\]) with Eq.(\[amp7\]), it is easily found that the annihilation contributions are almost helpless for $R_{00}^{\pi \pi}$ puzzle due to ${\cal A}_{B^0 \to \pi^+ \pi^-}^{\rm anni}$ $\simeq$ ${\cal A}_{B^0 \to \pi^0 \pi^0}^{\rm anni}$. So, the spectator scattering corrections, which play an important role in the color-suppressed coefficient $\alpha_2$ [@Beneke2; @Cheng1; @Cheng3], would be another important key for the good results of scenario I, especially for $B$ $\to$ $\pi\pi$ decays.
Within QCDF framework, besides $X_H$, the inverse moment $\lambda_B$ of $B$ wave function defined by Eq.(\[lamdef\]) is another important quantity in evaluating the contributions of spectator scattering. Unfortunately, its value is hardly to be obtained reliably with theoretical methods until now, for instance $350{\pm}150$ MeV (200 MeV in scenario S2) in Ref.[@Beneke2], $200^{+250}_{-0}$ MeV in Ref.[@Beneke4] and $300{\pm}100$ MeV in Ref.[@Cheng1], though QCD sum rule prefer $460{\pm}110$ MeV at the scale of 1 GeV [@Braun]. Experimentally, the upper limit on parameter $\lambda_B$ are set at the 90% C.L. via measurements on branching fraction of radiative leptonic $B$ $\to$ $\ell \bar{\nu}_{\ell} \gamma $ decay by BABAR collaboration, $\lambda_B$ $>$ 669 (591) MeV with different priors based on 232 million $B\bar{B}$ sample where the photon is not required to be sufficiently energetic in order not to sacrifice statistics [@BaBarBA1], and $\lambda_B$ $>$ 300 MeV based on 465 million $B\bar{B}$ pairs [@BaBarBA2]. Considering radiative and power corrections, an improved analysis is preformed in Ref.[@Beneke5] with the conclusion that present BABAR measurements cannot put significant constrains on $\lambda_B$ and that $\lambda_B$ $>$ 115 MeV from the experimental results [@BaBarBA2]. Anyway, the study of hadronic $B$ decays favors a relative small value of $\lambda_B$ $\approx$ 200 MeV to achieve a satisfactory description of color-suppressed tree decay modes [@lambda]. At the present time, the value of $\lambda_B$ is still a point of controversy. In the following analysis and evaluations, we treat $\lambda_B$ as a free parameter.
![The dependance of the direct $CP$ asymmetries $A_{CP}(B^- \to \pi^0 K^-)$, $A_{CP}(\bar{B}^0 \to \pi^+ K^-)$ and their difference $\Delta A$ on $\lambda_B$ (in unites of GeV) with the fitted annihilation parameters of scenario I (Part A). Their experimental results with $1\sigma$ error are shown by shaded bands with the same color as the lines.[]{data-label="LBpik"}](pikLB.pdf){width="30.00000%"}
\
$\rho_A^i$ $\phi_A^i[^{\circ}]$ $\rho_A^f$ $\phi_A^f[^{\circ}]$ $\lambda_B$ \[GeV\]
-------- ------------------------ ---------------------- ------------------------ ---------------------- ------------------------
Part A $2.88^{+1.52}_{-1.30}$ $-103^{+33}_{-40}$ $1.21^{+0.22}_{-0.25}$ $-40^{+12}_{-8}$ $0.18^{+0.11}_{-0.08}$
Part B $2.98^{+1.50}_{-1.40}$ $-106^{+35}_{-39}$ $2.78^{+0.29}_{-0.18}$ $165^{+4}_{-3}$ $0.19^{+0.09}_{-0.10}$
: Numerical results of annihilation parameters and moment parameter $\lambda_B$ in Scenario II.[]{data-label="pipikfit"}
To explicitly show the effects of spectator scattering contributions on $\pi K$ puzzle, dependance of $A_{CP}(B^- \to \pi^0 K^-)$, $A_{CP}(\bar{B}^0 \to \pi^+ K^-)$ and their difference $\Delta A$ on parameter $\lambda_B$ are displayed in Fig.\[LBpik\]. It is found that (1) observables of $A_{CP}(B^- \to \pi^0 K^-)$ and $\Delta A$ are more sensitive to variation of $\lambda_B$ than $A_{CP}(\bar{B}^0 \to \pi^+ K^-)$ in the region of $\lambda_B$ $\geq$ 100 MeV. The reason is aforementioned fact that coefficient $\alpha_2$ in amplitude ${\cal A}_{ B^- \to \pi^0 K^-}$ \[see Eq.(\[amp2\])\] receives significant spectator scattering corrections. A noticeable change of observables is easily seen in the low region of $\lambda_B$ because spectator scattering corrections are inversely proportional to $\lambda_B$ \[see Eq.(\[lamdef\]) and Eq.(\[hardblock\])\]. (2) a relative small value of $\lambda_B$ $\in$ \[150 MeV, 220 MeV\], as expected in [@lambda], is required to confront with available measurements. Especially, the value $\lambda_B$ $\approx$ 190 MeV provides a perfect description of the experimental data on $A_{CP}(B^- \to \pi^0 K^-)$, $A_{CP}(\bar{B}^0 \to \pi^+ K^-)$ and $\Delta A$ simultaneously. For $B$ $\to$ $\pi \pi$ decays, from Eqs.(\[amp5\]-\[amp7\]), it is easily seen that amplitude ${\cal A}_{ B^- \to \pi^- \pi^0 }$ $\propto$ $\alpha_1$ + $\alpha_2$, ${\cal A}_{ \bar{B}^0 \to \pi^+ \pi^- }$ $\propto$ $\alpha_1$, ${\cal A}_{ \bar{B}^0 \to \pi^0 \pi^0 }$ $\propto$ $\alpha_2$. The coefficient $\alpha_2$, corresponding to the color-suppressed tree contribution, its value is small relative to $\alpha_1$, so the experimental data on $R_{+-}^{\pi \pi}$ can be well explained with scenario S4 QCDF where $X_A^i$ $=$ $X_A^f$ and $\rho_A^{f,i}$ = 1 (see Table \[pikbr\]). But as to observable $R_{00}^{\pi \pi}$ or/and branching ratio ${\cal B}(\bar{B}^0 \to \pi^0 \pi^0)$, an enhanced $\alpha_2$ is desirable. Hence, the nonfactorizable spectator scattering contributions, which have significant effects on $\alpha_2$, would play an important role in studying the color-suppressed tree $B$ decays, and possibly provide a solution to the $\pi \pi$ puzzle. The dependencies of the branching fractions of $B$ $\to$ $\pi\pi$ decays and ratios $R_{+-}^{\pi\pi}$, $R_{00}^{\pi\pi}$ on $\lambda_B$ are shown in Fig.\[LBpipi\] where the fitted parameters of Part A in Table \[pikfit\] is used. It is interesting that beside a large value $\rho_H$, a small value of $\lambda_B$ $\sim$ 200 MeV is also required to confront with experimental data on ${\cal B}(B \to \pi \pi)$, $R_{+-}^{\pi\pi}$ and $R_{00}^{\pi\pi}$.
With the available experimental data on $B$ $\to$ $\pi \pi$, $\pi K$ and $K \bar{K}$ decays, we perform a comprehensive fit on both annihilation parameters ($\rho_{A}^{i,f}$, $\phi_{A}^{i,f}$) and $B$-meson wave function parameter $\lambda_B$. The allowed parameter spaces are shown in Fig.\[ParaSpacII\], and the corresponding numerical results are summarized in Table \[pipikfit\]. Like scenario I, there are two allowed spaces which are labelled by part A and B. It is easily found that (1) parameters $(\rho_A^i,\phi_A^i)$ $=$ $(\rho_H,\phi_H)$ are still required to have large values (see Table \[pipikfit\]), that is to say, it is necessary for penguin-dominated or color-suppressed tree $B$ decays to own large corrections from nonfactorizable annihilation and spectator scattering topologies. (2) There is still no overlap between the regions of $(\rho_A^f,\phi_A^f)$ and $(\rho_A^i,\phi_A^i)$ at the 95% confidence level. (3) The cental values of $\rho_A^{i,f}$ are a little larger than those in scenario I. The uncertainties on $(\rho_A^i,\phi_A^i)$ are a little smaller than those in scenario I, because more processes from $B$ $\to$ $\pi \pi$ decays are considered in fitting and the amplitudes for $B$ $\to$ $\pi \pi$ decays are sensitive to $X_A^i$ and $X_H$ rather than $X_A^f$. (4) A small value of parameter $\lambda_B$ $\leq$ 350 MeV at the 95% confidence level is strongly required to reconcile discrepancies between results of QCDF approach and available experimental data on $B$ $\to$ $\pi \pi$, $\pi K$ and $K \bar{K}$ decays.
The two solutions of scenario II, Part A and B, will give similar results, as discussed before. With the best fit parameters of Part A in Table \[pipikfit\], we present our evaluations on branching ratios, direct and mixing-induced $CP$ asymmetries for $B_{u,d}$ ${\to}$ $\pi K$, $K \bar{K}$, $\pi \pi$ decays in the “scenario II” column of Table \[pikbr\], \[pikdcp\] and \[pikmcp\], respectively. It is found that the central values of branching ratios for $B$ $\to$ $\pi \pi$, $\pi K$ and $K \bar{K}$ decays, expect $\bar{B}^0$ $\to$ $\pi^+ \pi^-$ decay, with the Part A parameters of scenario II, are a little larger than those of scenario I (see Table \[pikbr\]), because a bit larger values of $\rho_A^{i,f}$ and a bit smaller value of $\lambda_B$ than those of scenario I are taken in scenario II. Compared with results of scenario S4 QCDF, agreement between theoretical results within two scenarios and experimental measurements is improved, especially for the observables $\Delta A$, $R_{00}^{\pi \pi}$ and $A_{CP}(B^0 \to \pi \pi)$.
Scenario III {#sec0303}
------------
The above analyses and results are based on the assumption that $X_A^{i}$ $=$ $X_{H}$ (i.e. $(\rho_A^{i}, \phi_A^{i})$ $=$ $(\rho_H, \phi_H)$) for simplicity. While, there is no compellent requirement for such simplification, except for the fact that wave functions of $B$ mesons are involved in the convolution integrals of both spectator scattering and nonfactorizable annihilation corrections, but are irrelevant to the factorable annihilation amplitudes. So, as a general scenario (named scenario III), we would reevaluate the strength of annihilation and hard-spectator contributions without any simplification for the parameters $(\rho_A^{i}, \phi_A^{i})$, $(\rho_A^{i}, \phi_A^{i})$ and $(\rho_H, \phi_H)$.
![The allowed regions of annihilation and hard-spectator parameters ($\rho_A^{f}$, $\phi_A^{f}$), ($\rho_A^{i}$, $\phi_A^{i}$) and ($\rho_H$, $\phi_H$) at $68\%$ C.L.. The two solutions of ($\rho_A^{f}$, $\phi_A^{f}$) and ($\rho_A^{i}$, $\phi_A^{i}$) are labeled as Part A, B and $\rm A^{\prime}$, $\rm B^{\prime}$, respectively.[]{data-label="SpecFig"}](SpecFig.pdf){width="35.00000%"}
Considering the constraints from observables of $B_{u,d}$ ${\to}$ $K \bar{K}$, ${\pi}K$ and ${\pi}{\pi}$ decays, a fit for the annihilation and hard-spectator parameters is performed again. In this fit, $(\rho_A^{f}, \phi_A^{f})$, $(\rho_A^{i}, \phi_A^{i})$ and $(\rho_H, \phi_H)$ are treated as six free parameters. Moreover, from the hard-spectator corrections illustrated by Eq. (\[hardblock\]), it can be seen that $\lambda_B$ and $X_H$ are always combined together.
Although the inverse moment $\lambda_B$ of $B$ wave function could be determined or constricted by further experiments [@Beneke5; @BaBarBA1; @BaBarBA2; @lambda], $\lambda_B$ is more like a free parameter for the moment due to loose limitation on it. So it is impossible to strictly bound on $\lambda_B$ and $X_H$ simultaneously due to the interference effects between them. In our following fit, we will fix $\lambda_B=200\,{\rm MeV}$. Our fitting results at $68\%$ C.L. are presented in Fig. \[SpecFig\], where the range of ${\phi}$ ${\in}$ $[-360^{\circ},0^{\circ}]$ is assigned to illustrate their relative magnitude. Numerically, we get $$\begin{aligned}
&&(\rho_A^{f}, \phi_A^{f}[^{\circ}])=\left\{ \begin{array}{ll}
& (1.18^{+0.26}_{-0.23}, -40^{+12}_{-8})\qquad \text{Part A } \\
& (2.79^{+0.26}_{-0.20}, -196^{+5}_{-3})\qquad \text{Part B }
\end{array} \right. \\
&&(\rho_A^{i}, \phi_A^{i}[^{\circ}])=\left\{ \begin{array}{ll}
& (2.85^{+2.18}_{-1.92}, -103^{+52}_{-63})\qquad \text{Part ${\rm A}^{\prime}$ } \\
& (6.54^{+1.81}_{-3.30}, -206^{+23}_{-24})\qquad \text{Part ${\rm B}^{\prime}$ }
\end{array} \right. \\
&& (\rho_{H}, \phi_{H}[^{\circ}])=(3.09^{+1.64}_{-1.53}, -102^{+40}_{-31})\,.
\label{soluSIII}
\end{aligned}$$
It can be easily seen from Fig. \[SpecFig\] that: (1) for factorizable annihilation parameters $(\rho_A^{f}, \phi_A^{f})$, similar to scenarios I and II, there are two allowed regions (labelled by part A and B); (2) for nonfactorizable annihilation parameters $(\rho_A^{i}, \phi_A^{i})$, besides the solution similar to scenarios I and II (labelled by part ${\rm A}^{\prime}$), another solution (labelled by part ${\rm B}^{\prime}$) with a very large value of $\rho_A^{i}$ is gotten. (3) It is very intersting that the allowed space of $(\rho_{H}, \phi_{H})$ overlaps almost entirely with the “part ${\rm A}^{\prime}$” allowed space of $(\rho_A^{i}, \phi_A^{i})$. Moreover, their best-fit points $(\rho_A^{i}, \phi_A^{i})$ $=$ $(2.85, -103^{\circ})$ of “part ${\rm A}^{\prime}$” and $(\rho_{H}, \phi_{H})$ $=$ $(3.09, -102^{\circ})$ are very close to each other. It might imply that the assumption $X_A^{i}$ $(\rho_A^{i}, \phi_A^{i})$ $=$ $X_H$ $(\rho_H, \phi_H)$ used in scenarios I and II is a good simplification.
With the best fit parameters in scenarios III, either the small value of $\rho_A^{i}$ in “part ${\rm A}^{\prime}$” or the large value in “part ${\rm B}^{\prime}$”, our evaluations on branching ratios, direct and mixing-induced $CP$ asymmetries for $B_{u,d}$ ${\to}$ $\pi K$, $K \bar{K}$, $\pi \pi$ decays are similar to those given in our scenarios I and II, so no longer listed here. For the two solutions ${\rm A}^{\prime}$ and ${\rm B}^{\prime}$ of $(\rho_A^{i}, \phi_A^{i})$, it is expected by QCDF approach [@Beneke2] that the parameter $\rho_A^{i}$ should have a small value, which is also favored by our scenarios I and II fit. In fact, such two solutions lead to the same results of $A^{i}_{1,2}$, but the different ones of $A^{i}_{3}$, which principally provides an opportunity to refute one of them. However, because $A^{i}_{3}$ is numerically trivial due to $(r_\chi^{M_1}-r_\chi^{M_2}) \sim 0$ for the light mesons, such way is practically unfeasible for current accuracies of theoretical calculation and experimentally measurement.
Conclusions {#sec04}
===========
The recent CDF and LHCb measurements of large branching ratios for pure annihilation $\bar{B}_s^0$ $\to$ $\pi^+ \pi^-$ and $\bar{B}_d^0$ $\to$ $K^+ K^-$ decays imply possible large annihilation contributions, which induce us to modify the traditional QCDF treatment for annihilation parameters. Following the suggestion of Ref.[@zhu2], two sets of annihilation parameters $X_A^i$ and $X_A^f$ are used to parameterize the endpoint singularity in nonfactorizable and factorizable annihilation amplitudes, respectively. Besides annihilation effects, the resolution of so-called ${\pi}K$ and ${\pi}{\pi}$ puzzles also expect constructive contributions from spectator scattering topologies. With the approximation of $X_A^i$ $=$ $X_H$, we perform a global fit on both annihilation parameters ($\rho_{A}^{i,f}$, $\phi_{A}^{i,f}$) and $B$-meson wave function parameter $\lambda_B$ based on available experimental data for $B$ $\to$ $\pi \pi$, $\pi K$ and $K \bar{K}$ decays. Our main conclusions and findings are summarized as:
- The $95\%$ C.L. allowed region of $(\rho_A^i,\phi_A^i)$ is entirely different from that of $(\rho_A^f,\phi_A^f)$. This fact means that the traditional QCDF treatment $(\rho_A,\phi_A)$ as universal parameters for different annihilation topologies might be unapplicable to hadronic $B$ decays.
- The current experimental data on $B$ $\to$ $\pi \pi$, $\pi K$ and $K \bar{K}$ decays seems to favor a large value of $\rho_A^i$ $\sim$ 2.9, which corresponds to a sizable nonfactorizable annihilation contributions. But the range of $(\rho_A^i,\phi_A^i)$ is still very large, because the measurement precision of $CP$ asymmetries is low now.
- There are two possible choices for parameters $(\rho_A^f,\phi_A^f)$. One is $(\rho_A^f,\phi_A^f)$ $\sim$ $(1.1,-40^{\circ})$, the other is $(\rho_A^f,\phi_A^f)$ $\sim$ $(2.7,165^{\circ})$. These two choices correspond to similar factorizable annihilation contributions, although the QCDF approach tends to have a small value of $\rho_A^f$ [@Beneke2]. The space for $(\rho_A^f,\phi_A^f)$ is relatively tight due to the well measured branching ratios for $B$ $\to$ $\pi \pi$, $\pi K$ and $K \bar{K}$ decays.
- The spectator scattering corrections play an important role in resolving both $\pi K$ and $\pi \pi$ puzzles. Within QCDF approach, the spectator scattering amplitudes depend on parameters $(\rho_H,\phi_H)$ and $B$-meson wave function parameter $\lambda_B$. In our analysis, the approximation $(\rho_H,\phi_H)$ $=$ $(\rho_A^i,\phi_A^i)$ is assumed, which is proven to be a good simplification by a global fit in scenario III. A small value of $\lambda_B$ $\leq$ 350 MeV at the 95% C.L. is obtained by the global fit on $B$ $\to$ $\pi \pi$, $\pi K$ and $K \bar{K}$ decays, which needs to be further tested by future improved measurement on $B$ $\to$ $\ell \nu_\ell \gamma$ decays. An enhanced color-suppressed tree coefficient $\alpha_2$, which is supported by both large value of $\rho_H$ $\sim$ 2.9 and small value of $\lambda_B$ $\sim$ 200 MeV, is helpful to reconcile discrepancies on $\Delta A$ and $R_{00}^{\pi \pi}$ between QCDF approach and experiments.
The spectator scattering and annihilation contributions can offer significant corrections to observables of hadronic $B$ decays, and deserve intensive research especially when we apply the QCDF approach to the penguin-dominated, color-suppressed tree, and pure annihilation nonleptonic $B$ decays. As suggested in Ref.[@zhu1; @zhu2] and proofed by the pQCD approach [@pqcd], different parameters corresponding to different topologies should be introduced to regulate the endpoint divergences in spectator scattering and annihilation amplitudes within QCDF approach, even parameters reflecting the flavor symmetry-breaking effects should be considered for $B_{u,d,s}$ decays [@zhu1; @zhu2; @Cheng1; @Cheng2; @Cheng3; @Beneke2; @chang1]. This treatment might provide possible solution to “problematic” discrepancies between QCDF results and available measurements. Of course, a fine-tuning of these parameters is required to be compatible with the experimental constraints. With the running LHCb and the upcoming SuperKEKB experiments, more refined measurements on $B$-meson decays can be obtained, which will provide more powerful grounds to test various approach and confirm or refute some theoretical hypotheses.
Acknowledgments {#thanks .unnumbered}
===============
This work is supported by National Natural Science Foundation of China under Grant Nos. 11147008, 11105043 and U1232101, 11475055. Q. Chang is also supported by Foundation for the Author of National Excellent Doctoral Dissertation of P. R. China under Grant No. 201317, Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20114104120002 and Program for Science and Technology Innovation Talents in Universities of Henan Province (Grant No. 14HASTIT036).
Building blocks of annihilation and spectator scattering contributions {#app01}
======================================================================
The annihilation amplitudes for two-body nonleptonic $B$ $\to$ $M_{1}M_{2}$ decays (here $M_{i}$ denotes the light pseudoscalar meson) can be expressed as the following building blocks [@Beneke2], $$\begin{aligned}
A_1^i &=&
\pi \alpha_s \int_0^1 dx dy
\Big\{ \Phi_{M_2}^{a}(x) \Phi_{M_1}^{a}(y)
\Big[ \frac{1}{y(1-x \bar{y})}
+ \frac{1}{\bar{x}^2 y} \Big]
+ r_\chi^{M_1} r_\chi^{M_2}
\frac{2 \Phi_{M_2}^{p}(x) \Phi_{M_1}^{p}(y)}
{\bar{x}y} \Big\}
\label{ai1}, \\
%%%%%%%%%%%%%%%
A_2^i &=&
\pi \alpha_s \int_0^1 dx dy
\Big\{ \Phi_{M_2}^{a}(x) \Phi_{M_1}^{a}(y)
\Big[ \frac{1}{\bar x(1-x \bar{y})}
+ \frac{1}{\bar{x} y^2} \Big]
+ r_\chi^{M_1} r_\chi^{M_2}
\frac{2 \Phi_{M_2}^{p}(x) \Phi_{M_1}^{p}(y)}
{\bar{x} y} \Big\}
\label{ai2}, \\
%%%%%%%%%%%%%%%
A_3^i &=&
\pi \alpha_s \int_0^1 dx dy
\Big\{ r_\chi^{M_1}
\frac{2 \bar{y}\ \Phi_{M_2}^{a}(x) \Phi_{M_1}^{p}(y)}
{\bar{x}y(1-x\bar{y})}
- r_\chi^{M_2}
\frac{2x\ \Phi_{M_1}^{a}(y) \Phi_{M_2}^{p}(x)}
{\bar{x}y(1-x\bar{y})} \Big\}
\label{ai3}, \\
%%%%%%%%%%%%%%%
A_1^f &=& A_2^f =0
\label{af12}, \\
%%%%%%%%%%%%%%%
A_3^f &=&
\pi \alpha_s \int_0^1 dx dy
\Big\{ r_\chi^{M_1}
\frac{2(1+\bar x)\ \Phi_{M_2}^{a}(x) \Phi_{M_1}^{p}(y)}
{\bar{x}^2 y}
+ r_\chi^{M_2}
\frac{2(1+y)\ \Phi_{M_1}^{a}(y) \Phi_{M_2}^{p}(x)}
{\bar{x} y^2} \Big\}
\label{af3},
\end{aligned}$$ where the subscripts $k$ on $A^{i,f}_{k}$ correspond to three possible Dirac current structures, namely, $k$ $=$ $1$, $2$, $3$ for $(V-A)\otimes(V-A)$, $(V-A)\otimes(V+A)$, $-2(S-P)\otimes(S+P)$, respectively. $r_\chi^{M}$ $=$ $2m_{M}^{2}/m_{b}(m_{1}+m_{2})$, where $m_{1,2}$ are the current quark mass of the pseudoscalar meson with mass $m_{M}$. $\Phi_{M}^{a}$ and $\Phi_{M}^{p}$ are the twist-2 and twist-3 light-cone distribution amplitudes, respectively. Their asymptotic forms are $\Phi_{M}^{a}(x)$ $=$ $6x\bar{x}$ and $\Phi_{M}^{p}(x)$ $=$ $1$.
The spectator scattering corrections are given by [@Beneke2] $$H_i ( M_1 M_2) =
\left\{ \begin{array}{l}
\displaystyle
+\frac{B_{M_1 M_2}}{A_{M_1 M_2}}
{\int}_{0}^{1}d{\xi}
\frac{ \Phi_{B}(\xi) }{ \xi }
\int_0^1 dx dy \Big[
\frac{ \Phi_{M_2}^{a}(x) \Phi_{M_1}^{a}(y) }
{ \bar{x} \bar{y} }
+ r_\chi^{M_1}
\frac{ \Phi_{M_2}^{a}(x) \Phi_{M_1}^{p}(y) }
{ x \bar{y} } \Big],
\\ \qquad \qquad \qquad
\text{for }\, i=1,2,3,4,9,10
\\
\displaystyle
-\frac{B_{M_1 M_2}}{A_{M_1 M_2}}
{\int}_{0}^{1}d{\xi}
\frac{ \Phi_{B}(\xi) }{ \xi }
\int_0^1 dx dy \Big[
\frac{ \Phi_{M_2}^{a}(x) \Phi_{M_1}^{a}(y) }
{ x \bar{y} }
+ r_\chi^{M_1}
\frac{ \Phi_{M_2}^{a}(x) \Phi_{M_1}^{p}(y) }
{ \bar{x} \bar{y} } \Big],
\\ \qquad \qquad \qquad
\text{for }\, i=5,7
\\
0, \qquad \qquad \quad
\text{for }\, i=6,8
\end{array} \right.
\label{hardblock}$$ where the factorized matrix elements are parameterized as [@Beneke2] $$A_{M_{1}M_{2}} =
i\frac{G_{F}}{\sqrt{2}}
m_{B}^{2}F_{0}^{B{\to}M_{1}}f_{M_{2}},
\qquad \qquad
B_{M_{1}M_{2}} =
i\frac{G_{F}}{\sqrt{2}}
f_{B}f_{M_{1}}f_{M_{2}}.$$
Theoretical input parameters {#app02}
============================
For the CKM matrix elements, we adopt the fitting results for the Wolfenstein parameters given by the CKMfitter group [@CKMfitter] $$\begin{aligned}
\bar{\rho} = 0.140^{+0.027}_{-0.026}, \quad
\bar{\eta} = 0.343^{+0.015}_{-0.014}, \quad
A = 0.802^{+0.029}_{-0.011}, \quad
\lambda = 0.22543^{+0.00059}_{-0.00094}.
\end{aligned}$$
The pole masses of quarks are [@PDG12] $$\begin{aligned}
&&m_u=m_d=m_s=0, \quad
m_c=1.67 \pm 0.07 \, {\rm GeV},
\nonumber\\
&&m_b=4.78 \pm 0.06 \, {\rm GeV}, \quad
m_t=173.5 \pm 1.0\,{\rm GeV}
\label{polemass}.
\end{aligned}$$
The running masses of quarks are [@PDG12] $$\begin{aligned}
&&\frac{\bar{m}_s(\mu)}{\bar{m}_q(\mu)} = 27 \pm 1, \quad
\bar{m}_{s}(2\,{\rm GeV}) = 95 \pm 5 \,{\rm MeV}, \quad
\bar{m}_{c}(\bar{m}_{c}) = 1.275 \pm 0.025 \,{\rm GeV},
\nonumber \\
&&\bar{m}_{b}(\bar{m}_{b}) = 4.18 \pm 0.03 \,{\rm GeV}, \quad
\bar{m}_{t}(\bar{m}_{t}) = 160.0^{+4.8}_{-4.3}\,{\rm GeV}
\label{runningmass}.
\end{aligned}$$
The decay constants of $B$-meson and light mesons are [@PDG12] $$f_{B} = (0.190 \pm 0.013)\,{\rm GeV}, \quad
f_{\pi} = (130.4 \pm 0.2)\,{\rm MeV}, \quad
f_{K} = (156.1 \pm 0.8)\,{\rm MeV}.$$
We take the following heavy-to-light transition form factors [@BallZwicky] $$F^{B \to \pi }_{0}(0) = 0.258 \pm 0.031, \qquad
F^{B \to K }_{0}(0) = 0.331 \pm 0.041.$$ Moreover, for the Gegenbauer coefficients, we take [@BallG] $$a_{1}^{\pi}({\rm 2 GeV}) =0, \quad
a_2^{\pi}({\rm 2 GeV}) =0.17, \quad
a_{1}^{K}({\rm 2 GeV}) =0.05, \quad
a_{2}^{K}({\rm 2 GeV}) =0.17.$$
For the other inputs, such as the masses and lifetimes of mesons and so on, we take their central values given by PDG [@PDG12].
Fitting Approach {#app03}
================
Our fit is performed in a simple way, which is similar to the one adopted in Ref.[@Vernazza] based on the frequentist framework. Considering a set of $N$ observables $f_j$, the experimental measurements are assumed to be Gaussian distributed with the mean value $f_{j\,\rm exp}$ and error $\sigma_{j\,\rm exp}$. The theoretical prediction $f_{j\,\rm theo}$ for each observable could be treated as a function of a set of “unknown” free parameters $\{y_i\}$ (here $y_i$ $=$ $\rho_{A}^{i,f}$, $\phi_{A}^{i,f}$ and $\lambda_B$ in this paper). To estimate the values of “unknown” parameters $\{y_i\}$ and compare the theoretical results $f_{j\,\rm theo}$ with the experimental data $f_{j\,\rm exp}$, typically, it is need to construct a $\chi^2$ function as $$\chi^2(\{y_i\}) =
\sum\limits_{j=1}^N
\frac{(f_{j\,\rm theo}(\{y_i\})-f_{j\,\rm exp})^2}
{\sigma_{j\,\rm exp}^2}.
\label{chi2I}$$
In the evaluation of $f_{j\,\rm theo}$ for hadronic B decays, ones always encounter theoretical uncertainties induced by input parameters, like form factor and decay constant, whose probability distribution is unknown. Following the treatment of Rfit scheme [@CKMfitter; @Rfit] that input values are treated on an equal footing, irrespective of how close they are from the edge of the allowed range, the $\chi^2$ function is modified as [@Vernazza] $$\chi^2 = \sum_{j=1}^N
\left\{ \begin{array}{cl}
\displaystyle
\frac{([ f_{j\,\rm theo}-\delta_{j\,\rm theo,\,sub} ]
-f_{j\,\rm exp})^2}{\sigma_{j\,\rm exp}^2}
& \quad \text{if } f_{j\,\rm exp} <
[f_{j\,\rm theo}-\delta_{j\,\rm theo,\,sub}],
\\
\displaystyle
\frac{(f_{j\,\rm exp}-
[f_{j\,\rm theo}+\delta_{j\,\rm theo,\,sup}])^2}
{\sigma_{j\,\rm exp}^2}
& \quad \text{if } f_{j\,\rm exp} >
[f_{j\,\rm theo}+\delta_{j\,\rm theo,\,sup}],
\\
0 & \quad \text{otherwise}
\end{array} \right.
\label{chi2II}$$ where $\delta_{j\,\rm theo,\,sup}$ and $\delta_{j\,\rm theo,\,sub}$ denote asymmetric theoretical uncertainties, and are defined as $(f_{j\,\rm theo})^{+\delta_{j\,\rm theo,\,sup}}_{-\delta_{j\,\rm theo,\,sub}}$. As to the asymmetric experimental errors, we choose the larger one as weighting factor. Correspondingly, the confidence levels are defined by the function $${\rm CL}(\{y_i\}) = \frac{1}{\sqrt{2^{N_{\rm dof}}} \Gamma(N_{\rm dof}/2)}
\int_{\Delta \chi^2(\{y_i\})}^{\infty} e^{-t/2}t^{N_{\rm dof}/2 -1}dt
\label{CLfun},$$ with $\Delta \chi^2$ $=$ $\chi^2$ $-$ $\chi^2_{\rm min}$ and $N_{\rm dof}$ the number of degrees of freedom of free parameters.
With the input parameters summarized in Appendix \[app02\], we scan the space of the parameters $y_i$ and calculate the theoretical results $f_{j\,\rm theo}$. The $\chi^2 $ could be obtained with Eq.(\[chi2II\]). The numerical results at $1 \sigma$ and $2 \sigma$ confidence levels are gotten from Eq.(\[CLfun\]) by taking ${\rm CL}$ $=$ $1 - 68.27\%$ and ${\rm CL}$ $=$ $1 - 95.45\%$, respectively.
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| 0 |
---
abstract: |
This article is a brief review of “nonfreeness" and related measures of “correlation" for many-fermion systems.
The many-fermion states we deem “uncorrelated" are the gauge-invariant quasi-free states. Uncorrelated states of systems of finitely many fermions we call simply “free" states. Slater determinant states are free; all other free states are “substates" of Slater determinant states or limits of such.
The nonfreeness of a many-fermion state equals the minimum of its entropy relative to all free states. Correlation functionals closely related to nonfreeness can be defined in terms of Rényi entropies; nonfreeness is the one that uses Shannon entropy. These correlation functionals all share desirable additivity and monotonicity properties, but nonfreeness has some additional attractive properties.
author:
- 'Alex D. Gottlieb[^1] and Norbert J. Mauser'
title: 'Nonfreeness and related functionals for measuring correlation in many-fermion states'
---
Introduction
============
“Nonfreeness" is an entropy functional of states of many-electron systems. It was introduced as a “measure of electron correlation" [@GottliebMauser2007; @GottliebMauserArchived] that is purely a functional of the many-electron state, depending only on the structure of the state and not upon the physical circumstances attending it, e.g., the Hamiltonian operator for the system [@correlation].
By definition, the nonfreeness of a many-fermion state is its entropy relative to the unique gauge-invariant quasi-free state with the same $1$-particle density matrix ($1$-pdm). Gauge-invariant quasi-free (GIQF) states have $0$ nonfreeness by definition, but the nonfreeness of any other many-fermion state is positive, possibly infinite. Slater determinant states of $n$-fermions and “Fermi sea" states of infinitely many fermions are GIQF, as are restrictions of such states. Conversely, any GIQF state can be represented as restriction of a Slater determinant or Fermi sea state. These are the states we deem “uncorrelated."
In this article we shall mainly consider normal states of finite systems of fermions, identifying such states with the density operators that represent them on a fermion Fock space. Among such states we focus on those that have finite expected particle number. The GIQF states of finite average particle number we call simply “free" states.
For pure $n$-fermion states, the nonfreeness functional coincides with “particle-hole symmetric correlation entropy" [@Gori-GiorgiZiesche]. Particle-hole symmetric correlation entropy has been used to quantify electron correlation in the uniform electron gas [@Gori-GiorgiZiesche] and in short linear chains undergoing a Mott transition [@Bendazzoli...]. Particle-hole symmetric correlation entropy is defined only for pure states; nonfreeness is an extension of that functional to the domain of mixed many-fermion states, i.e., states that can be represented by density operators on the fermion Fock space.
A correlation functional for mixed states can be useful even if the system of interest is one of exactly $n$ fermions in a pure state, because open subsystems of the system of interest are typically in a mixed states, containing a random number of particles. Consider, for example, a system of fermions on a lattice. The fermions that occupy a given site or block of sites constitute a subsystem that is typically in a mixed state, and the von Neumann entropy of that local state can reflect physical properties such as quantum phase transitions [@VidalLatorreRicoKitaev; @GuDengLiLin; @LarssonJohannesson]. Indeed, nonfreeness has been used to quantify local correlation in a realistic tight-binding model of a transition metal oxide heterostructure [@MauserHeld]. The state of a many-fermion system determines the states of all its subsystems (e.g., local states in an extended system). The induced state of a subsystem may be called a “restriction" [@HainzlLewinSolovej] or “localization" [@LewinNamRougerie] of the state; we call it a “substate." Nonfreeness is monotone with respect to restriction of states: the nonfreeness of a substate is less than or equal to the nonfreeness of the state from which it is derived [@GottliebMauser2007]. Also, nonfreeness is additive over independent subsystems: when a many-fermion state is a product of statistically independent substates, its nonfreeness is the sum of its substates’ nonfreeness [@GottliebMauser2007].
The monotonicity and additivity properties of nonfreeness derive from its definition as a relative entropy. Correlation functionals closely related to nonfreeness can be defined using Rényi divergences instead of relative entropy. Rényi divergences also enjoy the properties of additivity and monotonicity, and so do the correlation functionals defined in terms of them. Indeed, the “new measure of electron correlation" that we proposed in Ref. [@GottliebMauser2005] is of this type. However, within this class of correlation functionals, the nonfreeness functional has a couple of additional attractive properties, presently to be stated precisely.
Suppose $\Delta$ is a density operator on a fermion Fock space that represents a state of finite average particle number, and let $\Gamma_\Delta$ denote the density operator of the unique free state with the same $1$-pdm as $\Delta$. The nonfreeness of $\Delta$, or of the state it represents, is defined to be $S(\Delta \|\Gamma_\Delta)$, the entropy of $\Delta$ relative to $\Gamma_\Delta$.
The nonfreeness of $\Delta$ is given by the following simple formula, provided its von Neumann entropy $S(\Delta) = -\Tr (\Delta \log\Delta)$ is finite: $$\begin{aligned}
\label{simple formula}
\lefteqn{ S(\Delta \| \Gamma_{\Delta}) \equals S(\Gamma_{\Delta}) - S(\Delta) }\nonumber \\
& = & - \sum p_j \log p_j - \sum (1-p_j) \log(1-p_j) \ - \ S(\Delta) \qquad\end{aligned}$$ where the $p_j$ denote the eigenvalues of the $1$-pdm of $\Delta$, its natural occupation numbers.
In any case, the nonfreeness of a many-fermion state is equal to the minimum of its entropy relative to all free states: $$\label{minimum property}
S(\Delta \|\Gamma_\Delta) \equals \min \big\{S(\Delta\|\Gamma ) : \ \Gamma \ \hbox{is free} \big\}\ .$$ Moreover, if the minimum in (\[minimum property\]) is finite, then $\Gamma = \Gamma_\Delta$ is the unique minimizer of $S(\Delta\|\Gamma )$ over all free states $\Gamma$.
The nonfreeness of a many-fermion state $\Delta$ is its entropy relative to the free “reference" state $\Gamma_\Delta$. Other authors have essayed similar relative entropy measures of correlation strength, using various other uncorrelated reference states chosen [*ad hoc*]{} on physical grounds [@Byczuk; @ByczukErrata]. They proposed to use judicious choices of “physically well-known uncorrelated states" $\Gamma$ as reference states, avowedly because [@Byczuk; @footnote] they did not know which choice of $\Gamma$ minimizes $S(\Delta\|\Gamma)$. Shortly afterward, Held and Mauser [@MauserHeld] pointed out that the minimizer is $\Gamma_{\Delta}$, and and argued that (\[minimum property\]) means that other choices of $\Gamma$ necessarily overestimate correlation. In this review we will present a very thorough proof of (\[minimum property\]).
$$\star$$
The rest of this article is organized as follows:
Section \[Density operators on Fock space\] presents the notation and terminology required for reading Sections \[Free states\] - \[Special properties of nonfreeness\].
In Section \[Free states\] we define free states. Prop. \[free at last\] there asserts that free states are substates of Slater determinant states or limits of such states.
In Section \[Relative entropy correlation functionals\] we discuss correlation functionals that are closely related to nonfreeness, focusing on properties they share.
In Section \[Special properties of nonfreeness\] we review special properties of nonfreeness. The simple formula (\[simple formula\]) for nonfreeness is developed in Prop. \[max ent prop\] and its corollary; and the minimum property (\[minimum property\]) of nonfreeness is proven in Prop. \[superproposition\].
In order not to impede the review while keeping the article as a whole fairly self-contained, many of the technical details and some of the proofs have been removed to Section \[Appendices\], which effectively consists of nine appendices.
Density operators on Fock space {#Density operators on Fock space}
===============================
Let $\HH$ denote a Hilbert space, the $1$-particle Hilbert space. Unit vectors in $\HH$ are called “orbitals." Although the notation we will use suggests that $\HH$ has countably infinite Hilbert dimension, this is not required; everything works for $\HH$ of any dimension.
The Hilbert space for finite systems of fermions in $\HH$ is the fermion Fock space over $\HH$, which we shall denote by $\FFF(\HH)$ or simply $\FFF$. Let $\hat{a}^*(f)$ and $\hat{a}(f)$ denote the creation and annihilation operators for $f \in \HH$, defined as bounded operators on the Fock space [@BratteliRobinson].
An $n$-fermion “Slater determinant" wave function can be identified with a Fock space vector $$\label{Slater determinant in Fock space}
|\Phi\rangle \equals \hat{a}^*(f_1)\hat{a}^*(f_2) \cdots \hat{a}^*(f_n)|\Omega \rangle$$ where $f_1,\ldots,f_n$ are orthonormal orbitals in $\HH$. We think of the vacuum vector $|\Omega \rangle$ as a $0$-particle Slater determinant wave function.
In this article, we are going to focus on states of many-fermion systems that can be represented by density operators on the Fock space, especially those that represent states of finite average particle number.
Let $\Delta$ be a density operator on $\FFF(\HH)$. The “$1$-particle density matrix" or “$1$-pdm" of $\Delta$ is the bounded operator $\gamma_\Delta$ on $\HH$ such that $$\label{defining property of $1$-pdm}
\langle g | \gamma_\Delta f \rangle \equals \Tr\big(\Delta \hat{a}^*(f)\hat{a}(g) \big)$$ for all $f,g \in \HH$. The preceding formula implies that $\gamma_{\Delta}$ is a positive-semidefinite contraction. Its eigenvectors are called “natural orbitals" of $\Delta$, and the corresponding eigenvalues are the “natural occupation numbers" of $\Delta$.
If $h \in \HH$ is a unit vector, the diagonal matrix element $ \langle h | \gamma_\Delta h \rangle$ of $\gamma_\Delta $ is the probability that the orbital $h$ is occupied when the system is in the state represented by $\Delta$. This is because the operator $\hat{a}^*(h)\hat{a}(h)$ corresponds to the physical observable of orbital $h$’s occupation, which takes the values $0$ and $1$. The expected value of this observable, the probability that orbital $h$ is occupied, is therefore $\Tr\big( \Delta \hat{a}(h)^*\hat{a}(h) \big)$, and this equals $\langle h|\gamma_{\Delta} h \rangle $ by definition.
We are especially interested in the class of density operators on $\FFF = \FFF(\HH)$ that represent states of finite average particle number. We shall denote this class by $\DD(\FFF)$. If $\Delta$ is a density operator on $\FFF$, then the average number of particles in the state represented by $\Delta$ equals the trace of its $1$-pdm. Thus, a density operator $\Delta$ belongs to $\DD(\FFF)$ if and only if $\Tr (\gamma_\Delta) < \infty$. Note that $\DD(\FFF)$ contains all the Slater determinant states $|\Phi\rangle\!\langle \Phi |$ where $|\Phi\rangle$ is a Slater determinant wave function (\[Slater determinant in Fock space\]) in $\FFF$.
Free states {#Free states}
===========
In this section we shall define and discuss the kind of many-fermion states we consider uncorrelated.
We restrict our attention to states that are represented by density operators on a fermion Fock space, and which have finite average particle number. Among such states, we consider Slater determinant states to be uncorrelated, as well as any state that can be represented as a “substate" or restriction of a Slater determinant state. A state can be represented as a substate of a Slater determinant state if and only if it is “gauge-invariant quasi-free" [@AlickiFannes] and its $1$-pdm has finite rank. We want limits of free states to be free, too. The smallest class of states containing all substates of Slater determinant states and limits of such is the class of “free" states, which we define as follows:
\[free def\] A density operator $\Gamma$ on a fermion Fock space is “free" if it represents a gauge-invariant quasi-free state and its $1$-pdm has finite trace.
Quasi-free states with finite expected particle number are called “generalized Hartree-Fock" states in Ref. [@BachLiebSolovej]. Accordingly, free states are gauge-invariant generalized Hartree-Fock states.
Gauge-invariant quasi-free states are discussed in Section \[Gauge-invariant quasi-free states\]. In Section \[Free states appendix\] we will prove that all free states are limits of substates of Slater determinant states:
\[free at last\] A density operator on a fermion Fock space is free if and only if (i) its $1$-pdm has finite trace, and (ii) it is a limit in trace norm of density operators that represent substates of Slater determinant states.
Free states whose natural occupation numbers are all strictly positive and less than $1$ have the form of (grand canonical) Gibbs states for noninteracting fermions [@GottliebMauserArchived]. Let $\hat{a}_1,\hat{a}_2,\ldots$ denote the fermionic annihilation operators associated to a complete orthonormal system of reference orbitals, so that $\hat{n}_i = \hat{a}^*_i \hat{a}_i$ represents the observable of “the number of fermions in the $i^{th}$ orbital" (either $0$ or $1$). Any density operator $\Gamma$ of the Gibbs form $$\Gamma \ \propto \ \exp\big( - \sum \lambda_i \hat{n}_i\big)
\label{Gibbs form}$$ is free. Formula (\[Gibbs form\]) defines a density operator if and only if $\sum e^{-\lambda_i} < \infty$ because the trace of the operator on the right-hand side of the formula equals $\prod\big(1+e^{-\lambda_i} \big)$. The reference orbitals of the density operator $\Gamma$ defined by (\[Gibbs form\]) are its natural orbitals. Its natural occupation numbers, the average values of the observables $\hat{n}_i$, are $ p_i = e^{-\lambda_i}/(1+e^{-\lambda_i}) $. For later use we note here that $\log\Gamma$ is the “quadratic Hamiltonian" [@BachLiebSolovej] operator $$\log \Gamma
\equals
\sum_i \big( \log(p_i) \hat{a}^*_i \hat{a}_i + \log(1-p_i) \hat{a}_i \hat{a}^*_i \big) \ .
\label{quadratic}$$
Free states are characterized by statistically independent occupation of their natural orbitals. In Section \[Free states appendix\] we show that a density operator $\Gamma$ on $\FFF(\HH)$ is free if and only if orthogonal natural orbitals are occupied independently of one another. For example, in the free Gibbs state (\[Gibbs form\]) the expected value of the occupation observables $\hat{n}_i$ and $\hat{n}_j$ are $p_i$ and $p_j$, respectively, while the expected value of $\hat{n}_i\hat{n}_j$, i.e., the probability that the $i^{th}$ and $j^{th}$ orbitals are both occupied, equals $p_ip_j$ (assuming $i \ne j$).
In order to define nonfreeness and related correlation functionals, we will require the following well-known fact [@BachLiebSolovej; @AlickiFannes], which we will also prove in Section \[Free states appendix\].
\[free existence remark\] Suppose $Q:\HH \longrightarrow \HH$ is a positive-semidefinite contraction operator with finite trace. Then there exists a unique free density operator on $\FFF(\HH)$ with $1$-pdm $Q$.
The von Neumann entropy of a free state is a simple function of its natural occupation numbers $p_i$. The following formula can be established using Proposition \[structural\] in Section \[Free states appendix\].
\[quasifree entropy remark\] If $\Gamma$ is a free state with natural occupation numbers $p_i$, then its von Neumann entropy is $$S(\Gamma) \ = \ -\sum_i p_i \log p_i -\sum_i (1-p_i) \log (1-p_i) \ .$$
Relative entropy correlation functionals {#Relative entropy correlation functionals}
========================================
Recall that $\DD(\FFF)$ denotes the set of density operators on the fermion Fock space $\FFF = \FFF(\HH)$ that represent states of finite average particle number. The $1$-pdm of a density operator $\Delta \in \DD(\FFF)$ is a positive-semidefinite contraction operator on $\HH$ with finite trace. By Remark \[free existence remark\], there exists a unique free density operator with the same $1$-pdm as $\Delta$. We denote this free density operator by $\Gamma_\Delta$. In other words, $\Gamma_\Delta$ denotes the unique free density operator such that $\gamma_{\Gamma_\Delta} = \gamma_\Delta$ (with the notation defined in formula (\[defining property of $1$-pdm\])).
The nonfreeness $\mathfrak{C}(\Delta)$ of $\Delta$ is defined to be the entropy of $\Delta$ relative to $\Gamma_\Delta$, that is, $$\label{nonfreeness intro def}
\mathfrak{C}(\Delta) \equals S(\Delta \|\Gamma_\Delta) \ .$$ This equals $\Tr(\Delta \log\Delta) - \Tr(\Delta \log\Gamma_{\Delta})$ provided that $\Tr(\Delta \log\Delta) > -\infty$.
Correlation functionals closely related to nonfreeness are obtained by using other “divergences" instead of the relative entropy to compare the states $\Delta$ and $\Gamma_\Delta$. Using a divergence that enjoys the properties of additivity and monotonicity will yield a correlation functional with those properties. We have in mind the Rényi divergences $$D_\alpha(\rho \| \sigma) \equals \frac{1}{\alpha - 1} \log \Tr (\rho^\alpha \sigma^{1-\alpha})$$ for $0 < \alpha \le 2$ and the “sandwiched" relative Rényi entropies [@Mueller-Lennert...; @WildeWinterYang; @FrankLieb] $$\widetilde{D}_\alpha(\rho \| \sigma) \equals \frac{1}{\alpha - 1} \log \Tr \Big(\big(\sigma^{\frac{1-\alpha}{2\alpha}} \rho \sigma^{\frac{1-\alpha}{2\alpha}}\big)^\alpha\Big)$$ for $\alpha \ge \tfrac12$. The divergences $D_1$ and $\widetilde{D}_1$ are defined by taking limits $\alpha \longrightarrow 1$ and both equal the relative entropy $S(\rho \| \sigma)$.
For values of $\alpha$ in the appropriate ranges, the correlation functionals $$\begin{aligned}
\mathfrak{C}_\alpha( \Delta ) &=& D_\alpha(\Delta \|\Gamma_{\Delta} ) \\
\widetilde{\mathfrak{C}}_\alpha( \Delta ) &=& \widetilde{D}_\alpha(\Delta \|\Gamma_{\Delta} ) \end{aligned}$$ all share the following properties with the nonfreeness functional $\mathfrak{C} = \mathfrak{C}_1 = \widetilde{\mathfrak{C}}_1$ :
[(i)]{} they take only non-negative values, possibly $+\infty$,
[(ii)]{} they assign the value $0$ to all Slater determinant states,
[(iii)]{} they are monotone with respect to restriction of states,
[(iv)]{} they are additive over independent subsystems, and
[(v)]{} they are invariant under changes of the $1$-particle basis.
The sandwiched relative Rényi entropy $\widetilde{D}_{1/2}$ equals twice the negative logarithm of “fidelity," and the corresponding correlation functional $\widetilde{\mathfrak{C}}_{1/2}$ is the “new measure" of correlation we proposed in Ref. [@GottliebMauser2005].
Special properties of nonfreeness {#Special properties of nonfreeness}
=================================
Due to its definition in terms of von Neumann entropy, nonfreeness has some intuitively appealing properties that the other relative-entropy-type correlation functionals do not share.
The nonfreeness of a many-fermion density operator $\Delta$ has been defined to be its entropy relative to the associated free state $\Gamma_\Delta$. Prop. \[max ent prop\] states that the nonfreeness of $\Delta$ equals the difference between the von Neumann entropies of $\Gamma_\Delta$ and $\Delta$. The nonfreeness $\Cor(\Delta)$ may be defined without direct reference to $\Gamma_\Delta$, by $$\Cor(\Delta) \equals \min \big\{S(\Delta\|\Gamma ) : \ \Gamma \ \hbox{is free} \big\}$$ because the minimum relative entropy is actually attained at $\Gamma=\Gamma_{\Delta}$, as shown in Prop. \[superproposition\] below.
Simple formulas for nonfreeness {#Simple formulas for nonfreeness}
-------------------------------
Recall that $\DD(\FFF)$ denotes the set of density operators on the fermion Fock space $\FFF$ that represent states of finite average particle number.
\[good lemma\] Suppose $\Delta \in \DD(\FFF)$ and let $\Gamma_{\Delta}$ denote the unique free state that has the same $1$-pdm as $\Delta$. If $\Gamma$ is free then $$\label{conclusion} - \Tr(\Delta \log\Gamma ) \equals - \Tr(\Gamma_\Delta \log\Gamma )\ . \$$
We prove this here for the case where $\Gamma$ is a free Gibbs state, i.e., when all natural occupation numbers $p_i$ of $\Gamma$ are strictly between $0$ and $1$. The proof is simple in this case because the operator $\log \Gamma$ is then quadratic in the creators and annihilators, while $\Delta$ and $\Gamma_\Delta$, having the same $1$-pdm, assign the same expectations to all such operators. The general case where some of the $p_i$ may equal $0$ or $1$ requires some care and is handled in Section \[Proof of Lemma 1\].
Suppose $\log \Gamma$ is the quadratic Hamiltonian operator (\[quadratic\]). By the defining property (\[defining property of $1$-pdm\]) of the $1$-pdm $\gamma_\Delta$, $$\begin{aligned}
- \Tr (\Delta \log \Gamma ) & = &
- \sum_i \big( \log(p_i) \Tr (\Delta \hat{a}^*_i \hat{a}_i) - \log(1-p_i) \Tr (\Delta \hat{a}_i \hat{a}^*_i) \big)
\\
& = &
- \sum_i \big( \log(p_i) \langle h_i | \gamma_\Delta h_i \rangle - \log(1-p_i) (1 - \langle h_i | \gamma_\Delta f_i \rangle ) \big).\end{aligned}$$ Since $\gamma_\Delta$ is also the $1$-pdm of $\Gamma_\Delta$, the conclusion (\[conclusion\]) follows.
When the von Neumann entropy $S(\Delta) = -\Tr (\Delta \log\Delta)$ is finite we may use the formula $$\label{we may use}
S(\Delta \| \Gamma_{\Delta}) \equals - \Tr(\Delta \log\Gamma_{\Delta}) - S(\Delta)$$ for the relative entropy. This leads to simple formulas for nonfreeness.
\[max ent prop\] Suppose $\Delta \in \DD(\FFF)$ satisfies $S(\Delta) < \infty$. Let $\Gamma_{\Delta}$ denote the unique free density operator with the same $1$-pdm as $\Delta$. Then, $$\label{basic formula}
S(\Delta\|\Gamma_{\Delta}) \equals S(\Gamma_\Delta)-S(\Delta) \ .$$
By Lemma \[good lemma\], $$-\Tr( \Delta \log \Gamma_{\Delta} ) \equals -\Tr( \Gamma_{\Delta} \log \Gamma_{\Delta} ) \equals
S(\Gamma_\Delta ).$$ Substituting $S(\Gamma_\Delta )$ for $-\Tr( \Delta \log \Gamma_{\Delta} )$ in equation (\[we may use\]) yields (\[basic formula\]).
By Remark \[quasifree entropy remark\], the von Neumann entropy of the free state $\Gamma_\Delta$ is a function of its natural occupation numbers. The natural occupation numbers of $\Gamma_{\Delta}$ are the same as those of $\Delta$, since they have the same $1$-pdm; therefore, using (\[basic formula\]) we obtain the following simple formula for nonfreeness:
\[explicit nonfreeness formula\] Suppose $\Delta \in \DD(\FFF)$ is a density operator on the fermion Fock space, and let $p_1,p_2,\ldots$ denote the eigenvalues of its 1-pdm. If $S(\Delta) < \infty$ then $$S( \Delta \| \Gamma_{\Delta} ) \equals - \sum p_j \log p_j - \sum (1-p_j) \log(1-p_j) - S(\Delta)\ .$$
Nonfreeness as relative entropy mimimum {#Nonfreeness as relative entropy mimimum}
---------------------------------------
The nonfreeness $\mathfrak{C}(\Delta)$ of a many-fermion state $\Delta$ is equal to the minimum of its entropy relative to all free reference states. To prove this we will use the inequality $$\label{an entropy inequality}
S( A ) \ \le\ -\Tr(A \log B)$$ for two density operators on the same Hilbert space. In case $S(A) < \infty$, then $$\label{if entropy is finite}
S( A \| B) = - \Tr ( A \log B ) - S(A)$$ and (\[an entropy inequality\]) follows immediately from the fact that $S( A \| B) \ge 0$.
\[superproposition\] Suppose $\Delta \in \DD(\FFF)$ and let $\Gamma_{\Delta}$ denote the unique free density operator with the same $1$-pdm as $\Delta$. Then, for all free density operators $\Gamma $, $$\label{superprop-equation}
S(\Delta\|\Gamma_{\Delta}) \ + \ S(\Gamma_{\Delta}\|\Gamma ) \ = \ S(\Delta\|\Gamma ),$$ and therefore $$\label{superprop-inequality}
S(\Delta\|\Gamma_{\Delta}) \ \le \ S(\Delta\|\Gamma )$$ with equality only if $S(\Delta\|\Gamma_\Delta) = \infty$ or $\Gamma=\Gamma_\Delta$.
The analog of Prop. \[superproposition\] for other Rényi divergences would be false. That is, if $\alpha \ne 1$ then $\Gamma_{\Delta}$ need not minimize $D_\alpha(\Delta \|\Gamma )$ or $\widetilde{D}_\alpha(\Delta \|\Gamma )$.
For example, let $\HH = \mathrm{span}\big\{| \uparrow\ \rangle,| \downarrow\ \rangle \big\}$ and let $\Delta$ denote the density operator that is entirely supported on the $1$-particle component of $\FFF(\HH)$, where it equals $\frac23 | \uparrow \ \rangle\!\langle \ \uparrow | + \tfrac13 | \downarrow \ \rangle\!\langle \ \downarrow |$. Then the minimum of $D_\alpha(\Delta \|\Gamma)$ or $\widetilde{D}_\alpha(\Delta \|\Gamma )$ is not attained at $\Gamma_\Delta$ unless $\alpha = 1$.
We first prove Prop. \[superproposition\] under the assumption that $S(\Delta) < \infty$, which allows us to use formula (\[if entropy is finite\]). Then we will relieve the assumption that $S(\Delta) < \infty$ by using the martingale property of relative entropy [@OhyaPetz].
Suppose that $S(\Delta) < \infty$. Then $$S( \Delta \| \Gamma ) \ = \ - \Tr ( \Delta \log \Gamma ) - S(\Delta)\ .$$ If $\Gamma$ is free, then $$\begin{aligned}
S(\Delta \| \Gamma) & = & -\Tr(\Delta \log \Gamma) - S(\Delta) \equals -\Tr( \Gamma_{\Delta} \log \Gamma ) - S(\Delta) \\
& \ge & S(\Gamma_{\Delta}) - S(\Delta) \equals -\Tr( \Delta \log \Gamma_{\Delta} ) - S(\Delta) \equals S(\Delta \| \Gamma_{\Delta} ) \ .\end{aligned}$$ The first and last equalities hold because $S(\Delta) < \infty$; the next-to-first and next-to-last equalities hold by Lemma \[good lemma\]; the inequality holds by (\[an entropy inequality\]). This establishes (\[superprop-inequality\]) when $S(\Delta) < \infty$.
If $S(\Delta \| \Gamma_{\Delta} ) = \infty$, then also $S(\Delta \| \Gamma)= \infty$, as we have just established, and equation (\[superprop-equation\]) holds trivially. On the other hand, if $S(\Delta \| \Gamma_{\Delta} )$ and $S(\Delta)$ are both finite, then Prop. \[max ent prop\] implies that $S(\Gamma_\Delta ) < \infty$ and $S(\Delta\|\Gamma_{\Delta}) = S(\Gamma_\Delta)-S(\Delta)$. By Lemma \[good lemma\], $$\begin{aligned}
S(\Delta \| \Gamma) - S(\Delta \| \Gamma_{\Delta} )
& = &
-\Tr( \Delta \log \Gamma ) - S( \Gamma_{\Delta} )
\nonumber \\
& = &
-\Tr( \Gamma_{\Delta} \log \Gamma ) - S( \Gamma_{\Delta} )
\equals
% \label{used good lemma again} \\
% & = &
S(\Gamma_{\Delta} \| \Gamma),
\nonumber
\end{aligned}$$ which is equivalent to equation (\[superprop-equation\]). Thus, equation (\[superprop-equation\]) holds whether or not $S(\Delta \| \Gamma_{\Delta} )$ is finite, provided $S(\Delta ) < \infty$.
Now assume that $S(\Delta) = \infty$. The symbol $\BB(\mathcal{X})$ in the sequel denotes the algebra of bounded operators on a Hilbert space $\mathcal{X}$.
Consider an increasing sequence of finite-rank projectors $P_n$ on $\HH$ that converges strongly to the identity, and let $\HH_n$ denote the range of $P_n$. The finite-dimensional von Neumann algebras $\BB(\FFF(\HH_n))$ can be embedded into $\BB(\FFF(\HH))$ as subalgebras, which we denote here by $\BB_n$. Let $\Delta_n$, $(\Gamma_{\Delta})_n$, and $\Gamma_n$ denote the density operators on $\FFF(\HH_n)$ that represent the restrictions of $\Delta$, $\Gamma_\Delta$, and $\Gamma$ to the corresponding substates delimited by $\HH_n$ (as defined in Section \[Substates of many-fermion states\]). The density operator $(\Gamma_{\Delta})_n$ is free because it is a substate of a free state (see Section \[Free states appendix\]) and it has the same $1$-pdm as $\Delta_n$, whence $(\Gamma_{\Delta})_n = \Gamma_{\Delta_n}$. Since $\Delta_n$ is a density operator on a finite-dimensional space, it has finite von Neumann entropy, and therefore $$\label{martingale}
S(\Delta_n\|\Gamma_{\Delta_n}) \ + \ S(\Gamma_{\Delta_n}\| \Gamma_n ) \ = \ S(\Delta_n \|\Gamma_n )$$ by (\[superprop-equation\]), as proven above. The norm closure of $\bigcup \BB_n$ is equal to the CAR algebra [@AlickiFannes Theorem 6.6] in its Fock representation as a subalgebra of $\BB(\FFF(\HH))$. The bi-commutant of $\bigcup \BB_n$, which equals that of its closure, is therefore all of $\BB(\FFF(\HH))$. The “filtration" $(\BB_n)_{n=1}^\infty$ thus satisfies the hypothesis of Cor. 5.12(iv) of Ref. [@OhyaPetz], and therefore the three terms in equation (\[martingale\]) converge to $S( \Delta \| \Gamma_{\Delta} )$, $S( \Gamma_\Delta \| \Gamma )$, and $S( \Delta \| \Gamma )$ as $n \longrightarrow \infty$. This establishes (\[superprop-equation\]) even when $S(\Delta )$ is infinite.
Appendices {#Appendices}
==========
The following nine appendices dilate on the technical background necessary for a thorough understanding of this review, and include a couple of deferred proofs. The appendices are titled:
\[Relative entropy for density operators\] Relative entropy for density operators
\[Fermion Fock spaces\] Fermion Fock spaces
\[Many-fermion states\] Many-fermion states
\[Substates of many-fermion states\] Substates of many-fermion states
\[1-particle density matrices\] $1$-particle density matrices
\[Gauge-invariant quasi-free states\] Gauge-invariant quasi-free states
\[Free states appendix\] Free states
\[Proof of Proposition 1\] Proof of Proposition \[free at last\]
\[Proof of Lemma 1\] Proof of Lemma \[good lemma\]
Relative entropy for density operators {#Relative entropy for density operators}
--------------------------------------
The general definition of relative entropy for normal states on von Neumann algebras requires some modular theory [@OhyaPetz]. However, for density operators on a Hilbert space $\mathcal{X}$, which represent normal states on the von Neumann algebra of bounded operators on $\mathcal{X}$, a more elementary definition of relative entropy is available.
Let $A$ and $B$ denote density operators on a Hilbert space $\mathcal{X}$. Let $\{\phi_1,\phi_2,\ldots\}$ and $\{\psi_1,\psi_2,\ldots\}$ be orthonormal bases of $\mathcal{X}$ consisting of eigenvectors of $A$ and $B$, respectively, with corresponding eigenvalues $p_i$ and $q_i$. We define $$\log A = \sum_{i: p_i > 0} \log(p_i) |\phi_i \rangle\!\langle \phi_i |\ ,$$ a negative-semidefinite, but generally unbounded, operator. Note that $\log A$ is defined so that $\ker(\log A)=\ker ( A )$. The von Neumann entropy of $A$ is defined to be $ S(A) = -\Tr ( A \log A ) = -\sum p_i \log(p_i)$. It may equal $+\infty$.
We define $-\Tr ( A \log B )$ to be $+\infty$ if $\ker B \not\subset \ker A$, otherwise, we define it by $$-\Tr ( A \log B ) \equals - \sum_i \sum_j | \langle \phi_i, \psi_j \rangle |^2 p_i \log q_j$$ as done in Ref. [@ArakiLieb]. We define the entropy of $A$ relative to $B$ by the formula $$S( A \| B ) \equals \sum_i \sum_j | \langle \phi_i, \psi_j \rangle |^2 ( p_i \log p_i - p_i \log q_j + q_j - p_i)$$ as done in Ref. [@Lindblad73]. The fact that the series defining $S( A \| B )$ has only nonnegative terms implies that $$\label{relative entropy}
S( A \| B) \ = \ - \Tr ( A \log B ) - S(A)$$ if $S(A) < \infty$, and that $$\label{entropy inequality}
S( A ) \ \le\ -\Tr(A \log B)$$ even if $S(A)$ is infinite. When $ S(A) = \infty$ formula (\[relative entropy\]) cannot be used and $S( A \| B )$ may still be finite.
Fermion Fock spaces {#Fermion Fock spaces}
-------------------
Let $\HH$ be a Hilbert space, the $1$-particle Hilbert space. Unit vectors in $\HH$ are called “orbitals."
The fermion Fock space over $\HH$, which we denote $\FFF(\HH)$, is the Hilbert space direct sum of alternating tensor powers of the $1$-particle Hilbert space $\HH$. That is, $$\label{Fock space}
\FFF(\HH) \equals \CCC \oplus \HH \oplus \wedge^2 \HH \oplus \cdots \oplus \wedge^m \HH \oplus \cdots \cdots$$ where $\wedge^m \HH $ denotes the $m^{th}$ exterior (alternating tensorial) power of $\HH$. The first component of $\FFF(\HH)$ contains a distinguished unit vector $|\Omega \rangle$ called the “vacuum vector."
The $m$-particle Hilbert space $\wedge^m \HH$ is the completion of the span of all tensor products $h_1 \wedge h_2 \wedge \ldots \wedge h_m$, where $h_1,\ldots,h_m$ are any $m>0$ vectors in $\HH$. The tensors $h_1
\wedge h_2 \wedge \ldots \wedge h_m$ are formally multilinear in $h_1,\ldots,h_m$ and satisfy $$h_1 \wedge \cdots \wedge h_i \wedge \cdots \wedge h_j \wedge \cdots \wedge h_m
\equals - h_1 \wedge \cdots \wedge h_j \wedge \cdots \wedge h_i \wedge \cdots \wedge h_m$$ for $1\le i < j \le m$. In the context of $n$-electron systems, wedge products are usually called “Slater determinants." The inner product of two Slater determinants is $$\langle h_1 \wedge \cdots \wedge h_{m} ,\ h'_1 \wedge \cdots \wedge h'_m \rangle
\equals
\det \big( \langle h_i, h'_j \rangle \big)_{ij=1}^{\ m}\ .$$ This extends to an inner product on the linear span of the Slater determinants, and the completion of this linear span is the Hilbert space $\wedge^m \HH$.
Let $\BB(\FFF)$ denote the space of bounded operators on $\FFF = \FFF(\HH)$. Creation and annihilation operators $\hat{a}^*(h)$ and $\hat{a}(h)$ on $\FFF$ may be defined for each $h \in \HH$ as in Refs. [@BratteliRobinson; @AlickiFannes]. These creation and annihilation operators (creators and annihilators) are bounded operators that satisfy the canonical anticommutation relations and generate the Fock representation of the CAR algebra. This is the uniform-norm closure in $\BB(\FFF)$ of the algebra of polynomials in the creators and annihilators. We shall denote the CAR algebra over $\HH$ by $\mathfrak{A}(\HH)$ and its Fock representation as a subalgebra of $\BB(\FFF)$ by $\pi( \mathfrak{A}(\HH) )$.
The Fock representation of $\mathfrak{A}(\HH)$ on $\FFF(\HH)$ is irreducible, i.e., the commutant of $\pi( \mathfrak{A}(\HH) )$ in $\BB(\FFF)$ is trivial. Therefore the bi-commutant of $\pi( \mathfrak{A}(\HH) )$, in which it is weakly dense, is all of $\BB(\FFF)$.
Given an ordered orthonormal basis $(h_1,h_2,\ldots)$ of $\HH$, one can build an orthonormal basis of $\FFF(\HH)$, called a “Fock basis" or “occupation number" basis, using the orbitals $h_i$ as “reference" orbitals. The Fock basis vectors represent configurations of particles in the reference orbitals and are indexed by “occupation lists" $$\bn = \big(\bn(1),\bn(2),\bn(3),\ldots \big)$$ such that $\sum \bn(i) < \infty$, that is, such that the total number of particles in the configuration is finite. The set $$\NN \equals \Big\{ \big(\bn(1),\bn(2),\bn(3),\ldots \big) : \bn(i) \in \{ 0,1\},\ \sum \bn(i) < \infty \Big\}$$ indexes the possible configurations of fermions in the modes $(h_1,h_2,\ldots)$. The occupation list ${\bf 0} = (0,0,0,\ldots)$ is the index of the vacuum vector $|\Omega \rangle$, i.e., $| {\bf 0}\rangle = |\Omega \rangle$. For $\bn \in \NN$ with $\sum \bn(i) > 0$, define the vector $$\label{Fock basis vector}
|\bn\rangle \equals \hat{a}^*(h_1)^{\bn(1)}\hat{a}^*(h_2)^{\bn(2)} \cdots |\Omega \rangle$$ (since the exponents $\bn(i)$ are eventually $0$, only finitely many creators appear to the left of $|\Omega\rangle$ in this formula). The orthonormal set $\big\{ |\bn\rangle : \bn \in \NN \big\}$ is an orthonormal basis of $\FFF(\HH)$. It is the Fock basis defined with respect to the ordered orthonormal basis $(h_1,h_2,\ldots)$ of reference orbitals.
Though we have written the occupation lists as if they are sequences, all that is really required is a well-ordering of the set of reference orbitals, to give a definite order to the creators in formula (\[Fock basis vector\]). Allowing a different kind of well-ordering facilitates the description of the isomorphism (\[basic isomorphism\]) below.
Many-fermion states {#Many-fermion states}
-------------------
We are considering many-fermion states that can be represented by density operators on the fermion Fock space $\FFF$. In the conventional formalism, physical observables correspond to self-adjoint operators on $\FFF$ and states correspond to certain linear functionals on $\BB(\FFF)$, the von Neumann algebra of bounded operators on $\FFF$. We are especially interested in the “normal" states on $\BB(\FFF)$. A normal state on $\BB(\FFF)$ is a $\sigma$-weakly continuous linear functional $\omega:\BB(\FFF) \longrightarrow \CCC$ such that $\omega(I) = 1$ and $\omega(B) \ge 0$ for all positive-semidefinite $B \in \BB(\FFF)$. A density operator $\Delta$ on $\FFF$ describes a normal state $\omega$ on $\BB(\FFF)$ via the formula $\omega(B) = \Tr(\Delta B) $. Conversely, any normal state is represented in this manner by a density operator.
Using the canonical anticommutation relations, polynomials in the creators and annihilators can be written as linear combinations of [*normally ordered*]{} monomials in the creators and annihilators. Since $\pi( \mathfrak{A}(\HH) )$ is $\sigma$-weakly dense in $\BB(\FFF)$, the correlations $$\label{correlations}
\Tr\big(\Delta\ \hat{a}^*(f_1)\cdots \hat{a}^*(f_n)\hat{a}(g_m)\cdots \hat{a}(g_1) \big)$$ for all $n,m \ge 0$ with $n+m>0$, and all $f_1,f_2,\ldots,f_n, g_1,\ldots,g_m \in \HH$, suffice to determine the density operator $\Delta$. That is, no other density operator can have all the same correlations (\[correlations\]) as $\Delta$.
A basic example of a many-fermion sate state is a Slater determinant state. Let $\Phi = h_1 \wedge h_2 \wedge \cdots \wedge h_n$ denote a Slater determinant vector in $\wedge^n \HH$, where $\{h_1 , h_2 ,\ldots, h_n\}$ is an orthonormal set in $\HH$. The density operator $$\label{Slater determinant state}
0_{\CCC} \oplus \cdots \oplus 0_{\wedge^{n-1} \HH } \oplus |\Phi\rangle\!\langle \Phi | \oplus 0_{ \wedge^{n+1} \HH } \oplus \cdots$$ defined relative to the decomposition (\[Fock space\]) of $\FFF$ represents an $n$-particle “Slater determinant state." We also think of the vacuum state $|\Omega \rangle\!\langle \Omega |$ as a $0$-particle Slater determinant state.
Substates of many-fermion states {#Substates of many-fermion states}
--------------------------------
If $\HH_1$ is a closed subspace of the $1$-particle space $\HH$ and $\HH_2$ is its orthogonal complement, then the Fock space over $\HH$ is isomorphic to the tensor product of the Fock spaces over $\HH_1$ and $ \HH_2$. That is, if $\HH \cong \HH_1 \oplus \HH_2$, then $$\label{basic isomorphism}
\FFF(\HH) \ \cong \ \FFF(\HH_1) \otimes \FFF(\HH_2)\ .$$ We shall write $\FFF_1$ for $\FFF(\HH_1)$, $\FFF_2$ for $\FFF(\HH_2)$, and $\FFF$ for $\FFF(\HH)$
An isomorphism (\[basic isomorphism\]) is easy to describe using Fock bases of $\FFF_1$ and $\FFF_2$. Let $(f_1,f_2,\ldots)$ and $(g_1,g_2,\ldots)$ denote ordered orthonormal bases of $\HH_1$ and $\HH_2$, respectively. Then $(f_1,f_2,\ldots,g_1,g_2,\ldots)$ is an ordered orthonormal basis of $\HH_1 \oplus \HH_2$. Occupation lists relative to $(f_1,f_2,\ldots,g_1,g_2,\ldots)$ are in one-to-one correspondence with pairs of occupation lists $(\bn_1,\bn_2) \in \NN_1 \times \NN_2$, where $\bn_1\in \NN_1$ is an occupation list relative to $(f_1,f_2,\ldots)$ and $\bn_2\in \NN_2$ is an occupation list relative to $(g_1,g_2,\ldots)$. The correspondence $$\label{isomorphism}
|\bn\rangle \ \longleftrightarrow\ |\bn_1\rangle \otimes |\bn_2\rangle$$ extends to an isomorphism.
The algebra $\BB(\FFF_1)$ is isomorphic to a subalgebra of $\BB(\FFF_1 \otimes \FFF_2) \cong \BB(\FFF)$ via the inclusion map $B \mapsto B \otimes I_2$, where $I_2$ denotes the identity operator on $ \FFF_2$. The embedding and isomorphism $$\BB(\FFF_1) \ \hookrightarrow \ \BB(\FFF_1 \otimes \FFF_2) \ \cong \ \BB(\FFF)\ ,
\label{embeddingandisomorphism}$$ map the creation and annihilation operators $\hat{a}^*(f),\hat{a}(f) \in \BB(\FFF_1)$, defined for vectors $f \in \HH_1$, to the creation and annihilation operators in $\BB(\FFF)$ denoted the same way. Let $\BB_1$ denote the isomorphic image of $\BB(\FFF_1)$ as a subalgebra of $\BB(\FFF)$. If $\dim(\HH_1) = d < \infty$ then $\BB_1$ is generated algebraically by the creators and annihilators and $\dim(\BB_1) = 2^d$. If $\HH_1$ is infinite-dimensional then $\BB_1$ is the bi-commutant and weak closure of the algebra generated by the creators and annihilators pertaining to $\HH_1$.
A state $\omega$ on the larger von Neumann algebra $\BB(\FFF)$ induces a state on the subalgebra $\BB_1 \cong \BB(\FFF_1)$. We call the induced substate on $\BB(\FFF_1)$ a “substate" of $\omega$, the substate “delimited by" the orbitals in the closed subspace $\HH_1$ of $\HH$. It may also be called a “restriction" [@HainzlLewinSolovej] or “localization" [@LewinNamRougerie] of $\omega$.
We are particularly interested in normal states on $
\BB(\FFF_1 \otimes \FFF_2) \cong \BB(\FFF)\ .
$ For each normal state $\omega$ there is a corresponding density operator $\Delta$ on $\FFF_1 \otimes \FFF_2$ such that $
\omega(A) = \Tr(\Delta A)
$ for all $A \in \BB(\FFF)$. The induced substate $B \mapsto \omega(B \otimes I_2)$ on $ \BB(\FFF_1) $ is also normal. It is represented by the partial trace of $\Delta$ with respect to $\FFF_2$, i.e., by the density operator $\Delta_1$ on $\FFF_1$ such that $$\label{partial trace 1}
\Tr(\Delta_1 B) \equals \Tr_{\FFF_2}(\Delta(B \otimes I_2))$$ for all bounded operators $B \in \BB(\FFF_1)$.
1-particle density matrices {#1-particle density matrices}
---------------------------
Consider the $n=m=1$ correlations $(\ref{correlations})$. The map $$\label{similartothis}
(g,f) \ \longmapsto \ \Tr\big(\Delta \hat{a}^*(f)\hat{a}(g)\big),$$ is a bounded conjugate-bilinear form, and therefore there exists a bounded operator $\gamma_\Delta$ on $\HH$ such that $$\label{defining property of $1$-pdm again}
\langle g | \gamma_\Delta f \rangle \equals \Tr\big(\Delta \hat{a}^*(f)\hat{a}(g) \big)$$ for all $f,g \in \HH$. We call $\gamma_\Delta$ the “$1$-particle density matrix" or “$1$-pdm" of $\Delta$. If $h \in \HH$ is any orbital, the diagonal matrix element $ \langle h | \gamma_\Delta h \rangle$ of the $1$-pdm is the probability that $h$ is occupied. The trace of $\gamma_\Delta $ is therefore the average [*total*]{} number of particles.
The eigenvectors of $\gamma_{\Delta}$ are called “natural orbitals" of $\Delta$, and the corresponding eigenvalues are the “natural occupation numbers" of $\Delta$. For example, the $1$-pdm of the Slater determinant state (\[Slater determinant state\]) is the orthogonal projector whose range is $\hbox{span}\{h_1 ,\ldots, h_n\}$. Thus, $n$ of the natural occupation numbers of that state are $1$ and the rest are $0$.
Let $\HH_1$ be a closed subspace of $\HH$, and let $\Delta_1$ be the substate of $\Delta$ defined in the preceding section. As noted there, the embedding and isomorphism (\[embeddingandisomorphism\]) map the creation and annihilation operators $\hat{a}^*(f),\hat{a}(f) \in \BB(\FFF_1)$ with $f \in \HH_1$ to the creators and annihilators on $\FFF$ denoted the same way. Therefore, the matrix elements (\[defining property of $1$-pdm again\]) of the $1$-pdm $ \gamma_{\Delta_1}$, defined for for all $f,g \in \HH_1$, are the same as the corresponding matrix elements of $ \gamma_\Delta$. In other words, $ \gamma_{\Delta_1}$ is the compression of $ \gamma_\Delta$ to $\HH_1 \subset \HH$.
Finally, we derive a formula for diagonal matrix elements of the $1$-pdm. Let $(h_1,h_2,\ldots)$ be an ordered orthonormal basis of $\HH$ and define the Fock basis with reference to this system of orbitals. Let $\hat{a}_i^*$ and $\hat{a}_i$ denote $\hat{a}^*(h_i)$ and $\hat{a}(h_i)$, respectively. Using the anticommutation relations, one can verify from (\[Fock basis vector\]) that $$\hat{a}_i^* \hat{a}_i |\bn\rangle \equals \bn(i) |\bn\rangle \ .$$ Therefore $$\begin{aligned}
\label{property of $1$-pdm}
\langle h_i|\gamma_{\Delta} h_i \rangle
& \stackrel{(\ref{defining property of $1$-pdm again})}{=} &
\Tr( \Delta \hat{a}_i^* \hat{a}_i)
\equals
\sum\limits_{\bn \in \NN} \langle \bn | \Delta \hat{a}_i^* \hat{a}_i |\bn\rangle
\equals
\sum\limits_{\bn \in \NN} \bn(i)\langle \bn | \Delta |\bn\rangle
\nonumber \\
&=&
\sum\limits_{\bn \in \NN :\ \bn(i)=1} \langle \bn | \Delta |\bn\rangle \ ,\end{aligned}$$ an expression for the probability that the $i^{th}$ reference orbital is occupied.
Gauge-invariant quasi-free states {#Gauge-invariant quasi-free states}
---------------------------------
Recall that $\mathfrak{A}(\HH)$ denotes the (abstract) CAR algebra over a Hilbert space $\HH$ and $\pi\big( \mathfrak{A}(\HH) \big)$ denotes its (Fock) representation as a subalgebra of $\BB(\FFF)$. A state $\omega$ on $\mathfrak{A}(\HH)$ is “quasi-free" if its $1$-particle correlations $\omega\big( \hat{a}^*(f)\hat{a}(g)\big) $ and “anomalous" correlations $\omega\big( \hat{a}(f)\hat{a}(g)\big)$ determine all of its higher correlations $$\omega\big( \hat{a}^*(f_1)\cdots \hat{a}^*(f_n)a(g_m)\cdots \hat{a}(g_1)\big)$$ via Wick’s formula, as in formula (2a.11) of Ref. [@BachLiebSolovej]. The anomalous correlations of a [*gauge-invariant*]{} state vanish, and Wick’s formula for gauge-invariant quasi-free states can be expressed compactly in terms of the state’s $1$-pdm:
A state $\omega$ on $\mathfrak{A}(\HH)$ is “gauge-invariant quasi-free" [@AlickiFannes] if there exists a bounded operator $Q$ on $\HH$ such that $$\label{giqf Wicks}
\omega\big(\ \hat{a}^*(f_1)\cdots \hat{a}^*(f_n)\hat{a}(g_m)\cdots a(g_1)\ \big)
\equals
\delta_{mn}
\det \big[ \langle g_i, Q f_j \rangle \big]_{i,j=1}^n$$ for all $f_1,\ldots,f_n,g_1,\ldots,g_m \in \HH$. $Q$ is what we call the $1$-pdm of $\omega$. Formula (\[giqf Wicks\]) in the case $m=n=1$ implies that $Q$ has to be a positive-semidefinite contraction. It is known that, conversely, for any positive-semidefinite contraction $Q$ on $\HH$, there exists a unique gauge-invariant quasi-free state satisfying (\[giqf Wicks\]).
Formula (\[giqf Wicks\]) also implies a couple of closure properties for gauge-invariant quasi-free (GIQF) states:
[1.]{} If a sequence of GIQF states converges (pointwise) to a state, the limit is also GIQF.
[2.]{} Let $\HH_1$ denote a closed subspace of $\HH$. The CAR algebra $\mathfrak{A}(\HH_1)$ may be identified with a C$^*$-subalgebra of $\mathfrak{A}(\HH)$, and states on the latter induce states on $\mathfrak{A}(\HH_1)$ by restriction. The restriction to $\mathfrak{A}(\HH_1)$ of a GIQF state on $\mathfrak{A}(\HH)$ is also GIQF.
We are particularly interested in states represented by density operators on the Fock space $\FFF$. The restriction of such a state to $\pi\big( \mathfrak{A}(\HH) \big) \subset \BB(\FFF)$ defines a state on the CAR algebra $\mathfrak{A}(\HH)$. We say that a density operator $\Gamma$ on $\FFF$, or the normal state corresponding to it, is GIQF if its restriction to the CAR subalgebra of $\BB(\FFF)$ is GIQF. Denoting the $1$-pdm of $\Gamma$ by $\gamma_\Gamma$, the Wick relations (\[giqf Wicks\]) for a GIQF density operator $\Gamma$ are that $$\label{Wicks}
\Tr\big(\ \Gamma \ \hat{a}^*(f_1)\cdots \hat{a}^*(f_n)a(g_m)\cdots a(g_1) \ \big)
\equals
\delta_{mn} \det \big[\langle g_i, \gamma_\Gamma f_j \rangle \big]_{i,j=1}^n$$ for all $m,n$ such that $m+n>0$ and all $f_1,\ldots,f_n,g_1,\ldots,g_n \in \HH$.
Free states {#Free states appendix}
-----------
By our definition, a many-fermion state is free if it is represented by a GIQF density operator on a fermion Fock space and has finite expected particle number. Since substates of GIQF states are GIQF, and since substates of states of finite expected particle number also have finite expected particle number, substates of free states are free.
The $1$-pdm of a free density operator on $\FFF(\HH)$ is positive-semidefinite contraction on $\HH$ with finite trace. Conversely, any positive-semidefinite contraction operator on $\HH$ is the $1$-pdm of a unique free state on $\FFF(\HH)$.
\[free existence\] Suppose $Q:\HH \longrightarrow \HH$ is a positive-semidefinite contraction operator with finite trace. Then there exists a unique free density operator on $\FFF(\HH)$ with $1$-pdm $Q$.
Since $Q$ is a positive-semidefinite contraction operator with finite trace, it has a spectral decomposition $$\label{1-matrix spectral}
Q \equals \sum p_i |h_i\rangle\!\langle h_i|$$ where $\{h_1,h_2,\ldots\}$ is an orthonormal basis of $\HH$ consisting of eigenvectors of $Q$. The corresponding eigenvalues $p_i$ all lie in the interval $[0,1]$ and their sum, the trace of $Q$, is finite.
Let $\big\{ |\bn\rangle : \bn \in \NN \big\}$ denote the Fock basis of $\FFF(\HH)$ defined with respect to the ordered basis $(h_1,h_2,\ldots)$ of reference orbitals, as in formula (\[Fock basis vector\]). Define $$\Gamma \equals \sum_{\bn \in \NN} \Big\{ \prod_i p_i^{\bn(i)} (1-p_i)^{1-\bn(i)} \Big\}|\bn \rangle\!\langle \bn |\ .$$ The off-diagonal matrix elements of the $1$-pdm $\gamma_{\Gamma}$ with respect to the basis $(h_1,h_2,\ldots)$ are all equal to $0$. Using formula (\[property of $1$-pdm\]) it is easy to show that the diagonal matrix element $\langle h_i|\gamma_{\Gamma} h_i \rangle$ equals $p_i$. Thus the $1$-pdm of $\Gamma$ equals $Q$. As $\Tr (Q) = \sum p_i$ is finite, $\Gamma$ has finite average particle number.
To show that $\Gamma$ is GIQF, we have to verify that Wick’s relations are satisfied. It is fairly straightforward to verify that the Wick’s relations (\[Wicks\]) are satisfied when $f_1,\ldots,f_n$ and $g_1,\ldots,g_m$ all belong to the set $\{h_1,h_2,\ldots\}$. This suffices to show that all relations (\[Wicks\]) are satisfied. For fixed $m$ and $n$, the left-hand and right-hand sides of (\[Wicks\]) are bounded multilinear forms in $f_1,\ldots,f_n$ and $g_1,\ldots,g_m$. Since these bounded multilinear forms agree when the $f$’s and $g$’s are all drawn from the same orthonormal basis of $\HH$, they must be equal.
Thus $\Gamma$ is GIQF and its $1$-pdm $Q$ has finite trace. This means that $\Gamma$ is a free density operator with $1$-pdm $Q$. No other GIQF density operator on $\FFF(\HH)$ can have the same $1$-pdm, since the $1$-pdm of a GIQF density operator determines all higher correlations via Wick’s relations (\[Wicks\]), and no other density operator on $\FFF(\HH)$ can have all the same correlations.
The proof of the preceding proposition can be modified to prove the following characterization of free states:
\[structural\] A density operator $\Gamma$ on the fermion Fock space $\FFF(\HH)$ is free if and only if there exists an ordered orthonormal basis $(h_1,h_2,\ldots)$ of $\HH$ and real numbers $p_i \in [0,1]$ with $\sum p_i < \infty$ such that $$\label{Free density}
\Gamma \equals \sum_{\bn \in \NN} \Big\{ \prod_i p_i^{\bn(i)} (1-p_i)^{1-\bn(i)} \Big\}|\bn \rangle\!\langle \bn |$$ when written in terms of the Fock basis vectors $|\bn \rangle$ that are indexed by the occupation numbers $ \bn = \big(\bn(1),\bn(2),\bn(3),\ldots \big) $ of the reference orbitals in $(h_1,h_2,\ldots)$.
Corollary \[structural\] provides us with a convenient structural formula for free states that we will use repeatedly in the sequel. The reference orbitals $h_i$ of the free density operator defined by formula (\[Free density\]) are its natural orbitals, and the $p_i$ are its natural occupation numbers.
Formula (\[Free density\]) shows clearly that, in a free state, the natural orbitals are occupied or unoccupied independently of one another. The free density operator (\[Free density\]) is a mixture of Fock states $|\bn \rangle\!\langle \bn |$, and the weight assigned to the configuration $\bn$ is the probability of obtaining the outcome $\bn$ in a sequence of independent Bernoulli trials for the occupations $\bn(i)$ of reference orbitals $h_i$. A calculation using (\[Free density\]) shows that the von Neumann entropy of a free state with natural occupation numbers $p_i$ is $$S(\Gamma) \ = \ -\sum_i p_i \log p_i -\sum_i (1-p_i) \log (1-p_i) \ .$$
Proof of Proposition 1 {#Proof of Proposition 1}
----------------------
We recall the statement of Proposition \[free at last\].
[ *A density operator on the fermion Fock space $\FFF(\HH)$ is free if and only if (i) its $1$-pdm has finite trace, and (ii) it is a limit in trace norm of density operators that represent substates of Slater determinant states.* ]{}
Slater determinants states are free: they are the free states whose $1$-matrices are finite-rank orthogonal projectors. Substates of Slater determinant states are free, because all substates of free states are free. Limits of GIQF states are also GIQF. Therefore, any density operator $\Gamma$ on $\FFF(\HH)$ that satisfies (ii) is GIQF. If, in addition, the $1$-pdm of $\Gamma$ has finite trace, then $\Gamma$ free. This proves the sufficiency of (i) and (ii).
To prove the necessity of conditions (i) and (ii), we show that any free state is a limit of free states whose $1$-matrices have finite rank, and that any free states whose $1$-matrix has finite rank is a substate of a Slater determinant state.
Let $\Gamma$ be a free density operator with spectral representation (\[Free density\]) and $1$-pdm (\[1-matrix spectral\]). Define $$\label{1-matrix compression}
Q_N \equals \sum_{i=1}^N p_i |h_i\rangle\!\langle h_i| \ .$$ Let $\Gamma_N$ be the unique free density with $1$-pdm $Q_N$. The probabilities $$\mathcal{P}_N(\bn) = \prod_{i=1}^N p_i^{\bn(i)} (1-p_j)^{1-\bn(i)}$$ converge for each $\bn \in \NN$ to the probabilities appearing as coefficients in (\[Free density\]). Since the probability measures $\mathcal{P}_N$ converge pointwise to a probability measure on $\NN$, they converge in $\ell^1(\NN)$, and the corresponding density operators $\Gamma_N$, which are all diagonal with respect to the same Fock basis, converge in trace norm to $\Gamma$.
To conclude the proof, we show that the free density operators $\Gamma_N$ can be represented as substates of a Slater determinant states. We shall construct a Slater determinant $\Phi $ out of vectors in a larger Hilbert space $\HH'$, such that $Q_N$ is the $1$-pdm of the substate delimited by the orbitals in the subspace $\HH$.
Let $\HH' = \HH \oplus \mathrm{span}\{ k'_1,k'_2,\ldots,k'_N\}$, where $\{ k'_1,k'_2,\ldots,k'_N\}$ is an orthonormal set of extraneous vectors, and define the Slater determinant $ \Phi \in \wedge^N \HH'$ by $$\Phi \equals ( \sqrt{p_1} k_1 + \sqrt{1- p_1} k'_1 ) \wedge ( \sqrt{p_2} k_2 + \sqrt{1- p_2} k'_2 ) \wedge \cdots \wedge ( \sqrt{p_N} k_N + \sqrt{1- p_N} k'_N ) \ .$$ The substate of $|\Phi \rangle\!\langle \Phi |$ delimited by the closed subspace $ \HH \cong \HH \oplus \{0\}\subset \HH'$ has $1$-pdm $Q_N$ of formula (\[1-matrix compression\]). Thus $\Gamma$ is a limit in trace norm of a sequence of density operators $\Gamma_N$ that represent substates of Slater determinant states.
Proof of Lemma 1 {#Proof of Lemma 1}
----------------
To prove the propositions in Sec. \[Special properties of nonfreeness\], we used the fact that $$\label{the fact}
- \Tr(\Delta \log\Gamma ) = - \Tr(\Gamma_\Delta \log\Gamma )$$ whenever $\Gamma$ is free. We proved this fact only in the case where all of the natural occupation numbers of $\Gamma$ lie strictly between $0$ and $1$.
General free states, where some of the $p_i$ may equal $0$ or $1$, are limits of Gibbs states (cf., Lemma 2.4 of Ref. [@BachLiebSolovej]). However, we prefer to deal with free states directly, rather than as limits of Gibbs states. To prove formula (\[the fact\]) in this spirit we have to keep an eye on the kernels of $\gamma_{\Delta}$ and $I-\gamma_\Gamma$.
\[lemma fermions\] Let $\Gamma,\Delta \in \DD(\FFF)$ be two density operators on the fermion Fock space $\FFF$ with $1$-matrices $\gamma_{\Delta}$ and $\gamma_{\Gamma}$. Suppose that $\Gamma$ is free.
The following are equivalent:\
[(i)]{}$\ker \Gamma \subset \ker \Delta$\
[(ii)]{}$\ker {\gamma_{\Gamma}} \subset \ker {\gamma_{\Delta}}$ and $\ker (I-\gamma_{\Gamma}) \subset \ker (I-\gamma_{\Delta})$
Consider a fermionic free density operator $\Gamma$, written as in formula (\[Free density\]). Let $J_1$ denote the set of indices $i$ for which $p_i=1$. Note that $J_1$ is a finite set, because $\sum p_i$ is assumed to be finite. Let $J_0$ denote the set of indices $j$ for which $p_j=0$. It may happen that $J_1 \cup J_0$ is the entire index set for the orbitals; in that case $\Gamma$ is a Slater determinant state or the vacuum state. Define $$\label{the notation for the index set: fermions}
\NN_{\Gamma} \equals \big\{ \bn : \ \bn(j) = 1 \hbox{ if } j \in J_1 \hbox{ and } \bn(j) = 0 \hbox{ if } j \in J_0 \big\}\ .$$ Let $\FFF_{\Gamma}$ denote the the closure of $\mathrm{span}\big\{ |\bn\rangle: \bn \in \NN_{\Gamma} \big\}$, a subspace of the fermion Fock space $\FFF(\HH)$. Then we can see from (\[Free density\]) that $$\label{Quasifree fermions rewritten in heaven}
\Gamma \equals \sum_{\bn \in \NN_{\Gamma}} \Big\{ \prod_{j \notin J_1 \cup J_0} p_j^{\bn(i)} (1-p_j)^{1-\bn(j)} \Big\}|\bn \rangle\!\langle \bn |$$ and $$\label{kernel of Gamma: fermions}
\ker \Gamma % \equals {\FFF_{\Gamma}}^{\ \perp}
\equals \overline{ \hbox{span} } \big\{ |\bn\rangle: \bn \notin \NN_{\Gamma} \big\} \ .$$ First we prove that (i) implies (ii).
Assume that $\ker \Gamma \subset \ker \Delta$.
If $\gamma_{\Gamma} f_j = 0$, then $\langle f_j | \gamma_{\Gamma}| f_j \rangle = 0$, and therefore, by (\[property of $1$-pdm\]), $\langle \bn | \Gamma | \bn\rangle = 0$ for all $\bn$ such that $\bn(j) = 1$. Therefore, if $\bn(j) = 1$, then $| \bn\rangle \in \ker \Gamma$ and hence also $| \bn\rangle \in \ker \Delta$. This implies that $\langle f_j | \gamma_{\Delta}| f_j \rangle = 0$, again by (\[property of $1$-pdm\]), and therefore $\gamma_{\Delta} f_j = 0$.
Similarly, if $(I-\gamma_{\Gamma} )f_j = 0$, then $\gamma_{\Gamma} f_j = f_j$ and therefore $1= \langle f_j | \gamma_{\Gamma}| f_j \rangle $. By (\[property of $1$-pdm\]), $\langle \bn | \Gamma | \bn\rangle = 0$ for all $\bn$ such that $\bn(j) = 0$. Since $\ker \Gamma \subset \ker \Delta$, also $\langle \bn | \Delta | \bn\rangle = 0$ for all $\bn$ such that $\bn(j) = 0$, and therefore $\langle f_j | \gamma_{\Delta}| f_j \rangle = 1$, or $(I-\gamma_{\Delta} )f_j = 0$.
The last few paragraphs establish (ii). Now we prove that (ii) implies (i).
Assume that $\ker {\gamma_{\Gamma}} \subset \ker {\gamma_{\Delta}}$ and $\ker (I-\gamma_{\Gamma}) \subset \ker (I-\gamma_{\Delta})$. We wish to prove that $\ker \Gamma \subset \ker \Delta$. By (\[kernel of Gamma: fermions\]) it suffices to show that every $|\bn\rangle$ with $ \bn \notin \NN_{\Gamma} $ is in the kernel of $ \Delta$. Every $ \bn \notin \NN_{\Gamma} $ has either $\bn(j)=1$ for some $j \in J_0$, or $\bn(j)=0$ for some $j \in J_1$. In both cases, $|\bn\rangle \in \ker \Delta$, as we now show.
Suppose $\bn(j)=1$ for some $j \in J_0$. Then $f_j \in \ker \gamma_{\Gamma}$ and, since $\ker {\gamma_{\Gamma}} \subset \ker {\gamma_{\Delta}}$, also $f_j \in \ker \gamma_{\Delta}$ and therefore $\langle f_j | \gamma_{\Delta}| f_j \rangle = 0$. By (\[property of $1$-pdm\]), $\langle \bn | \Delta | \bn\rangle = 0$ for all $\bn$ such that $\bn(j)=1$. Thus, $|\bn\rangle \in \ker \Delta$ if $\bn(j)=1$ for some $j \in J_0$.
Suppose $\bn(j)=0$ for some $j \in J_1$. Then $f_j \in \ker (I-\gamma_{\Gamma})$ and therefore, by assumption, $f_j \in \ker (I-\gamma_{\Delta})$. This implies that $\gamma_{\Delta}f_j = f_j$, $\langle f_j | \gamma_{\Delta}| f_j \rangle = 1$, and $\langle \bn | \Delta | \bn\rangle = 0$ for all $\bn$ such that $\bn(j)=0$. Thus, $|\bn\rangle \in \ker \Delta$ if $\bn(j)=0$ for some $j \in J_1$.
\[first cor\] $\ker \Gamma_\Delta \subset \ker \Delta$.
\[second cor\] If $\Gamma$ is free, then $\ker \Gamma \subset \ker \Delta$ if and only if $\ker \Gamma \subset \ker \Gamma_\Delta$.
Using these corollaries, we now complete the proof of Lemma 1. Recall that lemma:
[ *Suppose $\Delta \in \DD(\FFF)$ and let $\Gamma_{\Delta}$ denote the unique free state that has the same $1$-pdm as $\Delta$. If $\Gamma$ is free then $$\label{conclusion again} - \Tr(\Delta \log\Gamma ) \equals - \Tr(\Gamma_\Delta \log\Gamma )\ . \$$* ]{}
Recall the notation used in formulas (\[the notation for the index set: fermions\]) and (\[Quasifree fermions rewritten in heaven\]). The unbounded operator $\log \Gamma$, defined as in Section \[Relative entropy for density operators\], is $$\begin{aligned}
\lefteqn {\log \Gamma
\equals
\sum_{\bn \in \NN_\Gamma} \sum_{i \notin J_1 \cup J_0} \Big( \bn(i) \log(p_i)
+ (1-\bn(i)) \log(1-p_i) \Big) \ | \bn \rangle\!\langle\bn | } \\
& = &
\sum_{i \notin J_1 \cup J_0} \Bigg[ \log(p_i) \sum_{\bn \in \NN_\Gamma: \bn(i) = 1 } | \bn \rangle \! \langle\bn |
\ + \ \log(1-p_i) \sum_{\bn \in \NN_\Gamma: \bn(i) = 0 } | \bn \rangle\!\langle\bn | \Bigg] \ .\end{aligned}$$
If $\bn \notin \NN_\Gamma$ then $ | \bn\rangle \in \ker(\Gamma) \subset \ker \Delta$, and therefore $\langle \bn | \Delta | \bn\rangle = 0$. Thus, if $\ker \Gamma \subset \ker \Delta$, then $$\sum\limits_{\bn \in \NN_\Gamma:\ \bn(i) = 1 }\langle \bn | \Delta |\bn\rangle
\equals
\sum\limits_{\bn \in \NN: \bn(i) = 1} \langle \bn | \Delta |\bn\rangle
\equals
\langle f_i|\gamma_{\Delta} | f_i \rangle$$ by (\[property of $1$-pdm\]). Using this, we have that $$\begin{aligned}
\lefteqn{ - \Tr(\Delta \log\Gamma ) }
\nonumber \\
&= &
- \sum_{i \notin J_1 \cup J_0} \Bigg[ \log(p_i) \sum_{\bn \in \NN_\Gamma: \bn(i) = 1 }
\Tr\big( \Delta |\bn \rangle\!\langle\bn | \big)
\ + \ \log(1-p_i) \sum_{\bn \in \NN_\Gamma: \bn(i) = 0 }\Tr\big( \Delta |\bn \rangle\!\langle\bn | \big) \Bigg]
\nonumber \\
& = &
- \sum_{i \notin J_1 \cup J_0}
\Bigg[ \log(p_i) \sum_{\bn \in \NN_\Gamma: \bn(i) = 1 }
\langle \bn | \Delta |\bn\rangle
\ + \ \log(1-p_i) \sum_{\bn \in \NN_\Gamma: \bn(i) = 0 } \langle \bn | \Delta |\bn\rangle \Bigg]
\nonumber \\
& = &
- \sum_{i \notin J_1 \cup J_0} \Big[ \log(p_i)\langle f_i|\gamma_{\Delta}| f_i \rangle
+ \log(1-p_i) \big(1 - \langle f_i|\gamma_{\Delta} | f_i \rangle\big) \Big]
\label{maximizeme fermions}\end{aligned}$$ provided $\ker \Gamma \subset \ker \Delta$.
Formula (\[maximizeme fermions\]) is valid provided that $ \ker \Gamma \subset \ker \Delta$. By Corollary \[second cor\], if $ \ker \Gamma \subset \ker \Delta$ then also $\ker \Gamma \subset \ker \Gamma_{\Delta}$. Since $\Delta$ and $\Gamma_{\Delta}$ have the same $1$-pdm $\gamma_\Delta$, formula (\[maximizeme fermions\]) implies the conclusion (\[conclusion again\]) if $ \ker \Gamma \subset \ker \Delta$.
If $\ker \Gamma \not\subset \ker \Delta$, then $\ker \Gamma \not\subset \ker \Gamma_\Delta$ by Corollary \[first cor\], and both $-\Tr (\Delta \log \Gamma)$ and $-\Tr (\Gamma_\Delta \log \Gamma)$ equal $+\infty$ by definition. The conclusion (\[conclusion again\]) holds trivially in this case.
[**Acknowledgments:**]{}
This work has been supported by the Austrian Science Foundation (FWF) under grant F41 (SFB “VICOM") and W1245 (DK “Nonlinear PDEs"). This review has benefited from insightful suggestions by anonymous referees of our unpublished manuscript [@GottliebMauserArchived].
[99]{}
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[^1]: Wolfgang Pauli Institute c/o Fakultät f. Mathematik, Universität Wien, Oskar-Morgensternplatz 1, 1090 Vienna, Austria
| 0 |
---
abstract: 'An Euclidean first-passage percolation (FPP) model describing the competing growth between $k$ different types of infection is considered. We focus on the long time behavior of this multi-type growth process and we derive multi-type shape results related to its morphology.'
address: |
Institut de Mathématiques\
École Polytechinique Fédérale de Lausanne\
CH-1015 Lausanne\
Switzerland\
author:
- 'Leandro P. R. Pimentel'
title: 'Multi-type shape theorems for FPP models'
---
Introduction {#int}
============
In standard planar first-passage percolation [@HW65] each pair $\x$ and $\y$ of nearest-neighbor of $\bZ^2$ has an edge connecting them and each edge is equipped with a non-negative random variable (passage time) which may be interpreted as the time it takes for an infection to be transmitted from $\x$ to $\y$. We assume these random variables are i.i.d. with a continuous distribution $\bF$. The passage time $t(\gamma)$ for a nearest-neighbor path $\gamma$ is simply the sum of the passage times along the path. For $\x,\y\in\bZ^2$, the first-passage time from $\x$ to $\y$, which we denote $T(\x,\y)$, is the infimum of $t(\gamma)$ over all paths $\gamma$ from $\x$ to $\y$. For $t\geq 0$, let $\B(t)$ be the set of sites $\x$ reached from the origin $\0$ by time $t$, i.e. $T(\0,\x)\leq t$. One may think of sites in $\x\in \B(t)$ as infected and those in $\B(t)^c$ as healthy, and that at time $0$ the origin $\0$ is infected by some type of disease. The process $\big\{\B(t)\,:\,t\geq 0\big\}$ is then a model for the growth of an infection.
An interesting aspect of the evolution of the infection, namely the *tree of infection*, is constructed as follows. First notice that, since the passage time distribution is continuous, for all $\x,\y\in\bZ^2$ there is (almost surely) an unique time-minimizing path (or geodesic) from $\x$ to $\y$, which we denote $\rho(\x,\y)$, such that $T(\x,\y)=t\big(\rho(\x,\y)\big)$. Thus $\rho(\x,\y)$ may be interpreted as the path through which the infection was transmitted from $\0$ to $\x$. With this picture in mind, the tree of infection $\Gamma$ is defined by the union of edges $\e\in\rho(\0,\x)$ over all $\x\in\bZ^2$. Newman [@n95] has shown that the number $K(\Gamma)$ of topological ends of $\Gamma$, i.e. the number of semi-infinite self-avoiding paths in $\Gamma$, is infinite provided an exponential moment condition on $\bF$ and a certain hypothesis concerning the uniformly bounded curvature of the asymptotic shape of $\B(t)$. In spite of the curvature hypothesis is plausible it has so far not been proved.
In order to study the tree of infection, Hägggström and Pemantle [@hp98; @hp99] have introduced a multi-type growth model as follows. At time $0$ we start with $k$ different sites of $\bZ^2$, say $\x_1,\dots,\x_k$, each one representing a different type of infection. A site $\y\in\bZ^2$ is then infected at time $\min\big\{T(\x_1,\y),\dots,T(\x_k,\y)\big\}$ and it is acquired by the infection which first arrives there, i.e. by the unique type $j\in\{1,\dots,k\}$ such that $T(\x_j,\y)=\min\big\{T(\x_1,\y),\dots,T(\x_k,\y)\big\}$ (Figure \[f2\]) . It may happens that at some early stage one of the types of infection completely surrounds another one, which then is prevented to grow indefinitely. If this does not occur, or equivalently, if all types of infection grow unboundedly, we say that $k$-coexistence occurs.
![Growth and Competition[]{data-label="f2"}](tfgrow){width="50.00000%"}
Turning back to the question of topological ends of $\Gamma$, Häggström and Pemantle have noticed that if $k$-coexistence occurs with positive probability then $K(\Gamma)\geq k$ occurs with positive probability. They also have shown that, if one considers an exponential passage time distribution then $2$-coexistence occurs with positive probability, and thus $K(\Gamma)\geq 2$ occurs with positive probability. Later Garet and Marchand [@gm04] and Hoffman [@H05] have extended this last result for stationary and ergodic FPP models on $\bZ^d$.
In this work we focus on the long time behavior of this multi-type growth model. However, differently from the above mentioned authors, we choose a first-passage percolation set-up on a random Delaunay triangulation [@VW90] whose spherical symmetry (isotropy) ensures that the asymptotic shape of the corresponding growth process is an euclidean ball. This choice allows us to prove various statements concerning minimizing paths, such as $\bP\big(K(\Gamma)=\infty\big)=1$, who could mostly only be conjectured by Newman in the standard model. In this setting, the main results we will prove are the following:
- If a type of infection survives then the region it conquers is (asymptotically) a cone with a random angle (Theorem \[t1\], Remark \[rand+strai\]);
- If the $k$ initial sites form a regular polygon centered at the origin with radius $r$, then the probability that $k$ coexistence occurs tends to $1$ when $r$ tends to infinity. Moreover, for all $\epsilon>0$, the probability that for all $j\in\{1,\dots,k\}$ the region conquered by infection $j$ contains (asymptotically) the cone with axis through $\0$ and $\x_j$ and angle $\frac{\pi}{k}-\epsilon$ also tends to $1$ (Theorem \[t2\]).
The main idea to prove our results is to explore the relation between this multi-type growth model and the asymptotic behavior of $T(\x,\y_n)-T(\0,\y_n)$ when $\y_n$ goes to infinity along a ray of angle $\alpha$ (Theorem \[tBuse-1\] and Theorem \[tBuse-2\]). We also study some roughening aspects of the one-dimensional boundary between the infections, namely the *competition interface*, which were pointed out by physicists in numerical simulations [@dd91; @sk95] (Remark \[rand+strai\]). We note that analogous problems in the context of last-passage percolation and totally asymmetric exclusion processes were treated by Ferrari and Pimentel [@fp04-1] and Ferrari, Martin and Pimentel [@fp04-2]. Deijfen, Häggström and Bagley [@dhb03] have also considered isotropic multi-type growth models in $\bR^d$ where the growth is driven by outbursts in the infected region.
Multi-type growth process
-------------------------
Consider the random graph $\calD:=(\calD_v,\calD_e)$, named the *Delaunay triangulation*, constructed as follows. The vertex set $\calD_v\subseteq\bR^2$ is the set of points realized in a two-dimensional homogeneous Poisson point process with intensity $1$. To each vertex $\v$ corresponds an open and bounded polygonal region $\C_\v$ (the Voronoi tile at $\v$) consisting of the set of points of $\bR^2$ which are closer to $\v$ than to any other $\v'\in\calD_v$. The edge set $\calD_e$ consists of non oriented pairs $(\v,\v')$ such that $\C_\v$ and $\C_{\v'}$ share a one-dimensional edge (Figure \[f1\]). One can see that (with probability one) each Voronoi tile is a convex and bounded polygon, and the graph $\calD :=(\calD_v,\calD_e)$ is a triangulation of the plane [@M91].
![The Delaunay triangulation and the Voronoi tessellation[]{data-label="f1"}](vor2){width="30.00000%"}
The *Voronoi tessellation* $\calV:=(\calV_v,\calV_e)$ is defined by taking the vertex set $\calV_v$ equal to the set of vertices of the Voronoi tiles and the edge set $\calV_e$ equals to the set of edges of the Voronoi tiles.
Each edge $\e\in\calD_e$ is independently assigned a nonnegative random variable $\tau_\e$ from a common distribution $\bF$ (the passage time distribution) that is independent of the Poisson process $\calD_v$. We assume throughout that $\bF$ is continuous and that $$\label{a1}
\int e^{ax}\bF(dx)<\infty\,\mbox{ for some }\,a\in(0,\infty)\,.$$ We denote by $(\Omega,\calF,\bP)$ our underline probability space, i.e. from each realization $\omega\in\Omega$ one can determine the Poisson point process as well the passage time configuration. This model inherits the euclidean (translation and rotational) invariance of the Poisson point process.
The passage time $t(\gamma)$ of a path $\gamma$ in $\calD$ is the sum of the passage times of the edges in $\gamma$: $$t(\gamma):=\sum_{\e\in\gamma}\tau_\e \,.$$ The first-passage time between two vertices $\v$ and $\v'$ in $\calD_v$ is defined by $$T(\v,\v'):=\inf\big\{t(\gamma);\,\gamma\in\calC(\v,\v')\big\} ,$$ where $\calC(\v,\v')$ the set of all paths connecting $\v$ to $\v'$. We extend the first-passage time $T$ to $\x,\y\in\bR^2$ by setting $T(\x,\y):=T\big(\v(\x),\v(\y)\big)$, where $\v(\x)$ is the almost sure unique vertex $\v\in\calP$ with $\x\in\C_{\v}$. We say that $\rho(\v,\v')\in\calC(\v,\v')$ is a geodesic between $\v$ and $\v'$ if $t\big(\rho(\v,\v')\big)=T(\v,\v')$. For each $\x,\y\in\bR^2$ we denote $\rho(\x,\y):=\rho\big(\v(\x),\v(\y)\big)$. One can see that if $\bF$ is a continuous function then, almost surely, for all $\v,\v'\in\calD_v$ there exists a unique geodesic $\rho(\v,\v')$ [@p-105]. A self-avoiding and semi-infinite path $\rho=(\v_1,\v_2,\dots)$ in $\calD$ is called a semi-infinite geodesic if for all $\v_j,\v_k\in\rho$, the path $(\v_j,\v_{j+1},...,\v_k)$ is the unique geodesic connecting $\v_j$ to $\v_k$.
Given $k$ different points $\x_1,...,\x_k\in\bR^2$, the initial configuration of seeds, we define the multi-type growth process $\big\{(\B_{\x_1}(t),...,\B_{\x_k}(t))\,:\,t\geq 0\big\}$ by $$\B_{\x_j}(t):=\big\{\x\in\bR^2\,:\,\x\in c(\C_\v)\,\mbox{ for some }\,\v\in\calB_{\x_j}(t)\big\}\,,$$ where $$\calB_{\x_j}(t):=\big\{\v\in\calD_v\,:\,T(\x_j,\v)\leq t\,\mbox{ and }\,\min_{l=1,...,k}\{T(\x_l,\v)\}=T(\x_j,\v)\big\}\,,$$ and $c(\C_\v)$ denotes the closure of the tile $\C_\v$. If there exists $j<l$ such that $\v_{\x_j}=\v_{\x_l}$ then we set $\B_{\x_j}$ as before and $\B_{\x_l}(t)=\emptyset$.
When $k=1$ then we have a single growth process $\B_\x(t)$ which represents the set of points reached by time $t$ from the initial seed $\x$. For a continuous distribution $\bF$ satisfying the following shape theorem [@p-105; @VW92] holds: there exists a constant $\mu(\bF)\in(0,\infty)$, namely the time constant, such that for all $\epsilon>0$ $$\bP\big((1-\epsilon )t \D(1/\mu)\subseteq \B_\0(t)\subseteq (1+\epsilon )t
\D(1/\mu)\mbox{ eventually }\big)=1 \,,$$ where $\D(r):=\{\x\in\bR^2 \,:\, |\x|\leq r\}$ and $\0:=(0,0)$.
When $k\geq2$ the process $\big\{(\B_{\x_1}(t),...,\B_{\x_k}(t))\,:\,t\geq 0\big\}$ is a model for competing growth on the plane where each point $\x\in\bR^2$ is acquired by the specie $j\in\{1,\dots,k\}$ which first arrives there. The competition interface $\psi$ is the one-dimensional boundary between the species when $t=\infty$. This interface can be seen as a finite union of polygonal curves determined by edges in $\calV$ (the Voronoi tessellation) which are shared by tiles in different species. A branch of the competition interface is a self-avoiding path $\varphi=(\x_n)_{n\geq 1}$ in $\calV$ such that $\{\x_n\,:\,n\geq 1\}\subseteq\psi$.
For each $\alpha\in[0,2\pi)$ we say that a self-avoiding path $(\x_{n})_{n\geq 1}$, with vertices in $\bR^2$ and such that $|\x_n|\to\infty$ when $n\to\infty$, is a $\alpha$-path if $$\lim_{n\to\infty}\frac{\x_{n}}{|\x_{n}|}=e^{i\alpha}:=(\cos\alpha,\sin\alpha)\,.$$ In this case we also say that $(\x_n)_\bN$ has the asymptotic orientation $e^{i\alpha}$. This is equivalent to $$\lim_{n\to\infty}ang(\x_{n},e^{i\alpha})=0\,,$$ where $ang(\x,\y)$ denotes the angle in $[0,\pi]$ between the points $\x,\y\in\bR^2$. Thus, a sufficient condition for a path $(\x_{n})_{n\geq 1}$ to be a $\alpha$-path for some $\alpha\in[0,2\pi)$ is, for some fixed $\delta\in(0,1)$ and some constant $c>0$, for sufficiently large $n$ $$ang(\x_{n},\x_m)\leq |\x_n|^{-\delta}\,\mbox{ whenever }m > n\,,$$ which is the so called $\delta$-straightness property for semi-infinite paths introduced by Newman [@n95].
\[t1\] For $k\geq 2$ let $\Omega_k$ be the event that, for the competing growth model with $k$-different species, there exists a finite subset $\Theta:=\{\theta_1,...,\theta_m\}$ of $[0,2\pi)$ such that every branch $\varphi$ of the competition interface is a $\theta(\varphi)$-path for some $\theta\in\Theta$. Under , $\bP\big(\Omega_k\big)=1$.
\[rand+strai\] In Section \[geo\] (part \[pr-rand+strai\]) we will give a sketch of the proof that for all $\alpha\in[0,2\pi)$ $$\bP\big(\theta=\alpha\mbox{ for some }\theta\in\Theta\big)=0\,,$$ and that if $\xi\in(3/4,1)$ then, almost surely, for all branch $\varphi=(\x_{n})_{n\geq 1}$ of the competition interface there is a constant $c>0$ such that $$ang(\x_{n},e^{i\theta(\varphi)})\leq c|\x_n|^{\xi-1}\mbox{ eventually }\,.$$
Let $\x_1(r)=(0,r),\dots,\x_k(r)$ be the vertices of a regular polygon with $k$ sides and radius $r$. For each $j=1,...,k$ define the projection of the random set $\B_{j}^r:=\B_{\x_j(r)}(\infty)$ onto $\S^1$, the set of unit vectors $|\x|=1$, by $$\S_{j,r}:=\{\x = e^{i\alpha}\in \S^{1}\,:\,\bL_{s\x}(\alpha)\subseteq \B_{j}^{r}\,\mbox{ for some }\,s>0\}\,,$$ where $\bL_\x(\alpha)$ denotes the line starting from $\x$ and with direction $e^{i\alpha}$. For each $\epsilon\in (0,\pi/k)$ and $j\in\{1,\dots,k\}$ define $$\S_{j}(\epsilon):=\{\x\in\S^1\,:\, ang(\x,\x_j(r))\leq \frac{\pi}{k}-\epsilon\}\,.$$
\[t2\] Let $k\geq 2$. Under , for all $\epsilon>0$ $$\lim_{n\to\infty}\bP\big(\S_{j}(\epsilon)\subseteq \S_{j,n}\,\emph{ for all $j=1,\dots,k$ }\big)=1\,.$$
Busemann type asymptotics and the competition interface
-------------------------------------------------------
To illustrate the approach we follow in this work to study the competition interface assume that $k=2$. Consider the line $\bL_\0(\alpha)$ starting from the origin $\0$ and with direction $e^{i\alpha}$. Then we have three possibilities: i) either it intersets the competition interface infinitely many times; ii) or it is eventually contained in $\B_{\x_1}(\infty)$; iii) or it is eventually contained in $\B_{\x_2}(\infty)$. Notice that the former implies $$\liminf_{s\to\infty}\big(T(\x_{1},se^{i\alpha})-T(\x_{2} ,se^{i\alpha})\big)\leq
0\leq\limsup_{s\to\infty}\big(T(\x_{1},se^{i\alpha})-T(\x_{2},se^{i\alpha})\big)\,,$$ while the second implies $$\limsup_{s\to\infty}\big(T(\x_{1},se^{i\alpha})-T(\x_{2} ,se^{i\alpha})\big)\leq 0\,,$$ and the third implies $$0\leq \liminf_{s\to\infty}\big(T(\x_{1},se^{i\alpha})-T(\x_{2} ,se^{i\alpha})\big)\,.$$ It turns out that the above expressions resemble Busemann type asymptotics for $T$ (see Ballmann [@b95] for more details on this subject). Newman [@n95; @ln96] has shown for the lattice model that, under suitable assumptions on the curvature of the limit shape, $T(\x_1,\y_n)-T(\x_2,\y_n)$ attains eventually a nonzero value $H^{\alpha}(\x_1,\x_2)$, called the Busemann function. By following his method, and by taking profit of the isotropy in our model, we will show[^1]:
\[tBuse-1\] For $\alpha\in[0,2\pi)$ let $\Omega_0(\alpha)$ be the event that for all $\v,\bar{\v}\in\calD_v$, there exists $H^{\alpha}(\v,\bar{\v})$, nonzero for $\v\neq\bar{\v}$, such that $$\label{eBuse-1}
\lim_{{|\x|\to\infty}\atop{\x/|\x|\to e^{i\alpha}}}\big(T(\v,\x)-T(\bar{\v},\x)\big)=H^{\alpha}(\v,\bar{\v})\,.$$ Under , $\bP\big(\Omega_0(\alpha)\big)=1$.
For $\x ,\y\in\bR^2$ we set $H^\alpha(\x,\y):=H^\alpha\big(\v(\x),\v(\y)\big)$. It was conjectured by Howard and Newman [@hn01] that $$\lim_{n\to\infty}\frac{H^{\alpha}(n\vec{e}_1,\0)}{n}=-\mu(\bF)\cos\alpha\,,$$ where $\vec{e}_1:=(1,0)$. This observation is related to the asymptotic behavior of our multi-type growth model and the key result to show Theorem \[t2\] is the following theorem, which is a small step towards the above conjecture.
\[tBuse-2\] For $\alpha\in[0,\pi/2)$ let $\Omega_1(\alpha)\subseteq\Omega_0(\alpha)$ be the event that $$\label{eBuse-2}
-\mu(\bF)\leq \lim\inf_{n\to\infty}\frac{H^{\alpha}(n\vec{e}_{1},\0)}{n}\leq \limsup_{n\to\infty}\frac{H^{\alpha}(n\vec{e}_{1},\0)}{n}\leq -\mu(\bF)\frac{\cos\alpha}{1+\sin\alpha}\,.$$ Under , $\bP\big(\Omega_1(\alpha)\big)=1$. In particular, with probability one, $$\lim_{n\to\infty}\frac{H^{0}(n\vec{e}_{1},\0)}{n}=-\mu(\bF)\,.$$
Overview {#overview .unnumbered}
--------
In Section \[multi\] we will deduce Theorem \[t1\] and Theorem \[t2\] from Theorem \[tBuse-1\] and Theorem \[tBuse-2\]. In Section \[pre\] we will start by defining the probability space where our model takes place and we will show a modification lemma that will play an important rule in the study of coalescence of semi-infinite geodesics. After that we will study some geometrical aspects of Voronoi tilings. We note that in the Delaunay triangulation context some technical difficulties are imposed by its long range dependence. Some of them will be avoided by making references to results of a previous work of the author [@p-105; @p-205]. In the third part we will recall some geometrical lemmas concerning the $\delta$-straightness of semi-infinite paths. Finally, in Section \[geo\] will study existence and coalescence of semi-infinite geodesics to show Theorem \[tBuse-1\] and Theorem \[tBuse-2\]. It will largely parallel the analog study develop by Newman et al [@hn01; @ln96; @n95; @np95] in the lattice and in the Euclidean FPP models.
Proof of the multi-type shape theorems {#multi}
======================================
For each $j=1,...,k$, let $\S_{j}$ denote the set of unit vectors $e^{i\beta}$ such that $\bL_{se^{i\beta}}(\beta)\subseteq \B_{\x_j}(\infty)$ for some $s>0$ and let $$\S_{0}:=(\cup_{j=1}^{l}\S_{j})^{c}\,.$$ Let $$\D_n:=\{e^{i\beta}\,:\,\beta=2k\pi/2^{n}\mbox{ for some }1\leq k\leq 2^{n}\}\,$$ and $\D:=\cup_{n\geq 1}\D_n$. Consider the event $\cap_{\alpha\in\D}\Omega_0(\alpha)$ that for all $\alpha\in\D$ and $\v,\bar{\v}\in\calD_v$ there exists $H^{\alpha}(\v,\bar{\v})$, nonzero for $\v\neq\bar{\v}$, such that $$\lim_{{|\x|\to\infty}\atop{\x/|\x|\to e^{i\alpha}}}\big(T(\v,\x)-T(\bar{\v},\x)\big)=H^{\alpha}(\v,\bar{\v})\,.$$ By Theorem \[tBuse-1\], $\bP\big(\cap_{\alpha\in\D}\Omega_0(\alpha)\big)=1$.
We claim that, on this event, every branch of the competition interface is an $\theta$-path for some $\theta\in[0,2\pi)$. To see this, notice that if $e^{i\alpha}\in \S_{0}$ then for some $j_1 \ne j_2$, $\bL_{\0}(\alpha)$ intersects infinitely many times the region $\B_{\x_{j_1}}(\infty)$ and the region $\B_{\x_{j_2}}$. Thus $$\liminf_{s\to\infty}\big(T(\x_{j_1},se^{i\alpha})-T(\x_{j_2} ,se^{i\alpha})\big)\leq
0\leq\limsup_{s\to\infty}\big(T(\x_{j_1},se^{i\alpha})-T(\x_{j_2},se^{i\alpha})\big)\,,$$ which implies that $\D\cap\S_0=\emptyset$. Let $\C_k^n$ be the cone consisting of points $re^{i\beta}$ such that $r>0$ and $\beta\in(2\pi k/2^n,2\pi(k+1)/2^n)$. Now, if $\D\cap\S_0=\emptyset$ and $e^{i\beta}\in\D$ then every branch $\varphi$ of the competition interface can not intersect infinitely many times the line $\bL_{\0}(\beta)$. So, for each branch $\varphi$ of the competition interface we can find a sequence of cones $(\C^{n}_{k_{n}})_{n\geq1}$, with $n\to\infty$ and $\C^{n+1}_{k_{n+1}}\subseteq \C^{n}_{k_{n}}$, such that $\varphi$ is eventually contained in $C^{n}_{k_{n}}$. This implies that $\varphi$ must be a $\theta$-path for some $\theta\in[0,2\pi)$.
Since $$\bP\big(\S_{j}(\epsilon)\subseteq \S^{r}_{j}\big)=\bP\big(\S_{1}(\epsilon)\subseteq \S^{r}_{1}\big)$$ for all $j=1,...,k$, we only need to prove that $$\label{e1kshape}
\lim_{r\to\infty}\bP\big(\S_{1}(\epsilon)\subseteq \S^{r}_{1}\big)=1\,.$$ To do so, for each $j=1,\dots,k$ let $\alpha^k_j:=\pi(j-1)/k$, $\vec{e}^k_j:=e^{2i\alpha^k_j}$ and $A_{r}:=\cap_{j=1}^{k} A^{j}_{r}$, where $$A_{r}^{j}:= \cap_{l\ne j}\big[H^{\alpha^k_j}(r\vec{e}^k_l,r\vec{e}^k_j)>0\big]\,.$$ Let $\alpha^{+\epsilon}_{k}:=\frac{\pi}{k}-\epsilon$ and $\alpha^{-\epsilon}_{k}:= (2\pi-\frac{\pi}{k})+\epsilon$ and set $$B_{r}(\epsilon):=\cap_{j=2,\dots,k}\big[H^{\alpha^{+\epsilon}_{k}}(r\vec{e}^k_j,r\vec{e}^k_1)>0\mbox{ and }H^{\alpha^{-\epsilon}_{k}}(r\vec{e}^k_j,r\vec{e}^k_1)>0\big]\,.$$ By Theorem \[tBuse-2\] $$\label{e2kshape}
\lim_{r\to\infty}\bP\big(A_{r}\cap B_r(\epsilon)\big)=1\,.$$ The connectivity of the regions $\B_j^r$ yields that, on $A_r\cap B_r(\epsilon)$, $\S_1(\epsilon)\subseteq \S_1^r$. Together with , this yields and the proof of Theorem \[t2\] is complete.
Auxiliary results {#pre}
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The probability space {#pre-ps}
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During the subsequent proofs we will consider the following construction of $(\Omega,\mathcal{F},\bP)$, the underline probability space of our FPP model. Let $\u_{0}=(0,0),\u_{2},\dots$ be a ordering of $\bZ^{2}$ and for each $k\geq 1$ let $$\B_{k}:=\u_{k}+[-1/2,1/2]^{2}\,.$$ Consider $$\calN=\{N_{k}\,:\, k\geq 1\},$$ a collection of i.i.d. Poisson random variables with intensity $1$; $$\calU_k=\{U_{k,l}\,:\, l\geq 1\},$$ a collection of independent random points in the plane so that $U_{k,l}$ has an uniform distribution in the square box $B_{k}$; $$\calT_k=\{\tau_{k,l}^{m,n}\,:\, l\geq 1,m\geq k,n\geq 1\mbox{ and }n>l\mbox{
whenever }k=m\},$$ a collection of i.i.d. non negative random variables with common distribution $\bF$ (the passage time distribution). We also impose that all these collections are independent of each other.
To determine the vertex set $\calD_v=\calP$, at each square box $\B_{k}$ we put $N_{k}$ points given by $U_{k,1},...,U_{k,N_{k}}$. This procedure determines a Poisson point process $\calP$ from the collections $\calN$ and $\calU_k$ with $k\geq 1$. Given $\e\in\calD_e$ we know that there exist an unique pair $(U_{k,l},U_{m,n})$, where either $m>k$ or $m=k$ and $n>l$, so that $\e=(U_{k,l},U_{m,n})$. Set $\tau_{e}=\tau_{k,l}^{m,n}$.
For each $k\geq 1$ denote by $(\Omega^{k},\mathcal{F}^{k},\bP^{k})$ the probability induced by the random variable $N_k$ and the collections $\calU_k$, $\calT_k$. The probability space $(\Omega,\calF,\bP)$ is defined to be the product space of $(\Omega^{k},\calF^k,\bP^{k})$ over $k\geq 1$.
An important step in the construction of the Busemann function is the proof of the coalescence behavior of semi-infinite geodesics with the same asymptotic orientation. In this proof, we will use the following modification lemma. Let $\mathnormal{K}$ be the collection of all finite sequences $$\label{E:prescription}
I=\big((k_{j},l_{j},m_{j},n_{j})\big)_{j=1,...,q}\in(\bN^{4})^{q}$$ where $q\geq 1$, $(k_{j},l_{j},m_{j},n_{j})\ne(k_{i},l_{i},m_{i},n_{i})$ for $j\ne i$, $k_{1}\leq...\leq k_{q}$, and either $ k_{j}< m_{j}$ or $l_{j}<n_{j}$. To each $I\in\mathnormal{K}$ corresponds a random vector $(\tau_{k_{j},l_{j}}^{m_{j},n_{j}})_{j=1,...,q}$. We denote $(\Omega_{I},\calF_{I},\bP_{I})$ the probability space induced by this random vector. Let $$\hat{\Omega}_{I}:=\{\hat{\omega}_{I}\,:\,\exists\,\omega_{I}\in\Omega_{I}\mbox{ with }(\hat{\omega}_{I},\omega_{I})\in\Omega\}$$ and denote by $\hat{\bP}_I$ the probability law $\bP$ restricted to this subset. For each $I\subseteq\mathnormal{K}$, $A\subseteq\Omega$ and $\omega_{1}\in\hat{\Omega}_{I}$ define $$A_{I,\omega_{1}}:=\{\omega_{2}\in\Omega_{I}\,:\,\omega=(\omega_{1},\omega_{2})\in A\}\,.$$ Let $\{R_{I}\,:\,I\in\mathnormal{K}\}$ be a family of events $R_{I}\in\calF_{I}$ such that $\bP_{I}(R_{I})>0$ for all $I$. Then define the map on $\calF$ by $$\Phi_{I}(A):=\{\omega_{1}\in\hat{\Omega}_{I}\,:\,\bP_{I}(A_{I,\omega_{1}})>0\}\times R_{I}.$$ Suppose that $W(\omega)$ is a random element of $\mathnormal{K}$, which may be interpreted as the set of indexes (edges) whose passage time value will be modified. For $A\subseteq\Omega$, let $$\tilde{\Phi}(A):=\cup_{I\in\mathnormal{K}}\big[\{\omega_{1}\in\hat{\Omega}_{I}\,:\,A(I)_{I,\omega_{1}}\ne\emptyset\}\times R_{I}\big],$$ where $A(I):=A\cap [W=I]$.
\[lmodif\] For each $A\in\calF$, $\tilde{\Phi}(A)$ contains $\Phi(A)\in\calF$ defined as the following union $$\Phi(A):=\cup_{I\in\mathnormal{K}}\Phi_{I}\big(A(I)\big).$$ Furthermore, if $\bP(A)>0$ then $\bP\big(\Phi(A)\big)>0$.
If $\bP_{I}\big(A(I)_{I,w_{1}}\big)>0$ then $A(I)_{I,w_{1}}\ne\emptyset$ and so $\Phi(A)\subseteq\tilde{\Phi(A)}$. Since $\mathnormal{K}$ is countable and $A=\cup_{I\in\mathnormal{K}}A(I)$, if $\bP(A)>0$ then there exists $I\in\mathnormal{K}$ such that $\bP\big(A(I)\big)>0$. For this $I$, by Fubini’s theorem $$\label{emodi2}
0<\bP\big(A(I)\big)=\int_{\hat{\Omega}_{I}}\bP_{I}\big(A(I)_{I,w_{1}}\big)\hat{\bP}_{I}(dw_{1}).$$ Let $$\hat{A}_{I}:=\{ w_{1}\,:\,\bP_{I}\big(A(I)_{I,w_{1}}\big)>0\}\,.$$ By , $\hat{\bP}_{I}\big(\hat{A}_{I}\big)>0$. According to the definition of $\Phi_{I}$, $$\bP\Big(\Phi_{I}\big(A(I)\big)\Big)=\hat{\bP}_{I}(\hat{A}_{I})\bP_{I}(R_{I})>0\,.$$ Since $\Phi_{I}\big(A(I)\big)\subseteq\Phi(A)$, we conclude that $\bP\big(\Phi(A)\big)>0$.
Some geometrical aspects of Delaunay triangulations {#pre-geom}
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In this part we are going to study some geometrical aspects of Delaunay triangulations. Let $\x,\y\in\bR^2$ and construct a path $\gamma(\x,\y):=(\v_1,...,\v_{k})$ in $\calD$ connecting $\v(\x)$ to $\v(\y)$ as follows: set $\v_1:=\v(\x)$; if $\v_1\neq\v(\y)$ let $\v_{2}$ be the (almost-surely) unique nearest neighbor of $\v_1$ such that the edge of $\C_{\v_1}$ that is perpendicular to the line segment $[\v_1,\v_2]$ cross $[\x,\y]$; given $\v_l$ with $l\geq 1$, if $\v_{l}\neq\v(\y)$ then we set $\v_{l+1}$ to be the (almost-surely) unique nearest neighbor of $\v_l$, different from $\v_{l-1}$, such that the edge of $\C_{\v_l}$ that is perpendicular to $[\v_l,\v_{l+1}]$ cross $[\x,\y]$; otherwise we set $k:=l$ and the construction is finished. We denote $|\gamma(\0,n\vec{e}_1)|$ the number of edges in $\gamma(\0,n\vec{e}_1)$.
For $\z\in\bR^2$ and $L>0$ let $$\B_\z^{L}:=L\z+[-L/2,L/2]\,$$ For $n>0$ consider the set $\calE_n$ composed of edges $(\v,\bar{\v})\in\calD_e$ with $\C_\v\cap\B_\z^{1}\neq\emptyset$ or $\C_{\bar\v}\cap\B_\z^{1}\neq\emptyset$ for some $\z\in[\0,n\vec{e}_1]$. We denote $|\calE_n|$ the number of edges in $\calE_n$.
\[lgraph2\] There exists constants $z_j,c_j>0$ such that for all $n\geq 1$, $$\label{egraph2*}
\bP\big(|\gamma(\0,n\vec{e}_1)|\geq z n\big)\leq e^{-c_1 zn}\,\mbox{ whenever }z\geq z_0\,,$$ and $$\label{egraph3*}
\bP\big(|\calE_n|\geq z n\big)\leq e^{-c_2 zn}\,\mbox{ whenever }z\geq z_1\,.$$
The proof of this lemma is performed through renormalization ideas developed in [@p-205]. To avoid some repetitions we give a sketch of the proof and leave the details to the reader, which can be filled by following the arguments in the proof of Proposition 2.2 in [@p-205] (which is exactly the proof in ).
For $\z\in\bZ^2$ and $L>0$ divide a square box $\B_\z^{L}$ into thirty-six sub boxes of the same length, say $\B_1,\dots,\B_{36}$. We stipulate $\B$ is a full box if all those thirty-six sub boxes have at least one Poissonian point (Figure \[ffull\]).
![Renormalization: a full box[]{data-label="ffull"}](full){width="20.00000%"}
We say that $\Lambda:=(\B_{\z_1}^{L},\dots,\B_{\z_k}^{L})$ is a circuit of boxes if $(\z_1,\dots,\z_k)$ is a circuit in $\bZ^2$ (in the usual sense). Let $\lambda$ be the closed polygonal path composed by the line segments connecting $L\z_j$ to $L\z_{j+1}$, where $j=1,\dots,k-1$, together with $[\z_k,\z_1]$. To each circuit $\Lambda$ we associate two subsets of the plane: $\Lambda^{in}$ denotes the interior of the bounded component of $\bR^2\backslash\cup_{j=1}^k\B_{\z_j}^{L}$ while $\lambda^{in}$ will denote the interior of the bounded component of $\bR^2\backslash\lambda$. Now, assume that $\Lambda:=(\B_{\z_j}^{L})_{j=1}^k$ is a circuit composed by full boxes. By Lemma 2.1 in [@p-205], we have the following geometrical property: if $\C_{\v}\cap \Lambda^{in}\neq\emptyset$ then $\C_{\v}\subseteq \lambda^{in}$. One important consequence of this is that the set of vertices used by $\gamma(\0,n\vec{e}_1)$ or by $\calE_n$ are both contained in the region $\R_n$ limited by the smallest circuit of full boxes surrounding the line segment $[\0,n\vec{e}_1]$. Therefore to show Lemma \[lgraph2\] is enough to prove the analog decay for the number of Poissonian points in $\R_n$ [^2].
Notice that, since each box is full independently of each other and the probability that it occurs goes to $1$ when $L$ goes to infinity, for a fixed large $L_0>0$, the probability that $\R_n$ contains more than $zn$ boxes decays as $e^{-czn}$ (see for instance Grimmet [@G99]).
Now, the number of points in $\R_n$, say $R_n$, is the sum of independent Poisonian random variables. This is less or equal to $M_m$, the maximum of the number of points in $\R$ over all connected regions $\R$ intersecting at most $m$ boxes the origin $\0$. Thus, on the event that $\R_n$ contains less than $zn$ boxes, we have $R_n\leq M_{zn}$. On the other hand, $M_m$ can be seen as a Greedy lattice animal model and for such a model we can also show, for large $\bar{c}>0$, that the probability that $M_{m}\geq \bar{c}m$ decays as $e^{-cm}$ (Lemma 2.3 of [@p-205]).
By cooking together the arguments in these two last paragraphs one obtains that the probability that the number of points in $\R_n$ is greater than $zn$ also decays as $e^{-czn}$, for some constant $c>0$ and sufficiently large $z$.
Let $T_\calD$ denote the graph metric on $\calD$, i.e. for $\v,\bar{\v}\in\calD_v$, $T_\calD(\v,\bar{\v})$ is the minimum number of edges that one path should pass to go from $\v$ to $\bar{\v}$. Notice that $T_\calD(\v,\bar{\v})$ is the first-passage time between $\v$ and $\bar{\v}$ if one associates to each edge $\e$ the passage time value $1$. For each $\A,\B\subseteq\bR^2$ we set $T_\calD(\A,\B)$ to be the minimum of $T_\calD(\v,\tilde{\v})$ over all pairs $\v$ and $\tilde{\v}$ such that $\C_\v\cap\A\neq\emptyset$ and $\C_{\tilde{\v}}\cap\B\neq\emptyset$. By the shape theorem, we have:
\[lgraph1\] There exists $\nu\in(0,\infty)$ such that almost surely $$\lim_{n\to\infty}\frac{T_\calD(\A,ne^{i\alpha}+\B)}{n}=\nu\,.$$
We notice that $\nu$ does not depend either on $\A$ and $\B$ or on $\alpha\in[0,2\pi)$. One can also see that, if we denote $\lambda=\lambda(\bF)$ the supremum of the support of $\bF$ then $$\mu(\bF)\leq \bE(\tau_e)\nu <\lambda\nu\,$$ (one must assume that $\bF$ is not concentrate in one point, which is the case since $\bF$ is continuous).
We shall also use the following lemma, which is (5.2) of Lemma 5.2 in [@hn01]:
\[5.2\] For $\xi\in (0,1)$ and $r>0$ let $A_{\xi,r}$ be the event that there exists $\x\in\bR^2$ with $|\x|\leq 2r$ and $|\x-\v(\x)|\geq r^\xi$. Then, for some constant $c_1>0$, $$\bP\big(A_{\xi,r}\big)\leq c_1 e^{-r^{2\xi}}$$
$\delta$-straightness of semi-infinite paths {#pre-del}
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Recall that for $\alpha\in[0,2\pi)$ we have defined that a self-avoiding path $(\x_{n})_{n\geq 1}$, with vertices in $\bR^2$ and such that $|\x_n|\to\infty$ when $n\to\infty$, is a $\alpha$-path if $$\lim_{n\to\infty}\frac{\x_{n}}{|\x_{n}|}=e^{i\alpha}:=(\cos\alpha,\sin\alpha)\,,$$ and that a sufficient condition for a path $(\x_{n})_{n\geq 1}$ to be a $\alpha$-path for some $\alpha\in[0,2\pi)$ is that, for some fixed $\delta\in(0,1)$ and $c>0$, and for large enough $n$ $$ang(\x_{n},\x_m)\leq |\x_n|^{-\delta}\,\mbox{ whenever }m>n$$ ($\delta$-straightness). A sufficient condition for $\delta$-straightness is given by the next lemma, which is exactly Lemma 2.7 in [@hn01]:
\[hn-101\] If $(\x_n)_\bN$ is a sequence of points in $\bR^2$ with $|\x_n|\to\infty$ when $n\to\infty$, and such that for all large $n$ $$|\x_{n+1}-\x_n|\leq |\x_n|^{1-\delta}\mbox{ and }d(\x_n,[\x_1,\x_m])\leq |\x_m|^{1-\delta}\mbox{ for }m>n\,,$$ then there exist a contant $c>0$ such that for all $n$ sufficiently large, $$\label{eang}
ang(\x_n,\x_m)\leq c|\x_n|^{-\delta}\,\mbox{ whenever }m>n\,.$$
We also consider the $\delta$-straightness property for trees (we have the tree of infection in mind) as follows. For $\epsilon\in[0,\pi)$ let $$\C(\x,\epsilon):=\{\y\in\bR^{2}\backslash\{0\}\,:\,ang(\y,\x)\leq\epsilon\}\,.$$ If $\calT$ is a tree embedded in $\bR^{2}$, for each pair $\v,\tilde{\v}\in\calT$ let $\calR_{out}(v,\tilde{v})$ be the set of all $\hat{\v}\in\calT$ such that the unique path in $\calT$ connecting $\v$ to $\hat{\v}$ touches $\tilde{\v}$. For $\delta\in(0,1)$, define that $\calT$ is $\delta$-straight at $\v$ if, for all but finitely many $\tilde{\v}\in\calT$, $$\calR_{out}(\v,\tilde{\v})\subseteq \:\v+C(\tilde{\v}-\v,c|\tilde{\v}-\v|^{-\delta})\,.$$ We say that a subset $\calP$ of $\bR^2$ is omnidirectional if, for all $M>0$, the set composed of unit vectors $\v/|\v|$ with $\v\in\calP$ and $|\v|>M$ is dense in $\S^1$. The above lemma, which is Proposition 2.8 in [@hn01], states that $\delta$-straightness implies existence of an asymptotic orientation:
\[hn-201\] Assume that $\calT$ is a tree embedded in $\bR^2$, whose vertex set is locally finite but omnidirectional, and such that every vertex has finite degree. Assume further that for some vertex $\v$, $\calT$ is $\delta$-straight at $\v$. Then $\calT$ satisfies the following:
1. Every semi-infinite path in $\calT$ starting from $\v$ has an asymptotic orientation;\
2. For every $\alpha\in[0,2\pi)$ there exist at least one semi-infinite path in $\calT$ starting at $\v$ and with asymptotic orientation $e^{i\alpha}$.
3. Every semi-infinite path $(\v_n)_{n\geq 1}$ in $\calT$ starting from $\v$ is $\delta$-straight about its asymptotic orientation $e^{i\alpha}$, i.e. $ang(\v_n,e^{i\alpha})<c|\v_n|^{-\delta}$ eventually.
Semi-infinite geodesics and the Busemann function {#geo}
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Semi-infinite geodesics: existence
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Recall that a path $\rho=(\v_1,\v_2,\dots)$ in $\calD$ is a semi-infinite geodesic if for all $\v_j,\v_k\in\rho$, the path $(\v_j,\v_{j+1},...,\v_k)$ is the unique geodesic connecting $\v_j$ to $\v_k$. Semi-infinite geodesics starting from $\v\in\calD_v$ and with asymptotic orientation $e^{i\alpha}$ are denoted $\rho_{\v}(\alpha)$.
\[pexis\] Let $\Omega_2$ be the event that for all semi-infinite geodesic $\rho$ there exists $\alpha=\alpha(\rho )\in [0,2\pi)$ such that $\rho$ is a $\alpha$-path, and that for all $\alpha\in[0,2\pi)$ and for all $\v\in\calD_v$ there exists at least one geodesic starting from $\v$ and with asymptotic orientation $e^{i\alpha}$. Under , $\bP(\Omega_2)=1$.
The first step to show the existence of semi-infinite geodesics and its convergence is the following result on the fluctuations of $T$, which is exactly Corollary 1.1 in [@p-205]:
\[l2\] Under , for all $\kappa\in(1/2,1)$ there exist constants $\delta,c_j>0$ such that for all $r\geq 1$ and $s\in [c_1 (\log r)^{1/\delta},c_2 r^{\kappa}]$ $$\bP\big(|T(\0,r\vec{e}_1)-\mu r|\geq sr^{\kappa}\big)\leq e^{-c_3 s^{\delta}}\,.$$
The second step is to parallel Newman and Piza [@np95] to prove that the control of the fluctuations of $T$ can give the control of the fluctuations of a minimizing path connecting $\0$ to $r\vec{e}_1$ about the line segment $[\0,r\vec{e}_1]$. Precisely, for $\xi\in (0,1)$ let $$\C_r^\xi:=\{\x\in\bR^2\,:\,d(\x,[\0,r\vec{e}_1])\leq r^\xi\}\,,$$ where $[\x,\y]$ denotes the line segment connecting $\x$ to $\y$ and $d(\x,\A)$ denotes the euclidean distance between $\x$ and $\A\subseteq\bR^2$.
\[l1\] For all $\xi\in(3/4,1)$ there exist constants $c,\delta>0$ such that for all $r\geq 1$ $$\bP\big(\rho(\0,r\vec{e}_1)\not\subseteq\C_r^\xi\big)\leq e^{-cr^\delta}\,.$$
Let $\kappa\in(1/2,1)$, $\tilde{\kappa}\in(\kappa,1)$ and set $\xi\:=(\tilde{\kappa}+1)/2$. Let $$\C^{1,\xi}_{r}:=\{\x\in\bR^{2}\backslash \C^{\xi}_{r}\,:\, d(\x,\C^{\xi}_{r})<r^{\xi}\}\,.$$ Denote by $F_{r}$ the event defined by the following properties:
- $\v_{\0},\v_{r\vec{e}_1}\in \C_r^{\xi}$
- for all edges $\e=(\v,\tilde{\v})$ with $|\v|\leq 2r$ or $|\tilde{\v}|\leq 2r$ we have that $|\v-\tilde{\v}|\leq r^{\xi}$.
Notice that $F_r^c\subseteq A_{\xi,r/3}$ (as in Lemma \[5.2\]), and thus $$\label{HN1}
\bP\big(F_r^c\big)\leq c_1e^{-(r/3)^{2\xi}}\,.$$
For each $\z\in\bZ^2$ consider the random variable $$T_{\z}:=\max_{|\v-\z|\leq 1}\{T(\z,\v)\}\,.$$ We claim that, under , for some constants $c_2,c_3>0$ $$\label{e7t3}
\bP(T_{\z}\geq r^{\kappa})=\bP\big(T_{\0}\geq r^{\kappa}\big)\leq c_2e^{-c_3r^{\kappa}}\,.$$ To see this, notice that $T_\0\leq \sum_{\e\in\calE_1}\tau_\e$, where $\calE_1$ is the set of edges $\e=(\v,\bar{\v})$ in $\calD_e$ with $\C_{\v}\cap \B_{\0}^1\neq\emptyset$ or $\C_{\bar{\v}}\cap \B_{\0}^1\neq\emptyset$. By Lemma \[lgraph2\], $\bE\big(\exp(a|\calE_1|)\big)< \infty$ for some $a>0$. Combining this with assumption and the independence between the Poisson point process and the passage time distribution, one obtains .
Now, $$\big[\rho(\0,r\vec{e}_1)\not\subseteq\C_r^\xi\big]\cap F_{r}\subseteq$$ $$\label{e1t3}
\big[\exists v\in\calD_v\cap \C_{r}^{1,\xi}\,:\, T(0,\v)+T(\v,r\vec{e}_1)=T(0,r\vec{e}_1)\big]\subseteq A(r)\,,$$ where $$A(r):=\big[\exists \z\in\bZ^{2}\cap \C_{r}^{1,\xi}\,:\, T(\0,\z)+T(\z,r\vec{e}_1)\leq T(\0,r\vec{e}_1)+2T_{\z}\big]\,.$$ Let $$\Delta(\z,r\vec{e}_1):= \mu|\z-r\vec{e}_1|+\mu|\z|-\mu|r\vec{e}_1|\,.$$ Thus $$T(\0,\z)+T(\z,r\vec{e}_1)\leq T(\0,r\vec{e}_1)+2T_{\z}$$ if and only if, $$\Delta(\z,r\vec{e}_1)\leq \big(T(\0,r\vec{e}_1)-\mu r\big)+\big(\mu|\z|-T(\0,\z)\big)+$$ $$\big(\mu|\z-r\vec{e}_1|-T(\z,r\vec{e}_1)\big)+2T_{\z}\,.$$ This implies that $A(r)\subseteq\cup_{j=0}^{3}A_{j}(r)$, where $$A_{0}(r):=\big[\exists \z\in\bZ^{2}\cap \C_{r}^{1,\xi}\,:\,T_{\z}\geq \frac{\Delta(\z,r\vec{e}_1)}{8}\big]\,,$$ $$A_{1}(r):=\big[\exists \z\in\bZ^{2}\cap \C_{r}^{1,\xi}\,:\,|T(\z,r\vec{e}_1)-\mu|\z-r\vec{e}_1||\geq \frac{\Delta(\z,r\vec{e}_1)}{4}\big]\,,$$ $$A_{2}(r):=\big[\exists \z\in\bZ^{2}\cap \C_{r}^{1,\xi}\,:\,|T(\0,\z)-\mu|\z||\geq\frac{\Delta(\z,r\vec{e}_1)}{4}\big]\,,$$ $$A_{3}(r):=\big[\exists \z\in\bZ^{2}\cap \C_{r}^{1,\xi}\,:\,|T(\0,r\vec{e}_1)-\mu|r\vec{e}_1||\geq\frac{\Delta(\z,r\vec{e}_1)}{4}\big]\,.$$ Combining this with one gets that $$\label{e3t3}
\bP\big(\rho(\0,r\vec{e}_1)\not\subseteq\C_r^\xi\big)\leq \bP\big(F_{r}^{c}\big)+ \sum_{j=0}^{3}\bP\big(A_{j}(r)\big)\,.$$
Notice there exist constants $b_{1},b_{2}>0$ such that for sufficiently large $r>0$ and $\z\in\bZ^{2}\cap \C_{r}^{1,\xi}$ we have that $$\label{e4*t3}
b_{1}r^{\tilde{\kappa}}=b_{1}r^{2\xi-1}\leq \Delta(\z,r\vec{e}_1)\leq b_{2}r^{\xi}=b_{2}r^{\frac{\tilde{\kappa}+1}{2}}\,,$$ and $$\label{e4t3}
r^{\xi}\leq |\z|,|\z-r\vec{e}_1|\leq 2r\,.$$
Together with Lemma , and yield that for some constant $c_1 >0$ $$\label{e5t3}
\bP\big(A_{j}(r)\big)\leq e^{-c_{1}r^{\delta}}\,.$$ Combining with , , and one can finish the proof of this lemma.
For $\v\in\calD_v$ let $\calT_{\v}$ be the union over all $\tilde{\v}\in \calD_v$ of the unique geodesic between $\v$ and $\tilde{\v}$ (the tree of infection at $\v$). Therefore, $\calT_{\v}$ is a tree spanning all $\calD_v$. Thus, the third step is to use Lemma \[l1\] and the concept of *$\delta$-straightness* for trees discussed before.
Combining Lemma \[l1\] and Lemma \[5.2\] with the Borel-Cantelli’s lemma, one has that for all $\delta=1-\xi\in(0,1/4)$, almost surely, the assumptions of Lemma \[hn-101\] hold for all semi-infinite path (geodesic) $(\v_n)_{n\geq 1}$ in $\calT_\v$. So, $\calT_\v$ is $\delta$-straight at $\v$. Since, with probability one, a realization of the Poisson point process is omnidirectional, together with Lemma \[hn-201\] this yields Proposition \[pexis\].
\[r-straight\] Let $\xi\in(3/4,1)$. The almost sure $(1-\xi)$-straightness of the tree of infection also implies that for all $\alpha\in[0,2\pi)$, if $(\v_1,\v_2,\dots)$ is a semi-infinite geodesic with asymptotic orientation $e^{i\alpha}$ then $$ang(\v_n,e^{i\alpha})\leq c|\v_n|^{\xi-1}$$ for sufficiently large $n$.
Semi-infinite geodesics: uniqueness and coalescence
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Concerning uniqueness of semi-infinite geodesics we have:
\[puni\] For $\alpha\in[0,2\pi)$ let $\Omega_3(\alpha)$ be the event that for all $\v\in \calD_v$ there exists at most one geodesic starting from $\v$ and with asymptotic orientation $e^{i\alpha}$. Assume only that $\bF$ is continuous. Then $\bP\big(\Omega_3(\alpha)\big)=1$
For $(k,l)\in\bN^2$, let $A_\alpha(k,l)$ be the event that $U_{k,l}\in \calD_v$ (or equivalently, $N_k\geq l$) and there exists two semi-infinite geodesics starting from $\v=U_{k,l}$, with asymptotic orientation $e^{i\alpha}$, and such that after $\v$ they do not intersect each other. Thus, $$\big(\Omega_3(\alpha)\big)^c\subseteq \cup_{(k,l)\in\bN^2}A_\alpha(k,l)\,.$$ Now, semi-infinite geodesics starting from the same vertex are not allowed to cross each other and, if a semi-infinite geodesics is caught between two semi-infinite geodesics with the same asymptotic orientation $e^{i\alpha}$ then it must have the asymptotic orientation $e^{i\alpha}$ (by planarity). Therefore, if we denote by $d_{\v}$ the degree of the site $\v=U_{k,l}$ then $$|\{\alpha\in[0,2\pi)\,:\,\1_{A_\alpha(k,l)}(\omega)=1\}|\leq d_{\v}(\omega)\,.$$ ($|A|$ is the cardinality of the set $A$). In particular, almost surely, $$\int_{[0,2\pi)}\1_{A_\alpha(k,l)}d\alpha =0\,,$$ and so, by Fubini’s theorem, $$0\leq \int_{[0,2\pi)}\bP\Big(\big(\Omega_3(\alpha)\big)^c\Big)d\alpha=\int_{\Omega}\big(\int_{[0,2\pi)}\1_{\big(\Omega_3(\alpha)\big)^c}d\alpha\big)d\bP\leq$$ $$\int_{\Omega}\big(\int_{[0,2\pi)}\sum_{(k,l)}\1_{A_\alpha(k,l)}d\alpha\big)d\bP=\int_{\Omega}\big(\sum_{(k,l)}\int_{[0,2\pi)}\1_{A_\alpha(k,l)}d\alpha\big)d\bP=0\,.$$ Consequently, there exists $I\subseteq[0,2\pi)$ with total Lebesgue measure so that for all $\alpha\in I$, $\bP\big(\Omega_3(\alpha)\big)=1$. Since $\bP\big(\Omega_3(\alpha)\big)$ does not depend on $\alpha$, this yields Proposition \[puni\].
The last result we require to construct the Busemann function is the coalescence behavior of semi-infinite geodesics with the same asymptotic direction:
\[pcoal\] For $\alpha\in[0,2\pi)$ let $\Omega_4(\alpha)\subseteq\Omega_3(\alpha)$ be the event that for all $\v,\bar{\v}\in\calD_v$, if $\rho_{\v}(\alpha)$ and $\rho_{\bar{\v}}(\alpha)$ do exist (and are unique) then they must coalesce, i.e. there exists $\c=\c(\v,\bar{\v},\alpha)\in\calD_v$ such that $$\rho_{\v}(\alpha)=\rho(\v,\c)\cup\rho_{\c}(\alpha)\,\mbox{ and }\,\rho_{\bar{\v}}(\alpha)=\rho(\bar{\v},\c)\cup\rho_{\c}(\alpha)\,.$$ Assume only that $\bF$ is continuous. Then $\bP\big(\Omega_4(\alpha)\big)=1$.
We note that the almost sure statement in Proposition \[pcoal\] is for fixed $\alpha\in[0,2\pi)$. As we will see later, almost surely, there exists a random direction $\theta$ so that neither uniqueness nor coalescence hold. Indeed, we will show (in part \[pr-rand+strai\]) that every branch of the competition interface follows one of those random directions for which coalescence does not hold[^3].
Let $\calS(\alpha)$ denote the union over all $\v\in\calD_v$ of $\rho_{\v}(\alpha)$. Then $\calS(\alpha)$ is a forest with say $N(\alpha)$ disjoint trees. Notice that, on $\big[N(\alpha)\leq 1\big]\cap\Omega_3(\alpha)$, there are no site disjoint semi-infinite geodesic with asymptotic orientation $e^{i\alpha}$. So, Proposition \[pcoal\] will follow if we prove that $\bP\big(N(\alpha)\leq 1\big)=1$. As noted by Licea and Newman [@ln96], in this set up we can apply the Burton and Keanne [@bk89] method. This method requires several steps which we will be organized as independent claims. To state the first one, let $\delta\in\bQ$ (the set of rational numbers) and $\x_{i}=\big(x_i(1),x_i(2)\big),\tilde{\x}_{i}=\big(\tilde{x}_i(1),\tilde{x}_i(2)\big)\in\bQ^{2}$ for $i=1,...,j$ such that $x_1(2)\leq\dots\leq x_j(2)$ and $\tilde{x}_1(2)\leq\dots\leq\tilde{x}_j(2)$. Assume further that $x_i(1)\leq -\delta$ and that $\tilde{x}_i(1)\geq \delta$. Denote by $A_{\delta}(\x_{1},...,\x_{j},\tilde{\x}_{1},...,\tilde{\x}_{j})$ the event determined by the following:
- at each $D_{\delta}(\x_{i})$ and $D_{\delta}(\tilde{\x}_{i})$ there is an unique vertex $\v_{i}$ and $\tilde{\v}_{i}$ respectively;
- each $\e_{i}=(v_{i},\tilde{v}_{i})$ is an edge in $\calD_e$ and $\e_{i}\in\rho_{\v_{i}}(0)$;
- after $\v_{i}$, $\rho_{\v_{i}}(0)$ has vertices only with strictly positive coordinates;
- all $\rho_{\v_{i}}(0)$ are disjoint.
\[emodi1\] If $$\bP\big(N(0)\geq 2\big)>0\,$$ then $$\bP\big(A_{\delta}(\x_{1},\x_2,\x_3,\tilde{\x}_{1},\tilde{\x}_2,\tilde{\x}_{3})\big)>0\,,$$ for some $\delta\in\bQ$ and $\x_{i},\tilde{\x}_{i}\in\bQ^{2},i=1,2,3$.
Since $\bQ$ is enumerable, if $0<\bP\big(N(0)\geq 2\big)$ then there exist $\delta\in\bQ$ and $\x_{1},\x_{2},\tilde{\x}_{1},\tilde{\x}_{2}\in\bQ^{2}$ such that $$0<\bP\big(A_{\delta}(\x_{1},\x_{2},\tilde{\x}_{1},\tilde{\x}_{2})\big)\,.$$ Let $c_{n}$ be the maximum between the second coordinate of $\x_{2}$ and $\tilde{\x}_{2}$ and let $c_{s}$ be the minimum between the second coordinate of $\x_{1}$ and $\tilde{\x}_{1}$. Consider the rectangle $$\R_{0}:= [-\delta,\delta]\times (c_{s}-\delta,c_{n}+\delta)\,.$$ Let $\z_{0}$ be the circumcenter of the rectangle $\R_{0}$ and let $M_{0}$ be the vertical length of $\R_{0}$. For each $l\in\bZ$ set $\z_{l}:= \z_{0}+lM_0(0,1)$. Denote $\R_{l}:= z_{l}+\R_{0}$ and $$A(l):= A_{\delta}(\x^{l}_{1},\x^{l}_{2},\tilde{\x}^{l}_{1},\tilde{\x}^{l}_{2})\,,$$ where $\x^{l}_{j}:=\x_{j}+\z_{l}\in \R_l$ and $\tilde{\x}^{l}_{j}:=\tilde{\x}_{j}+\z_{l}$. Thus, $\bP\big(A(l)\big)=\bP\big(A(0)\big)$. By the Fatou’s lemma, $$0<\bP\big(A(0)\big)\leq \bP\big(\limsup_{l} A(l)\big)\leq \bP\big(\cup_{l_{1}\ne l_{2}}A(l_{1})\cap A(l_{2})\big)\,.$$ Therefore, there are $l_{1},l_{2}$ such that $$0<\bP\big(A(l_{1})\cap A(l_{2})\big)\,.$$
Without lost of generality assume that $l_1<l_2$. We claim that, in this case, the geodesic starting from $\v_{1}^{l_{1}}$ can not intersect either the geodesic starting from $\v_{1}^{l_{2}}$ or the geodesic starting from $\v_{2}^{l_{2}}$. This is so because otherwise (by planarity) the geodesic starting at $\v_{1}^{l_{1}}$ would intersect the geodesic starting from $\v_{2}^{l_{1}}$, which contradicts the definition of $A(l_{1})$. Thus, $$A(l_{1})\cap A(l_{2})\subseteq
A_{\delta}(\x_{1}^{l_{1}},\x_{1}^{l_{2}},\x_{2}^{l_{2}},\tilde{\x}_{1}^{l_{1}},\tilde{\x}_{1}^{l_{2}},\tilde{\x}_{2}^{l_{2}})\,$$ which yields Claim \[emodi1\].
The second step is given by the following claim: for $m,k\geq 0$ let $F_{m,k}$ be the event that some tree in $\calS(0)$ touches a vertex in the rectangle $$\R_{m,k}:=\big\{(x(1),x(2))\,:\,0\leq x(1)\leq m\mbox{ and }|x(2)|\leq k\big\},$$ but no other in $$\Q_{m}:=\big\{(x(1),x(2))\,:\,x(1)\leq m\big\}\backslash\R_{m,k}\,.$$
\[lmodi-0\] If for some $\delta\in\bQ$ and $\x_{i},\tilde{\x}_{i}\in\bQ^{2},i=1,2,3$ we have $$\bP\big(A_{\delta}(\x_{1},\x_2,\x_3,\tilde{\x}_{1},\tilde{\x}_2,\tilde{\x}_{3})\big)>0\,$$ then $$\bP\big(F_{m,k}\big)>0\,,$$ for some $m,k\geq 0$.
To prove this claim we shall use a local modification argument based on Lemma \[lmodif\], and we will divide this proof into two parts: in the first one we will assume that $\bF$ has unbounded support while in the second one we will assume that $\bF$ has bounded support.
#### **Part 1: $\bF$ has unbounded support.**
Let $\delta\in\bQ$ and $\x_{1},\x_{2},\x_{3},\tilde{\x}_{1},\tilde{\x}_{2},\tilde{\x}_{3}\in\bQ^2$ given by Claim \[emodi1\]. Let $\R_0:=[-\delta,\delta]\times[c_{s}-\delta,c_{n}+\delta]$, where $c_{n}$ be the maximum between the second coordinate of $\x_{3}$ and $\tilde{\x}_{3}$ and let $c_{s}$ be the minimum between the second coordinate of $\x_{1}$ and $\tilde{\x}_{1}$. Denote by $\Xi$ the set of edges which cross the rectangle $\R_0$ and the vertical coordinate axis. Then $\e_{i}:=(\v_{i},\tilde{\v}_{i})\in\Xi$ for all configurations in $A_{\delta}(\x_{1},\x_{2},\x_{3},\tilde{\x}_{1},\tilde{\x}_{2},\tilde{\x}_{3})$ (recall that $\x_{i}\in
C_{\v_{i}}$ and $\tilde{\x}_{i}\in C_{\tilde{\v}_{i}}$).
Define the event $B_{\lambda}$ by those configurations such that for all $\e=(\v_{1},\v_{2})\in\Xi$ there exists $\gamma$ with connecting $\v_{1}$ to $\v_{2}$, with $t(\gamma)<\lambda$, but not using edges in $\Xi$. Since $$\lim_{\lambda\to\infty}\bP\big(B_{\lambda}\big)=1\,,$$ we can choose a sufficiently large $\lambda>0$ such that $$\label{emod1}
\bP\big(A_{\delta}(\x_{1},\x_{2},\x_{3},\tilde{\x}_{1},\tilde{\x}_{2},\tilde{\x}_{3})\cap B_{\lambda}\big)>0\,.$$
Now we apply Lemma \[lmodif\]. To do so define $W(\omega)$, a random element of $\mathnormal{K}$, by the following procedure: given $\omega\in\Omega$ set $$W(\omega):=\big((k_{j},l_{j},n_{j},m_{j})\big)_{j=1,...,q}\,$$ by ordering all $(k,l,m,n)$ (according to ) so that $\e(\omega)=\big(U_{k,l}(\omega),U_{m,n}(\omega)\big)\in\Xi(\omega)$ and $\tau_\e\leq\lambda$. Thus $W$ is an ordered representation of the indexes of the edges $\e\in\Xi$ with $\tau_e\leq\lambda$.
For each $I\in\mathnormal{K}$ let $$R_{I}:=(\lambda,+\infty)^{q}\subseteq\Omega_{I}=\bR^q\,,$$ and let $$A:=A_{\delta}(\x_{1},\x_{2},\x_{3},\tilde{\x}_{1},\tilde{\x}_{2},\tilde{\x}_{3})\cap
B_{\lambda}\,$$ (given by ). Since $\bF$ has unbounded support, $\bP_{I}(R_{I})>0$ for all $I\in\mathnormal{K}$. By Lemma \[lmodif\], there exist a measurable $\Phi(A)\subseteq\tilde{\Phi}(A)$.
Now consider a configuration $\tilde{\omega}\in\Phi(A)\subseteq\tilde{\Phi}(A)$. By definition, there exists $I\in\mathnormal{K}$, $\omega_1\in\hat{\Omega}_I$, $\omega_2\in \Omega_I$ and $\tilde{\omega}_2\in R_I$ such that $\tilde{\omega}=(\omega_1,\tilde{\omega}_2)$ and $(\omega_1,\omega_2)\in A$. Since $\omega_2$ and $\tilde{\omega}_2$ concern only travel times which are associated to $I$ and $\omega_2\leq\tilde{\omega}_2$ (considering the canonical order in $\bR^q$), the paths $\rho_{\tilde{\v}_{i}}(0)(\omega_1,\omega_2)$ for $i=1,2,3$ remain disjoint geodesics, with asymptotic orientation $\vec{e}_1$, for the configuration $\tilde{\omega}=(\omega_1,\tilde{\omega}_2)$. By the same reason, $\tilde{\omega}\in B_\lambda$. On the other hand, since $\tilde{\omega}_2\in R_I$, we have that for all $\e\in\Xi$, $\tau_e(\tilde{\omega})>\lambda$ and thus no geodesic could have an edge in $\Xi$. Therefore $\Phi(A)\subseteq F_{m,k}$, where $k:=\max\{c_s,c_n\}$ and $m:=\delta + \max\{\tilde{x}_{1}(1),\tilde{x}_{2}(1),\tilde{x}_{3}(1)\}$. Since $\bP(A)>0$, we also have that $0<\bP\big(\Phi(A)\big)\leq \bP\big(F_{m,k}\big)$, which yields Claim \[lmodi-0\] when $\bF$ has unbounded support.
#### **Part 2: $\bF$ has bounded support.**
Consider again $\delta\in\bQ$ and $\x_{1},\x_{2},\x_{3},\tilde{\x}_{1},\tilde{\x}_{2},\tilde{\x}_{3}\in\bQ^2$ given by Claim \[emodi1\]. Let $\vec{e}_2:=(0,1)$, $\c_n:=(0,c_n)$ and $\c_s:=(0,c_s)$. For $\epsilon,\tilde{\epsilon}>0$ and $m>0$, let $$\Q_{m,\tilde{\epsilon}}:=m\vec{e}_1+[-\tilde{\epsilon}m\vec{e}_2,\tilde{\epsilon}\vec{e}_2]\,$$ and let $B^{\epsilon,\tilde{\epsilon}}_{m}$ be the event that for every $\z\in [\c_s,\c_n]$ and every $\u\in \Q_{m,\tilde{\epsilon}}$, $$\label{ecoal2}
T(z,u)< (\mu +\epsilon)m\,.$$ By the shape theorem, we have that for any $\epsilon>0$ and for sufficiently small $\tilde{\epsilon}$, $$\label{elimit3}
\lim_{m\to \infty}\bP\big(B^{\epsilon,\tilde{\epsilon}}_{m}\big)=1\,.$$
Denote by $C^{\tilde{\epsilon}}_{m,k}$ the event that for each $i=1,2,3$, $\rho_{\v_{i}}(0)$ touches the hyperplane with direction $\vec{e}_2$ and containing $(0,m)$ for the first time (coming from $\v_{i}$) within the vertical segment $\Q_{m,\tilde{\epsilon}}$. Since all those geodesics are $0$-paths, $$\label{elimit1}
\lim_{m\to \infty}\bP\big(C^{\tilde{\epsilon}}_{m}\big)=1$$ for all $\tilde{\epsilon}>0$.
For $m,k>0$ let $C_{m,k}$ denote the event that for each $i=1,2,3$, $\rho_{\v_{i}}(0)$ does not intersect the region consisting of points $(x(1),x(2))\in\bR^2$ such that $x(1)\in [0,m]$ and $|x(2)|>k$. Thus, for any fixed $m>0$, $$\label{elimit2}
\lim_{k\to \infty}\bP\big(C_{m,k}\big)=1\,$$ (by the same reason to obtain ).
Let $\x,\y\in\bR^2$ and recall the definition of the path $\gamma(\x,\y)$ given in Section \[pre\] (part \[pre-geom\]). By Lemma \[lgraph2\], $$\label{egraph2}
\lim_{n\to\infty}\bP\big(|\gamma(\0,n\vec{e}_1)|\geq c_1 n\big)=0\,,$$ for some contant $c_1>0$. We also have considered the graph metric $T_\calD$ and, by Lemma \[lgraph1\], $$\label{egraph1}
\lim_{n\to\infty}\frac{T_\calD\big([\c_s,\c_n],\Q_{m,\tilde{\epsilon}}\big)}{m}=\nu\,.$$
For each $i=1,2,3$, let $\rho_{i}$ denote the piece of $\rho_{\v_{i}}(0)$ between $\tilde{\v}_{i}$ and the first time it intersect $[m\vec{e}_1 -\tilde{\epsilon}m\vec{e}_2,m\vec{e}_1+\tilde{\epsilon}m\vec{e}_2]$, say at the point $\u_i$. For $\z\in[\c_s,\c_n]$ and $\u\in\Q_{m,\tilde{\epsilon}}$, let $\phi(\z,\u)$ be the path connecting $\z$ to $\u$, which first moves vertically by using $\gamma(\z,\v_1)$, then follows $\rho_{1}$, then moves vertically again by using $\gamma(\u_1,\u)$. Thus, on the intersection between $A_{\delta}(\x_{1},\x_{2},\x_{3},\tilde{\x}_{1},\tilde{\x}_{2},\tilde{\x}_{3})$, $C^{\tilde{\epsilon}}_{m}$, $ C_{m,k}$ and $B^{\epsilon,\tilde{\epsilon}}_{m}$, we have that $$t\big(\phi(z,u)\big)=t\big(\gamma(\z,\v_1)\big)+t\big(\rho_{1}\big)+t\big(\gamma(\u_1,\u)\big)\leq$$ $$\label{egraph3}
\lambda |\gamma(\c_s,\c_n)|+(\mu+\epsilon)m+\lambda|\gamma(m\vec{e}_1 -\tilde{\epsilon}m\vec{e}_2,m\vec{e}_1+\tilde{\epsilon}m\vec{e}_2 )|\,.$$
We also have that, by and (since $\mu(\bF)<\lambda(\bF)\nu$), there exists $\epsilon_{0},\tilde{\epsilon}_{0}>0$ such that for all $\epsilon<\epsilon_{0}$, $\tilde{\epsilon}<\tilde{\epsilon}_{0}$, $$\label{metric}
\lim_{m\to \infty}\bP\big(D(\lambda,\epsilon,\tilde{\epsilon})\big)=1\,.$$ where $D(\lambda,\epsilon,\tilde{\epsilon})$ is the event that $$\lambda |\gamma(\c_s,\c_n)|+(\mu+\epsilon)m+\lambda|\gamma(m\vec{e}_1 -\tilde{\epsilon}m\vec{e}_2,m\vec{e}_1+\tilde{\epsilon}m\vec{e}_2 )|$$ $$\leq (\lambda-\epsilon)T_\calD\big([\c_s,\c_n],\Q_{m,\tilde{\epsilon}}\big)\,.$$
Let $$A:=A_{\delta}(\x_{1},\x_{2},\x_{3},\tilde{\x}_{1},\tilde{\x}_{2},\tilde{\x}_{3})\cap C^{\tilde{\epsilon}}_{m}\cap C_{m,k}\cap B^{\epsilon,\tilde{\epsilon}}_{m}\cap D(\lambda,\epsilon,\tilde{\epsilon})\,.$$ Combining with , and , we get that $\bP\big(A\big)>0$ for sufficiently small $\epsilon>0$ and $\tilde{\epsilon}>0$ and for sufficiently large $m>0$ and $k>0$. Notice that for all configurations in $A$, and every $\z\in[\c_s,\c_n]$ and $\u\in\Q_{m,\tilde{\epsilon}}$ we must have that $$\label{emod2}
T(\z,\u)\leq t\big(\phi(\z,\u)\big)\leq (\lambda-\epsilon) T_\calD\big([\c_s,\c_n],\Q_{m,\tilde{\epsilon}}\big)\,.$$
Now we are able to use Lemma \[lmodif\] again. Let $\Xi$ be the set of edges in the interior of the region bounded by $\rho_{1}$, $\rho_{3}$, $[\c_s,\c_n]$ and $\Q_{m,\tilde{\epsilon}}$. Define $W(\omega)$ as follows: given $\omega\in\Omega$ we set $$W(\omega):=\big((k_{j},l_{j},m_{j},n_{j})\big)_{j=1,...,q}$$ by ordering all $(k,l,m,n)$ (according to ) so that $\e(\omega)=\big(U_{k,l}(\omega),U_{m,n}(\omega)\big)\in\Xi(\omega)$ with $\tau_\e\leq\lambda-\epsilon$. So $W$ represents the indexes of the edges $\e\in\Xi$ with $\tau_\e\leq\lambda -\epsilon$. For each $I\in\mathnormal{K}$, let $R_{I}:=(\lambda-\epsilon,\lambda)^{q}\subseteq\Omega_{I}$ and take $A$ above defined. Since $\bF(\lambda-\epsilon)<1$ then $\bP_{I}(R_{I})>0$. Thus, by Lemma there exists a measurable $\Phi(A)\subseteq\tilde{\Phi}(A)$.
Pick a configuration $\tilde{\omega}=(\omega_1,\omega_2)\in\tilde{\Phi}(A)$. By using the same argument we have done for the other case, one can see that the paths $\rho_{\tilde{\v}_{i}}(0)(\omega_1,\omega_2)$ for $i=1,3$ remain disjoint geodesics, with asymptotic orientation $\vec{e}_1$, for the configuration $\tilde{\omega}$. The same holds for $\rho_{\u_2}(0)$ and for the inequality . On the other hand, by , no path $\rho$ connecting $\z\in[\c_s,\c_n]$ to $\u\in\Q_{m,\tilde{\epsilon}}$ that is entirely containing in the region $\Xi$ can be a geodesic for the configuration $\tilde{\omega}$ because, otherwise, $$T(\z,\u)=t(\rho)>(\lambda-\epsilon) T_\calD\big([\c_s,\c_n],\Q_{m,\tilde{\epsilon}}\big)\,.$$ This allows us to conclude that $$\Phi(A)\subseteq\tilde{\Phi}(A)\subseteq F_{m,k}\,$$ (with $m,k>0$ given by the definition of $A$). Since $\bP\big(A\big)>0$ we have that $0<\bP\big(\Phi(A)\big)<\bP\big(F_{m,k}\big)$, which yields Claim \[lmodi-0\] when $\bF$ has bounded support.
The third and last step is:
\[lmodi-1\] $\bP\big(F_{m,k}\big)=0$ for all $m,k\geq 0$.
In fact, consider a rectangular array of non-intersecting translates $\R_{m,k}^{\z}$ of the basic rectangle $\R_{m,k}=\R^{\0}_{m,k}$ and of $\Q_m=\Q_m^{\0}$ indexed by $\z\in\bZ^{2}$, and also consider the corresponding event $F_{m,k}^{\z}$. Notice that if $F_{m,k}^{\z}$ and $F_{m,k}^{\tilde{\z}}$ occur, then the corresponding trees in $\calS(0)$ must be disjoint. Thus, if $N_{L}$ is the number of $\z\in [0,L]^{2}$ such that $F_{m,k}^{\z}$ occurs, then $$N_{L}\leq |\{\mbox{ edges crossing the boundary of }[0,L]^{2}\}|.$$ However, by Lemma \[lgraph2\], the expected value of the number of edges crossing the boundary of $[0,L]^{2}$ is of order $L$. By translation invariance, $$\bE\big(N_{L}\big)=n_{L}\bP\big(F_{M,k}\big),$$ where $n_{L}$ is the number of rectangles $\R_{m,k}^{z}$ intersecting $[0,L]^{2}$. Since $n_L$ is of order $L^2$, the assumption $\bP\big(F_{m,k}\big)> 0$ leads to a contradiction.
Now we are able to prove Proposition \[pcoal\]:
Combining Claim \[emodi1\] with Claim \[lmodi-0\] and Claim \[lmodi-1\] one obtains $$\label{lcoal}
\bP\big(N(\alpha)\leq 1\big)=\bP\big(N(0)\leq 1\big)=1\,.$$ By noticing that $\Omega_3\cap\big[N(\alpha\big)\leq 1\big]\subseteq \Omega_4(\alpha)$ one can see that Proposition \[pcoal\] follows from Proposition \[puni\] together with .
Existence and asymptotics for the Busemann function {#asyBuse}
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The idea to prove Theorem \[tBuse-1\] is to combine existence, uniqueness and coalescence of semi-infinite geodesics in a fixed direction $e^{i\alpha}$ to show that if $\z_n\to\infty$ along this direction then for sufficiently large $n$ we have $$T(\x,\z_n)-T(\y,\z_n)=T(\x,\c)-T(\y,\c)\,,$$ where $\c$ is coalescence point in direction $e^{i\alpha}$ (Proposition \[pcoal\]). We begin by introducing what we mean by convergence of paths. Assume that $(\gamma_n)_{n\geq 0} $ is a sequence of finite paths with vertices in $\bR^2$, and for each $n\geq 0$ denote $\gamma_n=(\z_0^n,\z_1^n,\dots,\z_{l_n}^n)$. We define that $\gamma_n$ converges to a semi-infinite path $\gamma=(\x_0,\x_1\,\dots)$, and we write $\gamma=\lim_{n\to\infty}\gamma_n$, if for all $k\geq 1$ there exists $n_k\geq 1$ so that $\gamma_n=(\x_0,\x_1,\dots,\x_k,\z_{k+1}^n,\dots,\z_{l_n}^n)$ for all $n\geq n_k$. For each sequence $(\z_n)_{n\geq 0}$ of vertices in $\bR^2$ with $|\z_n|\to\infty$ and $\z\in\bR^2$ we denote $\Pi\big(\z,(\z_n)_{n\geq 0}\big)$ the set of all semi-infinite paths $\rho$ so that there exists a subsequence $(n_j)_{j\geq 0}$ with $\lim_{j\to\infty} \rho(\z,\z_{n_j})=\rho$.
\[geoconv\] Let $\Omega_1$ be the event that, for all $\alpha\in[0,2\pi)$, if $(\z_n)_{n\geq 1}$ has the asymptotic orientation $e^{i\alpha}$ then: i) $\Pi\big(\z,(\z_n)_{n\geq 1}\big)\neq\emptyset$; ii) every $\rho\in\Pi\big(\z,(\z_n)_{n\geq 1}\big)$ is semi-infinite geodesic with the asymptotic orientation $e^{i\alpha}$. Under , $\bP\big(\Omega_1\big)=1$.
Let $\calT$ be the tree with vertex set $\cup_{n\geq 1} \rho(\z,\z_n)$ and oriented edges $(\u,\v)\in\calD_e$ (in the Delaunay triangulation) so that $\rho(\z,\u)\subseteq\rho(\z,\v)$. Notice that $\calT$ is an infinite tree. Since every vertex in the Delaunay triangulation has finite degree, the same is true for the vertices in $\calT$. Therefore, by a standard compactness argument, $\Pi\big(\z,(\z_n)_{n\geq 1}\big)\neq\emptyset$. To show that every $\rho\in\Pi\big((\z_n)_{n\geq 1}\big)$ has the asymptotic orientation $e^{i\theta}$ consider $\D\subseteq\S^1$ as in the proof of Theorem \[t1\]. By Proposition \[pexis\] and Proposition \[puni\], almost surely, for all $\beta\in[0,2\pi)$ such that $e^{i\beta}\in \D$ there exists an unique semi-infinite geodesic starting from $\v(\z)$ and with asymptotic orientation $e^{i\beta}$, which we have denoted by $\rho_{\z}(\beta)$. Now, let $\beta_1,\beta_2\in[0,2\pi)$ such that $e^{i\beta_1},e^{i\beta_2}\in \D$. Assume further that, by following the counter-clokwise orientation of $\S^1$, the unit vector $e^{i\alpha}$ is in between the unit vectors $e^{i\beta_1}$ and $e^{i\beta_2}$. Notice that the paths $\rho_{\z}(\beta_1)$ and $\rho_{\z}(\beta_2)$ bifurcate at some point $\v$ and have no further points in common. On the other hand, $(\z_n)_{n\geq 0}$ has the asymptotic orientation $e^{i\alpha}$. Therefore, once $k$ is large enough, $\rho(\z,\z_k)$ should be in between $\rho_{\z}(\beta_1)$ and $\rho_{\z}(\beta_2)$, and thus the same is true for any limit $\rho$. Since $\D$ is dense in $\S^1$, it follows that $\rho$ has the asymptotic orientation $e^{i\alpha}$.
Consider the intersection between $\Omega_1$ (path convergence, Lemma \[geoconv\]) and $\Omega_4(\alpha)$ (coalescence and uniqueness of semi-infinite geodesics, Proposition \[pcoal\]). In this case, if $(\z_n)_{n\geq 1}$ has the asymptotic orientation $e^{i\alpha}$ then $\lim_{n\to\infty}\rho(\x,\z_n)=\rho_\x(\alpha)$. Together with coalescence, this yields that for $\x,\y\in\bR^2$ there exists $\c=\c(\x,\y,\alpha)\in\calD_v$ and $n_0>0$ such that $$\rho(\x,\z_n)=\rho(\x,\c)\cup\rho(\c,\z_n)\mbox{ and
}\rho(\y,\z_n)=\rho(\y,\c)\cup\rho(\c,\z_n)\,$$ for all $n\geq n_0$, which implies that $$T(\x,\z_n)-T(\y,\z_n)=T(\x,\c)-T(\y,\c)\,$$ for all $n\geq n_0$.
Let $\bH_{r}^\alpha$ be the hyperplane that pass through $\a_r:=a_r e^{i\alpha}$ and $r\vec{e}_1$, where $a_r=r/\cos\alpha$. Let $\x_{r}$ be the crossing point between the linear interpolation of $\rho_{\0}(\alpha)$ and $\bH_{r}^\alpha$ that maximizes the distance from $\a_r $. We claim that $$\label{E:asycoal0}
-T(r\vec{e}_{1},\0)\leq H^{\alpha}(r\vec{e}_{1},\0)\leq T(r\vec{e}_{1},\x_{r})-T(\x_{r},\0)\,.$$ The left-hand side of follows directly from the triangle inequality for $T$, since $H^{\alpha}(r\vec{e}_{1},\0)=T(\x,\c_r)-T(\y,\c_r)$ (as in the proof of Theorem \[tBuse-1\]). To show the right-hand side, notice that if $\x_{r}\not\in\rho(\0,\c_r)$ then $\c_r \in\rho(\0,\x_{r})$ which implies that $\c_r\in\rho(r\vec{e}_{1},\x_{r})$. Thus $$H^{\alpha}(r\vec{e}_{1},\0)=T(r\vec{e}_{1},\c_r)-T(\0,\c_r)= T(r\vec{e}_{1},\x_{r})-T(\x_{r},\0)\,.$$ If $\x_{r}\in\rho(\0,\c_r)$ then $$T(\0,\c_r)=T(\0,\x_{r})+T(\x_{r},\c_r)\,.$$ Consequently, $$\label{asycoal0*}
H^{\alpha}(r\vec{e}_{1},\0)=T(r\vec{e}_{1},\c_r)-T(\0,\c_r)=\big(T(r\vec{e}_{1},\c_r)-T(\c_r,\x_{r})\big)-T(\0,\x_{r})\,.$$ Since (again the triangle inequality) $$T(r\vec{e}_{1},\c_r)-T(\c_r,\x_{r})\leq T(r\vec{e}_{1},\x_{r})\,,$$ yields .
Now, $$T(r\vec{e}_{1},\x_{r})-T(\x_{r},\0)=$$ $$\big(T(r\vec{e}_{1},\x_{r})-\mu|r\vec{e}_{1}-\a_{r}|\big)\Big(\,:=\,I_1(r)\,\Big)$$ $$+\big(\mu|\a_r|-T(\x_{r},\0)\big)\,\,\Big(\,:=\,I_2(r)\,\Big)$$ $$+\mu|r\vec{e}_{1}-\a_{r}|-\mu|\a_r|\,\,\Big(\,:=\,I_3(r)\,\Big)\,.$$ By Remark \[r-straight\], if we pick $\xi\in(3/4,1)$ then for some constant $c>0$, almost surely, $|\x_r-\a_r|\leq cr^\xi$ for sufficiently large $r$. On the other hand, by the triangle inequality, $$|T(\x_r,r\vec{e}_1)-T(\a_r,r\vec{e}_1)|\leq T(\x_r,\a_r)\mbox{ and }|T(\x_r,\0)-T(\a_r,\0)|\leq T(\x_r,\a_r)\,.$$ Thus $$\limsup_{r\to\infty}\frac{|I_1(r)|}{r}\leq \limsup_{r\to\infty}\frac{|T(r\vec{e}_{1},\a_{r})-\mu|r\vec{e}_{1}-\a_{r}||}{r}+\limsup_{r\to\infty}\frac{\max_{|\z-\a_r|\leq cr^\xi}\{T(\a_r,\z)\}}{r}$$ and $$\limsup_{r\to\infty}\frac{|I_2(r)|}{r}\leq \limsup_{r\to\infty}\frac{|T(\0,\a_{r})-\mu|\a_{r}||}{r}+\limsup_{r\to\infty}\frac{\max_{|\z-\a_r|\leq cr^\xi}\{T(\a_r,\z)\}}{r}\,.$$
Combining Lemma \[l2\] with translation invariance one gets that for all $\epsilon>0$ $$\sum_{r\geq 1}\bP\big(|T(r\vec{e}_{1},\a_{r})-\mu|r\vec{e}_{1}-\a_{r}||\geq \epsilon r\big)<\infty\mbox{ and }\sum_{r\geq 1}\bP\big(|T(\0,\a_{r})-\mu|\a_{r}||\geq \epsilon r\big)<\infty$$ Therefore, by Borel-Cantelli’s lemma, $$\limsup_{r\to\infty}\frac{|T(r\vec{e}_{1},\a_{r})-\mu|r\vec{e}_{1}-\a_{r}||}{r}=0\mbox{ and }\limsup_{r\to\infty}\frac{|T(\0,\a_{r})-\mu|\a_{r}||}{r}=0\,.$$ In [@p-205] (Lemma 4.3 there) it is proved that, for some constants $c_0,x_0>0$, if $x>x_0$ then $$\bP\big(T(\0,\z)>x|\z|)\leq e^{-c_0 x|\z|}\,.$$ By noticing that, with high probability, the number of vertices belonging to a ball of radius $cr^{\xi}$ is of order $r^{2\xi}$, one can get that, for all $\epsilon>0$, $$\sum_{r\geq 1}\bP\big(\max_{|\z|\leq cr^\xi}\{T(\0,\z)\}>\epsilon r\big)<\infty\,.$$ Thus, together with the Borel-Cantelli’s lemma (and translation invariance), this yields $$\limsup_{r\to\infty}\frac{\max_{|\z-\a_r|\leq cr^\xi}\{T(\a_r,\z)\}}{r}=0\,.$$ Consequently, $$\limsup_{r\to\infty}\frac{|I_1(r)|}{r}=\limsup_r\frac{|I_2(r)|}{r}=0\,.$$ Since $$\lim_{r\to\infty}\frac{I_3(r)}{r}=\mu\frac{\sin\alpha-1}{\cos\alpha}=-\mu\frac{\cos\alpha}{1+\sin\alpha}\,$$ we finally have that $$\lim_{r\to\infty}\frac{T(r\vec{e}_{1},\x_{r})-T(\x_{r},\0)}{r}=-\mu\frac{\cos\alpha}{1+\sin\alpha}\,.$$ Together with , this yields Theorem \[tBuse-2\].
Competition versus coalescence {#pr-rand+strai}
------------------------------
In this section we give a sketch of the proof of the statements in Remark \[rand+strai\]. Let $\varphi:=(\z_{1},\z_{2},\dots)$ be a branch of the competition interface. Thus this branch marks the boundary between two different species, say $j_1$ and $j_2$. Assume further that if one moves along $\z_{n},\z_{n+1},\dots$ then on the right hand side we always see species $j_1$ while on the left hand side one see species $j_2$. By Theorem \[t1\], this branch has the direction $e^{i\theta}$ for some $\theta=\theta(\varphi)$. For $l=1,2$, let $(\v_{n}^{l})_{n\geq 1}$ be the sequence of vertices in $\calD_v\cap\B_{\x_{j_l}}$, so that the tile $\C_{\v_{n}^{l}}$ has an edge boundary that belongs to $\varphi$ . Thus, we have that $\v_{n}^{l}$ has the asymptotic orientation $e^{i\theta(\varphi)}$ (since, by Lemma \[5.2\], the distance between $\v_n^{l}$ and the corresponding branch of the competition interface is small if compared with $|\v_n|$). Together with Lemma \[geoconv\], this yields that there exists a subsequence $(n_m)_{m\geq 1}$ and a semi-infinite geodesic $\rho_l$, with asymptotic orientation $\theta(\varphi)$, so that $\rho(\x_l,\v_{n_m}^l)\to \rho_l$. Since $\rho(\x_l,\v_n^l)$ is a geodesic connecting two points in $\B_{\x_{j_l}}(\infty)$, we have that $\rho(\x_l,\v_n^l)\subseteq\B_{\x_{j_l}}(\infty)$ and thus $\rho_l\subseteq \B_{\x_{j_l}}(\infty)$.
Consequently, we have two geodesics $\rho_1$ and $\rho_2$ with the same orientation $e^{i\theta(\varphi)}$, but which do not coalesce (because $\rho_i\subseteq\B_{\x_{j_l}}$ for $l=1,2$). By Proposition \[pcoal\], this occurs with zero probability which shows the first statement of Remark \[rand+strai\].
By Remark \[r-straight\], for all $\xi\in(3/4,1)$, $\rho_{1}$ and $\rho_{2}$ are $(1-\xi)$-straight about its asymptotic orientation $e^{i\theta(\varphi)}$. Since $\varphi$ is caught between $\rho_{1}$ and $\rho_{2}$, this also implies that $\varphi$ is $(1-\xi)$-straight about its asymptotic orientation $e^{i\theta(\varphi)}$, which shows the second statement of Remark \[rand+strai\].
#### **Acknowledgment**
This work was developed during my doctoral studies [@p04] at Impa and I would like to thank my adviser, Prof. Vladas Sidoravicius, for his dedication and encouragement during this period. I also thank Prof. Charles Newman for proposing me the problem studied here, Prof. Thomas Mountford for a careful reading and useful comments about a previous version of this work, and Prof. James Martin for providing me the numerical simulations in Figure \[f2\]. Finally, I thank the whole administrative staff of IMPA for their assistance and CNPQ for financing my doctoral studies, without which this work would have not been possible.
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[^1]: We also refer to [@hn97], where an analog result is proved in an Euclidean first-passage percolation set-up.
[^2]: Recall that, by the Euler formula, the number of edges and vertices in a triangulation have the same order.
[^3]: For more on the non coalescence of semi-infinite geodesics see Section 1.3 in [@hn01]
| 0 |
---
abstract: 'Our previous article \[Phys. Rev. Lett. **104**, 060401 (2010)\] predicted that Casimir forces induced by the material-dispersion properties of certain dielectrics can give rise to stable configurations of objects. This phenomenon was illustrated via a dicluster configuration of non-touching objects consisting of two spheres immersed in a fluid and suspended against gravity above a plate. Here, we examine these predictions from the perspective of a practical experiment and consider the influence of non-additive, three-body, and nonzero-temperature effects on the stability of the two spheres. We conclude that the presence of Brownian motion reduces the set of experimentally realizable silicon/teflon spherical diclusters to those consisting of layered micro-spheres, such as the hollow-core (spherical shells) considered here.'
author:
- Jaime Varela
- 'Alejandro W. Rodriguez'
- 'Alexander P. McCauley'
- 'Steven G. Johnson'
bibliography:
- 'photon.bib'
title: 'Casimir micro-sphere diclusters and three-body effects in fluids'
---
Introduction
============
![Schematic of two-sphere dicluster geometry consisting of two dielectric spheres of radii $R_1$ and $R_2$ separated by a center–center distance $d$ from each other, and suspended by heights $h_1$ and $h_2$, respectively, above a dielectric plate.[]{data-label="fig:fig1"}](sketch){width="0.7\columnwidth"}
In this paper, we investigate the influence of non-additive/three-body and nonzero-temperature effects on our earlier prediction that the Casimir force (which arises from quantum electrodynamic fluctuations [@casimir; @Lifshitz80; @milton04]) can enable dielectric objects (micro-spheres) with certain material dispersions to form stable non-touching configurations (diclusters) in fluids [@RodriguezMc10:PRL; @Rodriguez08:PRL]. Such micro-sphere interactions are predicted to possess a variety of unusual Casimir effects, including repulsive forces, [@Kenneth; @Munday09; @Dzyaloshinskii61] a strong interplay with material dispersion [@RodriguezMc10:PRL], and strong temperature dependences [@RodriguezW10:PRL], and may have applications in microfluidic particle suspensions [@RahiZa09:arxiv; @McCauleyRo10:PRA]. A typical situation considered in this paper is depicted in [Fig. \[fig:fig1\]]{}, consisting of silicon and teflon micro-spheres suspended in ethanol above a gold substrate. Although our earlier work considered pairs of micro-spheres suspended above a substrate in the additive/pairwise approximation, summing the exact two–body sphere–sphere and sphere–substrate interactions, in this paper we perform exact three-body calculations. In [Sec. \[sec:pairwise\]]{}, we explicitly demonstrate the breakdown of the pairwise approximation for sufficiently small spheres, in which an adjacent substrate modifies the equilibrium sphere separation, but we also identify experimentally relevant regimes in which pairwise approximations (and even a parallel-plate proximity-force/PFA approximation [@Derjaguin]) are valid. In [Sec. \[sec:temp\]]{}, we also consider temperature corrections to the Casimir interactions. Although a careful choice of materials can lead to a large temperature dependence stemming from the thermal change in the photon distribution [@Bostrom; @Bordag; @RodriguezW10:PRL], we find that such thermal-photon effects are negligible ($< 2$%) for the materials considered here. However, we show that substantial modifications to the objects separations occur due to Brownian motion of the micro-spheres. This effect can be reduced by lowering the temperature, limited by the freezing point of ethanol ($T \approx
159$ K ), or by increasing the sphere diameters. We propose experimentally accessible geometries consisting of hollow micro-spheres (which can be fabricated by standard methods [@Wilcox]) whose dimensions are chosen to exhibit a clear stable non-touching equilibrium in the presence of Brownian fluctuations. We believe that this work is a stepping stone to direct experimental observation of these effects.
In fluid-separated geometries the Casimir force can be repulsive, leading to experimental wetting effects [@wetting; @Capillarity; @Israelachvili] and even recent direct measurements of the repulsive force in fluids for sphere-plate geometries [@moh1; @Munday07; @Feiler]. In particular, for two dielectric/metallic materials with permittivity $\varepsilon_1$ and $\varepsilon_3$ separated by a fluid with permittivity $\varepsilon_2$, the Casimir force is repulsive when $\varepsilon_1 < \varepsilon_2 <
\varepsilon_3$ [@Dzyaloshinskii61]. More precisely, the permittivities depend on frequency $\omega$, and the sign of the force is determined by the ordering of the $\varepsilon_k(i\kappa)$ values at imaginary frequencies $\omega=i\kappa$ (where $\varepsilon_k$ is purely real and positive for any causal passive material [@Dzyaloshinskii61]). If the ordering changes for different values of $\kappa$, then there are competing repulsive and attractive contributions to the force. At larger or smaller separations, smaller or larger values of $\kappa$, respectively, dominate the contributions to the total force, and so the force can change sign with separation. For example, if $\varepsilon_1 <
\varepsilon_2 < \varepsilon_3$ for large $\kappa$ and $\varepsilon_1 <
\varepsilon_3 < \varepsilon_2$ for small $\kappa$, then the force may be repulsive for small separations and attractive for large separations, leading to a stable equilibrium at an intermediate nonzero separation. Alternatively, for a sphere–plate geometry in which the sphere is pulled downwards by gravity, a purely repulsive Casimir force (which dominates at small separations) will also lead to a stable suspension. These basic ideas were exploited in our previous work [@RodriguezMc10:PRL] to design sphere–sphere and sphere–plate geometries exhibiting a stable non-touching configuration. The effects of material dispersion are further modified by an interplay with geometric effects (which set additional length-scales beyond that of the separation), as well as by nonzero-temperature effects which set a Matsubara length-scale $2\pi
kT/\hbar$ [@Bordag] that can further interact with dispersion in order to yield strong temperature corrections [@RodriguezW10:PRL]. Experimentally, stable suspensions are potentially appealing in that one would be measuring static displacements rather than force between micro-scale objects. The stable configurations may be further modified, however, by three-body effects in sphere–sphere–plate geometries and by Brownian motion of the particles within the potential well created by the Casimir interaction, and these effects are studied in detail by the present paper.
Until the last few years, theoretical predictions of Casimir forces were limited to a small set of simple geometries (mainly planar geometries) amenable to analytical solution, but a number of computational schemes have recently been demonstrated that are capable of handling complicated (and, in principle, arbitrary) geometries and materials [@Rahi09:PRD; @Rodriguez07:PRA; @Homer09]. Here, since the geometries considered in this paper consist entirely of spheres and planes, we are able to adapt an existing technique [@Rahi09:PRD] based on Fourier-like (“spectral”) expansions that semi-analytically exploits the symmetries of this problem. This technique, formulated in terms of the scattering matrices of the objects in a basis of spherical or plane waves, was developed in various forms by multiple authors [@Emig07; @Rahi09:PRD; @Kenneth08], and we employ the generalization of [@Rahi09:PRD]. Although this process is described in detail elsewhere [@Rahi09:PRD] and is reviewed for the specific geometries of this paper in the appendix, the basic idea of the calculation is as follows. The Casimir energy can be expressed via path integrals as an integral $\int_0^\infty \log \det A(\kappa)
d\kappa$ over imaginary frequencies $\kappa$, where $A$ is a “T-matrix” related to the scattering matrix of the system. In particular, one needs to compute the scattering matrices relating outgoing spherical waves from each sphere (or planewaves from each plate) being reflected into outgoing spherical waves (or planewaves) from every other sphere (or plate), which can be expressed semi-analytically (as infinite series) by “translation matrices” that re-express a spherical wave (or planewave) with one origin in terms of spherical waves (or planewaves) around the origin of the new object [@Rahi09:PRD]. This formalism is exact (no uncontrolled approximations) in the limit in which an infinite number of spherical/plane waves is considered. To obtain a finite matrix $A$, the number of spherical waves (or spherical harmonics $Y_{\ell m}$) is truncated to a finite order $\ell$. Because this expansion converges exponentially fast for spheres [@Canaguier09; @Rahi09:PRD], we find that $\ell \leq 12$ suffices for $< 1$% errors with the geometries in this paper. (Conversion from planewaves to spherical waves is performed by a semi-analytical formula [@Rahi09:PRD] that involves integrals over all wavevectors, which was performed by a standard quadrature technique for semi-infinite integrals [@Genz83].) Although it is possible to differentiate $\log \det A$ analytically to obtain a trace expression for the force [@Homer09], in this paper we use the simple expedient of computing the energy and differentiating numerically via spline interpolation. Previously, [Ref. ]{} employed the same formalism in order to study a related geometry consisting of vacuum-separated perfect-metal spheres adjacent to a perfect-metal plate, where it was possible to employ the method of images to reduce the computational complexity dramatically. That work found a three-body phenomenon in which the presence of a metallic plate resulted on a stronger attractive interaction between the spheres, and that this effect becomes more prominent at larger separations [@Lopez09], related to an earlier three-body effect predicted for cylindrical shapes [@RahiRo07; @Rodriguez07:PRL]. Here, we examine dielectric spheres and plate immersed in a fluid and therefore cannot exploit the method of images for simplifying the calculation, which makes the calculation much more expensive because of the many oscillatory integrals that must be performed in order to convert between planewaves (scattering off of the plate) and spherical waves (see appendix). We also obtain three-body effects, in this case on the equilibrium separation distance, but find that the magnitude and sign of these effects depends strongly on the parameters of the problem.
Three-body Effects {#sec:pairwise}
==================
![Equilibrium separation $d_e(h) / d_e(\infty)$ between two $R=25$nm spheres suspended in ethanol as a function of their surface–surface separation $h$ from a plate (and normalized by the equilibrium separation for the case of two isolated spheres, i.e. $h
= \infty$). $d_e$ is plotted for various material combinations, denoted by the designation sphere–sphere–plate, e.g. a PS and silicon sphere suspended above a gold plate is denoted as PS–Si–Au. Solid/dashed lines correspond to stable/unstable equilibria. (In the case of a gold plate, the spheres are chosen to have $R = 50$nm.) The inset shows $d_e$ (in units of nm) for the case of two PS and silicon spheres ($R=57$nm) above a gold plate.[]{data-label="fig:fig2"}](dh_over_dinf){width="1.0\columnwidth"}
To quantify the strength of three-body effects in the sphere–sphere–plate system of [Fig. \[fig:fig1\]]{}, we begin by computing how the zero–temperature equilibrium sphere–sphere separation $d$ varies as a function of the sphere-plate separation $h$ for two equal-radius spheres, as plotted in [Fig. \[fig:fig2\]]{}. To start with, we consider very small spheres, with radius $R=25$ nm, for which the three-body effects are substantial. The separation $d_h$ at a given $h$ is normalized by $d_\infty$ ($d$ as $h\to\infty$, i.e. in the absence of the plate). Several different material combinations are shown (where $X$–$Y$–$Z$ denotes spheres of materials $X$ and $Y$ and a plate of material $Z$): polystyrene (PS), teflon (Tef), and silicon (Si) spheres with gold (Au), teflon (Tef), and vacuum (air) plates (the latter corresponding to a fluid-gas interface). Depending on the material combinations, we find that $d_h$ can either increase or decrease by as much as $15\%$ as the plate is brought into proximity with the spheres from $h=\infty$ to $h \approx R$. (We expect even larger deviations when $h < R$, but small separations are challenging for this computational method [@Rahi09:PRD] and our results for $h \geq R$ suffice here to characterize the general influence of three-body effects.)
Interestingly, depending on the material combination, the $d_h$ can either increase or decrease as a function of $h$: that is, the proximity of the plate can either increase or decrease the effective repulsion. This is qualitatively similar to previous results for vacuum-separated perfect-metal spheres/plates [@Lopez09] in the following sense. Previously, the attractive interaction between a sphere and a plate was in general found to enhance the attraction between two identical spheres as the plate became closer [@Lopez09].(There are certain regimes, not present here, where the attractive interaction decreases) Here, we observe that the sphere–plate interaction changes the sphere–sphere interaction with *the same sign* as $h$ becomes smaller: if the sphere–plate interaction is repulsive, the sphere–sphere interaction becomes more repulsive (larger $d$), and vice-versa for an attractive sphere–plate interaction. Since the spheres are not identical, the three-body effect is dominated by the sign of the stronger sphere–plate interaction out of the two spheres. Thus, examining the signs and magnitudes of the pairwise interactions in all cases of [Fig. \[fig:fig2\]]{} turns out to be sufficient to predict the sign of the three-body interaction, although we have no proof that this is a general rule. (In contrast, for non-spherical objects such as cylinders, there can be competing three-body effects that make the sign more difficult to predict, even in vacuum-separated geometries where all pairwise interactions are attractive, which can even lead to a non-monotonic effect [@RahiRo07; @Rodriguez07:PRL].)
[Figure \[fig:fig2\]]{} also exhibits the interesting phenomenon of bifurcations, in which stable equilibria (solid lines) and unstable equilibria (dashed lines) appear/disappear at some critical $h$ for certain materials and geometries, which is discussed in more detail in [Sec. \[sec:bifurcations\]]{}. As the sphere radius $R$ increases, all of these three-body effects rapidly decrease, eventually entering an additive regime in which three-body effects are negligible and in which a parallel-plate/PFA approximation eventually becomes valid, as described in [Sec. \[sec:additive\]]{}.
Bifurcations {#sec:bifurcations}
------------
![Equilibrium separation $d_e(\infty)$ (units of nm) between a Si sphere and either a teflon(Tef) or polystyrene(PS) sphere immersed in ethanol as a function of their equivalent radii $R$. Solid/dashed lines denote stable/unstable equilibria.[]{data-label="fig:fig3"}](new_Silicon-plsty_dvsR){width="1.0\columnwidth"}
In the case of PS and Si spheres suspended above either a gold or teflon plate, one can observe the emergence or disappearance of a stable (solid) and unstable (dashed) pair of equilibria as $h$ decreases from $h=\infty$, respectively, as evidenced by the blue curves in [Fig. \[fig:fig2\]]{} (teflon plate) and [Fig. \[fig:fig2\]]{} (inset) (Au plate). This can be qualitatively explained by the fact that the isolated sphere–sphere interactions exhibit a natural bifurcation for sufficiently-large spheres, in conjunction with the fact that the presence of the plate typically acts to either increase or decrease the sphere–sphere interaction, depending on the sign of the dominant sphere–plate interaction, as explained above.
In particular, [Fig. \[fig:fig3\]]{} shows the isolated Si–PS and Tef–Si sphere–sphere equilibrium separation $d_e$ as a function of the radius $R$ of the spheres. As a consequence of its material dispersion (similar to phenomena observed in [@RodriguezMc10:PRL]), the Si–PS combination exhibits a bifurcation at $R \approx 55$ nm where the stable and unstable equilibria, such that there is no equilibrium for larger $R$ (the interaction is purely attractive). The Tef–Si combination exhibits no such bifurcation (even if we extend the plot to $R=300$ nm), because it has no unstable equilibrium: the interaction is purely repulsive for small separations and attractive for large separations. Therefore, if the Si–PS radius is above or below the 55 nm bifurcation, the presence of the plate can shift this bifurcation and lead to a bifurcation as a function of $h$ as in [Fig. \[fig:fig2\]]{}, whereas no such bifurcation with $h$ appears for Tef–Si.
In the Si–PS–Au case of a gold plate with Si–PS spheres, the sphere–plate interactions turns out to be primarily repulsive, which should push the bifurcation in [Fig. \[fig:fig3\]]{} to the *right* (shrinking the attractive region) as $h$ decreases. Correspondingly, if we choose a radius $R=57$ nm just to the right of isolated-sphere bifurcation, then as $h$ decreases the Si–PS–Au combination should push the bifurcation past $R=57$ nm leading to the creation of a stable/unstable pair for small $h$, and precisely this behavior is observed in the inset of [Fig. \[fig:fig2\]]{}. Conversely, for the Si–PS–Tef case of a teflon plate with Si–PS spheres, the sphere–plate interaction is primarily attractive, and the opposite behavior occurs: choosing a radius $R=25$ nm to the left of the isolated-sphere bifurcation, decreasing $h$ increases the attraction and moves the bifurcation to the *left* in [Fig. \[fig:fig3\]]{}, eventually causing the disappearance of the stable/unstable equilibrium at $R=25$ nm. Correspondingly, for the Si–PS–Tef curve in [Fig. \[fig:fig2\]]{}, we see the disappearance of a stable/unstable pair for sufficiently small $h$.
The additive regime {#sec:additive}
-------------------
In general, three-body effects can expected to disappear in various regimes where key parameters of the interaction become small. First, for large radii, where $h$ (the sphere–plate separation) and $d$ (the sphere–sphere separation) become small compared to $R$, eventually the Casimir interaction is dominated by nearest-surface interactions, or the proximity-force approximation (PFA), in which the force can be approximated by additive surface–surface “parallel-plate” forces [@bordag01; @Gies; @Derjaguin]. In order to damp the Brownian fluctuations as described in the next section, we actually propose to use much larger ($R>5\,\mu$m) spheres, and we quantify the accuracy of PFA in this regime below. Second, as $h$ becomes large compared to $d$, the effect of the plate becomes negligible and three-body effects disappear; this is apparent in [Fig. \[fig:fig2\]]{}, where $d_e \to
d_e(\infty)$ when $h \gg d_e$. Third, in the limit where one of the spheres is much smaller than the other sphere, then the smaller sphere has a negligible effect on the sphere–plate interaction of the larger sphere, and at least some of the three-body effect disappear as described below. In fact, we find that even for a situation in which one sphere is only a few times smaller than the other, the three-body effects tend to be negligible. For the sphere-radius regime considered in our previous work, we argue below that equal-height suspension of the two spheres leads to a strong asymmetry in sphere radii that tends to eliminate three-body effects.
![Center–surface $L_e$ (solid lines) and surface–surface $h_e$ (dotted lines) equilibrium separation (units of nm) of a teflon (red) or Si hollowed sphere (shown on the inset) suspended in ethanol above a gold plate, as a function of radius $R$ (units of nm). The equilibria are plotted for different values of the fill-fraction $\alpha$, defined as the ratio of the spherical–shell thickness over the radius of the sphere. Solid/dashed lines correspond to stable/unstable equilibria.[]{data-label="fig:fig4"}](larger_alph_eqbms){width="1.0\columnwidth"}
To begin with, let us consider sphere radii on the order of $10^2$ nm, as in our previous work [@RodriguezMc10:PRL]. We wish to make a bound dicluster, at some separation $d$, of two spheres (Si and teflon) that are suspended above a gold substrate by Casimir repulsion in balance with gravity. Furthermore, suppose that we wish to suspend both spheres at the same equilibrium height $h_e$, and therefore choose the radii of the two spheres to equate their $h_e$ values. In [Fig. \[fig:fig4\]]{}, we plot $h_e$ as a function of radius $R$ for the isolated sphere–plate geometries ($d\to\infty$). For example, with an Si sphere of radius $R=100$ nm, the (stable) equilibrium height is $h_e = 298.17$ nm, whereas to obtain the same $h_e$ value for teflon one needs a much larger teflon sphere of radius $R=217.2$ nm, primarily because the Casimir repulsion is stronger for teflon. If, instead of a pairwise calculation, we perform an exact three-body calculation of the $h_e$ values for these radii at the equilibrium sphere–sphere separation $d_e = 92.8$ nm, we find that the $h_e$ values change by $< 1$%. Conversely, if we keep $h_e$ fixed and compute the three-body change in $d_e$ (compared to $h\to\infty$), again we find that the change is $<1$%. As mentioned above, the small size of the Si sphere makes it unsurprising that the Si sphere does not change the equilibrium $h_e$ of the much larger teflon sphere. Furthermore, the sensitivity of the sphere–sphere force $F_d$ to the teflon $h$ is equal to the sensitivity of the teflon sphere–plate force $F_h$ to $d$, thanks to the equivalence $\partial
F_d/\partial h = -\partial^2 U/\partial d \partial h = \partial F_h
/\partial d$ where $U$ is the energy. Therefore, one would also not expect the finite value of $h_e$ for the Si sphere to modify the equilibrium $d_e$. Size asymmetry alone, however, does not explain why the finite $h_e$ of the *teflon* sphere does not affect the sphere-plate interactions of the Si sphere. Even if the Si sphere were of infinitesimal radius, the Casimir–Polder energy the Si sphere would be determined by a Green’s function at the Si location [@Lifshitz80], and if the Si sphere is at comparable distance $d_e \sim h_e$ from both the teflon plate and the sphere, one would in general expect the response to a point-dipole source at the Si location (the Green’s function) to depend non-additively on the teflon sphere and the plate even for an infinitesimal Si sphere. However, in the present case we do not observe any non-additive effect on the Si-sphere $h_e$, because the factor of three(approximately) difference between $h_e$ and $d_e$ is already sufficient to eliminate three-body effects (as in [Fig. \[fig:fig2\]]{}).
[Figure \[fig:fig4\]]{} also exhibits a bifurcation of stable (solid lines) and unstable (dashed lines) equilibria that causes the stable $h$ equilibrium to vanish for Si at large radii. In order to utilize larger spheres to reduce the effects of Brownian motion in the next section, one can consider instead a geometry of hollow air-filled spherical shells with outer radius $R$ and shell thickness $\alpha R$ (so that $\alpha=1$ gives a solid sphere). Such hollow microspheres are readily fabricated with a variety of materials [@Wilcox]. As [Fig. \[fig:fig4\]]{} shows, decreasing the shell thickness $\alpha$ pushes the bifurcation to larger $R$, and also increase $h_e$ by making the sphere more buoyant. This modification allows us to consider $R\approx10\,\mu$m in the next section, where PFA should be accurate. For only $3\,\mu$m spheres and $500$ nm separations in fluids, we previously found that the correction to PFA (which scales as $d/R$ to lowest order [@gies06:PFA; @Maia-Neto; @Mazzitelli]) was only about 15%. For the three times larger radii and somewhat smaller separations in the next section the corrections to PFA are typically $< 5$%, sufficient for our current purposes.
Nonzero Temperature and Experiments {#sec:temp}
===================================
In this section, we address a number of questions of consequence to an experimental realization of the teflon/silicon two-sphere dicluster of [Fig. \[fig:fig1\]]{}. In particular, we consider several ways in which a nonzero temperature can disrupt the observation of stable equilibria. A nonzero temperature will manifest itself in at least two important ways. First, there will be a change in the Casimir force between the objects due to the presence of real (non-virtual) photons in the system. Second, the inclusion of nonzero temperature will cause the spheres to experience Brownian motion arising from the thermal agitations in the fluid [@Risken]. We consider the influence of both of these effects on the observability of stable particle clusters and suspensions.
At zero temperature, the Casimir force $F$ is determined by an integral $F = \int_0^\infty d\xi f(\xi)$ of a complicated integrand $f(\xi)$ evaluated at imaginary frequencies $\xi$ [@Lifshitz80]. At $T > 0$, the integral is replaced by a finite sum over Matsubara frequencies $\omega_n =2\pi n k T/\hbar$, arising from the poles of the $\coth$ photon distribution along the imaginary frequency axis [@schwinger78; @Bordag], leading to a force $F_T$ given by: $$F_T = \frac{2\pi k T}{\hbar} \left[\frac{f(0+)}{2} + \sum_0^\infty
f\left(\frac{2\pi kT}{\hbar}n\right)\right],$$ which is exactly a trapezoidal-rule approximation to the zero-temperature force with a discretization error determined by the Matsubara wavelength $\lambda_T = 2\pi c / \xi_T = \hbar /
kT$ [@boyd01:book]. Because the integrand $f(\xi)$ is smooth and typically varies on a scale much slower than $1/\lambda_T$, where $\lambda_T = 7.6\,\mu$m at room temperature $T=300$ K, the finite-$T$ correction to the zero-temperature Casimir force is often negligible [@milton04]. However, in fluids, as is the case here, larger temperature effects have been obtained [@RodriguezW10:PRL] by dispersion-induced oscillations in $f(\xi)$, and so we must check our previous zero-temperature predictions against finite-$T$ calculations. For the Tef–Si–Substrate case considered here, we find that $T>0$ corrections to the $T=0$ forces are no more than 2% over the entire range of separations considered here, and hence they can be neglected.
The presence of Brownian motion proves a much more difficult experimental complication to overcome. First, Brownian motion will lead to random fluctuations in the position of the spheres, making it hard to measure their stable separations in an experiment [@Risken]. Second, and more importantly, sufficiently large fluctuations can drive the Si sphere to “tunnel” past its unstable equilibrium position with the gold plate, leading to stiction [@Risken] since the Si–Au interaction is purely attractive for small separations. The remainder of this section will revolve around the question of how and whether one can overcome both of these difficulties to observe suspension in experiments. In particular, we consider observation of the average separation of the spheres over a sufficiently long time, but not so long that stiction occurs, and analyze the separation statistics and the stiction timescale. First, however, we describe how the parameters are chosen so that Brownian fluctuations are not so severe.
The sphere geometry that we consider is depicted in [Fig. \[fig:fig5\]]{}: a hollow spherical shell suspended by a surface–surface separation $h$ above a layered substrate, consisting of a thin indium tin oxide (ITO) film of thickness $H$ deposited on a gold substrate, where the purpose of the ITO layer is to eliminate the Si-sphere instability as explained below. The thickness of the shell is denoted as $t=\alpha
R$, where $\alpha$ is a convenient fill-fraction parameter. We consider hollow spheres in order to increase $R$ and thereby reduce Brownian fluctuations. In particular, both the Brownian fluctuations and the probability of stiction in the case of the silicon sphere are reduced by increasing the strength of the Casimir force, which can be achieved by increasing $R$ since the Casimir force scales roughly with surface area, and below we consider radii from 1 to 10 $\mu$m. In this regime, as quantified in the previous section, a simple PFA approximation is sufficient to accurately compute the forces and separations. However, because the gravitational force scales as $R^3$, for large $R$ the gravitational force will overcome the Casimir force and push the Si sphere past its unstable equilibrium into stiction. In order to reduce the gravitational force while keeping the surface area fixed, we propose using a hollow Si sphere. We find that in addition to hollowing the spheres, it is also beneficial to deposit a thin ITO film,o n top of the gold substrate (the permittivity of ITO is modeled via an empirical Drude model with plasma frequency $\omega_p=1.46\times 10^{15}$ rad/s and decay rate $\gamma = 1.53\times 10^{14}$ rad/s). The ITO layer acts to decrease the equilibria separations and therefore increase the Casimir interactions between the spheres and the substrate. However, because the Casimir force between teflon/silicon and ITO is attractive/repulsive at small separations, respectively, increasing $H$ pushes the silicon-substrate’s unstable equilibrium to smaller separations while *introducing* a teflon-substrate unstable equilibrium that gets pushed to larger separations. In what follows, we find that $H$ from 14–30 nm is sufficient to obtain experimentally feasible suspensions, although here we only consider the case of $H=15$ nm. The effect of hollowing the spheres is shown in the top panel of [Fig. \[fig:fig6\]]{}: smaller $\alpha$ values push the stable/unstable bifurcation of teflon out to larger $R$. Hollowing the silicon sphere is not necessary because silicon has no unstable equilibrium (in this configuration it is repulsive down to zero separation). However, as shown in the bottom panel of [Fig. \[fig:fig6\]]{}, hollowing the silicon sphere does change its $h_e$ at a given $R$. For example, one can choose a Tef $\alpha=0.142$ and a Si ($\alpha=0.14$) to obtain the same equilibrium surface-to-center height $L_e$ over a wide range of sphere radii, as shown in upper-right inset of [Fig. \[fig:fig6\]]{}. Alternatively, one can choose a hollow teflon sphere to match the equilibrium surface-surface separations $h_e$ for equal sphere radii, as shown in the lower-right inset of [Fig. \[fig:fig6\]]{}.
![Surface–surface equilibrium height $h_e$ (units of nm) for the hollowed–sphere geometry of [Fig. \[fig:fig5\]]{}, consisting of either a Si (top) or teflon (bottom) hollowed sphere (fill-fraction $\alpha$) suspended in ethanol above a $H = 15$ nm ITO layered gold plate, as a function of sphere radius $R$ (in units of $\mu$m). Solid/dashed lines correspond to stable/unstable equilibria. $h_e$ is plotted for different values of $\alpha$, denoted in the figure. The top inset plots the center–surface separation $L_e$ (in units of nm) as a function of $R$ of a hollowed teflon/Si (red/blue lines) sphere suspended again above a gold plate, for $\alpha = 0.14/0.142$. The lower inset shows $h_e$ for both teflon and Si spheres for $R \in [8.6,10]\mu$m.[]{data-label="fig:fig6"}](Figure6){width="1.0\columnwidth"}
![Average $\langle h \rangle$ (thick lines) and equilibrium $h_e$ (thin lines) height (in units of nm) of a hollowed teflon/Si (blue/red lines) sphere suspended above a $H = 15$ nm ITO layered gold plate, for $\alpha = 0.142/0.13$, as a function of sphere radius $R$ (in units of $\mu$m). Solid/dashed lines correspond to stable/unstable equilibria. The red/blue shaded regions indicate positions where the teflon/Si spheres are found with $95\%$ probability. The inset shows $\langle d \rangle $ (thick line) and $d_e$ (thin line) separations as a function of their radius for two equal radii teflon Si spheres. The gray shaded region indicate the separations which the teflon and Si spheres are found with $95\%$ probability.[]{data-label="fig:fig7"}](teflon_flucs_prime){width="1.0\columnwidth"}
Statistics of Brownian motion
-----------------------------
As mentioned above, Brownian motion will disturb the spheres by causing them to move randomly about their stable equilibrium positions, and this can cause the Si sphere to move past its unstable equilibrium point, inducing it to stick top the plate. To quantify the range of motion of both spheres about their equilibria, we consider the statistical properties of their fluctuations. In particular, we consider the average plate–sphere separations $\langle h \rangle_T$ and average sphere–sphere separations $\langle d \rangle_T$ near room temperature ($T=300$ K), determined by an ensemble average over a Boltzman distribution. For example, $\langle h \rangle_T$ is given by: $$\langle h \rangle_T = \dfrac{\int_0^\infty dz\, z
\exp\left(U(z)/kT\right)}{\int_0^\infty dz\,
\exp\left(U(z)/kT\right)},$$ where $U(z)$ is the total energy (gravity included) of the sphere–plate system at a surface–surface height $z$. (A similar expression yields $\langle d \rangle_T$). In the case of teflon, the short-range attraction means that the suspension is only metastable under fluctuations; here, we only average over separations prior to stiction by restriction $z$ to be $\geq$ the unstable equilibrium, and consider the stiction timescale separately below. In addition to the average equilibrium separations, we are also interested in quantifying the extent of the fluctuations of the spheres, which we do here by computing the 95% confidence interval $\{\sigma_{-},\sigma_{+}\}$, defined as the spatial region over which the sphere is found with 95% probability around the equilibria, where $\sigma_{\pm}$ denotes the lower/upper bound of that interval. These results are shown in [Fig. \[fig:fig7\]]{} for $h$, with $d$ shown in the inset, in which shaded regions indicate the confidence intervals, as a function of $R$ where $\alpha$ is chosen to yield approximately equal $h_e$ ($\alpha =
0.142$ for teflon and $\alpha=0.13$ for Si). (Note that the horizontal separation $\langle d \rangle$ is a purely Casimir interaction and the difference here from $\alpha=1$ is negligible in the PFA regime.) As predicted above, the Brownian fluctuations of the spheres vanish as $R \to \infty$ and are dramatically suppressed for $R \gtrsim 5\,\mu$m, where one finds $\langle h \rangle \approx
h_e$. In addition, we find that the teflon sphere can safely avoid the unstable equilibrium and stiction in the sense that the unstable equilibrium is far outside the confidence interval; the timescale of the stiction process is quantified below. The asymmetrical nature of the confidence interval results from the fact that the Casimir energy decreases as a function of $z$, and as a consequence the Brownian excursions favor the $+z$ direction. The fluctuations in $\langle d
\rangle$ are substantially larger than those in $\langle h \rangle$ (nor is there any obvious reason why they should be comparable, given that the nature of the sphere–sphere equilibrium is completely different from the sphere–plate equilibrium), making the precise value of $d_e$ potentially harder to observe.
![Average $ \langle h \rangle$ (thick line) and equilibrium $h_e$ (thin line) height (in units of nm) of a hollowed Si/teflon sphere (top/bottom) of radii $R = 10$/$9.915\mu$m suspended in ethanol above a $H = 15$ nm ITO layered gold plate, as a function of fill-fraction $\alpha$ (indicated in [Fig. \[fig:fig5\]]{}). Shaded regions indicate $h$ positions where the Si/teflon spheres are found with $95\%$ probability. Solid/dashed lines indicate stable/unstable equilibria. For reference, we state the equilibrium $d_e$ and average $\langle d \rangle$ horizontal seperations between $R =
10$/$9.915\mu$m Si/Tef spheres in the top figure.[]{data-label="fig:fig9"}](alpha_inset){width="1.0\columnwidth"}
Instead of considering the Brownian statistics as a function of $R$, we can instead consider the statistics as a function of $\alpha$ for fixed radii $\approx 10\mu$m (chosen to obtain nearly equal sphere-center heights $L_e$), as shown in [Fig. \[fig:fig9\]]{}. One key point is that there is a minimum allowed $\alpha$: if $\alpha$ is too small, the buoyant force (assuming an air-filled hollow sphere) will eventually become positive and the sphere will float, although this limitation is removed if one could infiltrate the hollow sphere with the fluid. For the teflon sphere, there is also an upper limit to $\alpha$ for a given $R$ to avoid stiction as discussed previously.
Stiction and tunnelling rates
-----------------------------
As mentioned above, the stable equilibrium for the teflon sphere is actually only metastable—because the Casimir force is attractive for small separations, given a sufficiently long observation time $\tau$ the sphere will “tunnel” (via Brownian fluctuations) past the energy barrier $\Delta$ posed by the unstable equilibrium, and stick to the plate (stiction). Given the energy barrier, the temperature $T$, and the viscous drag on the particle, we can apply standard methods [@Chandrekasar; @Risken; @Melnikov19911] to compute the timescale for stiction. This calculation, which is described in detail below, shows that for various values of the fill factor $\alpha$ the expected time $\tau$ to stiction (which increases exponentially with $\Delta/kT$) can vary dramatically, but can easily be made on the order of years.
![Energy barrier $\Delta / kT$ of a hollowed teflon sphere suspended in ethanol above a $H = 15$ nm ITO layered gold plate at $T=300$ K, as a function of sphere radius $R$ (in units of $\mu$m) and for different values of fill-fraction $\alpha$. The inset shows the energy landscape $U / k T$ as a function of the surface–surface height $h$ (units of nm) for a teflon sphere of radius $R=10\mu m$ with a fill fraction of $\alpha = 0.142$. []{data-label="fig:fig8"}](delta_kt_tef){width="1.0\columnwidth"}
The energy barrier $\Delta/kT$ is plotted versus the teflon sphere radius $R$ for various $\alpha$ in [Fig. \[fig:fig8\]]{}, and can easily be made $> 10$ to obtain a very long metastable lifetime. As we discussed earlier, the $\Delta$ increases with $R$ at first because this increases the Casimir force, but has a maximum at some $R$ where gravity begins to dominate. Decreasing $\alpha$ decreases the gravitational force and therefore increases both the maximum $\Delta$ and the corresponding $R$. A typical energy landscape $U(z)$ is shown in the inset, exhibiting a local minimum at a height $h_e$ and an unstable equilibrium (maximum) at $h_u$. Also noted on the inset is the “tunneling” height $h_* > h_e$ at which $U(h_*) =
U(h_u)$. [Figure \[fig:fig8\]]{} also shows the energy barrier $\Delta/kT$ of a silicon sphere ($\alpha = 0.1288 \approx \alpha_c$, $R=10~\mu$m) in the absence of the ITO layer ($H = 0$) to be significantly smaller than that of teflon. Of course $\Delta / kT$ in this case could be made larger merely by choosing $\alpha \approx \alpha_c$, but we find (below) that achieving experimentally realizable lifetimes severely limits the range of realizable $\alpha$, i.e. requires that the Si thickness be known to within a few nanometers.
Because $\Delta \gg kT$, the lifetime $\tau$ of a Brownian particle trapped around a local minimum of a potential $U(z)$ can be approximated by [@Melnikov19911]: $$\tau = e^{\Delta/kT}
\left[\left(1 +
\frac{\gamma}{4\omega^2}\right)^{1/2} -
\frac{\gamma}{2\omega}\right]^{-1}
\frac{2\pi}{\Omega}
\zeta\left(\frac{\gamma S}{kT}\right),
\label{eq:lifetime}$$ where $\gamma$ is the viscous drag coefficient (drag force = $-\gamma\,\mathrm{velocity}$), $\omega$ and $\Omega$ characterize the curvature of $U(z)$ at the energy maximum and minimum respectively \[as defined in [Eq. (\[eq:omega\])]{}\], $\zeta(\delta)$ is a transcendental function defined in [Eq. (\[eq:zeta\])]{}, and $S$ is an integral of the potential barrier defined by [Eq. (\[eq:S\])]{}. Let $m$ be the mass of the sphere. The drag coefficient for a sphere of radius $R$ in a fluid with viscosity $\eta$ is $\gamma = 6 \pi R \eta /
m$ [@Landau:fluid], where a typical viscosity is $\eta \approx
1.17 \pm 0.06$ mPas for ethanol [@rhodata]. The other quantities are given by: $$\begin{aligned}
S &= 2 \int_{h_{u}}^{h_{c}} \, dz \sqrt{-2 m U(z)} \label{eq:S} \\
\omega&=\sqrt{\frac{U''(h_u)}{m}} ; \Omega=\sqrt{\frac{U''(h_e)}{m}}
\label{eq:omega} \\ \zeta(\delta) &= \exp\left[-\frac{2}{\pi} \int_0^{\pi/2} \,dz
\ln\left(1 - e^{-\delta / 4 \cos^2 z}\right)\right]. \label{eq:zeta}\end{aligned}$$ Combining these formulas and choosing different values of $R$ and $\alpha$ to obtain different barriers $\Delta$ and landscapes $U(z)$ as in [Fig. \[fig:fig8\]]{}, the lifetime $\tau$ can be designed to take on a wide range of values. The exponential dependence on $\Delta$ means that $\tau$ rapidly transitions from very short to very long as $\alpha$ changes, but can easily be made large. For example, with $R=
8.5\mu$m and $\alpha < 0.15$, one obtains $\tau > 40$ days. (Conversely, for sufficiently large $\alpha$ one could design experiments where stiction occurs on an arbitrarily fast timescale, but in this $\Delta \sim kT$ regime the approximations of [Eq. (\[eq:lifetime\])]{} are no longer valid.)
Strictly speaking, this is a conservative estimate of the timescale because the drag coefficient $\gamma$ for a sphere above a plate is larger than that of an isolated sphere. As the sphere approaches the plate, the drag is dominated by the “lubrication” problem of the fluid squeezed between the sphere and the plate, and the drag increases dramatically [@Hamrock].
Conclusion {#sec:conc}
==========
Even including the thermal motion of the particles and the finite lifetime of metastable suspensions, the stable suspension and separation of particle diclusters appears to be experimentally feasible. In the experimentally relevant regimes, these effects consist primarily of pairwise sphere–sphere and sphere–plate interactions; while three-body effects become significant for smaller spheres, the increased Brownian fluctuations for small spheres makes such an experiment challenging. Although the systems considered here consisted of silicon and teflon spheres above layered substrate in ethanol, many other materials combinations could potentially be explored to modify these phenomena, including multi-material sphere systems such as multi-layer spheres or patterned substrates that could exhibit unusual effective dispersion phenomena. Although we considered hollow (air core) spheres, one could also use fluid-filled spheres or similar modifications in order to modify the effect of gravity. Alternatively, one could use non-spherical geometries such as disks, which have a both surface area and volume proportional to $R^2$ so that gravity does not dominate asymptotically. We have recently demonstrated computational methods capable of accurate modeling of such geometries, and find that the additional rotational degrees of freedom can lead to additional phenomena such as transitions in the stable orientation with separation [@Reid:arxiv]. In general, the possibility of both repulsion and stable equilibria in fluids (whereas the latter are not possible in vacuum [@Rahi10:PRL]but do exist in the critical casimir fluids [@Trondle; @Mohry:PRE]) opens the possibility of a rich and currently little explored territory for Casimir physics, and it is likely that many effects remain to be discovered.
Appendix {#appendix .unnumbered}
========
In what follows, we write down an expression for the Casimir energy of of the system in [Fig. \[fig:fig1\]]{}, in terms of the scattering and translation matrices of the individual objects (spheres and plates) of the geometry. A similar expression was derived in [@Lopez09] in the case of perfect-metal vacuum-separated objects, for which an additional simplification, based on the method of images, was possible [@Brown:1969]. Here, we consider the more general case of fluid-separated dielectric objects.
The starting point of the Casimir-energy expression is the well-known scattering-matrix formalism, derived in [@Emig07; @Rahi09:PRD], in which the Casimir energy $U$ between an arbitrary set of objects can be written as: $$U=\frac{\hbar c}{2 \pi}
\displaystyle\int_{0}^{\infty}d\kappa
\log{\det{\mathbb{M}\mathbb{M}_{\infty}^{-1}}},
\label{eq:caseq}$$ where $\mathbb{M}_{\infty}^{-1} = \mathrm{diag}(\mathbb{F}_{1} ,
\mathbb{F}_{2},...)$ and the matrix $\mathbb{M}$ is given by: $$\mathbb{M} =\left( \begin{array}{cccc} \mathbb{F}_{1}^{-1} &
\mathbb{X}^{12} & \mathbb{X}^{13} & ...\\ \mathbb{X}^{21} &
\mathbb{F}_2^{-1} & \mathbb{X}^{23} & ...\\ ... & ... & ... & ...\\
\end{array}
\right),
\label{eq:MMinf}$$ where $\mathbb{F}_i(\kappa)$ is the matrix of inside/outside scattering amplitudes of the $i$th object, and $\mathbb{X}^{ij}$ the translation matrix that relates the scattering matrix of the $i$th and $j$th objects, as described in [@Rahi09:PRD]. Here, the plate is labeled by the index $i=1$ whereas the left and right spheres are labeled as $i=2$ and $i=3$, respectively.
For computational convenience, the determinant in [Eq. (\[eq:caseq\])]{} can be re-expressed in terms of standard operations on the block matrices composing $\mathbb{M}$, and in this case we find that: $$\begin{gathered}
\det \mathbb{M}\mathbb{M}_{\infty} =
\det\left(\mathcal{I}-\mathcal{N}^{(1)}\right)
\det\left(\mathcal{I}-\mathcal{N}^{(2)}\right) \nonumber \\ \times
\det\left(\mathcal{I}-\left(\mathbb{I}-\mathcal{N}^{(2)}\right)^{-1}
\mathcal{A}\left(\mathcal{I}-\mathcal{N}^{(1)}\right)^{-1}\mathcal{B}\right),
\label{eq:detex}\end{gathered}$$ where $$\begin{aligned}
\mathcal{N}^{(2)} =
\mathbb{F}_3{\mathbb{X}}^{31}{\mathbb{F}}_1{\mathbb{X}}^{13}, \,\,\,
\mathcal{A}=F_{3}X^{32}-F_{3}X^{31}F_{1}X^{12};
\nonumber \\
\mathcal{B}=F_{2}X^{23}-F_{2}X^{21}F_{1}X^{13}, \,\,\,
\mathcal{N}^{(1)}=F_{2}X^{21}F_{1}X^{12},
\label{eq:matrices}\end{aligned}$$ where $(\mathcal{I}-\mathcal{N}^{(1)})$ and $(\mathcal{I}-\mathcal{N}^{(2)})$ yield the individual interaction energies of the left and right spheres with the plate, respectively. Because of the logarithm in [Eq. (\[eq:caseq\])]{}, it is possible to re-express the energy as: $$U = \mathcal{E}_{1}(h_1)+\mathcal{E}_{2}(h_2)+\mathcal{E}_{int}(h_1,h_2, d),
\label{eq:Newsum}$$ where, $$\begin{aligned}
\mathcal{E}_{1}(h_1) = \frac{\hbar c}{2 \pi}
\displaystyle\int_{0}^{\infty}d\kappa \log \det
\left(\mathcal{I}-\mathcal{N}^{(1)}\right) \nonumber \\
\mathcal{E}_{2}(h_2) = \frac{\hbar c}{2 \pi}
\displaystyle\int_{0}^{\infty}d\kappa \log \det
\left(\mathcal{I}-\mathcal{N}^{(2)}\right),
\label{eq:twobodyens}\end{aligned}$$ are the individual interaction energies of the left (1) and right (2) spheres above a plate, in the absence of the other sphere, and $\mathcal{E}_{int}(h_1,h_2,d)$ is a three-body interaction term given by: $$\begin{aligned}
\mathcal{E}_{int}=\frac{\hbar c}{2 \pi}\displaystyle \int d\kappa \log
\det&\left[\mathcal{I}-\left(\mathbb{I}-\mathcal{N}^{(2)}\right)^{-1}
\right. \nonumber \\ &\times
\left. \mathcal{A}\left(\mathcal{I}-\mathcal{N}^{(1)}\right)^{-1}\mathcal{B}\right],
\label{eq:threeterm}\end{aligned}$$
Finally, for completeness, we write down simplified expressions for the intermediate matrices $\mathcal{N}^{(i)}$, $\mathcal{A}$ and $\mathcal{B}$, in terms of appropriate and rapidly-converging multipole and Fourier basis, as explained in [@Rahi09:PRD]. The expression for $\mathcal{E}_{1,2}$ was derived in [@Rahi09:PRD] and thus here we can simply quote the result for the matrices $\mathcal{N}^{(1)}$ and $\mathcal{N}^{(2)}$. In particular, [@Rahi09:PRD] expresses the matrices in terms of a spherical multipole basis, indexed by the quantum numbers $l$, $m$, and $P$, corresponding to angular momentum, azithmutal angular momentum, and polarization \[TE ($P=E$) or TM ($P=M$)\]. The matrices $\mathcal{N}^{(i)}$ are given by: $$\begin{aligned}
\mathcal{N}^{(j)}_{lmP,l'm'P'} &= \delta_{m,m'}\mathcal{F}^{ee(j)}_{lmP,lmP} \nonumber \\
& \times\displaystyle\int_{0}^{\infty} \frac{k_{\bot}dk_{\bot}}{2 \pi}
\frac{e^{-2h_{j}\sqrt{{\mathbf{k}}_{\bot}^2+\kappa^2}}}{2 \kappa
\sqrt{k_{\bot}^2+\kappa^2}} \nonumber \\
& \times \displaystyle\sum_{Q}D_{lmP,k_{\bot}Q}
r^{Q}D^{\dag}_{k_{\bot}Q,l'm'P'} \left(2 \delta_{Q,P'}-1\right),
\label{eq:N}\end{aligned}$$ where ${\mathbf{k}}_{\bot}$ is the Fourier momentum parallel to the plate, the $\mathcal{F}^{ee(j)}_{lmP,lmP}$ are the outside scattering amplitudes of sphere $j$, $r^Q$ are the planar reflection coefficients (Fresnel reflection coefficients in the case of an isotropic plate), and $D_{lmP,k_{\bot m}}$ are conversion matrices: $$\begin{aligned}
D_{lmE,k_{\bot}E} &= D_{lmM,k_{\bot}M} =
\sqrt{\frac{4\pi(2l+1)(l-m)!}{l(l+1)(l+m)!}} \nonumber \\ &\times
\frac{|{\mathbf{k}}_{\bot}|}{\kappa} e^{-im\phi_{{\mathbf{k}}_{\bot}}}
P^{'m}_l\left(\sqrt{{\mathbf{k}}_{\bot}^2+\kappa^2}/\kappa\right)
\nonumber \\ D_{lmM,{\mathbf{k}}_{\bot}E} &= -D_{lmE,k_{\bot}M} =
-im\sqrt{\frac{4\pi(2l+1)(l-m)!}{l(l+1)(l+m)!}} \nonumber \\ &\times
\frac{\kappa}{{\mathbf{k}}_{\bot}} e^{-im\phi_{{\mathbf{k}}_{\bot}}}
P_l^m\left(\sqrt{{\mathbf{k}}_{\bot}^2+\kappa^2}/\kappa\right),
\label{eq:Dmats}\end{aligned}$$ given in terms of associated Legendre polynomials $P_l^m$ and their derivatives with respect to their corresponding argument $P^{'m}_l$.
Upon a number of algebraic manipulations, similar expressions can be obtained for the matrices $\mathcal{A}$ and $\mathcal{B}$, not found in previous works, and in particular we find that: $$\begin{aligned}
\label{eq:A}
-\mathcal{A}_{lmP,l'm'P'}
&=\mathcal{F}^{ee}_{R,lmP,lmP}\mathcal{U}^{23}_{lmP,l'm'P'}
\nonumber \\ &+ \left(-1\right)^{m'-m}
i^{m'-m}\mathcal{F}^{ee}_{R,lmP,lmP} \beta_{lmP, l'm' P'}
\\ -\mathcal{B}_{lmP,l'm'P'} &=
\mathcal{F}^{ee}_{L,lmP,lmP}\mathcal{U}^{32}_{lmP,l'm'P'} \nonumber
\\ &+ i^{m'-m}\mathcal{F}^{ee}_{L,lmP,lmP} \beta_{lmP.l'm'P'},
\label{eq:B}\end{aligned}$$ where $$\begin{aligned}
\beta_{lmP,l'm'P'} &= \displaystyle\int_{0}^{\infty}
\frac{k_{\bot}dk_{\bot}}{(2\pi)} J_{m'-m}\left(S k_{\bot}\right)
\nonumber \\ & \times
\frac{e^{-(h_2+h_3)\sqrt{k_{\bot}^2+\kappa^2}}}{2\kappa
\sqrt{k_{\bot}^2+\kappa^2}} \nonumber \\ & \times
\displaystyle\sum_{Q} D_{lmP,k_{\bot}Q} r^{Q}
D^{\dag}_{k_{\bot}Q,l'm'P'} \left(2 \delta_{Q,P'}-1\right),\end{aligned}$$ and where the $J_m(Sk_{\bot})$ is a Bessel function of the first kind evaluated at different values of $Sk_{\bot}$, where $S$ is given by the projection of the sphere center–center separation onto the plate axis: $$S=\sqrt{(d+R_1+R_2)^2-(h_1+R_1-h_2-R_2)^2}$$
From a numerical perspective, all that remains in order to obtain the Casimir energy in [Eq. (\[eq:caseq\])]{} is to evaluate the various matrix entries and perform standard numerical operations, such as inversion and multiplication, which we perform using standard free software [@GSL]. For the small matrices that we consider, most of the time is spent evaluating the various matrix elements, which can be numerically expensive due to the integration of the oscillatory Bessel functions in $\mathcal{A}$ and $\mathcal{B}$, although specialized methods for oscillatory and Bessel integrals are available that may accelerate the calculation [@Evans:1999; @Xiang:2007].
| 0 |
---
abstract: 'Abbott, Altenkirch, Ghani and others have taught us that many parameterized datatypes (set functors) can be usefully analyzed via container representations in terms of a set of shapes and a set of positions in each shape. This paper builds on the observation that datatypes often carry additional structure that containers alone do not account for. We introduce directed containers to capture the common situation where every position in a data-structure determines another data-structure, informally, the sub-data-structure rooted by that position. Some natural examples are non-empty lists and node-labelled trees, and data-structures with a designated position (zippers). While containers denote set functors via a fully-faithful functor, directed containers interpret fully-faithfully into comonads. But more is true: every comonad whose underlying functor is a container is represented by a directed container. In fact, directed containers are the same as containers that are comonads. We also describe some constructions of directed containers. We have formalized our development in the dependently typed programming language Agda.'
address:
- '[a]{}Laboratory for Foundations of Computer Science, School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, United Kingdom'
- 'Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia'
-
author:
- Danel Ahmana
- James Chapmanb
- Tarmo Uustaluc
title: 'When is a Container a Comonad?'
---
[^1]
Introduction {#sec:intro}
============
Containers, as introduced by Abbott, Altenkirch and Ghani [@Abbott2005] are a neat representation for a wide class of parameterized datatypes (set functors) in terms of a set of shapes and a set of positions in each shape. They cover lists, colists, streams, various kinds of trees, etc. Containers can be used as a “syntax” for programming with these datatypes and reasoning about them, as can the strictly positive datatypes and polynomial functors of Dybjer [@dybjer1997], Moerdijk and Palmgren [@moerdijk.palmgren], Gambino and Hyland [@gambino.hyland:poly], and Kock [@kock:polyfuncandtrees]. The theory of this class of datatypes is elegant, as they are well-behaved in many respects.
This paper proceeds from the observation that datatypes often carry additional structure that containers alone do not account for. We introduce directed containers to capture the common situation in programming where every position in a data-structure determines another data-structure, informally, the sub-data-structure rooted by that position. Some natural examples of such data-structures are non-empty lists and node-labelled trees, and data-structures with a designated position or focus (zippers). In the former case, the sub-data-structure is a sublist or a subtree. In the latter case, it is the whole data-structure but with the focus moved to the given position.
We show that directed containers are no less neat than containers. While containers denote set functors via a fully-faithful functor, directed containers interpret fully-faithfully into comonads. They admit some of the constructions that containers do, but not others: for instance, two directed containers cannot be composed in general. Our main result is that every comonad whose underlying functor is the interpretation of a container is the interpretation of a directed container. So the answer to the question in the title of this paper is: a container is a comonad exactly when it is a directed container. In more precise terms, the category of directed containers is the pullback of the forgetful functor from the category of comonads to that of set functors along the interpretation functor of containers. This also means that a directed container is the same as a comonoid in the category of containers.
In the core of the paper, we study directed containers on ${\mathbf{Set}}$. Toward the end of the paper we point it out that the development could also be carried out more generally in locally Cartesian closed categories (LCCCs) and yet more generally in categories with pullbacks.
In our mathematics, we use syntax similar to the dependently typed functional programming language Agda [@norell:thesis; @agda]. If some function argument will be derivable in most contexts, we mark it as implicit by enclosing it/its type in braces in the function’s type declaration and either give this argument in braces or omit it in the definition and applications of the function.
We have formalized the central parts of the theory presented in Agda. The development is available at <http://cs.ioc.ee/~danel/dcont.html>.
Structure of the Article {#structure-of-the-article .unnumbered}
------------------------
In Section \[sec:containers\], we review the basic theory of containers, showing also some examples. We introduce containers and their interpretation into set functors. We show some constructions of containers such as the coproduct of containers. In Section \[sec:dcontainers\], we revisit our examples and introduce directed containers as a specialization of containers and describe their interpretation into comonads. Our main result, that a container is a comonad exactly when it is directed, is the subject of Section \[sec:pullback\]. In Section \[sec:constructions\], we look at some constructions, in particular the cofree directed container and the focussed container (zipper) construction. In addition, we also introduce strict directed containers and construct the product of two strict directed containers in the category of directed containers. Intuitively, a strict directed container is a directed container where no position in a non-root subshape of a shape translates to its root. In Section \[sec:monads\], we ask whether a similar characterization is possible for containers that are monads and hint that this is the case. In Section \[sec:cointerp\], we show that interpreting the opposite of the category of directed containers into set functors gives monads. In Section \[sec:polycom\], we hint how the directed container theory (presented in the paper for ${\mathbf{Set}}$) could be developed in the more general setting of categories with pullbacks. We briefly summarize related work in Section \[sec:related\] and conclude with outlining some directions for future work in Section \[sec:concl\]. The proofs of the main results of Sections \[sec:dcontainers\] and \[sec:constructions\] appear in Appendices A and B.
We spend a section on the background theory of containers as they are central for our paper but relatively little known, but assume that the reader knows about comonads, monoidal categories and comonoids.
Differences from the FoSSaCS 2012 Conference Version {#differences-from-the-fossacs-2012-conference-version .unnumbered}
----------------------------------------------------
This article is a revised and expanded version of the FoSSaCS 2012 conference paper [@ACU]. We have added many of the proofs that were omitted from the conference version. We have rearranged the different constructions on directed containers into a separate section, namely Section \[sec:constructions\]. In Section \[sec:cofree\], we give a detailed discussion of cofree directed containers. In Section \[sec:products\], which is entirely new, we define strict directed containers and coideal comonads and give an explicit formula for the product of two strict directed containers.
Likewise entirely new are the sections on cointerpreting directed containers in Section \[sec:cointerp\] and directed containers in categories with pullbacks in Section \[sec:polycom\].
Containers {#sec:containers}
==========
We begin with a recap of containers. We introduce the category of containers and the fully-faithful functor into the category of set functors defining the interpretation of containers and show that these are monoidal. We also recall some basic constructions of containers. For proofs of the propositions in this section and further information, we refer the reader to Abbott et al. [@Abbott2005; @abbott:phd].
Containers {#containers}
----------
Containers are a form of “syntax” for datatypes. A *container* $S \lhd P$ is given by a set $S : {\mathsf{Set}}$ of *shapes* and a shape-indexed family $P : S \to {\mathsf{Set}}$ of *positions*. Intuitively, shapes are “templates” for data-structures and positions identify “blanks” in these templates that can be filled with data.
The datatype of lists is represented by $S \lhd P$ where the shapes $S
= {\mathsf{Nat}}$ are the possible lengths of lists and the positions $P\, s =
{\mathsf{Fin}}\, s = \{0,\ldots,s-1\}$ provide $s$ places for data in lists of length $s$. Non-empty lists are obtained by letting $S = {\mathsf{Nat}}$ and $P\, s = {\mathsf{Fin}}\, (s+1)$ (so that shape $s$ has $s+1$ rather than $s$ positions).
Streams are characterized by a single shape with natural number positions: $S = 1 = \{{{\ast}}\}$ and $P\, {{\ast}}= {\mathsf{Nat}}$. The singleton datatype has one shape and one position: $S = 1$, $P\, {{\ast}}= 1$.
A *morphism* between containers $S \lhd P$ and $S' \lhd P'$ is a pair $t \lhd q$ of maps $t : S \to S'$ and $q : \Pi {\{s : S\}}.\, P'\,
(t\, s) \to P\, s$ (the shape map and position map). Note how the positions are mapped backwards. The intuition is that, if a function between two datatypes does not look at the data, then the shape of a data-structure given to it must determine the shape of the data-structure returned and the data in any position in the shape returned must come from a definite position in the given shape.
- The head function, sending a non-empty list to a single data item, is determined by the maps $t : {\mathsf{Nat}}\to 1$ and $q : \Pi {\{s : {\mathsf{Nat}}\}}.\, 1 \to {\mathsf{Fin}}\, (s+1)$ defined by $t\, \_ =
{{\ast}}$ and $q\, {{\ast}}= 0$.
- The tail function, sending a non-empty list to a list, is represented by $t : {\mathsf{Nat}}\to {\mathsf{Nat}}$ and $q : \Pi {\{s :
{\mathsf{Nat}}\}}.\, {\mathsf{Fin}}\, s \to {\mathsf{Fin}}\, (s+1)$ defined by $t\, s = s $ and $q\,
p = p + 1$.
- For the function dropping every second element of a non-empty list, the shape and position maps $t : {\mathsf{Nat}}\to {\mathsf{Nat}}$ and $q
: \Pi {\{s : {\mathsf{Nat}}\}}.\, {\mathsf{Fin}}\, (s \div 2 + 1) \to {\mathsf{Fin}}\, (s + 1)$ are $t\, s = s \div 2$ and $q\, p = p * 2$.
- For self-append of a non-empty list, they are $t : {\mathsf{Nat}}\to
{\mathsf{Nat}}$ and $q : \Pi {\{s : {\mathsf{Nat}}\}}.\, {\mathsf{Fin}}\, (s * 2 + 2) \to {\mathsf{Fin}}\, (s +
1)$ defined by $t\, s = s * 2 + 1$ and $q\, \{s\}\, p = p \mod (s + 1)$.
- For reversal of non-empty lists, they are $t : {\mathsf{Nat}}\to {\mathsf{Nat}}$ and $q : \Pi {\{s :
{\mathsf{Nat}}\}}.\, {\mathsf{Fin}}\, (s + 1) \to {\mathsf{Fin}}\, (s + 1)$ defined by $t\, s = s$ and $q\, \{s\}\, p = s - p$.
(See Prince et al. [@Prince2008] for more similar examples.)
The *identity* morphism ${\mathsf{id}^\mathrm{c}}\{C\}$ on a container $C = S \lhd
P$ is defined by ${\mathsf{id}^\mathrm{c}}= {\mathsf{id}}\, {\{S\}} \lhd \lambda {\{s\}}.\, {\mathsf{id}}\,
{\{P\, s\}}$. The *composition* $h {\mathrel{\circ^\mathrm{c}}}h'$ of container morphisms $h = t \lhd q$ and $h' = t' \lhd q'$ is defined by $h {\mathrel{\circ^\mathrm{c}}}h' = t {\circ}t' \lhd \lambda {\{s\}}.\, q'\, {\{s\}} {\circ}q\, {\{t'\,
s\}}$. Composition of container morphisms is associative, identity is the unit.
Containers form a category ${\mathbf{Cont}}$.
Interpretation of Containers
----------------------------
To map containers into datatypes made of data-structures that have the positions in some shape filled with data, we must equip containers with a “semantics”.
For a container $C = S \lhd P$, we define its *interpretation* ${\llbracket C \rrbracket^\mathrm{c}} : {\mathsf{Set}}\to {\mathsf{Set}}$ on sets by ${\llbracket C \rrbracket^\mathrm{c}}\, X = \Sigma s: S.\,
P\, s \to X$, so that ${\llbracket C \rrbracket^\mathrm{c}}\, X$ consists of pairs of a shape and an assignment of an element of $X$ to each of the positions in this shape, reflecting the intuitive reading that shapes are “templates" for datatypes and positions identify “blanks" in these templates that can be filled in with data. The interpretation ${\llbracket C \rrbracket^\mathrm{c}} : \forall {\{X\}}, {\{Y\}}.\, (X \to Y) \to (\Sigma s: S.\,
P\, s \to X) \to \Sigma s: S.\, P\, s \to Y$ of $C$ on functions is defined by ${\llbracket C \rrbracket^\mathrm{c}}\, f\, (s, v) = (s, f {\circ}v)$. It is straightforward that ${\llbracket C \rrbracket^\mathrm{c}}$ preserves identity and composition of functions, so it is a set functor (as any datatype should be).
Our example containers denote the datatypes intended. If we let $C$ be the container of lists, we have ${\llbracket C \rrbracket^\mathrm{c}}\, X = \Sigma s : {\mathsf{Nat}}.\,
{\mathsf{Fin}}\, s \to X \cong {\mathsf{List}}\, X$. The container of streams interprets into $\Sigma {{\ast}}: 1.\, {\mathsf{Nat}}\to X \cong {\mathsf{Nat}}\to X \cong {\mathsf{Str}}\, X$. Etc.
A morphism $h = t \lhd q$ between containers $C = S \lhd P$ and $C =
S' \lhd P'$ is interpreted as a natural transformation between ${\llbracket C \rrbracket^\mathrm{c}}$ and ${\llbracket C' \rrbracket^\mathrm{c}}$, i.e., as a polymorphic function ${\llbracket h \rrbracket^\mathrm{c}}
: \forall {\{X\}}.\, (\Sigma s: S.\, P\, s \to X) \to \Sigma s' : S'.\,
P'\, s' \to X$ that is natural. It is defined by ${\llbracket h \rrbracket^\mathrm{c}}\, (s, v) =
(t\, s, v {\circ}q\, {\{s\}})$. ${\llbracket - \rrbracket^\mathrm{c}}$ preserves the identities and composition of container morphisms.
The interpretation of the container morphism $h$ for the list head function is ${\llbracket h \rrbracket^\mathrm{c}} : \forall{\{X\}}.\, (\Sigma s: {\mathsf{Nat}}.\,
{\mathsf{Fin}}\, (s+1) \to X) \to \Sigma {{\ast}}: 1.\, 1 \to X$ defined by ${\llbracket h \rrbracket^\mathrm{c}}\, (s, v) = ({{\ast}}, \lambda {{\ast}}.\, v\, 0)$.
${\llbracket - \rrbracket^\mathrm{c}}$ is a functor from ${\mathbf{Cont}}$ to ${[{\mathbf{Set}},{\mathbf{Set}}]}$.
Every natural transformation between container interpretations is the interpretation of some container morphism. For containers $C = S \lhd
P$ and $C' = S' \lhd P'$, a natural transformation $\tau$ between ${\llbracket C \rrbracket^\mathrm{c}}$ and ${\llbracket C' \rrbracket^\mathrm{c}}$, i.e., a polymorphic function $\tau :
\forall {\{X\}}.\, (\Sigma s : S.\, P\, s \to X) \to \Sigma s':
S'.\,P'\, s' \to X$ that is natural, can be “quoted” to a container morphism ${\ulcorner \tau \urcorner^\mathrm{c}} = t \lhd q$ between $C$ and $C'$ where $t : S \to
S'$ and $q : \Pi {\{s : S\}}.\, P'\, (t\, s) \to P\, s$ are defined by ${\ulcorner \tau \urcorner^\mathrm{c}} = (\lambda s.\, {\mathsf{fst}}\, (\tau\, {\{P\, s\}}\, (s, {\mathsf{id}})))
\lhd (\lambda {\{s\}}.\, {\mathsf{snd}}\, (\tau\, {\{P\, s\}}\, (s, {\mathsf{id}})))$.
For any container morphism $h$, ${\ulcorner {\llbracket h \rrbracket^\mathrm{c}} \urcorner^\mathrm{c}} = h$, and, for any natural transformation $\tau$ and $\tau'$ between container interpretations, ${\ulcorner \tau \urcorner^\mathrm{c}} = {\ulcorner \tau' \urcorner^\mathrm{c}}$ implies $\tau = \tau'$.
\[prop:csemfullyfaithful\] ${\llbracket - \rrbracket^\mathrm{c}}$ is fully faithful.
Monoidal Structure
------------------
We have already seen the *identity* container ${\mathsf{Id}^\mathrm{c}}= 1 \lhd \lambda {{\ast}}.\, 1$. The *composition* $C_0 {\mathrel{\cdot^\mathrm{c}}}C_1$ of containers $C_0 = S_0 \lhd P_0$ and $C_1 = S_1 \lhd P_1$ is the container $S \lhd P$ defined by $S = \Sigma s : S_0.\, P_0\, s \to
S_1$ and $P\, (s, v) = \Sigma p_0 : P_0\, s.\, P_1\, (v\, p_0)$. It has as shapes pairs of an outer shape $s$ and an assignment of an inner shape to every position in $s$. The positions in the composite container are pairs of a position $p$ in the outer shape and a position in the inner shape assigned to $p$. The (horizontal) composition $h_0 {\mathrel{\cdot^\mathrm{c}}}h_1$ of container morphisms $h _0 = t_0 \lhd
q_0$ and $h_1 = t_1 \lhd q_1$ is the container morphism $t \lhd q$ defined by $t\, (s, v) = (t_0\, s, t_1 {\circ}v {\circ}q_0\, {\{s\}})$ and $q\, {\{s, v\}}\, (p_0, p_1) = (q_0\, {\{s\}}\, p_0, q_1\, {\{v\,
(q_0\, {\{s\}}\, p_0)\}}\, p_1)$. The horizontal composition preserves the identity container morphisms and the (vertical) composition of container morphisms, which means that ${-} {\mathrel{\cdot^\mathrm{c}}}{-}$ is a bifunctor.
${\mathbf{Cont}}$ has isomorphisms $\rho : \forall {\{C\}}.\, C {\mathrel{\cdot^\mathrm{c}}}{\mathsf{Id}^\mathrm{c}}\to
C$, $\lambda : \forall {\{C\}}.\, {\mathsf{Id}^\mathrm{c}}{\mathrel{\cdot^\mathrm{c}}}C \to C$ and $\alpha :
\forall {\{C,C',C''\}}.\, (C {\mathrel{\cdot^\mathrm{c}}}C') {\mathrel{\cdot^\mathrm{c}}}C'' \to C
{\mathrel{\cdot^\mathrm{c}}}(C' {\mathrel{\cdot^\mathrm{c}}}C'')$, given by $\rho = \lambda (s, v).\, s \lhd
\lambda {\{s,v\}}.\, \lambda p.\, (p, {{\ast}})$, $\lambda = \lambda
({{\ast}},v).\, v\, {{\ast}}\lhd \lambda {\{{{\ast}},v\}}.\, \lambda p.\, ({{\ast}}, p)$ and $\alpha = \lambda ((s, v), v').\, (s , \lambda p.\, (v\, p, \lambda
p'.\, v'\, (p,p'))) \lhd \linebreak \lambda {\{(s, v), v'\}}.\, \lambda
(p,(p',p'')).\, ((p, p'), p'')$. They satisfy Mac Lane’s coherence conditions.
\[prop:monoidalcategory\] The category ${\mathbf{Cont}}$ is a monoidal category.
There are also natural isomorphisms ${\mathsf{e}}: {\mathsf{Id}}\to {\llbracket {\mathsf{Id}^\mathrm{c}}\rrbracket^\mathrm{c}}$ and ${\mathsf{m}}: \forall {\{C_0,C_1\}}.\, {\llbracket C_0 \rrbracket^\mathrm{c}} {\cdot}{\llbracket C_1 \rrbracket^\mathrm{c}}$ $\to$ ${\llbracket C_0 {\mathrel{\cdot^\mathrm{c}}}C_1 \rrbracket^\mathrm{c}}$ defined by ${\mathsf{e}}\, x = ({{\ast}},\lambda {{\ast}}.\, x)$ and ${\mathsf{m}}\, (s,v) = ((s,\lambda p.\, {\mathsf{fst}}\,
(v\, p)), \lambda (p,p').\, {\mathsf{snd}}\, (v\, p)\, p')$ satisfying the appropriate coherence conditions.
The functor ${\llbracket - \rrbracket^\mathrm{c}}$ is a monoidal functor.
Constructions of Containers
---------------------------
Containers are closed under various constructions such as products, coproducts and constant exponentiation, preserved by interpretation.
#### Products
For two containers $C_0 = S_0 \lhd P_0$ and $C_1 = S_1 \lhd
P_1$, their *product* $C_0 \times C_1$ is the container $S \lhd
P$ defined by $S = S_0 \times S_1$ and $P\, (s_0, s_1) = P_0\, s_0 +
P_1\, s_1$. It holds that ${\llbracket C_0 \times C_1 \rrbracket^\mathrm{c}} \cong {\llbracket C_0 \rrbracket^\mathrm{c}}
\times {\llbracket C_1 \rrbracket^\mathrm{c}}$.
#### Coproducts
The *coproduct* $C_0 + C_1$ of containers $C_0 = S_0 \lhd
P_0$ and $C_1 = S_1 \lhd P_1$ is the container $S \lhd P$ defined by $S = S_0 + S_1$, $P\, ({\mathsf{inl}}\, s) = P_0\, s$ and $P\, ({\mathsf{inr}}\, s) =
P_1\, s$. It is the case that ${\llbracket C_0 + C_1 \rrbracket^\mathrm{c}} \cong {\llbracket C_0 \rrbracket^\mathrm{c}} +
{\llbracket C_1 \rrbracket^\mathrm{c}}$.
#### Exponentials
For a set $K \in {\mathsf{Set}}$ and a container $C_0 = S_0 \lhd P_0$, the *exponential* $K \to C_0$ is the container $S \lhd P$ where $S
= K \to S_0$ and $P\, f = \Sigma k : K.\, P \,(f\, k)$. We have that ${\llbracket K \to C_0 \rrbracket^\mathrm{c}} \cong K \to {\llbracket C_0 \rrbracket^\mathrm{c}}$.
Directed Containers {#sec:dcontainers}
===================
We now proceed to our contribution, directed containers. We define the category of directed containers and a fully-faithful functor interpreting directed containers as comonads, and discuss some examples and constructions.
Directed Containers {#sec:dcont}
-------------------
Parametrized datatypes often carry some additional structure that is worth making explicit. For example, each node in a list or non-empty list defines a sublist (a suffix). In container terms, this corresponds to every position in a shape determining another shape, the subshape corresponding to this position. The theory of containers alone does not account for such additional structure. Directed containers, studied in the rest of this paper, axiomatize subshapes and translation of positions in a subshape into the global shape.
A *directed container* is a container $S \lhd P$ together with three operations
${{\mathrel{\downarrow}}} : \Pi s : S.\, P\, s \to S$ (the subshape corresponding to a position),
${\mathsf{o}}: \Pi {\{s : S\}}.\, P\, s$ (the root),
${{\mathrel{\oplus}}} : \Pi {\{s : S\}}.\, \Pi p : P\, s.\, P\, (s {\mathrel{\downarrow}}p) \to P\,
s$ (translation of subshape positions into positions in the global shape).
satisfying the following two shape equations and three position equations:
$\forall {\{s\}}.\, s {\mathrel{\downarrow}}{\mathsf{o}}= s$,
$\forall {\{s, p, p'\}}.\, s {\mathrel{\downarrow}}(p {\mathrel{\oplus}}p') = (s {\mathrel{\downarrow}}p) {\mathrel{\downarrow}}p'$,
$\forall {\{s, p\}}.\, p {\mathrel{\oplus}}{\{s\}}\, {\mathsf{o}}= p$,
$\forall {\{s, p\}}.\, {\mathsf{o}}\, {\{s\}} {\mathrel{\oplus}}p = p$,
$\forall {\{s, p, p', p''\}}.\, (p {\mathrel{\oplus}}{\{s\}}\, p') {\mathrel{\oplus}}p'' = p {\mathrel{\oplus}}(p' {\mathrel{\oplus}}p'')$.
(Using ${\mathrel{\oplus}}$ as an infix operation, we write the first, implicit, argument next to the operation symbol when we want to give it explicitly.) Modulo the fact that the positions involved come from different sets, laws 3–5 are the laws of a monoid. In the special case $S = 1$, we have exactly one set of positions, namely $P\, \ast$, and that is a monoid. If $S$ is general, but $s {\mathrel{\downarrow}}p$ does not depend on $p$ (in this case $s {\mathrel{\downarrow}}p = s$ thanks to law 1), then each $P\, s$ is a monoid. (One might also notice that laws 1–2 bear similarity to the laws of a monoid action. If none of $P\,
s$, ${\mathsf{o}}\, {\{s\}}$, $p \oplus\, {\{s\}}\, p'$ depends on $s$, then we have one single monoid and ${\mathrel{\downarrow}}$ is then a right action of that monoid on $S$.)
To help explain the operations and laws, we sketch in Fig. \[fig:dcontainer\] a data-structure with nested sub-data-structures.
$$\xymatrix@C=1.45em@R=1em{
& & & & & & & & \bullet \ar@(l,u)@{->}[]^(0.25){{\mathsf{o}}{\{s\}}} \ar@{-}[ddddddddllllllll] \ar@{-}[ddddddddrrrrrrrr]_>>>>>>{s = s {\mathrel{\downarrow}}{\mathsf{o}}{\{s\}}} \ar@{->}[dd]^(0.6){p = p {\mathrel{\oplus}}{\mathsf{o}}{\{s'\}} = {\mathsf{o}}{\{s\}} {\mathrel{\oplus}}p : P s} \ar@{->}@/_5pc/[dddd]_{p {\mathrel{\oplus}}p'} \ar@{->}@/_8.5pc/[ddddddd]_{(p {\mathrel{\oplus}}p') {\mathrel{\oplus}}p'' = p {\mathrel{\oplus}}(p' {\mathrel{\oplus}}p'')} \\
& & & & & & & & & \\
& & & & & & & & \bullet \ar@(l,u)@{.>}[]^(0.25){{\mathsf{o}}{\{s'\}}} \ar@{.>}[dd]^(0.6){p' : P s'} \ar@{.}[ddddddllllll] \ar@{.}[ddddddrrrrrr]_>>>>>>{s' = s {\mathrel{\downarrow}}p} \ar@{.>}@/^4.5pc/[ddddd]^(0.4){p' {\mathrel{\oplus}}p''} & & \\
& & & & & & & & & & & \\
& & & & & & & & \bullet \ar@{-->}[ddd]^(0.65){p'' : P s''} \ar@{--}[ddddllll]^>>>>>>{s'' = s {\mathrel{\downarrow}}(p {\mathrel{\oplus}}p') = s' {\mathrel{\downarrow}}p'} \ar@{--}[ddddrrrr]& & & & \\
& & & & & & & & & & & & & \\
& & & & & & & & & & & & & \\
& & & & & & & & \bullet & & & & & & \\
\ar@{-}[rrrrrrrrrrrrrrrr] & & & & & & & & & & & & & & & &
}$$
The global shape $s$ is marked with a solid boundary and has a root position ${\mathsf{o}}\, {\{s\}}$. Then, any position $p$ in $s$ determines a shape $s' = s {\mathrel{\downarrow}}p$, marked with a dotted boundary, to be thought of as the subshape of $s$ given by this position. The root position in $s'$ is ${\mathsf{o}}\, {\{s'\}}$. Law 3 says that its translation $p
{\mathrel{\oplus}}{\mathsf{o}}\, {\{s'\}}$ into a position in shape $s$ is $p$, reflecting the idea that the subshape given by a position should have that position as the root.
By law 1, the subshape $s {\mathrel{\downarrow}}{\mathsf{o}}\, {\{s\}}$ corresponding to the root position ${\mathsf{o}}{\{s\}}$ in the global shape $s$ is $s$ itself. Law 4, which is only well-typed thanks to law 1, stipulates that the translation of position $p$ in $s {\mathrel{\downarrow}}{\mathsf{o}}\, {\{s\}}$ into a position in $s$ is just $p$ (which is possible, as $P\, (s {\mathrel{\downarrow}}{\mathsf{o}}\, {\{s\}}) = P\, s$).
A further position $p'$ in $s'$ determines a shape $s'' = s'
{\mathrel{\downarrow}}p'$. But $p'$ also translates into a position $p {\mathrel{\oplus}}p'$ in $s$ and that determines a shape $s {\mathrel{\downarrow}}(p {\mathrel{\oplus}}p')$. Law 2 says that $s''$ and $s {\mathrel{\downarrow}}(p {\mathrel{\oplus}}p')$ are the same shape, which is marked by a dashed boundary in the figure. Finally, law 5 (well-typed only because of law 2) says that the two alternative ways to translate a position $p''$ in shape $s''$ into a position in shape $s$ agree with each other.
Lists cannot form a directed container, as the shape $0$ (for the empty list), having no positions, has no possible root position.
But the container of *non-empty lists* (with $S = {\mathsf{Nat}}$ and $P\,s
= {\mathsf{Fin}}\,(s +1)$) is a directed container with respect to *non-empty suffixes* as sublists. The subshape given by a position $p$ in a shape $s$ (for lists of length $s+1$) is the shape of the corresponding suffix, given by $s {\mathrel{\downarrow}}p = s - p$. The root ${\mathsf{o}}\, {\{s\}}$ is the position $0$ of the head node. A position in the global shape is recovered from a position $p'$ in the subshape of the position $p$ by $p {\mathrel{\oplus}}p' = p + p'$.
Fig. \[fig:dcontainernelist\] shows an example of the shape and positions of a non-empty list with length 6, i.e., with shape $s = 5$. This figure also shows that the subshape determined by a position $p = 2$ in the global shape $s$ is $s' = s {\mathrel{\downarrow}}p = 5 - 2 = 3$ and a position $p' = 1$ in $s'$ is rendered as the position $p {\mathrel{\oplus}}p' = 2 + 1 = 3$ in the initial shape.
$$\xymatrix@C=5em@R=1.2em{
& & & & & \\
& & & & & \\
\bullet \ar@{->}@/_0.8pc/[rr]_(0.8){p = 2} \ar@{->}@/^0.8pc/[rrr]^{p {\mathrel{\oplus}}p' = 2 + 1 = 3} \ar@{-}[uurr]_>{s = 5} \ar@{-}[ddrr] & \bullet & \bullet \ar@{.>}@/_0.8pc/[r]_>{p' = 1} \ar@{.}[uurr]_>{s' = s {\mathrel{\downarrow}}p = 5 - 2 = 3} \ar@{.}[ddrr] & \bullet & \bullet & \bullet \\
& & & & & \\
& & & & &
}$$
Clearly one could also choose prefixes as subshapes and the last node of a non-empty list as the root, but this gives an isomorphic directed container.
Non-empty lists also give rise to an entirely different directed container structure that has *cyclic shifts* as “sublists” (this example was suggested to us by Jeremy Gibbons). The subshape at each position is the global shape ($s {\mathrel{\downarrow}}p = s$). The root is still ${\mathsf{o}}\, {\{s\}} = 0$. The interesting part is that translation into the global shape of a subshape position is defined by $p {\mathrel{\oplus}}{\{s\}}\, p' = (p + p') \mod (s + 1)$, satisfying all the required laws.
$$\xymatrix@C=4.8em@R=1.2em{
& & & & & \\
& & & & & \\
\bullet \ar@{->}@/_1pc/[rr]_(0.8){p=2} \ar@{->}@/^1pc/[rrrr]^{p {\mathrel{\oplus}}p' = 2 + \, 2 = 4} \ar@{-}[uurr]_>{s = {{\ast}}} \ar@{-}[ddrr] & \bullet & \bullet \ar@{.>}@/_1pc/[rr]_(0.8){p' = 2} \ar@{.}[uurr]_>{s' = s {\mathrel{\downarrow}}p = {{\ast}}} \ar@{.}[ddrr] & \bullet & \bullet & \bullet & ... \\
& & & & & \\
& & & & &
}$$
The container of *streams* ($S = 1$, $P\, {{\ast}}= {\mathsf{Nat}}$) carries a very trivial directed container structure given by ${{\ast}}{\mathrel{\downarrow}}p =
{{\ast}}$, ${\mathsf{o}}= 0$ and $p {\mathrel{\oplus}}p' = p +
p'$. Fig. \[fig:dcontainerstream\] shows how a position $p = 2$ in the only possible global shape $s = {{\ast}}$ and a position $p' = 2$ in the equal subshape $s' = s {\mathrel{\downarrow}}p = {{\ast}}$ give back a position $p + p'
= 4$ in the global shape.
This directed container is nothing else than the monoid $({\mathsf{Nat}},
0, +)$ seen as a directed container.
Similarly to the theory of containers, one can also define morphisms between directed containers.
A *morphism* between two directed containers $(S \lhd P, {\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}})$ and $(S' \lhd P', {{\mathrel{\downarrow'}}}, {\mathsf{o'}},
{{\mathrel{\oplus'}}})$ is a morphism $t \lhd q$ between the containers $S \lhd P$ and $S' \lhd P'$ that satisfies three laws:
1. $\forall {\{s, p\}}.\, t\, (s {\mathrel{\downarrow}}q\, p) = t\, s {\mathrel{\downarrow'}}p$,
2. $\forall {\{s\}}.\, {\mathsf{o}}\, {\{s\}} = q\, ({\mathsf{o'}}\, {\{t\, s\}})$,
3. $\forall {\{s, p, p'\}}.\, q\, p {\mathrel{\oplus}}{\{s\}}\, q\, p' = q\, (p {\mathrel{\oplus'}}{\{t\, s\}}\, p')$.
In the special case $S = S' = 1$, laws 2 and 3 are the laws of a monoid morphism.
Recall the intuition that $t$ determines the shape of the data-structure that some given data-structure is sent to and $q$ identifies for every position in the data-structure returned a position in the given data-structure. These laws say that the positions in the sub-data-structure for any position in the resulting data-structure must map back to positions in the corresponding sub-data-structure of the given data-structure. This means that they can receive data only from those positions, other flows are forbidden. Morphisms between directed containers representing node-labelled tree datatypes are exactly upwards accumulations—this was one of the motivations for choosing the name ‘directed containers’.
The container representations of the head and drop-even functions for non-empty lists are directed container morphisms for the directed container of non-empty lists and suffixes (and the identity directed container). But those of self-append and reversal are not.
For the directed container of non-empty lists and cyclic shifts, not only the representations of the head and drop-even functions but also the self-append function are directed container morphisms.
The identities and composition of ${\mathbf{Cont}}$ can give the identities and composition for directed containers, since for every directed container $E = (C, {\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}})$, the identity container morphism ${\mathsf{id}^\mathrm{c}}\, {\{C\}}$ is a directed container morphism and the composition $h {\mathrel{\circ^\mathrm{c}}}h'$ of two directed container morphisms is also a directed container morphism.
Directed containers form a category ${\mathbf{DCont}}$.
Interpretation of Directed Containers
-------------------------------------
As directed containers are containers with some operations obeying some laws, a directed container should denote not just a set functor, but a set functor with operations obeying some laws. The correct domain of denotation for directed containers is provided by comonads on sets.
Given a directed container $E = (S \lhd P, {\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}})$, we define its *interpretation* ${\llbracket E \rrbracket^\mathrm{dc}}$ to be the set functor $D =
{\llbracket S \lhd P \rrbracket^\mathrm{c}}$ (i.e., the interpretation of the underlying container) together with two natural transformations $$\begin{array}{l}
{\varepsilon}: \forall {\{X\}}.\, (\Sigma s: S.\, P\, s \to X) \to X\\
{\varepsilon}\, (s, v) = v\, ({\mathsf{o}}\, {\{s\}})\\
{\delta}: \forall {\{X\}}.\, (\Sigma s: S.\, P\, s \to X) \to \Sigma s: S.\, P\, s \to \Sigma s': S.\, P\, s'\to X\\
{\delta}\, (s, v) = (s, \lambda p.\, (s {\mathrel{\downarrow}}p, \lambda p'.\, v\, (p {\mathrel{\oplus}}{\{s\}}\, p')))
\end{array}$$ The directed container laws ensure that the natural transformations ${\varepsilon}$, ${\delta}$ make the counit and comultiplication of a comonad structure on $D$.
Intuitively, the counit extracts the data at the root position of a data-structure (e.g., the head of a non-empty list), the comultiplication, which produces a data-structure of data-structures, replaces the data at every position with the sub-data-structure corresponding to this position (e.g., the corresponding suffix or cyclic shift).
The interpretation ${\llbracket h \rrbracket^\mathrm{dc}}$ of a morphism $h$ between directed containers $E = (C, {{\mathrel{\downarrow}}}, {\mathsf{o}}, {{\mathrel{\oplus}}})$, $E' = (C', {\mathrel{\downarrow'}},
{\mathsf{o'}}, {\mathrel{\oplus'}})$ is defined by ${\llbracket h \rrbracket^\mathrm{dc}} = {\llbracket h \rrbracket^\mathrm{c}}$ (using that $h$ is a container morphism between $C$ and $C'$). The directed container morphism laws ensure that this natural transformation between ${\llbracket C \rrbracket^\mathrm{c}}$ and ${\llbracket C' \rrbracket^\mathrm{c}}$ is also a comonad morphism between ${\llbracket E \rrbracket^\mathrm{dc}}$ and ${\llbracket E' \rrbracket^\mathrm{dc}}$.
Since the category ${\mathbf{Comonads({{\mathbf{Set}}})}}$ inherits its identities and composition from $[{\mathbf{Set}}, {\mathbf{Set}}]$, the functor ${\llbracket - \rrbracket^\mathrm{dc}}$ also preserves the identities and composition.
${\llbracket - \rrbracket^\mathrm{dc}}$ is a functor from ${\mathbf{DCont}}$ to ${\mathbf{Comonads({{\mathbf{Set}}})}}$. \[prop:dcsemfunctor\]
Similarly to the case of natural transformations between container interpretations, one can also “quote” comonad morphisms between directed container interpretations into directed container morphisms. For any directed containers $E = (C, {{\mathrel{\downarrow}}}, {\mathsf{o}}, {{\mathrel{\oplus}}})$, $E' =
(C', {{\mathrel{\downarrow'}}}, {\mathsf{o'}}, {{\mathrel{\oplus'}}})$ and any morphism $\tau$ between the comonads ${\llbracket E \rrbracket^\mathrm{dc}}$ and ${\llbracket E' \rrbracket^\mathrm{dc}}$ (which is a natural transformation between ${\llbracket C \rrbracket^\mathrm{c}}$ and ${\llbracket C' \rrbracket^\mathrm{c}}$), the container morphism ${\ulcorner \tau \urcorner^\mathrm{dc}} = {\ulcorner \tau \urcorner^\mathrm{c}}$ between the underlying containers $C$ and $C'$ is also a directed container morphism between $E$ and $E'$. The directed container morphism laws follow from the comonad morphism laws.
From what we already know about interpretation and quoting of container morphisms, it is immediate that ${\ulcorner {\llbracket h \rrbracket^\mathrm{dc}} \urcorner^\mathrm{dc}} = h$ for any directed container morphism $h$ and that ${\ulcorner \tau \urcorner^\mathrm{dc}} =
{\ulcorner \tau' \urcorner^\mathrm{dc}}$ implies $\tau = \tau'$ for any comonad morphisms $\tau$ and $\tau'$ between directed container interpretations.
\[prop:dcsemfullyfaithful\] ${\llbracket - \rrbracket^\mathrm{dc}}$ is fully faithful.
The *identity container* ${\mathsf{Id}^\mathrm{c}}= 1 \lhd \lambda {{\ast}}.\, 1$ extends trivially to an identity directed container whose denotation is isomorphic to the identity comonad. But, similarly to the situation with functors and comonads, composition of containers fails to yield a composition monoidal structure on ${\mathbf{DCont}}$.
We have elsewhere [@AU] shown that, similarly to the functors and comonads case [@Beck], the composition of the underlying containers of two directed containers carries a *compatible* directed container structure if and only if there is a *distributive law* between these directed containers. Compatible compositions of directed containers turn out to generalize Zappa-Szép products of monoids [@Zappa; @Brin], with distributive laws playing the role of matching pairs of mutual actions.
Containers n Comonads = Directed Containers {#sec:pullback}
-------------------------------------------
Since not every functor can be represented by a container, there is no point in asking whether every comonad can be represented as a directed container. An example of a natural comonad that is not a directed container is the cofree comonad on the finite powerset functor $\mathcal{P}_\mathrm{f}$ (node-labelled nonwellfounded strongly-extensional trees) where the carrier of this comonad is not a container ($\mathcal{P}_\mathrm{f}$ is also not a container). But, what about those comonads whose underlying functor is an interpretation of a container? It turns out that any such comonad does indeed define a directed container that is obtained as follows.
Given a comonad $(D, {\varepsilon}, {\delta})$ and a container $C = S \lhd P$ such that $D = {\llbracket C \rrbracket^\mathrm{c}}$, the counit ${\varepsilon}$ and comultiplication ${\delta}$ induce container morphisms $$\begin{array}{l}
{h^{\varepsilon}}: C \to {\mathsf{Id}^\mathrm{c}}\\
{h^{\varepsilon}}= {t^{\varepsilon}}\lhd {q^{\varepsilon}}= {\ulcorner {\mathsf{e}}{\circ}{\varepsilon}\urcorner^\mathrm{c}}\\
{h^{\delta}}:C \to C {\mathrel{\cdot^\mathrm{c}}}C \\
{h^{\delta}}= {t^{\delta}}\lhd {q^{\delta}}= {\ulcorner {\mathsf{m}}\, {\{C,C\}} {\circ}{\delta}\urcorner^\mathrm{c}}
\end{array}$$ using that ${\llbracket - \rrbracket^\mathrm{c}}$ is fully faithful. From $(D, {\varepsilon}, {\delta})$ satisfying the laws of a comonad we can prove that $(C, {h^{\varepsilon}}, {h^{\delta}})$ satisfies the laws of a comonoid in ${\mathbf{Cont}}$ (i.e., an object in $\mathbf{Comonoids}({\mathbf{Cont}})$). Further, we can define $$\begin{array}{l}
s {\mathrel{\downarrow}}p = {\mathsf{snd}}\, ({t^{\delta}}\, s)\, p\\
{\mathsf{o}}\, {\{s\}} = {q^{\varepsilon}}{\{s\}}\, {{\ast}}\\
p {\mathrel{\oplus}}{\{s\}}\, p' = {q^{\delta}}\, {\{s\}}\, (p, p')
\end{array}$$ and the comonoid laws further enforce the laws of the directed container for $(C, {{\mathrel{\downarrow}}}, {\mathsf{o}}, {{\mathrel{\oplus}}})$.
It may seem that the maps ${t^{\varepsilon}}$ and ${\mathsf{fst}}{\circ}{t^{\delta}}$ are not used in the directed container structure, but ${t^{\varepsilon}}: S \to 1$ contains no information ($\forall {\{s\}}.\, {t^{\varepsilon}}\, s = {{\ast}}$) and the comonad/comonoid right counital law forces that $\forall {\{s\}}.\, {\mathsf{fst}}\,
(t^{\delta}\, s) = s$, which gets used in the proofs of each of the five directed container laws. The latter fact is quite significant. It tells us that the comultiplication ${\delta}$ of any comonad whose underlying functor is the interpretation of a container preserves the shape of a given data-structure as the outer shape of the data-structure returned.
The situation is summarized as follows.
\[prop:comonad2dcontainer\] Any comonad $(D, {\varepsilon}, {\delta})$ and container $C$ such that $D
= {\llbracket C \rrbracket^\mathrm{c}}$ determine a directed container $\lceil (D, {\varepsilon},
{\delta}), C \rceil$.
\[prop:pullbacklaw2\] $\lceil {\llbracket C,{\mathrel{\downarrow}},{\mathsf{o}},{\mathrel{\oplus}}\rrbracket^\mathrm{dc}}, C \rceil = (C,{\mathrel{\downarrow}},{\mathsf{o}},{\mathrel{\oplus}})$.
\[prop:pullbacklaw1\] ${\llbracket \lceil (D, {\varepsilon}, {\delta}), C\rceil \rrbracket^\mathrm{dc}} = (D, {\varepsilon}, {\delta})$.
These observations combine into the following theorem.
\[prop:pullback\] The following is a pullback in $\mathbf{CAT}$: $$\xymatrix@C=4em@R=4em{
{\mathbf{DCont}}\ar[r]^{U} \ar[d]_{{\llbracket - \rrbracket^\mathrm{dc}}}^{\mathrm{f.f.}}
& {\mathbf{Cont}}\ar[d]_{{\llbracket - \rrbracket^\mathrm{c}}}^{\mathrm{f.f.}} \\
{\mathbf{Comonads({{\mathbf{Set}}})}} \ar[r]^{U}
& {[{\mathbf{Set}},{\mathbf{Set}}]}}$$
A structured way to prove this theorem is to first note that a pullback is provided by $\mathbf{Comonoids}({\mathbf{Cont}})$ and then verify that $\mathbf{Comonoids}({\mathbf{Cont}})$ is isomorphic to ${\mathbf{DCont}}$.
Sam Staton pointed it out to us that the proof of the first part only hinges on ${\mathbf{Cont}}$ and $[{\mathbf{Set}},{\mathbf{Set}}]$ being monoidal categories and ${\llbracket - \rrbracket^\mathrm{c}} : {\mathbf{Cont}}\to [{\mathbf{Set}},{\mathbf{Set}}]$ being a fully faithful monoidal functor. Thus we actually establish a more general fact, viz., that for any two monoidal categories $\mathcal{C}$ and $\mathcal{D}$ and a fully-faithful monoidal functor $F : \mathcal{C} \to \mathcal{D}$, the pullback of $F$ along the forgetful functor $U :
\mathbf{Comonoids}(\mathcal{D}) \to \mathcal{D}$ is $\mathbf{Comonoids}(\mathcal{C})$.
In summary, we have seen that the interpretation of a container carries the structure of a comonad exactly when it extends to a directed container.
Constructions of Directed Containers {#sec:constructions}
====================================
We now show some constructions of directed containers. While some standard constructions of containers extend to directed containers, others do not.
Coproducts of Directed Containers
---------------------------------
Given two directed containers $E_0 = (S_0 \lhd P_0, {\mathrel{\downarrow}}_0, {\mathsf{o}}_0,
{\mathrel{\oplus}}_0)$, $E_1= (S_1 \lhd P_1, {\mathrel{\downarrow}}_1, {\mathsf{o}}_1, {\mathrel{\oplus}}_1)$, their coproduct is $E = (S \lhd P, {\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}})$ where the underlying container $C = S
\lhd P$ is the coproduct of containers $C_0 = S_0 \lhd P_0$ and $C_1 =
S_1 \lhd P_1$. All of the directed container operations are defined either using ${\mathrel{\downarrow}}_0, {\mathsf{o}}_0, {\mathrel{\oplus}}_0$ or ${\mathrel{\downarrow}}_1, {\mathsf{o}}_1, {\mathrel{\oplus}}_1$ depending on the given shape. This means that the subshape operation is given by ${\mathsf{inl}}\, s {\mathrel{\downarrow}}p = {\mathsf{inl}}\, (s {\mathrel{\downarrow}}_0 p)$ and ${\mathsf{inr}}\, s {\mathrel{\downarrow}}p = {\mathsf{inr}}\, (s
{\mathrel{\downarrow}}_1 p)$, the root position is given by ${\mathsf{o}}\, {\{{\mathsf{inl}}\, s\}} = {\mathsf{o}}_0\,
{\{s\}}$ and ${\mathsf{o}}\, {\{{\mathsf{inr}}\, s\}} = {\mathsf{o}}_1\, {\{s\}}$ and the subshape position translation operation is given by $p {\mathrel{\oplus}}{\{{\mathsf{inl}}\, s\}}\, p' =
p {\mathrel{\oplus}}_0 {\{s\}}\, p'$ and $p {\mathrel{\oplus}}{\{{\mathsf{inr}}\, s\}}\, p' = p {\mathrel{\oplus}}_1 {\{s\}}\,
p'$. The interpretation of $E$ is isomorphic to the coproduct of comonads ${\llbracket E_0 \rrbracket^\mathrm{dc}}$ and ${\llbracket E_1 \rrbracket^\mathrm{dc}}$.
$E$ defined above is a coproduct of the given directed containers $E_0$ and $E_1$. It interprets to a coproduct of the comonads ${\llbracket E_0 \rrbracket^\mathrm{dc}}$ and ${\llbracket E_1 \rrbracket^\mathrm{dc}}$, whose underlying functor is isomorphic to ${\llbracket C_0 \rrbracket^\mathrm{c}}+{\llbracket C_1 \rrbracket^\mathrm{c}}$.
Products of (Strict) Directed Containers {#sec:products}
----------------------------------------
There is no general way to endow the product of the underlying containers of two directed containers $E_0 = (S_0 \lhd P_0, {\mathrel{\downarrow}}_0,
{\mathsf{o}}_0, {\mathrel{\oplus}}_0)$ and $E_1= (S_1 \lhd P_1, {\mathrel{\downarrow}}_1, {\mathsf{o}}_1, {\mathrel{\oplus}}_1)$ with the structure of a directed container. One can define $S = S_0 \times S_1$ and $P\, (s_0,s_1) = P_0\, s_0 + P_1\, s_1$, but there are two choices ${\mathsf{o}}_0$ and ${\mathsf{o}}_1$ for ${\mathsf{o}}$. Moreover, there is no general way to define $p {\mathrel{\oplus}}p'$. But this should not be surprising, as the product of the underlying functors of two comonads is not generally a comonad. Also, the product of two comonads would not be a comonad structure on the product of the underlying functors.
However, for monads it is known that, although the coproduct of two arbitrary monads may not always exist and is generally relatively difficult to construct explicitly [@kelly], there is a feasible explicit formula for the coproduct of two ideal monads [@ideal]. The duality with comonads gives a formula for the product of two coideal comonads.
A *coideal comonad* on ${\mathbf{Set}}$ is given by a functor $D^+ : {\mathbf{Set}}\to {\mathbf{Set}}$ and a natural transformation ${\delta^+}: D^+ \to D^+ {\cdot}D$ such that the diagrams below commute $$\xymatrix@C=3em@R=3em{
D^+ \ar@{=}[dr] \ar[r]^{{\delta^+}} & D^+ {\cdot}D \ar[d]^{D^+ {\cdot}{\varepsilon}}
& D^+ \ar[d]_{{\delta^+}} \ar[r]^{{\delta^+}} & D^+ {\cdot}D \ar[d]^{D^+ {\cdot}{\delta}} \\
& D^+
& D^+ {\cdot}D \ar[r]_{{\delta^+}{\cdot}D} & D^+ {\cdot}D {\cdot}D
}$$ for a functor $D : {\mathbf{Set}}\to {\mathbf{Set}}$ and natural transformations ${\varepsilon}: D
\to {\mathsf{Id}}$ and ${\delta}: D \to D {\cdot}D$ defined by
- $D X = X \times D^+ X$
- ${\varepsilon}: \forall{\{X\}} .\, X \times D^+ X \to X$\
${\varepsilon}= {\mathsf{fst}}$
- ${\delta}: \forall {\{X\}} .\, X \times D^+ X \to D\,X \times D^+\, (D\, X)$\
${\delta}= \langle {\mathsf{id}}, {\delta^+}{\circ}{\mathsf{snd}}\rangle$
The design of this definition ensures that the data $(D, {\varepsilon}, {\delta})$ make a comonad as soon as the data $(D^+, {\delta^+})$ satisfy the coideal comonad laws.[^2]
Given two coideal comonads $(D_0^+, {\delta^+_0})$ and $(D_1^+, {\delta^+_1})$, the functor $D$ given by
- $D\, X = {\overline{D^+_0}}\, X \times {\overline{D^+_1}}\, X$
where
- $({\overline{D^+_0}}\, X , {\overline{D^+_1}}\, X) = \nu (Z_0 , Z_1) .\, (D_0^+\, (X \times Z_1) , D_1^+\, (X \times Z_0))$
(assuming the existence of the final coalgebra) carries a coideal comonad structure that is a product, in the category of all comonads, of the given ones.
Next we define the corresponding specialization of directed containers and give an explicit product construction for this case. A strict directed container is, intuitively, a directed container where no position in a non-root subshape of a shape translates to its root, i.e., $p {\mathrel{\oplus}}p'$ should not be ${\mathsf{o}}$ when $p \neq {\mathsf{o}}$.
A *strict directed container* is specifiable by the data
- $S : {\mathsf{Set}}$
- $P^+ : S \to {\mathsf{Set}}$
- ${\mathrel{\downarrow^+}}: \Pi s : S .\, P^+\, s \to S$
- ${\mathrel{\oplus^+}}: \Pi {\{s : S\}}.\, \Pi p : P^+\, s.\, P^+\, (s {\mathrel{\downarrow^+}}p) \to P^+\, s$
satisfying the laws
1. $\forall {\{s,p,p'\}} .\, s {\mathrel{\downarrow^+}}(p {\mathrel{\oplus^+}}p') = (s {\mathrel{\downarrow^+}}p) {\mathrel{\downarrow^+}}p'$
2. $\forall {\{s,p,p',p''\}} .\, (p {\mathrel{\oplus^+}}{\{s\}}\, p') {\mathrel{\oplus^+}}p'' = p {\mathrel{\oplus^+}}(p' {\mathrel{\oplus^+}}p'')$
It induces a directed container $(S \lhd P , {\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}})$ via
- $P\, s = {\mathsf{Maybe}}\,(P^+ s)$
- $s {\mathrel{\downarrow}}{\mathsf{nothing}}= s$\
$s {\mathrel{\downarrow}}{\mathsf{just}}\, p = s {\mathrel{\downarrow^+}}p$
- ${\mathsf{o}}= {\mathsf{nothing}}$
- ${\mathsf{nothing}}{\mathrel{\oplus}}p = p$\
${\mathsf{just}}\, p {\mathrel{\oplus}}{\mathsf{nothing}}= {\mathsf{just}}\, p$\
${\mathsf{just}}\, p {\mathrel{\oplus}}{\mathsf{just}}\, p' = {\mathsf{just}}\, (p {\mathrel{\oplus^+}}p')$
Similarly to coideal comonads, the design of this definition also ensures that the data $(S \lhd P , {{\mathrel{\downarrow}}} , {\mathsf{o}}, {\mathrel{\oplus}})$ make a directed container as soon as the data $(S \lhd P^+ , {\mathrel{\downarrow^+}}, {\mathrel{\oplus^+}})$ satisfy the strict directed container laws.[^3]
Strict directed containers are the pullback of the interpretation of directed containers and the inclusion of coideal comonads into comonads.
Notice that the special case $S = 1$ describes monoids without right-invertible non-unit elements (such monoids are trivially also without left-invertible non-unit elements; they arise from adding a unit to a semigroup freely). For example, the datatype of lists and suffixes is a strict directed container; on the other hand, the datatype of lists and cyclic shifts is not.
We take inspiration from the construction of the product of two coideal comonads and construct the product of two strict directed containers.
Given two strict directed containers $E_0 = (S_0 \lhd P_0^+, {{\mathrel{\downarrow_0^+}}},
{{\mathrel{\oplus_0^+}}})$ and $E_1 = (S_1 \lhd P_1^+, {{\mathrel{\downarrow_1^+}}}, {{\mathrel{\oplus_1^+}}})$, we define the data $E = (S \lhd P^+, {{\mathrel{\downarrow^+}}}, {{\mathrel{\oplus^+}}})$ by
- $S = {\overline{S_0}}\times {\overline{S_1}}$\
where\
$({\overline{S_0}}, {\overline{S_1}}) = \nu (Z_0, Z_1).\, (\Sigma s_0 : S_0 .\, P_0^+ s_0 \to Z_1, \Sigma s_1 : S_1 .\, P_1^+ s_1 \to Z_0)$\
- $P^+ (s_0 , s_1) = {\overline{P_0^+}}s_0 + {\overline{P_1^+}}s_1$\
where\
$({\overline{P_0^+}}, {\overline{P_1^+}}) = \mu (Z_0, Z_1).\, \newline (\lambda (s_0 , v_0).\, \Sigma p_0 : P_0^+\, s_0.\, {\mathsf{Maybe}}\, (Z_1\, (v_0\, p_0)), \lambda (s_1 , v_1).\, \Sigma p_1 : P_1^+ s_1 .\, {\mathsf{Maybe}}\, (Z_0\, (v_1\, p_1)))$\
- ${\mathrel{\downarrow^+}}: \Pi s : S .\, P^+ s \to S$\
$(s_0, s_1) {\mathrel{\downarrow^+}}{\mathsf{inl}}\, p = s_0 {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}p$\
$(s_0, s_1) {\mathrel{\downarrow^+}}{\mathsf{inr}}\, p = s_1 {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}p$\
where\
${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}: \Pi s : {\overline{S_0}}.\, {\overline{P_0^+}}\, s \to S$\
${\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}: \Pi s : {\overline{S_1}}.\, {\overline{P_1^+}}\, s \to S$\
(by mutuual recursion)\
$(s_0 , v_0) {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p_0 , {\mathsf{nothing}}) = ((s_0 {\mathrel{\downarrow_0^+}}p_0 , \lambda p .\, v_0\, (p_0 {\mathrel{\oplus_0^+}}p)) , v_0\, p_0)$\
$(s_0 , v_0) {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p_0 , {\mathsf{just}}\, p) = v_0\, p_0 {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}p$\
$(s_1 , v_1) {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}(p_1 , {\mathsf{nothing}}) = (v_1\, p_1 , (s_1 {\mathrel{\downarrow_1^+}}p_1 , \lambda p .\, v_1\, (p_1 {\mathrel{\oplus_1^+}}p)))$\
$(s_1 , v_1) {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}(p_1 , {\mathsf{just}}\, p) = v_1\, p_1 {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}p$\
- ${\mathrel{\oplus^+}}: \Pi {\{s : S\}} .\, \Pi p : P^+ s . P^+ (s {\mathrel{\downarrow^+}}p) \to P^+ s$\
${\mathsf{inl}}\, p {\mathrel{\oplus^+}}p' = {\mathsf{inl}}\, (p {\mathrel{\overline{\oplus_0^+}}}p')$\
${\mathsf{inr}}\, p {\mathrel{\oplus^+}}p' = {\mathsf{inr}}\, (p {\mathrel{\overline{\oplus_1^+}}}p')$\
where\
${\mathrel{\overline{\oplus_0^+}}}: \Pi {\{s : {\overline{S_0}}\}} .\, \Pi p : {\overline{P_0^+}}\, s.\, P^+ (s {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}p) \to {\overline{P_0^+}}\, s$\
${\mathrel{\overline{\oplus_1^+}}}: \Pi {\{s : {\overline{S_1}}\}} .\, \Pi p : {\overline{P_1^+}}\, s .\, P^+ (s {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}p) \to {\overline{P_1^+}}\, s$\
(by mutual recursion)\
$(p_0 , {\mathsf{nothing}}) {\mathrel{\overline{\oplus_0^+}}}{\mathsf{inl}}\, (p_0' , p_1') = (p_0 {\mathrel{\oplus_0^+}}p_0' , p_1')$\
$(p_0 , {\mathsf{nothing}}) {\mathrel{\overline{\oplus_0^+}}}{\mathsf{inr}}\, p = (p_0 , {\mathsf{just}}\, p)$\
$(p_0 , {\mathsf{just}}\, p_1) {\mathrel{\overline{\oplus_0^+}}}p = (p_0 , {\mathsf{just}}\, (p_1 {\mathrel{\overline{\oplus_1^+}}}p))$\
$(p_1 , {\mathsf{nothing}}) {\mathrel{\overline{\oplus_1^+}}}{\mathsf{inr}}\, (p_1' , p_0') = (p_1 {\mathrel{\oplus_1^+}}p_1' , p_0')$\
$(p_1 , {\mathsf{nothing}}) {\mathrel{\overline{\oplus_1^+}}}{\mathsf{inl}}\, p = (p_1 , {\mathsf{just}}\, p)$\
$(p_1 , {\mathsf{just}}\, p_0) {\mathrel{\overline{\oplus_1^+}}}p = (p_1 , {\mathsf{just}}\, (p_0 {\mathrel{\overline{\oplus_0^+}}}p))$
\[prop:prod\] $E$ is a product, in the category of all directed containers, of the strict directed containers $E_0$ and $E_1$. It interprets to a product, in the category of all comonads, of their interpreting coideal comonads.
The definitions above a considerable amount of detail, but the intuition behind them is not difficult. The product of two strict directed containers generalizes the coproduct of two monoids without non-unit right-invertible elements. The elements of this monoid are finite alternating sequences of non-unit elements of the two given monoids. The definitions above arrange for alternations of a similar nature.
Cofree Directed Containers {#sec:cofree}
--------------------------
Given a container $C_0 = S_0 \lhd P_0$, let us define $E = (S \lhd P,
{\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}})$ by
- $S = \nu Z.\, \Sigma s: S_0. P_0\, s \to Z$
- $P = \mu Z.\,\lambda (s, v).\,1 + \Sigma p: P_0\, s.\, Z\, (v\, p)$
- (by recursion)\
$(s, v) {\mathrel{\downarrow}}{\mathsf{inl}}\, {{\ast}}= (s, v)$\
$(s, v) {\mathrel{\downarrow}}{\mathsf{inr}}\,(p, p')= v\, p {\mathrel{\downarrow}}p'$
- ${\mathsf{o}}\, {\{s, v\}} = {\mathsf{inl}}\, {{\ast}}$
- (by recursion)\
${\mathsf{inl}}\, {{\ast}}{\mathrel{\oplus}}{\{s, v\}}\,p'' = p''$\
${\mathsf{inr}}\, (p, p') {\mathrel{\oplus}}{\{s, v\}}\, p'' = {\mathsf{inr}}\, (p, p' {\mathrel{\oplus}}{\{v\, p\}}\, p'')$
\[prop:cofree\] $E$ is a cofree directed container on $C_0$. It interprets into a cofree comonad on the functor ${\llbracket C_0 \rrbracket^\mathrm{c}}$, which has its underlying functor isomorphic to $D\, X = \nu Z.\, X
\times {\llbracket C_0 \rrbracket^\mathrm{c}}\, Z$.
In the special case $S_0 = 1$, we get that $S = \nu Z.\ \Sigma {{\ast}}:
1.\, P_0\, {{\ast}}\to Z \cong 1$ and this example degenerates to the free monoid on a given set $P_0\, {{\ast}}$, i.e., the monoid of lists over $P_0\, {{\ast}}$ (with the empty list as the unit and concatenation as the multiplication operation). This directed container interprets into the comonad of nonwellfounded node-labelled $P_0\, {{\ast}}$-branching trees.
Cofree Recursive Directed Containers
------------------------------------
A recursive comonad is a coideal comonad $(D^+, {\delta}^+)$ such that, for any map $f : D^+\, (X \times Y) \to Y$, there exists a unique map $f^\dagger : D^+\, X \to Y$ such that $$\xymatrix@C=6pc{
D^+\, X \ar[r]^{f^\dagger} \ar[d]_{{\delta}^+}
& Y \\
D^+\, (D\, X) \ar[r]_{D^+\, (X \times f^\dagger)}
& D^+\, (X \times Y) \ar[u]_{f}
}$$
Recursive directed containers are the pullback of the interpretation of strict directed containers and the inclusion of recursive comonads into coideal comonads.
Now the cofree recursive directed container on a given container $C$ is obtained by replacing the $\nu$ in the definition of the shape set $S$ of the cofree directed container with $\mu$. The interpretation has its underlying functor isomorphic to $D\, X = \mu Z.\, X \times
{\llbracket C \rrbracket^\mathrm{c}}\, Z$, which is the cofree recursive comonad on ${\llbracket C \rrbracket^\mathrm{c}}$.
While cofree directed containers represent datatypes of node-labelled nonwellfounded trees, cofree recursive directed containers correspond to node-labelled wellfounded trees. The simplest interesting example is the datatype of non-empty lists (with its suffixes structure), which is represented by the cofree recursive directed container on the “maybe” container $1+1 \lhd \lambda \{({\mathsf{inl}}\, {{\ast}}) .\, 0 ~;~ ({\mathsf{inr}}\, {{\ast}}) .\, 1\}$, i.e., two shapes, one with no positions, the other with one position.
Data-structures with a Focus
----------------------------
Below we discuss directed containers equipped a notion of focus. We present a construction for turning any container into a directed container with a designated focus. We also show that the zipper types of Huet [@Huet1997] have a direct representation as directed containers.
### Focussing {#focussing .unnumbered}
Any container $C_0 = S_0 \lhd P_0$ defines a directed container $E = ({S
\lhd P}, {{\mathrel{\downarrow}}}, {\mathsf{o}}, {{\mathrel{\oplus}}})$ as follows. We take $S = \Sigma s :
S_0.\, P_0\, s$, so that a shape is a pair of a shape $s$, the “shape proper”, and an arbitrary position $p$ in that shape, the “focus”. We take $P\, (s, p) = P_0\, s$, so that a position in the shape $(s,
p)$ is a position in the shape proper $s$, irrespective of the focus. The subshape determined by position $p'$ in shape $(s, p)$ is given by keeping the shape proper but changing the focus: $(s, p) {\mathrel{\downarrow}}p' = (s,
p')$. The root in the shape $(s, p)$ is the focus $p$, so ${\mathsf{o}}\,
{\{s, p\}} = p$. Finally, we take the translation of positions from the subshape $(s, p')$ given by position $p'$ to shape $(s, p)$ to be the identity, by defining $p' {\mathrel{\oplus}}{\{s, p\}}\, p'' = p''$. All directed container laws are satisfied.
The directed container $E$ so obtained interprets into the canonical comonad structure on the functor $\partial {\llbracket C_0 \rrbracket^\mathrm{c}} \times {\mathsf{Id}}$, where $\partial F$ denotes the derivative of the functor $F$. (For derivatives of set functors and containers, see Abbott et al. [@abbott.altenkirch.ghani.mcbride:data].)
Differently from, e.g., the cofree directed container construction, this construction is not a functor from ${\mathbf{Cont}}$ to ${\mathbf{DCont}}$. Instead, it is a functor from the category of containers and Cartesian container morphisms (where position maps are bijections).
### Zippers {#zippers .unnumbered}
Inductive (tree-like) datatypes with a designated focus position are isomorphic to the zipper types of Huet [@Huet1997]. A zipper data-structure encodes a tree with a focus as a pair of a context and a tree. The tree is the subtree of the global tree rooted by the focus and the context encodes the rest of the global tree. On zippers, changing the focus is supported via local navigation operations for moving one step down into the tree or up or aside into the context.
Zipper datatypes are directly representable as directed containers. We illustrate this on the example of zippers for lists (which are, in fact, the same as zippers for non-empty lists, as one cannot focus on a position in the empty list).
A list zipper is a pair of a list (the context) and a non-empty list (the suffix determined by the focus position). Accordingly, by defining $S = {\mathsf{Nat}}\times {\mathsf{Nat}}$, the shape of a zipper is a pair $(s_0, s_1)$ where $s_0$ is the shape of the context and $s_1$ is the shape of the suffix. For positions, it is convenient to choose $P\, (s_0, s_1) = \{-s_0,\ldots, s_1\}$ by allocating the negative numbers in the interval for positions in the context and non-negative numbers for positions in the suffix. The root position is ${\mathsf{o}}\, {\{s_0, s_1\}} = 0$, i.e., the focus. The subshape for each position is given by $(s_0, s_1) {\mathrel{\downarrow}}p = (s_0 + p,
s_1 - p)$ and translation of subshape positions by $p {\mathrel{\oplus}}{\{s_0,
s_1\}}\, p' = p + p'$.
Fig. \[fig:dcontainernelistfocus\] gives an example of a non-empty list with focus with its shape fixed to $s = (5,6)$. It should be clear from the figure how the ${\mathrel{\oplus}}$ operation works on positions $p =
4$ and $p' = -7$ to get back the position $p {\mathrel{\oplus}}p' = -3$ in the initial shape. The subshape operation ${\mathrel{\downarrow}}$ works as follows: $s {\mathrel{\downarrow}}p$ gives back a subshape $s' = (9,2)$ and $s {\mathrel{\downarrow}}(p {\mathrel{\oplus}}p')$ gives $s'' = (2,9)$.
$$\xymatrix@C=1.55em@R=1.2em{
& & & & & & & & & & & & &\\
& & & & & & & & & & & & &\\
\bullet & \bullet & \bullet & \bullet & \bullet & & \bullet \ar@{->}@/_1pc/[rrrrr]_{p = 4} \ar@{->}@/^1pc/[llll]^>>>>>>>{p {\mathrel{\oplus}}p = + p' = -3} \ar@{-}[uurr]_>{s = (5,6)} \ar@{-}[ddrr] \ar@{-}[uull] \ar@{-}[ddll] & & \bullet & \bullet & \bullet & \bullet \ar@{.>}@/_1pc/[lllllllll]_>>>>>>>>{p' = -7} \ar@{.}[uurr]_>{s' = (9,2)} \ar@{.}[ddrr] \ar@{.}[uull] \ar@{.}[ddll] & \bullet & \bullet\\
& & & & & & & & & & & & &\\
& & & & & & & & & & & & &
}$$
The isomorphism of the directed container representation of the list zipper datatype and the directed container of focussed lists is $t : {\mathsf{Nat}}\times {\mathsf{Nat}}\to \Sigma s : {\mathsf{Nat}}.\, \{0, \ldots, s-1\}$, $t\,
(s_0, s_1) = (s_0 + s_1 + 1, s_0)$, $q : \Pi {\{(s_0, s_1) : {\mathsf{Nat}}\times {\mathsf{Nat}}\}}.\, \{0, \ldots, s_0 + s_1\} \to \{-s_0, \ldots,
s_1\}$, $q\, {\{s_0, s_1\}}\, p = p - s_0$.
We refrain here from delving deeper into the topic of derivatives and zippers, leaving this discussion for another occasion.
Containers n Monads = ? {#sec:monads}
=======================
Given that comonads whose underlying functor is the interpretation of a container are the same as directed containers, it is natural to ask whether a similar characterization is possible for monads whose underlying functor can be represented as a container. The answer is “yes”, but the additional structure is more involved than that of directed containers.
Given a container $C = S \lhd P$, the structure $(\eta, \mu)$ of a monad on the functor $T = {\llbracket C \rrbracket^\mathrm{c}}$ is interdefinable with the following structure on $C$
$\mathsf{e} : S$ (for the shape map for $\eta$),
$\bullet : \Pi s : S. (P\, s \to S) \to S$ (for the shape map for $\mu$),
${\nnwarrow} : \Pi {\{s : S\}}.\, \Pi v : P\, s \to S. P\, (s \bullet v) \to P\, s$ and
${\nnearrow} : \Pi {\{s : S\}}.\, \Pi v : P\, s \to S. \Pi p: P\, (s \bullet v).\, P\, (v\, (v \nnwarrow {\{s\}} p))$ (both for the position map for $\mu$)
subject to three shape equations and five position equations. Perhaps not unexpectedly, this amounts to having a monoid structure on $C$. We refrain from a more detailed discussion of this variation of the concept of containers.
To get some intuition, consider the monad structure on the datatype of lists. The unit is given by singleton lists and multiplication is flattening a list of lists by concatenation. For the list container $S
= {\mathsf{Nat}}$, $P\, s = {\mathsf{Fin}}\, s$, we get that $\mathsf{e} = 1$, $s \bullet
v = \sum_{p : {\mathsf{Fin}}\, s} v\, p$, $v\, \nnwarrow {\{s\}}\, p =
\mathrm{[greatest~} p': {\mathsf{Fin}}\, s \mathrm{~such~that~} \sum_{p'' :
{\mathsf{Fin}}\, p'} v\, p'' \leq p \mathrm{]}$ and $v\, \nnearrow {\{s\}}\,
p = p - \sum_{p'' : {\mathsf{Fin}}\, (v \nnwarrow {\{s\}}\, p)} v\, p''$. The reason is that the shape of singleton lists is $\mathsf{e}$ while flattening a list of lists with outer shape $s$ and inner shape $v\,
p$ for every position $p$ in $s$ results in a list of shape $s \bullet
v$. For a position $p$ in the shape of the flattened list, the corresponding positions in the outer and inner shapes of the given list of lists are $v\, \nnwarrow {\{s\}}\, p$ and $v \nnearrow {\{s\}}\,
p$.
Cointerpreting Directed Containers into Monads {#sec:cointerp}
==============================================
What we have just described is not the only way to relate containers to monads. In a recent work [@AU:updmon], we defined *cointerpretation* of containers as the functor ${\langle\!\langle - \rangle\!\rangle^\mathrm{c}} :
{\mathbf{Cont}}^{\mathrm{op}}\to [{\mathbf{Set}}, {\mathbf{Set}}]$ given by $${\langle\!\langle S \lhd P \rangle\!\rangle^\mathrm{c}} X = \Pi s : S.\, P s \times X \cong (\Pi s : S.\, P\, s)
\times (S \to X)$$ Differently from ${\llbracket - \rrbracket^\mathrm{c}}$, the functor ${\langle\!\langle - \rangle\!\rangle^\mathrm{c}}$ is neither full nor faithful. It also fails to be monoidal for the monoidal structure on ${\mathbf{Cont}}^{\mathrm{op}}$ (taken from ${\mathbf{Cont}}$). But it is lax monoidal.
It is straightforward that ${\mathbf{DCont}}^{\mathrm{op}}\cong ({\mathbf{Comonoids({{\mathbf{Cont}}})}})^{\mathrm{op}}\cong {\mathbf{Monoids({{\mathbf{Cont}}^{\mathrm{op}}})}}$. Lax monoidal functors send monoids to monoids. Hence ${\langle\!\langle - \rangle\!\rangle^\mathrm{c}}$ lifts to a functor ${\langle\!\langle - \rangle\!\rangle^\mathrm{dc}} :
{\mathbf{DCont}}^{\mathrm{op}}\to {\mathbf{Monads({{\mathbf{Set}}})}}$ that equips each set functor ${\langle\!\langle S
\lhd P \rangle\!\rangle^\mathrm{c}}$ with a monad structure $$\begin{array}{l}
\eta : \forall {\{X\}}.\, X \to \Pi s : S.\, P\, s \times X \\
\eta\, x\, s = ({\mathsf{o}}\, {\{s\}}, x) \\
\mu : \forall {\{X\}}.\,
(\Pi s : S.\ P\, s \times \Pi s' : S.\ P\, s' \times X)
\to \Pi s : S.\, P\, s \times X \\
\mu\, f\, s = \mathsf{let~} \{(p, g) = f\, s;\, (p', x) = g\, (s {\mathrel{\downarrow}}p)\}
\mathsf{~in~} (p {\mathrel{\oplus}}p', x)
\end{array}$$ Due to the resemblance to compatible compositions of reader and writer monads, we call monads in the image of this functor “dependently typed update monads”. It is instructive to think of shapes in $S$ as states, positions in $P s$ as updates applicable to a state $s$ (or programs safe to evaluate from state $s$), $s {\mathrel{\downarrow}}p$ as the result of applying an update $p$ to the state $s$ (or the result of evaluating $p$ from $s$), ${\mathsf{o}}\, {\{s\}}$ as the nil update in state $s$ and $p {\mathrel{\oplus}}p'$ as accumulation of two consecutive updates (skip and sequential composition).
The directed container for the nonempty list comonad, $S = {\mathsf{Nat}}$, $P\, s = [0..s]$, $s {\mathrel{\downarrow}}p = s - p$, ${\mathsf{o}}= 0$, $p {\mathrel{\oplus}}p' = p + p'$, gives us a monad on the set functor $T$ given by $T\, X = \Pi s:
{\mathsf{Nat}}.\, [0..s] \times X$. The states are natural numbers; the updates applicable to a state $s$ are numbers not greater than $s$; applying an update means decrementing the state.
We can see that directed containers are not more “comonadic” inherently than they are “monadic”. We see them first of all as an algebraic-like structure in their own right, a generalization of monoids.
Directed Containers in Categories with Pullbacks {#sec:polycom}
================================================
Container theory can be carried out in locally Cartesian closed categories (LCCCs)—the LCCC generalization of containers being well known under the name of polynomials [@gambino.hyland:poly; @kock:polyfuncandtrees]—and even more generally in categories with pullbacks [@weber]. It is natural to expect the same of directed container theory.
This is the case indeed. The proofs in this paper can be seen as having been carried out in the internal language of an LCCC (with the assumptions of existence of initial algebras and final coalgebras corresponding to assumptions about availability of W- and M-types).
In the weaker setting of a category with pullbacks, one has to be a lot more careful. It is possible to define the concepts required from the first principles.
We show the definitions of the counterparts of directed containers and directed container morphisms; we call them “directed polynomials” and “directed polynomial morphisms” in the local scope of this section.
In all diagrams below, bullet-labelled nodes with a pair of unlabelled outgoing arcs denote pullbacks defined by a pair of maps that are given directly or constructed. Dashed arrows denote unique maps into a pullback. The polygon actually required to commute is marked with a small circular arrow.
Given a category with pullbacks $\mathcal{C}$, a directed polynomial is given by
- two objects $S$ and $P$ (“sets” of shapes and positions) and an exponentiable map ${\mathsf{s}}$ (assigning every position a shape) $$\xymatrix@C=3em@R=3em{
P \ar[d]^{{\mathsf{s}}}
\\
S
}$$
- a morphism ${\mathrel{\downarrow}}$ picking out a shape for each position (the corresponding subshape) $$\xymatrix@C=3em@R=3em{
P \ar[r]_{{\mathrel{\downarrow}}} & S
}$$
- a map ${\mathsf{o}}$ picking out, for every shape, a position in that shape (the root position) $$\xymatrix@C=3em@R=3em{
P \ar[d]_{\mathsf{s}}&
\\
S \ar@{=}[r] \ar@{}[ur]|<<<<{\circlearrowright} & S \ar[ul]_{{\mathsf{o}}}
}$$
- a map ${\mathrel{\oplus}}$ sending a position in a given, global shape and a position in the corresponding to subshape to a position in the global shape (translation of the subshape position to the global shape) $$\xymatrix@C=3em@R=3em{
P \ar@/_1.5pc/[dddr]_{{\mathsf{s}}}
\\
& \bullet \ar[ul]_{{\mathrel{\oplus}}} \ar[d] \ar[r]
\ar@{}[dl]|(0.4){\circlearrowleft}
& P \ar[d]^{{\mathsf{s}}}
\\
& P \ar[r]_{{\mathrel{\downarrow}}} \ar[d]^{{\mathsf{s}}} & S
\\
& S
}$$
satisfying the following five laws:
1. $$\xymatrix@C=3em@R=3em{
P \ar[r]^{{\mathrel{\downarrow}}} & S \ar@{=}[d] \ar@{}[dl]|<<<<<{\circlearrowleft}
\\
& S \ar[ul]^{{\mathsf{o}}}
}$$
2. $$\xymatrix@C=3em@R=3em{
P \ar@/^1.5pc/[drrr]^{{\mathrel{\downarrow}}}
&
&
\\
& \bullet \ar@{}[ur]|(0.4){\circlearrowright} \ar[ul]^{{\mathrel{\oplus}}} \ar[d] \ar[r]
& P \ar[d]^{{\mathsf{s}}} \ar[r]_{{\mathrel{\downarrow}}}
& S
\\
& P \ar[r]_{{\mathrel{\downarrow}}}
& S
}$$
3. $$\xymatrix@C=3em@R=3em{
P \ar@{=}[dr] & \bullet \ar@/_1.5pc/[dd] \ar[r] \ar[l]_{{\mathrel{\oplus}}} \ar@{}[dl]|(0.3){\circlearrowright} & P \ar@/_1.5pc/[dd]_{{\mathsf{s}}}
\\
& P \ar@{.>}[u] & S \ar[u]_{{\mathsf{o}}}
\\
& P \ar[r]_{{\mathrel{\downarrow}}} \ar@{=}[u] & S \ar@{=}[u]
}$$
4. $$\xymatrix@C=3em@R=3em{
P \ar@{=}[dr] &
\\
\bullet \ar[d] \ar@/^1.5pc/[rr] \ar[u]^{{\mathrel{\oplus}}} \ar@{}[ur]|(0.3){\circlearrowleft} & P \ar@{.>}[l] & P \ar[d]^{{\mathsf{s}}} \ar@{=}[l]
\\
P \ar@/^1.5pc/[rr]^{{\mathrel{\downarrow}}} & S \ar[l]^{{\mathsf{o}}} \ar@{=}[r] & S
}$$
5. $$\xymatrix@C=4em@R=4em{
P \ar@{}[dr]|-{\circlearrowright}
& \bullet \ar[l]_{{\mathrel{\oplus}}} \ar@/_1.5pc/[ddd] \ar[r]
& P \ar@/_1.5pc/[ddd]_(0.47){{\mathsf{s}}}
\\
\bullet \ar[d] \ar@/^1.5pc/[rrr] \ar[u]^{{\mathrel{\oplus}}}
& \bullet \ar[d] \ar[r] \ar@{.>}[u] \ar@{.>}[l]
& \bullet \ar[d] \ar[r] \ar[u]_{{\mathrel{\oplus}}}
& P \ar[d]^{{\mathsf{s}}}
\\
P \ar@/^1.5pc/[rrr]^(0.47){{\mathrel{\downarrow}}}
& \bullet \ar[l]^{{\mathrel{\oplus}}} \ar[d] \ar[r]
& P \ar[r]_{{\mathrel{\downarrow}}} \ar[d]^{{\mathsf{s}}} & S
\\
& P \ar[r]_{{\mathrel{\downarrow}}}
& S
}$$
The data $S$, ${\mathrel{\downarrow}}$, ${\mathsf{o}}$, ${\mathrel{\oplus}}$ here correspond to the homonymous data of a directed container while $P$ and ${\mathsf{s}}$ together correspond to $P$. The five laws governing them correspond exactly to the five laws of a directed container.
A *morphism* between two directed polynomials $(S, P, {\mathsf{s}}, {\mathrel{\downarrow}}, {\mathsf{o}},
{\mathrel{\oplus}})$ and $(S', P', {\mathsf{s}}', {\mathrel{\downarrow'}}, {\mathsf{o'}}, {\mathrel{\oplus'}})$ is given by two maps $t$ and $q$ (of shapes and positions) $$\xymatrix@R=2em@C=2em{
P \ar[dd]_{{\mathsf{s}}}
&
& \\
\ar@{}[r]|-{\circlearrowright}
& \bullet \ar[ul]_{q} \ar[dl] \ar[dr]
& \\
S \ar[ddrr]_{t}
&
& P' \ar[dd]^{{\mathsf{s}}'} \\
&
& \\
&
& S'
}$$ satisfying the following three laws:
1. $$\xymatrix@R=2em@C=2em{
P \ar[rr]^{{\mathrel{\downarrow}}}
&
& S \ar[ddrr]^{t}
&
&\\
& \bullet \ar[ul]^{q} \ar[dl] \ar[dr] \ar@{}[rr]|-{\circlearrowright}
&
&
&\\
S \ar[ddrr]_{t}
&
& P' \ar[dd]^{{\mathsf{s}}'} \ar[rr]^{{\mathrel{\downarrow'}}}
&
& S' \\
&
&
&
&\\
&
& S'
&
&
}$$
2. $$\xymatrix{
P \\
\ar@{}[r]|-{\circlearrowright}
& \bullet \ar[ul]_{q} \ar[ddl] \ar[dr] \\
S \ar[uu]^{{\mathsf{o}}} \ar@{=}[d] \ar@{.>}[ur]
& & P' \ar@/_2pc/[ddd]_(0.4){{\mathsf{s}}'}\\
S \ar[ddrr]_{t} \\
& & S' \ar[uu]_{{\mathsf{o'}}} \ar@{=}[d]\\
& & S'
}$$
3. $$\xymatrix@R=2em@C=2em{
& & P \\
\\
& & \ar@{}[rrrd]|(0.4){\circlearrowright}
& & & \\
& & \bullet \ar[uuu]^{{\mathrel{\oplus}}} \ar[rr] \ar[dd]
& & P \ar[dd]^{{\mathsf{s}}}
& \bullet \ar[uuulll]_(0.35){q} \ar@/^2.3pc/[ddddlll] \ar[dddrrr]\\
\\
& & P \ar[rr]_{{\mathrel{\downarrow}}} \ar[dd]_{{\mathsf{s}}}
& & S \ar@/^1pc/[ddddddrrrrrr]^(0.4){t}\\
& & & & & \bullet \ar[dlll] \ar[dddrrr] \ar@{.>}[rr] \ar@{.>}[dd]
\ar@/^1pc/@{.>}[uuulll] \ar@{.>}[uuu]
& & \bullet \ar[uuulll]_(0.35){q} \ar[ulll] \ar[dddrrr]
& P' \ar@/_2pc/[ddddddd]_(0.45){{\mathsf{s}}'} \\
& & S \ar[ddddddrrrrrr]_{t}
& & & & & \\
& & & & & \bullet \ar[uuulll]^(0.35){q} \ar[ulll] \ar[dddrrr]
\\
& & & & & & & & \bullet \ar[uuu]_(0.4){{\mathrel{\oplus'}}} \ar[rr] \ar[dd]
& & P' \ar[dd]^{{\mathsf{s}}'}\\
\\
& & & & & & & & P' \ar[rr]_{{\mathrel{\downarrow'}}} \ar[dd]^{{\mathsf{s}}'}
& & S' \\
\\
& & & & & & & & S'
}$$
The data $t$, $q$ correspond to the homonymous data of a directed container morphism and the three laws to the three laws of a directed container morphism.
In the special case $\mathcal{C} = {\mathbf{Set}}$, the definitions of a directed polynomial and directed polynomial morphism are equivalent to those of a directed container and directed container morphism.
Remarkably, the definition of a directed polynomial is completely symmetric in ${\mathsf{s}}$ and ${\mathrel{\downarrow}}$—swapping them over we also get a directed polynomial. The definition of a directed polynomial morphism is symmetric, if $q$ is an isomorphism.
The definitions of the interpretation of a directed polynomial resp. directed polynomial morphism into a comonad resp. comonad morphism require using distributivity pullbacks in $\mathcal{C}$ (or pullbacks in its slice categories).
Related Work {#sec:related}
============
The core of this paper builds on the theory of containers as developed by Abbott, Altenkirch and Ghani [@Abbott2005; @abbott:phd] to analyze strictly positive datatypes. Some generalizations of the concept of containers are the indexed containers of Altenkirch and Morris [@altenkirch.morris:indexed; @morris:thesis] and the quotient containers of Abbott et al. [@abbott.altenkirch.ghani.mcbride:quotient]. In our work we look at a specialization of containers rather than a generalization. Recently [@AU], we have also studied compatible compositions of directed containers and how they generalize Zappa-Szép products [@Zappa] of two monoids.
Simple/indexed containers are intimately related to strongly positive datatypes/families and simple/dependent polynomial functors as appearing in the works of Dybjer [@dybjer1997], Moerdijk and Palmgren [@moerdijk.palmgren], Gambino and Hyland [@gambino.hyland:poly], Kock [@kock:polyfuncandtrees]. Girard’s normal functors [@girard:normal] and Joyal’s analytic functors [@joyal:analytic] functors are similar to containers resp.quotient containers, but only allow for finitely many positions in a shape. Gambino and Kock [@gambino.kock:monads] also treat polynomial monads.
Abbott, Altenkirch, Ghani and McBride [@abbott.altenkirch.ghani.mcbride:data] have investigated derivatives of datatypes. Derivatives provide a systematic way to explain Huet’s zipper type [@Huet1997].
Brookes and Geva [@bg92] and later Uustalu with coauthors [@essence; @atteval; @tt; @cell] have used comonads to analyze notions of context-dependent computation such as dataflow computation, attribute grammars, tree transduction and cellular automata. Uustalu and Vene’s [@uustalu.vene:comonadic] observation of a connection between bottom-up tree relabellings and containers with extra structure started our investigation into directed containers.
Conclusions and Future Work {#sec:concl}
===========================
We introduced directed containers as a specialization of containers for describing a certain class of datatypes (data-structures where every position determines a sub-data-structure) that occur very naturally in programming. It was a pleasant discovery for us that directed containers are an entirely natural concept also from the mathematical point of view: they are the same as containers whose interpretation carries the structure of a comonad. They also generalize monoids in an interesting way. In a recent piece of work [@AU:updlenses], we have witnessed that coalgebras of comonads interpreting directed containers are relevant for bidirectional transformations as a flavor of lenses (“dependently typed update lenses”).
As future work, we intend to take a closer look at focussing and related concepts, such as derivatives. A curious special case of directed containers supports translation of the root of a shape into every subshape. Such bidirectional containers include, e.g., focussed containers and generalize groups in the same way as directed containers generalize monoids. We would like to find out if this specialization of directed containers is an interesting and useful concept. We wonder whether our explicit formula for the product of two directed containers can be scaled to the general, non-strict, case. Last, we would like to analyze containers that are monads more closely.
### Acknowledgments {#acknowledgments .unnumbered}
We are indebted to Thorsten Altenkirch, Jeremy Gibbons, Peter Morris, and Sam Staton for comments and suggestions. We thank our anonymous referees for the useful feedback that helped us improve the article.
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Proofs for Section \[sec:dcontainers\]
======================================
Proof of Proposition \[prop:dcsemfunctor\] {#proof-of-proposition-propdcsemfunctor .unnumbered}
------------------------------------------
We must check that the interpretation ${\llbracket E \rrbracket^\mathrm{dc}} = (D, {\varepsilon},
{\delta})$ of the given directed container $E = (C, {\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}})$ is a comonad.
Proof of the right counital law: $$\begin{array}{cl}
&D\, {\varepsilon}\, ({\delta}\,(s,v)) \\
= & \quad\{\textrm{definition of $D$}\} \\
&(\lambda (s,v).\, (s,\lambda p.\, {\varepsilon}\,(v\, p)))\, ({\delta}\,(s,v))\\
=& \quad\{\textrm{definitions of ${\varepsilon}$, ${\delta}$}\}\\
&(s,\lambda p.\, v\,(p {\mathrel{\oplus}}\, {\mathsf{o}}))\\
=& \quad\{\textrm{directed container law 3}\}\\
&(s,v)
\end{array}$$ Proof of the left counital law: $$\begin{array}{cl}
&{\varepsilon}\, ({\delta}\, (s,v))\\
=& \quad\{\textrm{definitions of ${\varepsilon}$, ${\delta}$}\}\\
&(s {\mathrel{\downarrow}}{\mathsf{o}}, \lambda p'.\, v\,({\mathsf{o}}{\mathrel{\oplus}}\, p'))\\
=& \quad\{\textrm{directed containers laws 1 and 4}\}\\
&(s , v)
\end{array}$$ Proof of the coassociativity law: $$\begin{array}{cl}
& D\, {\delta}\, ({\delta}\,(s,v))\\
=& \quad\{\textrm{definition of $D$}\}\\
&(\lambda (s,v).\, (s,\lambda p.\, {\delta}\,(v\, p)))\, ({\delta}\,(s,v))\\
=& \quad\{\textrm{definition of ${\delta}$}\}\\
&(s,\lambda p.\, (s {\mathrel{\downarrow}}p,\lambda p'.\, ((s {\mathrel{\downarrow}}p) {\mathrel{\downarrow}}p', \lambda p''.\, v\,(p {\mathrel{\oplus}}\, (p' {\mathrel{\oplus}}p'')))))\\
=& \quad\{\textrm{directed container laws 2 and 5}\}\\
&(s,\lambda p.\, (s {\mathrel{\downarrow}}p,\lambda p'.\, (s {\mathrel{\downarrow}}(p {\mathrel{\oplus}}p'),\lambda p''.\, v\, ((p {\mathrel{\oplus}}p') {\mathrel{\oplus}}p''))))\\
= & \quad\{\textrm{definition of ${\delta}$}\}\\
&{\delta}\,({\delta}\,(s,v))
\end{array}$$
We must also verify that the interpretation ${\llbracket h \rrbracket^\mathrm{dc}} = \tau$ of a morphism $h = t \lhd q$ between two directed containers $E = (C, {\mathrel{\downarrow}},
{\mathsf{o}}, {\mathrel{\oplus}})$ and $E' = (C', {\mathrel{\downarrow}}', {\mathsf{o}}', {\mathrel{\oplus}}')$ is a comonad morphism between ${\llbracket E \rrbracket^\mathrm{dc}} = (D,{\varepsilon},{\delta})$ and ${\llbracket E' \rrbracket^\mathrm{dc}} = (D',{\varepsilon}',{\delta}')$.
Proof of the counit preservation law: $$\begin{array}{cl}
&{\varepsilon}\,(s,v) \\
=& \quad\{\textrm{definition of ${\varepsilon}$}\}\\
&v\, {\mathsf{o}}\\
=& \quad\{\textrm{directed container morphism law 2}\}\\
&v\,(q\, {\mathsf{o}}')\\
=& \quad\{\textrm{definitions of $\tau$, ${\varepsilon}'$}\}\\
&{\varepsilon}'\,(\tau\, (s,v))
\end{array}$$ Proof of the comultiplication preservation law: $$\begin{array}{cl}
&D\, \tau\, ( \tau\, ({\delta}\,(s,v)))\\
=& \quad\{\textrm{definition of $D$}\}\\
&(\lambda (s,v).\, (s,\lambda p.\, \tau\, (v\, p)))\, ( \tau\, ({\delta}\,(s,v)))\\
=& \quad\{\textrm{definitions of $\tau$, $\delta$}\}\\
&(t\, s,\lambda p.\, (t\, (s {\mathrel{\downarrow}}q\, p), \lambda p'.\, v\, (q\, p {\mathrel{\oplus}}q\, p')))\\
=& \quad\{\textrm{directed container morphism laws 1 and 3}\}\\
&(t\, s,\lambda p.\, (t\, s {\mathrel{\downarrow}}' p, \lambda p'.\, v\, (q\, (p {\mathrel{\oplus}}' p'))))\\
=& \quad\{\textrm{definitions of $\tau$, $\delta'$}\}\\
&{\delta}'\, (\tau\, (s,v))\rlap{\hbox to 268 pt{\hfill\qEd}}
\end{array}$$
Proof of Proposition \[prop:dcsemfullyfaithful\] {#proof-of-proposition-propdcsemfullyfaithful .unnumbered}
------------------------------------------------
From Proposition \[prop:csemfullyfaithful\], we know that the interpretation of containers is fully faithful. It remains to show that, for directed containers $E = (C, {\mathrel{\downarrow}}, {\mathsf{o}},
{\mathrel{\oplus}})$, $E' = (C', {\mathrel{\downarrow'}}, {\mathsf{o'}}, {\mathrel{\oplus'}})$ and a morphism $\tau$ between the comonads ${\llbracket E \rrbracket^\mathrm{dc}}$ and ${\llbracket E' \rrbracket^\mathrm{dc}}$, the container morphism $h =
t \lhd q = {\ulcorner \tau \urcorner^\mathrm{c}}$ between $C$ and $C'$ is also a directed container morphism between $E$ and $E'$.
The counit and comultiplication ${\varepsilon}$ and ${\delta}$ of the comonad ${\llbracket E \rrbracket^\mathrm{dc}}$ induce container morphisms $h^{\varepsilon}: C \to {\mathsf{Id}^\mathrm{c}}$ and $h^{\delta}: C \to C {\mathrel{\cdot^\mathrm{c}}}C$ by $h^{\varepsilon}= t^{\varepsilon}\lhd q^{\varepsilon}= {\ulcorner {\mathsf{e}}{\circ}{\varepsilon}\urcorner^\mathrm{c}}$, $h^{\delta}= t^{\delta}\lhd q^{\delta}= {\ulcorner {\mathsf{m}}{\circ}{\delta}\urcorner^\mathrm{c}}$. Similarly ${\varepsilon}'$ and ${\delta}'$ give us container morphisms $h^{{\varepsilon}'} : C' \to {\mathsf{Id}^\mathrm{c}}$ and $h^{{\delta}'} : C' \to C' {\mathrel{\cdot^\mathrm{c}}}C'$ by $h^{{\varepsilon}'} = t^{{\varepsilon}'} \lhd q^{{\varepsilon}'} = {\ulcorner {\mathsf{e}}{\circ}{\varepsilon}' \urcorner^\mathrm{c}}$, $h^{{\delta}'} = t^{{\delta}'} \lhd q^{{\delta}'} = {\ulcorner {\mathsf{m}}{\circ}{\delta}' \urcorner^\mathrm{c}}$.
Let us express $h^{\varepsilon}$ and $h^{\delta}$ directly in terms of ${\mathrel{\downarrow}}$, ${\mathsf{o}}$, ${\mathrel{\oplus}}$.
First, from the definitions of ${\varepsilon}$, ${\mathsf{e}}$ we get $$h^{\varepsilon}= {\ulcorner {\mathsf{e}}{\circ}{\varepsilon}\urcorner^\mathrm{c}} =
{\ulcorner \lambda (s, v).\, ({{\ast}}, \lambda {{\ast}}.\, v\, ({\mathsf{o}}\, {\{s\}})) \urcorner^\mathrm{c}}$$ The definition of ${\ulcorner - \urcorner^\mathrm{c}}$ further gives us $$\begin{aligned}
t^{\varepsilon}\, s & = & {{\ast}}\\
q^{\varepsilon}\, {\{s\}}\, {{\ast}}& = & {\mathsf{o}}\, {\{s\}}\end{aligned}$$
Second, the definitions of ${\delta}$, ${\mathsf{m}}$ dictate that $$h^{\delta}= {\ulcorner {\mathsf{m}}\, {\{C,C\}} {\circ}{\delta}\urcorner^\mathrm{c}} =
{\ulcorner \lambda (s, v).\, (s, \lambda p.\, s {\mathrel{\downarrow}}p),
\lambda (p, p').\, v\, (p {\mathrel{\oplus}}{\{s\}}\, p') \urcorner^\mathrm{c}}$$ The definition of ${\ulcorner - \urcorner^\mathrm{c}}$ allows us to infer that $$\begin{aligned}
t^{\delta}\, s & = & (s, \lambda p.\, s {\mathrel{\downarrow}}p) \\
q^{\delta}\, {\{s\}}\, (p, p') & = & p {\mathrel{\oplus}}{\{s\}}\, p' \end{aligned}$$ Analogous direct expressions in terms of ${\mathrel{\downarrow'}}$, ${\mathsf{o'}}$, ${\mathrel{\oplus'}}$ hold for $h^{{\varepsilon}'}$, $h^{{\delta}'}$.
Now, using $h^{\varepsilon}, h^{\delta}, h^{{\varepsilon}'}, h^{{\delta}'}$ above, we can repackage the two comonad morphism laws for $\tau = {\llbracket h \rrbracket^\mathrm{c}}$ in terms of container interpretations as depicted in the following two diagrams. $$\footnotesize
\xymatrix@C=0.8em@R=3em{
& {\llbracket {\mathsf{Id}^\mathrm{c}}\rrbracket^\mathrm{c}} \\
& {\mathsf{Id}}\ar[u]^{{\mathsf{e}}} \ar@{}[l]|(0.6){\textrm{def. $h^{{\varepsilon}'}$}} \ar@{}[r]|(0.6){\textrm{def. $h^{\varepsilon}$}} & \\
{\llbracket C \rrbracket^\mathrm{c}} \ar[rr]^{{\llbracket h \rrbracket^\mathrm{c}}} \ar[ur]^{{\varepsilon}} \ar@/^2pc/[uur]^{{\llbracket {h^{\varepsilon}}\rrbracket^\mathrm{c}}} \ar@{}[ur]_(0.4){\textrm{counit pres.}}
& & {\llbracket C' \rrbracket^\mathrm{c}} \ar[ul]_{{\varepsilon}'} \ar@/_2pc/[uul]_{{\llbracket {h^{\varepsilon'}}\rrbracket^\mathrm{c}}}
}
\hspace*{2cm}
\xymatrix@C=2.5em@R=3em{
& {\llbracket C {\mathrel{\cdot^\mathrm{c}}}C \rrbracket^\mathrm{c}}\ar[r]^{{\llbracket h {\mathrel{\cdot^\mathrm{c}}}h \rrbracket^\mathrm{c}}}
& {\llbracket C' {\mathrel{\cdot^\mathrm{c}}}C' \rrbracket^\mathrm{c}} \\
& {\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\llbracket C \rrbracket^\mathrm{c}} \ar[r]^{{\llbracket h \rrbracket^\mathrm{c}} {\cdot}{\llbracket h \rrbracket^\mathrm{c}}}\ar[u]^{{\mathsf{m}}\, {\{C, C\}}} \ar@{}[ur]|{\textrm{nat. ${\mathsf{m}}$}} \ar@{}[dr]|{\textrm{comult. pres.}} \ar@{}[dl]_(0.35){\textrm{def. ${h^{\delta}}$}}
& {{\llbracket C' \rrbracket^\mathrm{c}} {\cdot}{\llbracket C' \rrbracket^\mathrm{c}}}\ar[u]_{{\mathsf{m}}\, {\{C',C'\}}} \ar@{}[dr]^(0.35){\textrm{def. ${h^{\delta'}}$}}\\
& {\llbracket C \rrbracket^\mathrm{c}}\ar[r]^{{\llbracket h \rrbracket^\mathrm{c}}}\ar[u]^{\delta}\ar@/^4pc/[uu]^{{\llbracket {h^{\delta}}\rrbracket^\mathrm{c}}}
& {\llbracket C' \rrbracket^\mathrm{c}}\ar[u]_{{\delta}'} \ar@/_4pc/[uu]_{{\llbracket {h^{\delta'}}\rrbracket^\mathrm{c}}}
&
}$$
Going a step further, we can quote these two diagrams to get their reformulations in terms of containers, resulting in the two diagrams below. $$\footnotesize
\xymatrix@C=1.5em@R=3em{
& {\mathsf{Id}^\mathrm{c}}\\
C \ar[rr]^{h} \ar[ur]^{{h^{\varepsilon}}} & & C' \ar[ul]_{{h^{\varepsilon'}}}
}
\hspace*{2cm}
\xymatrix@C=2.5em@R=3em{
{C {\mathrel{\cdot^\mathrm{c}}}C} \ar[r]^{h {\mathrel{\cdot^\mathrm{c}}}h} & {C' {\mathrel{\cdot^\mathrm{c}}}C'}\\
C \ar[r]^{h} \ar[u]^{{h^{\delta}}} & C' \ar[u]_{{h^{\delta'}}}
}$$
We are now in a position to prove that $h = {\ulcorner \tau \urcorner^\mathrm{c}}$ satisfies directed container morphism laws.
From the counit preservation law by going clockwise we get that $$\footnotesize
\xymatrix@C=4em@R=1em{
s \ar@{|->}[r]& {t^{\varepsilon}}s\\
C \ar[r]^{h^{{\varepsilon}}} & {\mathsf{Id}^\mathrm{c}}\\
{q^{\varepsilon}}\, {\{s\}}\, {{\ast}}& \ar@{|->}[l] {{\ast}}}$$ and by going counter-clockwise $$\footnotesize
\xymatrix@C=4em@R=1em{
s \ar@{|->}[r] & t s \ar@{|->}[r] & {t^{\varepsilon'}}(t\, s)\\
C \ar[r]^{h} & C' \ar[r]^{h^{{\varepsilon}'}} & {\mathsf{Id}^\mathrm{c}}\\
q\, {\{s\}}\, ({q^{\varepsilon'}}\, {\{t\, s\}}\, {{\ast}}) & {q^{\varepsilon'}}\, {\{t\, s\}}\, {{\ast}}\ar@{|->}[l] & \ar@{|->}[l] {{\ast}}}$$ which gives us the second directed container morphism law: $${\mathsf{o}}\, {\{s\}}
= {q^{\varepsilon}}\, {\{s\}}\, {{\ast}}= q\, {\{s\}}\, ({q^{\varepsilon'}}\, {\{t\, s\}}\, {{\ast}})
= q\, {\{s\}}\, ({\mathsf{o}}'\, {\{t\, s\}})$$
Clockwise traversal of the comultiplication preservation law gives us that $$\footnotesize
\xymatrix@C=2.5em@R=1em{
s \ar@{|->}[r]
& {t^{\delta}}\, s \ar@{|->}[r]
& *\txt{$(t\, ({\mathsf{fst}}\, ({t^{\delta}}\, s)), $\\$\lambda p.\, t\, ({\mathsf{snd}}\, ({t^{\delta}}\, s) $\\$(q\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, p)))$}\\
C \ar[r]^{h^{\delta}}
& C {\mathrel{\cdot^\mathrm{c}}}C \ar[r]^{h {\mathrel{\cdot^\mathrm{c}}}h}
& C' {\mathrel{\cdot^\mathrm{c}}}C'\\
*\txt{${q^{\delta}}{\{s\}} $\\$(q\,{\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, p, $\\$q\, {\{{\mathsf{snd}}\, ({t^{\delta}}\, s)\, (q\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, p)\}}\, p')$}
& *\txt{$(q\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, p, $\\$q {\{{\mathsf{snd}}\, ({t^{\delta}}\, s)\, (q\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, p)\}}\, p')$} \ar@{|->}[l]
& (p,p') \ar@{|->}[l]
}$$ and counter-clockwise traversal that $$\footnotesize
\xymatrix@C=2.5em@R=1em{
s \ar@{|->}[r]
& t\, s \ar@{|->}[r]
& *\txt{${t^{\delta'}}\, (t\, s)$}\\
C \ar[r]^{h}
& C' \ar[r]^{h^{{\delta}'}}
& C' {\mathrel{\cdot^\mathrm{c}}}C'\\
q\, {\{s\}}\, ({q^{\delta'}}\, {\{t\, s\}}\, (p,p'))
& {q^{\delta'}}\, {\{t\, s\}}\, (p,p') \ar@{|->}[l]
& (p,p') \ar@{|->}[l]
}$$ from where we can derive both the first and the third directed container morphism laws: $$t\, (s {\mathrel{\downarrow}}q\, {\{s\}}\, p)
= t\, ({\mathsf{snd}}\, ({t^{\delta}}\, s)\, (q\, {\{s\}}\, p))
= t\, ({\mathsf{snd}}\, ({t^{\delta}}\, s)\, (q\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, p))$$$$= {\mathsf{snd}}\, ({t^{\delta'}}\, (t\, s))\, p
= t\, s {\mathrel{\downarrow'}}p$$ $$q\, {\{s\}}\, p {\mathrel{\oplus}}{\{s\}}\, q\, {\{s {\mathrel{\downarrow}}q\, {\{s\}}\, p\}}\, p'
={q^{\delta}}{\{s\}}\, (q\, {\{s\}}\, p, q\, {\{{\mathsf{snd}}\, ({t^{\delta}}\, s)\, (q\, {\{s\}}\, p)\}}\, p')$$$$={q^{\delta}}{\{s\}}\, (q\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, p, q\, {\{{\mathsf{snd}}\, ({t^{\delta}}\, s)\, (q\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, p)\}}\, p')$$$$=q\, {\{s\}}\, ({q^{\delta'}}\, {\{t\, s\}}\, (p,p'))
=q\, {\{s\}}\, (p {\mathrel{\oplus}}' {\{t\, s\}}\, p')\rlap{\hbox to 110 pt{\hfill\qEd}}$$
Proof of Proposition \[prop:comonad2dcontainer\] {#proof-of-proposition-propcomonad2dcontainer .unnumbered}
------------------------------------------------
We need to verify that $(C, {\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}})$ satisfies the directed container laws and can assume that $(D, {\varepsilon}, {\delta})$ satisfies the comonad laws.
The comonad laws can be rewritten in terms of container interpretations as outlined in the following commuting diagrams: $$\footnotesize
\xymatrix@C=3em@R=3em{
{\llbracket C {\mathrel{\cdot^\mathrm{c}}}C \rrbracket^\mathrm{c}} \ar[r]^{{\llbracket C {\mathrel{\cdot^\mathrm{c}}}{h^{\varepsilon}}\rrbracket^\mathrm{c}}}
& {\llbracket C {\mathrel{\cdot^\mathrm{c}}}{\mathsf{Id}^\mathrm{c}}\rrbracket^\mathrm{c}} \ar@/^2.5pc/[ddr]^{{\llbracket \rho \rrbracket^\mathrm{c}}}\\
\ar@{}[ur]|{\textrm{nat.\, ${\mathsf{m}}$}}
& {\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\llbracket {\mathsf{Id}^\mathrm{c}}\rrbracket^\mathrm{c}}\ar[u]_{{\mathsf{m}}\, {\{C,{\mathsf{Id}}\}}} \ar@{}[r]_(0.65){\textrm{mon.\ f.\ r.\ unit}}
& \\
{\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\llbracket C \rrbracket^\mathrm{c}} \ar[r]^{{\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\varepsilon}} \ar[uu]^{{\mathsf{m}}\, {\{C,C\}}} \ar@/^1pc/[ur]^{{\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\llbracket {h^{\varepsilon}}\rrbracket^\mathrm{c}}} \ar@{}[ur]|{\textrm{def. ${h^{\varepsilon}}$}} \ar@{}[uu]^(0.2){\textrm{def. ${h^{\delta}}$}}
& {\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\mathsf{Id}}\ar@{=}[r] \ar[u]_{{\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\mathsf{e}}}
& {\llbracket C \rrbracket^\mathrm{c}}\\
{\llbracket C \rrbracket^\mathrm{c}} \ar[u]^{{\delta}} \ar@/^3pc/[uuu]^{{\llbracket {h^{\delta}}\rrbracket^\mathrm{c}}} \ar@{=}[urr] \ar@{}[ur]^{\textrm{com. r.\ counit}}
}
\xymatrix@C=3em@R=3em{
& {\llbracket {\mathsf{Id}^\mathrm{c}}{\mathrel{\cdot^\mathrm{c}}}C \rrbracket^\mathrm{c}} \ar@/_2.5pc/[ddl]_{{\llbracket \lambda \rrbracket^\mathrm{c}}} & {\llbracket C {\mathrel{\cdot^\mathrm{c}}}C \rrbracket^\mathrm{c}} \ar[l]_{{\llbracket {h^{\varepsilon}}{\mathrel{\cdot^\mathrm{c}}}C \rrbracket^\mathrm{c}}} \\
& \ar@{}[l]^(0.65){\textrm{mon.\ f.\ l.\ unit}} \ar@{}[ur]|{\textrm{nat. ${\mathsf{m}}$}} {\llbracket {\mathsf{Id}^\mathrm{c}}\rrbracket^\mathrm{c}} {\cdot}{\llbracket C \rrbracket^\mathrm{c}} \ar[u]^{{\mathsf{m}}\, {\{{\mathsf{Id}}, C\}}}\\
{\llbracket C \rrbracket^\mathrm{c}}
& {\mathsf{Id}}{\cdot}{\llbracket C \rrbracket^\mathrm{c}} \ar@2{-}[l] \ar[u]^{{\mathsf{e}}{\cdot}{\llbracket C \rrbracket^\mathrm{c}}}
& {\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\llbracket C \rrbracket^\mathrm{c}} \ar[l]_{{\varepsilon}{\cdot}{\llbracket C \rrbracket^\mathrm{c}}} \ar[uu]_{{\mathsf{m}}\, {\{C,C\}}} \ar@/_1pc/[ul]_{{\llbracket {h^{\varepsilon}}\rrbracket^\mathrm{c}} {\cdot}{\llbracket C \rrbracket^\mathrm{c}}} \ar@{}[ul]|{\textrm{def. ${h^{\varepsilon}}$}} \ar@{}[uu]_(0.2){\textrm{def. ${h^{\delta}}$}} \\
& & {\llbracket C \rrbracket^\mathrm{c}} \ar[u]_{{\delta}} \ar@2{-}[ull] \ar@{}[ul]_{\textrm{com. l. counit}} \ar@/_3pc/[uuu]_{{\llbracket {h^{\delta}}\rrbracket^\mathrm{c}}}
}$$
$$\footnotesize
\xymatrix@C=2.5em@R=4em{
{\llbracket C \rrbracket^\mathrm{c}} \ar@{}[drr]_{\textrm{com. coass.}} \ar[rr]_{{\delta}} \ar[d]^{{\delta}} \ar@/^2pc/[rrr]^{{\llbracket {h^{\delta}}\rrbracket^\mathrm{c}}}_{\textrm{def. ${h^{\delta}}$}} \ar@/_3pc/[dd]_{{\llbracket {h^{\delta}}\rrbracket^\mathrm{c}}}
& & {\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\llbracket C \rrbracket^\mathrm{c}} \ar[r]^{{\mathsf{m}}\, {\{C,C\}}} \ar[d]_{{\delta}{\cdot}{\llbracket C \rrbracket^\mathrm{c}}} \ar@/^0.5pc/[dr]^{{\llbracket {h^{\delta}}\rrbracket^\mathrm{c}} {\cdot}{\llbracket C \rrbracket^\mathrm{c}}} \ar@{}[dr]_{\textrm{def. ${h^{\delta}}$}}
& {\llbracket C {\mathrel{\cdot^\mathrm{c}}}C \rrbracket^\mathrm{c}} \ar@/^3pc/[dd]^{{\llbracket {h^{\delta}}{\mathrel{\cdot^\mathrm{c}}}C \rrbracket^\mathrm{c}}} \\
{\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\llbracket C \rrbracket^\mathrm{c}} \ar@{}[u]^{\textrm{def. ${h^{\delta}}$}} \ar@{}[dr]^{\textrm{def. ${h^{\delta}}$}} \ar[d]^{{\mathsf{m}}\, {\{C,C\}}} \ar[r]^-{{\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\delta}} \ar@/_0.5pc/[dr]_{{\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\llbracket {h^{\delta}}\rrbracket^\mathrm{c}}}
& {\llbracket C \rrbracket^\mathrm{c}} {\cdot}({\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\llbracket C \rrbracket^\mathrm{c}}) \ar[d]^{{\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\mathsf{m}}\, {\{C,C\}}}
\ar@{}[drr]|{\textrm{mon. f. ass.}}
& ({\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\llbracket C \rrbracket^\mathrm{c}}) {\cdot}{\llbracket C \rrbracket^\mathrm{c}} \ar@{=}[l] \ar[r]_{{\mathsf{m}}\, {\{C,C\}} {\cdot}{\llbracket C \rrbracket^\mathrm{c}}}
& {\llbracket C {\mathrel{\cdot^\mathrm{c}}}C \rrbracket^\mathrm{c}} {\cdot}{\llbracket C \rrbracket^\mathrm{c}} \ar[d]_{{\mathsf{m}}\, {\{C {\mathrel{\cdot^\mathrm{c}}}C, C\}}} \ar@{}[u]|{\textrm{nat. ${\mathsf{m}}$}}\\
{\llbracket C {\mathrel{\cdot^\mathrm{c}}}C \rrbracket^\mathrm{c}} \ar@/_2pc/[rr]_{{\llbracket C {\mathrel{\cdot^\mathrm{c}}}{h^{\delta}}\rrbracket^\mathrm{c}}}
& {\llbracket C \rrbracket^\mathrm{c}} {\cdot}{\llbracket C {\mathrel{\cdot^\mathrm{c}}}C \rrbracket^\mathrm{c}} \ar[r]^{{\mathsf{m}}\, {\{C, C {\mathrel{\cdot^\mathrm{c}}}C\}}} \ar@{}[d]|(0.3){\textrm{nat. ${\mathsf{m}}$}}
& {\llbracket C {\mathrel{\cdot^\mathrm{c}}}(C {\mathrel{\cdot^\mathrm{c}}}C) \rrbracket^\mathrm{c}}
& {\llbracket (C {\mathrel{\cdot^\mathrm{c}}}C) {\mathrel{\cdot^\mathrm{c}}}C \rrbracket^\mathrm{c}} \ar[l]_{{\llbracket \alpha \rrbracket^\mathrm{c}}} \\
& &
}$$
Next we quote these three diagrams to get the comonad laws in terms of containers in the next three commuting diagrams. $$\footnotesize
\xymatrix@C=3.5em@R=3em{
C {\mathrel{\cdot^\mathrm{c}}}C \ar[r]^-{C {\mathrel{\cdot^\mathrm{c}}}{h^{\varepsilon}}} & C {\mathrel{\cdot^\mathrm{c}}}{\mathsf{Id}^\mathrm{c}}\ar[r]^{\rho} & C\\
C \ar[u]^{{h^{\delta}}} \ar@{=}[urr]
}
\quad
\xymatrix@C=3.5em@R=3em{
C & {\mathsf{Id}^\mathrm{c}}{\mathrel{\cdot^\mathrm{c}}}C \ar[l]_{\lambda} & C {\mathrel{\cdot^\mathrm{c}}}C \ar[l]_-{{h^{\varepsilon}}{\mathrel{\cdot^\mathrm{c}}}C} \\
& & C \ar[u]_{{h^{\delta}}} \ar@{=}[ull]
}$$ $$\footnotesize
\xymatrix@C=2em@R=3em{
C \ar[rr]^{{h^{\delta}}} \ar[d]_{{h^{\delta}}}
&
& C {\mathrel{\cdot^\mathrm{c}}}C \ar[d]^{{h^{\delta}}{\mathrel{\cdot^\mathrm{c}}}C} \\
C {\mathrel{\cdot^\mathrm{c}}}C \ar[r]^-{C {\mathrel{\cdot^\mathrm{c}}}{h^{\delta}}}
& C {\mathrel{\cdot^\mathrm{c}}}(C {\mathrel{\cdot^\mathrm{c}}}C)
& (C {\mathrel{\cdot^\mathrm{c}}}C) {\mathrel{\cdot^\mathrm{c}}}C \ar[l]_-{\alpha}
}$$
From the comonad right counital law we get by going clockwise $$\footnotesize
\xymatrix@C=2.5em@R=1em{
s \ar@{|->}[r] & {t^{\delta}}s \ar@{|->}[r]
& *\txt{$({\mathsf{fst}}\, ({t^{\delta}}\, s),$\\$\lambda\ \_.\, {{\ast}})$} \ar@{|->}[r]
& {\mathsf{fst}}\, ({t^{\delta}}\, s)\\
C \ar[r]^{{h^{\delta}}}
& C {\mathrel{\cdot^\mathrm{c}}}C \ar[r]^{C {\mathrel{\cdot^\mathrm{c}}}{h^{\varepsilon}}}
& C {\mathrel{\cdot^\mathrm{c}}}{\mathsf{Id}^\mathrm{c}}\ar[r]^{\rho} & C\\
*\txt{${q^{\delta}}{\{s\}}\, (p, $\\${q^{\varepsilon}}\, {\{{\mathsf{snd}}\, ({t^{\delta}}\, s)\, p\}}\, {{\ast}})$}
& *\txt{$(p, $\\${q^{\varepsilon}}\{{\mathsf{snd}}\, ({t^{\delta}}\, s)\, p\}\, {{\ast}})$} \ar@{|->}[l]
& (p,{{\ast}}) \ar@{|->}[l] & p \ar@{|->}[l]
}$$ from where it follows that ${\delta}$ preserves the shape of the given data-structure as the outer shape of the composite data-structure returned and that the third directed container law holds: $$s
= {\mathsf{fst}}\, ({t^{\delta}}\, s)$$ $$p
= {q^{\delta}}{\{s\}} (p, {q^{\varepsilon}}{\{{\mathsf{snd}}\, ({t^{\delta}}s)\, p\}}\, {{\ast}})
= p {\mathrel{\oplus}}{\{s\}}\, {\mathsf{o}}\, {\{s {\mathrel{\downarrow}}p\}}$$
Similarly, from the comonad left counital law we get by going counter-clockwise $$\footnotesize
\xymatrix@C=1.3em@R=1em{
s \ar@{|->}[r]
& {t^{\delta}}s \ar@{|->}[r]
& *\txt{$({{\ast}}, \lambda {{\ast}}.\, {\mathsf{snd}}\, ({t^{\delta}}\, s) $\\$({q^{\varepsilon}}\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}))$} \ar@{|->}[r]
& *\txt{${\mathsf{snd}}\, ({t^{\delta}}\, s) $\\$({q^{\varepsilon}}\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, {{\ast}})$}\\
C \ar[r]^{{h^{\delta}}}
& C {\mathrel{\cdot^\mathrm{c}}}C \ar[r]^{{h^{\varepsilon}}{\mathrel{\cdot^\mathrm{c}}}C}
& {\mathsf{Id}^\mathrm{c}}{\mathrel{\cdot^\mathrm{c}}}C \ar[r]^{\lambda}
& C\\
*\txt{${q^{\delta}}\, {\{s\}} $\\$({q^{\varepsilon}}\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, {{\ast}}, p)$}
& *\txt{$({q^{\varepsilon}}\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, {{\ast}}, $\\$p)$} \ar@{|->}[l]
& ({{\ast}},p) \ar@{|->}[l]
& p \ar@{|->}[l]
}$$ from where the first and fourth directed container laws follow: $$s
= {\mathsf{snd}}\, ({t^{\delta}}\, s)\, ({q^{\varepsilon}}\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, {{\ast}})
= {\mathsf{snd}}\, ({t^{\delta}}\, s)\, ({q^{\varepsilon}}\, {\{s\}})
= s {\mathrel{\downarrow}}{\mathsf{o}}\, {\{s\}}$$ $$p =
{q^{\delta}}{\{s\}}\, ({q^{\varepsilon}}\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, {{\ast}}, p)
= {q^{\delta}}{\{s\}}\, ({q^{\varepsilon}}\, {\{s\}}\, {{\ast}}, p)
= {\mathsf{o}}\, {\{s\}} {\mathrel{\oplus}}{\{s\}}\, p$$
The last two directed container laws are derivable from the comonad coassociativity law. By going clockwise we get $$\footnotesize
\xymatrix@C=2.8em@R=1em{
s \ar@{|->}[r]
& {t^{\delta}}s \ar@{|->}[r]
& *\txt{$({t^{\delta}}\, ({\mathsf{fst}}\, ({t^{\delta}}\, s)), $\\${\mathsf{snd}}\, ({t^{\delta}}\, s) {\circ}$\\$({q^{\delta}}\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}))$} \ar@{|->}[r]
& *\txt{$({\mathsf{fst}}\, ({t^{\delta}}\, ({\mathsf{fst}}\, ({t^{\delta}}\, s))), $\\$(\lambda p.\, {\mathsf{snd}}\, ({t^{\delta}}\, ({\mathsf{fst}}\, ({t^{\delta}}s)))\, p,$\\$\lambda p'.\, {\mathsf{snd}}\, ({t^{\delta}}\, s) $\\$({q^{\delta}}\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, (p , p'))))$}\\
C \ar[r]^{{h^{\delta}}}
& C {\mathrel{\cdot^\mathrm{c}}}C \ar[r]^{{h^{\delta}}{\mathrel{\cdot^\mathrm{c}}}C}
& (C {\mathrel{\cdot^\mathrm{c}}}C) {\mathrel{\cdot^\mathrm{c}}}C \ar[r]^{\alpha}
& C {\mathrel{\cdot^\mathrm{c}}}(C {\mathrel{\cdot^\mathrm{c}}}C)\\
*\txt{$({q^{\delta}}{\{s\}} $\\$({q^{\delta}}\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}} $\\$(p,p'),p'')$}
& *\txt{$({q^{\delta}}{\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}} $\\$(p,p'), p'')$} \ar@{|->}[l]
& ((p,p'),p'') \ar@{|->}[l]
& (p , (p' , p'')) \ar@{|->}[l]
}$$ and by going counter-clockwise we get $$\footnotesize
\xymatrix@C=3em@R=1em{
s \ar@{|->}[r] & {t^{\delta}}s \ar@{|->}[r]
& *\txt{$({\mathsf{fst}}\, ({t^{\delta}}\, s), $\\$(\lambda p.\, {\mathsf{fst}}\, ({t^{\delta}}\, ({\mathsf{snd}}\, ({t^{\delta}}\, s)\, p)), $\\$\lambda p'.\, {\mathsf{snd}}\, ({t^{\delta}}\, ({\mathsf{snd}}\, ({t^{\delta}}s)\, p))\, p'))$}\\
C \ar[r]^{{h^{\delta}}}
& C {\mathrel{\cdot^\mathrm{c}}}C \ar[r]^{C {\mathrel{\cdot^\mathrm{c}}}{h^{\delta}}} & C {\mathrel{\cdot^\mathrm{c}}}(C {\mathrel{\cdot^\mathrm{c}}}C)\\
*\txt{$({q^{\delta}}\, {\{s\}}\, (p, $\\${q^{\delta}}\, {\{{\mathsf{snd}}\, ({t^{\delta}}\, s)\, p\}}\, (p',p'')))$}
& *\txt{$(p, $\\${q^{\delta}}\, {\{{\mathsf{snd}}\, ({t^{\delta}}\, s)\, p\}}\, (p',p''))$} \ar@{|->}[l] & (p,(p',p'')) \ar@{|->}[l]
}$$ from where the second and fifth directed container laws follow $$s {\mathrel{\downarrow}}(p {\mathrel{\oplus}}p')
= {\mathsf{snd}}\, ({t^{\delta}}\, s)\, ({q^{\delta}}\, \{s\}\, (p , p'))
= {\mathsf{snd}}\, ({t^{\delta}}\, s)\, ({q^{\delta}}\, \{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}\, (p , p'))
=$$$${\mathsf{snd}}\, ({t^{\delta}}\, ({\mathsf{snd}}\, ({t^{\delta}}\, s)\, p))\, p'
= (s {\mathrel{\downarrow}}p) {\mathrel{\downarrow}}p'$$ $$(p {\mathrel{\oplus}}{\{s\}}\, p') {\mathrel{\oplus}}{\{s\}}\, p''
= {q^{\delta}}\, {\{s\}}\, ({q^{\delta}}\, {\{s\}}\, (p',p''),p'')
= {q^{\delta}}\, {\{s\}}\, ({q^{\delta}}\, {\{{\mathsf{fst}}\, ({t^{\delta}}\, s)\}}\, (p',p''),p'')
=$$$${q^{\delta}}\, {\{s\}}\, (p, {q^{\delta}}\, {\{{\mathsf{snd}}\, ({t^{\delta}}\, s)\, p\}}\, (p',p''))
= p {\mathrel{\oplus}}{\{s\}}\, (p' {\mathrel{\oplus}}{\{s {\mathrel{\downarrow}}p\}}\, p'')\rlap{\hbox to 75 pt{\hfill\qEd}}$$
Proof of Proposition \[prop:pullbacklaw2\] {#proof-of-proposition-proppullbacklaw2 .unnumbered}
------------------------------------------
By interpreting the given directed container $(C,{\mathrel{\downarrow}},{\mathsf{o}},{\mathrel{\oplus}})$ we get a comonad $(D,{\varepsilon},{\delta})$ whereby $D={\llbracket C \rrbracket^\mathrm{c}}$, ${\varepsilon}\, (s,v) = v\, ({\mathsf{o}}\,
{\{s\}})$ and ${\delta}\, (s,v) = (s , \lambda p.\, (s {\mathrel{\downarrow}}p,
\lambda p'.\, v\, (p {\mathrel{\oplus}}{\{s\}}\, p')))$.
From the comonad, we get a directed container $(C,{\mathrel{\downarrow}}',{\mathsf{o}}',{\mathrel{\oplus}}') =
\lceil (D,{\varepsilon},{\delta}), C \rceil$ by taking $s {\mathrel{\downarrow}}' p = {\mathsf{snd}}\, (t^{\delta}\,
s)\, p$, ${\mathsf{o}}'\, {\{s\}} = q^{\varepsilon}\, {\{s\}}\, {{\ast}}$, $p {\mathrel{\oplus}}' {\{s\}}\, p' =
q^{\delta}\, {\{s\}}\, (p, p')$.
This directed container must be equal to the original directed container $(C,{\mathrel{\downarrow}},{\mathsf{o}},{\mathrel{\oplus}})$, i.e., we need to prove that $s {\mathrel{\downarrow}}' p = s
{\mathrel{\downarrow}}p$ and ${\mathsf{o}}'\, {\{s\}} = {\mathsf{o}}\, {\{s\}}$ and $p {\mathrel{\oplus}}' p' = p {\mathrel{\oplus}}p'$.
By the definitions of ${\mathsf{e}}$, ${\mathsf{m}}$, ${\ulcorner - \urcorner^\mathrm{c}}$, for the container morphisms ${t^{\varepsilon}}\lhd {q^{\varepsilon}}= {\ulcorner {\mathsf{e}}{\circ}{\varepsilon}\urcorner^\mathrm{c}}$ and ${t^{\delta}}\lhd
{q^{\delta}}= {\ulcorner {\mathsf{m}}\, {\circ}\, {\delta}\urcorner^\mathrm{c}} $ we have that $${t^{\varepsilon}}\, s = {{\ast}}$$ $${q^{\varepsilon}}\, {\{s\}}\, {{\ast}}= {\varepsilon}\, (s, {\mathsf{id}})$$ $${t^{\delta}}\, s = ({\mathsf{fst}}\, ({\delta}\,(s, {\mathsf{id}})) , \lambda p.\, {\mathsf{fst}}\, ({\mathsf{snd}}\, ({\delta}\, (s, {\mathsf{id}}))\, p))$$ $${q^{\delta}}\, {\{s\}}\, (p,p') = {\mathsf{snd}}\, ({\mathsf{snd}}\, ({\delta}\, (s,{\mathsf{id}}))\, p)\, p'$$
Using the definitions of ${\mathrel{\downarrow}}'$, ${\mathsf{o}}'$, ${\mathrel{\oplus}}'$, ${\varepsilon}$, ${\delta}$, we calculate: $$s {\mathrel{\downarrow}}' p = {\mathsf{snd}}\, ({t^{\delta}}\, s)\, p
= {\mathsf{fst}}\, ({\mathsf{snd}}\, ({\delta}\, (s, {\mathsf{id}}))\, p))
= s {\mathrel{\downarrow}}p$$ $${\mathsf{o}}'\, {\{s\}} = {q^{\varepsilon}}\, {\{s\}}\, {{\ast}}= {\varepsilon}\, (s, {\mathsf{id}})
= {\mathsf{o}}\, {\{s\}}$$ $$p {\mathrel{\oplus}}' {\{s\}}\, p' = {q^{\delta}}\, {\{s\}}\, (p,p')
= {\mathsf{snd}}\, ({\mathsf{snd}}\, ({\delta}\, (s,{\mathsf{id}}))\, p)\, p'
= p {\mathrel{\oplus}}{\{s\}}\, p'\rlap{\hbox to 67 pt{\hfill\qEd}}$$
Proof of Proposition \[prop:pullbacklaw1\] {#proof-of-proposition-proppullbacklaw1 .unnumbered}
------------------------------------------
The comonad $(D, {\varepsilon}, {\delta})$ induces a directed container $(S \lhd
P,{\mathrel{\downarrow}},{\mathsf{o}},{\mathrel{\oplus}}) = \lceil (D, {\varepsilon}, {\delta}), S \lhd P\rceil$ whereby $$s {\mathrel{\downarrow}}p = {\mathsf{fst}}\, ({\mathsf{snd}}\, ({\delta}\, (s, {\mathsf{id}}))\, p)$$ $${\mathsf{o}}\, {\{s\}} = {\varepsilon}\, \{P\, s\}\, (s, {\mathsf{id}})$$ $$p {\mathrel{\oplus}}{\{s\}}\, p' = {\mathsf{snd}}\, ({\mathsf{snd}}\, ({\delta}\, (s,{\mathsf{id}}))\, p)\, p'$$
By interpreting this directed container, we get a comonad $(D', {\varepsilon}', {\delta}') = {\llbracket S \lhd P,{{\mathrel{\downarrow}}},{\mathsf{o}},{{\mathrel{\oplus}}} \rrbracket^\mathrm{dc}}$ by taking ${\varepsilon}'\, (s,v) = v\, ({\mathsf{o}}\,
{\{s\}})$ and ${\delta}'\, (s,v) = (s , \lambda p.\, (s {\mathrel{\downarrow}}p,
\lambda p'.\, v\, (p {\mathrel{\oplus}}{\{s\}}\, p')))$.
This comonad must equal $(D,{\varepsilon},{\delta})$, i.e., we need to prove that $D' = D$ and ${\varepsilon}'\, {\{X\}}\, (s,v) = {\varepsilon}\, {\{X\}}\, (s,v)$ and ${\delta}'\,
{\{X\}}\, (s,v)$ = ${\delta}\, {\{X\}}\, (s,v)$.
First of all, from the definition of directed container interpretation, we know that the underlying functors are equal: $D' =
{\llbracket S \lhd P \rrbracket^\mathrm{c}} = D$.
Using the definitions of ${\varepsilon}'$, ${\delta}'$, ${\mathrel{\downarrow}}$, ${\mathsf{o}}$, ${\mathrel{\oplus}}$ we can calculate $${\varepsilon}'\, {\{X\}} (s,v)
= v\, ({\mathsf{o}}\, {\{s\}})
= v\, ({q^{\varepsilon}}\, {\{s\}}\, {{\ast}})
= v\, ({\varepsilon}\, {\{P\, s\}}\, (s, {\mathsf{id}}))$$ $${\delta}'\, {\{X\}} (s,v)
= (s, \lambda p.\, s {\mathrel{\downarrow}}p\, ,\, \lambda p'.\, v\, (p {\mathrel{\oplus}}{\{s\}}\, p'))
=$$ $$= (s, \lambda p.\, ({\mathsf{fst}}\, ({\mathsf{snd}}\, {\delta}\, \{P\, s\}\, (s, {\mathsf{id}})\, p), \lambda p'.\, v\, ({\mathsf{snd}}\, ({\mathsf{snd}}\, {\delta}\, {\{P\, s\}}\, (s,{\mathsf{id}})\, p)\, p')))$$
Now, because of naturality of ${\varepsilon}$ and ${\delta}$ expressed in the diagrams $$\footnotesize
\xymatrix@C=5em@R=1.5em{
(s,{\mathsf{id}}) \ar@{|->}[r] \ar@{|->}@/_5pc/[dddd] & {\varepsilon}\, {\{P\, s\}}\, (s,{\mathsf{id}}) \ar@{|->}@/^5pc/[dddd] \\
\Sigma s : S . P\, s \to P\, s \ar[r]^-{{\varepsilon}\, {\{P\, s\}}} \ar[dd]_{\lambda (s,v').\, (s,v {\circ}v')} & P\, s \ar[dd]^{v}\\
& \\
\Sigma s : S . P\, s \to X \ar[r]_-{{\varepsilon}\, {\{X\}}} & X\\
(s,v) \ar@{|->}[r] & {\varepsilon}\, {\{X\}}\, (s,v) = v\, ({\varepsilon}\, {\{P\, s\}}\, (s,{\mathsf{id}}))
}$$
$$\footnotesize
\xymatrix@C=5em@R=1.5em{
(s,{\mathsf{id}}) \ar@{|->}[r] \ar@{|->}@/_6pc/[dddd] & {\delta}\, \{P s\}\, (s,{\mathsf{id}}) \ar@{|->}@/^12.5pc/[dddd] \\
\Sigma s : S . P\, s \to P\, s \ar[r]^-{{\delta}\, \{P\, s\}} \ar[dd]_{\lambda (s,v').\, (s,v {\circ}v')} & \Sigma s : S . P\, s \to \Sigma s' : S . P\, s' \to P\, s \ar[dd]^{\lambda (s,v').\, (s, \lambda p.\, ({\mathsf{fst}}\, (v'\, p), v {\circ}{\mathsf{snd}}\, (v'\, p)))}\\
& \\
\Sigma s : S . P\, s \to X \ar[r]_-{{\delta}\, {\{X\}}} & \Sigma s : S . P\, s \to \Sigma s' : S . P\, s' \to X \\
(s,v) \ar@{|->}[r] & *\txt{${\delta}\, {\{X\}}\, (s,v) = $\\$({\mathsf{fst}}\, ({\delta}\, {\{P\, s\}}\, (s,{\mathsf{id}})), $\\$\lambda p.\, ({\mathsf{fst}}\, ({\mathsf{snd}}\, {\delta}\, {\{P\, s\}}\, (s, {\mathsf{id}})\, p), $\\$\lambda p'.\, v\, ({\mathsf{snd}}\, ({\mathsf{snd}}\, {\delta}\, {\{P\, s\}}\, (s,{\mathsf{id}})\, p)\, p')))$}
}$$ it is evident that the counit and comultiplication of $(D,{\varepsilon},{\delta})$ and $(D',{\varepsilon}',{\delta}')$ are equal: $${\varepsilon}'\, {\{X\}}\, (s,v) = {\varepsilon}\, {\{X\}}\, (s,v)$$ $${\delta}'\, {\{X\}}\, (s,v) = {\delta}\, {\{X\}}\, (s,v)\eqno{\qEd}$$
Proofs for Section \[sec:constructions\]
========================================
Proof of Proposition \[prop:prod\] {#proof-of-propositionpropprod .unnumbered}
----------------------------------
We must show that the definitions yield a strict directed container that is a product of two given strict directed containers in the category of all directed containers.
We first check that $E$ is a strict directed container.
The data ${\mathrel{\downarrow^+}}$ and ${\mathrel{\oplus^+}}$ equip the container $S \lhd P^+$ with a strict directed container structure.
Auxiliary statements $s {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p {\mathrel{\overline{\oplus_0^+}}}p') = (s {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}p)
{\mathrel{\downarrow^+}}p'$ and $s {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}(p {\mathrel{\overline{\oplus_1^+}}}p') = (s {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}p) {\mathrel{\downarrow^+}}p'$ for law 1, by mutual induction on the two $p$s. We show only the cases for the first auxiliary statement; those of the second are symmetric.
Case $p=(p_0,{\mathsf{nothing}})$, $p' = {\mathsf{inl}}\,(p_0', {\mathsf{nothing}})$: $$\begin{array}{cl}
&(s_0, v_0) {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}((p_0, {\mathsf{nothing}}) {\mathrel{\overline{\oplus_0^+}}}{\mathsf{inl}}\,(p_0', {\mathsf{nothing}}))\\
=&\quad\{\textrm{definitions of ${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}$, ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&((s_0{\mathrel{\downarrow_0^+}}(p_0 {\mathrel{\oplus_0^+}}p_0'),
\lambda p.\,v_0\,((p_0 {\mathrel{\oplus_0^+}}p_0'){\mathrel{\oplus_0^+}}p)),
v_0\,(p_0 {\mathrel{\oplus_0^+}}p_0'))\\
=&\quad\{\textrm{strict directed container laws 1, 2} \}\\
&(((s_0{\mathrel{\downarrow_0^+}}p_0) {\mathrel{\downarrow_0^+}}p_0', \lambda p.\,v_0\,(p_0{\mathrel{\oplus_0^+}}(p_0'{\mathrel{\oplus_0^+}}p))),
v_0\,(p_0{\mathrel{\oplus_0^+}}p_0'))\\
=&\quad\{\textrm{definitions of ${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}$, ${\mathrel{\downarrow^+}}$} \}\\
&((s_0, v_0) {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p_0, {\mathsf{nothing}})) {\mathrel{\downarrow^+}}{\mathsf{inl}}\,(p_0', {\mathsf{nothing}})
\end{array}$$ Case for $p=(p_0, {\mathsf{nothing}})$, $p'={\mathsf{inl}}\,(p_0', {\mathsf{just}}\,p_1')$: $$\begin{array}{cl}
&(s_0, v_0) {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}((p_0, {\mathsf{nothing}}) {\mathrel{\overline{\oplus_0^+}}}{\mathsf{inl}}\,(p_0', {\mathsf{just}}\, p_1'))\\
=&\quad\{\textrm{definitions of ${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}$, ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&v_0\, (p_0 {\mathrel{\oplus_0^+}}p_0') {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}p_1'\\
=&\quad\{\textrm{definitions of ${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}$, ${\mathrel{\downarrow^+}}$} \}\\
&( (s_0,v_0){\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p_0, {\mathsf{nothing}})){\mathrel{\downarrow^+}}{\mathsf{inl}}\,(p_0',{\mathsf{just}}\, p_1')
\end{array}$$ Case $p = (p_0, {\mathsf{nothing}})$, $p' = {\mathsf{inr}}\,p'$: $$\begin{array}{cl}
& (s_0, v_0) {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}((p_0,{\mathsf{nothing}}) {\mathrel{\overline{\oplus_0^+}}}{\mathsf{inr}}\,p')\\
=&\quad\{\textrm{definitions of ${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}$, ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&v_0\, p_0 {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}p'\\
=&\quad\{\textrm{definitions of ${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}$, ${\mathrel{\downarrow^+}}$} \}\\
&((s_0,v_0) {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p_0,{\mathsf{nothing}})){\mathrel{\downarrow^+}}{\mathsf{inr}}\,p'
\end{array}$$ Case $p= (p_0,{\mathsf{just}}\,p_1)$: $$\begin{array}{cl}
&(s_0,v_0) {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}((p_0,{\mathsf{just}}\, p_1){\mathrel{\overline{\oplus_0^+}}}p')\\
=&\quad\{\textrm{definitions of ${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}$, ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
& v_0\, p_0 {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}(p_1 {\mathrel{\overline{\oplus_1^+}}}p') \\
=&\quad\{\textrm{inductive hypothesis for $p_1$}\}\\
& (v_0\, p_0 {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}p_1) {\mathrel{\downarrow^+}}p')\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}$}\}\\
&((s_0,v_0) {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p_0,{\mathsf{just}}\,p_1)) {\mathrel{\downarrow^+}}\,p'
\end{array}$$
Strict directed container law 1. Case $s = (s_0, s_1)$, $p = {\mathsf{inl}}\,
p$: $$\begin{array}{cl}
&(s_0,s_1) {\mathrel{\downarrow^+}}({\mathsf{inl}}\, p {\mathrel{\oplus^+}}p')\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus^+}}$}\}\\
&(s_0, s_1) {\mathrel{\downarrow^+}}{\mathsf{inl}}\, (p {\mathrel{\overline{\oplus_0^+}}}p')\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow^+}}$}\}\\
&s_0 {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p {\mathrel{\overline{\oplus_0^+}}}p')\\
=&\quad\{\textrm{first aux. statement}\}\\
&(s_0 {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}p) {\mathrel{\downarrow^+}}p'\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow^+}}$}\}\\
&((s_0, s_1) {\mathrel{\downarrow^+}}{\mathsf{inl}}\, p) {\mathrel{\downarrow^+}}p'
\end{array}$$ Case $p = {\mathsf{inr}}\, p$ is symmetric.
Auxiliary statements $(p{\mathrel{\overline{\oplus_0^+}}}p'){\mathrel{\overline{\oplus_0^+}}}p''=
p{\mathrel{\overline{\oplus_0^+}}}(p'{\mathrel{\oplus^+}}p'')$ and $(p{\mathrel{\overline{\oplus_1^+}}}p'){\mathrel{\overline{\oplus_1^+}}}p''= p{\mathrel{\overline{\oplus_1^+}}}(p'{\mathrel{\oplus^+}}p'')$ for law 2, by mutual induction on the two $p$s. We show only the cases of the first statement. Case $p=(p_0, {\mathsf{nothing}})$, $p'={\mathsf{inl}}\,(p_0',{\mathsf{nothing}})$, $p'' = {\mathsf{inl}}\,(p_0'', p_1'')$: $$\begin{array}{cl}
&((p_0,{\mathsf{nothing}}){\mathrel{\overline{\oplus_0^+}}}{\mathsf{inl}}\,(p_0',{\mathsf{nothing}})){\mathrel{\overline{\oplus_0^+}}}{\mathsf{inl}}\,(p_0'',p_1'')\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&((p_0{\mathrel{\oplus_0^+}}p_0'){\mathrel{\oplus_0^+}}p_0'',p_1'')\\
=&\quad\{\textrm{strict direct container law 2}\}\\
&(p_0{\mathrel{\oplus_0^+}}(p_0'{\mathrel{\oplus_0^+}}p_0''),p_1'')\\
=&\quad\{\textrm{definitions of ${\mathrel{\overline{\oplus_0^+}}}$, ${\mathrel{\oplus^+}}$}\}\\
&(p_0,{\mathsf{nothing}}){\mathrel{\overline{\oplus_0^+}}}({\mathsf{inl}}\,(p_0',{\mathsf{nothing}}){\mathrel{\oplus^+}}{\mathsf{inl}}\,(p_0'',p_1''))
\end{array}$$ Case $p=(p_0, {\mathsf{nothing}})$, $p'={\mathsf{inl}}\,(p_0',{\mathsf{nothing}})$, $p'' =
{\mathsf{inr}}\,p''$: $$\begin{array}{cl}
&((p_0,{\mathsf{nothing}}){\mathrel{\overline{\oplus_0^+}}}{\mathsf{inl}}\,(p_0',{\mathsf{nothing}})){\mathrel{\overline{\oplus_0^+}}}{\mathsf{inr}}\,p''\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&(p_0{\mathrel{\oplus_0^+}}p_0',{\mathsf{just}}\,p'')\\
=&\quad\{\textrm{definitions of ${\mathrel{\overline{\oplus_0^+}}}$, ${\mathrel{\oplus^+}}$}\}\\
&(p_0,{\mathsf{nothing}}){\mathrel{\overline{\oplus_0^+}}}({\mathsf{inl}}\,(p_0',{\mathsf{nothing}}){\mathrel{\oplus^+}}{\mathsf{inr}}\,p'')
\end{array}$$ Case $p=(p_0, {\mathsf{nothing}})$, $p'={\mathsf{inl}}\,(p_0',{\mathsf{just}}\,p_1')$: $$\begin{array}{cl}
&((p_0,{\mathsf{nothing}}){\mathrel{\overline{\oplus_0^+}}}{\mathsf{inl}}\,(p_0',{\mathsf{just}}\,p_1')){\mathrel{\overline{\oplus_0^+}}}p''\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&(p_0 {\mathrel{\oplus_0^+}}p_0',{\mathsf{just}}\,(p_1'{\mathrel{\overline{\oplus_1^+}}}p''))\\
=&\quad\{\textrm{definitions of ${\mathrel{\overline{\oplus_0^+}}}$, ${\mathrel{\oplus^+}}$}\}\\
&(p_0,{\mathsf{nothing}}){\mathrel{\overline{\oplus_0^+}}}({\mathsf{inl}}\,(p_0',{\mathsf{just}}\,p_1'){\mathrel{\oplus^+}}p'')
\end{array}$$ Case $p = (p_0, {\mathsf{nothing}})$, $p' = {\mathsf{inr}}\, p'$: $$\begin{array}{cl}
&((p_0,{\mathsf{nothing}}){\mathrel{\overline{\oplus_0^+}}}{\mathsf{inr}}\, p'){\mathrel{\overline{\oplus_0^+}}}p''\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&(p_0, {\mathsf{just}}\, (p' {\mathrel{\overline{\oplus_1^+}}}p'')) \\
=&\quad\{\textrm{definitions of ${\mathrel{\overline{\oplus_0^+}}}$, ${\mathrel{\oplus^+}}$}\}\\
&(p_0,{\mathsf{nothing}}){\mathrel{\overline{\oplus_0^+}}}({\mathsf{inr}}\, p'{\mathrel{\oplus^+}}p'')\\
\end{array}$$ Case $p= (p_0, {\mathsf{just}}\,p_1)$: $$\begin{array}{cl}
&((p_0,{\mathsf{just}}\,p_1){\mathrel{\overline{\oplus_0^+}}}p'){\mathrel{\overline{\oplus_0^+}}}p''\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&(p_0,{\mathsf{just}}\,((p_1{\mathrel{\overline{\oplus_1^+}}}p'){\mathrel{\overline{\oplus_1^+}}}p''))\\
=&\quad\{\textrm{inductive hypothesis for $p_1$}\}\\
&(p_0,{\mathsf{just}}\,(p_1{\mathrel{\overline{\oplus_1^+}}}(p'{\mathrel{\oplus^+}}p'')))\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&(p_0,{\mathsf{just}}\,p_1){\mathrel{\overline{\oplus_0^+}}}(p'{\mathrel{\oplus^+}}p'')\\
\end{array}$$
Strict directed container law 2. Case $p = {\mathsf{inl}}\, p$: $$\begin{array}{cl}
& ({\mathsf{inl}}\, p {\mathrel{\oplus^+}}p') {\mathrel{\oplus^+}}p''\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus^+}}$}\}\\
& {\mathsf{inl}}\, (p {\mathrel{\overline{\oplus_0^+}}}p') {\mathrel{\oplus^+}}p''\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus^+}}$}\}\\
& {\mathsf{inl}}\, ((p {\mathrel{\overline{\oplus_0^+}}}p') {\mathrel{\overline{\oplus_0^+}}}p')\\
=&\quad\{\textrm{first aux. statement}\}\\
& {\mathsf{inl}}\, (p {\mathrel{\overline{\oplus_0^+}}}(p' {\mathrel{\oplus^+}}p'')\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus^+}}$}\}\\
& {\mathsf{inl}}\, p {\mathrel{\oplus^+}}(p' {\mathrel{\oplus^+}}p'')
\end{array}$$ Case $p = {\mathsf{inr}}\, p$ is symmetric.
To check that $E$ is a product of $E_0$ and $E_1$ we can either verify it directly that it satisfies the required universal property or prove that it interprets to a product of the interpreting comonads. Here we have chosen to pursue the first route.
For $E$ to be a product of $E_0$ and $E_1$, it must come with directed container morphisms $\pi_0 = t^{\pi_0} \lhd
q^{\pi_0} : E_0 \to E$, $\pi_1 = t^{\pi_1} \lhd q^{\pi_1} : E_1 \to
E$. We claim that they can be defined by
- $t^{\pi_0} : S \to S_0$\
$t^{\pi_0}\, ((s_0 , v_0) , (s_1 , v_1)) = s_0$
- $t^{\pi_1} : S \to S_1$\
$t^{\pi_1}\, ((s_0 , v_0) , (s_1 , v_1)) = s_1$
- $q^{\pi_0} : \Pi {\{s : S\}} .\, P_0\, (t^{\pi_0}\, s) \to P\, s$\
$q^{\pi_0}\, {\mathsf{nothing}}= {\mathsf{nothing}}$\
$q^{\pi_0}\, ({\mathsf{just}}\, p) = {\mathsf{just}}\, ({\mathsf{inl}}\, (p , {\mathsf{nothing}}))$
- $q^{\pi_1} : \Pi {\{s : S\}} .\, P_1\, (t^{\pi_1}\, s) \to P\, s$\
$q^{\pi_1}\, {\mathsf{nothing}}= {\mathsf{nothing}}$\
$q^{\pi_1}\, ({\mathsf{just}}\, p) = {\mathsf{just}}\, ({\mathsf{inr}}\, (p , {\mathsf{nothing}}))$
Moreover, any directed container $E' = (S' \lhd P', {{\mathrel{\downarrow'}}}, {\mathsf{o'}},
{{\mathrel{\oplus'}}})$ with two directed container morphisms $f_0 = t^{f_0} \lhd
q^{f_0} : E' \to E_0$ and $f_1 = t^{f_1} \lhd q^{f_1}: E' \to E_1$ must jointly determine a unique directed container morphism $f = t^f
\lhd q^f : E' \to E$ such that the following two triangles commute. $$\tag{$\dag$}
\label{diag:products}
\begin{gathered}
\xymatrix@C=4.3em@R=4em{
& (S' \lhd P' , {\mathrel{\downarrow'}}, {\mathsf{o'}}, {\mathrel{\oplus'}}) \ar[dl]_{t^{f_0} \lhd q^{f_0}} \ar[dr]^{t^{f_1} \lhd q^{f_1}} \ar[d]_{t^f \lhd q^f}& \\
(S_0 \lhd P_0 , {\mathrel{\downarrow_0}}, {\mathsf{o}_0}, {\mathrel{\oplus_0}}) & \ar[l]^{t^{\pi_0} \lhd q^{\pi_0}} (S \lhd P , {\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}}) \ar[r]_{t^{\pi_1} \lhd q^{\pi_1}} & (S_1 \lhd P_1 , {\mathrel{\downarrow_1}}, {\mathsf{o}_1}, {\mathrel{\oplus_1}})
}
\end{gathered}$$ We claim that $f$ is given by
- $t^f : S' \to S$\
$t^f\, s = ({\overline{{t^{f_0}}}}\,s,{\overline{{t^{f_1}}}}\,s)$\
where\
${\overline{{t^{f_0}}}}: S' \to \overline{S_0}$\
${\overline{{t^{f_1}}}}: S' \to \overline{S_1}$\
(by mutual corecursion)\
${\overline{{t^{f_0}}}}\,s = ({t^{f_0}}\, s, \lambda p .\, {\overline{{t^{f_1}}}}\, (s {\mathrel{\downarrow'}}{q^{f_0}}\, ({\mathsf{just}}\, p)))$\
${\overline{{t^{f_1}}}}\,s = ({t^{f_1}}\, s, \lambda p .\, {\overline{{t^{f_0}}}}\, (s {\mathrel{\downarrow'}}{q^{f_1}}\, ({\mathsf{just}}\, p)))$\
- $q^f : \Pi {\{s : S'\}}.\, P\, (t^f\, s) \to P'\, s$\
$q^f\, {\mathsf{nothing}}= {\mathsf{o'}}$\
$q^f\, ({\mathsf{just}}\,({\mathsf{inl}}\,p)) = {\overline{{q^{f_0}}}}\,p$\
$q^f\, ({\mathsf{just}}\,({\mathsf{inr}}\,p)) = {\overline{{q^{f_1}}}}\,p$\
where\
${\overline{{q^{f_0}}}}: \Pi {\{s : S'\}}.\, {\overline{P_0^+}}\, ({\overline{{t^{f_0}}}}\, s) \to P'\, s$\
${\overline{{q^{f_1}}}}: \Pi {\{s : S'\}}.\, {\overline{P_1^+}}\, ({\overline{{t^{f_1}}}}\, s) \to P'\, s$\
(by mutual recursion)\
${\overline{{q^{f_0}}}}\, (p_0 , {\mathsf{nothing}}) = {q^{f_0}}\,({\mathsf{just}}\,p_0)$\
${\overline{{q^{f_0}}}}\, (p_0 , {\mathsf{just}}\, p_1) = {q^{f_0}}\,({\mathsf{just}}\, p_0) {\mathrel{\oplus'}}{\overline{{q^{f_1}}}}p_1$\
${\overline{{q^{f_1}}}}\, (p_1 , {\mathsf{nothing}}) = {q^{f_1}}\,({\mathsf{just}}\,p_1)$\
${\overline{{q^{f_1}}}}\, (p_1 , {\mathsf{just}}\, p_1) = {q^{f_1}}\,({\mathsf{just}}\, p_1) {\mathrel{\oplus'}}{\overline{{q^{f_0}}}}p_0$
The container morphisms $\pi_0$, $\pi_1$ are directed container morphisms.[^4]
We give the proof only for $\pi_0$. The proof for $\pi_1$ is symmetric.
Directed container morphism law 1. Case $p = {\mathsf{nothing}}$: $$\begin{array}{cl}
&t^{\pi_0}\,(s {\mathrel{\downarrow}}(q^{\pi_0}\, {\mathsf{nothing}}))\\
=&\quad\{\textrm{definition of $q^{\pi_0}$} \}\\
&t^{\pi_0}\,(s {\mathrel{\downarrow}}{\mathsf{nothing}})\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow}}$}\}\\
&t^{\pi_0}\,s\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow_0}}$}\}\\
&t^{\pi_0}\,s {\mathrel{\downarrow_0}}{\mathsf{nothing}}\end{array}$$ Case $p = {\mathsf{just}}\, p$: $$\begin{array}{cl}
&t^{\pi_0}\,(((s_0 , v_0) , (s_1 , v_1)) {\mathrel{\downarrow}}q^{\pi_0}\, ({\mathsf{just}}\, p))\\
=&\quad\{\textrm{definition of $q^{\pi_0}$} \}\\
&t^{\pi_0}\,(((s_0 , v_0) , (s_1 , v_1)) {\mathrel{\downarrow}}{\mathsf{just}}\, ({\mathsf{inl}}\, (p , {\mathsf{nothing}})))\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow}}$} \}\\
&t^{\pi_0}\,(((s_0 , v_0) , (s_1 , v_1)) {\mathrel{\downarrow^+}}{\mathsf{inl}}\, (p , {\mathsf{nothing}}))\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow^+}}$} \}\\
&t^{\pi_0}\,((s_0 , v_0) {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p , {\mathsf{nothing}}))\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}$} \}\\
&t^{\pi_0}\,((s_0 {\mathrel{\downarrow_0^+}}p , \lambda p'.\, v_0 \,(p {\mathrel{\oplus_0^+}}p')) , v_0\, p)\\
=&\quad\{\textrm{definition of $t^{\pi_0}$} \}\\
&s_0 {\mathrel{\downarrow_0^+}}p\\
=&\quad\{\textrm{definition of $t^{\pi_0}$} \}\\
&t^{\pi_0}\, ((s_0 , v_0) , (s_1 , v_1)) {\mathrel{\downarrow_0^+}}p\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow_0}}$} \}\\
&t^{\pi_0}\, ((s_0 , v_0) , (s_1 , v_1)) {\mathrel{\downarrow_0}}{\mathsf{just}}\, p
\end{array}$$
Directed container morphism law 2: $$\begin{array}{cl}
&q^{\pi_0}\,{\mathsf{o}_0}\\
=&\quad\{\textrm{definition of ${\mathsf{o}_0}$} \}\\
&q^{\pi_0}\,{\mathsf{nothing}}\\
=&\quad\{\textrm{definition of $q^{\pi_0}$} \}\\
&{\mathsf{nothing}}\\
=&\quad\{\textrm{definition of ${\mathsf{o}}$} \} \\
&{\mathsf{o}}\end{array}$$ Directed container morphism law 3. Case $p = {\mathsf{nothing}}$: $$\begin{array}{cl}
&q^{\pi_0}\,({\mathsf{nothing}}{\mathrel{\oplus_0}}p')\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus_0}}$} \}\\
&q^{\pi_0}\, p'\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$} \}\\
&{\mathsf{nothing}}{\mathrel{\oplus}}q^{\pi_0}\, p'
\end{array}$$ Case $p = {\mathsf{just}}\, p$, $p' = {\mathsf{nothing}}$: $$\begin{array}{cl}
&q^{\pi_0}\,({\mathsf{just}}\, p {\mathrel{\oplus_0}}{\mathsf{nothing}})\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus_0}}$} \}\\
&q^{\pi_0}\, ({\mathsf{just}}\, p)\\
=&\quad\{\textrm{definition of $q^{\pi_0}$} \}\\
&{\mathsf{just}}\, ({\mathsf{inl}}\, (p, {\mathsf{nothing}}))\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$} \}\\
&{\mathsf{just}}\, ({\mathsf{inl}}\, (p, {\mathsf{nothing}})) {\mathrel{\oplus}}{\mathsf{nothing}}\\
=&\quad\{\textrm{definition of $q^{\pi_0}$} \}\\
&q^{\pi_0}\, ({\mathsf{just}}\, p) {\mathrel{\oplus}}{\mathsf{nothing}}\end{array}$$ Case $p = {\mathsf{just}}\, p$, $p' = {\mathsf{just}}\, p'$: $$\begin{array}{cl}
&q^{\pi_0}\,({\mathsf{just}}\, p {\mathrel{\oplus_0}}{\mathsf{just}}\, p')\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus_0}}$}\}\\
&q^{\pi_0}\,({\mathsf{just}}\, (p {\mathrel{\oplus_0^+}}p'))\\
=&\quad\{\textrm{definition of $q^{\pi_0}$} \}\\
&{\mathsf{just}}\, ({\mathsf{inl}}\, (p {\mathrel{\oplus_0^+}}p' , {\mathsf{nothing}}))\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&{\mathsf{just}}\, ({\mathsf{inl}}\, ((p , {\mathsf{nothing}}) {\mathrel{\overline{\oplus_0^+}}}{\mathsf{inl}}\, (p' , {\mathsf{nothing}})))\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus^+}}$}\}\\
&{\mathsf{just}}\, ({\mathsf{inl}}\, (p , {\mathsf{nothing}}) {\mathrel{\oplus^+}}{\mathsf{inl}}\, (p' , {\mathsf{nothing}}))\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&{\mathsf{just}}\, ({\mathsf{inl}}\, (p , {\mathsf{nothing}}) {\mathrel{\oplus}}{\mathsf{just}}\, ({\mathsf{inl}}\, (p' , {\mathsf{nothing}}))\\
=&\quad\{\textrm{definition of $q^{\pi_0}$}\}\\
&q^{\pi_0}\,({\mathsf{just}}\, p) {\mathrel{\oplus}}q^{\pi_0}\, ({\mathsf{just}}\, p'))\rlap{\hbox to 187 pt{\hfill\qEd}}
\end{array}$$
The container morphism $f = t^f \lhd q^f$ is a directed container morphism.
Auxiliary statements $t^f\, (s {\mathrel{\downarrow'}}{\overline{{q^{f_0}}}}\, p) =
{\overline{{t^{f_0}}}}\, s {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}p$ and $t^f\, (s {\mathrel{\downarrow'}}{\overline{{q^{f_1}}}}\, p) =
{\overline{{t^{f_1}}}}\, s {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}p$ for law 1, by mutual induction on the $p$s, showing the cases of the first statement; those of the second one are symmetric. Case $p = (p_0, {\mathsf{nothing}})$. $$\begin{array}{cl}
&t^f\, (s {\mathrel{\downarrow'}}{\overline{{q^{f_0}}}}(p_0 , {\mathsf{nothing}}))\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$} \}\\
&t^f\, (s {\mathrel{\downarrow'}}{q^{f_0}}\, ({\mathsf{just}}\, p_0))\\
=&\quad\{\textrm{definition of $t^f$} \}\\
&({\overline{{t^{f_0}}}}\, (s {\mathrel{\downarrow'}}q^{f_0}\, ({\mathsf{just}}\, p_0)),
{\overline{{t^{f_1}}}}\, (s {\mathrel{\downarrow'}}q^{f_0}\, ({\mathsf{just}}\, p_0)))\\
=&\quad\{\textrm{definition of ${\overline{{t^{f_0}}}}$} \}\\
&((t^{f_0}\, (s {\mathrel{\downarrow'}}q^{f_0}\, ({\mathsf{just}}\, p_0)), \lambda p .\, {\overline{{t^{f_1}}}}\, ((s {\mathrel{\downarrow'}}q^{f_0}\, ({\mathsf{just}}\, p_0)) {\mathrel{\downarrow'}}q^{f_0}\, ({\mathsf{just}}\, p))) ,
{\overline{{t^{f_1}}}}\, (s {\mathrel{\downarrow'}}q^{f_0}\, ({\mathsf{just}}\, p_0)))\\
=&\quad\{\textrm{directed container law 2}\}\\
&((t^{f_0}\, (s {\mathrel{\downarrow'}}q^{f_0}\, ({\mathsf{just}}\, p_0), \lambda p .\, {\overline{{t^{f_1}}}}\, (s {\mathrel{\downarrow'}}(q^{f_0}\, ({\mathsf{just}}\, p_0) {\mathrel{\oplus}}' q^{f_0}\, ({\mathsf{just}}\, p))))) ,
{\overline{{t^{f_1}}}}\, (s {\mathrel{\downarrow}}' q^{f_0}\, ({\mathsf{just}}\, p_0)))\\
=&\quad\{\textrm{directed container morphism laws 1, 3}\}\\
&((t^{f_0}\, s {\mathrel{\downarrow_0}}{\mathsf{just}}\, p_0, \lambda p .\, {\overline{{t^{f_1}}}}\, (s {\mathrel{\downarrow'}}(q^{f_0}\, ({\mathsf{just}}\, p_0 {\mathrel{\oplus_0}}{\mathsf{just}}\, p))))) , {\overline{{t^{f_1}}}}\, (s {\mathrel{\downarrow'}}q^{f_0}\, ({\mathsf{just}}\, p_0)))\\
=&\quad\{\textrm{definitions of ${\mathrel{\downarrow_0}}$, ${\mathrel{\oplus_0}}$} \}\\
&((t^{f_0}\, s {\mathrel{\downarrow_0^+}}p_0, \lambda p .\, {\overline{{t^{f_1}}}}\, (s {\mathrel{\downarrow'}}(q^{f_0}\, ({\mathsf{just}}\, ( p_0 {\mathrel{\oplus_0^+}}p))))), {\overline{{t^{f_1}}}}\, (s {\mathrel{\downarrow'}}q^{f_0}\, ({\mathsf{just}}\, p_0)))\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}$} \}\\
&({t^{f_0}}\, s, \lambda p.\, {\overline{{t^{f_1}}}}\, (s {\mathrel{\downarrow'}}{q^{f_0}}\, ({\mathsf{just}}\, p)))
{\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p_0 , {\mathsf{nothing}})\\
=&\quad\{\textrm{definition of ${\overline{{t^{f_0}}}}$} \}\\
&{\overline{{t^{f_0}}}}\,s {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p_0 , {\mathsf{nothing}})
\end{array}$$ Case $p = (p_0 , {\mathsf{just}}\, p_1)$: $$\begin{array}{cl}
&t^f\, (s {\mathrel{\downarrow'}}{\overline{{q^{f_0}}}}\, (p_0 , {\mathsf{just}}\, p_1))\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$} \}\\
&t^f\, (s {\mathrel{\downarrow'}}(q^{f_0}\, ({\mathsf{just}}\, p_0) {\mathrel{\oplus'}}{\overline{{q^{f_1}}}}\, p_1))\\
=&\quad\{\textrm{directed container law 2 }\}\\
&t^f\, ((s {\mathrel{\downarrow'}}q^{f_0}\, ({\mathsf{just}}\, p_0)) {\mathrel{\downarrow'}}{\overline{{q^{f_1}}}}\, p_1)\\
=&\quad\{\textrm{inductive hypothesis for $p_1$}\}\\
&{\overline{{t^{f_1}}}}\, (s {\mathrel{\downarrow'}}q^{f_0}\, ({\mathsf{just}}\, p_0)) {\mathrel{\overline{{\mathrel{\downarrow}}_1^+}}}p_1\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}$}\}\\
&({t^{f_0}}\, s, \lambda p.\, {\overline{{t^{f_1}}}}\, (s {\mathrel{\downarrow'}}{q^{f_0}}\, ({\mathsf{just}}\, p)))
{\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p_0 , {\mathsf{just}}\, p_1)\\
=&\quad\{\textrm{definition of ${\overline{{t^{f_0}}}}$}\}\\
&{\overline{{t^{f_0}}}}\, s {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p_0 , {\mathsf{just}}\, p_1)
\end{array}$$
Directed container morphism law 1. Case $p = {\mathsf{nothing}}$: $$\begin{array}{cl}
&t^f\, (s {\mathrel{\downarrow'}}q^f\, {\mathsf{nothing}}) \\
=&\quad\{\textrm{definition of $q^f$} \}\\
&t^f\, (s {\mathrel{\downarrow'}}{\mathsf{o'}})\\
=&\quad\{\textrm{directed container law 1}\}\\
&t^f\, s\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow}}$} \}\\
&t^f\, s {\mathrel{\downarrow}}{\mathsf{nothing}}\end{array}$$ Case $p = {\mathsf{just}}\, ({\mathsf{inl}}\, p)$: $$\begin{array}{cl}
&t^f\, (s {\mathrel{\downarrow'}}q^f\, ({\mathsf{just}}\, ({\mathsf{inl}}\, p))) \\
=&\quad\{\textrm{definition of $q^f$} \}\\
& t^f\, (s {\mathrel{\downarrow'}}{\overline{{q^{f_0}}}}\, p)\\
=&\quad\{\textrm{aux. statement}\}\\
&{\overline{{t^{f_0}}}}\, s {\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}p \\
=&\quad\{\textrm{definition of $t^f$, ${\mathrel{\downarrow}}$} \}\\
&t^f\, s {\mathrel{\downarrow}}{\mathsf{just}}\, ({\mathsf{inl}}\, p)
\end{array}$$ Case $p = {\mathsf{just}}\, ({\mathsf{inr}}\, p)$ is symmetric.
Directed container morphism law 2: $$\begin{array}{cl}
&q^f\,{\mathsf{o}}\\
=&\quad\{\textrm{definition of ${\mathsf{o}}$} \}\\
&q^f\,{\mathsf{nothing}}\\
=&\quad\{\textrm{definition of $q^f$} \}\\
&{\mathsf{o'}}\end{array}$$
Auxiliary statements ${\overline{{q^{f_0}}}}\, (p {\mathrel{\overline{\oplus_0^+}}}p') =
{\overline{{q^{f_0}}}}\, p{\mathrel{\oplus'}}q^f\,({\mathsf{just}}\,p')$ and ${\overline{{q^{f_1}}}}\, (p {\mathrel{\overline{\oplus_1^+}}}p') =
{\overline{{q^{f_1}}}}\, p{\mathrel{\oplus'}}q^f\,({\mathsf{just}}\,p')$ for law 3, by mutual induction on the $p$s, showing the cases of the first statement. Case $p = (p_0,
{\mathsf{nothing}})$, $p' = {\mathsf{inl}}\,(p_0', {\mathsf{nothing}})$: $$\begin{array}{cl}
&{\overline{{q^{f_0}}}}\,((p_0, {\mathsf{nothing}}){\mathrel{\overline{\oplus_0^+}}}{\mathsf{inl}}\, (p_0',{\mathsf{nothing}}))\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&{\overline{{q^{f_0}}}}\,(p_0 {\mathrel{\oplus_0^+}}p_0', {\mathsf{nothing}})\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$}\}\\
&{q^{f_0}}\,({\mathsf{just}}\,(p_0{\mathrel{\oplus_0^+}}p_0'))\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus_0}}$}\}\\
&{q^{f_0}}\,({\mathsf{just}}\,p_0{\mathrel{\oplus_0}}{\mathsf{just}}\,p_0')\\
=&\quad\{\textrm{directed container morphism law 3}\}\\
&{q^{f_0}}\,({\mathsf{just}}\,p_0){\mathrel{\oplus'}}{q^{f_0}}\,({\mathsf{just}}\,p_0')\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$}\}\\
&{\overline{{q^{f_0}}}}\,(p_0,\,{\mathsf{nothing}}){\mathrel{\oplus'}}{\overline{{q^{f_0}}}}\,(p_0',\,{\mathsf{nothing}})\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&{\overline{{q^{f_0}}}}\,(p_0,\,{\mathsf{nothing}}){\mathrel{\oplus'}}q^f\,({\mathsf{just}}\,({\mathsf{inl}}\,(p_0',\,{\mathsf{nothing}})))
\end{array}$$ Case $p = (p_0, {\mathsf{nothing}})$, $p' = {\mathsf{inl}}\,(p_0', {\mathsf{just}}\,p_1')$: $$\begin{array}{cl}
&{\overline{{q^{f_0}}}}\,((p_0,\,{\mathsf{nothing}}){\mathrel{\overline{\oplus_0^+}}}{\mathsf{inl}}(p_0',{\mathsf{just}}\,p_1'))\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&{\overline{{q^{f_0}}}}\,(p_0 {\mathrel{\oplus_0^+}}p_0', {\mathsf{just}}\,p_1')\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$}\}\\
&{q^{f_0}}\,({\mathsf{just}}\,(p_0{\mathrel{\oplus_0^+}}p_0')) {\mathrel{\oplus'}}{\overline{{q^{f_1}}}}\,p_1'\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus_0}}$}\}\\
&{q^{f_0}}\,({\mathsf{just}}\,p_0{\mathrel{\oplus_0}}{\mathsf{just}}\,p_0') {\mathrel{\oplus'}}{\overline{{q^{f_1}}}}\,p_1'\\
=&\quad\{\textrm{directed container morphism law 3}\}\\
&({q^{f_0}}\,({\mathsf{just}}\,p_0){\mathrel{\oplus'}}{q^{f_0}}\,({\mathsf{just}}\,p_0')) {\mathrel{\oplus'}}{\overline{{q^{f_1}}}}\,p_1'\\
=&\quad\{\textrm{directed container law 5}\}\\
&{q^{f_0}}\,({\mathsf{just}}\,p_0){\mathrel{\oplus'}}({q^{f_0}}\,({\mathsf{just}}\,p_0') {\mathrel{\oplus'}}{\overline{{q^{f_1}}}}\,p_1')\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$}\}\\
&{\overline{{q^{f_0}}}}\,(p_0,\,{\mathsf{nothing}}){\mathrel{\oplus'}}{\overline{{q^{f_0}}}}\,(p_0',\,{\mathsf{just}}\,p_1')\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&{\overline{{q^{f_0}}}}\,(p_0,\,{\mathsf{nothing}}){\mathrel{\oplus'}}q^f\,({\mathsf{just}}\,({\mathsf{inl}}\,(p_0',\,{\mathsf{just}}\,p_1')))
\end{array}$$ Case $p = (p_0, {\mathsf{nothing}})$, $p' = {\mathsf{inr}}\,p'$: $$\begin{array}{cl}
&{\overline{{q^{f_0}}}}\,((p_0,\,{\mathsf{nothing}}){\mathrel{\overline{\oplus_0^+}}}{\mathsf{inr}}\,p')\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&{\overline{{q^{f_0}}}}\,(p_0,\,{\mathsf{just}}\,p')\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$}\}\\
&{q^{f_0}}\,({\mathsf{just}}\,p_0){\mathrel{\oplus'}}{\overline{{q^{f_1}}}}\,p'\\
=&\quad\{\textrm{definitions of ${\overline{{q^{f_0}}}}$ and $q^f$}\}\\
&{\overline{{q^{f_0}}}}\,(p_0,\,{\mathsf{nothing}}){\mathrel{\oplus'}}q^f\,({\mathsf{just}}\,({\mathsf{inr}}\,p'))
\end{array}$$ Case $p = (p_0, {\mathsf{just}}\, p_1)$: $$\begin{array}{cl}
&{\overline{{q^{f_0}}}}\,((p_0,\,{\mathsf{just}}\,p_1) {\mathrel{\overline{\oplus_0^+}}}p')\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&{\overline{{q^{f_0}}}}\,(p_0,\,{\mathsf{just}}\,(p_1 {\mathrel{\overline{\oplus_1^+}}}p'))\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$}\}\\
&{q^{f_0}}\,({\mathsf{just}}\,p_0){\mathrel{\oplus'}}{\overline{{q^{f_1}}}}\,(p_1{\mathrel{\overline{\oplus_1^+}}}p')\\
=&\quad\{\textrm{inductive hypothesis for $p_1$}\}\\
&{q^{f_0}}\,({\mathsf{just}}\,p_0){\mathrel{\oplus'}}({\overline{{q^{f_1}}}}\,p_1{\mathrel{\oplus'}}q^f\,({\mathsf{just}}\,p'))\\
=&\quad\{\textrm{directed container law 5}\}\\
&({q^{f_0}}\,({\mathsf{just}}\,p_0){\mathrel{\oplus'}}{\overline{{q^{f_1}}}}\,p_1){\mathrel{\oplus'}}q^f\,({\mathsf{just}}\,p')\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$}\}\\
&{\overline{{q^{f_0}}}}\,(p_0,\,{\mathsf{just}}\,p_1){\mathrel{\oplus'}}q^f\,({\mathsf{just}}\,p')
\end{array}$$
Directed container law 3. Case $p = {\mathsf{nothing}}$: $$\begin{array}{cl}
&q^f\,({\mathsf{nothing}}{\mathrel{\oplus}}p')\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$ }\}\\
&q^f\, p'\\
=&\quad\{\textrm{directed container law 4}\}\\
&{\mathsf{o'}}{\mathrel{\oplus'}}q^f\, p'\\
=&\quad\{\textrm{definition of $q^f$} \}\\
&q^f\, {\mathsf{nothing}}{\mathrel{\oplus'}}q^f\, p'\\
\end{array}$$ Case $p= {\mathsf{just}}\, p$, $p' = {\mathsf{nothing}}$: $$\begin{array}{cl}
&q^f\,({\mathsf{just}}\, p {\mathrel{\oplus}}{\mathsf{nothing}})\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$} \}\\
&q^f\, ({\mathsf{just}}\, p)\\
=&\quad\{\textrm{directed container law 3}\}\\
&q^f\, ({\mathsf{just}}\, p) {\mathrel{\oplus'}}{\mathsf{o'}}\\
=&\quad\{\textrm{definition of $q^f$} \}\\
&q^f\, ({\mathsf{just}}\, p) {\mathrel{\oplus'}}q^f\, {\mathsf{nothing}}\end{array}$$ Case $p = {\mathsf{just}}\, ({\mathsf{inl}}\, p)$, $p' = {\mathsf{just}}\, p'$: $$\begin{array}{cl}
&q^f\,({\mathsf{just}}\, ({\mathsf{inl}}\, p) {\mathrel{\oplus}}{\mathsf{just}}\, p')\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$ }\}\\
&q^f\,({\mathsf{just}}\, ({\mathsf{inl}}\, p {\mathrel{\oplus^+}}p'))\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus^+}}$ }\}\\
&q^f\,({\mathsf{just}}\, ({\mathsf{inl}}\, (p {\mathrel{\oplus_0^+}}p')))\\
=&\quad\{\textrm{definition of $q^f$} \}\\
&{\overline{{q^{f_0}}}}\, (p {\mathrel{\oplus}}p')\\
=&\quad\{\textrm{first aux. statement}\}\\
& {\overline{{q^{f_0}}}}\, p{\mathrel{\oplus'}}q^f\,({\mathsf{just}}\,p')\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$ }\}\\
&q^f\, ({\mathsf{just}}\, ({\mathsf{inl}}\, p)) {\mathrel{\oplus'}}q^f\, ({\mathsf{just}}\,p')
\end{array}$$ Case $p = {\mathsf{just}}\, ({\mathsf{inr}}\, p)$, $p' = {\mathsf{just}}\, p'$ is symmetric.
The product triangles $\eqref{diag:products}$ commute, i.e., $\pi_0
{\circ}f = f_0$ and $\pi_1 {\circ}f = f_1$.
We verify only the left triangle $\pi_0 {\circ}f = f_0$. The right triangle is symmetric.
Statement for shapes: $$\begin{array}{cl}
&t^{\pi_0}\,(t^f\,s)\\
=&\quad\{\textrm{definition of $t^{\pi_0}$} \}\\
&{\mathsf{fst}}\, ({\mathsf{fst}}\, (t^f\, s)) \\
=&\quad\{\textrm{definition of $t^f$} \}\\
&{\mathsf{fst}}\, ({\overline{{t^{f_0}}}}\, s)\\
=&\quad\{\textrm{definition of ${\overline{{t^{f_0}}}}$} \}\\
&t^{f_0}\,s
\end{array}$$
Statement for positions. Case $p = {\mathsf{nothing}}$: $$\begin{array}{cl}
&q^f(q^{\pi_0}\,{\mathsf{nothing}})\\
=&\quad\{\textrm{definition of $q^{\pi_0}$} \}\\
&q^f{\mathsf{nothing}}\\
=&\quad\{\textrm{definition of $q^f$} \}\\
&{\mathsf{o'}}\\
=&\quad\{\textrm{directed container morphism law 2} \}\\
&q^{f_0}\,{\mathsf{o}}_0\\
=&\quad\{\textrm{definition of ${\mathsf{o}}_0$} \}\\
&q^{f_0}\,{\mathsf{nothing}}\end{array}$$ Case $p = {\mathsf{just}}\,p$: $$\begin{array}{cl}
&q^f(q^{\pi_0}\,({\mathsf{just}}\,p))\\
=&\quad\{\textrm{definition of $q^{\pi_0}$} \}\\
&q^f\,({\mathsf{just}}\,({\mathsf{inl}}\,(p,\,{\mathsf{nothing}})))\\
=&\quad\{\textrm{definition of $q^f$} \}\\
&{\overline{{q^{f_0}}}}\, (p, {\mathsf{nothing}})\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$} \}\\
&q^{f_0}\,({\mathsf{just}}\,p)\rlap{\hbox to 217 pt{\hfill\qEd}}
\end{array}$$
The directed container morphism $f = t^f \lhd q^f$ is unique, i.e., if there is a directed container morphism $h =
t^h\lhd q^h : E' \to E$ such that $\pi_0 {\circ}h = f_0$ and $\pi_1
{\circ}h = f_1$, then $f = h$.
Auxiliary statements ${\overline{{t^{f_0}}}}\, s = {\mathsf{fst}}\, (t^h\,
s)$ and ${\overline{{t^{f_1}}}}\, s = {\mathsf{snd}}\, (t^h\, s)$ for shapes, by mutual coinduction, showing only the case of the first statement. $$\begin{array}{cl}
&{\overline{{t^{f_0}}}}\,s\\
=&\quad\{\textrm{definition of ${\overline{{t^{f_1}}}}$}\}\\
&(t^{f_0}\,s,\lambda p_0.\,{\overline{{t^{f_1}}}}\,(s{\mathrel{\downarrow'}}\,q^{f_0}\,({\mathsf{just}}\,p_0)))\\
=&\quad\{\textrm{assumption}\}\\
&(t^{\pi_0}\,(t^h\,s), \lambda p_0.\,{\overline{{t^{f_1}}}}\,(s{\mathrel{\downarrow'}}\,q^{f_0}\,({\mathsf{just}}\,p_0)))\\
=&\quad\{\textrm{coinductive hypothesis}\}\\
&(t^{\pi_0}\,(t^h\,s),\lambda p_0.\,{\mathsf{snd}}\,(t^h\,(s{\mathrel{\downarrow'}}\,q^h\,(q^{\pi_0}\,({\mathsf{just}}\,p_0))))\\
=&\quad\{\textrm{directed container morphism law 1}\}\\
&(t^{\pi_0}\,(t^h\,s),\lambda p_0.\,{\mathsf{snd}}\,(t^h\,s {\mathrel{\downarrow}}\, q^{\pi_0}\,({\mathsf{just}}\,p_0)))\\
=&\quad\{\textrm{definition of $q^{\pi_0}$}\}\\
&(t^{\pi_0}\,(t^h\,s),\lambda p_0.\,{\mathsf{snd}}\,(t^h\,s{\mathrel{\downarrow}}\,{\mathsf{just}}\,({\mathsf{inl}}\,(p_0,{\mathsf{nothing}}))))\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow}}$} \}\\
&(t^{\pi_0}\,(t^h\,s),\lambda p_0.\,{\mathsf{snd}}\,(t^h\,s{\mathrel{\downarrow^+}}{\mathsf{inl}}\,(p_0,{\mathsf{nothing}})))\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow^+}}$} \}\\
&(t^{\pi_0}\,(t^h\,s),\lambda p_0.\,{\mathsf{snd}}\,({\mathsf{fst}}\,(t^h\,s){\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}(p_0,{\mathsf{nothing}})))\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{{\mathrel{\downarrow}}_0^+}}}$} \}\\
&(t^{\pi_0}\,(t^h\,s),\lambda p_0.\,{\mathsf{snd}}\,({\mathsf{fst}}\,(t^h\,s))\,p_0)\\
=&\quad\{\textrm{definition of $t^{\pi_0}$}\}\\
&{\mathsf{fst}}\, (t^h\,s)
\end{array}$$
Statement for shapes, i.e., $t^f = t^h$: $$\begin{array}{cl}
& t^f\, s\\
=&\quad\{\textrm{definition of $t^f$}\}\\
& ({\overline{{t^{f_0}}}}\, s, {\overline{{t^{f_1}}}}\, s)\\
=&\quad\{\textrm{aux. statements}\}\\
& t^h\, s
\end{array}$$
Auxiliary statements ${\overline{{q^{f_0}}}}\, p = q^h\, ({\mathsf{just}}\,
({\mathsf{inl}}\, p))$ and ${\overline{{q^{f_1}}}}\, p = q^h\, ({\mathsf{just}}\, ({\mathsf{inr}}\, p))$ for positions, by mutual induction on the $p$s, showing the cases of the first statement. Case $p = (p_0,{\mathsf{nothing}})$: $$\begin{array}{cl}
&{\overline{{q^{f_0}}}}\,(p_0,{\mathsf{nothing}})))\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$}\}\\
&q^{f_0}\,({\mathsf{just}}\,p_0)\\
=&\quad\{\textrm{assumption}\}\\
&q^h\,(q^{\pi_0}\,({\mathsf{just}}\,p_0))\\
=&\quad\{\textrm{definition of $q^{\pi_0}$}\}\\
&q^h\,({\mathsf{just}}\,({\mathsf{inl}}\,(p_0,{\mathsf{nothing}})))\\
\end{array}$$ Case $p = (p_0,{\mathsf{just}}\,p_1)$: $$\begin{array}{cl}
& {\overline{{q^{f_0}}}}\, (p_0,{\mathsf{just}}\,p_1)))\\
=&\quad\{\textrm{definition of ${\overline{{q^{f_0}}}}$}\}\\
&q^{f_0}\,({\mathsf{just}}\,p_0){\mathrel{\oplus'}}{\overline{{q^{f_1}}}}\, p_1\\
=&\quad\{\textrm{assumption}\}\\
&q^h\,(q^{\pi_0}\, ({\mathsf{just}}\,p_0))){\mathrel{\oplus'}}{\overline{{q^{f_1}}}}\, p_1\\
=&\quad\{\textrm{inductive hypothesis for $p_1$}\}\\
&q^h\,(q^{\pi_0}\, ({\mathsf{just}}\,p_0))){\mathrel{\oplus'}}q^h\,({\mathsf{just}}\,({\mathsf{inr}}\,p_1))\\
=&\quad\{\textrm{definition of $q^{\pi_0}$}\}\\
&q^h\,({\mathsf{just}}\,({\mathsf{inl}}\,(p_0,{\mathsf{nothing}}))){\mathrel{\oplus'}}q^h\,({\mathsf{just}}\,({\mathsf{inr}}\,p_1))\\
=&\quad\{\textrm{directed container morphism law 3}\}\\
&q^h\,({\mathsf{just}}\,({\mathsf{inl}}\,(p_0,{\mathsf{nothing}})){\mathrel{\oplus}}{\mathsf{just}}\,({\mathsf{inr}}\,p_1))\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&q^h\,({\mathsf{just}}\,({\mathsf{inl}}\,(p_0,{\mathsf{nothing}}){\mathrel{\oplus^+}}{\mathsf{inr}}\,p_1))\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus^+}}$}\}\\
&q^h\,({\mathsf{just}}\,({\mathsf{inl}}\, ((p_0,{\mathsf{nothing}}){\mathrel{\overline{\oplus_0^+}}}{\mathsf{inr}}\,p_1)))\\
=&\quad\{\textrm{definition of ${\mathrel{\overline{\oplus_0^+}}}$}\}\\
&q^h\,({\mathsf{just}}\,({\mathsf{inl}}\,(p_0,{\mathsf{just}}\,p_1)))
\end{array}$$
Statement for positions, i.e., $q^f = q^h$. Case $p =
{\mathsf{nothing}}$: $$\begin{array}{cl}
&q^f\,{\mathsf{nothing}}\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&{\mathsf{o'}}\\
=&\quad\{\textrm{directed container morphism law 2}\}\\
&q^h\, {\mathsf{o}}\\
=&\quad\{\textrm{definition of ${\mathsf{o}}$}\}\\
&q^h\,{\mathsf{nothing}}\end{array}$$ Case $p = {\mathsf{just}}\, ({\mathsf{inl}}\, p)$: $$\begin{array}{cl}
&q^f\,({\mathsf{just}}\,({\mathsf{inl}}\,p)\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&{\overline{{q^{f_0}}}}\, p\\
=&\quad\{\textrm{first aux. statement}\}\\
&q^h\,({\mathsf{just}}\,({\mathsf{inl}}\,p))
\end{array}$$ Case $p = {\mathsf{just}}\, ({\mathsf{inr}}\, p)$ is symmetric.
Proof of Proposition \[prop:cofree\] {#proof-of-propositionpropcofree .unnumbered}
------------------------------------
We must prove that $E = (S \lhd P, {\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}})$ is a cofree directed container on the container $C_0 = S_0 \lhd P_0$.
The data ${\mathrel{\downarrow}}$, ${\mathsf{o}}$, ${\mathrel{\oplus}}$ provide a directed container structure on the container $C = S \lhd P$.
Directed container law 1: $$\begin{array}{cl}
&(s,v ) {\mathrel{\downarrow}}{\mathsf{o}}\, {\{s,v\}}\\
=&\quad\{\textrm{definition of ${\mathsf{o}}$}\}\\
&(s,v) {\mathrel{\downarrow}}({\mathsf{inl}}\, {{\ast}})\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow}}$}\}\\
&(s, v)
\end{array}$$
Directed container law 2 by induction on $p$. Case $p={\mathsf{inl}}\, {{\ast}}$: $$\begin{array}{cl}
&(s, v) {\mathrel{\downarrow}}({\mathsf{inl}}\, {{\ast}}{\mathrel{\oplus}}p')\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&(s, v) {\mathrel{\downarrow}}p'\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow}}$}\}\\
&((s, v) {\mathrel{\downarrow}}{\mathsf{inl}}\, {{\ast}}) {\mathrel{\downarrow}}p'
\end{array}$$ Case $p={\mathsf{inr}}\, (p , p')$: $$\begin{array}{cl}
&(s,v) {\mathrel{\downarrow}}({\mathsf{inr}}\, (p , p') {\mathrel{\oplus}}p'')\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&(s,v) {\mathrel{\downarrow}}({\mathsf{inr}}\, (p , p'{\mathrel{\oplus}}p''))\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow}}$}\}\\
&v\, p {\mathrel{\downarrow}}(p' {\mathrel{\oplus}}p'')\\
=&\quad\{\textrm{inductive hypothesis for $p'$}\}\\
&(v\, p {\mathrel{\downarrow}}p') {\mathrel{\downarrow}}p''\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow}}$}\}\\
&((s,v) {\mathrel{\downarrow}}{\mathsf{inr}}\, (p , p')) {\mathrel{\downarrow}}p''
\end{array}$$
Directed container law 3 by induction on $p$. Case $p={\mathsf{inl}}\, {{\ast}}$: $$\begin{array}{cl}
&{\mathsf{inl}}\, {{\ast}}{\mathrel{\oplus}}{\mathsf{o}}\, {\{(s,v) {\mathrel{\downarrow}}{\mathsf{inl}}\, {{\ast}}\}}\\
=&\quad\{\textrm{definitions of ${\mathrel{\oplus}}$, ${\mathrel{\downarrow}}$}\}\\
&{\mathsf{o}}\, {\{s,v\}}\\
=&\quad\{\textrm{definition of ${\mathsf{o}}$}\}\\
&{\mathsf{inl}}\, {{\ast}}\end{array}$$ Case $p={\mathsf{inr}}\, (p , p')$: $$\begin{array}{cl}
&{\mathsf{inr}}\, (p , p') {\mathrel{\oplus}}{\mathsf{o}}\, {\{(s, v) {\mathrel{\downarrow}}{\mathsf{inr}}\, (p , p')\}}\\
=&\quad\{\textrm{definitions of ${\mathrel{\oplus}}$, ${\mathrel{\downarrow}}$}\}\\
&{\mathsf{inr}}\, (p , p' {\mathrel{\oplus}}{\mathsf{o}}\, {\{v\, p {\mathrel{\downarrow}}p'\}})\\
=&\quad\{\textrm{inductive hypothesis for $p'$}\}\\
&{\mathsf{inr}}\, (p , p')
\end{array}$$
Directed container law 4. $$\begin{array}{cl}
&{\mathsf{o}}\, {\{s, v\}} {\mathrel{\oplus}}p\\
=&\quad\{\textrm{definition of ${\mathsf{o}}$} \}\\
&{\mathsf{inl}}\, {{\ast}}{\mathrel{\oplus}}p\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&p
\end{array}$$
Directed container law 5 by induction on $p$. Case $p={\mathsf{inl}}\, {{\ast}}$: $$\begin{array}{cl}
&({\mathsf{inl}}\, {{\ast}}{\mathrel{\oplus}}p') {\mathrel{\oplus}}p''\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&p' {\mathrel{\oplus}}p''\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$} \}\\
&{\mathsf{inl}}\, {{\ast}}{\mathrel{\oplus}}(p' {\mathrel{\oplus}}p'')
\end{array}$$ Case $p={\mathsf{inr}}\, (p , p')$: $$\begin{array}{cl}
&({\mathsf{inr}}\, (p , p') {\mathrel{\oplus}}p'') {\mathrel{\oplus}}p'''\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&{\mathsf{inr}}\, (p , p' {\mathrel{\oplus}}p'') {\mathrel{\oplus}}p'''\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&{\mathsf{inr}}\, (p , (p' {\mathrel{\oplus}}p'') {\mathrel{\oplus}}p''')\\
=&\quad\{\textrm{inductive hypothesis for $p'$} \}\\
&{\mathsf{inr}}\, (p , p' {\mathrel{\oplus}}(p'' {\mathrel{\oplus}}p'''))\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&{\mathsf{inr}}\, (p , p') {\mathrel{\oplus}}(p'' {\mathrel{\oplus}}p''')\rlap{\hbox to 179 pt{\hfill\qEd}}
\end{array}$$
That the directed container $E = (S \lhd P, {\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}})$ is cofree on the container $C_0 = S_0 \lhd P_0$ can be shown either directly or by proving that it interprets into a cofree comonad on ${\llbracket C_0 \rrbracket^\mathrm{c}}$. In the following, we illustrate the first route. This involves a fair amount of straightforward, but tedious inductive and coinductive reasoning in the lemmas below.
For the directed container $E$ to be cofree on the container $C_0$, there must be a container morphism ${\pi}= t^{{\pi}}\lhd q^{{\pi}} : S
\lhd P \to S_0 \lhd P_0$. This is defined by
- $t^{{\pi}} : S \to S_0$\
$t^{{\pi}}\,(s, v) = s$
- $q^{{\pi}} : \Pi {\{(s, v) : S\}}.\,
$P\_0(t\^(s, v))P(s, v)$\\
P_0\, s \to P\,(s, v)$\
$q^{{\pi}}\,p = {\mathsf{inr}}\,(p, {\mathsf{inl}}\,{{\ast}})$
The universal property of cofreeness states that, for any other directed container $E' = (S' \lhd P', {\mathrel{\downarrow'}}, {\mathsf{o'}}, {\mathrel{\oplus'}})$ and container morphism $f_0 = t^{f_0} \lhd q^{f_0} : S' \lhd P' \to S_0 \lhd P_0 $, there must exist a unique directed container morphism $f = t^f \lhd q^f : E' \to
E$ such that the following triangle commutes: $$\tag{$\ddag$}
\label{diag:cofreeness}
\begin{gathered}
\xymatrix@C=3.5em@R=3em{
S' \lhd P'\ar[d]_{f} \ar[rd]^{f_0} & \\
S \lhd P\ar[r]^{{\pi}}&S_0 \lhd P_0
}
\end{gathered}$$ We claim that this directed container morphism $f$ is given by
- $t^f : S' \to S$\
(by corecursion)\
$t^f\,s = (t^{f_0}\,s, \lambda p.\,t^f\,(s{\mathrel{\downarrow'}}q^{f_0}\,p))$
- $q^f : \Pi {\{s : S'\}}.\,P\,(t^{f_0}\,s, \lambda p.\,t^f\,(s{\mathrel{\downarrow'}}q^{f_0}\,p)) \to P'\,s$\
(by recursion)\
$q^f\, ({\mathsf{inl}}\,{{\ast}}) = {\mathsf{o'}}$\
$q^f\, ({\mathsf{inr}}\,(p, p')) = q^{f_0} p {\mathrel{\oplus'}}q^f\,p$
and prove it with the lemmas below.
The container morphism $f$ is a directed container morphism.
Directed container morphism law 1 by induction on $p$. Case $p
= {\mathsf{inl}}\,{{\ast}}$: $$\begin{array}{cl}
&t^f\, (s {\mathrel{\downarrow'}}q^f\, ({\mathsf{inl}}\,{{\ast}}))\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&t^f\, (s {\mathrel{\downarrow'}}{\mathsf{o'}})\\
=&\quad\{\textrm{directed container law 1}\}\\
&t^f\,s\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow}}$}\}\\
&t^f\,s{\mathrel{\downarrow}}{\mathsf{inl}}\, {{\ast}}\\
\end{array}$$ Case $p = {\mathsf{inr}}\,(p, p')$: $$\begin{array}{cl}
&t^f\, (s{\mathrel{\downarrow'}}q^f\,({\mathsf{inr}}\,(p, p')))\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&t^f\, (s{\mathrel{\downarrow'}}(q^{f_0}\, p {\mathrel{\oplus'}}q^f\, p))\\
=&\quad\{\textrm{directed container law 2}\}\\
&t^f\, ((s{\mathrel{\downarrow'}}q^{f_0}\, p) {\mathrel{\downarrow'}}q^f\, p')\\
=&\quad\{\textrm{inductive hypothesis for $p'$}\}\\
&t^f\, (s{\mathrel{\downarrow'}}q^{f_0}\,p){\mathrel{\downarrow}}p'\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow}}$}\}\\
&(t^{f_0}\,s, \lambda p.\,t^f\, (s{\mathrel{\downarrow'}}q^{f_0}\,p)){\mathrel{\downarrow}}{\mathsf{inr}}\,(p, p')\\
=&\quad\{\textrm{definition of $t^f$}\}\\
&t^f\,s{\mathrel{\downarrow}}{\mathsf{inr}}\,(p, p')
\end{array}$$
Directed container morphism law 2. $$\begin{array}{cl}
&q^f\,{\mathsf{o}}\\
=&\quad\{\textrm{definition of ${\mathsf{o}}$}\}\\
&q^f\,({\mathsf{inl}}\, {{\ast}})\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&{\mathsf{o'}}\end{array}$$
Directed container morphism law 3 by induction on $p$. Case $p =
{\mathsf{inl}}\,{{\ast}}$: $$\begin{array}{cl}
&q^f\,({\mathsf{inl}}\, {{\ast}}{\mathrel{\oplus}}p')\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&q^f p'\\
=&\quad\{\textrm{directed container law 4}\} \\
&{\mathsf{o'}}{\mathrel{\oplus'}}q^f\,p'\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&q^f\,({\mathsf{inl}}\,{{\ast}}){\mathrel{\oplus'}}q^f\,p'
\end{array}$$ Case $p = {\mathsf{inr}}\, (p, p')$: $$\begin{array}{cl}
&q^f\,({\mathsf{inr}}\, (p, p'){\mathrel{\oplus}}p'')\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&q^f\,({\mathsf{inr}}\,(p, p'{\mathrel{\oplus'}}p''))\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&q^{f_0}\,p {\mathrel{\oplus'}}q^f(p' {\mathrel{\oplus'}}p'')\\
=&\quad\{\textrm{inductive hypothesis for $p'$}\}\\
&q^{f_0}\,p{\mathrel{\oplus'}}(q^f\,p' {\mathrel{\oplus'}}q^f\,p'')\\
=&\quad\{\textrm{directed container law 5}\}\\
&(q^{f_0}\,p {\mathrel{\oplus'}}q^f\,p') {\mathrel{\oplus'}}q^f\,p''\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&q^f\,({\mathsf{inr}}\, (p, p')) {\mathrel{\oplus'}}q^f\,p''\rlap{\hbox to 179 pt{\hfill\qEd}}
\end{array}$$
The cofreeness triangle commutes, i.e., ${\pi}{\circ}f = f_0$.
Statement for shapes, i.e., $t^{{\pi}}{\circ}t^f = t^{f_0}$: $$\begin{array}{cl}
&t^{{\pi}}\,(t^f\,s)\\
=&\quad\{\textrm{definition of $t^f$}\}\\
&t^{{\pi}}\,(t^{f_0}\,s,\lambda p.\,t^f\,(s{\mathrel{\downarrow'}}q^{f_0}\,p))\\
=&\quad\{\textrm{definition of $t^{{\pi}}$}\}\\
&t^{f_0}\,s
\end{array}$$ Statement for positions, i.e., $q^f {\circ}q^{{\pi}} = q^{f_0}$: $$\begin{array}{cl}
&q^f\,(q^{{\pi}}\,p)\\
=&\quad\{\textrm{definition of $q^{{\pi}}$}\}\\
&q^f\,({\mathsf{inr}}\,(p, {\mathsf{inl}}\,{{\ast}}))\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&q^{f_0}\,p{\mathrel{\oplus'}}q^f\,({\mathsf{inl}}\,{{\ast}})\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&q^{f_0}\,p{\mathrel{\oplus'}}{\mathsf{o'}}\\
=&\quad\{\textrm{directed container law 3}\}\\
&q^{f_0}\,p\rlap{\hbox to 254 pt{\hfill\qEd}}
\end{array}$$
The directed container morphism $f$ is unique, i.e., if there is a directed container morphism $h = t^h \lhd q^h: E' \to E$ such that ${\pi}{\circ}h = f_0$, then $f = h$.
Statement for shapes, i.e., $t^f = t^h$, by coinduction. $$\begin{array}{cl}
&t^f\,s\\
=&\quad\{\textrm{definition of $t^f$}\}\\
&(t^{f_0}\,s, \lambda p.\,t^f\,(s{\mathrel{\downarrow'}}q^{f_0}\, p))\\
=&\quad\{\textrm{coinductive hypothesis}\}\\
&(t^{f_0}\,s, \lambda p.\,t^h\,(s{\mathrel{\downarrow'}}q^{f_0}\, p))\\
=&\quad\{\textrm{assumption, i.e., $t^{{\pi}}{\circ}t^h = t^{f_0}$ and $q^h{\circ}q^{{\pi}} = q^{f_0}$}\}\\
&(t^{\pi}\,(t^h\,s), \lambda p.\,t^h\,(s{\mathrel{\downarrow'}}q^h\, (q^{{\pi}}\, p)))\\
=&\quad\{\textrm{directed container morphism law 1}\}\\
&(t^{\pi}\,(t^h\,s), \lambda p.\,t^h\,s{\mathrel{\downarrow}}q^{{\pi}} p)\\
=&\quad\{\textrm{definitions of $t^{{\pi}}$, $q^{{\pi}}$}\}\\
&({\mathsf{fst}}\,(t^h\,s), \lambda p.\,t^h\,s{\mathrel{\downarrow}}{\mathsf{inr}}\, (p, {\mathsf{inl}}\,{{\ast}}))\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow}}$}\}\\
&({\mathsf{fst}}\,(t^h\,s), \lambda p.\,{\mathsf{snd}}\,(t^h\,s)\,p {\mathrel{\downarrow}}{\mathsf{inl}}\,{{\ast}})\\
=&\quad\{\textrm{definition of ${\mathrel{\downarrow}}$}\}\\
&t^h\,s
\end{array}$$
Statement for positions, i.e., $q^f = q^h$, by induction on position $p$. Case $p = {\mathsf{inl}}\,{{\ast}}$: $$\begin{array}{cl}
&q^f\,({\mathsf{inl}}\,{{\ast}})\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&{\mathsf{o}}'\\
=&\quad\{\textrm{directed container morphism law 2}\}\\
&q^h\,{\mathsf{o}}\\
=&\quad\{\textrm{definition of ${\mathsf{o}}$}\}\\
&q^h\,({\mathsf{inl}}\,{{\ast}})
\end{array}$$ Case $p = {\mathsf{inr}}\,(p, p')$: $$\begin{array}{cl}
&q^f\,({\mathsf{inr}}\,(p, p'))\\
=&\quad\{\textrm{definition of $q^f$}\}\\
&q^{f_0}\,p{\mathrel{\oplus'}}q^f\,p'\\
=&\quad\{\textrm{inductive hypothesis for $p'$}\}\\
&q^{f_0}\,p{\mathrel{\oplus'}}q^h\,p'\\
=&\quad\{\textrm{assumption for positions, i.e., $q^h {\circ}q^{{\pi}} = q^{f_0}$}\}\\
&q^h\, (q^{{\pi}}\,p) {\mathrel{\oplus'}}q^h\,p'\\
=&\quad\{\textrm{directed container morphism law 3}\}\\
&q^h\, (q^{{\pi}}\,p {\mathrel{\oplus}}p')\\
=&\quad\{\textrm{definition of $q^{{\pi}}$}\}\\
&q^h\, ({\mathsf{inr}}\,(p,{\mathsf{inl}}\,{{\ast}}){\mathrel{\oplus}}p')\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&q^h\, ({\mathsf{inr}}\,(p,{\mathsf{inl}}\,{{\ast}}{\mathrel{\oplus}}p')\\
=&\quad\{\textrm{definition of ${\mathrel{\oplus}}$}\}\\
&q^h\, ({\mathsf{inr}}\,(p, p'))\rlap{\hbox to 258 pt{\hfill\qEd}}
\end{array}$$
[^1]: The authors were supported by the ERDF funded CoE project EXCS, the Estonian Ministry of Education and Research target-financed theme no. 0140007s12, and the Estonian Science Foundation grants no. 9219 and 9475.
[^2]: The term ‘coideal comonad’ is motivated by $(D^+, {\delta^+})$ being a right comodule of the comonad $(D, {\varepsilon}, {\delta})$. For the same concept, also the term ‘ideal comonad’ has been used.
[^3]: You may notice a small “mismatch” between the definitions of strict directed containers and coideal comonads. We have given ${\mathrel{\oplus^+}}$ the type $\Pi {\{s : S\}}.\, \Pi p:
P^+\, s.\, P^+\, (s {\mathrel{\downarrow^+}}p) \to P^+\, s$ while the ${\delta^+}$ has type $D^+ {\cdot}D \to D^+$, not $D^+ {\cdot}D^+ \to D^+$. The reason is that the first option for the type of ${\delta^+}$ is more general and really the “correct” one for comonads. For comonads whose underlying functors are containers, however, the corresponding type $\Pi {\{s : S\}}.\, \Pi p: P^+\, s.\, P\, (s {\mathrel{\downarrow^+}}p) \to P^+\, s$ buys no additional generality.
[^4]: They are in fact strict directed container morphisms, but we will not prove this here, as we have not defined this concept.
| 0 |
---
abstract: 'In this paper we prove that the spherical subalgebra $eH_{1,\tau}e$ of the double affine Hecke algebra $H_{1,\tau}$ is an integral Cohen-Macaulay algebra isomorphic to the center $Z$ of $H_{1,\tau}$, and $H_{1,\tau}e$ is a Cohen-Macaulay $eH_{1,\tau}e$-module with the property $H_{1,\tau}=End_{eH_{1,\tau}e}(H_{1,\tau}e)$. In the case of the root system $A_{n-1}$ the variety $Spec(Z)$ is smooth and coincides with the completion of the configuration space of the relativistic analog of the trigomonetric Calogero-Moser system. This implies the result of Cherednik that the module $eH_{1,\tau}$ is projective and all irreducible finite dimensional representations of $H_{1,\tau}$ are regular representation of the finite Hecke algebra.'
address: 'Department of Mathematics, MIT, 77, Massachusetts Ave., Cambridge, MA 02139, USA.'
author:
- Alexei Oblomkov
title: ' Double affine Hecke algebras and Calogero-Moser spaces'
---
Introduction {#introduction .unnumbered}
============
Ivan Cherednik in his pioneering paper [@Ch3] introduced the double affine Hecke algebras. These algebras play a crucial role in the proof of Macdonald Conjectures [@Ch4] and are a natural generalization of affine Hecke algebras, which are an object of great importance in representation theory.
In the paper [@EG] Pavel Etingof and Victor Ginzburg studied the rational degeneration of a double affine Hecke algebra. They discovered that in the case when this algebra has a nontrivial center, the spectrum of the center is isomorphic to the so called Calogero-Moser space, and this isomorphism respects the Poisson structure. The Calogero-Moser space first appeared in [@KKS] as a completed configuration space for the Calogero-Moser integrable system. Recently attention to this object was aroused by the paper [@W].
The isomorphism between the spectrum of the center of the degenerate double affine Hecke algebra and the Calogero-Moser space gives an interpretation of the degenerate double affine Hecke algebra as an Azumaya algebra in the case when the Calogero-Moser space is smooth.
In the present paper we study the double affine Hecke algebra $H$ with $q=1$. In this case the algebra has a nontrivial center. We establish a Poisson isomorphism between the spectrum of the center $Z(H)$ of $H$ and a relativistic analog of the Calogero-Moser space in the case of the root system $A_{n-1}$. The relativistic analog of the Calogero-Moser space is a completed configuration space for the so called Ruijsenaars-Shneider (or briefly RS) integrable system [@RS].
For the general algebra $H$ (with $q=1$) we prove that the ring $Z(H)$ has no zero divisors, and that it is a normal, Cohen-Macaulay ring isomorphic to the spherical subalgebra $eHe$ (where $e$ is the symmetrizer in the finite Hecke algebra). We also prove the equality $H=End_{eHe}(He)$, which allows us, in the case of the root system $A_{n-1}$, to interpret $H$ as an Azumaya algebra.
The techniques of the paper work also in the degenerate case and furnish a simpler proof of the results of [@EG]. Furthermore, there exists an intermediate degeneration of the double affine Hecke algebra which lies between the double affine Hecke algebra and the rational degeneration of this algebra. We call this algebra the trigonometric degeneration of the double affine Hecke algebra. The corresponding degeneration of the Calogero-Moser space yields the configuration space for the trigonometric Calogero-Moser system (sometimes this space is called the trigonometric Calogero-Moser space). The results of the paper hold for this intermediate degeneration and are given in the last section.
[**Acknowledgments.**]{} I’d like to thank my adviser Pavel Etingof for posing the problem, multiple explanations and help with proving many statements from the paper. I am especially grateful for the proof of Theorem 2. I am very grateful to Ivan Cherednik who helped me with the proof of Lemma \[lcom\]. I am also grateful to Victor Ostrik for very useful consultations about Hecke algebras and to Yuri Berest for his consultations on the Calogero-Moser space.
Definitions
===========
Definition of the double affine Hecke algebra corresponding to $GL(n,{\mathbb C})$
----------------------------------------------------------------------------------
We denote this algebra by the symbol $H_{q,t}$. It is generated by the elements $T_i$, $1\le i\le n-1$, $\pi$, $X_i^{\pm 1}$, $1\le i\le n$ with relations $$\begin{gathered}
X_iX_j=X_jX_i,\quad (1\le i,j\le n),\label{dbre1}\\ T_i X_i
T_i=X_{i+1},\quad (1\le i<n),\\ T_iX_j=X_jT_i,\mbox{ if } j-i\ne
0,1\\ [T_i,T_j]=0,\mbox{ if } |i-j|>1\\ T_i T_{i+1} T_i=T_{i+1}T_i
T_{i+1}, (1\le i<n),\\ \pi X_i=X_{i+1}\pi \quad (1\le i\le
n-1),\quad \pi X_n=q^{-1}X_1\pi,\\ \pi T_i=T_{i+1}\pi,\quad \pi^n
T_j=T_j\pi^n,\quad (1\le i<n-1, 1\le j<n)
,\label{dbre2}\\(T_i-\tau)(T_i+\tau^{-1})=0,\quad (1\le i\le
n).\label{T^2}\end{gathered}$$
To identify this definition with the standard definition from the papers of Cherednik one should replace $\tau$ by $t^{\frac12}$ and $q$ by $q^{\frac12}$. Also, some definitions use the element $T_0=\pi T_{n-1}\pi^{-1}$.
The double affine Hecke algebra corresponding to $SL(n,{\mathbb C})$ is a quotient of the subalgebra of $H_{q,\tau}$ generated by $X_i/X_{i+1}$, $T_i$, $\pi$, $1\le i\le n-1$, by one extra relation: $$\pi^n=1.$$
Definition of the Calogero-Moser space {#CMsec}
--------------------------------------
Let $E$ be an $n$-dimensional vector space (over ${\mathbb C}$). We denote by the symbol $CM'_\tau$ the subset of $GL(E)\times GL(E) \times
E\times E^*$ consisting of the elements $(X,Y,U,V)$ satisfying the equation $$\label{CMeq}X^{-1}Y^{-1}XY\tau-\tau^{-1}=U\otimes V.$$ Obviously it is an affine variety.
The group $GL(n,\mathbb C)=GL(E)$ acts on it by conjugation: $$(X,Y,U,V)\to (gXg^{-1},gYg^{-1},gU,Vg^{-1}),\quad g\in GL(E).$$ Later we will show that this action is free if $\tau$ is not a root of unity. So the naive quotient by the action (i.e. the spectrum of the ring of $GL(E)$ invariant functions) yields an affine variety, and the quotient is nonsingular if $CM'_\tau$ is.
The quotient of $CM'_\tau$ by the action $GL(E)$ is called the Calogero-Moser space. We use the notation $CM_\tau$ for this space.
Below we always suppose that $\tau$ is not a root of unity.
Properties of the Calogero-Moser space
======================================
The goal of this section is to prove that $CM_\tau$ is a smooth irreducible algebraic variety of dimension $2n$. We also introduce coordinates on its dense subset. The methods of this section are analogous to the ones from the paper [@W]. In principal smoothness of $CM_\tau$ follows from the results of the paper [@FR], the authors of [@FR] use the moduli space of the vector bundles on the punctured torus. For convenience of reader we give a direct elementary proof.
Smoothness of the Calogero-Moser space
--------------------------------------
First we prove a simple lemma on which all the following statements are based.
If $(X,Y,U,V)\in CM'_\tau$ and $[R,X]=[R,Y]=0$, $R\in\mathfrak{gl}(E)$ then $R=\lambda Id$ for some $\lambda\in
\mathbb C$.
Let $W\subset E$ be a nonzero subspace which is invariant under the action of $X$, $Y$ and $R$. We denote by $\bar{X}$ and $\bar{Y}$ the restriction of the operators $X$, $Y$ to this subspace. It follows from equation (\[CMeq\]) that there are two possibilities.
In the first case $W\subset V^\perp$, where $V^\perp$ is the notation for the annihilator. In this case (\[CMeq\]) implies $$\bar{X}^{-1}\bar{Y}^{-1}\bar{X}\bar{Y}=\tau^{-2}Id.$$ But the determinant of LHS is equal to $1$, hence we get a contradiction.
In the second case $W\nsubseteq V^\perp,$ $U\in W$. In this case (\[CMeq\]) implies $$\bar{X}^{-1}\bar{Y}^{-1}\bar{X}\bar{Y}-U\bar{V}=\tau^{-2}Id,$$ where $0\ne \bar{V}$ is the restriction of $V$ to the subspace $W$. Since $det(\bar{X}\bar{Y}\bar{X}^{-1}\bar{Y}^{-1})=1$, the last equation implies that there is a basis in $W$ in which $\bar{X}^{-1}\bar{Y}^{-1}\bar{X}\bar{Y}$ is diagonal with the spectrum $\tau^{-2},\tau^{-2},\dots,\tau^{-2},\tau^{2k}$ where $k=\dim W$. But we know from equation (\[CMeq\]) that the spectrum of $X^{-1}Y^{-1}XY$ is equal to $\tau^{-2},\tau^{-2},
\dots,\tau^{2n}$. Thus we get $W=E$.
The fact that the only common nonzero invariant subspace of $X,Y$ and $R$ is the whole $E$ immediately implies the statement of the lemma. Indeed, let $\lambda$ be an eigenvalue of $R$, then the corresponding eigenspace $W_\lambda$ is invariant under the action of $X$ and $Y$, hence it coincides with $E$.
The action of $GL(E)$ on $CM'_{\tau}$ is free.
$CM'_\tau$ is smooth.
Let us introduce the map $\Psi$: $GL(E)\times GL(E)\times E\times
E^* \to \mathfrak{gl}(E)$: $$\Psi(X,Y,U,V)=X^{-1}Y^{-1}XY-U\otimes
V.$$
It is enough to show that $d\Psi$ is epimorphic at a point $(X,Y,U,V)\in CM'_\tau$. Let $x,y\in\mathfrak{gl}(E)$, $u\in
E,v\in E^*$ and $X(t)=Xe^{xt}$, $Y(t)=Ye^{yt}$, $U(t)=U+tu$, $V(t)=V+tv$. Then $$\begin{gathered}
d\Psi_{(X,Y,U,V)}(x,y,u,v)=
\frac{d}{dt}(\Psi(X(t),Y(t),U(t),V(t))|_{t=0}=\\
-xX^{-1}Y^{-1}XY+X^{-1}Y^{-1}XxY-X^{-1}yY^{-1}XY+
X^{-1}Y^{-1}XYy-U\otimes v-u\otimes V.\end{gathered}$$
If $d\Psi$ is not an epimorphism, then there exists $0\ne
R\in\mathfrak{gl}(E)$ such that $$tr(d\Psi_{(X,Y,U,V)}(x,y,u,v)R)=0$$ for all $x,y\in\mathfrak{gl}(E)$, $u\in E,v\in E^*$. Using the cyclic invariance of the trace, we can rewrite the last condition in the form: $$\begin{gathered}
tr(x(YRX^{-1}Y^{-1}X-X^{-1}Y^{-1}XYR))+\\
tr(y(RX^{-1}Y^{-1}XY-Y^{-1}XYRX^{-1}))-
v(RU)-VR(u)=0.\end{gathered}$$ As the bilinear form $tr(xy)$ is nondegenerate, the last equation implies $$\begin{gathered}
YRX^{-1}Y^{-1}X-X^{-1}Y^{-1}XYR=0,\label{def1}\\
RX^{-1}Y^{-1}XY-Y^{-1}XYRX^{-1}=0,\label{def2}\\ RU=0,\quad
VR=0.\label{def3}\end{gathered}$$ These equations together with equation (\[CMeq\]) imply $[R,X]=[R,Y]=0$. Indeed, let us derive the first equation.
Multiplying on the right formula (\[CMeq\]) by $R$ we get $$\label{multR}
X^{-1}Y^{-1}XYR=\tau^{-2}R.$$ Hence $$\begin{gathered}
\tau^{-2}XRX^{-1}=Y^{-1}XYRX^{-1}=RX^{-1}Y^{-1}XY=\\R(\tau^{-1}U\otimes
V+\tau^{-2} Id)=\tau^{-2}Id,\end{gathered}$$ here the first equation uses (\[multR\]), second (\[def2\]), third (\[CMeq\]) and fourth (\[def3\]).
By the previous lemma $R=\lambda Id$ and finally from (\[def3\]) we get $R=0$.
\[sm2n\] $CM_\tau$ is smooth algebraic variety, and all its irreducible components have dimension $2n$.
Local coordinates on $CM_\tau$ {#coor}
-------------------------------
It is easy to see that matrices $X,Y\in\mathfrak{gl}(n,\mathbb
C)$, $$\begin{gathered}
\label{repf}
X=diag(\lambda_1,\dots,\lambda_n), \\ \label{repm}Y_{ii}=q_i,\quad
i=1,\dots,n,\\ Y_{ij}=\frac{(\tau-\tau^{-1})q_i\lambda_j}
{(\tau\lambda_i-\tau^{-1}\lambda_j)},\quad 1\le i\ne j\le
n,\label{repl}\end{gathered}$$ satisfy the equation $$\label{eqXY-YX} rk(\tau
XY-\tau^{-1}YX)= 1,$$ for all $\lambda\in(\mathbb
C^*)^n\setminus D_\tau$, $q\in(\mathbb{C}^*)^n$ where $$D_\tau=
\{\lambda|\delta_\tau(\lambda)=\prod_{i\ne
j}(\tau\lambda_i-\tau^{-1}\lambda_j)=0\}.$$
There is a well known formula: if $M=(M_{ij})$, where $M_{ij}=(\lambda_i-\mu_j)^{-1},$ $ 1\le i,j\le n$, then $$\det(M)=\frac{\prod_{i<j}(\lambda_i-\lambda_j)(\mu_j-\mu_i)}
{\prod_{i,j}(\lambda_i-\mu_j)}.$$ To prove this formula one can proceed by the induction on $n$ using the Gaussian method of calculation of the determinant for the step of the induction.
Applying the last formula to the matrix $Y$ we see that $\det(Y)$ is nonzero if and only if $\lambda_i\ne\lambda_j$, $i\ne j$.
Let us denote by $\pi'_{12}$: $CM'_\tau\to GL(E)\times
GL(E)$ the projection on the first two coordinates. The previous reasoning shows that $(X,Y)\in \pi_{12}(CM'_\tau)$, for $\lambda\in(\mathbb C^*)^n\setminus (D_\tau\cup D)$, $q\in(\mathbb
C^*)^n$ where $$D= \{\lambda|\delta(\lambda)=\prod_{i< j}
(\lambda_i-\lambda_j)=0\}.$$
Now we can state
\[propcoor\] Let $(X,Y,U,V)\in CM'_\tau$ and $X$ be diagonalizable with the different eigenvalues $\lambda_i$, $i=1,\dots,n$ such that $\tau\lambda_i\ne \tau^{-1}\lambda_j$. Then the $GL(n,\mathbb C)$ orbit of $(X,Y,U,V)$ contains a representative satisfying equations $V=\lambda^t$ and (\[repf\]-\[repl\]) for some $q\in(\mathbb C^*)^n$. Such a representative is unique up to (simultaneous) permutation of the parameters $(\lambda_i,q_i)$.
Equation (\[eqXY-YX\]) is equivalent to the system $$\label{orb}
\frac{(\tau\lambda_i-\tau^{-1}\lambda_j)Y_{ij}}{\tau-\tau^{-1}}=p_is_j,\quad
1\le i,j\le n,$$ if $X=diag(\lambda_1,\dots,\lambda_n).$ If there exists $i$ such that $s_i=0$ then $Y_{ij}=0$, $j=1,\dots,n$ and $\det(Y)=0$. Thus we have $s_i\ne 0$. Analogously we get $p_i\ne 0$.
Let us fix a solution of (\[orb\]) lying in the $GL(n,E)$ orbit of $(X,Y,U,V)$. Putting $q_i=p_is_i/\lambda_i$ we get the desired representative with $X$ given by formula (\[repf\]), $Y$ by formulas (\[repm\]),(\[repl\]) and $U=(\tau-\tau^{-1})X^{-1}Y^{-1}q$.
This proposition together with Corollary \[sm2n\] implies that $(\lambda,q)$ are local coordinates on the open subset $\bold{U}
\subset CM_\tau$. In the next section we show that this subset is dense.
Irreducibility of $CM_\tau$
---------------------------
In this subsection we prove
\[irrCM\] The variety $CM_\tau$ is irreducible.
Let us consider the projection on the first component $\pi'_1$: $CM'_\tau\to GL(E)$. After the taking the quotient by the action of $GL(E)$ this map becomes a map $\pi_1$: $CM_\tau\to JNF$, where $JNF$ is a stack but we can think about it as the set of Jordan normal forms of matrices (we do not need the stack structure).
Inside $JNF$ there is an open part $\tilde{U}$ corresponding to diagonal matrices with eigenvalues $\{\lambda_1,\dots,\lambda_n\}$ such that $\lambda_i\ne\lambda_j$, $\tau\lambda_i\ne\tau^{-1}\lambda_j$ for $i\ne j$. The subset $\pi_1^{-1}(\tilde{U})$ was described in the previous section. It is obviously connected. If we show that $\dim
\pi^{-1}(JNF\setminus \tilde{U})<2n$ then Corollary \[sm2n\] implies the irreducibility.
Let us denote by $J_k(\lambda)$ the Jordan block of size $k$ with the eigenvalue $\lambda$ and by the symbol $J_{\vec{k}}(\lambda)$ the matrix $diag(J_{k^1}(\lambda),\dots,J_{k^t}(\lambda))$, $\vec{k}\in\mathbb N^t$ and $k^i\ge k^{i+1}$, $i=1,\dots,t-1$. Let us formulate without a proof an elementary statement from linear algebra.
The dimension of $$Stab(J_{\vec{k}}(\lambda))=
\{X\in GL(n,\mathbb C)|[X,J_{\vec{k}}(\lambda)]=0\}$$ is equal to $\sum_{1\le i,j\le t} min\{k^i,k^j\}$.
Let us denote by $J_{\bold{k}}(\lambda)$ the matrix $$diag(J_{\vec{k_1}}(\lambda),J_{\vec{k_2}}(\lambda\tau^2),\dots,
J_{\vec{k_r}}(\lambda\tau^{2r})),$$ $\vec{k_i}\in\mathbb N^{t_i}$. We use notations $|\vec{k}_i|=\sum_{j=1}^{t_i} k_i^j$, $|\bold{k}|=\sum_{j=1}^r|\vec{k}_j|$.
Let $\lambda_1,\dots,\lambda_s\in\mathbb C$ be such that $\lambda_i/\lambda_j\ne \tau^{2c}$, $c\in\mathbb Z$, $|c|\le n$ and $$\label{J}
J=
diag(J_{\bold{k}_1}(\lambda_1),\dots,J_{\bold{k}_s}(\lambda_s)).$$
We denote by $\pi'_{34}$: $CM'_\tau\to E\times E^*$ the slightly modified projection on the last two components: $\pi'_{34}(X,Y,U,V)=(YXU,V)$. The fiber of the map $\pi'_{34}$ over the point $(U,V)$ of the subset $\hat{J}=\pi'_{34}((\pi'_1)^{-1}(J))$ consists of the points $(J,Y+F,J^{-1}(Y+F)^{-1}U,V)$ where $F$ is an element of the kernel of the linear map: $$S_J(F)=\tau J F-\tau^{-1} F J,\quad F\in\mathfrak{gl}(E),$$ $Y+F$ is invertible, and $(J,Y,J^{-1}Y^{-1}U,V)\in CM'_\tau$. Obviously $(\pi'_{34})^{-1}(U,V)$ is a Zariski open nonempty subset inside $ker(S_J)$ hence they have the same dimension.
First let us study the map $S_J$ in the simple case when in the equation (\[J\]) we have $s=1$ and $\bold{k}_1=\bold{k}=
(\vec{k}_1,\dots,\vec{k}_r)$, $\vec{k_i}\in\mathbb N^{d_i}$, $1\le
i\le r$. In this situation we denote by $F^{st}_{ij}\in
Mat(k^i_s,k^j_t)$, $1\le s,t\le r$ the matrix with the entries $F^{st}_{ij;pq}=F_{p'q'}$, $p'=\sum_{l=1}^{s-1}|\vec{k_l}|+
\sum_{l=1}^{i-1} k_s^l+p$, $q'=
\sum_{l=1}^{t-1}|\vec{k_l}|+\sum_{l=1}^{j-1} k_s^l+ q$. In these notations the following lemma holds
\[kerSJ\] Let $J$ be the matrix given by (\[J\]) with $s=1$ and $\bold{k}_1=\bold{k}=(\vec{k}_1,\dots,\vec{k}_r)$. Then $F\in
\ker S_J$ if and only if $$\begin{gathered}
F^{st}_{ij}=0, \mbox{ if } t-s\ne 1,\\ F^{s,s+1}_{ij}=
(\sum_{l=0}^{k_s^i-1}
c^{s}_{ij;l}J_{k_s^i}^l(0))D^{k_s^i,k_{s+1}^j}_\tau \mbox{ if }
k_s^i\le k_{s+1}^j\label{ker1},\\ F^{s,s+1}_{ij}=
D^{k_s^i,k_{s+1}^j}_\tau(\sum_{l=0}^{k_{s+1}^j-1}
c^{s}_{ij;l}J_{k_{s+1}^j}^l(0)) \mbox{ if } k_s^i>
k_{s+1}^j\label{ker2},\end{gathered}$$ where $c^{s}_{ij;l}\in \mathbb C$, $J^l_{k_s^i}(0)$ ( and $J^l_{k_{s+1}^j}(0)$) is the $l$-th power of the Jordan block matrix, and $D^{k_s^i,k_{s+1}^j}\in Mat(k_s^i,k_{s+1}^j)$ is given by formula $$\begin{gathered}
D^{k_s^i,k_{s+1}^j}_{\tau;pq}=\delta_{p+k_{s+1}^j,q+k_s^i}\tau^{2-2p}
\mbox{ if } k_s^i\le k_{s+1}^j\\
D^{k_s^i,k_{s+1}^j}_{\tau,pq}=\delta_{p,q}\tau^{2-2p} \mbox{ if }
k_s^i> k_{s+1}^j.\end{gathered}$$
The system of linear equations $S_J(F)=0$ is equivalent to the collection of linear systems: $$\tau J_{k_s^i}(\lambda\tau^{2s-2})
F^{st}_{ij}-\tau^{-1}F^{st}_{ij}
J_{k_t^j}(\lambda\tau^{2t-2})=0,\quad 1\le s,t \le r,$$ because $J$ has a block structure. The equations for the entries of $F^{st}_{ij}$ are of the simple form: $$\label{blok}
F^{st}_{ij;pq}\lambda(\tau^{2s-1}-\tau^{2t-3})=
\tau(\delta_{p,k^s_i}-1)F^{st}_{ij;p+1,q}-
\tau^{-1}(\delta_{q,1}-1)F^{st}_{ij;p,q-1}.$$
First consider the case $t-s\ne 1$. Then $\tau^{2s-1}-\tau^{2t-3}\ne 0$ and equations (\[blok\]) express the entries of the $i$-th diagonal through the entries of the $(i-1)$-th diagonal. It easy to see that in this case (\[blok\]) implies $F^{st}_{ij;k_s^i,1}=0$, that is, the first diagonal is zero. Moving from the left to the right we get that all the diagonals of $F^{st}_{ij}$ are zero.
If $s+1=t$ then equation (\[blok\]) is a linear relation between the neighboring entries on the diagonal. It is easy to derive equations (\[ker1\]), (\[ker2\]) from this fact.
Indeed, let us consider the case $k_s^i\le k_{s+1}^j$. Then equation (\[blok\]) for $p=k_i^s$, $1< q\le k_i^s$ says $F^{s,s+1}_{ij;k_i^s,q-1}=0$. Moving along the diagonal from the bottom to the top and using equation (\[blok\]) we get that the first $k_i^s-1$ diagonals of the matrix $F^{s,s+1}_{ij}$ are zero. For the rest of the diagonals equation (\[blok\]) implies $F^{s,s+1}_{ij;p+1,q+p}=F^{s,s+1}_{ij;1,q}\tau^{-2p}$. Putting $c^s_{ij;l}=F^{s,s+1}_{ij;1,l+k_{s+1}^j-k_s^i+1}$ we get equation (\[ker1\]).
Obviously $Z\in Im S_J$ if and only if $tr(ZF)=0$ for all $F\in
ker \bar{S}_J$, $\bar{S}_J(F)=\tau^{-1} JF-\tau F J$. The space $ker \bar{S}_J$ has a description similar to the one of $ker S_J$ (to get $ker\bar{S}_J$ from $ker S_J$ it is enough to change the order of the Jordan blocks in $J$) and one can easily derive
$Z\in Im S_J$ if and only if following equations hold $$\sum^{u-1}_{l=0}Z_{ij;k_s^i-l,u-l}^{s,s+1}\tau^{2l}=0, \quad
u=1,\dots,min\{k^i_s,k^j_{s+1}\},$$ where $s=1,\dots,r-1$.
The lowest nonzero diagonal of a rank one matrix contains only one nonzero entry. As $\hat{J}\subset Im S_J\cap \{\mbox{ matrices of
rank } 1\}$ the following statement holds
\[Im\] $(U,V)\in \hat{J}=\pi_{34}'((\pi'_1)^{-1}(J))$ if and only if $Z=U\otimes V$ satisfies the equation $$Z_{ij;pq}^{s,s+1}=0 \mbox{ if } p-q\ge
min\{0,k_s^i-k_{s+1}^j\},\quad s=1,\dots,r-1.$$
Lemma \[kerSJ\] gives us the formula for the dimension of the kernel $$\dim \ker S_J=\sum_{s=1}^{r-1}\sum_{i,j}
min\{k^i_s,k^j_{s+1}\}.$$
We know that $GL(n,E)$ acts on $CM'_\tau$ freely. Hence if we want to estimate the dimension of the fiber of $\pi_{34}$ over $\hat{J}$ we should estimate $\dim Stab(J)-\dim ker S_J$. This difference is positive:
\[ineq\] Let $k_s\in \mathbb{N}^{d_s}$, $s=1,\dots,r$, $k^i_s\ge k^{i+1}_{s}$ then the following inequality holds $$\sum_{s=1}^r\sum_{i,j} min\{
k_s^i,k_s^j\}-\sum_{s=1}^{r-1}\sum_{i,j} min\{
k_s^i,k_{s+1}^j\}>0,$$ if there exists $s$ such that $k_s\ne 0$.
Because of the inequality $k^i_s\ge k^{i+1}_s$ we can rewrite LHS of the inequality in the form $$\begin{gathered}
\sum_{\nu=1}\left(\sum_{s=1}^r (x^{\nu}_s)^2- \sum_{s=1}^{r-1}
x^{\nu}_sx^{\nu}_{s+1}\right), \\ x^\nu_s=\#\{i\in\mathbb
N|k_s^i\ge\nu\}.\end{gathered}$$ But the first expression is a sum of positive definite quadratic forms. Thus we get the lemma.
The following statement is crucial for estimating of $\dim(\pi^{-1}_1(JNF\setminus\tilde{U}))$:
If $J$ is given by (\[J\]) with $s=1$ and $\bold{k}_1=
\bold{k}=(\vec{k}_1,\dots,\vec{k}_r)$, then $\dim
\pi_1^{-1}(J)<2n-1$ when either $r>1$ or $k_1^1>1$.
In the case $r>1$ Corollary \[Im\] implies that $\dim
\pi_{34}'((\pi'_1)^{-1}(J))\le 2n-1$. The theorem on the dimension of the fibers and previous reasoning imply: $$\dim\pi_1^{-1}(J)\le \dim\pi'_{34}({\pi'}_1^{-1}(J))+\dim \ker
S_J- \dim Stab(J).$$ Together with the inequality from Lemma \[ineq\] it proves the statement.
Another case (i.e. $\bold{k}=\vec{k}_1$) is even easier because in this case we have $$\dim\pi_1^{-1}(J)\le 2n -\dim Stab(J)<2n-1.$$
The case when in formula (\[J\]) $s>1$ can be easily reduced to the previous case. For that let us introduce the embedding $i_l$: $\mathfrak{gl}(|\bold{k}_l|,\mathbb C)\to
\mathfrak{gl}(n,\mathbb C)$ and the projection $pr_l$: $\mathfrak{gl}(n,\mathbb C)\to \mathfrak{gl}(|\bold{k}_l|,\mathbb
C)$: $i_l(Y)_{p',q'}=Y_{pq}$, $pr_l(Y)_{pq}=Y_{p',q'}$, $p'=p+\sum_{m=1}^{l-1}|\vec{k}_m|$, $q'=q+\sum_{m=1}^{l-1}|\vec{k}_m|$, $0\le p,q\le |\vec{k}_l|$, and $i_l(Y)_{ij}=0$ for the rest of the entries of $i_l(Y)$.
Using arguments analogous to the ones from Lemma \[kerSJ\] one gets
Let $J$ be given by formula (\[J\]). Then
1. $ker S_J=\oplus_{l=1}^s i_l(ker
S_{J_{\bold{k}_l}}(\lambda_i))$
2. for $l=1,\dots,s$, $pr_l(Im S_J)\subset Im S_{J_{\bold{k}_l}}$.
This lemma immediately implies
\[dimfib\] Let $J$ be given by formula (\[J\]) and exists $l$, $1\le l\le s$ such that $|\bold{k}_l|>1$ then $\dim\pi_1^{-1}(J)<2n-s$.
And we eventually achieved the goal of the subsection:
Indeed Proposition \[dimfib\] implies $\dim\pi_1^{-1}(JNF\setminus\tilde{U})<2n$. Hence by Corollary \[sm2n\] $\pi_1^{-1}(JNF\setminus\tilde{U})$ lies inside the Zariski closure of $\pi_1^{-1}(\tilde{U})$. But $\pi_1^{-1}(\tilde{U})$ is irreducible.
The Poisson structure on the CM space {#Pois}
-------------------------------------
In the paper [@FR] the Poisson structure on the space $CM_\tau$ was constructed. This Poisson structure on $CM_\tau$ yields the RS integrable system which is the relativistic analog of the trigonometric Calogero-Moser system.
On the open part $\bold{U}$ of $CM_\tau$ described in the subsection \[coor\] the Poisson bracket $\{\cdot,\cdot\}_{FR}$ takes the form (see Appendix of [@FR] for the proof): $$\begin{gathered}
\{\lambda_i,\lambda_j \}_{FR}=0,\quad
\{\lambda_j,q_i\}_{FR}=\lambda_i q_i\delta_{ij},\\
\{q_i,q_j\}_{FR}=
\frac{(\tau-\tau^{-1})^2q_iq_j(\lambda_i+\lambda_j)\lambda_i\lambda_j}
{(\tau\lambda_i-\tau^{-1}\lambda_j)(\tau\lambda_j-\tau^{-1}\lambda_i)
(\lambda_i-\lambda_j)}.\end{gathered}$$
The formulas in [@FR] contain the misprint, the authors lost the factor $(\tau^2-1)^2$ in the expression for $\{
q_i,q_j\}_{FR}$.
Using the Hamiltonian reduction on the combinatorial model of the space of flat connections on the torus without a point the authors of [@FR] prove that the Poisson structure $\{\cdot,\cdot\}_{FR}$ has a holomorphic extension from $\bold{U}$ to the whole $CM_\tau$, and this Poisson structure is nondegenerate (i.e. $CM_\tau$ is a symplectic variety). Another way to see this Poisson structure is to use Quasi-Poisson reduction [@AS]. In this picture the Poisson structure is the result of the reduction of the natural Quasi-Poisson structure on the product $GL(n,{\mathbb C})\times GL(n,{\mathbb C})$ and it is immediate that this Poisson structure is symplectic.
Finite dimensional representation of $H_{1,\tau}$
=================================================
In this subsection we construct a family of finite dimensional representations of $H_{1,\tau}$. Later we will show that this family forms an open dense set inside the space of all finite dimensional representations. The main tool of this section is the faithful representation of $H_{1,\tau}$ which is the quasiclassical limit of the standard realization of $H_{q,\tau}$ as a subring of the ring of reflection difference operators [@Ch4].
Limit of the Lusztig-Demazure operators
---------------------------------------
Let us introduce the ring $\tilde{R}={\mathbb C}[P_1^{\pm 1}, \dots,
P_n^{\pm 1},X_1^{\pm 1},\dots,X_n^{\pm 1}]_{\delta(X)}\# S_n$, where the subscript $\delta(X)$ means localization by the ideal generated by $\delta(X)=\prod_{1\le i<j\le n}(X_i-X_j)$ and $\#$ is a notation for the smash product. Let us explain what the smash product is. For brevity we will use notation ${\mathbb C}[P^{\pm 1},
X^{\pm 1}]$ for the ring $\mathbb C[P_1^{\pm 1}, \dots, P_n^{\pm
1},X_1^{\pm 1},\dots,X_n^{\pm 1}]$
An element of the ring $\tilde{R}$ has the form $\sum_{w\in S_n}
F_w(P,X) w$. The group $S_n$ acts on the ring $R=\mathbb C[P^{\pm
1},X^{\pm 1}]_{\delta(X)}$ by the formulas $$P_i^w=P_{w^{-1}(i)},\quad X_i^w=X_{w^{-1}(i)},$$ and $$F(P,X)wF'(P,X)w'=F(P,X)(F')^{w}(P,X)ww'.$$
[@Ch4] The following formulas give an injective homomorphism of $H_{1,\tau}\to \tilde{R}$: $$\begin{gathered}
X^\mu\mapsto X^\mu,\\ T_i\mapsto \tau s_i+ \frac{\tau-\tau^{-1}}
{X_{i}/X_{i+1}-1}(s_i-1), \quad i=1,\dots,n-1,\\ \pi\mapsto
P_1^{-1}c,\end{gathered}$$ where $s_i=(i,i+1)\in S_n$ is a transposition and $c\in S_n$ is a cyclic transformation: $c(i)=i+1, i=1,\dots,n-1$, $c(n)=1$.
The homomorphism from the proposition is a quasiclassical limit of the Lusztig-Demazure representation [@Ch4]. For brevity we call this homomorphism the Lusztig-Demazure representation.
Actually the paper [@Ch4] contains the proof for the case $q\ne 1$. The proof in the case $q=1$ can be obtained from this proof by mechanical replacement of shifts operators $\tau(\lambda)$, $\lambda\in {\mathbb Z}^n$ by their quasiclassical limits $P^\lambda$. The reader may do this operation with Lecture 5 from the exposition [@K].
The representation $V_{\mu,\nu}$ {#Vml}
--------------------------------
Let $(\mu,\nu)\in (\mathbb C^*)^{2n}$ and $\chi_{\mu,\nu}\simeq
\mathbb C$ be a one dimensional $R$-module (character): $\chi_{\mu,\nu}(R(P,X))=R(\mu,\nu)$. We can induce a finite dimensional module $V_{\mu,\nu}$ from this module: $$V_{\mu,\nu}=\tilde{R}\otimes_{R}\chi_{\mu,\nu}.$$
This module has a $\mathbb C$ basis $w\otimes 1$, $w\in S_n$, hence $\dim V_{\mu,\nu}=n!$.
If $\nu_i\ne \nu_j$, $i\ne j$ then the $H_{1,\tau}$-module $V_{\mu,\nu}$ is irreducible.
The module $V_{\mu,\nu}$ has a natural $H_{\delta(X)}$ module structure. From the Lusztig-Demazure representation we see that $H_{\delta(X)}\simeq {\mathbb C}[P^{\pm 1}, X^{\pm 1}]_{\delta(X)}\#S_n$. The group $S_n$ acts freely on the variety $Spec({\mathbb C}[P^{\pm 1},
X^{\pm 1}]_{\delta(X)})$ hence the algebra $H_{\delta(X)}$ is Morita equivalent to the algebra ${\mathbb C}[P^{\pm 1}, X^{\pm
1}]^{S_n}_{\delta(X)}$. In particular, the module $V_{\mu,\nu}$ corresponds to the one-dimensional representation: $P\mapsto
P(\mu,\nu)$. Thus $V_{\mu,\nu}$ is an irreducible $H_{\delta(X)}$-module and hence an irreducible $H$-module.
The action of the finite Hecke algebra {#fHecke}
--------------------------------------
The elements $T_i$, $i=1,\dots,n-1$ generate an algebra of dimension $n!$ which is called the finite Hecke algebra. We will denote it by the symbol $A^n_\tau$.
If $\rm{e}$ is the unit in $S_n$ then by the action of elements $T_i$ we can get from the vector $\rm{e}\otimes 1$ the whole space $V_{\mu,\nu}$. Hence the map $j$: $A^n_\tau\to V_{\mu,\nu}$, $j(T_{i_1}\dots T_{i_k})=T_{i_1}\dots T_{i_k} {\rm e}\otimes 1$ is an isomorphism of (left) $A^n_\tau$ modules.
We denote the subset of all finite dimensional irreducible $H_{1,\tau}$-modules which are regular $A^n_\tau$-modules by the symbol $Irrep^{n!}$.
Let us denote the subset of $Irrep^{n!}$ consisting of $V_{\mu,\nu}$ $\mu,\nu\in ({\mathbb C}^*)^n$, $\delta(\nu)\ne 0$ by $\mathcal U$. Later we will show that all finite dimensional irreducible modules are from $Irrep^{n!}$.
The $GL(2,\mathbb Z)$ action on double affine Hecke algebras {#Four}
------------------------------------------------------------
One of the most important properties of the double affine Hecke algebra $H_{q,\tau}$ is the existence of the action of $GL(2,\mathbb Z)$ [@Ch4]. To explain how this group acts on the double affine Hecke algebra we need to introduce pairwise commutative elements $Y_i\in H_{q,\tau}$: $$\label{Y}
Y_i=T_1\dots T_{n-i}\pi^{-1} T^{-1}_{n-i+1}\dots T^{-1}_{n-1},
\quad i=1,\dots,n-1.$$ These elements satisfy the relations $$\begin{gathered}
\label{YT1}
T_i Y_{i+1} T_i= Y_i, \quad (1\le i<n)\\ T_i Y_j=Y_jT_i,\mbox{ if
} j-i\ne 0,1.\label{YT2}\end{gathered}$$
The group $GL(2,\mathbb Z)$ is generated by the elements: $$\varepsilon=\begin{pmatrix}0&-1\\-1&0\end{pmatrix}, \quad
\sigma=\begin{pmatrix}1&1\\0&1\end{pmatrix}.$$
These generators act by the following formulas: $$\begin{gathered}
\varepsilon: X_i\mapsto Y_i, Y_i\mapsto X_i, T_i\mapsto
T_i^{-1},\\ \sigma: X_i\mapsto X_i, Y_i\mapsto X_iY_iq^{-1},
T_i\mapsto T_i,\end{gathered}$$ where $\varepsilon: H_{q,\tau}\to H_{q^{-1},\tau^{-1}}$, $\sigma:
H_{q,\tau}\to H_{q,\tau}$. The transformation $\varepsilon$ is called the Fourier-Cherednik transform.
Using these transformations we can construct new finite dimensional representations. Indeed if $\gamma\in GL(2,\mathbb
Z)$ is such that $\gamma(H_{1,\tau})=H_{1,\tau'}$ and $\phi':
H_{1,\tau'}\to GL(V'_{\mu,\nu})$ is the corresponding representation of $H_{1,\tau'}$ (here $\tau'$ is either $\tau$ or $\tau^{-1}$ ) then the map $\phi'\circ\gamma$ is a representation of $H_{1,\tau}$. We denote the set of such representations by $\gamma(\mathcal U)$.
The map from $Irrep^{n!}$ to $CM_\tau$ {#constr}
======================================
In this section we construct a map $\Phi$: $Irrep^{n!}\to
CM_\tau$. Later we will show that it is an isomorphism. Constructions of this section generalize constructions of section 11 of [@EG].
Construction of the map
-----------------------
Let us denote by $A^{n-1}_\tau\subset A^n_\tau$ the subalgebra generated by the elements $T_{2},\dots, T_{n-1}$. It is the finite Hecke algebra of rank $n-2$. The element $v$ of an $A^n_\tau$-module is said to be $A^{n-1}_\tau$-invariant if $xv=\tau v$ for all $x\in A^{n-1}_{\tau}.$
The $H_{1,\tau}$-module $V\in Irrep^{n!}$ by definition is a regular $A^n_\tau$-module. Hence the space $V^{A^{n-1}_\tau}$ of $A^{n-1}_\tau$-invariants has dimension $n$. The relations inside $H_{1,\tau}$ and (\[YT2\]) imply that $X_1$ and $Y_1$ commute with the action of $A^{n-1}_\tau$. Thus if we fix a basis in $V$ we get $X_1|_{V^{A^{n-1}_\tau}}, Y_1|_{V^{A^{n-1}_\tau}}\in
GL(n,\mathbb C)$. The following statement is a key statement of the section.
\[rk1\] Let $V\in Irrep^{n!}$ then the operators $\bar{X}_1=X_1|_{V^{A^{n-1}_\tau}}$, $\bar{Y}_1=Y_1|_{V^{A^{n-1}_\tau}}$ satisfy the equation: $$rk(\bar{X}_1\bar{Y}_1\bar{X}^{-1}_1\bar{Y}^{-1}_1-\tau^{-2} Id)=1.$$
Obviously the space $CM_\tau$ is isomorphic to the quotient of the space of solutions of (\[eqXY-YX\]) by the action of $GL(n,{\mathbb C})$. Thus the last proposition proves that the map $\Phi$: $Irrep^{n!}\to CM_\tau$, $\Phi(V)=(\bar{X}_1,\bar{Y}_1)$ is well defined.
In the rest of the section we prove Proposition \[rk1\]. It is done in two steps. First we prove
\[lcom\] The elements $X_1,Y_1\in H_{1,\tau}$ satisfy the relation $$\label{com}
X_1Y_1X^{-1}_1Y^{-1}_1=T_1T_2\dots T_{n-2}T_{n-1}^2T_{n-2}\dots
T_1.$$
This is done in the next subsection using the geometric interpretation of the double affine Hecke algebra. The proof of this lemma was communicated to the author by Ivan Cherednik. The last step is the analysis of the LHS of (\[com\]) using the quasiclassical limit $\tau\to 1$. It is done in the last subsection.
The double affine braid group
-----------------------------
The double affine Hecke algebra admits a simple topological interpretation [@Ch3]. This construction is especially simple in the case $q=1$. In this case the algebra $H_{1,\tau}$ is a quotient of the so called double affine braid group $\mathfrak
{B}_n$.
For better understanding of this group the reader may have in mind the picture analogous to the geometric interpretation of the usual braid group but in the case when the points live on the two dimensional torus.
In this picture the elements $T_i$, $i=1,\dots,n-1$ correspond to the paths which permute of the $i$-th and the $i+1$-th nearby points (just like in the case of the usual braid group) and $X_i$, $Y_i$ correspond to the paths in which the $i$-th point goes along the parallel, respectively the meridian of the torus. In this geometric setting it is obvious that formula (\[com\]) holds in $\mathfrak{ B}_n$. The double affine Hecke algebra is a quotient of $\mathfrak{B}_n$ by the relations (\[T\^2\]). Hence (\[com\]) holds in $H_{1,\tau}$. Below we give formal definitions to justify this reasoning.
Let $$U=\{z\in\mathbb C^n| z_k-z_l\notin \mathbb Z+ {\rm
i}\mathbb Z, k\ne l \},$$ and $\bar W=S_n\ltimes(\mathbb
Z\oplus\mathbb Z {\rm i})$ acts on $z\in{U}$ by the formula: $$\bar{w}({z})=w(z+a+{\rm i} b), \quad \bar{w}=w(a+{\rm i}b), \quad
a,b\in \mathbb Z^n,\quad w\in S_n.$$
We fix a point $z^0$ such that its real and imaginary part is sufficiently small.
Paths $\gamma\subset U$ joining $z^0$ with points from $\{ \bar{w}(z^0),\bar{w}\in \bar{W}\}$ modulo homotopy and the action of $\bar{W}$ form the double affine braid group $\mathfrak{
B}_n$ with the multiplication induced by the usual composing operation for the paths.
This group is generated by the elements: $$\begin{gathered}
T_j=t_j(\psi)=z^0+(exp(\pi {\rm
i}\psi)-1)(z_j-z_{j+1})(e_j-e_{j+1})\\ X_j=x_j(\psi)=z^0+\psi
e_j,\quad Y_j=y_j(\psi)=z^0+\psi e_j {\rm i},\\ \pi=T_1\dots
T_{n-1} Y_1,\end{gathered}$$ where $0\le\psi\le 1$.
[@Ch3] The group $\mathfrak{ B}_n$ is generated by elements $X_i$, $Y_i$, $i=1,\dots,n$, $T_j$, $j=1,\dots,n-1$ (and $\pi$) with defining relations (\[dbre1\]-\[dbre2\]), (\[YT1\]-\[YT2\]) with $q=1$.
Thus $H_{1,\tau}$ is a quotient of $\mathfrak{B}_n$ by relations (\[T\^2\]) and Lemma \[lcom\] follows.
The spectrum of $Z=T_1\dots T_{n-2}T_{n-1}^2 T_{n-2}\dots T_1$
--------------------------------------------------------------
For a representation $V$ is from $Irrep^{n!}$ there is an isomorphism $V\simeq A^n_\tau$ of left $A^n_\tau$-modules. Hence the right multiplication on $A^n_\tau$ induces a structure of a right $A^n_\tau$-module on $V$ and as a consequence on $V^{A^{n-1}_\tau}$.
The right $A^{n}_\tau$-module $V^{A^{n-1}_\tau}$ is a sum of the $n-1$ dimensional vector representation and one-dimensional representation because it is true for $\tau=1$. Obviously, the operator $Z$ (acting by the left multiplication) commutes with the right action of $A^n_\tau$. Hence by the Schur lemma $Z$ acts by a constant on $A^n_\tau$-irreducible components of the right $A^n_\tau$-module $V^{A^{n-1}_\tau}$. That is, there exists a basis in the module in which $Z$ is diagonal and of the form $diag(\lambda_1(\tau),\lambda_2(\tau),\dots,\lambda_2(\tau))$. Thus we only need to calculate $\lambda_1(\tau)$, $\lambda_2(\tau)$.
The module $V^{A^{n-1}_\tau}$ exists for all $\tau\ne 0$. As the operator $Z$ is invertible for all nonzero values of $\tau$, we have $\lambda_1(\tau)\ne 0$, $\lambda_2(\tau)\ne 0$.
The functions $\lambda_i(\tau)$ are single valued. Indeed for $n=2$ it is obvious. So let us suppose $n>2$, then the eigenvalues $\lambda_i(\tau)$ have the different multiplicities. Hence the Galois group of the extension of the field of rational functions by $\lambda_1(\tau)$, $\lambda_2(\tau)$ is trivial because it cannot exchange $\lambda_1$ and $\lambda_2$. Thus the functions $\lambda_i(\tau)$ are rational and we have $\lambda_i(\tau)=C_i
\tau^{k_i}$, $i=1,2$.
When $\tau=1$, the algebra $A^n_\tau$ becomes the group algebra of $S_n$, and $Z=1$. Thus we have $C_1=C_2=1$. The calculation of $k_1,k_2$ uses the quasiclassical limit reasoning.
If $\tau=e^h$ then we can write the expansion of $T_i$ in terms of $h$ $$T_i=s_i+h \tilde{s}_i+O(h^2),\quad i=1,\dots,n-1,$$ where $s_i=(i,i+1)$ is a usual transposition. Relation (\[T\^2\]) inside $H_{1,\tau}$ implies $$s_i\tilde{s}_i+\tilde{s}_i s_i=2s_i, \quad i=1,\dots,n-1.$$
Let us calculate the first nontrivial term $\tilde{Z}$ of the expansion of $Z=1+h\tilde{Z}+O(h^2)$: $$\tilde{Z}=\sum_{i=1}^{n-1} s_1\dots s_{i-1}(\tilde{s}_i
s_i+\tilde{s}_i s_i)s_{i-1}\dots s_1= 2\sum_{i=1}^{n-1} s_{1i},$$ where $s_{1i}=s_1\dots s_{i-1}s_is_{i-1}\dots s_1$ is a permutation of $1$ and $i$.
The operator $\tilde{Z}/2$ acts on $\mathbb C[S_n]^{S_{n-1}}$ (by the left multiplication) and in the basis $e_i=(\sum_{w'\in
S_{n-1}}w')s_{1i}$ it has the matrix $J-Id$, $J_{ij}=1$, $1\le i,j
\le n.$ Hence $Spec(\tilde{Z}/2)=(n-1,-1,\dots,-1)$. On the other hand $Spec(\tilde{Z})=(k_1,k_2,\dots,k_2)$. Thus $k_1=2(n-1)$, $k_2=-2$ and we proved Proposition \[rk1\].
The map $\Phi$ on the subset $\mathcal{U}\subset Irrep^{n!}$ {#PhiU}
------------------------------------------------------------
It is possible to calculate $\Phi(V_{\mu,\nu})$ explicitly. Indeed let us fix a basis in $V_{\mu,\nu}^{A^{n-1}_\tau}$: $e_i=(\sum_{w'\in S_{n-1}}w')s_{1i}$, $i=1,\dots,n$.
For the matrices of the operators $\bar{X}_1$ and $\bar{Y}_1$ written in the basis $e_i$ the following equations hold $$\begin{gathered}
\bar{X}_1=diag(\nu_1,\dots,\nu_n)\\ \bar{Y}_{ii} =\mu_i
\prod_{j\ne i}\frac{(\tau^{-1}\nu_j-\tau\nu_i)}{(\nu_j-\nu_i)},
\quad i=1,\dots,n.\end{gathered}$$
The first equation is obvious. The second formula is a result of direct calculation using formulas (\[Y\]) for $Y_1$ and explicit formulas for $T_i$.
Indeed let make this calculation for $i=1$. The expansion of the product expression for $Y_1$ consists of the terms of the form $s_{i_1,j_1}\dots s_{i_r,j_r}c^{-1}F(X)P_1$, where $i_l<j_l$, $j_m<i_{m+1}$, $l=1,\dots,r$, $m=1,\dots,r-1$ and $F\in
{\mathbb C}[X^{\pm 1}]_{\delta(X)}$. We know that $Y_1e_1$ is a linear combination of $e_i$, $i=1,\dots,n$. The terms of the expansion of $Y_1 e_1$ which contribute to the coefficient before $e_1$ satisfy the equation $s_{i_1,j_1}\dots s_{i_r,j_r}c^{-1}(1)=1$. This is possible only in the case $r=1$, $i_1=1,j_1=n$. Thus rewriting $T_i$ in the form: $$T_i=\frac{(\tau
X_i-\tau^{-1}X_{i+1})}{X_i-X_{i+1}}s_i+\frac{X_{i+1}(\tau^{-1}-\tau)}
{X_i-X_{i+1}},$$ we see that $$Y_1e_1=
\left(\prod_{i=1}^{n-1}\frac{(\tau
X_i-\tau^{-1}X_{i+1})}{X_i-X_{i+1}}s_i\right) c^{-1}e_1+R,$$ where $R$ is a linear combination of $e_j$ with $j>1$. This formula immediately implies the last formula from the proposition for $i=1$.
Let $D_\tau$ be a subset of $\mathcal{U}$ consisting of the representations of the form $V_{\mu,\nu}$ such that $\delta_\tau(\nu)=\prod_{i,j}(\tau \nu_i-\tau^{-1}\nu_j)=0$.
It is actually not easy to compute all coefficients $\bar{Y}_1$ using explicit formulas for $Y_1$ and $T_i$ but we do not need them. Because by proposition \[propcoor\], if the pair $(X,Y)$ satisfies equation (\[eqXY-YX\]) and $X$ is diagonal with eigenvalues satisfying the conditions of Proposition \[propcoor\] then the corresponding $GL(E)$-orbit is uniquely determined by the diagonal elements of $X$ and $Y$ (because the stabilizer of $X$ consists of diagonal matrices which do not change diagonal elements of $Y$ and we can extract $q$ from these elements). This reasoning implies
\[isoopen\] The map $\Phi$ is an isomorphism on the subset $\mathcal{U}_0=\mathcal{U}\setminus D_\tau$, and local coordinates $\lambda,q$ on $CM_\tau$ are expressed through coordinates $\mu,\nu$ on $\mathcal{U}_0\subset Irrep^{n!}$ by the formulas $$\begin{gathered}
\lambda_i=\nu_i,\quad q_i=\mu_i \prod_{j\ne
i}\frac{(\tau^{-1}\nu_j-\tau\nu_i)}{(\nu_j-\nu_i)}.\end{gathered}$$
Results on the general double affine Hecke algebra
===================================================
Let $R=\{\alpha\}$ be a root system (possibly nonreduced) of type $A,$ $B,$ $BC,\dots,F,$ $G$, $W$ the Weyl group generated by the reflections $s_\alpha$, $\alpha\in R$. The extended affine Weyl group $\tilde{W}$ is a semidirect product $W\ltimes P$, where $P$ is a weight lattice (i.e. $b\in P$ if $2(b,\alpha)/(\alpha,\alpha)\in \mathbb Z$ for all $\alpha\in R$).
The affine Hecke algebra $\hat{H}_\tau$ is a deformation of the group algebra ${\mathbb C}[\tilde{W}]$ with deformation parameters $\tau_\alpha$, $\tau_{w(\alpha)}=\tau_\alpha$, $\alpha\in R$, $w\in W$ (for the exact definition of the affine Hecke algebra see [@EG]). The double affine Hecke algebra $H_{q,\tau}$ is a nontrivial extension of the affine Hecke algebra $\hat{H}_\tau$ by the group algebra ${\mathbb C}[P^{\vee}]$ of the coweight lattice $P^{\vee}$ ($b\in P^{\vee}$ if $(b,\alpha)\in{\mathbb Z}$ for all $\alpha\in R$). This extension has one parameter $q$ which is the shift parameter in the Lusztig-Demazure representation of this algebra. We consider algebras with $q=1$ and we denote them by $H$. For the exact definition of the double affine Hecke algebra and formulas for the Lusztig-Demazure representation see the original paper [@Ch3] or survey [@K].
We use the notation $\delta(X)$ for the Weyl denominator for the root system $R$. By symbol ${\mathbb C}[X^{\pm 1}]$ we denote the group algebra of the weight lattice $P$ lying inside the affine Hecke algebra $\hat{H}_\tau$ and by symbol ${\mathbb C}[Y^{\pm 1}]$ we denote group algebra ${\mathbb C}[P^{\vee}]\subset H$ which extends $\hat{H}_\tau$.
There is an injective homomorphism $g$: $H\to{\mathbb C}[P^{\pm 1} ,X^{\pm
1} ]_{\delta(X)}\#W$ via the quasiclassical Lusztig-Demazure operators $w\mapsto w, X_b\mapsto X^b,
T_{s_\alpha}=T_{\alpha}\mapsto D_\alpha,$ where $\alpha\in R$, $b\in P.$ The formulas for the embedding are very similar to the formulas from the previous section. Let $A$ be the corresponding finite Hecke algebra, and $e$ the symmetrizer in $A$: $$e=\sum_{w\in W}\tau^{l(w)} T_w/(\sum_{w\in W}\tau^{2l(w)}),$$ where $T_w=T_{i_1}\dots T_{i_{l(w)}}$ if $w=s_{i_1}\dots
s_{i_{l(w)}}$ is a reduced expression for $w$.
In this section we will need the following PBW type result
\[PBW\][@Ch4] Each element $h\in H$ can be uniquely presented in the forms: $$\begin{gathered}
h=\sum_{w\in W} f_w(X) T_w
g_w(Y),\\ h=\sum_{w\in W} g'_w(Y) T_w f'_w(X).\end{gathered}$$
Formulation of the theorem
--------------------------
The goal of this section is to study the center $Z$ of $H$ and corresponding scheme $Spec(Z)$. It turns out that $Z$ is isomorphic to the subalgebra $eHe$ and we can reduce the study of $Z$ to the study of $eHe$.
We remind the definition of a Cohen-Macaulay algebra.
[@CG] A finitely generated commutative ${\mathbb C}$-algebra $A$ is called [*Cohen-Macaulay*]{} if it contains a subalgebra of the form $\mathcal{ O}(V)$ such that $A$ is a free $\mathcal
O(V)$-module of finite rank, and $V$ is a smooth affine algebraic variety.
For the definition of a Cohen-Macaulay module see [@Ser] (Chapter 4 p. 18). In this section we prove the following
\[general\] For any double affine Hecke algebra $H$ the following is true:
1. \[comm\] $eHe$ is commutative.
2. \[Cohen\] $M=Spec(eHe)$ is an irreducible Cohen-Macaulay and normal variety.
3. \[CohenM\] The right $eHe$ module $He$ is Cohen-Macaulay.
4. \[eHe-eH\]The left action of $H$ on $He$ induces an isomorphism of algebras $H\simeq End_{eHe}(He)$.
5. \[Sat\]The map $\eta: z\to ze$ is an isomorphism $Z\to eHe$. Thus, $M=Spec( Z)$.
We call the isomorphism $\eta$ the Satake isomorphism (by analogy with [@EG]).
Proofs of theorem \[general\]
-----------------------------
$Z$ contains ${\mathbb C}[X^{\pm 1}]^W\otimes {\mathbb C}[Y^{\pm 1}]^W$.
${\mathbb C}[X^{\pm 1}]^W$ clearly lies in the center of ${\mathbb C}[P^{\pm 1},X^{\pm 1}]_{\delta^{-1}(X)}\#W$, and therefore in the center of $H$. The fact that ${\mathbb C}[Y^{\pm 1}]^W$ is contained in $Z$ follows from the existence of the Fourier-Cherednik morphism [@Ch4] (i.e. the action of the element $\varepsilon$ of $GL(2,{\mathbb Z})$ which is described in the previous section in the case of the root system $A_{n-1}$).
Indeed, the morphism $\varepsilon$ is an isomorphism between the double affine Hecke algebra $H'$ with parameter $\tau^{-1}$ and the double affine Hecke algebra $H$. This morphism maps the subring ${\mathbb C}[X^{\pm 1}]^W\subset H'$ onto the subring ${\mathbb C}[Y^{\pm
1}]^W$ of $H$.
Now the statement follows from the PBW theorem for $H$.
\[int\] $eHe$ is commutative, without zero divisors.
Let us prove that the subalgebra $eH_{\delta(X)}e$ of $H_{\delta(X)}\simeq{\mathbb C}[P^{\pm 1},X^{\pm 1}]_{\delta(X)}\# W$ is commutative and without zero divisors. Obviously it implies the statement.
An element $z\in H_{\delta(X)}$ has a unique representation in the form $z=\sum_{w\in W} Q_w T_w$; that is, $H_{\delta(X)}$ is isomorphic to ${\mathbb C}[P^{\pm 1},X^{\pm 1}]_{\delta(X)}\otimes A$ as a right $A$-module. If $z\in eH_{\delta(X)}e$ then $zT_{\alpha}=\tau_\alpha z$ for all $\alpha\in R$ because $eT_{\alpha}=\tau_\alpha e$. Hence $z$ is an $A$-invariant element of the right $A$-module $eH_{\delta(X)}e\subset{\mathbb C}[P^{\pm
1},X^{\pm 1}]_{\delta(X)}\otimes A$. As ${\mathbb C}(P,X)\otimes A$ is a regular $A$-module (over the field ${\mathbb C}(P,X)$) ${\mathbb C}(P,X)\otimes e$ is a unique copy of the trivial representation. It implies that $z=Qe$, $Q\in {\mathbb C}[P^{\pm 1},X^{\pm 1}]_{\delta(X)}$.
Finally for $z=Qe\in eH_{\delta(X)}e$ we have $(T_\alpha-\tau_\alpha)Qe=0$. The simple calculation using the explicit expression for $T_\alpha$ yields: $$(T_\alpha-\tau_\alpha)Qe=P_\alpha(s_\alpha-1)Qe=P_\alpha
(s_\alpha(Q)-Q)e,$$ where $P_\alpha\in {\mathbb C}[X^{\pm 1}]_{\delta(X)}$ and $\alpha$ is a simple root. This implies $Q\in {\mathbb C}[P^{\pm
1},X^{\pm 1}]^W_{\delta(X)}e$ and $eH_{\delta(X)}e\simeq{\mathbb C}[P^{\pm
1},X^{\pm 1}]^W_{\delta(X)}$.
The algebra $H$ has a natural ${\mathbb C}[X^{\pm 1}]^W\otimes {\mathbb C}[Y^{\pm
1}]^W$-module structure: the element $p\otimes q$ acts on $x\in H$ by the formula $(p\otimes q)x=pxq$.
\[proj\] $H$ is a projective finitely generated ${\mathbb C}[X^{\pm 1}]^W\otimes {\mathbb C}[Y^{\pm 1}]^W$-module.
Let us first show that ${\mathbb C}[X^{\pm 1}]$ is a projective finitely generated ${\mathbb C}[X^{\pm 1}]^W$ module. Finite generation is clear, since $W$ is a finite group. Also, it is well known that ${\mathbb C}[X]^W$ is a polynomial ring (it is generated by the characters of the fundamental representations of the corresponding simply connected group). Since ${\mathbb C}[X^{\pm 1}]$ is a regular ring, by Serre’s theorem ([@Ser], chapter 4, p. 37, proposition 22) ${\mathbb C}[X^{\pm 1}]$ must be locally free over ${\mathbb C}[X^{\pm 1}]^W$ (in fact, by Steinberg-Pittie [@St] theorem it is free, but we will not use it). For the same reasons ${\mathbb C}[Y^{\pm 1}]$ is locally free over ${\mathbb C}[Y^{\pm 1}]^W$.
Now the claim follows from the PBW factorization from Proposition \[PBW\] $H={\mathbb C}[X^{\pm 1}]\otimes A\otimes {\mathbb C}[Y^{\pm
1}]$.
\[projeH\] $He$ and $eHe$ are projective finitely generated modules over ${\mathbb C}[X^{\pm 1}]^W\otimes {\mathbb C}[Y^{\pm 1}]^W$.
The finite generation follows from the Hilbert-Noether lemma and Lemma \[proj\]. The projectivity is true because $He$ and $eHe$ are direct summands in $H$.
\[delta\]
1. $H_{\delta(X)}\simeq {\mathbb C}[X^{\pm 1},P^{\pm 1}]_{\delta(X)}\# W$
2. The map $\eta$: $Z(H_{\delta(X)})\to {\mathbb C}[P^{\pm 1},
X^{\pm 1}]^W_{\delta(X)}e$, induced by multiplication by $e$ is an isomorphism.
3. The left $H_{\delta(X)}$-action on $H_{\delta(X)}$ induces the isomorphism $H_{\delta(X)}\simeq
End_{eH_{\delta(X)}e}(H_{\delta(X)})$.
The first and second items of the lemma follow from the representation of $H$ by the quasiclassical Lusztig-Demazure operators. The third item is equivalent to the isomorphism $${\mathbb C}[P^{\pm 1},X^{\pm}]_{\delta(X)}\# W\simeq End_{{\mathbb C}[P^{\pm
1},X^{\pm 1}]^W_{\delta(X)}}({\mathbb C}[P^{\pm 1},X^{\pm
1}]_{\delta(X)}).$$ We will proceed analogously to the proof of theorem 1.5 from [@EG].
If $a$: ${\mathbb C}[P^{\pm 1},X^{\pm 1}]_{\delta(X)}\to{\mathbb C}[P^{\pm
1},X^{\pm 1}]_{\delta(X)}$ is ${\mathbb C}[P^{\pm 1},X^{\pm
1}]^W_{\delta(X)}$-linear then it defines a ${\mathbb C}(P,X)^W$-linear map ${\mathbb C}(P,X)\to {\mathbb C}(P,X)$. The isomorphism ${\mathbb C}(P,X)\# W\simeq
End_{{\mathbb C}(P,X)^W}({\mathbb C}(P,X))$ implies $a=\sum_{w\in W} a_w w$, $a_w\in {\mathbb C}(P,X)$. It is clear that the functions $a_w$ are regular on $({\mathbb C}^*)^n\times ({\mathbb C}^*\setminus D)^n\setminus \Delta$ where $\Delta$ is the subset of the points of $({\mathbb C}^*)^n\times({\mathbb C}^*)^n$ with a nontrivial stabilizer in $W$ and $D=\{X\in{\mathbb C}^*|\delta(X)=0\}$. But $\Delta\subset D$, hence $a_w\in {\mathbb C}[P^{\pm 1},X^{\pm 1}]_{\delta(X)}$.
The first item follows from Lemma \[int\].
[**Proof of (\[Cohen\]):**]{} $M=Spec(eHe)$ is an irreducible affine variety by Lemma \[int\]. The subalgebra $({\mathbb C}[X^{\pm 1}]^W\otimes
{\mathbb C}[Y^{\pm 1}]^W)e$ is polynomial. Hence to prove that $M$ is Cohen-Macaulay it is sufficient to show that $eHe$ is a locally free module of finite rank over its subalgebra $({\mathbb C}[X^{\pm
1}]^W\otimes {\mathbb C}[Y^{\pm 1}]^W)e$. But the module is projective and finitely generated by Lemma \[projeH\].
It is easy to see by localizing with respect to $e\delta(X)$ or $e\delta(Y)$ that $M$ is smooth away from a codimension 2 subset. Indeed, by the first item of Lemma \[delta\] after localizing with respect to $e\delta(X)$ the image of $eHe$ under the injection $g$ becomes $e{\mathbb C}[X,Y]_{\delta(X)}e\simeq
{\mathbb C}[X,Y]_{\delta(X)}^We$, which is the ring of regular functions on a smooth affine variety. The statement for the localization with respect to $e\delta(Y)$ follows from the existence of the Fourier-Cherednik transform. But an irreducible Cohen-Macaulay variety that is smooth outside of a codimension 2 subset is normal ([@CG],2.2).
[**Proof of (\[CohenM\]):**]{} $eHe$ is finitely generated over ${\mathbb C}[X^{\pm 1}]^W\otimes{\mathbb C}[Y^{\pm 1}]^W$. Hence by Theorem 2.1 of [@BBG] $eH$ is Cohen-Macaulay over $eHe$ if and only if it is Cohen-Macaulay over ${\mathbb C}[X^{\pm 1}]^W\otimes{\mathbb C}[Y^{\pm 1}]^W$.
We know that $He\simeq {\mathbb C}[X^{\pm 1},Y^{\pm 1}]$ as a ${\mathbb C}[X^{\pm
1}]^W\otimes{\mathbb C}[Y^{\pm 1}]^W$-module and $He$ is projective over ${\mathbb C}[X^{\pm 1}]^W\otimes{\mathbb C}[Y^{\pm 1}]^W$. As ${\mathbb C}[X^{\pm
1}]^W\otimes{\mathbb C}[Y^{\pm 1}]^W$ is a polynomial ring, the module ${\mathbb C}[X^{\pm 1},Y^{\pm 1}]$ is Cohen-Macaulay if and only if it is projective. So Lemma \[projeH\] implies the statement.
[**Proof of (\[eHe-eH\]):**]{} We have an obvious homomorphism $f: H\to End_{eHe}He$. It is clearly injective because it is injective after localization by the ideal $(\delta(X))$.
Let us denote $End_{eHe}(He)$ by $H'$. Regard $H'\supset H$ as ${\mathbb C}[X^{\pm 1}]^W\otimes {\mathbb C}[Y^{\pm 1}]^W$-modules. $H'$ is torsion free because $He$ is a torsion free ${\mathbb C}[X^{\pm 1}]^W\otimes{\mathbb C}[Y^{\pm 1}]^W$-module (by the PBW theorem). As $He$ is finitely generated over $eHe$, $H'$ is a finitely generated ${\mathbb C}[X^{\pm 1}]^W\otimes{\mathbb C}[Y^{\pm
1}]^W$-module. Also, $H$ is finitely generated projective, and $H'/H$ is supported in codimension 2. Indeed, the last part of Lemma \[delta\] implies that $H_{\delta(X)}$ is isomorphic to $H'_{\delta(X)}$ as a $e H_{\delta(X)} e$ module. Similarly, the module $H_{\delta(Y)}$ is isomorphic to $H'_{\delta(Y)}$ as a $eH_{\delta(Y)}e$-module because we can use (the same way as in the proof of Lemma \[int\]) the Fourier-Cherednik transform $\varepsilon$ from subsection \[Four\].
The module $H'$ represents some class in $Ext^1(H'/H,H)$, which must be zero since $H'/H$ is finitely generated and lives in codimension 2 and $H$ is projective. Thus, $H'=H\oplus H'/H$ and the summand $H'/H$ is torsion. But $H'$ is a torsion free $eHe$ module, hence $H'/H=0$ and $H'=H$.
[**Proof of (\[Sat\]):**]{} It is clear that $\eta$ is injective, by looking at the Lusztig-Demazure representation. Indeed the equation $ze=0$ implies $zp=0$ for any $p\in {\mathbb C}[X]^W$, hence by the PBW theorem $z=0$.
It remains to show that $\eta$ is surjective. Since $eHe$ is commutative, every element $a\in eHe$ defines an endomorphism of $He$ over $eHe$ (by right multiplication). So by statement (\[eHe-eH\]) $a$ defines an element $z_a\in H$. This element commutes with $H$. Indeed, the right multiplication by $a$ is an endomorphism of the right $eHe$-module which commutes with left multiplication by elements of $H$ hence by the fourth part of the theorem $[z_a,h]=0$ for all $h\in H$. For any $x\in H$, $z_axe=xa$, so $xz_ae=xa$, i.e. $x(z_ae-a)=0$. Since $eHe$ has no zero divisors, we find $\eta(z_a)=a$, as desired.
The results in the case of the root system $A_{n-1}$
====================================================
In this section $H=H_{1,\tau}$ is the double Hecke algebra corresponding to $GL(n,{\mathbb C})$.
A point $(\mu,\nu)\in ({\mathbb C}^*)^n\times (({\mathbb C}^*)^n\setminus D)$ defines a ${\mathbb C}[P^{\pm 1},X^{\pm 1}]_{\delta(X)}^{S_n}$-character $\chi_{(\mu,\nu)}$: $\chi_{(\mu,\nu)}(Q(P,X))=Q(\mu,\nu)$. The embedding $Z\hookrightarrow Z_{\delta(X)}\simeq{\mathbb C}[P^{\pm
1},X^{\pm 1}]_{\delta(X)}^{S_n}$ allows us to restrict this character to $Z$. We use the same notation for this character.
For any point $(\mu,\nu)\in ({\mathbb C}^*)^n\times (({\mathbb C}^*)^n\setminus D)$ we have $$He\otimes_{eHe}\chi_{(\mu,\nu)}\simeq V_{\mu,\nu}.$$
The $H$-module $V_{\mu,\nu}$ has a natural structure of an $H_{\delta(X)}$-module. Let us study finite dimensional irreducible $H_{\delta(X)}$-modules.
By Lemma \[delta\] the ring $eH_{\delta(X)}e$ is a regular ring. As the action of $S_n$ on $({\mathbb C}^*)^n\times
({\mathbb C}^*\setminus D) $ is free, the ring ${\mathbb C}[P^{\pm 1},X^{\pm
1}]_{\delta(X)}\simeq H_{\delta(X)} e$ is a projective $eH_{\delta(X)}e$-module and defines the vector bundle $F$ over $
{\mathbb C}^n\times ({\mathbb C}^n\setminus D)=Spec(eH_{\delta(X)}e)$. Hence by the last item of Lemma \[delta\] $H_{\delta(X)}=End(F)$ is an Azumaya algebra and by the basic property of Azumaya algebras any irreducible $H_{\delta(X)}$-module is of the form $H_{\delta(X)}e\otimes_{eH_{\delta(X)}e} \chi_{(\mu',\nu')}$ for some point $(\mu',\nu')\in {\mathbb C}^n\times ({\mathbb C}^n\setminus D)$.
Obviously any irreducible $H_{\delta(X)}$-module is irreducible as an $H$-module. Also we have an obvious isomorphism of $H$ modules $H_{\delta(X)}e\otimes_{eH_{\delta(X)}e}\chi_{(\mu',\nu')}\simeq
He\otimes_{eHe}\chi_{(\mu',\nu')}$. Thus the previous paragraph implies $V_{\mu,\nu}\simeq He\otimes_{eHe}\chi_{(\mu',\nu')}$. Comparing the action of the center on the both sides yields the statement.
The previous lemma implies that there is a map $\Upsilon$ from the open part $Spec(Z_{\delta(X)})$ of $Spec(Z)$ to the space $CM_\tau$: $\Upsilon(\mu,\nu)=\Phi(V_{\mu,\nu})$, where $\Phi$ is the map constructed at the section \[constr\]. As $Spec(Z_{\delta(X)})$ is an open dense subset in $Spec(Z)$, we can define a rational map $\Upsilon$: $Spec(Z)\dasharrow CM_\tau$.
The map $\Upsilon$: $Spec(Z)\dasharrow CM_\tau$ is a regular isomorphism of the algebraic varieties. In particular $Spec(Z)$ is smooth.
The previous lemma and Corollary \[isoopen\] imply that $\Upsilon$ is a regular isomorphism on $Spec(Z_{\delta(X)\delta_\tau(X)})$. The Fourier-Cherednik transform from the section \[Four\] allows us to state the same for the open subset $Spec(Z_{\delta(Y)\delta_\tau(Y)})$.
Indeed, the Fourier-Cherednik transform $\varepsilon$ maps the double affine Hecke algebra $H_{1,\tau}$ to $H_{1,\tau^{-1}}$ and it induces the map $\varepsilon_{CM}$: $CM_{\tau}\to
CM_{\tau^{-1}}$, $\varepsilon_{CM}(X,Y,U,V)=(Y,X,-Y^{-1}X^{-1}YXU,V)$. By the construction we have $\varepsilon_{CM}\circ\Upsilon=\Upsilon\circ\varepsilon$. Thus the restriction of the morphism $\varepsilon_{CM}^{-1}\circ\Upsilon\circ\varepsilon$ to $Spec(Z_{\delta(Y)\delta_\tau(Y)})$ is a regular isomorphism.
Now, we know from the Theorem \[general\] that $Spec(Z)$ is normal. As the complement of $Spec(Z_{\delta(X)\delta_\tau(X)})\cup
Spec(Z_{\delta(Y)\delta_\tau(Y)})$ has codimension $2$ (because $Spec(Z)$ is irreducible by Theorem \[general\]), we can extend $\Upsilon$ to a regular map on the whole $Spec(Z)$. The extended map is dominant because by Proposition \[irrCM\] the variety $CM_\tau$ is irreducible.
Thus $\Upsilon$ is a regular birational map which is an isomorphism outside of the subset of codimension $2$. But we know that $CM_\tau$ is smooth and $Spec(Z)$ is normal, hence (by theorem 5 section 5 of chapter 2 of [@Sh]) the map $\Upsilon^{-1}$ is regular and as a consequence is an isomorphism.
$He$ is a projective $eHe$-module.
We proved for any $R$ that $He$ is a Cohen-Macaulay module over $eHe$. Since $M=Spec(eHe)$ is smooth, the result follows from corollary 2 from chapter 4 of [@Ser].
Thus $He$ defines the vector bundle $E$ on $Spec(eHe)$, with fibers of dimension $n!$.
\[Azum\] For the double affine Hecke algebra $H=H_{1,\tau}$ the following is true:
1. $H=End
E$ where $E$ is a vector bundle over $Spec(Z)$ i.e. $H$ is an Azumaya algebra.
2. Every irreducible representation of $H$ is of the form $V_z=He\otimes_{eHe}\chi_z$, $z\in M=Spec(Z)$.
3. $V_z$ has dimension $n!$ and is a regular representation of $A^n_{\tau}$.
The first item follows from Theorem \[general\]. The second item is a general property of Azumaya algebras. The third item follows from the fact that it is true for the generic point $z\in Spec(Z)$.
This corollary was proved in 2000 by Cherednik using different methods [@Chpriv].
The ring $Z\simeq eH_{1,\tau}e$ has a natural noncommutative deformation $e H_{q,\tau}e$. Hence this ring has a natural Poisson structure $\{\cdot,\cdot\}$. The variety $CM_\tau$ also has a Poisson structure described in subsection \[Pois\]. It turns out that the isomorphism $\Phi$ respects these Poisson structures.
The isomorphism $\Phi$ is an isomorphism of Poisson varieties, that is the following formula holds $$\{\cdot,\cdot\}_{FR}=\{\cdot,\cdot\}.$$
It is enough to prove that it is an isomorphism of Poisson varieties on the open set $\mathcal{U}$. For $q=e^h\ne 1$ we have an embedding $g_q$; $H_{q,\tau}\to \mathbb{D}_q\# S_n$ via Lusztig-Demazure reflection difference operators. Here $\mathbb{D}_q$ is a localization of the Weyl algebra with generators $X_i^{\pm 1}$, $\hat{P}^{\pm 1}_i$, $i=1,\dots,n$ and relations: $$[X_i,X_j]=0,\quad [\hat{P}_i,\hat{P}_j]=0,\quad
X_j\hat{P}_i-q^{\delta_{ij}}\hat{P}_iX_j=0,$$ by the ideal $(\delta(X))$. When $q=1$, the noncommutative ring $\mathbb{D}_q$ becomes the commutative ring ${\mathbb C}[P^{\pm 1},X^{\pm
1}]_{\delta(X)}$ and the corresponding Poisson structure on this ring is given by the formulas: $$\{ X_i,X_j\}=0,\quad\{ P_i,P_j\}=0,\quad
\{X_i,P_j\}=\delta_{ij}X_iP_j.$$
The $H_{1,\tau}$-module $V_{\mu,\nu}$ has a natural ${\mathbb C}[P^{\pm 1}
,X^{\pm 1}]_{\delta(X)}\#S_n$ structure. It is easy to see that in the basis $1\otimes w$, $w\in W$ operators $P_i$, $X_j$ are diagonal. In particular $P_i(1\otimes {\rm e})=\mu_i(1\otimes{\rm e})$ and $X_i(1\otimes{\rm e})=\nu_i(1\otimes{\rm e})$, hence we have the following Poisson bracket on $\mathcal{U}$: $$\label{mnPoi}
\{\nu_i,\nu_j\}=0,\quad \{\mu_i,\mu_j\}=0\quad
\{\nu_i,\mu_j\}=\delta_{ij}\nu_i\mu_j.$$ The comparison of the formulas for the Poisson bracket on $\bold{U}\subset CM_\tau$ from subsection \[Pois\] and explicit formulas for the map $\Phi|_{\mathcal{U}}$ from subsection \[PhiU\] give the formula. Indeed, we can express the functions $\lambda_i$, $q_k$ through the functions $\mu_s,\nu_t$ and using (\[mnPoi\]) calculate the Poisson brackets $\{\lambda_i,\lambda_k\}$, $\{\lambda_i,q_k\}$, $\{q_i,q_k\}$. We prove a formula for the last bracket: $$\begin{gathered}
\{q_i,q_k\}=q_iq_k\left(\nu_k\frac{\partial \ln(q_i)}{\partial
\nu_k}-\nu_i\frac{\partial\ln(q_k)}{\partial\nu_i}\right)=\\
q_iq_k\left(\nu_k\left(-\frac{\tau}{\tau^{-1}\nu_i-\tau\nu_k}+\frac{1}{\nu_i-\nu_k}\right)-
\nu_i\left(-\frac{\tau}{\tau^{-1}\nu_k-\tau\nu_i}+
\frac{1}{\nu_k-\nu_i}\right)\right)=\\
\frac{(\tau^{-1}-\tau)^2q_iq_k(\nu_k+\nu_i)\nu_i\nu_k}{(\nu_i-\nu_k)
(\tau^{-1}\nu_k-\tau\nu_i)(\tau^{-1}\nu_i-\tau\nu_k)}.\end{gathered}$$
The rational and trigonometric cases
====================================
In this section we explain how to degenerate results from the main body of the paper to obtain an easier proof of the results of [@EG] on the rational double affine Hecke algebra. We also give the version of the results of the paper for the trigonometric Hecke algebra and explain how to modify the proof from the paper for this case.
We give the modifications of the results from the main body of the text only for the root system $A_{n-1}$ but similar things can be done for any root system $R$. Moreover, in the rational case we can replace the Weyl group $W$ by a finite Coxeter group (see [@EG]). Proofs of these results almost identically repeat proofs for (nondegenerate) double affine Hecke algebras.
Definition of the rational and trigonometric double affine Hecke algebras
-------------------------------------------------------------------------
Below we give a definition of the rational and trigonometric double affine Hecke algebra.
[@EG; @ChM] The rational double affine Hecke algebra $H^{rat}_{t,c}$ is generated by elements $s_{ij}$, $1\le i\ne j\le n$, $x_i,y_j$, $1\le i,j\le
n$. The elements $s_{ij}$, $1\le i,j\le n$ generate the subalgebra inside $H^{rat}_{t,c}$ isomorphic to the group algebra of the symmetric group $S_n$, and $s_{ij}$ corresponds to the transposition $(ij)$. In addition generators of $H^{rat}_{t,c}$ satisfy the relations $$\begin{gathered}
x_is_{ij}=s_{ij}x_j,\quad y_is_{ij}=s_{ij}y_j,\quad 1\le i,j\le
n,\\ [x_k,s_{ij}]=0,\quad [y_k,s_{ij}]=0,\quad k\notin \{i,j\},
\quad 1\le i,j,k\le n,\\ [y_i,x_j]= cs_{ij},\quad 1\le i\ne j\le
n\\ [x_i,x_j]=0=[y_i,y_j],\quad 1\le i, j\le n,\\
[y_k,x_k]=t-c\sum_{i\ne k}s_{ik},\quad 1\le k\le n.\end{gathered}$$
The trigonometric double affine Hecke algebra $H^{trig}_{t,c}$ is generated by elements $s_{ij}$, $1\le i\ne j\le n$, $X^{\pm 1}
_i,y_j$, $1\le i,j\le n$. The elements $s_{ij}$, $1\le i,j\le n$ generate the subalgebra inside $H^{rat}_{t,c}$ isomorphic to the group algebra of the symmetric group $S_n$, and $s_{ij}$ corresponds to the transposition $(ij)$. In addition the generators of $H^{trig}_{t,c}$ satisfy the relations $$\begin{gathered}
X_is_{ij}=s_{ij}X_j,\quad 1\le i,j\le n,
\\ s_{ij}y_i-y_js_{ij}=c \mbox{ if } j>i,\quad
s_{ij}y_i-y_js_{ij}=-c\mbox{ if } j<i,\\ [X_k,s_{ij}]=0,\quad
[y_k,s_{ij}]=0 \mbox{ if } k\notin \{i,j\}, \quad 1\le i,j,k\le
n,\\ [X_i,X_j]=0= [y_i,y_j],\quad 1\le i, j\le n,\\
X_j^{-1}y_iX_j-y_i=cs_{ij}\mbox{ if } j>i,\quad
X_j^{-1}y_iX_j-y_i=X_iX_j^{-1}cs_{ij}\mbox{ if } j<i,\\
X_k^{-1}y_kX_k-y_k=t-c(\sum_{i<k
}s_{ik}+\sum_{i>k}X_iX_k^{-1}s_{ik}),\quad 1\le k\le n.\end{gathered}$$
Let $\hat{H}$ be the ${\mathbb C}[c,t][[h]]$-algebra topologically generated (in the $h$-adic topology) by $X_i$, $y_i$, $s_{i,i+1}$ with $T_i=s_{i,i+1}e^{chs_{i,i+1}}$, $i=1,\dots,n-1$, $Y_i=e^{hy_i}$, $X_i$, $i=1,\dots,n$ satisfying the relations for the double affine Hecke algebra $H_{q,\tau}$, $q=e^{th},\tau=e^{ch}$. It coincides with an appropriate completion of the double affine Hecke algebra $H_{q,\tau}$, in the $h$-adic topology. Moreover, one can show that $\hat{H}$ is flat over ${\mathbb C}[[h]]$ and $\hat{H}/h\hat{H}=H^{trig}_{t,c}$. Analogously, if $\hat{H}^{trig}$ is the ${\mathbb C}[c,t][[h]]$-algebra topologically generated by by $s_{ij}, y_i, x_j$, $1\le i\le n$ with $s_{ij}, y_i,X_j=e^{hx_j},$ $i,j=1,\dots,n,$ satisfying the relations for the trigonometric double affine Hecke algebra $H^{trig}_{ht,hc}$ then the algebra $\hat{H}^{trig}$ is flat over ${\mathbb C}[[h]]$ and $H^{rat}_{t,c}=\hat{H}^{trig}/h\hat{H}^{trig}$.
Representation by Dunkl operators
---------------------------------
Let $\mathcal{D}^{rat}_t$ be the localization of the $n$-dimensional Weyl algebra $\mathcal{A}^{rat}_t$ by the ideal generated by $\delta(x)$. The Weyl algebra $\mathcal{A}^{rat}_t$ is generated by elements $x_i,p_i$, $1\le i\le n$ modulo relations: $$[x_i,x_j]=0=[p_i,p_j],\quad [x_i,p_j]=t\delta_{ij},\quad 1\le
i,j\le n.$$
Let us denote by $\mathcal{D}_t^{trig}$ the trigonometric version of the algebra $\mathcal{D}^{rat}_t$. This algebra is localization by $(\delta(X))$ of the algebra $\mathcal{A}_t^{trig}$ with generators $p_i, X^{\pm 1}_i$, $i=1,\dots,n $ modulo relations: $$\label{AA}
[X_i,X_j]=0=[p_i,p_j],\quad [X_i,p_j]=t\delta_{ij}X_i,\quad 1\le
i,j\le n.$$ It is easy to see that the ring $\mathcal{A}_t^{trig}$ is isomorphic to the ring of differential operators on the torus $({\mathbb C}^*)^n$.
\[Dunklpr\] The homorphisms $g^{rat}$: $H^{rat}_{t,c}
\to D^{rat}_t \#S_n$, $g^{trig}$: $H^{trig}\to
\mathcal{D}^{trig}_t\#S_n$ defined by the formulas $$\begin{gathered}
g^{rat}(y_i)=p_i+c\sum_{j\ne i}\frac1{x_i-x_j}(s_{ij}-1),\\
g^{rat}(x_i)= x_i, \quad
g^{rat}(w)= w,\\
g^{trig}(y_i)= p_i+
c\sum_{j<i}\frac{X_i}{X_i-X_j}(s_{ij}-1)+c\sum_{j>i}
\frac{X_j}{X_i-X_j}(s_{ij}-1),\\ g^{trig}(X_i)=X_i, \quad
g^{trig}(w)= w,\end{gathered}$$ ($i=1,\dots,n$) is injective.
This proposition allows to prove the PBW type result for these algebras.
Calogero-Moser spaces
---------------------
In this subsection we give a definition of the Calogero-Moser space in the rational and trigonometric cases.
Let $CM'_{rat}$ be the subset of $\mathfrak{gl}(n,{\mathbb C})\times\mathfrak{gl}(n,{\mathbb C})$ consisting of the elements $(x,y)$ satisfying the equation $$rk([x,y]+Id)=1.$$ By $CM'_{trig}\subset GL(n,{\mathbb C})\times \mathfrak{gl(n,{\mathbb C})}$ we denote the subset of pairs $(X,y)$ satisfying: $$rk(X^{-1}yX-y+Id)=1.$$
The group $GL(n,{\mathbb C})$ acts on the spaces $CM'_{rat}$ and $CM'_{trig}$ by conjugation. This action is free.
The quotient of $CM'_{rat}$ ($CM'_{trig}$) by the action of $GL(n,{\mathbb C})$ is called the rational (trigonometric) Calogero-Moser space. We use the notation $CM_{rat}$ (respectively $CM_{trig}$) for this space.
The rational (trigonometric) Calogero-Moser space $CM_{rat}$ ($CM_{trig}$) is an irreducible smooth variety of dimension $2n$.
For the rational Calogero-Moser space this statement is proved in section 1 of [@W]. The proof in the trigonometric case almost identically repeats the proof in the rational case.
The Calogero-Moser spaces $CM_{rat}$ and $CM_{trig}$ are the configuration spaces for the rational and trigonometric integrable Calogero-Moser systems. The Poisson structures corresponding to these systems are the results of the Hamiltonian reduction of the natural Poisson structures on the spaces $\mathfrak{gl}(n,{\mathbb C})\oplus \mathfrak{gl}^*(n,{\mathbb C})$ and $T^*GL(n,{\mathbb C})$ (see [@OP]).
The main result for the rational and trigonometric double-affine Hecke algebras
-------------------------------------------------------------------------------
As we mentioned in the first subsection, the algebras $H^{rat}_{0,c}$ $H^{trig}_{0,c}$ are in some sense quasiclassical limits of the double affine Hecke algebra $H_{1,\tau}$. Naturally, the theorems from the previous section have their rational and trigonometric analogs:
Let $H$ be one of three described algebras: $H_{1,\tau}$, $H^{trig}_{0,c}$, $H^{rat}_{0,c}$, $CM$ is the corresponding Calogero-Moser space, and $e$ is the symmetrizer (in the finite Hecke algebra if $H=H_{1,q}$ and in the symmetric group otherwise). Then the following is true:
1. The map $h$: $z\to z e$ is an isomorphism between $Z(H)$ and $eHe$.
2. $Spec(Z(H))$ is an irreducible smooth variety naturally isomorphic to $CM$.
3. The Poisson structure on $CM$ which comes from the noncommutative deformation $eH_{q,\tau}e$ ($eH^{trig}_{t,c}e$, $eH^{rat}_{t,c}e$ respectively) of $eHe$ coincides (up to a constant) with the (Quasi) Poisson structure on $CM$ coming from the (Quasi) Hamiltonian reduction.
4. The left $eHe$-module $He$ is projective and $H=End_{eHe}(He)$.
In particular the algebras $H^{rat}_{0,c}$ and $H^{trig}_{0,c}$ are Azumaya algebras and for these algebras the statement of Corollary \[Azum\] holds with $A^n_\tau$ replaced by $S_n$.
The proof of the theorem in the case $H=H^{rat}_{0,c}$ is completely parallel to the case $H=H_{1,\tau}$.
In the trigonometric case the only difficulty is that the group $GL(2,{\mathbb Z})$ does not act on $H_{0,c}^{trig}$ and we do not have any analog of the Fourier-Cherednik transform. But instead of the Fourier-Cherednik transform one can use the faithful representation $\bar{g}^{trig}$ of $ H_{0,c}^{trig}$. The representation $\bar{g}^{trig}$ is the “bispectral dual” to $g^{trig}$; that is, the role of $X_i$, $1\le i\le n$ is played by $y_i$, $1\le i\le n$.
Let us describe the representation $\bar{g}^{trig}$. The homomorphism $\bar{g}^{trig}: H_{t,c}^{trig}\to {\mathbb C}[P^{\pm
1},y]_{\delta(y)}\# S_n$ is defined by the formulas $$\begin{gathered}
s_{i,i+1}\mapsto
\bar{T}_i=s_{i,i+1}+\frac{c}{y_i-y_{i+1}}(s_{i,i+1}-1),\quad 1\le
i\le n-1,
\\ y_i\mapsto y_i,\quad 1\le i\le n,\\
X_i\mapsto \bar{T}_1\dots \bar{T}_{n-i}wP_1\bar{T}_{n-i+1}\dots
\bar{T}_{n-1},\quad 1\le i\le n,\end{gathered}$$ where $w\in S_n$, $w(1)=n$, $w(i)=i-1$, $i=2,\dots,n$.
[99]{}
I.Cherednik, Double affine Hecke algebras, KZ equations and Macdonald operators, IMRN (1992), no. 9, 171–180.
I.Cherednik, Macdonald’s evaluation conjectures and difference Fourier transform, Invent. Math. vol. 122 (1995), no.1, 191–216.
P.Etingof, V.Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math., vol.147 (2002), no. 2, 243–348.
D.Kazhdan, B.Kostant, S.Sternberg, Hamiltonian group action and dynamic systems of Callogero type, Comm. Pure Appl. Math, vol. 31 (1978), 484-507.
G.Wilson, Collision of Calogero-Moser particles and adelic Grassmanian, Invent. Math. vol. 133 (1998), no.1, 1-41.
S.N.M.Ruijsenaars and H.Shneider, A new class of integrable systems and its relations to solitons, Ann. Phys., vol. 170 (1986), no. 2, 370–405.
V.V.Fock, A.A.Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and $r$-matrices, Moscow Seminar in Mathematical Physics, 67–86, Amer. Math. Soc. Transl. Ser. 2, 191, Amer. Math. Soc., Providence, RI, 1999.
A.Alexeev, Y.Kosmann-Schwarzbach and E.Mienrenken, Quasi-Poisson Manifolds, Can. Journal of Math., vol. 54 (2002), no.1, 3–29
A.Kirillov Jr., Lectures on affine Hecke algebras and Macdonald conjectures, Bull. of AMS, vol.34 (1997), no.3, 251–292.
J.P.Serre, Algebre locale, multiplicites, Lecture Notes in Math., vol 11, 1965, Springer.
R.Steinberg, On the theorem of Pittie, Topology, vol.14 (1975), 173–177.
N.Chriss, V.Ginzburg, Representation theory and complex geometry, Birkhauser Boston, 1997.
J.Bernstein, A.Braverman, D.Gaitsgory, The Cohen-Macaulay property of the category of $(\mathfrak{g},K)$-modules. Selecta Math. (N.S.), vol.3 (1997), no. 3, 303–314.
M.A.Olshanetsky, A.M.Perelomov, Integrable systems and Lie algebras. Mathematical physics reviews, Vol. 3, 151–220, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., 3, Harwood Academic, Chur, 1982.
I.R.Shafarevich, Basic algebraic geometry, Springer, 1994
I. Cherednik, private communication, 2000.
I. Cherednik, Y.Markov, Hankel trasform via double Hecke algebra, math.QA/0004116.
| 0 |
---
abstract: 'Mobile edge clouds (MECs) bring the benefits of the cloud closer to the user, by installing small cloud infrastructures at the network edge. This enables a new breed of real-time applications, such as instantaneous object recognition and safety assistance in intelligent transportation systems, that require very low latency. One key issue that comes with proximity is how to ensure that users always receive good performance as they move across different locations. Migrating services between MECs is seen as the means to achieve this. This article presents a layered framework for migrating active service applications that are encapsulated either in virtual machines (VMs) or containers. This layering approach allows a substantial reduction in service downtime. The framework is easy to implement using readily available technologies, and one of its key advantages is that it supports containers, which is a promising emerging technology that offers tangible benefits over VMs. The migration performance of various real applications is evaluated by experiments under the presented framework. Insights drawn from the experimentation results are discussed.'
author:
- 'Andrew Machen, Shiqiang Wang, Kin K. Leung, Bong Jun Ko and Theodoros Salonidis [^1] [^2] [^3] [^4]'
bibliography:
- 'bibliography.bib'
title: Live Service Migration in Mobile Edge Clouds
---
Cloudlet, containers, edge/fog computing, service migration, virtualization
Introduction
============
Cloud-based mobile applications have become increasingly popular over the recent years [@dinh2013survey]. One key issue therein is to ensure that services are always delivered with good performance. The current centralized structure of the cloud has led to a generally large geographical separation between the users and the cloud infrastructure. In such a setting, end-to-end communication between user and cloud can involve many network hops resulting in high latency; the ingress bandwidth to the cloud may also suffer from saturation as the cloud infrastructure is accessed on a many-to-one basis [@m.satyanarayanan2015].
A promising approach for resolving the above problems is to install computing infrastructures at the network edge. Particularly for real-time applications such as instantaneous object recognition [@ha2013WearableCognitiveAssistance] and safety assistance in intelligent transportation systems (ITS) [@VehicularFogComputing], service applications have to remain in relatively close proximity to their end users in order to ensure low latency and high bandwidth connectivity. This is captured by the newly emerged concept of *mobile edge clouds (MECs)* [@ETSIWhitepaper], as well as similar concepts such as cloudlet [@m.satyanarayanan2015], fog computing [@VehicularFogComputing], follow-me cloud [@FollowMeMagazine], mobile micro-cloud [@wang2015dynamic], and small cell cloud [@becvar2014pimrc]. While these different concepts may propose slightly different implementations, they all suggest placing small cloud infrastructures at the network edge so that users can have seamless connection to cloud services. In particular, MECs are typically placed only one or a few network hops away from the mobile user, thus the communication latency can be kept very low. The ingress traffic into the backhaul network can also be reduced, because large amounts of data can be processed directly at the edge. It is envisioned that MECs will co-exist and work in unison with the existing centralized cloud that we have today, as shown in Figure \[fig:architecture\].
![Mobile edge cloud (MEC) architecture[]{data-label="fig:architecture"}](architecture){width="0.9\columnwidth"}
To maintain the benefits of running services close to the user, when a mobile user moves away from its original location, its service may need to be *migrated* to a new MEC server that is near the current user location [@FollowMeMagazine]. This article focuses on systems aspects of live service migration in MEC environments. The main challenge is how to maintain relatively low service downtime and overall migration time. We address this challenge with a layered framework which decomposes a cloud application into multiple layers so that only those layers that are missing at the destination are transferred. Our framework applies to both virtual machines and containers and can be readily implemented with existing tools. To the best of our knowledge, this is the first systematic study on live migration of MEC applications in a container-based environment.
In the rest of this article, we first summarize the motivation and background, then propose our layered framework for live service migration and its experimentation results.
Live Service Migration: Motivation and Background
=================================================
The need for live service migration in MECs can be illustrated with the example shown in Figure \[fig:architecture\]. Here, mobile user 2 is initially connected to its service running on MEC 1. The direct connection between user 2 and MEC 1 ensures low-latency access to the service. However, after some time, user 2 moves to a location that does not have direct connection to MEC 1; it has direct connection to MEC 2 instead. As shown in [@ha2015adaptive], connecting to MEC 1 in such cases would incur a significantly higher latency (due to backhaul network communication) than connecting to MEC 2 directly. It can be therefore beneficial to migrate user 2’s service from MEC 1 to MEC 2, so that user 2 can continue enjoying low-latency access to its service. Such live migrations may be frequently required, especially in vehicular applications where it is likely that the users have high mobility.
Service migration can be classified into stateful and stateless migrations. Stateless migration does not move application running states, it only redirects the user requests to a new server with a separate instance of the service running. This is applicable for applications which do not keep states for users. However, for interactive services that are becoming increasingly popular today, such as active safety warning, mobile multimedia, and mobile online gaming, it is very likely that the application needs to keep some state for each user. We therefore focus on *stateful* migration in this article, which involves moving running states of applications. We consider the stateful migration of a guest operating system (OS) hosting service applications, where a user receives service for a continuous period of time, and the service application may need to keep some internal state for the user (e.g., some intermediate data processing results). After migration is completed, programs resume exactly where they left off before migration, thus the migration is classified as *live*. The user starts to receive service before migration occurs, and it continues receiving service after migration.
### Optimization of Migration Decisions {#optimization-of-migration-decisions .unnumbered}
Migration can incur service interruption as well as computation and communication resource overheads. Therefore, the decision on whether, when, and where to migrate depends on many aspects, such as user mobility, communication channel characteristics, resource availability at MECs, etc., which is a sophisticated optimization problem. In essence, there is a trade-off between the cost of migration and the benefit after migration. Algorithms for making migration decisions need to balance this trade-off. They usually need to predict the future service demands with some accuracy or buffer service requests in queues so that they can be served in batches possibly after migration. Readers are referred to [@wang2015dynamic; @ceselli2015cloudlet] on how live migration decisions can be formulated as optimization problems and how to solve these problems.
### Execution of Live Migration {#execution-of-live-migration .unnumbered}
An active systems research challenge is how to efficiently execute live migration in a practical cloud system containing MECs. We first recall that services running on a cloud platform are likely to have OS and application dependencies that need to be met by their host system. Therefore, a service application is often encapsulated into its own self-sufficient and pre-configured environment for easy distribution. Current examples of such an environment are the well established hypervisor-based *virtual machines (VMs)*, or the relatively new technique, *containers*. Both technologies allow the creation and running of multiple isolated guest OSs on top of a host OS. The main difference between the two technologies is that VMs fully emulate the OS kernel and hardware, whereas containers directly share the hardware and kernel with their host machines. As a result, containers occupy much less resources and have lower virtualization overhead than VMs, but are less adaptable, e.g., a Linux container cannot run on a Windows server.
Recent effort towards the implementation of service migration in MEC environments has focused on VM migration [@ha2015adaptive]. Container migration is a relatively new area which has not been systematically studied in the literature. As containers usually have a much smaller size than VMs, it can be very beneficial to run container-based applications on MECs that have limited storage and processing capability. Thus, a natural question to ask is how to support the live migration of containers. We particularly would like to support container live migration without drastically changing the existing container implementations, so that minimal effort is required to add this functionality to existing systems.
We should also note that there are existing VM live migration methods for cloud environments [@LiveMigrationSurvey]. However, most of them are built for data centers, requiring the use of storage area networks (SANs) and shared storage. Moreover, these methods are usually specific to the underlying virtualization technology. The method presented in this article is designed to work over wide area networks (WANs) which is envisioned to be the way that MECs are interconnected, and it is a generic mechanism that applies for different types of containers and VMs.
A Layered Framework for Live Migration
======================================
We present a generic layered migration framework using incremental file synchronization, which *works with both containers and VMs*. A benefit of this framework is that it is built based on readily available functionalities in most container and VM technologies that are popular today, which means that one does not need modify the internals of container and VM implementation in order to apply this framework.
We focus on LXC ([linuxcontainers.org](linuxcontainers.org)) and KVM ([www.linux-kvm.org](www.linux-kvm.org)) as representative technologies for containers and virtual machines, respectively. LXC and KVM are chosen for their popularity, and their ability to run Linux-built applications without modification. We note that this article focuses on the migration of back-end application components between different MECs, where LXC and KVM are applicable because the server often has a Linux-based operating system. The core idea of the method we use for live migration is derived from [@tychoandersen2014], which proposed an LXC live migration mechanism. We have largely extended [@tychoandersen2014] so that our approach works with multiple layers (see below), applies (with minor alterations) to the live migration of KVM and undoubtedly other container and VM technologies as well.
In the following, we first describe a basic procedure that we have developed for performing stateful live migrations in MEC environments. Then we describe our layered framework built on top of this basic procedure that optimizes live migration time further.
Basic Procedure of Stateful Live Migration in MECs
--------------------------------------------------
To migrate an application, the in-memory state of a running guest OS is recorded, transferred, and then recreated at the destination. The in-memory state includes the applications, system processes, and resources currently loaded into memory for quick access. It contains the progress (state) of running applications, including any data that the application is currently working on. Transferring the in-memory state makes it possible to restore an application exactly from where it was suspended.
The migration framework we present uses accessible tools that already exist in container and VM technology. In LXC, this goes by the name of *checkpointing*, and in KVM, *saving*. Both methods suspend the guest OS (at which point the service is temporarily stopped) and save down the in-memory state of the guest OS into one or more files that can be easily transferred. Complementary tools exist to restore the guest OS from their checkpoint (or save).
After the in-memory state is saved into files, the next step is to use a file transfer protocol to transfer the guest OS’s filesystem (i.e., all files saved on the hard disk of the guest OS) and saved in-memory state[^5]. For this we use incremental file synchronization, namely in the form of *rsync* – a well-known file syncing tool ([rsync.samba.org](rsync.samba.org)), to compress and transfer files. A major difference of incremental file synchronization over other basic file transfer protocols is its ability to identify and transfer only those files, or parts thereof, which are different from those already located at the destination. This can substantially reduce the amount of data that needs to be transferred, particularly with our layered framework presented next.
Layering
--------
(base) \[process, minimum height=3.5cm, align=center, rounded corners=0.5cm, fill=blue!30\] [Base\
(e.g. Ubuntu Server)]{}; (app1) \[process, right of=base, yshift=1cm, minimum height=1.6cm, align=center, rounded corners=0.2cm, fill=blue!10\] [Application 1\
(e.g. Game Server)]{}; (app2) \[process, right of=base, yshift=-1cm, align=center, minimum height=1.6cm, rounded corners=0.2cm, fill=blue!10\] [Application 2\
(e.g. Face Detection)]{}; (instance1) \[process, right of=app1, yshift=0.5cm, minimum height=0.7cm, rounded corners=0.2cm, fill=blue!3\] [Instance 1]{}; (instance2) \[process, right of=app1, yshift=-0.5cm, minimum height=0.7cm, rounded corners=0.2cm, fill=blue!3\] [Instance 2]{}; (instance3) \[process, right of=app2, minimum height=0.7cm, rounded corners=0.2cm, fill=blue!3\] [Instance 3]{}; (baseLayer) \[draw=none, above of=base, yshift=-0.9cm, font=**\] [Base Layer]{}; (appLayer) \[draw=none, above of=app1, yshift=-1.9cm, font=**\] [Application Layer]{}; (instanceLayer) \[draw=none, above of=instance1, yshift=-2.4cm, font=**\] [Instance Layer]{}; (1.5,-2) – (1.5,2.5); (1.5,2.5) – (4.7,2.5); (4.7,-2) – (4.7,2.5); (1.5,-2) – (4.7,-2); (base) – (app1); (base) – (app2); (app1) – (instance1); (app1) – (instance2); (app2) – (instance3);******
(start) \[startstop\] [Source:\
Start migration]{}; (instanceFound) \[decision, below of=start, yshift=-2cm\] [Destination: Instance found]{}; (appFound) \[decision, below of=instanceFound, yshift=-1cm\] [Destination: App found]{}; (baseFound) \[decision, below of=appFound, yshift=-1cm\] [Destination: Base found]{}; (cloneBase) \[process, minimum width=3.5cm, text width=3.5cm,left of=baseFound, xshift=-3.5cm\] [Destination:\
Clone base as app]{}; (rsyncBase) \[process, minimum width=3.5cm, text width=3.5cm,below of=cloneBase, yshift=0.5cm\] [Source to destination:\
*rsync* base filesystem]{}; (cloneApp) \[process, minimum width=3.5cm, text width=3.5cm,left of=appFound, xshift=-3.5cm\] [Destination:\
Clone app as instance]{}; (rsyncApp) \[process, minimum width=3.5cm, text width=3.5cm,above of=cloneBase, yshift=-0.5cm\] [Source to destination:\
*rsync* app filesystem]{}; (checkpoint) \[process, minimum width=3.5cm, text width=3.5cm,left of=instanceFound, xshift=-3.5cm\] [Source:\
Suspend instance]{}; (rsyncInstance) \[process, minimum width=3.5cm, text width=3.5cm, above of=checkpoint, yshift=-0cm, align=center\] [Source to destination:\
*rsync* instance filesystem & in-memory state]{}; (end) \[startstop, left of=start, xshift=-3.5cm\] [Destination:\
Restore instance]{}; (start) – (instanceFound); (instanceFound) – node\[anchor=west\] [N]{} (appFound); (appFound) – node\[anchor=west\] [N]{} (baseFound); (baseFound) |- node\[anchor=west\] [N]{} (rsyncBase); (instanceFound) – node\[anchor=south\] [Y]{} (checkpoint); (appFound) – node\[anchor=south\] [Y]{} (cloneApp); (baseFound) – node\[anchor=south\] [Y]{} (cloneBase); (rsyncBase) – (cloneBase); (cloneBase) – (rsyncApp); (rsyncApp) – (cloneApp); (cloneApp) – (checkpoint); (checkpoint) – (rsyncInstance); (rsyncInstance) – (end); (3,-6) – (3,-9.2); (-7.5,-6) – (-7.5,-9.2); (3,-9.2) – (-7.5,-9.2); (3,-6) – (-7.5,-6);
The problem with migrating an encapsulated service directly using the above basic live migration procedure is that the “package” contains the guest OS, virtualization data, and, for VMs, the system kernel, which are required to make the service self-contained. A base package with no services installed can have a size that tips the scale at about $400$ MB and $2.7$ GB for LXC and KVM, respectively. Our experiments using the live migration method discussed above have shown that, for a $100$ Mbps bandwidth connection, the average migration time for a base package is $25$ seconds and $160$ seconds for LXC and KVM, respectively, during which time the service is down and may appear as an unresponsive or frozen application.
In our approach, we aim to reduce service downtime (i.e., time of service interruption) and overall migration time (i.e., time from the beginning till the end of the whole migration process) through the use of layers. Abstractly, we can separate the base package (that includes the guest OS, kernel, etc., but with no applications installed) into a layer separate and distinct from its service applications. We call this package the *base layer*. This base layer is generic and all MECs shall have a copy of it. The service applications and their running states are placed within a separate layer called the *instance layer*. Assuming the base layer exists at the destination, when we want to migrate a service, we first suspend the service and then transfer only the instance layer to the destination. A running service can be reconstructed from a combination of the base and instance layers. By removing the need to transfer the base package in every migration, we can drastically cut down the amount of data that needs to be transferred, and in turn reduce the service downtime and overall migration time.
We extend this *two-layer* approach further, by splitting out the application from the instance layer into an intermediate *application layer* that contains an idle version of the application and any application-specific data. The instance layer now only needs to contain the running state (i.e., in-memory state) of a service. When we want to migrate a service, the application layer can be migrated first whilst the service is running, then the service is suspended and the instance layer is transferred to the destination. A running service can be reconstructed from a combination of the base layer, instance layer, and associated application layer. By doing this, we are able to transfer the majority of an encapsulated service’s program and data before suspending the service, leaving only the running state transfer to count towards the service downtime. This three-layer setup is shown in Figure \[fig:threeLayerModel\], and is referred to as the *three-layer model*.
The intermediate application layer of the three-layer model has advantages other than improvements to service downtime. A distinct application layer allows the application and its related data to be distributed independently of a running service. Through smart pre-distribution of services, possibly by application caching based on demand prediction, the overall migration time can be reduced to only the time it takes to migrate the instance layer.
In the current implementation, a layer contains its unique data (including files and possibly in-memory state) plus all the data from its preceding layers. We achieve pseudo-incremental layering for migration, by first cloning a lower layer (e.g., application layer), and then using incremental file synchronization to transfer only the difference between that and the higher layer (e.g., instance layer) we want to recreate. This process of cloning and incremental file synchronization is repeated depending on what layers are missing at the destination, the complete migration process is shown in Figure \[fig:migrationMechanism\]. The time to start the migration and the destination of migration can be determined by an optimization algorithm, see [@wang2015dynamic; @ceselli2015cloudlet] for details.
Experimentation Results and Discussions
=======================================
We study the live migration performance of our layered framework for a variety of applications. We ran experiments in three “host” VMs connected to an emulation framework, where the emulation framework is the open-source Common Open Research Emulator (CORE) [@core]. The host VMs were each given $2$ virtual CPU cores and $2$ GB of virtual memory from a physical machine with $2.6$ GHz Intel Core i7 and a total of $16$ GB $1600$ MHz DDR3 memory. Two host VMs acted as MECs, between which migration was carried out, and the third host VM acted as a user requiring MEC service. Unless otherwise specified, the bandwidth between MECs was configured as $100$ Mbps with only system-inherited latency and jitter. The connection between each MEC and the user was configured with $100$ Mbps bandwidth, $25$ ms latency, and $5$ ms jitter. These connection specifications are typical for wide-area wired networks and local wireless networks, on which MECs and their users operate. They are much more inferior than network connections in data centers. Migration times were calculated by a migration script at each stage of the migration process. We use *rsync* for incremental file synchronization and the amount of transferred data was measured using Wireshark.
Nested KVM and LXC, which contain running applications, were run inside the two host VMs that mimic MECs. The guest OS of the nested KVM and LXC was Ubuntu 15.10. Note that the base OS image sizes for KVM and LXC are different due to their different virtualization mechanisms, but this has no bearing in our results since, in our experiments, the base layer resides in every MEC and is not included in the migration. As we see below, the installation footprint of applications can be larger for LXC than for KVM, because the Ubuntu installation of LXC has fewer packages included as standard. As such, the installation of any missing packages is counted into the LXC application size.
We studied the migration of the following applications using our migration framework presented above.
1. *Game Server* runs the *sauerbraten-server* package, a server for the online game Cube 2. It sends and receives regular packets related to player location and other match statistics. The installation footprint is very small ($0.7$ MB), as is the memory requirement (approximately $1$ MB).
2. *RAM Simulation* is a simple script (approximately $0.1$ MB) that consumes a large amount of RAM, where the exact RAM consumption is user-defined and the RAM contents keep changing over time. It represents memory intensive applications, such as those that process large data sets or perform complex calculations (e.g., big data analysis, training of deep neural networks, etc.). Unless otherwise specified, RAM utilization is maintained at around $330$ MB.
3. *Video Streaming* uses the *vlc-nox* package (approximately $280$ MB for LXC, $230$ MB for KVM) to stream video to a user. A $50$ MB video file is stored with the application at the MEC. Video Streaming has a low memory requirement (approximately $30$ MB).
4. *Face Detection* uses the *OpenCV* library to process an incoming video stream. It detects faces in the video received, and sends the detection result, embedded into each video frame, back to the user. This application has a very large installation footprint (approximately $655$ MB for LXC, $565$ MB for KVM), and a moderate memory requirement of approximately $100$ MB.
5. *No Application* is a guest OS with no applications installed, and therefore requires no additional resources. It represents the overhead, or minimum bound, on migration.
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The experimentation results (averaged over $10$ independent experiments in each case) are shown in Table \[tab:threeLayers\] and Figures \[fig:lxcMigrationTimesforApps\] and \[fig:ramUsage\]. For a particular setup, the service downtime is always smaller than the total migration time, because the total migration time also includes the time taken for cloning and data transfer (during which the service remains running). The time required for different migration stages are unequal for different applications, as shown in Figure \[fig:lxcMigrationTimesforApps\], due to the different installation and RAM occupation sizes. This aligns with the layered abstraction discussed earlier. For example, the time required for *rsync*ing the application filesystem is related to the application installation size, and the time required for *rsync*ing the instance in-memory state is related to the application’s RAM usage.
Container vs. Virtual Machine
-----------------------------
Although it is understood that containers are more lightweight than VMs, a quantitative view on the difference in their migration performance does not exist in the literature. The container-supported migration framework presented above allows us to provide such a quantitative view and draw further insights.
We can see from the experimentation results that for all the example applications shown, LXC has a clear advantage over KVM in terms of total migration time, service downtime, and amount of transferred data. This is mainly because containers are more compact and the filesystem and in-memory contents of a container is mostly relevant to the application; whereas the filesystem and in-memory contents of a VM can be related to many other background processes irrelevant to the considered application, and *rsync* (or any other incremental file synchronization mechanism) needs to remotely compare a larger amount of data and may not be able to filter out everything that does not belong to the application.
From Figure \[fig:ramUsage\] (top), we see that for the RAM Simulation application, the migration times of both LXC and KVM are approximately linear in RAM utilization. This linear relationship is because the amount of in-memory state data that needs to be transferred is proportional to the RAM usage. The lower bound on RAM usage represents the overhead required to migrate LXC or KVM. Depending on the application’s RAM utilization, we can migrate approximately up to $19\mathsf{x}$ the number of containers compared to VMs within the same amount of time. That said, the relative advantage of LXC over KVM is reduced as the RAM utilization increases. At $400$ MB of RAM usage, LXC is $5\mathsf{x}$ faster to migrate than KVM; whereas at $600$ MB of RAM usage, LXC is only $4\mathsf{x}$ faster to migrate than KVM.
Figure \[fig:ramUsage\] (bottom) shows that the relative advantage of LXC over KVM increases as bandwidth increases. At $1$ Mbps, LXC is $2\mathsf{x}$ faster to migrate than KVM, whereas at $10$ Mbps LXC is $4\mathsf{x}$ faster. We found in our experiments that the data transmission rate is capped by how fast *rsync* is able to compare files and compress data. When the bandwidth is over $50$ Mbps, the migration time remains about the same, where LXC is $8\mathsf{x}$ faster to migrate than KVM.
Containers also have their shortcomings though. By relying on the host system for both the hardware and kernel, they are less adaptable than VMs. They can be nested within VMs to regain their adaptability, but this is likely to degrade performance. Instead, it is advisable that MEC infrastructure is chosen to ensure container compatibility, in as much as the OS (most likely a Linux derivative) being consistent across the MEC network.
There are also considerations other than performance that will need to be addressed, notably security, in order for containers to be recommended over VMs in all scenarios. Security, a pre-requisite for any enterprise software, is a serious concern with containers, as unlike their VM counterparts they are not fully isolated from their host system, and can therefore be more susceptible to attack from a compromised host.
Two Layers vs. Three Layers
---------------------------
From the experimentation results, we see that most applications respond positively towards the three-layer model. For LXC, when the application has been found at the migration target, Table \[tab:threeLayers\] and Figure \[fig:lxcMigrationTimesforApps\] show that the Video Streaming and Face Detection applications have significant reductions in the total migration time compared to using only two layers, in the orders of $3\mathsf{x}$ for both. Even more significant is the reduction in service downtime for these applications, in the orders of $7\mathsf{x}$ and $13\mathsf{x}$, respectively. This service downtime reduction is particularly important for a seamless experience for the end user. The most significant is the reductions in the amount of data transferred over the network for migration, which are respectively in the orders of $24\mathsf{x}$ and $36\mathsf{x}$. This is particularly important for bandwidth or time constrained connections. Total migration times and data transfers for the Game Server, RAM Simulation, and No Application do not improve significantly, but are also not worse than with two layers when the application is found at the migration target. The relative reductions in total migration time and data transfers are smaller for KVM, because the application and associated data count for a smaller proportion of the total size of a KVM virtual machine.
We note that Video Streaming and Face Detection both have large filesystems compared to RAM Simulation and Game Server, and it is from this the three-layer model derives its benefit. The larger the application or associated data, the greater the benefit from introducing an intermediate layer. This can be seen in Figure \[fig:lxcMigrationTimesforApps\], which shows how the migration of application data can account for a significant portion of the total time it takes to migrate an application.
The three-layer model does have a downside, in that if the application has not been found at the migration destination, the total migration time can be longer than with only two layers, because additional time is required for cloning the application into an instance and *rsync*ing the third layer (see Figure \[fig:lxcMigrationTimesforApps\]). However, the service downtime (which is usually the more critical factor for MEC applications) remains the same no matter whether the application is found at the destination or not, because the service is suspended only after the application layer (containing an idle application) has been transferred.
Therefore, it is apparent that the benefits of the three-layer model need to be balanced with any trade-offs incurred. Factors such as how frequently the application is used and how much service downtime impacts the user experience should be taken into account when deciding whether to use two or three layers. Such decisions have to be made based on application characteristics. For example, it may be better for applications with small installation sizes and intensive RAM utilization to use only two layers for quicker migration.
A practical implementation of an MEC system can support both two-layer and three-layer models, since their underlying mechanisms are similar. The system can perform some simple profiling of applications and historical migration performance, and it can decide whether to use two or three layers on a case-by-case basis in real time.
Open Issues
===========
Our current implementation of containers uses the default directory backing store, and as a result, each additional layer duplicates the entire filesystem of the previous layer plus the new data. For a layered setup like the one we demonstrated, a more ideal solution would be to try using an overlay filesystem such as overlayFS, which allows the sharing of lower-layer files with different upper layers. This would provide a much more efficient usage of storage, and is much closer to the abstract model. However, LXC currently does not support overlayFS yet (as of version 1.1.5).
Another way of potentially improving performance, especially service downtime, would be to investigate the use of iterative migration. Iterative migration is the process of transferring memory pages whilst the service is running, so that when the service is finally suspended and migrated, in theory, only a small portion of the remaining in-memory state needs to be transferred.
For our experiments, LXC offered the greatest ease for running different applications without specialist knowledge in order to set them up. Other container technologies exist, among them Docker ([www.docker.com](www.docker.com)) which has become the de facto standard for containerization. Future research should investigate whether Docker or other container technologies offer a better overall solution. To answer this question, consideration should be given as to what technology is likely to receive wide-spread adoption from industry, and therefore have the greatest impact. An interesting development to this is the work being carried out by the Open Container Initiative ([www.opencontainers.org](www.opencontainers.org)) which aims to create an open industry standard around container formats and run-times. Extensive industrial backing and a foundation based on Docker’s format and run-times makes this a group to follow.
Besides migrating back-end applications between different MEC servers, another interesting aspect that is worth studying is the “vertical” migration of code and data between the mobile device and MEC, to strike a balance among device resource consumption and service quality. This is also known as application/computation offloading [@mobileOffload]. The challenge here is that the mobile device’s operating system is often different from the server’s operating system. A proper encapsulation mechanism that works on both platforms is needed to facilitate the migration. Since the layered framework presented in this article is applicable to a general class of encapsulation methods that supports the suspension of applications, we envision that a similar approach can be applied to live migration between mobile device and MEC. A detailed study on the vertical migration can be conducted in the future.
Future work should also study the live migration performance under large-scale networked MEC systems. The performance of migration decision making in large-scale MEC environments has been mainly studied using simulations, where no real application migration is carried out [@wang2015dynamic; @ceselli2015cloudlet]. We have focused on the other end of the problem in this article, namely the implementation of live migration itself. In the future, it is worthwhile to study the migration of real applications in a realistic, large-scale networked system, which would connect the theoretical results in [@wang2015dynamic; @ceselli2015cloudlet] with the systems work in this article.
Summary
=======
We have presented a layered framework for service migration in MECs. The framework supports both container and VM technologies, and it can be easily implemented using existing functionalities of popular container and VM implementations. Extensive experimentation results on the performance of different approaches for various applications under different scenarios have been presented. In general, the three-layer model with a container-based encapsulation environment gives the best overall performance, but other options may be preferred in specific cases, as discussed. The three-layer model also allows the pre-caching of popular applications at MECs, so that the time required for future instantiation of such applications can be shortened. In addition, as migration is performed on the entire container or VM, the underlying service applications do not need to be specifically modified to support migration. This makes it easy to run existing applications in the migration framework. A future implementation may also specify an optional interface between the framework and the application, so that the application can announce when it prefers or does not prefer to be migrated, thereby improving the migration performance.
Acknowledgement {#acknowledgement .unnumbered}
===============
Some preliminary results only on the three-layer model were presented as a 2-page poster abstract in [@machen2016migration]. This article provides more comprehensive discussion and experimentation results.
Andrew Machen’s contribution to this work was performed while he was affiliated with Imperial College London and IBM U.K.
This research was sponsored in part by the US Army Research Laboratory and the UK Ministry of Defense and was accomplished under Agreement Number W911NF-16-3-0001. The views and conclusions contained in this document are those of the author(s) and should not be interpreted as representing the official policies, either expressed or implied, of the US Army Research Laboratory, the US Government, the UK Ministry of Defense or the UK Government. The US and UK Governments are authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon.
[^1]: A. Machen was with Imperial College London and IBM, United Kingdom, when this work was performed. Email: [email protected]
[^2]: S. Wang, B. J. Ko, and T. Salonidis are with IBM T. J. Watson Research Center, Yorktown Heights, NY, USA, Email: {wangshiq, bongjun\_ko, }@us.ibm.com
[^3]: K. K. Leung is with Imperial College London, United Kingdom. Email: [email protected]
[^4]: This is the author’s version of the paper accepted for publication in IEEE Wireless Communications. 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
[^5]: Note that besides the filesystem and in-memory state, we also need to transfer any additional files/data related to the container or VM virtualization itself. We do not specifically discuss those additional portions of data, and only emphasize on the main part of data being transferred. Also, the filesystem and in-memory state may be saved into a single file (in the case of a VM image) or multiple files (in the case of containers). We do not separately discuss them since *rsync* can find differences in two single files as well as multiple files.
| 0 |
---
abstract: |
Any infinite uniformly recurrent word ${\bf u}$ can be written as concatenation of a finite number of return words to a chosen prefix $w$ of ${\bf u}$. Ordering of the return words to $w$ in this concatenation is coded by derivated word $d_{\bf u}(w)$. In 1998, Durand proved that a fixed point ${\bf u}$ of a primitive morphism has only finitely many derivated words $d_{\bf u}(w)$ and each derivated word $d_{\bf u}(w)$ is fixed by a primitive morphism as well. In our article we focus on Sturmian words fixed by a primitive morphism. We provide an algorithm which to a given Sturmian morphism $\psi$ lists the morphisms fixing the derivated words of the Sturmian word ${\bf u} = \psi({\bf u})$. We provide a sharp upper bound on length of the list.
*Keywords:* Derivated word, Return word, Sturmian morphism, Sturmian word
*2000MSC:* 68R15
address:
- 'Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19, Prague 1, Czech Republic'
- 'Department of Applied Mathematics, Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, 160 00, Prague 6, Czech Republic'
author:
- Karel Klouda
- Kateřina Medková
- Edita Pelantová
- Štěpán Starosta
title: Fixed points of Sturmian morphisms and their derivated words
---
[^1]
Introduction
============
Sturmian words are probably the most studied object in combinatorics on words. They are aperiodic words over a binary alphabet having the least factor complexity possible. Many properties, characterizations and generalizations are known, see for instance [@BeSe_Lothaire; @Be_survey_corr; @BaPeSta2].
One of their characterizations is in terms of return words to their factors. Let $\uu = u_0u_1u_2 \cdots$ be a binary infinite word with $u_i \in \{0,1\}$. Let $w = u_iu_{i+1} \cdots u_{i+n-1}$ be its factor. The integer $i$ is called an *occurrence* of the factor $w$. A return word to a factor $w$ is a word $u_iu_{i+1} \cdots u_{j-1}$ with $i$ and $j$ being two consecutive occurrences of $w$ such that $i < j$. In [@Vu], Vuillon showed that an infinite word $\uu$ is Sturmian if and only if each nonempty factor $w$ has exactly two distinct return words. A straightforward consequence of this characterization is that if $w$ is a prefix of $\uu$, we may write $$\uu = r_{s_0}r_{s_1}r_{s_2}r_{s_3}\cdots$$ with $s_i \in \{0,1\}$ and $r_0$ and $r_1$ being the two return words to $w$. The coding of these return words, the word $d_{\uu}(w) = s_0s_1s_2 \cdots$ is called the *derivated word of $\uu$ with respect to $w$*, introduced in [@Durand98]. A simple corollary of the characterization by return words and a result of [@Durand98] is that the derivated word $d_{\uu}(w)$ is also a Sturmian word (see ). This simple corollary follows also from other results. For instance, it follows from [@AraBru05], where the authors investigate the derivated word of a standard Sturmian word and give its precise description. It also follows from the investigation of a more general setting in [@BuLu], which may in fact be used to describe derivated words of any episturmian word — generalized Sturmian words [@GlJu].
By the main result of [@Durand98], if $\uu$ is a fixed point of a primitive morphism, the set of all derivated words of $\uu$ is finite (the result also follows from [@HoZa99]). In this case, again by [@Durand98], a derivated word itself is a fixed point of a primitive morphism.
In this article we study derivated words of fixed points of primitive Sturmian morphisms. By the results of [@MiSe93], any primitive Sturmian morphism may be decomposed using elementary Sturmian morphisms — generators of the Sturmian monoid. In , we describe the relation between the set of derivated words of a Sturmian sequence $\uu$ and the set of derivated words of $\varphi(\uu)$, where $\varphi$ is a generator of the Sturmian monoid.
The main result of our article is an exact description of the morphisms fixing the derivated words $d_{\uu}(w) $ of $\uu$, where $\uu$ is fixed by a Sturmian morphism $\psi$ and $w$ is its prefix. For this purpose, we introduce an operation $\Delta$ acting on the set of Sturmian morphisms with unique fixed point, see . Iterating this operation we create the desired list of the morphisms as stated in . The Sturmian morphisms with two fixed points are treated separately, see .
We continue our study by counting the number of derivated words, in particular by counting the distinct elements in the sequence $\bigl(\Delta^k(\psi)\bigr)_{k \geq 1}$. This number depends on the decomposition of $\psi$ into the generators of the special Sturmian monoid, see below in .
Using this decomposition, provide the exact number of derivated words for two specific classes of Sturmian morphisms.
For a general Sturmian morphism $\psi$, gives a sharp upper bound on their number. The upper bound depends on the number of the elementary morphisms in the decomposition of $\psi$. In the last section, we give some comments and state open questions.
Preliminaries
=============
An *alphabet* $\mathcal{A}$ is a finite set of symbols called *letters*. A *finite word* of length $n$ over $\mathcal {A}$ is a string $u=u_0u_1\cdots u_{n-1}$, where $u_i \in \mathcal{A}$ for all $i=0,1,\ldots, n-1$. The *length* of $u$ is denoted by $|u| = n$. By $|u|_a$ we denote the number of copies of the letter $a$ used in $u$, i.e. $|u|_a= \# \{i \in \mathbb{N} \colon i< n, u_i = a\}$. The set of all finite words over $\mathcal{A}$ together with the operation of concatenation forms a monoid $\mathcal{A}^*$. Its neutral element is the *empty word* $\varepsilon$ and $\A^+ = \A^* \setminus \left\{ \varepsilon \right\}$. On this monoid we work with two operations which preserve the length of words. The *mirror image* or *reversal* of a word $u=u_0u_1\cdots u_{n-1} \in \mathcal{A}^*$ is the word $\overline{u} = u_{n-1}u_{n-2}\cdots u_{1}u_0$. The *cyclic shift* of $u$ is the word $$\label{eq:def_of_cyc}
{\rm cyc}(u) = u_{1}u_{2}\cdots u_{n-1}u_0.$$
An *infinite word* over $\mathcal {A}$ is a sequence $\uu = u_0u_1u_2\cdots = \left(u_i\right)_{i\in \mathbb{N}} \in \mathcal{A}^{\mathbb{N}}$ with $u_i \in \mathcal{A}$ for all $i \in \N = \left\{ 0,1,2, \ldots \right \}$. Bold letters are systematically used to denote infinite words throughout this article.
A finite word $p \in \mathcal{A}^*$ is a *prefix* of $u=u_0u_1\cdots u_{n-1}$ if $p= u_0u_1u_2\cdots u_{k-1}$ for some $k\leq n$, the word $u_k u_{k+1} \cdots u_{n-1}$ is denoted $p^{-1}u$. Similarly, $p \in \mathcal{A}^*$ is a prefix of $\uu= u_0u_1u_2\cdots$ if $p=u_0u_1u_2\cdots u_{k-1}$ for some integer $k$. We usually abbreviate $u_0u_1u_2\cdots u_{k-1} = \uu_{[0,k)}$.
A finite word $w$ is a *factor* of $\uu= u_0u_1u_2\cdots$ if there exists an index $i$ such that $w$ is a prefix of the infinite word $u_iu_{i+1}u_{i+2}\cdots$. The index $i$ is called an *occurrence* of $w$ in $\uu$. If each factor of $\uu$ has infinitely many occurrences in $\uu$, the word $\uu$ is *recurrent*.
The *language $\mathcal{L}(\uu)$ of an infinite word $\uu$* is the set of all its factors. The mapping $\mathcal{C}_{\uu}: \mathbb{N}\mapsto \mathbb{N}$ defined by $\mathcal{C}_{\uu}(n) = \#\{w \in \mathcal{L}(\uu): |w|=n\}$ is called the *factor complexity* of the word $\uu$.
An infinite word $\uu$ is *eventually periodic* if $\uu = wvvvvv\cdots $ for some $v,w \in \mathcal{A}^*$. If $w$ is the shortest such word possible, we say that $|w|$ is the *preperiod* of $\uu$; if $v$ is the shortest possible, we say that $|v|$ is the *period* of $\uu$. If $\uu$ is not eventually periodic, it is *aperiodic*. A factor $w$ of $\uu$ is a *right special* factor if there exist at least two letters $a,b \in \mathcal{A}$ such that $wa, wb$ belong to the language $\mathcal{L}(\uu)$. A *left special* factor is defined analogously.
An infinite word $\uu$ is eventually periodic if and only if $\mathcal{L}(\uu)$ contains only finitely many right special factors. Equivalently, $\uu$ is eventually periodic if and only if its factor complexity $\mathcal{C}_{\uu}$ is bounded. On the other hand, the factor complexity of any aperiodic word satisfies $\mathcal{C}_{\uu}(n)\geq n+1$ for every $n \in \mathbb{N}$.
An infinite word $\uu$ with $ \mathcal{C}_{\uu}(n)= n+1$ for each $n \in \mathbb{N}$ is called Sturmian. A Sturmian word is *standard* (or *characteristic*) if each of its prefixes is a left special factor.
Derivated words {#SecDerivatedWords}
---------------
Consider a prefix $w$ of an infinite recurrent word $\uu$. Let $i<j$ be two consecutive occurrences of $w$ in $\uu$. The string $u_iu_{i+1}\cdots u_{j-1}$ is a *return word* to $w$ in $\uu$. The set of all return words to $w$ in $\uu$ is denoted by $\mathcal{R}_{\uu}(w)$. Let us suppose that the set of return words to $w$ is finite, i.e. $\mathcal{R}_{\uu}(w) = \{r_0, r_1, \ldots, r_{k-1}\}$. The word $\uu$ can be written as unique concatenation of the return words $\uu = r_{s_0}r_{s_1}r_{s_2}\cdots$. The *derivated* word of $\uu$ with respect to the prefix $w$ is the infinite word $$d_{\uu}(w) = s_0s_1s_2 \cdots$$ over the alphabet of cardinality $ \#\mathcal{R}_{\uu}(w)=k $. In his original definition, Durand [@Durand98] fixed the alphabet of the derivated word to the set $\{0,1, \ldots, k-1\}$. Moreover, Durand’s definition requires that for $i< j$ the first occurrence of $r_i$ in $\uu$ is less than the first occurrence of $r_j$ in $\uu$. In particular, a derivated word always starts with the letter $0$. In the article [@AraBru05], where derivated words of standard Sturmian words are studied, the authors required that the starting letters of the original word and its derivated word coincide. For our purposes, we do not need to fix the alphabet of derivated words: two derivated words which differ only by a permutation of letters are identified one with another.
In the sequel, we work only with infinite words which are *uniformly recurrent*, i.e. each prefix $w$ of $\uu$ occurs in $\uu$ infinitely many times and the set $\mathcal{R}_{\uu}(w)$ is finite. Our aim is to describe the set $$\Der(\uu) = \{ d_{\uu}(w) \colon w \text{ is a prefix of } \uu\}.$$ Clearly, if a prefix $w$ is not right special, then there exists a unique letter $x$ such that $wx \in \mathcal{L}(\uu)$. Thus the occurrences of $w$ and $wx$ coincide, $\mathcal{R}_{\uu}(w) = \mathcal{R}_{\uu}(wx)$ and $d_{\uu}(w)$ = $d_{\uu}(wx)$. If $\uu$ is not eventually periodic, then $w$ is a prefix of a right special prefix of $\uu$. Therefore for an aperiodic uniformly recurrent word $\uu$ we have $$\Der(\uu) = \{ d_{\uu}(w) \colon w \text{ is a right special prefix of } \uu \}.$$
Sturmian words
--------------
Any Sturmian word $\uu$ can be identified with an upper or lower mechanical word. A mechanical word is described by two parameters: slope and intercept. The slope is an irrational number $\gamma \in (0,1)$ and the intercept is a real number $\rho \in [0,1)$. To define the lower mechanical word ${\bf s}(\gamma, \rho) = \left(s_{n} (\gamma, \rho) \right)_{n \in \N}$ we put $I_0 = [0,1-\gamma)$. The $n^{th}$ letter of ${\bf s}(\gamma, \rho)$ is as follows: $$s_n(\gamma, \rho) =
\begin{cases}
0 & \text{if the number } \gamma n + \rho \mod 1 \text{ belongs to } I_0, \\
1 & \text{otherwise.}
\end{cases}$$ The definition of the upper mechanical word ${\bf s}'(\gamma, \rho) = \left(s'_n(\gamma, \rho) \right)_{n \in \N}$ is analogous, it just uses the interval $I_0 = (0,1-\gamma]$. Let us stress that $s_n(\gamma, \rho) \neq s'_n(\gamma, \rho)$ for at most two neighbouring indices $n$ and $n+1$. All upper and lower mechanical words with irrational slope are Sturmian and any Sturmian word equals to a lower or to an upper mechanical word. Let us stress that one-sided Sturmian words with irrational slope are always uniformly recurrent. The language of a Sturmian word depends only on $\gamma$. The number $\gamma$ is in fact the density of the letter $1$, i.e., $\gamma = \lim\limits_{n\to \infty}\tfrac1n \# \left \{i\in \mathbb{N} \colon i < n, s_i(\gamma, \rho) = 1 \right \} $. Consequently, $1-\gamma$ is the density of the letter $0$.
For any irrational $\gamma \in (0,1)$ there exists a unique mechanical word ${\bf c}(\gamma)$ with slope $\gamma$ such that both $0{\bf c}(\gamma)$ and $1{\bf c}(\gamma)$ are Sturmian. The word ${\bf c}(\gamma)$ is a standard Sturmian word and ${\bf c}(\gamma) ={\bf s}(\gamma, \gamma) ={\bf s}'(\gamma, \gamma)$. Many further properties of Sturmian words can be found in [@Lo83; @BeSe_Lothaire].
For our study of derivated words, the following result of Vuillon from [@Vu] is important: a word $\uu$ is Sturmian if and only if any prefix of $\uu$ has exactly two return words. By combining this result with [@Durand98], we obtain an essential observation about derivated words of Sturmian words, which also follows from [@AraBru05].
\[thm:derstst\] If $\uu$ is a Sturmian word and $w$ is a prefix of $\uu$, then its derivated word $d_{\uu}(w)$ is Sturmian as well.
Set ${\bf v} = d_{\bf u}(w)$. Let $p$ be a prefix of ${\bf v}$. Due to Proposition 2.6 in [@Durand98], there exists a prefix $q$ of $\uu$ such that $d_{\bf v}(p) = d_{\bf u}(q) $. By Vuillon’s characterization of Sturmian words, the word $ d_{\bf u}(q)$ is binary. It means that any prefix $p$ of ${\bf v} $ has two return words in ${\bf v} $ and so ${\bf v} $ is Sturmian.
The Sturmaian words (sequences) were originally defined by Hedlund and Morse in [@HeMo]. Their definition is more general as they consider also biinfinite words and (in terms of our definition above) rational slopes. Hence their Sturmian words may not be recurrent. For details on the history of definition of Sturmian words see [@Fogg], especially the historical remark at page 146. Interestingly enough, the term *derivated* sequence is also used in [@HeMo], however, its definition differs from our one (as taken from [@Durand98]): Using again our terminology, their derivated word is a derivated word with respect to a one-letter word in a biinfinite Sturmian word.
Sturmian morphisms {#sec:stu_mor}
------------------
A *morphism* over $\mathcal{A}^*$ is a mapping $\psi : \mathcal{A}^* \mapsto \mathcal{A}^*$ such that $\psi(vw) = \psi(v)\psi(w)$ for all $v,w \in \mathcal{A}^*$. The domain of the morphism $\psi$ can be naturally extended to $\mathcal{A}^{\mathbb{N}}$ by $$\psi(u_0u_1u_2\cdots ) = \psi(u_0)\psi(u_1) \psi(u_2)\cdots .$$ A morphism $\psi$ is *primitive* if there exists a positive integer $k$ such that the letter $a$ occurs in the word $\psi^k(b)$ for each pair of letters $a,b \in \mathcal{A}$. A *fixed point* of a morphism $\psi$ is an infinite word $\uu$ such that $\psi(\uu) = \uu$.
A morphism $\psi$ is a *Sturmian morphism* if $\psi(\uu)$ is a Sturmian word for any Sturmian word $\uu$. The set of Sturmian morphisms together with composition forms the so-called *Sturmian monoid* usually denoted [*St*]{}. We work with these four elementary Sturmian morphisms: $$\varphi_a: \begin{cases} 0 \to 0 \\ 1 \to 10 \end{cases} \quad
\varphi_b: \begin{cases} 0 \to 0 \\ 1 \to 01 \end{cases} \quad
\varphi_\alpha: \begin{cases} 0 \to 01 \\ 1 \to 1 \end{cases} \quad
\varphi_\beta: \begin{cases} 0 \to 10 \\ 1 \to 1 \end{cases}$$ and with the monoid $\mathcal{M}$ generated by them, i.e. $\mathcal{M} = \langle \varphi_a, \varphi_b, \varphi_\alpha, \varphi_\beta \rangle $. The monoid $\mathcal{M}$ is also called *special Sturmian monoid*. For a nonempty word $u = u_0\cdots u_{n-1}$ over the alphabet $\{a,b,\alpha,\beta\}$ we put $$\varphi_u = \varphi_{u_0} \circ \varphi_{u_1} \circ \cdots \circ \varphi_{u_{n-1}}.$$
The monoid $\mathcal{M}$ is not free. It is easy to show that for any $k \in \N$ we have $$\varphi_{\alpha a^k\beta } = \varphi_{\beta b^k\alpha}\quad \text{ and } \quad \varphi_{a\alpha^kb} = \varphi_{b\beta^ka}.$$ We can equivalently say that the following rewriting rules hold on the set of words from $\{a,b,\alpha,\beta\}^*$: $$\label{eq:relations}
\alpha a^k\beta = \beta b^k\alpha \quad \text{ and } \quad a\alpha^kb = b\beta^ka\qquad \text{ for any $k \in \N$ }.$$
In [@See91], the author reveals a presentation of the Sturmian monoid which includes the special Sturmian monoid $\mathcal{M}= \langle \varphi_a, \varphi_b,\varphi_\alpha, \varphi_\beta\rangle$. A presentation of the special Sturmian monoid follows from this result. It is also given explicitly in [@ReKa07]:
\[thm:relations\] Let $w,v\in \{a,b,\alpha,\beta\}^*$. The morphism $\varphi_w$ is equal to $\varphi_{v}$ if and only if the word $v$ can be obtained from $w$ by applying the rewriting rules .
Note that the presentation of a generalization of the Sturmian monoid, the so-called *episturmian monoid*, is also known, see [@Ri03]. The next lemma summarizes several simple and well-known properties of Sturmian morphisms we exploit in the sequel.
\[lem:properties\_of\_sturm\_morph\] Let $w \in \{a,b,\alpha,\beta\}^+$.
(i) The morphism $\varphi_w$ is primitive if and only if $w$ contains at least one Greek letter $\alpha$ or $\beta$ and at least one Latin letter $a$ or $b$.
(ii) If $\varphi_w$ is primitive, then each of its fixed points is aperiodic and uniformly recurrent.
(iii) If $\varphi_w$ is primitive, then it has two fixed points if and only if $w$ belongs to $\{a,\alpha\}^* $.
For $w \in \{a,b,\alpha,\beta\}^*$ the rules preserve positions in $w$ occupied by Latin letters $ \{a,b\}$ and positions occupied by Greek letters $ \{\alpha,\beta\}$. We define that $a<b$ and $\alpha < \beta$ which allows the following definition.
Let $w \in \{a,b,\alpha,\beta\}^*$. The lexicographically greatest word in $ \{a,b,\alpha,\beta\}^*$ which can be obtained from $w$ by application of rewriting rules is denoted $N(w)$. If $\psi = \varphi_w$, then the word $N(w)$ is the *normalized name* of the morphism $\psi$ and it is also denoted by $N(\psi) = N(w)$.
The next lemma is a direct consequence of Theorem \[thm:relations\].
\[lem:normalized\_words\] Let $w \in \{a, b, \alpha, \beta\}^*$. We have $w = N(w)$ if and only if $w$ does not contain $\alpha a^k\beta$ or $a\alpha^kb$ as a factor for any $k \in \mathbb{N}$. In particular, if $w \in \{ a, b, \alpha, \beta\}^* \setminus \{a, \alpha\}^*$, the normalized name $N(w)$ has prefix either $a^i\beta$ or $\alpha^ib$ for some $i \in \mathbb{N}$.
Since $\psi = \varphi_a\varphi_b\varphi_\alpha\varphi_b = \varphi_b\varphi_a\varphi_\alpha\varphi_b = \varphi_b\varphi_b\varphi_\beta\varphi_a$, the normalized name of $\psi$ is $N(\psi) = bb\beta a$.
The morphism $E: 0\to 1, 1\to 0$ which exchanges letters in words over $\{0,1\}$ cannot change the factor complexity of an infinite word. Thus, $E$ is clearly a Sturmian morphism. But $E$ does not belong to the monoid $\mathcal{M} = \langle \varphi_a,\varphi_b,\varphi_\alpha,\varphi_\beta\rangle $. In fact, $E$ is the only missing morphism. More precisely, any Sturmian morphism $\psi$ either belongs to $\mathcal{M}$ or $\psi = \eta\circ E$, where $\eta \in \mathcal{M}$ (see [@MiSe93]). To generate the whole monoid of Sturmian morphisms $St$, one needs only three morphisms, say $E$, $\varphi_a$ and $\varphi_b$ (see [@Lo83]). We have $$\label{zamena}
\varphi_\alpha = E\varphi_aE \quad \text{ and } \quad \varphi_\beta = E\varphi_bE .$$
Our aim is to study derivated words of fixed points of Sturmian morphisms. If $\uu$ is a fixed point of $\psi$, it is also a fixed point of $\psi^2$. Due to , the square $\psi^2$ always belongs to $\mathcal{M}$. To illustrate why this is true, assume, e.g., that $\psi \in St = \langle E, \varphi_a, \varphi_b \rangle$ equals $\psi = \varphi_a E \varphi_b \varphi_a$. Using and the fact that $E^2$ is the identity morphism, we have $$\psi = \varphi_a E \varphi_b E E \varphi_a E E = \varphi_a \varphi_\beta \varphi_\alpha E$$ and hence $$\psi^2 = \varphi_a \varphi_\beta \varphi_\alpha E \varphi_a \varphi_\beta \varphi_\alpha E = \varphi_a \varphi_\beta \varphi_\alpha E \varphi_a E E\varphi_\beta E E\varphi_\alpha E = \varphi_a \varphi_\beta \varphi_\alpha \varphi_\alpha \varphi_b \varphi_a \in \mathcal{M}.$$ Therefore we may restrict ourselves to fixed points of morphisms from the special Sturmian monoid $\mathcal{M}$. Note that this would not be true if we consider only the morphisms from $\langle \varphi_a, \varphi_b \rangle$, see also .
\[fibonacci1\] The Fibonacci word is the fixed point of the morphism $\tau: 0\to 01, 1\to 0$. The morphism $\tau$ is Sturmian, but $\tau \notin \mathcal{M}$. We see that $\tau = \varphi_b\circ E$ and by the relations we have $\tau^2 =\varphi_b\varphi_\beta $.
\[nerozlisuj\] Two infinite words $\uu$ and $E(\uu)$ over the alphabet $\{0,1\}$ coincide up to a permutation of the letters $0$ and $1$. If a word $\uu$ is a fixed point of a morphism $\varphi_w$, then $E(\uu)$ is a fixed point of the morphism $E\circ \varphi_w\circ E = \varphi_v$ for some $v$. By , the word $v$ is obtained from $w$ by exchange of letters $a\leftrightarrow \alpha$ and $b\leftrightarrow\beta$. Therefore we introduce the following morphism $F: \{a,b,\alpha,\beta\}^* \mapsto \{a,b,\alpha,\beta\}^* $ by $$\label{eq:definition_of_F}
F(a)=\alpha, \quad F(\alpha)=a,\quad F(b)=\beta, \quad F(\beta)=b.$$ This notation enables us to formulate two useful facts on composition of $E$ with morphisms from $\mathcal{M}$. Namely, $$\label{skladani} E\circ \varphi_w\circ E = \varphi_{F(w)} \qquad \text{ and} \qquad (\varphi_w\circ E)^2 = \varphi_{wF(w)}\,.$$
Later on we will need the following statement on the morphism $F$. First we recall two classical results on word equations:
\[lem:Lyndon1\] Let $y \in \A^*$ and $x,z \in \A^+$. Then $xy = yz$ if and only if there are $u, v \in \A^*$ and $\ell \in \N$ such that $x = uv, z = vu$ and $y = (uv)^\ell u$.
\[lem:Lyndon2\] Let $x, y \in \A^+$. The following three conditions are equivalent:
(i) $xy = yx$;
(ii) There exist integers $i, j > 0$ such that $x^i = y^j$;
(iii) There exist $z \in \A^+$ and integers $p, q > 0$ such that $x = z^p$ and $y = z^q$.
With these two lemmas we prove the following result on word equations involving the morphism $F$. Note that this result is within the general setting considered in [@JoFlMaNo], however we give an explicit solution of cases that we need later.
\[lem:words\_equations\_F\] Let $z$ and $p$ be nonempty words from $\{a,b,\alpha,\beta\}^+$.
(i) If $zp = F(p)F(z)$, then there is $x \in \{a,b,\alpha,\beta\}^+$ such that
$z = x\bigl(F(x)x\bigr)^i$ and $p = \bigl(F(x)x\bigr)^jF(x)$ for some $i,j \in \N$.
(ii) If $zp = pF(z)$, then there is $x \in \{a,b,\alpha,\beta\}^+$ such that
$z = \bigl(F(x)x\bigr)^i$ and $p =\bigl(F(x)x\bigr)^jF(x)$ for some $i,j \in \N$.
We prove Item $(i)$ by induction on $|zp| \geq 2$. If $|z| = |p|$, then $z = F(p)$ and the statement is true for $x = z$ and $i = j = 0$.
Assume $|z| > |p|$ (the case of $|z| < |p|$ is analogous). There must be a nonempty word $q$ such that $z = F(p)q$ and this yields $qp = F(z) = pF(q)$. By there are words $u$ and $v$ and $\ell \in \N$ such that $q = uv, p = (uv)^\ell u$ and $F(q) = vu$. This implies that $vu = F(u)F(v)$ and we can apply the induction hypothesis as $|uv| < |pz|$. Therefore, there are $x$ and $s,r \in \N$ such that $v = x\bigl(F(x)x\bigr)^s$ and $u = \bigl(F(x)x\bigr)^tF(x)$. Putting this altogether we obtain $$\begin{aligned}
q & = uv =\bigl(F(x)x\bigr)^tF(x)x\bigl(F(x)x\bigr)^s = \bigl(F(x)x\bigr)^{t+s+1}, \\
p & = (uv)^\ell u = \bigl(F(x)x\bigr)^jF(x), \quad \text{with} \ j={\ell(t+s+1) + t} \\
z & = F(p)q =x \bigl(F(x)x\bigr)^i, \quad \text{with} \ i=\ell(t+s+1) + 2t + s + 1.
\end{aligned}$$
To prove Item $(ii)$, we apply on $zp = pF(z)$. We have $z = uv, F(z) = vu$ and $p = (uv)^\ell u$ for some words $u$ and $v$ and $\ell \in \N$. It follows that $vu = F(u)F(v)$ and so, by Item $(i)$, there is $x$ such that $v = x\bigl(F(x)x\bigr)^i$ and $u = \bigl(F(x)x\bigr)^jF(x)$ for some $i,j \in \N$. Using all these equations we finish the proof by stating that $$z = uv = \bigl(F(x)x\bigr)^jF(x)x\bigl(F(x)x\bigr)^i = \bigl(F(x)x\bigr)^{j+i+1} \quad \text{and} \quad p = (uv)^\ell u = \bigl(F(x)x\bigr)^{\ell(j+i+1) + j}F(x). \qedhere$$
Derivated words of Sturmian preimages {#sec:der_wo_St_preim}
=====================================
In this section we study relations between derivated words of a Sturmian word and derivated words of its preimage under one of the morphisms $\varphi_a, \varphi_b, \varphi_\alpha$ and $\varphi_\beta$. We prove that the set of all derivated words of these two infinite words coincide up to at most one derivated word, see Theorems \[thm:preimages\_fi\_b\] and \[thm:preimages\_fi\_a\]. This will be crucial fact for proving the main results of this paper. Because of , the roles of $\varphi_a$ and $\varphi_\alpha$ and, analogously, the roles of $\varphi_b$ and $\varphi_\beta$ are symmetric. Therefore we can restrict the statements and proofs in this section to the morphisms $\varphi_a$ and $\varphi_b$ with no loss of generality. Again we use results from [@Lo83], in particular this slightly modified Proposition 2.3.2:
Let $\bf x$ be an infinite word.
(i) If $\varphi_b(\bf x)$ is Sturmian, then $\bf x$ is Sturmian.
(ii) If $\varphi_a(\bf x)$ is Sturmian and $\bf x$ starts with the letter $1$, then $\bf x$ is Sturmian.
\[thm:preimages\_fi\_b\] Let $\uu$ and $\uu'$ be Sturmian words such that $\uu = \varphi_{b}(\uu')$. Then the sets of their derivated words satisfy $$\Der(\uu) = \Der(\uu') \cup \{ \uu'\}\,.$$
The proof of the previous theorem is split into two parts: In Proposition \[prop:der\_of\_images\_fi\_b\], Item (i) says $ \{ \uu'\} \subset \Der(\uu)$ and Item (ii) says $ \Der(\uu) \subset \Der(\uu') \cup \{ \uu'\}.$ Proposition \[prop:der\_of\_preimage\_fi\_b\] says $\Der(\uu') \subset \Der(\uu) $. Proofs of these propositions use the following simple property of the injective morphism $\varphi_{b}$.
\[lem:unique\_preimage\_under\_fi\_b\] Let $\uu = \varphi_{b}(\uu')$ be a Sturmian word. If $p0 \in \mathcal{L}(\uu)$ and $0$ is a prefix of $p$, then there exists a unique factor $p'\in \mathcal{L}(\uu')$ such that $p0= \varphi_b(p')0$.
\[prop:der\_of\_images\_fi\_b\] Let $\uu$ and $\uu'$ be Sturmian words such that $\uu = \varphi_{b}(\uu')$ and let $w$ be a prefix of $\uu$.
(i) If $|w|=1$, then $d_{\uu}(w) = \uu'$ (up to a permutation of letters). \[it:der\_of\_images\_fi\_b\_1\]
(ii) If $|w|>1$, then there exists a prefix $w'$ of $\uu'$ such that $|w'|<|w|$ and $d_{\uu}(w) = d_{\uu'}(w')$ (up to a permutation of letters). \[it:der\_of\_images\_fi\_b\_2\] Moreover, if $w$ is right special, $w'$ is right special as well.
Since $\varphi_b(0) =0$ and $\varphi_b(1)=01$, the word $\uu = \varphi_{b}(\uu')$ has a prefix $0$ and the letter $1$ is in $\uu$ separated by blocks $0^k$ with $k\geq 1$. Therefore, the two return words in $\uu$ to the word $w=0$ are $r_0=0$ and $r_1=01$. We may write $\uu = r_{s_0} r_{s_1} r_{s_2} \cdots$, where $r_{s_j} \in \{r_0,r_1\}$ and thus $ d_{\uu}(w) = s_0s_1s_2\cdots$. Since $r_0=\varphi_b(0)$ and $r_1=\varphi_b(1)$, we obtain also $\varphi_b(\uu') = \uu = \varphi_b(s_0) \varphi_b(s_1) \varphi_b(s_2) \cdots = \varphi_b(s_0s_1s_2\cdots)$. The statement in \[it:der\_of\_images\_fi\_b\_1\] now follows from injectivity of $\varphi_b$.
Now suppose that the prefix $w$ of $\uu$ is of length $>1$. As explained earlier, it suffices to consider right special prefixes. Since the letter $1$ is always followed by $0$, each right special factor must end in $0$. So the first and the last letter of $w$ is $0$, hence by Lemma \[lem:unique\_preimage\_under\_fi\_b\] there is a unique prefix $w'$ of $\uu'$ such that $\varphi_b(w')0 = w$. Let $r_0$ and $r_1$ be the two return words to $w$ and let $\uu = r_{s_0}r_{s_1}r_{s_2}\cdots$. Since the first letter of both $r_0$ and $r_1$ is $0$, there are uniquely given $r'_0$ and $r'_1$ such that $r_0=\varphi_b(r'_0)$ and $r_1=\varphi_b(r'_1)$ and $\uu' = r'_{s_0}r'_{s_1}r'_{s_2}\cdots$.
Clearly $w'$ is a prefix of $r'_{s_j}r'_{s_{j+1}}r'_{s_{j+2}}\cdots$ for all $j \in \N$ and so the number $|r'_{s_0}r'_{s_1}\cdots r'_{s_k}|$ is an occurrence of $w'$ in $\uu'$ for all $k \in \N$. Let $i > 0$ be an occurrence of $w'$ in $\uu'$. It follows that $\varphi_b\bigl(\uu'_{[0,i)}\bigr)w$ is a prefix of $\uu$ and $|\varphi_b\bigl(\uu'_{[0,i)}\bigr)|$ is an occurrence of $w$ in $\uu$. There must be $j \in \N$ such that $\varphi_b\bigl(\uu'_{[0,i)}\bigr) = r_{s_0}r_{s_1}\cdots r_{s_j}$ and hence, by injectivity of $\varphi_b$, $\uu'_{[0,i)} = r'_{s_0}r'_{s_1}\cdots r'_{s_j}$ and $i = |r'_{s_0}r'_{s_1}\cdots r'_{s_j}|$.
We have proved that the numbers $0$ and $|r'_{s_0}r'_{s_1}\cdots r'_{s_j}|, j = 0,1,\ldots$, are all occurrences of $w'$ in $\uu'$. It follows that $r'_0$ and $r'_1$ are the two return words to $w'$ in $\uu'$ and $$d_{\uu'}(w') = s_0s_1s_2 \cdots = d_{\uu}(w).$$
Since $w = \varphi_b(w')0$ is a right special factor, we must have that both $\varphi_b(w')00$ and $\varphi_b(w')01$ are factors of $\uu$. It follows that both $w'0$ and $w'1$ are factors of $\uu'$ and $w'$ is right special.
\[prop:der\_of\_preimage\_fi\_b\] Let $\uu$ and $\uu'$ be Sturmian words such that $\uu = \varphi_{b}(\uu')$ and let $w'$ be a nonempty right special prefix of $\uu'$. Then $d_{\uu'}(w') = d_{\uu}(w)$, where $w = \varphi_{b}(w')0$.
Let $r_0'$ and $r_1'$ be the two return words to $w'$ in $\uu'$ and $d_{\uu'}(w') = s_0s_1s_2\cdots$. Put $w=\varphi_b(w')0$, $r_0=\varphi_b(r'_0)$ and $r_1=\varphi_b(r'_1)$. We obtain $$\uu = \varphi_b(\uu') = \varphi_b(r'_{s_0}r'_{s_1}r'_{s_2}\cdots ) = r_{s_0}r_{s_1}r_{s_2}\cdots$$ Clearly, $w$ is prefix of $r_{s_k}r_{s_{k+1}}r_{s_{k+2}}\cdots$ for all $k \in \N$ and $|r_{s_0}r_{s_1}\cdots r_{s_j}|$ is an occurrence of $w$ in $\uu$ for all $j \in \N$.
Assume now $i > 0$ is an occurrence of $w$ in $\uu$. This means that $\uu_{[0,i)}w$ is a prefix of $\uu$ and hence, by Lemma \[lem:unique\_preimage\_under\_fi\_b\] (note that $w$ begins with $0$), there must be $p'$ a prefix of $\uu'$ such that $\varphi_b(p') = \uu_{[0,i)}$ and $p'w'$ is a prefix of $\uu'$. Since $|p'|$ is an occurrence of $w'$ in $\uu'$, there is $j \in \N$ such that $p' = r'_{s_0}r'_{s_1}\cdots r'_{s_j}$. It follows that $$\uu_{[0,i)} = \varphi_b(r'_{s_0}r'_{s_1}\cdots r'_{s_j}) = r_{s_0}r_{s_1}\cdots r_{s_j}$$ and $i = |r_{s_0}r_{s_1}\cdots r_{s_j}|$.
So, again as in the previous proof, we have shown that the numbers $0$ and $|r_{s_0}r_{s_1}\cdots r_{s_j}|, j = 0,1,\ldots$, are all occurrences of $w$ in $\uu$. It follows that $r_0$ and $r_1$ are the two return words to $w$ in $\uu$ and $$d_{\uu}(w) = s_0s_1s_2 \cdots = d_{\uu'}(w'). \qedhere$$
\[thm:preimages\_fi\_a\] Let $\uu$ and $\uu'$ be Sturmian words such that $\uu$ starts with the letter $1$ and $\uu = \varphi_{a}(\uu')$. Then $\uu'$ starts with $1$ and the sets of their derivated words coincide, i.e., $$\Der(\uu) = \Der(\uu')\,.$$ In particular, for any prefix $w$ of $\uu$ there exists a prefix $w'$ of $\uu'$ such that $|w'| \leq |w|$ and $d_{\uu}(w) = d_{\uu'}(w')$ (up to a permutation of letters). Moreover, if $w$ is right special, $w'$ is right special as well.
The morphisms $\varphi_a$ and $\varphi_b$ are conjugate, that is, $0\varphi_a(x)=\varphi_b(x)0$ for each word $x$. This means that for any prefix $u'_0u'_1u'_2 \cdots u'_k$ of $\uu'$ we have $0\varphi_a(u'_0u'_1u'_2 \cdots u'_k) = \varphi_b(u'_0u'_1u'_2 \cdots u'_k)0$. As this holds true for each $k$, we obtain $0\uu = 0\varphi_a(\uu') =\varphi_b(\uu')$.
Denote ${\bf v} = v_0v_1v_2\cdots = 0u_0u_1u_2 \cdots$. We have $v_i = u_{i-1}$ for each $i\geq 1$. Let $w$ be a nonempty prefix of $\uu$ and $(i_n)$ be the increasing sequence of its occurrences in $\uu$. Note that $w$ starts with the letter $1$. This letter is in $\uu$ surrounded by $0$’s. Thus the sequence $(i_n) $ is also the sequence of occurrences of $0w$ in ${\bf v}$ and thus $d_{\bf v}({0w}) = d_{\bf u}({w})$. It follows that $$\Der(\uu)=\{d_{\bf v}(v) \colon v \text{ is a prefix of $\bf v$ and } |v|>1\}\,.$$ We finish the proof by applying Theorem \[thm:preimages\_fi\_b\] and Proposition \[prop:der\_of\_images\_fi\_b\] to the word ${\bf v} =\varphi_b(\uu')$.
The only case which is not treated by Theorems \[thm:preimages\_fi\_b\] and \[thm:preimages\_fi\_a\], namely the case when $\uu = \varphi_a(\uu')$ and $\uu$ begins with $0$, can be translated into one of the previous cases.
\[lem:prevod\_fi\_a\_na\_fi\_b\] Let $\uu$ be a Sturmian word such that $\uu$ starts with the letter $0$ and $ \uu = \varphi_a(\uu')$ for some word $\uu'$. Then there exists a Sturmian word $\bf v$ such that $\uu' = 0{\bf v}$ and $\uu = \varphi_b({\bf v})$.
Since $\uu$ starts with $0$, the form of $\varphi_a$ implies that $\uu' = 0{\bf v}$ for some Sturmian word $\bf v$. As $0\varphi_a(x)=\varphi_b(x)0$ for each word $x$, we have $$\uu = \varphi_a(\uu') = \varphi_a(0{\bf v}) = 0 \varphi_a({\bf v}) = \varphi_b({\bf v}). \qedhere$$
To sum up the results of this section, let us assume we have a sequence of Sturmian words $\uu_0, \uu_1, \uu_2, \ldots$ such that $\uu = \uu_0$ and for every $i \in \N$ one of the following is true:
(i) $\uu_i = \varphi_b(\uu_{i+1})$ or $\uu_i = \varphi_\beta(\uu_{i+1})$, \[it:seq\_pos\_1\]
(ii) $\uu_i$ begins with $1$ and $\uu_i = \varphi_a(\uu_{i+1})$, \[it:seq\_pos\_2\]
(iii) $\uu_i$ begins with $0$ and $\uu_i = \varphi_\alpha(\uu_{i+1})$. \[it:seq\_pos\_3\]
If \[it:seq\_pos\_1\] holds for $\uu_i$, then by Theorem \[thm:preimages\_fi\_b\] $$\mathrm{Der}(\uu_i) = \mathrm{Der}(\uu_{i+1}) \cup \{\uu_{i+1}\},$$ moreover, $\uu_{i+1}$ is the derivated word of the first letter of $\uu_i$. This first letter is also the shortest right special prefix. If \[it:seq\_pos\_2\] or \[it:seq\_pos\_3\] holds for $\uu_i$, then by Theorem \[thm:preimages\_fi\_a\] $$\mathrm{Der}(\uu_i) = \mathrm{Der}(\uu_{i+1}).$$
The crucial assumption, namely the existence of the above described sequence $(\uu_k)_{k\geq 0}$, is guaranteed by the well-known fact on the desubstitution of Sturmian words (see, e.g., [@JuPi] and [@HeMo] and also ). Here we formulate this fact as the following theorem:
An infinite binary word $\uu$ is Sturmian if and only if there exists an infinite word ${\bf w}=w_0w_1w_2\cdots$ over the alphabet $\{a,b,\alpha, \beta\}$ and an infinite sequence $(\uu_i)_{i\geq 0}$, such that $\uu = \uu_0$ and $\uu_{i} = \varphi_{w_i}(\uu_{i + 1})$ for all $i \in \mathbb{N}$.
In the following section we work only with the sequence $(\uu_i)_{i \geq 0}$ corresponding to a fixed point $\uu$ of a Sturmian morphism $\psi$. The next lemma provides us a simple technical tool for a description of the elements $\uu_i$ as fixed points of some Sturmian morphisms.
\[lem:rotace\_morfizmu\] Let $\xi$ and $\eta$ be Sturmian morphisms and $\uu = \bigl(\xi \circ \eta\bigr) (\uu)$. If $\uu = \xi(\uu')$ for some $\uu'$, then $\uu'$ is the fixed point of the morphism $\eta \circ \xi$, i.e. $\uu' = \bigl( \eta \circ \xi\bigr)(\uu')$.
For any Sturmian morphism $\xi$, the equation $\xi(\bf x) = \xi (\bf y)$ implies that ${\bf x}= { \bf y}$. We deduce that $$\xi(\uu') = \uu = \bigl(\xi \circ \eta\bigr) (\uu) = \bigl(\xi \circ \eta\bigr) \bigl(\xi (\uu')\bigr) = \bigl( \xi \circ \eta\circ \xi\bigr) (\uu')\,,$$ and so $\uu' = \bigl(\eta \circ \xi\bigr)(\uu')$.
Derivated words of fixed points of Sturmian morphisms {#sec:der_wo_of_fixed_points}
=====================================================
Let $\uu$ be an fixed point of a primitive Sturmian morphism (note that if the morphism is primitive, all its fixed points are aperiodic). It is known due to Durand [@Durand98] that the set $\mathrm{Der}(\uu)$ is finite (as the morphism is primitive). Put $$\mathrm{Der}(\uu) = \{\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_\ell\}.$$ Our main result is an algorithm that returns a list of Sturmian morphisms $\psi_1, \psi_2, \ldots, \psi_\ell$ such that $\mathbf{x}_i$ is a fixed point of $\psi_i$ (up to a permutation of letters) for all $i$ such that $1 \leq i \leq \ell$.
As we have noticed before, we can restrict ourselves to the morphisms belonging to the monoid $\mathcal{M} = \langle \varphi_a, \varphi_b, \varphi_\alpha, \varphi_\beta \rangle $. Let us recall (see Lemma \[lem:properties\_of\_sturm\_morph\]) that a morphism from $\langle \varphi_a, \varphi_b \rangle $ or from $\langle \varphi_\alpha, \varphi_\beta \rangle $ is not primitive and has no aperiodic fixed point. Thus we consider only morphisms $\varphi_w$ whose normalized name $w$ contains at least one Latin and one Greek letter.
We will treat two cases separately. The first one is the case when the morphism $\varphi_w$ has only one fixed point. Lemma \[lem:properties\_of\_sturm\_morph\] says that in such a case $w \notin \{a, \alpha\}^*$. In the second case, when $w \in \{a, \alpha\}^*$, the morphism $\varphi_w$ has two fixed points.
Morphisms with unique fixed point
---------------------------------
Let $\psi \in \langle\varphi_a, \varphi_b, \varphi_\alpha, \varphi_\beta\rangle$ and $ N(\psi) = w \in \{a,b,\alpha,\beta\}^*\setminus \{a, \alpha\}^*$ be the normalized name of the morphism $\psi$. By Lemma \[lem:normalized\_words\] the word $w$ has a prefix $a^k\beta$ or $\alpha^kb$ for some $k\in \mathbb{N}$. This property enables us to define a transformation on the set of morphisms from $\mathcal{M} \setminus \langle \varphi_a, \varphi_\alpha\rangle$. As we will demonstrate later, this transformation is in fact the desired algorithm returning the morphisms $\psi_1, \psi_2, \ldots, \psi_\ell$ mentioned above.
\[def:delta\] Let $w\in \{a,b,\alpha,\beta\}^*\setminus \{a,\alpha\}^*$ be the normalized name of a morphism $\psi$, i.e., $\psi = \varphi_{w}$. We put $$\Delta(w) =
\begin{cases}
N( w'a^k\beta) & \text{ if \ } w = a^k\beta w', \\
N(w'\alpha^kb) & \text{ if \ } w = \alpha^kb w'
\end{cases}$$ and, moreover, $\Delta (\psi) = \varphi_{\Delta(w)}$.
\[example:degAnedeg\] Consider the morphism $\psi = \varphi_w$, where $w = \beta \alpha aa \alpha$, and apply repeatedly the transformation $\Delta$ on $\psi$. $$\begin{aligned}
\psi & = \varphi_{\beta \alpha a a \alpha} \quad \text{ and } \quad N(\psi)= w =\beta \alpha aa \alpha \\
\Delta (\psi) & = \varphi_{\alpha a a \alpha \beta} \quad \text{ and } \quad N\bigl(\Delta (\psi)\bigr)= \beta bb \alpha \alpha \\
\Delta^2 (\psi) & = \varphi_{ bb \alpha \alpha \beta} \quad \text{ and } \quad N\bigl(\Delta^2 (\psi)\bigr) = bb \beta \alpha \alpha \\
\Delta^3 (\psi) & = \varphi_{b \beta \alpha \alpha b} \quad \text{ and } \quad N\bigl(\Delta^3(\psi)\bigr) = b \beta \alpha \alpha b \\
\Delta^4 (\psi) & = \varphi_{\beta \alpha \alpha b b} \quad \text{ and } \quad N\bigl(\Delta^4 (\psi)\bigr) = \beta \alpha \alpha bb \\
\Delta^5 (\psi) & = \varphi_{\alpha \alpha bb \beta} \quad \text{ and } \quad N\bigl(\Delta^5 (\psi)\bigr) = \alpha \alpha bb \beta \\
\Delta^6 (\psi) & = \Delta^3 (\psi)
\end{aligned}$$ In what follows we prove that the five fixed points of morphisms $\Delta (\psi), \Delta^2(\psi), \Delta^3(\psi), \Delta^4(\psi), \Delta^5(\psi)$ are exactly the five derivated words of the fixed point of $\psi$.
\[lem:tvar\_prefixu\] Let $\uu$ be a fixed point of a morphism $\psi$ and $ N(\psi) = w \in \{a,b,\alpha,\beta\}^*$ be the normalized name of the morphism $\psi$. If one of the following condition is satisfied
(i) $\uu$ starts with $0$ and $w$ starts with $a$,
(ii) $\uu$ starts with $1$ and $w$ starts with $\alpha$,
then $w \in \{a, \alpha\}^*$.
We consider only the case $(i)$, the case $(ii)$ is analogous. Let us assume $w \notin \{a, \alpha\}^*$. According to Lemma \[lem:normalized\_words\], the word $w$ has a prefix $a^k\beta$, for some $k \geq 1$. Consequently, the morphism $\psi$ equals $\varphi_a^k \circ \varphi_\beta\circ \eta$ for some morphism $\eta$. Any morphism of this form maps $0$ to $1w_1 $ and $1$ to $1w_2$ for some words $w_1$ and $w_2$. Therefore, the fixed point starts with the letter $1$, which is a contradiction.
The following theorem along with provide the algorithm which to a given Sturmian morphism $\psi$ lists the morphisms fixing the derivated words of the Sturmian word ${\bf u} = \psi({\bf u})$.
\[thm:main\_result\_vetsina\] Let $\psi \in \langle\varphi_a, \varphi_b, \varphi_\alpha, \varphi_\beta\rangle$ be a primitive morphism and $ N(\psi) = w \in \{a,b,\alpha,\beta\}^* \setminus \{a, \alpha\}^*$ be its normalized name. Denote $\uu$ the fixed point of $\psi$. Then $\mathbf{x}$ is (up to a permutation of letters) a derivated word of $\uu$ with respect to one of its prefixes if and only if $\mathbf{x}$ is the fixed point of the morphism $\Delta^j(\psi)$ for some $j \geq 1$.
Denote $\mathbf{x}_j$ the fixed point of $\Delta^j(\psi), j = 1,2,\ldots$ and assume that $v$ is a right special prefix of $\uu$. We will prove that if $|v| = 1$, then $\mathrm{d}_\uu(v) = \mathbf{x}_1$, and if $|v| > 1$, then there is a right special prefix $v'$ of $\mathbf{x}_1$ such that $|v'| < |v|$ and $\mathrm{d}_\uu(v) = \mathrm{d}_{\mathbf{x}_1}(v')$. We can repeat this proof for the prefix $v'$ of $\mathbf{x}_1$ and eventually prove that $\mathrm{d}_\uu(v) = \mathbf{x}_j$ for some $j$ and that for any $j$ there is a right special prefix $v$ of $\uu$ so that $\mathrm{d}_\uu(v) = \mathbf{x}_j$.
Without loss of generality we assume that the normalized name of $\psi$ is $w= a^k\beta z$. This means that $\Delta(\psi) = \varphi_{z}\circ \varphi_{a^k\beta}$.
First we assume $|v| = 1$. If $k > 0$, then the first letter of $\uu$ is $1$ which is not a right special factor. This implies that $k = 0$. Hence we have that $\uu = \varphi_\beta (\uu')$, where $\uu'= \varphi_{z}(\uu)$. By of we obtain $\mathrm{d}_\uu(v) = \mathbf{\uu'}$. Lemma \[lem:rotace\_morfizmu\] says the word $\uu'$ is fixed by the morphism $\varphi_{z}\circ \varphi_{\beta} = \Delta(\psi)$, which implies $\uu' = \mathbf{x}_1$.
Now assume $|v| > 1$. If $k = 0$, then by of there is a right special prefix $v'$ of $\uu'= \varphi_{z}(\uu)$ such that $|v'| < |v|$ and $\mathrm{d}_\uu(v) = \mathrm{d}_{\uu'}(v')$. Again by Lemma \[lem:rotace\_morfizmu\] we obtain $\uu' = \mathbf{x}_1$.
Let $k > 0$. For $i =0, 1, \ldots, k$ we define $\uu^{(i)} = \varphi_{a^{k-i}\beta z}(\uu)$. By Lemma \[lem:tvar\_prefixu\], the words $\uu^{(i)}$ all start with the letter $1$. Obviously, $\uu^{(0)} = \uu$ and $\uu^{(i)} = \varphi_a\bigl(\uu^{(i+1)}\bigr)$ for $i =0,1,\ldots, k-1$. By Theorem \[thm:preimages\_fi\_a\], there are factors $v^{(i)}$ with $i =0, 1, \ldots, k$ such that $v^{(i)}$ is a right special prefix of $\uu^{(i)}$, $$|v| = |v^{(0)}| \geq |v^{(1)}| \geq |v^{(2)}| \geq \cdots \geq |v^{(k)}|$$ and $$\mathrm{d}_{\uu}(v) = \mathrm{d}_{\uu^{(1)}}(v^{(1)}) = \mathrm{d}_{\uu^{(2)}}(v^{(2)}) \cdots = \mathrm{d}_{\uu^{(k)}}(v^{(k)})\, .$$ Define $\uu' = \varphi_{z}(\uu)$. Then $\uu^{(k)} = \varphi_{\beta z}(\uu) = \varphi_\beta (\uu')$ and by of there is a right special prefix $v'$ of $\uu'= \varphi_{z}(\uu)$ such that $|v'| < |v^{(k)}|$ and $\mathrm{d}_{\uu^{(k)}}(v^{(k)}) = \mathrm{d}_{\uu'}(v')$. According to Lemma \[lem:rotace\_morfizmu\], the word $\uu'$ is fixed by the morphism $\varphi_{z}\circ \varphi_{a^k\beta} = \Delta(\psi)$. Thus, we have again proved that there is a prefix $v'$ of $\uu' = \mathbf{x}_1$ such that $|v'| < |v|$ and $\mathrm{d}_\uu(v) = \mathrm{d}_{\uu'}(v')$.
In Example \[example:degAnedeg\] we considered the morphism $\psi = \varphi_w$, where $w = \beta \alpha aa \alpha$. We have found only five different morphisms $\Delta^i (\psi)$ for $i=1,\ldots,5$. The sixth morphism $\Delta^6 (\psi)$ already coincides with $\Delta^3 (\psi)$. As it follows from the proofs of Theorems \[thm:preimages\_fi\_b\] and \[thm:main\_result\_vetsina\], the fixed points of $\Delta^3 (\psi)$, $\Delta^4 (\psi)$ and $\Delta^5 (\psi)$ represent the derivated words of $\uu$ to infinitely many prefixes of $\uu$. Whereas the fixed point of $\Delta (\psi)$ or $\Delta^2(\psi)$ is a derivated word of $\uu$ to only one prefix of $\uu$.
\[ex:derivated\_words\_of\_Fibon\] As explained in Example \[fibonacci1\], to find the derivated words of the Fibonacci word we consider the morphism $\psi = \tau^2 =\varphi_b\varphi_\beta $. We have $\Delta(\psi) = \varphi_\beta\varphi_b$ and $\Delta^2(\psi) = \psi$. But these two morphisms are equal up to a permutation of letters, as $E\psi E = \Delta(\psi)$. This means that all derivated words of the Fibonacci word are the same and coincide with the Fibonacci word itself.
Morphisms with two fixed points
-------------------------------
Let us now consider a Sturmian morphism $\psi$ which has two fixed points. Let us denote $\uu^{(0)}$ and $\uu^{(1)}$ the fixed points of $\psi$ starting with $0$ and $1$, respectively. Clearly, $\psi(0)$ starts with $0$ and $\psi(1)$ with $1$. Since the morphism $\psi$ has to belong to the monoid $\langle \varphi_a, \varphi_\alpha \rangle$, the transformation $\Delta$ cannot be applied on it. However, we will show that there is a morphism from $\langle \varphi_a, \varphi_\beta\rangle$ (or $ \langle \varphi_b, \varphi_\alpha\rangle$) with a unique fixed point $\mathbf{v}$ such that the set of derivated words of $\uu^{(0)}$ (or $\uu^{(1)}$) equals to $\{{\bf v}\} \cup {\rm Der}(\bf v)$. And since $\mathbf{v}$ is a fixed point of some morphism from $\langle \varphi_a, \varphi_b, \varphi_\beta, \varphi_\alpha \rangle \setminus \langle \varphi_a, \varphi_\alpha \rangle$, the set ${\rm Der}(\bf v)$ can be described using .
Here we give results only for the case when the normalized name $w \in\{a,\alpha\}^*$ of the morphism begins with $a$. The case when the first letter is $\alpha$ is completely analogous. It suffices to exchange $a \leftrightarrow b$ and $\alpha \leftrightarrow \beta$ in the statements and proofs.
\[lem:jinam\] Let $w \in\{a,\alpha\}^*$ be the normalized name of a morphism starting with the letter $a$. Then the normalized name $N(wb)$ has a prefix $b$ and a suffix $a$, the word $v=b^{-1}N(wb)$ belongs to $\{a,\beta\}^*$, and $|v|_{\beta} = |w|_{\alpha}$.
First, we consider the special case when $w=a^k\alpha^\ell$, with $k \geq 1$ and $\ell \geq 0$. By the relation , $N(wb)=ba^{k-1}\beta^\ell a$ and the statement is true.
Let $w \in\{a,\alpha\}^*$ be arbitrary. It can be decomposed to several blocks of the form $a^k\alpha^\ell$ with $k \geq 1$, $\ell \geq 0$. Now the proof can be easily finished by induction on the number of these blocks.
\[lem:revers\_of\_standard\] Let $w \in\{a,\alpha\}^*$ be the normalized name of a primitive morphism $\psi$ and let $a$ be its first letter.
(i) Let $\uu$ be the fixed point of $\psi$ starting with $0$. Denote $v =b^{-1}N(wb)\in \{a,\beta\}^*$. Then $ {\rm Der}(\uu) = \{{\bf v}\} \cup {\rm Der}(\bf v)$, where ${\bf v }$ is the unique fixed point of the morphism $\varphi_v$.
(ii) Let $\uu$ be the fixed point of $\psi$ starting with $1$. Put $v ={\rm cyc} (w)$ (see). Then ${\rm Der}(\uu) = {\rm Der}({\bf v})$, where ${\bf v }$ is the fixed point of the morphism $\varphi_v$.
Let us start with proving $(i)$. Let ${\bf v}$ be the infinite word given by Lemma \[lem:prevod\_fi\_a\_na\_fi\_b\]. Then $$\varphi_b({\bf v}) = \uu = \psi(\uu) = \varphi_w(\uu) = \bigl(\varphi_w \circ \varphi_b\bigr)({\bf v}) = \varphi_{wb}({\bf v})= \varphi_{N(wb)}({\bf v}) \,.$$ By definition of $v$ we have $N(wb) = bv$ and thus $$\varphi_b({\bf v}) = \varphi_{bv}({\bf v}) = \varphi_b \bigl( \varphi_v({\bf v}) \bigr).$$ This implies that ${\bf v} = \varphi_v({\bf v}) \,.$ Since $v \notin \{ a, \alpha \}^*$, the morphism $\varphi_v$ has a unique fixed point, namely the word $\bf v$. By Theorem \[thm:preimages\_fi\_b\], ${\rm Der}(\uu) = \{{\bf v}\} \cup {\rm Der}(\bf v)$ as stated in $(i)$.
Statement $(ii)$ is a direct consequence of Theorem \[thm:preimages\_fi\_a\] and Lemma \[lem:rotace\_morfizmu\].
Bounds on the number of derivated words {#sec:number_der_wo}
=======================================
In this section we study the relation between the normalized name $w$ of a primitive morphism $\psi = \varphi_w$ and the number of distinct return words to its fixed point. We restrict ourselves to the case when $w \notin \{a, \alpha\}^*$, as the case $w \in \{a, \alpha\}^*$ is treated in the next section.
Theorem \[thm:main\_result\_vetsina\] says that the number of derivated words of $\uu$ cannot exceed the upper bound: $$\text{number of distinct words in the sequence } \left(\Delta^k(w)\right)_{k \geq 1}.$$ Since the words $\Delta^k(w) \in \{a,b,\alpha,\beta\}^*$ are all of the same length and $\Delta^{k+1}(w)$ is completely determined by $\Delta^{k}(w)$, the sequence $\left(\Delta^k(w)\right)_{k \geq 1}$ is eventually periodic.
The number of distinct elements in $\left(\Delta^k(w)\right)_{k \geq 1}$ is only an upper bound on the number of derivated words of $\uu$. As we have already mentioned in Remark \[nerozlisuj\], fixed points of morphisms corresponding to the names $v$ and $F(v)$ coincide up to exchange of letters $0$ and $1$ and hence define the same derivated word. On the other hand, if $v$ and $v'$ are normalized names with $|v| = |v'|$ and fixed points of $\varphi_{v}$ and $\varphi_{v'}$ coincide (up to exchange of letters), then either $v'=v$ or $ v'= F(v)$.
First we look at two examples that illustrate some special cases of the general on the period and preperiod of the sequence $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$.
\[exa:pomoc6\] Consider a word $w$ of length $n$ in the form $w=b^{n-2}\beta a$. The sequence of $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$ is eventually periodic. Its preperiod equals $n-2$ and is given by the words $b^{n-k} \beta b^{k-2} a$, for $ k=3,4,\ldots,n$. The period equals $n-1$ and is given by the words $b^{n-k} a\beta b^{k-2} $, for $ k=2,3, \ldots,n$.
Let us stress that for any $v \in \{a,b,\alpha,\beta\}^*$ the equation $v' = F(v)$ implies $|v|_a = |v'|_\alpha$ and $|v|_b = |v'|_\beta$. Since all words $\Delta^k(w)$ we listed above contain one letter $a$ and no letter $\alpha$, we can conclude that the morphism $\varphi_w$ has $2n-3$ distinct derivated words.
\[exa:pomoc7\]
Consider a normalized name $w$ in which the letter $b$ is missing and $w$ contains all the three remaining letters. Necessarily $w$ has the form $$\beta^{\ell_1} a^{k_1}\beta^{\ell_2}a^{k_2} \cdots \beta^{\ell_s}a^{k_s}\alpha^j,$$ where $s\geq 1$, $\ell_i\geq 1$ for all $i= 2,\ldots ,s$ and $k_i\geq 1$ for all $i=1,2,\ldots ,s-1$ and $j\geq 1$. It is easy to see that the normalized names of words obtained by repeated application of the mapping $\Delta$ are $$\Delta^{\ell_1}(w)= a^{k_1}\beta^{\ell_2}a^{k_2} \cdots \beta^{\ell_s}a^{k_s}\beta^{\ell_1} \alpha^j\qquad \text{and} \qquad \Delta^{\ell_1+1}(w)= \beta^{\ell_2-1}a^{k_2} \cdots \beta^{\ell_s}a^{k_s}\beta^{\ell_1+1}\alpha^{j-1} b^{k_1}\alpha$$ We see that the $(\ell_1+1)^{st}$ iteration already contains all four letters.
\[prop:upper\_bound\] Let $w \in \{a,b,\alpha,\beta\}^* \setminus \{a,\alpha\}^*$ be the normalized name of a primitive Sturmian morphism $\psi =\varphi_w$. Then the sequence $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$ is eventually periodic and:
(i) If it is purely periodic, then its period is at most $|w|$, otherwise, its period is at most $|w| -1$.
(ii) If both $b$ and $\beta$ occur in $w$, then the preperiod is at most $|w|-2$, otherwise the preperiod is at most $2|w|-3$.
By Lemma \[lem:normalized\_words\], the word $w$ (and all the elements of the sequence $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$) has the form $w =a^i\beta w' $ or $w =\alpha^ib w' $ for some $ i\geq 0$. In this proof we distinguish three cases such that exactly one of them is valid for all $\Delta^k(w), k = 1,2, \ldots$ The first two cases correspond to the “periodic” part of the sequence $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$.
[**Case 1**]{}: If $w$ has a suffix $\beta$ or $b$, then the word $\Delta(w)$ equals to $w' a^i\beta $ or $w' \alpha^ib$ and thus has again a suffix $\beta$ or $b$. Indeed, since $N(w) = w$, the words $\alpha a^j \beta$ and $a \alpha^j b$ are not factors of $w$ and so they are not even factors of $w'$. As the last letter of $w'$ is $b$ or $\beta$, neither $\alpha a^j \beta$ nor $a \alpha^j b$ is a factor of $w' a^i\beta$ and hence $\Delta(w) = N(w' a^i\beta) = w' a^i\beta$. This means that for any $k$ the word $\Delta^k(w)$ is just a cyclic shift of $w$ (see). Therefore, $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$ is purely periodic and its period is given by the number of letters $\beta$ and $b$ in $w$ which is clearly at most $|w|$. Moreover, the word $w$ belongs to the sequence $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$ and the fixed point $\uu$ of $\psi$ itself is a derivated word of $\uu$.
Without loss of generality we assume that $w =a^i\beta w'$; the case of $w =\alpha^ib w'$ can be treated in the same way, it suffices to exchange letters $a \leftrightarrow b$ and $\alpha \leftrightarrow \beta$. Denote $p$ the longest suffix of $w$ such that $p \in \{a,\alpha\}^*$. It remains to consider only the case of nonempty $p$.
[**Case 2**]{}: If $p=a^j$ for some $j\geq 1$, then $w'$ has a suffix $ba^j$ or $\beta a^j$. No rewriting rule from can be applied to $w'a^i\beta$, hence, $\Delta(w)= w'a^i\beta$ has a suffix $\beta$. So, we can apply the reasoning from Case 1 on the word $\Delta(w)$ and hence the sequence $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$ is purely periodic. As $w$ contains at least one letter $a$ as a suffix, the period is shorter than $|w|$ and $w$ itself does not occur in $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$.
[**Case 3**]{}: Now assume that the letter $\alpha$ occurs in $p$. We split this case into three subcases and show that if one of these subcases is valid for a word $\Delta^k(w)$, then this word belongs to the “preperiodic” part of $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$. These three subcases (for word $w$) read:\
(i) $w$ begins with the letter $a$, i.e., $i\geq 1$;\
(ii) $w$ has a prefix $\beta$ and $p$ has a factor $\alpha a$;\
(iii) $w$ has a prefix $\beta$ and $p = a^j\alpha^s$ for $j\geq 0$ and $s\geq 1$.
**(i)** Since we assume that $\alpha$ occurs in $p$, a suffix of $p$ has a form $\alpha a^t$ for some $t \geq 0$. It follows that $w'a^i\beta$, has a suffix $\alpha a^{t+i}\beta$. After applying the rewriting rules to $w'a^i\beta$ we obtain the normalized name $\Delta(w)$ which has a suffix $b\alpha$.
**(ii)** A suffix of $w$ can be expressed in the form $\alpha a^r\alpha^sa^t$, where $r\geq 1$ and $s, t\geq 0$. Therefore $w'\beta$ has a suffix $\alpha a^r\alpha^sa^t\beta$. After normalization we get that $\Delta(w)$ has a suffix in the form of $b\alpha^\ell$ for some $\ell\geq 1$.
**(iii)** As $w = \beta w'$ has a suffix $\beta a^j\alpha^s$ or $ba^j\alpha^s$, the word $N(w'\beta)$ has a suffix $ \beta \alpha^s$.
All the three discussed subcases share the following property: The longest suffix $p'\in \{a,\alpha\}^*$ of the normalized name $v = \Delta(w)$ is of the form $p'=\alpha^m$, for some $m\geq 1$. It means that Case 3 (ii) is not applicable in the second iteration of $\Delta$.
By , the word $v$ has a prefix $a^n\beta $ or $\alpha^n b, n \geq 0$.
If the prefix of $v$ is of the form $\alpha^n b$, then the word $\Delta(v) = \Delta^2(w)$ belongs to Case 2. This means that $v$ is the last member of the preperiodic part.
If the prefix of $v$ is of the form $a^n\beta$, then we must apply either Case 3 (i) or 3 (iii) which means that $\alpha$ is again a suffix of the word obtained in the next iteration of $\Delta$.
Let us give a bound on the number of times that we have to use Case 3 (i) or 3 (iii) before we reach Case 2.
If $w$ contains both $\beta$ and $b$, then the number of times of using Case 3 (i) or 3 (iii) is at most the number of letters $\beta$ occurring in $w$ before the first occurrence of $b$. Thus there are at most $|w|-2$ such letters since $w$ contains $\beta$, $b$ and $\alpha$.
If $w$ does not contain $b$, then $w$ must contain besides the letters $\beta$ and $\alpha$ also the letter $a$; otherwise the morphism $\varphi_w$ would be not primitive (see ). The word $w$ has a form described in Example \[exa:pomoc7\] and thus $\Delta^{\ell_1+1}(w)$ contains both letter $b$ and $\beta$ (for the meaning of $\ell_1$ see Example \[exa:pomoc7\]). For this word we can apply the reasoning from the previous paragraph, meaning that after $\ell_1+1$ iterations we need at most $|w| -2$ further iterations before reaching the periodic part of $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$. Since $\ell_1\leq |w|-2$, we get that the preperiod is at most $2|w|-3$.
Example \[exa:pomoc6\] illustrates that in the case that $w$ contains the letters $b$ and $\beta$ the upper bounds on preperiod and period provided by the previous proposition are attained. The following example proves that the bound from Proposition \[prop:upper\_bound\] for $w$ which does not contain both letters $b$ and $\beta$ is attained as well.
Let us consider the normalized name $w =\beta^{n-2}a\alpha$. It is easy to evaluate iterations of the operator $\Delta$: $$\begin{aligned}
\Delta^{n-2}(w) & = a\beta^{n-2}\alpha \\
\Delta^{n-1}(w) & = \beta^{n-2}b\alpha \\
\Delta^{2n-3}(w) & = b\beta^{n-2}\alpha \\
\Delta^{2n-2}(w) & = \beta^{n-2}\alpha b \quad \text{--- the first member of the periodic part of $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$} \\
\Delta^{3n-4}(w) & = \alpha b\beta^{n-2} \quad \text{--- the last member of the periodic part of $\bigl(\Delta^k(w)\bigr)_{k\geq 1}$}\\
\Delta^{3n-3}(w) & = \Delta^{2n-2}(w).\end{aligned}$$
In we showed that for the Fibonacci word the derivated words to all prefixes coincide. There are infinitely many words with this property:
Consider $w=a^{n-1}\beta$ and the morphism $\psi = \varphi_w$. Then $\Delta(\psi) = \psi$ and thus the fixed point $\uu$ of $\psi$ is the derivated word to any prefix of $\uu$.
Combining and the last two examples we can give an upper and lower bound on the number of distinct derivated words.
\[coro:bounds\_on\_nr\_of\_der\_words\] Let $w \in \{a,b,\alpha,\beta\}^* \setminus \{a,\alpha\}^*$ be normalized name of a primitive Sturmian morphism $\psi =\varphi_w$ and $\uu$ be a fixed point of $\psi$. Then $$\label{meze}
1\leq \#\Der(\uu) \leq 3|w| -4\,.$$ Moreover, for any length $n\geq 2$ there exist normalized names $w', w'' \in \{a,b,\alpha,\beta\}^* \setminus \{a,\alpha\}^*$ of length $n$ such that
(i) $\varphi_{w'}$ and $\varphi_{w''}$ are not powers of other Sturmian morphisms,
(ii) for the fixed points $\uu'$ and $\uu''$ of the morphism $\varphi_{w'}$ and $\varphi_{w''}$, the lower resp. the upper bound in is attained.
Standard Sturmian morphisms and their reversals
===============================================
In this section we provide precise numbers of distinct derivated words for these three types of morphisms:
1. $\psi$ is a standard morphism from $\mathcal{M}$, i.e. $\psi\in \langle \varphi_b,\varphi_\beta\rangle$,
2. $\psi$ is a standard morphism from $\mathcal{M}\circ E$, i.e. $\psi\in \langle \varphi_b,\varphi_\beta\rangle\circ E$,
3. $\psi$ is a morphism from $\langle \varphi_a,\varphi_\alpha\rangle$.
First we explain the title of this section and the fact that the fourth type of Sturmian morphism, namely a Sturmian morphism from $ \langle \varphi_a,\varphi_\alpha\rangle \circ E$, is not considered at all.
A standard Sturmian morphism is a morphism fixing some standard Sturmian word. A reversal morphism $\overline{\psi}$ to a morphism $\psi$ is defined by $\overline{\psi}(0) = \overline{\psi(0)}$ and $\overline{\psi}(1) = \overline{\psi(1)}$. Since $\varphi_a = \overline{\varphi_b}$ and $\varphi_\alpha = \overline{\varphi_\beta}$, any morphism in $\langle \varphi_a,\varphi_\alpha\rangle$ is just a reversal of a morphism in $\langle \varphi_b,\varphi_\beta\rangle$.
Due to the form of the morphisms $\varphi_a$ and $\varphi_\alpha$, any morphism $\eta \in \langle \varphi_a,\varphi_\alpha\rangle$ satisfies that the letter $0$ is a prefix of $\eta(0)$ and the letter $1$ is a prefix of $\eta(1)$. As any morphism $\xi \in \langle \varphi_a,\varphi_\alpha\rangle \circ E$ can be written in the form $\xi(0)= \eta(1)$ and $\xi(1)= \eta(0)$ for some $\eta \in \langle \varphi_a,\varphi_\alpha\rangle$, the morphism $\xi$ cannot have any fixed point.
The normalized name $w$ of a standard morphism from $\mathcal{M}$ is composed of the letters $b$ and $\beta$ only. Thus $\Delta(w) = {\rm cyc}(w)$ (see ).
To describe all standard morphisms we have to take into account also the morphisms of the form $\psi = \varphi_w\circ E$. In this case $\psi^2 \in \langle \varphi_b, \varphi_\beta \rangle$, in particular $\psi^2 = \varphi_{wF(w)}$. To describe the derivated words of fixed points of these standard morphisms, we need the notation $${\rm cyc_F}(w_1w_2w_3\cdots w_n) = w_2w_3\cdots w_nF(w_1) \,.$$
\[cor:standard\_sturmian\] Let $\uu$ be a fixed point of a standard Sturmian morphism $\psi$ which is not a power of any other Sturmian morphism.
(i) If $\psi = \varphi_w$, then $\uu$ has $|w|$ distinct derivated words, each of them (up to a permutation of letters) is fixed by one of the morphisms $$\varphi_{v_0}, \varphi_{v_1} , \varphi_{v_2}, \ldots, \varphi_{v_{|w|-1}}, \quad \text{where } v_k ={\rm cyc}^k(w) \text{ for } k = 0, 1, \ldots, |w|-1.$$ \[cor:standard\_sturmian\_1\]
(ii) If $\psi = \varphi_w\circ E$, then $\uu$ has $|w|$ distinct derivated words, each of them (up to a permutation of letters) is fixed by one of the morphisms $$\varphi_{v_0}\circ E, \varphi_{v_1} \circ E, \varphi_{v_2}\circ E, \ldots, \varphi_{v_{|w|-1}}\circ E, \quad \text{where } v_k ={\rm cyc}_F^k(w) \text{ for } k= 0, 1, \ldots, |w|-1. \label{cor:standard_sturmian_2}$$
\(i) Since $\psi = \varphi_w$ is a standard morphism, its normalized name $w$ belongs to $\{b, \beta\}^*$ and $\Delta(w) = {\rm cyc}(w)$. By Theorem \[thm:main\_result\_vetsina\], all derivated words of $\uu$ are fixed by one of the morphisms listed in (i). We only need to show that fixed points of the listed morphisms differ. More precisely, we need to show that $v_s \neq v_t$ and $v_s\neq F(v_t)$ for all $0\leq t<s\leq |w|-1$. Here the assumption that $\psi$ is not a power of any other Sturmian morphism is crucial.
Let us recall simple facts about powers of morphisms: For any $\ell = 1, 2, \ldots$ and $u \in \{b,\beta\}^+$ we have $$(\varphi_{u})^{\ell} = \varphi_{u^\ell} \ , \quad (\varphi_{u} \circ E)^{2\ell} = \varphi_{(uF(u))^\ell} \quad \text{and} \quad (\varphi_{u} \circ E)^{2\ell + 1} = \varphi_{(uF(u))^\ell u} \circ E.$$ If $\psi = \varphi_w$ is not a power of any Sturmian morphism, we have $$\label{nepower}
w \neq u^{\ell} \ \quad \text{and }\quad w\neq \bigl(uF(u)\bigr)^{k}\quad \text{for any} \ u \in \{b, \beta\}^+ \ \text{and any }\ \ell, k \in \mathbb{N}, \ell\geq 2, k\geq 1.$$
Lemma \[lem:Lyndon2\] implies that equation ${\rm cyc}^s(w) = {\rm cyc}^t(w)$ has no solution if $w\neq u^\ell$ and $0\leq t<s\leq |w|-1$. Therefore all the normalized names $v_0,v_1, \ldots, v_{|w|-1}$ are distinct.
Now assume that $v_s = {\rm cyc}^{s}(w) = F\bigl({\rm cyc}^{t}(w) \bigr) = F(v_t)$, where $0\leq t < s\leq |w|-1$.
Let $z$ and $p$ be the words such that ${\rm cyc}^{s}(w) = zp$, where $|z| = s - t$. We have $zp = F(p)F(z)$ and by there is $x$ such that ${\rm cyc}^{s}(w) = zp = x(F(x)x)^i(F(x)x)^jF(x) = (xF(x))^{i + j + 1}$ for some non-negative integers $i, j$. This implies that there is a factor $y$ of $xF(x)$ such that $|y| = |x|$ and $w = (yF(y))^{i + j + 1}$ which is a contradiction with .
\(ii) If we apply Theorem \[thm:main\_result\_vetsina\] to the morphism $\bigl(\varphi_w\circ E\bigr)^2 = \varphi_{wF(w)}$, we obtain the list of $2|w|$ normalized names ${\rm cyc}^{s}(wF(w))$, with $s=0,1, \ldots, 2|w|-1$. As ${\rm cyc}^{|w|+i}(wF(w)) = {\rm cyc}^{i}(F(w)w)$, all the derivated words are given by the fixed points of morphisms $$\varphi_{v_0F(v_0)}, \varphi_{v_1F(v_1)}, \varphi_{v_2F(v_3)}, \ldots, \varphi_{v_{|w|-1}F(v_{|w|-1})}$$ that are just squares of morphisms listed in Item (ii) of the proposition. To finish the proof, we need to show that the fixed points of the listed morphisms do not coincide nor coincide after exchange of the letters $0\leftrightarrow 1$. In other words we need to show $v_sF(v_s) \neq v_tF(v_t)$ and $v_sF(v_s) \neq F(v_t)v_t$.\
Assume the contrary. Then $v_s = v_t$ or $v_s=F(v_t)$ for some $t<s$. If we put $k=s-t$, then $v_s ={\rm cyc}_F^k(v_t)$. Let $v_t = zp$, where $|z| = k$, then $v_s = pF(z)$. Since the morphism $\psi = \varphi_w\circ E$ is not a power of other morphism we know that $$\label{nepower2}
w\neq \bigl(uF(u)\bigr)^{\ell}u \quad \text{for any} \ u \in \{b, \beta\}^+ \ \text{and any }\ \ell \in \mathbb{N}, \ell\geq 1.$$
Two cases $v_s = v_t$ and $v_s=F(v_t)$ will be discussed separately.
- If $v_s = v_t$, then $zp = pF(z)$ and says there is $x$ so that $v_t = zp = (F(x)x)^{i + j}F(x)$, which contradicts .
- If $v_s = F(v_t)$, then $zp = F(p)z$ and by there is $x$ so that $v_t = zp = (F(x)x)^{i + j}F(x)$ which is again a contradiction with .
\[prop:number\_for\_aalpha\] Let $w \in \{\alpha, a\}^*$ be the normalized name of a primitive morphism $\psi$ such that the letter $a$ is a prefix of $w$. Moreover, assume that $\psi$ is not a power of any other Sturmian morphism.
(i) The fixed point of $\psi$ starting with $0$ has exactly $1+|w|_\alpha$ distinct derivated words.
(ii) The fixed point of $\psi$ starting with $1$ has exactly $1+|w|_a$ distinct derivated words.
We prove only Item $(i)$, the proof of $(ii)$ is analogous. Let $\uu$ denote the fixed point starting with $0$.
Proposition \[lem:revers\_of\_standard\] says that we have to count elements in the set $ \{{\bf v}\} \cup {\rm Der}(\bf v)$, where $\bf v$ is a fixed point of $\varphi_v$ with the normalized name $v=b^{-1}N(wb)$. By Lemma \[lem:jinam\], the word $v \in\{a, \beta\}^*$. This property of $v$ implies that $\Delta^k (v)$ is equal to some cyclic shift ${\rm cyc}^j(v)$ having a suffix $\beta$. There are $|v|_\beta$ cyclic shifts of $v$ with this property and hence this number is an upper bound for the period of the sequence $\bigl(\Delta^k(v)\bigr)_{k\geq1}$. By Lemma \[lem:jinam\], the normalized name $v$ has a suffix $a$ and thus the word $v$ itself does not appear in $\bigl(\Delta^k(v)\bigr)_{k\geq1}$. We can conclude that $\uu$ has at most $1+|w|_\alpha$ derivated words.
For each $k$ the iteration $\Delta^k (v)$ belongs to $\{a, \beta\}^*$ and consequently $F\bigl(\Delta^i(v)\bigr)$ belongs to $ \{a,b\}^*$. Therefore, $\Delta^j(v)\neq F\bigl(\Delta^i(v)\bigr)$ for any pair of positive integers $i,j$.
As $\psi$ is not a power of any other morphism, we can use the same technique as in the proof of to show that ${\rm cyc}^i(v) \neq {\rm cyc}^j(v)$ for $i, j=1,\ldots, |v|$, $i\neq j$. This means that the period of the sequence $\bigl(\Delta^k(v)\bigr)_{k\geq1}$ is indeed equal to $|w|_\alpha$ and its preperiod is zero.
Comments and conclusions {#sec:comments}
========================
1. In [@AraBru05], the authors studied derivated words only for standard Sturmian words ${\bf c}(\gamma)$.
However, they did not restrict their study to words fixed by a primitive morphism.
Let us show an alternative proof of their result.
The proof is a direct corollary of our and the following result of [@BeSe_Lothaire]:
\[zLothaira\]
For any irrational $\gamma \in (0,1)$ we have $$\varphi_b(\mathbf{c}(\gamma)) = \mathbf{c}\left({\frac{\gamma}{1+\gamma}}\right).$$
As we have already mentioned, the authors of [@AraBru05] required that any derivated word $\mathrm{d}_\uu(v)$ to a prefix $v$ of a Sturmian word $\uu$ starts with the same letter as the word $\uu$.
By interchanging letters $0\leftrightarrow 1$ in a characteristic word $\mathbf{c}(\gamma)$, we obtain the characteristic word $ \mathbf{c}(1-\gamma)$. If $\gamma <\tfrac12$ , then the continued fraction of $\gamma$ is of the form $ [0, c_1+1, c_2, c_3, \ldots]$ with $c_1>0$ and the continued fraction of $1-\gamma$ equals $[0, 1, c_1, c_2, c_3, \ldots]$. Clearly, $ \Der({\bf c}(\gamma))$ and $ \Der({\bf c}(1-\gamma))$ coincide up to a permutation of letters. Without loss of generality we state the next theorem for the slope $\gamma <\tfrac12$ only.
Let ${\bf c}(\gamma$) be a standard Sturmian word and $\gamma = [0, c_1+1, c_2, c_3, \ldots]$ with $c_1>0$. Then $$\Der({\bf c}(\gamma)) = \left \{{\bf c}(\delta) \colon \delta = [0, c_k+1-i, c_{k+1},c_{k+2}, \ldots] \text{ with } 0 \leq i\leq c_{k}-1 \text{ and } (k,i) \neq (1,0) \right \}.$$
Let $\delta = [0, d_1+1, d_2, d_3, \ldots]$ with $d_1>0$. Set $\delta' =\frac{\delta}{1-\delta}$. It is easy to see that $\delta'= [0,d_1, d_2, d_3, \ldots]$. Since $\delta' \in (0,1)$ and $\delta = \frac{\delta'}{1+\delta'}$, implies that ${\bf c}(\delta) = \varphi_b({\bf c}(\delta'))$. Applying we obtain that $\Der({\bf c}(\delta)) = \{{\bf c}(\delta')\} \cup \Der({\bf c}(\delta'))$. We have transformed the original task to the task to determine the set of derivated words of the standard sequence ${\bf c}(\delta')$. If $\delta' < \tfrac12$, i.e., $d_1 > 1$, we repeat this procedure with $\delta'$. If $d_1 = 1$, i.e., $\delta' > \tfrac12$, we use the fact that $\Der({\bf c}(\delta))$ and $ \Der({\bf c}(1-\delta))$ coincide, and replace $\delta'$ by $1-\delta'$ and repeat the procedure with its continued fraction $[0, d_2+1, d_3, d_4, \ldots]$.
In the terms of corresponding continued fractions, one step of the described procedure can be represented as $$[0, d_1+1, d_2, d_3, \ldots] \mapsto \begin{cases}
[0,d_1, d_2, d_3, \ldots] & \text{if } d_1 > 1, \\
[0, d_2+1, d_3, d_4, \ldots] & \text{if }d_1 = 1.
\end{cases}$$ We conclude that the set $\Der({\bf c}(\gamma))$ is in the form given in the theorem.
2. In case that $\uu$ is a fixed point of a standard Sturmian morphisms, we have determined the exact number of distinct derivated words of $\uu$, see . Let us mention that this result can be inferred from [@AraBru05]. We also have provided the exact number of derivated words when $\uu$ is a fixed point of a Sturmian morphisms which has two fixed points, see .
For fixed points of other Sturmian morphisms we only gave an upper bound on the number of their distinct derivated words, see . To give an exact number, one needs to describe when the normalized name $w \in \{a,b,\alpha, \beta\}^*$ corresponds to some power of a Sturmian morphism. Clearly, $w$ may be a normalized name of a power of a Sturmian morphism without $w$ being a power of some other word from $ \{a,b,\alpha, \beta\}^*$. For example, if $v=\alpha ba\alpha\alpha = N(v)$, then the normalized name of $v^3$ is the primitive word $ N(v^3)=\alpha bb\beta\beta\beta ba\beta\beta\beta aa \alpha\alpha$.
3. The key tool we used to determine the set $\Der({\bf u})$ is provided by . We believe that an analogue of these theorems can be found also for Arnoux–Rauzy words over multiliteral alphabet. For definition and properties of these words see [@Be_survey_corr; @GlJu].
In [@CaLaLe17], the authors described a new class of ternary sequences with complexity $2n+1$. These sequences are constructed from infinite products of two morphisms. The structure of their bispecial factors suggests that due to result of [@BaPeSt], any derivated word of such a word is over a ternary alphabet. Probably, even for these words an analogue of can be proved. Other candidates for generalization of Theorems \[thm:preimages\_fi\_b\] and \[thm:preimages\_fi\_a\] seem to be the infinite words whose language forms tree sets as defined in [@tree_sets].
Acknowledgements {#acknowledgements .unnumbered}
================
K.M., E.P. and Š.S. acknowledge financial support by the Czech Science Foundation grant GAČR 13-03538S. K. M. also acknowledges financial support by the Czech Technical University in Prague grant SGS17/193/OHK4/3T/14.
[^1]: *E-mail:* [[email protected]]([email protected]) (K. Medková)
| 0 |
---
abstract: 'We show uniqueness and stability in $L^2$ and for all time for piecewise-smooth solutions to hyperbolic balance laws. We have in mind applications to gas dynamics, the isentropic Euler system and the full Euler system for a polytropic gas in particular. We assume the discontinuity in the piecewise smooth solution is an extremal shock. We use only mild hypotheses on the system. Our techniques and result hold without smallness assumptions on the solutions. We can handle shocks of any size. We work in the class of bounded, measurable solutions satisfying a single entropy condition. We also assume a strong trace condition on the solutions, but this is weaker than $BV_{\text{loc}}$. We use the theory of a-contraction (see Kang and Vasseur \[[*Arch. Ration. Mech. Anal.*]{}, 222(1):343–391, 2016\]) developed for the stability of pure shocks in the case without source.'
address: |
Department of Mathematics\
The University of Texas at Austin\
Austin, TX 78712\
USA
author:
- 'Sam G. Krupa'
bibliography:
- 'references.bib'
date: 'April 20th, 2019'
title: 'Criteria for the a-contraction and stability for the piecewise-smooth solutions to hyperbolic balance laws'
---
Introduction
============
We consider an $n\times n$ system of balance laws, $$\begin{aligned}
\label{system}
\begin{cases}
\partial_t u + \partial_x f(u)=G(u(\cdot,t))(x),\mbox{ for } x\in\mathbb{R},\mbox{ } t>0,\\
u(x,0)=u^0(x) \mbox{ for } x\in\mathbb{R}.
\end{cases}\end{aligned}$$
For a fixed $T>0$ (including possibly $T=\infty$), the *unknown* is $u\colon\mathbb{R}\times[0,T)\to \mathbb{M}^{n\times 1}$. The function $u^0\colon\mathbb{R}\to\mathbb{M}^{n\times 1}$ is in $L^\infty(\mathbb{R})$ and is the *initial data*. The function $f\colon\mathbb{M}^{n\times 1}\to\mathbb{M}^{n\times 1}$ is the *flux function* for the system. The *source term* $G\colon (L^2(\mathbb{R}))^n\to (L^2(\mathbb{R}))^n$ is translation invariant. We also ask that $G$ be Lipschitz continuous from $(L^2(I))^n\to (L^2(I))^n$ for every interval $I\subseteq\mathbb{R}$, with a Lipschitz constant uniform in $I$. In other words, there exists $C_G>0$ such that $$\begin{aligned}
\label{G_acts_like}
\norm{G(g_1)-G(g_2)}_{L^2(I)}\leq C_G \norm{g_1-g_2}_{L^2(I)},\end{aligned}$$ for every $g_1,g_2\in(L^2(\mathbb{R}))^n$ and for every interval $I\subseteq\mathbb{R}$. Furthermore, we require that $G$ is bounded on $(L^\infty(\mathbb{R}))^n$: $$\begin{aligned}
\label{G_acts_like_2}
\norm{G(g)}_{L^\infty(\mathbb{R})}\leq C_G \norm{g}_{L^\infty(\mathbb{R})},\end{aligned}$$ for every $g\in(L^\infty(\mathbb{R}))^n$.
We assume the system is endowed with a strictly convex entropy $\eta$ and associated entropy flux $q$. Note the system will be hyperbolic on the state space where $\eta$ exists. We assume the functions $f, \eta$, and $q$ are defined on an open convex state space $\mathcal{V}\subset\mathbb{R}^n$. We assume $f,q\in C^2(\mathcal{V})$ and $\eta \in C^3(\mathcal{V})$. By assumption, the entropy $\eta$ and its associated entropy flux $q$ verify the following compatibility relation: $$\begin{aligned}
\label{compatibility_relation_eta_system}
\partial_j q =\sum_{i=1}^n \partial_i\eta\partial_j f_i,\hspace{.25in} 1\leq j \leq n.\end{aligned}$$ By convention, the relation is rewritten as $$\begin{aligned}
\nabla q = \nabla \eta \nabla f,\end{aligned}$$ where $\nabla f$ denotes the matrix $(\partial_j f_i)_{i,j}$.
For $u\in\mathcal{V}$ where $\eta$ exists , the system is hyperbolic, and the matrix $\nabla f(u)$ is diagonalizable, with eigenvalues $$\begin{aligned}
\lambda_1(u)\leq \ldots \leq \lambda_n(u),\end{aligned}$$ called *characteristic speeds*.
We consider both bounded *classical* and bounded *weak* solutions to . A weak solution $u$ is bounded and measurable and satisfies in the sense of distributions. I.e., for every Lipschitz continuous test function $\Phi:\mathbb{R}\times[0,T)\to \mathbb{M}^{1\times n}$ with compact support, $$\begin{aligned}\label{u_solves_equation_integral_formulation_chitchat}
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty} \Bigg[\partial_t\Phi u + \partial_x\Phi f(u) \Bigg]\,dxdt +\int\limits_{-\infty}^{\infty} \Phi(x,0)u^0(x)\,dx
\\
=-\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\Phi G(u(\cdot,t))(x)\,dxdt.
\end{aligned}$$
We only consider solutions $u$ which are entropic for the entropy $\eta$. That is, they satisfy the following entropy condition: $$\begin{aligned}
\label{entropy_condition_distributional_system_chitchat}
\partial_t \eta(u)+\partial_x q(u) \leq \nabla\eta(u)G(u(\cdot,t))(x),\end{aligned}$$ in the sense of distributions. I.e., for all positive, Lipschitz continuous test functions $\phi:\mathbb{R}\times[0,T)\to\mathbb{R}$ with compact support: $$\begin{aligned}\label{u_entropy_integral_formulation_chitchat}
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\Bigg[\partial_t\phi\big(\eta(u(x,t))\big)+&\partial_x \phi \big(q(u(x,t))\big)\Bigg]\,dxdt+ \int\limits_{-\infty}^{\infty}\phi(x,0)\eta(u^0(x))\,dx\geq
\\
&-\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\phi\nabla\eta(u(x,t))G(u(\cdot,t))(x)\,dxdt.
\end{aligned}$$
For $u_L,u_R\in\mathbb{R}^n$, the function $u:\mathbb{R}\times[0,\infty)\to\mathbb{R}^n$ defined by $$\begin{aligned}
\label{shock_solution_system}
u(x,t)\coloneqq
\begin{cases}
u_L &\mbox{ if } x<\sigma t ,\\
u_R &\mbox{ if } x>\sigma t
\end{cases}\end{aligned}$$ is a weak solution to if and only if $u_L,u_R$, and $\sigma$ satisfy the Rankine-Hugoniot jump compatibility relation: $$\begin{aligned}
\label{RH_jump_condition}
f(u_R)-f(u_L)=\sigma (u_R-u_L),\end{aligned}$$ in which case is called a *shock* solution.
Moreover, the solution will be entropic for $\eta$ (according to ) if and only if, $$\begin{aligned}
\label{entropic_shock_condition_system}
q(u_R)-q(u_L)\leq \sigma (\eta(u_R)-\eta(u_L)).\end{aligned}$$ In this case, $(u_L,u_R,\sigma)$ is an *entropic Rankine–Hugoniot discontinuity*.
For a fixed $u_L$, we consider the set of $u_R$ which satisfy and for some $\sigma$. For a general $n\times n$ strictly hyperbolic system of conservation laws endowed with a strictly convex entropy , we know that locally this set of $u_R$ values is made up of $n$ curves (see for example [@lefloch_book p. 140-6]).
The present paper concerns the finite-time stability of piecewise-smooth solutions to , working in the $L^2$ setting. We work in a very general setting. Our techniques are based on the theory of shifts as developed by Vasseur within the context of the relative entropy method (see [@VASSEUR2008323]). We consider systems of the form , with minimal assumptions on the shock families. We ask that the extremal shock speeds (1-shock and n-shock speeds) are separated from the intermediate shock families. If we want to consider 1-shocks, we ask that the 1-shock family satisfy the Liu entropy condition (shock speed decreases as the right-hand state travels down the 1-shock curve), and we ask that the shock strength increase in the sense of relative entropy (an $L^2$ requirement) as the right-hand state travels down the 1-shock curve. If we want to consider n-shocks, we ask for similar requirements on the n-shock family.
The intermediate wave families have far fewer requirements. The intermediate shock curves might not even be well-defined and characteristic speeds might cross.
In particular, the results in this article apply to both the isentropic Euler system and the full Euler system for a polytropic gas, viewing both systems in Eulerian coordinates.
We study solutions $\bar{u}$ which are piecewise-Lipschitz continuous in the space variable $x$. We study the stability and uniqueness of these solutions among a large class of weak solutions $u$ which are bounded, measurable, entropic for at least one strictly convex entropy, and verify a strong trace condition (weaker than $BV_{\text{loc}}$). We do not make small data assumptions. We require the piecewise-smooth $\bar{u}$ contain a single shock of extremal family. However, the rougher solutions $u$ which we compare to this solution $\bar{u}$ may have shocks of any type or family.
Previous results in the theory of stability and a-contraction have only been able to consider initial data which is pure shock (piecewise constant). This present paper extends the ideas in the theory of a-contraction (in particular as developed in [@MR3519973]).
![In this paper, we study the stability of solutions $u$ (to ) which are $L^2$ perturbations of a piecewise-smooth solution $\bar{u}$, as shown in this schematic. The nonlinearity in the solution $\bar{u}$ causes significant technical challenges not present in the piecewise-constant case (for the piecewise-constant case, see [@Leger2011; @MR3519973]).[]{data-label="figure_diagram"}](art/move_entire_solution_systems){width="\textwidth"}
The point of the present article is this: As discussed for the case of nonlocal scalar balance laws in [@scalar_move_entire_solution], when studying the stability up to a translation in space of solutions piecewise-constant in space, we can view the shift function which is doing the translation as simply determining at which points do we want to see the left hand state of our solution, and at which points do we want to see the right hand state of our solution. However, for piecewise-*smooth* data, the shift function cannot be viewed like this. Instead, the shift function is viewed as artificially translating in space our solution. If the solution is non-constant away from the discontinuity, this artificial translation creates a linear term in the entropy dissipation (see ), which we cannot Gronwall in comparison with the quadratic terms. The answer is to create a shift function which not only neutralizes entropy production at the discontinuity of the solution, but also creates additional negative entropy (see ) we can use to cancel out the linear term in the Gronwall argument (see ). Regarding the idea of additional negative entropy caused by a shift, see [@2017arXiv171207348K].
This work is related to the *generalized Riemann problem*, which concerns solutions with initial data which is piecewise-smooth instead of simply piecewise-constant across a single jump discontinuity. For existence and uniqueness results for the generalized Riemann problem, see [@MR2070131; @MR1680921]. However, these results have small data limitations.
Previous results in this direction include Chen, Frid, and Li [@MR1911734] where for the full Euler system, they show uniqueness and long-time stability for perturbations of Riemann initial data among a large class of entropy solutions (locally $BV$ and without smallness conditions) for the $3\times3$ Euler system in Lagrangian coordinates. They also show uniqueness for solutions piecewise-Lipschitz in $x$. For an extension to the relativistic Euler equations, see Chen and Li [@MR2068444]. However, these papers do not give $L^2$ stability results for all time.
We study the stability in $L^2$ of piecewise-smooth solutions to the system of balance laws . The study of piecewise-smooth solutions takes us a step beyond the classical Riemann problem, which considers piecewise-constant initial data. Furthermore, when the system has the source term $G$, it is important to study piecewise-smooth solutions and just not piecewise-constant, for the source term may mean that even pure shock wave initial data evolves into something more complicated. For a nonlocal example of this phenomenon, consider the solution to the Riemann problems for the Burgers–Hilbert equation, which is Burgers equation with a nonlocal source term [@MR3248030; @MR3605552; @MR3348783; @MR2982741].
Our method is the relative entropy method, a technique created by Dafermos [@doi:10.1080/01495737908962394; @MR546634] and DiPerna [@MR523630] to give $L^2$-type stability estimates between a Lipschitz continuous solution and a rougher solution, which is only weak and entropic for a strictly convex entropy (the so-called *weak-strong* stability theory). For a system endowed with an entropy $\eta$, the technique of relative entropy considers the quantity called the *relative entropy*, defined as $$\begin{aligned}
\eta(u|v)\coloneqq \eta(u)-\eta(v)-\nabla\eta(v)\cdot (u-v).\end{aligned}$$
Similarly, we define relative entropy-flux, $$\begin{aligned}
q(u;v)\coloneqq q(u)-q(v)-\nabla\eta(v)\cdot (f(u)-f(v)).\end{aligned}$$
Remark that for any constant $v\in\mathbb{R}^n$, the map $u\mapsto\eta(u|v)$ is an entropy for the system , with associated entropy flux $u\mapsto q(u;v)$. Furthermore, if $u$ is a weak solution to and entropic for $\eta$, then $u$ will also be entropic for $\eta(\cdot|v)$. This can be calculated directly from and – note that the map $u\mapsto\eta(u|v)$ is basically $\eta$ plus a linear term.
Moreover, by virtue of $\eta$ being *strictly* convex, the relative entropy is comparable to the $L^2$ distance, in the following sense:
\[entropy\_relative\_L2\_control\_system\] For any fixed compact set $V\subset\mathcal{V}$, there exists $c^*,c^{**}>0$ such that for all $u,v\in V$, $$\begin{aligned}
c^*\abs{a-b}^2\leq \eta(u|v)\leq c^{**}\abs{a-b}^2.\end{aligned}$$ The constants $c^*,c^{**}$ depend on $V$ and bounds on the second derivative of $\eta$.
This lemma follows from Taylor’s theorem; for a proof see [@Leger2011; @VASSEUR2008323].
Given a Lipschitz solution $\bar{u}$ to , and a weak, entropic solution $u$, the method of relative entropy gives estimates on the growth in time of the quantity $$\begin{aligned}
\norm{\bar{u}(\cdot,t)-u(\cdot,t)}_{L^2(\mathbb{R})}\end{aligned}$$ by studying the time derivative $\partial_t\int\eta(u|\bar{u})\,dx$ and using the entropy inequality . By , we get $L^2$-type stability estimates.
Introducing a discontinuity into $\bar{u}$ causes difficulties in the method of relative entropy. In particular, simple examples for the scalar conservation laws show that a discontinuity in $\bar{u}$ prevents stability between $\bar{u}$ and $u$ in the form of the classical weak-strong estimates.
However, by allowing the discontinuity in $\bar{u}$ to move with an artificial speed which depends on $u$, we can recover weak-strong type estimates. Within the context of the relative entropy method, this theory of stability up to a shift was initiated in [@VASSEUR2008323] by Vasseur. Over the last decade, this theory of stability up to a shift has been matured and developed by Vasseur and his team. The first result was for pure shock wave initial data for the scalar conservation laws [@Leger2011_original]. Further results include work on the scalar viscous conservation laws in both one space dimension [@MR3592682] and multiple [@multi_d_scalar_viscous_9122017]. Recently, work on the scalar conservation laws has allowed for many discontinuities to exist in the otherwise smooth $\bar{u}$ – with each discontinuity shifted in such a way as to maintain $L^2$ stability between $\bar{u}$ and an arbitrary weak solution $u$ entropic for at least one entropy. With this, it is possible to make comparisons between two solutions which satisfy only one entropy condition, and thus show that one entropy condition is enough for uniqueness. See [@2017arXiv170905610K] (and the references therein) for more details. To study the $L^2$ stability of pure shock wave initial data in the systems case, the technique of a-contraction was introduced [@MR3519973; @MR3479527; @MR3537479; @serre_vasseur; @Leger2011]. For a general overview of theory of shifts and the relative entropy method, see [@MR3475284 Section 3-5]. By considering stability up to a shift, the method of relative entropy can also be used to study the asymptotic limit when the limit is discontinuous (see [@MR3333670] for the scalar case, [@MR3421617] for systems). There is a long history of using the relative entropy method to study the asymptotic limit. However, without the theory of shifts, it appears that only limits which are Lipschitz continuous can be studied (see [@MR1842343; @MR1980855; @MR2505730; @MR1115587; @MR1121850; @MR1213991; @MR2178222; @MR2025302] and [@VASSEUR2008323] for a survey).
The present article is a further step in the program of stability up to a shift.
In this paper, we continue the ideas introduced in [@scalar_move_entire_solution]. In [@scalar_move_entire_solution], it is shown that the generalized characteristics of $u$ can be used as shift functions to kill growth in $L^2$ between a piecewise smooth solution $\bar{u}$ and weak solution to entropic for the entropy $\eta$. Further, using the generalized characteristic as a shift function provides various benefits over using the previous shift function constructions, as discussed in [@scalar_move_entire_solution].
In this paper, we bring novel ideas from the scalar case in [@scalar_move_entire_solution] to the systems case. In the systems case, we need to use the theory of a-contraction.
For the case of scalar, the generalized characteristics for $u$ are the natural shift functions to be using. In the systems case, we use a shift function which again is based on the generalized characteristics, but with a correction where the shift travels at greater-than-characteristic-speed due to a-contraction and the existence of multiple shock families in the systems case.
On top of the benefits for generalized-characteristic-based shifts mentioned in [@scalar_move_entire_solution] (such as simplicity of analysis, ease of construction, enhanced control on the shifts, and strictly negative entropy creation) the use of generalized-characteristic-based shifts for the *systems case* allows for simplified proofs compared to the previous state-of-the-art a-contraction result, [@MR3519973]. By having very obvious control on the speed of generalized-characteristic-based shifts, we are able to obviate the need for many of the computations in the foregoing analysis [@MR3519973].
For systems of conservation laws in one space dimension such as (including the scalar conservation laws), we have non-uniqueness for solutions. We impose entropy conditions such as , motivated by physics, to try to weed out “nonphysical” solutions which have physical entropy decreasing (or according to , mathematical entropy increasing). Remark that requiring more than one entropy condition (for more than one entropy) is impractical – many systems only admit a single nontrivial entropy. In the scalar case, this approach has had tremendous success. In fact, requiring solutions satisfy the entropy condition for at least one strictly convex entropy in $C^1$ is enough to get uniqueness for solutions (see [@panov_uniquness; @delellis_uniquneness; @2017arXiv170905610K]). However, even for the scalar case proving uniqueness with a single entropy condition has proved difficult. The first result [@panov_uniquness] was not until 1994. Furthermore, the first two results [@panov_uniquness; @delellis_uniquneness] use techniques limited to the scalar case. They use the special connection between scalar conservation laws in one space dimension and Hamilton–Jacobi equations: the space derivative of the solution to a Hamilton–Jacabi equation is formally the solution to the associated scalar conservation law. Notably, [@2017arXiv170905610K] gives a proof of the single entropy condition for scalar conservation laws which works directly on the conservation law and utilizes the theory of shifts. Moreover, progress for uniqueness of entropic solutions to *systems* of conservation laws has been slow. The best theory so far is the Bressan, Crasta, and Piccoli $L^1$ theory [@MR1686652] for uniqueness in the class of solutions with small total variation. It would be interesting however to study the uniqueness of these solutions amongst a larger class. For example, existence of solutions with large data is known for the $2\times2$ Euler system – but the uniqueness theory for such solutions with large data lags behind.
The situation for the hyperbolic conservation laws in multiple space dimensions is even more dire – there is non-uniqueness for entropic solutions to incompressible and compressible Euler by virtue of the many highly oscillatory solutions created via convex integration or related techniques. For incompressible Euler, see two papers by De Lellis and Székelyhidi [@MR2600877; @MR2564474]. For compressible Euler, see [@MR3352460; @MR3269641; @MR3744380].
However, there is still the possibility of pushing forward the theory of *uniqueness* for hyperbolic systems of conservation laws in one space dimension. The current paper is a step in that direction – utilizing the $L^2$-type relative entropy method and the constantly evolving theory of shifts.
In this article, we use the method of relative entropy, the theory of shifts and a-contraction. These theories are not perturbative. They enable us to get results without small data limitations. Further, by the nature of these theories, we only use a single entropy condition.
We present our main and most important theorem regarding $L^2$-type stability and uniqueness results. The hypotheses $(\mathcal{H})$ and $(\mathcal{H})^*$ in the theorem depend only on the hyperbolic part of the system and the fixed piecewise-smooth solution $\bar{u}$. The hypotheses are related to conditions on 1-shocks and n-shocks and in particular are satisfied by the isentropic Euler and full Euler systems. These hypotheses are explained in detail in .
\[local\_stability\_systems\]
Fix $R,T>0$.
Fix $i\in\{1,n\}$. Assume that $u,\bar{u}\in L^\infty(\mathbb{R}\times[0,T))$. If $\bar{u}$ contains a 1-shock, assume the hypotheses $(\mathcal{H})$ hold. Likewise, if $\bar{u}$ contains an n-shock, assume the hypotheses $(\mathcal{H})^*$ hold. Assume that $u$ and $\bar{u}$ are entropic for the entropy $\eta\in C^3(\mathbb{R}^n)$. Assume that $\bar{u}$ is Lipschitz continuous on $\{(x,t)\in\mathbb{R}\times[0,T) | x<s(t)\}$ and on $\{(x,t)\in\mathbb{R}\times[0,T) | x>s(t)\}$, where $s:[0,T)\to\mathbb{R}$ is a Lipschitz function . Assume also that $u$ verifies the strong trace property (). Assume also that there exists $\rho>0$ such that for all $t\in[0,T)$ $$\begin{aligned}
\label{gap_local_case}
\abs{\bar{u}(s(t)+,t)-\bar{u}(s(t)-,t)}>\rho.\end{aligned}$$
Then there exists a Lipschitz continuous function $X:[0,T)\to\mathbb{R}$ with $X(0)=0$ and constants $\mu_1,\mu_2,r>0$ such that,
$$\begin{aligned}
\label{main_local_stability_result}
\int\limits_{-R+s(0)}^{R+s(0)}\abs{u(x,t_0)-\bar{u}(x+X(t_0),t_0)}^2\,dx\leq \mu_2 e^{\mu_1 t_0}\int\limits_{-R-rt_0+s(0)}^{R+rt_0+s(0)}\abs{u^0(x)-\bar{u}^0(x)}^2\,dx ,\end{aligned}$$
for all $t_0\in[0,T)$.
Moreover, we have control on $X$: $$\begin{aligned}
\label{L2_control_shift_piecewise_systems}
\int\limits_0^{t_0} (\dot{X}(t))^2\,dt\leq \mu_2(1+e^{\mu_1 t_0})\int\limits_{-R-rt_0+s(0)}^{R+rt_0+s(0)}\abs{u^0(x)-\bar{u}^0(x)}^2\,dx.\end{aligned}$$
- The constants $\mu_1,\mu_2>0$ depend on $a$, $\rho$, $\norm{u}_{L^\infty}$, $\norm{\bar{u}}_{L^\infty}$, and bounds on the derivatives of $\eta$ on the range of $u$ and $\bar{u}$. In addition, $\mu_1$ depends on $C_G$ (see ), $\mbox{Lip}[\bar{u}]$, $R$, $T$, and bounds on the derivatives of $f$ on the range of $u$ and $\bar{u}$. Note that $r$ only depends on bounds on the derivatives of $f$ and $\eta$ (on the range of $u$ and $\bar{u}$).
- As opposed to , the proof of will in fact go through whenever we have an estimate of the form $$\begin{aligned}
\label{systems_new_estimate_remark_more_general}
\abs{\int\limits_{x_1}^{x_2} \nabla\eta(u(x,t)|\bar{u}(x+X(t),t))G(u(\cdot,t))(x)\,dx} \leq C \int\limits_{x_1}^{x_2}\abs{\nabla\eta(u(x,t)|\bar{u}(x+X(t),t))}\,dx,\end{aligned}$$ for $x_1,x_2\in\mathbb{R}$ and some constant $C>0$. Note that $u\in L^\infty$ and implies .
- Note that Hölder’s inequality and give control on the shift in the form of $$\begin{aligned}
\frac{1}{t_0}\int\limits_0^{t_0}\abs{\dot{X}(t)}\,dt \leq \frac{\sqrt{\mu_2(1+e^{\mu_1 t_0})}}{\sqrt{t_0}}\norm{u^0(\cdot)-\bar{u}^0(\cdot)}_{L^2(-R-rt_0+s(0),R+rt_0+s(0))}.\end{aligned}$$
- Note that by Property (b) of $(\mathcal{H}1)$ or $(\mathcal{H}1)^*$, condition is equivalent to the existence of a $\tilde{\rho}>0$ such that for all $t\in[0,T)$ $$\begin{aligned}
\label{gap_local_case}
r(t)>\tilde{\rho},\end{aligned}$$ where $r(t)$ satisfies $S^i_{\bar{u}(s(t)-,t)}(r(t))=\bar{u}(s(t)+,t)$.
The outline of the paper is as follows: in , we give our hypotheses on the system. In , we present technical lemmas. In , we construct the shift with the additional entropy dissipation. Finally, in we prove the main theorem by using the additional entropy dissipation from the shift to translate in $x$ the piecewise-smooth solution artificially.
Hypotheses on the system {#hypotheses_on_system}
========================
We will consider the following structural hypotheses $(\mathcal{H})$, $(\mathcal{H})^*$ on the system ,
regarding the 1-shock and n-shock curves (they are closely related to hypotheses in [@Leger2011] and [@MR3519973]). For a fixed piecewise smooth solution $\bar{u}$ (as in the context of the main theorem ):
- $(\mathcal{H}1)$: (Family of 1-shocks verifying the Liu condition) There exists $r_0>0$ such that for all $u_L\in\{\bar{u}(s(t)-,t) | t\in[0,T)\}\coloneqq I_{-}$, and for all $u\in B_{r_0}(u_L)$, there is a 1-shock curve (issuing from $u$) $S_u^1\colon[0,s_u)\to \mathcal{V}$ (possibly $s_u=\infty$) parameterized by arc length. Moreover, $S_u^1(0)=u$ and the Rankine-Hugoniot jump condition holds: $$\begin{aligned}
f(S_u^1(s))-f(u)=\sigma^1_u(s)(S_u^1(s)-u),\end{aligned}$$ where $\sigma^1_u(s)$ is the velocity function. The map $u\mapsto s_u$ is Lipschitz on $\mathcal{V}$. Further, the maps $(s,u)\mapsto S_u^1(s)$ and $(s,u)\mapsto \sigma^1_u(s)$ are both $C^1$ on $\{(s,u)|s\in [0,s_u), u\in\mathcal{V}\}$, and the following conditions are satisfied: $$\begin{aligned}
&\mbox{(a) (Liu entropy condition) } \frac{\mbox{d}}{\mbox{d}s} \sigma^1_u(s) <0,\hspace{.2in} \sigma^1_u(0)=\lambda_1(u),
\\&\mbox{(b) (shock ``strengthens'' with $s$) } \frac{\mbox{d}}{\mbox{d}s}\eta(u|S_u^1(s))>0, \hspace{.2in}\mbox{for all } s>0,
\\&\mbox{(c) (the shock curve cannot wrap tightly around itself)}
\\&\hspace{.5in}\mbox{For all $R>0$, there exists $\tilde{S}>0$ such that}
\\
&\hspace{.8in}\Big\{S^1_{u}(s) \Big| s\in[0.s_u), \abs{u}\leq R \mbox{ and } \abs{S^1_u(s)}\leq R\Big\} \subseteq \Big\{S^1_u(s) \Big| \abs{u} \leq R \mbox{ and } s\leq \tilde{S}\Big\}.\end{aligned}$$
- $(\mathcal{H}2)$: If $(u_L,u_R)$ is an entropic Rankine-Hugoniot discontinuity with shock speed $\sigma$, then $\sigma> \lambda_1(u_R)$.
- $(\mathcal{H}3)$: If $(u_L,u_R)$ (with $u_L\in B_{r_0}(\tilde{u}_L)$, for $\tilde{u}_L\in I_{-}$) is an entropic Rankine-Hugoniot discontinuity with shock speed $\sigma$ verifying $$\begin{aligned}
\sigma\leq\lambda_1(u_L),\end{aligned}$$ then $u_R$ is in the image of $S_{u_L}^1$. In other words, there exists $s_{u_R}\in[0,s_{u_L})$ such that $S_{u_L}^1(s_{u_R})=u_R$ (and by implication, $\sigma=\sigma^1_{u_L}(s_{u_R})$).
Similarly, we will consider the following structural hypotheses $(\mathcal{H})^*$ on the system , regarding the n-shock curves:
- $(\mathcal{H}1)^*$: (Family of n-shocks verifying the Liu condition) There exists $r_0>0$ such that for all $u_R\in\{\bar{u}(s(t)+,t) | t\in[0,T)\}\coloneqq I_{+}$, and for all $u\in B_{r_0}(u_R)$, there is an n-shock curve (issuing from $u$) $S_u^n\colon[0,s_u)\to \mathcal{V}$ (possibly $s_u=\infty$) parameterized by arc length. Moreover, $S_u^n(0)=u$ and the Rankine-Hugoniot jump condition holds: $$\begin{aligned}
f(S_u^n(s))-f(u)=\sigma^n_u(s)(S_u^n(s)-u),\end{aligned}$$ where $\sigma^n_u(s)$ is the velocity function. The map $u\mapsto s_u$ is Lipschitz on $\mathcal{V}$. Further, the maps $(s,u)\mapsto S_u^n(s)$ and $(s,u)\mapsto \sigma^n_u(s)$ are both $C^1$ on $\{(s,u)|s\in [0,s_u), u\in\mathcal{V}\}$, and the following conditions are satisfied: $$\begin{aligned}
&\mbox{(a) (Liu entropy condition) } \frac{\mbox{d}}{\mbox{d}s} \sigma^n_u(s) >0,\hspace{.2in} \sigma^n_u(0)=\lambda_n(u),
\\&\mbox{(b) (shock ``strengthens'' with $s$) } \frac{\mbox{d}}{\mbox{d}s}\eta(u|S_u^n(s))>0, \hspace{.2in}\mbox{for all } s>0,
\\&\mbox{(c) (the shock curve cannot wrap tightly around itself)}
\\&\hspace{.5in}\mbox{For all $R>0$, there exists $\tilde{S}>0$ such that}
\\
&\hspace{.8in}\Big\{S^n_{u}(s) \Big| s\in[0.s_u), \abs{u}\leq R \mbox{ and } \abs{S^n_u(s)}\leq R\Big\} \subseteq \Big\{S^n_u(s) \Big| \abs{u} \leq R \mbox{ and } s\leq \tilde{S}\Big\}.\end{aligned}$$
- $(\mathcal{H}2)^*$: If $(u_R,u_L)$ is an entropic Rankine-Hugoniot discontinuity with shock speed $\sigma$, then $\sigma< \lambda_n(u_L)$.
- $(\mathcal{H}3)^*$: If $(u_R,u_L)$ (with $u_R\in B_{r_0}(\tilde{u}_R)$, for $\tilde{u}_R\in I_{+}$) is an entropic Rankine-Hugoniot discontinuity with shock speed $\sigma$ verifying $$\begin{aligned}
\sigma\geq\lambda_n(u_R),\end{aligned}$$ then $u_L$ is in the image of $S_{u_R}^n$. In other words, there exists $s_{u_L}\in[0,s_{u_R})$ such that $S_{u_R}^n(s_{u_L})=u_L$ (and by implication, $\sigma=\sigma^n_{u_R}(s_{u_L})$).
See [@Leger2011; @MR3519973] for remarks on these hypotheses. We include them here for completeness. In particular,
- Note that the system verifies the hypotheses $(\mathcal{H}1)$-$(\mathcal{H}3)$ on the 1-shock family if and only if the system $$\begin{aligned}
\begin{cases}
\partial_t u - \partial_x f(u)=G(u(\cdot,t))(x),\mbox{ for } x\in\mathbb{R},\mbox{ } t>0,\\
u(x,0)=u^0(x) \mbox{ for } x\in\mathbb{R}.
\end{cases}\end{aligned}$$ verifies the properties $(\mathcal{H}1)^*$-$(\mathcal{H}3)^*$ for the n-shock family. It is in this way that $(\mathcal{H}1)$-$(\mathcal{H}3)$ are dual to $(\mathcal{H}1)^*$-$(\mathcal{H}3)^*$.
- On top of the Liu entropy condition (Property (a) in $(\mathcal{H}1)$), we also assume Property (b), which says that the 1-shock strength grows along the 1-shock curve $S^1_{u_L}$ when measured via the pseudo-distance of the relative entropy (recall that the map $(u,v)\mapsto\eta(u|v)$ measures $L^2$-distance somehow – see ). This growth condition arises naturally in the study of admissibility criteria for systems of conservation laws. In particular, Property (b) ensures that Liu admissible shocks are entropic for the entropy $\eta$ even for moderate-to-strong shocks (see [@MR1600904; @MR0093653; @MR2053765]).
In [@MR3338447], Barker, Freistühler, and Zumbrun show that stability and in particular contraction fails to hold for the full Euler system if we replace Property (b) with $$\begin{aligned}
\frac{\mbox{d}}{\mbox{d}s}\eta(S_u^1(s))>0,\hspace{.2in} s>0.\end{aligned}$$ This shows that it is better to measure shock strength using the relative entropy rather than the entropy itself.
- Recall the famous Lax E-condition for an i-shock $(u_L,u_R,\sigma)$, $$\begin{aligned}
\lambda_i(u_R)\leq\sigma\leq\lambda_i(u_L).\end{aligned}$$ The hypothesis $(\mathcal{H}2)$ is implied by the first half of the Lax E-condition along with the hyperbolicity of the system . In addition, we do not allow for right 1-contact discontinuities.
- The hypothesis $(\mathcal{H}3)$ is a statement about the well-separation of the 1-shocks from all other Rankine-Hugoniot discontinuities entropic for $\eta$; the 1-shocks do not interfere with any other shocks. In particular, $(\mathcal{H}3)$ will hold for any strictly hyperbolic system in the form if all Rankine-Hugoniot discontinuities $(u_L,u_R,\sigma)$ entropic for $\eta$ lie on an i-shock curve for some $i$ and the extended Lax admissibility condition holds: $$\begin{aligned}
\label{extended_lax_admissibility_condition}
\lambda_{i-1}(u_L) \leq \sigma \leq \lambda_{i+1} (u_R),\end{aligned}$$ where $\lambda_0\coloneqq -\infty$ and $\lambda_{n+1}\coloneqq\infty$. Moreover, we only use the first inequality in and the fact that $\lambda_1(u)\leq \lambda_{i-1}(u)$ for all $u\in\mathcal{V}$ and for all $i>1$.
Furthermore, note that for *any* strictly hyperbolic system in the form , if $u_R$ and $u_L$ live in a fixed compact set, then there exists $\delta>0$ such that will hold if $\abs{u_R-u_L}\leq\delta$. Similarly, for any strictly hyperbolic system endowed with a strictly convex entropy, all Rankine-Hugoniot discontinuities $(u_L,u_R,\sigma)$ entropic for $\eta$ will locally be in the form $S^i_{u_L}(s)=u_R$ for some $s>0$, and where $S^i_{u_L}$ is the i-shock curve issuing from $u_L$. See [@lefloch_book Theorem 1.1, p. 140] and more generally [@lefloch_book p. 140-6]. For the full Euler system , $(\mathcal{H}3)$ will hold regardless of the size of the shock $(u_L,u_R)$.
- Fix $B,\rho>0$. Then, for all $u\in\mathcal{V}$ with $\abs{u}\leq B$ and for all $s\in[\rho,B]$, we have $$\begin{aligned}\label{s_shock_strength_comparable_part1}
(s-\rho)\inf_{{\substack{u\in\mathcal{V},\hspace{.02in}\abs{u}\leq B\\t\in[\rho,B]}}}\frac{\mbox{d}}{\mbox{d}t}\eta(u|S_u^1(t))&\leq\eta(u|S_u^1(s))\\
&=\int\limits_\rho^s \frac{\mbox{d}}{\mbox{d}t}\eta(u|S_u^1(t))\,\mbox{d}t \leq (s-\rho)\sup_{{\substack{u\in\mathcal{V},\hspace{.02in}\abs{u}\leq B\\t\in[\rho,B]}}}\frac{\mbox{d}}{\mbox{d}t}\eta(u|S_u^1(t)).
\end{aligned}$$ Note that $$\begin{aligned}
0<\inf_{{\substack{u\in\mathcal{V},\hspace{.02in}\abs{u}\leq B\\t\in[\rho,B]}}}\frac{\mbox{d}}{\mbox{d}t}\eta(u|S_u^1(t))\end{aligned}$$ and $$\begin{aligned}
0<\sup_{{\substack{u\in\mathcal{V},\hspace{.02in}\abs{u}\leq B\\t\in[\rho,B]}}}\frac{\mbox{d}}{\mbox{d}t}\eta(u|S_u^1(t))<\infty\end{aligned}$$ due to $(\mathcal{H}1)$.
Recall also that by hypothesis $(\mathcal{H}1)$, $S_u^1$ is parameterized by arc length. Thus, $\abs{S_u^1(s)-u}\leq B$ for all $s\in[0,B]$. We can then use and to get, $$\begin{aligned}
\label{shock_strength_comparable_s_systems1}
(s-\rho)d_1\leq\abs{u-S_u^1(s)}^2\leq (s-\rho) d_2\end{aligned}$$ for all $u\in\mathcal{V}$ with $\abs{u}\leq B$ and for all $s\in[\rho,B]$. The constants $d_1,d_2>0$ depend only on $B$ and $\rho$. This says that $s-\rho$ is comparable to the shock strength $\abs{u-S_u^1(s)}^2$.
- On the state space $\mathcal{V}$ where the strictly convex entropy $\eta$ is defined, the system is hyperbolic. Further, by virtue of $f\in C^2(\mathcal{V})$, the eigenvalues of $\nabla f (u)$ vary continuously on the state space $\mathcal{V}$. Further, if the eigenvalue $\lambda_1(u)$ ($\lambda_n(u)$) is simple for $u\in\mathcal{V}$ (such as when the system is strictly hyperbolic), the map $u\mapsto \lambda_1(u)$ ($u\mapsto \lambda_n(u)$) will be in $C^1(\mathcal{V})$ due to the implicit function theorem.
We study solutions $u$ to among the class of functions verifying a strong trace property (first introduced in [@Leger2011]):
\[strong\_trace\_definition\] Fix $T>0$. Let $u\colon\mathbb{R}\times[0,T)\to\mathbb{R}^n$ verify $u\in L^\infty(\mathbb{R}\times[0,T))$. We say $u$ has the *strong trace property* if for every fixed Lipschitz continuous map $h\colon [0,T)\to\mathbb{R}$, there exists $u_+,u_-\colon[0,T)\to\mathbb{R}^n$ such that $$\begin{aligned}
\lim_{n\to\infty}\int\limits_0^{t_0}\operatorname*{ess\,sup}_{y\in(0,\frac{1}{n})}\abs{u(h(t)+y,t)-u_+(t)}\,dt=\lim_{n\to\infty}\int\limits_0^{t_0}\operatorname*{ess\,sup}_{y\in(-\frac{1}{n},0)}\abs{u(h(t)+y,t)-u_-(t)}\,dt=0\end{aligned}$$ for all $t_0\in(0,T)$.
Note that for example a function $u\in L^\infty(\mathbb{R}\times[0,T))$ will satisfy the strong trace property if for each fixed $h$, the right and left limits $$\begin{aligned}
\lim_{y\to0^{+}}u(h(t)+y,t)\hspace{.7in}\mbox{and}\hspace{.7in}\lim_{y\to0^{-}}u(h(t)+y,t)\end{aligned}$$ exist for almost every $t$. In particular, a function $u\in L^\infty(\mathbb{R}\times[0,T))$ will have strong traces according to if $u$ has a representative which is in $BV_{\text{loc}}$. However, the strong trace property is weaker than $BV_{\text{loc}}$.
Technical Lemmas {#technical_lemmas}
================
Throughout this paper, we use the following definition for the relative flux $$\begin{aligned}
\label{Z_def}
f(a|b)\coloneqq f(a)-f(b)-\nabla f (b)(a-b),\end{aligned}$$ and the relative $\nabla\eta$: for $a,b\in\mathbb{M}^{n\times 1}$, $$\begin{aligned}
\nabla\eta(a|b)\coloneqq \nabla\eta(a)-\nabla\eta(b)-[a-b]^T\nabla^2\eta(b).\end{aligned}$$.
The following lemma from [@MR3537479] describes how the relative entropy obeys a sort of triangle inequality:
\[triangle\_inequality\_systems\_entropy\] For any $u,v,w\in\mathcal{V}$, we have $$\begin{aligned}
\eta(u|w)+\eta(w|v)=\eta(u|v)+(\nabla\eta(w)-\nabla\eta(v))\cdot(w-u),\end{aligned}$$ and $$\begin{aligned}
q(u;w)+q(w;v)=q(u;v)+(\nabla\eta(w)-\nabla\eta(v))\cdot(f(w)-f(u)).\end{aligned}$$
Thus, for any $\sigma\in\mathbb{R}$, $$\begin{aligned}\label{48_kang_vasseur_a_contraction}
q(u;v)-\sigma\eta(u|v)=&(q(u;w)-\sigma\eta(u;w))+(q(w;v)-\sigma\eta(w|v))
\\
&-(\nabla\eta(w)-\nabla\eta(v))\cdot(f(w)-f(u)-\sigma(w-u)).
\end{aligned}$$
The proof of follows immediately from the definition of $q(\hspace{.015in}\cdot\hspace{.015in};\hspace{.015in}\cdot\hspace{.015in})$ and $\eta(\hspace{.015in}\cdot\hspace{.015in}|\hspace{.015in}\cdot\hspace{.015in})$. In particular, see [@MR3519973 p. 360-1] for a simple proof.
\[a\_cond\_lemma\_itself\] Fix $B>0$. Then there exists a constant $C>0$ depending on $B$ such that the following holds:
If $u_L,u_R\in\mathcal{V}$ with $\abs{u_L},\abs{u_R}\leq B$, then whenever $\alpha,\theta\in(0,1)$ verify $$\begin{aligned}
\label{cond_a}
\alpha<\frac{\theta^2}{C},\end{aligned}$$ then $R_a\coloneqq\{u | \eta(u|u_L)\leq a\eta(u|u_R)\}\subset B_{\theta}(u_L)$ for all $0<a<\alpha$.
The set $R_a$ is compact.
The proof of is found in the proof of Lemma 4.3 in [@MR3519973]. We repeat the proof in for the reader’s convenience.
The following Lemma gives the entropy dissipation caused by changing the domain of integration, translating the solution $\bar{u}$ in $x$ (by a function $X(t)$), and from the source term $G$.
\[local\_entropy\_dissipation\_rate\_systems\] Fix $T>0$. Let $u,\bar{u}\in L^\infty(\mathbb{R}\times[0,T))$ be weak solutions to . Assume that $u$ and $\bar{u}$ are entropic for the entropy $\eta$. Assume that $\bar{u}$ is Lipschitz continuous on $\{(x,t)\in\mathbb{R}\times[0,T) | x<s(t)\}$ and on $\{(x,t)\in\mathbb{R}\times[0,T) | x>s(t)\}$, where $s:[0,T)\to\mathbb{R}$ is a Lipschitz function . Assume also that $u$ verifies the strong trace property (). Let $h_1,h_2, X:[0,T)\to\mathbb{R}$ be Lipschitz continuous functions with the property that there exists $\delta>0$ such that $h_2(t)-h_1(t)\geq \delta$ for all $t\in[0,T)$. Assume also that for all $t\in[0,T)$, $s(t)-X(t)$ is not in the open set $(h_1(t),h_2(t))$.
Then, $$\begin{aligned}\label{local_compatible_dissipation_calc}
&\int\limits_{0}^{t_0} \bigg[q(u(h_1(t)+,t);\bar{u}((h_1(t)+X(t))+,t))-q(u(h_2(t)-,t);\bar{u}((h_2(t)+X(t))-,t))
\\
&\hspace{.7in}+\dot{h}_2(t)\eta(u(h_2(t)-,t)|\bar{u}((h_2(t)+X(t))-,t))
\\
&\hspace{.7in}-\dot{h}_1(t)\eta(u(h_1(t)+,t)|\bar{u}((h_1(t)+X(t))+,t))\bigg]\,dt
\\
&\hspace{.5in}\geq
\int\limits_{h_1(t_0)}^{h_2(t_0)}\eta(u(x,t_0)|\bar{u}(x+X(t_0),t_0))\,dx
-\int\limits_{h_1(0)}^{h_2(0)}\eta(u^0(x)|\bar{u}^0(x))\,dx
\\
&\hspace{.7in}+\int\limits_{0}^{t_0}\int\limits_{h_1(t)}^{h_2(t)}\Bigg(\partial_x \bigg|_{(x+X(t),t)}\hspace{-.45in} \nabla\eta(\bar{u}(x,t))\Bigg)f(u(x,t)|\bar{u}(x+X(t),t))
\\
&\hspace{.7in}+\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\dot{X}(t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
\\
&\hspace{.7in}-\nabla\eta(u(x,t)|\bar{u}(x+X(t),t))G(u(\cdot,t))(x)
\\
&+
\Bigg(G(\bar{u}(\cdot,t))(x+X(t))-G(u(\cdot,t))(x)\Bigg)^T\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]\,dxdt.
\end{aligned}$$
This proof is based on a similar argument in [@scalar_move_entire_solution].
We show that for all positive, Lipschitz continuous test functions $\phi:\mathbb{R}\times[0,T)\to\mathbb{R}$ with compact support and that vanish on the set $\{(x,t)\in\mathbb{R}\times[0,T) | x=s(t)-X(t)\}$, we have $$\begin{aligned}\label{combined1}
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty} [\partial_t \phi \eta(u(x,t)|\bar{u}(x+X(t),t))+\partial_x \phi q(u(x,t);\bar{u}(x+X(t),t))]\,dxdt\hspace{1.5in} \\
+ \int\limits_{-\infty}^{\infty}\phi(x,0)\eta(u^0(x)|\bar{u}^0(x))\,dx\hspace{3in} \\
\geq
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\phi\Bigg[\Bigg(
\partial_x \bigg|_{(x+X(t),t)} \hspace{-.45in}\nabla\eta(\bar{u}(x,t))\Bigg)f(u(x,t)|\bar{u}(x+X(t),t))
\\
+\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\dot{X}(t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
\\
-\nabla\eta(u(x,t)|\bar{u}(x+X(t),t))G(u(\cdot,t))(x)
\\
+
\Bigg(G(\bar{u}(\cdot,t))(x+X(t))-G(u(\cdot,t))(x)\Bigg)^T\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
\Bigg]\,dxdt.
\end{aligned}$$
Note that is the analogue in our case of the key estimate used in Dafermos’s proof of weak-strong stability, which gives a relative version of the entropy inequality (see equation (5.2.10) in [@dafermos_big_book p. 122-5]). The proof of is based on the famous weak-strong stability proof of Dafermos and DiPerna [@dafermos_big_book p. 122-5]. To take into account the entropy production due to translating the solution $\bar{u}$ by the function $X$, we use the argument introduced in [@scalar_move_entire_solution].
Note that on the complement of the set $\{(x,t)\in\mathbb{R}\times[0,T) | x=s(t)\}$, $\bar{u}$ is smooth and so we have the exact equalities, $$\begin{aligned}
\partial_t\bigg|_{(x,t)}\hspace{-.21in}\big(\bar{u}(x,t)\big)+\partial_x\bigg|_{(x,t)}\hspace{-.21in}\big(f(\bar{u}(x,t))\big)&=G(\bar{u}(\cdot,t))(x),\label{solves_equation}\\
\partial_t\bigg|_{(x,t)}\hspace{-.21in}\big(\eta(\bar{u}(x,t))\big)+\partial_x\bigg|_{(x,t)}\hspace{-.21in}\big(q(\bar{u}(x,t))\big)&=\nabla\eta(\bar{u}(x,t))G(\bar{u}(\cdot,t))(x).\label{solves_entropy}\end{aligned}$$
Thus for any Lipschitz continuous function $X: [0,T)\to\mathbb{R}$ with $X(0)=0$ we have on the complement of the set $\{(x,t)\in\mathbb{R}\times[0,T) | x=s(t)-X(t)\}$, $$\begin{aligned}\label{solves_equation_shift}
\partial_t\bigg|_{(x,t)}\hspace{-.21in}&\big(\bar{u}(x+X(t),t)\big)+\partial_x\bigg|_{(x,t)}\hspace{-.21in}\big(f(\bar{u}(x+X(t),t))\big)=
\\
&\hspace{1.5in}\Bigg(\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\big(\bar{u}(x,t)\big)\Bigg)\dot{X}(t)+G(\bar{u}(\cdot,t))(x+X(t)),
\end{aligned}$$ and $$\begin{aligned}\label{solves_entropy_shift}
\partial_t\bigg|_{(x,t)}\hspace{-.21in}&\big(\eta(\bar{u}(x+X(t),t))\big)+\partial_x\bigg|_{(x,t)}\hspace{-.21in}\big(q(\bar{u}(x+X(t),t))\big)=
\\
&\nabla\eta(\bar{u}(x+X(t),t))\Bigg(\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\big(\bar{u}(x,t)\big)\Bigg)\dot{X}(t)+\nabla\eta(\bar{u}(x+X(t),t))G(\bar{u}(\cdot,t))(x+X(t)).
\end{aligned}$$
We can now imitate the weak-strong stability proof in [@dafermos_big_book p. 122-5], using and instead of and .
Recall , which says
$$\begin{aligned}
f(u|\bar{u})\coloneqq f(u)-f(\bar{u})-\nabla f (\bar{u})(u-\bar{u}).\end{aligned}$$
Remark that $f(u|\bar{u})$ is locally quadratic in $u-\bar{u}$.
Fix any positive, Lipschitz continuous test function $\phi:\mathbb{R}\times[0,T)\to\mathbb{R}$ with compact support. Assume also that $\phi$ vanishes on the set $\{(x,t)\in\mathbb{R}\times[0,T) | x=s(t)-X(t)\}$. Then, we use that $u$ satisfies the entropy inequality in a distributional sense: $$\begin{aligned}\label{u_entropy_integral_formulation}
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\Bigg[\partial_t\phi\big(\eta(u(x,t))\big)+&\partial_x \phi \big(q(u(x,t))\big)\Bigg]\,dxdt+ \int\limits_{-\infty}^{\infty}\phi(x,0)\eta(u^0(x))\,dx
\\
&\geq-\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\phi\nabla\eta(u(x,t))G(u(\cdot,t))(x)\,dxdt.
\end{aligned}$$
We also view as a distributional equality: $$\begin{aligned}\label{solves_entropy_shift_integral_formulation}
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\Bigg[\partial_t\phi\big(\eta(\bar{u}(x+&X(t),t))\big)+\partial_x \phi \big(q(\bar{u}(x+X(t),t))\big)\Bigg]\,dxdt+ \int\limits_{-\infty}^{\infty}\phi(x,0)\eta(\bar{u}^0(x))\,dx
\\
&=-\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\phi\Bigg[\nabla\eta(\bar{u}(x+X(t),t))\Bigg(\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\big(\bar{u}(x,t)\big)\Bigg)\dot{X}(t)
\\
&\hspace{.5in}+\nabla\eta(\bar{u}(x+X(t),t))G(\bar{u}(\cdot,t))(x+X(t))\Bigg]\,dxdt.
\end{aligned}$$
To get , we do integration by parts twice on the right hand side of . Once on the domain $\{(x,t)\in\mathbb{R}\times[0,T) | x<s(t)-X(t)\}$ and once on the domain $\{(x,t)\in\mathbb{R}\times[0,T) | x>s(t)-X(t)\}$. We don’t have a boundary term along the set $\{(x,t)\in\mathbb{R}\times[0,T) | x=s(t)-X(t)\}$ because $\phi$ vanishes on this set.
We subtract from , to get
$$\begin{aligned}\label{difference_entropy_equations}
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty} [\partial_t \phi \eta(u(x,t)|\bar{u}(x+X(t),t))+\partial_x \phi q(u(x,t),\bar{u}(x+X(t),t))]\,dxdt \hspace{.5in}\\
+ \int\limits_{-\infty}^{\infty}\phi(x,0)\eta(u^0(x)|\bar{u}^0(x))\,dx \hspace{2.5in}\\
\geq -\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty} \Big(\partial_t \phi\nabla\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
\\
+\partial_x \phi \nabla\eta(\bar{u}(x+X(t),t))[f(u(x,t))-f(\bar{u}(x+X(t),t))]\Big)\,dxdt
\\
- \int\limits_{-\infty}^{\infty}\phi(x,0)\nabla\eta(\bar{u}^0(x))[u^0(x)-\bar{u}^0(x)]\,dx
\\
+
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\phi\Bigg[\nabla\eta(\bar{u}(x+X(t),t))\Bigg(\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\big(\bar{u}(x,t)\big)\Bigg)\dot{X}(t)
\\
+\nabla\eta(\bar{u}(x+X(t),t))G(\bar{u}(\cdot,t))(x+X(t))-\nabla\eta(u(x,t))G(u(\cdot,t))(x)\Bigg]\,dxdt.
\end{aligned}$$
The function $u$ is a distributional solution to the system of conservation laws. Thus, for every Lipschitz continuous test function $\Phi:\mathbb{R}\times[0,T)\to \mathbb{M}^{1\times n}$ with compact support, $$\begin{aligned}\label{u_solves_equation_integral_formulation}
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty} \Bigg[\partial_t\Phi u + \partial_x\Phi f(u) \Bigg]\,dxdt +\int\limits_{-\infty}^{\infty} \Phi(x,0)u^0(x)\,dx
\\
=-\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\Phi G(u(\cdot,t))(x)\,dxdt.
\end{aligned}$$
We also can rewrite in a distributional way, for $\Phi$ which have the additional property of vanishing on $\{(x,t)\in\mathbb{R}\times[0,T) | x=s(t)-X(t)\}$: $$\begin{aligned}\label{shift_solves_equation_integral_formulation}
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty} \Bigg[\partial_t\Phi \bar{u}(x+X(t),t) + \partial_x\Phi f(\bar{u}(x+X(t),t)) \Bigg]\,dxdt +\int\limits_{-\infty}^{\infty} \Phi(x,0)\bar{u}^0(x)\,dx
\\
=-\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\Phi \Bigg[\Bigg(\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\big(\bar{u}(x,t)\big)\Bigg)\dot{X}(t)+G(\bar{u}(\cdot,t))(x+X(t))\Bigg]\,dxdt.
\end{aligned}$$ To prove , on the right hand side of we again do integration by parts twice. Once on the domain $\{(x,t)\in\mathbb{R}\times[0,T) | x<s(t)-X(t)\}$ and once on the domain $\{(x,t)\in\mathbb{R}\times[0,T) | x>s(t)-X(t)\}$. We lose the boundary terms along $\{(x,t)\in\mathbb{R}\times[0,T) | x=s(t)-X(t)\}$ because $\Phi$ vanishes there.
Then, we can choose $$\begin{aligned}
\label{our_choice_for_Phi}
\phi\nabla\eta(\bar{u}(x+X(t),t))\end{aligned}$$ as the test function $\Phi$, and subtract from . We can extend the function to the set $\{(x,t)\in\mathbb{R}\times[0,T) | x=s(t)-X(t)\}$ by defining it to be zero. This extension is still Lipschitz continuous.
This yields, $$\begin{aligned}\label{difference_of_solutions_equation}
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty} \Bigg[\partial_t[\phi\nabla\eta(\bar{u}(x+X(t),t))][u(x,t)- \bar{u}(x+X(t),t)] \hspace{2in}
\\
+\partial_x[\phi\nabla\eta(\bar{u}(x+X(t),t))][f(u(x,t))-f(\bar{u}(x+X(t),t))] \Bigg]\,dxdt \\
+\int\limits_{-\infty}^{\infty} \phi(x,0)\nabla\eta(\bar{u}^0(x))[u^0(x)-\bar{u}^0(x)]\,dx\hspace{1.42in}
\\
=\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\phi\nabla\eta(\bar{u}(x+X(t),t)) \Bigg[\Bigg(\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\big(\bar{u}(x,t)\big)\Bigg)\dot{X}(t)\\
+G(\bar{u}(\cdot,t))(x+X(t))-G(u(\cdot,t))(x)\Bigg]\,dxdt.
\end{aligned}$$
Recall $\bar{u}$ is a classical solution on the complement of the set $\{(x,t)\in\mathbb{R}\times[0,T) | x=s(t)\}$ and verifies . Thus, on the complement of the set $\{(x,t)\in\mathbb{R}\times[0,T) | x=s(t)-X(t)\}$, $$\begin{aligned}\label{notice_this}
&\partial_t\bigg|_{(x,t)}\hspace{-.21in}\big(\nabla\eta(\bar{u}(x+X(t),t))\big)=\Bigg(\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\dot{X}(t)+\partial_t\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))
\\
&\hspace{1in}=
\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\dot{X}(t)
-\partial_x \bigg|_{(x+X(t),t)}\hspace{-.45in} \bar{u}^T(x,t)\big[\nabla f(\bar{u}(x+X(t),t))\big]^T\\
&\hspace{1.5in}+G^T(\bar{u}(\cdot,t))(x+X(t))\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))
\\
&\hspace{1in}=
\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\dot{X}(t)
+G^T(\bar{u}(\cdot,t))(x+X(t))\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))
\\
&\hspace{1.5in}-\partial_x \bigg|_{(x+X(t),t)}\hspace{-.45in} \bar{u}^T(x,t)\nabla^2\eta(\bar{u}(x+X(t),t))\nabla f(\bar{u}(x+X(t),t)),
\end{aligned}$$ because $\big[\nabla f(\bar{u})\big]^T\nabla^2\eta(\bar{u})=\nabla^2\eta(\bar{u})\nabla f(\bar{u})$.
Thus, by and the definition of the relative flux in ,
$$\begin{aligned}\label{5.2.9}
\partial_t\bigg|_{(x,t)}\hspace{-.21in}\big(\nabla\eta(\bar{u}(x+X(t),t))\big)[u(x,t)-\bar{u}(x+X(t),t)]\hspace{2.5in}
\\
+\partial_x\bigg|_{(x,t)}\hspace{-.21in}\big(\nabla\eta(\bar{u}(x+X(t),t))\big)[f(u(x,t))-f(\bar{u}(x+X(t),t))]\hspace{1in}
\\
=
\partial_x \bigg|_{(x+X(t),t)}\hspace{-.45in} \bar{u}^T(x,t)\nabla^2\eta(\bar{u}(x+X(t),t)) f(u(x,t)|\bar{u}(x+X(t),t))
\\
+\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\dot{X}(t)
+G^T(\bar{u}(\cdot,t))(x+X(t))\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)].
\end{aligned}$$
We combine , , and to get $$\begin{aligned}\label{combined}
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty} [\partial_t \phi \eta(u(x,t)|\bar{u}(x+X(t),t))+\partial_x \phi q(u(x,t);\bar{u}(x+X(t),t))]\,dxdt\hspace{1in} \\
+ \int\limits_{-\infty}^{\infty}\phi(x,0)\eta(u^0(x)|\bar{u}^0(x))\,dx\hspace{2in} \\
\geq
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\phi\Bigg[\nabla\eta(\bar{u}(x+X(t),t))\Bigg(\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\big(\bar{u}(x,t)\big)\Bigg)\dot{X}(t)
\\
+\nabla\eta(\bar{u}(x+X(t),t))G(\bar{u}(\cdot,t))(x+X(t))-\nabla\eta(u(x,t))G(u(\cdot,t))(x)
\\
+\Bigg(\partial_x \bigg|_{(x+X(t),t)}\hspace{-.45in} \bar{u}^T(x,t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t)) f(u(x,t)|\bar{u}(x+X(t),t))
\\
+\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\dot{X}(t)
\\
+G^T(\bar{u}(\cdot,t))(x+X(t))\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
\\
-\nabla\eta(\bar{u}(x+X(t),t)) \Big[\Bigg(\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\big(\bar{u}(x,t)\big)\Bigg)\dot{X}(t)+G(\bar{u}(\cdot,t))(x+X(t))-G(u(\cdot,t))(x)\Big]
\Bigg]\,dxdt
\\
=
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\phi\Bigg[-\nabla\eta(u(x,t))G(u(\cdot,t))(x)
\\
+\Bigg(\partial_x \bigg|_{(x+X(t),t)}\hspace{-.45in} \bar{u}^T(x,t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t)) f(u(x,t)|\bar{u}(x+X(t),t))
\\
+\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\dot{X}(t)
\\
+G^T(\bar{u}(\cdot,t))(x+X(t))\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
\\
-\nabla\eta(\bar{u}(x+X(t),t)) \Big[-G(u(\cdot,t))(x)\Big]
\Bigg]\,dxdt.
\end{aligned}$$
Note that we can add zero, to get $$\begin{aligned}\label{note_that}
-\nabla\eta(u(x,t))G(u(\cdot,t))(x)+G^T(\bar{u}(\cdot,t))(x+X(t))\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
\\
-\nabla\eta(\bar{u}(x+X(t),t)) \Big[-G(u(\cdot,t))(x)\Big]\hspace{3in}
\\
=
-G^T(u(\cdot,t))(x)\Bigg(\big(\nabla\eta(u(x,t))\big)^T-\big(\nabla\eta(\bar{u}(x+X(t),t))\big)^T
\\
-\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]\Bigg)
\\
+
\Bigg(G^T(\bar{u}(\cdot,t))(x+X(t))-G^T(u(\cdot,t))(x)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
\\
=
-G^T(u(\cdot,t))(x)(\nabla\eta(u(x,t)|\bar{u}(x+X(t),t)))^T\hspace{.84in}
\\
+
\Bigg(G^T(\bar{u}(\cdot,t))(x+X(t))-G^T(u(\cdot,t))(x)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
\\
=
-\nabla\eta(u(x,t)|\bar{u}(x+X(t),t))G(u(\cdot,t))(x)\hspace{1.15in}
\\
+
\Bigg(G^T(\bar{u}(\cdot,t))(x+X(t))-G^T(u(\cdot,t))(x)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)].
\end{aligned}$$ This calculation is from [@VASSEUR2008323].
Then, from and , we get .
Choose $0<\epsilon<\min\{T-t_0,\frac{1}{2}\delta\}$.
We apply the test function $\omega(t)\chi(x,t)$ to , where
$$\begin{aligned}
\omega(t)\coloneqq
\begin{cases}
1 & \text{if } 0\leq t< t_0\\
\frac{1}{\epsilon}(t_0-t)+1 & \text{if } t_0\leq t < t_0+\epsilon\\
0 & \text{if } t_0+\epsilon \leq t,
\end{cases}\end{aligned}$$
and $$\begin{aligned}
\chi(x,t)\coloneqq
\begin{cases}
0 & \text{if } x<h_1(t)\\
\frac{1}{\epsilon}(x-h_1(t)) & \text{if } h_1(t)\leq x < h_1(t)+\epsilon\\
1 & \text{if } h_1(t)+\epsilon\leq x \leq h_2(t) -\epsilon\\
-\frac{1}{\epsilon}(x-h_2(t)) & \text{if } h_2(t)-\epsilon<x\leq h_2(t)\\
0 & \text{if } h_2(t)<x.
\end{cases}\end{aligned}$$
The function $\omega$ is modeled from [@dafermos_big_book p. 124]. The function $\chi$ is from [@Leger2011_original p. 765]. We get,
$$\begin{aligned}\label{local_plugged_test}
&\int\limits_{0}^{t_0} \Bigg[-\int\limits_{h_1(t)}^{h_1(t)+\epsilon}\frac{1}{\epsilon}\dot{h}_1(t)\eta(u(x,t)|\bar{u}(x+X(t),t))\,dx
+\int\limits_{h_1(t)}^{h_1(t)+\epsilon}\frac{1}{\epsilon}q(u(x,t);\bar{u}(x+X(t),t))\,dx
\\
&+\int\limits_{h_2(t)-\epsilon}^{h_2(t)}\frac{1}{\epsilon}\dot{h}_2(t)\eta(u(x,t)|\bar{u}(x+X(t),t))\,dx-\int\limits_{h_2(t)-\epsilon}^{h_2(t)}\frac{1}{\epsilon}q(u(x,t);\bar{u}(x+X(t),t))\,dx\Bigg]\,dt
\\
&+
\int\limits_{h_1(0)}^{h_2(0)}\eta(u^0(x)|\bar{u}^0(x))\,dx
-
\int\limits_{t_0}^{t_0+\epsilon}\frac{1}{\epsilon}\int\limits_{h_1(t)}^{h_2(t)}\eta(u(x,t)|\bar{u}(x+X(t),t))\,dxdt
+\mathcal{O}(\epsilon)
\\
&&\hspace{-2in}\geq
\int\limits_{0}^{t_0}\int\limits_{h_1(t)}^{h_2(t)}\mbox{RHS}\,dxdt,
\end{aligned}$$
where RHS represents everything being multiplied by $\phi$ in the integral on the right hand side of .
We let $\epsilon\to0$ in . We use dominated convergence, the Lebegue differentiation theorem, and recall that $u$ satisfies the strong trace property (). This yields,
$$\begin{aligned}
\int\limits_{0}^{t_0} \bigg[q(u(h_1(t)+,t);\bar{u}((h_1(t)+X(t))+,t))-q(u(h_2(t)-,t);\bar{u}((h_2(t)+X(t))-,t))
\\
+\dot{h}_2(t)\eta(u(h_2(t)-,t)|\bar{u}((h_2(t)+X(t))-,t))\hspace{2in}
\\
-\dot{h}_1(t)\eta(u(h_1(t)+,t)|\bar{u}((h_1(t)+X(t))+,t))\bigg]\,dt\hspace{1.76in}
\\
\geq
\int\limits_{h_1(t_0)}^{h_2(t_0)}\eta(u(x,t_0)|\bar{u}(x+X(t_0),t_0))\,dx
\\
-\int\limits_{h_1(0)}^{h_2(0)}\eta(u^0(x)|\bar{u}^0(x))\,dx
\\
+\int\limits_{0}^{t_0}\int\limits_{h_1(t)}^{h_2(t)}\mbox{RHS}\,dxdt,
\end{aligned}$$
where we also used the convexity of $\eta$ to take the limit of the term $$\begin{aligned}
\int\limits_{t_0}^{t_0+\epsilon}\frac{1}{\epsilon}\int\limits_{h_1(t)}^{h_2(t)}\eta(u(x,t)|\bar{u}(x+X(t),t))\,dxdt\end{aligned}$$ for every $t_0$ and not just almost every $t_0$.
We receive .
Construction of the shift {#construction_of_the_shift}
=========================
In this section, we prove
\[systems\_entropy\_dissipation\_room\]
Fix $T>0$. Assume $u$ is a bounded weak solution to . Assume $u$ is entropic for the entropy $\eta$, and $u$ has strong traces (). Fix $i\in\{1,n\}$. Then let $(\bar{u}_+(t),\bar{u}_-(t),\dot{s}(t))$ be an i-shock for all $t\in[0,T)$, where $s:[0.T)\to\mathbb{R}$ is a Lipschitz continuous function. Assume also that the map $t\mapsto(\bar{u}_+(t),\bar{u}_-(t))$ is bounded. For $i=1$, assume the hypotheses $(\mathcal{H})$ hold. Likewise, if $i=n$, assume the hypotheses $(\mathcal{H})^*$ hold.
Assume also that there exists $\rho>0$ such that for all $t\in[0,T)$ $$\begin{aligned}
\label{gap_local_case}
r(t)>\rho,\end{aligned}$$ where $r(t)$ satisfies $S^1_{\bar{u}_-(t)}(r(t))=\bar{u}_+(t)$.
Then, there exists a constant $a>0$ and a Lipschitz continuous map $h: [0,T)\to\mathbb{R}$ with $h(0)=s(0)$ and such that for almost every $t$, $$\begin{aligned}\label{dissipation_negative_claim}
a\big(q(u_+;\bar{u}_+(t))&-\dot{h}(t)\eta(u_+|\bar{u}_+(t))\big)-q(u_-;\bar{u}_-(t))+\dot{h}(t)\eta(u_-|\bar{u}_-(t)) \leq \\
& -c \abs{\dot{s}(t)-\dot{h}(t)}^2,
\end{aligned}$$ where $u_{\pm}\coloneqq u(u(h(t)\pm,t)$. The constants $c,a>0$ depend on $\norm{u}_{L^\infty}$, $\norm{\bar{u}_+(\cdot)}_{L^\infty([0,T))}$, $\norm{\bar{u}_-(\cdot)}_{L^\infty([0,T))}$, and $\rho$.
The proof of uses
\[dissipation\_negative\_theorem\] Assume the hypotheses $(\mathcal{H})$ hold.
Let $B,\rho>0$. Then there exists a constant $a_*\in(0,1)$ depending on $B$ and $\rho$ such that the following is true:
For any $a\in(0,a_*)$, there exists a constant $c_1$ depending on $B$, $\rho$, and $a$ such that $$\begin{aligned}\label{dissipation_negative}
a\big(q(S^1_u(s);S^1_{u_L}(s_R))&-\sigma^1_u(s)\eta(S^1_u(s)|S^1_{u_L}(s_R))\big)-q(u;u_L)+\sigma^1_u(s)\eta(u|u_L) \leq \\
& -c_1\abs{\sigma^1_{u_L}(s_R)-\sigma^1_u(s)}^2,
\end{aligned}$$ for all $u_L\in\mathcal{V}$ with $\abs{u_L}\leq B$, all $u\in\{u | \eta(u|u_L)\leq a\eta(u|S^1_{u_L}(s_R))\}$, any $s\in[0,B]$, and any $s_R\in[\rho,B]$.
Moreover, $$\label{dissipation_negative_boundary_of_convex_set}
a\big(q(u;S^1_{u_L}(s_R))-\lambda_1(u)\eta(u|S^1_{u_L}(s_R))\big)-q(u;u_L)+\lambda_1(u)\eta(u|u_L)\leq -c_1,$$ for all $u\in\{u | \eta(u|u_L)\leq a\eta(u|S^1_{u_L}(s_R))\}$ and for the same constant $c_1$.
The proof of holds when we only have $\eta\in C^2$.
uses ideas from the proof of Lemma 4.3 in [@MR3519973], but to prove we keep careful track of the dependencies on the constants and make sure in our calculations to leave some extra negativity in the entropy dissipation lost at the shock $(u_L,u_R,\sigma_{L,R})$ (thus we have a negative right hand side in our and ). The idea of the extra negativity in the entropy dissipation is similar to the work [@2017arXiv171207348K; @scalar_move_entire_solution].
To prove , we will need
\[entropy\_lost\_right\_side\_1\_shock\] Assume the system satisfies the hypothesis $(\mathcal{H}1)$. Fix $B,\rho>0$. Then there exists $k,\delta_0>0$ depending on $B$ and $\rho$ such that for any $\delta\in(0,\delta_0]$, $u\in\mathcal{V}\cap B_{r_0}({I_{-}})$ with $\abs{u}\leq B$ and for any $s_0\in(\rho,B)$ and $s\geq0$, $$\begin{aligned}\label{entropy_lost_right_side_1_shock_inequalities}
&q(S_u^1(s);S_u^1(s_0))-\sigma_u^1(s)\eta(S_u^1(s)|S_u^1(s_0))\leq -k\abs{\sigma_u^1(s)-\sigma_u^1(s_0)}^2,\hspace{.2in}\mbox{for } \abs{s-s_0}<\delta,\\
&q(S_u^1(s);S_u^1(s_0))-\sigma_u^1(s)\eta(S_u^1(s)|S_u^1(s_0))\leq -k\delta\abs{\sigma_u^1(s)-\sigma_u^1(s_0)},\hspace{.2in}\mbox{for } \abs{s-s_0}\geq\delta.
\end{aligned}$$
The formulas and are modifications on a key lemma due to DiPerna [@MR523630]. Our proof of is based on the proof of a very similar result in [@MR3519973 p. 387-9]. We modify the proof in [@MR3519973 p. 387-9] – being careful to keep the constants $k$ and $\delta_0$ uniform in $s_0$ and $u$.
The proof of is based on the formulas, and this is where the negative right hand sides in and come from.
itself follows from giving us an explicit formula for the entropy lost at an entropic i-shock $(u,S^i_u(s))$, for any i-family:
\[entropy\_lost\_i\_shock\_lemma\] For any i-shock ($i\in\{1,\ldots,n\}$) $(u,S^i_u(s),\sigma^i_u(s))$ and any $v\in\mathbb{R}^n$, $$\begin{aligned}
\label{dissipation_formula_1}
q(S^i_u(s);v)-\sigma^i_u(s)\eta(S^i_u(s)|v)=q(u;v)-\sigma^i_u(s)\eta(u|v)+\int\limits_0^s\frac{\mbox{d}}{\mbox{dt}}\sigma^i_u(t)\eta(u|S^i_u(t))\,\mbox{d}t.\end{aligned}$$
Therefore, for any $s\geq0, s_0>0$, $$\begin{aligned}
\label{dissipation_formula_2}
q(S^i_u(s);S^i_u(s_0))-\sigma^i_u(s)\eta(S^i_u(s)|S^i_u(s_0))=\int\limits_{s_0}^s\frac{\mbox{d}}{\mbox{dt}}\sigma^i_u(t)\Big(\eta(u|S^i_u(t))-\eta(u|S^i_u(s_0))\Big)\,\mbox{d}t.\end{aligned}$$
See Lax [@MR0093653] for the formula . For a proof of , see [@MR3537479]. Note that and hold for a shock $(u,S^i_u(s),\sigma^i_u(s))$ from any $i$-family, $i=1,2,\ldots,n$, and not just extremal families (1-family or n-family) – the relation is a direct consequence of the Rankine-Hugoniot condition. Further, comes from applying twice.
Proof of
---------
This is based on the proof of a similar result in [@MR3519973 p. 387-9]. Define $$\begin{aligned}
&M\coloneqq \sup_{s\in(0,B),\hspace{.05in}\abs{u}\leq B} \frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s),\\
&P\coloneqq \inf_{s\in(\rho,B),\hspace{.05in}\abs{u}\leq B} \frac{\mbox{d}}{\mbox{d}s}\eta(u|S^1_u(s)).\end{aligned}$$
Note that by Property (a) of $(\mathcal{H}1)$ $M<0$ and by Property (b) of $(\mathcal{H}1)$ $P>0$. Furthermore, note that $M$ and $P$ depend only on the system , , $B$ and $\rho$.
Then by uniform continuity on the compact set $\{(s,u)|s\in[0,B] \mbox{ and } \abs{u}\leq B\}$, there exists $\delta_0>0$ such that for all $s_0\in(\rho,B)$ and for all $s\geq0$ with $\abs{s_0-s}\leq \delta_0$, $$\begin{aligned}
&\abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s)-\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s_0)}\leq \frac{1}{2}\abs{M},\\
&\abs{\frac{\mbox{d}}{\mbox{d}s}\eta(u|S^1_u(s))-\frac{\mbox{d}}{\mbox{d}s}\eta(u|S^1_u(s_0))}\leq \frac{1}{2}P,\\
\end{aligned}$$ Note that $\delta_0$ only depends on the system , , $B$ and $\rho$.
In particular, $$\begin{aligned}\label{RHS_entropy_dissipation_step1}
&\abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s)-\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s_0)}\leq \frac{1}{2}\abs{M}\leq\frac{1}{2}\abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s_0)},\\
&\abs{\frac{\mbox{d}}{\mbox{d}s}\eta(u|S^1_u(s))-\frac{\mbox{d}}{\mbox{d}s}\eta(u|S^1_u(s_0))}\leq \frac{1}{2}P\leq\frac{1}{2}\abs{\frac{\mbox{d}}{\mbox{d}s}\eta(u|S^1_u(s_0))}.
\end{aligned}$$
From , we get the estimates $$\begin{aligned}\label{RHS_entropy_dissipation_step2}
&\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s)=-\abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s)}\leq-\frac{1}{2}\abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s_0)},\\
&\frac{\mbox{d}}{\mbox{d}s}\eta(u|S^1_u(s))=\abs{\frac{\mbox{d}}{\mbox{d}s}\eta(u|S^1_u(s))}\geq\frac{1}{2}\abs{\frac{\mbox{d}}{\mbox{d}s}\eta(u|S^1_u(s_0))}.
\end{aligned}$$
We use and to get for all $s$ with $\abs{s-s_0}<\delta_0$, $$\begin{aligned}
q(S_u^1(s);S_u^1(s_0))-\sigma^1_u(s)\eta(S_u^1(s)|S_u^1(s_0))
&=
\int\limits_{s_0}^s \frac{\mbox{d}}{\mbox{d}t}\sigma_u^1(t)\Big(\eta(u|S_u^1(t))-\eta(u|S^1_u(s_0))\Big)\,dt\\
&\leq -\frac{1}{4}\abs{\frac{\mbox{d}}{\mbox{d}t}\sigma_u^1(s_0)}\frac{\mbox{d}}{\mbox{d}t}\eta(u|S_u^1(s_0))\int\limits_{s_0}^s (t-s_0)\,dt\\
&=-\frac{1}{8}\abs{\frac{\mbox{d}}{\mbox{d}t}\sigma_u^1(s_0)}\frac{\mbox{d}}{\mbox{d}t}\eta(u|S_u^1(s_0))\abs{s-s_0}^2.\end{aligned}$$
Note that due to , $$\begin{aligned}
\abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s)}\leq\frac{3}{2}\abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s_0)}.\end{aligned}$$ Thus, $$\begin{aligned}
\abs{\sigma_u^1(s)-\sigma_u^1(s_0)}\leq\frac{3}{2}\abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s_0)}\abs{s-s_0},\end{aligned}$$ which gives us that for all $s$ verifying $\abs{s-s_0}<\delta_0$, $$\begin{aligned}
q(S_u^1(s);S_u^1(s_0))-\sigma_u^1(s)\eta(S_u^1(s)|S_u^1(s_0))\leq - k_1\abs{\sigma_u^1(s)-\sigma_u^1(s_0)}^2,\end{aligned}$$ where we define $$\begin{aligned}
k_1\coloneqq \frac{1}{18}P \inf_{s\in(0,B),\hspace{.05in}\abs{u}\leq B} \abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s)}^{-1}.\end{aligned}$$ Note that $k_1$ only depends on $B$ and $\rho$.
On the other hand, we now show for $\abs{s-s_0}\geq\delta_0$. For all $s$ verifying $s\leq s_0-\delta_0$, we get from $$\begin{aligned}\label{k_2_step_1}
q(S^1_u(s);S^1_u(s_0))-\sigma^1_u(s)\eta(S^1_u(s)|S^1_u(s_0))&=\int\limits_{s}^{s_0-\delta_0}\frac{\mbox{d}}{\mbox{dt}}\sigma^1_u(t)\Big(\eta(u|S^1_u(s_0))-\eta(u|S^1_u(t))\Big)\,\mbox{d}t\\
&\hspace{.3in}+
\int\limits_{s_0-\delta_0}^{s_0}\frac{\mbox{d}}{\mbox{dt}}\sigma^1_u(t)\Big(\eta(u|S^1_u(s_0))-\eta(u|S^1_u(t))\Big)\,\mbox{d}t\\
&\coloneqq I_1+I_2.
\end{aligned}$$
Note that for a positive constant $c_1$ satisfying $$\begin{aligned}
\label{c_1_system}
c_1\leq \inf_{s_0\in[\delta_0,B]\text{ and } \abs{u}\leq B} \Big(\eta(u|S^1_u(s_0))-\eta(u|S^1_u(s_0-\delta_0))\Big),\end{aligned}$$ then we have (recalling Property (a) of hypothesis $(\mathcal{H}1)$) $$\begin{aligned}\label{k_2_step_2}
I_1&\leq \int\limits_{s}^{s_0-\delta_0}\frac{\mbox{d}}{\mbox{dt}}\sigma^1_u(t)\Big(\eta(u|S^1_u(s_0))-\eta(u|S^1_u(s_0-\delta_0))\Big)\,\mbox{d}t\\
&\leq -c_1\abs{\sigma_u^1(s_0-\delta_0)-\sigma_u^1(s)}\\
&\leq -c_1\abs{\sigma_u^1(s_0)-\sigma_u^1(s)}+c_1\abs{\sigma_u^1(s_0)-\sigma_u^1(s_0-\delta_0)}\\
&\leq -c_1\abs{\sigma_u^1(s_0)-\sigma_u^1(s)}+c_1\delta_0\sup_{s\in(0,B),\hspace{.05in}\abs{u}\leq B} \abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s)}.
\end{aligned}$$
Recall that $\delta_0$ depends only on $B$ and $\rho$. Thus, we can find a $c_1$ which satisfies and depends only on $B$ and $\rho$. In particular, note that $$\begin{aligned}
\label{note_that_k2}
\delta_0 P \leq\inf_{s_0\in[\delta_0,B]\mbox{ and } \abs{u}\leq B} \Big(\eta(u|S^1_u(s_0))-\eta(u|S^1_u(s_0-\delta_0))\Big).\end{aligned}$$ Note that for $t\in(s_0-\delta_0,s_0)$, $$\begin{aligned}
\eta(u|S^1_u(s_0))-\eta(u|S^1_u(t))&=\int\limits_t^{s_0} \frac{\mbox{d}}{\mbox{d}s}\eta(u|S^1_u(s))\,ds\\
&\geq P (s_0-t).\end{aligned}$$ Thus, $$\begin{aligned}\label{k_2_step_3}
I_2&\leq PM\int\limits_{s_0-\delta_0}^{s_0}(s_0-t)\,dt\\
&=\frac{\delta_0^2 PM}{2}.
\end{aligned}$$ Recall $M<0$.
Pick $$\begin{aligned}
c_1\coloneqq -\delta_0 k_2,\end{aligned}$$ where $$\begin{aligned}
k_2\coloneqq \min\Bigg\{\frac{PM}{2\sup_{s\in(0,B),\hspace{.05in}\abs{u}\leq B} \abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s)}},P\Bigg\}.\end{aligned}$$ Note that $k_2$ depends only on $B$ and $\rho$.
Then from ,, , and , we get $$\begin{aligned}
q(S^1_u(s);S^1_u(s_0))-\sigma^1_u(s)\eta(S^1_u(s)|S^1_u(s_0))\leq -\delta_0 k_2\abs{\sigma_u^1(s_0)-\sigma_u^1(s)}.\end{aligned}$$
The case for $s>s_0+\delta_0$ is analogous to the case for $s\leq s_0-\delta_0$: For $s>s_0+\delta_0$, consider a constant $c_2>0$ such that $$\begin{aligned}
\label{c_2_system}
c_2\leq \inf_{s_0\in[\rho,B]\text{ and } \abs{u}\leq B} \Big(\eta(u|S^1_u(s_0+\delta_0))-\eta(u|S^1_u(s_0))\Big),\end{aligned}$$ Note that $\delta_0$ only depends on $B$ and $\rho$. Thus, we can find a constant $c_2$ verifying and depending only on $B$ and $\rho$. In particular, note that $$\begin{aligned}
\label{note_that_k3}
\delta_0 P \leq\inf_{s_0\in[\rho,B]\text{ and } \abs{u}\leq B} \Big(\eta(u|S^1_u(s_0+\delta_0))-\eta(u|S^1_u(s_0))\Big).\end{aligned}$$
Then write (recalling ), $$\begin{aligned}\label{k_3_step_1}
q(S^1_u(s);S^1_u(s_0))-\sigma^1_u(s)\eta(S^1_u(s)|S^1_u(s_0))&=\int\limits_{s_0}^{s_0+\delta_0}\frac{\mbox{d}}{\mbox{dt}}\sigma^1_u(t)\Big(\eta(u|S^1_u(t))-\eta(u|S^1_u(s_0))\Big)\,\mbox{d}t\\
&\hspace{.3in}+
\int\limits_{s_0+\delta_0}^{s}\frac{\mbox{d}}{\mbox{dt}}\sigma^1_u(t)\Big(\eta(u|S^1_u(t))-\eta(u|S^1_u(s_0))\Big)\,\mbox{d}t\\
&\coloneqq J_1+J_2.
\end{aligned}$$
Then, $$\begin{aligned}\label{k_3_step_2}
J_2&\leq \int\limits_{s_0+\delta_0}^{s}\frac{\mbox{d}}{\mbox{dt}}\sigma^1_u(t)\Big(\eta(u|S^1_u(s_0+\delta_0))-\eta(u|S^1_u(s_0))\Big)\,\mbox{d}t\\
&\leq c_2 \int\limits_{s_0+\delta_0}^{s}\frac{\mbox{d}}{\mbox{dt}}\sigma^1_u(t)\,\mbox{d}t
\\
\mbox{Then, by Property (a) of hypothesis $(\mathcal{H}1)$,}
\\
&=-c_2\abs{\sigma_u^1(s)-\sigma_u^1(s_0+\delta_0)}\\
&\leq -c_2\abs{\sigma_u^1(s)-\sigma_u^1(s_0)}+c_2\abs{\sigma_u^1(s_0+\delta_0)-\sigma_u^1(s_0)}\\
&\leq-c_2\abs{\sigma_u^1(s)-\sigma_u^1(s_0)}+ c_2\delta_0\sup_{s\in(0,B),\hspace{.05in}\abs{u}\leq B} \abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s)}.
\end{aligned}$$
Note that for $t\in(s_0,s_0+\delta_0)$, $$\begin{aligned}
\eta(u|S^1_u(t))-\eta(u|S^1_u(s_0))&=\int\limits_{s_0}^{t} \frac{\mbox{d}}{\mbox{d}s}\eta(u|S^1_u(s))\,ds\\
&\geq P (t-s_0).\end{aligned}$$ Thus, $$\begin{aligned}\label{k_3_step_3}
J_1&\leq PM\int\limits_{s_0}^{s_0+\delta_0}(t-s_0)\,dt\\
&=\frac{\delta_0^2 PM}{2}.
\end{aligned}$$ Recall $M<0$.
Pick $$\begin{aligned}
c_2\coloneqq -\delta_0 k_3,\end{aligned}$$ where $$\begin{aligned}
k_3\coloneqq \min\Bigg\{\frac{PM}{2\sup_{s\in(0,B),\hspace{.05in}\abs{u}\leq B} \abs{\frac{\mbox{d}}{\mbox{d}s}\sigma^1_u(s)}},P\Bigg\}.\end{aligned}$$ Note that $k_3$ depends only on $B$ and $\rho$.
Then from ,, , and , we get $$\begin{aligned}
q(S^1_u(s);S^1_u(s_0))-\sigma^1_u(s)\eta(S^1_u(s)|S^1_u(s_0))\leq -\delta_0 k_3\abs{\sigma_u^1(s_0)-\sigma_u^1(s)}.\end{aligned}$$
Note that in hypothesis $(\mathcal{H}1)$, we assume the 1-shock curve $S^1_u$ is parameterized by arc length. Thus, if $s<B$ then $\abs{S^1_u(s)}<B$.
Proof of
---------
This proof is based on the proof of Lemma 4.3 in [@MR3519973].
In what follows, we use $C$ to denote a generic constant which only depends on $B$ and $\rho$.
Also, for convenience we define $$\begin{aligned}
&u_R\coloneqq S^1_{u_L}(s_R)\\
&R_a\coloneqq \{u | \eta(u|u_L)\leq a\eta(u|u_R)\}.\end{aligned}$$
We first need to show that for any fixed $\sigma_0\in\mathbb{R}$ such that $\lambda_1(u_L)>\sigma_0$, there exists $\beta,\epsilon_0>0$ such that $$\begin{aligned}
\label{sigma_0_ineq}
-q(u;u_L)+\sigma_0\eta(u|u_L) \leq -\beta \eta(u|u_L),\end{aligned}$$ for all $u\in B_{\epsilon_0}(u_L)$.
The difference between $\lambda_1(u_L)$ and $\sigma_0$ will power the proof of . We will choose a $\sigma_0$ later.
We use Taylor expansion to prove : $$\begin{aligned}
\label{taylor_expanse}
-q(u;u_L)+\sigma_0\eta(u|u_L) =(u-u_L)^T\nabla^2\eta(u_L)(\sigma_0I-\nabla f (u_L))(u-u_L)+\mathcal{O}(\abs{u-u_L}^3)\end{aligned}$$
Due to the strict convexity of $\eta$, $\nabla^2\eta(u_L)$ is symmetric and strictly positive definite. Also, by assumption $\nabla^2\eta(u_L) \nabla f(u_L)$ is symmetric. Thus these two matrices are diagonalizable in the same basis. We receive, $$\begin{aligned}
\label{matrix_ordering}
\nabla^2\eta(u_L) \nabla f(u_L) \geq \lambda_1(u_L)\nabla^2\eta(u_L).\end{aligned}$$
Let $C_1>0$ be a constant such that the term $\mathcal{O}(\abs{u-u_L}^3)$ in satisfies $\mathcal{O}(\abs{u-u_L}^3)\leq C_1 \abs{u-u_L}^3$ for all $\abs{u_L}\leq B$ and all $u\in B_{1}(u_L)$. Note $C_1$ depends only on $B$. Let $$\begin{aligned}
C_2\coloneqq \inf_{\abs{x}=1\hspace{.07cm},\hspace{.07cm}\abs{u_L}\leq B}x^T\nabla^2\eta(u_L)x.\end{aligned}$$ Note that because $\eta$ is strictly convex, $C_2>0$. Note $C_2$ depends only on $B$.
Then, for all $$\begin{aligned}
\epsilon_0<\min\{\frac{C_2}{2C_1}(\lambda_1(u_L)-\sigma_0),1\}\end{aligned}$$ and for all $u\in B_{\epsilon_0}(u_L)$, we have from and because $\lambda_1(u_L)>\sigma_0$, $$\begin{aligned}
-q(u;u_L)+\sigma_0\eta(u|u_L) &\leq -(\lambda_1(u_L)-\sigma_0)(u-u_L)^T\nabla^2\eta(u_L)(u-u_L)+\mathcal{O}(\abs{u-u_L}^3)\\
&\leq -\frac{(\lambda_1(u_L)-\sigma_0)}{2}(u-u_L)^T\nabla^2\eta(u_L)(u-u_L)\\
&\leq -C\frac{(\lambda_1(u_L)-\sigma_0)}{2}\eta(u|u_L)\end{aligned}$$ by . This proves , with $$\begin{aligned}
\label{beta_value}
\beta=C\frac{(\lambda_1(u_L)-\sigma_0)}{2}.\end{aligned}$$
We can now compute to show .
In the context of , we can use the same value of $B$ in as in . In , we have constants $k$ and $\delta_0$. Note that these constants depend on $B$ and $\rho$. In the context of , we are allowed to choose $\delta$ as long as it is sufficiently small. Choose $$\begin{aligned}
\label{choose_delta}
\delta\coloneqq \min\{\delta_0,\frac{s_R}{2}\}\end{aligned}$$ for the $\delta$ in . Note that $\delta$ depends on $B$ and $\rho$. Then, define $$\begin{aligned}
k^*\coloneqq \min\{\delta k,k\}.\end{aligned}$$
Note that $k^*$ depends on $B$ and $\rho$.
Define the following quantities, $$\begin{aligned}
M&\coloneqq \sup_{0\leq s \leq B\hspace{.07cm},\hspace{.1cm}\abs{u}\leq B+1} \frac{d}{ds}\sigma_u^1(s)\label{M_def},
\shortintertext{where the constant $M$ exists and satisfies $M<0$ because by the hypotheses $(\mathcal{H}1)$, $(s,u)\mapsto \sigma_u^1(s)$ is $C^1$ and $\frac{d}{ds}\sigma_u^1(s)<0$. We further define,}
L&\coloneqq \sup_{\abs{u}\leq B+1}\norm{\nabla \lambda_1},\\
\sigma_0&\coloneqq \lambda_1(u_L)+\frac{k^{*}M}{16C_3}\frac{s_R}{2}\label{sigma_0_def},\end{aligned}$$ where $C_3$ will appear later, in – and $C_3$ will depend on $B$. The constant $L$ exists because by assumption the flux $f\in\ C^2(\mathcal{V})$ (see the remarks after the hypotheses $(\mathcal{H})$ and $(\mathcal{H})^*$). Note $M$ and $L$ depend only on $B$.
We choose $\epsilon_0$ such that $$\begin{aligned}
\label{condition_1_epsilon0}
\epsilon_0<\min\Bigg\{-\frac{k^{*}M}{16C_3}\frac{s_R}{2}\frac{1}{L},-\frac{C_2}{C_1}\frac{k^{*}M}{16C_3}\frac{s_R}{2}\frac{1}{L},-\frac{C_2}{C_1}\frac{k^{*}M}{16C_3}\frac{s_R}{2},1\Bigg\}.\end{aligned}$$ Note the right hand side of depends on $B$ and $\rho$. We also need to make sure that $a_*$ is small enough such that $R_a\subset B_{\epsilon_0}(u_L)$ for all $0<a<a_*$. Recall .
We claim that for all $u\in B_{\epsilon_0}(u_L)$, $$\begin{aligned}
\sigma_u^1(s)\leq \sigma_0, \hspace{.1in} \mbox{for } s\geq \frac{s_R}{2},\label{claim_1}\end{aligned}$$ and $$\begin{aligned}
\lambda_1(u)-\sigma_0\leq \frac{k^{*}}{8C_3}\abs{\sigma_u^1(\frac{s_R}{2})-\sigma_u^1(s_R)}.\label{claim_2}\end{aligned}$$
We show : for $s\geq \frac{s_R}{2}$, $$\begin{aligned}
\sigma_u^1(s) &\leq \sigma_u^1(0)+sM\\
&=\lambda_1(u)+sM\\
&\leq \lambda_1(u_L)-\frac{k^*M}{16C_3}\frac{s_R}{2}+sM\\
&=\lambda_1(u_L)+M(s-\frac{k^*}{16C_3}\frac{s_R}{2})\\
&\leq \lambda_1(u_L)+M(\frac{s_R}{2}-\frac{k^*}{16C_3}\frac{s_R}{2})\\
&= \lambda_1(u_L)+M\frac{s_R}{2}(1-\frac{k^*}{16C_3})\\
&< \sigma_0,\end{aligned}$$ where to get the last inequality we can make $C_3$ larger if necessary such that $\frac{k^*}{16C_3}<\frac{1}{2}$, noting $C_3$ will then depend on $\rho$ and $B$.
We now show : $$\begin{aligned}
\lambda_1(u)-\sigma_0 &\leq \lambda_1(u_L)-\frac{k^*M}{16C_3}\frac{s_R}{2} -\sigma_0\\
&= \lambda_1(u_L)-\frac{k^*M}{16C_3}\frac{s_R}{2}-\lambda_1(u_L)-\frac{k^*M}{16C_3}\frac{s_R}{2}\\
&=-\frac{k^*M}{8C_3}\frac{s_R}{2} \leq \frac{k^*}{8C_3}\abs{\sigma_u^1(\frac{s_R}{2})-\sigma_u^1(s_R)},\end{aligned}$$ by definition of $M$.
**To prove , we consider two cases: $s\geq \frac{s_R}{2}$ and $s < \frac{s_R}{2}$.**
We first consider $s\geq \frac{s_R}{2}$. From , we get
$$\begin{aligned}
q(&S^1_u(s);u_R)-\sigma^1_u(s)\eta(S^1_u(s)|u_R)
=
-\big(q(u_R;S_u^1(s_R))-\sigma^1_u(s)\eta(u_R|S^1_u(s_R))\big)\\
&+\big(q(S^1_u(s);S^1_u(s_R))-\sigma^1_u(s)\eta(S^1_u(s)|S^1_u(s_R))\big)\\
&+\big(\nabla\eta(u_R)-\nabla\eta(S^1_u(s_R))\big)\big(f(u_R)-f(S^1_u(s))-\sigma_u^1(s)(u_R-S^1_u(s))\big).\end{aligned}$$
By using the Rankine-Hugoniot jump compatibility conditions $$\begin{aligned}
f(u_R)-f(u_L)=\sigma_{u_L}^1(s_R)(u_R-u_L),\\
f(S^1_u(s))-f(u)=\sigma_u^1(s)(S^1_u(s)-u),\end{aligned}$$ we can rewrite
$$\begin{aligned}
q(&S^1_u(s);u_R)-\sigma^1_u(s)\eta(S^1_u(s)|u_R)
=
-\big(q(u_R;S_u^1(s_R))-\sigma^1_u(s)\eta(u_R|S^1_u(s_R))\big)\\
&+\big(q(S^1_u(s);S^1_u(s_R))-\sigma^1_u(s)\eta(S^1_u(s)|S^1_u(s_R))\big)\\
&+\big(\nabla\eta(u_R)-\nabla\eta(S^1_u(s_R))\big)\big(f(u_L)-f(u)-\sigma_u^1(s)(u_L-u)\\
&\hspace{.8in}+(\sigma_{u_L}^1(s_R)-\sigma_u^1(s))(u_R-u_L)\big)\\
&\coloneqq I_1+I_2+I_3.\end{aligned}$$
To estimate $I_2$ and $I_3$, we use the following rough estimates. In these estimates, the constants are uniform in $u_L$ (with $\abs{u_L}\leq B$) and $s_R\in[\rho,B]$. The estimates hold for any $u\in B_{\epsilon_0}(u_L)$ (recall by , $\epsilon_0<1$). Recall that by the hypothesis $(\mathcal{H}1) $, $(s,u)\mapsto S_u^1(s)$ is $C^1$. Then, $$\begin{aligned}\label{estimates}
\abs{\eta(S^1_{u_L}(s_R)|S^1_{u}(s_R))}&\leq C \abs{S^1_{u_L}(s_R)-S^1_{u}(s_R)}^2\leq C \abs{u_L-u}^2,
\\
&\hspace{-1.7in}\mbox{because $\eta\in C^2$ and by \Cref{entropy_relative_L2_control_system}, $\eta(a|b)$ is locally quadratic in $a-b$. Continuing,}
\\
\abs{(q(S^1_{u_L}(s_R);S^1_{u}(s_R))}&\leq C \abs{S^1_{u_L}(s_R)-S^1_{u}(s_R)}^2\leq C \abs{u_L-u}^2,
\\
&\hspace{-1.7in}\mbox{because $q\in C^2$ and $q(a;b)$ is locally quadratic in $a-b$. Further,}
\\
\abs{\nabla\eta(S^1_{u_L}(s_R))-\nabla\eta(S^1_{u}(s_R))}&\leq C \abs{S^1_{u_L}(s_R)-S^1_{u}(s_R)}\leq C \abs{u_L-u},
\\
&\hspace{-1.7in}\mbox{because $\eta\in C^2(\mathcal{V})$. Lastly,}
\\
\abs{\sigma^1_{u_L}(s_R)-\sigma_u^1(s_R)}&\leq C \abs{u_L-u},
\\
&\hspace{-1.7in}\mbox{because by the hypothesis $(\mathcal{H}1) $, $(s,u)\mapsto \sigma_u^1(s)$ is $C^1$.}
\end{aligned}$$
Then, from the estimates , we get $$\begin{aligned}\label{I_1_systems_estimate}
I_1&=-q(u_R;S_u^1(s_R))+\sigma^1_u(s)\eta(S^1_{u_L}(s_R)|S^1_u(s_R))\\
&=-q(u_R;S_u^1(s_R))+\sigma^1_u(s_R)\eta(S^1_{u_L}(s_R)|S^1_u(s_R))+(\sigma^1_u(s)-\sigma^1_u(s_R))\eta(S^1_{u_L}(s_R)|S^1_u(s_R))\\
&\leq C\abs{u_L-u}^2(1+\abs{\sigma^1_u(s)-\sigma^1_u(s_R)}),
\end{aligned}$$ and $$\begin{aligned}\label{I_3_systems_estimate}
I_3&=\big(\nabla\eta(u_R)-\nabla\eta(S^1_u(s_R))\big)\big(f(u_L)-f(u)-\sigma_u^1(s)(u_L-u)\\&\hspace{.8in}+(\sigma_{u_L}^1(s_R)-\sigma_u^1(s))(u_R-u_L)\big)\\
&\leq C\abs{u_L-u}(\abs{u_L-u}+\abs{\sigma^1_u(s)-\sigma^1_u(s_R)}\abs{u_L-u}+\abs{\sigma^1_u(s)-\sigma^1_u(s_R)}).
\end{aligned}$$
To control $I_2$, we use . Note first that
$$\begin{aligned}\label{estimates1}
\abs{\sigma_u^1(s)-\sigma_{u_L}^1(s_R)}^2&\leq \Big(\abs{\sigma_u^1(s)-\sigma_{u}^1(s_R)}+\abs{\sigma_u^1(s_R)-\sigma_{u_L}^1(s_R)}\Big)^2\\
&\leq \Big(\abs{\sigma_u^1(s)-\sigma_{u}^1(s_R)}+C\abs{u-u_L}\Big)^2\\
&=\abs{\sigma_u^1(s)-\sigma_{u}^1(s_R)}^2+2C\abs{u-u_L}\abs{\sigma_u^1(s)-\sigma_{u}^1(s_R)}+C^2\abs{u-u_L}^2.
\end{aligned}$$
Then, for $\abs{s-s_R}<\delta$ we use and above: $$\begin{aligned}
I_2&=q(S^1_u(s);S^1_u(s_R))-\sigma^1_u(s)\eta(S^1_u(s)|S^1_u(s_R))\\
&\leq -k^* \abs{\sigma_u^1(s)-\sigma_u^1(s_R)}^2\\
&=-\frac{k^*}{2} \abs{\sigma_u^1(s)-\sigma_u^1(s_R)}^2-\frac{k^*}{2} \abs{\sigma_u^1(s)-\sigma_u^1(s_R)}^2\\
&\leq -\frac{k^*}{2} \abs{\sigma_u^1(s)-\sigma_u^1(s_R)}^2 -\frac{k^*}{2}\abs{\sigma_u^1(s)-\sigma_{u_L}^1(s_R)}^2 \\
&\hspace{.5in}+Ck^*\abs{u-u_L}\abs{\sigma_u^1(s)-\sigma_u^1(s_R)}+\frac{k^*}{2}C^2\abs{u-u_L}^2\\
&= -\frac{k^*}{2} \abs{\sigma_u^1(s)-\sigma_u^1(s_R)}^2 -\frac{k^*}{2}\abs{\sigma_u^1(s)-\sigma_{u_L}^1(s_R)}^2 \\
&\hspace{.5in}+C\abs{u-u_L}\abs{\sigma_u^1(s)-\sigma_u^1(s_R)}+C\abs{u-u_L}^2,\end{aligned}$$ where in the last equality we just absorb some constants into the $C$.
Then, if $\abs{s-s_R}<\delta$, we use our estimates on $I_1, I_2$, and $I_3$ to get $$\begin{aligned}\label{estimates2}
q(S^1_u(s);u_R)-&\sigma^1_u(s)\eta(S^1_u(s)|u_R)
\leq
-\frac{k^*}{2} \abs{\sigma_u^1(s)-\sigma_u^1(s_R)}^2 -\frac{k^*}{2}\abs{\sigma_u^1(s)-\sigma_{u_L}^1(s_R)}^2 \\&+C\abs{u-u_L}\abs{\sigma_u^1(s)-\sigma_u^1(s_R)}+C\abs{u-u_L}^2\abs{\sigma_u^1(s)-\sigma_u^1(s_R)}+C\abs{u-u_L}^2\\
&\leq -\frac{k^*}{2}\abs{\sigma_u^1(s)-\sigma_{u_L}^1(s_R)}^2 +C(\abs{u-u_L}^2+\abs{u-u_L}^4),
\\
&\hspace{-1.04in}\mbox{where we have used the version of Young's inequality with $\epsilon$. Continuing,}
\\
&\leq -\frac{k^*}{2}\abs{\sigma_u^1(s)-\sigma_{u_L}^1(s_R)}^2 +C\abs{u-u_L}^2,
\end{aligned}$$ because $u\in B_{\epsilon_0}(u_L)$ and by , $\epsilon_0<1$.
Thus, putting everything together, we have for $s\geq\frac{s_R}{2}$ and $\abs{s-s_R}<\delta$, $$\begin{aligned}
a&\big(q(S^1_u(s);u_R)-\sigma^1_u(s)\eta(S^1_u(s)|u_R)\big)-q(u;u_L)+\sigma^1_u(s)\eta(u|u_L)\\
&\leq aC\abs{u-u_L}^2 -\frac{ak^*}{2}\abs{\sigma_u^1(s)-\sigma_{u_L}^1(s_R)}^2 -q(u;u_L)+\sigma_0\eta(u|u_L),
\shortintertext{by \eqref{estimates2} and \eqref{claim_1}. Continuing,}
&\leq aC\abs{u-u_L}^2 -\frac{ak^*}{2}\abs{\sigma_u^1(s)-\sigma_{u_L}^1(s_R)}^2 -\beta\eta(u|u_L),\end{aligned}$$ by . We recall , and choose $a_*$ small enough such that $aC\abs{u-u_L}^2- \beta\eta(u|u_L)\leq 0$ for all $u$. As always, we also require that $a_*$ is small enough such that $R_a\subset B_{\epsilon_0}(u_L)$ for all $0<a<a_*$ (recall the condition ). This proves .
When $s\geq \frac{s_R}{2}$ and $\abs{s-s_R}>\delta$, using and our estimates on $I_1$ and $I_3$ and ,
$$\begin{aligned}\label{estimates1_redo_control}
q(S^1_u(s);u_R)-&\sigma^1_u(s)\eta(S^1_u(s)|u_R)
\leq
-k^* \abs{\sigma_u^1(s)-\sigma_u^1(s_R)} +C\abs{u-u_L}\abs{\sigma_u^1(s)-\sigma_u^1(s_R)}\\&+C\abs{u-u_L}^2\abs{\sigma_u^1(s)-\sigma_u^1(s_R)}+C\abs{u-u_L}^2\\
&= -\frac{k^*}{2} \abs{\sigma_u^1(s)-\sigma_u^1(s_R)} -\frac{k^*}{2}\abs{\sigma_u^1(s)-\sigma_{u}^1(s_R)} \\&+C\abs{u-u_L}\abs{\sigma_u^1(s)-\sigma_u^1(s_R)}+C\abs{u-u_L}^2\abs{\sigma_u^1(s)-\sigma_u^1(s_R)}+C\abs{u-u_L}^2\\
&\leq-\frac{k^*}{2} \abs{\sigma_u^1(s)-\sigma_u^1(s_R)} +C\abs{u-u_L}^2,
\end{aligned}$$
because $u\in B_{\epsilon_0}(u_L)$ and we pick $\epsilon_0$ even smaller such that $\epsilon_0<\min\{\frac{k^*}{4C},1\}$. Recall we require that $a_*$ is small enough such that $R_a\subset B_{\epsilon_0}(u_L)$ for all $0<a<a_*$ (see ).
Putting everything together, for $s \geq \frac{s_R}{2}$ and $\abs{s-s_R}>\delta$, $$\begin{aligned}
a&\big(q(S^1_u(s);u_R)-\sigma^1_u(s)\eta(S^1_u(s)|u_R)\big)-q(u;u_L)+\sigma^1_u(s)\eta(u|u_L)\\
&\leq aC\abs{u-u_L}^2 -\frac{ak^*}{2}\abs{\sigma_u^1(s)-\sigma_{u}^1(s_R)} -q(u;u_L)+\sigma_0\eta(u|u_L)
\shortintertext{by \eqref{estimates1_redo_control} and \eqref{claim_1}. Continuing,}
&\leq aC\abs{u-u_L}^2 -\frac{ak^*}{2}\abs{\sigma_u^1(s)-\sigma_{u}^1(s_R)} -\beta\eta(u|u_L)\end{aligned}$$ by . We again recall , and choose $a_*$ small enough such that $aC\abs{u-u_L}^2- \beta\eta(u|u_L)\leq 0$ for all $u$. Recall, we always require that $a_*$ is small enough such that $R_a\subset B_{\epsilon_0}(u_L)$ for all $0<a<a_*$ (use condition ). Again note that with $\sigma_0$ defined in and $\beta$ defined in , $\beta=Cs_R$. Finally, we get the right hand side of by noting that $\abs{\sigma_u^1(s)-\sigma_{u}^1(s_R)}$ will be uniformly bounded from below for all $\abs{s-s_R}>\delta$ (with $s\in[0,B]$ and $s_R\in[\rho,B]$), because by Property (a) of $(\mathcal{H}1)$, $\frac{d}{ds}\sigma^1_u(s)<0$. Furthermore, the term $\abs{\sigma^1_{u_L}(s_R)-\sigma^1_u(s)}^2$ on the right hand side of is bounded (with the bound depending on $B$). Thus, by making $c_1$ sufficiently small, this proves . Recall also that $\delta$ depends on $B$ and $\rho$. Thus, $c_1$ depends on $B$ and $\rho$.
On the other hand, we now consider $s< \frac{s_R}{2}$. From , we have $\delta<\frac{s_R}{2}$. Thus when $s< \frac{s_R}{2}$, $\abs{s-s_R}>\delta$.
The computations in apply exactly. We get again,
$$\begin{aligned}
q(S^1_u(s);u_R)-\sigma^1_u(s)\eta(S^1_u(s)|u_R)
\leq -\frac{k^*}{2} \abs{\sigma_u^1(s)-\sigma_u^1(s_R)} +C\abs{u-u_L}^2,\end{aligned}$$
again because $u\in B_{\epsilon_0}(u_L)$ and $\epsilon_0$ verifies $\epsilon_0<\frac{k^*}{4C}$.
Then, because by the assumptions $(\mathcal{H})$ $\frac{d}{ds}\sigma^1_u(s)<0$, we have for all $s<\frac{s_R}{2}$, $$\begin{aligned}
q(S^1_u(s);u_R)-\sigma^1_u(s)\eta(S^1_u(s)|u_R)
&\leq -\frac{k^*}{2} \abs{\sigma_u^1(\frac{s_R}{2})-\sigma_u^1(s_R)} +C\abs{u-u_L}^2
\shortintertext{Then, for $\epsilon_0$ small enough such that $C\epsilon_0^2\leq\frac{k^*Ms_R}{8}$ (where $M$ is from \eqref{M_def}),}
&\leq -\frac{k^*}{4} \abs{\sigma_u^1(\frac{s_R}{2})-\sigma_u^1(s_R)}. \end{aligned}$$ Recall we also need $a_*$ sufficiently small so that $R_a\subset B_{\epsilon_0}(u_L)$ for all $0<a<a_*$. See .
To control the left hand side of the entropy dissipation in , we estimate $$\begin{aligned}\label{c_2_eq}
-q(u;u_L)+\sigma^1_u(s)\eta(u|u_L)&\leq -q(u;u_L)+\lambda_1(u)\eta(u|u_L),
\\
&\hspace{-1.55in}\mbox{because by the assumptions $(\mathcal{H})$ $\frac{d}{ds}\sigma^1_u(s)<0$ and $\sigma^1_u(0)=\lambda_1(u)$. Continuing, }
\\
&= -q(u;u_L)+\sigma_0\eta(u|u_L)+(\lambda_1(u)-\sigma_0)\eta(u|u_L)\\
&\leq (\lambda_1(u)-\sigma_0)\eta(u|u_L),
\\
&\hspace{-1.55in}\mbox{by \eqref{sigma_0_ineq}. Continuing, }
\\
&\leq a(\lambda_1(u)-\sigma_0)\eta(u|u_R),
\\
&\hspace{-1.55in}\mbox{because $u\in R_a\subset B_{\epsilon_0}(u_L)$. Continuing, recall $\epsilon_0<1$ by \eqref{condition_1_epsilon0}. Furthermore,}\\
&\hspace{-1.55in}\mbox{recall \Cref{entropy_relative_L2_control_system}, $\abs{u_L}\leq B$, $s_R\leq B$, and $S^1_{u_L}$ is parameterized by arc length.}\\
&\hspace{-1.55in}\mbox{Then, we get\hspace{5in} }
\\
&\leq aC_3(\lambda_1(u)-\sigma_0).
\end{aligned}$$ Note $C_3$ is a constant which depends on $B$.
Putting everything together, for all $s<\frac{s_R}{2}$, $$\begin{aligned}\label{ending_computation}
a&\big(q(S^1_u(s);u_R)-\sigma^1_u(s)\eta(S^1_u(s)|u_R)\big)-q(u;u_L)+\sigma^1_u(s)\eta(u|u_L)\\
&\leq -a\Big(\frac{k^*}{4} \abs{\sigma_u^1(\frac{s_R}{2})-\sigma_u^1(s_R)}-C_3(\lambda_1(u)-\sigma_0)\Big)\\
&\leq-a\Big(\frac{k^*}{8} \abs{\sigma_u^1(\frac{s_R}{2})-\sigma_u^1(s_R)}\Big),
\\
\mbox{by \eqref{claim_2}. Continuing,}
\\
&\leq \frac{aMk^* s_R}{16},
\end{aligned}$$ where $M$ is from . Recall $M<0$.
Note that the term $\abs{\sigma^1_{u_L}(s_R)-\sigma^1_u(s)}^2$ on the right hand side of is bounded (with the bound depending on $B$), so we get the right hand side of by making $c_1$ smaller if necessary. Note that in making this adjustment to $c_1$, $c_1$ will depend on $B$ and $\rho$. This proves .
Lastly, we get by the same computation and taking $s=0$. Recall that by the hypothesis $(\mathcal{H}1)$, $\sigma^1_u(0)=\lambda_1(u)$.
Proof of
---------
By the remark about taking the negative of the flux ($-f$) if necessary, we can assume that $(\bar{u}_+(t),\bar{u}_-(t),\dot{s}(t))$ is a 1-shock.
We will use . The 1-shock $(\bar{u}_+(t),\bar{u}_-(t),\dot{s}(t))$ in will play the role of $(u_L,S^1_{u_L}(s_R))$ in . Take $R\coloneqq \max\{\norm{u}_{L^\infty},\norm{\bar{u}_-(\cdot)}_{L^\infty([0,T))}\}$ and then take the $\tilde{S}$ corresponding to this $R$ as in Property (c) of $(\mathcal{H}1)$. Define the $B$ in to be $B\coloneqq \max\{R,\tilde{S},\norm{\bar{u}_+(\cdot)}_{L^\infty([0,T))}\}$. Then, we have that for all $(u_-,u_+,\sigma)$ 1-shock with $u_+,u_- < R$, there exists $s\in(0,B)$ such that $u_+=S^1_{u_-}(s)$. Further, note that $B$ depends on $\norm{u}_{L^\infty}$ and $\norm{\bar{u}_-(\cdot)}_{L^\infty([0,T))}$.
Then, pick $0<a<1$ as in . Here, $a$ is playing the same role as the $a$ in .
Throughout this proof, $c$ denotes a generic constant that depends on $\norm{u}_{L^\infty}$, $\rho$, $\norm{\bar{u}_+(\cdot)}_{L^\infty([0,T))}$, $\norm{\bar{u}_-(\cdot)}_{L^\infty([0,T))}$, and $a$.
Note by , the constant $a$ depends on $\norm{u}_{L^\infty}$, $\norm{\bar{u}_-(\cdot)}_{L^\infty([0,T))}$,$\norm{\bar{u}_+(\cdot)}_{L^\infty([0,T))}$, and $\rho$.
We now show that for any $\gamma_0>0$,
$$\begin{aligned}
\label{bound_on_inf}
\inf \eta(u|u_L)- a\eta(u|u_R) \geq c_4\gamma_0^2\end{aligned}$$
for a constant $c_4>0$, where the infimum runs over all $(u,u_L,u_R)$ such that $\mbox{dist}(u,\{w|\eta(w|u_L)\leq a\eta(w|u_R)\})\geq \gamma_0$ and $\abs{u_L},\abs{u_R}\leq B$. Here, $B$ is from and the distance $\mbox{dist}(x,A)$ between a point $x$ and a set $A$ is defined in the usual way, $$\begin{aligned}
\mbox{dist}(x,A) \coloneqq \inf_{y\in A} \abs{x-y}.\end{aligned}$$
Consider any triple $(u,u_L,u_R)$ such that $\mbox{dist}(u,\{w|\eta(w|u_L)\leq a\eta(w|u_R)\})\geq \gamma_0$ and $\abs{u_L},\abs{u_R}\leq B$.
By , the set $\{w|\eta(w|u_L)\leq a\eta(w|u_R)\}$ is compact. Thus, there exists $w_0\in\{w|\eta(w|u_L)\leq a\eta(w|u_R)\}$ such that $$\begin{aligned}
\abs{u-w_0}= \mbox{dist}(u,\{w|\eta(w|u_L)\leq a\eta(w|u_R)\}).\end{aligned}$$
We Taylor expand the function $$\begin{aligned}
\Gamma(u)\coloneqq \eta(u|u_L)- a\eta(u|u_R)\end{aligned}$$ around the point $w_0$:
$$\begin{aligned}
\Gamma(u)=\Gamma(w_0)+\nabla\Gamma(w_0)(u-w_0)+\int\limits_0^1 (1-t)(u-w_0)^{T}\nabla^2\Gamma(w_0+t(u-w_0))(u-w_0)\,dt.\end{aligned}$$
By definition of $w_0$, we must have $\Gamma(w_0)=0$ and $\nabla\Gamma(w_0)(u-w_0)\geq0$.
Note that $\nabla^2\Gamma=(1-a)\nabla^2\eta$. Thus, by strict convexity of $\eta$ and because $0<a<1$, we have $\nabla^2\Gamma\geq cI$ for some constant $c>0$.
We then calculate, $$\begin{aligned}
&\int\limits_0^1 (1-t)(u-w_0)^{T}\nabla^2\Gamma(w_0+t(u-w_0))(u-w_0)\,dt\\
&\geq \int\limits_0^{.5} (1-t)(u-w_0)^{T}\nabla^2\Gamma(w_0+t(u-w_0))(u-w_0)\,dt,
\shortintertext{where we have changed the limits of integration. Continuing,}
&\geq .5 c \abs{u-w_0}^2 \geq .5 c \gamma_0^2,\end{aligned}$$ where the last inequality comes from $\mbox{dist}(u,\{w|\eta(w|u_L)\leq a\eta(w|u_R)\})\geq \gamma_0$. This proves .
We choose $$\begin{aligned}
\label{epsilon_0_def}
\gamma_0\coloneqq \frac{c_1}{2L_*},\end{aligned}$$ where $c_1$ is from and $L_*$ is the Lipschitz constant of the map $$\begin{aligned}
(u,u_L,u_R)\mapsto a\big(q(u;u_R)-\lambda_1(u)\eta(u|u_R)\big)-q(u;u_L)+\lambda_1(u)\eta(u|u_L).\end{aligned}$$
Define $$\begin{aligned}
\label{V_def_systems}
V(u,t)\coloneqq \lambda_{1}(u)-C_*\mathbbm{1}_{\{u|a\eta(u|\bar{u}_+(t))<\eta(u|\bar{u}_-(t))\}}(u),\end{aligned}$$ where $C_*>0$ is a large constant, which we can pick to be $$\begin{aligned}
\label{C_star_def}
C_*\coloneqq \frac{1}{c_4\gamma_0^2}\Bigg(\sup_{u,u_L,u_R\in B_{B}(0)}\abs{aq(u;u_R)-q(u;u_L)}+1\Bigg) + 2\sup_{u\in B_{B}(0)}\abs{\lambda_1(u)},\end{aligned}$$ where $c_4$ is from .
We solve the following ODE in the sense of Filippov flows, $$\begin{aligned}
\begin{cases}\label{ODE}
\dot{h}(t)=V(u(h(t),t),t)\\
h(0)=s(0),
\end{cases}\end{aligned}$$
The existence of such an $h$ comes from the following lemma,
\[Filippov\_existence\] Let $V(u,t):\mathbb{R}^n \times [0,\infty)\to\mathbb{R}$ be bounded on $\mathbb{R}^n \times [0,\infty)$, upper semi-continuous in $u$, and measurable in $t$. Let $u$ be a bounded, weak solution to , entropic for the entropy $\eta$. Assume also that $u$ verifies the strong trace property (). Let $x_0\in\mathbb{R}$. Then we can solve $$\begin{aligned}
\begin{cases}\label{ODE}
\dot{h}(t)=V(u(h(t),t),t)\\
h(0)=x_0,
\end{cases}\end{aligned}$$ in the Filippov sense. That is, there exists a Lipschitz function $h:[0,\infty)\to\mathbb{R}$ such that $$\begin{aligned}
\mbox{Lip}[h]\leq \norm{V}_{L^\infty},\label{fact1}\\
h(0)=x_0,\label{fact2}
\shortintertext{and}
\dot{h}(t)\in I[V(u_+,t),V(u_-,t)],\label{fact3}\end{aligned}$$ for almost every $t$, where $u_\pm\coloneqq u(h(t)\pm,t)$ and $I[a,b]$ denotes the closed interval with endpoints $a$ and $b$.
Moreover, for almost every $t$, $$\begin{aligned}
f(u_+)-f(u_-)=\dot{h}(u_+-u_-),\label{fact4}\\
q(u_+)-q(u_-)\leq\dot{h}(\eta(u_+)-\eta(u_-)),\label{fact5}\end{aligned}$$ which means that for almost every $t$, either $(u_+,u_-,\dot{h})$ is an entropic shock (for $\eta$) or $u_+=u_-$.
The proof of , , and is very similar to the proof of Proposition 1 in [@Leger2011]. A proof of , , and is included in for the reader’s convenience.
It is well known that and are true for any Lipschitz continuous function $h:[0,\infty)\to\mathbb{R}$ when $u$ is BV. When instead $u$ is only known to have strong traces (), then and are given in Lemma 6 in [@Leger2011]. We do not prove and here; their proof is in the appendix in [@Leger2011].
Note that $V$ (see ) is upper semi-continuous in $u$ because indicator functions of open sets are lower semi-continuous and the negative of a lower semi-continuous function is upper semi-continuous.
Let $u_{\pm}\coloneqq u(u(h(t)\pm,t)$.
Note that by , $$\begin{aligned}
\label{where_h_lives}
\dot{h}(t)\in I \Bigg[\lambda_{1}(u_+)-C_*\mathbbm{1}_{\{u|a\eta(u|\bar{u}_+(t))<\eta(u|\bar{u}_-(t))\}}(u_+),\\
\lambda_{1}(u_-)-C_*\mathbbm{1}_{\{u|a\eta(u|\bar{u}_+(t))<\eta(u|\bar{u}_-(t))\}}(u_-)\Bigg].\end{aligned}$$
We are now ready to show .
For each fixed time $t$, we have 4 cases to consider to prove :*Case 1* $$\begin{aligned}
a\eta(u_-|\bar{u}_+(t))<\eta(u_-|\bar{u}_-(t)),\\
a\eta(u_+|\bar{u}_+(t))<\eta(u_+|\bar{u}_-(t)).\end{aligned}$$ *Case 2* $$\begin{aligned}
a\eta(u_-|\bar{u}_+(t))<\eta(u_-|\bar{u}_-(t)),\\
a\eta(u_+|\bar{u}_+(t))\geq\eta(u_+|\bar{u}_-(t)).\end{aligned}$$ *Case 3* $$\begin{aligned}
a\eta(u_-|\bar{u}_+(t))\geq\eta(u_-|\bar{u}_-(t)),\\
a\eta(u_+|\bar{u}_+(t))<\eta(u_+|\bar{u}_-(t)).\end{aligned}$$ *Case 4* $$\begin{aligned}
a\eta(u_-|\bar{u}_+(t))\geq\eta(u_-|\bar{u}_-(t)),\\
a\eta(u_+|\bar{u}_+(t))\geq\eta(u_+|\bar{u}_-(t)).\end{aligned}$$
Note that we allow for $u_+=u_-$.
We start with
*Case 1*
In this case, by , , and we know that $$\begin{aligned}\label{control_h}
\dot{h}(t)\leq -\frac{1}{c_4\gamma_0^2}\Bigg(\sup_{u,u_L,u_R\in B_{B}(0)}\abs{aq(u;u_R)-q(u;u_L)}+1\Bigg)-\sup_{u\in B_{B}(0)}\abs{\lambda_1(u)}
\\
<\inf_{u\in B_{B}(0)} \lambda_1(u).
\end{aligned}$$
If $u_+\neq u_-$, then we have and . But then, contradicts $(\mathcal{H}2)$. Thus, $u_+= u_-$.
Let $v\coloneqq u_+=u_-$.
If $\mbox{dist}(v,\{w|\eta(w|\bar{u}_-(t))\leq a\eta(w|\bar{u}_+(t))\})\geq \gamma_0$, then
$$\begin{aligned}\label{dissipation_negative_claim_proof_case1_1}
&a\bigg(q(u_+;\bar{u}_+(t))-\dot{h}(t)\eta(u_+|\bar{u}_+(t))\bigg)-q(u_-;\bar{u}_-(t))+\dot{h}(t)\eta(u_-|\bar{u}_-(t)) \\
&=a\bigg(q(v;\bar{u}_+(t))-\dot{h}(t)\eta(v|\bar{u}_+(t))\bigg)-q(v;\bar{u}_-(t))+\dot{h}(t)\eta(v|\bar{u}_-(t))\\
&=aq(v;\bar{u}_+(t))-q(v;\bar{u}_-(t))-\dot{h}(t)\big(a\eta(v|\bar{u}_+(t))-\eta(v|\bar{u}_-(t))\big)\\
&\leq -1,
\end{aligned}$$
because of and . Because the term $\abs{\dot{s}(t)-\dot{h}(t)}^2$ on the right hand side of is bounded due to and $s$ being Lipschitz, we have proven by choosing $c$ sufficiently small.
If on the other hand, $\mbox{dist}(v,\{w|\eta(w|\bar{u}_-(t))\leq a\eta(w|\bar{u}_+(t))\})< \gamma_0$, then
$$\begin{aligned}\label{dissipation_negative_claim_proof_case1_2}
&a\bigg(q(u_+;\bar{u}_+(t))-\dot{h}(t)\eta(u_+|\bar{u}_+(t))\bigg)-q(u_-;\bar{u}_-(t))+\dot{h}(t)\eta(u_-|\bar{u}_-(t)) \\
&=a\bigg(q(v;\bar{u}_+(t))-\dot{h}(t)\eta(v|\bar{u}_+(t))\bigg)-q(v;\bar{u}_-(t))+\dot{h}(t)\eta(v|\bar{u}_-(t)) \\
&=aq(v;\bar{u}_+(t))-q(v;\bar{u}_-(t))-\dot{h}(t)\big(a\eta(v|\bar{u}_+(t))-\eta(v|\bar{u}_-(t))\big)\\
&\leq a\bigg(q(v;\bar{u}_+(t))-\lambda_1(v)\eta(v|\bar{u}_+(t))\bigg)-q(v;\bar{u}_-(t))+\lambda_1(v)\eta(v|\bar{u}_-(t)),\\
&\mbox{because $\eta(v|\bar{u}_-(t))-a\eta(v|\bar{u}_+(t))\geq 0$ and $\dot{h}\leq -\sup_{u\in B_{B}(0)}\abs{\lambda_1(u)}$. Continuing,} \\
&\mbox{we get}
\\
&\leq -\frac{1}{2}c_1,
\end{aligned}$$
from , the definition of $\gamma_0$ , the assumption that $$\begin{aligned}
\mbox{dist}(v,\{w|\eta(w|\bar{u}_-(t))\leq a\eta(w|\bar{u}_+(t))\})< \gamma_0\end{aligned}$$ and the assumption that $r(t)>\rho$ for all $t$, where $r(t)$ satisfies $S^1_{\bar{u}_-(t)}(r(t))=\bar{u}_+(t)$. Again because the term $\abs{\dot{s}(t)-\dot{h}(t)}^2$ on the right hand side of is bounded due to and $s$ being Lipschitz, we have proven by choosing $c$ sufficiently small. Note $c$ will depend on $\rho$.
*Case 2*
In this case, we must have $u_-\neq u_+$. Recall also that is hyperbolic. Furthermore, we have from that $\dot{h}\in \Bigg[-\frac{1}{c_4\gamma_0^2}\Bigg(\sup_{u,u_L,u_R\in B_{B}(0)}\abs{aq(u;u_R)-q(u;u_L)}+1\Bigg)-\sup_{u\in B_{B}(0)}\abs{\lambda_1(u)},\lambda_1(u_+)\Bigg]$. However, this implies that $(u_+,u_-,\dot{h})$ is a right 1-contact discontinuity (see [@dafermos_big_book p. 274]). This contradicts the hypothesis $(\mathcal{H}2)$ on the shock $(u_+,u_-,\dot{h})$, which is entropic for $\eta$ because of and . The hypothesis $(\mathcal{H}2)$ forbids right 1-contact discontinuities. Thus, we conclude that this case (*Case 2*) cannot actually occur.
*Case 3*
In this case, we have from that $$\begin{aligned}
\dot{h}\in \Bigg[-\frac{1}{c_4\gamma_0^2}\Bigg(\sup_{u,u_L,u_R\in B_{B}(0)}\abs{aq(u;u_R)-q(u;u_L)}+1\Bigg)-\sup_{u\in B_{B}(0)}\abs{\lambda_1(u)},\lambda_1(u_-)\Bigg].\end{aligned}$$ By the hypothesis $(\mathcal{H}3)$, along with , , we have that $(u_+,u_-,\dot{h})$ must be a 1-shock. Also, $u_-$ verifies $a\eta(u_-|\bar{u}_+(t))\geq\eta(u_-|\bar{u}_-(t))$. Thus, we can apply . Recall that $r(t)>\rho$ for all $t$, where $r(t)$ satisfies $S^1_{\bar{u}_-(t)}(r(t))=\bar{u}_+(t)$. We receive .
*Case 4*
In this case, we have from that $\dot{h}\in I[\lambda_1(u_+),\lambda_1(u_-)]$. Then, by the hypothesis $(\mathcal{H}2)$, along with , , we know that we cannot have $$\begin{aligned}
\label{this_would_imply_bad}
I[\lambda_1(u_+),\lambda_1(u_-)]=(\lambda_1(u_-),\lambda_1(u_+))\end{aligned}$$ because then would imply that $(u_+,u_-,\dot{h})$ is a right 1-contact discontinuity. However, $(\mathcal{H}2)$ prevents right 1-contact discontinuities. Recall $(\mathcal{H}3)$. We conclude that $(u_+,u_-,\dot{h})$ is a 1-shock. Moreover, $u_-$ verifies $a\eta(u_-|\bar{u}_+(t))\geq\eta(u_-|\bar{u}_-(t))$. We can now apply . Recall that $r(t)>\rho$ for all $t$, where $r(t)$ satisfies $S^1_{\bar{u}_-(t)}(r(t))=\bar{u}_+(t)$. This gives .
Proof of main theorem {#proof_main_theorem}
======================
Note that if $\bar{u}$ contains an n-shock, then the solution $(x,t)\mapsto \bar{u}(-x,t)$ to the system $\partial_t u -f(u)=G(u)$ will have 1-shock for this system. Thus, we can always assume $\bar{u}$ has a 1-shock.
Let $h$ be as in .
Define $$\begin{aligned}\label{h_defs}
h_1(t)\coloneqq -R+s(0)+r(t-t_0),\\
h_2(t)\coloneqq R+s(0)-r(t-t_0),
\end{aligned}$$ where $r>0$ verifies $$\begin{aligned}
\label{r_def}
\abs{q(u;\bar{u})}\leq r \eta(u|\bar{u}).\end{aligned}$$ Such an $r>0$ exists because $u$ and $\bar{u}$ are bounded, $q(a;b)$ and $\eta(a|b)$ are both locally quadratic in $a-b$, and $\eta$ is strictly convex.
Then we apply to $h_1$ and $h$. This yields, $$\begin{aligned}\label{local_compatible_dissipation_calc_left}
\int\limits_{0}^{t_0} \bigg[q(u(h_1(t)+,t);\bar{u}((h_1(t)+X(t))+,t))-q(u(h(t)-,t);\bar{u}((h(t)+X(t))-,t))\hspace{.5in}
\\
+\dot{h}(t)\eta(u(h(t)-,t)|\bar{u}((h(t)+X(t))-,t))\hspace{2in}
\\
-\dot{h}_1(t)\eta(u(h_1(t)+,t)|\bar{u}((h_1(t)+X(t))+,t))\bigg]\,dt\hspace{1.57in}
\\
\geq
\int\limits_{h_1(t_0)}^{h(t_0)}\eta(u(x,t_0)|\bar{u}(x+X(t_0),t_0))\,dx
-\int\limits_{h_1(0)}^{h(0)}\eta(u^0(x)|\bar{u}^0(x))\,dx
\\
+\int\limits_{0}^{t_0}\int\limits_{h_1(t)}^{h(t)}\Bigg(\partial_x \bigg|_{(x+X(t),t)}\hspace{-.45in} \bar{u}^T(x,t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t)) f(u(x,t)|\bar{u}(x+X(t),t))
\\
+\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\dot{X}(t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
\\
-\nabla\eta(u(x,t)|\bar{u}(x+X(t),t))G(u(\cdot,t))(x)
\\
+
\Bigg(G(\bar{u}(\cdot,t))(x+X(t))-G(u(\cdot,t))(x)\Bigg)^T\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]\,dxdt,
\end{aligned}$$ where $$\begin{aligned}\label{defs_z_X}
f(u|\bar{u})\coloneqq f(u)-f(\bar{u})-\nabla f (\bar{u})(u-\bar{u}),\\
X(t)\coloneqq s(t)-h(t).
\end{aligned}$$ Similarly, we apply to $h$ and $h_2$. This yields, $$\begin{aligned}\label{local_compatible_dissipation_calc_right}
\int\limits_{0}^{t_0} \bigg[q(u(h(t)+,t);\bar{u}((h(t)+X(t))+,t))-q(u(h_2(t)-,t);\bar{u}((h_2(t)+X(t))-,t))\hspace{.5in}
\\
+\dot{h}_2(t)\eta(u(h_2(t)-,t)|\bar{u}((h_2(t)+X(t))-,t))\hspace{2in}
\\
-\dot{h}(t)\eta(u(h(t)+,t)|\bar{u}((h(t)+X(t))+,t))\bigg]\,dt\hspace{1.95in}
\\
\geq
\int\limits_{h(t_0)}^{h_2(t_0)}\eta(u(x,t_0)|\bar{u}(x+X(t_0),t_0))\,dx
-\int\limits_{h(0)}^{h_2(0)}\eta(u^0(x)|\bar{u}^0(x))\,dx
\\
+\int\limits_{0}^{t_0}\int\limits_{h(t)}^{h_2(t)}\Bigg(\partial_x \bigg|_{(x+X(t),t)}\hspace{-.45in} \bar{u}^T(x,t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t)) f(u(x,t)|\bar{u}(x+X(t),t))
\\
+\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\dot{X}(t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
\\
-\nabla\eta(u(x,t)|\bar{u}(x+X(t),t))G(u(\cdot,t))(x)
\\
+
\Bigg(G(\bar{u}(\cdot,t))(x+X(t))-G(u(\cdot,t))(x)\Bigg)^T\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]\,dxdt.
\end{aligned}$$
We combine and $a$ multiples of . This gives,
$$\begin{aligned}\label{local_compatible_dissipation_calc_left_and_right}
\int\limits_{0}^{t_0} \bigg[a\Bigg(q(u(h(t)+,t);\bar{u}((h(t)+X(t))+,t))-\dot{h}(t)\eta(u(h(t)+,t)|\bar{u}((h(t)+X(t))+,t))\Bigg)
\\
+\dot{h}(t)\eta(u(h(t)-,t)|\bar{u}((h(t)+X(t))-,t))
-q(u(h(t)-,t);\bar{u}((h(t)+X(t))-,t))
\\
+aq(u(h_1(t)+,t);\bar{u}((h_1(t)+X(t))+,t))
-a\dot{h}_1(t)\eta(u(h_1(t)+,t)|\bar{u}((h_1(t)+X(t))+,t))
\\
-q(u(h_2(t)-,t);\bar{u}((h_2(t)+X(t))-,t))
+\dot{h}_2(t)\eta(u(h_2(t)-,t)|\bar{u}((h_2(t)+X(t))-,t))\bigg]\,dt
\\
\geq
\Bigg[a\int\limits_{h_1(t_0)}^{h(t_0)}\eta(u(x,t_0)|\bar{u}(x+X(t_0),t_0))\,dx+\int\limits_{h(t_0)}^{h_2(t_0)}\eta(u(x,t_0)|\bar{u}(x+X(t_0),t_0))\,dx\Bigg]
\\
-\Bigg[a\int\limits_{h_1(0)}^{h(0)}\eta(u^0(x)|\bar{u}^0(x))\,dx+\int\limits_{h(0)}^{h_2(0)}\eta(u^0(x)|\bar{u}^0(x))\,dx\Bigg]
\\
+\int\limits_{0}^{t_0}\int\limits_{\mathbb{R}}\mathbbm{1}_a(x)\Bigg[\Bigg(\partial_x \bigg|_{(x+X(t),t)}\hspace{-.45in} \bar{u}^T(x,t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t)) f(u(x,t)|\bar{u}(x+X(t),t))
\\
+\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\dot{X}(t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
-
\\
\nabla\eta(u(x,t)|\bar{u}(x+X(t),t))G(u(\cdot,t))(x)
\\
+
\Bigg(G(\bar{u}(\cdot,t))(x+X(t))-G(u(\cdot,t))(x)\Bigg)^T\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]\Bigg]\,dxdt,
\end{aligned}$$
where $$\begin{aligned}
\mathbbm{1}_a(x)\coloneqq a\mathbbm{1}_{\{x|h_1(t)<x<h(t)\}}(x)+\mathbbm{1}_{\{x|h(t)<x<h_2(t)\}}(x).\end{aligned}$$
We estimate the last term on the right hand side of , which is of the form $$\begin{aligned}
\int\limits_{0}^{t_0}\int\limits_{\mathbb{R}}\overbracket[.5pt][7pt]{\mathbbm{1}_a(x)}^{L^\infty(\mathbb{R})}\overbracket[.5pt][7pt]{\Bigg[\cdots\Bigg]}^{L^1([h_1(t),h_2(t)])}\,dxdt,\end{aligned}$$ using the indicated Hölder dualities.
We then want to estimate from above the term $$\begin{aligned}\label{big_term_estimate_above}
&\int\limits_{h_1(t)}^{h_2(t)}\Bigg|\overbracket[.5pt][7pt]{\Bigg(\partial_x \bigg|_{(x+X(t),t)}\hspace{-.45in} \bar{u}^T(x,t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))}^{L^\infty([h_1(t),h_2(t)])}\overbracket[.5pt][7pt]{f(u(x,t)|\bar{u}(x+X(t),t))}^{L^1([h_1(t),h_2(t)])}
\\
&+\dot{X}(t)\overbracket[.5pt][7pt]{\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\Bigg)}^{L^2([h_1(t),h_2(t)])}\overbracket[.5pt][7pt]{\nabla^2\eta(\bar{u}(x+X(t),t))}^{L^\infty([h_1(t),h_2(t)])}\overbracket[.5pt][7pt]{[u(x,t)-\bar{u}(x+X(t),t)]}^{L^2([h_1(t),h_2(t)])}
\\
&-\overbracket[.5pt][7pt]{\nabla\eta(u(x,t)|\bar{u}(x+X(t),t))}^{L^1([h_1(t),h_2(t)])}\overbracket[.5pt][7pt]{G(u(\cdot,t))(x)}^{L^\infty([h_1(t),h_2(t)])}
\\
&+
\overbracket[.5pt][7pt]{\Bigg(G(\bar{u}(\cdot,t))(x+X(t))-G(u(\cdot,t))(x)\Bigg)^T}^{L^2([h_1(t),h_2(t)])}\overbracket[.5pt][7pt]{\nabla^2\eta(\bar{u}(x+X(t),t))}^{L^\infty([h_1(t),h_2(t)])}\overbracket[.5pt][7pt]{[u(x,t)-\bar{u}(x+X(t),t)]}^{L^2([h_1(t),h_2(t)])}\Bigg|\,dx.
\end{aligned}$$
We use the Hölder dualities indicated above. In particular, recall that $f(a|b)$ is locally quadratic in $a-b$ and that $\partial_x \bar{u}\in L^\infty(\mathbb{R}\times[0,T))$ due to $\bar{u}$ being Lipschitz continuous.
Note that from $G:(L^2(\mathbb{R}))^n\to (L^2(\mathbb{R}))^n$ being translation invariant and from , we have $$\begin{aligned}\label{control_on_difference_G}
&\norm{G(\bar{u}(\cdot,t))(\cdot+X(t))-G(u(\cdot,t))(\cdot)}_{L^2([h_1(t),h_2(t)]} \\
&\hspace{1in}=\norm{G(\bar{u}(\cdot+X(t),t))(\cdot)-G(u(\cdot,t))(\cdot)}_{L^2([h_1(t),h_2(t)])}
\\
&\hspace{1in}\leq
C_G\norm{\bar{u}(\cdot+X(t),t)-u(\cdot,t)}_{L^2([h_1(t),h_2(t)])},
\end{aligned}$$
where $C_G$ is from .
Recall also .
Note also that we can estimate, $$\begin{aligned}
\label{estimate_partial_x_u_bar}
\norm{\partial_x\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}
\leq
\sqrt{2(R+rT)}\norm{\partial_x\bar{u}}_{L^\infty(\mathbb{R}\times[0,T))}=\sqrt{2(R+rT)}\mbox{Lip}[\bar{u}].\end{aligned}$$
For $\norm{\partial_x\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}\norm{\nabla^2\eta(\bar{u})}_{L^\infty}\neq0$ we have, from using the ‘Young’s inequality with $\epsilon$,’ $$\begin{aligned}\label{youngs_inequality_part1}
&\abs{\dot{X}(t)}\norm{u(\cdot,t)-\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}
\\
&\hspace{.3in}\leq\frac{c}{4\norm{\partial_x\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}\norm{\nabla^2\eta(\bar{u})}_{L^\infty}}(\dot{X}(t))^2
\\
&\hspace{.3in}+\frac{\norm{\partial_x\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}\norm{\nabla^2\eta(\bar{u})}_{L^\infty}}{c} \norm{u(\cdot,t)-\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}^2,
\end{aligned}$$ where $c$ is from the right hand side of . Note that $c$ depends on $\rho$, $\norm{u}_{L^\infty}$, $\norm{\bar{u}(s(t)+,t)}_{L^\infty([0,T))}$, $\norm{\bar{u}(s(t)-,t)}_{L^\infty([0,T))}$, and $a$. From , we get $$\begin{aligned}\label{youngs_inequality}
&2\abs{\dot{X}(t)}\norm{\partial_x\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}\norm{\nabla^2\eta(\bar{u})}_{L^\infty}\norm{u(\cdot,t)-\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}
\\
&\leq\frac{c}{2}(\dot{X}(t))^2+\frac{2\norm{\partial_x\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}^2\norm{\nabla^2\eta(\bar{u})}_{L^\infty}^2}{c} \norm{u(\cdot,t)-\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}^2.
\end{aligned}$$ If for some $t$, $\norm{\partial_x\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}\norm{\nabla^2\eta(\bar{u})}_{L^\infty}=0$, then we don’t have to estimate the term $$\begin{aligned}
\dot{X}(t)\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)].\end{aligned}$$
Recall and . Note in particular we have $\dot{h}_1=r$ and $\dot{h}_2=-r$. Then from (in ) and , we get
$$\begin{aligned}\label{local_compatible_dissipation_calc_left_and_right_one_piece}
&-\int\limits_{0}^{t_0}\int\limits_{\mathbb{R}}\Bigg[
\Bigg(2\partial_x\bigg|_{(x+X(t),t)}\hspace{-.45in}\bar{u}^T(x,t)\dot{X}(t)\Bigg)\nabla^2\eta(\bar{u}(x+X(t),t))[u(x,t)-\bar{u}(x+X(t),t)]
\Bigg]\,dxdt
\\
&\hspace{.5in}+
\int\limits_{0}^{t_0} \bigg[a\Bigg(q(u(h(t)+,t);\bar{u}((h(t)+X(t))+,t))-\dot{h}(t)\eta(u(h(t)+,t)|\bar{u}((h(t)+X(t))+,t))\Bigg)
\\
&\hspace{.5in}+\dot{h}(t)\eta(u(h(t)-,t)|\bar{u}((h(t)+X(t))-,t))
-q(u(h(t)-,t);\bar{u}((h(t)+X(t))-,t))
\\
&\hspace{.5in}+aq(u(h_1(t)+,t);\bar{u}((h_1(t)+X(t))+,t))
-a\dot{h}_1(t)\eta(u(h_1(t)+,t)|\bar{u}((h_1(t)+X(t))+,t))
\\
&\hspace{.5in}-q(u(h_2(t)-,t);\bar{u}((h_2(t)+X(t))-,t))
+\dot{h}_2(t)\eta(u(h_2(t)-,t)|\bar{u}((h_2(t)+X(t))-,t))\bigg]\,dt
\\
&\hspace{.2in}\leq
\int\limits_0^{t_0} -\frac{c}{2}(\dot{X}(t))^2
\\ &\hspace{.2in}+
\frac{2\norm{\partial_x\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}^2\norm{\nabla^2\eta(\bar{u})}_{L^\infty}^2}{c} \norm{u(\cdot,t)-\bar{u}(\cdot+X(t),t)}_{L^2([h_1(t),h_2(t)])}^2\,dt.
\end{aligned}$$
Recall , , and . Recall also and . Further, recall from that $h(0)=s(0)$. Recall also that from , we know the constant $c$ depends on $\rho$, $\norm{u}_{L^\infty}$, and $\norm{\bar{u}}_{L^\infty}$. Lastly, recall that $f(a|b)$, $\eta(a|b)$, and $\nabla\eta(a|b)$ are locally quadratic in $a-b$ (recall $\eta\in C^3(\mathbb{R}^n)$), and from the strict convexity of $\eta$ we in fact have . Then, from , we receive
$$\begin{aligned}\label{right_before_Gronwall}
\mu_1\int\limits_0^{t_0}\int\limits_{h_1(t)}^{h_2(t)}\abs{u(x,t)-\bar{u}(x+X(t),t)}^2\,dxdt
+\mu_2\int\limits_{-R-rt_0+s(0)}^{R+rt_0+s(0)}\abs{u^0(x)-\bar{u}^0(x)}^2\,dx
\\
-\frac{1}{\mu_2}\int\limits_0^{t_0} (\dot{X}(t))^2\,dt
\geq
\int\limits_{-R+s(0)}^{R+s(0)}\abs{u(x,t_0)-\bar{u}(x+X(t_0),t_0)}^2\,dx
\end{aligned}$$
for all $t_0\in[0,T)$, where $\mu_1,\mu_2>0$ are constants depending on $a$, $\rho$, $\norm{u}_{L^\infty}$, $\norm{\bar{u}}_{L^\infty}$, and bounds on the derivatives of $\eta$ on the range of $u$ and $\bar{u}$. Furthermore, $\mu_1$ also depends on $C_G$ (see and ), $\mbox{Lip}[\bar{u}]$, $\rho$, $R$, $T$, and bounds on the derivatives of $f$ on the range of $u$ and $\bar{u}$. Note that $r$ (see ) only depends on bounds on the derivatives of $f$ and $\eta$ on the (range of $u$ and $\bar{u}$). The constant $a$ then itself depends on $\rho$, $\norm{u}_{L^\infty}$, and $\norm{\bar{u}}_{L^\infty}$ (see ).
We can drop the last term on the left hand side of , to get $$\begin{aligned}\label{right_before_Gronwall462019}
\mu_1\int\limits_0^{t_0}\int\limits_{h_1(t)}^{h_2(t)}\abs{u(x,t)-\bar{u}(x+X(t),t)}^2\,dxdt
+\mu_2\int\limits_{-R-rt_0+s(0)}^{R+rt_0+s(0)}\abs{u^0(x)-\bar{u}^0(x)}^2\,dx
\\
\geq
\int\limits_{-R+s(0)}^{R+s(0)}\abs{u(x,t_0)-\bar{u}(x+X(t_0),t_0)}^2\,dx.
\end{aligned}$$
We then apply the Gronwall inequality to . This yields, $$\begin{aligned}
\label{Gronwall_in_proof_piecewise_systems}
&\int\limits_{-R+s(0)}^{R+s(0)}\abs{u(x,t_0)-\bar{u}(x+X(t_0),t_0)}^2\,dx
\\
&\hspace{1in}\leq \mu_2 e^{\mu_1 t_0}\Bigg(\int\limits_{-R-rt_0+s(0)}^{R+rt_0+s(0)}\abs{u^0(x)-\bar{u}^0(x)}^2\,dx\Bigg).\end{aligned}$$
From , we get .
We now show . From , we get $$\begin{aligned}\label{right_before_Gronwall_for_control_462019}
&\mu_1\int\limits_0^{t_0}\int\limits_{h_1(t)}^{h_2(t)}\abs{u(x,t)-\bar{u}(x+X(t),t)}^2\,dxdt
+\mu_2\int\limits_{-R-rt_0+s(0)}^{R+rt_0+s(0)}\abs{u^0(x)-\bar{u}^0(x)}^2\,dx
\\
&\hspace{3in}\geq
\frac{1}{\mu_2}\int\limits_0^{t_0} (\dot{X}(t))^2\,dt.
\end{aligned}$$ Then we bootstrap, and use to estimate the term $$\begin{aligned}
\int\limits_{h_1(t)}^{h_2(t)}\abs{u(x,t)-\bar{u}(x+X(t),t)}^2\,dx\end{aligned}$$ in . This gives .
This proves .
Appendix
========
Proof of {#appendix_a_cond_lemma_itself}
---------
Throughout this proof, $C$ will denote a generic constant depending only on $B$.
We will first show that for $0<a<1$, the set $R_a$ is convex.
For $a<1$, we can rewrite $$\begin{aligned}
\eta(u|u_L)\leq a\eta(u|u_R)\end{aligned}$$ as $$\begin{aligned}
\label{convex_rewrite}
\eta(u)\leq \frac{1}{1-a}(\eta(u_L)-a\eta(u_R)-\nabla\eta(u_L)\cdot u_L +a\nabla\eta(u_R)\cdot u_R+(\nabla\eta(u_L)-a\nabla\eta(u_R))\cdot u).\end{aligned}$$ The right hand side of is (affine) linear in $u$. Thus the convexity of $\eta$ implies that $R_a= \{u | \eta(u|u_L)\leq a\eta(u|u_R)\}$ is convex.
For $a<\frac{1}{2}$, we can rewrite to get $$\begin{aligned}
\eta(u|u_L)&\leq \frac{a}{1-a}(\eta(u_L)-\eta(u_R)-\nabla\eta(u_L)\cdot u_L +\nabla\eta(u_R)\cdot u_R+(\nabla\eta(u_L)-\nabla\eta(u_R))\cdot u)\\
&\leq Ca(1+\abs{u}).\end{aligned}$$
We combine this with to get that for all $u\in R_a\cap B_{\theta}(u_L)$ (recalling $\theta<1$), $$\begin{aligned}
\label{proto_cond_a}
\abs{u-u_L}^2\leq Ca(1+\abs{u}) \leq Ca.\end{aligned}$$
Thus, when $\alpha$ satisfies with $C$ as in , and $0<a<\alpha$, we have $$\begin{aligned}
\abs{u-u_L}^2\leq Ca < \frac{\theta^2}{2}.\end{aligned}$$ Thus $R_a\cap B_{\theta}(u_L)$ is strictly contained in $B_{\theta}(u_L)$. As we have shown, the set $R_a$ is convex. Thus $R_a$ is also connected, which implies that $$\begin{aligned}
R_a=R_a\cap B_{\theta}(u_L).\end{aligned}$$ We conclude that $R_a\subset B_{\theta}(u_L)$ for all $0<a<\alpha$. This completes the proof.
Proof of {#Filippov_existence_section}
---------
The following proof of , , and is based on the proof of Proposition 1 in [@Leger2011], the proof of Lemma 2.2 in [@serre_vasseur], and the proof of Lemma 3.5 in [@2017arXiv170905610K]. We do not prove or here; these properties are in Lemma 6 in [@Leger2011], and their proofs are in the appendix in [@Leger2011].
Define
$$\begin{aligned}
v_n(x,t)\coloneqq \int\limits_0^1 V\bigg(u(x+\frac{y}{n},t),t\bigg)\,dy.\end{aligned}$$
Let $h_{n}$ be the solution to the ODE: $$\begin{aligned}
\begin{cases}\label{n_ode}
\dot{h}_n(t)=v_n(h_n(t),t),\mbox{ for }t>0\\
h_n(0)=x_0.
\end{cases}\end{aligned}$$
The $v_n$ are uniformly bounded in $n$ because by assumption $V$ is bounded ($\norm{v_n}_{L^\infty}\leq \norm{V}_{L^\infty}$). The $v_n$ are measurable in $t$, and due to the mollification by $\frac{1}{n}$ are also Lipschitz continuous in $x$. Thus has a unique solution in the sense of Carathéodory.
The $h_n$ are Lipschitz continuous with Lipschitz constants uniform in $n$, due to the $v_n$ being uniformly bounded in $n$. Thus, by Arzelà–Ascoli the $h_n$ converge in $C^0(0,T)$ for any fixed $T>0$ to a Lipschitz continuous function $h$ (passing to a subsequence if necessary). Note that $\dot{h}_n$ converges in $L^\infty$ weak\* to $\dot{h}$.
We define $$\begin{aligned}
V_{\mbox{max}}(t)\coloneqq \max\{V(u_-,t),V(u_+,t)\},\\
V_{\mbox{min}}(t)\coloneqq \min\{V(u_-,t),V(u_+,t)\},\end{aligned}$$ where $u_\pm \coloneqq u(h(t)\pm,t)$.
To show , we will first prove that for almost every $t>0$ $$\begin{aligned}
\lim_{n\to\infty}[\dot{h}_n(t)-V_{\mbox{max}}(t)]_+=0,\label{limit_1}\\
\lim_{n\to\infty}[V_{\mbox{min}}(t)-\dot{h}_n(t)]_+=0,\label{limit_2}\end{aligned}$$ where $[\hspace{.1cm}\cdot\hspace{.1cm}]_+\coloneqq\max(0,\cdot)$.
The proofs of and are similar; we only show the first one.
$$\begin{aligned}
[\dot{h}_n(t)-V_{\mbox{max}}(t)]_+\\
=\Bigg[\int\limits_0^1 V\bigg(u(h_n(t)+\frac{y}{n},t),t\bigg)\,dy-V_{\mbox{max}}(t)\Bigg]_+\\
=\Bigg[\int\limits_0^1 V\bigg(u(h_n(t)+\frac{y}{n},t),t\bigg)-V_{\mbox{max}}(t)\,dy\Bigg]_+\\
\leq\int\limits_0^1 \Big[V\bigg(u(h_n(t)+\frac{y}{n},t),t\bigg)-V_{\mbox{max}}(t)\Big]_+\,dy\\
\leq\operatorname*{ess\,sup}_{y\in(0,\frac{1}{n})} \Big[V\bigg(u(h_n(t)+y,t),t\bigg)-V_{\mbox{max}}(t)\Big]_+\\
\leq\operatorname*{ess\,sup}_{y\in(-\epsilon_n,\epsilon_n)} \Big[V\bigg(u(h(t)+y,t),t\bigg)-V_{\mbox{max}}(t)\Big]_+,\label{last_ineq_Filippov}\end{aligned}$$
where $\epsilon_n\coloneqq \abs{h_n(t)-h(t)}+\frac{1}{n}$. Note $\epsilon_n\to0^+$.
Fix a $t\geq0$ such that $u$ has a strong trace in the sense of . Then because the map $u\mapsto V(u,t)$ is upper semi-continuous, $$\begin{aligned}
\label{esssuplim_is_zero}
\lim_{n\to\infty}\operatorname*{ess\,sup}_{y\in(0,\frac{1}{n})} \Big[V\bigg(u(h(t)\pm y,t),t\bigg)-V\big(u_\pm,t\big)\Big]_+=0,\end{aligned}$$ where $u_\pm \coloneqq u(h(t)\pm,t)$. Recall that the map $u\mapsto V(u,t)$ being upper semi-continuous at the point $u_0$ means that $$\begin{aligned}
\limsup_{u\to u_0} V(u,t) \leq V(u_0,t).\end{aligned}$$
From , we get $$\begin{aligned}
\label{esssuplim_is_zero2}
\lim_{n\to\infty}\operatorname*{ess\,sup}_{y\in(0,\frac{1}{n})} \Big[V\bigg(u(h(t)\pm y,t),t\bigg)-V_{\mbox{max}}(t)\Big]_+=0.\end{aligned}$$
We can control from above by the quantity $$\begin{aligned}\label{esssuplim_is_zero3}
\operatorname*{ess\,sup}_{y\in(-\epsilon_n,0)} \Big[V\bigg(u(h(t)+ y,t),t\bigg)-V_{\mbox{max}}(t)\Big]_++\\
\operatorname*{ess\,sup}_{y\in(0,\epsilon_n)} \Big[V\bigg(u(h(t)+ y,t),t\bigg)-V_{\mbox{max}}(t)\Big]_+.
\end{aligned}$$
By , we have that goes to $0$ as $n\to\infty$. This proves .
Recall that $\dot{h}_n$ converges in $L^\infty$ weak\* to $\dot{h}$. Thus, due to the convexity of the function $[\hspace{.1cm}\cdot\hspace{.1cm}]_+$, $$\begin{aligned}
\int\limits_0^T[\dot{h}(t)-V_{\mbox{max}}(t)]_+\,dt\leq \liminf_{n\to\infty}\int\limits_0^T[\dot{h}_n(t)-V_{\mbox{max}}(t)]_+\,dt.\end{aligned}$$
By the dominated convergence theorem and , $$\begin{aligned}
\liminf_{n\to\infty}\int\limits_0^T[\dot{h}_n(t)-V_{\mbox{max}}(t)]_+\,dt=0.\end{aligned}$$
We conclude, $$\begin{aligned}
\int\limits_0^T[\dot{h}(t)-V_{\mbox{max}}(t)]_+\,dt=0.\end{aligned}$$
From a similar argument, $$\begin{aligned}
\int\limits_0^T[V_{\mbox{min}}(t)-\dot{h}(t)]_+\,dt=0.\end{aligned}$$
This proves .
| 0 |
---
abstract: 'In this paper we consider the branched transportation problem in 2D associated with a cost per unit length of the form $1+\alpha m$ where $m$ denotes the amount of transported mass and $\alpha>0$ is a fixed parameter (notice that the limit case $\alpha=0$ corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ($\{{\mathcal{F}}_{\varepsilon}\}_{{\varepsilon}>0}$) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the $\Gamma$-convergence of $\{{\mathcal{F}}_{\varepsilon}\}$ as ${\varepsilon}\downarrow0$. Our functionals are modeled on the Ambrosio-Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method.'
author:
- 'A. Chambolle'
- 'B. Merlet'
- 'L. Ferrari'
bibliography:
- 'bib.bib'
title: 'A simple phase-field approximation of the Steiner problem in dimension two'
---
Introduction
============
In this paper, we introduce a phase-field approximation of a branched transportation energy for lines in the plane. Our main goal is to derive a computationally tractable approximation of the Steiner problem (of minimizing the length of lines connecting a given set of points) in a phase-field setting. Similar results have recently be obtained by [@Bon_Lem_San], however we believe our approach is slightly simpler and numerically easier to implement. We show that we can modify classical approximations for free discontinuity problems [@Mod_Mort; @Amb_Tort1; @Iur; @ContiFocardiIurlano] to address our specific problem, where the limiting energy is concentrated only on a singular one-dimensional network (and roughly measures its length). Numerical results illustrate the behaviour of these elliptic approximations. In this first study, we limit ourselves to the two-dimensional case, as in that case lines can locally be seen as discontinuities of piecewise constant functions, so that our construction is deriving in a quite simple ways from the above mentioned previous works on free discontinuity problems. Higher dimension is more challenging, from the topological point of view; an extension of this approach is currently in preparation.
We now introduce precisely our mathematical framework. Let ${\Omega}\subset {\mathbf{R}}^2$ be a convex, bounded open set. We consider measures $\s\in \M({\overline}{{\Omega}},{\mathbf{R}}^2)$ that write $$\s=\t\xi\cdot\H^1{ \!\!\mbox{{ \Large$\llcorner$}} }M,$$ where $M$ is a $1$-dimensional rectifiable set orientated by a Borel measurable mapping $\xi:M{\rightarrow}\S^1$ and $\t:M{\rightarrow}{\mathbf{R}}_+$ is a Borel measurable function representing the multiplicity. Such measure is called a rectifiable measure. We follow the notation of [@OS2011] and write $\s=U(M,\t,\xi)$. Given a cost function $f\in C({\mathbf{R}}_+,{\mathbf{R}}_+)$, we introduce the functional defined on $\M({\overline}{\Omega},{\mathbf{R}}^2)\to{\mathbf{R}}_+\cup\{+\infty\}$ as $${\mathcal{E}}_f(\s):=
\begin{dcases}
\int_Mf(\t)\dH^1&\text{if }\s=U(\t,\xi,M),\\
\quad +{\infty}&\text{in the other cases.}
\end{dcases}$$ Given a sequence of $N+1$ distinct points $S=(x_0,\dots,x_N)\in {\Omega}^{N+1}$, we consider the minimization of ${\mathcal{E}}_f(\s)$ for $\s\in \M({\overline}{\Omega},{\mathbf{R}}^2)$ satisfying the constraint $$\label{eq:divconstraint1}
{\nabla}\cdot\s=N\d_{x_0}-\sum_{i=1}^N\d_{x_i}\quad\mbox{ in }{\mathcal{D}}'({\mathbf{R}}^2).$$ The distributional support of such $\sigma$ connects the source in $x_0$ to the sinks in $x_1,\cdots,x_N$. In general, a model for branched transport connecting a set of sources to a set of sinks (represented by two discrete probabilistic measures supported on a set of points in ${\Omega}$) is obtained by choosing $f(\t)=|\t|^\a$ with $0<\a<1$ and minimizing the associated functional under a divergence constraint similar to equation . The direct numerical optimization of the functional ${\mathcal{E}}_f$ is not easy because we do not know [*a priori*]{} the topological properties of the tree $M$. For this reason it is interesting to optimize an “approximate" functional defined on more flexible objects such as functions. Such approximate model has been introduced in [@OS2011] where the authors study the $\Gamma$-convergence (see [@Bra2]) of a family of functionals inspired by the well known work of Modica and Mortola [@Mod_Mort]. Another effort in this direction can be found in the work [@Bon_Lem_San] where is studied an approximation to the Steiner Minimal Tree problem ([@Gilb_Poll], [@Ambr_Tilli] and [@Paol_Step]) by means of analogous techniques.
Here, we consider variational approximations of some energies of the form ${\mathcal{E}}_f$ through a family of functionals modeled on the Ambrosio-Tortorelli functional [@Amb_Tort1]. For being more precise, we need to introduce some material. Let $\rho\in C^\infty_c({\mathbf{R}}^2,{\mathbf{R}}_+)$ be a classical radial mollifier with $\operatorname{supp}\rho\subset B_1(0)$ and $\int \rho=1$. For ${\varepsilon}\in (0,1]$, we set $\rho_{\varepsilon}(x)={\varepsilon}^{-2}\rho({\varepsilon}^{-1}x)$ and we define the space $V_{\varepsilon}({\Omega})$ of square integrable vector fields whose weak divergence satisfy the constraint $$\label{eq:divconstraint2}
{\nabla}\cdot\s_{\varepsilon}\, =\, {\left}(N\delta_{x_0} - \sum_{j=1}^N \delta_{x_j} {\right})\ast \rho_{\varepsilon}.$$ For $\eta=\eta({\varepsilon})>0$, we note $$\label{eq:Weps}
W_{\varepsilon}({\Omega})\,=\,{\left}\{\phi\in H^1({\Omega})\,:\, \eta \leq\phi\leq 1\mbox{ in }{\Omega},\ \phi\equiv 1\mbox{ on }\de{\Omega}{\right}\}.$$ Then we define the energy ${\mathcal{F}}_{\varepsilon}:\M({\Omega},{\mathbf{R}}^2)\times L^1({\Omega}){\rightarrow}[0,+{\infty}]$ as $$\label{eq:functional}
{\mathcal{F}}_{\varepsilon}(\s,\phi):=
\begin{dcases}
\;\int_{{\Omega}} \frac{1}{2{\varepsilon}}\phi^2 |\s|^2 \dx+ \int_{{\Omega}} \frac{{\varepsilon}}{2}|{\nabla}\phi|^2 + \frac{(1-\phi)^2}{2{\varepsilon}} \dx, & \text{if } (\s,\phi)\in V_{\varepsilon}({\Omega})\times W_{\varepsilon}({\Omega}),\\
\quad+{\infty}& \text{in the other cases.}
\end{dcases}$$ The first integral in the definition of the energy will be refered to as the “constraint component” while the second integral will be regarded as the “Modica-Mortola component”. Let us briefly describe the qualitative properties of the associated minimization problem. First notice that the constraint enforces $\s$ to be non zero on a set connecting $S$. Next, the constraint component of the energy strongly penalizes $\phi^2 |\s|^2$ so that $\phi$ should be small in the region where $|\sigma|$ is large. On the other hand the behavior of $\phi$ is controlled by the Modica Mortola component that forces $\phi$ to be close to $1$ away from a one-dimensional set as ${\varepsilon}$ converges to 0. As a consequence, we expect the support of $\sigma$ and the energy to concentrate on a one dimensional set connecting $S$. The main part of the paper consists in making rigorous and quantitative this analysis.
From now on, we assume that there exists some $\a\geq 0$ such that $$\label{eq:limalpha}
\dfrac\eta{{\varepsilon}}\ \stackrel{{\varepsilon}\downarrow0}\longrightarrow\ \a.$$ We note $\M_S({\overline}{\Omega})$ the set of ${\mathbf{R}}^2$-valued measures $\s\in \M({\mathbf{R}}^2,{\mathbf{R}}^2)$ with support in ${\overline}{{\Omega}}$ such that the constaint holds. We define the limit energy ${\mathcal{E}}_\a:\M({\overline}{\Omega},{\mathbf{R}}^2)\times L^1({\Omega}){\rightarrow}[0,+{\infty}]$ as $$\label{eq:limit}
{\mathcal{E}}_\a(\s,\phi)=
\begin{dcases}
\;\int_{M}(1+\a\,\t)\dH^1 & \text{if } \phi\equiv1,\ \s\in \M_S({\overline}{\Omega})\mbox{ and }\s=U(M,\t,\xi),\\
~ \quad+{\infty}&\text{in the other cases.}
\end{dcases}$$ We prove the $\Gamma$-convergence of the sequence $({\mathcal{F}}_{\varepsilon})$ to the energy ${\mathcal{E}}_\a$ as ${\varepsilon}{\downarrow}0$. More precisely the convergence holds in $\M({\overline}{\Omega},{\mathbf{R}}^2)\times L^1({\Omega})$ where $\M({\overline}{\Omega},{\mathbf{R}}^2)$ is endowed with the weak star topology and $L^1({\Omega})$ is endowed with its classical strong topology.\
We establish the following lower bound.
\[teo:sigma\_liminf\] For any sequence $(\s_{\varepsilon},\phi_{\varepsilon})\subset\M({\Omega},{\mathbf{R}}^2)\times L^1({\Omega})$ such that $\s_{\varepsilon}\rwstar\s$ and $\phi_{\varepsilon}{\rightarrow}\phi$ in the $L^1({\Omega})$ topology, with $(\s,\phi)\in\M({\Omega},{\mathbf{R}}^2)\times L^1({\Omega})$ $$\liminf_{k{\rightarrow}+{\infty}}{\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon})\geq {\mathcal{E}}_\a(\s,\phi).$$
In this statement and throughout the paper, we make a small abuse of language by noting $(a_{\varepsilon})_{{\varepsilon}\in(0,1]}$ and calling sequence a family $\{a_{\varepsilon}\}$ labeled by a continuous parameter ${\varepsilon}\in(0,1]$. In the same spirit, we call subsequence of $(a_{\varepsilon})$, any sequence $(a_{{\varepsilon}_j})$ with ${\varepsilon}_j{\rightarrow}0$ as $j{\rightarrow}+{\infty}$.\
To complete the $\Gamma$-convergence analysis, we establish the matching upper bound.
\[teo:sigma\_limsup\] For any $(\s,\phi)\subset\M({\Omega},{\mathbf{R}}^2)\times L^1({\Omega})$ there exists a sequence $(\s_{\varepsilon},\phi_{\varepsilon})$ such that $\s_{\varepsilon}\rwstar\s$ and $\phi_{\varepsilon}{\rightarrow}\phi$ in the $L^1({\Omega})$ topology and $$\limsup_{k{\rightarrow}+{\infty}}{\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon})\leq {\mathcal{E}}_\a(\s,\phi).$$
Moreover, under the assumption $\a>0$ we prove the equicoercivity of the sequence $({\mathcal{F}}_{\varepsilon})$.
\[teo:sigma\_equicoercive\] Assume $\a>0$. For any sequence $(\s_{\varepsilon},\phi_{\varepsilon})_{{\varepsilon}\in(0,1]}\subset\M({\Omega},{\mathbf{R}}^2)\times L^1({\Omega})$ with uniformly bounded energies, [*i.e.*]{} $$\sup_{{\varepsilon}} {\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon})\ <\ +{\infty},$$ there exist a subsequence ${\varepsilon}_j{\downarrow}0$ and a measure $\s\in\M_S({\overline}{\Omega},{\mathbf{R}}^2)$ such that $\s_{{\varepsilon}_j}\rightarrow \s$ with respect to the weak-$*$ convergence of measures and $\phi_{{\varepsilon}_j} \rightarrow 1$ in $L^1({\Omega})$. Moreover, $\s$ is a rectifiable measure ([*i.e.*]{} it is of the form $\s=U(M,\t,\xi)$).
Observe that letting $\a=0$ in equation we obtain ${\mathcal{E}}_0(\sigma)=\H^1(\{x\in M\,:\, \t(x)>0\})$ where $\s=U(M,\t,\xi)$. This is exactly the functional associated with the Steiner Minimal Tree problem. Unfortunately, the hypothesis $\a>0$ is necessary in the compactness Theorem \[teo:sigma\_equicoercive\].
The fact that we are working in dimension $2$ is fundamental for the proof of Theorem \[teo:sigma\_liminf\] as it allows to locally rewrite the vector field $\s_{\varepsilon}$ as the rotated gradient of a function.
**Structure of the paper:** In Section \[sec:Not\] we introduce some notation and several tools and notions on $SBV$ functions and vector field measures. In Section \[sec:LocRes\] we study a first family of energies obtained by substituting ${\nabla}u$ for $\s$ in the definition of ${\mathcal{F}}\e$. In Section \[sec:Comp\] we prove the equicoercivity result, Theorem \[teo:sigma\_equicoercive\] and we establish the lower bound stated in Theorem \[teo:sigma\_liminf\]. In Section \[sec:UppBound\] we prove the upper bound of Theorem \[teo:sigma\_limsup\]. Finally, in the last section, we present and discuss various numerical simulations.
Notation and Preliminary Results {#sec:Not}
================================
In the following ${\Omega}\subset\subset\hat{\Omega}\subset{\mathbf{R}}^d$ are bounded open convex sets. Given $X\subset{\mathbf{R}}^d$ (in practice $X={\Omega}$ or $X=\hat{\Omega})$, we denote by $\A(X)$ the class of all open subsets of $X$ and by $\A_S(X)$ the subclass of all simply connected open sets $O\subset X$ such that $\overline{O}\cap S=\varnothing$. We denote by $(e_1,\dots,e_d)$ the canonical orthonormal basis of ${\mathbf{R}}^d$, by $|\cdot|$ the euclidean norm and by $\langle\cdot,\cdot\rangle$) the euclidean scalar product in ${\mathbf{R}}^d$. The open ball of radius $r$ centered at $x\in{\mathbf{R}}^d$ is denoted by $B_r(x)$. The $(d-1)$-dimensional Hausdorff measure in ${\mathbf{R}}^d$ is denoted by $\H^{d-1}$. We write $|E|$ to denote the Lebesgue measure of a measurable set $E\subset {\mathbf{R}}^d$. When $\mu$ is a Borel meaure and $E\subset {\mathbf{R}}^d$ is a Borel set, we denote by $\mu{ \!\!\mbox{{ \Large$\llcorner$}} }E$ the measure defined as $\mu{ \!\!\mbox{{ \Large$\llcorner$}} }E(F)=\mu(E\cap F)$.\
Let us remark that from Section \[sec:Comp\] onwards, we work in dimension $d=2$.\
For $O\in\A({\Omega})$, the functional ${\mathcal{F}}_{\varepsilon}(\cdot,\cdot;O)$ is the functional obtained by substuting $O$ for ${\Omega}$ in the definition of ${\mathcal{F}}_{\varepsilon}$ (see ). Similarly we define the local version ${\mathcal{E}}_\a(\cdot,\cdot;O)$ of ${\mathcal{E}}_\a$.
BV(${\Omega}$) functions and Slicing
------------------------------------
BV(${\Omega}$) is the space of functions $u\in L^1({\Omega})$ having as distributional derivative $Du$ a measure with finite total variation. For $u\in BV({\Omega})$, we denote by $S_u$ the complement of the Lebesgue set of $u$, that is $x\not \in S_u$ if and only if $$\lim_{\rho{\rightarrow}0^+}\frac{1}{|B_\rho(x)|}\int_{B_\rho(x)}|u(y)-z|\dy=0$$ for some $z\in{\mathbf{R}}$. We say that $x$ is an approximate jump point of $u$ if there exist $\xi \in \S^{d-1}$ and distinct points $a,b \in {\mathbf{R}}$ such that $$\lim_{\rho{\downarrow}0}\frac{1}{|B^+_\rho(x,\xi)|}\int_{B^+_\rho(x,\xi)}|u(y)-a|\dy=0\qquad\text{and}\qquad\lim_{\rho{\downarrow}0}\frac{1}{|B^-_\rho(x,\xi)|}\int_{B^+_\rho(x,\xi)}|u(y)-b|\dy=0,$$ where $B^\pm_\rho(x,\xi):=\{y\in B_\rho(x)\;:\;\pm\langle y-x,\xi\rangle\geq0\}$. Up to a permutation of $a$ and $b$ and a change of sign of $\xi$, this characterizes the triplet $(a,b,\xi)$ which is then denoted by $(u^+,u^-,\nu_u)$. The set of approximate jump points is denoted by $J_u$. The following theorem holds [@Am_Fu_Pal].
\[teo:federer\] The set $S_u$ is countably $\H^{d-1}$-rectifiable and $\H^{d-1}(S_u\setminus J_u)=0$. Moreover $Du{ \!\!\mbox{{ \Large$\llcorner$}} }J_u=(u^+-u^-)\nu_u\H^{d-1}{ \!\!\mbox{{ \Large$\llcorner$}} }J_u$ and $$Tan^{d-1}(J_u,x)=\nu_u(x)^\perp$$ for $\H^{d-1}$-a.e. $x\in J_u$.
We write the Radon-Nikodym decomposition of $Du$ as $Du= {\nabla}u \dx+D^s u$. Setting $D^c u=D^s u { \!\!\mbox{{ \Large$\llcorner$}} }({\Omega}\setminus S_u)$ we get the decomposition $$Du={\nabla}u \dx+ (u^+-u^-)\nu_u\H^{d-1}{ \!\!\mbox{{ \Large$\llcorner$}} }J_u+D^c u.$$ Moreover the Cantor part is such that if $\H^{d-1}(E)<\infty$, we have $|D^cu|(E)=0$ [@Am_Fu_Pal Thm. 3.92]. In particular, we have the following useful consequence $$\label{BVproperty}
\H^{d-1}(E)=0\quad\Longrightarrow\quad |Du|(E)=0.$$ We frequently use the notation $[u]$ for the jump function $(u^+-u^-):J_u{\rightarrow}{\mathbf{R}}$.\
When $d=1$ we use the symbol $u'$ in place of ${\nabla}u$ and $u(x^\pm)$ to indicate the right and left limits of $u$ at $x$.
Let us introduce the space of special functions of bounded variation and a variant: $$SBV({\Omega}):=\{u\in BV({\Omega}):D^cu=0\},$$ $$GSBV({\Omega}):=\{u\in L^1({\Omega}):\max(-T,\min(u,T))\in SBV({\Omega})\;\forall T\in{\mathbf{R}}\}.$$ Eventually, in Section \[sec:LocRes\], the following space of piecewise constant functions will be useful. $$\P({\Omega})=\{u\in GSBV({\Omega}):{\nabla}u=0\}.$$ To conclude this section we recall the slicing method for functions with bounded variation. Let $\xi\in\S^{d-1}$ and let $$\Pi_\xi:=\{y\in{\mathbf{R}}^d:\langle y,\xi\rangle=0\}.$$ If $y\in\Pi_\xi$ and $E\in{\mathbf{R}}^d$, we define the one dimensional slice $$E_{\xi,y}:=\{t\in{\mathbf{R}}: y+t\xi\in E\}.$$ For $u:{\Omega}{\rightarrow}{\mathbf{R}}$, we define $u_{\xi,y}:{\Omega}_{\xi,y}{\rightarrow}{\mathbf{R}}$ as $$u_{\xi,y}(t):=u(y+t\xi), \quad t\in {\Omega}_{\xi,y}.$$ Functions in $GSBV({\Omega})$ can be characterized by one-dimensional slices (see [@Bra1 Thm. 4.1])
\[teo:braides\] Let $u\in GSBV({\Omega})$. Then for all $\xi \in\S^{d-1}$ we have $$u_{\xi,y}\in GSBV({\Omega}_{\xi,y})\quad \text{for } \H^{d-1}\text{-a.e. }y\in\Pi_\xi.$$ Moreover for such $y$, we have $$\begin{gathered}
u'_{\xi,y}(t)=\langle{\nabla}u(y+t\xi),\xi\rangle\quad \text{for a.e. } t\in{\Omega}_{\xi,y},\\
J_{u_{\xi,y}}=\{t\in {\mathbf{R}}:y+t\xi\in J_u\},\end{gathered}$$ and $$u_{\xi,y}(t^\pm)=u^\pm(y+t\xi)\quad \text{or}\quad u_{\xi,y}(t^\pm)=u^\mp(y+t\xi)$$ according to whether $\langle\nu_u,\xi\rangle>0$ or $\langle\nu_u,\xi\rangle<0$. Finally, for every Borel function $g:{\Omega}{\rightarrow}{\mathbf{R}}$, $$\label{eq:braidesricostruzione}
\int_{\Pi_\xi}\sum_{t\in J_{u_{\xi,y}}} g_{\xi,y}(t)\dH^{d-1}(y)=\int_{J_u}g|\langle\nu_u,\xi\rangle|\dH^{d-1}.$$ Conversely if $u\in L^1({\Omega})$ and if for all $\xi \in\{e_1,\dots,e_d\}$ and almost every $y\in\Pi_\xi$ we have $u_{\xi,y}\in SBV({\Omega}_{\xi,y})$ and $$\int_{\Pi_\xi}|Du_{\xi,y}|({\Omega}_{\xi,y})\dH^{d-1}(y)<+{\infty}$$ then $u\in SBV({\Omega})$.
Rectifiable vector Measures
---------------------------
Let us introduce the linear operator $\perp$ that associates to each vector $v=(v_1,v_2)\in {\mathbf{R}}^2$ the vector $v^\perp=(-v_2,v_1)$ obtained via a $90^\circ$ counterclockwise rotation of $v$. Notice that the $\perp$ operator maps divergence free ${\mathbf{R}}^2$-valued measures onto curl free ${\mathbf{R}}^2$-valued measures. Let $O\subset{\mathbf{R}}^2$ be a simply connected and bounded open set. By Stokes Theorem, for any divergence free measure $\s\in\M(O,{\mathbf{R}}^2)$ there exists a function $u\in BV({\Omega})$ with zero mean value such that $\s= Du^\perp$. On the other hand for $u\in \P({\Omega})$ $\sigma:=Du^\perp$ is divergence free and by Theorem \[teo:federer\], $\sigma =(u^+-u^-)\nu_u^\perp \H^1=U(J_u,[u],\nu_u^\perp)$.
Let us now produce an elementary example of measure $\g$ of the form $U(M,\t,\xi)$.
\[ex:div2punti\] Given two points $x,y\in{\Omega}$ we consider the smooth path from $x$ to $y$ defined as $$r(t):=x+t\frac{y-x}{|x-y|}\quad \mbox{ for } t\in [0,|x-y|].$$ We define the measure $\g\in \M({\Omega},{\mathbf{R}}^2)$ by $$(\phi,\g)\,:=\,\int\langle\phi(r(t)),\dot{r}(t)\rangle\dt\qquad\mbox{for any } \phi\in \C(\overline{{\Omega}},{\mathbf{R}}^2).$$ We then have $\g=U([x,y],1,\xi)$ with $\xi=\frac{y-x}{|y-x|}$. Notice that ${\nabla}\cdot\g=\d_{x}-\d_y$.\
Similarly we obtain a measure satisfying this property by substituting for $r$ any Lipschitz path from $x$ to $y$.
We will make use of the following construction.
\[lemma:completediv\] Given $S=(x_0,\cdots,x_N)\in {\Omega}^{N+1}$ a sequence of $N+1$ distinct points, there exist a vector measure $\g= U(M_\g,\t_\g,\xi_\g)$ and a finite partition $({\Omega}_i)\subset \A({\Omega})$ of ${\Omega}$ such that
1. ${\nabla}\cdot\g=-N\d_{x_0}+\sum_{i=1}^{N} \d_{x_i}$,
2. $\t_\g:M_\g{\rightarrow}\{1,N\}$,
3. each ${\Omega}_i$ is a polyhedron,
4. $ M_\g\subset \bigcup_i \de{\Omega}_i$,
5. ${\Omega}_i$ is of finite perimeter for each $i$ and ${\Omega}_i\cap{\Omega}_j=\emptyset$ for $i\neq j$,
6. $\L^2({\Omega}\sm\cup_i{\Omega}_i)=0$.
Moreover if $M$ is a $1$ dimensional countably rectifiable set, we can choose $\gamma$ and $({\Omega}_i)$ such that $\H^1(M\cap \bigcup_i \de{\Omega}_i)=0$.
Let us fix a point $p\in{\Omega}\setminus S$. By Example \[ex:div2punti\] we can construct a measure $\g_i$ with ${\nabla}\cdot\g_i=\d_{x_i}-\d_{p}$ for $i\in\{0,\cdots,N\}$. We define $$\g=-N \g_0 +\sum_{i=1}^N\g_i.$$ By construction $(a)$ holds true. Moreover, up to a small displacement of $p$ we may assume that $[p,x_i]\cap[p,x_j]=\{p\}$ for $i\neq j$ so that $(b)$ holds.\
Next, let $D_j$ be the straight line supporting $[p,x_j]$. We define the sets $({\Omega}_i)$ as the connected components of ${\Omega}\setminus \left(D_0 \cup\cdots\cup D_N\right)$. We see that $(c,d,e,f)$ hold true.\
For the last statement, we observe that by the coarea formula, we have $\H^1(M\cap \bigcup_i \de{\Omega}_i)=0$ for a.e. choice of $p$.
![Example of the construction of the $\H^1$-rectifiable measure $\g$ (red) and of the partition $\{{\Omega}_i\}$ (gray) in the case $\s$ (green) is being an $\H^1$-rectifiable vector measure.[]{data-label="fig:partizione"}](partizione){width=".4\textwidth"}
Local Result {#sec:LocRes}
============
In this section we introduce a localization of the family of functionals $({\mathcal{F}}_{\varepsilon})$ (see ). We establish a lower bound and a compactness property for these local energies.
\[procedure\] Let $O\in A_S({\Omega})$, for $u{\varepsilon}\in H^1(O)$ and $\phi_{\varepsilon}\in H^1(O)$, we define $$\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O)\ :=\ {\mathcal{F}}_{\varepsilon}({\nabla}u_{\varepsilon},\phi_{\varepsilon};O).$$ Notice that for ${\varepsilon}<d(O,S)$, we have ${\nabla}\cdot \s_{\varepsilon}\equiv 0$ in $O$ for any $\s_{\varepsilon}\in V_{\varepsilon}({\Omega})$. By the Stokes theorem we have $Du_{\varepsilon}=\s_{\varepsilon}^\perp$ for some $u\in H^1(O)$ and we have $${\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon};O)=\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O).$$
The remaining of the section is devoted to the proof of
\[teo:localResult\] Let $(u_{\varepsilon})_{{\varepsilon}\in(0,1]}\subset H^1(O)$ be a family of functions with zero mean value and let $(\phi_{\varepsilon})\subset H^1(O)$ such that $\phi_{\varepsilon}\in H^1(O,[\eta({\varepsilon}),1])$. Assume that $c_0:=\sup_{\varepsilon}\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O)$ is finite, then there exist a subsequence ${\varepsilon}_j$ and a function $u\in BV({\Omega})$ such that
a) $\phi_{{\varepsilon}_j} {\rightarrow}1$ in $L^2(O)$,
b) $u_{{\varepsilon}_j}{\rightarrow}u$ with respect to the weak-$*$ convergence in $BV$,
c) $u\in \P(O)$.
Furthermore for any $u\in\P(O)$ and any sequence $(u_{\varepsilon},\phi_{\varepsilon})$ as above such that $u_{\varepsilon}\rwstar u$, we have the following lower bound of the energy: $$\liminf_{{\varepsilon}{\rightarrow}0}\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O)\geq \int_{J_u\cap O}{\left}[1+\alpha|[u]|{\right}]\dH^{d-1}.$$
The proof is achieved in several steps and mostly follows ideas from [@Iur] (see also [@ContiFocardiIurlano]). In the first step we obtain *(a)* and *(b)*. In steps 2. we prove *(c)* and the lower bound for one dimensional slices. Finally in step 3. we prove *(c)* and the lower bound in dimension $d$. The construction of a recovery sequence that would complete the $\Gamma$-limit analysis is postponed to the global model in Section \[sec:UppBound\].
*Step 1.* Item *(a)* is a straightforward consequence of the definition of the functional. Indeed, we have $$\int_O(1-\phi_{\varepsilon})^2\, \dx\, \leq {\varepsilon}\, \LF_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon})\, \leq\, c_0\, {\varepsilon}\, \stackrel{{\varepsilon}{\downarrow}0}\longrightarrow \ 0.$$ For *(b)*, since $(u_{\varepsilon})$ has zero mean value, we only need to show that $ \sup_{\varepsilon}\{|Du_{\varepsilon}|(O): k\in{\mathbf{N}}\}<+\infty$. Using Cauchy-Schwarz inequality we get $$\label{eq:compactness1}
{\left}[|Du_{\varepsilon}|(O){\right}]^2={\left}(\int_{O}|{\nabla}u_{\varepsilon}|{\right})^2\leq {\left}({\varepsilon}\int_{O}\frac{1}{\phi^2\e}{\right}){\left}(\frac{1}{{\varepsilon}}\int_{O}\phi^2\e|{\nabla}u_{\varepsilon}|^2{\right}).$$ By assumption, the second therm in the right hand side of is bounded by $2c_0$. In order to estimate the first term we split $O$ in the two sets $\{\phi_{\varepsilon}<1/2\}$ and $\{\phi_{\varepsilon}\geq1/2\}$. We have,$${\varepsilon}\int_{O}\frac{1}{\phi\e}\,=\, {\varepsilon}\int_{\{\phi_{\varepsilon}<1/2\}}\frac{1}{\phi^2\e}+{\varepsilon}\int_{\{\phi\e\geq1/2\}}\frac{1}{\phi^2\e}.$$ Since $\phi_{\varepsilon}\geq \eta$ and $(1-t)^2$ is decreasing in the interval $O$ the following inequalities hold $$\begin{aligned}
\int_{\{\phi\e<1/2\}}\frac{1}{\phi^2\e}&\leq& \frac{2 {\varepsilon}}{\eta^2(1-1/2)^2}\int_{O}\frac{(1-\phi\e)^2}{2{\varepsilon}}\,\leq\,\frac {8{\varepsilon}}\eta^2 c_0,\\
\nonumber\int_{\{\phi_{\varepsilon}\geq1/2\}}\frac{1}{\phi^2\e}&\leq &\int_{O}\frac{1}{(1/2)^2} \, =\, 4|O|.\end{aligned}$$ Combining these estimates with we obtain $$\label{eq:limitatezzagradu}
{\left}[|Du_{\varepsilon}|(O){\right}]^2\, \leq\, \frac{{\varepsilon}^2}{\eta^2}\:16c_0^2 + 8{\varepsilon}|O| c_0\,\stackrel{{\varepsilon}{\downarrow}0}\longrightarrow\, \frac{16c_0^2}{\a^2}\,<\,\infty.$$ This establishes *(b)*.
*Step 2.* In this step we suppose $O$ to be an interval of ${\mathbf{R}}$. We first prove that $u$ is piecewise constant. The idea is that in view of the constraint component of the energy, variations of $u_{\varepsilon}$ are balanced by low values of $\phi_{\varepsilon}$. On the other hand the Modica-Mortola component of the energy implies that $\phi_{\varepsilon}\simeq 1$ in most of the domain and that transitions from $\phi_{\varepsilon}\simeq 1$ to $\phi_{\varepsilon}\simeq 0$ have a constant positive cost (and therefore can occur only finitely many times).
*Step 2.1. proof of $u\in \P(O)$.* Let us define $$\label{def:insiemi}
B_{\varepsilon}:= {\left}\{x\in O:\phi_{\varepsilon}(x)<\frac{3}{4}{\right}\}\,\supset\, A_{\varepsilon}:= {\left}\{x\in O:\phi_{\varepsilon}(x)<\frac{1}{2}{\right}\},$$ and let $$\label{def:C}
C_{\varepsilon}= \{I \text{ connected component of } B_{\varepsilon}: I\cap A_{\varepsilon}\neq \varnothing \}.$$ Let us show that the cardinality of $C_{\varepsilon}$ is bounded by a constant independent of ${\varepsilon}$. Let ${\varepsilon}$ be fixed and consider an interval $I\in C_{\varepsilon}$. Let $a,b\in \bar I$ such that $\{\phi_{\varepsilon}(a),\phi_{\varepsilon}(b)\}=\{1/2,3/4\}$. Using the usual Modica-Mortola trick, we have $$\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};I)\,\geq\,\int_I {\varepsilon}|\phi'_{\varepsilon}|^2 + \frac{(1-\phi_{\varepsilon})^2}{4{\varepsilon}} \dx \geq \int_{(a,b)} |\phi'_{\varepsilon}|(1-\phi_{\varepsilon})\dx\,\geq\, \int_{1/2}^{3/4} (1-t)\,dt \, =\, \dfrac3{2^5}.$$ Since all the elements of $C_{\varepsilon}$ are disjoint, we deduce from the energy bound that $$\# C_{\varepsilon}\leq2^5c_0/3,$$ where we note $\# C_{\varepsilon}$ the cardinality of $C_{\varepsilon}$. Next, up to extraction we can assume that $\# C_{\varepsilon}=N$ is fixed. The elements of $ C_{\varepsilon}$ are written on the form $I^{\varepsilon}_i=(m^{\varepsilon}_i-w_i^{\varepsilon},m^{\varepsilon}_i+w_i^{\varepsilon})$ for $i=1,\cdots,N$ with $m^{\varepsilon}_i<m^{\varepsilon}_{i+1}$. Since $\phi_{\varepsilon}{\rightarrow}1$ in $L^1(O)$ we have $$\label{misura:zero}
\sum_{I^{\varepsilon}_i\in C_{\varepsilon}}|I^{\varepsilon}_i|= \sum_i2w^{\varepsilon}_i {\rightarrow}0.$$ Up to further extraction, we can assume that each sequence $(m_i^{\varepsilon})$ converges in ${\overline}O$. We call $m_1\leq m_2 \leq \dots \leq m_N$ their limits. We now prove that $|D u|(O\setminus \{m_i\}_{i=1}^N) =0$. For this, we fix $x\in O\setminus\{m_i\}_{i=0}^N$ and establish the existence of a neighborhood $B_\delta(x)$ of $x$ for which $|Du|(B_\delta(x))=0$. Let $0<\delta\leq 1/2\min_i|x-{m}_i|$. Equation ensures that for ${\varepsilon}$ small enough $B_\delta(x)\cap C_{\varepsilon}=\varnothing$. Notice that from the definitions in and we have that $\phi_{\varepsilon}\geq 1/2$ outside $C_{\varepsilon}$. Hence, using Cauchy-Scwarz inequality, we have for ${\varepsilon}$ small enough, $${\left}(\int_{B_\delta(x)}| u'_{\varepsilon}|\dx{\right})^2\leq 2\delta \int_{B_\delta(x)}| u'_{\varepsilon}|^2\dx\leq (2\delta)(2{\varepsilon}) 4{\left}(\frac{1}{2{\varepsilon}}\int_{B_\delta(x)}\phi_{\varepsilon}|^2 u'_{\varepsilon}|^2\dx{\right})\, \leq\, 16c_0{\varepsilon}\delta \,\stackrel{{\varepsilon}{\downarrow}0}\longrightarrow\,0.$$ By lower semicontinuity of the total variation on open sets we conclude that $|Du|(B_\delta(x))=0$. This establishes $u\in \P(O)$ with $J_u\subset \{m_1,\cdots,m_N\}$.
*Step 2.2. Proof of the lower bound.* Without loss of generality, we prove the lower bound in the case $J_u=\{0\}$ and $ D:=u(0^+)=-u(0^-)>0$. Using the same argument as in [@Iur Pag. 7] for any $0<d<D$ there exist six points $y_1<x^1_{\varepsilon}\leq\tilde{x}^1_{\varepsilon}<\tilde{x}^2_{\varepsilon}\leq x^2_{\varepsilon}<y_2$ such that $$\begin{gathered}
\lim_{{\varepsilon}{\rightarrow}0}\phi_{\varepsilon}(y_1)= \lim_{{\varepsilon}{\rightarrow}0} \phi_{\varepsilon}(y_2) = 1,\\
\lim_{{\varepsilon}{\rightarrow}0}\phi_{\varepsilon}(x^1_{\varepsilon})= \lim_{{\varepsilon}{\rightarrow}0} \phi_{\varepsilon}(x^2_{\varepsilon}) =0,\\
u_{\varepsilon}(\tilde{x}^1_{\varepsilon})= -D+d,\quad\quad \; u_{\varepsilon}(\tilde{x}^2_{\varepsilon})= D-d.\end{gathered}$$ Using the Modica-Mortola trick in the intervals $(y_1,x^1_{\varepsilon})$ and $(x^2_{\varepsilon},y_2)$ as above, we compute: $$\label{dis1:liminf}
\liminf_{{\varepsilon}{\downarrow}0}\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};(y_1,x^1_{\varepsilon})\cup(x^2_{\varepsilon},y_2))\geq\liminf_{{\varepsilon}{\downarrow}0}\int^{x^1_{\varepsilon}}_{y_1}(1-\phi_{\varepsilon})|\phi'_{\varepsilon}| \dx+\int_{x^2_{\varepsilon}}^{y_2}(1-\phi_{\varepsilon})|\phi'_{\varepsilon}| \dx\,\geq\,1.$$ For the estimation on the interval $I_{\varepsilon}=(\txk^1,\txk^2)$ let us introduce: $$\begin{gathered}
G_{\varepsilon}:= {\left}\{ w\in H^1(I_{\varepsilon}) : w(\txk^1)=-D+d,\, w(\txk^2)=D-d {\right}\},\\
Z_{\varepsilon}:= {\left}\{ z\in H^1(I_{\varepsilon}) :\eta\leq z\leq1 \text{ a.e. on } I_{\varepsilon}{\right}\},\\
H_{\varepsilon}(w,z):=\int_{I_{\varepsilon}}{\left}(\frac{1}{2{\varepsilon}}z^2|w'|^2+\frac{(1-z)^2}{2{\varepsilon}}{\right})\dx,\\
h_{\varepsilon}(z)= \inf_{w\in W_{\varepsilon}}H_{\varepsilon}(w,z) \text{ for } z\in Z_{\varepsilon}.\end{gathered}$$ Note that by Cauchy-Schwarz inequality, we have for $w\in G_{\varepsilon}$ and $z\in Z_{\varepsilon}$, $$\int_{I_{\varepsilon}}z^2|w'|^2\, \geq \, \left(\int_{I_{\varepsilon}}|w'|\dx\right)^2{\left}(\int_{I_{\varepsilon}}\frac{1}{z^2}{\right})^{-1}\, \geq\, 4(D-d)^2{\left}(\int_{I_{\varepsilon}}\frac{1}{z^2}{\right})^{-1}.$$ We deduce the lower bound $$\label{eq:minimo}
h_{\varepsilon}(z) \, \geq \, 4(D-d)^2 {\left}(2{\varepsilon}\int_{I_{\varepsilon}}\frac{1}{z^2}\dx{\right})^{-1}+ \int_{I_{\varepsilon}}{\left}(\frac{(1-z)^2}{2{\varepsilon}}{\right})\dx.$$ Let us remark that optimizing $H_{\varepsilon}(w,z)$ with respect to $w\in G_{\varepsilon}$ we see that this inequality is actually an equality.\
Consider for $0 <\lambda<1$ the inequalities: $$\int_{\{x\in I_{\varepsilon}: \phi_{\varepsilon}\geq \lambda\}}\frac{1}{\phi^2_{\varepsilon}}\leq \frac{\L^1(I_{\varepsilon})}{\lambda^2}\qquad\mbox{ and }\qquad\int_{\{x\in I_{\varepsilon}: \phi_{\varepsilon}< \lambda\}}\frac{1}{\phi^2_{\varepsilon}}\leq \frac{1}{(1-\lambda)^2}\frac{2{\varepsilon}}{\eta^2}{\left}(\int_{I_{\varepsilon}}\frac{(1-\phi_{\varepsilon})^2}{2{\varepsilon}}\dx{\right}).$$ Applying both of them in we obtain $$\begin{aligned}
\nonumber
\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon},I_{\varepsilon})&\geq h_{\varepsilon}(\phi_{\varepsilon})\\ \nonumber
&\geq\frac{2(D-d)^2}{\frac{{\varepsilon}\L^1(I_{\varepsilon})}{\lambda^2}+\frac{1}{(1-\lambda)^2}\frac{2{\varepsilon}^2}{\eta^2}{\left}(\int_{I\e}\frac{(1-\phi\e)^2}{2{\varepsilon}}\dx{\right})}+ \int_{I\e}{\left}(\frac{(1-\phi\e)^2}{2{\varepsilon}}{\right})\dx \\
&\geq 2(1-\lambda)\frac{\eta}{{\varepsilon}}(D-d)-(1-\lambda)^2\frac{\eta^2}{2{\varepsilon}}\frac{\L^1(I\e)}{\lambda^2} \label{dis2:liminf}\end{aligned}$$ where the latter is obtained by minimizing the function: $$t \mapsto \frac{2(D-d)^2}{\frac{{\varepsilon}\L^1(I_{\varepsilon})}{\lambda^2}+\frac{1}{(1-\lambda)^2}\frac{2{\varepsilon}^2}{\eta^2}t}+ t.$$ Therefore we can pass to the limit in and obtain: $$\liminf_{{\varepsilon}{\downarrow}0}\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon},I_{\varepsilon})\geq(1-\lambda)\a\,2(D-d).$$ Sending $\lambda$ and $d$ to $0$ and recalling the estimation in we get $$\label{eq:liminf1d}
\liminf_{{\varepsilon}{\downarrow}0}\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon},(y_1,y_2))\, \geq\, 1+\a\,2D \, =\, 1+\a|u(0^+)-u(0^-)|.$$ *Step 3.* Using Fubini’s decomposition we can rewrite the energy obtaining for every $\xi\in \S^{d-1}$, $$ \LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O)\,\geq\,\int_{\Pi_\xi}{\left}( \int_{O^\xi_y}\frac{1}{2{\varepsilon}}(\phi^2_{\varepsilon})^\xi_y |(u'_{\varepsilon})^\xi_y|^2 + \frac{{\varepsilon}}{2}|(\phi'_{\varepsilon})^\xi_y |^2 + \frac{(1-(\phi_{\varepsilon})^\xi_y )^2}{2{\varepsilon}} \dt{\right})\dH^{d-1}(y).$$ Since $\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O)$ is bounded, for $\H^{d-1}$ almost every $y\in \Pi_\xi$, the inner integral in the latter is also bounded, furthermore it corresponds to the functional on the one dimensional slice $O^\xi_y$ studied in the previous step evaluated on the couple $((u_{\varepsilon})^\xi_y,(\phi_{\varepsilon})^\xi_y)$. Taking the $\liminf$ of the above quantity, using , we get by Fatou’s lemma that for any $\xi\in \S^{d-1}$ and $\H^{d-1}$ almost every $y\in {\Omega}_\xi$ $$\int_{\Pi_\xi}\sum_{m_i\in(J_{u})^\xi_y}{\left}[1+\a|u^\xi_y(m_i^+)-u^\xi_y(m_i^-)| {\right}]\mathrm{d}\H^{d-1}(y)\leq \liminf_{{\varepsilon}{\downarrow}0} \LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O).$$ Therefore in force of Theorem \[teo:braides\] we have $u\in SBV(O)$. Moreover, since $(u')^\xi_y=0$ on each slice, we have $u\in \P(O)$. Applying identity we get $$\label{eq:liminfxi}
\liminf_{{\varepsilon}{\rightarrow}0}\LF_{\varepsilon}(u_{\varepsilon}, \phi_{\varepsilon};O) \geq \int_{J_u\cap O}|\nu_u\cdot \xi|{\left}[1+\a|[u]|{\right}]\dH^{d-1}.$$ In order to conclude, we use the following localization method stated by Braides in [@Bra1 Prop. 1.16].
\[lem:braides\] Let $\mu:\A(X){\rightarrow}[0,+{\infty})$ be a superadditive set function and let $\lambda$ be a positive measure on $X$. For any $i\in{\mathbf{N}}$ let $\psi_i$ be a Borel function on $X$ such that $\mu(A)\geq\int_A \psi_i\dif \lambda$ for all $A\in\A(X)$. Then $$\mu(A)\geq\int_A\psi\dif\lambda$$ where $\psi:=\sup_i\psi_i$.
We introduce the superadditive increasing set function $\mu$ defined on $\A(O)$ by $$\mu(A):=\Gamma-\liminf_{{\varepsilon}{\rightarrow}0} \LF_{\varepsilon}(u;)),\quad\text{for any }A\in\A(O)$$ and we let $\lambda$ be a Radon measure defined as $$\lambda:=[1+\a|u(x^+)-u(x^-)|]\H^{d-1}{ \!\!\mbox{{ \Large$\llcorner$}} }J_u.$$ Fix a sequence $(\xi_i)_{i\in{\mathbf{N}}}$ dense in $\S^{d-1}$. By we have $$\mu(O)\geq \int_O \psi_i\dif \lambda, \quad i\in {\mathbf{N}},$$ where $$\psi_i(x):=
\begin{dcases}
|\langle\nu_u(x),\xi_i\rangle|&\text{if } x\in J_u,\\
0&\text{if } x\in O\setminus J_u.
\end{dcases}$$ Hence by Lemma \[lem:braides\] we finally obtain $$\liminf_{{\varepsilon}{\rightarrow}0}\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O)\geq \int_O \sup_i\psi_i(x)\dif \mu =\int_{J_u\cap O}{\left}[1+\a|[u]|{\right}]\dH^{d-1}.$$
Equicoercivity and $\Gamma$-liminf {#sec:Comp}
==================================
We first prove the compactness property stated in the introduction. Let us consider a sequence $(\s_{\varepsilon},\phi_{\varepsilon})\in\M({\Omega},{\mathbf{R}}^2)$ uniformly bounded in energy by $c_0<+{\infty}$, $$\label{eq:unifbound}
0\leq{\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon})\leq c_0\qquad\mbox{for } {\varepsilon}\in(0,1].$$
First observe that by definition and equation , we have $\s_{\varepsilon}\in V_{\varepsilon}({\Omega})$ and $\phi_{\varepsilon}\in W_{\varepsilon}({\Omega})$.
Next, substituting $|\s_{\varepsilon}|$ for $|{\nabla}u_{\varepsilon}|$ in the argument of *Step 1.* of the proof of Theorem \[teo:localResult\], inequality reads $$ |\s_{\varepsilon}|({\Omega})\leq\sqrt{16\frac{{\varepsilon}^2}{\eta^2}\:c_0^2 + 8{\varepsilon}|{\Omega}| c_0} \,\stackrel{{\varepsilon}{\downarrow}0}\longrightarrow\, \frac{4 c_0}{\a}\,<\,\infty.$$ Thus the total variation of $(\s_{\varepsilon})$ is uniformly bounded and there exists $\s\in \M_S({\overline}{{\Omega}})$ such that up to extraction $\s_{\varepsilon}\to\s$ weakly-$*$ in $\M({\overline}{\Omega})$.
Now, considering the last term in the energy we have $$\int_{\Omega}(1-\phi_{\varepsilon})^2 \dx\leq 2{\varepsilon}\;{\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon})\leq 2{\varepsilon}\;c_0 \rightarrow 0.$$ Hence, $\phi_{\varepsilon}\to1$ in $L^2({\Omega})$.
Let us now study the structure of the limit measure $\s$. Let us recall that $\hat{\Omega}$ is a bounded convex open set such that ${\overline}{\Omega}\subset\hat{\Omega}$ and let us extend $\s_{\varepsilon}$ by 0 and $\phi_{\varepsilon}$ by 1 in $\hat{\Omega}\setminus{\overline}{\Omega}$. Obviously we have ${\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon};\hat{\Omega})={\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon};{\Omega})$, therefore for any $O\in\A_s(\hat{\Omega})$ applying the localization described in Section \[sec:LocRes\] we can associate to each $\s_{\varepsilon}$ a function $ u_{\varepsilon}\in H^1(O)$ with mean value $0$ such that $\s_{\varepsilon}={\nabla}^\perp u_{\varepsilon}$ in $O$. By Theorem \[teo:localResult\] there exists $u\in\P(O)$ such that up to extraction $u_{\varepsilon}\rwstar u$. Eventually, by uniqueness of the limit, we get $$\s{ \!\!\mbox{{ \Large$\llcorner$}} }O=-[u]\nu_{J_u}^\perp\H^1{ \!\!\mbox{{ \Large$\llcorner$}} }(J_u\cap O).$$ Since we can cover ${\overline}{\Omega}\setminus S$ by finitely many sets $O\in \A_s(\hat{\Omega})$, this shows that $\s$ decomposes as $$\s\, =\, U(M_\s,\t_\s,\xi_\s)+\underbrace{\sum_{j=0}^N c_j \delta_{x_j}}_{\mu}.$$ By Lemma \[lemma:completediv\] there exists a rectifiable measure $\g=U(M_\g,\t_\g,\xi_\g)$ such that ${\nabla}\cdot(\s+\g)=0$ and $\H^1(M_\g\cap M_\s )=0$. Then there exists $u\in BV({\Omega})$ such that $Du =\s^{\perp}+\g^{\perp}$. From , we deduce $|Du|(S)=0$ which implies $|\mu|(S)=\sum|c_j|=0$. Hence $c_j=0$ for $j=0,\dots,N$ and $\s$ writes in the form $U(M_\s,\t_\s,\xi_\s)$.
Let us now use the local results of Section \[sec:LocRes\] to prove the lower bound.
Let $(\s_{{\varepsilon}},\phi_{{\varepsilon}})$ as in the statement of the theorem. Without loss of generality, we can suppose that ${\mathcal{F}}_{{\varepsilon}}(\s_{{\varepsilon}},\phi_{{\varepsilon}})<+{\infty}$. Theorem \[teo:sigma\_equicoercive\] then ensures the existence of a rectifiable measure $\s=U(M_\s,\t_\s,\xi_\s)$ with $\operatorname{supp}(\s)\subset\overline{\Omega}$ such that $\s_{{\varepsilon}}\rwstar\s$.\
Let $\hat{\Omega}$ be as in the previous proof and let us define $\mu=\Gamma-\liminf_{{\varepsilon}}{\mathcal{F}}_{{\varepsilon}}(\s_{{\varepsilon}},\phi_{{\varepsilon}})$ and $\lambda= \a|\s|+\H^{1}{ \!\!\mbox{{ \Large$\llcorner$}} }M_\s$. Consider the countable family of sets $\{O_{i}\}\subset\A_{S}(\hat{\Omega})$ made of the open rectangles $O_i\subset \hat{\Omega}\setminus S$ with vertices in ${\mathbf{Q}}^{2}$ and let $\psi_{i}:=1_{O_{i}}$. The local result stated in Theorem \[teo:localResult\] gives for any $i\in{\mathbf{N}}$ $$\mu(A)\geq\mu(O_{i}\cap A)\geq\lambda(O_{i}\cap A)=\int_{A}\psi_{i}\dif \lambda.$$ Therefore Lemma \[lem:braides\] gives $$\begin{aligned}
\Gamma-\liminf_{{\varepsilon}{\downarrow}0}{\mathcal{F}}_{{\varepsilon}}(\s_{{\varepsilon}},\phi_{{\varepsilon}})&=\mu(\hat{\Omega})\geq\lambda(\hat{\Omega})=\a|\s|(\overline{\Omega})+\H^{1}(M_\s) \end{aligned}$$ since $\sup_{i}\psi_{i}$ is the constant function $1$.
Upper bound {#sec:UppBound}
===========
A density result
----------------
In order to obtain the upper bound we first provide a density lemma. We show that measures which have support contained in a finite union of segments, are dense in energy.\
Without loss of generality let us assume that $\s\in \M_S({\overline}{\Omega})$ is such that ${\mathcal{E}}_\alpha(\s)<\infty$. In particular $\s=U(M_\s,\t_\s,\xi_\s)$ is a $\H^1$-rectifiable measure. Applying Lemma \[lemma:completediv\] we obtain a $\H^1$-rectifiable measure $\g=U(M_\g,\t_\g,\xi_\g)$ and a partition of ${\Omega}$ made of polyhedrons $\{{\Omega}_i\}$ such that $M_\g\subset \cup_i\de{\Omega}_i$, $\H^1(M_\s \cap\cup_i\de{\Omega}_i)=0$ and $\s+\g$ is divergence free.\
From the above properties, we can write $$\s^\perp+\g^\perp=Du$$ for some $u\in\P({\Omega})$. Our strategy is the following, using existing results [@Bel_Cham_Gold], we build an approximating sequence for $u$ on each ${\Omega}_j$ whose gradient is supported on a finite union of segments. We then glue these approximations together to obtain a sequence $(w_j)$ approximating $u$ in $\hat{\Omega}$. The main difficulty is to establish that $Dw_j { \!\!\mbox{{ \Large$\llcorner$}} }[\cup_i\de{\Omega}_i]$ is close to $Du{ \!\!\mbox{{ \Large$\llcorner$}} }[\cup_i\de{\Omega}_i]=\g^\perp$.
\[lem:uapproximation\] There exists a sequence $(w_j)\subset\P(\hat{\Omega})$ with the following properties:
1. $w_j\to u$ weakly in $BV(\hat{\Omega})$,
2. $\operatorname{supp}w_j\subset {\overline}{\Omega}$,
3. $\limsup_{j{\rightarrow}{\infty}}{\mathcal{E}}_\a(w_j,1)\leq{\mathcal{E}}_\a(u,1)$,
4. $J_{w_j}$ is contained in a finite union of segments for any $j\in {\mathbf{N}}$,
5. $|Dw_j-Du|(\cup \de{\Omega}_i)\to0$.
*Step 1.* In order to apply the results of [@Bel_Cham_Gold], we first need to modify $u$ and the energy. Let us note the energy density function $f(t)=1+\a t$ and for $k\geq0$ and $t\geq 0$ let us introduce the approximation $$\label{eq:definizionefk}
f_k(t):=\min\{(2^{k/2}+\a 2^{-k/2})\sqrt{t},f(t)\}.$$
![Graph of f and two of its approximations $f_{k_1} $ and $f_{k_2}$ with $k_1<k_2$.[]{data-label="fig:approssimazionef"}](approssimazionef){width=".3\textwidth"}
We have $0\leq f_k\leq f$ and $f_k\equiv f$ on $[2^{-k},+\infty)$. Notice that $f_k$ is continuous, sub-additive and increasing on $[0,+{\infty})$ and that $f_k(0)=0$ with $\lim_{t{\rightarrow}0} \frac{f_k(t)}{t}=+{\infty}$. We define the associated energy for functions $v\in\P(\hat{\Omega})$ as ${\mathcal{E}}_{f_k}(v,\hat {\Omega}):=\int_{J_{v}\cap\hat{\Omega}}f_k([v])\dH^1$.\
Now we note $\P_k(\hat{\Omega})$ the set of functions $v\in \P(\hat{\Omega})$ such that $v(\hat{\Omega})\subset 2^{-k}{\mathbf{Z}}$. For these functions we have $|v^+(x)-v^-(x)|\geq 2^{-k}$ for $\H^1$-almost every $x\in J_v$. Consequently, there holds $${\mathcal{E}}_{f_k}(v)\, =\, {\mathcal{E}}_f(v).$$ For each fixed $k\geq0 $, let us introduce the function $$u_k=2^{-k}\lfloor2^k u\rfloor$$ where $\lfloor t\rfloor$ denotes the integer part of the real $t$. Note that $u_k\in \P_k(\hat{\Omega})$ with $J_{u_k}\subset J_u$ and $\|u-u_k\|_\infty\leq 2^{-k}$. Notice also since ${\left}|(u_k^+-u_k^-)-(u^+-u^-){\right}|\leq 2^{-k}$ we have $$\label{eq:diffuuk}
|Du_k-Du|(\hat{\Omega})\, \leq\, 2^{-k}\H^1(J_u)$$ In particular $u_k\to u$ strongly in $BV(\hat{\Omega})$. Moreover, we see that $$\label{eq:Ef=Efk}
{\mathcal{E}}_{f_k}(u_k) \, =\, {\mathcal{E}}_f(u_k) \, \leq\ \, {\mathcal{E}}_f(u) + \a 2^{-k}\H^1(J_u).$$
*Step 2.* Let us approximate the function $u_k$. Let us fix $k\geq 0$ and ${\Omega}_i$. We can apply Lemma 4.1 of [@Bel_Cham_Gold] to the function $u_k{ \!\!\mbox{{ \Large$\llcorner$}} }{{\Omega}_i}$ and to the energy ${\mathcal{E}}_{f_k}(\cdot,{\Omega}_i)$. We obtain a sequence $(w^i_j)$ which enjoys the following properties: $$\label{eq:properties}
\begin{aligned}
& w^i_j({\Omega}_i)\subset u_k({\Omega}_i)\subset 2^{-k}{\mathbf{Z}},\quad \forall j\in{\mathbf{N}}, \mbox{ hence }w^i_j\in \P_k(\hat{\Omega}), \\
& w^j_i\to u_k \mbox{ in } L^1({\Omega}_i)\mbox{ as }j{\rightarrow}+{\infty}, \\
& \lim_{j{\rightarrow}+{\infty}} {\mathcal{E}}_{f_k}(w^i_j,{\Omega}_i)= \lim_{j{\rightarrow}+{\infty}} {\mathcal{E}}_{f}(w^i_j,{\Omega}_i)={\mathcal{E}}_{f}(u_k,{\Omega}_i),\\
& J_{w^j_i} \mbox{ is contained in a finite union of segments for any } j\in {\mathbf{N}},\\
& \int_{\de {\Omega}_i}|Tw^i_j-Tu_k|\dH^1{\rightarrow}0\mbox{ where } T:BV({\Omega}_i){\rightarrow}L^1(\de {\Omega}_i) \mbox{ denotes the trace operator}.\\
\end{aligned}$$ Let us now define globally $$w_j:=\sum_j w_j^i 1_{{\Omega}_i}.$$ From the above properties, we have $w_j\rwstar u_k$, $$\label{eq:convEfw}
\lim{\mathcal{E}}_{f}(w^i_j,\hat {\Omega})={\mathcal{E}}_{f}(u_k,{\Omega}_i)$$ and $$\label{eq:diffukwj}
|D w_j - D u_k|(\cup_i\de{\Omega}_i)\, \to\, 0\quad\mbox{as $j\to\infty$}.$$ Eventually, using a diagonal argument, we have proved the existence of a sequence $(w_j)\subset \P(\hat{\Omega})$ complying to items *(a)*, *(b)* and *(d)* of the lemma. Moreover, item *(c)* is the consequence of and and item *(e)* follows from and .
Going back to the $\H^1$-rectifiable measures $\s=U(M_\s,\t_\s,\xi_\s)$, we define the sequence $$\s_j:=-Dw^\perp_i-\g.$$ We recall that $\g=U(M_\g,\t_\g,\xi_\g)$ with $M_\g\subset \cup \de {\Omega}_i$. In particular $\g=-Du^\perp{ \!\!\mbox{{ \Large$\llcorner$}} }(\cup_i \de{\Omega}_i)$. We deduce from the previous lemma:
\[lem:density\] There exists a sequence $(\s_j)\in\M_S({\overline}{\Omega})$ with the properties:
1. $\s_j{\rightarrow}\s$ with respect to weak-$*$ convergence of measures,
2. $\s_j= U(M_{\s_j},\t_{\s_j},\xi_{\s_j})$ with $M_{\s_j}$ contained in a finite union of segments,
3. $\limsup_{j{\rightarrow}{\infty}}{\mathcal{E}}_\a(\s_j,1)\leq{\mathcal{E}}_\a(\s,1)$.
Construction of a recovery sequence
-----------------------------------
Let us prove the $\Gamma$-limsup inequality stated in Theorem \[teo:sigma\_limsup\]. Recall that the latter consists in finding a sequence $(\s\e,\phi\e)$ for any given couple $(\s,\phi)\in\M(\overline{\Omega},{\mathbf{R}}^2)\times L^1({\Omega})$ such that $\s\e\rwstar\s$, $\phi\e{\rightarrow}\phi$ in $L^1({\Omega})$ and $$\label{eq:limsup}
\limsup_{{\varepsilon}{\downarrow}0}{\mathcal{F}}\e(\s\e,\phi\e)\leq{\mathcal{E}}_\a(\s,\phi).$$ When ${\mathcal{E}}_\a(\s,\phi)=+\infty$ the inequality is valid for any sequence therefore by definition we can assume $\s=U(M,\t,\xi)$ and $\phi=1$. Furthermore by density Lemma \[lem:density\] is sufficient to consider measures of the form $$\label{eq:strutturadense}
\s=\sum_{i=1}^n U(M_i,\t_i,\xi_i),$$ where $M_i$ is a segment, $\t_i\in{\mathbf{R}}_+$ is $\H^1$-a.e. constant and $\xi_i$ is an orientation of $M_i$ for each $i$. Without loss of generality we can suppose that for each couple of segments $M_i$, $M_j$, for $i \neq j$, the intersection $M_i\cap M_j$ is at most a point (called branching point) not belonging to the relative interior of $M_i$ and $M_j$. We firstly produce the estimate for $\s$ composed by a single segment thus let us assume $\s=\t e_1\cdot \H^1{ \!\!\mbox{{ \Large$\llcorner$}} }(0,l)\times\{0\}$.
*Notation:* Let us fix the values $$\label{eq:infinitesimi}
a_{\varepsilon}:=\begin{dcases}
\frac{\t\a\,{\varepsilon}}{2} &\mbox{ if } \a>0\\
\quad{\varepsilon}& \mbox{ if } \a=0
\end{dcases},
\qquad\qquad
b_{\varepsilon}:={\varepsilon}\ln{\left}(\frac{1-\eta}{{\varepsilon}}{\right})
\qquad\mbox{ and }\qquad
r\e=\max\{{\varepsilon},a\e\}.$$ Let $d_\infty(x,S)$ be the distance function from $x$ to the set $S\subset {\Omega}$ relative to the infinity norm on ${\mathbf{R}}^2$ and $Q_r(P)=\{x\in{\mathbf{R}}^2: d_\infty(x,P)\leq r\}$ the square centered in $P$ of size $2r$ and sides parallel to the axes. Introduce the sets $$\begin{aligned}
\label{def:rectangular}
I_{a\e}&:=\{x\in{\mathbf{R}}^2: d_\infty(x,[0,l]\times\{0\})\leq a\e\}\cup Q_{r\e}(0,0)\cup Q_{r\e}(l,0),\\
I_{b\e}&:=\{x\in{\mathbf{R}}^2: d_\infty(x,I_{a\e})\leq b\e\},\\
I_{c\e}&:=\{x\in{\mathbf{R}}^2: d_\infty(x,(I_{a\e}\cup I_{b\e}))\leq {\varepsilon}\},\\
I_{d\e}&:={\Omega}\setminus(I_{a\e}\cup I_{b\e}\cup I_{c\e}),\end{aligned}$$ and define $R\e= I_{a\e}\setminus(Q_{r\e}(0,0)\cup Q_{r\e}(l,0))$.
![Example of the neighborhoods of the segment $[0,l]\times\{0\}$. On the left the case $r\e={\varepsilon}$ on the right the case in which $r\e=a\e>{\varepsilon}$. The stripped region is $R\e$ and $I_{a\e}=R\e\cup(Q_{r\e}(0,0)\cup Q_{r\e}(l,0))$. Remark that $\operatorname{supp}(\rho\e)=B(0,{\varepsilon})$.[]{data-label="fig:intorninuovo"}](nuovointorni){width="\textwidth"}
*Costruction of $\s\e$:* We build $\s\e$ as a vector field supported on $I_{a\e}$. In particular we add together three different constructions performed respectively on $R\e$, $Q_{r\e}(0,0)$ and $ Q_{r\e}(l,0)$. Let $r=r_{\varepsilon}/{\varepsilon}$ and consider the problem
$$\begin{dcases}
\Delta u=\pm\t\delta_{x_0}*\rho &\mbox{ on } Q_{r}(0,0),\\
\frac{\de u}{\de \nu}=\frac{\pm\t}{\H^1(\Sigma)} &\mbox{ on } \Sigma^\pm=\{x\in{\mathbf{R}}^2: x_1=\pm1,\;|x_2|\leq\frac{\t\a}{2}\}.
\end{dcases}$$
{width=".6\textwidth"}
Let $u^+$ be the solution relative to the problem in which every occurrence of $\pm$ is replaced by $+$ and let $u^-$ defined accordingly. Then set $$\label{def:se}
\s\e=
\begin{dcases}
\frac{{\nabla}u^+(x/{\varepsilon})}{{\varepsilon}} &\mbox{ on } Q_{r\e}(0,0),\\
\quad\frac{\t}{2a_{\varepsilon}}\cdot e_1 &\mbox{ on } R\e,\\
\frac{{\nabla}u^-((x-(l,0))/{\varepsilon})}{{\varepsilon}} &\mbox{ on } Q_{r\e}(l,0).
\end{dcases}$$ By construction we have that ${\nabla}\cdot\s\e=\t(\d_{(0,0)}-\d_{(l,0)})\ast\rho\e$ and $\s\e\rwstar\s$. Let us point out as well that there exists a constant $c(\a,\t)$ such that $$\label{eq:energiapunti}
c(\a,\t):=\int_{Q_{r\e}(l,0)}|\s\e|^2\dx=\int_{Q_{r\e}(0,0)}|\s\e|^2\dx=\int_{Q_{r}(0,0)}{\left}|{\nabla}u^+(x){\right}|^2\dx=\int_{Q_{r}(0,0)}{\left}|{\nabla}u^-(x){\right}|^2\dx.$$
*Costruction of $\phi\e$:* Most of the properties of $\phi\e$ are a consequence of the inequalities obtained in Theorem \[teo:localResult\] and the structure of $\s\e$. On one hand we need $\phi\e$ to attain the lowest value possible on $I_{a\e}$ in order to compensate the concentration of $\s\e$ in this set, on the other, as shown in inequality , we need to provide the optimal profile for the transition from this low value to $1$. For this reasons we are led to consider the following ordinary differential equation associated with the optimal transition $$\label{eq:ode}
\begin{dcases}
w'_{\varepsilon}= \frac{1}{{\varepsilon}}(1-w_{\varepsilon}), \\
w_{\varepsilon}(0)= \eta. \\
\end{dcases}$$ Observe that $w_{\varepsilon}=1-(1-\eta)\exp{\left}(\frac{-t}{{\varepsilon}}{\right})$ is the explicit solution of equation and set $$\label{def:phie}
\phi_{\varepsilon}(x):=\begin{dcases}
\eta&\mbox{ if } x\in I_{a_{\varepsilon}},\\
w_{\varepsilon}(d_\infty(x, I_{a\e})) &\mbox{ if } x\in I_{b_{\varepsilon}},\\
d_\infty(x, I_{b\e})-{\varepsilon}+1&\mbox{ if } x\in I_{c_{\varepsilon}},\\
1&\mbox{ otherwise.}
\end{dcases}$$ The choice of the behavior in the region $I_{c\e}$ is given by the fact that following the optimal profile we will reach the value $1$ only at $+\infty$ thus a linear correction on a small set ensures that this value is achieved with a small cost in energy.
*Evaluation of ${\mathcal{F}}\e(\s\e,\phi\e)$:* We prove inequality for the sequence we have produced. Since the sets $I_{a\e}$, $I_{b\e}$, $ I_{c\e}$ and $I_{d\e}$ are disjoint we can split the energy as follows $$\label{dislimsup0}
{\mathcal{F}}\e(\s_{\varepsilon},\phi\e)={\mathcal{F}}\e(\s_{\varepsilon},\phi\e;I_{a\e})+{\mathcal{F}}\e(\s_{\varepsilon},\phi\e;I_{b\e})+{\mathcal{F}}\e(\s_{\varepsilon},\phi\e;I_{c\e})+{\mathcal{F}}\e(\s_{\varepsilon},\phi\e;I_{d\e})$$ and evaluate each component individually. Since $\s\e$ is null and $\phi\e$ is constant and equal to $1$ in $I_{d\e}$ we have that ${\mathcal{F}}\e(\s,\phi\e;I_{d\e})=0$. For the other components we strongly use the definitions in and . Firstly we split again the energy on the set $I_{a\e}$ as following $${\mathcal{F}}\e(\s\e,\phi\e;I_{a\e})={\mathcal{F}}\e(\s\e,\phi\e;R\e)+{\mathcal{F}}\e(\s\e,\phi\e;Q_{r\e}(0,0))+{\mathcal{F}}\e(\s\e,\phi\e;Q_{r\e}(l,0)).$$ Now identity leads to the estimate $${\mathcal{F}}\e(\s\e,\phi\e;Q_{r\e}(0,0))={\mathcal{F}}\e(\s\e,\phi\e;Q_{r\e}(l,0))=\frac{\eta^2}{2{\varepsilon}}c(\a,\t)+\frac{(1-\eta)^2}{2{\varepsilon}}\;r\e^2$$ and $${\mathcal{F}}\e(\s\e,\phi\e;R\e)={\left}[\frac{1}{2{\varepsilon}}\eta^2 {\left}|\frac{\t}{2a\e}{\right}|^2+\frac{(1-\eta)^2}{2{\varepsilon}}{\right}]|R\e|\leq {\left}[\frac{(\t\eta)^2}{8{\varepsilon}a\e^2}+\frac{1}{2{\varepsilon}}{\right}]2a\e l.$$ Then passing to the limsup we obtain $$\label{dislimsup1}
\limsup_{{\varepsilon}{\downarrow}0}{\mathcal{F}}\e(\s\e,\phi\e;I_{a\e})\leq \t\a l= \t_{}\a\H^1([0,l]\times\{0\}).$$ To obtain the inequality on the sets $I_{b\e}$ and $I_{c\e}$ we are going to apply the Coarea formula therefore let us observe that for both $d_{\infty}(x, I_{a\e})$ and $d_{\infty}(x, I_{b\e})$ there holds $|{\nabla}d_{\infty}(x,\cdot)|=1$ for a.e. $x\in{\Omega}$ and that there exist a constant $k=k(\a,\t)$ such that the level lines $\{d_{\infty}(x,\cdot)=t\}$ have $\H^1$ length controlled by $2l+kt$. In force of these remarks we obtain $$\begin{aligned}
\label{dislimsup2}
{\mathcal{F}}\e(\s\e,\phi\e;I_{b\e})&=\int_{I_{b\e}} {\left}[\frac{{\varepsilon}}{2}|{\nabla}\phi_{\varepsilon}|^2 + \frac{(1-\phi_{\varepsilon})^2}{2{\varepsilon}} {\right}]|{\nabla}d_{\infty}(x,I_{a\e})|\dx\nonumber\\
&=\int_{0}^{b_{\varepsilon}}{\left}[\frac{(1-w_{\varepsilon}(t))^2}{2{\varepsilon}}+\frac{{\varepsilon}}{2}|w'_{\varepsilon}(t)|^2 {\right}] \H^1(\{d_\infty(\cdot, I_{a\e})=t\})\;\dt\nonumber\\
&\leq(2l+k{\varepsilon}){\left}[\frac{1}{2}(1-w_{\varepsilon}(t))^2{\right}]_{0}^{b_{\varepsilon}}\nonumber\\
&={\left}(l-\frac{k{\varepsilon}}{2}{\right}){\left}[(1-\eta)^2-{\varepsilon}^2{\right}]\xrightarrow [{\varepsilon}{\downarrow}0]{} l= \H^1([0,l]\times\{0\})\end{aligned}$$ and $$\begin{aligned}
\label{dislimsup3}
{\mathcal{F}}\e(\s\e,\phi\e;I_{c\e})&=\int_{I_{c\e}} {\left}[\frac{{\varepsilon}}{2}|{\nabla}\phi_{\varepsilon}|^2 + \frac{(1-\phi_{\varepsilon})^2}{2{\varepsilon}} {\right}]|{\nabla}d_{\infty}(x,I_{b\e})|\dx\nonumber\\
&=\int_{0}^{{\varepsilon}}{\left}[\frac{(1-t+{\varepsilon}-1)^2}{2{\varepsilon}}+\frac{{\varepsilon}}{2} {\right}] \H^1(\{d_\infty(\cdot, I_{b\e}\cup I_{a\e})=t\})\;\dt\nonumber\\
&\leq(2l+k{\varepsilon}){\left}[\frac{(t-{\varepsilon})^3}{6{\varepsilon}}+\frac{{\varepsilon}}{2} t{\right}]_{0}^{{\varepsilon}}\nonumber\\
&=(2l+k{\varepsilon})\,{\varepsilon}^2\,\frac{2}{3}\xrightarrow [{\varepsilon}{\downarrow}0]{} 0.\end{aligned}$$ Finally adding up equations , , and we obtain $$\limsup_{{\varepsilon}{\downarrow}0}{\mathcal{F}}\e(\s_{\varepsilon},\phi\e)\leq (1+\a\;\t)\; \H^1([0,l]\times\{0\}).$$
*Case $\s$ of the form :*
Let us call $\s\e^i$, $\phi\e^i$ the functions obtained above for each $\s_i=\t_i\xi_i \H^1{ \!\!\mbox{{ \Large$\llcorner$}} }M_i$ and set $$\s\e=\sum_{i=1}^n \s\e^i,\qquad\qquad \phi\e=\min_{i}\;\phi\e^i.$$ Let us remark that in force of the local construction we have made at the ending points of each segment and since $\s$ satisfies equation for each ${\varepsilon}$ there holds $${\nabla}\cdot\s_{\varepsilon}\, =\, {\left}(N\delta_{x_0} - \sum_{j=1}^N \delta_{x_j} {\right})\ast \rho_{\varepsilon}.$$ We now prove inequality . The following inequality holds true $$\begin{aligned}
{\mathcal{F}}\e(\s\e,\phi\e)&=\int_{\Omega}\frac{1}{2{\varepsilon}}|\min_i \phi\e^i|^2|\sum_{i=1}^n \s^i\e|^2 +\frac{{\varepsilon}}{2}|{\nabla}(\min_i \phi\e^i)|^2 + \frac{(1-\min_i \phi\e^i)^2}{2{\varepsilon}}\dx\nonumber\\
&\leq\int_{\Omega}\frac{1}{2{\varepsilon}}|\min_i \phi\e^i|^2|\sum_{i=1}^n \s^i\e|^2 \dx+\sum_{i=1}^n\int_{\Omega}\frac{{\varepsilon}}{2}|{\nabla}\phi\e^i|^2 + \frac{(1-\phi\e^i)^2}{2{\varepsilon}}\dx,\label{dis:decomposizione}\end{aligned}$$ therefore we look into an estimation of the first integral in the latter. Observe that for ${\varepsilon}$ sufficiently small we can assume that all the $R\e^i$ are pairwise disjoint thus we study the behavior in the squares. Let $M_{i_1},\dots,M_{i_{m_P}}$ be the segments meeting at a branching point $P$. For $j=i_1,\dots,i_{m_P}$ let us call $Q_{r\e^j}(P)$ the squared neighborhood of $P$ relative to the segment $M_j$ as constructed previously. Let us recall that by definition $\phi\e$ is constant and equal to $\eta$ on $\cup_{j=i_1}^{m_P} Q_{r\e^j}(P)$ then we have the estimation $$\begin{aligned}
\int\limits_{\cup_{j=i_1}^{m_P} (R\e^j\cup Q_{r\e^j}(P))}\frac{\phi^2\e}{2{\varepsilon}}\,|\s\e|^2\dx&=\sum_{j=i_1}^{m_P}\int_{R\e^j}\frac{\phi^2\e}{2{\varepsilon}}\,|\s\e|^2\dx+\int_{\cup_{i=i_1}^{m_P} Q_{r\e^j}(P)}\frac{\phi^2\e}{2{\varepsilon}}\,|\sum_{j=i_1}^{m_P}\s^j\e|^2\dx\nonumber\\
&\leq \sum_{j=i_1}^{m_P}\int_{R\e^j}\frac{\phi^2\e}{2{\varepsilon}}\,|\s\e|^2\dx+m_P\,\frac{\eta^2}{2{\varepsilon}}\sum_{j=i_1}^{m_P}\int_{Q_{r\e^j}(P)}|\s^j\e|^2\dx\nonumber\\
&\leq \sum_{j=i_1}^{m_P}\int\limits_{(R\e^j\cup Q_{r\e^j}(P))}\frac{1}{2{\varepsilon}}|\phi\e^j|^2|\s^j\e|^2\dx+\underbrace{(m_P-1)\,{\left}(\sum_{j=i_1}^{i_{m_P}}{c(\a,\t_j)}{\right})\frac{\eta^2}{2{\varepsilon}}}_{c(m_P,\a,\t_{i_1},\dots,\t_{i_{m_P}}){\varepsilon}}.\label{dis:punti}\end{aligned}$$ Applying inequality on each branching point in equation and recomposing the integral gives $$\begin{aligned}
\limsup_{{\varepsilon}{\downarrow}0}{\mathcal{F}}\e(\s\e,\phi\e)&\leq\limsup_{{\varepsilon}{\downarrow}0}\sum_{i=1}^n{\mathcal{F}}\e(\s^i\e,\phi^i\e) +n\; c(n,\a,\t_i,\dots,\t_n){\varepsilon}\\
&\leq \sum_{i=1}^n(1+\a\;\t_i)\; \H^1(M_i)\\
&=\int_{\operatorname{supp}(\s)}(1+\a\,\t)\dH^1 ={\mathcal{E}}_\a(\s,1)\end{aligned}$$ which ends the proof.
Numerical Approximation {#sec:NumApprox}
=======================
Equations
---------
In this section we present some numerical simulations of the $\Gamma$-convergence result we have shown. The first issue we address is how to impose the divergence constraint. To this aim is convenient to introduce the following notation $$\begin{aligned}
f\e&= {\left}(N\delta_{x_0} - \sum_{j=1}^N \delta_{x_j} {\right})\ast \rho_{\varepsilon},\\
G\e(\s,\phi)&=\begin{dcases}
\int_{{\Omega}} {\left}[\frac{1}{2{\varepsilon}}|\phi|^2 |\s|^2 {\right}]\dx& \text{if } \s\in V\e,\\
+{\infty}&\mbox{otherwise in } L^2({\Omega},{\mathbf{R}}^2),
\end{dcases}\\
\Lambda\e(\phi)&=\begin{dcases}
\int_{{\Omega}}{\left}[ \frac{{\varepsilon}}{2}|{\nabla}\phi|^2 +\frac{(1-\phi^2)}{2{\varepsilon}} {\right}]\dx& \text{if } \phi\in W\e,\\
+{\infty}&\mbox{otherwise in } L^1({\Omega}).
\end{dcases}\end{aligned}$$ Then let us observe that the following equality holds $$\min_{\s\in L^2({\Omega},{\mathbf{R}}^2)} G\e(\s,\phi)= \inf_{\s\in L^2({\Omega},{\mathbf{R}}^2)}{\left}\{ \sup_{u\in H^1({\Omega})}\int_{{\Omega}} \frac{1}{2{\varepsilon}}|\phi|^2 |\s|^2 +u({\nabla}\cdot\s-f\e)\dx{\right}\}.$$ By Von Neumann’s min-max Theorem [@Att_Hed_Butt Thm. 9.7.1] we can exchange inf and sup obtaining for each ${\varepsilon}>0$ and $\phi \in W\e$ $$\begin{aligned}
\min_{\s} G\e(\s,\phi)&=\sup_{u}\inf_{\s}\int_{{\Omega}} \frac{1}{2{\varepsilon}}|\phi|^2 |\s|^2 -({\nabla}u\s+u f\e)\dx\\
&=-\min_{u}\int_{{\Omega}} \frac{{\varepsilon}|{\nabla}u|^2}{2|\phi|^2} +uf\e\dx=-\min_u G'\e(u,\phi).\end{aligned}$$ With the relation $\s=\frac{{\varepsilon}{\nabla}u}{\phi^2}$. This naturally leads to the following alternate minimization problem: given an initial guess $\phi_0$ we define $$\begin{aligned}
\s_j&:=\frac{{\varepsilon}{\nabla}u_j}{\phi_j^2} \quad\mbox{ where } \quad u_j:=\operatorname{argmin}G'\e(u,\phi_j), \\
\phi_{j+1}&:=\operatorname{argmin}G\e(\s_j,\phi)+\Lambda\e(\phi).\end{aligned}$$ We supplement the alternate minimization with a third step where we optimize the component $\Lambda\e$ with respect to a deformation of the domain. Let us describe this step. For $T:{\Omega}{\rightarrow}{\Omega}$ a smooth map we define $$\phi_T=\phi\circ T(x) \quad\mbox{and}\quad\Lambda\e(T)=\Lambda\e(\phi_T).$$ By a change of variables we get $$\Lambda\e(T)=\int_{\Omega}{\left}[\frac{{\varepsilon}}{2}|({\nabla}T\circ T^{-1}){\nabla}\phi|^2 +\frac{(1-\phi^2)}{2{\varepsilon}}{\right}]\det({\nabla}T^{-1})\dy$$ In particular we choose $T$ to be of the form $x+V(x)$ and evaluate the gradient obtaining $$\langle d\Lambda\e(T),W\rangle=\int_{{\Omega}}{\left}[{\varepsilon}({\nabla}\phi_T;{\nabla}W{\nabla}\phi_T)-\frac{{\varepsilon}}{2}|{\nabla}\phi_T|^2{\nabla}\cdot W-\frac{1}{2{\varepsilon}}(1-\phi_T)^2{\nabla}\cdot W{\right}]\dx$$ Representing in $H^1({\Omega},{\Omega})$ the gradient of the functional $\Lambda\e$ evaluated for $T(x)=x$ obtains the elliptic problem $$\int_{\Omega}{\left}({\nabla}V,{\nabla}W{\right})\dx+\int_{{\Omega}}{\left}[{\varepsilon}({\nabla}\phi;{\nabla}W{\nabla}\phi)-\frac{{\varepsilon}}{2}|{\nabla}\phi|^2{\nabla}\cdot W-\frac{1}{2{\varepsilon}}(1-\phi)^2{\nabla}\cdot W{\right}]\dx=0$$ This method enhances the length minimization process since, as we already pointed out, $\Lambda\e$ is a variation of Modica-Mortola’s functional.
Discretization
--------------
We define a circular domain ${\Omega}$ containing the points in $S$ endowed with a uniform mesh and four values $\a$, ${\varepsilon}_{in}, {\varepsilon}_{end}$ and $N_{iter}$ and a gaussian convolution kernel $\rho_{{\varepsilon}_{end}}$ in order to define $f\e$. For the discrete spaces we have chosen for $u$, $\phi$ and the vector field $V$ to be piecewise polynomials of order $1$. This leads to the following algorithm
$S=\{x_0,\ldots,x_N\}$, ${\varepsilon}_{in},\quad{\varepsilon}_{end}$,$N_{iter}$,$\a$, index. Set $f\e=(N\d_{x_0}-\sum_{i=1}^N \d_{x_i})*\rho_{{\varepsilon}_{end}}$ and $\phi_0=1$ ${\varepsilon}_j= {\left}(\frac{j-N_{iter}}{N_{iter}}{\right}){\varepsilon}_{in}-{\left}(\frac{j}{N_{iter}}{\right}){\varepsilon}_{end}$ $\tilde\phi\leftarrow L^1$-projection of $\phi_{j-1}^2$ Set $u_j$ as the minimizer of $G'_{{\varepsilon}_j}(\cdot,\phi_{j-1})$ Set $\s_j=\frac{{\varepsilon}_j {\nabla}u_j}{\tilde\phi_{j-1}}$ Set $\phi_j$ as the minimizer of $G_{{\varepsilon}_j}(\s_j,\cdot)+\Lambda\e(\cdot)$ Solve $\langle d\Lambda_{{\varepsilon}_j}(T),W\rangle=0$ Set $\phi_j=\phi_j(x+T)$ Set $\phi_j=\max\{\eta,\phi_j\}$ $\phi_{N_{iter}}$, $\s_{N_{iter}}$.
We have implemented the algorithm in FREEFEM++. In the next figures we show the graphs obtained for the couple $(\s_{N_{iter}},\phi_{N_{iter}})$ via the approximation algorithm with the choices $\a=0.05$, ${\varepsilon}_{in}=0.5$, ${\varepsilon}_{end}=0.05$, $\a=0.05$, $N_{iter} = 500$ and $index = 300$. We have chosen to make simulations for points located on the vertices of regular polygons of respectively 3, 4, 5 and 6 vertices. This choice allows a direct visual perception of the results.
![Graph of the couple $(\s_{N_{iter}},\phi_{N_{iter}})$ obtained via Algorithm \[alg:1\] in the case of 3, 4, 5 and 6 points located on the vertices of a regular polygon.[]{data-label="fig:simulazione1"}](3puntiFA.png "fig:"){width="3cm"}![Graph of the couple $(\s_{N_{iter}},\phi_{N_{iter}})$ obtained via Algorithm \[alg:1\] in the case of 3, 4, 5 and 6 points located on the vertices of a regular polygon.[]{data-label="fig:simulazione1"}](4puntiFA.png "fig:"){width="3cm"}![Graph of the couple $(\s_{N_{iter}},\phi_{N_{iter}})$ obtained via Algorithm \[alg:1\] in the case of 3, 4, 5 and 6 points located on the vertices of a regular polygon.[]{data-label="fig:simulazione1"}](5puntiFA "fig:"){width="3cm"}![Graph of the couple $(\s_{N_{iter}},\phi_{N_{iter}})$ obtained via Algorithm \[alg:1\] in the case of 3, 4, 5 and 6 points located on the vertices of a regular polygon.[]{data-label="fig:simulazione1"}](6puntiFA "fig:"){width="3cm"} ![Graph of the couple $(\s_{N_{iter}},\phi_{N_{iter}})$ obtained via Algorithm \[alg:1\] in the case of 3, 4, 5 and 6 points located on the vertices of a regular polygon.[]{data-label="fig:simulazione1"}](3puntiA "fig:"){width="3cm"}![Graph of the couple $(\s_{N_{iter}},\phi_{N_{iter}})$ obtained via Algorithm \[alg:1\] in the case of 3, 4, 5 and 6 points located on the vertices of a regular polygon.[]{data-label="fig:simulazione1"}](4puntiA "fig:"){width="3cm"}![Graph of the couple $(\s_{N_{iter}},\phi_{N_{iter}})$ obtained via Algorithm \[alg:1\] in the case of 3, 4, 5 and 6 points located on the vertices of a regular polygon.[]{data-label="fig:simulazione1"}](5puntiA "fig:"){width="3cm"}![Graph of the couple $(\s_{N_{iter}},\phi_{N_{iter}})$ obtained via Algorithm \[alg:1\] in the case of 3, 4, 5 and 6 points located on the vertices of a regular polygon.[]{data-label="fig:simulazione1"}](6puntiA "fig:"){width="3cm"}
Finally let us point out the need of the third minimization step. In the following figure we have the graph of the solution obtained for a simulation in which the third step is omitted. Even from visual perception is possible to recognize that the solution differs both from the solution of the Steiner Tree and the minimizer of the ${\mathcal{E}}_\a$ energy as evident from the figure. Furthermore we do not obtain the classical straight segments we would expect in studying geodesic in the euclidean metric. We suppose that these alterations are a consequence of the alternate minimization method that could not lead to a global minimum and therefore we introduced the third step in the algorithm in order to perturbate local solutions.
![On the left: Graph of $\phi$ obtained via Algorithm \[alg:1\] in which the gradient descend method is omitted. On the right: in red, one of the solutions to the Steiner problem for four points on the vertices of a square, while in blue, a minimizer of the energy ${\mathcal{E}}_\a$ associated to the same constraint.[]{data-label="fig:4sbagliato"}](4sbagliato "fig:"){width="3.5cm"}![On the left: Graph of $\phi$ obtained via Algorithm \[alg:1\] in which the gradient descend method is omitted. On the right: in red, one of the solutions to the Steiner problem for four points on the vertices of a square, while in blue, a minimizer of the energy ${\mathcal{E}}_\a$ associated to the same constraint.[]{data-label="fig:4sbagliato"}](soluzionisteinerapprossimate "fig:"){width="3cm"}
To ensure that this step is reasonable we have studied several experiments and plotted the numerical energy of each experiment and observed that we are always led to a lower energy. The following plot shows the behavior of the energy for the iterations concerning the third step for the first two solutions in figure \[fig:simulazione1\]. Is possible to observe that although there are increments
![Behavior of the estimated energy of the last 200 iterations of Algorithm \[alg:1\] referring to the first two figures in figure \[fig:simulazione1\] .[]{data-label="fig:grafici_energia"}](grafici_energia){width=".6\textwidth"}
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors have been supported by the ANR project Geometrya, Grant No. ANR-12-BS01-0014-01. A.C. also acknowledges the hospitality of Churchill College and DAMTP, U. Cambridge, with a support of the French Embassy in the UK, and a support of the Cantab Capital Institute for Mathematics of Information.
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abstract: 'Recent experimental progress with Alkaline-Earth atoms has opened the door to quantum computing schemes in which qubits are encoded in long-lived nuclear spin states, and the metastable electronic states of these species are used for manipulation and readout of the qubits. Here we discuss a variant of these schemes, in which gate operations are performed in *nuclear-spin-dependent* optical lattices, formed by near-resonant coupling to the metastable excited state. This provides an alternative to a previous scheme \[A. J. Daley, M. M. Boyd, J. Ye, and P. Zoller, Phys. Rev. Lett **101**, 170504 (2008)\], which involved independent lattices for different *electronic* states. As in the previous case, we show how existing ideas for quantum computing with Alkali atoms such as entanglement via controlled collisions can be freed from important technical restrictions. We also provide additional details on the use of collisional losses from metastable states to perform gate operations via a lossy blockade mechanism.'
author:
- 'Andrew J. Daley'
- Jun Ye
- Peter Zoller
title: 'State-dependent lattices for quantum computing with alkaline-earth-metal atoms'
---
Introduction
============
There has been a lot of recent experimental progress in cooling and manipulating alkaline-earth and alkaline-earth-like atoms in the laboratory, especially in the context of optical clocks with Strontium Atoms [@AEatom1; @zeemanshift1; @Boyd07; @ludlow08; @Boydthesis], and the production of Bose-Einstein condensates and degenerate Fermi gases of Ytterbium [@takasu03; @takasu07; @Takahashi], Calcium [@Kraft2009] and Strontium [@Stellmer2009; @deEscobar2009; @DeSalvo2010; @Tey2010dgb]. The control that has been developed over these atoms makes them an extremely interesting candidate for the implementation of quantum information processing [@aeshort; @Gorshkov2009; @hayes07; @reichenbach07]. This is especially true in light of the laser stability achieved in optical clock experiments [@Boyd07; @Boydthesis], which is reminiscent of the development path towards quantum computing taken in the case of trapped ions [@ions; @ions2].
![Level structure for Alkaline-earth-like atoms. (a) These atoms possess a singlet-triplet transition with long-lived metastable $^3$P$_0$ and $^3$P$_2$ levels. (b) Adiabatic dressed potentials can be created by a resonant coupling on the clock transition with a sinusoidally varying Rabi frequency $\Omega_i(x)$, producing dressed states of the $^1$S$_0$ and $^3$P$_0$ levels. (c) The differential Zeeman shift for different nuclear spin levels can be used to produce nuclear spin-dependent lattices, by driving the transition between the $^1$S$_0$ and $^3$P$_0$ levels resonantly at the different resonant frequencies for different nuclear spin states in a magnetic field. Here we show some of the $m_I$ states for an atom with nuclear spin $I=9/2$, such as $^{87}$Sr.[]{data-label="fig:levelstructure"}](levelstructure){width="8.5cm"}
The key new feature of alkaline earth atoms in comparison with alkali atoms is the singlet-triplet metastable transition, with the $^1$S$_0$ – $^3$P$_0$ transition being used as the clock transition (see Fig. \[fig:levelstructure\]a). In particular, for $^{87}$Sr, the $^3$P$_0$ manifold has a measured lifetime of $\tau \sim 30$s, and the $^3$P$_2$ levels have even longer predicted lifetimes. In addition, for species with non-zero nuclear spin, this spin can be decoupled from the electronic state on the clock transition [@hayes07; @reichenbach07; @yi08; @aeshort; @Gorshkov2009], especially in the presence of a large magnetic field. The use of this nuclear spin for storage of quantum information would then be ideal, as the nuclear spin is much less sensitive to magnetic fields than electron spins, and thus much less susceptible to decoherence from magnetic field fluctuations than qubits stored on electronic states. This has lead to a series of proposals [@aeshort; @Gorshkov2009; @Shibata] in which the electronic state is used for access to and manipulation of the qubit [@dereviankoaddressing], and the nuclear spin state is used for qubit storage.
In previous work [@aeshort] we developed a scheme for quantum computing with alkaline earth atoms that was based on *electronic-state-dependent* lattices in which independent control over lattices for the metastable excited $^3$P$_0$ and ground $^1$S$_0$ levels is obtained by using light of different wavelengths. This is made possible by the fact that these levels are optically separated, providing very different AC polarisabilities for the states as a function of the wavelength. We showed how these two independent lattices could be used as a *storage* lattice for qubits encoded on the nuclear spin state, and a *transport* lattice to manipulate the qubits and perform gate operations [@aeshort]. A key theme in this context is that many schemes and concepts developed for alkali atoms, including certain techniques that have already been demonstrated in proof-of-principle alkali experiments simply work quantitatively better for alkaline earth atoms, where they are freed from important technical restrictions. In this sense, alkaline earth atoms represent an extremely important technological advance in various settings.
Here we present an alternative scheme to this previous proposal, in which we show that near-resonant coupling on the clock transition can produce frequency selective *nuclear-spin-dependent* lattices (see Fig. \[fig:levelstructure\]b,c). As in the case of electronic-state-dependent lattices, this makes it possible to produce state-dependent lattices without the restriction of having to tune couplings between fine-structure states [@spinlattice1; @spinlattice2], which can lead to large heating and decoherence from spontaneous emissions in the case of alkali atoms. In addition, spin-dependent lattices made in this way can be easily generated so that motion of the two potentials is independent in 2D. Below we discuss this implementation in detail, developing a complete proposal for quantum computing with alkaline-earth(-like) atoms, including methods for production of a quantum register and for performing single-qubit operations. Qubit readout with individual addressing can be performed in a similar manner to the case of electronic-state-dependent lattices [@aeshort], using magnetic gradient fields to shift the energy of states in the $^3$P$_2$ manifold. In nuclear-spin-dependent lattices, the large two-body loss rates from metastable $^3$P$_2$ levels can also be used to perform two-qubit gate via a lossy blockade mechanism as an alternative to the implementation of gates via controlled collisions [@spinlattice1]. This was originally discussed for electron-state-dependent lattices [@aeshort], and we provide further details of this mechanism below.
This alternative scheme with nuclear-spin-dependent lattices has the advantage of not requiring additional lasers to trap the $^1$S$_0$ and $^3$P$_0$ lattices independently, and can be performed with a strong laser on the clock transition. At the same time, this method is somewhat sensitive to collisional losses when two atoms in the $^3$P$_0$ manifold collide, as the near-resonant lattices will always produce admixtures of this state. However, this is only an issue during the short times involved in gate operations, and is strongly suppressed in a realistic setup where atoms are also AC-Stark shifted, as discussed below. Nuclear-spin-dependent lattices would also have immediate applications in quantum simulation [@Cazalilla; @Gorshkov; @Hermele2009; @FossFeig; @Gerbier2010]. In particular, the degeneracy in models with SU(N) symmetry [@Cazalilla; @Gorshkov; @Hermele2009] (which can be studied using alkaline-earth-like atoms by making use of the symmetry for interactions of atoms in different nuclear spin levels) could be deliberately broken and restored by applying these nuclear-spin-dependent potentials.
The rest of this article is organised as follows: We first discuss the formation of near-resonant spin-dependent optical lattices in more detail in Sec II, together with preparation of atomic registers in these lattices. In Sec. III we then discuss means for readout of individual qubits, and in Sec IV we treat ideas for gate schemes to entangle two qubits, including making use of lossy blockade mechanisms. In Sec. V we present a summary and outlook.
Spin-dependent Adiabatic potentials {#sec:potentials}
===================================
In the following, we discuss a quantum register formed by one atom trapped every site of a deep optical lattice, where tunnelling of atoms between sites can be neglected on the timescale of the experiment. As discussed above, we identify hyperfine states with two chosen nuclear spin states, and we would like to create spin-dependent potentials in order to move these qubit states independently. We will make use of these in the two-qubit gate operations that we discuss in Sec. IV.
A novel method of forming optical lattices for alkaline earth atoms is to make use of a near-resonant optical coupling directly on the clock transition, which will produce adiabatic dressed potentials [^1]. In the case that the coupling field is a standing wave, the Rabi frequency, and thus the final dressed potential, will be sinusoidally varying, providing an optical lattice for dressed states that are superposition of states in the $^1$S$_0$ and $^3$P$_0$ levels (as shown in Fig. \[fig:levelstructure\]b). In a large magnetic field, there is a differential Zeeman shift $\Delta E_Z$ between the $^1$S$_0$ and $^3$P$_0$ states (109 Hz/G for $^{87}$Sr [@Boydthesis]), meaning that a direct coupling preserving the nuclear spin (with $\pi$-polarised light) will be resonant at substantially different frequencies for different nuclear spin states (see Fig. \[fig:levelstructure\]c). We can then drive each transition independently with Rabi frequencies $\Omega_0=\Omega_{\pm}$, as shown in Fig. \[fig:levelstructure\]b. Provided that the shift $\Delta E_Z \gg \Omega_{\pm}$, we will then obtain independent two-level systems for each $m_I$ state for which we apply the appropriate coupling frequency. For example, if we choose $\Omega_0\sim 100$ kHz, then for $^{87}$Sr, we would like to apply a field $\gtrsim 1000$ G in order to obtain shifts between neighbouring states $\gtrsim 100$ kHz (however, states separated further in $m_I$ could also be used to reduce the required field - see below). In this way, we can choose, e.g., two $m_I$ states as our two qubit states, ${|0 \rangle}$ and ${|1 \rangle}$, and create independent potentials for these two states. At the same time, because the frequency differences between lattices for different $m_I$ states will of the order of 1 MHz, the lattice laser wavenumber $k_l$ is approximately the same for the two species - in fact the resulting lattice potentials will overlap for the order of millions of periods.
Dressed potentials for a two-level system
-----------------------------------------
We will now discuss the form of the dressed potentials for a single nuclear spin state, identified with qubit state $i=0$ or $i=1$, and discuss the case where we have multiple nuclear spin states below. We can first write the Hamiltonian for a two-state atom, with states ${|g,i \rangle}\equiv{|^1{\rm S}_0,m_I=i \rangle}$ and ${|e,i \rangle}\equiv{|^3{\rm P}_0,i \rangle}$ as ($\hbar\equiv 1$) $$\hat H=\hat H_M+\hat
H_0,$$ where $\hat{H}_M=\hat{\mathbf{p}}^2/2m$ is the kinetic energy, and $$\hat H_0=-\delta_i{|e,i \rangle}{\langlee,i |}+(\Omega_i(\mathbf{x})/2){|e,i \rangle}{\langleg,i |}+{\rm h.c.}$$ describes the near-resonant coupling field with $\Omega_i(\mathbf{x})$ and $\delta_i$ the Rabi frequency and detuning respectively.
Generation of adiabatic dressed potentials is then based on the validity of a Born-Oppenheimer-type assumption, where we assume that the kinetic energy of the atoms is small on a scale given by the separation of the resulting adiabatic potentials. The wavefunction $|\Phi(t)\rangle$ of a single atom satisfies the Schrödinger equation, $$i\hbar\frac{\partial}{\partial t}|\Phi(t)\rangle=(H_M+H_0)|\Phi(t)\rangle.$$ If we omit the kinetic energy term from the Hamiltonian, we obtain an equation for adiabatic eigenstates, ${|\Psi_{\pm}(t) \rangle}$, $$H_0(\mathbf{x})|\Psi_{\pm}\rangle=V^{\pm}(\mathbf{x})|\Psi_{\pm}\rangle,$$ Note here that as $H_0(\mathbf{x})$ is time-independent, there are only two such eigenstates ${|\Psi_{\pm} \rangle}$. If we consider the 1D case, and set $\Omega_i(x)=\Omega_i \sin(k_l x + \phi)$, representing the field of a standing wave (with $k_l$ the laser wavenumber and $\phi$ a phase), we find the adiabatic potentials $V^{\pm}({x})=(-\delta_i\pm\sqrt{\delta_i^{2}+\Omega_i(x)^{2}})/2$. These are shown schematically in Fig. \[fig:levelstructure\]b). The complete wavefunction can then be expanded in a basis of these adiabatic eigenstates, which play the role of Born-Oppenheimer channel functions, $$|\Phi(t)\rangle=c_+ (x,t) |\Psi_{+}\rangle+c_- (x,t) |\Psi_{-}\rangle,$$ resulting in the equation $$i\hbar\frac{\partial}{\partial t}c_{\pm}(x,t)=[H_M+V^{\pm}(x)]c_{\pm}(x,t)+H_{M}^{\pm}c_{\mp}(x,t),$$ where $H_{M}^{\pm}=\langle\Psi_{\pm}(x,t)|H_M|\Psi_{\mp}(x,t)\rangle$ gives the non-adiabatic couplings between the dressed states due to the motion of the atom. Provided these latter terms are small, the two equations decouple and the atoms remain in a single dressed state. In our case, there will be no non-adiabatic loss of atoms in this sense, provided that they are loaded into low energy states of the lower of the two adiabatic potentials ($V^-$). If atoms are loaded into the higher energy dressed potential, loss of atoms into the continuum states of the $V^-$ potential can occur, however this is exponentially suppressed as the separation between adiabatic potentials is increased, with the loss rate $\Gamma_l \sim \Gamma_0 \exp(-\delta_i/\omega)$ [@yi08], where $\Gamma_0$ is a prefactor that we do not compute in detail here, and $\omega$ is the trap frequency in an individual site of the dressed potential.
Dressed potentials for independent states
-----------------------------------------
When these dressed potentials are created independently for different $m_I$ levels, the result is that we can form two independent but almost identical potentials. These can, e.g., be shifted with respect to each other in a 2D plane using interferometrically stable methods, e.g., by adding path length to an interferometer arm in which the light is frequency-shifted in order to produce one of the trapping frequencies. As discussed above, this means of creating spin-dependent lattices has substantial advantages over spin-dependent lattices for Alkali atoms, where the lifetime is limited by the need to tune the lattice beams to a frequency in the middle of the fine structure splitting. Here, the lifetime will be controlled by the lifetime of the $^3$P$_0$ level (which is many seconds), or by off-resonant couplings to shorter lived states (but these will typically be many tens of nanometers detuned). In the presence of a second frequency (e.g., due to the laser creating the lattice for the second internal state), atoms can also be lost from the lower adiabatic potential, essentially being coupled out of the lattice into the continuum. However, due to the large momenta in the resulting state, this rate is suppressed exponentially in the ratio of the separation between manifolds in a Floquet basis and the trapping frequency in the lattice, $\Gamma_l\sim \tilde \Gamma_0 \exp(-\omega_{\rm diff} / \omega)$ [@yi08], where $\tilde \Gamma_0$ is a prefactor [@yi08] $\omega_{\rm diff}$ is the frequency difference of the lattices for the two qubit states. If we operate in a field $\sim 5000$ G, then $\omega_{\rm diff}\sim 2\pi \times 550$kHz, and if we choose the Rabi Frequency $\Omega \sim 120$ kHz, then $\omega\sim 15$ kHz. We can also reduce the required field strength by choosing $m_I$ levels that are further separated (in $^{87}$Sr we can reduce the required field strength by a factor of 9 by choosing $m_l=-9/2$ and $m_l=+9/2$ as the two trapped states).
Combining resonant and off-resonant potentials {#sec:combining}
----------------------------------------------
In practice, strong coupling at intensity $I$ on the clock transition will also give rise to off-resonant AC-Stark shifts $\Delta E_{AC}^e$ and $\Delta E_{AC}^g$ of the states ${|e,i \rangle}$ and ${|g,i \rangle}$ from coupling to other manifolds (e.g., $^1$P$_1$ and $^3$S$_1$) in addition to the resonant couplings between the two levels. These must be added to the Hamiltonian, as $H=H_M+H_0+\Delta E_{AC}^e |e,i\rangle \langle e,i|+\Delta E_{AC}^g |g,i\rangle \langle g,i|$. As $\Delta E_{AC}^{e,g} \propto I$ and $\Omega\propto \sqrt{I}$, the off-resonant contributions will become more important as the intensity of the applied field becomes larger. For $^{87}$Sr, the shifts from the AC-Stark shift become of the same order as the AC-Stark splitting due to resonant coupling at relatively high fields, with $I \sim 50$kW/cm$^2$ [@Boydthesis; @yi08]). At higher fields, the potentials $V^{\pm}$ will be modified by these shifts, but can still be made spin-dependent if the detunings and Rabi frequencies of the lattice beams are chosen carefully. This is illustrated in Fig. \[fig:combinedpotentials\], where we show the lower adiabatic potential for each of the two nuclear spin states for a selection of different phases $\phi$ between the potentials. We see that at relative phase $\phi=0$ the potentials for different nuclear spin states are identical, and are given by a combination of the resonant and off-resonant contributions. At phase $\phi=\pi/2$, however, the off-resonant contributions from the two coupling frequencies, which are independent of the nuclear spin state, become spatially homogeneous due to the addition of the two spatially shifted contributions. At this point the sinusoidal form of the lattice potentials is due solely to the resonant contribution. It can be seen that the lattice, also in between, will be modified in such a way that the atoms will be transported through the lattice spin-dependently.
![a) Lower dressed potentials for the ${|0 \rangle}$ qubit state (solid lines) and the ${|1 \rangle}$ qubit state (dashed lines) formed by combining resonant and off-resonant contributions. These are plotted as a function of position for varying phase offsets $\phi$ between the coupling fields for the ${|0 \rangle}$ and ${|1 \rangle}$ states. For $^{87}$Sr at the wavelength of the clock transition, we obtain $\Delta E_{AC}^e \approx 3 \Delta E_{AC}^g$, and we choose the intensity so that $\Omega=4 \Delta E_{AC}^g$ (ca. 10 kW/cm$^2$). Here, $\delta=-3\Omega/4$. b) Projection on the excited state of the dressed state corresponding to the lower dressed potential, plotted for the same values of $\Omega$, $\Delta E_{AC}^{g}$, and $\Delta E_{AC}^{e}$ as in (a) as a function of $\phi$. The different lines correspond to varying $\delta$, from top to bottom, $\delta=-\Omega/2$, $-3\Omega/4$, $-\Omega$, and $-5\Omega/4$.[]{data-label="fig:combinedpotentials"}](combined){width="8.5cm"}
We also note that two important characteristics relating to the shape of the lattice and the form of the dressed states changes as a function of $\phi$. Firstly, the lattice depth changes, because for $\phi=0$ the effect of the resonant and off-resonant contributions to the lower dressed potential are summed, whereas for $\phi=\pi/2$, the lattice is formed solely by a resonant contribution. This is shown in Fig. \[fig:combinedpotentials\]a. In addition, for $\phi=0$ the off-resonant potentials shift the coupling out of resonance, changing the adiabatic dressed states. As a result, the admixture of the excited internal state in the lower dressed level is relatively small. As the lattices are shifted, and the resonant contribution dominates, the admixture of the excited state increases. In Fig. \[fig:combinedpotentials\]b, we plot the admixture of the $^3$P$_0$ level, averaged over one period of the lower dressed potential for different values of the detuning $\delta$. We note that for the detuning values we choose here, this value is always small. This will lead to a significant reduction in collisional loss rates due to $^3$P$_0$-$^3$P$_0$ collisions when two atoms are on the same lattice site.
Loading a quantum register
--------------------------
In order to produce a quantum register with one atom in every lattice site, we begin from a spin-polarised gas of fermionic alkaline earth-like atoms, produced by optical pumping. This should be a degenerate Fermi gas so that the densities are sufficiently high to load a single atom per lattice site. Note that we choose Fermions here because for Yb and Sr, it is the fermionic isotopes that have non-zero nuclear spin, and thus allow us to encode qubits using this degree of freedom. In the case that we have sufficient intensity to produce a large AC Stark shift at the same frequency as the final lattice, we can first load the gas into an off-resonance optical lattice in the $^1$S$_0$ state, and then adiabatically tune the coupling closer to resonance with the $^3$P$_0$ state in order to load the gas carefully into the lower dressed potential. A high-fidelity quantum register can then be formed by creating a band-insulator state [@esslinger04], and we gain substantially over the case where bosons would be used for a quantum register, as the temperature need only be substantially smaller than the bandgap, and not an interaction energy for the band insulator to form. In addition, if a harmonic trapping potential is added to the system, most defects in the state will be localised near the edges of the trap [@calarco04]. The resulting state can be further improved upon by applying additional techniques, such as filtering of the state to improve the fidelity [@rablloading] or fault-tolerant loading of atoms by transfer of atoms between two internal states, one trapped by the lattice and the other not [@agloading].
Single qubit addressing via the $^3$P$_2$ level
===============================================
We would like to be able to read out the state of a single qubit, or alternatively perform gate operations on a single qubit. The has been enormous recent process in individual addressing of sites in an optical lattice via optical means [@addweiss; @addchin; @addgreiner; @addgreiner2; @addbloch; @addbloch2; @addott; @addmeschede]. However, it would also be useful to be able to address individual qubits without the use of these techniques and the corresponding overheads in experiments. Such addressing can be achieved by coupling our dressed state qubit-selectively to states in the metastable $^3$P$_2$ level, and then detecting whether the atom is indeed present in the $^3$P$_2$ manifold. For the purpose of readout it is only necessary to be able to couple one of our two qubit states, e.g., the ${|0 \rangle}$ state (which could be represented, e.g., by $m_I=-9/2$ in $^{87}$Sr) to an auxilliary level ${|0x \rangle}$ in the $^3$P$_2$ level (e.g., the ${|^3{\rm P}_2,F=13/2,m_F=-13/2 \rangle}$ state, where $F$ is the total angular momentum quantum number $F$ and $m_F$ is the magnetic quantum number). The readout process is depicted schematically in Fig. \[fig:readoutlevels\]
![Schematic diagram of qubit readout. Qubits are stored in dressed states ${|0 \rangle}$ and ${|1 \rangle}$, which are dressed superpositions of states in the $^1$S$_0$ and $^3$P$_0$ manifolds with a definite values of $m_I$. These can be coupled via off-resonant Raman processes to long-lived auxilliary states ${|0x \rangle}$ and ${|1x \rangle}$ in the $^3$P$_2$ manifold for the purpose of readout. In order to read out a particular qubit state, this state should be coupled to the $^3$P$_2$ manifold. It can then be detected by fluorescence on the cycling transition $^3$P$_2$-$^3$D$_3$.[]{data-label="fig:readoutlevels"}](readoutlevels){width="8.5cm"}
Because of their non-zero electron spin, states in the long-lived $^3$P$_2$ manifold are much more sensitive to magnetic fields than the $^3$P$_0$ and $^1$S$_0$ level, and we can use these shifts to make possible a spatially-dependent readout of spin states by applying a magnetic gradient field, in a manner first mentioned in Ref. [@dereviankoaddressing]. In applying such a field, the $^3$P$_2$ level can be significantly shifted, whilst the $^1$S$_0$ and $^3$P$_0$ states are not substantially shifted, and thus the form of the dressed lattice potential is not substantially changed. In particular, a gradient field of 1 G/cm will provide an energy gradient of 4.1 MHz/cm for the ${|^3{\rm P}_2,F=13/2,m_f=-13/2 \rangle}$ state, or an energy difference of about 15 kHz between atoms in neighbouring sites for a field gradient of 100 Gauss/cm. Atoms in the dressed lattice can then be selectively transferred via a Raman process connecting off-resonantly via the $^3$S$_1$ manifold to the $^3$P$_2$ manifold, on a timescale limited by the frequency shift between neighbouring sites.
This assumes, of course, that the state in the $^3$P$_2$ manifold to which we couple, ${|0x \rangle}$ is trapped in a lattice, preferably in a lattice at the same position as our qubit states ${|0 \rangle}$ and ${|1 \rangle}$. Thus, the most favourable states are those with a significant negative $AC$-polarisability $\alpha$ at the wavelength of the clock transition, as the potential they experience due to the AC-Stark shift will have minima in the same places as the lower dressed state generated by the same lattice laser. We have computed the polarisability from known data of the states in the $^3$P$_2$ manifold of $^{87}$Sr, and have found that they vary substantially due to a large tensor shift. We write the shift $\Delta_E$ from linearly polarised light as $$\begin{aligned}
h\Delta_E&=&-\frac{1}{2}\alpha E^2,\\
&=&-\left[\alpha^{\rm scalar} +\alpha^{\rm tensor}\frac{3m_F^2-F(F+1)}{F(2F-1)}\right]\frac{E^2}{2},\end{aligned}$$ where we have separated the coefficients of the scalar and tensor shifts [@boyd07; @Boydthesis], and we obtain total polarisabilities at the clock transition frequency as shown in Fig. \[fig:starkshifts\]. Here we note that light polarised along the quantisation axis will give rise to a negative polarisability for the $F=13/2$, $m_F=-13/2$ state. This state is thus trapped by the same field creating the dressed lattice. We can couple from the $m_I=-9/2$ states in the dressed lattice via a Raman process directly into the $F=13/2$, $m_F=-13/2$ state of the $^3$P$_2$ manifold, making this state ideal for use as the ${|0x \rangle}$ state in readout operations. A qubit could be read out by choosing the detuning of a Raman coupling between the $|^3$P$_2,\, F=13/2, \, m_F=-13/2\rangle$ state (auxiliary state ${|0x \rangle}$) and the $|-\rangle$ dressed state with $m_I=-9/2$ (qubit state ${|0 \rangle}$) so that it is in resonance at only one site as a result of a gradient field shifting the energy of the $|^3$P$_2,\, F=13/2, \, m_F=-13/2\rangle$ state. Coupling of the $^3$P$_2$ level to a second qubit state with $m_I=-7/2$ would not occur as the $m_F=-11/2$ state is not trapped (if for a different species the equivalent state was trapped, then the large tensor shift would probably result in this transition being anyway out of resonance). The occupation of the $^3$P$_2$ level can then be determined by fluorescence measurements, e.g., using the cycling transition $^3$P$_2$-$^3$D$_3$, independent of the atoms remaining in the $^1$S$_0$ and $^3$P$_0$ levels. Note that the timescale for this readout process $\tau_{\rm readout}$ is limited by the trapping frequency in the dressed lattice potential, $\tau_{\rm readout} \gg 2\pi /\omega$. This requirement must be fulfilled so that the atom is not coupled to excited Bloch bands of the lattice. It is also desirable for this coupling to have similar trapping frequencies for the lattices trapping ${|0 \rangle}$ and ${|0x \rangle}$, so maximising spatial overlap of the wavefunctions. Again, the $F=13/2$, $m_F=-13/2$ state of the $^3$P$_2$ manifold is favourable for this, as the polarisability indicates that the lattice depth will be around $150$ kHz for $I\sim3$kW/cm$^3$, which is a similar depth to that of the lattice for the dressed levels at the same lattice intensity (assuming that the detuning of the resonant coupling lasers, $\delta_i$ is small).
Note that one could equally use states with $|m_I|<-13/2$ in this process if one stores the qubit states in the upper dressed potential. This is disadvantageous, because a large detuning $\delta$ must be chosen for the lattice lasers to prevent non-adiabatic loss of atoms from the potential [@yi08]. Alternatively, an additional standing wave at a different frequency could be added to trap states in $^3$P$_2$ manifold via an additional AC-Stark shift.
An alternative to using magnetic field gradients for addressing would be to apply a laser with spatially varying intensity at the magic wavelength (for equal shifts of the $^3$P$_0$ and $^1$S$_0$ levels. This would provide a position-dependent differential AC-Stark shift between the qubit states and the $^3$P$_2$ level, without affecting the relative energy of the $^3$P$_0$ and $^1$S$_0$ levels, and thus the dressed lattice.
![AC-Polarisabilities for $^{87}$Sr in the $^3$P$_2$ manifold with $F=13/2$ at the frequency of the clock transition.[]{data-label="fig:starkshifts"}](starkshifts){width="8.5cm"}
\[fig:polarisability\]
Quantum Gates in spin-dependent potentials
==========================================
Single-qubit gates can be performed in one of two ways in this scheme. The simplest means to obtain a global rotation of many qubits is to directly couple the dressed states for two nuclear spins via a Raman process. Alternatively, different nuclear spin (qubit) states can be alternately coupled to auxiliary states in the $^3$P$_2$ level in order to provide individual addressing for single-qubit rotations using the techniques described in the previous section. Such coupling requires the use of a trapped state in the $^3$P$_2$ manifold that can be coupled to both qubit states. For $^{87}$Sr, such addressing for single-qubit operations would thus mean either using an auxiliary lattice to trap states from the $^3$P$_2$ manifold, or using the upper dressed states for qubit storage.
Two-qubit gates can, in principle, be performed similarly to exisiting schemes for alkali atoms, making use of the spin-dependent potentials. In particular, exisiting schemes for controlled collisions can be used to produce controlled-phase gates for atoms in neighbouring sites [@spinlattice1]. This has been implemented experimentally in a proof-of-principle experiment with alkali atoms [@spinlattice2], but here we could take advantage of the 2D spin-dependent lattices without having to tune trapping lasers between fine-structure states.
These schemes can be seen to implement controlled-phase gates in three steps:
1. The spin-dependent lattices for each state are shifted relative to each other so that atoms at a chosen distance, e.g., in neighbouring lattice sites, will come together on the same site if and only if they were originally in a specific combination of qubit states. For example, if we write the state of a pair of neighbouring qubits as ${|q_1 q_2 \rangle}$, where $q_1$ is the state of the first qubit and $q_2$ is the state of the second qubit in the pair, then atoms in the state ${|01 \rangle}$ are brought together, whilst ${|00 \rangle}$, ${|11 \rangle}$ and ${|10 \rangle}$ remain separated (see Fig. \[fig:spindep\]).
2. A phase shift is generated conditioned on whether two atoms are on the same site or not.
3. The atoms are returned to their initial positions.
The phase in step two can be generated in a number of different methods, including via direct collisional phase shifts, or the use of blockade mechanisms. These different mechanisms are discussed in the following two subsections.
Phase for two-qubit gates: controlled collisions
------------------------------------------------
![If the qubits are trapped in a spin-dependent lattice, it is possible to shift the lattice for one qubit state by one site, so that neighbouring atoms are brought together only if the qubit to the left was in state ${|0 \rangle}$ and the qubit to the right was in state ${|1 \rangle}$. This can be used to aid in producing two-qubit quantum gates (see text for details).[]{data-label="fig:spindep"}](spindep){width="8.5cm"}
For alkali atoms, the phase in step 2 is generated by collisional interactions between atoms. This could be performed directly if the atoms used have a relatively large scattering length in the $^1$S$_0$ manifold (e.g., $^{87}$Sr). For other species and isotopes such as $^{171}$Yb, this could also be achieved using optical Feshbach resonances [@srfeshbach; @ybfeshbach] to enhance the otherwise very weak collisional interaction. The speed of such gates is limited by the strength of the on-site interaction between atoms, which for a single-band model is limited by the trap frequency in each lattice site, $\omega$.
However, the existence of weak collisional interactions for certain isotopes also motivates us to look at other gate schemes, particularly using excitations to states in the $^3$P$_2$ manifold.
Use of $^3$P$_2$ levels
-----------------------
We would again like to make use of states in the $^3$P$_2$ manifold to which our dressed qubit states (for a fixed nuclear spin) can be coupled, and which are trapped in the same locations as our qubits. This time we will assume that we have two such auxiliary levels, ${|0x \rangle}$ and ${|1x \rangle}$, as depicted in Fig. \[fig:readoutlevels\].
### Phase for two-qubit gates: dipole blockade mechanism
For sufficiently large onsite dipole-dipole interactions, which provide a energy shift between $^3$P$_2$-$^3$P$_2$ collisional interactions and $^3$P$_0$/$^1$S$_0$-$^3$P$_2$ corresponding to a large frequency shift $\Delta$, we can use a dipole blockade mechanism to produce a $\pi$ phase shift, as proposed, e.g., for Rydberg atoms [@int:rydberg]. This is illustrated in Fig. \[fig:lossyblockade\], and consists of 3 steps:
1. Excite all ${|0 \rangle}$ qubit states to an auxillary level ${|0x \rangle}$ with a $\pi$-pulse
2. Couple all ${|1 \rangle}$ qubit states to an auxillary level ${|1x \rangle}$ with a $2\pi$-pulse at Rabi frequency $\Omega$, assuming that there is no collisional interaction between the ${|0x \rangle}$ state and either ${|1 \rangle}$ or ${|1x \rangle}$ (i.e., the pulse duration $T$ is given by $\Omega T=2\pi$. In the ideal case, if the two atoms are on the same site (as will happen for an initial state ${|0,1 \rangle}$, this step should be blocked by collisional interactions, which detune the coupling by a frequency $\Delta$.
3. Return the ${|0x \rangle}$ state to the ${|0 \rangle}$ state with a $\pi$ pulse.
Assuming there is no coupling of the qubit state ${|1 \rangle}$ to the auxillary ${|1x \rangle}$ when the atom is on the same site as an already excited ${|0x \rangle}$ state (i.e., the blocking is perfect), the states of the two-qubit system after each step of this protocol are given in table \[table1\].
Initial State After Step 1 After Step 2 After Step 3
------------------ ----------------------- ---------------------- -------------------
${|0,0 \rangle}$ $ - {|0x,0x \rangle}$ $- {|0x,0x \rangle}$ ${|0,0 \rangle}$
${|0,1 \rangle}$ $ -i {|0x,1 \rangle}$ $-i{|0x,1 \rangle}$ $-{|0,1 \rangle}$
${|1,0 \rangle}$ $ -i {|1,0x \rangle}$ $i {|1,0x \rangle}$ ${|1,0 \rangle}$
${|1,1 \rangle}$ $ {|1,1 \rangle}$ $-i{|1x,1x \rangle}$ ${|1,1 \rangle}$
: The state of a two-qubit system after each step of the protocol for a blockade gate.[]{data-label="table1"}
In practice, the state ${|0,1 \rangle}$ will collect a small additional phase $\phi\sim \Omega/\Delta$, where $\Omega$ is the coupling Rabi frequency and $\Delta$ the detuning from the excited state, generated by the difference between $^3$P$_0$.
![Schematic diagram of a blockade gate including loss. (a) Operations performed on the individual qubit states ${|0 \rangle}$ and ${|1 \rangle}$. (see text for details) (b) Comparison of the operations for initial two-qubit states ${|0,1 \rangle}$ and ${|1,0 \rangle}$ in neighbouring qubits, showing the two-qubit levels. []{data-label="fig:lossyblockade"}](lossyblockade){width="8.5cm"}
### Lossy blockade mechanism
It was shown by C. Greene and his collaborators [@greene1; @greene2] that, in fact, two-body collisions of atoms in the $^3$P$_2$ level lead to large inelastic loss. However, this loss can actually help us in producing the blockade effect, as large losses involving coupling to the continuum at a rate $\Gamma$ from a given level can also dynamically suppress occupation of that level, as is well known from the physics of a two-level system. In the limit where $\Delta \ll \Gamma$, this would even produce a blockade gate based entirely on a lossy blockade mechanism. In this way we can turn an apparent problem into a feature of the system. Such ideas have also been proposed in the context of quantum simulation with cold atoms in optical lattices, where three-body losses can be used to prepare interesting many-body states via a similar mechanism [@daley3body; @cirac3body]
The key characteristics of the inelastic loss processes that make this possible are:
- [The energy change in the inelastic collision is larger than the lattice depth, so that the energy carried away as kinetic energy is sufficient to couple the atoms into the continuum of motional states.]{}
- [The length scale on which the physics of the inelastic collision takes place is smaller than the confinement length in a lattice site, so we do not expect the loss process to be substantially modified by the presence of the lattice]{}
- [The rates for loss are large, and could reach of the order of $\Gamma=2\pi\times20$kHz for lattice densities up to $10^{16}$cm$^{-3}$.]{}
In the presence of loss, the basic physics of the second step of the protocol, as illustrated in Fig. \[fig:lossyblockade\] then reduces to a two level system, where the state with one atom in $^1$S$_0$ and one in $^3$P$_2$ playing the role of a lossless “ground” state and that with two atoms in $^3$P$_2$ the role of the lossy excited state. If we write these states as a spin-1/2 system, the Hamiltonian reduces to $$H=\frac{\Omega}{2} (\sigma^+ + \sigma^-) -\frac \Delta 2 \sigma^z$$ where $\sigma^+=|e\rangle\langle g|$, $\sigma^-=|g\rangle\langle e|$ and $\sigma^z=|e\rangle\langle e|-|g\rangle\langle g|$ are the usual spin operators for our two-level system with lossy excited state $|e\rangle$ and lossless “ground” state $|g\rangle$, $\Omega$ is the Rabi frequency for the coupling laser, and $\Delta$ is the effective detuning from the excited state, which can be induced by interaction between two atoms when they are both in the $^3$P$_2$ manifold. Including the loss, this system is described by the master equation $$\dot \rho =-i [H,\rho] -\frac{\Gamma}{2} \left[\sigma^+ \sigma ^- \rho + \rho \sigma^+ \sigma^- - 2\sigma^- \rho \sigma^+ \right]. \label{mastereq}$$ In the limit $\Delta, \Gamma \gg \Omega$ we can describe the time evolution of a system initially prepared in the ground state in perturbation theory, giving the probability that no decay has occurred at short times $t$ as $$p={\rm e}^{-\Gamma_{\rm eff}t},$$ with $$\Gamma_{\rm eff}\approx \frac{\Omega^2}{4(\Delta^2+\Gamma^2/4)}\Gamma \approx \frac{\Omega^2}{\Gamma}$$ in the limit that $\Gamma \ll \Delta$. For our lossy blockade gate this is the worst-case scenario for loss events. We immediately see that the ratio of the loss time to the gate time (determined by $\Omega$) is given by $\tau_{\rm loss}/\tau_{\rm gate}=\Omega/\Gamma$. This will give the fidelity of the lossy blockade gate.
The blockade mechanism is illustrated in Fig. \[fig:twolevel\], where we plot the decay probability as a function of time $t$, and then at fixed time $\Omega t =2\pi$ for varying $\Gamma/\Omega$.
![Loss from a two-level system prepared in the stable state and coupled to the lossy state computed via integration of eq. (\[mastereq\]). (a) The probability that the system has undergone a loss event as a function of time when prepared in the ground state, with $\Delta=0$, for varying values of $\Gamma/\Omega$. (b) The probability that a system has undergone a loss event by time $t\Omega=2\pi$. These values represent the gate fidelity of a lossy blockade gate with $\Delta=0$, or with a combined blockade generated by interactions and loss with $\Delta\neq 0$. []{data-label="fig:twolevel"}](twolevelblockade){width="8.5cm"}
Other Two-qubit gates
=====================
It would also be possible to make use of exchange interactions for fermions [@exchangegate], but we will not discuss this in detail because it does not make specific use of the spin-dependent potentials, and does not, in its original form, take specific advantage of the properties of alkaline earth atoms.
Another possibility is the direct use of Rydberg gates [@int:rydberg], which have been recently demonstrated for trapped alkali atoms [@Urban09; @Gaetan09]. The separate hierarchy of Rydberg states for the singlet and triplet manifolds could give advantages for Rydberg excitations in alkaline earth atoms, especially facilitating easier state-dependent excitation. These could also be performed together with gradient addressing, exciting the Rydberg state from the $^3$P$_2$ manifold.
Decoherence/loss Mechanisms
===========================
There will be a number of possible sources of decoherence within this setup, all of which should be controllable in the experiment. These include magnetic field fluctuations, decoherence due to frequency noise of the lasers, spontaneous emissions, and collisional losses. We briefly summarise the role of these key sources of decoherence below.
Magnetic field fluctuations
---------------------------
Magnetic field fluctuations will probably still constitute the largest source of decoherence, however, this is reduced by almost 3 orders of magnitude compared with qubits encoded on an electron spin. Decoherence from local fluctuations in the magnetic field will contribute both due to direct shifts of the energy of the qubit states, and from the modification to the dressed potentials due to the differential shift between the $^1$S$_0$ and $^3$P$_0$ levels.
Stability of the trapping lasers
--------------------------------
A finite laser linewidth for the dressing laser creating the potentials will give rise to fluctuations $\Delta \delta_i$ in the detuning $\delta_i$, and therefore the energy of atoms trapped in the dressed potential. However, these fluctuations will lead to the same fluctuation in lattice depth for the two qubits, $\Delta \delta_0=\Delta \delta_1$ . The resulting ground state energy will shift by different amounts, as the lattice periods are different. However, if the corresponding wavelengths are $\lambda$ and $(1+\varepsilon)\lambda$, then the difference in trap frequencies for the two qubit states, $\Delta_\omega$ is given in terms of the depth fluctuations $\Delta V$ by $$\frac{\Delta_\omega}{2} = \sqrt{\Delta V \frac{4\pi^2 \hbar^2}{2m\lambda^2} }-\sqrt{\frac{\Delta V 4\pi^2 \hbar^2}{2m\lambda^2 (1 + \varepsilon)^2} }\approx -\varepsilon\sqrt{\delta V \frac{4\pi^2 \hbar^2}{2m\lambda^2}}$$ Thus, as $\varepsilon\sim 10^{-8}$, this decoherence mechanism will be strongly suppressed, and for laser linewidths of the order of tens of Hz, dephasing times can be many minutes. On the other hand, the resulting noise $\Delta V$ on the depth of the lattice could give rise to heating of the particles to higher oscillator levels, if appropriate frequency components are present in the noise in order to drive these coupling.s This would lead to imperfect couplings for gate and readout operations. Such heating rates can be estimated [@heating1; @heating2] as giving an energy increase $\langle \dot E \rangle=\Gamma_{\rm heat} \langle E \rangle$ with rate $\Gamma_{\rm heat}= \pi^2 \omega^2 S_e(2\omega)/2$, where $S_e(2\omega)$ is the one-sided power spectrum of the trap amplitude noise at twice the trap frequency $\omega$. In our case, as for $\Omega_i \gg \delta_i$ $\Delta V \approx \Delta \delta_i^2/\Omega^2$, this is also suppressed by an extra factor of $\Delta \delta/\Omega$. In addition, other sources of heating, such as intensity noise on the lasers creating the lattice, or shaking of the lattice potential (due, e.g., to vibrating optical components) will have a similar effect [@heating1; @heating2].
Spontaneous emissions
---------------------
Qubits can decohere or be destroyed (the atoms lost from the lattice) by spontaneous emission events. These can come from two sources: the finite lifetime of the $^3$P$_0$ state, and off-resonant coupling to states with a short lifetime induced by the lattice lasers (storage) or coupling lasers (during gate and readout operations). However, the lifetime of $^3$P$_0$ is many seconds, and this source of atom loss can be further suppressed by using the resonant lattices only for spin-dependent transfer, and storing the atoms at other times in the $^1$S$_0$ state (see previous subsection).
Collisional losses from $^3$P$_0$
---------------------------------
Measurements of collisional losses between atoms in the $^3$P$_0$ manifold are currently underway in several groups, in order to determine what the collisional lifetime is when two atoms are present in these states at the typical lattice densities that will be encountered here (ca. $10^{14}$cm$^{-3}$-$10^{15}$cm$^{-3}$ onsite). Effects of these losses have been observed recently, e.g., in samples of Strontium atoms confined in 1D tubes [@bishof2011]. However in our case, during storage, readout, and single-qubit operations, the atoms are anyway isolated by the lattice, and two atoms will not collide. Thus, the only time that two atoms with components of states from the $^3$P$_0$ manifold are present on the same site is during two-qubit gate operations. If these take place on a timescale ca. $1$ms, then we would require collisional stability of our atoms for timescales longer than $100$ms in order to achieve gate fidelities larger than $99$% if both atoms were in the $^3$P$_0$ manifold. However, as shown in Sec. \[sec:combining\], the combination of resonant and off-resonant lattices mean that the amplitude for atoms to be in the $^3$P$_0$ manifold is small for all stages of operation except during transport of atoms. Gate schemes can be made more immune to these losses by using larger intensity trapping lasers, and thus introducing a larger component from the off-resonant lattice (see Sec. \[sec:combining\]). This will ensure that when the lattices for the two qubit states overlap that the dressed states are dominated by off-resonant lattices for $^1$S$_0$, and that the admixture of the $^3$P$_0$ state is small. If the probability to find a single atom on a given site in the $^3$P$_0$ manifold is $\varepsilon_3$ for each of the qubit states, then the onsite loss rate will be suppressed by a factor $\sim \varepsilon_3^2$.
Summary and outlook
===================
In summary, the quantum computing scheme we presented based on nuclear-spin-depenent lattices with near-resonant coupling on the clock transition for alkaline-earth(-like) atoms has several advantages over schemes with alkali atoms. The use of nuclear spins for qubit storage makes this scheme relatively robust against decoherence due to magnetic field fluctuations, and coupling to the $^3$P$_2$ manifold provides high-resolution individual qubit addressing with a magnetic gradient field. There are also possibilities here to perform gates based on transfer of states to long-lived metastable excited levels (e.g., $^3$P$_2$), including the new mechanism of lossy blockade gates. In comparison with a scheme presented previously using electronic-state-dependent lattices, this scheme does not require lasers that independently trap the $^1$S$_0$ and $^3$P$_0$ manifolds. This method is more sensitive to collisional losses between two atoms in the $^3$P$_0$ manifold, although this only affects short periods of time during the gate operations. While we have focused here on gates based on state-dependent lattices, other schemes, including Rydberg gates will benefit from the unique properties of alkaline-earth-like atoms. In particular, state-selective excitation to a Rydberg state would be simplified, e.g., by exciting one nuclear spin state to the $^3$P$_2$ manifold first.
The key experimental requirements for implemention of these methods are: (i) Large, stable magnetic fields (to provide the differential Zeeman shifts allowing spin-dependent lattices for different nuclear spin states; (ii) High-intensity stable laser on the clock transition (to provide a deep optical lattice whilst avoiding decoherence due to noise on the detuning $\delta_0$.); and (iii) Control over magnetic field gradients (to allow for either large parallel operations or individual addressing with qubits operations involving coupling to $^3$P$_2$ (although single-qubit gates could also be done directly in parallel, and the use of $^3$P$_2$ is only necessary in two-qubit gates in the case that the scattering lengths for the clock states are not sufficiently large).
Nuclear-spin-dependent lattices also have immediate possible application for quantum simulation with alkaline-earth-metal atoms. In particular, the dependence on the nuclear spin state could be used to break the degeneracy in models with SU(N) symmetry [@Cazalilla; @Gorshkov].
We thank Martin Boyd for close collaboration on the previously presented scheme for quantum computing with alkaline earth atoms (Ref. [@aeshort]), and for estimates of polarizabilities used in Fig. \[fig:polarisability\]. We thank L.-M. Duan, A. Gorshkov, C. Greene, and M. Lukin for stimulating discussions. J. Y. and P. Z. thank Caltech for hospitality during the course of this work, and A.J.D. thanks the Institute for Quantum Information at Caltech for hospitality. Work at JILA is supported by DARPA, NIST and NSF. Work in Innsbruck was supported by the Austrian Science Foundation (FWF) through SFB F40 FOQUS and EuroQUAM\_DQS (I118-N16) as part of the ESF EuroQUAM network, and also by the EU Networks NAMEQUAM and AQUTE.
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abstract: 'We explore Casimir effect on an interacting Bose-Einstein condensate (BEC) inside a cylindrical tube. The Casimir force for the confined BEC comprises of (i) a mean-field part arising from the spatial inhomogeneity of the condensate order parameter, and (ii) a quantum fluctuation part arising from the confinement of Bogoliubov excitations in the condensate. Our analytical result predicts Casimir force on a cylindrical shallow of $^4$He well below the $\lambda$-point, and can be tested experimentally.'
author:
- Shyamal Biswas$^1$
- Saugata Bhattacharyya$^2$
- Amit Agarwal$^3$
title: 'Casimir effect for a Bose-Einstein condensate inside a cylindrical tube'
---
Introduction {#Sec1}
============
Casimir (like) or fluctuation induced force arises due to confinement of fluctuations, either classical or quantum, and depends upon the confining geometry of the system [@Casimir; @Krech; @Golestanian; @Gambassi]. Casimir effect due to the confinement of classical (critical) fluctuations is found to be stronger than the corresponding effect due to the confinement of the quantum (vacuum) fluctuations [@Mohideen; @Chan2; @Hertlein]. Systems with different geometry of the confining plates, eg. sinusoidally corrugated geometry, eccentric-cylindrical geometry, grating geometry, parabolic geometry, concentric-cylindrical geometry, etc, have also been studied both experimentally and theoretically, in the context of the quantum Casimir force [@Emig; @Chen; @Dalvit; @Lambrecht; @Graham; @Teo]. Casimir effects for classical and quantum fluids (specially for superfluid $^4$He) in slab geometry have been the subject of a number of experimental and theoretical works [@Krech; @Chan2; @Biswas2010; @Biswas; @Dohm2014; @Diehl2014]. In this context, Casimir force in a silicon micro-mechanical chip has also been demonstrated, opening up the possibility of tailoring the Casimir force by lithographically made components [@chip]. Since the Casimir effect is strongly affected by the confining geometry of the plates, we naturally take up the study of the Casimir force due to the confinement of quantum fluctuations in a cylindrical geometry.
Casimir effect due to confinement of quantum (vacuum) fluctuations of electromagnetic field in a hollow cylindrical tube was studied initially by DeRaad Jr. and Milton in Ref. [@Milton1981], and later on by a number of authors [@Gosdzinsky; @Milton1999; @Milton2010; @Straley; @Milton2004]. In this article, we consider a Bose-Einstein condensate confined to the cylindrical geometry, and explore the Casimir force arising from the presence of the condensate inside the cylinder. For a confined BEC, while the Bogoliubov excitations in the condensate are responsible for the quantum fluctuation part of the Casimir force, the confinement of the condensate wave function, which obeys Gross-Pitaevskii equation, gives rise to the mean-field part of Casimir force. We explicitly compute these two contributions to the Casimir effect.
We begin this article by revisiting the calculation of the Casimir force due to the confinement of electromagnetic field in a hollow conducting circular-cylinder. Bessel functions \[$J_\sigma(\alpha_{\sigma,n}\rho/R)$ and $J_\sigma(\beta_{\sigma,n}\rho/R)$\], Bessel zeros ($\alpha_{\sigma,n};~\sigma=0,\pm1,\pm2,...;~n=1,2,3,...$), and zeros of the first derivatives of the Bessel functions of the first kind ($\beta_{\sigma,n}$) naturally appear in the modes of quantum fluctuations and in the expression of the vacuum energy. To get the Casimir force from the vacuum energy we apply zeta regularization technique prescribed by Gosdzinsky and Romeo [@Gosdzinsky]. Then by applying these techniques, we study the Casimir effect filling the hollow cylinder by an interacting Bose-Einstein condensate for $T \to 0$. For this case, we begin with the calculation of the grand canonical energy of the interacting BEC in terms of its mean-field and quantum fluctuation (discrete Bogoliubov excitation) parts. We explicitly calculate the contribution to the Casimir force arising from the mean-field and quantum fluctuations parts for both the Dirichlet and the Neumann boundary conditions. Thus we study condensate density dependence of the Casimir force. We apply dimensional regularization cum Chowla-Selberg lattice summation technique [@Milton2010] for the higher orders in condensate density. Finally, we summarize our results in the conclusion.
Casimir effect on the hollow cylinder {#Sec2}
=====================================
Before discussing the Casimir effect for a Bose-Einstein condensate enclosed in a circular-cylinder geometry, let us review the conventional Casimir effect on a hollow circular-cylinder. This will also enable us to establish the condensate-density dependence on the Casimir force, and the validity of our approximations later. Let us consider an infinitely long hollow circular-cylinder of radius $R$ and length $L$, whose axis is along the $z$-direction, and whose radial and angular coordinates are represented by $(\rho, \theta)$. The zero point or vacuum energy of the electromagnetic field inside the cylinder is given by [@Casimir] $$\begin{aligned}
\label{eqn:11}
E=\frac{1}{2}\sum_{\bm{k}}\hbar\omega_{\bm{k}}~,\end{aligned}$$ $\bm{k}$ represents all possible (TM and TE) modes of the vacuum fluctuations compatible to the boundary conditions imposed by the waveguide *i.e.* the circular-cylindrical conductor. While in free space or in the space between two parallel conductors, the two polarization states of the electromagnetic field are equivalent; in the space inside a cylindrical waveguide, the two polarization states known as transverse magnetic (TM) and transverse electric (TE) modes are not equivalent. For the conducting cylindrical shell surface, the TM modes of electromagnetic fluctuations ($B_z=0$ and $E_z|_{\rho=R}=0$) which obey Dirichlet boundary condition are given by the Bessel functions (so that $E_z=J_\sigma(\alpha_{\sigma,n}\rho/R)\text{e}^{i\sigma\theta}\text{e}^{ikz}$), and the Bessel zeros specified by $\alpha_{\sigma,n}$, with $\sigma=0,\pm1,\pm2,...$ and $n=1,2,3,...$, naturally appear in the expression of the vacuum energy [@Jackson]. On the other hand, the TE modes ($E_z=0$ and $\frac{\partial B_z}{
\partial\rho}|_{\rho=R}=0$) inside the cylinder are compatible with the Neumann boundary condition, and they are given by $B_z=J_\sigma(\beta_{\sigma,n}\rho/R)\text{e}^{i\sigma\theta}\text{e}^{ikz}$ where $\beta_{\sigma,n}$s are positive zeros of the first derivatives of the Bessel functions of the first kind [@Jackson]. The two distinct class of modes, as shown in Fig. \[fig1\], contribute simultaneously to the vacuum energy of the conducting cylinder, which can be obtained from Eq. (\[eqn:11\]) to be [@Gosdzinsky; @Milton1999; @Jackson] $$\begin{aligned}
\label{eqn:12}
E= \frac{\hbar c L}{2}\sum_{\sigma,n} \int_{-\infty}^\infty \frac{dk}{2\pi}\bigg[\sqrt{\frac{\alpha_{\sigma,n}^2}{R^2}+k^2}+\sqrt{\frac{\beta_{\sigma,n}^2}{R^2}+k^2}\bigg],\end{aligned}$$ where the first and second terms in the square bracket represent contributions for the TM modes and TE modes respectively. Note that the integral over $k$ in the above expression has a logarithmic divergence which may not be easily removed by applying standard analytic continuation techniques. What work here are ultraviolet cutoff technique with Green’s dyadic formulation for the field strengths [@Milton1981], zeta regulation technique [@Gosdzinsky], mode-by-mode summation technique [@Milton1999], and, to some extent, dimensional regularization technique with Chowla-Selberg lattice summation [@Milton2010].
Integrals in Eq. can be evaluated within dimensional regularization scheme. The first integral can be recast with the substitution $q^2=\frac{\alpha_{\sigma,n}^2}{R^2}+k^2$, as $I_1=2\int_{\alpha_{\sigma,n/R}}^\infty \frac{dq}{2\pi}q^2(q^2-\frac{\alpha_{\sigma,n}^2}{R^2})^{-1/2}$, which for the sake of dimensional regularization, can be further recast as $I_1=2\int_{\alpha_{\sigma,n/R}}^\infty \frac{dq}{2\pi}q^{-2s}(q^2-\frac{\alpha_{\sigma,n}^2}{R^2})^{-d}$ where $s=-1$ and $d=1/2$. It is easy to check that, this integral diverges for $s=-1$ and $d=1/2$. This divergence can be avoided within dimensional regularization scheme by substituting $d=1/2+\epsilon$ (where $\epsilon\rightarrow0$) and $s=-1+\delta$ (where $\delta\rightarrow0$), and by subsequently performing analytic continuation of gamma and zeta functions for $\Re[s]+d\le1/2$. In a similar way, we can regularize the second integral in Eq. . Thus within the dimensional regularization scheme, Eq. can be recast as $$\begin{aligned}
\label{eqn:13o0}
E=L\hbar c[D^{(-2s)}+N^{(-2s)}], \end{aligned}$$ where keeping $s\rightarrow-1$ unaltered we put $d=1/2$ after evaluating both the integrals in Eq. , as, $$\begin{aligned}
\label{eqn:13o1}
D^{(-2s)}=\frac{1}{4\sqrt{\pi}}\sum_{\sigma=-\infty,n=1}^{\infty,\infty}\left(\frac{\alpha^2_{\sigma,n}}{R^2}\right)^{-s}\frac{\Gamma(s)}{\Gamma(\frac{1+2s}{2})},\end{aligned}$$ and $$\begin{aligned}
\label{eqn:13o2}
N^{(-2s)}=\frac{1}{4\sqrt{\pi}}\sum_{\sigma=-\infty,n=0}^{\infty,\infty}\left(\frac{\beta^2_{\sigma,n}}{R^2}\right)^{-s}\frac{\Gamma(s)}{\Gamma(\frac{1+2s}{2})}.\end{aligned}$$ Here ‘D’ and ‘N’ respectively stand for Dirichlet and Neumann boundary conditions. Regularization of $D^{(-2s)}$ in Eq. and that of $N^{(-2s)}$ in Eq. , with the use of Chowla-Selberg lattice-sum formula [@Chowla-Selberg], needs exact values of $\alpha_{\sigma,n}$ and $\beta_{\sigma,n}$ which although appear in closed forms for triangular-cylinder, square-cylinder, *etc*, unfortunately do not appear in closed forms for circular-cylinder. For more details regarding the Chowla-Selberg lattice summation, see– Appendix-A of Ref. [@Milton2010], and also our Appendix-\[A\]. Another problem with the Chowla-Selberg formula for circular-cylinder is that, it is not even applicable for the McMahon asymptotic forms of $\alpha_{\sigma,n}\rightarrow\pi(\sigma/2+n-1/4)$ and $\beta_{\sigma,n}\rightarrow\pi(\sigma/2+n+1/4)$ [@McMahon] as the ‘determinant’[^1] $\triangle\rightarrow0$. Exact renormalized values of $D^{(2)}$ and $N^{(2)}$ were obtained, within the integral forms of the lattice summations (which do not at all need the values of $\alpha_{\sigma,n}$ and $\beta_{\sigma,n}$) with further subtraction of homogeneous energy eigenvalue $\omega_{\bm k}|_{R\rightarrow\infty}$ from each energy eigenvalue $\omega_{\bm k}$ in Eq. , as [@Milton1999; @Gosdzinsky] $$\begin{aligned}
\label{eqn:13o3}
D^{(2)}=0.000614794033/R^2=0.0019314324176/\pi R^2,\end{aligned}$$ and $$\begin{aligned}
\label{eqn:13o4}
N^{(2)}=-0.01417613719/R^2=-0.0445356485/\pi R^2.\end{aligned}$$ Thus we get the total Casimir (or renormalized vacuum) energy as [@Milton1981; @Gosdzinsky] $$\begin{aligned}
\label{eqn:18}
E_c=L\hbar c[D^{(2)}+N^{(2)}]=-0.013561343{\small{~}}L\hbar c/R^2,\end{aligned}$$ and the Casimir force ($f_R=-\frac{\partial E_c}{\partial R}$) as $$\begin{aligned}
\label{eqn:19a}
f_R=-0.027122686{\small{~}}L\hbar c/R^2.\end{aligned}$$ The Casimir pressure ($p_R=f_R/2\pi RL$) is now given by $p_R\approx-0.0043\frac{\hbar c}{R^4}$, which is reasonably strong as $R$ approaches the atomic dimension. For example, if $R=10~\text{\AA}$, the Casimir pressure is $p_R\approx-1\times10^{8}$ N/m$^2$. Such a strong attractive Casimir pressure can possibly be tested for cylindrical graphene sheet, carbon nano-tube, etc. For a circular waveguide of radius $R=0.1$ mm, on the other hand, the Casimir pressure is $p_R\approx-1\times10^{-12}$ N/m$^2$. It is interesting to note that confinement of TE modes dominates that of TM modes, and that $N^{(2)}$ per cross-sectional area of a square-cylinder ($-0.0429968$ [@Milton2010]) is close to that of the circular-cylinder in Eq. with deviation less than $4\%$. This brings a hope that the Casimir effect per cross-sectional area of circular-cylinder would be close to that of a square-cylinder as both are topologically same, and as the later specially is having 4-fold symmetry.
Casimir effect for a BEC inside the circular-cylinder {#Sec3}
=====================================================
The Casimir force on an interacting Bose-Einstein condensate was studied by Biswas *et al* in Ref. \[\] for a film geometry. Here we generalize the same (in a nontrivial manner) for an interacting BEC confined to a cylindrical geometry.
Elementary excitations in the BEC inside the cylindrical tube
-------------------------------------------------------------
Let us consider a Bose gas of $N$ identical particles be kept inside the same cylinder of radius $R$ and length $L$, and take the same set of cylindrical coordinates ($\rho,\theta,z$) to specify position $\textbf{r}$ as in the previous section. For $T\rightarrow 0$, almost all the particles ($N_0\approx N$) form the condensate, and the elementary (phononic as well as Bogoliubov) excitations in the condensate can be treated perturbatively. The elementary excitations of a BEC are generally described by expanding the Hamiltonian up-to the second order in quantum fluctuations over the BEC order parameter ($\sqrt{N_0}\phi_0(\bf r)$), via a standard perturbative approach [@Biswas2010]. We follow a similar approach here for obtaining the elementary excitations for the cylindrical geometry.
Let the position vector and mass of a single Bose particle be denoted as ${\bf r}$ and $M$ respectively. The grand canonical Hamiltonian operator for such a system of interacting Bose gas is given by [@Biswas2010] $$\begin{aligned}
\label{eqn:61}
\hat{\mathcal H}&=&\int\hat{\Psi}^{\dagger}({\bf r})\bigg(-\frac{\hbar^2}{2M}\nabla^2-\mu\bigg)\hat{\Psi}({\bf r})\text{d}^3{\bf r}\nonumber \\ &+&\frac{1}{2}\int\int\bigg(\hat{\Psi}^{\dagger}({\bf r})\hat{\Psi}^{\dagger}({\bf r'})V({\bf r-r'})\hat{\Psi}({\bf r})\hat{\Psi}({\bf r'})\bigg)\text{d}^3{\bf r}\text{d}^3{\bf r'},~~~\end{aligned}$$ where $\mu$ is the chemical potential, $V({\bf r-r'})$ is the inter-particle interaction potential, and $\hat{\Psi}({\bf r})$ is the field operator for Bose particles. For the simplest case, the interaction potential can be considered as $V({\bf r-r'})=g\delta^3({\bf r-r'})$, where $g=4\pi\hbar^2a_s/M$ is the coupling constant and $a_s$ is the s-wave scattering length [@Biswas2010]. The field operator can be expressed in terms of the single particle orthonormal wave functions $\{\phi_i({\bf r})\}$, as $\hat{\Psi}({\bf r})=\sum_{i=0}^{\infty}\phi_i({\bf r})\hat{a}_i$, where $\hat{a}_i$ ($\hat{a}_i^{\dagger}$) annihilates (creates) a Bose particle in the state $\phi_i({\bf r})$. Within a perturbative approach, the grand canonical Hamiltonian can be expressed in terms of the fluctuations (excitations) $\delta\hat{\Psi}({\bf r})=\hat{\Psi}({\bf r})-\sqrt{N_0}\phi_0({\bf r})$, and up-to quadratic order in $\delta \hat{\Psi}$, it is given by [@Biswas2010] $$\begin{aligned}
\label{eqn:62}
\hat{\mathcal H}&=&\Omega_0+\int d^3{\bf r} \left[\delta\hat{\Psi}^{\dagger}({\bf r})\bigg(-\frac{\hbar^2}{2M}\nabla^2\bigg)\delta\hat{\Psi}({\bf r})\text{d}^3{\bf r} + \frac{g\bar{n}}{2} \right. \nonumber \\
&\times&\left. \bigg(2\delta\hat{\Psi}^{\dagger}({\bf r})\delta\hat{\Psi}({\bf r})+\delta\hat{\Psi}^{\dagger}({\bf r})\delta\hat{\Psi}^{\dagger}({\bf r})
+ \delta\hat{\Psi}({\bf r})\delta\hat{\Psi}({\bf r})\bigg) \right],~~~\end{aligned}$$ where $\bar{n}\approx N_0\phi_0^2(\textbf{r})$ denotes the bulk particle density, $\mu\approx g\bar{n}$, and $\Omega_0$ is the grand potential for the condensate, which is given by [@Biswas2010] $$\begin{aligned}
\label{eqn:63}
\frac{\Omega_0}{N_0} = \int\bigg(\frac{\hbar^2}{2M}|\nabla\phi_0(\textbf{r})|^2-\mu|\phi_0(\textbf{r})|^2+\frac{gN_0}{2}|\phi_0(\textbf{r})|^4\bigg)\text{d}^3{\textbf{r}}.\end{aligned}$$ In a Bose-Einstein condensate low energetic excitations are probabilistically favored as temperature of the system decreases. We already have experienced in the previous section that, while the Casimir energy for the TM modes (which obey Dirichlet boundary condition) are positive, that for the TE modes (which obey Neumann boundary condition) are negative. Thus, the natural boundary condition for the elementary excitations in the condensate in an infinitely long cylinder, is expected to be of Neumann type. However, by the definition of the quantum vacuum energy, contribution of the elementary excitations of Dirichlet type is also unavoidable even for $T\rightarrow0$ unless any filtering of modes is engineered (e.g. like that of a resonant cavity) at the two ends of the cylinder. The quantum fluctuations (*i.e.* the elementary excitations) for the Neumann boundary on a cylindrical tube can be expressed as $\delta\hat{\Psi}(\textbf{r})= L (2 \pi \hbar)^{-1} \sum_{\sigma=-\infty,
n=1}^{\infty}\int_{-\infty}^\infty \phi_{p_z,\sigma,n} \hat{a}_{p_z,\sigma,n} {d}p_z$, where $\hat{a}_{p_z,\sigma,n}$ annihilates a Bose particle specified by the state $\phi_{p_z,\sigma,n}(z,\rho,\theta)=\sqrt{\frac{2}{R^2L}}\text{e}^{i\frac{p_zz}{\hbar}}\frac{e^{i\sigma\theta}}{\sqrt{2\pi}}\frac{J_\sigma(\beta_{\sigma,n}\rho/R)}{J_{\sigma+1}(\beta_{\sigma,n})}$ [@Cavity-Modes], whose momentum in the $z$-direction is $p_z$ and whose energy is $\xi=\frac{p_z^2}{2M}+\frac{\hbar^2\beta_{\sigma,n}^2}{2MR^2}$. The Hamiltonian in Eq. can now be diagonalized in terms of the phononic operators through the usual Bogoliubov transformation: [@Roberts] $\hat{a}_{p_z,\sigma,n}=u_{p_z,\sigma,n}\hat{b}_{p_z,\sigma,n}+v_{p_z,\sigma,n}\hat{b}_{-p_z,\sigma,n}^{\dagger}$ , where $u_{p_z,\sigma,n}=\big(\frac{\xi^2+g\bar{n}}{2\epsilon_N(|p_z|,\sigma,n)}+\frac{1}{2}\big)^{1/2}$, $v_{p_z,\sigma,n}=-\big(\frac{\xi^2+g\bar{n}}{2\epsilon_N(|p_z|,\sigma,n)}-\frac{1}{2}\big)^{1/2}$, and $$\label{eqn:64}
\epsilon_N(|p_z|,\sigma,n)=\bigg[\frac{g\bar{n}}{M}\bigg(p_z^2+\frac{\hbar^2\beta_{\sigma,n}^2}{R^2}\bigg)\bigg(1+\frac{p_z^2+\frac{\hbar^2\beta_{\sigma,n}^2}{R^2}}{4Mg\bar{n}}\bigg)\bigg]^{\frac{1}{2}}. \\$$
For the TM modes (i.e. for the Dirichlet boundary condition), on the other hand, the Bogoliubov excitation energy would be $\epsilon_D(|p_z|,\sigma,n)$ which is almost like $\epsilon_N(|p_z|,\sigma,n)$ in Eq. except $\beta_{\sigma,n}$ be replaced by $\alpha_{\sigma,n}$ everywhere. For the TM modes, we also call the phononic annihilation and creation operators by $\hat{d}_{p_z,\sigma,n}$ and $\hat{d}_{p_z,\sigma,n}^{\dagger}$ respectively instead of $\hat{b}_{p_z,\sigma,n}$ and $\hat{b}_{p_z,\sigma,n}^{\dagger}$ used for the TE modes.
Thus from Eqs. and , and also from the subsequent discussion about the TM modes, we have the diagonalized Hamiltonian in terms of both the (Neumann and Dirichlet) types of the Bogoliubov excitations, as $$\begin{aligned}
\label{eqn:65}
\hat{{\mathcal H}}&=& \Omega_0+{\mathcal E}_0(R,\bar{n}) + \sum_{p_z,\sigma,n}\epsilon_N(|p_z|,\sigma,n)\hat{b}_{p_z,\sigma,n}^{\dagger}\hat{b}_{p_z,\sigma,n}\nonumber\\&&+ \sum_{p_z,\sigma,n}\epsilon_D(|p_z|,\sigma,n)\hat{d}_{p_z,\sigma,n}^{\dagger}\hat{d}_{p_z,\sigma,n}, \end{aligned}$$ where $$\begin{aligned}
{\mathcal E}_0(R,\bar{n})&=&\frac{1}{2}\sum_{p_z,\sigma,n}\bigg[ \epsilon_N(|p_z|,\sigma,n)-\bigg(\frac{p_z^2}{2M}+\frac{\hbar^2\beta_{\sigma,n}^2}{2MR^2}\bigg)-g\bar{n}\bigg]\nonumber\\&+&\frac{1}{2}\sum_{p_z,\sigma,n}\bigg[\epsilon_D(|p_z|,\sigma,n)-\bigg(\frac{p_z^2}{2M}+\frac{\hbar^2\alpha_{\sigma,n}^2}{2MR^2}\bigg)-g\bar{n}\bigg]\nonumber\\\end{aligned}$$ is the contribution to the grand canonical potential due to the quantum fluctuations of the phonon field. For $T\rightarrow 0$, there are no phonons, and consequently, the Hamiltonian of the system for the vacuum of the phonons is specified by $ {\mathcal H}=\Omega_0+{\mathcal E}_0(R,\bar{n}) $. Roberts and Pomeau also deduced a similar term for a homogeneous condensate in Ref. [@Roberts] based on the original work of Lee, Huang and Yang [@Lee].
Quantum fluctuation part of the Casimir force
---------------------------------------------
In this subsection we calculate the contribution to the Casimir force arising from the second term in Eq. , [*i.e. *]{} the quantum (vacuum) fluctuation part (${\mathcal E}_0$) of the grand canonical Hamiltonian ($\hat{{\mathcal H}}$). Before proceeding with further calculations, we anticipate that the Casimir force due to vacuum fluctuations of the phonon field ($\delta\hat{\Psi}$) would be similar to that due to the photon (electromagnetic) field in Eq. with the replacement of the speed of light by the speed of phonon (sound), $c \to v(\bar{n})=\sqrt{g\bar{n}/M}$. Thus, using Eqs. , , and , we expect the phonon Casimir force to be $$\begin{aligned}
\label{eqn:67}
f_R^{(\text{phn})}=\frac{2L\hbar v(\bar{n})}{R}[N^{(2)}+D^{(2)}]\approx-0.027122686\frac{\hbar v(\bar{n})L}{R^3}.~~\end{aligned}$$
Let us now go back to the Eq. whose second term ${\mathcal E}_0(R,\bar{n})$, gives the quantum vacuum fluctuations as well as the quantum Casimir force for phonons. Expanding $\epsilon_N(|p_z|,\sigma,n)$ in terms of $p=\sqrt{p_z^2+\frac{\hbar^2\beta_{\sigma,n}^2}{R^2}}$ and $\epsilon_D(|p_z|,\sigma,n)$ in terms of $p'=\sqrt{p_z^2+\frac{\hbar^2\alpha_{\sigma,n}^2}{R^2}}$, we have $$\begin{aligned}
\label{eqn:68}
{\mathcal E}_0(R,\bar{n})&=&\frac{1}{2}\sum_{p_z,\sigma,n}\bigg(v(\bar{n})p\left[1-\frac{p}{\sqrt{4Mg\bar{n}}}+\frac{p^2}{8Mg\bar{n}} \right.\nonumber\\
&-& \left. \frac{p^4}{128M^2g^2\bar{n}^2}+...\right]-g\bar{n}\bigg)\nonumber\\&&+\frac{1}{2}\sum_{p_z,\sigma,n}\bigg(v(\bar{n})p'\left[1-\frac{p'}{\sqrt{4Mg\bar{n}}}+\frac{p'^2}{8Mg\bar{n}} \right.\nonumber\\
&-& \left. \frac{p'^4}{128M^2g^2\bar{n}^2}+...\right]-g\bar{n}\bigg).\end{aligned}$$ Here the first and the second summations correspond to the TE and TM modes respectively. Now, replacing the sum over $p_z$ by an integration over $p_z$, we can rewrite Eq. as $$\begin{aligned}
\label{eqn:69}
{\mathcal E}_0(R,\bar{n})&=&L\bigg(v(\bar{n})\bigg[\hbar (N^{(2)}+D^{(2)})-\frac{\hbar^{2}(N^{(3)}+D^{(3)})}{\sqrt{4Mg\bar{n}}}\nonumber\\
&&+\frac{\hbar^{3}(N^{(4)}+D^{(4)})}{8Mg\bar{n}}-\frac{\hbar^{5}(N^{(6)}+D^{(6)})}{128M^2g^2\bar{n}^2}+...\bigg]\nonumber\\
&&-g\bar{n}(N^{(1)}+D^{(1)})\bigg).\end{aligned}$$ where $q=p/\hbar$, $N^{(-2s)}=\frac{1}{2\pi}\sum_{\sigma,n}\int_{\frac{\beta_{\sigma,n}}{R}}^\infty q^{-2s}(q^2-\beta_{\sigma,n}^2/R^2)^{-d}= \frac{1}{4\sqrt{\pi}}\sum_{\sigma,n}\left(\frac{\beta^2_{\sigma,n}}{R^2}\right)^{-s}\frac{\Gamma(s)}{\Gamma(\frac{1+2s}{2})}$, $q'=p'/\hbar$, $D^{(-2s)}=\frac{1}{2\pi}\sum_{\sigma,n}\int_{\frac{\alpha_{\sigma,n}}{R}}^\infty q'^{-2s}(q'^2-\alpha_{\sigma,n}^2/R^2)^{-d}= \frac{1}{4\sqrt{\pi}}\sum_{\sigma,n}\left(\frac{\beta^2_{\sigma,n}}{R^2}\right)^{-s}\frac{\Gamma(s)}{\Gamma(\frac{1+2s}{2})}$, $d\rightarrow1/2$, and $s\rightarrow-1/2,-1,-3/2,-2,-5/2,...$. We already have mentioned various regularization method for $N^{(-2s)}$ and $D^{(-2s)}$ in Sec. \[Sec2\]. Among such integrals cum lattice summations, $N^{(2)}$ and $D^{(2)}$ can be exactly obtained within zeta regularization scheme as we have already mentioned: $N^{(2)}=-0.01417613719/R^2$ and $D^{(2)}=0.000614794033/R^2$ [@Gosdzinsky]. It is absolutely not easy to obtain regularized values of $N^{(-2s)}$ and $N^{(-2s)}$ for $s\le0$ generalizing the works of the authors of Refs. [@Gosdzinsky] and [@Milton1999]. At the end of Sec. \[Sec2\], we have argued that the Casimir force per cross-sectional area of circular-cylinder would be close to that of a square-cylinder as both are topologically same, and as the later specially is having 4-fold symmetry. Evaluation of $N^{(-2s)}$ is reasonably easy for square-cylinder within dimensional regularization and subsequent Chowla-Selberg lattice summation technique [@Milton2010]. See– Appendix-\[A\] for the use of Chowla-Selberg lattice summation in this subsection too. Following this technique we get $N^{(4)}=0.00914223/R^4$, $D^{(4)}=-0.000263472/R^4$, $N^{(6)}=-0.0185855/R^6$, $D^{(6)}=-0.000389195/R^6$, $N^{(8)}=0.0745135/R^8$, $D^{(8)}=0.00400674/R^8$, and $N^{(1)}=D^{(1)}=N^{(3)}=N^{(D)}=N^{(5)}=D^{(5)}=0$. Substituting these values of $N^{(-2s)}$ and $D^{(-2s)}$ in Eq. , we have $$\begin{aligned}
\label{eqn:70}
{\mathcal E}_0(R,\bar{n})&=&-\frac{0.013561343~\hbar v(\bar{n})L}{R^2}\bigg(1\nonumber\\
&-&\frac{0.008878758}{0.013561343\times4\tilde{\rho}(R,g\bar{n})} \\
&-&\frac{0.018974695}{0.013561343\times32\tilde{\rho}^2(R,g\bar{n})}+...\bigg),~ \nonumber \end{aligned}$$ where $\tilde{\rho}(R,g\bar{n})=\frac{g\bar{n}2MR^2}{\hbar^2}$ is the dimensionless density. The quantum (vacuum) fluctuation part of the Casimir force, $f_R^{\text{(qf)}}=-\frac{\partial}{\partial R}{\mathcal E}_0(R,\bar{n})$, is now easily obtained from Eq. , as $$\begin{aligned}
\label{eqn:71}
f_R^{(\text{qf})}=-0.019\frac{\hbar^2L}{MR^4}\tilde{\rho}^{1/2}\bigg(1-\frac{0.655}{\tilde{\rho}}-\frac{0.262}{\tilde{\rho}^2}+...\bigg).~~\end{aligned}$$ Note that the first term of the Eq. is the most dominating term for the quantum phonon fluctuations part of the Casimir force, and it is identical to Eq. , which is just based on an intuitive similarity to the case of Casimir force arising the TE modes of the vacuum fluctuations of the electromagnetic field.
Mean field part of the Casimir force
------------------------------------
In this subsection we calculate the contribution to the Casimir force arising from the first term in Eq. , [*i.e. *]{} the mean field part ($\Omega_0$) of the grand canonical Hamiltonian (${\mathcal H}$). Mean-field part of the Casimir force arises due to inhomogeneity of the BEC order parameter \[$m(\textbf{r})=\sqrt{N_0}\phi_0(\textbf{r})$\] in $\Omega_0$. If the order parameter is complex, it can be expressed as $m(\textbf{r})=m(\rho)\text{e}^{i(\sigma\theta+kz)}$ where the amplitude $m(\rho)$ takes care of inhomogeneity of the condensate density ($n_0(\textbf{r})=|m(\textbf{r})|^2$) along the radial direction and the phase $\sigma\theta+kz$ accounts for the flow of the condensate either rotationally along the $\hat{\theta}$ direction or translationally along the $\hat{z}$ direction. For simplicity, we consider the condensate to be static inside the cylinder so that the condensate order parameter can be written as a real function of the radial coordinate as $m(\textbf{r})=m(\rho)$. We can express the Ginzburg-Landau form in Eq. in terms of the BEC order parameter $m(\rho)$ as, $$\begin{aligned}
\label{eqn:mf1}
\Omega_0=\frac{\hbar^2N_0}{M}\int\bigg[\frac{1}{2}[\nabla m(\rho)]^2-\frac{|a|}{2}m^2(\rho)+\frac{b}{4}m^4(\rho)\bigg]\text{d}^3\textbf{r},~\end{aligned}$$ where we have defined, ${a} = \frac{2M\mu}{\hbar^2}=\frac{2Mg\bar{n}}{\hbar^2}$ and $ b = \frac{2MgN_0}{\hbar^2}>0$. The minimization of the energy in Eq. with respect to the order parameter $m(\rho)$, [*i.e. *]{}$ \frac{\delta\Omega_0}{\delta m}=0$, leads to the following equation for the profile of the order parameter, $$\label{eqn:39}
-\nabla^2m-|a|m+bm^3=0~,$$ which can be rewritten in terms of the cylindrical polar coordinates as $$\label{eqn:40}
-\frac{\text{d}^2m}{\text{d}\rho^2}-\frac{1}{\rho}\frac{\text{d}m}{\text{d}\rho}-|a|m+bm^3=0~.$$ Note that, for a large system, the order parameter is expected to be homogeneous, and in the bulk limit we have $m \to m_0=\sqrt{|a|/b}$.
The mean-field component of the force is given by $F_R^{(\text{mf})}=-\frac{\delta \Omega_0}{\delta R}$. It is not necessary that the boundary condition on the condensate has to be the same as that assigned to the elementary excitation in the condensate as they are independent. In a realistic situation of a superfluid/condensate in contact with a metallic plate the boundary condition is considered to be of Dirichlet type [@Chan; @Krech; @Biswas]. For Dirichlet boundary condition on a cylindrical surface and the ground sate of the system, [*i.e. *]{} for $ m(R)=0, \frac{dm}{d\rho}|_{\rho=0}=0$[^2], we obtain, $$\begin{aligned}
\label{eqn:42}
F_R^{(\text{mf})}=\frac{\pi\hbar^2N_0}{M} LR\bigg(\frac{\text{d}m}{\text{d}\rho}\bigg)^2\bigg|_{\rho=R}.\end{aligned}$$ Eq. is physically appealing as it relates the energy density at the boundary ($\propto (d_\rho m|_R)^2$) to the mean-field part of the Casimir pressure, and a similar form is also expected in other confined geometries [@Biswas2010]. The value of $\big(\frac{\text{d}m}{\text{d}\rho}\big)\big|_{\rho=R}$ is to be obtained from the non-linear Eq. which does not have a closed form analytical solution to the best of our knowledge. However, we can obtain an approximate form for the same. Multiplying Eq. by $2\frac{\text{d}m}{\text{d}\rho}$ and integrating over $\rho$, we get $$\begin{aligned}
\label{eqn:43}
\left(\frac{\text{d}m}{\text{d}\rho}\right)^2+\int_0^\rho\frac{2}{\rho}\left(\frac{\text{d}m}{\text{d}\rho}\right)^2\text{d}\rho&=&-|a|\left[m^2-m^2(0)\right]\nonumber\\&+&\frac{b}{2}\left[m^4-m^4(0)\right].\end{aligned}$$ The second term in the LHS in Eq. , can be integrated by parts: $\int_0^\rho\frac{2}{\rho}\big(\frac{\text{d}m}{\text{d}\rho}\big)^2\text{d}\rho=2\big(\frac{\text{d}m}{\text{d}\rho}\big)^2|_{0}^\rho-\{\frac{\text{d}}{\text{d}\rho}\big(\frac{1}{\sqrt{\rho}}\frac{\text{d}m}{\text{d}\rho}\big)^2\}\rho^2|_{0}^\rho+\int_0^\rho\{\frac{\text{d}^2}{\text{d}\rho^2}\big(\frac{1}{\sqrt{\rho}}\frac{\text{d}m}{\text{d}\rho}\big)^2\}\rho^2\text{d}\rho$. Now as $\rho\rightarrow0$, $\frac{1}{\sqrt{\rho}}\frac{\text{d}m}{\text{d}\rho}\rightarrow0$ for the ground state. Additionally, the derivatives of $\frac{1}{\sqrt{\rho}}\frac{\text{d}m}{\text{d}\rho}$ can also be neglected in comparison to the first term as the order parameter for the ground state is smoothest. Under these approximations, $\int_0^\rho\frac{2}{\rho}\big(\frac{\text{d}m}{\text{d}\rho}\big)^2\text{d}\rho \approx 2\big(\frac{\text{d}m}{\text{d}\rho}\big)^2$. Thus Eq. (\[eqn:43\]) can be rewritten as $-\int_{m(0)}^0\frac{\text{d}m}{\sqrt{-|a|\big(m^2-m^2(0)\big)+\frac{b}{2}\big(m^4-m^4(0)\big)}}\approx\int_0^R\frac{\text{d}\rho}{\sqrt{3}}$, which in turn can be expressed as $$\label{eqn:44}
\frac{\sqrt{3}}{\sqrt{1-\eta}}K\big(\eta/(1-\eta)\big)=R\sqrt{|a|}~,$$ where $K(x)=(\pi/2)[1+x/4+(9/64)x^2+(25/256)x^3+...]$ is the complete elliptic integral of the first kind, and $\eta=bm^2(0)/2|a|\le1/2$. Note that $\eta \le 1/2$, just implies the physical requirement that $m(0)^2 \le m_0^2 = |a|/b$. Based on the approximations above, the mean-field force in Eq. (\[eqn:42\]) can be recast in terms of $\eta$ as $$\begin{aligned}
\label{eqn:45}
F_R^{(\text{mf})}=\frac{2\pi\hbar^2N_0LR}{3M}\frac{\eta|a|^2}{b}(1-\eta),~~\end{aligned}$$ where $\eta$ is to be obtained from the graphical solution of Eq. , which does not appear in closed form. However, an approximate closed form of $\eta$ was obtained by Biswas *et al* in the context of film geometry [@Biswas]. Following their approach, for cylindrical geometry, we get $$\label{eqn:46}
\eta(x)=\frac{1}{2}\text{tanh}^2\bigg(\frac{1}{\sqrt{6}}\sqrt{x^2-\frac{3\pi^2}{4}}\bigg)~,$$ where $x = R \sqrt{|a|}$. We emphasis that Eq. gives a very close approximation [^3] of the actual roots of Eq. . It is clear from Eq. (\[eqn:46\]) that $\eta\rightarrow1/2$ as $R\rightarrow\infty$, and $\eta=0$ for $R\sqrt{|a|}\le\sqrt{3}\pi/2$. Thus we get the mean-field bulk pressure exerted by the condensate on the cylindrical wall as $F_R^{(\text{mf})}/2\pi RL|_{R\rightarrow\infty}=g\bar{n}^2/6$ in comparison to $g\bar{n}^2/2$ for the film geometry [@Biswas2010]. Now, it is easy to obtain the mean-field Casimir force, $f_R^{(\text{mf})}=F_R^{(\text{mf})}-F_\infty^{(\text{mf})}$, from Eqs. -, and it is given by $$\label{eqn:47}
f_R^{(\text{mf})}=-\frac{2\pi\hbar^2N_0LR|a|^2}{3Mb}\left[\frac{1}{4}-\eta(1-\eta)\right].$$ We can rewrite the mean-field force in Eq. in terms of the dimensionless density ($\tilde{\rho}(R,g\bar{n})=\frac{g\bar{n}2MR^2}{\hbar^2}$) and substituting the original expressions of $a$ ($=\frac{2Mg\bar{n}}{\hbar^2}$), $b$ ($=\frac{2MgN_0}{\hbar^2}$) and the coupling constant g ($=\frac{4\pi\hbar^2a_s}{M}$), as $$\begin{aligned}
\label{eqn:72}
f_R^{(\text{mf})}=-\frac{1}{48}\frac{\hbar^2L}{MR^4}\frac{R}{a_s}\tilde{\rho}^2~\text{sech}^4\bigg(\frac{\sqrt{\frac{4\tilde{\rho}}{3}-\pi^2}}{2\sqrt{2}}\bigg).\end{aligned}$$
Total Casimir force
--------------------
The total Casimir force is now simply given by the sum of the terms on the RHS of Eq. and Eq. , as $$\begin{aligned}
\label{eqn:73}
f_R=f_R^{(\text{mf})}+f_R^{(\text{qf})}~.\end{aligned}$$ We emphasis that the mean-field part of the Casimir force is a direct consequence of the inhomogeneity of the condensate, and it dominates over the phononic contribution for low densities. For high densities, the condensate inside the cylinder becomes essentially homogeneous, and the phononic contribution to the Casimir force dominate the mean-field part of the Casimir force for typical values of the parameters. We plot this force in units of $\frac{\hbar^2L}{MR^4}$ in Fig. \[fig2\], for different values of interaction strength - $a_s/R$.
While the Casimir-Polder force between a cigar shaped BEC and a flat surface has been observed to be attractive [@Harber], the Casimir effect for a BEC inside a circular-cylinder although has not been observed; yet is predicted, according to Eq. , to be attractive. This prediction is potentially relevant for atomic waveguides, i.e. geometries in which BECs can be transported. If the velocity of the BEC in the atomic waveguide is less than the Landau critical velocity beyond which the BEC can be destroyed by inelastically interacting with the wall, then the Bogoliubov excitations in the moving condensate would be unaltered from that for the static case. So, the Casimir force for the slowly moving condensate in an atomic waveguide is expected to be attractive along the radial ($\hat{\rho}$) direction, and it may not affect the flow along the perpendicular ($z$) direction. On the other hand, if the velocity of the condensate is more than the Landau critical velocity, then the condensate can inelastically interact with the wall. Since the attractive Casimir force between any two parts (semi-cylindrical walls) of the cylindrical tube may induce shrinking of its cross-section, there would be an enhancement of inelastic collisions with the cylindrical wall. Thus, the attractive Casimir force may slowdown the condensate if it is strong enough to shrink the cross-section. This is possible for a sufficiently small radius of a cylindrical waveguide made up of a flexible wall. However, such forces are usually not strong enough to deform cylindrical wall of a metallic waveguide.
Conclusion {#Sec4}
==========
In this article, we have presented theoretical calculation on quantum Casimir force (and pressure) exerted by a self-interacting Bose-Einstein condensate confined to a circular-cylinder by applying zeta regularization technique introduce be Gosdzinsky and Romeo [@Gosdzinsky] for the leading order in condensate density, and dimensional regularization cum Chowla-Selberg lattice summation technique [@Milton2010] for the higher orders in condensate density. In this regard, we have approximately computed the (grandcanonical) self-energy of the confined condensate up-to the quadratic order in fluctuations. Both the inhomogeneity of the order parameter of the confined BEC and the vacuum fluctuations of the phonon field in the BEC contribute to the Casimir force.
Our analytic results, for the Casimir force, are valid only for the repulsive interaction ($a_s>0$). For $a_s<0$, the condensate becomes unstable beyond a critical number of particles [@Biswas2009]. The Casimir force exerted by the condensate would be minimum for $\tilde{\rho} = 3 \pi^2/4$. For high density, the Bose condensate inside the cylinder essentially becomes homogeneous, and the quantum fluctuation of the phonon field become important. For noninteracting case, there will not be any Bogoliubov excitation. So, fluctuation part of the Casimir force would be zero. But, the mean field part of the Casimir force is not necessarily zero, as because, the profile of the order parameter is still given by $m_{p_z,0,1}(z,\rho,\theta)=\sqrt{\frac{2}{R^2L}}\text{e}^{i\frac{p_zz}{\hbar}}\frac{J_0(\alpha_{0,1}\rho/R)}{J_{1}(\alpha_{0,1})}$[^4] whose derivative at the cylindrical surface ($\rho=R$) is nonzero. Our present calculation can be extended for finite temperature by further consideration of the confinement of thermal fluctuations of the thermal cloud over the condensate [@Biswas2007].
In a realistic experimental scenario, liquid $^4$He can be placed inside a cylindrical shell. Well below the $\lambda$-point ($T_\lambda=2.18$ K), the liquid can become a Bose-Einstein condensate, and the condensate would exert Casimir force on the cylindrical surface. For 1 $\mu$m radius of the shallow of the ultra-cold $^{4}$He liquid ($a_s=2.5\text{\AA}$ [@Leggett]), where there is no classical critical fluctuations, the Casimir pressure is predicted from Eq. to be of the order of $-20\times10^{-12}$ N/m$^2$ for the unit dimensionless density ($\tilde{\rho}\rightarrow1$).
While McMahon asymptotic expansion of roots of the Bessel functions and their first derivative fails to reproduce the fluctuation part of the Casimir force as one can obtain from the prediction of Milton, DeRaad Jr., Gosdzinsky and Romeo [@Milton1981; @Gosdzinsky], analytic continuation of $\sum_{n=1}^\infty\alpha^{-2}_{\sigma,n}$ from the exact result ($1/4(\sigma+1)$ [@Sneddon1960]) may open a door to the exact evaluation of $\sum_{\sigma,n}\alpha^{2}_{\sigma,n}$ and to reproduce the Dirichlet b.c. part. This method is to be generalized for the Neumann b.c. part to predict the actual scenario from the roots of the first derivative of the Bessel functions of the first kind.
Casimir effect could have been further studied by considering a small gap between the condensate and the cylindrical surface generalizing the effect for two concentric cylinders [@Teo]. Instead of keeping the BEC inside the cylinder, we could keep a Fermi liquid (say $^3$He). How to calculate the Casimir force, for a Fermi liquid in the confined geometry, is an open question.
S. Biswas and A.A. gratefully acknowledge funding from the INSPIRE Faculty Award by the DST (Govt. of India). S. Biswas further acknowledges the hospitality of the Department of Physics, IIT-Kanpur during the initial phase of this work. A.A. also acknowledges funding from the Faculty Initiation Grant of IIT-Kanpur, India. Several useful discussions with J.K. Bhattacharjee (HRI, India) and S. Dutta Gupta (University of Hyderabad, India) are gratefully acknowledged. Useful comments of K.A. Milton (University of Oklahoma, USA) prior to the preparation of the present form of the manuscript is also gratefully acknowledged.
[99]{} H. B. G. Casimir, Proc. K. Ned. Akad. W. **51**, 193 (1948). M. Krech and S. Dietrich, [](http://dx.doi.org/10.1103/PhysRevA.46.1886). M. Kardar and R. Golestanian, [](http://dx.doi.org/10.1103/RevModPhys.71.1233). A. Gambassi, [J. Phys.: Conf. Ser. **161**, 012037 (2009)](http://dx.doi.org/10.1088/1742-6596/161/1/012037). U. Mohideen and A. Roy, [](http://dx.doi.org/10.1103/PhysRevLett.81.4549); G. Bressi, G. Carugno, R. Onofrio and G. Ruoso, [](http://dx.doi.org/10.1103/PhysRevLett.88.041804). A. Ganshin, S. Scheidemantel, R. Garcia, and M. H. W. Chan, [](http://dx.doi.org/10.1103/PhysRevLett.97.075301). C. Hertlein, L. Helden, A. Gambassi, S. Dietrich and C. Bechinger, [](http://dx.doi.org/10.1038/nature06443). T. Emig, A. Hanke, R. Golestanian, and M. Kardar, [](http://dx.doi.org/10.1103/PhysRevLett.87.260402). F. Chen, U. Mohideen, G. L. Klimchitskaya, and V. M. Mostepanenko, [](http://dx.doi.org/10.1103/PhysRevLett.88.101801). D. A. R. Dalvit, F. C. Lombardo, F. D. Mazzitelli, and R. Onofrio, [](http://dx.doi.org/10.1103/PhysRevA.74.020101). A. Lambrecht and V. N. Marachevsky, [J. Phys.: Conf. Ser. **161**, 012014 (2009)](http://iopscience.iop.org/1742-6596/161/1/012014); F. Impens, A. M. C. Reyes, P. A. Maia Neto *et al*, [](http://dx.doi:10.1209/0295-5075/92/40010). N. Graham, A. Shpunt, T. Emig, S. J. Rahi, R. L. Jaffe, and M. Kardar, [](http://dx.doi.org/10.1103/PhysRevD.83.125007). L. P. Teo, [](http://dx.doi:10.1209/0295-5075/96/10006). S. Biswas, J. K. Bhattacharjee, D. Majumder *et al*, [](http://dx.doi.org/10.1088/0953-4075/43/8/085305). S. Biswas, J. K. Bhattacharjee, H. S. Samanta, S. Bhattacharyya, and B. Hu, [](http://dx.doi.org/10.1088/1367-2630/12/6/063039). V. Dohm, [](http://dx.doi.org/10.1103/PhysRevE.90.030101). H. W. Diehl *et al*, [](http://dx.doi.org/10.1103/PhysRevE.89.062123). J. Zou, Z. Marcet *et al*, [Nat. Commun. **4**, 1845 (2013)](http://dx.doi.org/10.1038/ncomms2842). L. L. DeRaad Jr. and K. A. Milton, [Ann. Phys. **136**, 229 (1981)](http://dx.doi.org/10.1016/0003-4916(81)90097-X). P. Gosdzinsky and A. Romeo, [](http://dx.doi.org/10.1016/S0370-2693(98)01164-2). K. A. Milton, A. V. Nesterenko, and V. V. Nesterenko, [](http://dx.doi.org/10.1103/PhysRevD.59.105009). E. K. Abalo, K. A. Milton, and L. Kaplan, [](http://dx.doi.org/10.1103/PhysRevD.82.125007). J. P. Straley, G. A. White, and E. B. Kolomeisky, [](http://dx.doi.org/10.1103/PhysRevA.87.022503). For a review, see – K. A. Milton, [](http://dx.doi.org/10.1088/0305-4470/37/38/R01). J. D. Jackson, *Classical Electrodynamics*, 3rd Ed., Sec. 8.2 & 8.6 (New York: John Wiley, 1999). S. Chowla and A. Selberg, [Proc. Natn’l. Acad. Sc. (USA) **35**, 371 (1949)](http://dx.doi.org/10.1073/pnas.35.7.371). For higher orders of the asymptotic expansion, see – J. McMahon, [Ann. Math. **9**, 23 (1895)](http://dx.doi.org/10.2307/1967501). For the cylindrical cavity modes ($\phi_{p_z,\sigma,n}(z,\rho,\theta)$) of the scalar field inside the circular-cylinder, specially for Dirichlet boundary condition, see – S. Bhattacharyya and J. K. Bhattacharjee, [](http://dx.doi.org/10.1103/PhysRevB.60.R746). For Neumann boundary condition, Bessel zeros ($\alpha_{\sigma,n}$) in $\phi_{p_z,\sigma,n}(z,\rho,\theta)$ are to be replaced by the zeros of the first derivative ($\beta_{\sigma,n}$) of the Bessel functions of the first kind. See – Refs. [@Jackson; @Gosdzinsky] for similar replacement specially for electromagnetic field inside a conducting circular-cylinder. D. C. Roberts and Y. Pomeau, [](http://dx.doi.org/10.1103/PhysRevLett.95.145303); [arXiv:cond-mat/0503757](http://arxiv.org/abs/cond-mat/0503757). T. D. Lee, K. Huang, and C. N. Yang, [](http://dx.doi.org/10.1103/PhysRev.106.1135). R. Garcia and M. H. W. Chan, [](http://dx.doi.org/10.1103/PhysRevLett.83.1187). D. M. Harber, J. M. Obrecht, J. M. McGuirk, and E. A. Cornell, [](http://dx.doi.org/10.1103/PhysRevA.72.033610). S. Biswas, [Eur. Phys. J. D [**55**]{}, 653 (2009)](http://dx.doi.org/10.1140/epjd/e2009-00221-7). S. Biswas, [](http://dx.doi.org/10.1088/1751-8113/40/33/002); S. Biswas, [](http://link.springer.com/article/10.1140%2Fepjd%2Fe2007-00007-y). A. J. Leggett, [](http://dx.doi.org/10.1103/RevModPhys.73.307). I. N. Sneddon, Proc. Glasgow Math. Assoc. **4**, 144 (1960).
Chowla-Selberg Lattice Summation {#A}
================================
Chowla-Selberg lattice summation formula involves evaluation of Epstein zeta function [@Chowla-Selberg] $$\begin{aligned}
\label{eqn:ap1}
Z(s)=\Sigma^{'} (am^2+bmn+cn^2)^{-s},\end{aligned}$$ where $s$ is a complex number, and the summation is for all integers $m,n$ (each going from $-\infty$ to $+\infty$), while the prime indicates that $m=n=0$ is excluded from the summation; further $a>0$ and $c>0$, while $b$ is real number and subject to the condition that the ‘determinant’ $\triangle=4ac-b^2>0$. This is called lattice summation because the summation is over the lattice points ($\{m,n\}$) on the two-dimensional $m-n$ plane. The Epstein zeta function $Z(s)$ expressed in terms of the lattice summation over $m$ and $n$ in Eq. , is defined for $\Re[s]>1$, and can be continued analytically over the whole s-plane, and satisfies a functional equation similar to the one satisfied by the Riemann zeta Function [@Chowla-Selberg]. Such an analytic continuation, in the context of Casimir effect on different cylindrical geometries, was greatly exercised by Abalo-Milton-Kaplan in Ref. [@Milton2010]. For the evaluation of the Epstein zeta function, we have Chowla-Selberg lattice summation formula, as [@Chowla-Selberg; @Milton2010] $$\begin{aligned}
\label{eqn:ap2}
Z(s)&=&\Sigma^{'} (am^2+bmn+cn^2)^{-s}\nonumber\\&=&2a^{-s}\zeta(2s)+\frac{2^{2s}\sqrt{\pi}a^{s-1}}{\Gamma(s)\triangle^{s-1/2}}\zeta(2s-1)\zeta(s-1/2)\nonumber\\&&+\frac{2^{s+3/2}\pi^s}{\Gamma(s)\triangle^{s/2-1/4}\sqrt{a}}\sum_{n=1}^\infty n^{s-1/2}\sigma_{1-2s}(n)\nonumber\\&&\times\cos(n\pi b/a)2K_{1/2-s}(n\pi\sqrt{\triangle}/a)~, \end{aligned}$$ where $K_{\nu}$ is the modified Bessel function of the second kind of order $\nu$, and $\sigma_{k}(n)$ is a divisor function, which is defined as sum of $k$-th powers of the divisors of $n$, as $$\begin{aligned}
\label{eqn:ap3}
\sigma_k(n)=\sum_{d|n}d^k~;\end{aligned}$$ e.g. divisors of $n=6$ are $1,2,3,6$, so that $\sigma_0(6)=1^0+2^0+3^0+6^0=4$, $\sigma_1(6)=1^1+2^1+3^1+6^1=12$, $\sigma_2(6)=1^2+2^2+3^2+6^2=50$, etc.
For a square-cylinder of cross-sectional area $\tilde{L}^2=\pi R^2$, which is equal to that of the circular-cylinder of radius $R$, $\alpha_{\sigma,n}^2$ in Eq. would be $\alpha_{\sigma,n}^2=(\sigma^2+n^2)\pi^2$ where $\sigma=1,2,3,...$ and $n=1,2,3,...$. The space of the lattice summation over $\sigma$ and $n$ is now one fourth of that over $\sigma$ and $n$ in Eq. except the summation over the axes ($\sigma=0$ & $n\neq0$ and $\sigma\neq0$ & $n=0$) on the $\sigma-n$ plane. Thus we recast Eq. , for $s\rightarrow-1$, as $$\begin{aligned}
\label{eqn:ap4}
D^{(-2s)}&=&\frac{1}{4\sqrt{\pi}}\sum_{\sigma=1,n=1}^{\infty,\infty}\left(\frac{\alpha^2_{\sigma,n}}{\tilde{L}^2}\right)^{-s}\frac{\Gamma(s)}{\Gamma(\frac{1+2s}{2})}\nonumber\\&=&\frac{1}{4\sqrt{\pi}}\frac{\Gamma(s)}{\Gamma(\frac{1+2s}{2})}\bigg(\frac{1}{\pi R^2}\bigg)^{-s}\sum_{\sigma=1,n=1}^{\infty,\infty}\left(\alpha^2_{\sigma,n}\right)^{-s}\nonumber\\&=&\frac{\pi^2}{4\sqrt{\pi}}\frac{\Gamma(s)}{\Gamma(\frac{1+2s}{2})}\frac{1}{(\pi R^2)^{-s}}\frac{1}{4}\bigg[\Sigma'\left(\sigma^2+n^2\right)^{-s}\nonumber\\&&-4\zeta(2s)\bigg].~~~~~~\end{aligned}$$ Analytic continuation of Eq. can be obtained by using the lattice summation formula in Eq. and the reflection property of the zeta function, $$\begin{aligned}
\label{eqn:ap5}
\Gamma(s)\zeta(2s)=\Gamma\big(\frac{1-2s}{2}\big)\zeta(1-2s)\pi^{2s-1/2},\end{aligned}$$ for $s\rightarrow-1$, as [@Milton2010] $$\begin{aligned}
\label{eqn:ap6}
D^{(2)}=0.00483155/\pi R^2=0.00153793/R^2.\end{aligned}$$
For Neumann boundary condition on the square-cylinder, on the other hand, we have $\beta_{\sigma,n}^2=\alpha_{\sigma,n}^2$ except this time $\sigma=0,1,2,...$ and $n=0,1,2,...$. So, this times, half of the two axes of $\sigma-n$ plane would contribute to the lattice summation unlike that in Eq. . This would result the only difference between $N^{(2)}$ and $D^{(2)}$. Thus $N^{(2)}$ in Eq. , for the square-cylinder, would be [@Milton2010] $$\begin{aligned}
\label{eqn:ap7}
N^{(2)}=D^{(2)}-\frac{\zeta(3)}{8\pi^2R^2}=-\frac{0.0429968}{\pi R^2}=-\frac{0.0136863}{R^2}.~~~\end{aligned}$$
Eq. obtained for the square-cylinder of cross-sectional area $\pi R^2$ is not only true for $s\rightarrow-1$ but also for any $\Re[s]\le0$ as far as dimensional regularization and subsequent analytic continuation is concerned. Thus, following the similar steps, as described from Eq. to Eq. , we get $N^{(4)}=0.00914223/R^4$, $D^{(4)}=-0.000263472/R^4$, $N^{(6)}=-0.0185855/R^6$, $D^{(6)}=-0.000389195/R^6$, $N^{(8)}=0.0745135/R^8$, $D^{(8)}=0.00400674/R^8$, etc. On the other hand, for negative half-integral values of $s$, since $\Gamma(\frac{1+2s}{2})\rightarrow\infty$ in Eq. , we have $D^{(-2s)}=N^{(-2s)}=0$.
[^1]: If $\alpha^2_{\sigma,b}=\pi^2(a\sigma^2+b\sigma n+cn^2)$, then the ‘determinant’ is defined as $\
triangle=4ac-b^2$.
[^2]: For the ground state, $m(\rho)=\sum_{n=1}^\infty a_n\cos(\frac{n\pi r}{2R})$, which leads to $\frac{\text{d}m}{\text{d}\rho}|_{\rho=R}=-\frac{dm}{dR}$.
[^3]: For a comparison see– Fig. 1 in Ref. [@Biswas]
[^4]: If the coupling constant ($g$) be zero, there will be no difference between the order parameter and the ground state cavity mode $\phi_{p_z,0,1}(z,\rho,\theta)$.
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abstract: 'In order to understand the physical mechanisms underlying non-steady stellar spiral arms in disk galaxies, we analyzed the growing and damping phases of their spiral arms using three-dimensional $N$-body simulations. We confirmed that the spiral arms are formed due to a swing amplification mechanism that reinforces density enhancement as a seeded wake. In the damping phase, the Coriolis force exerted on a portion of the arm surpasses the gravitational force that acts to shrink the portion. Consequently, the stars in the portion escape from the arm, and subsequently they form a new arm at a different location. The time-dependent nature of the spiral arms are originated in the continual repetition of this non-linear phenomenon. Since a spiral arm does not rigidly rotate, but follows the galactic differential rotation, the stars in the arm rotate at almost the same rate as the arm. In other words, every single position in the arm can be regarded as the co-rotation point. Due to interaction with their host arms, the energy and angular momentum of the stars change, thereby causing the radial migration of the stars. During this process, the kinetic energy of random motion (random energy) of the stars does not significantly increase, and the disk remains dynamically cold. Owing to this low degree of disk heating, the short-lived spiral arms can recurrently develop over many rotational periods. The resultant structure of the spiral arms in the $N$-body simulations is consistent with some observational nature of spiral galaxies. We conclude that the formation and structure of spiral arms in isolated disk galaxies can be reasonably understood by non-linear interactions between a spiral arm and its constituent stars.'
author:
- 'Junichi <span style="font-variant:small-caps;">Baba</span>, Takayuki R. <span style="font-variant:small-caps;">Saitoh</span>, and Keiichi <span style="font-variant:small-caps;">Wada</span>'
bibliography:
- 'ms.bib'
title: 'Dynamics of Non-Steady Spiral Arms in Disk Galaxies'
---
Introduction
============
The spiral arms of disk galaxies are the most prominent structures, and the arms are formed due to gravitationally driven variations in the surface density in the stellar disk [@RixZaritsky1995; @Grosbol+2004; @Zibetti+2009; @Elmegreen+2011]. It is known that the spiral arms of disk galaxies can be excited by tidal interactions with nearby companion galaxies [e.g., @Oh+2008; @Dobbs+2010; @Struck+2011], as well as by the central stellar bar [e.g., @SellwoodSparke1988]. In addition, spiral arms can also be self-induced and maintained without these external gravitational perturbations, and further, they can propagate stationary density waves in globally stable disks, as hypothesized by @LinShu1964. However, the physical origin and dynamical evolution of spiral arms in disk galaxies have thus far not been fully understood. In this study, we focus on the dynamics of stellar spiral arms and stars around them in a disk galaxy without external perturbations and a bar structure.
The Lin-Shu theory posits that the stellar spiral arms can be interpreted as (quasi-) stationary density waves with a constant pattern speed [@LinShu1964; @BertinLin1996]; the theory has gained wide acceptance as providing a clear explanation regarding the dynamics of spiral arms. However, as pointed out in the study by @Toomre1969, the quasi-stationary hypothesis has a serious limitation because of a dispersal nature of the tight winding spiral waves, they radially propagate with the group velocity, and are consequently absorbed at the Lindblad resonances [@LyndenBellKalnajs1972]. Therefore, the stationary waves that last for a long duration require some amplification mechanisms such as WASER [@Mark1976] and a feedback cycle that involves the reflection of the inward propagating wave into an outward propagating one at the $Q$-barrier [@Bertin+1989a; @Bertin+1989b].
However, nearly all the previous time-dependent simulations that have been executed thus far have been unable to prove the existence of stationary density waves in a disk galaxy without external perturbations and a bar structure [^1]. @SellwoodCarlberg1984 claimed that self-induced stellar spiral structures in stellar disks are not stationary and fade away within an interval of about ten rotational periods. Their study stresses on the significance of cooling mechanisms, such as dissipation by the interstellar medium (ISM), that are required to maintain stellar spiral arms. More recently, @Fujii+2011 used 3-D $N$-body simulations to show that although stellar spiral arms are short-lived, they are also recurrently formed, and as a result, the spiral features are maintained over 10 Gyr [see also @Sellwood2011]. This is because of negative feedback that causes the dynamical heating of stars in the disk galaxies due to spirals. The non-steady spirals are also subject to a similar phenomenon when the dynamics of the ISM are self-consistently solved with stars in disk galaxies. Both the stellar spirals and the ISM undergo motion in the wake of the the galactic rotation [see also @Grand+2012]. In fact, the spirals can be considered as being “wound”. The ISM forms dense regions associated with the non-steady stellar spirals; however, these regions are not the conventional “galactic shocks”[@Wada+2011].
The existence of non-steady stellar spirals has been suggested by certain observations in our Galaxy. For example, the age-velocity dispersion relation of stars in the solar neighborhood [e.g., @Holmberg+2007] most naturally accounts for the existence of the non-steady spirals [@CarlbergSellwood1985; @BinneyLacey1988; @JenkinsBinney1990; @deSimone+2004]. @Baba+2009 analyzed the kinematics of star-forming regions using numerical simulation data of a barred galaxy such as the Milky Way galaxy [@Baba+2010], and they found that the kinematics of these regions are consistent with the observed peculiar motions of maser sources. These studies support the view that the Galactic stellar spiral arms are non-steady instead of stationary density waves.
However, the current theoretical understanding of the dynamics of non-steady spirals is thus far insufficient. It has been suggested that a key factor towards understanding their dynamics is the orbital evolution of stars. In this study, we analyze the orbits of stars associated with non-steady spirals in the growth and damping phases.
In Section \[sec:Method\], we summarize the numerical model and the method, which are the same as reported by @Wada+2011, except that the present model does not include the ISM, i.e., our study involves pure $N$-body simulations. We examine the global evolution of the stellar disk in Section \[sec:GlobalMorphology\], in which we show the non-steady nature of spirals developed in the galactic disk. In Section \[sec:SpiralDynamics\], we focus on a typical spiral arm and describe its dynamical evolution, i.e., the amplification (Section \[sec:AmplifySpiralArm\]) and destruction (Section \[sec:DampingSpiralArm\]) processes. Section \[sec:StarAroundSpiralArm\] examines the evolution of stars around the growing spiral arm from the viewpoint of the angular momentum-energy space. Finally, in Section \[sec:Discussion\], we compare our simulation results with observations, and we provide a perspective view on grand-design spirals. The grand-design spirals (i.e., $m=2$ spirals) [^2] will be the subject of our future studies.
Models and Methods {#sec:Method}
==================
The simulations of the formation and evolution of spiral arms were carried out using a three-dimensional $N$-body simulation. The initial point of our simulation was a disk galaxy in a nearly equilibrium state. The density profile of the stellar disk in this case is given by $$\rho_{\rm sd}(R,z) = \frac{M_{\rm sd}}{4\pi {R_{\rm sd}}^2 z_{\rm sd}}
\exp(-R/R_{\rm sd}) {\rm sech}^2(z/z_{\rm sd}),\label{eq:stellardisk}$$ where $M_{\rm sd}$ is the mass of the stellar disk, $R_{\rm sd}$ is the radial scale-length, and $z_{\rm sd}$ is the vertical scale-length. Since the distribution function of a stellar disk is unknown, we generated the equilibrium state of the stellar disk by using an empirical method: We first generated a stellar disk in a near-equilibrium state based on a Maxwellian approximation [@Hernquist1993], and we subsequently allowed the disk to evolve for 6 Gyr under the constraint of axisymmetry [@McMillanDehnen2007; @Fujii+2011].
We modeled a fixed potential of the dark matter (DM) halo whose density profile follows the NFW profile [@Navarro+1997]: $$\rho_{\rm h}(r)=\frac{\rho_0}{r/r_{\rm s}(1+r/r_{\rm s})^2},$$ where $$\rho_0 = \frac{M_{\rm h}}{4\pi {R_{\rm h}}^3}
\frac{{C_{\rm NFW}}^3}{\ln(1+C_{\rm NFW})+C_{\rm NFW}/(1+C_{\rm NFW})},$$ $$C_{\rm NFW} = R_{\rm h}/r_{\rm s},$$ an the terms $C_{\rm NFW}$, $M_{\rm h}$, and $R_{\rm h}$ denote the halo concentration parameter of the halo profile, the DM halo mass, and the virial radius, respectively. The model parameters adopted here are summarized in Table \[tbl:models\]. The circular velocity, velocity dispersions, $Q$-values and $\Gamma$-profile of the stellar disk are shown in Figure \[fig:rotcurve\]. Here, $\Gamma$ denotes a dimensionless shear rate of the disk as defined by the expression $-d\ln\Omega/d\ln R$, where $\Omega$ denotes the angular frequency.
We used the simulation code [ASURA]{} [@Saitoh+2008; @Saitoh+2009], where the self-gravity of stellar particles is calculated by the Tree with the GRAPE method [@Makino1991]. We used a software emulator of GRAPE, the Phantom-GRAPE [@Tanikawa+2012]. The opening angle was set to $0.5$ with the center-of-mass approximation. To perform a time integration, we used a leapfrog integrator with variable and individual time-steps. We used a Plummer softening length of $\epsilon = 30$ pc. This value is sufficiently small to resolve the three-dimensional structure of a disk galaxy [@Hernquist1987; @Bottema2003]. It is to be noted that the formation of the spiral arms is inhibited if the adopted softening length is considerably larger than the disk thickness ($\sim 300$ pc). The number of particles is 300 million.
Component Parameter Value
---------------------- ------------------------------- ---------------------------------
Dark halo Mass ($M_{\rm h}$) $6.3\times 10^{11}$ M$_{\odot}$
(Rigid) Radius ($R_{\rm h}$) $122$ kpc
Concentration ($C_{\rm NFW}$) $5.0$
Initial stellar disk Mass ($M_{\rm sd}$) $3.2\times 10^{10}$ M$_{\odot}$
(Live) Scale length ($R_{\rm sd}$) $4.3$ kpc
Scale height ($z_{\rm sd}$) $0.3$ kpc
: Model parameters for each mass component in $N$-body simulation
\[tbl:models\]
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Global Evolution of Spirals {#sec:GlobalMorphology}
===========================
Figure \[fig:snapshotsMS\] shows the time evolution of the stellar disk. Hereafter, we use a rotational period measured at $R = 2 R_{\rm sd} (=8.6~\rm kpc)$, $T_{\rm rot}$, which corresponds to 325 Myr. The top panels show the face-on views of the stellar disk. The middle panels show the radial distributions of the spiral modes which are analyzed by performing a one-dimensional Fourier decomposition of the disk surface density using the polar coordinates $(R,\phi)$: $$\begin{aligned}
\frac{\Sigma(R,\phi)}{\Sigma_{\rm 0}(R)}
= \sum_{m=0}^{\infty} A_m(R)\cos\{m[\phi-\phi_m(R)]\}.\end{aligned}$$ Here, $A_m$ and $\phi_m(R)$ denote the Fourier amplitude and phase angle for the $m$-th mode, respectively [e.g, @RixZaritsky1995]. Upon execution of the simulation, spiral patterns initially developed from the noise with an $e$-folding time of about 4 galactic rotations ($T_{\rm rot} \sim 4$), and subsequently, the patterns settled to a nearly constant level, consistent with the report by @Fujii+2011. While the global features were in the quasi-steady state, the local spiral features were not static. The most prominent mode always changed on a rotational time scale, and it also showed radial dependence.
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An important dynamical feature of the time-dependent multi-arm spirals is that the spirals “co-rotate” with the galactic rotation. Figure \[fig:PatternSpeed\] shows the angular pattern speed of a dominant stellar spiral ($m=4$). The angular frequency is analyzed by the rate of change of phase $\phi_{\rm m=4}(R)$ during two galactic rotation periods. The pattern speed is not constant. In fact, the pattern speed decreases with radius in a manner similar to galactic rotation for any radii. This result shows that there is no “single” co-rotating point in the disk, in contrast to stationary spiral waves (or bar) with a rigid rotation. Therefore, no single spiral can last for more than one rotational period (see Section \[sec:StarAroundSpiralArm\]). @Wada+2011 and @Grand+2012 have also reported that the spiral arms in their simulations are co-rotating, winding and short-lived. Observationally, the phenomenon of co-rotating spiral arms has been confirmed in at least a few of the nearby spiral galaxies such as M51, NGC 1068, M101, IC 342, NGC 3938, and NGC 3344 using the Tremaine-Weinberg (TWR) analysis [@Merrifield+2006; @Meidt+2008; @Meidt+2009].
It is noteworthy that the pattern speed curve seems slightly flatter (particularly when $R \sim 10-15$ kpc) than the circular rotation curve. In order to investigate the cause of the flattening, we divided this period into five periods and performed the same analysis for each period (Figure \[fig:PatternSpeedAppendix\]). The pattern speeds in the all periods decrease with increasing radius in a manner similar to galactic rotation for any radii; however, they show slightly flatter distributions at certain instants (e.g., $T_{\rm rot}= 12.4-12.8$). This flatter distribution lasts for less than one galactic rotation period ($\Delta T_{\rm rot}<1$). Thus, this structure is not a long-lasting structure.
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Further, it is significant that the gravitational scattering of stars by the spiral arms is not sufficiently large to erase all the spiral features. The curves in Figure \[fig:rotcurve\]c indicate that the value of $Q$ increases from an initial value of $1.2$ to a final value of $1.4$ around $R \approx 2 R_{\rm sd}$ at $T_{\rm rot} = 15$. However, the spirals are not completely erased due to dynamical heating [see @Fujii+2011]. It is to be noted that heating effect along the vertical direction to the disk plane is negligibly small (bottom panels in Figure \[fig:snapshotsMS\]), which is consistent with the results of previous studies [e.g., @BinneyLacey1988; @JenkinsBinney1990].
Growth and Damping Phases of Stellar Spiral Arms {#sec:SpiralDynamics}
================================================
In this section, we investigate the dynamical evolution of co-rotating spiral arms in detail. Figures \[fig:snapshotModelMSgrowth\] and \[fig:snapshotModelMSdamping\] show the time evolution of a spiral arm in the co-rotating frame. A weak density enhancement can be observed around $\phi \simeq 240-300^{\circ}$ and $R \simeq 7-10$ kpc for $T_{\rm rot} = 11.7$. This density enhancement causes the growth of a prominent spiral arm until $T_{\rm rot} = 12.0$, and the arm has a maximum amplitude around $T_{\rm rot} \sim 12.2-12.3$. Beyond $T_{\rm rot} \simeq 12.3$, this spiral arm rapidly fades out ($T_{\rm rot} = 12.3-12.5$). The arm merges with with a neighboring weak spiral arm ($T_{\rm rot} = 12.5-12.7$), and the peak density contrast again becomes $\delta \sim 1$ when $T_{\rm rot} = 12.8$. In the following sections, we discuss the growth phase (Figure \[fig:snapshotModelMSgrowth\]) and the damping phase (Figure \[fig:snapshotModelMSdamping\] ) of the spiral separately.
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Growth Phase {#sec:AmplifySpiralArm}
------------
In the growth phase of the spiral arm ($T_{\rm rot} \simeq 12.0-12.2$), the pitch angle of the arm reduces due to the differential rotation of the disk. This is clearly observed in the right hand panels of Figures \[fig:snapshotModelMSgrowth\] and \[fig:snapshotModelMSdamping\]; the two peaks at $2R_{\rm sd}+1.5$ kpc and $2R_{\rm sd}-1.5$ kpc show increasing separation in terms of $\phi$. More quantitatively, Figure \[fig:PitchAngleAmplitude\] shows the evolution of the spiral arm along the pitch angle - density contrast ($i-\bar{\delta}$) plane. Here, we calculated the pitch angle ($i$) of the spiral arms using the relation $$\begin{aligned}
\tan i = \frac{1}{R}\frac{\Delta R}{\Delta\phi}, \end{aligned}$$ where $R=2 R_{\rm sd}$ (= 8.6 kpc), $\Delta R = 3$ kpc, and $\Delta \phi$ is the azimuthal angle difference between the contrast peaks at $R=2 R_{\rm sd} - 1.5$ kpc (dot-dashed lines) and $R=2 R_{\rm sd} + 1.5$ kpc (dotted lines) shown in Figures \[fig:snapshotModelMSgrowth\] and \[fig:snapshotModelMSdamping\]. Here, we evaluated the positions of the contrast peaks at each radius by visual inspection. The arm density contrast, $\bar{\delta}$, is calculated by averaging the corresponding density contrast over the radial range in Figures \[fig:snapshotModelMSgrowth\] and \[fig:snapshotModelMSdamping\]. As the pitch angle of the spiral arm decreases from $i \simeq 40^\circ$ ($T_{\rm rot} =12.0$) to $i \simeq 32^\circ$ ($T_{\rm rot} =12.20$), the density contrast increases to a maximum, and subsequently, it decreases with increase in the pitch angle. Thus, the spiral arm has a maximum amplitude when $i \sim 32^\circ$.
This amplification process associated with the galactic shear motion is known as the swing amplification [@Toomre1981; @GoldreichLynden-Bell1965; @JulianToomre1966; @ToomreKalnajs1991] [^3]. @Fuchs2001a calculated grid models of swing amplification process by varying the shear rate $\Gamma$ for $Q=1.4$. In his work, he derived a fitting empirical equation for swing amplification (eq.98 in his paper) as the following: $$\begin{aligned}
\tan i_{\rm th}
= 1.932 - 5.186 \left(\frac{1}{2}\Gamma \right) + 4.704 \left(\frac{1}{2}\Gamma \right)^2,\end{aligned}$$ where $i_{\rm th}$ is the pitch angle at which the spiral arm reaches the maximum amplitude. We adopt this equation to evaluate the predicted pitch angle $i_{\rm th}$. In our model, $\Gamma \simeq 0.8$ around $R = 2R_{\rm sd}$ (Figure \[fig:rotcurve\]c). Thus, substituting $\Gamma \simeq 0.8$ in eq. (7), we obtain $i_{\rm th} \simeq 32^\circ$. This value is consistent with the evolution of the spiral arm with the maximum contrast (hatched region in Figure \[fig:PitchAngleAmplitude\]).
![ Evolution of spiral arm on $i-\bar{\delta}$ plane for $T_{\rm rot} = 12.0 - 12.5$. The hatched region corresponds to the predicted maximum pitch angle around the analyzed region ($Q \simeq 1.4$ and $\Gamma \simeq 0.75-0.85$) due to swing amplification (refer to equation (7)). []{data-label="fig:PitchAngleAmplitude"}](fig7.eps){width=".45\textwidth"}
Damping Phase {#sec:DampingSpiralArm}
-------------
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The swing amplification mechanism can explain certain aspects of the evolution of spiral arms, i.e., the amplification (excitation) of density enhancement. However, the destruction process of the non-steady spirals, as seen in Figure \[fig:snapshotModelMSdamping\] [see also @Fujii+2011; @Wada+2011], cannot be understood only by the swing amplification mechanism. The top right panels in Figure \[fig:VelocityProfile\] show the time evolution of the relative velocities ($\Delta v_\eta$ and $\Delta v_\xi$) in the co-rotating frame of the spiral arm. Upon considering the coordinates $(\eta, \xi)$, i.e., the $\eta$- and $\xi$-axes are perpendicular and parallel to the spiral arm, respectively, the velocity component along each is given by $$\begin{aligned}
&& \Delta v_{\eta} \equiv v_{R} \cos{i} + (v_{\phi} - v_{\rm cir})\sin{i},\\
&& \Delta v_{\xi} \equiv -v_{R} \sin{i} + (v_{\phi} - v_{\rm cir})\cos{i},\end{aligned}$$ where $i$ is the pitch angle of the spiral arm, and $v_{\rm cir}$ is the circular velocity determined by the azimuthally averaged gravitational field. For this definition, inflow to the arm corresponds to $\Delta v_{\eta} > 0$ for $\eta < 0$, and $\Delta v_{\eta} < 0$ for $\eta > 0$. In the amplification phase ($T_{\rm rot} \lesssim 12.2$), we see a clear inflow motion to the spiral arm along both sides of the chosen strip (solid lines). In the initial stages of the damping phase ($T_{\rm rot} \simeq 12.3$), the streaming velocity around the spiral arm gradually transits from inflow to outflow. At the end of the damping phase ($T_{\rm rot} \simeq 12.5$), it is clear that the stars in the spiral arm move away from the arm along both its sides (dot-dashed lines). During the dynamical evolution, the parallel component of the velocity, $\Delta v_{\xi}$, does not change its sign or show a decrease its magnitude. The left panels of Figure \[fig:VelocityProfile\] show a schematic view of the time evolution of the non-circular velocity field associated with the non-steady spiral arm.
The middle right panels in Figure \[fig:VelocityProfile\] show the time evolution of the non-axisymmetric gravitational forces (i.e., spiral perturbation) involved in the damping phase. The component of the non-axisymmetric gravitational force perpendicular to the spiral arm ($\Delta F_{\rm grav,\eta}$), which is stronger than the parallel component ($\Delta F_{\rm grav,\xi}$), is always directed towards the arm ($\Delta F_{\rm grav,\eta} > 0$ for $\eta < 0$, and $\Delta F_{\rm grav,\eta} < 0$ for $\eta > 0$) during the dynamical evolution. The bottom right panels in Figure \[fig:VelocityProfile\] show the time evolution of the net force (i.e., non-axisymmetric gravitational force plus Coriolis force perturbation). The component of the net force perpendicular to the spiral arm ($\Delta F_{\rm grav+cori,\eta}$) evolves in the same manner as $\Delta F_{\rm grav,\eta}$ because there is a strong density gradient along the $\eta$-direction. The parallel component $\Delta F_{\rm grav,\xi}$ is almost zero during the growth and damping phases and does not change. This is because the density gradient along the $\xi$-direction is small and almost unchanged. However, the parallel component of the net force $\Delta F_{\rm grav+cori,\xi}$ changes its $\eta$-dependence from the growing phase ($T_{\rm rot} = 12.1$) to the damping phase ($T_{\rm rot} > 12.3$) according to the change of the sign of the perpendicular velocity ($\Delta v_{\eta}$), since the Coriolis force works perpendicular to the direction of the velocity.
This indicates that the Coriolis force exerted on the stars in the damping phase exceeds the non-axisymmetric gravitational force due to the spiral perturbation. This causes stars to “escape” from the spiral perturbation, and eventually the spiral arm itself begins to thin out and fade.
The above argument suggests that the non-steady nature of stellar spirals is originated in the evolution of the orbits of the stars in the spirals. The phenomenon of swing amplification [@Toomre1981] is a part of this non-linear coupling between particles and waves, but it does not describe all observed phenomena. In the next section, we explore the orbital evolution of stars associated with the spiral arm in detail.
Orbital Evolution of Stars Around Spiral Arms {#sec:StarAroundSpiralArm}
=============================================
Figure \[fig: orbit\_linear\] shows the evolution of stars along the $\phi-R$ plane in the early phase of spiral development. At this stage of the simulation, we selected 15 particles associated with one of the three weak spiral arms that had evolved at $T_{\rm rot} = 4.0$. The figure shows that stars with epicycle motion are captured by the density enhancement ($T_{\rm rot} = 3.6-4.0$), and further, these stars are dissociated from the original arm ($T_{\rm rot} = 4.2-4.6$). This behavior is similar to that of the “density wave” from a certain viewpoint, but the spiral arms are short-lived and co-rotating (left panel of Figure \[fig:PatternSpeed\]). Thus, even if the arms appeared in the early linear phase, they are not completely explained by the picture of stationary density waves. The middle column of images in Figure \[fig: orbit\_linear\] shows the plots of the azimuth angle ($\phi$) versus the angular momentum $L_{z}$ curve instead of the $\phi-R$ plot. It is clear that all the stars along the $\phi-L_{z}$ plane oscillate horizontally when the angular momentum of each star is conserved. The images in the right column of Figure \[fig: orbit\_linear\] show the so-called Lindblad diagram, where the angular momentum $L_{z}$ of each star is plotted against its total energy $E$. Most of the stars show no significant movement from their original position, thereby suggesting that their energy is also conserved.
On the other hand, the behavior of the stars is very different in the non-linear phase, where the spiral arms are well-developed and non-steady (refer to previous sections). Figure \[fig:EvolutionSpiral\] shows images identical to those in Figure \[fig: orbit\_linear\]; however, in this set of images, the stars are in motion due to change in their angular momenta in the non-linear phase of orbital evolution (see also the supplementary video). When the stars are captured by the density enhancement ($T_{\rm rot} \simeq 11.8-12.0$), they radially migrate along the spiral arms. The stars approaching from behind the spiral arm (i.e., inner radius) tend to attain increased angular momenta via acceleration along the spiral arm, whereby they move to the disk’s outer radius. In contrast, the stars approaching ahead of the spiral arm (i.e., outer radius) tend to lose their angular momenta via deceleration along the spiral arm, and they move to the disk’s inner radius. Along the $\phi-L_{z}$ plane, the stars oscillate both horizontally as well as vertically. Moreover, the guiding centers of the oscillations do not remain constant at the same value of $L_{z}$. This is essentially different from the epicycle motion in which $L_{z}$ is conserved.
The Lindblad diagram in Figure \[fig:EvolutionSpiral\] shows that the stars oscillate along the curve of circular motion by undergoing change in terms of both angular momentum and energy. The oscillating stars successively undergo aggregation and disaggregation along the curve, thereby leading to the formation of structures referred to as “swarms of stars” along the $\phi-L_{z}$ and $R-\phi$ planes. The non-steady nature of the spiral arms originates in the dynamical interaction between these “swarming” stars with a [*non-linear*]{} epicycle motion and the high-density regions, i.e., the spiral arms moving with the galactic rotation. This is similar to the wave-particle interaction described previously; however, in this case, the high-density regions are not “waves”. Since the non-steady spirals move at the rate of the local galactic rotational speed (Section 3), in contrast to a spiral perturbation with a single pattern speed, co-rotating points are found everywhere on the spiral arms. Therefore, the motion of stars along the $E-L_{z}$ plane can be naturally understood as due to their scattering around co-rotating spirals [^4]. This behavior is similar to reported in previous studies [e.g., @SellwoodBinney2002; @Grand+2012; @Bird+2011; @Grand+2012b]. Furthermore, we observe that the $E-L_{z}$ curve of each star changes over a large range of radii. This is entirely different from what is expected in stationary density waves, where these changes are limited to the Lindblad resonances [@LyndenBellKalnajs1972]. Further, it is noteworthy that the structures formed self-induced in the angular momentum space; this is a property similar to that of the “groove” mode hypothesized by @SellwoodLin1989. This point is beyond the scope of this paper. We will investigate this elsewhere.
The time evolutions of the angular momentum of stars associated with a spiral arm at $T_{\rm rot} \simeq 4.0$ and $T_{\rm rot} \simeq 12.2$ are plotted in the left top panels of Figures \[fig:L-t04\] and \[fig:L-t12\], respectively. In the early phase ($T_{\rm rot} \simeq 4.0$), both the angular momentum and random energy do not change by more than 10% during one rotational period (top right panel of Figure \[fig:L-t04\]). On the other hand, in the non-linear phase ($T_{\rm rot} \simeq 12.2$), the fraction of the angular momentum changes by $\sim$ 50% (top right panel of Figure \[fig:L-t12\]). It is clear that the angular momentum of the stars changes significantly due to their scattering by a well-developed spiral at $T_{\rm rot} = 12.2$. This corresponds to the hypothesis that the guiding center of the epicycle motion of each star undergoes radial motion.
The change in the normalized random energy shown in the left bottom panels of Figures \[fig:L-t04\] and \[fig:L-t12\] indicates that certain stars with relatively small initial random energies experience a large energy change $\Delta E_{\rm rand}$ after interaction with a spiral arm. Here, the random energy is calculated as the difference between the total energy and the circular energy, i.e., $E_{\rm rand} = E - E_{\rm cir}(R_{\rm gc})$. It is noteworthy that the random energy change is [*not*]{} always positive; a significant fraction of stars [*lose*]{} their random energy. This is because the perturbation from the spiral arm shifts the guiding center of the stars’ epicycle motion without increase in orbital eccentricity. It is to be noted that this argument is rigorously correct; in fact, Figure \[fig:L-t12\] (bottom left panel) shows the changes in the random energy of the stars as a function of change in the angular momentum. We can see a weak trend: the outer migrators (i.e., $L_{\rm fin}-L_{\rm ini}>0$) undergo a decrease in their random energy ($E_{\rm rand,fin}-E_{\rm rand,ini}<0$), and vice versa. A similar effect of the radial migration of stars around spiral arms upon disk heating has been noticed in recent numerical simulations [@Grand+2012; @Roskar+2011; @Grand+2012b; @Minchev+2012].
In summary, the gravitational interaction between the stars in the spiral arm and the spiral density enhancement changes the angular momentum and random energy of the stars, and this process in turn changes the structure of the spirals. During this process, the random energy of individual stars in the system does [*not*]{} increase monotonically. In other words, local interactions between the non-steady arms and stars increase or decrease the total energy of individual stars locally; however, the energy remains around its value for circular motion with the occurrence of a small dispersion. This is because the interaction causes the migration of the guiding centers of the stars without increasing their eccentricity or random energy. This “dynamical cooling” mechanism is essential to prevent heating of the stellar disk and erasure of the spiral arms, and the mechanism produces “swarms” of stars moving between non-steady spirals. The non-linear epicycle motion of the stars and their non-linear coupling with the density perturbation is the fundamental physics of the recurrently formed, non-steady spiral arms in a stellar disk.
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Discussion {#sec:Discussion}
==========
Figure \[fig:spiralproperties\] compares the morphological properties (the pitch angle $i$ and the amplitude $|A_{m}|$) of the stellar spiral arms between simulations and observations. For the comparison, we used a 2D fast Fourier transform (2D FFT) analysis with logarithmic spirals [for details, see for e.g., @PuerariDottori1992]. The plotted points in green and red symbols represent the time evolution of the spiral arms for the two models that are essentially identical (see footnote). We found that the simulated spiral arms exhibit a maximum amplitude around $i \sim 30^{\circ}$. Further, we found that the spiral arms tend to be weaker in the pure stellar disk model (model MS) than the model with the ISM (model MSG) [^5]. This distribution which has a peak at around $i \sim 20-30^\circ$ is qualitatively consistent with the the prediction of the swing amplification mechanism (see section 4.1). We compared the simulated distribution with the observed one. The dependence of $|A_{m}|$ on the pitch angle in the models with/without the ISM is consistent with observations at least for the range of values corresponding to $i < 30^{\circ}$. This comparison suggests that the self-induced spiral arms in differentially rotating disks and their time evolution can fairly consistently account for the morphological diversity seen in nearby spiral galaxies.
However, this comparison between the simulated and observed distributions requires further examination. Firstly, the determination of the pitch angle is somewhat uncertain. The analyzed pitch angles differ between studies for the same galaxy. For example, the pitch angles of NGC 3054 are given by $33^\circ$, $43^\circ$, and $12^\circ$ in @Grosbol+2004, @Seigar+2006, and @Davis+2012, respectively. Secondly, there is a lack of statistical studies on the relation between the pitch angle and the amplitude of the stellar spiral arms. Although some statistical studies have used 2D FFT to analyze the pitch angle of the spiral arms, they do not include examination of the amplitude or contrast of the spiral arms [e.g., @Seigar+2005; @Seigar+2006; @Davis+2012]. Thus, further progress in understanding the spiral dynamics requires additional statistical and robust observation data of the morphological properties (both the pitch angle and amplitude) of the stellar spiral arms. This relation between the pitch angle and amplitude will form one of the tests for spiral genesis theories along with the pitch angle-shear rate (or Hubble-type) relation [@Roberts+1975; @SeigarJames1998; @Hozumi2003; @Grand+2012c], existence of systematic angular offset between the young stellar component and the stellar spiral arm [@Fujimoto1968; @Roberts1969; @Egusa+2009; @Foyle+2011; @Ferreras+2012], and the radial dependence of the pattern speed [@Meidt+2008; @Meidt+2009].
Finally, we comment on grand-design spirals. In a series of studies [@Fujii+2011; @Wada+2011] including this one, we have examined the stellar dynamics of non-steady stellar spirals, as well as the interactions between them and the ISM. However, these studies have focused more on multi-armed spirals than grand-design spirals (i.e., $m=2$ spirals) whose fraction is more than $\sim 50\%$ in nearby spiral galaxies [@Grosbol+2004; @Kendall+2011]. It has been observed that grand-design spirals are associated with bars or companions [@KormendyNorman1979; @SeigarJames1998; @Salo+2010; @Kendall+2011], and this observation is consistent with the results of many numerical simulations of bar-driven spirals [e.g., @SellwoodSparke1988; @Bottema2003] and tidally-driven spirals [e.g., @Oh+2008; @Dobbs+2010; @Struck+2011]. We will focus on bar-driven spirals and tidally-induced spirals in forthcoming studies.
![ Pitch angle-amplitude correlation. Morphological properties (amplitudes $A_{\rm m}$ and pitch angles $i$) of stellar spiral arms are derived using a 2D Fourier fitting to the range of radii given by $R_{\rm sd} < R < 3R_{\rm sd}$. Here, $|A_{m}|$ values for the $m=4$ mode in the simulations are shown. Model MSG is nearly identical to Model MS; however, model MSG also includes the gaseous component with a mass fraction that is 10% of the stellar disk mass [refer to @Wada+2011]. The filled triangles represent the observational data obtained from the study by @Grosbol+2004. []{data-label="fig:spiralproperties"}](fig13.eps){width=".4\textwidth"}
Summary
=======
The $N$-body simulations of an isolated disk galaxy show the formation of self-induced, non-steady multi-arm spirals that follow the differential galactic rotation. We found that the swing amplification mechanism causes the development of the spirals. When a spiral undergoes the damping phase, the Coriolis force dominates the gravitational perturbation exerted by the spiral, and as a result, stars escape from the spirals, and join a new spiral at a different position. This process uniformly for a given spiral, thereby resulting in the formation of bifurcating and merging spiral arms; therefore, the dominant spiral modes always show change in their radii over time. We confirmed that this phenomenon originates due to the changing orbital properties of stars. The angular momentum and energy of each star undergo changes due to the star’s interaction with the spiral arms. As a result, the epicycling stars radially migrate; in other words, their guiding centers also undergo motion. Interestingly, the movement of groups of stars with similar orbital properties causes the appearance of “swarming”. In the non-linear phase of development of spiral instability, the swarming stars cause complicated morphological changes in the spiral arms.
During this process, the random energy of individual stars (or orbital eccentricity) does [*not*]{} increase monotonically. In fact, a significant fraction of stars even lose their random energy. This “dynamical cooling” due to the mechanism like the wave-particle interaction can explain why the short-lived spiral arms are self-induced over several rotational periods despite the absence of dissipative component in the disk [@Fujii+2011].
In the above process, it is essential that spiral arms mostly follow the galactic rotation at any radius; in other words, the “co-rotating points” are required to be ubiquitous in the differentially rotating galactic disk. The conclusions of other previous studies also indicate the possibility that the co-rotation resonance affects stellar motions more in terms of radial migration than heating up of the disk [@LyndenBellKalnajs1972; @SellwoodBinney2002; @Grand+2012; @Grand+2012b; @Roskar+2011; @Minchev+2012].
We conclude that the non-linear epicycle motions and self-gravity in the differential rotation of stellar disks are essential for the recurrent amplification and destruction processes of the spiral arms. In other words, the issue of the so-called “winding dilemma” is no longer a problem at least in multi-armed spiral galaxies.
The authors are grateful to the anonymous referee for his/her valuable comments. We would like to thank Michiko S. Fujii, Junichiro Makino, Shunsuke Hozumi, and Daisuke Kawata. Numerical simulations were performed using the Cray XT-4 at the Center for Computational Astrophysics (CfCA), National Astronomical Observatory of Japan. This research was supported by the HPCI Strategic Program Field 5 “The Origin of Matter and the Universe.” This work was supported in part by Grant-in-Aid for Scientific Research (C)23540267.
[^1]: @Thomasson+1990 and @ElmegreenThomasson1993 have suggested that the feedback cycle caused by the reflection at the $Q$-barrier causes the long life of the $m=2-3$ spiral arms. However, their $N$-body spirals are not stationary; instead, a time-dependent evolution of the spiral features is observed. See also @Sellwood2011.
[^2]: Although the term grand-design spiral indicates a reasonably coherent and extensive spiral arm in the stellar mass distribution [See @ElmegreenElmegreen1982 for a more precise definition], in a majority of cases, this means that the galaxy has $m=2$ spiral arms [@Kendall+2011].
[^3]: @DOnghia+2012 have investigated the growth of spiral arms via swing-amplification, and their nonlinear evolution is not fully consistent with the classic swing-amplification picture proved by @JulianToomre1966.
[^4]: Stars around the co-rotation point change their angular momenta without increasing their random energy [@LyndenBellKalnajs1972].
[^5]: Model MSG is the same model as that presented by @Wada+2011. This model is mostly identical to model MS; however, in model MSG, the ISM, star formation, and supernova feedback are also taken into account. The hydrodynamics of the model is solved by the smoothed particle hydrodynamics (SPH) method. The initial gas mass fraction is 10 % of the stellar disk mass, and the initial density profile follows an exponential profile with a scale-length twice that of the stellar disk. Refer to @Wada+2011 for details.
| 0 |
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abstract: 'We consider theoretically effects of random charged impurity disorder on the [*quality*]{} of high-mobility two dimensional (2D) semiconductor structures, explicitly demonstrating that the sample mobility is not necessarily a reliable or universal indicator of the sample quality in high-mobility modulation-doped 2D GaAs structures because, depending on the specific system property of interest, mobility and quality may be controlled by different aspects of the underlying disorder distribution, particularly since these systems are dominated by long-range Coulomb disorder from both near and far random quenched charged impurities. We show that in the presence of both channel and remote charged impurity scattering, which is a generic situation in modulation-doped high-mobility 2D carrier systems, it is quite possible for higher (lower) mobility structures to have lower (higher) quality as measured by the disorder-induced single-particle level broadening. In particular, we establish that there is no reason to expect a unique relationship between mobility and quality in 2D semiconductor structures as both are independent functionals of the disorder distribution, and are therefore, in principle, independent of each other. Using a simple, but reasonably realistic, “2-impurity” minimal model of the disorder distribution, we provide concrete examples of situations where higher (lower) mobilities correspond to lower (higher) sample qualities. We discuss experimental implications of our theoretical results and comment on possible strategies for future improvement of 2D sample quality.'
address: |
$^1$Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111\
$^2$SKKU Advanced Institute of Nanotechnology and Department of Physics, Sungkyunkwan University, Suwon, 440-746, Korea
author:
- 'S. Das Sarma$^1$ and E. H. Hwang$^{1,2}$'
title: Mobility versus quality in 2D semiconductor structures
---
introduction
============
One of the most significant materials developments in the fundamental quantum condensed matter physics, which is not universally known outside the 2D community, has been the astonishing 3,000-fold increase in the low temperature electron mobility of GaAs-based 2D confined quantum systems from $ \sim 10^4$ cm$^2$/Vs in the first modulation-doped GaAs-AlGaAs 2D heterostructures [@stormer] in 1978 to the current world-record mobility of $\sim 3-4 \times 10^7$ cm$^2$/Vs in the best available modulation doped GaAs-AlGaAs quantum wells of today. [@heiblum] This represents a truly remarkable more than three orders of magnitude enhancement in the low temperature ($\sim 1$K) electron mean free path from a rather short microscopic length $\sim 50$ nm in 1978 to the essentially macroscopic length scale of $\sim 0.2$ mm in 2010. This incredible 3,000-fold enhancement of the 2D carrier mean free path, although much less well-known than the celebrated Moore’s law in the Si electronics industry, is actually quantitatively on par with Moore’s law increase in the microprocessor performance. Unlike Moore’s law in Si microelectronics performance, where the motivation has been technological, however, the drive for the mobility enhancement in 2D GaAs structures has been motivated entirely by fundamental physics considerations. Indeed, this increase in the 2D mobility has been accompanied by some of the most spectacular experimental discoveries in modern physics including, for example, the fractional quantum Hall effect (FQHE) [@tsuiprl1982], the even-denominator FQHE [@willettprl1987], the bilayer half-filled FQHE [@shayeganprl1992], the anisotropic stripe and bubble phases [@lillyprl], and many other well-known novel 2D phenomena. (As an aside we point out that, qualitatively similar to the situation in the Moore’s law, the exponential enhancement in the 2D mobility of semiconductor heterostructures has slowed down considerably in the recent years with only a 30% increase in the mobility during the $2003-2013$ period, from $\sim 3\times 10^7$ to $\sim 4 \times 10^7$ cm$^2$/Vs, after roughly a factor of 1,000 increases during $1978-2003$.)
We mention right in the beginning that our interest here is obviously the $T=0$ transport properties (or low-temperature transport properties) with the temperature being much smaller than both the Fermi temperature and the Bloch-Grüneisen (BG) temperature of the 2D system so that all thermal effects have saturated, and we do not need to account for either phonon scattering or finite temperature effects in the Fermi distribution function. The high mobility of 2D semiconductor structures applies only to this low-temperature situation, and at higher temperatures ($>10$K), the mobility is dominated by phonon scattering, a situation already well-studied in the literature[@stormerbg]. Our theory is just restricted only to $T=0$ impurity-scattering-limited 2D transport properties, which limit the ultimate achievable mobility in these systems. It is also important to emphasize that we ignore all weak localization aspects of 2D transport properties, restricting entirely to the semiclassical transport behavior where the concept of a mobility is valid. Thus, the theory applies only at densities where weak-localization behavior does not manifest itself at the experimental temperatures ($\sim 25$ mK $- 2$ K). At a fixed density, this limits our theory to a temperature high enough ($>1$ mK) so that the phase breaking length is shorter than the semiclassical mean free path. The physics of mobility/quality dichotomy discussed in this paper satisfies all of these constraints (i.e. no phonon scattering, no weak localization, temperature much lower than the fermi temperature) very well.
Although the [*mobility*]{} enhancement of 2D systems has generally led to the improvement of sample [*quality*]{} on the average over the years as manifested in the observation of new phenomena, it has been known from the early days that the connection between [*mobility*]{} and [*quality*]{} is at best a statistically averaged statement over many samples and is not unique, i.e., a sample with higher mobility than another sample may not necessarily have a higher quality with respect to some specific property (e.g., the existence or not of a particularly fragile fractional quantum Hall plateau). Thus, higher (lower) mobility does not necessarily always translate into higher (lower) quality for specific electronic properties. Since 2D carrier mobility is a function of carrier density [@dassarmaprb2013], in fact, it is, in principle, possible for a sample to have a higher (lower) mobility than another sample at higher (lower) carrier density, implying that the measured mobility at some fixed high density is not (always) even a good indicator of transport quality itself as a function of carrier density, let alone being an indicator for the quality of other electronic properties!
The reason for the above-mentioned mobility/quality dichotomy is rather obvious to state, but not easy to quantify. Both mobility and quality (e.g., the measured activation gap for a specific FQHE state or some other specified electronic property) depend on the full disorder distribution affecting the system which is in general both unknown and complex, and depends also on the sample carrier density in a complicated manner. The disorder distribution is characterized by many independent parameters, and therefore, all physical properties of the system, being unique functionals of the disorder distribution, are independent of each other. In particular, the dc conductivity $\sigma$, which determines the density-dependent mobility $\mu = \sigma/ne$ where $n$ is the 2D carrier density (and $e$ the magnitude of electron charge), is determined by essentially an integral over the second moment of the disorder distribution whereas other physical properties (i.e., the quality, although there could, in principle, be many independent definitions of sample quality depending on independent experimental measurements of interest) could be determined by other functionals of the disorder distribution. Thus, for any realistic disorder distribution, we do not expect any unique relationship between mobility and quality, and it should be possible, in principle, for samples of different mobility to have the same quality or vice versa (i.e., samples of different quality to have the same 2D mobility).
There is still the vague qualitative expectation, however, that if the sample mobility is enhanced by improving the sample purity (i.e., suppressing disorder), then this should automatically also improve the sample quality (perhaps not necessarily by the same quantitative factor) since reduced disorder should enhance quality. We will show below that this may not always be the case since mobility and quality (for a specific property) may be sensitive to completely different aspects of disorder and therefore enhancing mobility by itself may do nothing to improve quality in some situations. On the other hand, when mobility and quality are both determined by exactly the same microscopic disorder in the sample, increasing (decreasing) one would necessarily improve (suppress) the other although we will see later in this work \[see Fig. \[fig01\](c)\], for example) this is not necessarily true if mobility and quality are both determined by remote dopant scattering arising from long-range Coulomb disorder.
It is important to emphasize a subtle aspect of the mobility/quality dichotomy with respect to experimental samples. Theoretically, we can consider a hypothetical system with continuously variable disorder (i.e., the parameters characterizing the disorder distribution such as the quenched charged impurity density and their strength as well as their spatial locations including possible spatial correlations in the impurity distribution can all be varied at will). Experimentally, however, the situation is qualitatively different. One does not typically change the impurity distribution in a sample in a controlled manner and make measurements as a function of disorder distribution. Experimentally, measurements are made in [*different*]{} samples and compared, and in such a situation there is no reason to expect two samples with identical mobility at some specified carrier density to have identical impurity distributions. The impurity distributions in different samples can be considered to be identical only if the full density-dependent mobility $\mu(n)$, or equivalently the density-dependent conductivity $\sigma(n)$, are identical in all the samples. Such a situation of course never happens in practice, and typically when experimental mobilities in different samples are quoted to be similar in magnitudes, one is referring to either the maximum mobility (occurring typically at different carrier densities in different samples) or the mobility at some fixed high carrier density (and not over a whole range of carrier density). If two samples happen to have the same maximum mobility or the same mobility at one fixed density, there is no reason to expect them to have the same quality with respect to all experimental properties at arbitrary densities. This obvious aspect of mobility versus quality dichotomy has not much been emphasized in the literature.
It should be clear from the discussion above that in the ‘trivial’ (and experimentally unrealistic in 2D semiconductor structures) situation of the system having just one type of impurities uniquely defining the applicable impurity distribution, both mobility and quality, by definition, would be determined by exactly the same impurity configuration since there is just one set of impurities by construction. Such a situation is, in fact, common in 3D semiconductors where the applicable disorder is almost always described by a random uniform background of uncorrelated quenched charged impurity centers, which can be uniquely characterized by a single 3D impurity density $n_i$. Obviously, in this situation both mobility and quality are uniquely defined by $n_i$, and thus increasing (decreasing) $n_i$ would decrease (increase) both mobility and quality (although not necessarily by the same quantitative factor since mobility and quality are likely to be different functions of system parameters in general). In such a simplistic situation, mobility and quality are likely to monotonically connected by a unique relationship, and hence enhancing system mobility should always improve the system quality since the same disorder determines both properties. In fact, this simple situation always applies if the mobility/quality are both limited by purely short-range disorder in the system.
By contrast, 2D semiconductor structures almost always have several qualitatively distinct disorder mechanisms affecting transport and other properties arising from completely different physical origins. For example, it is well-known [@andormp] that Si-MOSFETs have at least three different operational disorder mechanisms: random charged impurities in the insulating SiO$_2$ oxide layer, in the bulk Si itself, and random short-range surface roughness at the Si-SiO$_2$ interface. There may still be other distinct scattering mechanisms associated with still different disorder sources in Si-MOSFETs such as neutral defects or impurities, making the whole situation quite complex. In MOSFETs, low-density carrier transport is controlled by long-range charged impurity scattering whereas the high-density carrier transport is controlled by short-range surface roughness scattering, and this dichotomy may very well lead to situations where a measured electronic property (i.e., “quality”) does not necessarily correlate with the high-density maximum mobility of the system. In high-mobility 2D GaAs-AlGaAs-based systems of interest in the current work, there are at least six distinct scattering mechanisms of varying importance arising from different physical sources of disorder in the system. These are: Unintentional background charged impurities in the 2D GaAs conducting layer; remote dopant impurities in the insulating AlGaAs layer (which are necessary for introducing 2D carriers to form the 2DEG); short-range interface roughness at the GaAs-AlGaAs interfaces; short-range disorder in the insulator AlGaAs layer arising from alloy disorder (and neutral defects); unintentional background charged impurities in the insulating AlGaAs barrier regime; random charged impurities at the GaAs-AlGaAs interface. This is obviously a highly complex situation where the complete disorder distribution will have many independent parameters, and in general, there is no reason to expect a unique relationship between mobility and quality since mobility could be dominated by one type of disorder (e.g., background unintentional charged impurities) and quality may be dominated by a different type of disorder (e.g., remote dopants far from the 2D layer).
It is clear from the above discussion that the minimal disorder model capable of capturing the mobility versus quality dichotomy in high-quality 2D semiconductor structures is a “2-impurity” model with one type of impurity right in the 2D layer itself (arising, for example, from the unintentional background impurities in the system) and the other type of impurity being a remote layer separated by a distance ‘$d_R$’ from the 2D electron layer. This minimal 2-impurity model is characterized by three independent parameters: $n_R$ and $d_R$, denoting respectively the 2D charged impurity density in the remote dopant layer separated by a distance $d_R$ from the 2D carriers, and $n_B$, the 2D impurity density corresponding to the unintentional background impurities in the 2D layer with $d=0$. It is easy to go beyond this minimal model and consider the remote impurities to be distributed over a finite distance (rather than simply being placed in a $\delta$-function like layer located at a distance $d_R$ from the 2D system) or assume the background impurities to be distributed three-dimensionally (rather than in a 2D plane at $d=0$), but such extensions do not modify any of our qualitative conclusion in the current paper as we have explicitly checked numerically. Also, we discuss our theory assuming a strict 2D model (with zero thickness) for the electron layer because the finite quasi-2D layer thickness has no qualitative effect on the physics of quality versus mobility being discussed in this work. Many of our numerical results are, however, obtained by incorporating the appropriate quantitative effects of the quasi-2D layer thickness of the 2D carriers in calculating the mobility and the quality of the system within the 2-impurity model.
One last issue we need to discuss in the Introduction is the question of how to define the sample quality since it is obviously not a unique property and depends on the specific experiment being carried out. In order to keep things both simple and universal, we have decided to use the level broadening or the Dingle temperature as a measure of the sample quality. The level broadening $\Gamma$ is defined as $\Gamma = \hbar/2\tau_q$, where $\tau_q$ is the quantum scattering time (or the single particle relaxation time) in contrast to the mobility scattering time (or the transport relaxation time) $\tau_t$ which defines the conductivity ($\sigma$) or the carrier mobility ($\mu$) through $\sigma = ne^2\tau_t/m$ or $\mu =
ne \tau_t$. In general, $\tau_q \leq \tau_t$ with the equality holding for purely short-range disorder scattering. Thus, for pure $s$-wave $\delta$-function short-range scattering model, mobility and quality are identical for obvious reasons. For a strict 1-impurity model of underlying sample disorder, mobility ($\tau_t$) and quality ($\tau_q$) are both affected by the same impurity density, and hence they must behave monotonically (but not necessarily identically) with changing disorder, i.e., if the impurity density is decreased (increased) at a fixed carrier density, both mobility and quality must increase (decrease) as well.
As emphasized already, however, the 2-impurity model (near and far impurities or background and remote impurities) introduces a new element of physics by allowing for the possibility that mobility and quality could possibly be affected more strongly by different types of disorder, for example, mobility (quality) could be dominated by near (far) impurities in the 2-impurity disorder model, thus allowing, in principle, the possibility of mobility and quality being completely independent physical properties of realistic 2D semiconductor samples at least in some situations. We find this situation to be quite prevalent in very high-mobility 2D semiconductor structures where the mobility (quality) seems to be predominately determined by near (far) impurities. In low mobility samples, on the other hand, the situation is simpler and both mobility and quality are typically determined by the same set of impurities (usually the charged impurities close to the 2D layer itself).
In section II we describe our model giving the theoretical formalism and equations for the 2-impurity model. In section III we provide detailed results for mobility and quality along with discussion. We conclude in section IV with a summary of our findings along with a discussion of our approximations and of the open questions.
model and theory
================
We assume a 2D electron (or hole) system at $T=0$ located at the $z=0$ plane with the 2D layer being in the x-y plane in our notation. The 2D system is characterized entirely by a carrier effective mass ($m$) defining the single-particle kinetic energy ($E_{\bf k}=\hbar^2k^2/2m$ with [**k**]{} as the 2D wave vector), a background lattice dielectric constant ($\kappa$) defining the 2D Coulomb interaction \[$V({\bf q}) = 2\pi
e^2/\kappa q$ with [**q**]{} as the 2D wave vector), and a 2D carrier density ($n$).
As discussed in the Introduction, we use a minimal 2-impurity model for the static disorder in the system characterized by three independent parameters: $n_R$, $d_R$, $n_B$. Here $n_R$ ($n_B$) is the effective 2D charged impurity density for the remote (background) impurities with the remote (background) impurities being distributed randomly in the 2D x-y plane at a distance ‘$d$’ from the 2D electron system in the $z$-direction with $d=d_R$ (0) for the remote (background) impurities. We assume the random quenched charged impurities to all have unit strength (i.e., having an elementary charge of $\pm e$ each) with no loss of generality.
Our model thus has four independent parameters with dimensions of length: $n^{-1/2}$, $n_R^{-1/2}$, $n_B^{-1/2}$, $d_R$. In addition to these (experimentally variable) parameters, we also have $m$ and $\kappa$ defining the material system which is fixed in all samples for a given material. In principle, two additional materials parameters should be added to describe the most general situation, namely, the spin ($g_s$) and the valley ($g_v$) degeneracy, but we assume $g_s=2$ and $g_v=1$ throughout the current work (and for all our numerical results) since our interest here is entirely focussed on high-mobility n- and p-GaAs 2D systems where the mobility/quality dichotomy has mostly been discussed. Additional experimentally relevant (but, theoretically non-essential) parameters, such as a finite width of the 2D electron system (instead of the strict 2D limit) and/or a 3D distribution of the random impurities (instead of the 2D distribution assumed above), are straightforward to include in the model and are not discussed further in details.
The mobility (quality) is now simply defined by the characteristic scattering time $\tau_t$ ($\tau_q$) as given by the following equations in our leading-order transport theory (i.e., Boltzmann transport plus Born approximation for scattering): $$\begin{aligned}
\frac{1}{\tau_t(k)} = \frac{2\pi}{\hbar} \sum_{\bf k'} \sum_l &
& \int_{-\infty}^{\infty} dz N_i^{(l)}(z) \left |u_{\bf k-k'}(z) \right
|^2 \nonumber \\
& &\times (1-\cos\theta) \delta(E_{\bf k}-E_{\bf k'}),
\label{eq1}\end{aligned}$$ and $$\begin{aligned}
\frac{1}{\tau_q(k)} = \frac{2\pi}{\hbar} \sum_{\bf k'} \sum_l &
& \int_{-\infty}^{\infty} dz N_i^{(l)}(z) \left |u_{\bf k-k'}(z) \right
|^2 \nonumber \\
& &\times \delta(E_{\bf k}-E_{\bf k'}),
\label{eq2}\end{aligned}$$ where $N_i^{(l)}(z)$ is the 3D impurity distribution for the $l$-th kind of disorder in the system, and $u_{\bf q}(z)$ is the screened electron-impurity Coulomb interaction given by: $$u_{\bf q}(z) = \frac{V_{\bf q}(z)}{\varepsilon(q)} = \frac{2\pi e^2}{\kappa q}
\frac{e^{-qz}}{\varepsilon(q)},
\label{eq3}$$ with $\varepsilon(q)$ being the static RPA dielectric function for the 2D electron system. We note that $V_q(z) = V(q) e^{-qz} = \frac{2\pi
e^2}{\kappa q} e^{-qz}$ is simply the 2D Fourier transform of the 3D $1/r$ Coulomb potential, which explicitly takes into account the fact that a spatial separation of ‘$z$’ may exist between the 2D electron layer and the charged impurities. The 2D static RPA dielectric function or the screening function is given by $$\varepsilon(q) = 1 + \frac{2\pi e^2}{\kappa q} \Pi(q),
\label{eq4}$$ where the static 2D electronic polarizability function $\Pi(q)$ is given by $$\Pi(q) = N_F \left [ 1- \theta(q-2k_F) \sqrt{1-(2k_F/q)^2} \right ],
\label{eq5}$$ where $N_F = m/\pi \hbar^2$ and $\theta(x) = 0$ (1) for $x <0$ ($x>0$) is the Heaviside step (or theta) function and the 2D Fermi wave vector $k_F$ is determined by the 2D carrier density through the formula $k_F
= (2\pi n)^{1/2}$. We note that the 2D Fermi energy ($E_F$) is given by $E_F = \hbar^2k_F^2/2m = \pi \hbar^2 n/m$. The RPA screening function $\varepsilon(q)$ can be expressed in the convenient form $$\varepsilon(q) = 1 + q_s/q,$$ which is exactly equivalent to Eqs. (\[eq4\]) and (\[eq5\]) if we define the 2D screening wave vector $q_s$ to be $$q_s = q_{TF} \left [ 1- \theta(q-2k_F) \sqrt{1-(2k_F/q)^2} \right ],
\label{eq8}$$ where the 2D Thomas-Fermi wave vector $q_{TF}$ is defined to be $$q_{TF} = 2 m e^2/(\kappa \hbar^2).$$ We note that (1) the Thomas-Fermi wave vector $q_{TF}$ is proportional to the 2D density of states at the Fermi energy $N_F = m/\pi \hbar^2$ (and is inversely proportional to the effective background lattice dielectric constant $\kappa$), and (2) screening is constant in 2D for $0 \leq q \leq 2k_F$ \[see Eq. (\[eq8\])\] because of the constant energy-independent 2D electronic density of states. Since the $\delta$-functions (necessary for energy conservation during the impurity-induced elastic scattering of an electron from the momentum state $|{\bf k}\rangle$ to the momentum state $|{\bf k}' \rangle$ with a net wave vector transfer of ${\bf q} = {\bf k} - {\bf k}'$) in Eqs. (\[eq1\]) and (\[eq2\]) restrict the wave vector transfer $0
\leq q \leq 2k_F$ range (this is simply because we are at $T=0$ so that the maximum possible scattering wave vector is $2k_F$ corresponding to the pure back-scattering of an electron from $+k_F$ to $-k_F$ while obeying energy conservation), the relevant screening wave vector for our problem is purely the Thomas-Fermi wave vector $q_s = q_{TF}$ as follows from Eq. (\[eq8\]) for $q\le 2k_F$.
We can, therefore, rewrite Eq. (\[eq3\]) as $$u_q(z) = \frac{2\pi e^2}{\kappa (q+ q_{TF})} e^{-qz},
\label{eq10}$$ giving the effective screened Coulomb interaction (in the 2D momentum space) between an electron in the 2D layer and a random quenched charged impurity located a distance ‘$z$’ away.
Finally, our 2-impurity disorder model is given by $$N_i(z) = n_R \delta(z-d_R) + n_B \delta(z),
\label{eq11}$$ with three independent parameters $n_R$, $d_R$, and $n_B$ completely defining the underlying disorder. We note that writing the second term in Eq. (\[eq11\]) as $n_B \delta(z-d_B)$, and thus introducing an additional length parameter into the model, is completely unnecessary since, as we will see below, the physics of the mobility versus quality duality in high-quality 2D structures is entirely dominated by scattering from near impurities (controlling mobility) and far impurities (controlling quality), which allows us to put $d_B=0$ in the minimal model (thus making the near impurities very near indeed). Putting Eqs. (\[eq3\]) – (\[eq11\]) in Eqs. (\[eq1\]) and (\[eq2\]), we can combine them into a single 2D integral, obtaining $$\begin{aligned}
\frac{1}{\tau_{t,q}} = \frac{2\pi}{\hbar} \left (\frac{2\pi
e^2}{\kappa} \right )^2 & &
\int\frac{d^2k'}{(2\pi)^2} \frac{\delta(E_{\bf k}-E_{\bf k'})}{(q_{TF}
+ |{\bf k}-{\bf k'}| )^2}
f_{t,q}(\theta) \nonumber \\
& &\times \left \{ n_R e^{-2|{\bf k}-{\bf k'}| d_R} + n_B \right \},
\label{eq12}\end{aligned}$$ where $f_t(\theta) = 1- \cos \theta_{kk'}$ and $f_q(\theta) =1$. For completeness, we mention that $E_{\bf k} = \hbar^2 k^2/2m$.
Before proceeding further, we emphasize that the “only" difference between mobility (i.e., $\tau_t$) and quality (i.e., $\tau_q$) is the appearance (or not) of the angular factor $(1-\cos \theta)$ inside the double integral in Eq. (\[eq12\]) for $\tau_t$ ($\tau_q$). This arises from the fact that the mobilty or the conductivity is unaffected by forward scattering (i.e., $\theta \approx 0$ or $\cos
\theta \approx 1$) whereas the single-particle level-broadening ($\Gamma \sim
\hbar/\tau_q$) is sensitive to scattering through all angles. Technically, the ($1-\cos\theta$) factor arises from the impurity scattering induced vertex correction in the 2-particle current-current correlation function representing the electrical conductivity whereas the single-particle scattering rate ($\tau_q^{-1}$) is given essentially by the imaginary part of the impurity scattering induced 1-particle electronic self-energy which does not have any vertex correction in the leading-order impurity scattering strength. The absence (presence) of the vertex correction in $\tau_q^{-1}$ ($\tau_t^{-1}$) makes the relevant scattering rate sensitive (insensitive) to forward scattering, leading to a situation where $\tau_q$ and $\tau_t$ could be very different from each other if forward (or small angle) scattering is particularly important in a system as it would be for long range disorder potential.
This could happen in 3D systems if the scattering potential is strongly spatially asymmetric (or non-spherical) for some reason. It was pointed out [@dassarmaPRB1985] a long time ago and later experimentally verified [@exp11] that such a strongly non-spherically symmetric scattering potential arises naturally in 2D modulation-doped structures from random charged impurities placed very far ($k_Fd \gg 1$) away from the 2D electron system due to the influence of the exponential $e^{-q d}$ factor in the Coulomb potential which restricts much of the scattering to $q \ll 1/z$, thus exponentially enhancing the importance of small-angle (i.e., small scattering wave vector) scattering. This then leads to $\tau_t \gg
\tau_q$ for scattering dominated by remote dopants (the two scattering times could differ by more than two orders of magnitude in high-mobility modulation-doped structures where $k_F d_R \gg 1$ is typically satisfied due to the far-away placement of the remote dopants in order to minimize large-angle resistive scattering processes) in 2D systems, but for the unintentional background impurities, which always satisfy $k_F d_B \ll 1$ by definition (since $d_B \approx 0$), the two scattering times are approximately equal. We note that the finite layer thickness of the 2D system puts a lower bound on how small $d_B$ can be, but this has no qualitative significance for our consideration.
Now, we immediately realize the crucial relevance of the 2-impurity model in distinguishing mobility (i.e., $\tau_t$) and quality (i.e., $\tau_q$) in 2D systems since it now becomes possible for one type of disorder (e.g., remote impurities) to control the quality (i.e., $\tau_q$) and the other type to control the mobility (i.e., $\tau_t$). Of course, whether such a distinction actually applies to a given situation or not depends entirely on the details of the sample parameter (i.e., the specific values of $n_R$, $d_R$, $n_B$) as well as the carrier density $n$, but the possibility certainly exists for $d_R$ to be large enough so that the remote impurity scattering is almost entirely small-angle scattering (thus only adversely affecting $\tau_q$ in an appreciable way) whereas the background impurity scattering determines $\tau_t$. If this happens, then mobility ($\tau_t$) and quality ($\tau_q$) could very well be very different in 2D samples, and may have little to do with each other.
To bring out the above physical picture explicitly, we now provide some analytical calculations for the integrals in Eq. (\[eq12\]) defining mobility ($\tau_t$) and quality ($\tau_q$). We rewrite Eq. (\[eq12\]) as $$\tau^{-1}_{t,q} = I^{(R)}_{t,q} + I^{(B)}_{t,q},
\label{eq16}$$ where $$\begin{aligned}
I^{(R)}_{t,q} = n_RV_0 \frac{2\pi}{\hbar} & &
\int\frac{d^2k'}{(2\pi)^2} \frac{\delta(E_{\bf k}-E_{\bf k'})}{(q_{TF} + |{\bf k}-{\bf k'}| )^2}
\nonumber \\
& &\times f_{t,q}(\theta) e^{-2|{\bf k}-{\bf k'}| d_R},
\label{eq17}\end{aligned}$$ and $$I^{(B)}_{t,q} = n_B V_0 \frac{2\pi}{\hbar}
\int\frac{d^2k'}{(2\pi)^2} \frac{\delta(E_{\bf k}-E_{\bf k'})}{(q_{TF} + |{\bf k}-{\bf k'}| )^2}
f_{t,q}(\theta),
\label{eq18}$$ where $V_0=(2\pi e^2/\kappa)^2$. Eqs. (\[eq17\]) and (\[eq18\]) can be rewritten in dimensionless forms as $$I^{(R)}_{t,q} = \left ( \frac{mn_R V_0}{2\pi \hbar^3 k_F^2} \right ) \int_0^1 dx \frac{g_{t,q}(x) e^{-2 x a_R}}{\sqrt{1-x^2}(x+s)^2}
\label{eq19}$$ and $$I^{(B)}_{t,q} = \left ( \frac{mn_B V_0}{2\pi \hbar^3 k_F^2} \right )
\int_0^1 dx \frac{g_{t,q}(x)}{\sqrt{1-x^2}(x+s)^2}
\label{eq20}$$ where $a_R \equiv 2k_F d_R$ and $s=q_{TF}/2k_F$, and $g_t(x) = 2x^2$, $g_q(x) = 1$.
To proceed further analytically, we now make the assumption that, by definition, the remote dopants are far enough that the condition $a_R = 2k_F d_R \gg 1$ is satisfied. We note that for any arbitrarily large value of $d_R$, this “remote" impurity condition breaks down at a low enough carrier density such that $n \alt 1/ 8\pi d_R^2$. Thus, the distinction between ‘far’ and ‘near’ impurities in our 2-impurity model starts disappearing at very low carrier density. Putting $a_R \gg 1$ as well as $s=q_{TF}/2k_F \gg 1$ we can obtain the following asymptotic expressions for $I_{t,q}^{(R)}$ $$I^{(R)}_{t} = \left ( \frac{m n_R}{2\pi \hbar^3 k_F^2} \right ) \left ( \frac{2\pi e^2}{\kappa} \right )^2
\left ( \frac{2}{3s^2} \right ) \frac{1}{a_R^3},
\label{eq24}$$ $$I^{(R)}_{q} = \left ( \frac{m n_R}{2\pi \hbar^3 k_F^2} \right ) \left ( \frac{2\pi e^2}{\kappa} \right )^2
\frac{1}{a_R s^2},
\label{eq25}$$ $$I^{(B)}_{t} = \left ( \frac{m n_B}{2\pi \hbar^3 k_F^2} \right ) \left ( \frac{2\pi e^2}{\kappa} \right )^2
\frac{2\pi}{s^2},
\label{eq26}$$ $$I^{(B)}_{q} = \left ( \frac{m n_B}{2\pi \hbar^3 k_F^2} \right ) \left ( \frac{2\pi e^2}{\kappa} \right )^2
\frac{2\pi}{s^2},
\label{eq27}$$ with $\tau_{t,q}^{-1} = I_{t,q}^{(R)} + I_{t,q}^{(B)}$. Equations (\[eq24\]) – (\[eq27\]) provide us with approximate analytical expressions for the contributions $I_{t,q}^{(R)}$ and $I_{t,q}^{(B)}$ by the remote and the background scattering respectively to the transport and quantum scattering rates. We note that in these asymptotic limits ($a_R \gg 1$ and $s \gg 1$) $$\frac{I_t^{(B)}}{I_t^{(R)}} = 3 \pi a_R^3 \frac{n_B}{n_R},
\label{eq29}$$ and $$\frac{I_q^{(R)}}{I_q^{(B)}} = \frac{1}{2\pi a_R}\frac{n_R}{n_B}.
\label{eq30}$$ In addition, $${I_q^{(R)}}/{I_t^{(R)}} = 3 a_R^2/2,
\label{eq31}$$ $${I_q^{(B)}}/{I_t^{(B)}} = 1.
\label{eq32}$$ We note, therefore, that $I_q^{(R)} \gg I_t^{(R)}$ (since $a_R \gg 1$) as is already well established, and that $I_t^{(B)} > I_t^{(R)}$ if $3\pi a_R^3 > n_R/n_B$, whereas $I_q^{(R)} > I_q^{(B)}$ if $n_R/n_B >
2 \pi a_R$. Consistency demands that $3\pi a_R^3 \gg 2 \pi a_R$, i.e., $a_R^2 \gg 2/3$, which is guaranteed since $a_R \gg 1$.
It is thus possible for the $B$-scatterers to dominate $\tau_t^{-1}$, i.e., $I_t^{(B)} \gg I_t^{(R)}$, and $R$-scatterers to dominate $\tau_q^{-1}$, i.e., $I_q^{(R)} \gg I_q^{(B)}$ if the following conditions are both satisfied $$\begin{aligned}
3 \pi a_R^3 & \gg & n_R/n_B, \nonumber \\
2 \pi a_R & \ll & n_R/n_B.
\label{eq35}\end{aligned}$$ The two inequalities defined by Eq. (\[eq35\]) are not mutually exclusive if $n_R \gg n_B$ and $$n_R/2\pi n_B \gg a_R \gg (n_R/3\pi n_B)^{1/3}.
\label{eq36}$$ It is easy to see the Eq. (\[eq36\]) is consistent as long as $n_R
\gg 5.13 n_B$, a perfectly reasonable scenario! In fact, we expect this condition to be extremely well-satisfied in high-quality 2D systems where $n_B$ is very small, but $n_R \approx n$ due to modulation doping and charge neutrality.
The above analytic considerations lead to the conclusion that it is possible for $\tau_t^{-1}$ to be dominated by background impurities, and at the same time for $\tau_q^{-1}$ to be dominated by the remote impurities provided the necessary conditions $a=2k_Fd_R \gg 1$ and $n_R \gg 5.13 n_B$ are obtained. We emphasize that these are only [ *necessary*]{} conditions, and not sufficient conditions. Whether the 2-impurity model indeed leads to mobility (i.e., $\tau_t$) and quality (i.e., $\tau_q$) being controlled by physically distinct disorder mechanisms in real 2D semiconductor structures can only be definitely established through explicit numerical calculations of $\tau_{t,q}^{-1}$ for specific disorder configurations, which we do in the next section of this article. The general analytical theory developed above also explicitly shows that there can be no mobility/quality dichotomy if there is only one type of disorder mechanism operational in the sample since both quality and mobility will then be controlled by exactly the same disorder parameters (unlike the situation discussed above).
It may be useful to write a full analytical expression for $\tau_t^{-1}$ and $\tau_q^{-1}$ (in the $a_R \gg 1$ limit) combining all the expressions given above to show how the disorder parameters $n_R$, $d_R$, and $n_B$ enter the expressions for mobility and quality: $$\tau_t^{-1} = I_t^{(R)} + I_t^{(B)} = A_t^{(R)}n_R/a_R^3 + A_t^{(B)} n_B,
\label{eq37}$$ $$\tau_q^{-1} = I_q^{(R)} + I_q^{(B)} = A_q^{(R)}n_R/a_R + A_q^{(B)} n_B,
\label{eq38}$$ where $$\begin{aligned}
A_t^{(R)} & = & \left ( \frac{m}{2\pi\hbar^3} \right ) \left ( \frac{2\pi e^2}{\kappa} \right )^2 \left ( \frac{8}{3 q_{TF}^2} \right ) \nonumber \\
A_t^{(B)} & = & \left ( \frac{m}{2\pi \hbar^3} \right ) \left ( \frac{2\pi e^2}{\kappa} \right )^2 \left ( \frac{8\pi}{q_{TF}^2} \right ),
\label{eq39}\end{aligned}$$ $$\begin{aligned}
A_q^{(R)} & = & \left ( \frac{m}{2\pi\hbar^3} \right ) \left ( \frac{2\pi e^2}{\kappa} \right )^2 \left ( \frac{4}{q_{TF}^2} \right ) \nonumber \\
A_q^{(B)} & = & \left ( \frac{m}{2\pi \hbar^3} \right ) \left ( \frac{2\pi e^2}{\kappa} \right )^2 \left ( \frac{8\pi}{q_{TF}^2} \right ).
\label{eq40}\end{aligned}$$ Equations (\[eq37\]) and (\[eq38\]) immediately lead to the approximate sufficient conditions for mobility and quality to be controlled by background and remote impurities, respectively: $$n_B \gg n_R/a_R^3, \;\; i.e., \; n_R \ll n_B a_R^3,
\label{eq41}$$ and $$n_R/a_R \gg n_B, \;\; i.e., \; n_R \gg n_B a_R.
\label{eq42}$$ Since $a_R \gg 1$ by definition, the two conditions, Eqs. (\[eq41\]) and (\[eq42\]), can simultaneously be satisfied if $$a_R^3 \gg n_R/n_B \gg a_R,
\label{eq43}$$ with $a_R \gg 1$. Equation (\[eq43\]) gives the sufficient condition for the existence of a mobility/quality dichotomy in 2D semiconductor structures. Since the unintentional background charged impurity concentration is typically very low in high-mobility 2D GaAs structures and since the remote charged dopant density is typically (at least) equal to the carrier density, the condition $a_R^3 \gg
n_R/n_B \gg a_R \gg 1$ can certainly be satisfied in some 2D samples (but obviously not in all samples). For example, a typical modulation-doped high mobility 2D GaAs-Al$_x$Ga$_{1-x}$As structure may have $n \approx n_R \approx 3 \times 10^{11}$ cm$^{-2}$; $d_R
\approx 1000$ Å; $n_B \approx 10^8$ cm$^{-2}$ (corresponding roughly to a 3D bulk background charged impurity density of $3\times 10^{13}$ cm$^{-3}$ for a 300 Å wide quantum well structure). These system parameters satisfy the constraint defined by Eq. (\[eq43\]) with $k_Fd_R \approx 15$, i.e., $a_R \approx 30$; $n_R/n_B \approx
3000$. We emphasize the obvious role of carrier density here, i.e., lowering the carrier density decreases $k_F d$ (and hence $a_R$), and eventually at low enough carrier density, $\tau_t^{-1}$ and $\tau_q^{-1}$ are determined by the same disorder parameters!
In concluding this theoretical section, let us consider a concrete numerical example of the mobility/quality dichotomy using two hypothetical samples (1 and 2) with the following realistic sample parameters: $$\begin{aligned}
{\rm Sample \; 1:}& & \; d_R^{(1)} = 500 \AA; \; n_B^{(1)} = 10^7 cm^{-2} \nonumber \\
{\rm Sample \;2:}& & \; d_R^{(1)} = 1000 \AA; \; n_B^{(2)} = 10^8 cm^{-2}.
\label{eq44}\end{aligned}$$ We assume the sample carrier density to be the same in both cases (so that “an apple-to-apple" comparison in being made): $n=4\times 10^{11}$ cm$^{-2}$. For the purpose of keeping the number of parameters a minimum we assume $n_R = n = 4\times 10^{11}$ cm$^{-2}$ also for both samples. Using the analytical theory developed above (or by direct numerical calculations), we find $$\tau_t^{(1)}/\tau_t^{(2)} \approx 5; \; \tau_q^{(1)}/\tau_q^{(2)} \approx 2.
\label{eq45}$$ This means that sample 1 (with $n_B^{(1)} < n_B^{(2)}$) has a five times higher mobility than sample 2 whereas sample 2 (with $d_R^{(2)}
> d_R^{(1)}$) has two times higher ‘quality’ than sample 1, i.e., sample 2 has a single-particle level broadening $\Gamma$ (or “Dingle temperature") which is half of that of sample 1 although both samples have exactly the same carrier density! This realistic example shows that it is generically possible in the 2-impurity model for $\mu_1 >
\mu_2$ and $\Gamma_1 > \Gamma_2$, with the conclusion that a higher mobility does not necessarily ensure a higher quality. We emphasize that (1) this would not be possible within the 1-impurity model, and (2) this conclusion is density dependent – for much lower carrier density, where $k_F d \gg 1$ condition cannot be satisfied for the remote dopants, mobility and quality will again be closely connected since at sufficiently low carrier density, the 2-impurity model effectively reduces to an 1-impurity model.
Before presenting our realistic numerical results for 2D GaAs-AlGaAs structures (including both the quasi-2D finite well-width effect and a 3D distribution of the background unintentional charged impurity distribution within the well) using the ‘near’ and ‘far’ 2-impurity model in the next section (Sec. III), we conclude the current theory section by showing some numerical results for $\tau_t$ and $\tau_q$ using the idealized 2-impurity model \[i.e., Eq. (\[eq11\])\] and the strict 2D model for the semiconductor structure. These results presented in Figs. \[fig01\] – \[fig04\] explicitly visually demonstrate that $\tau_t$ and $\tau_q$ cannot be single-valued functions of each other as long as the underlying disorder consists of (at least) two distinct scattering mechanisms as operational within the 2-impurity model. The results shown in Figs. \[fig01\] – \[fig04\] also serve to establish the validity of the analytical theory we provided above in this section.
In Fig. \[fig01\] we show the dependence of the calculated $\tau_t$ and $\tau_q$ on the individual scattering mechanism (i.e., the near-impurity scattering strength defined by the background impurity concentration $n_B$ or the far-impurity scattering strength defined by either $n_R$ or $d_R$) assuming that the other mechanism is absent (i.e., just an effective 1-impurity model applies). Results in Fig. \[fig01\] should be compared with the corresponding results in Figs. \[fig02\] – \[fig04\] where both scattering mechanisms are operational to clearly see that $\tau_t$ and $\tau_q$ are manifestly not unique functions of each other by any means and a given value of $\tau_t$ (or $\tau_q$) could lead to distinct values of $\tau_q$ (or $\tau_t$) depending on the details of the disorder distribution. Thus, mobility ($\tau_t$) and quality ($\tau_q$) are not simply connected.
![(a) The scattering times ($\tau_t$ and $\tau_q$) and (b) the ratio ($\tau_t/\tau_q$) as a function of the background impurity density $n_B$ for $n=3\times 10^{11} cm^{-2}$ and $n_R=0$. (c) The scattering times and (d) the ratio ($\tau_t/\tau_q$) as a function of the remote impurity density $n_R$ for $n=n_R$, $n_B= 0$, and $d_R=80$nm. (e) The scattering times and (f) the ratio ($\tau_t/\tau_q$) as a function of the impurity location $d_r$ for $n=n_r=3\times 10^{11} cm^{-2}$ and the background impurity density $n_B=0$. Here the $\delta$-layer is considered (i.e., $a=0$). []{data-label="fig01"}](fig01.eps){width="1.\columnwidth"}
![(a) The scattering times ($\tau_t$ and $\tau_q$) and (b) the ratio ($\tau_t/\tau_q$) as a function of the background impurity density $n_B$ for $n=n_R=3\times 10^{11} cm^{-2}$ and $d_R=80$ nm. (c) The scattering times and (d) the ratio ($\tau_t/\tau_q$) as a function of the remote impurity density $n_R$ for $n=n_R$, $n_B= 10^8$ cm$^{-2}$, and $d_R=80$nm. (e) The scattering times and (f) the ratio ($\tau_t/\tau_q$) as a function of the impurity location $d_R$ for $n=n_R=3\times 10^{11} cm^{-2}$ and the background impurity density $n_B=0$. Here the $\delta$-layer is considered (i.e., $a=0$). []{data-label="fig02"}](fig02.eps){width="1.\columnwidth"}
In presenting the results for Figs. \[fig01\] – \[fig04\] we first note that $\tau_{t,q} = \tau(n,n_B,n_R,d_R)$ even within the simple 2-impurity model. Since the carrier density dependence of $\tau$ is not the central subject matter of our interest in the current work (and has been discussed elsewhere by us [@dassarmaprb2013], we simplify the presentation by assuming $n_R=n$ in Figs. \[fig01\] – \[fig04\] which also assures a straightforward charge neutrality. This, however, has important implications since the dependence on $n_R$ and $n$ now become compounded rather than being independent. Thus the $n_R$-dependence of $\tau$ shown in Figs. \[fig01\] – \[fig04\] is not the trivial $\tau_{t,q} \sim n_R^{-1}$ behavior (as it is for the $n_B$-dependence, where $\tau_{t,q} \sim n_B^{-1}$) since the $n_R =
n$ condition synergistically combines both $n_R$ and $n$ dependence. We mention that in ungated samples with fixed carrier density, the condition $n=n_R$ is perfectly reasonable, and therefore, the results shown in Figs. \[fig01\] – \[fig04\] apply to modulation-doped samples with fixed carrier density $n=n_R$.
![(a) The scattering times and (b) the ratio ($\tau_t/\tau_q$) as a function of the remote impurity density $n_R$ for $n=n_R$, $n_B= 10^9$ cm$^{-2}$, and $d_R=80$nm. (c) The scattering times and (d) the ratio ($\tau_t/\tau_q$) as a function of the impurity location $d_R$ for $n=n_R=3\times 10^{11} cm^{-2}$ and the background impurity density $n_B=10^9$ cm$^{-2}$. Here the $\delta$-layer is considered (i.e., $a=0$). []{data-label="fig03"}](fig03.eps){width="1.\columnwidth"}
![ (a) The scattering times and (b) the ratio ($\tau_t/\tau_q$) as a function of the remote impurity density $n_R$ for $n=n_r$, $n_B= 10^{10}$ cm$^{-2}$, and $d_R=80$nm. (c) The scattering times and (d) the ratio ($\tau_t/\tau_q$) as a function of the impurity location $d_R$ for $n=n_r=3\times 10^{11} cm^{-2}$ and the background impurity density $n_B=10^{10}$ cm$^{-2}$. Here the $\delta$-layer is considered (i.e., $a=0$). []{data-label="fig04"}](fig04.eps){width="1.\columnwidth"}
From Fig. \[fig01\], we immediately conclude the obvious: The functional relationship between $\tau_t$ and $\tau_q$ depends entirely on which disorder parameter is being varied – in fact Figs. \[fig01\](a), (c), (e) give three completely distinct functional relationships between $\tau_t$ and $\tau_q$ depending on whether $n_B$, $n_R$, or $d_R$ is being varied, respectively. From the corresponding values of $\tau_t/\tau_q$, as shown in Figs. \[fig01\](b), (d), (f) respectively we clearly see that $\tau_t$ and $\tau_q$ variations with individual disorder parameters $n_B$, $n_R$, $d_R$ are very different indeed. We point out an important qualitative aspect of Fig. \[fig01\](c) which has not been explicitly discussed in the literature and which has important implications for the mobility/quality dichotomy. Here the mobility (i.e., $\tau_t$) increases with increasing $n_R = n$, but the quality (i.e., $\tau_q$) decreases with increasing $n=n_R$. This is a peculiar feature of long range Coulomb scattering by remote dopants.
Figs. \[fig02\] – \[fig04\] explicitly show how the 2-impurity model can very strongly modify the 1-impurity model functional dependence of $\tau_{t,q}$ on the disorder parameters $n_B$, $n_R$ ($=n$), and $d_R$. Clearly, depending on the specific 2D samples, $\tau_t$ and $\tau_q$ could behave very differently as already established in our analytical theoretical results given above. For example, in contrast to Fig. \[fig01\](a), where both $\tau_t$ and $\tau_q$ decrease monotonically (and trivially as $n_B^{-1}$) with increasing amount of unintentional background impurity density $n_B$, Fig. \[fig02\](a) shows that $\tau_q$ is essentially a constant whereas $\tau_t$ decreases with increasing $n_B$, thus demonstrating a specific example of how the effective quality (i.e. $\tau_q$) remains the same although the effective mobility decreases by more than an order of magnitude due to the variation in the background disorder. Similarly, in Fig. \[fig01\](c), increasing the remote dopant separation $d_R$ (keeping $n=n_R$ fixed, and $n_B=0$) increases both $\tau_t$ and $\tau_q$ monotonically (with $\tau_t$ increasing as $d^3$ in contrast to $\tau_q$ increasing as $d$ for large $d$), but Fig. \[fig02\](e), \[fig03\](c), \[fig04\](c) show that, depending on the background impurity scattering strength, $\tau_t$ basically saturates for larger $d$ since it is then dominated by the unintentional background impurities rather than by the remote dopants whereas $\tau_q$ continues to be limited by the remote dopants. This leads to an interesting nonmonotonicity in $\tau_t/\tau_q$ as a function of $d_R$ in Figs. \[fig02\] – \[fig04\] in contrast to Fig. \[fig01\](c) where $\tau_t/\tau_q \sim d_R^2$ keeps on increasing forever in the absence of background scattering. The realistic dependence of $\tau_{t,q}$ on $n_R$ (with $n=n_R$) in the presence of fixed $n_B$ and $d_R$ remains qualitatively similar in Figs. \[fig01\] – \[fig04\] although there could be large quantitative differences, indicating that increasing $n_R$ ($=n$) would typically by itself tend to enhance (suppress) $\tau_t$ ($\tau_q$), but the effect becomes whether for larger (smaller) values of $n_B$ ($d_R$). This is an important result of our paper.
The analytical and numerical results presented in this section establish clearly that $\tau_t$ and $\tau_q$ can essentially be independent functions of the disorder parameters in the 2-impurity model, and thus, mobility and quality could, in principle, have little to do with each other in realistic 2D semiconductor structures. We make this point even more explicit by carrying out calculations in experimentally realistic samples in the next section of this article.
Numerical results and discussions
=================================
We begin presenting our realistic numerical results for 2D transport properties (both $\tau_t$ or mobility and $\tau_q$ or quality) without any reference to the analytical asymptotic theoretical results presented in the last section by showing in Fig. \[fig1\] the calculated $T=0$ mobility ($\mu$), transport scattering time ($\tau_t$) and the single-particle (or quantum) scattering time ($\tau_q$) for a 2D GaAs-AlGaAs sample at a fixed carrier density ($n=3\times 10^{11}$ cm$^{-2}$) in the presence of two types of disorder: a remote charged impurity sheet ($n_i = 1.5 \times 10^{11}$ cm$^{-1}$) placed at a distance ‘$d$’ from the edge of the quantum well (i.e. from the GaAs-AlGaAs interface) and a background 3D charged impurity density ($n_{iw}$) which is distributed uniformly throughout the inside of the GaAs quantum well (which has a well thickness of 300 Å). Results presented in Fig. \[fig1\] involve no approximation other than assuming a uniform random distribution of the quenched charged impurities (2D distribution with a fixed 2D impurity density $n_i$ for the remote impurities placed at a distance $d$ from the quantum well and 3D distribution with a variable 3D impurity density $n_{iw}$ for the unintentional background impurities inside quantum well) as we include the full quantitative effect of the quasi-2D nature of the quantum well width in the calculation and calculate all the integrals \[Eqs. (\[eq1\]) and (\[eq2\])\] for $\tau_t^{-1}$ and $\tau_q^{-1}$ numerically exactly. Of course, our basic theory is a leading-order theory in the impurity scattering strength which should be an excellent approximation at the high carrier density of interest in the current work where our focus is on very low-disorder and high-quality 2D semiconductor systems. The use of the realistic 3D background impurity distribution is easily reconciled with our minimal model in section II by using: $n_i=n_R$, $d_R = d +a/2$ and $n_B \approx n_{iw} a$, where $a$ ($=300$ Å in Fig. \[fig1\]) is the quantum well-width.
![(a) Calculated mobilities as a function of background 3D charged impurity density $n_{iw}$ for fixed remote impurity density, $n_i=1.5\times 10^{11}cm^{-2}$, and electron density, $n=3\times 10^{11}cm^{-2}$. The red dashed line indicates $\mu \sim n_{iw}$. The numbers indicate the location of remote impurities which is measured from the interface of quantum well. (b) The calculated scattering times and their ratio $\tau_t/\tau_q$ as a function of background impurity density using the same parameters of (a) and $d=800$ Å. The dashed line represents the ratio of $\tau_t$ to $\tau_q$. A quantum well with the thickness of $a=300$ Å is used in this calculation. []{data-label="fig1"}](fig1.eps){width="1.\columnwidth"}
Results in Fig. \[fig1\] are quite revealing of the physics discussed already in section II. First, we clearly see in Fig. \[fig1\](a) the trend that for large (small) $d$, the mobility is determined by the background (remote) impurities, and hence for $d=800$ ($100$) Å, the mobility depends strongly (weakly) on the background impurity density (until it becomes very large, leading to $\mu < 10^6$ cm$^2$/Vs which is no longer a high-quality situation). In particular, for $d=800$ Å, $\mu^{-1}$ ($\propto
\tau_t^{-1}$) $\propto n_{iw}$ approximately, implying that $\tau_t^{-1}$ is dominated almost entirely (for $d=800$ Å) by the background impurities in the quantum well. By contrast, for $d=100$ Å, $\mu$ is almost independent of $n_{iw}$ for $n_{iw} \alt 10^{15}$ cm$^{-3}$ (corresponds to $n_B \sim 3 \times 10^9$ cm$^{-2}$), indicating that $\tau_t^{-1}$ is dominated almost entirely by the “remote" dopant scattering. In Fig. \[fig1\](a) we focus on the interesting $d=800$ Å situation where the 2-impurity model might apply – obviously, for $d=100$ Å, the remote dopants dominate both $\tau_t^{-1}$ and $\tau_q^{-1}$ rendering the 2-impurity model inapplicable since $k_F d < 1$ for both “remote" and “background" impurities for small ‘$d$’.
![ (a) Calculated scattering times and (b) mobilities as a function of carrier density. Here $\tau_{iq}$, $\mu_i$ ($\tau_{dq}$, $\mu_d$) indicate the single particle relaxation time and mobility due to interface impurities at $d=0$ (remote impurities at finite $d=800$ Å), respectively, and $\tau
=(\tau_{iq}^{-1} + \tau_{dq}^{-1})^{-1}$, $\mu =(\mu_i^{-1}+\mu_d^{-1})^{-1}$. The Green line indicates $\tau_q = 0.92/n$ in units of ps with $n$ measured by $10^{10}cm^{-2}$. The crossing point between green line and blue line represents $\Gamma=E_F$. The following parameters are used: $n_i (d=0) = 10^8 cm^{-2}$, $n_d(d=800\AA) =10^{10}cm^{-2}$ and quantum well width $a=300$ Å. At a density $n=10^{11}cm^{-2}$ we have $\mu=35.7\times 10^6 cm^2/Vs$ and $\tau_q=25.5$ ps. The critical density (i.e. the crossing point) is $n_c=0.24 \times 10^{10}cm^{-2}$. []{data-label="fig2"}](fig2.eps){width="1.\columnwidth"}
In Fig. \[fig1\](b) we show as a function of background disorder the calculated $\tau_q$ (as well as $\tau_t$) for $d = 800$ Å, and it is clear that for $5\times10^{13} cm^{-3} < n_{iw} < 2 \times
10^{15} cm^{-3}$ (i.e., over a factor of 40 increase in the background disorder!) $\tau_q$ (i.e., quality) remains almost a constant whereas $\tau_t$ (i.e., mobility) decreases approximately by a factor of 40 in this regime. Combining Figs. \[fig1\](a) and (b), we then conclude that there could be an infinite series of samples, where the mobility decreases from $\sim 20 - 40 \times 10^6$ cm$^2$/Vs to below $10^6$ cm$^2$/Vs as $n_{iw}$ increases from $5\times 10^{13}$ cm$^{-3}$ to $2\times 10^{15}$ cm$^{-3}$, with all of them having essentially the same quality as characterized by the quantum scattering time $\tau_q \sim 3$ ps, corresponding to a quantum level broadening of $\Gamma \sim \hbar/2\tau_q \sim 0.1$ meV. Results shown in Fig. \[fig1\], which are completely realistic, clearly bring out the fact that, when the underlying disorder has a basic 2-impurity model structure (one type of impurity with $k_Fd \gg1$ and the other type with $k_F d \ll 1$), mobility and quality of individual samples may very well be completely independent quantities.
We believe that the results of Fig. \[fig1\] completely qualitatively (perhaps even quantitatively) explain the recent “puzzling" finding[@gamez] that the fragile 5/2 fractional quantum Hall effect (FQHE), which is traditionally only studied in the highest quality samples with $\mu >
10^7$ cm$^2$/Vs, can actually be observed in much lower mobility samples with $\mu \sim 10^6$ cm$^2$/Vs since, under suitable circumstances (as in the results of Fig. \[fig1\]), it is possible for samples with orders of magnitude different mobilities (i.e. values of $\tau_t$) to have more or less the same “quality" (i.e., the same value of $\tau_q$).
![ The same as Fig. \[fig2\] with following parameters: $n_i (d=0) = 3\times
10^8 cm^{-2}$, $n_d(d=500\AA) =3\times 10^{9}cm^{-2}$, and quantum well width $a=300$ Å. At a density $n=10^{11}cm^{-2}$ we have $\mu=14.1\times 10^6 cm^{2}/Vs$ and $\tau_q=42$ ps. The critical density $n_c=0.161 \times 10^{10} cm^{-2}$. []{data-label="fig3"}](fig3.eps){width="1.\columnwidth"}
In Figs. \[fig2\] – \[fig5\], we make the above issue very clear by showing realistic transport calculation results (for both $\tau_t$ and $\tau_q$) in various situations within the 2-impurity model. In each case, the high carrier density mobility is determined by the background impurity scattering whereas the quality, i.e., the quantum lifetime $\tau_q$ (or equivalently the single-particle level broadening $\Gamma \sim \tau_q^{-1}$) is determined by remote impurity scattering, creating a clear dichotomy where mobility and quality are disconnected and the high-density mobility by itself does not provide a unique characterization of the sample quality.
To make the physical implication of the mobility/quality dichotomy very explicit, we have shown in each figure the carrier density where the Ioffe-Regel criterion for strong localization, $\Gamma = E_F$, is satisfied in each of these samples.[@IRC] (We mention that Figs. \[fig2\] – \[fig5\] should be thought of as representing five distinct 2D samples with fixed bare disorder each as described in each figure caption, but with variable 2D carrier density, as for example, can be implemented experimentally using an external back gate.) This $\Gamma
= E_F$ Ioffe-Regel point should be thought of as the critical density below (above) which the system behaves insulating (metallic) as has recently been discussed by us in details elsewhere [@IRC]. Such an apparent disorder driven effective 2D metal-insulator transition (2D MIT) has been extensively studied in the literature [@review], and is usually discussed in terms of the maximum mobility of the sample at high carrier density.
![(a) The same as Fig. \[fig2\] with following parameters: $n_i (d=0) = 3\times
10^8 cm^{-2}$, $n_d(d=500\AA) =5\times 10^{9}cm^{-2}$, and quantum well width $a=300$ Å. At a density $n=10^{11}cm^{-2}$ we have $\mu=13.3\times 10^6 cm^{2}/Vs$ and $\tau_q=28.8$ ps. The critical density $n_c =0.215 \times 10^{10}cm^{-2}$. []{data-label="fig4"}](fig4.eps){width="1.\columnwidth"}
One can think of the Ioffe-Regel criterion induced critical density $n_c$ to be an approximate quantitative measure of the “sample quality" with $n_c$ decreasing (increasing) as the quality improves. The corresponding approximate measure of the sample mobility has traditionally been the so-called “maximum mobility ($\mu_m$)", or equivalently for high-mobility GaAs-based modulation-doped structures, the measured mobility at the highest possible carrier density (since for modulation-doped high-mobility structures, as can be seen in Figs. \[fig2\] – \[fig5\] and as has been extensively experimentally observed over the last 20 years, the sample mobility decreases with decreasing carrier density and the typically quoted sample mobility is always the one measured at the highest carrier density). The näive expectation is that higher (lower) the maximum mobility, lower (higher) should be the critical density for 2D MIT since the sample quality should improve with sample mobility.
Figure $\mu_m $ ($10^6$ cm$^2$/Vs) $ n_c$ ($10^{10}$ cm$^{-2}$)
--------------- ----------------------------- ------------------------------
Fig. \[fig2\] 35.7 0.24
Fig. \[fig3\] 14.1 0.16
Fig. \[fig4\] 13.3 0.22
Fig. \[fig5\] 11.7 0.33
: $\mu_m$ is the mobility calculated at $n=10^{11}$ cm$^{-2}$ and $n_c $ represents the critical density calculated from $\Gamma = E_F$.
Using $\mu_m$ to be the mobility at $n=10^{11}$ cm$^{-2}$ in Figs. \[fig2\] – \[fig5\], we conclude that the sample quality, if it is indeed determined entirely by the maximum mobility (or the mobility at a very high carrier density), should decrease monotonically as we go from the sample of Fig. \[fig2\] ($\mu_m =
35.7 \times 10^6$ cm$^{-2}$/Vs) to that of Fig. \[fig5\] ($\mu_m =
11.7\times 10^6$ cm$^{-2}$/Vs). We list below in Table I the calculated critical density for each sample in Figs. \[fig2\] – \[fig5\] noting also the mobility $\mu_m$ at $n=10^{11}$ cm$^{-2}$.
![(a) The same as Fig. \[fig2\] with following parameters: $n_i (d=0) = 3\times
10^8 cm^{-2}$, $n_d(d=500\AA) =10^{10}cm^{-2}$, and quantum well width $a=300$ Å. At a density $n=10^{11}cm^{-2}$ we have $\mu=11.7\times 10^6 cm^{2}/Vs$ and $\tau_q=16.2$ ps. The critical density $n_c =0.326 \times 10^{10}cm^{-2}$. []{data-label="fig5"}](fig5.eps){width="1.\columnwidth"}
We conclude from Table I that there is simply no one-to-one relationship between mobility and quality in these numerical transport results based on the 2-impurity model. For example, although the “lowest mobility" sample (Fig. \[fig5\], $\mu_m=11.7\times10^6$ cm$^2$/Vs) does indeed have the “lowest quality" as reflected in the highest value of $n_c$ ($\sim 0.33 \times 10^{10}$ cm$^{-2}$), the highest mobility sample (with almost three times the mobility of all the other samples) has the second highest value of $n_c$ ($\sim 0.24
\times 10^{11}$ cm$^{-2}$) instead of having the lowest $n_c$ as it would if quality is determined exclusively by mobility. The other two intermediate mobility samples with mobilities $14.1\times 10^6$ cm$^{-2}$/Vs and $13.3\times 10^6$ cm$^{-2}$/Vs also have their $n_c$ values “reversed" ($0.16\times 10^{10}$ cm$^{-2}$ and $0.22 \times
10^{10}$ cm$^{-2}$, respectively) compared with what they should be if the mobility really determined quality. We note that the samples of Figs. \[fig2\] and \[fig4\] have almost identical quality (i.e., essentially the same values of $n_c$) although the sample of Fig. \[fig2\] has almost three times the high-density mobility as that of Fig. \[fig4\]!
We do mention that the values of $n_c$ ($\sim 2 \times 10^9$ cm$^{-2}$) we obtain in our Figs. \[fig2\] – \[fig5\] are consistent with the observed 2D MIT critical density in ultra-high-mobility 2D GaAs structures where $n_c \sim 2 \times 10^9$ cm$^{-2}$ has been reported for $\mu_m \sim 10^7$ cm$^2$/Vs for $n
\alt 10^{11}$ cm$^{-2}$. [@lilly]
The last set of numerical results we show for the 2-impurity model is based on HIGFET (heterojunction-insulator-gated-field-effect-transistor) structures (in contrast to MODFET structures or modulation-doped-field-effect-transistors, which we have discussed so far in this paper) and is motivated by the recent experimental work by Pan and his collaborators on the effect of disorder on the observation, existence, and stability of the 5/2 FQHE in high-mobility GaAs-AlGaAs HIGFET structures [@panprl2011]. This work of Pan [*et al*]{}. is closely related to similar work by Gamez and Muraki [@gamez] and by Samkharadze [*et al.*]{} [@samkharadze] who also studied disorder effects on the stability of the 5/2 FQHE in modulation doped GaAs-AlGaAs 2D systems. All three of these experimental studies conclude, using different phenomenology and methodology, that the quality of the observed 5/2 FQHE in 2D systems is not directly connected in an one-to-one manner with the sample mobility, and it is possible to find robust 5/2 FQHE in samples with mobility in the $\agt 10^6$ cm$^2$/Vs range whereas much of the earlier work [@choi; @dean] had to use ultra-high mobility ($ >
10^7$ cm$^2$/Vs) for the observation of stable 5/2 FQHE. This observation by these three experimental groups of the mobility/quality dichotomy is very similar to the theory being developed in the current work with the only difference being that our work specifically focuses on the quality being associated with the single-particle quantum scattering rate $\tau_q^{-1}$ or the collisional level-broadening $\Gamma \sim \tau_q^{-1}$ rather than the 5/2 FQHE gap since we do not know of any quantitative microscopic theory which directly connects FQHE gap values with disorder. We comment further on this feature below in our discussion after presenting our HIGFET numerical results.
A HIGFET system is different from modulation-doped quantum well structures we considered so far in our work with the important qualitative difference being that HIGFETs are undoped (except, of course, for unintentional background charged impurities as represented by $n_B$, which are unavoidable in a semiconductor) with no remote modulation doping layer present in the system. Instead, the 2D carriers are induced in the GaAs surface layer at the AlGaAs-GaAs interface by a remote heavily doped gate placed very far from the GaAs-AlGaAs interface. [@kane] Thus, HIGFETs are basically the GaAs version of Si-MOSFETs (metal-oxide-semiconductor-field-effect transistors) with the insulator being the AlGaAs layer instead of SiO$_2$. An additional difference between HIGFETs and modulation-doped quantum wells is that the quasi-2D carrier confinement in the HIGFET is in an asymmetric triangular potential well (similar to MOSFETs [@andormp]) in contrast to the symmetric square well confinement in the AlGaAs-GaAs quantum well system.
![ (a) The calculated mobility as a function of background impurity density $n_B$ in a GaAs HIGFET structure. Here $n_R=10^{13}$ cm$^{-2}$, $d_R=630$ nm, and the carrier density $n=4.7 \times 10^{11}$ cm$^{-2}$ are used. []{data-label="fig6"}](fig6){width="0.70\columnwidth"}
Given that HIGFETs have no intentional modulation doping, it may appear that the 2-impurity model is simply inapplicable here since the background unintentional charged impurities seem to be the only possible type of Coulomb disorder in the system so that the system should belong to a 1-impurity disorder description (i.e., just the unintentional background random charged impurities). This is, however, incorrect because the presence of the far-away gate, which induces the 2D carriers in the HIGFET, introduces remote charged disorder (albeit at a very large value of $d$) arising from the gate charges which must be present due to the requirement of charge neutrality. We, therefore, use exactly the same minimal 2-impurity model for the HIGFETs that we have used for the modulation doped systems assuming $n_R$ to be the charged impurity density on the far away gate at a very large distance $d_R$ away from the induced 2D electron layer on the GaAs side of the GaAs-ALGaAs interface. (Later in this section we will present results for a realistic 3-impurity model in order to provide a quantitative comparison with the HIGFET data of Pan [*et al*]{}. [@panprl2011].)
![ The calculated (a) mobility, $\mu$, (b) level broadening, $\Gamma$, (c) transport lifetime, $\tau_t$ and (d) quantum lifetime, $\tau_q$, as a function of carrier density in a GaAs HIGFET structure with parameters $n_R=10^{13}$ cm$^{-2}$, $d_R=630$ nm, and $n_B=1.69
\times 10^8$ cm$^{-2}$. []{data-label="fig7"}](fig7){width="1.0\columnwidth"}
In Figs. \[fig6\] – \[fig8\] we show our full numerical results of a n-GaAs HIGFET structure using the 2-impurity model. The specific HIGFET structure used for our numerical calculations is motivated by the sample used in Ref. \[\], but we do not attempt any quantitative comparison with the experimental transport results, which necessitates a 3-impurity model to be described later. At this stage, i.e., for Figs. \[fig6\] – \[fig8\], our goal is to establish the mobility/quality dichotomy for HIGFET 2D systems based on our minimal 2-impurity model.
In Fig. \[fig6\] we show the calculated mobility as a function of $n_B$, the background impurity density and in Fig. \[fig7\] we show the mobility ($\mu$), the level broadening ($\Gamma$), the transport lifetime ($\tau_t$), and the quantum lifetime ($\tau_q$) as a function of the 2D carrier density $n$ in a GaAs HIGFET structure using the 2-impurity model with $n_R =
10^{13}$ cm$^{-2}$; $d_R = 630$ nm; $n_B = 1.69 \times 10^8$ cm$^{-2}$. We note that $\mu = ne\tau_t$ and $\Gamma = \hbar/2\tau_q$ are simple measures of mobility and quality which are directly linearly connected to $\tau_t$ and $\tau_q^{-1}$. Our choice of $d_R =
630$ nm is specifically aimed at the sample of Ref. \[\] where the gate is located 630 nm away from the GaAs-AlGaAs interface. Our choice of $n_R=10^{13}$ cm$^{-2}$ and $n_B=1.69 \times 10^8$ cm$^{-2}$ is arbitrary at this stage (and the precise choice here is irrelevant with respect to our qualitative conclusions) except that this combination of a large (small) $n_R$ ($n_B$) is the appropriate physical situation in high-quality HIGFETs. Our choice of $n_R$, $n_B$, and $d_R$ (which we get from the actual experimental system) gives the correct 2D “maximum" mobility of $\mu = 14 \times 10^6$ cm$^2$/Vs at a 2D carrier density of $n=4.7
\times 10^{11}$ cm$^{-2}$ consistent with the experimental sample in Ref. \[\] as shown in Fig. \[fig6\].
![ The calculated (a) mobility and (b) level broadening as a function of remote impurity density in a GaAs HIGFET structure with parameters $n=1.8 \times 10^{11}$ cm$^{-2}$, $d_R=630$ nm, and $n_B=1.69 \times
10^{10}$ cm$^{-2}$. []{data-label="fig8"}](fig8){width="1.0\columnwidth"}
The calculated mobility in Fig. \[fig6\] decreases monotonically with increasing $n_B$, and we choose $n_B = 1.69 \times 10^8$ cm$^{-2}$ to get the correct maximum mobility of $14 \times 10^6$ cm$^2$/Vs reported in Ref. \[\] with the corresponding value of the level broadening being 0.638 meV at the same density. We emphasize that the level broadening (or equivalently, $\tau_q$) here is determined entirely by the remote scattering from the gate in spite of the gate being an almost macroscopic distance ($\sim 0.6 \mu m$) away from the 2D electrons – changing $n_B$ by even a factor of 100 does not change the the value of $\Gamma$ or $\tau_q^{-1}$ (but does change mobility $\mu$ or $\tau_t^{-1}$ by a factor of 100) whereas the mobility is determined entirely by the background scattering (and therefore changing $n_R$ does not affect the mobility).
In Fig. \[fig7\] we show the calculated $\mu$, $\Gamma$, $\tau_t$, and $\tau_q$ (remembering $\mu = ne \tau_t$ and $\Gamma =
\hbar/2\tau_q$) as a function of 2D carrier density $n$ for fixed $n_R$, $d_R$, $n_B$ as shown. These results clearly show the mobility/quality dichotomy operational within the 2-impurity model in this particular HIGFET structure (for the chosen realistic disorder parameters $n_R$, $d_R$, $n_B$). For $n \agt 2\times 10^{10}$ cm$^{-2}$, the mobility is determined essentially by the background impurity scattering (i.e. $n_B$) whereas the level broadening or the quantum scattering rate is determined entirely by the remote scattering for the entire density range ($10^9
cm^{-2} < n < 10^{12} cm^{-2}$) shown in Fig. \[fig7\]. At low carrier density ($n \alt 10^{10}$ cm$^{-2}$) $k_Fd_R$ ($\alt 10$) is no longer very large, and given the rather large value of $n_R$ ($=10^{13}$ cm$^{-2}$) corresponding to the remote gate charges, the scattering by $n_R$ starts affecting the mobility. But, the high density mobility (determined by $n_B$) and the quality at all density (determined by $n_R$) are still completely independent quantities, and therefore it is possible for the quality (e.g., the FQHE gap at high density) to be completely independent of the mobility as found experimentally in Ref. \[\].
![ (a) The calculated mobility as a function of background impurity density $n_B$ in a GaAs MODFET structure with a well width $a=300$Å. Here $n=n_R=1.8\times 10^{11}$ cm$^{-2}$ and $d_R=2000$ nm are used. []{data-label="fig9"}](fig9){width="0.70\columnwidth"}
This is demonstrated explicitly in Fig. \[fig8\] where we show that the variation in the mobility is essentially non-existent for four orders of magnitude changes in $n_R$ whereas $\Gamma$ changes essentially by four orders of magnitude. Similarly, Fig. \[fig7\] indicates that (since $\tau_{qB}^{-1} \propto \Gamma \propto n_B$), a 2 orders of magnitude change in $n_B$ will hardly change $\Gamma$, but $\mu$ will change by 2 orders of magnitude (due to a 2-orders of magnitude change in $n_B$) at high carrier density.
We believe that our Figs. \[fig6\] – \[fig8\] provide a complete explanation for the puzzling observation in Ref. \[\] where a drop in the mobility of the sample at high carrier density hardly affected its quality as reflected in the measured 5/2 FQHE energy gap. This is because the high carrier density mobility is determined by background impurity density $n_B$ which does not affect the quality at all whereas the quality is affected by remote scattering which does not much affect the mobility at high carrier density. For the sake of completeness, and to make connection with the interesting recent works of Refs. \[\], who also independently conclude in agreement with Pan [*et al*]{}.[@panprl2011] that very high mobility ($>10^7$ cm$^2$/Vs) is not necessarily required for the experimental observation of a robust 5/2 FQHE in standard modulation-doped GaAs quantum wells (in contrast to Pan’s usage of undoped HIGFETs), we show in Figs. \[fig9\] – \[fig11\] ( which correspond to the HIGFET results shown in Figs. \[fig6\] – \[fig8\] respectively) our calculated transport results for a modulation-doped quantum well structure with a high-density mobility identical (i.e., $14\times 10^6$ cm$^2$/Vs) to the HIGFET structure considered in Figs. \[fig6\] – \[fig8\].
![ The calculated (a) mobility, $\mu$, (b) level broadening, $\Gamma$, (c) transport lifetime, $\tau_t$ and (d) quantum lifetime, $\tau_q$, as a function of carrier density in a GaAs MODFET structure with a well width $a=300$ Å. Here the parameters $n_R=n$, $d_R=200$ nm, and $n_B=6.8 \times 10^8$ cm$^{-2}$ are used. []{data-label="fig10"}](fig10){width="1.0\columnwidth"}
The main differences between the 2D systems for Figs. \[fig6\] – \[fig8\] (HIGFET) and Figs. \[fig9\] – \[fig11\] (MODFET) are the following: (1) the HIGFET has a triangular quasi-2D confinement potential (determined self-consistently by the carrier density) and the MODFET has a square-well confinement imposed by the MBE-grown AlGaAs-GaAs-AlGaAs structure with a given confinement width ($a=30$ nm in Figs. \[fig9\] – \[fig11\]); (2) the 2D carriers are induced by a very far away gate in the HIGFET whereas it is induced by the remote dopants (we choose $n_R=n$ in Figs. \[fig9\] – \[fig11\]) in the modulation doping layer (we choose $d_R=200$ nm in Figs. \[fig9\] – \[fig11\]); (3) the specific necessary values of $n_B$ are somewhat different in the two systems in order to produce the same high-density maximum mobility. The quantitative differences described in items (1) - (3) above are sufficient to produce substantial differences between the numerical results in the HIGFET and the MODFET system as can easily be seen by comparing the results of Figs. \[fig6\] – \[fig8\] with those of Figs. \[fig9\] – \[fig11\], respectively, although we ensured that both have exactly the same high-density mobility ($\mu_m = 1.4
\times 10^7$ cm$^2$/Vs). However, qualitatively the two sets of results shown in Figs. \[fig6\] – \[fig8\] and \[fig9\] – \[fig11\] are similar in that the mobility (quality) at high carrier density ($>10^{10}$ cm$^{-2}$) is invariably determined by the background (remote) scattering respectively, leading to the possibility that a substantial change in mobility (quality) by changing $n_B$ ($n_R$) respectively may not at all affect quality (mobility), and thus it is possible at high carrier density for a system to have a modest mobility ($\sim 10^6$ cm$^2$/Vs) by having a large $n_B$ with little adverse effect on quality (i.e., $\Gamma$). Thus, the experimental observations in Refs. \[\] are all consistent with our theoretical results.
![ The calculated (a) mobility and (b) level broadening as a function of remote impurity density in a GaAs MODFET structure with a well width $a=300$ Å. The parameters $n=1.8 \times 10^{11}$ cm$^{-2}$, $d_R=200$ nm, and $n_B=6.8 \times 10^{10}$ cm$^{-2}$ are used. []{data-label="fig11"}](fig11){width="1.0\columnwidth"}
Finally, we show in Figs. \[fig12\] and \[fig13\] the numerical transport results for the HIGFET structure (of Figs. \[fig6\] – \[fig8\]) using a more realistic 3-impurity model going beyond the 2-impurity model mostly used in our current work. The 3-impurity model is necessary for obtaining agreement between experiment [@panprl2011] and theory since experimentally the measured mobility, $\mu(n)$, as a function of carrier density manifests non-monotonicity with a maximum in the mobility around $n \sim 2\times
10^{11}$ cm$^{-2}$. Such a non-monotonicity, where $\mu$ increases (decreases) with increasing $n$ at low (high) carrier density, is common in Si-MOSFETs [@andormp], and is known to arise from short-range interface scattering which becomes stronger with increasing carrier density as the self-consistent confinement of the 2D carriers becomes stronger and narrower pushing the electrons close to the interface and thus increasing the short-range interface roughness scattering as well as the alloy disorder scattering in AlGaAs as the confining wave function tail of the 2D electrons on the GaAs side pushes into the Al$_x$Ga$_{1-x}$As side of the barrier. We include this realistic short-range scattering effect, which becomes important at higher carrier density leading to a decrease of the mobility at high density (as can be seen in Fig. \[fig12\](a)). Importantly, however, this higher-density suppression of mobility (by a factor of 3 in Fig. \[fig12\](a) consistent with the observation of Pan [*et al*]{}. [@panprl2011]) has absolutely no effect on the quality (see Fig. \[fig12\](b)) with the level broadening $\Gamma$ decreasing monotonically with increasing carrier density (since $\Gamma$ is determined essentially entirely by the remote scattering – see Fig. \[fig12\](d)). Thus, we see an apparent paradoxical situation (compare Figs. \[fig12\](a) and (b)) where the mobility decreases at higher carrier density, but the quality keeps on improving with increasing carrier density! This is precisely the phenomenon observed by Pan [*et al*]{}. [@panprl2011] who found that, although the mobility itself decreased in their sample by a factor of 3 at higher density, the sample quality, as measured by the 5/2 FQHE gap, improved with increasing density precisely as we predict in our work. In Fig. \[fig13\] we show that the 3-impurity model, except for allowing the mobility to decrease at high carrier density due to the increasing dominance of short-range scattering (thus bringing experiment and theory into agreement at high density in contrast to the 2-impurity results), has no effect on the basic quality/mobility dichotomy being discussed in this work – for example, Fig. \[fig13\] shows that while quality decreases (i.e., $\Gamma$ increases) monotonically with increasing remote scattering, nothing basically happens to the mobility!
![ The calculated (a) mobility and (b) level broadening with long range remote impurity at $d_R$, short range impurities at the interface, and background short range impurities. We assume that the density dependence of scattering time with interface short range impurities is $\tau_{qs}^{-1} \propto n^{\alpha}$. In (a) the blue line is calculated with $n_B = 0.8 \times 10^8
cm^{-2}$ and $\tau_{qs}^{-1} = 5\times 10^7 n$. The green line is calculated with $n_B = 0.9 \times 10^8
cm^{-2}$ and $\tau_{qs}^{-1} = 1.85\times 10^7 n^{1.3}$. The black line is calculated with $n_B = 1.08 \times 10^8
cm^{-2}$ and $\tau_{qs}^{-1} = 0.19\times 10^7 n^2$. The red dots are experimental data from Pan [*et al.*]{}[@panprl2011] In (c) and (d) we show the total scattering times as well as the individual scattering time corresponding to the each scattering source. Here $n_B
= 0.8 \times 10^8
cm^{-2}$ and $\tau_{qs}^{-1} = 5\times 10^7 n$ are used. []{data-label="fig12"}](fig12){width="1.0\columnwidth"}
Before concluding this section, we provide a critical and quantitative theoretical discussion of two distinct experiments (one from 1993 [@duprl1993] and the other from 2011 [@samkharadze]), separated by almost 20 years in time, involving high-mobility 2D semiconductor structures in the context of the mobility versus quality question being addressed in the current work. Our reason for focusing on these two papers is because both report $\tau_t$ and $\tau_q$ for the samples used in these experimental studies, thus enabling us to apply our theoretical analyses quantitatively to these samples.
In ref. \[\], two GaAs-AlGaAs heterojunctions were used (samples A and B) with the following characteristics [@duprivate]: Sample A: $n=1.1 \times 10^{11}$ cm$^{-2}$; $\mu = 6.8
\times 10^6$ cm$^2$/Vs; $\tau_t=270$ ps; $\tau_q = 9$ ps, and Sample B: $n=2.3 \times 10^{11}$ cm$^{-2}$; $\mu = 12
\times 10^6$ cm$^2$/Vs; $\tau_t=480$ ps; $\tau_q = 4.5$ ps. Both samples A and B have the same setback distance of $d_R=80$ nm for the remote dopants, but we should consider $d_A > d_B \agt 80$ nm since sample A has a lower carrier density and therefore the quasi-2D layer thickness for sample A must be slightly higher since the self-consistent confinement potential must be weaker in A than in B due to its lower density. We note that sample A and B indeed manifest the mobility/quality dichotomy in that A (B) has higher (lower) quality (i.e. $\tau_q$), but lower (higher) mobility!
We start by assuming the absence of any background impurity scattering ($n_B=0$), then the asymptotic formula for $k_F d_R \gg 1$ applies to both samples, giving, $\tau_t \sim (k_Fd_R)^3/n_R$; $\tau_q \sim
(k_Fd_R)/n_R$. Making the usual assumption $n_R = n$, since no independent information is available for $n_R$, we conclude that the theory predicts $\tau_t^B/\tau_t^A = \sqrt{n_B/n_A} (d_B/d_A)^3
\approx 1.4$ assuming $d_B \approx d_A$, and $\tau_q^B/\tau_q^A =
\sqrt{n_A/n_B} d_B/d_A \approx 0.7$ assuming $d_B \approx
d_A$. Experimentally, A and B samples satisfy: $\tau_t^B/\tau_t^A
\approx 1.8$; $\tau_q^B/\tau_q^A =0.5$. Thus, just the consideration of only remote dopant scattering which must always be present is all modulation-doped samples already gives semi-quantitative agreement between theory and experiment including an explanation of the apparent paradoxical finding that the sample B with higher mobility has a lower quality! The key here is that the higher density of sample B leads to a higher mobility, but also leads to a higher values of $\tau_q^{-1}$ (and hence lower quality) by virtue of higher carrier density necessitating a higher value of $n_R$ leading to a lower value of $\tau_q$ \[see, for example, Fig. \[fig01\](c) where increasing $n=n_R$ leads to increasing (decreasing) $\tau_t$ ($\tau_q$)\].
We can actually get essentially precise agreement between theory and experiment for the dichotomy in samples A and B of Ref. \[\], with $\tau_t$ higher (lower) in sample B (A) and $\tau_q$ higher (lower) in sample A (B) by incorporating the fact that a higher (by a factor of 2) carrier density in sample B compared with sample A makes $d_A>d_B$ due to self-consistent confinement effect in heterostructures and hence the theoretical ratios of $\tau^A$ and $\tau^B$ change from the values given above to $\tau_t^B/\tau_t^A <1.4$ and $\tau_q^B/\tau_q^A \approx 0.5$ (i.e., $<0.7$). This means that while the quality ratio $\tau_q^B/\tau_q^A$ of samples A and B can be understood quantitatively on the basis of remote scattering (which determines the quality almost exclusively in high-mobility modulation-doped structures), the mobility ratio $\tau_t^B/\tau_t^A$ is not determined exclusively by remote dopant scattering. Inclusion of somewhat stronger background disorder scattering in sample A compared with sample B immediately gives $\tau_t^B/\tau_t^A =1.8$ in agreement with experiment. Thus, we see as asserted by us theoretically, quality and mobility are mainly controlled by distinct scattering mechanisms (quality by remote scattering and mobility by background scattering) in the data of ref. \[\] providing an explicit example of the mobility/quality dichotomy as far back as in 1993 when this dichotomy was not discussed at all in the literature.
 are used. []{data-label="fig13"}](fig13){width="1.0\columnwidth"}
Considering now the samples used in ref. \[\], there are again two distinct modulation-doped quantum well samples with the following sample specifications: Sample A: $a=56$ nm; $d_R =
320$ nm; $n=8.3 \times 10^{10}$ cm$^{-2}$; $\mu = 12\times 10^6$ cm$^2$/Vs; $\Gamma = 0.24$ K, and Sample B: $a=30$ nm; $d_R =
78$ nm; $n=2.78 \times 10^{11}$ cm$^{-2}$; $\mu = 11\times 10^6$ cm$^2$/Vs; $\Gamma = 2.04$ K. Thus, in this case [@samkharadze], although the two samples have almost identical mobilities, the lower-density sample A has almost 8 times higher quality with $\Gamma_B/\Gamma_A = \tau_q^A/\tau_q^B \approx 8$. We note that the lower quality sample has three times the carrier density, and going back to our Figs. \[fig01\] – \[fig04\], we see that a higher carrier density $n$ ($=n_R$) always leads to higher mobility and lower quality since the quality (i.e., $\tau_q$) is determined mostly by long-range remote scattering whereas the mobility is determined by a combination of both remote and background scattering with the background scattering often dominating the mobility. The fact that $d_R^A \gg d_R^B$ considerably improves the quality of sample A with respect to sample B without much affecting the mobility since the quality (mobility) is limited by remote (background) scattering.
Using the asymptotic formula (for $k_Fd \gg 1$), $\tau_q \propto k_F d
/n_R$ and $n_R \approx n$, we conclude for the comparative quality of the two samples: $\tau_q^A/\tau_q^B = \Gamma_B/\Gamma_A \approx
\sqrt{n_B/n_A} d_A/d_B \approx 8$ where we use $d_A = 348$ nm and $d_B
= 93$ nm by taking into account their differences in both the set back distances and the well thickness. Experimentally, $\Gamma_B/\Gamma_A
\approx 8.5$ in excellent agreement with the theoretical estimate. The fact that the mobilities of A and B are similar is easily explained by their similarity with respect to background disorder with sample B having somewhat larger value of unintentional background impurity density than sample A. Thus, our mobility/quality theoretical dichotomy is in perfect accord with the data of ref. \[\].
We now conclude this section by mentioning that we have used the quantum lifetime (or the single particle scattering time) $\tau_q$ (or equivalently $\Gamma \propto \tau_q^{-1}$) as a measure of the quality because it is well-defined and theoretically calculable. Experimentally, the quality can be defined in a number of alternative ways as , for example, done in the recent experiments [@panprl2011; @gamez; @samkharadze] where the 5/2 FQHE gap is used as a measure of the quality. There is no precise microscopic theory for calculating disorder effects on the FQHE gap, but there are strong indications [@dean; @morf; @samkharadze] that the FQHE gap $\Delta_{\Gamma}$ in the presence of finite disorder scales approximately as $$\Delta_{\Gamma} \approx \Delta_0 - \Gamma,
\label{eq48}$$ where $\Gamma$ is indeed the quantum level broadening we use in our current work as the measure of quality and $\Delta_0$ is the FQHE gap in the absence of any disorder. If this is even approximately true (as it seems to be on empirical grounds), then our current theoretical work shows complete consistency with the recent experimental results concerning the dichotomy between mobility and FQHE gap values in the presence of disorder. In this context, it may be worthwhile to emphasize an often overlooked fact: the mobility itself (i.e., $\tau_t^{-1}$ and [*not*]{} $\tau_q^{-1}$) can be converted into an energy scale by writing (for GaAs) $$\Gamma_{\mu} = \frac{\hbar}{2\tau_t} \approx (10^{-4}/\tilde{\mu}) \;
{\rm meV} \; \approx (.01/\tilde{\mu}) \; {\rm K},$$ where $\tilde{\mu} = \mu/(10^7 cm^2/Vs)$. Thus, a mobility of $10^7$ cm$^2$/Vs corresponds only to a broadening of 10 mK which is miniscule compared with the theoretically calculated [@morf; @morf2] 5/2 FQHE gap of $2-3$ K! Even a mobility of $10^6$ cm$^2$/Vs corresponds to a mobility broadening of only 100 mK, which is much less than the expected 5/2 FQHE gap. Thus, the quality of the 5/2 FQHE cannot possibly be determined directly by the mobility value (unless the mobility is well below $10^6$ cm$^2$/Vs) and there must be some other factor controlling the quality, which we take to be the quantum level broadening in this work. It must be emphasized here that the mobility/quality dichotomy obviously arises from the underlying disorder in high-mobility semiconductor structures being long-ranged. If both mobility and quality are dominated by short-range disorder, then $\tau_q \approx
\tau_t$, and a mobility of $10^6$ cm$^2$/Vs with $\Gamma \approx 100$ mK will be a high-quality sample!
In concluding this section, we should mention that the very first experimental work we know of where the mobility/quality dichotomy was demonstrated and noted explicitly in the context of FQHE physics is a paper by Sajoto et al. [@sajorto] from the Princeton group which appeared in print an astonishing 24 years ago! In this work, (see the “Note added in proof" in Ref. \[\]), it was specifically stated that the samples used by Sajoto et al. manifested as strong FQHE states as those observed in other samples from other groups with roughly $5-10$ times the mobility of the Sajoto et al. samples, thus providing a clear and remarkable early example of the mobility/quality dichotomy much discussed during the last couple of years in the experimental 2D literature. We note that the samples used by Sajoto et al. [@sajorto] had unusually large set-back distances ($d_R \sim 270$ nm), leading to rather small values of $\tau_q^{-1}$ and $\Gamma$ corresponding to our theory although the mobility itself, being limited by background impurity scattering (i.e. by $n_B$), was rather poor ($\sim 10^6$ cm$^2$/Vs). We believe that the reason the samples of Sajoto et al. had such high quality in spite of having rather modest mobility is the mobility/quality dichotomy studied in our work where the mobility determined by background scattering is disconnected from the quality determined by the remote dopant scattering.
summary and conclusion
======================
In summary, we have theoretically discussed the important issue of mobility versus quality in high-mobility 2D semiconductor systems such as modulation-doped GaAs-AlGaAs quantum wells and GaAs undoped HIGFET structures. We have established, both analytically (section II) and numerically (section III), that modulation-doped (or gated) 2D systems should generically manifest a mobility/quality dichotomy, as often observed experimentally, due to the simple fact that mobility and quality are often determined by different underlying disorder mechanisms in 2D semiconductor structures – in particular, we show definitively that in many typical situations, the mobility (quality) is controlled by near (far) quenched charged impurities, particularly at higher carrier density and higher mobility samples. We show that often the 2D mobility (or equivalently, the 2D transport scattering time) is controlled by the unintentional background charged impurities in the 2D layer whereas the quality, which we have parameterized throughout our work by the quantum single-particle scattering time (or equivalently, the quantum level broadening), is controlled by the remote charged impurities in the modulation doping layer whose presence is necessary for inducing carriers in the 2D layer. Somewhat surprisingly, we show that the same mobility/quality dichotomy could actually apply to undoped HIGFET structures where the charges on the far-away gate play the role of remote scattering mechanism. Quite unexpectedly, we show that a very far away gate (located almost $10^{-4}$ cm away from the 2D layer) can still completely dominate the quantum level broadening, while at the same time having no effect on the mobility. We develop a minimal 2-impurity model (near and far or background and remote) which is sufficient to explain all the observed experimental features of the mobility/quality dichotomy. The key physical point here is that the dimensionless parameter ‘$k_F d$’, where $k_F \propto \sqrt{n}$ is the Fermi wave number of the 2D electron system and ‘$d$’ is the distance of the relevant charged impurities from the 2D system completely controls the mobility/quality dichotomy. Impurities with $k_F d \gg 1$ ($\ll 1$) could totally dominate quality (mobility) without affecting the other property at all. We give several examples of situations where identical or very similar sample mobilities at high carrier density could lead to very different sample qualities (i.e., quantum level broadening differing by large factors) and vice versa. The mobility/quality dichotomy in our minimal 2-impurity model arises from the exponential suppression of the large angle scattering by remote charged impurities which leads to the interesting situation that remote scattering contributes little to the resistivity, but a lot to the level broadening through the accumulation of substantial small angle scattering. We emphasize that the mobility/quality dichotomy arises entirely from the long-range nature of the underlying disorder, and would disappear completely if the dominant disorder in the system is short-ranged.
It is important to realize that $\tau_t$ (mobility) and $\tau_q$ (quality) both depend not only on the disorder strength, but also on the carrier density, i.e., $\tau_{t,q} \equiv
\tau_{t,q}(n,n_R,d_R,n_B)$. Thus, even within the 2-impurity model (parameterized by disorder parameters $n_R$, $d_R$, $n_B$), $\tau_{t,q}$ are both functions of carrier density. For very low carrier density, the dimensionless parameter $k_Fd$ may be small for all relevant impurities in the system, and eventually our 2-impurity model will then fail since at such a low carrier density, all impurities are essentially near impurities with the distinction between R-impurities and B-impurities being merely a semantic distinction with no real difference. Mobility and quality at such low densities then will behave similarly. The same situation may also apply as a matter of principle at very high carrier densities (i.e., very large $k_F$) where all impurities may satisfy $k_F d \gg 1$ and thus act as far impurities, again leading to a breakdown of the 2-impurity model. This density dependence of the 2-impurity model with respect to mobility/quality dichotomy is, however, a non-issue for our current work since (1) typically, samples are characterized by their mobility values at some fixed high (but not too high) carrier density ($n \sim 10^{11} - 4 \times 10^{11}$ cm$^{-2}$), and (2) the very low and high density regimes where the 2-impurity model is no longer operational are completely out of the experimentally relevant density range of interest in high-mobility 2D semiconductor structures for the physics (e.g., FQHE) studied in this context. Assuming a high-mobility modulation doped GaAs quantum well of thickness $200 - 400$ Å and a set-back distance of $600 - 2000$ Å (these are typical numbers for high-mobility 2D GaAs structures), the 2-impurity model should be well-valid in a wide range of carrier density $5\times 10^9$ cm$^{-2}
\alt n \alt 5\times 10^{11}$ cm$^{-2}$, which is the applicable experimental regime of interest. Thus, the applicability of the 2-impurity model for considering the mobility/quality dichotomy is not a serious issue of concern.
A second concern could be the validity (or not) of our theoretical approximation scheme for calculating $\tau_{t,q}$, where we have used the zero-temperature Boltzmann theory and the leading-order Born approximation for obtaining the scattering rates. The $T=0$ approximation is excellent as long as $T \ll T_F = E_F/k_B$, which is valid in all systems of interest in this context. For high-mobility 2D semiconductor structures of interest in the current work, where the issue of the dichotomy of mobility/quality is relevant (since in low-mobility samples, typically $\tau_t \approx \tau_q$), the leading order theory (in the disorder strength) employed in our approximation scheme should, however, be excellent since the conditions $n\gg n_B$ and $n \gg n_Re^{-2k_Fd_R}$ are both satisfied making Born approximation essentially an exact theory in this manifestly very weak disorder situation (consistent with the high carrier mobility under consideration). An equivalent way of asserting the validity of Born approximation in our theory is to note that the conditions $E_F \gg
\Gamma$ and $k_F l \gg 1$ are always satisfied in the regime of our interest (with $\Gamma$ and $l$ being the quantum level broadening and the transport mean free path respectively). A related issue, which is theoretically somewhat untractable, is the possible effect of impurity correlation effects [@correlation] on the mobility versus quality question in 2D semiconductor structures. It is straightforward to include impurity correlation effects among the dopant ions in our transport theory, but unfortunately no sample-dependent experimental information is available on impurity correlations for carrying out meaningful theoretical calculations. We have carried out some representative numerical calculations assuming model inter-impurity correlations among the remote dopants, finding that such correlations enhance both $\tau_{t}$ and $\tau_q$, as expected (with $\tau_q$ being enhanced more than $\tau_t$ in general), compared with the completely random impurity configuration results presented in the current article, but our qualitative conclusions about the mobility/quality dichotomy remain unaffected since the fact that $\tau_t$ and $\tau_q$ are controlled respectively by background and remote scattering in high-mobility modulation-doped structures continues to apply in the presence of impurity correlation effects. We therefore believe that our current theory involving Born approximation assuming weak leading order disorder scattering from random uncorrelated quenched charged impurities in the environment (both near and far) is valid in the parameter regime of our interest.
Finally, we comment on the possibility of future experimental work to directly verify (or falsify) our theory. Throughout this paper, we have, of course, made extensive contact with the existing experimental results which, in fact, have motivated our current theoretical work on the mobility versus quality dichotomy. For a direct future experimental test of the theory, it will be necessary to produce a large number of high-mobility 2D semiconductor structures with different fixed carrier densities and with varying values of the remote dopant setback distance, and then measure the values of $\tau_{t,q}$ in a large set of samples which are all characterized by their high-density mobility. The measurement of the transport relaxation time $\tau_t$ is simple since it is directly connected to the carrier mobility $\mu$ (or conductivity $\sigma$): $\tau_t = m
\sigma/ne^2 = m\mu/e$. The measurement of the single-particle relaxation time (or the quantum scattering time) $\tau_q$ is, however, not necessarily trivial although its theoretical definition is very simple. In particular, the Dingle temperature or equivalently the Dingle level broadening $\Gamma_D$ obtained from the measured temperature dependence of the amplitude of the 2D SdH oscillations may not necessarily give the zero-field quantum scattering time $\tau_q$ defining the sample quality in our theoretical considerations (i.e. $\Gamma_D = \hbar/2\tau_q$ may not necessarily apply to the 2D SdH measurements) because of complications arising from the quantum Hall effect and inherent spatial density inhomogeneities (associated with MBE growth) in the 2D sample. Since our theory is explicitly a zero-magnetic field theory, it is more appropriate to obtain $\tau_q$ simply by carefully monitoring low-field magneto-resistance oscillations finding the minimum magnetic field $B_0$ where the oscillations disappear. The corresponding cyclotron energy $\omega_0
= e B_0/mc$ then defines the single particle level-broadening $\Gamma
\sim \hbar \omega_0$, providing $\tau_q = 1/2\omega_0$. An advantage of this method of obtaining $\tau_q$ is that one is necessarily restricted to the low magnetic field regime in high-mobility systems (i.e., $E_F \gg \hbar \omega_0$), where our theory should be applicable. A much stronger advantage be applicable. A much stronger advantage of using this proposed definition (i.e., the disappearance of magneto-resistance oscillations at the lowest experimental temperature) for the experimental determination of $\tau_q$ is that this is much easier to implement in the laboratory than the full measurement of the Dingle temperature which requires accurate measurements of the temperature dependent SdH amplitude oscillations. We therefore suggest low-temperature measurements of $\mu$ and $\omega_0$ to obtain $\tau_t$ and $\tau_q$ respectively in a large number of modulation doped samples with varying $n$, $n_R$, $d_R$, and $n_B$ in order to carry out a quantitative test of our theory. A large systematic data base of both $\tau_t$ and $\tau_q$ in many different samples should manifest poor correlations between these two scattering times (i.e., the mobility/quality dichotomy) provided the samples are high-mobility samples dominated by long-range charged impurity disorder. As emphasized (and as is well-known) throughout this work, if one type of disorder completely dominates both $\tau_t$ and $\tau_q$, then they will obviously be correlated, but this should be more an exception than the rule in high-mobility modulation-doped 2D semiconductor structures, where both near (“$n_B$”) and far (“$n_R$") impurities should, in general, play important roles in manifesting the mobility/quality dichotomy.
One important open question is whether the mobility/quality dichotomy we establish in the current work can be extended to other definitions of quality beyond our definition of quality in terms of the single-particle scattering rate or quantum level-broadening. The advantage of using the quantum scattering rate as the measure of sample quality is that this definition is generic, universal, and simple to calculate (and to measure). However, obviously, given an arbitrary disorder distribution involving long-range Coulomb-disorder, there are many possible definitions of quality involving many different moments of the effective Coulomb disorder. It will be very interesting for future work to choose alternate possible definitions of sample quality to establish whether our finding of the mobility/quality dichotomy applies to all possible definitions of sample quality.
This work is supported by LPS-CMTC, IARPA-ARO, Microsoft Q, and Basic Science Research Program through the National Research Foundation of Korea Grant funded by the Ministry of Science, ICT & Future Planning (2009-0083540).
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| 0 |
---
abstract: 'Motivated by string constructions, we consider a variant on the Type II see-saw mechanism involving the exchange of triplet representations of $SU(2)_L$ in which this group arises from a diagonal embedding into $SU(2)_A \times SU(2)_B$. A natural assignment of Standard Model lepton doublets to the two underlying gauge groups results in a bimaximal pattern of neutrino mixings and an inverted hierarchy in masses. Simple perturbations around this leading-order structure can accommodate the observed pattern of neutrino masses and mixings.'
author:
- Paul Langacker
- 'Brent D. Nelson'
title: 'String-Inspired Triplet See-Saw from Diagonal Embedding of $SU(2)_L \subset SU(2)_A \times SU(2)_B$'
---
Introduction {#introduction .unnumbered}
============
Observations by a variety of experimental collaborations have now firmly established the hypothesis that neutrino oscillations occur and that they are the result of non-vanishing neutrino masses and mixing angles [@Maltoni:2003da; @Bahcall:2004ut; @Fogli:2005cq]. While our knowledge of neutrino mass-differences and mixings has continued to improve over recent years, there continues to be as yet no consensus on the correct mechanism for generating the quite small neutrino masses implied by the experimental data. In some respects this is similar to the case of quark masses and mixings: despite having access to even more of the relevant experimental data for an even longer period of time, no compelling model of the hierarchies of masses and mixings in the quark sector has emerged either. But most theoretical effort in the area of neutrinos goes beyond the simple Dirac-mass Yukawa operator by introducing new structures in the superpotential to account for neutrino masses, such as the see-saw mechanism (in one of its various forms, to be defined more precisely below).[^1] Thus neutrinos are likely to be very special in the Standard Model – and its supersymmetric extensions – and may thereby provide a unique window into high-scale theories that the quark sector fails to illuminate.
It has thus far been mostly in vain that we might look to string theory for some guidance in how to approach the issue of flavor. In part this is because of the vast number of possible vacua in any particular construction, each with its own set of fields and superpotential couplings between them. On the other hand the problem of generating small neutrino masses may be one of the most powerful discriminants in finding realistic constructions. This was one of the conclusions of a recently completed survey [@Giedt:2005vx] of a large class of explicit orbifold compactifications of the heterotic string for the standard (or “Type I”) see-saw in its minimal form. The fact that no such viable mechanism was found may suggest that often-neglected alternatives to the standard see-saw may have more theoretical motivation than considerations of simplicity, elegance, or GUT structure would otherwise indicate.
In this work we study the properties of a new construction of see-saw mechanisms that is motivated by known string constructions. The mechanism is an example of the Higgs triplet or “Type II” seesaw [@Lazarides:1980nt; @Mohapatra:1980yp; @Schechter:1981cv; @Ma:1998dx; @Hambye:2000ui; @Rossi:2002zb], but the stringy origin has important implications for the mixings and mass hierarchy that distinguish it from conventional “bottom-up” versions of the triplet model. After outlining the model in a general way below we will motivate its plausibility in string theory by considering a particular $\mathbb{Z}_3 \times
\mathbb{Z}_3$ orbifold of the heterotic string [@Font:1989aj], where several of the properties needed for a fully realistic model are manifest.
General Features of $SU(2)$ Triplet Models {#sec:general}
==========================================
Let us briefly review the form of the effective neutrino mass matrix to establish our notation and to allow the contrast between models involving triplets of $SU(2)_L$ and those involving singlets to be more apparent. While models of neutrino masses can certainly be considered without low-energy supersymmetry, our interest in effective Lagrangians deriving from string theories which preserve $N=1$ supersymmetry leads us to couch our discussion in a supersymmetric framework. Then the effective mass operator involving only the light (left-handed) neutrinos has mass dimension five. Once the Higgs fields acquire vacuum expectation values (vevs) the effective neutrino masses are given by $$%
(\mathbf{m}_{\nu})_{ij} = (\lambda_{\nu})_{ij} \frac{v_2^2}{M} ,
%
\label{mnu}$$ where $v_2$ is the vev of the Higgs doublet $H_2$ with hypercharge $Y=+1/2$. The $3 \times 3$ matrix of couplings $\lambda_{\nu}$ is necessarily symmetric in its generation indices $i$ and $j$.
Such an operator can be induced through the exchange of heavy singlet (right-handed) neutrinos $N_R$ – as in the standard or “Type I” see-saw approach [@GRS; @Yanagida; @Valle1] – or through the exchange of heavy triplet states $T$ [@Lazarides:1980nt; @Mohapatra:1980yp; @Schechter:1981cv; @Ma:1998dx; @Hambye:2000ui; @Rossi:2002zb], or both. In either case, the mass scale $M$ is given by the scale at which lepton number is broken (presumably the mass scale of the heavy state being exchanged). In the presence of both contributions to the light neutrino masses we have the general mass matrix $$%
\Lag = \frac{1}{2} \( \oline{\nu}_L \; \; \oline{N}_L^c \) \(
\begin{array}{cc} \mathbf{m}_{T} & \mathbf{m}_{D} \\ \mathbf{m}_{D}^{T} &
\mathbf{m}_{S} \end{array} \) \( \begin{array}{c} \nu_R^c \\ N_R
\end{array} \) + \hc .
%
\label{general}$$ Each of the four quantities in (\[general\]) are understood to be $3 \times 3$ matrices in flavor space. That is, we imagine a model with one [*species*]{} of lepton doublet $L$ (with three generations) and, if present, one [*species*]{} of right-handed neutrino field (with three generations).
We will use the name “triplet models” to refer to any model which uses electroweak triplet states alone to generate neutrino masses. That is, such a model dispenses with right-handed neutrinos altogether, and the effective neutrino mass in (\[mnu\]) is then simply identified with the entry $\mathbf{m}_{T}$ in (\[general\]). A supersymmetric extension of the MSSM capable of giving small effective masses to left-handed neutrinos would involve two new sets of fields $T_i$ and $\oline{T}_i$ which transform as triplets under $SU(2)_{L}$ and have hypercharge assignments $Y_{T} = +1$ and $Y_{\oline{T}} =
-1$, respectively [@Hambye:2000ui; @Rossi:2002zb]. For the time being we will consider just one pair of such fields, which couple to the Standard Model through the superpotential $$\begin{aligned}
%
W_{\nu} &=& (\lambda_{T})_{ij} L_i T L_j + \lambda_1 H_1 T H_1 +
\lambda_2 H_2 \oline{T} H_2 \nonumber \\
%
& & + M_T T \oline{T} + \mu H_1 H_2 ,
%
\label{Wnu} \end{aligned}$$ where the $SU(2)$ indices on the doublets and triplet have been suppressed. Strictly speaking, the coupling $\lambda_1$ is not necessary to generate the required neutrino masses, but given the Standard Model charge assignments of the fields $T$ and $\oline{T}$ there is no [*a priori*]{} reason to exclude this coupling. The mass scale $M$ in (\[mnu\]) is to be identified with $M_T$ in this case, and the matrix $\lambda_{T}$ is symmetric in its generation indices. From the Lagrangian determined by (\[Wnu\]), it is clear that should the auxiliary fields of the chiral supermultiplets for the triplets vanish in the vacuum $\lang F^T \rang = \lang F^{\oline{T}} \rang = 0$, and we assume no vevs for the left-handed sneutrino fields, then there is a simple solution for the vevs of the neutral components of the triplet fields $$%
\lang T \rang = -\frac{\lambda_2 \lang H_2 \rang ^2}{M_T} \; ;
\quad \lang \oline{T} \rang = -\frac{\lambda_1 \lang H_1 \rang
^2}{M_T} ,
%
\label{Tvevs}$$ implying $m_{\nu} = \lambda_T \lang T \rang$.
Recent models of the Type II variety [@King:2003jb; @DeGouvea:2005gd; @Langacker:2004xy; @Altarelli:2004za; @Mohapatra:2004vr] would typically retain the right-handed neutrinos and utilize all the components in the mass matrix of (\[general\]) to explain the neutrino masses and mixings. These examples are often inspired by SO(10) GUT considerations or are couched in terms of left-right symmetry more generally. The latter commonly employ additional Higgs fields transforming as $(1,\mathbf{3})$ under $SU(2)_L
\times SU(2)_R$, which acquire vevs to break the gauge group to the Standard Model.
Instead we imagine a process by which the $SU(2)$ of the Standard Model emerges as the result of a breaking to a [*diagonal*]{} subgroup $SU(2)_A \times SU(2)_B \to SU(2)_L$ at a very high energy scale. Furthermore, while we will employ two conjugate triplet representations which form a vector-like pair under the Standard Model gauge group with $Y = \pm 1$, we do not seek to embed this structure into a left-right symmetric model.
Diagonal Embedding of $SU(2)_L$ {#sec:diagonal}
===============================
In attempting to embed the framework of the previous section in a model of the weakly coupled heterotic string we immediately encounter an obstacle: the simplest string constructions contain in their massless spectra only chiral superfields which transform as fundamentals or (anti-fundamentals) of the non-Abelian gauge groups of the low-energy theory. Scalars transforming as triplets of $SU(2)$ simply do not exist for such affine level one constructions [@Gross:1985fr; @Font:1990uw]. Indeed, scalars transforming in the adjoint representation appear only at affine level two, while representations such as the $\mathbf{120}$ and $\mathbf{210}$ of $SO(10)$ appear only at affine level four - and the $\mathbf{126}$ of $SO(10)$, which contains triplets of the Standard Model $SU(2)_L$, has been shown to never appear in free-field heterotic string constructions [@Dienes:1996yh].
Directly constructing four-dimensional string compactifications yielding higher affine-level gauge groups has proven to be a difficult task. But a group factor $\mathcal{G}$ can be effectively realized at affine level $k = n$ by simply requiring it to be the result of a breaking of $\mathcal{G}_1 \times
\mathcal{G}_2 \times \cdots \times \mathcal{G}_n$ to the overall diagonal subgroup. In fact, these two ways of understanding higher affine levels – picking a particular set of GSO projections in the underlying string construction and the low energy field theory picture of breaking to a diagonal subgroup – are equivalent pictures [@Dienes:1996yh]. With this as motivation, let us consider an appropriate variation on the superpotential of (\[Wnu\]). The breaking of the gauge group $SU(2)_A \times
SU(2)_B$ to the diagonal subgroup, which we identify as $SU(2)_L$, can occur through the vacuum expectation value of a field in the bifundamental representation of the underlying product group via an appropriately arranged scalar potential. For the purposes of our discussion here we will need only assume that this breaking takes place at a sufficiently high scale, say just below the string compactification scale. As such ideas for product group breaking have been considered in the past [@Barbieri:1994jq], we will not concern ourselves further with this step.
Any additional bifundamental representations will decompose into triplet and singlet representations under the surviving $SU(2)_L$. Gauge invariance of the underlying $SU(2)_A \times SU(2)_B$ theory then [*requires*]{} that the neutrino-mass generating superpotential coupling involving lepton doublets and our $SU(2)$ triplet now be given by $$%
W_{\nu} = (\lambda_{T})_{ij} L_i T L'_j ,
%
\label{WT}$$ where the field $T$ is the $SU(2)_L$ triplet representation and $L$ and $L'$ are two [*different species*]{} of doublets under the $SU(2)_L$ subgroup. That is, we denote by $L$ fields which have representation $(\mathbf{2},1)$ and by $L'$ fields which have representation $(1, \mathbf{2})$ under the original $SU(2)_A
\times SU(2)_B$ gauge theory. These two sets of lepton doublets, each of which may carry a generation index as determined by the string construction, arise from different sectors of the string Hilbert space. Once the gauge group is broken to the diagonal subgroup this distinction between the species is lost [*except for the pattern of couplings represented by the matrix $\mathbf{\lambda}_{T}$.*]{} The indices $i$ and $j$ carried by the lepton doublets represent internal degeneracies arising from the specific construction. It is natural to identify these indices with the flavor of the charged lepton (up to mixing effects, which we assume to be small).
In a minimal model, with only three lepton doublets charged under the Standard Model $SU(2)_L$, we are obliged to separate the generations, with two arising from one sector of the theory and one from the other. The precise form of the effective neutrino mass matrix will depend on this model-dependent identification, but one property is immediately clear: [*the effective neutrino mass matrix will necessarily be off-diagonal in the charged lepton flavor-basis.*]{}
We will restrict our study to the case of one triplet state with supersymmetric mass $M_T$ as in (\[Wnu\]). If we separate the doublet containing the electron from the other two, by defining $L_i = L_e = (\mathbf{2},1)$ and $L'_j = L_{\mu}, L_{\tau} = (1,
\mathbf{2})$ under $SU(2)_A \times SU(2)_B$, then the matrix of couplings $\lambda_T$ is (to leading order) $$%
\lambda_T = \lambda_0 \(\begin{array}{ccc} 0 & a & b \\ a & 0 & 0
\\ b & 0 & 0 \end{array} \) .
%
\label{lambdaT1}$$ It is natural to assume that the overall coefficient $\lambda_0$ in (\[lambdaT1\]) is of order unity. In fact if we now return to a string theory context, particularly that of the heterotic string with orbifold compactification, then the fact that the two generations of $L'_j$ in (\[WT\]) arise from the same sector of the string Hilbert space (i.e., the same fixed point location under the orbifold action) suggests that we should identify the coupling strengths: $a=b$.
Neutrino mass matrices based on the texture in (\[lambdaT1\]) with $a=b$ are not new to this work, but were in fact considered not long ago as a starting point for the bimaximal mixing scenario [@Barger:1998ta; @Barbieri:1998mq; @Altarelli:1998nx; @Jezabek:1998du]. In fact, the form of (\[lambdaT1\]) can be derived from a bottom-up point of view by first postulating a new symmetry based on the modified lepton number combination $L_e - L_{\mu} -
L_{\tau}$ [@Mohapatra:1999zr; @Babu:2002ex]. Indeed, the operator in (\[WT\]) with the identification of $L = L_e$ and $L' = L_{\mu}, L_{\tau}$ does indeed conserve this quantum number. However, in the string-theory motivated (top down) approach this conserved quantity arises as an [*accidental*]{} symmetry pertaining to the underlying geometry of the string compactification. It reflects the different geometrical location of the fields (in terms of orbifold fixed points) of the electron doublet from the muon and tau doublets.
To make contact with data it is necessary to consider the Yukawa interactions of the charged leptons as well. To that end, our string-inspired model should have a superpotential of the form $$\begin{aligned}
%
W &=& \lambda_T L T L' + \lambda_1 H_1 T H'_1 + \lambda_2 H_2
\oline{T} H'_2 \nonumber \\
%
& & + \lambda_3 S_3 T \oline{T} + \lambda_4 S_4 H_1 H_2 +
\lambda_5 S_5 H'_1 H'_2 \nonumber \\
%
& & + \wh{\lambda}_4 \wh{S}_4 H_1 H'_2 +
\wh{\lambda}_5 \wh{S}_5 H'_1 H_2 \nonumber \\
%
& & + \lambda_6 S_6 L H_1 + \lambda_7 S_7 L' H'_1 \nonumber \\
%
& & + \wh{\lambda}_6 \wh{S}_6 L H'_1 + \wh{\lambda}_7 \wh{S}_7 L' H_1 ,
%
\label{Wfull} \end{aligned}$$ where generation indices have been suppressed. The terms proportional to the couplings $\lambda_1$ and $\lambda_2$ in (\[Wnu\]) must now be modified to reflect the fact that the Higgs doublets must also come from two different species. These are denoted in the same manner as the lepton doublets: $H_{1,2}$ for $(\mathbf{2},1)$ representations and $H'_{1,2}$ for $(1,\mathbf{2})$ representations. The second line in (\[Wfull\]) are the dynamically-generated supersymmetric mass terms, with $\lambda_3 \lang S_3 \rang \equiv M_T$. The fields $S_3$, $S_4$ and $S_5$ are singlets under $SU(2)_A \times SU(2)_B$ with hypercharge $Y=0$, while $\wh{S}_4$ and $\wh{S}_5$ are $SU(2)_L$ singlets with $Y=0$ transforming as $(\mathbf{2},\mathbf{2})$ under $SU(2)_A \times SU(2)_B$. We anticipate a large vev for $S_3$. The fields $S_4, S_5, \wh{S}_4$, and $\wh{S}_5$ may acquire TeV scale vevs from supersymmetry breaking, leading to generalized $\mu$ terms[^2], or some could have vevs near the string scale (or at an intermediate scale), projecting some of the Higgs states out of the low energy theory. From the point of view of both $SU(2)_L$ as well as the underlying $SU(2)_A \times
SU(2)_B$ it is not necessary that $S_4$ and $S_5$ be distinct fields; there may be string selection rules forbidding their identification in an explicit construction, however. Similar statements apply to $\wh{S}_4$ and $\wh{S}_5$. Of course, some of these fields could be absent.
The final line of (\[Wfull\]) represents the Dirac mass couplings of the left-handed leptons with their right-handed counterparts. Again, the fields $S_6$ and $S_7$, both singlets under $SU(2)_A \times SU(2)_B$, carrying only hypercharge $Y=+1$, may or may not be identified depending on the construction, while $\wh{S}_6$ and $\wh{S}_7$, which may or may not be distinct, transform as $(\mathbf{2},\mathbf{2})$. Some of these fields may be absent. We assume that $S_6$, $S_7$, $\wh{S}_6$, and $\wh{S}_7$ do not acquire vevs. Charged lepton masses are then determined by some combination of the coupling matrices $\lambda_6$, $\lambda_7$, $\wh{\lambda}_6$ and $\wh{\lambda}_7$ (and possibly higher-order terms that connect the two sectors) as well as appropriate choices of Higgs vevs for the neutral components of the four Higgs species.
Making Contact with Experimental Data {#sec:data}
=====================================
Having laid out the framework for our string-based model, we now wish to ask how well such a structure can accommodate the measurements of neutrino mixing angles and mass differences that have been made, and what sort of predictions (if any) might this framework make in terms of future experimental observations. We use a convention in which the solar mixing data defines the mass difference between $m_2$ and $m_1$ with $m_2 > m_1$. Then the eigenvalue $m_3$ relevant for the atmospheric data is the “isolated” eigenvalue.
The current experimental picture is summarized by the recent three neutrino global analysis in [@Fogli:2005cq]. For the solar neutrino sector we take $$\begin{aligned}
%
\Delta m_{12}^{2} & = & 7.92 (1\pm 0.09) \; \times 10^{-5} {\rm
eV}^2 \label{dm12} \\
%
\sin^{2}\theta_{12} & = & 0.314(1_{-0.15}^{+0.18}) ,
\label{theta12}
%\end{aligned}$$ where all measurements are $\pm 2 \sigma$ (95% C.L.). The last measurement implies a value for the mixing angle $\theta_{12}$ itself of $\theta_{12} \simeq 0.595_{-0.052}^{+0.060}$, well below the maximal mixing value $\theta_{12}^{max}=\pi/4$. We take the upper bound on $\theta_{13}$ to be $$%
\sin^{2} \theta_{13} =0.9_{-0.9}^{+2.3} \times 10^{-2} \; \quad
\Rightarrow |\theta_{13}| < 0.18
%
\label{theta13}$$ at the 2$\sigma$ level. For the atmospheric oscillations $$\begin{aligned}
%
|\Delta m^2_{23}|& =& 2.4 (1_{-0.26}^{+0.21}) \times 10^{-3} {\rm
eV}^{2} \label{dm23} \\
%
\sin^{2}\theta_{23}& =& 0.44 (1_{-0.22}^{+0.41}),
%
\label{theta23} \end{aligned}$$ consistent with maximal mixing ($\sin^2 \theta_{23}^{\rm
max}=0.5$).
With this in mind, let us consider the general off-diagonal Majorana mass matrix $$%
m_{\nu} = \( \begin{array}{ccc} 0 & a & b \\ a & 0 & \epsilon \\ b
& \epsilon & 0
\end{array} \) = m_{\nu}^{T}
%
\label{genoff}$$ with $\det m_{\nu} = -2ab\epsilon$, and where we imagine the entry $\epsilon$ to be a small perturbation around the basic structure of (\[lambdaT1\]), which can arise from higher-order terms in $W$. Without loss of generality we can redefine the phases of the lepton doublets $L_i$ and $L'_i$ such that the entries $a$, $b$ and $\epsilon$ are real and $m_{\nu} = m_{\nu}^{\dagger}$. This implies $$%
U_{\nu}^{\dagger}m_{\nu} U_{\nu} = {\rm diag}(m_1, m_2, m_3)
\equiv m_{\rm diag} .
%
\label{Unu}$$ We also have $\Tr \; m_{\rm diag} = m_1 + m_2 + m_3 = \Tr \;
m_{\nu} = 0$, where the various eigenvalues $m_i$ are real but can be negative.
If we begin by first ignoring the solar mass difference, and take the atmospheric mass difference to be given by (\[dm23\]), then there is no way to accommodate the “normal” hierarchy while maintaining the requirement that $\epsilon \ll a,b$. For the inverted hierarchy (in the same approximation of vanishing solar mass difference) we would require $m_2 = -m_1 = 0.049 \; {\rm eV}$ with $m_3 =0$. This could derive from (\[genoff\]) if $\sqrt{a^2
+ b^2} = m_2$ and $\epsilon=0$. In this case $\sum_i |m_i| = 0.098
\; {\rm eV}$. This is clearly in line with the form of (\[WT\]) and implies a triplet mass of order $$%
M_T = 2.0 \times \lambda_2 \lambda_T \(\frac{v_2 v'_2}{(100
\GeV)^2}\) \times 10^{14} \GeV
%
\label{MT}$$ where we have defined $v_2 = \lang (h_2)^0 \rang$ and $v'_2 =
\lang (h'_2)^0 \rang$. The solar mass difference (\[dm12\]) can be restored in this case by taking $\epsilon \simeq \frac{1}{43}$ in the mass matrix given by $$%
m_{\nu} = \sqrt{\frac{|\Delta m^2_{23}|}{2}} \(
\begin{array}{ccc} 0 & -1 & -1 \\ -1 & 0 & \epsilon
\\ -1 & \epsilon & 0 \end{array} \) .
%
\label{perturb}$$ This value is particularly encouraging for theories motivated by the weakly coupled heterotic string compactified on orbifolds. Such theories generally give rise to an Abelian gauge factor with non-vanishing trace anomaly. This anomaly is cancelled by the Green-Schwarz mechanism, which involves a Fayet-Iliopoulos (FI) term $\xi_{\rm FI}$ in the 4D Lagrangian [@Dine:1987xk; @Dine:1987gj; @Atick:1987gy]. In general, at least one field $X$ of the massless spectrum, charged under this anomalous $U(1)$ factor, will receive a vev $X \simeq
\sqrt{\xi_{\rm FI}}$ so as to ensure $\lang D_X \rang = 0$ below the scale $\xi_{\rm FI}$. Explicit orbifold constructions suggest that $0.09 \leq r_{\rm FI} = \sqrt{|\xi_{\rm FI}|}/M_{\PL} \leq
0.14$ for $g^2 \simeq 1/2$ [@Giedt:2001zw]. Thus the perturbation $\epsilon$ could be the result of non-renormalizable operators in the superpotential of relative low-degree – perhaps involving only one or two powers of such a field vev, depending on the size of the dimensionless Yukawa couplings involved.
Considering the underlying $SU(2)_A \times SU(2)_B$ theory, fields bifundamental under both $SU(2)$ factors will decompose into a triplet and a singlet under the breaking to the diagonal subgroup. Let us denote this singlet representation as $\psi$. Then terms at dimension four in the superpotential that can populate the vanishing entries in (\[lambdaT1\]) include $$\begin{aligned}
%
\Delta W &=& \frac{\lambda_{11}}{M_{\PL}} L_1 (\mathbf{2}, 1) T
(\mathbf{2}, \mathbf{2}) \psi (\mathbf{2}, \mathbf{2}) L_1
(\mathbf{2}, 1) \nonumber \\
%
& & +\frac{\lambda_{ij}}{M_{\PL}}L'_i (1, \mathbf{2})
T(\mathbf{2},\mathbf{2}) \psi (\mathbf{2},\mathbf{2}) L'_j
(1,\mathbf{2}) ,
%
\label{psi} \end{aligned}$$ where $i, j = 2, 3$ and we denote the representations under $SU(2)_A \times SU(2)_B$ for convenience. The singlet field $\psi$ must have vanishing hypercharge, so it cannot be the singlet component of the same bifundamental that led to $T$ and $\oline{T}$, though it may be the singlet component of some bifundamental representation that served to generate the breaking to the diagonal subgroup in the first place, or could be identified with $\wh{S}_4$ or $\wh{S}_5$ of (\[Wfull\]). To the extent that string models seldom give self-couplings at such a low order in the superpotential, we might expect $\lambda_{11} =
\lambda_{22} = \lambda_{33} =0$, thereby generating (\[perturb\]) at roughly the correct order of magnitude.
Now let us consider the leptonic (PMNS) mixing matrix defined by $U_{\rm PMNS} = U_e^{\dagger}U_{\nu}$, where $U_{\nu}$ is the matrix in (\[Unu\]) and $U_e$ is the analogous matrix for the charged leptons. Most of the earlier studies [@DeGouvea:2005gd; @Langacker:2004xy; @Altarelli:2004za; @Mohapatra:2004vr; @Barger:1998ta; @Barbieri:1998mq; @Altarelli:1998nx; @Jezabek:1998du; @Mohapatra:1999zr; @Babu:2002ex] of the texture in (\[lambdaT1\]) assumed that this form holds in the basis for which $U_e=1$. In that case, one has an inverted hierarchy and $U_{\rm PMNS} = U_{\nu}$ is bimaximal for $a=b$, i.e., $\theta_{12}=\theta_{23}=\pi/4$, while $\theta_{13}=0$. For $|a|\ne |b|$ the solar mixing remains maximal while the atmospheric mixing angle is $|\tan \theta_{23}|=|b|/|a|$. It is now well established, however, that the solar mixing is not maximal, i.e., $\pi/4-\theta_{12}=0.19^{+0.05}_{-0.06}$ [@Fogli:2005cq], where the quoted errors are 2$\sigma$. It is well-known that reasonable perturbations on this texture (still with $U_e=1$) have difficulty yielding a realistic solar mixing and mass splitting. To see this, let us add a perturbation $\delta$ to the 13-entry of (\[perturb\]) and perturbations $\epsilon_{ii}$ to the diagonal entries. To leading order, $\delta$ only shifts the atmospheric mixing from maximal (to $\theta_{12}\sim \pi/4-\delta/2$). $\epsilon_{22}$ and $\epsilon_{33}$ large enough to affect the Solar mixing tend to give too large contributions to $|\theta_{13}|$, so we will ignore them (their inclusion would merely lead to additional fine-tuned parameter ranges). One then finds $$%
\frac{\pi}{4} - \theta_{12} \simeq 0.19 \simeq
\frac{1}{4}(\epsilon - \epsilon_{11}) ,
%
\label{solardev}$$ whereas the solar mass difference is $$%
\frac{\Delta m^2_{12}}{\sqrt{2}|\Delta m_{\rm atm}^2|} \simeq
\frac{1}{43} \simeq (\epsilon + \epsilon_{11}) .
%
\label{ratio}$$ Satisfying these constraints would require a moderate tuning of $\epsilon$ and $\epsilon_{11}$. Moreover, they would each have to be of order 0.4 in magnitude, somewhat large to be considered perturbations.
On the other hand, a simple and realistic pattern emerges when we instead allow for small departures from $U_e \propto
\mathbf{1}$ [@Frampton:2004ud; @Rodejohann:2003sc; @Petcov:2004rk; @Altarelli:2004jb; @Romanino:2004ww; @Datta:2005ci], and for the general superpotential in (\[Wfull\]) there is no reason for such mixings to be absent.[^3] For example, starting from (\[perturb\]) a Cabibbo-sized 12-entry in the charged lepton mixing matrix $$%
U_e^{\dagger} \sim \left( \begin{array}{ccc} 1 &
-\sin\theta_{12}^e & 0
\\ \sin\theta_{12}^e & 1 & 0 \\ 0 & 0 &1 \end{array}\right) ,
%$$ leads to $$%
\frac{\pi}{4} - \theta_{12} \simeq \frac{\sin\theta_{12}^e}{\sqrt{2}},%$$ which is satisfied for $\sin\theta_{12}^e\simeq
0.27^{+0.07}_{-0.08}$. This mixing also leads to the prediction of a large $$%
\sin^2 \theta_{13}\simeq \frac{\sin^2\theta_{12}^e}{{2}}
\simeq(0.017-0.059)
%$$ (the range is $\pm 2\sigma$), close to the current experimental upper limit of 0.032. Finally, this model implies $$%
m_{\beta\beta} \simeq m_2 (\cos^2 \theta_{12}-\sin^2
\theta_{12})\simeq 0.018 {\rm \ eV}
%$$ for the effective mass relevant to neutrinoless double beta decay. This is the standard result for the inverted hierarchy, with the minus sign due to the opposite signs of $m_1$ and $m_2$. Such a value should be observable in planned experiments [@DeGouvea:2005gd; @Langacker:2004xy; @Altarelli:2004za; @Mohapatra:2004vr].
Realization in Heterotic String Models {#sec:string}
======================================
Having outlined in a broad manner the elementary requirements for phenomenological viability of any triplet-based model with a structure dictated by the superpotential in (\[Wfull\]), we might now wish to ask whether such a set of fields and couplings really does arise in explicit string constructions as we have been assuming. Rather than build all possible constructions of a certain type for a dedicated scan – an undertaking that would undoubtedly produce interesting results in many areas, but which we reserve for a future study – we will here choose one particular example as a case study. The $\mathbb{Z}_3 \times
\mathbb{Z}_3$ orbifold construction of Font et al. [@Font:1989aj] begins with a non-standard embedding that utilizes two shift vectors and one Wilson line in the first complex plane. This Wilson line breaks the observable sector gauge group from $SO(10)$ to $SU(3) \times SU(2)_A \times SU(2)_B$. The massless spectrum of this model contains 75 species of fields. Those from the untwisted sectors have a multiplicity of one, while twisted sectors have a multiplicity of three or nine, depending on the representation. It is natural to consider this multiplicity factor as a generation index.
There are three species of fields which are bifundamental under the observable sector $SU(2)_A \times SU(2)_B$ (one in the untwisted sector and two in various twisted sectors), five doublets under $SU(2)_A$ and eight doublets under $SU(2)_B$. There were also 17 species that were singlets under all non-Abelian groups. So the minimal set of fields needed to generate the superpotential of (\[Wfull\]) are present, as well as an additional bifundamental representation that may be used to break the product group to the diagonal subgroup and/or generate the needed higher-order corrections in (\[psi\]). We note that there are additional species that have non-trivial representations under the non-Abelian groups of [*both*]{} the observable and hidden sectors. In order to avoid potential complications should any of these hidden sector groups undergo confinement we have not considered these in what follows.
From the selection rules given in [@Font:1989aj] it is possible to construct all possible dimension three (renormalizable) and dimension four (non-renormalizable) superpotential couplings consistent with gauge invariance. Considering only the 33 relevant fields mentioned in the previous paragraph, the selection rules and gauge invariance under the observable and hidden sector non-Abelian groups allow 32 and 135 terms at dimension three and four, respectively. Requiring in addition gauge invariance under the six $U(1)$ factors (one of which being anomalous) reduces these numbers to a tractable 15 and 8, respectively.
To ascertain which of the terms in (\[Wfull\]) can be identified from the above it is necessary to choose a linear combination of the five non-anomalous $U(1)$ factors to be identified as hypercharge, and then determine the resulting hypercharges of the bifundamentals, doublets and singlets under this assignment. Our algorithm was to begin with the two bifundamentals of the twisted sector, as the untwisted bifundamental had no couplings to $SU(2)$ doublets at dimension three or four. These two twisted sector fields had a selection-rule allowed coupling to a non-Abelian group singlet at the leading (dimension three) order, which could therefore play the role of $S_3$ in (\[Wfull\]). Requiring these two fields to carry hypercharge $Y= \pm 1$ (and thus automatically ensuring that the candidate $S_3$ have vanishing hypercharge) placed two constraints on the allowed hypercharge embedding.
We then proceed to the coupling (\[WT\]), or $\lambda_T$ in (\[Wfull\]) for the $Y=+1$ species. Each bifundamental had several couplings of this form to various pairs of $SU(2)$ doublets, at both dimension three and dimension four. By considering all possible pairs and requiring that the doublets involved be assigned $Y= -1/2$ places two more constraints on the allowed hypercharge embedding. Finally, we proceed to the equally critical $\lambda_2$ coupling in (\[Wfull\]) for the oppositely charged $Y=-1$ species. Again by considering all possible pairs of doublets with this coupling, and requiring that both have hypercharge $Y=+1/2$ we typically constrained the hypercharge embedding to a unique embedding. The hypercharges of all the states in the theory are then determined. Not all will have Standard Model hypercharges, and thus most will have fractional or non-standard electric charges and must be discarded as “exotics.” From the set with Standard Model hypercharge assignments we can identify the surviving couplings of (\[Wfull\]). In all, this process resulted in 35 distinct field assignment possibilities, each having as a minimum the couplings $\lambda_T$, $\lambda_2$ and $\lambda_3$ – the minimum set to generate the triplet see-saw and the mass pattern of (\[lambdaT1\]). Though these couplings are not enough to generate the perturbations on the bimaximal texture, nor do they include the couplings needed to generate charged lepton masses or $\mu$-terms to break electroweak symmetry, they still represent a complete set of needed couplings to explain the smallness of neutrino masses generally – something that a more exhaustive search of a whole class for the “standard” seesaw failed to achieve [@Giedt:2005vx].
None of the 35 possibilities allowed for all of the couplings of (\[Wfull\]), and 12 had no other couplings than the minimal set. This is yet another example of how selection rules of the underlying conformal field theory often forbid operators that would otherwise be allowed by gauge invariance in the 4D theory. Rather than present the various features of all of these assignments, we instead point out a few particular cases. One successful hypercharge assignment allows for a superpotential of the form $$\begin{aligned}
%
W &=& \lambda_T L T L' + \lambda_2 H_2 \oline{T} H'_2 + \lambda_3
S_3 T \oline{T} \nonumber \\
%
& & + \lambda_5 S_5 H'_1 H'_2 + \lambda_7 E_R L' H'_1 ,
%
\label{W2} \end{aligned}$$ where in this case $L$, $L'$, $H_2$ and $H'_2$ all have multiplicity three, $H'_1$ has multiplicity one and there is no species with the correct hypercharge to be identified as $H_1$. In this case, identifying $L$ with the doublet containing the electron leaves the electron massless after electroweak symmetry breaking up to terms of dimension five in the superpotential.
Alternatively, one can obtain candidates for all six species of doublets, such as an example in which the allowed superpotential is given by $$\begin{aligned}
%
W &=& \lambda_T L T L' + \lambda_1 H_1 T H'_1 + \lambda_2 H_2
\oline{T} H'_2 \nonumber
\\
%
& & + \lambda_3 S_3 T \oline{T} + \lambda_7 E_R L' H'_1 .
%
\label{W3} \end{aligned}$$ All doublets except $H_2$ in this case arise from twisted sectors, so have multiplicity three. It is interesting to note that in several of the 35 cases the hypercharge embedding assigned $Y=0$ to the bifundamental representation of the untwisted sector, suggesting it could play the role of breaking the product group to the diagonal subgroup. Couplings of the form of (\[psi\]), however, were forbidden by the string theoretic selection rules through dimension four.
Of course none of these cases are truly realistic in the sense of what is needed to explain the observed neutrino data as outlined in the previous section, and it would have been naive to have expected any to be in the first place. The above examples are instead meant to demonstrate the plausibility of this new realization of a triplet-induced seesaw from a string-theory viewpoint by means of a ready example from the literature. Having introduced the concept, defined a basic structure as in (\[Wfull\]) and demonstrated that the structure may in fact be realized in the context of tractable string constructions, it becomes reasonable to propose a dedicated search over a wide class of constructions for precisely this model – a search that would necessarily be a separate research project in its own right but which would complement well the analysis already performed in [@Giedt:2005vx].
The $\mathbb{Z}_3 \times \mathbb{Z}_3$ construction is often considered because it, like its $\mathbb{Z}_3$ cousin, generates a three-fold redundancy for most of the massless spectrum in a relatively straightforward way. But a minimal model would presumably prefer to break away from the three-fold degeneracy on every species, but not the requirement of three generations globally. For example, it is possible to imagine a model in which there are only three “lepton” doublets of $SU(2)_L$ once we break to the diagonal subgroup. Since species in orbifold models (and orientifold models of open strings as well) are defined by fixed point locations (i.e., geometrically) this is not unreasonable to imagine – in fact, precisely such a separation of the three “generations” occurs in the recent $\mathbb{Z}_2\times
\mathbb{Z}_3$ construction of Kobayashi et al. [@Kobayashi:2004ud]. Nevertheless, there is no getting around the need for at least an extra pair of one, if not both, of the Higgs doublets of the MSSM. As discussed following (\[Wfull\]), it is possible that the extra doublets are projected out near the string scale (e.g., if some of the $S_{4,5}$ and $\wh{S}_{4,5}$ are associated with the Fayet-Iliopoulos terms) or at an intermediate scale. It is also possible that one or more extra doublets survives to the TeV scale, in which case there are potential implications for FCNC [@Sher:1991km; @Atwood:1996vj] and CP violation [@Hall:1993ca], as well as for the charged lepton mixing generated from (\[Wfull\]). A more detailed study of such issues is beyond the scope of this paper.
Conclusions {#conclusions .unnumbered}
===========
We have presented a new construction of Type II seesaw models utilizing triplets of $SU(2)_L$ in which that group is realized as the diagonal subgroup of an $SU(2)_A \times SU(2)_B$ product group. The triplets in this construction begin as bifundamentals under the two original $SU(2)$ factors, and this identification immediately leads to a bimaximal mixing texture for the effective neutrino mass matrix provided generations of lepton doublets are assigned to the two underlying $SU(2)$ factors in the appropriate way. The observed atmospheric mass difference can be accommodated if the triplets obtain a mass of order $10^{14} \GeV$, and the solar mass difference can easily be incorporated by a simple perturbation arising at dimension four or five in the superpotential. The observed deviation of the Solar mixing from maximal can be accommodated by a small (Cabibbo-like) mixing in the charged lepton sector, leading to predictions for $\theta_{13}$ and neutrinoless double beta decay.
We were led to consider this construction by imagining the simplest possible requirements for generating a triplet of $SU(2)_L$ from string constructions – particularly the weakly coupled heterotic string, though the model can be realized in other constructions as well. Though inspired by string theory, the model is not itself inherently stringy and is interesting in its own right. Some of the properties of this model are known to phenomenologists, who have arrived at a similar mass matrix from other directions. Interestingly, however, to the best of our knowledge the particular texture has not emerged from other versions of heavy triplet models, e.g., motivated by grand unification or left-right symmetry. The simplest version of the construction requires at least one additional pair of Higgs doublets, which may or may not survive to the TeV scale.
Having laid out a concrete model as a plausible alternative to the standard Type I seesaw in string-based constructions, it is now possible to examine large classes of explicit string models to search for both types of neutrino mass patterns. Given the difficulty in finding a working example of the minimal Type I seesaw in at least one otherwise promising class of string construction, having alternatives with clear “signatures” (in this case, at least two $SU(2)$ factors, with at least two fields bifundamental under both, capable of forming a hypercharge-neutral mass term) is welcome.
We wish to thank Joel Giedt and Boris Kayser for helpful discussions and advice. This work was supported by the U.S. Department of Energy under Grant No. DOE-EY-76-02-3071.
[99]{}
[^1]: For some recent reviews of theoretical models of neutrino masses and mixings, see [@King:2003jb; @DeGouvea:2005gd; @Langacker:2004xy; @Altarelli:2004za; @Mohapatra:2004vr] and references therein.
[^2]: The $\mu$ parameters of the Higgs scalar potential could also arise as effective parameters only after SUSY breaking via the Giudice-Masiero mechanism [@Giudice:1988yz].
[^3]: The relatively large value required for $\sin\theta_{12}^e$ compared to $\sqrt{m_e/m_\mu}\sim 0.07$ suggests an asymmetric charged lepton mass matrix, but this would not be unexpected.
| 0 |
---
abstract: 'We derive an $S=1$ spin polaron model which describes the motion of a single hole introduced into the $S=1$ spin antiferromagnetic ground state of Ca$_2$RuO$_4$. We solve the model using the self-consistent Born approximation and show that its hole spectral function qualitatively agrees with the experimentally observed high-binding energy part of the Ca$_2$RuO$_4$ photoemission spectrum. We explain the observed peculiarities of the photoemission spectrum by linking them to two anisotropies present in the employed model—the spin anisotropy and the hopping anisotropy. We verify that these anisotropies, and *not* the possible differences between the ruthenate ($S=1$) and the cuprate ($S=1/2$) spin polaron models, are responsible for the strong qualitative differences between the photoemission spectrum of Ca$_2$RuO$_4$ and of the undoped cuprates.'
author:
- 'Adam Kł‚osiń„ski'
- 'Dmitri V. Efremov'
- Jeroen van den Brink
- Krzysztof Wohlfeld
date:
-
-
title: |
Photoemission Spectrum of Ca$_2$RuO$_4$:\
Spin Polaron Physics in an $S=1$ Antiferromagnet with Anisotropies
---
Introduction. {#sec:introduction}
=============
Understanding the strongly correlated physics of the transition metal oxides constitutes a nontrivial task [@Dagotto1994; @Imada1998; @Lee2006; @Khomskii2010; @Khomskii2014]. On one hand, this is due to the fact that even the simplest, but still realistic, effective models may have to contain several degrees of freedom. On the other, this is due to the fact that such relatively simple models are often not solvable in the thermodynamic limit. That is why examples when a theoretical model can be solved without too drastic approximations [*and*]{} explain the experimentally observed features of a correlated oxide are of interest. One such case, known already since the end of the 80s of the last century, is the so-called spin polaron problem [@Schmitt1988; @Martinez1991; @Ramsak1998; @Manousakis2007; @Wang2015; @Grusdt2018; @Bieniasz2019], which explains the peak dispersion found in the photoemission spectrum of the (spin $S=1/2$) antiferromagnetically ordered and Mott insulating copper oxides [@Wells1995; @LaRosa1997; @Kim1998; @Damascelli2003; @Shen2007]: It turns out that the peak dispersion found in the spectra of the ‘parent compounds’ to the high-temperature superconductors can be well explained using a $t$–$J$ or Hubbard model that is mapped onto a (spin) polaron problem.
Interestingly, despite the still unresolved mysteries associated with high-temperature superconductivity, the copper oxides are the simplest class of oxides to model. This is basically due to the fact that the uncorrelated part of their physics can be effectively described using a single-band picture [@Zhang1988; @Lee2006]. Such situation is not realised in many other oxides, such as the manganites, vanadates, nickelates—or the ruthenates studied here [@Imada1998; @Khomskii2014]. In this case the effective models are far more involved and are never of the single-band $t$–$J$ or Hubbard variety. Can one thus expect that some of the principles of the spin polaron physics do hold there?
To investigate this rather general problem, we take a closer look at one of the intensively investigated transition metal oxides—Ca$_2$RuO$_4$ [@Alexander1999; @Nakatsuji2000; @Mizokawa2001; @Lee2002; @Gorelov2010; @Kunkemoeller2015; @Fatuzzo2015; @Kunkemoeller2017; @Jain2017; @Sutter2017; @Zhang2017; @Das2018; @Ricco2018; @Pincini2019; @Gretarsson2019] which is a spin $S=1$ antiferromagnetically ordered [@Mizokawa2001; @Kunkemoeller2015; @Kunkemoeller2017; @Das2018; @Pincini2019] and Mott insulating analogue of the unconventional superconductor Sr$_2$RuO$_4$ [@Maeno1994]. The other reason for choosing this system is that recently its detailed photoemission spectrum was not only studied experimentally but also successfully modelled using a multiband Hubbard model [@Sutter2017]. Nevertheless, as the multiband Hubbard model was solved using the single-site dynamical mean-field theory approach, it is not clear to what extent the spin polaron physics is present there.
In this paper we concentrate on the origin of the incoherent and almost momentum-independent Ca$_2$RuO$_4$ photoemission spectrum, that is present at the high-binding energy part \[see the yellow rectangle of Fig. \[result\](a)\] and is associated with the hole motion in the $xz$ and $yz$ orbitals [@Sutter2017]. The reason for leaving the $xy$ orbital out of our analysis is that the highly dispersive, quasiparticle-like part of the photoemission spectrum stretching from low- to high-binding energy \[see Fig. \[result\](a)\] can be easily understood as the free hole motion on the $xy$ orbitals. This follows straightforwardly from, on the one hand, the relatively large energy gap between the $xy$ and $xz/yz$ orbitals leading to the $xy$ orbital being fully occupied and, on the other hand, the absence of any mixing between $xy$ and $xz/yz$ orbitals, be it from hopping or the Coulomb interaction. Thus, including the $xy$ orbitals in our analysis would not much enrich our understanding of the physics at work in this compound.
Altogether, our aim here is twofold: First, we want to model the high-binding energy part of the photoemission spectrum of Ca$_2$RuO$_4$ by using a realistic [*$S=1$ spin polaron*]{} model, specifically derived for this case. Second, we wish to understand to what extent such an $S=1$ spin polaron problem is different from the ‘standard’ (i.e. $S=1/2$) spin polaron model that is well-known from the cuprate studies.
The paper is organized as follows. In Sec. \[sec:tjmodel\] we introduce the $t$-$J$ model that describes the motion of a single hole in the ground state of Ca$_2$RuO$_4$. Next, in Sec. \[sec:fromto\] we perform a mapping of the $t$-$J$ model onto a S=1 spin polaron model. The latter is solved using the self-consistent Born approximation (SCBA) in Sec. \[sec:mandr\]. Finally, we discuss the obtained results in Sec. \[sec:discussion\] and end the paper with conclusions in Sec. \[sec:conclusions\]. The paper is supplemented by an Appendix which contains some details of the mapping from the $t$-$J$ to the polaron model.
![Comparison between the experimental and theoretical spectral functions of Ca$_2$RuO$_4$: (a) Angle resolved photoemission (ARPES) spectrum obtained for Ca$_2$RuO$_4$ and published in Ref. ; (b) Hole spectral function $A({\bf k}, \omega)$ calculated for the spin $S=1$ $t$–$J$ Hamiltonian (\[model1\]-\[model2\]) with ${\bf e}_{xz} = \hat{x}$, ${\bf e}_{yz}=\hat{y}$ and using the mapping onto the spin-polaronic model (\[finalhamiltonian\]) and the SCBA method (see text); model (\[finalhamiltonian\]) parameters: $t = 22 J$, $\epsilon = 5.6 J$, $\gamma = 0.25 J$, $J = 5.6 $ meV, numerical broadening of $A({\bf k}, \omega)$ $\delta = 1.1 J $; (c) Hole spectral function $A({\bf k}, \omega)$ calculated as in (b) but convoluted with a Gaussian with the half-width at half maximum equal to $0.5t$, simulating the experimental resolution of ARPES on Ca$_2$RuO$_4$ [@Sutter2017]. The yellow rectangles mark the high binding energy parts of the spectra that are incoherent and almost momentum-independent, are identified in ARPES as having a dominant $xz$/$yz$ orbital character [@Sutter2017], and are theoretically modelled by panel (b) \[The dispersive branch visible in (a), both inside and outside of the yellow rectangle and not discussed here, is associated with the $xy$ orbital [@Sutter2017]. See main text of the paper\]. Theoretical spectra (b-c) are normalised in the same manner as the ARPES spectrum \[(a)\] of Ref. . []{data-label="result"}](fig1.png){width="50.00000%"}
${\bf t}$-${\bf J}$ model. {#sec:tjmodel}
==========================
In order to model (the high-binding energy part of) the photoemission spectrum of Ca$_2$RuO$_4$ we follow the scheme that was already developed ca. 30 years ago and, as mentioned in the introduction, was successfully used to describe, [*inter alia*]{}, the photoemission spectra of several undoped copper oxides [@Wells1995; @LaRosa1997; @Kim1998; @Damascelli2003; @Shen2007]. Thus, we consider an appropriate $t$–$J$-like Hamiltonian constructed as a sum of two parts ${\mathcal H} = {\mathcal H}_J+ {\mathcal H}_t$.
The first part, ${\mathcal H}_J$, describes the low energy physics of the Mott insulating Ca$_2$RuO$_4$ in terms of the interaction between the localised $S=1$ magnetic moments. The relevant Hamiltonian is well-known in this case and reads [@Kunkemoeller2015; @Kunkemoeller2017; @Jain2017], $$\label{model1}
{\mathcal H}_J = J \sum\limits_{\langle {\bf i}, {\bf j} \rangle} \mathbf{S_{\bf i}} \cdot \mathbf{S_{\bf j}} + \epsilon \sum\limits_{\bf i} \left( S_{\bf i}^z \right)^2 + \gamma \sum\limits_{\bf i} \left( S_{\bf i}^x \right)^2,$$ where the summation runs over all nearest-neighbour pairs on a 2D square lattice, $J$ is the spin exchange constant, and $\mathbf{S_{\bf i}}$ are the spin $S=1$ operators. As already discussed in Refs. the spin model is highly anisotropic, with the suggested values of the $\hat{z}$ ($\hat{x}$) axis anisotropy being equal to $\epsilon = 5.6 J$ ($\gamma = 0.25 J$), respectively, with $J = 5.6$ meV reproducing the spin wave dispersion observed in the inelastic neutron scattering experiment [@Kunkemoeller2015; @Kunkemoeller2017; @Jain2017]. We note that such a large spin anisotropy originates in the large spin-orbit coupling on the ruthenium ions which, however, is quite widely considered as not strong enough to stabilise the $S=0$ ground state [@Mizokawa2001; @Kunkemoeller2015; @Kunkemoeller2017; @Das2018; @Pincini2019]. Although the latter result can naively be understood as a consequence of the crystal field splitting (between the $xz, yz$ and the $xy$ orbitals) being about twice larger than the spin-orbit coupling [@Das2018] and therefore the spin $S=0$ states having considerably higher energy than the spin $S=1$ states, see Fig. S1 of [@Jain2017], it has been postulated [@Khaliullin2013; @Akbari2014; @Jain2017; @Gretarsson2019] that nevertheless the ‘excitonic magnetism’ can be at play here.
The second part of the Hamiltonian, ${\mathcal H}_t$, is the ‘kinetic’ term. This term is introduced, in order to describe the motion of a single hole created in the ruthenium oxide plane in the photoemission experiment. We restrict our model to the $xz$ and $yz$ orbitals, for, as discussed in the introduction, we are solely interested in the part of the photoemission spectrum associated with these orbitals [@Sutter2017]. Altogether, we end up with, $$\begin{aligned}
\label{model2}
\begin{split}
{\mathcal H}_t &= - t \sum\limits_{{\bf i}, \sigma} \left( \tilde{c}_{{\bf i}+{{{\bf e}_{xz}}},xz,\sigma}^{\dag} \tilde{c}_{{\bf i},xz,\sigma} +\tilde{c}_{{\bf i}+{{\bf e}_{yz}},yz,\sigma}^{\dag} \tilde{c}_{{\bf i},yz,\sigma} +h.c.\right),\\
\end{split}\end{aligned}$$ where the first (second) term describes the hoppings of an electron with spin $\sigma$ between the nearest neighbor ruthenium $xz$ ($yz$) orbitals along the ${\bf e}_{xz}=\hat{x}$ (${\bf e}_{yz}=\hat{y}$) direction in the 2D square lattice, respectively. Such effectively one-dimensional (‘directional’) hoppings follow from the Slater-Koster scheme [@Slater1954] applied to the square lattice geometry of the ruthenium oxide plane and is, in fact, a common feature of systems with active $\{xz, yz\}$ orbital degrees of freedom [@Harris2004]. As for the value of the hopping element $t$ in Ca$_2$RuO$_4$ we choose $t= 123$ meV [@Gorelov2010], i.e. $t = 22J$ for the above chosen realistic value of $J=5.6$ meV. We note that to simplify the analysis we skip here the spin-orbit coupling between holes in the $xz$ and $yz$ orbitals. Such a simplification is not [*a priori*]{} justified for a realistic situation in Ca$_2$RuO$_4$ but is rationalised by the intuitive understanding of its spectral functions presented in Ref. , which relies on the Hund’s coupling and does not include the spin-orbit coupling as an essential part. Moreover, a (surprisingly) good agreement between the theoretical results presented below and the experimental results, cf. Fig. \[result\], [*a posteriori*]{} legitimizes this assumption further. Finally, it will not affect the study of the possible intrinsic differences between the $S=1/2$ and $S=1$ spin polaron models.
There are two projections in place in the kinetic Hamiltonian (\[model2\]). First, due to the strong on-site Coulomb repulsion $U$–and since we confine ourselves to the low energy physics valid for energies smaller than the ‘Hubbard’ $U$–we restrict the hole motion to the Hilbert space spanned by the $d^2$ or $d^1$ multiplets on the single ruthenium ions. \[Since the $xy$ orbital is considered to be ‘always’ occupied by two electrons in the studied model [@Mizokawa2001; @Gorelov2010; @Sutter2017], the $xy$ electrons are integrated out and effectively the nominal occupancy of the ruthenium ions is not $d^4$ ($d^3$) but $d^2$ ($d^1$) in the undoped (single-hole) case, respectively.\] As typical to any $t$–$J$-like model [@Chao1977] to formally denote such a constraint we use the tildas above the electron creation and annihilation operators. Second, just as in the case of the ground state (see discussion above), we project the spin $S=0$ states out of the Hilbert space and, formally, Hamiltonian (\[model2\]) contains such projections. We will not, however, keep them explicit in the formulae below.
Finally, as we are interested in the photoemission spectrum, we define the following orbitally-resolved hole spectral function, $$\begin{aligned}
A_{\alpha}({\bf k}, \omega)\!= \!- \frac{1}{\pi}
{\rm Im} \Big\langle 0 \Big| \tilde{c}^\dag_{{\bf k}, \alpha, \sigma} \frac{1}{\omega - {\mathcal H} + E_0 +i \delta } \tilde{c}_{{\bf k}, \alpha, \sigma} \Big| 0 \Big\rangle,\end{aligned}$$ where $| 0 \rangle$ is the ground state of the undoped $t$–$J$ model (\[model1\]-\[model2\]) with energy $E_0$, $\delta$ is the infinitesimally small broadening that is nevertheless finite in the numerical calculations below, and we explicitly keep the orbital index $\alpha \in \{ xz, yz \}$ but suppress the spin index $\sigma$ (the spectral function is spin-independent). In what follows we are also interested in the orbitally-integrated spectral function which is defined in the usual way: $A({\bf k}, \omega)=\sum_{\alpha}A_{\alpha}({\bf k}, \omega)$.
From ${\bf t}$-${\bf J}$ to polaron model. {#sec:fromto}
==========================================
Stimulated by the successful description of the photoemission spectra of the undoped cuprates [@Wells1995; @LaRosa1997; @Kim1998; @Ramsak1998; @Damascelli2003; @Shen2007; @Manousakis2007; @Wang2015] and to gain a better insight into the physics of the photoemission problem, we perform a mapping of the $S=1$ $t$–$J$ problem onto an $S=1$ spin polaron problem. This is done in two steps:
First, we introduce the slave fermions, $$\label{slavefermions}
\begin{array}{l}
\tilde{c}_{{\bf i},\alpha,\uparrow} \rightarrow h_{{\bf i},\alpha}^{\dag}, \quad \quad
\tilde{c}_{{\bf i},\alpha,\downarrow} \rightarrow \hat{A} \: h_{{\bf i},\alpha}^{\dag} \: S_{\bf i}^+,\\
\end{array}$$ where $h_{{\bf i},\alpha}^{\dag}$ is the creation operator for a spinless hole on site $i$ and orbital $\alpha$, $S_{\bf i}^+$ is the spin $S=1$ operator on site ${\bf i}$ and $\hat{A}$ is an operator yet to be determined. It can be shown that in the Hilbert space being considered, that is with the $S=0$ states projected out, the $\hat{A}$ operator is diagonal and an explicit expression for it can be found (see Sec. \[sec:appendix\] for details). Second, we we rotate spins on one of the antiferromagnetic sublattices and express the spin operators through bosonic operators by way of the Holstein-Primakoff transformation. Finally, we use the linear spin wave approximation and the Bogoliubov transformation to diagonalize the resulting spin Hamiltonian–see Sec. \[sec:appendix\] for details.
In the end we are left with a diagonal magnon term and a vertex coupling spinless holes to magnons in the following $S=1$ spin polaron Hamiltonian: $$\begin{aligned}
\label{finalhamiltonian}
H &= H_t + H_J \approx \sum\limits_{\bf q} \: {\Omega}_{\bf q} \: {\beta}_{\bf q}^{\dag} {\beta}_{\bf q} + E_0 \nonumber \\
&+\frac{\sqrt{2} \: t}{\sqrt{N}} \sum\limits_{\textbf{k},\textbf{q}}\Big[\left({\gamma}_{k_{x}}v_{\textbf{q}}+{\gamma}_{k_{x}-q_{x}}u_{\textbf{q}}\right)
h_{\textbf{k},xz}^{\dag}h_{\textbf{k}-\textbf{q},xz} {\beta}_{\textbf{q}}\nonumber \\
&+ \left({\gamma}_{k_{y}}v_{\textbf{q}}+{\gamma}_{k_{y}-q_{y}}u_{\textbf{q}}\right)
h_{\textbf{k},yz}^{\dag}h_{\textbf{k}-\textbf{q},yz}{\beta}_{\textbf{q}}+h.c. \Big],\end{aligned}$$ where $\gamma_{k_i} = \cos(k_i)$ and $\beta_q$ are the Bogoliubov boson (magnon) annihilation operators. $u_{\bf q}, v_{\bf q}$ are the Bogoliubov coefficients - see Sec. \[sec:appendix\] for details. The above transformations also lead directly to the expression for the spectral function in terms of the spinless hole Green’s function (see Sec. \[sec:appendix\] for details): $$\begin{aligned}
A_{\alpha}({\bf k}, \omega)\! =\! - \frac{1}{\pi}
{\rm Im} \Big\langle 0 \Big| {h}_{{\bf k}, \alpha} \frac{1}{\omega - H + E_0 +i \delta } {h}^\dag_{{\bf k}, \alpha} \Big| 0 \Big\rangle.\end{aligned}$$
Methods and results. {#sec:mandr}
=====================
We calculate the hole spectral function $A_{\alpha}({\bf k}, \omega)$ using the self-consistent Born approximation (SCBA), see Ref. [@Martinez1991]. Such approach has been widely-successful in obtaining the cuprate spectral functions [@Martinez1991; @Ramsak1998; @Manousakis2007; @Wang2015; @Bieniasz2018] and amounts to neglecting the so-called crossing diagrams and summing all the other (‘rainbow diagrams’) to infinite order. The resulting self-consistent expressions for the self-energies and the Green’s function are given in Sec. \[sec:appendix\]. These equations are then solved numerically on a finite lattice of $36 \times 36$ ${\bf k}$ points. The resulting orbitally-integrated hole spectral function $A({\bf k}, \omega)$ is calculated for the realistic parameters of the model (see above) and is shown in Fig. \[result\](b).
The calculated hole spectral function qualitatively reproduces the incoherent and almost momentum-independent spectrum observed in the high binding energy part of Ca$_2$RuO$_4$ photoemission results found in Ref. and reproduced in Fig. \[result\](a). Although the onset of several horizontal ‘stripes’ in the theoretical spectrum (see below) make the similarities between the theoretical and experimental spectral functions less apparent at first sight \[Fig. \[result\](b)\], a convolution of the theoretical spectral function with the available experimental resolution of ca. $0.5t$ yields a spectrum \[Fig. \[result\](c)\] which surprisingly well resembles the high binding energy part of the observed experimental spectrum \[Fig. \[result\](a)\]: both spectra have an incoherent character, without a clear quasiparticle band emerging, and only very weak dependence of its intensity on the momentum. (A weak dependence on the momentum of the intensities in the high binding energy part of the observed experimental spectrum \[Fig. \[result\](a)\] originates in the $xy$ orbital spectral function, not considered here but explained in detail in Ref. .) What is more, the low- and high-energy edges of both broad spectra are basically momentum-independent. Finally, also the overall energy scale, which is given by the width of the broad spectrum estimated at the half maximum intensity, is of the same order of magnitude in both cases and amounts to about $0.5$ eV.
Discussion. {#sec:discussion}
===========
What might be the origin of the onset of such an incoherent, almost momentum-independent and, apart from the horizontal ‘stripes’, rather featureless spectrum of Fig. \[result\](b)? Looking first at model (\[model1\]-\[model2\]) we can immediately note what distinguishes it from the ‘standard’ $S=1/2$ $t$–$J$ model, that has been widely used to describe the photoemission spectra of the undoped cuprates [@Schmitt1988; @Martinez1991; @Ramsak1998; @Manousakis2007; @Wang2015] and for which such a broad and flat incoherent band has not been observed. The most apparent are the two anisotropies. The spin anisotropy reflects the distortion of the lattice and leads to the $\gamma$ and $\epsilon$ terms in Eq. (\[model1\]). The perfect hopping anisotropy, on the other hand, which has its origin in the nominal valence of the ruthenium ions and the geometry of the ruthenium-oxide plane, leads to an effectively one-dimensional hole motion, cf. Eq. (\[model2\]). On top of that, a more subtle distinction is related to the larger value of the spin $S=1$ in the studied model. The latter leads to the onset of additional projection operators in the hopping part of the Hamiltonian.
![The hole spectral functions $A({\bf k}, \omega)$ obtained for distinct versions of the relevant $t$–$J$ models and calculated by mapping onto the spin-polaronic model and using the SCBA method (see text): (a) the ‘standard’ spin $S=1/2$ $t$–$J$ model, cf. Ref. ; (b) spin $S=1$ $t$–$J$ model with neither the spin nor the hopping anisotropy, i.e. model (\[model1\]-\[model2\]) with $\varepsilon=\gamma \equiv 0$ and ${\bf e}_{xz}\equiv{\bf e}_{yz} \in \{\hat{x}, \hat{y}\}$; (c) the spin $S=1$ $t$–$J$ model with the hopping anisotropy as suggested for Ca$_2$RuO$_4$ but no spin anisotropy, i.e. model (\[model1\]-\[model2\]) with ${\bf e}_{xz} = \hat{x}$ and ${\bf e}_{yz}=\hat{y}$; and (d) the spin $S=1$ $t$–$J$ model with both anisotropies present as suggested for Ca$_2$RuO$_4$ and equivalent to Fig. \[result\](b), i.e. model (\[model1\]-\[model2\]) with ${\bf e}_{xz} = \hat{x}$, ${\bf e}_{yz}=\hat{y}$, and all model parameters as in Fig. \[result\](b). All spectra normalised as Fig. \[result\]. []{data-label="fourplots"}](fig2.png){width="50.00000%"}
We explore the above-listed differences in detail by comparing the spectral function $A({\bf k}, \omega)$ calculated for the distinct versions of the relevant $t$–$J$ models, cf. Fig. \[fourplots\]. Firstly, it is evident that the effects of the spin anisotropy are profound, see Figs. \[fourplots\] (c) and (d). On the one hand, it makes the spectrum less dispersive and in general more featureless; on the other it leads to the formation of a horizontal stripes superimposed onto the otherwise featureless spectrum. Both the former and the latter can be understood, when one considers the fact that the very large anisotropy limit leads, in this case, to the dominant Ising-like interactions between spins. This triggers the hole confinement in a linear string potential and leads to a well-known ladder-like spectrum with the horizontal ‘stripes’ [@Martinez1991; @Dagotto1994; @Bieniasz2018].
![The one dimensional character of the orbitally-resolved hole spectral function: (a) Constant-energy cut of the spectral function $A_{xz} ({\bf k}, \omega_c)$ for a hole introduced into the $xz$ orbital ($\omega_c = -1.9 $ eV); (b) Constant-energy cut of the spectral function $A_{yz} ({\bf k}, \omega_c)$ for a hole introduced into the $yz$ orbital ($\omega_c = -1.9 $ eV); (c-d) A schematic view of the ruthenium-oxygen plane explaining the dominant one-dimensional character of the electronic hopping processes on the single-particle level that is also inherited by the many-body hopping processes of Eq. (\[model2\]): For the $xz$ ($yz$) orbital, only hopping in the $\hat{x}$ ($\hat{y}$) direction is possible, cf. panel (c) \[(d)\] [@Slater1954; @Harris2004]. The oxygen (ruthenium) orbitals are shown in blue (red).[]{data-label="oned"}](fig3.png){width="50.00000%"}
![A schematic view of the possible nearest neighbor hole hoppings in the $S=1$ and $S=1/2$ antiferromagnet (AF): (Top panel) A hopping process in the $S=1$ antiferromagnet that, according to the here studied $t$–$J$ Hamiltonian (\[model1\]-\[model2\]), leads to the creation of one magnon in the effective $S=1$ spin polaron model (\[finalhamiltonian\]); (Middle panel) An analogous hopping process as above but in the $S=1/2$ antiferromagnet which, according to the ‘standard’ $t$–$J$ Hamiltonian [@Chao1977], leads to the creation of one magnon in the spin polaron model of Ref. ; (Bottom panel) A hopping process in the $S=1$ antiferromagnet that, according to the here studied $t$–$J$ Hamiltonian (\[model1\]-\[model2\]), leads to the creation of three magnons and is [*neglected*]{} in the $S=1$ spin polaron model (\[finalhamiltonian\]) for it goes beyond the linear spin wave approximation.[]{data-label="hoppingprocesses"}](fig4.png){width="50.00000%"}
Secondly, one can see that the one-dimensional hole motion completely changes the character of the spectral function, cf. Fig. \[fourplots\](b) and (c). In order to better understand why this happens, in Fig. \[oned\] we present the constant energy cuts of two spectral functions–one describing a hole in the $xz$ orbital, the other a hole in the $yz$ orbital. We see that the one-dimensional hole motion—a consequence of the geometry of the ruthenium-oxygen plane and the vanishing of the transfer integrals between the oxygen $p$ orbitals and some of the $t_{2g}$ orbitals [@Harris2004]—is reflected in the hole spectral functions. They both show a manifestly one-dimensional dispersion, very much unlike what we see, for instance, in the copper oxides [@Wells1995; @LaRosa1997; @Kim1998; @Damascelli2003; @Shen2007]. We note that, while including a finite spin-orbit coupling for holes in the $xz$ or $yz$ orbitals would naturally lead to the ‘mixing’ between the one-dimensional bands, a good agreement between the theoretical and experimental spectra suggests that such an effect should be small in Ca$_2$RuO$_4$.
Finally, with all other parameters equal, the fact that we do not consider here a spin $S=1/2$ (which would be formed by a single hole or electron per site) but a spin $S=1$ antiferromagnet (two holes on each site) does not influence the spectral function qualitatively–thus, the difference between these two cases is purely quantitative, cf. Fig. \[fourplots\](a) and (b). To understand why it is so, let us compare the possible hole hopping processes in the $S=1/2$ and $S=1$ antiferromagnet, which are represented schematically in Fig. \[hoppingprocesses\]. What we can conclude by looking at the process represented on the bottom panel is that all the more complex processes, which have no analog in the single hole per site case, involve more than one magnon. In fact they involve either three or five magnons, which is why we exclude them in the spin wave approximation employed here and why they are absent from Hamiltonian (\[finalhamiltonian\]). Consequently, only the simplest process remains, the one analogous to the only process possible in the spin $S=1/2$ case, cf. the first two panels of Fig. \[hoppingprocesses\]. We stress that such a similarity between the hole moving in the $S=1/2$ and the $S=1$ antiferromagnet would not be achieved in the classical double exchange picture [@Zener1951], for the latter one would not allow for the existence of the $|1, 0 \rangle$ states on any site.
Conclusions. {#sec:conclusions}
============
In this work we showed how a relatively simple spin $S=1$ $t$–$J$ model, that was mapped onto an $S=1$ spin polaron model, can qualitatively reproduce the high-binding energy part of the observed Ca$_2$RuO$_4$ photoemission spectrum. In particular, we were able to explain the observed incoherent and almost momentum-independent photoemission spectrum by linking these peculiar features of the spectrum to two anisotropies present in the employed spin polaron model—the spin anisotropy [@Kunkemoeller2017] and the hopping anisotropy [@Harris2004; @Sutter2017].
Interestingly, the differences between the spectral functions of the ‘standard’ spin polaron model well-known from the cuprates (i.e. spin $S=1/2$) and the model for Ca$_2$RuO$_4$ should not be regarded as being intrinsic to the $S=1$ spin polaron model: They are all solely related to the above-mentioned strong anisotropies present in the model suggested for Ca$_2$RuO$_4$ and [*not*]{} to the potential differences between the hole moving in the $S=1/2$ and the $S=1$ antiferromagnet. These turn out to be basically irrelevant in the linear spin wave approximation. Such a result can naturally be expected following basic quantum mechanics but would not be achieved in the well-known double exchange picture [@Zener1951], for in that classical approach the hole would not be able to hop at all in the $S=1$ antiferromagnet.
Acknowledgments. {#acknowledgments. .unnumbered}
================
We are very grateful to David Sutter and Johan Chang for sharing the experimental data that was presented in Ref. and which allowed us to plot Fig. \[result\](a). We thank Eugenio Paris for insightful discussions. A.K. thanks the IFW Dresden for the kind hospitality. A.K. and K.W. (K.W.) acknowledge(s) support by Narodowe Centrum Nauki (NCN, Poland) under Projects No. 2016/22/E/ST3/00560 (2016/23/B/ST3/00839), respectively. J.v.d.B. acknowledges financial support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) via SFB1143 project A5 and the Würzburg-Dresden Cluster of Excellence ct.qmat.
APPENDIX: Derivation of the Polaron model {#sec:appendix .unnumbered}
=========================================
Mapping onto a polaronic model: Spin Hamiltonian ($\mathcal{H}_J$)
------------------------------------------------------------------
To diagonalize the spin $S=1$ Hamiltonian discussed in the paper \[cf. Eq. (\[model1\]) of the main text of the paper\] we start by performing two rotations of spins. First, we make a different choice of the spin quantization axis $\hat{z}$ - in this case we pick the axis without an anisotropy term. Second, we perform a $\pi$ rotation of spins around the $\hat{x}$ axis on one of the two sublattices—such transformation maps the anticipated antiferromagnetic ground state onto a ferromagnetic state. The result is the rotated spin Hamiltonian $$\begin{aligned}
\label{eq:1}
\mathcal{\tilde{H}}_J &= \sum\limits_{\left< {\bf i,j} \right>} J \: \left[ -\: S_{\bf i}^z S_{\bf j}^z + \frac{1}{2} \left( S_{\bf i}^+ S_{\bf j}^+ + S_{\bf i}^- S_j^- \right) \right] \nonumber \\
&+ \gamma \sum\limits_{\bf i}\left(S_{\bf i}^y\right)^2 + \epsilon \sum\limits_{\bf i}\left(S_{\bf i}^x\right)^2.\end{aligned}$$
The next step is to utilize the Holstein-Primakoff transformation and the linear spin wave approximation using the assumption that the ground state is ferromagnetic and dressed with magnons. The thus obtained Hamiltonian is then diagonalized using the successive Fourier and Bogoliubov transformations. The resulting spin Hamiltonian reads $$\begin{aligned}
\label{eq:1}
{H}_J &= \sum_{\bf q} {\Omega}_{\bf q} \beta^\dag_{\bf q} \beta_{\bf q} \; ,\end{aligned}$$ where the magnon energies $ {\Omega}_{\bf q}$ are given by $${\Omega}_{\bf q} = Y \sqrt{1 - \left( \frac{Z_{\bf q}}{Y} \right)^2},$$ with $$\label{hj solved}
\begin{array}{l}
Y = 4J+{\epsilon}+{\gamma},\\
Z_{\bf q} = 4J{\gamma}_{\bf q}+{\epsilon}-{\gamma},\\
\end{array}$$ with $$\gamma_{\bf q} = \frac{1}{2} \left( \cos(q_x) + \cos(q_y) \right).$$ The Bogoliubov coefficients $u_{\bf q}$ and $v_{\bf q}$, which describe the connection between the bosons before and after the Bogoliubov transformation and are needed to correctly express the kinetic part of the Hamiltonian in the polaronic language (see next section), are expressed through the magnon energies $ {\Omega}_{\bf q}$ via the standard formulae, cf. Ref. .
Mapping onto a polaronic model: Kinetic Hamiltonian ($\mathcal{H}_t$)
---------------------------------------------------------------------
### Dealing with projection operators
The implicit projection operators in Eq. (\[model2\]) in the main text have a relatively complex form. The easiest way to deal with them is to construct the projected form of the two single orbital, two-site kinetic Hamiltonians $$\begin{aligned}
\label{hamt}
(H_t^{xz})_{{\bf i},{\bf i}+{\bf e}_{xz}} &= -t \sum\limits_{\sigma} \tilde{c}_{{\bf i},xz,\sigma}^{\dag}\tilde{c}_{{\bf i}+{\bf e}_{xz},xz,\sigma} + h.c.,\\
\label{hamt2}
(H_t^{yz})_{{\bf i},{\bf i}+{\bf e}_{yz}} &= -t \sum\limits_{\sigma} \tilde{c}_{{\bf i},yz,\sigma}^{\dag}\tilde{c}_{{\bf i}+{\bf e}_{yz},yz,\sigma} + h.c.,\end{aligned}$$ step by step, that is by considering every possible hopping process and the matrix element associated with it.
We consider a single hole introduced into the half-filled ground state and $H_t$ conserves the number of electrons in the system. Moreover, the ‘no double occupancy’ constraint implies that there can never be 3 electrons on one site in the $xz/yz$ orbitals. As a result, a single hole introduced into the system propagates leaving the number of electrons in the $xz/yz$ orbitals on all other sites unchanged and equal to two. It follows that the only nonzero matrix elements of the two-site Hamiltonians (\[hamt\]-\[hamt2\]) describe processes in which a hole hops between a doubly occupied site and a singly occupied site. Furthermore, these hoppings obey two rules:
1. The hole can hop to the neighbouring site, albeit only to an unoccupied orbital.
2. The hole can hop to the neighbouring site, albeit only if the resulting on-site wavefunction is not a spin singlet.
The first rule is simply the ‘no double occupancy’ constraint and the second rule follows from the exclusion of the $S=0$ sector of the Hilbert space discussed in the main text.
All of the above means that on each site we have either the spin triplet or one spin $S=1/2$ fermion on the $xz/yz$ orbitals. This in turn implies that for each pair of neighbouring sites ${\bf i,j}$ we can use the following basis $$\label{site basis}
\left\{ \Big| {^{\uparrow}_{\_}} \Big>_{\bf i}, \Big| {^{\downarrow}_{\_}} \Big>_{\bf i}, \Big| {^{\_}_{\uparrow}} \Big>_{\bf i}, \Big| {^{\_}_{\downarrow}} \Big>_{\bf i} \right\} \; \otimes \; \left\{ \Big| {^{\downarrow}_{\downarrow}} \Big>_{\bf j}, \Big| {^{\uparrow}_{\uparrow}} \Big>_{\bf j}, \frac{1}{\sqrt{2}} \left( \Big| {^{\uparrow}_{\downarrow}} \Big> + \Big| {^{\downarrow}_{\uparrow}} \Big> \right)_{ \bf j} \right\},$$ where the upper and lower positions are the two ($xz/yz$) orbitals, the arrows show spin and the two particle states form the standard triplet. The main task now is to consider the matrix elements of (\[hamt\]-\[hamt2\]) in the basis (\[site basis\]).
Because the Hilbert space is 12-dimensional there are 144 different matrix elements. However, it is easy to see that only a small number of them is nonzero. In order to find them one can use a graphical technique ilustrated in Table \[tab:hop\].
The matrix elements listed above and their complex conjugates constitute all nonzero matrix elements. In total there is 32 of them. There are, however, some symmetries that help us write the Hamiltonian in a simpler form.
First, $(M_1,M_2,M_5,M_6,M_9,M_{10},M_{13},M_{14})$ and conjugates come from $(H_t^{xz})_{{\bf i},{\bf i}+{\bf e}_{xz}}$, the rest comes from $(H_t^{yz})_{{\bf i},{\bf i}+{\bf e}_{yz}}$. We only need to consider one of these Hamiltonians as the matrix elements of other one can be obtained in an analogous way. From now on we will only consider $(H_t^{xz})_{{\bf i},{\bf i}+{\bf e}_{xz}}$.
Second, let us look at $M_4$ and $M_{11}^*$ or $M_{11}$ and $M_{4}^*$, where a star denotes complex conjugation. They represent exactly the same process but happening in the opposite direction on the lattice. $M_4$ and $M_{11}$ on the other hand represent the inverse processes happening in opposite directions on the lattice. This means we need only calculate one of these matrix elements. The same is true for the pair $(M_8,M_{16})$ and their conjugates. This leaves us with $(M_3,M_4,M_7,M_8,M_{12},M_{15})$.
Third, $(M_3,M_7)$ are the same but have all spins flipped, which is a symmetry of the Hamiltonian. The same is true for $(M_4,M_8)$ and $(M_{12},M_{15})$. Altogether, this leaves us with the classification of the nonzero matrix elements shown in Table \[tab:second\].
In order to obtain the second quantised form of (\[hamt\]-\[hamt2\]) we need to calculate all matrix elements after which we can construct the projected two-site Hamiltonian using $$\label{conc ham}
(H_t^{xz})_{{\bf i},{\bf i}+{\bf e}_{xz}} = \sum\limits_{k,l} (H_t^{xz})_{{\bf i},{\bf i}+{\bf e}_{xz}}^{k,l} \left| k \right> \left< l \right|,$$ where the vectors $\left| k \right>, \left| l \right>$ are written in the second quantised form. $(H_t^{xz})_{{\bf i},{\bf i}+{\bf e}_{xz}}^{k,l} \equiv M_j$ are the matrix elements that have been discussed above and need to be explicitly calculated, see below.
### Calculation of the matrix elements
The matrix elements $M_j$ can be calculated in a straightforward way. As an example we calculate $M_3$: $$\begin{aligned}
M_{3} & =\bigg(\left< 0 \right| \tilde{c}_{{\bf i},yz,\uparrow}\tilde{c}_{{\bf j},xz,\uparrow}\tilde{c}_{{\bf j},yz,\uparrow} \bigg) \bigg( -t \: \tilde{c}_{{\bf j},xz,\uparrow}^{\dag}\tilde{c}_{{\bf i},xz,\uparrow} \bigg) \nonumber \\
& \times \bigg( \tilde{c}_{{\bf j},yz,\uparrow}^{\dag}\tilde{c}_{{\bf i},yz,\uparrow}^{\dag}\tilde{c}_{{\bf i},xz,\uparrow}^{\dag}\left| 0 \right>\bigg)\nonumber \\
& =t \: \left< 0 | 0 \right>.\end{aligned}$$ $M_4$ and $M_{15}$ are calculated in a similar manner. The result is shown in Table \[tab:result\].
Once all the matrix elements are calculated one can easily write down the projected form of Eq. (\[model2\]) in the main text in the second quantized form.
### Polaronic mapping
In order to map our model onto a polaronic one, we need to introduce slave fermions (cf. Ref ) using a general mapping $$\label{slavefermions}
\begin{array}{l}
\tilde{c}_{{\bf i},\alpha,\uparrow} \rightarrow h_{{\bf i},\alpha}^{\dag}, \quad \quad
\tilde{c}_{{\bf i},\alpha,\downarrow} \rightarrow \hat{A} \: h_{{\bf i},\alpha}^{\dag} \: S_{\bf i}^+,\\
\end{array}$$ where $\hat{A}$ is an operator to be determined. The spinless hole operators $h_{{\bf i},\alpha}$ obey the Pauli exclusion principle and the standard anticommutation relations. The spin $S=1$ operators obey the standard commutation relations. Finally, the spinles hole operators commute with the spin operators, which introduces an extra term in the Hamiltonian (see below).
### Finding ${\bf \hat{A}}$
After transformation (\[slavefermions\]) the on-site Fock basis consists of the spinless holes with two orbital flavors $xz, yz$ and three eigenvalues of the projection of the spin $S=1$ onto the $\hat{z}$ axis ($S_z$). Thus, we label these states by the number of spinless holes on each orbital $n_{xz}, n_{yz} \in \{ 0,1\}$ and the $S_z \in \{-1,0,1\}$ spin quantum number and write the basis states as $\left\{ \left| n_{xz}, n_{yz}, S_z \right> \right\}$.
To see how the transformation works let us look at the state $\left| {^-_{\downarrow}} \right>$ ($\left| 0 \right> \: \equiv \: \left| 1,1,1 \right>$ defines the vacuum): $$\begin{aligned}
\label{example state}
\Big| {^-_{\downarrow}} \Big> &= \sqrt{2} \tilde{c}_{xz , \uparrow} \left( \frac{1}{\sqrt{2}}S^{-} \right) \tilde{c}_{yz , \uparrow}^{\dag} \tilde{c}_{xz , \uparrow}^{\dag} \left| 0 \right> \nonumber\\
&= \sqrt{2} h_{xz}^{\dag} \left( \frac{1}{\sqrt{2}}S^{-} \right) h_{yz}h_{xz} \left| 1,1,1 \right> = \sqrt{2} \left| 1,0,0 \right>.
\end{aligned}$$ Thus, the state $\Big| {^-_{\downarrow}} \Big>$ maps to the state $\sqrt{2} \left| 1,0,0 \right>$—a state of one hole and one magnon. Similarly, one can determine the other states $$\label{basis map}
\begin{array}{ll}
\Big| 0 \Big> \equiv \left| 1,1,1 \right>, \; & \Big| {^{\uparrow}_-} \Big> \equiv \left| 0,1,1 \right>,\\
\Big| {^{\uparrow}_{\uparrow}} \Big> \equiv \left| 0,0,1 \right>, \; & \Big| {^-_{\uparrow}} \Big> \equiv \left| 1,0,1 \right>,\\
\Big| {^{\uparrow}_{\downarrow}} \Big> \equiv \left| 0,0,0 \right>, \; & \Big| {^{\downarrow}_-} \Big> \equiv \sqrt{2} \left| 0,1,0 \right>,\\
\Big| {^{\downarrow}_{\downarrow}} \Big> \equiv \left| 0,0,-1 \right>, \; & \Big| {^-_{\downarrow}} \Big> \equiv \sqrt{2} \left| 1,0,0 \right>.\\
\end{array}$$ We stress that in the above notation the first two quntum numbers are the number of holes on each orbital. \[For example, the state $\left| 1, 1, 1 \right>$ is the vacuum (no electrons) and the state $\left| 0, 0, 0 \right>$ is the $S_z = 0$ two electron state.\]. The operator $\hat{A}$ is necessary to normalize the states in the new basis. In (\[basis map\]) we see that two states are not normalized and acquire a factor of $\sqrt{2}$, which is a consequence of the projection onto the $S_z=0$ triplet state. It is easy to check that consequently $\hat{A} = 1$ for $\{ \left|1,0,0\right>, \left|0,1,0\right> \}$ and $\hat{A} = \frac{1}{\sqrt{2}}$ otherwise.
### Restricting the Hilbert space
Looking at (\[basis map\]) again, we observe that there are four states that do not map to any states in the old basis, namely $$\left\{ \left| 1,0,-1 \right> , \left| 0,1,-1 \right> , \left| 1,1,0 \right> , \left| 1,1,-1 \right> \right\}.$$ Evidently, these need to be projected out. One could achieve this using projection operators, but it would complicate the formula for the Hamiltonian. Another approach, presented in Ref. for the $S=1/2$ case, is to include an extra term in the Hamiltonian with a very large coupling constant $\zeta > 0$, in the spirit of the Lagrange multipliers. In our case this term takes the form $$\begin{aligned}
\label{constraint}
\begin{split}
H_{\zeta} &= \zeta \sum\limits_{\bf i} \Big[ \left( h_{{\bf i} , xz}^{\dag}h_{{\bf i} , xz} h_{{\bf i} , yz}^{\dag}h_{{\bf i} , yz} \left( S_{\bf i}^z \right)^2 \right) +\\
&+ \left( h_{{\bf i} , xz}^{\dag}h_{{\bf i} , xz} + h_{{\bf i} , yz}^{\dag}h_{{\bf i} , yz} \right) \left( S_{\bf i}^z - 1 \right) S_{\bf i}^z \Big].\\
\end{split}\end{aligned}$$ Following the authors of Ref. we will neglect this part of the Hamiltonian. It is clear that this is not without consequence. For simpler models it was shown [@Bieniasz2018] that including such constraints in the diagrammatic expansion of the Dyson equation leads to quantitative differences. We believe that the same situation happens for the $S=1$ case studied here.
### The linear spin wave (LSW) approximation
To arrive at the formula (5) in the main text we need to find the expressions for the operators $\big| k \big> \big< l \big|$ appearing in Eq. (\[conc ham\]). We look for them in the LSW approximation.
After introducing magnons via the Holstein-Primakoff transformation, the spin quantum number $S_z$ maps onto the number of magnons quantum number $n_{mag}$: $$S_z= 1,0,-1 \rightarrow n_{mag}= 0,1,2 \ ,$$ respectively, while the fermionic quantum numbers $\{ n_{xz}, n_{yz} \}$ remain the same.
In this basis, let us examine the projection operator ($P_3$) associated with the matrix element $M_3$ that was discussed above (the other cases are analogous, see below). After the sublattice rotation we obtain $$\begin{aligned}
P_{3} &= \Big| {^{-}_{\uparrow}} \Big>_{{\bf i}} \Big| {^{\downarrow}_{\downarrow}} \Big>_{{\bf j}} \Big< {^{\uparrow}_{\uparrow}} \Big|_{{\bf i}} \Big< {^{-}_{\downarrow}} \Big|_{{\bf j}} = \nonumber \\
&= \sqrt{2} \: \left| 1,0,0 \right>_{\bf i} \left| 0,0,2 \right>_{{\bf j}} \left< 0,0,0 \right|_{\bf i} \left< 1,0,1 \right|_{{\bf j}} = \nonumber \\
&= \sqrt{2} \: \left( h_{{\bf j},yz} h_{{\bf j},xz} h_{{\bf i},yz} a_{{\bf j}}^{\dag}a_{{\bf j}}^{\dag} \left| 1,1,0 \right>_{\bf i} \left| 1,1,0 \right>_{{\bf j}} \right) \nonumber \\
&\otimes \left( \left< 1,1,0 \right|_{{\bf i}} \left< 1,1,0 \right|_{{\bf j}} a_{{\bf j}} h_{{\bf j},yz}^{\dag} h_{{\bf i},xz}^{\dag} h_{{\bf i},yz}^{\dag} \right). \end{aligned}$$ First, we notice that the projection onto the double vacuum state, which represents two empty sites, is obsolete. Indeed, if a state survives the action of the spinless fermion creation operators on the right it survives it as one of two states:
1. A state with four spinless holes, two on each site, in which case the projection is obsolete as this is a unique property of the vacuum,
2. A state with three spinless holes, two on the $i$-th site and one on the $i+1$-st site. In this case the annihilation operators on the left annihilate it, because two of them act on the $i+1$-st site.
It is easy to see that the same is true for any of the 16 operators multiplying the matrix elements in Table \[tab:hop\] and their Hermitean conjugates.
Using this we can write $P_3$ as $$\begin{aligned}
P_{3} &= \sqrt{2} \: h_{{\bf j},yz} h_{{\bf j},xz} h_{{\bf i},yz} h_{{\bf j},yz}^{\dag} h_{{\bf i},xz}^{\dag} h_{{\bf i},yz}^{\dag} a_{{\bf j}}^{\dag}a_{{\bf j}}^{\dag} a_{{\bf j}} \nonumber \\
& \approx \sqrt{2} \: h_{{\bf i},xz}^{\dag} h_{{\bf j},xz} a_{{\bf j}}^{\dag}a_{{\bf j}}^{\dag} a_{{\bf j}},\end{aligned}$$ where we have neglected the normal ordered terms with three or more spinless hole operators which go beyond our diagrammatic expansion (see section D).
We see that $P_3$ is of order three in the bosonic operators. Performing similar calculations for the other 15 operators one can show that they can be divided into three groups:
1. of order one in bosonic operators,
2. of order three in bosonic operators,
3. of order five in bosonic operators.
In the LSW approximation we only consider the first group of terms. Consequently, only four amongst the 16 operators are non-negligible. These are $ \{ P_2 , P_4, P_{10}, P_{16} \} $. Together with their respective matrix elements the four operators and their Hermitean conjugates give the projected kinetic Hamiltonian in the LSW approximation, see last two lines of Eq. (\[finalhamiltonian\]) in the main text.
Mapping onto a polaronic model: Spectral functions
--------------------------------------------------
As discussed in the main text of the paper we are interested in calculating the following spectra function $$\begin{aligned}
&{A}_{\alpha}({\bf k},\omega) = -\frac{1}{\pi} {\rm Im}\left\{{G}_{\alpha}({\bf k},\omega) \right\} = \nonumber \\
&= -\frac{1}{\pi} {\rm Im} \left\langle 0 \right| \tilde{c}^{\dag}_{{\bf k},\alpha,\sigma} \frac{1}{\omega - \mathcal{H} + E_0 + i \delta} \tilde{c}_{{\bf k},\alpha\sigma} \left| 0 \right\rangle,\end{aligned}$$ where $\tilde{c}_{{\bf k},\alpha,\sigma} = c_{{\bf k},\alpha,\sigma}(1-c^{\dag}_{{\bf k},\alpha,{\bar \sigma}}c_{{\bf k},\alpha,{\bar \sigma}})$ are the restricted hole annihilation operators. It is therefore not a one particle Green’s function.
The relation between the above-defined hole spectral function and the spinless hole spectral function is nontrivial, cf. Appendix of Ref. . The latter one, that is natural to the polaronic language, is calculated from the single-particle spinless hole Green’s function and reads $$\begin{aligned}
\label{spectral ok}
&A_{\alpha}({\bf k},\omega) = -\frac{1}{\pi} {\rm Im}\left\{ G_{\alpha}({\bf k},\omega) \right\} = \nonumber \\
&=-\frac{1}{\pi} {\rm Im } \left\langle 0 \right| h_{{\bf k},\alpha} \frac{1}{\omega - H +E_0 + i \delta} h^{\dag}_{{\bf k},\alpha} \left| 0 \right\rangle.\end{aligned}$$ Fortunately, it was shown that for the $S=1/2$ $t$-$J$ model the spinless hole spectral function and the hole spectral function almost coincide [@Wang2015]. We assume that the same also holds also for the $S=1$ $t$-$J$ model investigated here.
The self-consistent Born approximation to the Dyson equation {#sec:scba}
------------------------------------------------------------
To obtain the Green’s function $G_{\alpha}({\bf k},\omega)$ of Eq. (\[spectral ok\]), and thus calculate the spectral function $A_{\alpha}({\bf k},\omega)$, we use the Dyson equation that reads $$G_{\alpha}({\bf k},\omega) = G^{0}_{\alpha}({\bf k},\omega) + G^{0}_{\alpha}({\bf k},\omega) \Sigma_{\alpha}({\bf k},\omega) G_{\alpha}({\bf k},\omega),$$ where $\alpha$ is an orbital index. The self energy $\Sigma_{\alpha}({\bf k},\omega)$ is defined as the sum of all non-reducible diagrams starting and ending with the same vertex with an external line representing a spinless hole with momentum ${\bf k}$ and orbital index $\alpha$.
We calculate the self energy $\Sigma_{\alpha}({\bf k},\omega)$ approximately, using the self-consistent Born approximation (SCBA): $$\begin{aligned}
\label{scba diag}
\begin{split}
&\Sigma_{\alpha}({\bf k},\omega) \approx \\
&= \int\limits_{-\infty}^{\infty} d \omega' \sum\limits_{{\bf q}} \; D^{0}(\omega') G_{\alpha}({\bf k} - {\bf q}, \omega - \omega') V_{\alpha} ({\bf k}, {\bf q}) V_{\alpha} ({\bf k}, {\bf q})^*,
\end{split}\end{aligned}$$ where the vertex is defined as $$\begin{aligned}
\label{vertex}
\begin{split}
V_{\alpha} \left( {\bf k}, {\bf q} \right) &= \frac{\sqrt{2} \: t}{\sqrt{N}} \
\left({\gamma}_{{\bf k} \cdot {\bf e}_{\alpha}} v_{\textbf{q}}+{\gamma}_{({\bf k-q}) \cdot {\bf e}_{\alpha}} u_{\textbf{q}} \right), \\
\end{split}\end{aligned}$$ and the magnon Green’s function is $$\label{magnongreen}
D^{0}(\omega) = \delta(\omega - \Omega_{\bf q}).$$
Using Eqs. (\[vertex\]) and (\[magnongreen\]) we obtain the self-consistent equation for the self energy (\[scba diag\]) in the SCBA approximation $$\begin{aligned}
\label{scba}
&\Sigma_{\alpha}({\bf k},\omega) = \sum\limits_{{\bf q}} \; G_{\alpha}({\bf k} - {\bf q}, \omega - \Omega_{\bf q}) V_{\alpha} ({\bf k}, {\bf q}) V_{\alpha} ({\bf k}, {\bf q})^* \nonumber
\\
&= \sum\limits_{{\bf q}} \; \frac{V_{\alpha} ({\bf k}, {\bf q}) V_{\alpha} ({\bf k}, {\bf q})^*}{\omega + J - \Omega_{\bf q} - \Sigma_{\alpha}({\bf k}-{\bf q},\omega - \Omega_{\bf q})}.\end{aligned}$$ Finally, the above equation is solved numerically for the self energy $\Sigma_{\alpha}({\bf k},\omega)$ on a finite mesh of ${\bf k}$ and $\omega$ points (see main text of the paper).
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abstract: 'Hyperfine structure (HFS) of atomic energy levels arises due to interactions of atomic electrons with a hierarchy of nuclear multipole moments, including magnetic dipole, electric quadrupole and higher rank moments. Recently, a determination of the magnetic octupole moment of the $^{173}\mathrm{Yb}$ nucleus was reported from HFS measurements in neutral ${}^{173}\mathrm{Yb}$ \[PRA 87, 012512 (2013)\], and is four orders of magnitude larger than the nuclear theory prediction. Considering this substantial discrepancy between the spectroscopically extracted value and nuclear theory, here we propose to use an alternative system to resolve this tension, a singly charged ion of the same $^{173}\mathrm{Yb}$ isotope. Utilizing the substantial suite of tools developed around $\mathrm{Yb}^+$ for quantum information applications, we propose to extract nuclear octupole and hexadecapole moments from measuring hyperfine splittings in the extremely long lived first excited state ($4f^{13}(^2\!F^{o})6s^2$, $J=7/2$) of $^{173}\mathrm{Yb}^+$. We present results of atomic structure calculations in support of the proposed measurements.'
author:
- Di Xiao
- 'Jiguang Li (李冀光)'
- 'Wesley C. Campbell'
- Thomas Dellaert
- Patrick McMillin
- Anthony Ransford
- Conrad Roman
- Andrei Derevianko
date:
title: 'Hyperfine structure of $^{173}\mathrm{Yb}^+$: toward resolving the $^{173}\mathrm{Yb}$ nuclear octupole moment puzzle'
---
[UTF8]{}[gkai]{}
Introduction {#Sec:Intro}
============
While the size of an atomic nucleus is far too small to image its features directly with a microscope, the interaction of an atomic nucleus with electrons bound to it will leave signatures of the size and shape of the nucleus on the the resulting atom in the form of hyperfine structure (HFS). In particular, $P$ and $T$ symmetries dictate that the distribution of protons leads to even-rank $(k\!=\!0,2,4...)$ electric $2^k$-pole moments (e.g., monopole, quadrupole, and hexadecapole) and the distribution of currents and magnetic moments leads to odd-rank $(k\!=\!1,3,5...)$ magnetic moments (e.g. dipole, octupole, and 32-pole) that interact with the electrons to shift their energies. In this sense, when combined with accurate atomic structure calculations, a measurement of the HFS of an atom constitutes an electron scattering experiment on the nucleus that allows us to “see” the distribution of its nucleons by observing how these well-characterized electrons scatter from it.
In general, the dominant contributions to HFS come from (nuclear) magnetic dipole and electric quadrupole interactions. Presently, the nuclear magnetic dipole ($\mu$) and electric quadrupole ($Q$) moments of most nuclei are well established (see e.g., compilation [@Stone2005]). This is largely because the HFS signatures of higher-order moments only appear on electronic states with sufficiently high multiplicity $(2J \ge k)$ and the magnitude of the energy shift tends to decrease with increasing rank $k$. The measurement of HFS signatures of high rank $(k \ge 3)$ multipoles, therefore, requires a well-controlled atom in a high angular momentum state for precision and state of the art atomic structure theory for accuracy.
Here, we focus on the potential for measuring the rarely observed nuclear octupole ($\Omega, k\!=\!3$) and hexadecapole ($\Pi, k\!=\!4$) moments. These moments have been deduced for only a handful of nuclei and, in most cases, are in tension with nuclear theory (see Table \[table:NuclearMoments\]). For example, in $^{133}$Cs, the extracted [@GerDerTan03] nuclear octupole moment is 40 times larger that the nuclear theory value. This paper is motivated by the even more substantial disagreement for $^{173}\mathrm{Yb}$. Recently, @Singh reported a measurement of the nuclear octupole moment from their measurements of HFS in the ${^3P_2}$ state of neutral $^{173}\mathrm{Yb}$. However, this value $\Omega = -34.4 \,{\mathrm{b{\mu_N}}}$ is $10^4$ times larger than the nuclear theory prediction, $\Omega =0.003\,{\mathrm{b{\mu_N}}}$ [@Williams1962]. This striking four orders of magnitude disagreement calls for an independent measurement and analysis. Here, we investigate the prospects for extracting $\Omega$ and higher rank nuclear multipole moments of ytterbium-173 by a combined theoretical and experimental investigation of the hyperfine level splittings in the first excited state ($4f^{13}(^2\!F^{o})6s^2$, $J=7/2$) of the $^{173}\mathrm{Yb}^{+}$ ion. This ${}^2\!F^o_{7/2}$ state is metastable, contains six $m_F=0$ states that will be first-order insensitive to magnetic fields, and easily state-selectively coupled to the ground state for precision spectroscopy.
-------------------------------------- ------------------- ----------------------------- ------------------- ----------- ---------
$^{87}\mathrm{Rb}$ [@Gerginov2009] $\frac{3}{2}^{-}$ $p_{3/2}, \mathrm{proton}$ $^2\!P_{{3}/{2}}$ $-0.58$ $0.30$
$^{113}\mathrm{In}$ [@EckKus57] $\frac{9}{2}^{+}$ $g_{9/2}, \mathrm{proton}$ $^2\!P_{3/2}$ $0.574$ $0.99$
$^{115}\mathrm{In}$ [@EckKus57] $\frac{9}{2}^{+}$ $g_{9/2}, \mathrm{proton}$ $^2\!P_{{3/2}}$ $0.565$ $1.00$
$^{133}\mathrm{Cs}$ [@GerDerTan03] $\frac{7}{2}^{+}$ $g_{7/2}, \mathrm{proton}$ $^{2}\!P_{{3/2}}$ $0.82$ $0.022$
$^{137}\mathrm{Ba^{+}}$ [@Lewty2013] $\frac{3}{2}^{+}$ $d_{3/2}, \mathrm{neutron}$ $^{2}\!D_{{3/2}}$ $-0.0629$ $0.039$
$^{155}\mathrm{Gd}$ [@Unsworth1969] $\frac{3}{2}^{-}$ $p_{3/2}, \mathrm{neutron}$ $^9\!D_3$ $-1.66$ $-0.29$
$^{165}\mathrm{Ho}$ [@DanFerGeb74] $\frac{7}{2}^{-}$ $f_{7/2}, \mathrm{proton}$ $^4\!I_{{15/2}}$ $0.75$ $1.0$
$^{173}\mathrm{Yb}$ [@Singh] $\frac{5}{2}^{-}$ $f_{5/2}, \mathrm{neutron}$ $^{3}\!P_2$ $-34.4$ $0.15$
-------------------------------------- ------------------- ----------------------------- ------------------- ----------- ---------
: Compilation of spectroscopic determinations of nuclear octupole moments. $\Omega^\mathrm{emp}$ are the empirical moments derived from the combination of spectroscopic HFS measurements and electronic structure calculations. $\Omega^\mathrm{SP}$ are octupole moments predicted by the single-particle model [@sch55]. All octupole moments are in units $\mathrm{barn}\times \text{nuclear magneton} (\mathrm{b\times{\mu_N}})$. All listed isotopes are stable except for $^{115}\mathrm{In}$ and $^{87}\mathrm{Rb}$ with half-lives of $4.4\times10^{14}$ and $4\times10^{10}$ years, respectively [@Stone2005]. Values of $\Omega$ have also been reported for about 20 additional nuclei from nuclear scattering experiments [@Fuller1976]. []{data-label="table:NuclearMoments"}
Most of the previous spectroscopic determinations of high-order moments of nuclei focused on extraction of octupole moments, and are compiled in Table \[table:NuclearMoments\]. This Table also lists the nuclear single-particle model [@sch55] values for the nuclear octupole moments. In addition to the listed spectroscopic determinations, the experiments were carried out in Eu [@Chi91] and Hf [@JinWakIna95]. However, due to the complexity of electronic structure calculations, these experiments only determined ratios of nuclear octupole moments between different isotopes, $\Omega(^{151}\mathrm{Eu})/\Omega(^{153}\mathrm{Eu})$ and $\Omega(^{177}\mathrm{Hf})/\Omega(^{179}\mathrm{Hf})$.
Beyond octupole order, the hexadecapole moment $\Pi$ has been spectroscopically determined for only one species: $^{165}$Ho [@DanFerGeb74]. Access to the hexadecapole moments requires $J \ge 2$ and $I \ge 2$. For example, although $^{133}$Cs nucleus has $I=7/2$ and thereby possesses hexadecapole moment, this moment can not be determined from the measured HFS of the $6\,{p_{3/2}}$ state [@GerDerTan03]. This argument prohibits the extraction of nuclear hexadecapole moments from the structure of the states used to measure magnetic octupole moments for all but two exceptions in Table \[table:NuclearMoments\]: $^{165}$Ho and $^{173}$Yb. The value for $\Pi$ that was extracted from spectroscopic measurements in $^{165}$Ho was found to be larger than the nuclear theory value by an order of magnitude [@DanFerGeb74]. In principle, one could extract $\Pi$ from the measurements made by Singh *et al.* in neutral $^{173}\mathrm{Yb}$ [@Singh], but its contribution was neglected in that work. Here, in order to leverage the considerable experimental toolbox built around $\mathrm{Yb}^+$ for quantum information applications, we evaluate the necessary electronic structure factors for $^{173}$Yb$^{+}$ needed to enable extraction of the hexadecapole moment of this isotope from future spectroscopic measurements.
The $\mathrm{Yb}^{+}$ ground state hyperfine structure is among the most precisely measured and easiest to control of all the HFS in atomic physics owing to its $m_F=0$ “clock states” and its readily available state preparation and readout schemes. The ${}^2S_{1/2}$ HFS of ${}^{171}\mathrm{Yb}^+$ has been used for decades for frequency standards and quantum information processors, and its splitting has been known to $\mathrm{mHz}$ precision for many years [@Fisk1997]. The long coherence time of clock-state qubits defined on this hyperfine splitting (recently shown to exceed 10 minutes [@Wang2017]) has made ${}^{171}\mathrm{Yb}^{+}$ a premier qubit host for quantum computing and quantum simulation [@Olmschenk2007; @Figgatt2019; @Zhang2017; @Landsman2019; @Wright2019]. Likewise, the metastable $^2F_{7/2}^o$ electronic state of ${}^{171}\mathrm{Yb}^+$ lives for years, and the E3 transition on ${}^2S_{1/2} \! \leftrightarrow {}^2F_{7/2}^o$ is used as an optical frequency standard [@Huntemann2016], where hyperfine structure within these states allows control of systematics. Some of the current best limits on the time variation of fundamental constants are based on precision measurements between specific hyperfine components of this E3 transition [@Godun2014; @Huntemann2014b].
As the experimental progress with this species continues to achieve higher accuracy and precision [@Huntemann2016; @Sanner2018], theoretical work is needed in parallel with these improvements to understand contributions to systematics. Ref. [@Beloy2008a] has shown how to calculate the second-order energy correction due to hyperfine interaction for the alkali-atoms in the first excited state, and gives the theoretical basis for higher-order terms calculated in this paper. The values of hyperfine constants $A$ and $B$ for $^{171}\mathrm{Yb}^{+}$ and $^{173}\mathrm{Yb}^{+}$ in the ${}^2F_{7/2}^o$ state are also given in Ref. [@Dzuba2016a], and are used as a comparison to our values.
The paper is organized as follows. In Sec. \[Sec:General-theory\], we review the theory of hyperfine structure. Based on this general theory, we derive the first- and second-order corrections to the HFS of $^{173}\mathrm{Yb^{+}}$ in the first excited state. In Sec. \[Sec:electronic-reduced-ml\], we compute $^{173}\mathrm{Yb^{+}}$ electronic-structure factors required for extracting nuclear moments. We discuss the importance of correlation effects in Sec. \[Sec:electron-correlation-eff\]. Finally, we estimate theoretical accuracy and consider its implications on the extraction of octupole and hexadecapole moments in Sec. \[sec:Discussion\]. Unless specified otherwise, atomic units are used throughout.
Review of the theory of hyperfine structure {#Sec:General-theory}
===========================================
The hyperfine interaction can be decomposed into the magnetic dipole (M1), electric quadrupole (E2), magnetic octupole (M3), electric hexadecapole (E4), and higher rank contributions. We start by expressing the hyperfine Hamiltonian in irreducible tensor form [@sch55; @Johnson2007] $$H_\mathrm{HFI}=\sum\limits_{k,\mu}(-1)^{\mu}T_{k,\mu}^eT_{k,-\mu}^n \,,\ \label{Eq:HF-general}$$ where rank-$k$ tensors $T_{k,\mu}^e$ act in the electron space, and $T^n_{k,-\mu}$ — in the nuclear space. The many-electron operators are $T_{k,\mu}^e=\sum\limits_{i}t_{k,\mu}^e(i)$, where the summation is over all the atomic electrons. The single-electron operators $t_{k,\mu}^e(i)$ can be divided into two groups [@Johnson2007], $$t_{k,\mu}^e(i)= \begin{cases}
-\frac{1}{r^{k+1}}C_{k,\mu}(\hat{r}), & \text{electric (even $k$)}\,, \\
-\frac{i}{r^{k+1}}\sqrt{\frac{k+1}{k}} \bm{\alpha} \cdot{\bm{C}^{(0)}_{k,\mu}}(\hat{r}), & \text{magnetic (odd $k$)}.
\end{cases}\label{Eq:single-electron-operator}$$ Here, $\bm{\alpha}$ is the Dirac matrix, $r$ is the radial coordinate, $C_{k,\mu}$ are normalized spherical harmonics, and $\bm{C}_{k,\mu}^{(0)}$ are normalized vector spherical harmonics.
The first-order energy correction due to hyperfine interaction, Eq. (\[Eq:HF-general\]), in the basis of coupled nuclear and atomic states is [@Beloy2008a] $$\begin{split}
W^{(1)}_F = \langle \gamma{IJFM_F}|H_{\mathrm{HFI}}|\gamma IJFM_F\rangle = (-1)^{I+J+F} \\
\times\sum\limits_{k}\begin{Bmatrix}
F & J & I \\
k & I & J
\end{Bmatrix}\langle \gamma{J}||T^e_k||\gamma{J}\rangle\langle I||T^{n}_k||I\rangle\,,
\end{split}\label{Eq:1st-order-ME}$$ where $I$ is the nuclear spin, $J$ is the total electronic angular momentum, $F$ is the grand total angular momentum $\bm{F}=\bm{J} +\bm{I}$, and $\gamma$ stands for remaining quantum numbers.
The first-order energy corrections are conventionally expressed as linear combinations of HFS constants $A$, $B$, $C$, $D$...($k$=1,2,3,4...). The first four constants are defined as [@Beloy2008a] $$\begin{aligned}
\label{eq:hpf-constants}
A&=&\frac{1}{IJ}\langle{T_1^n}\rangle_{I}\langle{T_1^e}\rangle_{J}=\frac{1}{IJ}\mu\langle{T_1^e}\rangle_J\,,\nonumber \\
B&=&4\langle{T_2^n}\rangle_I\langle{T_2^e}\rangle{_J}=2Q\langle{T_2^e}\rangle_{J}\,,\\
C&=&\langle{T_3^n}\rangle_I\langle{T_3^e}\rangle{_J}=-\Omega\langle{T_3^e}\rangle{_J}\,,\nonumber\\
D&=&\langle{T_4^n}\rangle{_I}\langle{T_4^e}\rangle{_J}=\Pi{\langle{T_4^e}}\rangle{_J} \,.\nonumber
\end{aligned}$$ Here, the stretched matrix element $\langle T_k^e \rangle_J$ is defined as $\langle T_k^e \rangle_J ={\ensuremath{\begin{pmatrix}J&k&J\\#4&0&J\end{pmatrix}}}{\ensuremath{\langle \gamma{J}|}}|T_{k}^e|{\ensuremath{|\gamma{J}\rangle}}$. Nuclear stretched matrix elements are proportional to the nuclear moments: $\langle{T_1^n}\rangle_I=\mu$, $\langle{T_2^n}\rangle_I=Q/2$, $\langle{T_3^n}\rangle_I=-\Omega$, and $\langle{T_4^n}\rangle_I=\Pi$.
The second-order energy correction due to hyperfine interaction reads
$$\begin{aligned}
\label{eq:2nd-order-general}
W^{(2)}_{F}&=&\sum_{\gamma{'}J^{'}}\frac{\langle\gamma{IJFM_F}|H_{\mathrm{HFI}}|\gamma^{'}IJ'FM_F\rangle\langle\gamma^{'}IJ'FM_F|H_{\mathrm{HFI}}|\gamma{IJFM_F}\rangle}{E_{\gamma{J}}-E_{\gamma{'}J'}}.
\end{aligned}$$
This equation reduces to $$\label{eq:2nd-order}
\begin{split}
W_F^{(2)} = \sum\limits_{\gamma'J'}\frac{1}{E_{\gamma{J}}-E_{\gamma{'}J'}}\sum\limits_{k_1,k2}{\ensuremath{\begin{Bmatrix}I&J&F\\#4&I&k_1\end{Bmatrix}}}{\ensuremath{\begin{Bmatrix}I&J&F\\#4&I&k_2\end{Bmatrix}}} \times \\
{\ensuremath{\langle I|}} |T_{k_1}^n|{\ensuremath{|I\rangle}}{\ensuremath{\langle I|}} |T_{k_2}^n|{\ensuremath{|I\rangle}}{\ensuremath{\langle \gamma{J}|}}|T_{k_1}^e|{\ensuremath{|\gamma'J'\rangle}}{\ensuremath{\langle \gamma J|}}|T_{k_2}^e| {\ensuremath{|\gamma' J'\rangle}},
\end{split}$$ where primed quantities refer to intermediate states; $E_{\gamma{J}}$ and $E_{\gamma{'}J{'}}$ are the HFI-unperturbed energy levels.
Based on the general theory, in the next section we investigate the hyperfine structure of $^{173}\mathrm{Yb^{+}}$ in the first excited state.
Hyperfine structure of $\mathrm{Yb}^{+}$ in the first excited state {#Sec:HFI-YbII}
===================================================================
The first excited state of $\mathrm{Yb^{+}}$ has the electronic configuration $4f^{13}(^2\!F^o)6s^2$ with electronic angular momentum $J$ equal to $7/2$. Since $^{173}\mathrm{Yb}$ has nuclear spin of $5/2$, the grand total angular momentum $F$ is an integer in the interval $[1,6]$. The $^{173}\mathrm{Yb}$ isotope possesses five distinct nuclear electromagnetic moments. The nucleus has an unpaired valence neutron in the $f_{5/2}$ state. The observed [@Stone2005] nuclear magnetic dipole $\mu$ and electric quadrupole moments $Q$ are equal to $-0.680\,\mathrm{\mu_N}$ and $2.80\mathrm{\,b^2}$, respectively. The nuclear single-particle shell model is not adequate for this isotope as it predicts zero value for the quadrupole moment (the valence nucleon is a neutron for this isotope, whereas the electric moments arise from the distribution of protons in the core). This discrepancy points to a strong nuclear deformation of $^{173}\mathrm{Yb}$. Following the theoretical proposal [@BelDerJoh08], the value for the octupole moment was deduced [@Singh] from the HFS in neutral $^{173}\mathrm{Yb}$ atom in the metastable $6s6p\,^3\!P_2$ state. However, the deduced value, $\Omega = -34.4 \,{\mathrm{b\times{\mu_N}}}$, is $\sim 200$ times larger and of opposite sign compared to the prediction of the single-particle nuclear shell model [@Schwartz1955]. A more sophisticated nuclear structure calculation [@Williams1962] (axially-symmetric collective model in strong coupling) yields $\Omega =0.003\,{\mathrm{b\times{\mu_N}}}$, bringing the discrepancy with the spectroscopic determination in neutral Yb to four orders of magnitudes. As to the electric hexadecapole moment $\Pi$, the single-particle nuclear shell model again predicts zero (similar to $Q$) because the valence nucleon is electrically neutral. We are not aware of any nuclear structure calculations for $\Pi$ of $^{173}\mathrm{Yb}$. We estimate $\Pi \approx Q^2 \approx 9 \, \mathrm{b^2}$ as both $Q$ and $\Pi$ arise due to nuclear deformation; we will take this value as fiducial in further computations.
From Eqs. (\[Eq:1st-order-ME\],\[eq:hpf-constants\]), we obtain the following first-order energy corrections, $$\begin{aligned}
W_6^{(1)}&=&\frac{35}{4}A+\frac{1}{4}B+C+D \, , \nonumber \\
W_5^{(1)}&=&\frac{11}{4}A-\frac{37 }{140}B-\frac{109}{35}C-\frac{41}{7}D \, ,\nonumber \\
W_4^{(1)}&=&-\frac{9}{4}A-\frac{3}{10}B+\frac{46}{35}C+12 D\, , \nonumber \\
W_3^{(1)}&=&-\frac{25}{4}A-\frac{1}{14}B+\frac{22}{7}C-\frac{44}{7}D \, , \label{Eq:W1-individual}\\
W_2^{(1)}&=&-\frac{37}{4}A+\frac{1}{4}B+\frac{11}{35}C-11D \, ,\nonumber \\
W_1^{(1)}&=&-\frac{45}{4}A+\frac{15}{28}B-\frac{33}{7}C+\frac{99}{7}D \, \nonumber .\end{aligned}$$
The second-order corrections are computed from Eqs. (\[eq:2nd-order-general\],\[eq:2nd-order\]), where we keep magnetic dipole and electric quadrupole contributions. To streamline the notation, we introduce dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole constants. These are defined for individual intermediate states $|\gamma'J'\rangle$,
$$\begin{aligned}
\label{eq:2nd-order-term}
\eta_{\mu\mu}[\gamma'J']&=&\frac{(I+1)(2I+1)}{I}\frac{\mu^2{\ensuremath{\langle \gamma J|}} |T_1^e|{\ensuremath{|\gamma' {J{'}}\rangle}}^2}{E_{\gamma{J}}-E_{\gamma' J' }}\,,\nonumber \\
\eta_{\mu{Q}}[\gamma'J']&=&{\frac{(I+1)(2I+1)}{I}}\sqrt{\frac{2I+3}{2I-1}}\times\frac{\mu{Q}{\ensuremath{\langle \gamma{J}|}}|T_1^e|{\ensuremath{|\gamma'{J'}\rangle}}{\ensuremath{\langle \gamma{J}|}}|T_2^e|{\ensuremath{|\gamma'{J'}\rangle}}}{E_{\gamma{J}}-E_{\gamma'{J'}}}\,, \\
\eta_{QQ}[\gamma'J']&=&\frac{(2I+1)(I+1)(2I+3)}{4I(2I-1)}\frac{Q^2{\ensuremath{\langle \gamma{J}|}}|T_2^e|{\ensuremath{|\gamma' {J'}\rangle}}^2}
{E_{\gamma{J}}-E_{\gamma' {J'}}}\,.\nonumber\end{aligned}$$
Eq. (\[eq:2nd-order\]) shows that we need to sum over all possible intermediate states obeying both the parity and the angular selection rules - that is, the parity of the ${\ensuremath{|\gamma{J}\rangle}}$ and ${\ensuremath{|\gamma{'}{J'}\rangle}}$ states has to be the same and $|J+J'|\geq{k}\geq|J-J'|$. Thus, for dipole-dipole and dipole-quadrupole terms, there are three possible $J'$ values, while for quadrupole-quadrupole term, there are five possible $J'$ values. Among the ${\ensuremath{|\gamma{'}{J'}\rangle}}$ intermediate states, the dominant contribution comes from the configuration $4f^{13}(^2\!F^{o})6s^2$ with $J=5/2$. The electronic matrix elements of other possible intermediate states are small enough to be neglected, or the energy denominators are large. With the single intermediate state fixed, we rewrite Eq. (\[eq:2nd-order\]) as $$\begin{aligned}
W_F^{(2)}&\approx &C_{\mu{\mu}}[J',F]\times\eta_{\mu{\mu}}[\gamma{'}J']+ \nonumber \\
&& C_{\mu{Q}}[J',F]\times\eta_{\mu{Q}}[\gamma'J'] + \label{Eq:W2-individual} \\
&&C_{QQ}[J',F]\times\eta_{QQ}[\gamma{'}J']\,,\nonumber\end{aligned}$$ where the angular factors are $$\begin{aligned}
C_{{\mu\mu}}[J{'},F]&=&{\ensuremath{\begin{Bmatrix}I&J&F\\#4&I&1\end{Bmatrix}}}^2\,,\nonumber\\
C_{{\mu{Q}}}[J{'},F]&=&{\ensuremath{\begin{Bmatrix}I&J&F\\#4&I&1\end{Bmatrix}}}{\ensuremath{\begin{Bmatrix}I&J&F\\#4&I&2\end{Bmatrix}}}\,,\label{eq:2nd-order-coeff}\\
C_{{{Q}{Q}}}[J{'},F]&=&{\ensuremath{\begin{Bmatrix}I&J&F\\#4&I&2\end{Bmatrix}}}^2 \,.\nonumber\end{aligned}$$ Adding the first-order, Eq. (\[Eq:W1-individual\]) and second-order, Eq (\[Eq:W2-individual\]), corrections for individual hyperfine levels, we arrive at $$\begin{aligned}
\label{Eq:W12-individual}
W_6^{(1+2)}&=&W_6^{(1)}+0\times{\eta_{\mu\mu}}+0\times{\eta_{\mu{Q}}}+0\times\eta_{QQ}\,,\nonumber\\
W_5^{(1+2)}&=&W_5^{(1 )}+\frac{1}{98}\eta_{\mu\mu}+\frac{\sqrt\frac{5}{6}}{98}\eta_{\mu{Q}}+\frac{5}{588}\eta_{QQ}\,,\nonumber\\
W_4^{(1+2)}&=&W_4^{(1)}+\frac{11}{882}\eta_{\mu\mu}+0\times\eta_{\mu{Q}}+0\times\eta_{QQ}\,,\nonumber\\
W_3^{(1+2)}&=&W_3^{(1)}+\frac{1}{98}\eta_{\mu\mu}-\frac{\sqrt{\frac{2}{15}}}{49}{\eta_{\mu{Q}}}+\frac{4}{735}\eta_{QQ}\,,\\
W_2^{(1+2)}&=&W_2^{(1)}+\frac{3}{490}\eta_{\mu\mu}-\frac{\sqrt{\frac{3}{10}}}{70}\eta_{\mu{Q}}+\frac{1}{100}\eta_{QQ}\,,\nonumber\\
W_1^{(1+
2)}&=&W_1^{(1)}+\frac{1}{441}\eta_{\mu\mu}-\frac{1}{{49}\sqrt{30}}{\eta_{\mu{Q}}}+\frac{3}{490}\eta_{QQ}\,.\nonumber
\end{aligned}$$
Experimentally relevant quantities are the HFS energy intervals $\Delta{W}_F = W_{F+1} - W_F$. Explicitly,
$$\begin{aligned}
\label{eq:DeltaW12}
\Delta{W}_5^{(1+2)}&=&6 A+\frac{18 }{35}B+\frac{144 }{35}C+\frac{48 }{7}D-\frac{1}{98}\eta_{\mu\mu} -\frac{1}{98} \sqrt{\frac{5}{6}} \eta_{\mu{Q}}-\frac{5}{588}\eta_{{Q}{Q}} \,, \nonumber\\
\Delta{W}_4^{(1+2)}&=&5 A+\frac{1}{28}B-\frac{31 }{7}C-\frac{125 }{7}D-\frac{\eta_{\mu\mu}}{441} +\frac{1}{98} \sqrt{\frac{5}{6}} \eta_{\mu{Q}}+\frac{5}{588}\eta_{{Q}{Q}}\,,\nonumber\\
\Delta{W}_3^{(1+2)}&=&4 A-\frac{8 }{35}B-\frac{64 }{35}C+\frac{128 }{7}D+\frac{\eta_{\mu\mu}}{441}+\frac{1}{49} \sqrt{\frac{2}{15}} \eta_{\mu{Q}} -\frac{4}{735}\eta_{{Q}{Q}} \,,\\
\Delta{W}_2^{(1+2)}&=&3 A-\frac{9 }{28}B+\frac{99 }{35}C+\frac{33 }{7}D+\frac{1}{245} \eta_{\mu\mu}+(\frac{1}{70} \sqrt{\frac{3}{10}} -\frac{1}{49} \sqrt{\frac{2}{15}}) \eta_{\mu{Q}}-\frac{67}{14700}\eta_{{Q}{Q}}\,,\nonumber\\
\Delta{W}_1^{(1+2)}&=&2 A-\frac{2 }{7}B+\frac{176 }{35}C-\frac{176 }{7}D+\frac{17 }{4410}\eta_{\mu\mu} +(\frac{1}{49 \sqrt{30}}-\frac{1}{70} \sqrt{\frac{3}{10}}) \eta_{\mu{Q}}+\frac{19}{4900}\eta_{{Q}{Q}}\,.\nonumber
\end{aligned}$$
To determine the HFS constants $A,B,C$, and $D$ from experimental measurements of $\Delta{W}_F$, in Sec. \[Sec:electronic-reduced\] we compute the second-order corrections. Further, to find the values of nuclear octupole and hexadecapole moments from $C$ and $D$ we need electronic form-factors; these are also computed in Sec. \[Sec:electronic-reduced\]. We neglect contributions of one remaining HFS constant $E$ arising from the $2^5$-pole nuclear magnetic moment. This contribution is expected to be strongly suppressed compared to the contribution of the octupole moment (see Sec. \[sec:Discussion\]).
Calculations of electronic structure factors {#Sec:electronic-reduced}
============================================
Dirac-Hartree-Fock calculations {#Sec:electronic-reduced-ml}
-------------------------------
$\mathrm{Yb^{+}}$ ion in the first excited state contains thirteen $4f$ electrons and two $6s$ electrons. In this section, we start our calculation of the electronic wave functions by employing the frozen core Dirac-Hartree-Fock (DHF) approximation. In this approximation, we compute the DHF orbitals of the $\mathrm{YbIII}\,([\mathrm{Xe}]4f^{14})$ core. Then the valence (outside the $\mathrm{[Xe]}4f^{14}$ core) orbitals are computed using the DHF potential of the core. The many-body wave function $\psi_{J\,,M}$ can be approximated as $$\begin{aligned}
\label{eq:multi-electron-wf}
&&{\ensuremath{|\psi_{J,\,M}\rangle}}\simeq \frac{1}{2} (-1)^{7/2-M}\times\nonumber\\
&&(\sum\limits_{m}(-1)^{m-1/2}{a_{6s_{1/2,m}}^{\dagger}a_{6s_{1/2,-m}}^{\dagger})\,a_{4f_{7/2,-M}}}{\ensuremath{|{0_c}\rangle}}\,,\end{aligned}$$where $a_{6s_{1/2},m}^{\dagger}$ are creation operators with magnetic quantum number $m$ equal to either $-1/2$ or $1/2$, $a_{4f_{7/2,\,M}}$ is an annihilation operator for the $4f_{7/2}$ orbital, and ${\ensuremath{|0_c\rangle}}$ represents the $\mathrm{[Xe]}4f^{14}$ core. The phase factor $(-1)^{7/2-M}$ is generated after moving the hole operator from the core state [@Johnson2007]. The two $6s_{1/2}$ orbitals are coupled so that the $6s^2$ valence shell has zero value of angular momentum. Using Wick’s theorem, we write the matrix element (\[Eq:single-electron-operator\]) in the multi-electron state as an expectation value in the hole orbital (see Appendix [\[app:2nd-quantization\]]{} for derivation) $${\ensuremath{\langle \psi_{J,\,M}|}} T^e_{k,\,\mu} {\ensuremath{|\psi_{J,\,M}\rangle}} = -{\ensuremath{\langle \phi_{J,\,-M}|}}t_{k,\,\mu}^e {\ensuremath{|\phi_{J,\,-M}\rangle}},\label{eq:multi-electron-state}$$ where ${\ensuremath{|\phi_{J,\,-M}\rangle}}$ represents the $4f_{7/2}$ hole orbital with $J$ and $-M$ being the electron’s angular momentum and magnetic quantum number. The electronic tensors $T_{k,\,\mu}^e$ are given by Eq. (\[Eq:HF-general\]). In Appendix [\[app:2nd-quantization\]]{}, we show that the reduced matrix elements are related as $$\begin{aligned}
\label{eq:relation_multi_hole}
{\ensuremath{\langle \psi_{J}|}}|T^e_{k,\mu}|{\ensuremath{|\psi_{J}\rangle}}=(-1)^{k+1}{\ensuremath{\langle \phi_{J}|}}|t_k^e|{\ensuremath{|\phi_{J}\rangle}}\,,\end{aligned}$$ with reduced matrix elements specified in Appendix \[app:matrix-element\]. The transition from a multi-electron state to the single-electron hole orbital greatly simplifies our calculation since it only requires the one-electron $4f_{7/2}$ orbital, which can be easily obtained self-consistently with the DHF method. Our computed values of the first- and second-order hyperfine constants are listed in the first row of Table \[Table:HFS\].
[cccccccc]{} Method & $A$ & $B$& $C/{\Omega}$ & $D/{\Pi}$ & $\eta_{\mu\mu}$& $\eta_{\mu{Q}}$ &$\eta_{QQ}$\
\
DHF & $-239$ & $-5330$ & $4.53\times 10^{-4} $ & $2.00\times{10^{-4}}$ & $-1.50\times{10^{-2}}$ & $-0.112$ & $-0.209$\
DHF (GRASP) & $-252$ & $-5622$ & $4.83 \times 10^{-4}$ & $2.25\times{10^{-4}}$ & $-1.91\times{10^{-2}}$ &$-0.124$ &$-0.200$\
MCDHF & $-241$ & $-5061$ & $-6 \times 10^{-4}$ & $2.35\times{10^{-4}}$ & & &\
\
CI+MBPT, Ref. [@Dzuba2016a] & $-240$ & $-4762$ & & & &&\
[MCDHF]{}, Ref [@Petrasiunas2012] & $-304$ &$-3680$ & & & & &\
Electron correlation effects {#Sec:electron-correlation-eff}
----------------------------
We employ the multi-configuration Dirac-Hartree-Fock (MCDHF) method [@Grant2007; @FroeseFischer2016a] to capture the main electron correlations in the Yb$^+$ ion. In this approach, an atomic state wave-function (ASF) is represented as a linear combination of configuration state functions (CSFs) with the same parity, total angular momentum, and its component along the quantization axis. The CSFs are generated by single (S) and double (D) substitutions of orbitals occupied in the reference configurations with virtual orbitals. The reference configurations constitute the dominant CSFs of the ASF concerned. The MCDHF calculation starts from the optimization on occupied orbitals in the reference configurations. By contrast to Sec \[Sec:electronic-reduced-ml\], all of these orbitals are generated in the self-consistent field procedure. Virtual orbitals are augmented layer by layer in order to monitor the convergence of level energies and other atomic properties. Each layer includes orbitals with different angular symmetries. In addition, only the virtual orbitals in the latest added layer are variable. The details of computational strategies can be found in Ref. [@Bieron2009; @Li2012].
In our calculations, [ we adopt the extended optimal level (EOL) scheme to optimize the two states of the \[Xe\]$4f^{13}6s^2$ configuration simultaneously.]{} The electron correlations in the $4f$ and $6s$ valence subshells and the correlations between electrons in the valence and $n=3,4$ core subshells were accounted for by CSFs generated by the SD replacement of the $n \ge 3$ occupied orbitals in the reference configuration with the virtual orbitals. The double replacements were restricted to only a single electron of the core subshells being promoted into the virtual orbitals at a time. The final set of virtual orbitals is composed of five orbitals per each of the $s, p, d, f, g, h, i$ angular momenta. The magnetic octupole and electric hexadecapole hyperfine interaction constants were calculated by an extended version [@JGLiunpublished] of the HFS92 code [@HFS92] based on the GRASP package [@FroeseFischer2018b] Our results, labelled as MCDHF, are presented in Table \[Table:HFS\].
Evaluation of theoretical uncertainties {#Sec:TheoreticalErrors}
---------------------------------------
We start with a comparison of our computed values for $A$ and $B$ HFS constants with the previously published results and then assess our theoretical accuracy.
Comparing our computed values (see Table \[Table:HFS\]) with theoretical values by @Dzuba2016a, we observe that our $A$ values match, while there is a roughly $10\%$ discrepancy in values of $B$. Itano (cited in Ref. [@Petrasiunas2012]) has previously computed the $A$ and $B$ constants for the $4f^{13}6s^2$ ($J=7/2$) state in $^{171}\mathrm{Yb}^+$ and $^{173}\mathrm{Yb}^+$. Itano has also used the MCDHF method, but his results are markedly different from ours. Since there are no details of calculations given in Ref. [@Petrasiunas2012], it is difficult to assess the reasons for this difference. We, however, point out that our MCDHF results are in a better agreement with experimental values. For example, the deviation is about 20% between his result and the experimental value $A(^{171}\textrm{Yb}) = 905$ MHz [@Taylor1999]. Multiplying our $A$ constant for $^{173}$Yb by the ratio $\mu(^{171}\textrm{Yb})I(^{173}\textrm{Yb})/\mu(^{173}\textrm{Yb})I(^{171}\textrm{Yb})$, we obtained $A=882$ MHz for $^{171}$Yb, which differs from the measurement [@Taylor1999] by only 3%.
Based on these comparisons we conservatively estimate the uncertainty of our MCDHF calculations to be $\sim 10\%$ for the magnetic dipole and electric quadrupole hyperfine interaction constants. This estimate is also consistent with that of Ref. [@Dzuba2016a], where they claimed a similar 10% theoretical uncertainty for these two constants using a different computational method. We assign a 10% theoretical uncertainty to the $D$ constant due to its stable convergence trend with the increasing size of the virtual orbital set. However, it is difficult to evaluate the theoretical uncertainty for the $C/\Omega$ constant since it strongly depends on the computational model, as discussed below. We are, however, confident in the sign and order of magnitude of this octupole constant.
The magnetic octupole HFS constant has proven to be sensitive to the electron correlations, as they flip the sign of the DHF result. [We systematically investigated the dependence of the calculated $C/\Omega$ values on the size of computational model space, see Table \[CV\_effects\]. For example, in this table, the results in the “no opened subshells" row demonstrate the effect of correlation between electrons in the valence subshells. Because the octupole coupling operator has high multiplicity (tensor of rank 3) and we are interested in the properties of the $l=3$ $f$-state hole, we include up to $l=6$ virtual orbitals in each layer. The results including the valence-valence correlation show a good convergence pattern (first row of Table \[CV\_effects\]). However, the convergence pattern worsens when we start including core-valence correlations by opening core subshells (columns of Table \[CV\_effects\]). While the results show some degree of convergence, results from an even larger model space would have been more conclusive. Unfortunately, the largest computation model that we employed already pushes the limits of computational power at our disposal. Considering the convergence trends of Table \[CV\_effects\], we believe that the sign and the order of magnitude of the computed $C/\Omega$ constant would not change with increasing model space. We carried out additional convergence tests that support this conclusion. For example, trends in Table \[CV\_effects\] indicate that opening the $3p$ and $4p$ subshells substantially modify the result; so it is plausible that opening the subshell of the same angular momentum, $2p$ subshell, might modify the result further. To test this hypothesis, we opened the $2p$ subshell for a small model space and found this effect to be negligible. We take the result obtained with the largest model space as our final value, $C/\Omega = -6 \times 10^{-4} \, \mathrm{MHz/(b\times{\mu_N}})$. ]{}
[cdddddd]{} & & & & &\
no opened subshells & 0.578 & 0.630 & 0.649 & 0.652 & 0.652\
$~5s$ & 1.692 & 2.857 & 2.625 & 2.708 & 2.646\
$+5p$ & 2.453 & 3.633 & 3.360 & 3.439 & 3.366\
$+4d$ & 1.986 & 3.072 & 2.778 & 2.822 & 2.749\
$+4p$ & 2.014 & 2.823 & 2.148 & 1.904 & 1.685\
$+4s$ & 2.239 & 3.126 & 2.380 & 2.058 & 1.792\
$+3d$ & 2.131 & 2.956 & 2.173 & 1.822 & 1.541\
$+3p$ & 2.007 & 2.208 & 0.941 & 0.276 & -0.198\
$+3s$ & 1.932 & 2.037 & 0.688 & -0.040 & **[-0]{}.**[558]{}\
****
As to the second-order corrections $\eta_X$, these are proportional to various products of electronic matrix elements of magnetic-dipole and electric-quadrupole hyperfine interactions. Based on our accuracy estimates for $A$ and $B$, we conservatively assign $\sim 10\%$ theoretical uncertainty to such matrix elements. Thereby, we expect a $\sim 10\%$ theoretical uncertainty in the second-order HFS constants. In addition, the second-corrections contain summation over intermediate states; in our calculations we truncated the entire sum to a single contribution from the lowest-energy $F_{5/2}$ state. We examined contributions from other 12 lowest-energy intermediate states and found that $\eta_{\mu\mu}$, $\eta_{\mu{Q}}$, $\eta_{QQ}$ are modified by less than 4%, 8%, and 20%, respectively. Thus the overall theoretical uncertainty in second-order corrections is in the order of 10%.
Projected Experimental Accuracy {#sec:Experimental}
===============================
Experimental Procedure
----------------------
The measurement of the hyperfine intervals $\Delta W_i$ of ${}^{173}\mathrm{Yb}^+({}^2\!\!\;F_{7/2}^o)$ can be accomplished via microwave Ramsey spectroscopy on a single trapped ion. A pure state can be prepared by beginning with optical pumping on the narrow-band (E2) ${}^2\!\:D_{5/2} \!\leftarrow \! {}^2\!\:S_{1/2} $ transition at $411 \mbox{ nm}$, which will spontaneously decay mainly to ${}^2\!F_{7/2}^o$ via the allowed E1 transition at $\lambda \! = \! 3.4\mbox{ }\mu\mbox{m}$. By restricting the E2 transition to drive only ${}^2\!D_{5/2}(F\!=\!0) \!\leftarrow \! {}^2\!S_{1/2}(F\!=\!2) $, the $F\!=\!1$ hyperfine level in ${}^2\!F_{7/2}^o$ will be populated. Following this optical pumping step, resonant microwaves can be used to drive $2\!\leftarrow \! 1$ at $\approx 1\mbox{ GHz}$, followed by de-shelving of the remaining $F\!=\!1$ population in ${}^2\!F_{7/2}^o$ back to ${}^2\!S_{1/2}$ via the E2 transition ${}^1\![3/2]_{3/2}^o \! \leftarrow \! {}^2\!F_{7/2}^o$ at $\lambda\!=\!760 \mbox{ nm}$. An ion in the ground state can be distinguished from a ${}^2\!F_{7/2}^o$ ion via the appearance or lack of laser-induced fluorescence on ${}^2\!P_{1/2}^o \leftrightarrow \!{}^2\!S_{1/2}$. By observing how the microwave resonance frequency depends upon the magnetic field in the trap, the $M_F\!=\!0 \! \leftrightarrow \! 0$ transition can be isolated, permitting preparation of the ${}^2\!F_{7/2}^o(F\! = \!2, M_F\! = \!0)$ single quantum state. From there, stepwise microwave excitation through the hyperfine structure can be used to complete the spectroscopy. In all cases, read-out is accomplished by observing whether the $760 \mbox{ nm}$ transition de-shelved the ion back to the ground state manifold.
Precision
---------
Since the lifetimes of the states in ${}^{173}\mathrm{Yb}^+(^2\!F_{7/2}^o)$ are all expected to on the order of 1 day or longer [@Dzuba2016a], the achievable precision of these measurements is likely to be limited by practical considerations (as opposed to $T_1$). In particular, since a small magnetic field will be used to isolate the $M_F\!=\!0 \! \leftrightarrow \!0$ transitions, second-order Zeeman shifts of the clock states can lead to decoherence. Based on the experimenally determined coherence time of the Zeeman-sensitive hyperfine transitions in the ground state of ${}^{171}\mathrm{Yb}^+$ that we have achieved, we anticipate that $1\mbox{ Hz}$ precision can be obtained by keeping the effective magnetic sensitivity of the “clock transitions” ($M_F\!=\!0 \! \leftrightarrow \!0$) in $\mathrm{Yb}^+({}^2F_{7/2}^o)$ below $10^{-3} \mu_\mathrm{B}$. The offset field required to accomplish this will depend upon how close the zero-field hyperfine states with $\Delta F \! = \! \pm 1$ are to degeneracy. Assuming there is a pair with significantly smaller zero-field splitting $\Delta W$ than the rest, the effective magnetic moment associated with an offset field $B_\mathrm{o}$ scales as $\mu_\mathrm{eff} \sim B_\mathrm{o}\mu_\mathrm{B}^2/\Delta W$. Barring any “accidental” near-degeneracies ($\Delta W\! < \!10\mbox{ MHz}$), a precision of $1 \mbox{ Hz}$ should be achievable with our current level of magnetic field control.
Accuracy
--------
The potential systematic effects that are expected for this system can be divided into those that will be common to measurements of ground state splittings, and those that are unique to the ${}^2\!\!\;F_{7/2}^o$ state. The former group includes the nonlinear Zeeman shifts from static magnetic fields, differential Stark shifts from the trap fields, blackbody and time-dilation shifts, off-resonant shifts of the levels being measured due to the microwave probe field, and hyperfine-induced third-order corrections [@Safronova2010b]. Since ground-state splittings have been measured below the target precision of $1\mbox{ Hz}$ for many years [@Fisk1997; @Werth1987], the techniques to avoid effects such as these have already been demonstrated and are expected to be sufficient for reaching the comparatively modest target accuracy of $1\mbox{ Hz}$. In particular, taking the expected zero-field splittings from the coefficients in Table \[Table:HFS\] suggests that the largest second- and fourth-order Zeeman shifts will be on the $F =4 \!\leftrightarrow \!3$ transition, which will contribute a systematic shift of less than $1\mbox{ Hz}$ at $B_\mathrm{o} =5\mbox{ mG}$.
For systematics that are unique to the ${}^2\!\!\;F_{7/2}^o$ state, the largest is anticipated to be the energy shifts from the electronic electric quadrupole interacting with static electric field gradients in the trap. The diagonal contributions to the shifts are be given by $$\begin{split}
E^{(e.q.)}_{F,M_F}= -e \sum_\mu T_{2,\mu}(\nabla \mathbf{E}) \,\langle \gamma{IJF}|T_{2,-\mu}({{\mbox{\boldmath$ {\Theta} $}}})|\gamma IJF\rangle\\
= - e \,T_{2,0}(\nabla \mathbf{E})\,\frac{2(3M_F^2 - F(F+1))}{\sqrt{(2F+3)(2F+2)(2F+1)2F(2F-1)}}\\
\times (-1)^{I+J+F} (2F+1)
\frac{\begin{Bmatrix}
J & F & I \\
F & J & 2
\end{Bmatrix}}{
{\ensuremath{\begin{pmatrix}J&2&J\\#4&0&J\end{pmatrix}}}}\Theta(\gamma J) ,
\end{split}\label{Eq:DiagonalQuad}$$ where the quadrupole moment has been measured to be $\Theta({}^2\!\!\;F_{7/2}^o) = -0.041(5) a_\mathrm{o}^2$ [@Huntemann2012]. These contribute sub-Hz shifts for an electric field gradient of $1\mbox{ kV}/\mbox{cm}^2$, which is significantly larger than the gradient in our current trap. There are also potentially, off-resonant shifts due to the Paul trap’s radiofrequency drive if pairs of states happen to be split by a frequency near the rf drive, in which case the rf drive frequency may need to be changed. We are therefore not aware of any barriers to achieving a precision of $1 \, \mathrm{ Hz}$ for this measurement.
Discussion {#sec:Discussion}
==========
Equations (\[eq:DeltaW12\]) provide the relationship between the 5 quantities that will be measured experimentally (the $\Delta W^{(1+2)}_F$) and the 7 parameters to be determined, $A$-$D$ and the $\eta_{mn}$. However, since all of the terms included in our model are tensors of rank $k\!\le\!4$, there is a degeneracy in Eqs. (\[eq:DeltaW12\]) and a proper linear combination of any four of the measurements can be used to predict the fifth. While this reduces the number of experimentally determined quantities to $k_\mathrm{max}\!=\!4$, it will provide a test of the model presented above and way to detect and reject systematic effects in the experiment.
Within the 3 second order terms ($\eta_{mn}$), since the energy difference between the ${}^2\!F_{J^\prime\!=\!5/2}^o$ and ${}^2\!F_{J\!=\!7/2}^o$ states is known, if we assume that these are the only terms that contribute, they contain only 2 unknowns: $\mu \langle \gamma J || T^e_1 || \gamma J^\prime \rangle$ and $Q \langle \gamma J || T^e_2 || \gamma J^\prime \rangle$. Further, the coefficient $A$ can be determined from existing experimental data [@Fisk1997; @Werth1987; @Taylor1999], $$\label{eq:Acoeffs}
A^{(173)}_{^2F_{7/2}^o} = \frac{A^{(173)}_{^2S_{1/2}}\,\, A^{(171)}_{^2F_{7/2}^o}}{A^{(171)}_{^2S_{1/2}}} = -250 \mbox{ MHz}.$$ Here, we have extracted $A_{{}^2F_{7/2}^o}^{(171)}$ from the measured energy splitting $\Delta W_3^{(171)}=3.620\mbox{ GHz}$ [@Taylor1999] via $$A_{{}^2F_{7/2}}^{(171)} = \frac{\Delta W_3^{(171)}}{4} + \frac{1}{144}\left( \frac{\mu^{(171)}}{\mu^{(173)}}\right)^2 \eta_{\mu\mu} \approx \frac{\Delta W_3^{(171)}}{4}$$ and therefore neglected the contribution (tens of Hz) of the second order correction to the hyperfine splitting of ${}^{171}\mathrm{Yb}^+({}^2\!F_{7/2}^o)$ since it is not expected to contribute at the current level of experimental precision. This term should of course be included in a full treatment when experimental precision reaches the $100\mbox{ Hz}$ level, and adding it does not increase the number of unknowns in the system of equations (\[eq:DeltaW12\]). The two ground state $A$ coefficients in (\[eq:Acoeffs\]) are known to sub-Hz precision [@Fisk1997; @Werth1987], and the limiting measurement is $\Delta W_3^{(171)}$, the ${}^2F_{7/2}^o$ HFS splitting in ${}^{171}\mathrm{Yb}^+$ [@Taylor1999]. Using essentially the same procedure as described below, this splitting in ${}^{171}\mathrm{Yb}^+$ can be measured to the same precision (if not better) than the $\Delta W_i$ in ${}^{173}\mathrm{Yb}^+$. This leaves Eqs. (\[eq:DeltaW12\]) with 5 unknowns ($B$, $C$, $D$, $\mu \langle \gamma J || T^e_1 || \gamma J^\prime \rangle$, and $Q \langle \gamma J || T^e_2 || \gamma J^\prime \rangle$).
Because the experimental uncertainty can reach $\sim 1 \, \mathrm{ Hz}$, we expect that the dominant error in extracting first-order HFS constants is due to theoretical uncertainty in the second-order corrections $\eta_{X}$ (see Sec. \[Sec:TheoreticalErrors\]). One of the possibilities is to determine the second-order corrections directly from the experimental data, but the system of effectively 4 equations and 5 unknowns here will not allow unambiguous extraction of all 5 unknown parameters.
Instead, we solve Eqs. (\[eq:DeltaW12\]) for the first four HFS splittings $\Delta W_{F}$ for the HFS constants, $A$, $B$, $C$, and $D$. Each of the resulting equations contains a contribution from the second-order corrections. In particular, the induced variation in $D$ is $\delta D \approx 3.4 \times 10^{-4} \, \delta \eta_{QQ}$. As discussed in Sec. \[Sec:HFI-YbII\], the fiducial value of the hexadecapole moment $\Pi \sim 9 \, \mathrm{b}^2$, leading, in combination with results in Table \[Table:HFS\], to the expected value of $D\approx 2 \, \mathrm{kHz}$. Since $\eta_{QQ}\approx- 200 \, \mathrm{kHz}$, even a 100% error in $\eta_{QQ}$ would lead to only 3% error in the extracted value of $D$. Estimating the induced uncertainty in $C$ is more involved: $\delta C = -1.6 \times 10^{-3} \, \delta\eta_{\mu Q}
+ 8.9 \times 10^{-4} \, \delta\eta_{QQ}$. If we assume a 10% error in both $\eta_{\mu Q}$ and $\eta_{QQ}$ per Sec. \[Sec:TheoreticalErrors\], then the induced uncertainty in $C$ is $30\, \mathrm{Hz}$. Meanwhile, the expected values of $C$ depends substantially on the assumed value of the octupole moment $\Omega$. If we take $\Omega$ from the spectroscopic determination [@Singh] in neutral Yb, the resulting value of $C \approx 21 \, \mathrm{kHz}$; the nuclear shell model value of $\Omega$ (see Table \[table:NuclearMoments\]) yields $C \approx 90 \, \mathrm{Hz}$, and the more sophisticated nuclear model [@Williams1962] reduces $C$ to $2 \, \mathrm{Hz}$. It is clear that for the latter case the uncertainties in the second order correction would mask the contribution of $C$ to the hyperfine splittings and only an upper limit on $\Omega$ can be placed. In such a scenario, one could still determine $D$ and extract the hexadecapole moment, as the value of $D$ is several orders of magnitude larger than $C$.
Given that the well-controlled electronic structure of the ${}^2\!\!\;F_{7/2}^o$ state of $\mathrm{Yb}^+$ should allow for the extraction of measurable, high-order spectroscopic multipole moments, it is possible that even finer detail may be possible. While nuclear theory suggests that the magnetic multipole moments may be difficult to discern, the electric moments from deformed cores appear straightforward to measure. In particular, the radioactive ${}^{169}\mathrm{Yb}$ nuclide has spin $I\!=\!7/2$ and a half-life of $\approx \!32$ days, suggesting that precision spectroscopy of the ${}^2\!\!\;F_{7/2}^o$ state of ${}^{169}\mathrm{Yb}^+$ may reveal signatures of its electric 64-pole moment. The calculation of more 2nd-order correction terms as well as 3rd-order corrections would be required to extract this moment from the data, but we see no fundamental barriers to future studies along these lines.
We would like to thank V. Dzuba for discussions. This work was supported in part by the U.S. National Science Foundation (Award Numbers 1912555 and 1912465). JGL is grateful to the University of Nevada, Reno for hospitality and acknowledged the financial support by the National Natural Science Foundation of China (Grant No. 11874090).
Relation between multi-electron and single-electron matrix elements {#app:2nd-quantization}
===================================================================
In this Appendix, we prove Eqs. (\[eq:multi-electron-state\], \[eq:relation\_multi\_hole\]).
The operator $T_{k,\mu}^e$ in the second quantized form reads [@lindgren1986atomic] $$\begin{aligned}
\label{eq:T_in_2nd_quantization}
T_{k,\mu}^e=\sum\limits_{i,j}{\ensuremath{:\! a_i^{\dagger}a_j\!:}}{\ensuremath{\langle i|}}t_{k,\mu}^e{\ensuremath{|j\rangle}},\,\end{aligned}$$ where $i$ and $j$ represent either core or virtual orbitals, ${\ensuremath{\langle i|}}t_{k,\mu}^e{\ensuremath{|j\rangle}}$ is the matrix element, and ${\ensuremath{:\! a_i^{\dagger}a_j\!:}}$ are products of creation and annihilation operators in the normal form. We would like to evaluate the expectation value of the operator in Eq. (\[eq:T\_in\_2nd\_quantization\]) in the many-body state [$|\psi_{J,\,M}\rangle$]{}, Eq. (\[eq:multi-electron-wf\]). The intermediate result for the expectation value can be obtained using the Wick’s theorem [@lindgren1986atomic], $$\begin{aligned}
\label{eq:kronecker_delta}
\langle0_c|a_{h'}^{\dagger}a_{v'}a_{w'}{\ensuremath{:\! a_i^{\dagger}a_j\!:}}a_w^{\dagger}a_v^{\dagger}a_h|0_c\rangle &=&\nonumber\\
-\delta_{ih}\delta_{h'j}(\delta_{vv'}\delta_{ww'}&-&\delta_{v'w}\delta_{w'v})\,\nonumber\\
+\delta_{jw}\delta_{h'h}(\delta_{vv'}\delta_{iw'}&-&\delta_{iv'}\delta_{w'v})\nonumber\\
-\delta_{jv}\delta_{hh'}(\delta_{v'w}\delta_{w'i}&-&\delta_{iv'}\delta_{ww'})\,,\end{aligned}$$ where $h(h')$ stands for the $4f$ hole orbital and $v(v')$ and $w(w')$ represent the $6s$ orbitals.
Then we immediately obtain $$\label{eq:appendix_multi_to_hole_state}
{\ensuremath{\langle \psi_{J,\,M}|}} T^e_{k,\mu} {\ensuremath{|\psi_{J,\,M}\rangle}} = -{\ensuremath{\langle h|}}t_{k,\mu}^e {\ensuremath{|h\rangle}}\,,$$ where ${\ensuremath{|\psi_{J,M}\rangle}}$ is the multi-electron state of $^{173}\mathrm{Yb}^{+}$, Eq. [(\[eq:multi-electron-wf\])]{}. The reason that the $6s$ orbitals do not contribute to Eq. (\[eq:appendix\_multi\_to\_hole\_state\]) is that the operator is non-scalar and the $6s^2$ shell has zero total angular momentum by construction of the multi-electron state [(\[eq:multi-electron-wf\])]{}.
In general, Eq. (\[eq:appendix\_multi\_to\_hole\_state\]) works for any non-scalar one-body operator. If we replace $T_{k,\mu}^{e}$ and $t_{k,\mu}^e$ with the $z$ components of the angular momentum operators $J_z$ and $j_z$ respectively in Eq. (\[eq:appendix\_multi\_to\_hole\_state\]), we obtain the magnetic quantum number of the hole state, $m_h$ equal to $-M$.
Then, we rewrite Eq. (\[eq:appendix\_multi\_to\_hole\_state\]) as follows, $$\label{eq:app_final_multi_single_mel}
{\ensuremath{\langle \psi_{J,\,M}|}} T^e_{k,\mu} {\ensuremath{|\psi_{J,\,M}\rangle}} = -{\ensuremath{\langle \phi_{J,\,-M}|}}t_{k,\mu}^e {\ensuremath{|\phi_{J,\,-M}\rangle}}\,,$$ where $\phi_{J,-M}$ is the orbital of the hole-state electron. This proves Eq. (\[eq:multi-electron-state\]) of the main text. Applying the Wigner-Eckart theorem and setting $\mu=0$ on each side of Eq. (\[eq:app\_final\_multi\_single\_mel\]), we obtain, $$\begin{aligned}
\label{apd:apd1}
{\ensuremath{\langle \psi_{J,\,M}|}}T^e_{k,0
}{\ensuremath{|\psi_{J,\,M}\rangle}}=&& \nonumber \\
(-1)^{J-M}\begin{pmatrix}
J & k & J \\
-M & 0 & M
\end{pmatrix} &\langle \psi_{J}||T^e_{k}||\psi_{J}\rangle& \,,
\end{aligned}$$ $$\begin{aligned}
\label{apd:apd2}
-{\ensuremath{\langle \phi_{J,\,-M}|}}t^e_{k,0}{\ensuremath{|\phi_{J,\,-M}\rangle}}
= &&
\nonumber\\ -(-1)^{J+M}\begin{pmatrix}
J & k &J \\
M & 0 & -M
\end{pmatrix}& \langle \phi_{J}||t^e_{k}(i)||\phi_{J}\rangle&\,.
\end{aligned}$$ Since $\begin{pmatrix}
J & k & J \\
-M & 0 & M
\end{pmatrix} =(-1)^{2J+k}\begin{pmatrix}
J & k & J \\
M & 0 & -M
\end{pmatrix}$, the reduced matrix elements satisfy the following identity, $$\begin{aligned}
\label{apd:apd3}
{\ensuremath{\langle \psi_{J}|}}|T^e_k|{\ensuremath{|\psi_{J}\rangle}} &=&(-1)^{1+2M+2J+k} {\ensuremath{\langle \phi_{J}|}}|t^e_{k}|{\ensuremath{|\phi_{J}\rangle}}\,\nonumber\\ &=&(-1)^{k+1}{\ensuremath{\langle \phi_{J}|}}|t^e_{k}|{\ensuremath{|\phi_{J}\rangle}}.
\end{aligned}$$ Eq. ([\[apd:apd3\]]{}) suggests that when evaluating the reduced matrix elements of even-$k$ operators with multi-electron states, one needs to add an extra negative sign to the single-electron reduced matrix elements. The sign of odd-$k$ reduced matrix elements is unaffected. This proves Eq.(\[eq:relation\_multi\_hole\]) of the main text.
Now we generalize these identities to the off-diagonal reduced matrix elements entering the second-order corrections. As discussed in Sec. [\[Sec:HFI-YbII\]]{}, the dominant intermediate state is the $4f^{13}5s^2\,^2\!F_{5/2}$ state denoted as ${\ensuremath{|\psi_{J'M'}\rangle}}$. The many-body state ${\ensuremath{|\psi_{J'M'}\rangle}}$ has a similar form as Eq. (\[eq:multi-electron-wf\]) but differs in the phase factor, $(-1)^{5/2-M'}$ and the annihilation operator $a_{4f_{5/2,-M'}}$. It can be shown that the relation in Eq. (\[apd:apd3\]) still holds for the reduced matrix element, $$\begin{aligned}
\label{apd:apd4}
{\ensuremath{\langle \psi_{J}|}}|T^e_k|{\ensuremath{|\psi_{J'}\rangle}} =(-1)^{k+1}{\ensuremath{\langle \phi_{J}|}}|t^e_{k}|{\ensuremath{|\phi_{J'}\rangle}}.\end{aligned}$$
Reduced matrix elements of hyperfine interaction {#app:matrix-element}
================================================
Formally, the one-electron wave function is represented by Dirac bi-spinor $${\ensuremath{|nj\kappa m\rangle}}=\begin{pmatrix}
iP_{n\kappa}(r)\Omega_{\kappa,m}(\hat{r}) \\
Q_{n\kappa}(r)\Omega_{-\kappa,m}(\hat{r})
\end{pmatrix},$$ where $P$ and $Q$ are the large and small components of one-electron wave function and $\kappa$ is the relativistic quantum number ($\kappa=\mp{j+\frac{1}{2}}$ for $j=l\pm{\frac{1}{2}}$). The reduced matrix elements of the electronic part of hyperfine interaction are explicitly [@Johnson2007]
$$\begin{aligned}
\langle n'\kappa'|| t_k^e|| n\kappa \rangle&=&
\begin{cases}
-{\ensuremath{\langle \kappa^{'}|}}|C_{k}|{\ensuremath{|\kappa\rangle}} \int_{0}^{\infty}\frac{dr}{r^{k+1}}(P_{n'\kappa'}P_{n,\kappa}+Q_{n'\kappa'}Q_{n,\kappa}), & \text{odd k}\,, \\
{\ensuremath{\langle \kappa^{'}|}}|C_{k}|{\ensuremath{|-\kappa\rangle}}\frac{\kappa'+\kappa}{k}\int_{0}^{\infty}\frac{dr}{r^{k+1}}(P_{n'\kappa'}Q_{n,\kappa}+Q_{n'\kappa'}P_{n,\kappa}), & \text{even k},
\end{cases}\label{eq:reduced-matrix-element}
\end{aligned}$$
where we suppressed $j$ for brevity. The odd and even $k$ sub-cases correspond to electric and magnetic interactions respectively.
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| 0 |
---
abstract: 'We use a recently found parametrization of the solutions of the inverse Frobenius-Perron problem within the class of complete unimodal maps to develop a Monte-Carlo approach for the construction of one-dimensional chaotic dynamical laws with given statistical properties, i.e. invariant density and autocorrelation function. A variety of different examples are presented to demonstrate the power of our method.'
author:
- |
F.K. Diakonos\
Department of Physics, University of Athens\
GR-15771, Athens, Greece\
D.Pingel and P.Schmelcher\
Theoretische Chemie, Physikalisch–Chemisches Institut\
Universität Heidelberg, INF 229, D-69120 Heidelberg\
Federal Republic of Germany
title: '**[A Stochastic Approach to the Construction of One-Dimensional Chaotic Maps with Prescribed Statistical Properties]{}**'
---
Introduction
============
Chaotic dynamical systems show a rich diversity of possible behaviour with respect to their statistical properties. In recent years much work has been devoted to the understanding of how these statistical properties emerge from the dynamics. The most popular area for such investigations are unimodal 1-d maps [@Col80; @DynSta; @Gros87; @Cso93]. They became widespread due to two main advantages: their dynamics can be calculated efficiently enough to make extensive numerical investigations and the variety of statistical properties within this class of dynamical laws is very large. Certain features of special systems can even be investigated analytically. The interplay between the dynamical and statistical behaviour becomes more transparent if we consider the inverse problem, i.e. given the statistical behaviour of a system how could we extract relevant information on the possible dynamics. This subject has recently been addressed and discussed in the literature [@InvPro; @Dia96; @PiSchmeDia; @BaraDa; @Bollt; @Gora] and from a communications point of view in refs. [@Abel; @Broom]. The purpose is then to construct an one-dimensional fully chaotic (and ergodic) dynamical law for a given invariant density or/and time autocorrelation function. Given the latter we can on the one hand calculate all expectation values, i.e. statistical averages, of measurable observables depending exclusively on the single dynamical variable. On the other hand the given autocorrelation function provides us with valuable information on the dynamics of the trajectories of the system. It represents an important quantity strongly related to the physics described by the dynamical system. The construction of a dynamical system with prescribed statistical properties can have the background of either some experimentally given data for which a dynamical system should be modelled or it can be justified by the fact that a definite behaviour of the correlation function is required in the context of the control of a physical system [@Bollt; @Gora].
Restricting ourselves to the first part of the problem, i.e. the construction of a map with a given invariant density (the so called inverse Frobenius-Perron problem (IFPP)) we have recently found a general and for practical purposes very helpful representation of the solution of the IFPP within the class of smooth complete and unimodal maps [@PiSchmeDia]. However the combined problem (both the invariant density and time autocorrelation function are given) is much more complicated. Some progress in this direction can be made if one uses the class of piecewise linear Markov maps as a basis for this search. Constructing such a map with a given autocorrelation function one can then obtain the proper invariant density performing a suitable conjugation (coordinate) transformation [@BaraDa]. A substantial disadvantage of this method is that the changes of the autocorrelation function in the course of the conjugation transformation are not fully under control. Another rather anaesthetic aspect is the fact that in many cases the solution found to the inverse problem does not fulfill certain smoothness criteria.
The purpose of this letter is to present a new approach to the inverse problem including the autocorrelation function. It is based on the general representation of solutions of the IFPP found in [@PiSchmeDia] for complete smooth unimodal maps. We suitably parametrize the key function $h_f$ occurring in this approach and develop a stochastic (Monte-Carlo) algorithm to determine the optimal solution such that the $\chi^2$ deviation of the resulting autocorrelation from a desired function is minimized. The paper is organized as follows: In section 2 we briefly review the representation of our solution to the IFPP. In section 3 we present the Monte-Carlo algorithm used to perform the $\chi^2$ minimization with respect to the time autocorrelation function. Finally in section 4 we provide several applications in the framework of our Monte-Carlo scheme and discuss the convergence properties of our approach.
The general solution of the IFPP for smooth unimodal 1-d maps
=============================================================
In a recent paper [@PiSchmeDia] we investigated the problem of designing dynamical systems which possess an arbitrary but fixed invariant density. The solution to this inverse problem is of great interest for the numerical simulation of real physical systems as well as for the understanding of the relationship between the functional form of the map and the statistical features of the corresponding dynamics. We were able to derive a general representation of all ergodic and chaotic complete unimodal maps with a given invariant density. This corresponds to the general solution of the so called inverse Frobenius-Perron problem [@DynSta; @InvPro; @BaraDa; @Gross; @Ghik; @Pala; @Hunt] within this class of maps. We review here the basic aspects of our approach.
As a starting-equation for the construction of the map we use the Frobenius-Perron equation: $$\rho(y)\,|dy|=\sum\limits_{x_i=f^{-1}(y)}\rho(x_i)|dx_i|
\label{froperequ}$$ where the summation runs over all preimages of $y$ (for unimodal maps $i=L(left),R(right)$). For any given complete unimodal $f(x)$ the right preimage of $y$ is determined if the left preimage is given and vice versa. The essential feature of our approach is the following: A prescribed relation between the two preimages reduces the number of independent differentials on the rhs of eq.(\[froperequ\]) and allows to integrate the Frobenius-Perron equation. Such a relation is given by the function $h_f(x)$ which maps the position of the left preimage onto the position of the right one:
$$\begin{aligned}
h_f&:&[0,x_{\max}]\longrightarrow[x_{\max},1] \nonumber \\
x_R&=&h_f(x_L)\;\;\;\;\;\;\mbox{with}\;\;f(x_L)=f(x_R)
\label{hfprei}\end{aligned}$$
where $x_{\max}$ is the position of the maximum of the map. $h_f(x)$ is a monotonously decreasing function on the defining interval and it is differentiable with the exception of a finite number of points. It obeys the equations:
$$\begin{aligned}
h^\prime_f(x)<0 &&\;\;\;\;x\in[0,x_{\max}] \nonumber \\
h_f(0)=1 &&\;\;\;\;h_f(x_{\max})=x_{\max}
\label{hfprop}\end{aligned}$$
In terms of $h_f$ we obtain the general solution to IFPP within the class of unimodal maps as: $$f(x)=\mu^{-1}\left(1-|\mu(x)-\mu(H_f(x))|\right)
\label{funx}$$ where $H_f(x)$ is: $$H_f(x)=\left\{ \begin{array}{l@{\quad;\quad}l}
h_f(x)&0 \leq x<x_{\max}\\
h^{-1}_f(x)&x_{\max}\leq x \leq 1\\
\end{array}\right.
\label{genhf}$$ and $\mu(x)=\displaystyle{\int_0^x} \rho(y) dy$ is the corresponding invariant measure. All unimodal maps with prescribed invariant density $\rho(x)$ are given by (\[funx\]), $h_f(x)$ taking on all possible functional forms obeying (\[hfprei\]) and (\[hfprop\]) (for more details on the above derivation as well as its relation to the standard conjugation procedure we refer the reader to [@PiSchmeDia]).
The above shows that fixing the invariant measure is a relatively weak constraint in the framework of the inverse problem and that there is still a considerable freedom to model the mapping. In the next section we will use this freedom and suitably parametrize $h_f(x)$ such that, by tuning appropriately the corresponding parameters, we get a map with an autocorrelation function possessing a minimum deviation (to be defined below) from a desired correlation function.
Stochastic optimization as a tool for the construction of a map
===============================================================
Before presenting our search method for the optimal map in the sense of some desired statistical properties let us specify in more detail what we want to achieve and what our input for the problem at hand is. Consider a given invariant density $\rho(x)$ (corresponding to a finite measure) which arises in the asymptotic limit ($t \to \infty$) from the dynamics of some unknown $1-d$ fully chaotic single humped map. Consider also as given the first $m$ values $\{C(1), C(2),..,C(m)\}$ of the time autocorrelation function $C(n)$ (note that $C(0)$ is determined entirely through $\rho(x)$). We are seeking a map $f(x)$ which possesses the above statistical properties ($\rho(x), C(n) (n=0,...,m)$). We focus in the following on the representation (\[funx\]) for all admissible maps where the auxiliary functions $h_f(x)$ fulfill the requirements (\[hfprei\],\[hfprop\]). Using this expression for $f(x)$ it is guaranteed that the map we are looking for possesses the invariant density $\rho(x)$ (provided of course that $\mu(x)=\int_0^x \rho(t)~dt$ in (\[funx\]) exists). All our freedom in modelling the map $f(x)$ is now contained in the auxiliary function $h_f(x)$. We will use a suitable representation of $h_f(x)$ to parametrize the map $f(x)$. This parametrization is the starting point for a stochastic optimization procedure to obtain a map with the desired autocorrelation function $C(n)$ at times $n=1,2,..,m$. One of the simplest ansatz would be to write down a piecewise linear expression for $h_f$. Our knowledge of the properties and dynamics of $1-d$ maps however suggests that the local behaviour of the map in the neighbourhood of special points, like for example a marginal unstable fixed point in the case of intermittent dynamics (see ref.[@Ott] and references therein), plays a crucial role in determining the statistical properties of the map. A more intuitive ansatz is therefore needed. Here we will proceed as follows.\
We define a $1-d$ lattice with $N+1$ points in the interval $[0,x_{\max}]$. The coordinates of the lattice points are $x_p(i),~i=0,1,..,N$ with $x_p(0)=0$ and $x_p(N)=x_{\max}$. The function $h_f(x)$ is piecewice defined. We will call in the following the expression of $h_f$ in the $i-th$ interval $[x_p(i-1),x_p(i))$ an [**[element]{}**]{} and use the notation $h_{f,i}$ for the $i-th$ element. In practice one is free to choose many different expressions for the $i$-th element. Constraints on $h_f$ as for example continuity in $[0,x_{\max}]$ together with the conditions (\[hfprei\],\[hfprop\]) however restrict the possible forms. A desirable but not necessary requirement is that $h_{f,i}$ should be analytically invertible in order to easily extend the solution in $[0,x_{\max}]$ to the interval $(x_{\max},1]$ (see (\[genhf\])). Here we will investigate two different choices for the element $h_{f,i}$. In both cases continuity of $h_f$ is guaranteed. The first case, refered to in the following as model I, is obtained by choosing the order of $h_{f,i}$ at the left point $x_p(i-1)$ in the interval $[x_p(i-1),x_p(i))$. This parametrization allows us to fit the derivative of the map at $x=0$ and has the advantage of using a minimal set of parameters in describing the element $h_{f,i}$. The disadvantage of this choice is that one cannot fit independently the order of the maximum of the map at the right point $x_p(N)$ of the last interval $[x_p(N-1),x_p(N)]$. This is the reason for considering also a second case (model II). Here we use for the element $h_{f,i}$ an expression representing expansions around both limiting points $x_p(i-1),x_p(i)$ which match together at some point of the interval $[x_p(i-1),x_p(i)]$. An additional parameter is therefore required in order to satisfy continuity for $h_{f,i}$ and its first derivative in $(x_p(i-1),x_p(i))$. Due to this fact model II uses 2 more parameters than model I for the description of a single element $h_{f,i}$.
Let us now discuss the two models in more detail. For model I the element $h_{f,i}$ is given by: $$\begin{aligned}
h_{f,i}&=&(y_p(i)-y_p(i-1)) \left(\frac{x-x_p(i-1)}
{x_p(i)-x_p(i-1)}\right)^{\alpha(i)} + y_p(i-1)
\nonumber \\
~~~x&\in&[x_p(i-1),x_p(i))~~;~~~i=1,2,..,N
\label{hfexp}\end{aligned}$$ where $y_p(i-1),y_p(i)$ are the values of $h_f$ at the points $x_p(i-1),x_p(i)$ respectively, while the power $\alpha(i)$ determines the local behaviour of $h_f$ around $x_p(i-1)$. Due to the constraints (\[hfprei\],\[hfprop\]) we get: $y_p(0)=1$ and $y_p(N)=x_{\max}$. The monotony of $h_f$ implies that $y_p(i-1) > y_p(i)$ for $i=1,2,..,N$. The above ansatz determines the map $f(x)$ in the interval $[0,x_{\max}]$. To find $f(x)$ in the remaining interval $[x_{\max},1]$ we take advantage of eq.(\[funx\]) and therefore have to invert the auxiliary function $h_f(x)$. It is straightforward to show that the expression (\[hfexp\]) can be inverted analytically leading to a closed form for $h_f^{-1}(x)$. The above defined ansatz for $h_{f}$ is in general piecewise smooth, i.e. smooth with the exception of a finite number of points.
In model II we use the following ansatz for the element $h_{f,i}$: $$\begin{aligned}
h_{f,i}(x)&=&y_p(i-1) - c_L(i) (x-x_p(i-1))^{\alpha_L(i)}~;~
x \in [x_p(i-1),x_s(i)) \label{hf2eq1}\\
h_{f,i}(x)&=&y_p(i) + c_R(i) (x_p(i)-x)^{\alpha_R(i)}~~~~;~~~~
x \in [x_s(i),x_p(i)] \label{hf2eq2}\end{aligned}$$ where we have introduced four new parameters $c_L(i),c_R(i),\alpha_R(i),x_s(i)$ and a new point $x_s(i) \in [x_p(i-1),x_p(i))$ for each $i=1,...,N$. Two of these parameters, namely $c_L(i)$ and $c_R(i)$, can be fixed demanding continuity of $h_f$ and its first derivative at $x=x_s(i)$. We arrive then at the following generating formula for $h_f$: $$\begin{aligned}
h_{f,i}(x)=y_p(i-1) - &S_i& \alpha_R(i)(x_s(i)-x_p(i-1))^{1 - \alpha_L(i)}
(x-x_p(i-1))^{\alpha_L(i)}~; \nonumber \\
x &\in& [x_p(i-1),x_s(i)) \nonumber \\
h_{f,i}(x)=y_p(i) +
&S_i& \alpha_L(i)(x_p(i)-x_s(i))^{1 - \alpha_R(i)}
(x_p(i)-x)^{\alpha_R(i)}~; \nonumber \\
x &\in& [x_s(i),x_p(i))
\label{hfexpn}\end{aligned}$$ with: $$S_i=\frac{(y_p(i-1)-y_p(i))}
{\alpha_R(i) (x_s(i)-x_p(i-1)) + \alpha_L(i) (x_p(i)-x_s(i))}
\label{shel}$$ $h_f(x)$ given in eq.(\[hfexpn\]) can also be inverted analytically to obtain $h_f^{-1}$ defined in $[x_{\max},1]$. Comparing eqs.(\[hfexp\],\[hfexpn\]) we see immediately that the element of model II has 2 more parameters than the element of model I.
In the remaining part of this section we will discuss the algorithm which allows us to obtain dynamical systems possessing certain statistical properties based on the above ansatz for the auxiliary function $h_f(x)$. To this end we will concentrate on the parametrization of model I. One can then directly apply these ideas to the case of model II. We have determined the ansatz of $h_f(x)$ and therefore also of $f(x)$ in terms of the parameters $\{x_p(i),y_p(i),\alpha(i)\}$. Once the values of these parameters are chosen the map is completely specified. Due to its appearance (see eq.(\[funx\])) we automatically know its invariant density (measure) and expectation values of observables. The corresponding autocorrelation function can be obtained via its defining formula $$C(n)=\int_{0}^{1}x f^{(n)}(x) d\mu(x) - \left(\int_{0}^{1}x d\mu(x)\right)^2
\label{corfct}$$ which requires a numerical integration. Here $f^{(n)}$ is the $n-th$ iterate of the map $f$.
To proceed with our central subject, the inverse problem, we assume that the first $m$ values of the autocorrelation function are given. This can be due to some experimentally given data for which a dynamical system should be modelled or due to the fact that a definite behaviour of the correlation function is required in the context of the control of a physical system. As discussed above each set $\{x_p(i),y_p(i),\alpha(i)\},~i=1,2,..,N$ respecting the constraint of monotony determines uniquely the autocorrelation function $C_{h_f}(n)$ of the corresponding map $f(x)$. One can now ask for the best set $\{x_p(i),y_p(i),\alpha(i),i=1,...,N\}$ in the sense that the resulting $C_{h_f}(n)$ possesses the least possible deviation from the given autocorrelation $C(n)$. As a measure for the above-mentioned deviation one can use a $\chi^2$-like cost function. We therefore introduce the following quantity: $$K[h_f]=\sqrt{\sum_{j=1}^{m} \left( \frac{C_{h_f}(j)-C(j)}{C(j)} \right)^2}
\label{chi2}$$ The functional $K[h_f]$ is a highly nonlinear function of the parameters $\{x_p(i),y_p(i),\alpha(i)\}$ and we are looking for the global minimum of this function. To perform the minimization of $K[h_f]$ we use a Monte-Carlo (MC) approach based on the Metropolis algorithm [@Metro].
The minimization is performed in several steps, increasing in each step the number of elements $h_{f,i}$ used for the determination of $h_f$. We start with a lattice consisting of only two points (the origin $x_p(0)=0$ and the position of the maximum $x_p(1)=x_{\max}$). This means that only one element $h_{f,1}$ is needed for the specification of $h_f$. The parameters to be fitted in this case are only two: $x_{\max}$ ($x_p(1)$) and the power $\alpha(1)$ determining the behaviour of $h_f(x)$ in the neighbourhood of the origin. The first step ends when the MC minimization has converged to some optimal values for the two fit-parameters. In the second step we use a lattice with three points $x_p(0),x_p(1),x_p(2)$ with $x_p(0)=0,~x_p(1)=\frac{x_{\max}}{2},~x_p(2)=x_{\max}$. Now we need two elements $h_{f,1}$ and $h_{f,2}$ to determine $h_f$. We do not keep the values of the old fit-parameters ($x_{\max}$,$\alpha(1)$) fixed in the second step. Instead we use Gaussian distributed random variables for the choice of the old fit-parameters. The mean values of these Gaussians are the optimum values obtained for these parameters in the previous step and the corresponding widths are taken small enough to allow only weak fluctuations ($\approx 10\%$) around the mean values. Again we perform a MC optimization to obtain optimal values for the two new parameters. This procedure is repeated until the desired convergence is achieved. In each step the lattice size is increased by one point while we include two new parameters.
We use the Metropolis algorithm to find the optimal values for $x_p(i)$ and $\alpha(i+1)$ in each step. For every trial in the $i$-th turn we assume that $x_p(i)$ and $\alpha(i+1)$ follow a uniform distribution in $(0,1)$ and $(0,\infty)$, respectively (in practice the interval $(0,\infty)$ is replaced by the finite one $(0,c)$ with an upper cutoff $c>>1$). The annealing is introduced through a thermalized probability distribution of the type: $P=e^{-K[h_f]/T}$ to avoid the trapping into local minima. The parameter $T$, playing the role of the temperature, is positive and has to be tuned adiabatically to smaller and smaller values such that the global minimum is reached asymptotically.
We do not keep the values of the old fit-parameters fixed in the following step. Instead we relax this constraint using Gaussian distributed random variables for the choice of the old fit-parameters. The mean values of these Gaussians are the optimum values obtained for these parameters in the previous step and the corresponding widths are taken small enough to allow only weak fluctuations ($\approx 10\%$) around the mean values.
Adding new fit parameters allows us to determine $h_f$ in more and more detail and improves the convergence of the autocorrelation function of the model dynamical system to the given (experimental) autocorrelation function. Indeed, as we shall see below, only a few elements $h_{f,i}$ are required to achieve a rather good convergence. In the next section we give some examples demonstrating how the above-described method can be applied to construct a (piecewise) smooth map simulating a system with given time correlations.
Numerical examples and discussion
=================================
Before turning to the examples and results of our computational method let us provide some additional aspects concerning the determination of the correlation function. We restrict our investigations to a rather small set of values for the correlation function $C(n)$, more precisely to the set $\{C(0),...,C(5)\}$. The reason for this restriction is that the exact calculation of the correlation function which has to be accomplished for each single step of the Monte-Carlo scheme is computationally very intensive (see below, the total amount of CPU of our calculations on a powerful workstation was approximately three months). The reliable evaluation of the correlation function is by no means trivial. As demonstrated for example in ref.[@SPDan] the results obtained for the correlation function calculated with the trajectories of the dynamical system are, in many cases, not reliable and cannot be improved by going to longer propagation times. Therefore other methods for the calculation of the time correlations are needed. Here we use a numerical approach to eq.(\[corfct\]). It is based on the extraction of the monotony intervals for the $n$-th iterate. The endpoints of the monotony intervals are given as the preimages of the maximum $x_{\max}$. The integration is then performed for each monotony interval separately. This ensures an accurate although very CPU time consuming evaluation of the correlation function. The latter is related to the exponential growth ($2^n$) of the number of monotony intervals with increasing $n$.
We apply now the stochastic method described in the previous section to several examples to demonstrate its capability of providing dynamical systems with prescribed statistical properties. The four different cases we study here can be summarized with respect to their statistical properties as follows:
- [exponentially decaying autocorrelation function and uniform density]{}
- [exponentially decaying autocorrelation function and linear density]{}
- [oscillatory decaying autocorrelation function and power-law density]{}
- [power-law decaying autocorrelation function and uniform density]{}
Figs.(1-4) illustrate the corresponding results of our Monte-Carlo approach. The subfigures (a,c) within each figure show the resulting maps for model I and model II, respectively. Each of the subfigures (b,d) within each figure (1-4) shows both the prescribed data for the autocorrelation signal as well as the result of our optimization approach for model I and II, respectively. In comparing the given autocorrelation data and the results of our optimization scheme we deduce in the following a ’mean relative error’ per individual data point $\frac{K[h_f]}{\sqrt{m}}$ of the autocorrelation function.
Since it is not our goal to provide as precise data as possible for the optimized autocorrelation functions but to demonstrate the feasibility of our approach with a good accuracy for a few data points we restrict ourselves to auxiliary functions $h_f$ composed of a few elements $h_{f,i}$. For the case of model I we use (with one exception) four points, i.e. three elements $h_{f,i}$ for the decomposition of the auxiliary function $h_f$. For model II we use only two points, i.e. one element $h_{f,i}$. Typically a few ten thousand Monte-Carlo steps are performed. The most time consuming part of the algorithm is the calculation of the autocorrelation function which has to be done for each MC step.
First we consider a simple example for which the maps we are looking for are well-known. We use an exponential decay as a prescribed behaviour for the autocorrelation signal while the invariant density of the dynamical process is assumed to be uniform in $[0,1]$. A corresponding family of maps, the nonsymmetric tent maps, has been studied in some detail in ref.[@Mori]. They are the nonsymmetric tent maps. The resulting maps of our Monte-Carlo optimization are illustrated for model I in fig.1(a) and for model II in fig.1(c). They show only minor differences, i.e. the outcome of approach I is almost the same as for approach II, and both are to a very good approximation an asymmetric tent map. Both models lead also to a very good approximation of the corresponding prescribed correlation function with an error of only $2.2\%$.
For the second example the two models lead to quantitatively different results. The invariant density is supposed to be linear while the autocorrelation function shows an exponential decay similar to the previous case. The results of the stochastic minimization for this case are illustrated in figure 2. The two obtained dynamical systems (figs.2(a,c)) have a very different appearance reflecting the fact that our modelling procedure allows for various dynamical systems with the same invariant density and autocorrelation. Let us briefly address the main features of the resulting maps. The map of model I possesses an almost marginal unstable fixed point at the origin and an almost vertical derivative at $x=1$. Additionally it possesses two obvious points of noncontinuous derivatives located on the left and right branch of the map, respectively. The one on the left branch has a nonvanishing right derivative which, via the function $h_f(x)$, is mapped to a point with almost vertical left derivative in the right branch of $f(x)$. The reader should note that the latter point coincides with the nonzero fixed point of the map ! The map of model II (fig.2(c)) is almost straighlined on its left branch and shows an almost vertical right derivative at the maximum and an almost vertical left derivative at $x=1$. In general the map of model II is ’smoother’ than the map of model I, which is an overall tendency to be observed in any of our examples. It can be viewed as a result of the additional flexibility within model II which allows to independently adapt the left and right derivative in a given interval thereby joining them smoothly together (see the above description for the ansatz of $h_f$ in model II). The prescribed and optimized autocorrelation data for model I are illustrated in subfigure 2(b): They show a deviation of $129 \%$ which is predominantly due to the inability of reproducing the single point $C(1)$ within the approach of model I. Although hardly visible in fig.2(b) the prescribed and optimized data coincide very well for $\{C(n),2\le n \le5\}$. The optimized correlation function for model II leads to a much better approximation to the prescribed exponential decay and yields an error of only $5.4 \%$ (see fig.2(d)).
As a third example we use an invariant density obeying a power law with an exponent $\beta=-0.5$ while the autocorrelation function is chosen to possess a decaying oscillatory behaviour. The corresponding minimization results are shown in fig.3. Both maps (figs.3(a,c)) show an almost horizontal derivative at $x=1$ and a strong cusp at $x_{\max}$. For the map of model II the left derivative at the maximum is almost vertical. Again the appearance of the map of model II is much smoother compared to the map of model I. Regarding the autocorrelation function we have the opposite situation compared to the previous example. Within model I (fig.3(b)) we obtain a relative error of $22 \%$ for the autocorrelation data while model II (fig.3(d) provides an error of $45 \%$. Obviously model I is advantageous in the present case (at least for the present number of grid points chosen for the ansatz of $h_f$): it nicely reproduces the oscillations of the correlation function.
Finally we study the case that the autocorrelation function decays algebraically with the exponent $\gamma=-2.5$, i.e. we encounter the case of long-range correlations. The invariant density is chosen to be uniform. Both resulting maps (fig.4(a,c)) possess a large but finite derivative at $x=1$. The map of model I possesses an almost vertical derivative at a single point on its right branch. In contrast to our second example (see above) this point does not coincide with the fixed point. It is conjectured that this single point with almost vertical derivative is responsible for the observed power law decay of the correlation function (fig4(b)) (see also ref.[@Hor]). Analogously the map of model II (fig.4(c)) shows a very well-pronounced vertical derivative at the cusp. It is interesting to observe that in this case both models I and II provide very satisfactory approximations to the prescribed autocorrelation data (fig4(b,d)), i.e. a relative error of only $13 \%$.
Summary
=======
We have introduced a stochastic Monte-Carlo based approach to the inverse problem which uses both the invariant density as well as a finite number of points of the autocorrelation function as prescribed statistical quantities. A key ingredient for this approach was a recently found representation for one-dimensional complete chaotic and single-humped dynamical system in terms of an auxiliary function $h_f$. This representation is a formal solution of the inverse Frobenius-Perron problem and provides the dynamical system $f(x)$ explicitly as a function of the given measure and the function $h_f$. $h_f$ therefore reflects the freedom of changing the map without changing its invariant measure (density). In order to quantify the freedom available for the determination of $h_f$, i.e. to parametrize its functional space, we have introduced two different models which allow to vary $h_f$ extensively. The parameters involved are then used within our stochastic minimization procedure to obtain a correlation function with least deviation from a prescribed autocorrelation signal. Through a number of examples we have demonstrated that our approach possesses an enormous flexibility allowing for a large variety of qualitatively different behaviour of the density and correlation function. To our knowledge there is in general no unique map which belongs to a given density and correlation function. This fact has to be seen in the context of the present investigation as an advantage since it allows for a great flexibility and possible variety with respect to the underlying dynamical systems.
We would like to mention that the above-discussed prescribed behaviour of the autocorrelation data (exponential, oscillating, power law decay) is, strictly speaking, enforced only for the first five points included in our Monte-Carlo optimization. In principle it is imaginable that this behaviour represents a transient and the asymptotics of the correlation function might show a different behaviour. To determine exact asymptotic properties of certain dynamical systems is however not the issue of the present paper. Our goal is to extend the inverse problem by including the correlation function in terms of a few (experimentally) available data points, thereby enabling us to design a dynamical system with desired statistical properties. Furthermore our approach might be suggestive in terms of influencing or controlling dynamical systems [@Bollt; @Gora] in a certain way.
Acknowledgements
================
One of the authors (P.S.) thanks the Max-Planck Institute for Physics of Complex Systems in Dresden for its kind hospitality. D.P. acknowledges financial support by the Deutsche Forschungsgemeinschaft and the Landesgraduiertenförderungsgesetz (LGFG).
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[[**[FIGURE CAPTIONS]{}**]{}]{}
[**Figure 1:**]{} The stochastic minimization results for a dynamical system with uniform invariant density and exponentially decaying autocorrelation function. (a) The resulting map using model I. (b) The autocorrelation function for the map of model I (solid line) and the corresponding prescribed data (full circles). (c) The resulting map using model II. (d) The autocorrelation function for the map of model II (solid line) and the corresponding prescribed data (full circles).
[**Figure 2:**]{} Same as in figure 1 but for prescribed linear invariant density and exponentially decaying autocorrelation function.
[**Figure 3:**]{} Same as in figure 1 but for prescribed power-law invariant density ($\beta=-0.5$) and oscillatory decaying autocorrelation function.
[**Figure 4:**]{} Same as in figure 1 but for prescribed uniform invariant density and power-law decaying autocorrelation function (with exponent -2.5).
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abstract: 'This paper presents a model based on an hybrid system to numerically simulate the climbing phase of an aircraft. This model is then used within a trajectory prediction tool. Finally, the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) optimization algorithm is used to tune five selected parameters, and thus improve the accuracy of the model. Incorporated within a trajectory prediction tool, this model can be used to derive the order of magnitude of the prediction error over time, and thus the domain of validity of the trajectory prediction. A first validation experiment of the proposed model is based on the errors along time for a one-time trajectory prediction at the take off of the flight with respect to the default values of the theoretical BADA model. This experiment, assuming complete information, also shows the limit of the model. A second experiment part presents an on-line trajectory prediction, in which the prediction is continuously updated based on the current aircraft position. This approach raises several issues, for which improvements of the basic model are proposed, and the resulting trajectory prediction tool shows statistically significantly more accurate results than those of the default model.'
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title: Online Learning for Ground Trajectory Prediction
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Air Traffic Control, Trajectory Prediction, Hybrid System, Total-Energy Model, Black-Box Optimization, Machine Learning
Introduction {#sec:intro}
============
Trajectory Prediction (TP) is the core component of automated systems in Air Traffic Control (ATC). Many functionalities of the Decision Support Tool directly rely on an accurate TP: controller posting, workload estimation, arrival sequencing, loss of separation detection, and conflict resolution, to name the most prominent ones. As a consequence, TP is the weakness of the current automation ATC systems and a major issue in the ATC research community, even more with the new paradigmatic shift toward 4D trajectories in both the SESAR Joint Undertaking and the NextGen project. The challenge is to reduce the uncertainty of the prediction of the aircraft states on a temporal horizon of at least 20 minutes. To achieve this, the information of the current state of the aircraft and its environment shall be reliable. As a matter of fact, the Flight Management System (FMS) has access to the measurements from the sensors of the aircraft and creates its own TP, which is updated frequently. Therefore, we should expect that this TP is the most accurate one. Thanks to the Data Link, it should be possible for the ground control to receive data from this on-board TP. Unfortunately, today, this promising technology is not implemented in all aircrafts. Moreover, a decision support tool requires the capacity to efficiently generate ’what-if’ scenarios. Today, it is unrealistic to receive many trajectories from the on-board system of every aircraft in the airspace and merge them into different airspace scenarios in real-time. Therefore, the ground TP is still an essential component of the future air traffic control systems.
A trajectory can roughly be separated in three phases: climbing, level flight, and descent. Phases with altitude changes are the most difficult to handle due to the differences of aircraft performances and the effect of the weather conditions. Operationally, the controllers of approach control centers, who deal with climbing and descent phases, must have a good representation of the vertical evolution of the aircraft to ensure their separations. In most airspace, this separation shall be at least 10 flight levels. For these reasons, this study will focus mainly on the prediction in the vertical axis.
The paper is organized as follows: Section \[sec:relatedWork\] surveys some recent works in the domain of trajectory prediction. Section \[Model\] details the proposed model. The validation experiments are described in Section \[sec:modelValidation\], while the on-line prediction method is experimented with in Section \[sec:Online\]. Section \[conclusion\] summarizes the paper and gives some hints for further research in the area of trajectory prediction.
Related Work {#sec:relatedWork}
============
Research on TP includes a set of methodologies for its specification, implementation and evaluation. First, the Eurocontrol Specification for Trajectory Prediction gives the requirements for an operational TP and a validation methodology useful for the Air Navigation Service Providers in their choice of a new ATM system compatible with SESAR. This document covers many functionalities like flight plan and clearance processing, airspace constraints and real-time monitoring. Unfortunately, it does not specify the accuracy requirements. Besides, [@Musialek2010] provides an important literature survey of trajectory prediction technology with 282 reviewed documents and 20 selected for further studies. From the selected set, many implement the point-mass model where the rotational moments are not modeled. This is an acceptable assumption for airline aircraft. Also, [@Coppenbarger1999] and [@Swierstra2003] enumerate the principal difficulties inherent to TP, i.e. the uncertainty on the input data, the controller and pilot intents, and quantify the errors accordingly. The input data mainly refers to aircraft characteristics which are given in the [*Base of Aircraft DAta*]{} (BADA) Aircraft Performance Model. Still, these values are only nominal and can be different from the real situation. Therefore, the SESAR project 5.5.2 [@SesarTP2012] concludes that sharing airline operational control data, like the mass and the speed schedule, could provide quick improvement to ground trajectory prediction, with limited investment. As a matter of fact, these parameters are determinants for computing the TP with a point-mass model.
More generally, a parametric approach refers to a model based on flight equations and aircraft characteristics. The point-mass model is an example of a parametric approach. As we will see, many parameters can be used to tune the model to reality. To generate trajectories, [@Gallo] uses six ordinary differential integrators depending on the longitudinal motion instruction. Similarly, [@Glover2004] and [@Kamgarpour2011] use hybrid systems to model the change of differential equations according to the control law and the aircraft states. A discrete space, e.g. an automaton, is defined to represent the modes of the system. Every mode defines the differential equations and creates trajectories in a continuous space. As seen in [@Glover2004], this model is well-suited for implementing BADA. [@Alligier] exposes a technique to find a general thrust settings, i.e. a control law, that could be used in such framework. The idea of fitting the mass parameter of BADA on a few past points is used. Results on the accuracy of this generic control law in the vertical plane should be expected in a near future.
A common flaw in parametric approaches concerns the nominal values used for every parameters. As shown on figure \[fig:massEffect\], the effect of the mass parameter clearly shows the importance of tuning these parameters. As an example, 400 seconds after the takeoff, there is already a difference of approximately 250 flight levels between the minimal and the maximum mass which is enormous for an application like the TP. This can easily become a burden when the model is rather complex. To overcome this inherent difficulty, non-parametric approaches are studied in order to obtain an aircraft model from the past trajectories. Non-parametric approaches rely on machine learning and statistical inference: [@LeFablec1999] uses neural networks, [@M2R_Richard_ALLIGIER] uses genetic programming in order to learn the structure of the variables of a linear regression and [@M2R_GHASEMI] uses fuzzy regression with k-nearest neighbor. The main drawback of non-parametric approaches is that it requires lots of historical data and the model is learned for a specific context and can hardly be generalized because of the airspace constraints, e.g. aircraft following a Standard Instrument Departure.
![The Effect of the mass parameter[]{data-label="fig:massEffect"}](./img/massEffect.pdf){width="2.5in"}
Combining both approaches is an interesting research question addressed by [@Alligier] and [@Crisostomi2008]. The latter combines Monte-Carlo Simulation and worst-case scenario for modifying the parameters of BADA while integrating a wind model. However, this work is limited to the descent phase and the experiments are performed on trajectories obtained by simulation. The main contribution of our work on this question is to show that tuning the parameters of BADA during the progress of a flight is a way to overcome the difficulties of parametric approach and avoiding to learn a model from the historical data. To the best of our knowledge, no work has answered to this question with results of the vertical plane.
Model {#Model}
=====
The basic idea is to create an hybrid system to generate a trajectory in the vertical plane using BADA. The system describes the transitions between the different modes in function of the states and the control laws. Following the mode, a given differential equation is integrated to obtain the trajectory. This section states the assumptions, defines the model and gives an integration schema for generating the trajectories.
Assumptions
-----------
Depending on the requirements of the simulation, one may choose different levels of complexity for the coordinate systems and the aircraft control system. Since we consider only the climbing phase, the system will evolve on a short period of time on a limited geographical area. From these considerations, we assume that the flat earth model is a reasonable approximation (cf. [@Hull2007]). This model assumes that the earth is flat, non-rotating and can be defined as an inertial reference frame. Also, the gravitational force is constant and perpendicular to the ground. The atmosphere is at rest relative to the earth and atmospheric properties solely depend on the altitude. With these assumptions, one can define the differential equations of kinematics and dynamics. For the kinematics, the equations imply solely that the displacement is proportional to the speed projected on the vertical or horizontal plan. For the dynamics, we need to model the thrust, the drag, the lift and the weight. To our knowledge, BADA is the most complete model describing the capabilities of many aircraft types.
Total-Energy Model
------------------
This section relies on the BADA 3.10 User Manual [@Bada2012]. BADA is based on the total-energy model where the rate of work is equal to the rate of potential and kinetic energy. Here, the rate of climb is obtained by controlling the speed and the throttle. From [@Bada2012] p.14:eq.3.2-7 and by transforming the quantities into functions of altitude $h$, speed $V$ and mode $q$, the rate of climb is defined by: $$\label{eq:roc}
\dot h(V,q) = \frac{T(h) - \Delta T}{T(h)} \left[ \frac{(Thr(h) - D(V,h)) \cdot V}{m \cdot g} \right] \cdot f(V,h,q)$$ where the function $T(h)$ is defined in BADA 3.10 User Manual [@Bada2012] p. 10, $Thr(h)$ at p.22, $D(V,h)$ at p.20 and $f(V,h,q)$ at p.15-16. As a first approximation, the mass $m$, the gravitational acceleration $g$ and the temperature differential $\Delta T$ are constant during the climb phase. According to [@Hull2007], this is a reasonable assumption since the climb phase lasts around 20 minutes. The terms $h$, $V$ and $q$ are variables that evolve with the system and one must specify the evolution of the two remaining independent variables. However, the acceleration in function of the aircraft dynamic is not specified in BADA. From [@Hull2007], it is given by: $$\label{eq:acceleration}
\dot V(h, \dot h) = \frac{1}{m}(Thr(h) - D(V,h) - m \cdot g \cdot \sin(\gamma))$$ where $\sin ( \gamma )=\frac{\dot h}{V}$. So, from eq. \[eq:acceleration\], we can see that the acceleration evolves independently of the mode given the rate of climb. Also, when $\dot h$ is high, $\sin( \gamma )$ is high and the acceleration $\dot V$ is low. This goes along with the hypothesis the total-energy model. Finally, the next section will present the evolution of the variable $q$.
Mode Definition
---------------
In this section, we define the modes and their transitions based on BADA 3.10 User Manual [@Bada2012]. First, let $Q = \{CAS,MACH\} \times \{LOW,HIGH\} \times \{DEC,CST,ACC\}$ be the mode space. The first two modes, transition altitude and tropopause, depend solely on the altitude: $$q_1(h) = \left\{
\begin{array}{l l}
\text{CAS} & \quad \text{if $h \le H_{trans}$}\\
\text{MACH} & \quad \text{otherwise}\\
\end{array} \right.$$
$$q_2(h) = \left\{
\begin{array}{l l}
\text{LOW} & \quad \text{if $h \le H_{trop}$}\\
\text{HIGH} & \quad \text{otherwise}\\
\end{array} \right.$$ where $H_{trans}$ is the transition altitude and $H_{trop}$ is the tropopause geopotential pressure altitude. Finally, the last feature $q_3$ is the mode of acceleration. The simplest controller of this mode is given by: $$q_3(V,h) = \left\{
\begin{array}{l l}
\text{ACC} & \quad \text{if $V \le V^*(h) - \epsilon$}\\
\text{DEC} & \quad \text{if $V \ge V^*(h) + \epsilon$}\\
\text{CST} & \quad \text{otherwise}
\end{array} \right.$$ where ACC is acceleration, DEC is deceleration and CST is constant speed and $V^*$ be a target speed at altitude $h$ where $V$ tends to converge. $V^*$ can be chosen according to the speed schedule defined in the Airline Procedures Model of BADA 3.10 User Manual [@Bada2012] p.29, where three target speeds $(V_1,V_2,M)$ are required as parameters. The nominal values can be found in the airline procedure files of BADA. Finally, $\epsilon \in \mathbb{R}$ is a threshold value to avoid jitter. Next, the energy share factor function $f$ takes its values according to the following flight conditions:
1. Constant $V_M$ above tropopause,
2. Constant $V_M$ below tropopause,
3. Constant $V_{CAS}$ above tropopause,
4. Constant $V_{CAS}$ below tropopause,
5. Acceleration in climb,
6. Deceleration in climb
where $V_M$ is the Mach speed and $V_{CAS}$ is the calibrated speed. These flight conditions can be defined in terms of conjunctions of $q_1$, $q_2$ and $q_3$. From [@Bada2012] p.15-16, $f$ is discontinuous when $q$ jumps from one mode to another. Also, $f$ is bounded and therefore, with the flight envelope, one can also bound $\dot h$ and $\dot V$. One must pay attention to the Zeno behavior (cf. [@Lygeros2010]) where a system can make an infinite number of jumps in a finite amount of time. With these considerations in minds, one can use a numerical procedure to integrate eq.\[eq:roc\] and eq.\[eq:acceleration\].
Trajectory Generation
---------------------
In order to generate the trajectory, one must specify eq.\[eq:roc\] and eq.\[eq:acceleration\] as functions of time. With respect to BADA, let $Thr(h(t),t) = Thr(h(t))$, $D(h(t), V(t),t)=D(h(t),V(t))$ and $T(h(t),t) = T(h(t))$, that is the evolution of the thrust, the temperature and the drag are time-invariant. Moreover, the aircraft dynamic functions shall be specified with respect to the flight envelope constraints. In this study, we choose a nominal thrust function. In eq.\[eq:roc\], $Thr$ is replaced by the maximum thrust $Thr_{max}$ and the whole equation is multiplied by a reduced climb power coefficient $C_{red}$, which is supposed to give realistic profiles (cf. [@Bada2012] p.24). To simulate the system, a common fourth-order Runge-Kutta method is used. Let $f_1(t,h,V,q) = \dot h(V,q)$ and $f_2(t,V,h, \dot h) = \dot V(h,\dot h)$. Then, one obtains the following integration scheme: $$\begin{aligned}
dh_1 &=& f_1(t_n,h_n,V_n,q(V_n,h_n)) \nonumber \\
dv_1 &=& f_2(t_n,V_n,h_n, \dot h_n) \nonumber \\
dh_2 &=& f_1( t_n + \frac{\Delta t}{2}, h_n + dh_1\frac{\Delta t}{2}, V_n + dv_1\frac{\Delta t}{2}, \nonumber\\
&& \quad q(V_n + dv_1\frac{\Delta t}{2},h_n + dh_1\frac{\Delta t}{2})) \nonumber\\
dv_2 &=& f_2(t_n + \frac{\Delta t}{2}, V_n + dv_1\frac{\Delta t}{2},h_n + dh_1\frac{\Delta t}{2}, dh_2) \nonumber \\
dh_3 &=& f_1(t_n + \frac{\Delta t}{2}, h_n + dh_2\frac{\Delta t}{2}, V_n + dv_2\frac{\Delta t}{2}, \nonumber\\
&& \quad q(V_n + dv_2\frac{\Delta t}{2},h_n + dh_2\frac{\Delta t}{2})) \nonumber\\
dv_3 &=& f_2(t_n + \frac{\Delta t}{2},V_n + dv_2\frac{\Delta t}{2},h_n + dh_2\frac{\Delta t}{2}, dh_3) \nonumber \\
dh_4 &=& f_1(t_n + \Delta t, h_n + dh_3 \Delta t, V_n + dv_3 \Delta t, \nonumber\\
&& \quad q(V_n + dv_3 \Delta t,h_n + dh_3 \Delta t)) \nonumber\\
dv_4 &=& f_2(t_n + \Delta t, V_n + dv_3 \Delta t,h_n + dh_3 \Delta t, dh_4) \nonumber \\
h_{n+1} &=& h_n + \frac{\Delta t}{6} \left( dh_1 + 2dh_2 + 2dh_3 + dh_4 \right) \nonumber \\
V_{n+1} &=& V_n + \frac{\Delta t}{6} \left( dv_1 + 2dv_2 + 2dv_3 + dv_4 \right) \nonumber \\
t_{n+1} &=& t_n + \Delta t \nonumber\end{aligned}$$ where the initial conditions $t_0$, $q_0$, $h(t_0,q_0)=h_0 $, $V(t_0,h_0)=V_0$ are given. The choice of the Runge-kutta method is justified by the jumps in the function $f$. As a matter of fact, this method will minimize the impact of a jump during a timestep by averaging the variations at the beginning, twice at the middle point and at the end of the timestep (cf. [@Fortin2011]). Nevertheless, stability questions shall be addressed in the later in order to validate the approach.
Parameter Tuning
----------------
Now that the trajectory generator is defined, we would like to tune the model parameters according to observations. First, there should be a trade-off between the number of parameters and the capacity of the model to approximate real trajectories. As a bad example, instead of using the default thrust controller defined in [@Bada2012], we tried to find the optimal controller for the acceleration mode for a given trajectory. The resulting controller was a Bang-Bang control function where we have observed that the mode switches between accelerating and decelerating at every timestep. Clearly, this function can approximate any real trajectories, but does not generalize from one trajectory to another and does not reflect the real behavior of the aircraft. So, from eq.\[eq:roc\] and eq.\[eq:acceleration\] and the speed schedule $V^*$, the parameters $m$, $V_1$, $V_2$ ,$M$ and $\Delta T$ are good candidates since they are time-invariant, contrary to a discrete control law. Moreover, their values are fixed to nominal values and BADA explicitly suggests to tune them.
One way to tune the parameters is to minimize the position errors between the TP and the real trajectory. This leads to an optimization problem where the function to minimize is expressed by the trajectory generation scheme. Here, we decide to use a black-box optimization algorithm that will perform parameter estimation on the hybrid system. Due to the relations between the parameters and the complexity of the BADA models, we assume that parameter estimation is a non-trivial optimization problem.
Black-Box Optimization
----------------------
As already mentioned, parameter tuning pertains to non-convex black-box optimization, and several methods could be used to tackle it. Furthermore, no information whatsoever is available regarding the modality of the objective function with five parameters. As a matter of fact, it is unimodal when the mass parameter or the differential temperature are the only ones to be tuned. These have an effect on the whole trajectory. But, the speed parameters have a local effect depending on the speed schedule. Moreover, the differences between the speed parameters will induce acceleration phases that will transform the trajectory. Finding the best value for a speed parameter to fit locally the trajectory will create a local optimum that could be worse than finding the two speed parameters that will avoid an acceleration phase that is not undertaken in reality. Even if we did not prove that the objective function is multimodal, we can think that it can possibly be the case. Finally, the objective is non-differentiable (or at least the analytical derivative is out of reach). Hence general-purpose derivative-free optimization method is required.
The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [@hansenTutorial2005] is today a state-of-the-art derivative-free optimization method that has demonstrated outstanding performances for problems up to a few hundred variables, in several official comparisons (see, among others, the CEC 2005 challenge [@continuousCEC05], and both Black Box Optimization Benchmark workshops at ACM-GECCO 2009 [@BBOB09] and 2010 [@BBOB10]), as well as on a large number of real-world applications [@CMA-ESApplications]. CMA-ES is an Evolution Strategy [@Rechenberg; @Schwefel] that uses Gaussian mutation with adaptive parameter setting. A Gaussian mutation is defined here by its step-size and its covariance matrix. The step-size is increased (resp. decreased) if the cumulated path of the current best solution is smaller (resp. larger) than that of a random walk, and in the original version [@Hansen:ICEC96; @Hansen:ECJ01], the covariance matrix was updated by adding a rank-one matrix with eigenvector the direction of progress. An improved version with rank-$\mu$ update was later proposed [@Hansen:ECJ03], and several additional variant made it more and more powerful. The most recent version is the so-called bi-pop-CMA-ES [@bipopCMA2009], that evolves both a large and a small population and outperforms previous versions in case of multi-modal functions. All source code is available on the author’s web page (<http://www.lri.fr/~hansen/index.html>), in different programming languages, including the bi-pop version. Using CMA-ES for parameter estimation is then straightforward, and amounts to interfacing the objective function for BADA TP with the core CMA-ES program, after eventually normalizing its parameters.
Model Validation {#sec:modelValidation}
================
This section deals both with the empirical validation of the nominal model and the estimation algorithm used to fit the parameters on the whole trajectory.
Dataset {#subsec:dataset}
-------
To validate our model, we will use a dataset composed of 262 real departure trajectories of A320. These trajectories have been recorded via a radar systems during one month. For one trajectory, there is a data vector at every 5 seconds composed of the aircraft position, the rate of climb and the true airspeed. Then, the top of climb is calculated as the first highest point of the trajectory. Also, these trajectories do not have a rate of climb that is equal to zero for more than 30 seconds, they shall reach a cruise level at least of 300 FL and their durations are at least of 1100 seconds. These filtering conditions permits to keep trajectories that are not affected by a level flight clearance.
Methodology {#subsec:method}
-----------
To assess the model, we evaluate the position error between the TP and the real trajectory. Let $\mathcal{H}(\theta;s_0)$ be the sequence of altitudes generated by the simulation of the hybrid system with the parameters $\theta = \left[ m \quad \Delta T \quad V_1 \quad V_2 \quad V_M \right]^T$ and the initial condition vector $s_0 = \left[ t_0 \quad q_0 \quad h_0 \quad V_0 \right]^T$. Then, the measure is simply the sum of absolute errors. In the case of parameter estimation, we search in the feasible space of parameters $\Theta$, the point $\theta \in \Theta$ that minimizes this measure. So, we have the optimization problem: $$\label{eq:opt}
\theta_{i,j}^* = \underset{\theta \in \Theta}{\operatorname{argmin}} \sum_{n=i}^j \left| \mathcal{H}(\theta;s_0)_n - T_n \right|$$ where $\mathcal{H}(\theta,s_0)$ is the sequence of ordered altitudes obtained for given parameters $\theta$ and initial conditions $s_0$ and $T$ is the sequence of observed altitudes from a trajectory of the dataset. We suppose that $(j-i) \leq |\mathcal{H}(\theta,s_0)| \leq |T|$ and that the timestamp associated to the point $\mathcal{H}(\theta,s_0)_n$ is the same than $T_n$. When fitting the whole climbing phase, some difficulties with eq.\[eq:opt\] may arise. First, we need to give a termination criterion when simulating the hybrid system that is to stop when the TP reaches the level flight. Depending on the parameters, the cardinal of the resulting sequence of points will be different. So, it might be necessary to add points to the real trajectory T since the TP can reach the top of climb after T, e.g. a high value for the mass parameter. To compute this error, we simply add points at the level flight until the TP reaches it. Inversely, we add these points to the TP if it reaches the top of climb before the real trajectory. Besides, $i < j$ are bounds to restrain the optimization on any subset of contiguous points of the trajectory. Thereafter, this notation will be useful for the online predictor.
Time after takeoff Nominal (FL) Tuned (FL)
-------------------- ------------------ -----------------
2min. 4.9195 (3.1422) 3.0929 (2.3133)
5min. 7.1416 (4.8556) 2.5496 (2.5282)
10min. 9.6714 (6.6146) 1.4057 (1.7441)
15min. 10.9441 (9.0016) 2.1957 (2.2600)
20min. 11.8008 (8.8068) 2.0546 (2.1367)
: \[tab:badaVad\] Modelization Errors - Mean and Standard Deviation
Results {#subsec:result}
-------
Table \[tab:badaVad\] shows the evolution of the mean errors and the standard deviation with time for both models: with nominal values and with tuned values. The first important evidence is the inaccuracy of the Total-Energy Model to model the positions at the beginning of the trajectory. As a matter of fact, for the tuned values, the errors at 2 minutes are the highest. An explanation consists in the fact that the aircraft states (position, speed, heading) change rapidly at the beginning of the trajectory and the selected five parameters are not sufficient to capture this complexity. Furthermore, the optimization of the equation eq.\[eq:opt\] has a global scope and so, the errors generated by local behaviors of the aircraft are ignored in favor of the common behavior. This depends directly on the selected values of $i$,$j$ in $\theta_{i,j}^*$, which in this case are $i=0$ and $j= t_{toc}$ where $t_{toc}$ is the time at top of climb. From our dataset, this common behavior happens around 10 minutes where the errors are the smallest after an acceleration phase which happens around 5 minutes. As an example, on the figure \[fig:traj\], we can distinguish three main behaviors: the initial climb from 0s. to approximately 200s, a short acceleration phase from 200s. to 300s. and the common behavior after 300s. From figure \[fig:roc\], we can see that the initial climb is characterized with high variability in the rate of climb and an acceleration phase between 50s. and 100s., shown by a huge decrease in the rate of climb. During this phase, the predicted rate of climb is far from reality for both parameter sets. Thereafter, both models capture the acceleration phase and finally, they average the rate of climb during the common behavior, which fits well the positions as shown on figure \[fig:traj\].
Another interesting result is the evolution of the standard deviation, which is in parentheses, for the model with nominal values. It increases with time from 2 min. to 15 min. and afterwards, it seems to stabilize around 9 FL. This can be interpreted as an uncertainty cone, which is often used in the air traffic community, but here, we can see that the cone stops to grow at 15 minutes and becomes a corridor of uncertainty. First, we understand that the cone grows with the flight envelope, but afterwards, the uncertainty is bounded by the fact that the trajectories, as functions of time, are strictly increasing because of the filtering conditions of the dataset (cf. Section \[subsec:dataset\]) and the upper bound that is the cruise level.
![Trajectory Fitting[]{data-label="fig:traj"}](./img/traj.pdf){width="2.5in"}
![Rate of Climb Modeling[]{data-label="fig:roc"}](./img/roc.pdf){width="2.5in"}
Online Trajectory Predictor {#sec:Online}
===========================
In this section, we present an online trajectory prediction which uses the observed positions of a current flight to tune at the same time the parameters of the model used to predict the rest of the trajectory. Different kind of algorithms can be used to undertake this task. Traditionally, a probabilistic approach is used in this online configuration, e.g. Kalman Filter, where the uncertainty is explicitly modeled. These performs very well on short periods of time but, for longer period like in this context, the linearity of the model could be too limited and should be subject to future experiments. In our approach, we use the BADA model in conjunction with the optimization algorithm CMA-ES in order to fit the parameters of the model with an objective function that defines a distance between the observed positions and the modeled one. As we will see, the problem of overfitting, which is studied by statistics and machine learning, arises in our context. The current section presents the difficulties and the solutions chosen with the associated results.
Design of Experiment
--------------------
In order to verify that the idea of an online predictor, as the one mentioned previously, is valid, we must do an empirical evaluation of the chosen algorithm by replicating the same context. The main hypothesis is that from the observation set, we can determine the values of parameters that will be fitted to the current flight. So the trajectory is separated in two subsets: the observed altitudes from the beginning to the present and the future altitudes from the present to the top of climb. As in the subsection \[subsec:result\], we use metering points to evaluate the quality of the prediction by computing the errors between the predicted altitudes and the real ones. To distinguish if a set of errors is statistically greater than the other one, we use a Wilcoxon signed-rank test. The null hypothesis of this test is that two related paired samples come from the same distribution. In our case, this test is a relevant choice because two approaches are tested on the same trajectory dataset. As usual, we reject the null hypothesis if the p-value is lower than 0.05.
Methodology {#methodology}
-----------
The most naive way to learn the parameters of the model from the observed altitudes is to directly apply Eq.\[eq:opt\] from subsection \[subsec:method\] and to apply them to generate the rest of the trajectory. By doing so, the default model is always better than the fitted one with a significant p-value. The reason behind this result is simply that the fitted model does not generalize over all the behaviors of the aircraft. In other words, it is fitted only to the behavior captured in the observations. This problem is referred to overfitting in Machine Learning litterature. To circumvent this problem, we must think of the trajectory as a time serie where the observations arrive with an determined order, i.e. the temporal order. So, at first, we will always observe the initial climb where we know from Table \[tab:badaVad\] that the inaccuracy of the model is the greatest. Furthermore, we are more interested by the parameter values that fit better the positions near the present time than at the beginning of the trajectory. To this end, we will add a weighting vector $\alpha$ that will penalize more the errors that are near the present time. But still, this is not sufficient because, depending on the present time, some parameters will not have any effect on the trajectory. As a matter of fact, from 0 to FL60, a predefined schedule is applied and only the mass parameter has an effect in BADA. The scope of the parameter $V_1$ is from FL60 to FL100, the scope of $V_2$ is from FL100 to the transition altitude (around FL277) and finally, the scope of $V_m$ is over the transition altitude. Furthermore, we add the constraint that $V_2$ is greater than $V_1$ to the optimization problem. To avoid that the optimization algorithm assigns them some arbitrary values resulting in unrealistic trajectories, we use a regulation method that penalizes any deviation from the default parameters. A meta-parameter $\lambda$ is associated to the weight of the penalty, which controls the tradeoff between exploration and exploitation. Consequently, the resulting objective function is:
$$\label{eq:learning}
\theta_{0,t}^* = \underset{\theta \in \Theta}{\operatorname{argmin}} \left[ \sum_{i=0}^t \alpha_i \left| \mathcal{H}(\theta;s_0)_i - O_i \right| + \lambda \sum_{i=0}^{|\theta|} \left| \theta_i - \theta_i^d \right| \right]$$
where $\theta^d$ is the default parameters of BADA. Notice that the penalty occurs on the parameters space, but could also be applied on the trajectory space because the mapping from the parameter space to the trajectory space is not linear in the sum of differences of altitudes.
Finally, in order to set the value of $\lambda$, we use a cross-validation approach where we partition the observation set in two: the learning set and the validation set. We choose the validation set to be just before the current altitude. One must notice that the samples are not independent and identically distributed and that we create a bias in favor of the points located just after the current altitude. Because of our extrapolation context, a bias is inevitable and this one seems the most justifiable one in order to gain accuracy in predicting the future positions. Figure \[fig:online\] shows the partition of the trajectory. The cross-validation technique used in this study consists in learning the parameters of the model on the learning set with the objective function and to use these parameters for generating the points on the validation intervals. Then, we compare these points with the real ones. We do it for multiple values of $\lambda$ and we choose the parameter values where the validation error is the lowest to generate the rest of the trajectory.
![Online Prediction[]{data-label="fig:online"}](./img/online.pdf){width="2.5in"}
Results {#results}
-------
The approach is validated on the same dataset than subsection \[subsec:dataset\]. We choose three different time slices in order to represent the online aspect of the method. The validation set size is fixed to 36 points or 180 seconds. This choice must do the trade-off between the validation purpose of avoiding overfitting which could lead to a large validation set size and the learning purpose of finding the best parameter values which could lead to a large learning set size and therefore, a small validation set size. At least, the validation set size must be higher than the acceleration phase, where the local behavior is the most different from the global one. In the learning objective function (cf. Eq.\[eq:learning\]), we choose a linear weight function where $\alpha_i = \frac{i}{t-1}$. The initial lambda value is arbitrarily set to 100 and are doubled until the penalty is high enough so that the fitted values equal the default ones. Then, the parameter values generating the lowest validation error are chosen.
Also, to avoid that the algorithm changes the parameter values based on a poor learning performance, we arbritrarily set a threshold error at 5 FL, which is higher than the results at subsection \[subsec:result\]. When the threshold is exceeded, the BADA default values are chosen.
Table \[tab:onlineVadP400\], \[tab:onlineVadP500\] and \[tab:onlineVadP600\] show the results of the proposed methodology for each time slice. At $P=400$s, the two model performances are not significantly different as shown with the high p-values. We can see that our approach increases the accuracy by 1 FL in average for the metering points at 2 minutes and 5 minutes after the current time slice with a p-value significantly under 0.05. For the metering points at 10 minutes, the p-value is higher than 0.05 and the difference is not statistically significant due to the high standard deviation values. This can be explained by the fact that the model is not very accurate during initial climb (cf. model validation section) and the validation set covers the acceleration phase. Also, because the learning error is too high, the algorithm can choose the BADA default values. So, the default choice ratio is around $20\%$ which is rather high.
At $P=500$s, the fitted model performs better at 2 and 5 minutes after the current position with small p-values. For 10 minutes, the two models are not significantly different because of the high value of the standard deviation. In fact, this can be interpreted as the two models are equally affected by the uncertainty around the possible maneuvers of the aircraft. In order to perform better, more information on the flight intents are required to reduce the variability in the trajectories. Here, the ratio of the default choice is $16\%$.
At $P=600$s, the results are similar to $P=500$s. The reason is that the aircraft keeps the same behavior between 500s. and 600s. which is different from the behavior at 400s. There is some kind of regularity that explains the fact that the prediction is enhanced up to 5 minutes. This regularity is captured more easily by the learning algorithm when the behavior is stable during the validation interval. In this case, the ratio of the default choice is $14\%$.
Time after takeoff Nominal (FL) Tuned (FL) p-value
-------------------- ----------------- ----------------- ---------
2min. 3.3029 (2.6698) 3.1699 (2.6740) 0.3401
5min. 6.7553 (5.6084) 6.5518 (5.6578) 0.6726
10min. 8.7851 (7.0757) 9.1846 (7.5687) 0.4541
: \[tab:onlineVadP400\] Comparison of Online Models at $P=400$s
Time after takeoff Nominal (FL) Tuned (FL) p-value
-------------------- ----------------- ----------------- ----------
2min. 4.0406 (3.2758) 3.2834 (2.7237) 5.612e-4
5min. 8.1290 (6.0885) 7.0567 (4.8281) 0.02049
10min. 9.0872 (7.0085) 9.4205 (6.7658) 0.7939
: \[tab:onlineVadP500\] Comparison of Online Models at $P=500$s
Time after takeoff Nominal (FL) Tuned (FL) p-value
-------------------- ----------------- ----------------- -----------
2min. 4.5110 (3.4354) 3.5912 (2.4845) 1.289e-05
5min. 6.7936 (4.9209) 5.7231 (4.0456) 1.289e-03
10min. 8.5131 (6.6410) 9.4992 (7.8805) 0.09098
: \[tab:onlineVadP600\] Comparison of Online Models at $P=600$s
Conclusions and Future Work {#conclusion}
===========================
In conclusion, this article presents a flight model for the climbing phase defined as a hybrid system based on BADA. An integration scheme is defined in order to generate the trajectories from this system. Then, tuning parameters are identified in order to be used in the online context. The method is validated through measuring the vertical errors between real trajectories and generated ones: both for default and fitted parameter values. In the validation context, fitting is done on the entire trajectory i.e. with total information. The measured errors are considered as the accuracy limit of this five parameters model. Afterwards, the model is applied in the online context, which evolves with time. Known altitudes are used to fit the parameters and then, these are used to predict the remaining points. To avoid overfitting, the known points are partitioned in a learning set and a validation set. The validation set is used to fit the regulation parameter, which penalizes the deviation from the default values. Results shows that the initial climb, which is before the main acceleration phase, is not modeled accurately in order to fit the parameters. On the contrary, when the flights adopt a common behavior after this acceleration phase, the online learning method increases the accuracy of the trajectory prediction. The gain is about 1 FL for 2 minutes and 5 minutes after the current time. After that, the two models are not significantly different because of the huge uncertainty on the trajectory. This study shows that the uncertainty becomes too important between 5 and 10 minutes with minimal information. Consequently, without further information on the flight intents and airspace constraints, ground trajectory prediction is not accurate enough for automated tasks such as conflict resolution. Furthermore, this study confirms the need to use the aircraft derived data to feed the BADA model in order to build a ground trajectory prediction as the foundation stone of automated Air Traffic control systems. The next step should be the identification of the relevant onboard data improving significantly the trajectory predictability on ground.
Acknowledgments
===============
This author Ga[é]{}tan Marceau is funded by the scholarship CIFRE 710/2012 established between Thales Air Systems and INRIA-Saclay, and the scholarship 141138/2011 from the [*Fonds de Recherche du Qu[é]{}bec - Nature et Technologies*]{}.
| 0 |
---
abstract: |
Linear polymers adsorbing on a wall can be modelled by half-space self-avoiding walks adsorbing on a line in the square lattice, or on a surface in the cubic lattice. In this paper a numerical approach based on the GAS algorithm is used to approximately enumerate states in the partition function of this model. The data are used to approximate the free energy in the model, from which estimates of the location of the critical point and crossover exponents are made. The critical point is found to be located at $$a_c^+ =
\cases{
1.779 \pm 0.003, & \hbox{in the square lattice}; \\
1.306 \pm 0.007, & \hbox{in the cubic lattice}.
}$$ These results are then used to estimate the crossover exponent $\phi$ associated with the adsorption transition, giving $$\phi =
\cases{
0.496 \pm 0.009, & \hbox{in two dimensions}; \\
0.505 \pm 0.006, & \hbox{in three dimensions}.
}$$ In addition, the scaling of these thermodynamic quantities is examined using the numerical data, including the scaling of metric quantities, and the partition and generating functions. In all cases results and numerical values of exponents were obtained which are consistent with estimates in the literature.
address: '$^1$Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada\'
author:
- 'E.J. Janse van Rensburg$^1$'
bibliography:
- 'References.bib'
title: 'Microcanonical Simulations of Adsorbing Self-Avoiding Walks'
---
Introduction {#section1}
============
The adsorption of a linear polymer on an attractive surface is a conformational rearrangement of the polymer to a state where it explores conformations which remain near or on the surface. This is the so-called polymer adsorption transition, and the many models of this phenomenon (see, for example, reference [@JvR15]) remain a rich source of mathematical and numerical studies. Polymer adsorption is a phase transition [@BDG83], and the properties of the adsorbed polymer have been examined both experimentally (see, for example, references [@DR71; @BDD95; @MBM14]) and theoretically [@deG79]. These models include directed path models of adsorbing polymers [@PFF88; @W98], as well as self-avoiding walk models [@HTW82], and these have received considerable attention in the literature [@HG94], using both rigorous methods [@RW11] and numerical methods (for example, the Monte Carlo simulation of adsorbing self-avoiding walks [@JvRR04]).
In this study, a new Monte Carlo method for sampling adsorbing self-avoiding walks is proposed and implemented. In particular, the GAS algorithm [@JvRR09] is generalised to sample adsorbing walks in the microcanonical ensemble, and the data obtained are analysed to estimate the locations of critical points, and the values of critical exponents and scaling of adsorbing walks, in the square and cubic lattices. The algorithm is related to the Rosenbluth algorithm [@RR55] and to the GARM algoritm [@RJvR08]. The GARM algorithm is related to the PERM algorithm [@G97; @HG11], and flat histogram implementations of PERM [@PK04] have been used to sample states from a flat histogram over state space in a variety of different models of interacting walks, including collapsing walks [@PO00] and adsorbing walks [@KPOL04].
figure1-.tex
The GAS algorithm was introduced in reference [@JvRR09] and was used in several studies as an algorithm to approximately enumerate walks or polygons [@JvRR11; @JvRR11A]. However, it was not clear how to generalise the algorithm to sample states in models of interacting walks. In this paper our purpose is (1) to generalise the algorithm to a model of adsorbing walks, (2) to examine the behaviour of the algorithm by computing critical points and exponents of the walk, and to compare this to results found elsewhere, and (3) to use our data to examine scaling in adsorbing walks by computing critical exponents and examining the scaling of thermodynamic functions.
Adsorbing walks {#subsection1.2}
---------------
An adsorbing self-avoiding walk in the square lattice is illustrated in figure \[figure0\]. Let $\mathL^d$ denote the $d$-dimensional hypercubic lattice and denote a unit length edge with endpoints $\vec{x}$ and $\vec{y}$ in $\mathL^d$ by $\edge{\vec{x}}{\vec{y}}$. The hypercubic half-lattice $\mathL^d_+$ is defined by $$\mathL^d_+ =
\{ \edge{\vec{x}}{\vec{y}} \in \mathL^d \vv
\hbox{$\vec{x}(d) \geq 0$ and $\vec{y}(d) \geq 0$} \},$$ where $\vec{x}(d)$ is the $d$-th Cartesian component of $\vec{x}$, and $\vec{y}(d)$ is the $d$-th Cartesian component of $\vec{y}$. The boundary of $\mathL^d_+$ is given by $$\partial\mathL^d_+ =
\{ \edge{\vec{x}}{\vec{y}} \in \mathL^d_+ \vv
\hbox{$\vec{x}(d)=0$ and $\vec{y}(d)=0$} \},$$ and it is isomorphic to $\mathL^{d-1}$ if $d\geq 2$. Please note that these definitions, and the definitions of additional functions and quantities, are listed in table \[Defs\].
The number of self-avoiding walks of length $n$ from the origin in $\mathL^d$ is denoted by $c_n$. The *growth constant* of the self-avoiding walk [@H57; @H60] is defined by the limit $$\lim_{n\to\infty} c_n^{1/n} = \mu_d,$$ and $\kappa_d = \log \mu_d$ is the connective constant of the self-avoiding walk. This shows that $c_n = \mu_d^{n+o(n)}$. Self-avoiding walks from the origin in $\mathL^d_+$ are *positive walks*. The number of positive walks of length $n$ is denoted by $c_n^+$, and it is known that $c_n^+ = \mu_d^{n+o(n)}$ [@HTW82].
If positive walks are counted with respect to the number of vertices in $\partial\mathL^d_+$ (these are *visits*), then the walks are *adsorbing walks*. For example, the walk in figure \[figure0\] is an adsorbing walk with $6$ visits.
Let $c_n^+(v)$ be the number of adsorbing walks of length $n$ from the origin in $\mathL^d_+$, with $v$ visits to $\partial \mathL^d_+$. The canonical partition function of adsorbing walks is obtained by introducing an activity $a$ conjugate to the number of visits: $$Z_n(a) = \sum_{v=0}^n c_n^+(v)\,a^v .
\label{eqnZ} $$ When $a$ is large, then $Z_n(a)$ is dominated by walks with a large number of visits, and if $a$ is small (but positive), then $Z_n(a)$ is dominated by walks with a small number of visits.
The *finite size free energy* of these models is computed from the partition function $Z_n(a)$ (see equation ), and is given by $$\C{F}_n(a) = \Sfrac{1}{n} \log \sum_v c_n^+(v)\, a^v .
\label{eqn23} $$ The *limiting free energy* of the model is given by the thermodynamic limit in the model: $$\C{F}(a) = \lim_{n\to\infty} \Sfrac{1}{n} \log \sum_v c_n^+(v)\, a^v .
\label{eqn25F} $$ This limit exists (see reference [@HTW82], and also, for example, reference [[@JvR15]]{}), and it is a convex function of $\log a$ with a singular point at $a=a_c^+$ (which is the *adsorption critical point* in the model). For $a<a_c^+$ the model is in a *desorbed state*, and for $a>a_c^+$ the model is in an *adsorbed state*. It is known that $a_c^+>1$ [@JvR98], and $a_c^+ < \sfrac{\mu_d}{\mu_{d-1}}$ [@HTW82], and $$\C{F}(a) \cases{
= \log \mu_d, & if $a\leq a_c^+$; \\
> \log \mu_d, & if $a>a_c^+$.
}$$ Since $\C{F}(a)$ is a convex function of $\log a$, it is differentiable for almost all $a>0$, and it follows that the density of visits to the adsorbing plane is $a\sfrac{d}{da}\C{F}(a) = 0$ if $a<a_c^+$ (this is the *desorbed phase*), and $a\sfrac{d}{da} \C{F}(a) >0$ if $a>a_c^+$ $\alsu$ (whenever $\C{F}(a)$ is differentiable). This is the *adsorbed phase*. In the desorbed phase the walk tends to make few returns to the adsorbing plane (walks of length $n$ will return, on average, $o(n)$ times to the adsorbing plane in the desorbed phase). Thus, a desorbed walk will tend to move away from the boundary into the bulk of $\mathL^d_+$. An adsorbed walk, on the other hand, is expected to have a positive density of returns to the adsorbing plane. This implies that the walk will remain near the adsorbing plane, and so have the properties of a walk which is stretched out in $(d\minus 1)$ dimensions near $\partial\mathL^d_+$ (and compressed in the $d$-th dimension). Separating these two regimes is the adsorption critical point $a_c^+$.
The finite size scaling of the free energy $\C{F}_n(a)$ is given by $$\C{F}_n(a) \simeq \log \mu_d + (a\minus a_c^+)^{2-\alpha}\,f(n^\phi(a\minus a_c^+)),
\label{eqn24} $$ where $f$ is a scaling function, $\alpha$ is the specific heat exponent, and $\phi$ is the finite size crossover exponent. The exponents $\alpha$ and $\phi$ are related by the hyperscaling relation $$2-\alpha=\Sfrac{1}{\phi} .
\label{eqn25} $$
The bulk entropy contribution to $\C{F}_n(a)$ is $\log\mu_d$ in $\mathL^d_+$ (where $\mu_d$ is the growth constant of self-avoiding walks in the square lattice). Slightly redefining the scaling function, it is found that $$\C{F}_n(a) \simeq \log \mu_d + \Sfrac{1}{n}\, g(n^\phi (a\minus a_c^+)).
\label{eqn26} $$ By plotting $n(\C{F}_n(a) \minus \log\mu_d)$ against $n^\phi(a\minus a_c^+)$, the function $g$ can be uncovered (for $n$ large and $|a\minus a_c^+|$ small).
Taking derivatives of $\C{F}_n(a)$ to $\log a$ gives the energy (density) $\C{E}_n(a)$ and specific heat $\C{C}_n(a)$ of the model. The scaling of these quantities follows directly from equation : $$\C{E}_n(a) \simeq n^{\phi-1}\,h_e(n^\phi(a\minus a_c^+)),
\q \hbox{and}\q
\C{C}_n(a) \simeq n^{\alpha\phi}\,h_c(n^\phi(a\minus a_c^+)),
\label{eqn24EC} $$ for some scaling functions $h_e$ and $h_c$. In the limit as $n\to\infty$, $\C{E}_n(a) \to \C{E}(a)$ (the limiting energy density) and $\C{C}_n(a)\to\C{C}(a)$ (the limiting specific heat). Existence of these limits (almost everywhere) is a consequence of the convexity properties of the limiting free energy (see for example reference [@JvR15]). Physically, $\C{E}(a)$ is the density of visits per unit length, and $\C{C}(a)$ is the rate of change in $\C{E}(a)$ as a function of changes in $\log a$ (it has a maximum at $a_c^+$).
For adsorbing walks it is thought that $\phi=\sfrac{1}{2}$ in all dimensions $d\geq 2$ [@BEG89; @BY95], and numerical evidence supporting this in dimensions lower than $d=4$ (the upper critical dimension) are available in references [@LM88A; @ML88A; @BWO99; @JvRR04; @KPHMBS13]. If $\phi = \sfrac{1}{2}$, then $\alpha = 0$, so, for example, the specific heat has scaling $\C{C}_n(a) = h_c(n^\phi(a\minus a_c^+))$, and plotting measurements of $\C{C}_n(a)$ against the rescaled variable $\tau=n^\phi(a\minus a_c^+)$ for small values of $\tau$ should collapse the curves to a limiting curve (with some finite size corrections to scaling), exposing the scaling function $h_c$.
The partition function has a more complicated scaling law (see, for example, equation (23) in reference [@JvRR04], or section 4.2.2 in reference [@JvR15]). In the high temperature (or small $a$) regime, the partition function scales as $$Z_n(a) \simeq B_\lambda n^{\gamma_t -1} h_\lambda(n^\phi |t|)\,
\kappa_-^{n\,|t|^{1/\phi}},\q
\hbox{if $a< a_c^+$},
\label{eqn3} $$ where $t = (a\minus a_c^+)$ and $\lambda$ denotes the high temperature regime, and where $h_\lambda(x) \simeq |x|^{(\gamma_1-\gamma_t)/\phi}$ and $\log \kappa_- \simeq |a\minus a_c^+|^{-1/\phi}\log \mu_d$. Putting $a=1$, for example, and adsorbing constants and functions of $t$ into $B_\lambda$, give $$Z_n(1) \simeq B_\lambda n^{\gamma_1-1} \mu_d^n,
\q\hbox{since $\C{F}(a) = \log \mu_d$ if $a<a_c^+$},$$ and the exponent $\gamma_1$ is the entropic exponent of half-space walks, namely $c_n^+ \sim n^{\gamma_1-1} \mu_d^n$.
At the critical adsorption point $a_c^+$, the above scaling is modified to $$Z_n(a_c^+) \simeq B_c n^{\gamma_t -1}\,\mu_d^n ,\q
\hbox{if $a= a_c^+$},
\label{eqn15} $$ where $\gamma_t$ is the entropic exponent associated with adsorbing walks at the critical adsorption point. The ensemble of half-space walks at the critical point has associated *surface entropic exponent* $\gamma_s$, and this is related to $\gamma_t$ by $\gamma_t
= \gamma_s$ (see for example section 9.1.3 in reference [@JvR15]).
The scaling in the adsorbed phase is similar to the above, but now with different exponents $$Z_n(a) \simeq B_{\tau_0} n^{\gamma_t -1} h_{\taus0} (n^\phi |t|)\,
\kappa_+^{n\,|t|^{1/\phi}},\q
\hbox{if $a> a_c^+$},$$ where $h_{\taus0} (x) \simeq |x|^{(\gamma_+ -\gamma_t)/\phi}$, and where the subscript $\taus0$ denotes the low temperature (and large $a$) scaling. Since $\sfrac{1}{n} \log Z_n(a) = \C{F}(a)\,(1\plus o(1))$, it follows that $\log \kappa_+ \simeq |a\minus a_c^+|^{-1/\phi} \C{F}(a)$. This, in particular, gives the scaling $$\C{F}_s(a) \sim |a \minus a_c^+|^{1/\phi}$$ for the singular part of the free energy in the adsorbed phase (consistent with the hyperscaling relation ). The scaling of $Z_n(a)$ simplifies here to $$Z_n(a) \simeq B_{\tau_0} n^{\gamma_+-1} \; e^{n\thin\C{F}(a)} ,\q
\hbox{if $a>a_c^+$}.$$ The exponent $\gamma_+$ should be that of adsorbed walks, and so given by $\gamma_+ = \gamma^{(d-1)}$, the entropic exponent of walks in one dimension lower.
Function Definition
-------------------------- ------------------------------------------------------------------------------------------
$\mathL^d$ The hypercubic lattice
$\mathL^d_+$ The half-hypercubic lattice
$\partial\mathL^d_+$ The boundary of $\mathL^d_+$ (it is isomorphic is $\mathL^{d-1}$)
$\mu_d$ The growth constant of self-avoiding walks in $\mathL^d$
$c_n^+(v)$ The number of positive walks of length $n$ and $v$ visits from $\vec{0}$ in $\mathL^d_+$
$Z_n(a)$ Partition function of positive adsorbing walks of length $n$ and activity $a$
$\C{F}_n(a)$ Finite size free energy: $\displaystyle F_n(a) = \sfrac{1}{n} \log Z_n(a)$
$\C{F}(a)$ Limiting free energy: $\displaystyle \C{F}(a) = \lim_{n\to\infty} \C{F}_n(a)$
$\C{E}_n(a)$, $\C{E}(a)$ Energy density and limiting energy density
$\C{C}_n(a)$, $\C{C}(a)$ Specific heat and limiting specific heat
$H_n$ The mean height of the endpoint of the walk (a function of $a$)
$R_n^2$ The mean square radius of gyration of the walk (a function of $a$)
$P^+(\eps)$ The microcanonical density function (see equation )
$a_c^+$ The adsorption critical point
$\phi$ The crossover exponent
$\alpha$ The specific heat exponent
$\gamma_1$ The half-space entropic exponent, see also $\gamma_s$ (the surface exponent)
$\nu$ The metric exponent
$G(a,t)$ Generating function of $Z_n(a)$ (see equation )
$G_N(a,t)$ Truncated generating function (see equation )
$t_c^+(a)$ The radius of convergence of $G(a,t)$
: Short list of definitions[]{data-label="Defs"}
Organisation of the manuscript
------------------------------
This paper is a report on two aspects of the Monte Carlo simulation of adsorbing walks. The first is the generalisation of the GAS algorithm to a model of interacting walks, and in particular, an implementation of this algorithm to achieve flat histogram sampling over state space of adsorbing square and cubic lattice walks. The second aspect of the paper is a report on the properties of adsorbing self-avoiding walks in the square and cubic lattices. The aim here is to verify the results obtained in the Multiple Markov Chain Monte Carlo study in reference [@JvRR04], and also to use the data generated here to test the scaling of the thermodynamic and metric quantities of adsorbing walks.
The model of adsorbing walks is defined in section \[subsection1.2\], and its partition function and free energy are discussed. The limiting free energy of this model exists, and its properties have been examined elsewhere [@HTW82]; see, for example, reference [@JvR15]. The basic scaling relations for the free energy, energy, and specific heat were introduced above, and the scaling of the partition function was briefly reviewed.
In section \[section2\] the GAS algorithm [@JvRR09] is reviewed and then generalised to interacting models. The algorithm normally has only one set of parameters (associated with the size of the walks), but it is shown here that introducing a second set of parameters (associated with the energy of the walks) can give an algorithm which samples effectively in both length and energy. It is shown that the algorithm can be tuned to give flat histogram sampling in the spirit of the PERM algorithm [@G97; @HG11] (but without using enrichment or pruning of states).
Numerical results for adsorbing walks are analysed in section \[section3\]. The location of the adsorption critical point is determined from the mean energy of adsorbing walks, giving $$a_c^+ =
\cases{
1.779 \pm 0.003, & \hbox{in the square lattice}; \\
1.306 \pm 0.007, & \hbox{in the cubic lattice}.
}$$ These results are then used to estimate the crossover exponent $\phi$ associated with the adsorption transition, giving $$\phi =
\cases{
0.496 \pm 0.009, & \hbox{in two dimensions}; \\
0.505 \pm 0.006, & \hbox{in three dimensions}.
}$$ The microcanonical density function of adsorbing walks is determined as well, and shown to have properties consistent with the location of the critical points above. In addition, the specific heat of the model is determined, and it is found that it has scaling behaviour consistent with the estimates of the critical points above.
It is also shown that the adsorption transition is seen in a change in the metric scaling of walks at the critical point. The desorbed phase is a phase of positive walks with the scaling properties of self-avoiding walks in a good solvent, while the adsorbed walk has the metric properties of a walk in one dimension less; that is, of walks adhering to the adsorbing surface.
Scaling of the generating and partition functions are found to be consistent with the exact values of critical exponents determined elsewhere in two dimensions [@BEG89; @BY95], and with the value $\phi=\shalf$ in three dimensions [@HG94; @JvRR04; @ML88A]. Similary, the data are also consistent with estimates for the surface exponent $\gamma_s$; this is seen particularly in the scaling of the partition function (see equation ).
The paper is concluded with a few brief remarks and a summary in section \[section4\].
figure2-.tex
GAS Sampling of self-avoiding walks {#section2}
===================================
The GAS algorithm is a generalisation of kinetic growth algorithms. It is designed to sample along weighted sequences in state space in such a way that the ratios of average weights of sequences ending in walks of length $n$ and $m$ are estimates of the ratio of the numbers of walks of lengths $n$ and $m$. In this section I show how to generalise this algorithm so that it can be used to estimate the number of walks of length $n$ and *energy* $m$. That is, the algorithm will be used to sample walks in the microcanonical ensemble.
Let $w=\LA \omega_0,\omega_1,\ldots,\omega_n\RA$ be a self-avoiding walk of length $n$ from its source vertex $\omega_0 =\vec{0}$ at the origin, to its terminal vertex $\omega_n$, giving $n$ steps $\LA \omega_{j-1},\omega_j\RA$ for $j=1,2,\ldots,n$. The walk $w$ may be made longer by adding a step $\LA \omega_n,\omega_{n+1}\RA$ to $\omega_n$, or it may be made shorter by removing its last step. These two operations compose an end-point elementary move for sampling self-avoiding walks, as illustrated in figure \[figure1\]: A *positive elementary move* is the addition of an edge to the endpoint of a growing walk. The reverse of a positive elementary move is a *negative elementary move*, namely the deletion of the least edge in a walk. Notice that every positive move is immediately reversible by a negative elementary move.
Endpoint elementary moves have been used widely in the simulation of self-avoiding walks (for example the Rosenbluth algorithm [@RR55], and the Beretti-Sokal algorithm [@BS85]). In what follows the discussion will be restricted to endpoint elementary moves; however, the algorithms generalise directly if other elementary moves, such as BFACF elementary moves [@BF81; @AA83], or generalised atmospheric moves [@RJvR08], are used instead.
figure3-.tex
As an example, consider the elementary move in figure \[figure1\], which is an implementation of an endpoint elementary move on a self-avoiding walk in the positive half-lattice $\mathL^+_2$ (where $\vec{w}(2)$ is the $y$-coordinate of $\vec{w}$). The set of lattice edges in $\mathL^+_2$ which may be appended to the walk $w$ to extend it by one step is the *positive atmosphere* of $w$, and the number of edges in the positive atmosphere is denoted by $a^+(w)$. For example, for the walk on the left in figure \[figure1\], $a^+(w)=2$.
Similarly, the set of edges which may be removed from the endpoint of a walk to decrease its length by one, is the *negative atmosphere* of the walk. For endpoint elementary moves, the last step is always the sole negative atmospheric edge, so that the size of the negative atmosphere for endpoint elementary moves is always $a^-(w)=1$, if $w$ is not the trivial walk of length $0$.
An elementary move may change the energy of a walk. For example, in a model of adsorbing walks in $\mathL^+_2$, the energy is the number of *visits* of the walk to the adsorbing line $\partial\mathL^+_2$ (the boundary of the half-lattice $\mathL^+_2$). The positive move in figure \[figure1\] does not change the number of visits, but the move in figure \[figure2\] increases the number of visits (and so the energy) by $1$. The negative move in figure \[figure2\] similarly reduces the number of visits by $1$.
figure4-.tex
A similar situation arises if a model of collapsing walks with energy given by nearest neighbour *contacts* between vertices in the walk which are adjacent in $\mathL$, but not in the walk. This is illustrated in figure \[figure3\]; the positive elementary move creates $2$ new contacts in the walk, and so it changes the energy of the walk by $2$.
More general elementary moves (for example, the BFACF elementary moves) may contribute to the atmospheric statistics $a^+$ and $a^-$ in various ways, and may even give rise to *neutral atmospheres* which do not change the length of the walk (but which may change the energy of the walk).
Thus, in what follows, let $a^+_v(w)$ be the size of the positive atmosphere of a walk $w$ of length $\ell_n=|w|$, and which changes the energy of $w$ by $v$ units. For example, the walk on the left in figure \[figure1\] has $a^+_{0}(w)=1$ and $a^+_{1}(w)=1$, and if this was a walk in the three dimensional half-lattice $\mathL_3^+$, then $a^+_{0}(w)=3$ and $a^+_{1}(w)=1$. The walk on the left in figure \[figure3\] (with energy given by the number of contacts), has, in a similar way, $a^+_{0} (w)=2$, $a^+_{1}(w)=0$ and $a^+_{2} (w)=1$.
In exactly the same way one may define the size of the negative atmosphere of a walk $w$ of length $\ell$ which changes the energy by $v$, denoted by $a^-_{v} (w)$.
The neutral atmosphere (an elementary move which does not change the length of the walk) of a walk $w$ is similarly given by $a^0_{v} (w)$, if it changes the energy of a walk by $v$. If the endpoint elementary move in figure \[figure1\] is used, then $a^0_{v} (w)=0$ by default (since there are no neutral elementary moves implemented), but in general assume that a more general set of elementary moves (for example BFACF elementary moves) is used to sample walks, and in that case the neutral atmosphere may have positive size.
Implementation of GAS-sampling
------------------------------
Suppose that an elementary move is implementated on the state space of self-avoiding walks from the origin, and assume the implementation is irreducible (that is, the elementary move gives a connected graph on the state space of walks).
Suppose that the sequence $$\phi_N = \LA w_0,w_1,w_2,\ldots,w_n,\ldots,w_N\RA
\label{eqn1} $$ is realised after $N$ steps and that the atmospheres of the states $w_n$ have sizes (or *statistics*) $a_{v}^+(w_n)$, $a_{v}^0(w_n)$ and $a_{v}^-(w_n)$. These elementary moves may be classified as follows, depending on whether they increase or decrease the lengths of the walks (or are neutral), or whether they increase or decrease, or leave unchanged, the energy of the walk. This is done by defining atmospheric statistics as follows: Let the states $w_n$ be walks of length $\ell_n = |w_n|$. Define $$\begin{aligned}
\alpha_n^{--} = \sum_{v< 0} a_{v}^- (w_n),\qq &
\alpha_n^{-0} = \sum_{v=0} a_{v}^-(w_n), \qq
\alpha_n^{-+} = \sum_{v>0} a_{v}^-(w_n); \\
\alpha_n^{0-} = \sum_{v< 0} a_{v}^0 (w_n),\qq &
\alpha_n^{00} = \sum_{v=0} a_{v}^0(w_n), \qq
\alpha_n^{0+} = \sum_{v>0} a_{v}^0(w_n); \\
\alpha_n^{+-} = \sum_{v< 0} a_{v}^+ (w_n),\qq &
\alpha_n^{+0} = \sum_{v=0} a_{v}^+(w_n), \qq
\alpha_n^{++} = \sum_{v>0} a_{v}^+(w_n). \end{aligned}$$ For example, $\alpha_n^{--}$ is the number of negative elementary moves which also *decreases* the energy of the walk, and $\alpha_n^{-+}$ is the number of negative elementary moves which also *increases* the energy of the walk. The rest of the $\alpha$’s are similarly defined.
With these atmospheric statistics defined, a rule needs to be constructed in order to realise the sequence $\phi_N$ in equation .
The endpoint elementary moves in figures \[figure2\] and \[figure3\] have the property that no positive elementary move can decrease the energy, and no negative elementary move can increase the energy. Moreover, there are no neutral moves amongst the elementary moves. Thus, assume that $$\alpha_n^{-+} = 0,\q\alpha_n^{+-}=0,\q\alpha_n^{0-}=0,\q\alpha_n^{00}=0
\q\hbox{and}\; \alpha_n^{0+} = 0.$$ The algorithm can be modified appropriately to account for such transitions in models where this is not the case. This assumption leaves the following atmospheric statistics: $\{\alpha_n^{--},\alpha_n^{-0},\alpha_n^{+0},\alpha_n^{++}\}$. That is, distinguish between negative moves which decrease the energy, or negative moves which leave the energy unchanged, or positive moves which leave the energy unchanged, and positive moves which increase the energy.
Introduce parameters $\{\beta_{\ell,u}\}$ to control positive elementary moves on walks of length $\ell$ and energy $u$, and which leave the energy unchanged (that is, the elementary moves contributing to $\alpha_n^{+0}$). Similarly, introduce the parameters $\{\gamma_{\ell,u}\}$ to control positive elementary moves which increase the energy on walks of length $\ell$ and energy $u$.
The transition probabilities of positive elementary moves which leave the energy unchanged will be proportional to $\beta_{\ell,u}$; if the state has length $\ell$ and energy $u$. For example, since $\ell=16$ and $u=2$ in the walk in figure \[figure1\], the transition probability of the positive move in that figure is proporsional to $\beta_{16,2}$. Similarly, the positive elementary move in figure \[figure2\] increases the energy; so here the transition probability is proportional to $\gamma_{16,2}$, instead. In figure \[figure3\] the positive elementary move also increases the energy, and so its transition probability is proportional to $\gamma_{16,4}$.
Thus, if $w_n$ is the current state (of length $\ell_n$ and energy $u_n$), and $w_{n+1}$ is the next state (of length $\ell_{n+1}$ and energy $u_{n+1}$), then define the change in length by $\Delta_n = \ell_{n+1}\minus \ell_n$, and the change in energy by $\delta_n = u_{n+1} \minus u_n$. Notice that $\Delta_n=\pm 1$ for endpoint elementary moves, and that $\delta_n = \pm 1$, or $\delta_n = 0$, for the model of adsorbing walks in figures \[figure1\] and \[figure2\] (but these quantities may take on other values in the model in figure \[figure3\]). The transition probabilities are chosen such that $$\Pr(w_n\to w_{n+1}) \propto
\cases{
\beta_{\ell,u}, & \hbox{if $\Delta_n = +1$ and $\delta_n=0$;} \\
1 & \hbox{if $\Delta_n = -1$ and $\delta_n=0$},
}$$ where $\ell=\ell_n$ and $u=u_n$ are functions of $n$. In the case that the energy is changed, then the transition probability is constructed such that $$\Pr(w_n\to w_{n+1}) \propto
\cases{
\gamma_{\ell,u}, & \hbox{if $\Delta_n=+1$ and $\delta_n>0$;} \\
1 & \hbox{if $\Delta_n=-1$ and $\delta_n < 0$}.
}$$ Normalising the transition probabilities gives $$\fl
\Pr(w_n \to w_{n+1})
= \cases{
\frac{\beta_{\ell,u}}{\alpha_n^{--} + \alpha_n^{-0} + \alpha_n^{+0}\thin\beta_{\ell,u}
+ \alpha_n^{++}\thin\gamma_{\ell,u}}, & \hbox{if $\Delta_n=1$ and $\delta_n=0$}; \\
\frac{\gamma_{\ell,u}}{\alpha_n^{--} +\alpha_n^{-0} + \alpha_n^{+0}\thin\beta_{\ell,u}
+ \alpha_n^{++}\thin\gamma_{\ell,u}}, & \hbox{if $\Delta_n=1$ and $\delta_n>0$}; \\
\frac{1}{\alpha_n^{--} +\alpha_n^{-0} + \alpha_n^{+0}\thin\beta_{\ell,u}
+ \alpha_n^{++}\thin\gamma_{\ell,u}}, & \hbox{if $\Delta_n=-1$ and $\delta_n\leq 0$}; \\
}
\label{eqn4} $$ The probability for any type of move can be explicitly computed for any given state $w_n$ (of length $\ell$ and energy $u$) by computing the $\alpha$’s. For example, the probability for a positive elementary move which leaves the energy unchanged is $\alpha_n^{+0}\thin\beta_{\ell,u}/(\alpha_n^{--} \plus \alpha_n^{+0}\thin\beta_{\ell,u} \plus \alpha_n^{++}\thin\gamma_{\ell,u})$. For the state on the left in figure \[figure1\] this becomes $\beta_{16,2}/(1+\beta_{16,2}+\gamma_{16,2})$ since $\alpha_{16}^{--}=1$, $\alpha_{16}^{+0} = 1$ and $\alpha_{16}^{++}=1$.
GAS-weights
-----------
Let $w_0$ be a starting state (possibly the walk consisting of a single vertex at the origin, of length $\ell_0=0$ and energy $u_0=0$). Implement the endpoint elementary moves on $w_0$ by computing its atmosphere recursively and updating it using the transition probabilities in equation , starting at $n=0$. This generates a Markov Chain of states in a sequence $\phi_N$ (see equation ).
Assume that the parameters $\{\beta_{n,u},\gamma_{n,u}\}$ are known and fixed, so that the sampling can be implemented by simply computing the transtition probabilities and selecting positive and negative elementary moves with appropriate probabilities.
The probability of the sequence $\phi_N$ is given by $$\fl
\Pr (\phi_N) = \prod_{n=0}^{N-1}
\frac{1}{\alpha_n^{--} + \alpha_n^{-0}
+ \alpha_n^{+0}\thin\beta_{\ell,u}
+ \alpha_n^{++}\thin \gamma_{\ell,u}}
\; {\prod}_m^\prime \beta_{\ell,u}
\; {\prod}_k^{\prime\prime} \gamma_{\ell,u}
\label{eqn6} $$ where the primed product ${\prod}^\prime$ is over all the $\beta_{\ell,u}$ for transitions through positive elementary moves leaving the energy unchanged, and the double primed product ${\prod}^{\prime\prime}$ is over all the $\gamma_{\ell,u}$ where the transition is a positive elementary move increasing the energy. In this expression the $\ell$ and $u$ are functions of $n$, $m$, and $k$ in each of the products.
A weight $W(\phi_N)$ will be assigned to the sequence $\phi_N$. In order to compute the weight, define $$\fl
\sigma(j,j\plus 1) = \sigma(w_j\to w_{j+1}) =
\cases{
-1, &\hbox{if $\Delta_j=+1$ and $\delta_j=0$}; \\
+1, &\hbox{if $\Delta_j=-1$ and $\delta_j=0$}; \\
\hspace{0.8em} 0, &\hbox{otherwise}. \\
}$$ Similarly, define $$\fl
\rho (j,j\plus 1) = \rho (w_j\to w_{j+1}) =
\cases{
-1 , &\hbox{if $\Delta_j=+1$ and $\delta_j>0$}; \\
+1, &\hbox{if $\Delta_j=-1$ and $\delta_j<0$}; \\
\hspace{0.8em} 0, &\hbox{otherwise}. \\
}$$ That is, the function $\sigma(j,j\plus 1)$ tracks the negative and positive moves along $\phi_N$ where the energy is not changed, and the function $\rho(j,j\plus 1)$ tracks the negative and positive transitions along $\phi_N$ where the energy is also changed.
Assign the weight $$\fl
W(\phi_N) = \LB \Sfrac{\alpha^{--}_0 + \alpha^{-0}_0 + \alpha^{+0}_0 \beta_0
+ \alpha^{++}_0 \gamma_0}{
\alpha^{--}_N +\alpha^{-0}_N +\alpha^{+0}_N \beta_N+ \alpha^{++}_N \gamma_N} \RB
\prod_{j=0}^{N-1} \beta_{\ell,u}^{\sigma(j,j{+}1)}
\prod_{j=0}^{N-1} \gamma_{\ell,u}^{\rho(j,j{+}1)}
\label{eqn9} $$ to the sequence $\phi_N$.
The expected value of the weight over sequences of length $N$ from state $w_0$ to state $w_N$ is $$\LA W(w_0\to w_N) \RA_N = \sum_{\phi:w_0 \to w_N} \Pr(\phi) W(\phi) .
\label{eqn10} $$ Inserting equations and in this, and simplifying, gives $$\fl
\LA W(w_0\to w_N) \RA_N = \sum_{\phi:w_0\to w_N}
\prod_{j=1}^N
\frac{1}{\alpha_j^{--} \plus \alpha_j^{-0} \plus \alpha_j^{+0}\thin\beta_{\ell,u}
\plus \alpha_j^{++}\thin \gamma_{\ell,u}}
\; {\prod}^{n} \beta_{\ell,u}
\; {\prod}^{nn} \gamma_{\ell,u}
\label{eqn11} $$ where the product ${\prod}^{n}$ is over all the $\beta_{\ell,u}$ for transitions through *negative* elementary moves leaving the energy unchanged, and the product ${\prod}^{nn}$ is over all $\gamma_{\ell,u}$ where the transition is a *negative* elementary move decreasing the energy. As before, the $\ell
\equiv \ell(j)$ and $u\equiv u(j)$ are functions of $j$ (in other words, functions of the states $w_j$ in the sequence $\phi$).
Reverse all the sequences in equation so that the starting state is $w_N$ and the final state is $w_0$. Under this reversal all negative elementary moves become positive elementary moves and vice versa. That is, equation becomes $$\fl
\LA W(w_0\to w_N) \RA_N = \sum_{\psi:w_N\to w_0}
\prod_{j=1}^N
\frac{1}{\alpha_j^{--} \plus \alpha_j^{-0} \plus \alpha_j^{+0}\thin\beta_{\ell,u}
\plus \alpha_j^{++}\thin \gamma_{\ell,u}}
\; {\prod}^{n} \beta_{\ell,u}
\; {\prod}^{nn} \gamma_{\ell,u} ,
\label{eqn12} $$ where, as before, the product ${\prod}^{n}$ is over all the $\beta_{\ell,u}$ for transitions through *negative* elementary moves along the *reverse sequence* $\psi$ leaving the energy unchanged, and the product ${\prod}^{nn}$ is over all $\gamma_{\ell,u}$ where the transition is a *negative* elementary move along the *reverse sequence* $\psi$ decreasing the energy.
For example, consider the model of collapsing walks in figure \[figure3\] and suppose the sequence $\phi$ is realised, where
at -25 5 at 0 0 at 20 0 40 0 50 0 / at 40 0 50 0 / at 70 0 90 0 100 0 100 10 / at 90 0 100 0 100 10 / at 120 0 140 0 150 0 150 10 140 10 / at 140 0 150 0 150 10 140 10 / at 170 0 190 0 200 0 200 10 / at 190 0 200 0 200 10 / at 220 0 240 0 250 0 250 10 240 10 / at 240 0 250 0 250 10 240 10 / at 270 0 300 0 310 0 310 10 300 10 290 10 / at 300 0 310 0 310 10 300 10 290 10 / at 320 5 at 0 -10 at 45 -10 at 95 -10 at 145 -10 at 195 -10 at 245 -10 at 300 -10 at 0 -20
The probability of this sequence is $$\fl
\Pr(\phi) = \Sfrac{\beta_{0,0}}{4\beta_{0,0}}
\Sfrac{\beta_{1,0}}{(1+3\thin\beta_{1,0})}
\Sfrac{\gamma_{2,0}}{(1+2\thin\beta_{2,0}+\gamma_{2,0})}
\Sfrac{1}{(1+2\thin\beta_{3,1})}
\Sfrac{\gamma_{2,0}}{(1+2\thin\beta_{2,0}+\gamma_{2,0})}
\Sfrac{\beta_{3,1}}{(1+2\thin\beta_{3,1})}$$
The weight of $\phi$ can similarly be computed from equation . This gives $$W(\phi) = \Sfrac{4\thin\beta_{0,0}}{1+3\thin\beta_{4,1}}
\times \LB \beta_{0,0}^{-1}\,\beta_{1,0}^{-1}\,\gamma_{2,0}^{-1}
\,\gamma_{3,1}^{+1}\,\gamma_{2,0}^{-1}\,\beta_{3,1}^{-1}\RB .$$ The consequence is that $$\fl
\Pr(\phi)W(\phi) =
\Sfrac{1}{(1+3\thin\beta_{4,1})}
\Sfrac{1}{(1+2\thin\beta_{3,1})}
\Sfrac{1}{(1+2\thin\beta_{2,0}+\gamma_{2,0})}
\Sfrac{\gamma_{3,1}}{(1+2\thin\beta_{3,1})}
\Sfrac{1}{(1+2\thin\beta_{2,0}+\gamma_{2,0})}
\Sfrac{1}{(1+3\thin\beta_{1,0})} ,$$ and this is the probability that a sequence $\psi$, starting in the state $w_6$ and terminating in the state $w_0$, is realised by the algorithm.
The same observation is true generally for equation : The summand is the probability that a particular sequence $\psi$ from state $w_N$ to $w_0$ is realised by the algorithm, and the summation is over all such sequences $\psi$. That is, $\LA W(w_0\to w_N) \RA_N = \Pr(w_N\to w_0)$ is the probability that the algorithm realises a sequence $\psi$ of length $N$ from state $w_N$ to state $w_0$.
The sequence $\psi$ is a Markov Chain, and if it is aperiodic and irreducible, and if $w_0$ is a recurrent state, then asymptotically (for large $N$) the probability that the sequence hits the state $w_0$ is positive and independent of the starting state $w_N$. That is, $\Pr(w_N\to w_0) \to C(w_0)>0$ as $N\to\infty$ where $C(w_0)$ is dependent on the parameters of the algorithm, and independent of $w_N$. Thus, the average weight $\LA W(w_0\to w_N) \RA_N \to C(w_0)$ as $N\to\infty$. Summing $\LA W(w_0\to w_N) \RA_N$ over all the states $w_N$ of length $n = \ell(w_N)$ and energy $u$, shows that the average weights of sequences of length $N$ ending in walks of length $n$ and energy $u$ is $$\LA W_{n,u} \RA_N = \sum_{w_N:\,n,u} \LA W(w_0\to w_N) \RA_N \to c_n(u)\, C(w_0) ,$$ where the summation is over all walks of length $n$ and energy $u$ (and $c_n(u)$ is the number of walks of length $n$ and energy $u$).
Taking ratios of average weights give $$\frac{\LA W_{n,u} \RA_N}{\LA W_{m,v} \RA_N} \to \frac{c_n(u)}{c_m(v)} .$$ That is, if $c_n(u)$ is known for some values of $n$ and $u$, then the ratios of average weights can be used to estimate $c_m(v)$.
$n\backslash u$ 0 1 2 3 4 5 6 7 8 9 10
----------------- ------- ------- ------- ------- ------- ------- ------- ------- -------- ------- -------
0 1.000
1 0.333 1.000
2 0.428 0.250 1.001
3 0.368 0.499 0.200 1.000
4 0.387 0.380 0.416 0.166 1.000
5 0.373 0.396 0.427 0.429 0.142 1.000
6 0.386 0.375 0.377 0.359 0.438 0.124 1.000
7 0.377 0.392 0.384 0.401 0.348 0.445 0.111 1.000
8 0.383 0.378 0.388 0.356 0.368 0.327 0.449 0.100 0.999
9 0.378 0.386 0.385 0.395 0.370 0.368 0.308 0.454 0.0908 0.999
10 0.381 0.379 0.383 0.373 0.368 0.352 0.364 0.289 0.458 0.083 0.999
: Numerical estimates of $\beta_{n,u}$ for adsorbing walks in $\mathL^2_+$[]{data-label="betanu"}
Sampling with GAS
-----------------
The algorithm is implemented by choosing a starting state $w_0$ and then sampling along a sequence $\phi$ of length $N$. The weight is updated along $\phi$, and collected into bins for walks of length $n$ and energy u. Once the sequence is completed, then the average of each bin is computed, giving the average weight $W_{n,u}$ of walks of length $n$ and energy $u$ seen along $\phi$.
This process is repeated $M$ times, so that $M$ sequences of length $N$, denoted by $\LA \phi_1, \phi_2,\ldots,\phi_M\RA$ are realised, and for each sequence $\phi_j$ the average weight $W_{n,u}^{(j)}$ is calculated. The estimated average weight is computed over the $M$ sequences: $$[W_{n,u}]^{est}_{N,M} = \Sfrac{1}{M} \sum_{j=1}^M W_{n,u}^{(j)} .$$ The estimated weight $[W_{n,u}]_{N,M}^{est}$ is an estimator of $\LA W_{n,u}\RA_N$, and so as $M\to\infty$ and $N\to\infty$, it is expected that $[W_{n,u}]_{N,M}^{est}
\to c_n(u)\,C(w_0)$. Taking ratios give $$\frac{[W_{n,u}]_{N,M}^{est}}{[W_{m,v}]_{N,M}^{est}} \to \frac{c_n(u)}{c_m(v)} .$$
The sampling requires that both $N$ and $M$ are sufficiently large, and there is a trade-off between these quantities. $M$ should be large enough to have sufficient independent estimates of the weights to compute sound averages, and $N$ should be large enough to have sufficiently long sequences to have sampled large enough regions of state space.
$n\backslash u$ 0 1 2 3 4 5 6 7 8 9 10
----------------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- ---------
0 0\. 499
1 0\. 499 1\. 000
2 0\. 374 1\. 001 0\. 999
3 0\. 436 0\. 801 0\. 998 0\. 999
4 0\. 451 0\. 668 0\. 832 0\. 998 1\. 000
5 0\. 461 0\. 751 0\. 857 0\. 855 1\. 000 0\. 999
6 0\. 463 0\. 716 0\. 721 0\. 873 0\. 874 0\. 999 0\. 999
7 0\. 470 0\. 734 0\. 767 0\. 845 0\. 887 0\. 889 0\. 998 1\. 000
8 0\. 470 0\. 728 0\. 709 0\. 775 0\. 834 0\. 898 0\. 899 0\. 999 1\. 00
9 0\. 473 0\. 741 0\. 721 0\. 807 0\. 835 0\. 845 0\. 908 0\. 907 1\. 000 0\. 999
10 0\. 474 0\. 736 0\. 699 0\. 752 0\. 795 0\. 836 0\. 853 0\. 915 0\. 915 1\. 000 0\. 999
: Numerical estimates of $\gamma_{n,u}$ for adsorbing walks in $\mathL^2_+$[]{data-label="gammanu"}
There remains the additional issue of the GAS parameters $\beta_{n,u}$ and $\gamma_{n,u}$. These can be estimated using training runs prior to the simulation. Best results are obtained when the sampling is *flat*, so that states of size and energy $\{n,u\}$ are sampled uniformly in $\{n,u\}$. This is best achieved when the GAS sequences are random walks on $n$ and $u$. Thus, the probability of a positive move should be, on average, equal to the probability of a negative move. A good choice for $\beta_{n,u}$ is $$\beta_{n,u} = \frac{\LA \alpha_{n,u}^{-0}\RA}{\LA \alpha_{n,u}^{+0}\RA }
\label{eqn40C} $$ where $\LA\alpha_{n,u}^{-0}\RA$ is the average negative atmosphere which does not decrease the energy, and $\LA\alpha_{n,u}^{+0}\RA$ is the average positive atmosphere which does not increase the energy, of walks of length $n$ and energy $u$.
A similar argument shows that a good choice for $\gamma_{n,u}$ is $$\gamma_{n,u} = \frac{\LA \alpha_{n,u}^{--}\RA}{\LA \alpha_{n,u}^{++}\RA }
\label{eqn41C} $$ where $\LA\alpha_{n,u}^{--}\RA$ is the average negative atmosphere which decreases the energy, and $\LA\alpha_{n,u}^{++}\RA$ is the average positive atmosphere which increases the energy, of walks of length $n$ and energy $u$.
$n\backslash u$ 0 1 2 3 4 5 6 7 8 9 10
----------------- ------ ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
0 8658
1 8616 13208
2 9642 7711 13261
3 8971 9622 7144 13012
4 8919 9576 9261 6528 12883
5 8957 8947 9501 9388 5998 13072
6 8982 8736 9220 9019 9281 6055 13263
7 9059 8562 9026 9095 8711 9733 5812 13278
8 8996 8398 9030 8844 8568 8914 9493 5728 13362
9 9169 8162 8788 8772 8579 8843 8417 9689 5678 13204
10 9134 8182 8580 8434 8559 8898 8390 8221 10101 5477 12883
: The frequency of states sampled to $n=10$ and $u=10$ for adsorbing walks in $\mathL^2_+$[]{data-label="statessampled"}
Computed values for $\beta_{n,u}$ and $\gamma_{n,u}$ for a square lattice adsorbing walk model in figure \[figure1\] are given in tables \[betanu\] and \[gammanu\] for $0\leq n\leq 10$ and $0 \leq u \leq 10$. These data are chopped at an accuracy of three decimal places.
Observe that $\beta_{n,n} = 1$ (when rounded) and $\gamma_{n,n}=1$. This is expected, since there are exactly two states if $n>0$ and $n=u$ (a completely adsorbed walk which never leaves the adsorbing line).
With these values of $\beta_{n,u}$ and $\gamma_{n,u}$ the sampling is reasonably flat, as shown in table \[statessampled\]. The data in table \[statessampled\] is the number of times a sequence of length $10^6$ visited states of length $n$ and energy $u$. The maximum length was set at $n=500$, so that the number of pairs $(n,u)$ is $124750$; thus, the expected number of visits to states of length $n$ and energy $u$ is roughly $8000$. The data in table \[statessampled\] are spread around this number, and the distribution is a reasonably flat histogram. There are exceptions for data along the diagonal where a larger number is seen. The explanation for this is that the sequence can only visit states along the diagonal if it starts in $(n,u)=(0,0)$ and then stay on the diagonal (doing a random walk on states of length and energy both equal to $n$) – the elementary moves chosen in the simulation do not include neutral moves, and so the sequence cannot enter the diagonal except at $n=0$.
Observe that since $c_n^+(n)=2$ for $n>0$ in this model, that these states are rare, but are sampled frequently by the algorithm. This is an example of *rare event* sampling, where an algorithm spends significant time sampling rare states in the tails of a distribution in order to get good estimates of microcanonical quantities.
Simulations were performed by collecting data over $500$ realised sequences, each of length $10^9$. Numerical estimates of $c_{n,u}$ are shown in table \[countsads2d\]. The data in table \[countsads2d\] are rounded to the nearest integer. These data can be compared to exact counts to verify the algorithm and its implementation.
$n\backslash u$ 0 1 2 3 4 5 6 7 8 9 10
----------------- ------ ------ ------ ------ ----- ----- ----- ---- ---- --- ----
0 1
1 1 2
2 3 2 2
3 7 8 2 2
4 19 16 10 2 2
5 49 42 24 12 2 2
6 131 106 56 28 14 2 2
7 339 283 148 76 32 16 2 2
8 897 720 385 193 92 36 18 2 2
9 2338 1905 990 543 249 110 40 20 2 2
10 6178 4932 2571 1372 672 298 130 44 22 2 2
: Estimates of $c_n^+(u)$ to $n=10$ and $u=10$ for adsorbing walks in $\mathL^2_+$[]{data-label="countsads2d"}
Partition functions (see equation ) can be directly estimated from the microcanonical data in table \[countsads2d\], for example, for $n=6$ the partition function is approximated by $$Z_6(a) = 131 + 106\thin a+ 56\thin a^2 + 28\thin a^3 + 14\thin a^4 + 2\thin a^5 + 2\thin a^6$$ where $a=e^{\upsilon/kT}$ is a Boltzmann factor ($\upsilon$ is the interaction energy associated with a single visit).
Additional data were collected in the square and cubic lattices. For example, the average mean square radius of gyration of walks of length $n$ and energy $u$ were also determined, as was the average height of the endpoint of the walks. In addition, data were also collected on the average positive and negative atmospheric statistics from which estimates of the $\beta_{n,u}$ and $\gamma_{n,u}$ were made (see equations and ).
Numerical Results {#section3}
=================
Adsorbing walks in the square lattice
-------------------------------------
The finite size free energy $\C{F}_n(a)$ was determined from the data and is plotted in figure \[figure4\] for walks of lengths $n=50\thin N$ with $=1,2,3,\ldots,10$. $\C{F}_n(a)$ is a function of the combined variable $\tau=n^{1/2}(a\minus a_c^+)$ (see equations and , and note that the crossover exponent is $\phi=\sfrac{1}{2}$). Plotting the free energy $\C{F}_n(a)$ against $\tau$ should collapse the data to a single underlying curve (near the critical point $a_c^+$; that is, for small values of $\tau$) which exposes the scaling function $g$ in equation . This is displayed in figure \[figure4B\].
figure5-.tex
figure6-.tex
The energy $\C{E}_n(a)$ and specific heat $\C{C}_n(a)$ (see equation ) can be determined by differentiation of $\C{F}_n(a)$ and is plotted in figure \[figure5\] against $\log a$ and in figure \[figure6\] against $\tau$. These curves clearly show the adsorption transition in the model at a critical value of $a$.
figure7-.tex
figure8-.tex
Closer inspection of the specific heat curves in figure \[figure5\] shows that they intersect each other close to a fixed point. To the left of this point the curves decrease with increasing $n$ to zero, and to the right of this point the curves increase with increasing $n$. The common point of intersection is located approximatedly at the critical adsorption point $a_c^+$ in the model. In figure \[figure5B\] the specific heat curves are magnified in a region close to the point where they intersect.
figure9-.tex
### Location of the critical point $a_c^+$: {#section311}
In order to determine $a_c^+$, consider the finite size energy density $\C{E}_n(a) = a\sfrac{d}{da} \C{F}(a)$ per edge in the model. It is known that (see, for example, equation and reference [@JvRR04]) $$\C{E}_n(a) = n^{\phi-1}\,h_n(\tau)$$ where $h_n$ is a scaling function. From this equation, construct the ratio $$\frac{\log (n\, \C{E}_n(a))}{\log (m\,\C{E}_m(a))}
= \frac{\phi\log n + \log h_n(\tau)}{\phi\log m + \log h_m(\tau)}$$ If $a=a_c^+$, then $\tau = 0$, and the above simplifies to $$\frac{\log (n\, \C{E}_n(a_c^+))}{\log (m\,\C{E}_m(a_c^+))}
= \frac{\phi\log n + \log h_n(0)}{\phi\log m + \log h_m(0)} .$$ For large values of $n$ and $m$, $\log h_n(0)$ and $\log h_m(0)$ approaches the same constant $C$, so that the above becomes $$\frac{\log (n\,\C{E}_n(a_c^+))}{\log (m\, \C{E}_m(a_c^+))}
= \frac{\phi\log n + C}{\phi\log m + C} + \hbox{small correction}.$$ When $C$ is small compared to $\log m$, the right hand side may be expanded to obtain $$\hspace{-1cm}
\frac{\log (n\,\C{E}_n(a_c^+))}{\log (m\, \C{E}_m(a_c^+))}
= \Sfrac{\log n}{\log m} + \LH \Sfrac{C}{\phi \log m}
- \Sfrac{C \log n}{\log^2 m} \RH -
\Sfrac{C^2}{\phi^2\log^2 m} + \hbox{small correction}.$$ The signs of the two terms in square brackets are opposite, and for $n$ and $m$ not too far apart, these terms grow about at the same rate as the last term $\sfrac{C^2}{\phi^2\log^2 m}$. That is, the approximation $$\hspace{-1cm}
\frac{\log (n\,\C{E}_n(a_c^+))}{\log (m\, \C{E}_m(a_c^+))}
= \Sfrac{\log n}{\log m} -
\Sfrac{C_0}{\phi^2\log^2 m} + \hbox{small correction},$$ where $C_0$ is a constant, should be accurate at the critical point. Divide both sides by $\sfrac{\log n}{\log m}$ to obtain $$\hspace{-1cm}
P_{n,m}(a_c^+)
= \frac{\log (n\,\C{E}_n(a_c^+))}{\log (m\, \C{E}_m(a_c^+))}\frac{\log m}{\log n}
= 1 - \Sfrac{C^2}{\phi^2\log n \log m} + \hbox{small correction},
\label{eqn44C} $$ Solving for the critical point by inverting $P_{n,m}$ gives the solution $a_{n,m}^+$ as an approximation of the critical point, namely $a_{n,m}^+
= P_{n,m}^{-1} (1 \minus \sfrac{C^2}{\phi^2\log n \log m} \plus \hbox{small correction})$ which may be expanded to $$a_{n,m}^+ = P_{n,m}^{-1}(1) - \Sfrac{C_1}{\log n \log m} +
\hbox{small correction},
\label{eqn44D} $$ where $C_1$ is a constant. That is, for given values of $n$ and $m$, an estimate of $a_c^+$ can be obtained by determining the solution of $P_{n,m}(a) = 1$, or $a_{n,m}^+$ is estimated obtained by solving for $a$ in $$\hspace{-1cm}
\frac{\log (n\,\C{E}_n(a))}{\log (m\, \C{E}_m(a))}\frac{\log m}{\log n}
= 1.$$ In the above, $m$ was put equal to $n\minus 100$, and $n$ was assigned values starting at $n=200$ to $n=500$ in steps of $1$. The estimates $a_n^+ \equiv a_{n,n-100}^+$ showed no systematic dependence on $n$, and a simple average over all $n\in[200,500]$ gives the best value $$a_c^+ = 1.7788 \pm 0.0029 .
\label{abest} $$ The confidence interval is obtained by doubling the square root of the variance of the estimates $a_n^+$. This result compares well with the result in reference [@BGJ12], namely $a_c^+ = 1.77564$ (obtained by the exact enumeration of adsorbing walks), and also with $a_c^+ = 1.759\pm 0.018$ in reference [@JvRR04] (obtained by using a Multiple Markov Chain implementation of the Berretti-Sokal algorithm [@BS85]).
The estimate can be used to determine the crossover exponent $\phi$. By equation , the specific heat scales as $\C{C}_n(a_c^+) \sim n^{\alpha\phi} h_c(0)$ when $a=a_c^+$. That is, an estimate of $\alpha\phi$ is obtained by computing $\sfrac{\log \C{C}_n(a_c^+)}{\log n}$. Computing this for $100 \leq n \leq 500$, and taking the mean as the best estimate (and estimating a confidence interval by doubling the square root of the variance of the estimates), gives $\alpha\phi = -0.0091 \pm 0.0162$. Determining $\phi$ by using equation then gives the best estimate for $\phi$: $$\phi = 0.4955 \pm 0.0081 .
\label{phibest} $$ This result is consistent with $\phi=\shalf$ [@BY95; @BWO99; @BEG89], and compares well with other estimates in the literature (for example, $\phi=0.501\pm 0.014$ in reference [@JvRR04]).
figure10-.tex
### The critical point $a_c^+$ and the specific heat $\C{C}_n(a)$:
The best estimates above may be compard to estimates obtained from the specific heat curves in figure \[figure5\]. These curves intersect each other near $a_c^+$, and the region containing the intersections (in figure \[figure5B\]) is magnified in figure \[figure5C\].
In general the location of the intersection between $\C{C}_n(a)$ and $\C{C}_m(a)$ is a function of $n$ and $m$. The location of the critical point $a_c^+$ can be estimated by extrapolating this dependence. Consider for example the intersections between the curves $\C{C}_n(a)$ and $\C{C}_{n+100}(a)$. The location of these intersections are plotted against $\sfrac{1}{\sqrt{n}}$ in figure \[figure5C\], where $n=2\, N$ and $N\in [23,200]$. The data lie along a straight line, except for a few points at the largest values of $n$ (where the data is more uncertain). The best line through the data can be extrapolated to its intersection with the vertical axis (where $\sfrac{1}{\sqrt{n}} = 0$). This gives an estimate of $a_c^+$ as being located in the interval $[1.77,1.79]$. Using a linear least squares model for $n\geq 10$ gives the (extrapolated) estimate $a_c^+ \approx 1.7839$. By examining the spread of the data in figure \[figure5C\], a confidence interval can be estimated. The result is $$a_c^+ = 1.784\pm 0.010 .
\label{eqn29} $$ This estimate is consistent with the best estimate in equation .
An alternative approach to determining the critical point $a_c^+$, is to consider the scaling of the specific heat in equation . Taking ratios for $n$ and $m$, and then logarithms, give $$\log \LB \Sfrac{\C{C}_n(a)}{\C{C}_m(a)} \RB
= \alpha \phi \log \LB \Sfrac{n}{m} \RB
+ \log \LB \Sfrac{h_c(n^\phi(a\minus a_c^+))}{h_c(m^\phi(a\minus a_c^+))} \RB .
\label{eqn27} $$ Observe that the last term is zero when $a=a_c^+$. Since, in addition, $\alpha = 2\minus \sfrac{1}{\phi}=0$ in this model, this shows that an estimate of the critical point is given by the solution of $$\log \LB \Sfrac{\C{C}_n(a)}{\C{C}_m(a)} \RB = 0 .$$ Solving this for $150\leq n \leq 500$ (and $n$ a multiple of $10$), and for $m=n\minus k$ where $k\in\{10,20,\ldots,100\}$ gives a large collection of estimates of $a_c^+$, with mean $$a_c^+ = 1.762 \pm 0.016 .
\label{eqn27AA} $$ The confidence interval is one-half of difference between the maximum and minimum estimates of $a_c^+$. This estimate is slightly less than,but still consistent with, the results in equation and equation . Recall that it is also predicated on the assumption that $\alpha=0$ (or $\phi = \sfrac{1}{2}$).
figure11-.tex
### The microcanonical density function:
The microcanonical density function of visits in adsorbing positive walks is determined from the microcanonical data in the model, and is given by $$P^+(\eps) =\lim_{n\to\infty} (c_n^+(\lfl \eps n \rfl))^{1/n} = \lim_{n\to\infty} P_n^+(\eps),
\label{eqn59X} $$ where $P_n^+(\eps) = (c_n^+(\lfl \eps n\rfl))^{1/n}$ is a finite size approximation to $P^+(\eps)$. Existence of the limit is known (see for example reference [@JvR15]), and $\log P^+(\eps)$ is a concave function of $\eps$.
$P^+(\eps)$ can be estimated by interpolating the finite size approximations $P_n^+(\eps)$ and then extrapolating to $n=\infty$ by fitting a least squares model to the data. In figure \[figure6-B\] the data for the extrapolated function $P^+(\eps)$ is plotted together with $P_n^+(\eps)$ for $n=100$ and $n=500$.
A least squares fit of a quadratic to $\log P^+(\epsilon)$ for $\eps\in[0,0.1]$ gives the $\log P^+(\eps) \approx 0.97007\minus 0.58190\eps\minus 0.13030\eps^2$, and by taking the right derivative and then taking $\eps\to 0^+$, an estimate for the critical point is obtained: $$a_c^+ \approx 1.789 .
\label{eqn27BB} $$ This is close to the estimates obtained in equations and , showing consistency in the data and the analysis above.
The free energy $\C{F}(a)$ is the Legendre transform of $\log P^+ (\epsilon)$. This may be estimated by fitting a polynomial to $\log P^+(\epsilon)$. If a cubic polynomial in $\eps$ is fitted to $\log P^+(\eps)$ for $0 \leq \eps \leq 0.5$, then the estimated free energy for $a>a_c^+$ is approximately $$\fl
F (a) \approx 1.0376 -0.1153\log a - (0.3306 \minus 0.5693\log a)
\sqrt{-1.3603 \plus 2.3422 \log a}.$$ The critical point can be estimated as that location where the square root in the above is zero. This gives $a_c^+ \approx 1.787$. Similarly, the factor $(0.3306-0.5693\log a)$ vanishes when $a_c^+ \approx 1.787$.
### Metric data:
The mean square radius of gyration $R_n^2$, and the mean height $H_n$ of the endpoint of the walk, are functions of the adsorption activity $a$. In the desorbed phase (for $a<a_c^+$) it is expected that $R_n^2 \sim n^{2\nu}$, and $H_n \sim n^\nu$, where $\nu = \sfrac{3}{4}$ is the metric exponent [@D89C]. This scaling changes in the adsorbed phase (when $a>a_c^+$) to $R_n^2 \sim n^2$ and $H_n \sim \hbox{constant}$. These expectations are confirmed by the data, as seen, for example, in figure \[figure8\], where data for the mean square radius of gyration are normalised and then plotted as a function of $a$. These graphs clearly show two scaling regimes, namely a high temperature phase (when $a< a_c^+$) where the walk has bulk critical exponents and is desorbed, and a low temperature phase where the walk stays near the adsorbing boundary and has critical exponents of a linear object.
figure12-.tex
In general, the metric exponent associated with $R^2_n$ is a function of $a$, and it will be denoted by $\nu_a$, where $\nu_a=\sfrac{3}{4}$ in the desorbed phase, and $\nu_a=1$ in the adsorbed phase. This exponent may be estimated from the mean square radius of gyration $R_n^2$ data by examining the ratio $$2\,\nu_{n,m}(a) = \frac{\log ( R_n^2/R_m^2 )}{\log (n/m)}.
\label{eqn41} $$ Here, $\nu_{n,m}(a)$ is a function of $n$ and $m$. By averaging over $m$, the estimate $\nu_n(a) = \LA \nu_{n,m}(a)\RA_m$ may be determined. In particular, fixing $n$ and taking the average over $m$ for $100 \leq m \leq 500$ in multiples of $5$ (and for $m\not=n$) gives estimates of $\nu_n(a)$. These results are plotted in figure \[figure8-nu\] for $n\in\{50,100,150,\ldots,500\}$. The data for $a\leq 1.5$ gives $\nu \approx 0.747$, and for $a \geq 1.95$, $\nu \approx 1.01$. These results are evidence for the exact value $\nu_a = \sfrac{3}{4}$ in the desorbed phase, and $\nu_a=1$ in the adsorbed phase.
figure13-.tex
The function $\nu_n(a)$ should scale with the combined variable $\tau = n^\phi(a\minus a_c^+)$. That is, one may expect that $\nu_n (a) = \hbox{\Large$\nu$}(\tau)$, where is a scaling function. In figure \[figure6-D\] the data in figure \[figure8-nu\] are rescaled by plotting against $\tau$ to uncover the scaling function .
figure14-.tex
The (normalised) average height of the endpoint of the walk is plotted as a function of $a$ in figure \[figure8HH\]. These data show a clear transition where the scaling of $H_n$ changes. That is, the metric exponent associated with $H_n$ is a function of $a$, and is denoted by $\nu_a^\perp$, where $\nu_a^\perp = \nu= \sfrac{3}{4}$ if $a<a_c^+$, and $\nu_a^\perp = 0$ if $a>a_c^+$, so that $H_n \sim n^{\nu_a^\perp}$. The graph of $n^{-3/4}H_n$ contains a set of curves which decreases with increasing $a$. These curves intersect each other close to $a_c^+$.
figure15-.tex
figure16-.tex
An approach similar to the ratio method in equation may be used to estimate $\nu_a^\perp$ (which will be referred to as the *vertical metric exponent*): $$\nu^{\perp}_{n,m} (a) = \frac{\log (H_n/H_m)}{\log (n/m)}.
\label{eqn42} $$ This estimate of $\nu_a^\perp$ is a function of $n$ and $m$, and may be averaged over $m$ to obtain $\nu^{\perp}_n(a) = \LA \nu_{n,m}(a)\RA_n$. Taking the average for $100\leq m \leq 500$ in multiples of $5$ to estimate $\nu^\perp_n(a)$ gives the curves in figure \[figure6-nuH\] when plotted against the combined variable $\tau=n^\phi(a\minus a_c^+)$. It is seen in the graph that if $a<a_c^+$, then $\nu_n (a) \approx \sfrac{3}{4}$, but for $a>a_c^+$, $H_n \simeq \hbox{const}$ so that $\nu_n^{\perp}(a) \approx 0$.
figure17-.tex
Since $R^2_n \sim n^{2\nu_a}$ where $\nu_a = \nu = \sfrac{3}{4}$ if $a<a_c^+$, and $\nu_a = 1$ if $a>a_c^+$, ratios of $R^2_n$ may be defined by $$\frac{R^2_{n}}{R^2_{2n}} \approx 2^{-2\nu_a} ,\q\hbox{or}\q
\frac{2^{2\nu}R^2_{n}}{R^2_{2n}} \approx
\cases{
1, & if $a< a_c^+$; \\
2^{2\nu - 2\nu_a} = 2^{-1/2} , & if $a>a_c^+$.
}
\label{eqn58} $$ In figure \[figure8RR\] the quantity $\sfrac{2^{2\nu}R_{n}^2}{R_{2n}^2}$ is plotted as a function of the rescaled variable $\tau = n^{1/2}(a\minus a_c^+)$ for $n$ from $25$ to $250$ in steps of $25$. In the desorbed phase this ratio should be equal to $1$, but, in the adsorbed phase, it should be equal to $2^{1.5-2.0} \approx 0.71$. This is clearly seen in the graph. The curves coincide well over the entire range of $n$ and $\tau$, and decreases sharply close to the critical adsorption point at $\tau=0$. A similar approach using the heights of the endpoint involves the ratios $$\frac{H_{n}}{H_{2n}} \approx 2^{-\nu_a^\perp} ,\q\hbox{or}\q
\frac{2^\nu H_{n}}{H_{2n}} \approx
\cases{
1, & if $a< a_c^+$; \\
2^{\nu-\nu^\perp} = 2^{3/4}, & if $a>a_c^+$.
}
\label{eqn59} $$ The quantity $\sfrac{2^\nu H_{n}}{H_{2n}}$ is plotted against $\tau$ on the right panel in figure \[figure8RR\]. In the desorbed phase this ratio should be equal to $1$, as seen in the graph. In adsorbed phase the scaling of $H_n$ changes, and the ratio should be equal to $2^{3/4} \approx 1.68$, as seen in the graph.
figure18-.tex
The generating function
-----------------------
The generating function is given by the series $$G (a,t) = \sum_{n=0}^\infty \sum_{v=0}^n c_n^+(v)\,a^vt^n .
\label{eqn45} $$ Approximations of $G(a,t)$ is given by the truncated sum $$G_N(a,t) = \sum_{n=0}^N \sum_{v=0}^n c_n^+(v)\,a^vt^n ,
\label{eqn46a} $$ where $c_n^+(v)$ is approximately enumerated by the GAS algorithm. In this study, $G_{500}(a,t)$ was estimated using the approximate values of $c_n^+(v)$ obtained in the simulations.
![A plot of $\log G_{500}(a,t)$ as a function of $(a,t)$ for adsorbing walks in $\mathL_2^+$. The critical curve is the black curve along the surface, and the critical point separating the $\tau_0$ and $\lambda$ curves (see figure \[figure4-6\]) is denoted. For $t<t_c^+(a)$ the generating $G(a,t)$ is finite, and for $t>t_c^+(a)$ it is divergent. These two regimes are clearly visible.[]{data-label="figureAdsrb-2"}](figure19-.jpg)
The *critical curve* $t_c^+(a)$ of $G(a,t)$ is its radius of convergence as a function of $a$. By equation and by equation , $$\log t_c^+(a) = - \C{F}(a) = - \lim_{n\to\infty} \sfrac{1}{n} \log Z_n(a) .
\label{eqn46A} $$ The critical curve is shown schematically in figure \[figure4-6\]. $G(a,t)$ is singular when $t=t_c^+(a)$, and if $t>t_c^+(a)$, then $G(a,t)$ is divergent.
The critical curve is parametrized by the scaling fields $(\sigma,g)$ as shown in figure \[figure4-6\]. The critical point when $a=a_c^+$ is located at $(a_c^+,t_c^+)$, where $t_c^+\equiv t_c^+(a_c^+)$, and it divides the critical curve $t_c^+(a)$ into two parts. The part marked by $\tau_0$ corresponds to a transition to desorbed walks, so that $t_c^+(a)=1/\mu_2$ in this regime (which is a transition to a high temperature phase). For $a>a_c^+$ the approach to $t_c^+(a)$ is to adsorbed walks, along the critical curve marked by $\lambda$ (which is a transition to a low temperature phase).
The phase diagram in figure \[figure4-6\] may be described in terms of the coordinates $g=(\sfrac{1}{\mu_2} \minus t)$ and $\sigma = (a\minus a_c^+)$. The behaviour of $G(a,t)$ along its singular points along the critical curve is described by $$G(a,t) \sim \cases{
g^{-\gamma_1}, & \hbox{along $\tau_0$;} \\
g^{-\gamma_s}, & \hbox{at the critical point $a=a_c^+$;}\\
g^{-\gamma_+}, & \hbox{along $\lambda$.}
}
\label{eqn46} $$ In two dimensions exact values are known for the exponents: The exponent $\gamma_1 = \sfrac{61}{64}$ [@C87] is the entropic exponent of half-space walks, and $\gamma_s = \sfrac{93}{64}$ [@BEG89] is the surface exponent of adsorbing half-space walks at the critical point $a_c^+$. The exponent $\gamma_+$ is the entropic exponent of adsorbed walks, and is given by the entropic exponent of self-avoiding walks in one dimension lower. In one dimension, this is $\gamma_+=1$.
A plot of $G_N(a,t)$ is shown in figure \[figureAdsrb-2\]. The horizontal plane is the $(a,t)$-plane, and the critical curve in figure \[figure4-6\] is shown as a black curve with the critical point shown. Below the critical curve $G(a,t)$ is finite, and approximated well by $G_N(a,t)$ (for large values of $N$ not too close to the critical curve). Above the critical curve $G(a,t)$ is divergent, while $G_N(a,t)$, which is a polynomial, is finite.
figure20-.tex
The exponent $\gamma_1$ can be estimated by putting $a=1$ so that $G(1,t) \sim g^{-\gamma_1}$ where $g=(t_c^+(1)\minus t)$ and $t_c^+(1) = \sfrac{1}{\mu_2}$ (and where $\mu_2$ is the growth constant of the walks in two dimensions). Thus, estimate $\gamma_1$ by noting that $$\frac{\log G(1,t)}{\log g} = - \gamma_1 + \Sfrac{C_1}{\log g} + \Sfrac{C_2}{\log^2 g}
+ \ldots .
\label{eqn47} $$ Proceed by approximating $G(1,t)$ by $G_N(1,t)$ (with $t < t_c^+(1)$ so that $G_{500}(1,t)$ is a good approximation of $G(1,t)$). A least squares fit of the ratio on the left to a quadratic in $\sfrac{1}{\log g}$ gives the estimate $$\gamma_1=0.952\ldots .$$ This result is very close to the exact value $\sfrac{61}{64} = 0.953\ldots$. Similar analysis for $a>1$ and $a<a_c^+$ gives results slightly larger, since the critical point at $a_c^+$ influences the data in its vicinity for finite values of $N$.
figure21-.tex
A similar analysis with $a=a_c^+$ gives the estimate $$\gamma_s=1.429\ldots,$$ for the surface exponent at the critical adsorption point. This is close in value to the exact result $\sfrac{93}{64} = 1.453\ldots$.
The case that $a>a_c^+$ may also be analysed. The adsorbed walk should have the statistics of the self-avoiding walk in one dimension, so that the entropic exponent is $\gamma_+ = 1$ (this is the value of the exponent $\gamma$ in one dimension). Putting $a=3.5$ and plotting $Z_n^{1/n}(3.5)$ against $n$ gives an estimate for the critical value $t_c^+(3.5)$. In this case the data quickly converges to $t_c^+(3.5)=0.260\ldots$ (to three decimal places). Assuming that $t_c^+(3.5)=0.260$ and choosing the scaling field $g=(0.260 \minus t)$ gives the model $$\Sfrac{\log G(3.5,t)}{\log g} = - \gamma_+ + \Sfrac{C_1}{\log g} + \Sfrac{C_2}{\log^2 g}
+ \ldots ,
\label{eqn48} $$ similar to the above, but now with the exponent $\gamma_+$. Plotting the left hand side as a function of $\kappa=\sfrac{1}{\log g}$, and fitting the data to a quadratic in $\kappa$, give the estimate $\gamma_+ = 1.00\ldots$. There remains, however, some curvature in the model for small values of $g$ (and of $\sigma$), so that there may remain strong systematic corrections to this result (however, $\gamma_+ = 1$ is consistent with this result).
In the vicinity of the critical point $(a_c^+,t_c^+)$ the generating function should exhibit scaling given by $$G(a,t) \sim g^{-\gamma_s} \,f(g^{-\phi} \sigma),
\label{eqnGscale} $$ where $\phi=\sfrac{1}{2}$ is the crossover exponent and $f$ is a scaling function. That is, plotting $g^{\gamma_s}\,G(a,t)$ against the combined variable $g^{-\phi} \sigma$ should expose the scaling function $f$. In figure \[figureGscale\] this is done by plotting $\log (g^{\gamma_s}\,G(a,t) )$ against $g^{-1/2}\sigma$ for $g=\sfrac{1}{\mu_2}\minus t \in [0.01,0.05]$ and $\sigma=a\minus a_c^+
\in [-\sfrac{1}{2}g^{1/2},g^{1/2}]$.
The partition function (see equation ) should also exhibit scaling for large $n$, given by $$Z_n(a) \sim n^{\gamma_t-1}\,h(n^\phi (a_c^+ \minus a))\,\mu_a^n,
\label{eqn53} $$ where $\log \mu_a = \C{F}(a)$ and $\log \mu_a = - \log t_c^+(a)$. The exponent $\gamma_t$ can be related to the $\gamma$-exponents as $a\to a_c^+$ (that is, as $\sigma\to 0$), and namely to the surface exponent $\gamma_s$, where the walk is critical with respect to the adsorption transition. By noting that $G(a,t) = \sum_n Z_n(a)\,t^n$, and approximating the summation by an integral, it follows that $G(a,t) \sim
g^{-\gamma_t} h(0)$ if $a=a_c^+$ and $g=t_c^+\minus t$. This shows that $$\gamma_t-1 = \gamma_s - 1 = \Sfrac{93}{64} - 1 = \Sfrac{29}{64}.
\label{eqn71g} $$ That is, when $a$ is close to $a_c^+$ (and $a<a_c^+$), the partition function has asymptotic behaviour $$Z_n(a) \sim n^{29/64} h(n^\phi(a_c^+\minus a))\, \mu_a^n .
\label{eqnZscaling} $$ This result may be tested by plotting $n^{-29/64} Z_n(a) \, (t_c^+(a))^n$ against $|\tau | = n^{1/2} |a\minus a_c^+|$. All the data should collapse to the same universal curve exposing the scaling function $h$. This is done with $a<a_c^+$ (and $t_c^+(a) = \sfrac{1}{\mu_2}$) in figure \[figureZscale\] for $n\in\{50,100,150,\ldots,500\}$ and $0 \leq a < a_c^+$.
figure22-.tex
Adsorbing walks in the cubic lattice
------------------------------------
The (finite size) free energy $\C{F}_n(a)$ is a function of the combined variable $\tau=n^{1/2}(a\minus a_c^+)$ (see equations and ; note that $\phi=\sfrac{1}{2}$ for adsorbing walks in three dimensions [@HG94]). Plotting $\C{F}_n(a)$ against $\tau$ for data in the cubic lattice gives a graph similar to figure \[figure4B\]. In figure \[figure4B-3\] the scaled free energy $n(\C{F}_n(a)\minus\log\mu_3)$ is plotted against $\tau$. The data shows a clear transition in the model from a desorbed to an adsorbed phase.
Derivatives of the free energy to $\log a$ gives the (finite size) energy density $\C{E}_n(a)$ and (finite size) specific heat $\C{C}_n(a)$. These are plotted in figure \[figure1-3a\] against $\log a$ and in figure \[figure4-3a\] against $\tau$. In these plots, as in figure \[figure4B-3\], the critical point was approximated by $a_c^+ = 1.31$. This is a close approximation of the best estimate for the critical point from our data (see equation ).
figure23-.tex
figure24-.tex
### Location of the critical point $a_c^+$: {#location-of-the-critical-point-a_c}
The location of the critical adsorption point can be determined using the same analysis as in section \[section311\], and in particular using equation as a starting point. That, is for given values of $n$ and $m$, an estimate $a_{n,m}^+$ of $a_c^+$ can be obtained by solving for $a$ in $$\hspace{-1cm}
\frac{\log (n\,\C{E}_n(a))}{\log (m\, \C{E}_m(a))}\frac{\log m}{\log n}
= 1.$$ Here, the choice $m=n\minus 100$ worked well, and $n$ was assigned values starting at $n=200$ to $n=500$ in steps of $1$. The estimates $a_n^+ \equiv a_{n,n-100}^+$ showed a dependence on $n$, systematically decreasing with increasing $n$. The best estimate is obtained by using the model $a_n^+ = a_c^+ - \sfrac{c_1}{\log^2 n}$ suggested by equation . A least squares fit for all $n\in [200,500]$ gives the best estimate $$a_c^+ = 1.3055 \pm 0.0061 .
\label{abest3} $$ The confidence interval is obtained by doubling the square root of the variance of the estimates $a_n^+$. This result is slightly smaller than the result in reference [@JvRR04], namely $a_c^+ = 1.334\pm 0.027$ (obtained by using a Multiple Markov Chain implementation of the Berretti-Sokal algorithm [@BS85]).
The estimate can be used to determine the crossover exponent $\phi$. This is again done by considering the scaling of the specific heat (equation ). It is expected that $\C{C}_n(a_c^+)
\sim n^{\alpha\phi} h_c(0)$ when $a=a_c^+$. An estimate of $\alpha\phi$ is obtained by computing $\sfrac{\log \C{C}_n(a_c^+)}{\log n}$ for a range of values of $n$ (in this case $100 \leq n \leq 500$). The average is taken as the best estimate and a confidence interval is estimated by doubling the square root of the variance of the estimates. This gives $\alpha\phi = 0.0106 \pm 0.0116$. Determine the best estimate for $\phi$ by using equation : $$\phi = 0.5053 \pm 0.0053 .
\label{phibest3} $$ This result compares well with the estimate $\phi=0.5005\pm 0.0036$ for adsorbing walks in reference [@JvRR04].
### The critical point $a_c^+$ and the specific heat $\C{C}_n(a)$:
The best estimate for $a_c^+$ above (see equation ) should be examined by comparing it to estimates obtained from the specific heat curves in figure \[figure1-3a\]. These curves intersect each other near $a_c^+$, and the region containing the intersections is magnified in figure \[figure2-3a\].
figure25-.tex
The locations of the intersections between the specific heat curves in figure \[figure2-3a\] is a function of $n$ and are estimates of the critical adsorption point $a_c^+$. By plotting the intersections between $\C{C}_n(a)$ and $\C{C}_{n+100}(a)$ against $\sfrac{1}{\sqrt{n}}$ (see figure \[figure3-3a\], where $n=2\, N$ and $N\in [23,200]$); it is seen that the intersections fall approximately along a curve, which may be extrapolated to its intersection with the vertical axis. This gives a rough estimate of the critical point $\log a_c^+$ as being located in the the interval $[1.33,1.35]$. A more accurate extrapolation is done by using a linear least squares model to extrapolate to $n=\infty$. Fitting to the model $a_c^+\plus \sfrac{a_0}{\sqrt{n}}
\plus \sfrac{a_1}{n}$, for all $n \geq 50$, gives the estimate $\log a_c^+ \approx 1.337$. By examining the spread of the data in figure \[figure3-3a\], a confidence interval can be estimated: $$\log a_c^+ = 1.337\pm 0.020 .
\label{eqn29-3} $$ This estimate is slightly larger than the estimate in equation , but is consistent with the estimate $a_c^+ = 1.334$ in reference [@JvRR04]. However, the noise in the data in figure \[figure3-3a\] makes this a less reliable estimate.
figure26-.tex
Equation is equally valid for adsorbing walks in the cubic lattice. The last term on the right hand side is equal to zero when $a=a_c^+$. Thus, by plotting $\log (\C{C}_n(a)/\C{C}_m(a))$ against $\log (n/m)$, a set of curves should be seen which intersect when $a=a_c^+$. At this point the coefficient of $\log \sfrac{n}{m}$ is an estimate of $\alpha\phi$. Since $\alpha=0$ in this model (and $\phi=\sfrac{1}{2}$), the critical point can also be determined by solving for $a$ in $$\log \LB \Sfrac{\C{C}_n(a)}{\C{C}_m(a)} \RB = 0 .$$ Solving this for $n\in[150,500]$ and $m\in[n\minus 100,n\minus 10]$ gives a large collection of estimates. The average is $$a_c^+ = 1.324 \pm 0.012,
\label{eqn27CC} $$ where the confidence interval is one-half of the largest difference between two estimates in the collection. This result is smaller than the estimate in equation , and larger than the best estimate in equation .
These results indicate that there may remain sources of systematic errors in the data and in the analysis, and that the estimates for $a_c^+$ should be considered in this context.
figure27-.tex
### The microcanonical density function:
The microcanonical density function of visits in adsorbing positive walks is determined from the microcanonical data in the model, and is given by $$P^+(\eps) =\lim_{n\to\infty} (c_n^+(\lfl \eps n \rfl))^{1/n} = \lim_{n\to\infty} P_n^+(\eps),$$ where $P_n^+(\eps) = (c_n^+(\lfl \eps n\rfl))^{1/n}$ is a finite size approximation to $P^+(a)$. Existence of $P^+(\eps)$ can be shown (see for example reference [@JvR15]), and $\log P^+(\eps)$ is a concave function of $\eps$.
$P^+(\eps)$ can be determined by interpolating the finite size approximations $P_n^+(\eps)$ and then extrapolating to $n=\infty$ by fitting a least squares model to the data. In figure \[figure9-3\] the data for the extrapolated function $P^+(\eps)$ is plotted together with $P_n^+(\eps)$ for $n=100$ and $n=500$.
A least squares fit of a quadratic to $\log P^+(\epsilon)$ for $\eps\in[0,0.1]$ gives $\log P^+(\eps) \approx 1.54378\minus 0.28704\eps\minus 0.08784\eps^2$. By taking the right derivative and then taking $\eps\to 0^+$, an estimate for the critical point is obtained: $$a_c^+ \approx 1.332 .
\label{eqn27EE} $$ The free energy is the Legendre transform of $\log P^+ (\epsilon)$. This may be estimated by fitting a polynomial to $\log P^+(\epsilon)$. If a cubic polynomial in $\eps$ is fitted to $\log P^+(\eps)$ for $0 \leq \eps \leq 0.5$, then the estimated free energy for $a>a_c^+$ is approximately $$\fl
F (a) \approx 1.6526 -0.4240\log a - (0.3184 \minus 1.319\log a)
\sqrt{-0.2438 \plus 1.0105 \log a}.$$ The critical point can be estimated as that location where the square root in the above is zero. This gives $a_c^+ \approx 1.273$. Similarly, the factor $(0.3184-1.319\log a)$ vanishes when $a_c^+ \approx 1.273$. These estimates are far less secure than the estimates above, and are also smaller.
figure28-.tex
figure29-.tex
### Metric data:
The mean square radius of gyration $R_n^2$ and mean height $H_n$ of the endpoint of the walk can be calculated as a function of $a$. In the desorbed phase (for $a<a_c^+$) it is expected that $R_n^2 \sim n^{2\nu}$, and $H_n \sim n^\nu$, where $\nu = 0.587\ldots$ [@C10] is the metric exponent. This scaling changes in the adsorbed phase (when $a>a_c^+$); in this phase it should be the case that $R_n^2 \sim n^{3/2}$ and $H_n \sim \hbox{constant}$, since an adsorbed walk in the cubic lattice should have the statistics of a walk in one dimension lower.
These expectations are supported by the data, as seen, for example, in figure \[figure8-3\], where data for the mean square radius of gyration are plotted as a function of $a$. These graphs clearly show two scaling regimes, namely a high temperature phase (when $a< a_c^+$) where the walk has bulk critical exponents and is desorbed, and a low temperature phase where the walk stays near the adsorbing boundary and has critical exponents of a walk in one dimension lower.
The metric exponent $\nu$ may be estimated from $R_n^2$ by examining the ratios $$2\,\nu_{n,m}(a) = \frac{\log ( R_n^2/R_m^2 )}{\log (n/m)}.
\label{eqn41-3} $$ Here, $\nu_{n,m}(a)$ is a function of $n$ and $m$. By averaging over $m$, the estimate $\nu_n(a) = \LA \nu_{n,m}\RA_m$ can be determined. Taking the average for $100\leq m \leq 500$ in multiples of $5$ (and for $m\not= n$) gives an estimate for $\nu_n(a)$. The results are plotted in figure \[figure8-nu3\] for $n\in\{50,100,150,\ldots,500\}$. The data for $a\leq 1.2$ give $\nu \approx 0.592$, and for $a \geq 1.7$, $\nu \approx 0.740$.
The scaling of $\nu_n(a)$ as a function of $\tau=n^\phi(a\minus a_c^+)$ can be uncovered by plotting the data in figure \[figure8-nu3\]. This gives a set of curves which are very close to one another, uncovering a scaling function where $\nu_n(a) =\hbox{\Large$\nu$}(\tau)$.
figure30-.tex
The average height of the endpoint of the walk is plotted as a function of $a$ in figure \[figure8HH3\]. The left panel displays the height normalised by division with $n^\nu$ and gives a set of curves which increase with $n$ for $a< a_c^+$, and decrease with $n$ for $a>a_c^+$. The curves intersect close to the critical adsorption point, and the limiting curve (in the $n\to\infty$ limit) should be a step function with critical point at $a=a_c^+$.
figure31-.tex
The vertical metric exponent $\nu^\perp$ can be estimated from $H_n$, by using a method similar to equation , namely an approximation by examing the ratios of $H_n$: $$\nu^{\perp}_{n,m} (a) = \frac{\log (H_n/H_m)}{\log (n/m)}.
\label{eqn42-3} $$ The exponent is approximated by $\nu^{\perp}_n(a) = \LA \nu_{n,m}(a)\RA_n$. Taking the average for $100\leq m \leq 500$ for fixed $n$ gives esimates for $\nu^\perp_n(a)$. If $a<a_c^+$, then $\nu^\perp_n(a)$ should have value approximately equal to $\nu$; that is, $\nu_n (a) \approx 0.58\ldots$, and for $a>a_c^+$, $H_n \simeq \hbox{const}$ so that $\nu_n^{\perp}(a) \approx 0$ in this phase. The results are plotted in figure \[figure6-nuH3\] against $\tau = n^\phi(a\minus a_c^+)$.
figure32-.tex
Since $R^2_n \sim n^{2\nu_a}$ where $\nu_a = \nu \approx 0.588\ldots$ if $a<a_c^+$, and $\nu_a = \sfrac{3}{4}$ if $a>a_c^+$, the ratio of $R^2_{2n}$ and $R^2_n$ is given by $$\frac{R^2_{n}}{R^2_{2n}} \approx 2^{-2\nu_a} ,\q\hbox{or}\q
\frac{2^{2\nu}R^2_{n}}{R^2_n} \approx
\cases{
1, & if $a< a_c^+$; \\
2^{2\nu - 2\nu_a} \approx 2^{-0.32}, & if $a>a_c^+$.
}$$ In figure \[figure8RR3\] this is plotted as a function of the rescaled variable $\tau = n^{1/2}(a\minus a_c^+)$ for $n$ from $25$ to $250$ in steps of $25$. The curves coincide well with increasing $n$ and signals a transition when $a=a_c^+$ from the desorbed scaling regime into the adsorbed scaling regime. When $a>a_c^+$, $2^{-0.32} = 0.801\ldots$, as shown in the graph. A similar approach using the heights of the endpoint would involve plotting $$\frac{H_{n}}{H_{2n}} \approx 2^{-\nu_a} ,\q\hbox{or}\q
\frac{2^\nu H_{n}}{H_{2n}} \approx
\cases{
1, & if $a< a_c^+$; \\
2^{\nu - \nu^\perp} \approx 2^{0.588}, & if $a>a_c^+$,
}$$ where $\nu^\perp$ is the vertical metric exponent. When $a>a_c^+$, $2^{0.588} = 1.503\ldots$, as shown in figure \[figure8RR3\].
figure33-.tex
### The generating function:
The generating function of adsorbing walks in the cubic lattice is given by equation , where $c_n^+(v)$ is again the number of walks from the origin of length $n$ in $\mathL^3_+$, and with $v$ visits to the adsorbing boundary $\partial\mathL^3_+$. Approximations to $G(a,t)$ are given by $G_N(a,t)$ in equation , and $G_{500}(a,t)$ was calculated using the approximate values of $c_n^+(v)$ obtained by sampling with the GAS algorithm. The critical curve is given by equation (see figure \[figure4-6\]). $G(a,t)$ is singular when $t=t_c^+(a)$, and if $t>t_c^+(a)$, then $G(a,t)$ is divergent. In figure \[figureAdsrb-3\] the approximation $G_{500}(a,t)$ is plotted, with the location of the critical curve, and critical point $(a_c^+,t_c^+)$ indicated (where, as before, $t_c^+ = t_c^+(a_c^+)$). The critical curve is similar to the critical curve in figure \[figure4-6\], and the critical point divides the critical curve into two curves, namely a curve where the transition is a high temperature or desorbed walk marked by $\tau_0$, and a curve where the transition is to a low temperature or adsorbed walk marked by $\lambda$. Along $\tau_0$ the critical curve is given by $t_c^+(a) =
\sfrac{1}{\mu_3}$, for $a\leq a_c^+$ and where $\mu_3$ is the growth constant of self-avoiding walks in the cubic lattice.
![$G_{500}(a,t)$ as a function of $(a,t)$ for adsorbing walks in $\mathL_3^+$. The critical curve $t_c(a)$ is the black curve, and the location of the critical point at $a=a_c^+$ is indicated. Below the critical curve $G(a,t)$ is finite, and approximated well by $G_{N}(a,t)$ (for large $N$ and not too close to the critical curve). Above the critical curve $G(a,t)$ is divergent, while $G_N(a,t)$ is finite.[]{data-label="figureAdsrb-3"}](figure34-.jpg)
The critical curve is parametrized by the scaling fields $(\sigma,g)$ as shown in figure \[figure4-6\]. Here, the scaling fields are given by $g = t_c^+\minus t$, and $\sigma=a\minus a_c^+$. The singular points in $G(a,t)$ along $t_c^+(a)$ are described by the scaling assumptions shown in equation . The exponent $\gamma_1$ can be estimated by putting $a=1$ so that $G(1,t) \sim g^{-\gamma_1}$. By using the model in equation , the estimate $$\gamma_1=0.725\ldots$$ is obtained. This result is close to the estimate $0.697(2)$ in reference [@HG94]. A similar analysis with $a=a_c^+$ gives the estimate $$\gamma_s=1.203\ldots
\label{eqn71} $$ for the surface exponent at the critical adsorption point. This is slightly smaller than the estimate $1.304(16)$ in reference [@ML88A].
figure35-.tex
The situation is less clear in the adsorbed phase. The adsorbed walk should have the scaling of a self-avoiding walk in two dimensions, so that $\gamma_+$ is given by the entropic exponent of walks in $d=2$ ($\gamma_+=\sfrac{43}{32}$). Putting $a=4$ and plotting $Z_n^{1/n}(4)$ against $n$ gives an estimate for the critical value $t_c^+(4)$. In this case the data quickly converges to $t_c^+(4)=0.0926\ldots$ (to four decimal places). Assuming that $t_c^+(4)=0.0926$ and choosing the scaling field $g=(0.0926 \minus t)$ gives a model similar to equation . Plotting the left hand side as a function of $\sfrac{1}{\log g}$, and fitting it to a quadratic for $t\in[0,0.06]$, gives the estimate $\gamma_+ \approx 1.2\ldots$, still well below the expected result $\sfrac{43}{32} = 1.34375$. Examination of the data shows strong dependence of this result on the range of $g$ in the model. For example, a fit with $t\in[0,0.09]$ gives a smaller value $\gamma_+ \approx 1.1\ldots$. These variable results indicate that $G_{500}(a,t)$ is not a good approximation to $G(a,t)$ near the critical curve $\lambda$ in figure \[figure4-6\] for adsorbing walks in the cubic lattice.
In the vicinity of the critical point $(a_c^+,t_c^+)$ the generating function should exhibit scaling given by equation . Plotting $g^{\gamma_s}\,G(a,t)$ (with $\gamma_s=1.203$) against the combined variable $g^{-1/2}\sigma$ should expose the scaling function $f$. In figure \[figureGscale-3\] this is done by plotting $\log (g^{\gamma_s}\,G(a,t) )$ against $g^{-1/2}\sigma$ for $g=\sfrac{1}{\mu_3}\minus t \in [0.01,0.05]$ and $\sigma
=(a\minus a_c^+) \in [-\sfrac{1}{2}g^{1/2},\sfrac{1}{2}g^{1/2}]$.
The partition function (see equation ) also exhibit scaling for large $n$. The scaling assumption in equation applies here as well, where $\log \mu_a = \C{F}(a)$ and $\log \mu_a = - \log t_c(a)$. As before, the exponent $\gamma_t$ is related to the $\gamma_s$-exponent as in equation : $$\gamma_t-1 = \gamma_s - 1 \approx 1.203 - 1 = 0.203.$$ That is, when $a$ is close to $a_c^+$ (and $a<a_c^+$), then the partition function has asymptotic behaviour $$Z_n(a) \simeq n^{0.203} h(n^\phi (a_c^+\minus a))\, \mu_a^n .
\label{eqn73} $$ This result may be tested by plotting $n^{0.203} Z_n(a_c^+) \, (t_c(a))^n$ against $\tau = n^\phi (a\minus a_c^+)$. This scaling is seen in figure \[figureZscale3\] for $a<a_c^+$ and $n\in\{50,100,150,\ldots,500\}$; all the data accumulate along a single curve, exposing the scaling function $h$.
figure36-.tex
Conclusions {#section4}
===========
The adsorbing self-avoiding walk is a classical model in rigorous and numerical statistical mechanics, and have received considerable attention in the physics and mathematics literature [@deG79; @HTW82; @LM88A; @HG94; @JvRR04].
In this paper the feasibility of collecting data in the microcanonical ensemble on adsorbing walks using a flat histogram implementation of the GAS algorithm [@JvRR09] was considered. This is an approximate enumeration algorithm, and the data can be used to directly estimate partition and generating functions, from which thermodynamic functions such as the free energy and specific heat can be determined (see, for example, reference [@JvR15]).
The implemementation of the algorithm was done using endpoint elementary moves on half space self-avoiding walks, and the algorithm sampled from a flat histogram with reasonable success in both length and energy in the square and cubic lattices. Analysis of the data gives good results, better than previous Monte Carlo simulations in, for example, references [@HG94; @JvRR04], but not as good as exact enumeration data in references [@BWO99; @BGJ12]. A significant advance of this algorithm is that its produces a large amount of microcanonical data. Modifications to obtained data with respect to other quantities are trivial, and can easily be implemented. The simulations reported here were done on a Dell Inspiron 530 desktop machine, but note that the algorithm can be implemented in parallel on a cluster with each cluster generating an independent sequence. This should give radically improved statistical data.
The success of the implementation suggests that this numerical method may be used on other models (collapsing self-avoiding walks [@ML89; @TJvROW96], for example). However, it may be necessary to extend the method by introducing, in addition to the sets of parameters denoted by $\{\beta_{\ell,u}\}$ and $\{\gamma_{\ell,u}\}$, additional sets of parameters which are conjugate to classes of elementary moves. For example, in the model of collapsing walks (see figure \[figure3\]), the energy may be changed by $\Delta u \in \{-2d\plus 1,-2d\plus 2, \ldots,2d\minus 1\}$ by an elementary move, and parameters may be introduced for each value of $\Delta u$ to achieve flat histogram sampling (in a way similar to the introduction of $\gamma_{\ell,u}$ for elementary moves increasing the energy of the walk. This will increase the complexity of the implementation, but with the result that flat histogram sampling will be easier to achieve.
In the models of square and cubic lattice adsorbing walks, the algorithm produced data which gave good estimates of the locations of the critical adsorption point. The best estimates are obtained from equations and , namely $$a_c^+ =
\cases{
1.779 \pm 0.003, & \hbox{in the square lattice}; \\
1.306 \pm 0.007, & \hbox{in the cubic lattice}.
}$$ These results can be used to estimate the crossover exponent $\phi$ association with the adsorption transition, and our best esimates are seen in equations and : $$\phi =
\cases{
0.496 \pm 0.009, & \hbox{in two dimensions}; \\
0.505 \pm 0.006, & \hbox{in three dimensions}.
}$$ In addition, other quantities from which $a_c^+$ and $\phi$ can be estimated were examined, and results largely consistent with the above values were obtained (see, for example, equations , and for square lattice results, and equations , and for cubic lattice results). These numerical estimates are in good agreement with those presented in reference [@JvRR04], and also in reference [@BGJ12] in the case of the square lattice. The estimate in this reference, obtained from exact series data, namely $a_c^+ = 1.77564$, agrees with the estimate above to two decimal places. In the cubic lattice the estimate for $a_c^+$ above is slightly smaller than the estimates $a_c^+=1.338
\pm 0.005$ [@ML88A] and $a_c^+ = 1.334 \pm 0.027$ in reference [@JvRR04] (rounding up of this last error bar gives a confidence interval which includes $1.306$).
The signature of the adsorption transition in the metric quantities of the model was also examined. The scaling of these quantities with $\tau = n^{\phi}(a\minus a_c^+)$ were plotted in figures \[figure6-D\], \[figure6-nuH\] and \[figure8RR\] in the square lattice, and in figures \[figure6-D3\], \[figure6-nuH3\] and \[figure8RR3\] in the cubic lattice. These results show a transition strongly characterised by changes in metric scaling and verify the value of the metric exponent and its finite size scaling through the critical point.
Finally, the scaling of the generating function and partition function in these models were examined. Our results strongly supports the conventional properties of the model, and the values of the exponents $\{\gamma_1,\gamma_s,\gamma_+\}$ estimated here are consistent with exact values and other estimates in the literature.
The results in the square lattice are consistent with the exact values of $\gamma_1$ and the surface exponent $\gamma_s$, and the generating and partition partition function exhibit scaling consistent with the value of $\gamma_s$, as shown in figures \[figureGscale\] and \[figureZscale\]. In the cubic lattice our data gave the estimates $\gamma_1 \approx 0.725$ and $\gamma_s \approx 1.203$. These values are in addition to estimates elsewhere in the literature (see references [@HG94], [@ML88A]), and although the estimates here may be improvements on previous esimates, they remain uncertain. However, scaling of the generating function in figure , and of the partition function in figure \[figureZscale3\], is some evidence that the esimate for $\gamma_s$ is at least consistent with the scaling in the model.
[**Acknowledgements:**]{} EJJvR acknowledges financial support from NSERC (Canada) in the form of a Discovery Grant.
[**References**]{}
| 0 |
---
abstract: 'The lensing cross section of triaxial halos depends on the relative orientation between a halo’s principal axes and its line of sight. Consequently, a lensing subsample of randomly oriented halos is not, in general, randomly oriented. Using an isothermal mass model for the lensing galaxies and their host halos, we show that the lensing subsample of halos that produces doubles is preferentially aligned along the lines of sight, whereas halos that produce quads tend to be projected along their middle axes. These preferred orientations result in different projected ellipticity distributions for quad, doubles, and random galaxies. We show that $\approx 300$ lens systems must be discovered to detect this effect at the $95\%$ confidence level. We also investigate the importance of halo shape for predicting the quad-to-double ratio and find that the latter depends quite sensitively on the distribution of the short-to-long axis ratio, but is otherwise nearly independent of halo shape. Finally, we estimate the impact of the preferred orientation of lensing galaxies on their projected substructure mass fraction, and find that the observed alignment between the substructure distribution and the mass distribution of halos result in a negligible bias.'
author:
- 'Eduardo Rozo, Jacqueline Chen, Andrew R. Zentner'
bibliography:
- 'mybib.bib'
title: Biases in the Gravitational Lens Population Induced by Halo and Galaxy Triaxiality
---
Introduction
============
Statistics of lensing galaxies have been used as cosmological and galaxy formation probes since early in the modern history of gravitational lensing [@turneretal84]. Lensing rates can be used to constrain dark energy [@fukugitaetal92; @chae03; @mitchelletal05; @chae07; @ogurietal07], to probe the structure of lensing galaxies [@keeton01d; @kochanekwhite01; @chae05], and to probe galaxy evolution [@chaemao03; @ofeketal03; @rusinkochanek05]. While the use of lensing statistics as a cosmological probe has had mixed success, particularly early on, it remains a unique probe with entirely different systematics from more traditional approaches. Consequently, lensing statistics are likely to remain a fundamental cross-check of our understanding of cosmology and galaxy evolution.
One of the difficulties that confronts the study of lensing statistics is that, in general, the halo population that produces gravitational lenses can in fact be a highly biased subsample of the general halo population. For instance, it has long been known that while early type galaxies compose only $\approx 30\%$ of all luminous galaxies, the majority of lensing galaxies are in fact early type since these tend to be more massive and reside in more massive halos than their late counterparts. By the same token, lensing early type galaxies tend to have higher luminosity and velocity dispersions than non-lensing early type galaxies [@moelleretal06; @boltonetal06]. Overall, then, when interpreting lensing statistics, one ought to always remember that by selecting lensing galaxies one is automatically introducing an important selection effect that can significantly bias the distribution of any galaxy observable that has an impact on the lensing probabilities. Here, we consider one such source of bias, the triaxiality of galaxy halos.[^1]
That halo triaxiality can have important consequences for lensing statistics has been known for several years. For instance, @ogurikeeton04 have shown that triaxiality can significantly enhance the optical depth of large image separation lenses. Similar conclusions have been reached concerning the formation of giant arcs by lensing clusters [see e.g. @ogurietal03; @rozoetal06c; @hennawietal07 and references therein]. Curiously, however, little effort has gone into investigating how observational properties of lensing galaxies can be different from those of the galaxy population as a whole due to the triaxial structure of galactic halos. This work addresses this omission.
The first observable we consider is the projected axis ratio of lensing galaxies. Roughly speaking, given that non-zero ellipticities are needed in order to produce quad systems, one would generically expect lenses that lead to this image configuration to be more elliptical than the overall galaxy population. Likewise, lensing galaxies that produce doubles should, on average, be slightly more circular than a random galaxy. There can, however, be complications for these simple predictions due to halo triaxiality. For instance, given a prolate halo, projections along the long axis of the lens will result in highly concentrated, very circular profiles. Will the increase in Einstein radius of such projections compensate for the lower ellipticity of the system, implying most quads will be projected along their long axis, or will it be the other way around? Clearly, the relation between ellipticity and lensing cross sections is not straightforward once triaxiality of the lensing galaxies is taken into account, but it seems clear that there should be some observable difference between the ellipticity distribution of lensing galaxies and that of all early types. Interestingly, no such difference has been observed [@keetonetal97; @rusintegmark01], which seems to fly in the face of our expectations [though see also the discussion in @keetonetal98]. Is this actually a problem, or will a quantitative analysis show that the consistency of the two distributions is to be expected? Here, we explicitly resolve this question, and demonstrate that current lens samples are much too small to detect the expected differences.
Having considered the ellipticity distribution of random and lensing galaxies, it is then a natural step to investigate the impact of halo triaxiality on predictions of the quad-to-double ratio. Specifically, it is well known that the quad-to-double ratio is sensitive to the ellipticity distribution of lensing galaxies [@keetonetal97], so if lensing can bias the distribution of ellipticities in lensing galaxies, then it should also affect the predicted quad-to-double ratios. This is an important point because it has been argued that current predictions for the quad-to-double ratio are at odds with observations. More specifically, the predicted quad-to-double ratio for the CLASS [Cosmic Lens All-Sky Survey, @myersetal03; @browneetal03] sample of gravitational lenses is too low relative to observations [@rusintegmark01; @hutereretal05]. Curiously, however, recent work on the quad-to-double ratio observed in the SQLS [Sloan Digital Sky Survey Quasar Lens Search, @ogurietal06; @inadaetal07]. suggests that the exact opposite is true for the latter sample, namely, theoretical expectations are too high relative to observations [@oguri07]. In either case, it is of interest to determine how exactly does triaxiality affects theoretical predictions, especially since the aforementioned difficulties with the CLASS sample has led various authors to offer possibilities as to how one might boost the expected quad-to-double ratios. Specifically, one can boost the quad-to-double ration in the class sample either from the effect of massive satellite galaxies near the lensing galaxies [@cohnkochanek04], or through the large-scale environment of the lensing galaxy [@keetonzabludoff04]. Clearly, we should determine whether halo triaxiality can be added to this list.
This brings us then to the final problem we consider here, namely whether the substructure population of lensing galaxies is different from that of non-lensing galaxies. Specifically, we have argued that lensing galaxies will not be isotropically distributed in space. Since the substructure distribution of a dark matter halo is typically aligned with its parent halo’s long axis [@zentneretal05; @libeskindetal05; @agustssonbrainerd06; @azzaroetal06], it follows that the projected distribution of substructures for lensing galaxies may in fact be different for lensing halos than for non-lensing halos. Such an effect could be quite important given the claimed tension between the Cold Dark Matter (CDM) predictions for the substructure mass fraction of halos [see @maoetal04] and their observed values [@dalalkochanek02a; @kochanekdalal04]. Likewise, such a bias would impact the predictions for the level of astrometric and flux perturbations produced by dark matter substructures in gravitational lenses [@rozoetal06; @chenetal07]. Here, we wish to estimate the level at which the projected substructure mass fraction of lensing halos could be affected due to lensing biasing.
The paper is organized as follows: in section \[sec:biases\] we derive the basic equations needed to compute how observable quantities will be biased in lensing galaxy samples due to halo triaxiality. Section \[sec:model\] presents the model used in this work to quantitatively estimate the level of these biases, and discusses how lensing halos are oriented relative to the line of sight as a function of the halos’ axes ratios. Section \[sec:axis\] investigates the projected axis ratio distributions of lensing versus non-lensing galaxies, and demonstrates that present day lensing samples are too small to detect the triaxiality induced biases we have predicted. Section \[sec:ratio\] discusses the problem of the quad to double ratio, and section \[sec:subs\] demonstrates that halo triaxiality biases the projected substructure mass fraction in lensing halos by a negligible amount. Section \[sec:caveats\] discusses a few of the effects we have ignored in our work and how these may alter our results, and finally section \[sec:summary\] summarizes our work and presents our conclusions.
Lens Biases Induced by Triaxiality {#sec:biases}
==================================
We begin by deriving the basic expressions on which we rely to estimate the effects of halo triaxiality on the observed properties of lensing galaxies. In particular, we show that since the lensing cross section for triaxial lenses is in general not spherically symmetric, this implies that a population of randomly oriented halos produces a non-random lens population. Finally, we show that the induced non-randomness of the lensing halo population can alter the mean observational properties of these halos relative to the general halo population.
The Lensing Cross Section {#sec:cs}
-------------------------
Let ${\mathbf{p}}$ be a set of parameters that characterizes the projected gravitational potential of a halo. For instance, ${\mathbf{p}}$ can be the Einstein radius of the lens, its ellipticity, and so on. Given a background source density $n_s(z_s)$ and a halo density $n_h({\mathbf{p}},z_h)$, and in the absence of a flux limit, the mean number of lensing events per unit redshift per area is given by $$\frac{dN_{lenses}}{dz_sdz_hd\Omega} = n_s(z_s) n_h({\mathbf{p}},z_h)
\frac{d\chi}{dz_s}\frac{d\chi}{dz_h}\sigma({\mathbf{p}},z_h,z_s)$$ where $\chi$ is the comoving distance to the appropriate halo or source redshift, and $$\sigma({\mathbf{p}},z_h,z_s) = \int_{lensing} d^2{\mathbf{y}}.Ä
\label{eq:cs}$$ The integral is over all regions of the source plane that produce lensed images of interest. For instance, if one were interested in quadruply imaged sources, the integral would be over all source positions that result in four image lenses. The quantity $\sigma$ is called the *lensing cross section, and of particular interest to us will be the cross sections $\sigma^{(N)}$ for producing $N$-image systems.*
In reality, one always has some flux limit $F_{min}$ which corresponds to a minimum source luminosity $L_{min}$. Fortunately, the above argument is easily generalized: let $dn_s(L,z_s)/dL$ be the number density of background sources with luminosity $L$. Then, the mean number of lensing events becomes $$\frac{dN_{lenses}}{dz_sdz_hd\Omega} = n_h\frac{d\chi}{dz_s}\frac{d\chi}{dz_h}
\int d^2{\mathbf{y}} \int_{L_{min}/\mu({\mathbf{y}})}^\infty dL\ \frac{dn_s(L,z_s)}{dL}.$$ If the source luminosity function can be approximated by a power law $dn_s(L,z_s)/dL = AL^{-\alpha}$ (note both $A$ and $\alpha$ can depend on $z_s$), the above expression reduces to $$\frac{dN_{lenses}}{dz_sdz_hd\Omega} = n_b(>L_{min}) n_h
\frac{d\chi}{dz_s} \frac{d\chi}{dz_h} \sigma_B({\mathbf{p}},\alpha,z_h,z_s)
\label{eq:bcs}$$ where $n_b(>L_{min},z_s)$ is the number density of sources above the flux limit *in the absence of lensing, and $\sigma_B$ is given by $$\sigma_B ({\mathbf{p}},z_h,z_s,\alpha)= \int d^2{\mathbf{y}}\ \mu({\mathbf{y}})^{\alpha-1}$$ where $\mu({\mathbf{y}})$ is the total magnification of a source at position ${\mathbf{y}}$. Following @hutereretal05, we call $\sigma_B$ the *biased cross section. Indeed, since the distribution of magnifications $p(\mu)$ among all lensing events is given by $$p(\mu) = \frac{1}{\sigma}\int d^2{\mathbf{y}}\ \delta(\mu({\mathbf{y}})-\mu)$$ where $\sigma$ is the (unbiased) lensing cross section defined in Eq. \[eq:cs\], then we can rewrite Eq. \[eq:bcs\] as $$\sigma_B = {\left\langle \mu^{\alpha-1} \right\rangle}\sigma,$$ where $${\left\langle \mu^{\alpha-1} \right\rangle} = \int d\mu\ p(\mu)\mu^{\alpha-1}.$$ Thus, the net effect of gravitational magnification on the frequency of lensing events can be summarized as a biasing factor ${\left\langle \mu^{\alpha-1} \right\rangle}$ that multiplies the unbiased lensing cross section $\sigma$.**
Triaxiality and Lensing Biasing
-------------------------------
Let ${\mathbf{P}}$ characterize the mass distribution of a triaxial halo, and let ${\mathbf{\hat n}}$ be the orientation of the halo’s long axis relative to the line of sight. The halo’s two dimensional potential is then characterized by a new set of parameters ${\mathbf{p}}({\mathbf{P}},{\mathbf{\hat n}})$ which depend on the halo properties ${\mathbf{P}}$ and the particular line of sight ${{\mathbf{\hat n}}}$ along which the halo is being viewed. For instance, the vector ${\mathbf{P}}$ can include such halo properties as halo mass and axis ratios, whereas ${\mathbf{p}}$ could include parameters such as the Einstein radius of the projected mass distribution as well as the projected axis ratio.
As discussed above, the mean number of lensing events per unit redshift by a halo along a given line of sight is given by Eq. \[eq:bcs\]. For convenience, we define the halo and source surface densities $d\Sigma_h/d{\mathbf{P}}$ and $d\Sigma_s/dz_s$ via $$\begin{aligned}
\frac{d\Sigma_h}{d{\mathbf{P}}dz_h}\ & =\ & \frac{dn_h}{d{\mathbf{P}}}\frac{d\chi}{dz_h} \\
\frac{d\Sigma_s}{dz_s} \ & =\ & n_s(>L_{min})\frac{d\chi}{dz_s}.\end{aligned}$$ In terms of these surface densities, and assuming a randomly-oriented distribution of halos, the mean number of lenses per unit area as a function of their orientation ${{\mathbf{\hat n}}}$ is given by $$\frac{dN_{lenses}}{d{\mathbf{P}}d{{\mathbf{\hat n}}}dz_sdz_hd\Omega} = \frac{1}{2\pi}\frac{d\Sigma_s}{dz_s}\frac{d\Sigma_h}{d{\mathbf{P}}dz_h}
\sigma_B({\mathbf{p}}({\mathbf{P}},{{\mathbf{\hat n}}}),z_h,z_s,\alpha).
\label{eq:numlens}$$ The prefactor of $1/(2\pi)$ arises from the fact that $dn_h/d{\mathbf{P}}d{{\mathbf{\hat n}}}= (dn_h/d{\mathbf{P}})/(2\pi)$ due to our assumption of randomly oriented halos.[^2] We emphasize that Eq. \[eq:numlens\] characterizes the number of lenses *as a function of the relative orientation ${{\mathbf{\hat n}}}$ between the halo’s major axis and the line of sight. Thus, to compute the total number of lenses irrespective of halo orientation, we would simply integrate the above expression over all lines of sight ${{\mathbf{\hat n}}}$.*
There is an absolutely key point to be made concerning Eq. \[eq:numlens\], which provides the motivation behind this work. Specifically, we note that the number of lenses is proportional to $\sigma_B({\mathbf{p}}({\mathbf{P}},{{\mathbf{\hat n}}}))$. This implies that even though the overall halo population does not have a preferred orientation in space, [*the lens population is not randomly oriented*]{}, a fact which can have observable consequences. In particular, given an observable halo property $f({\mathbf{P}},{{\mathbf{\hat n}}})$ that depends on the line of sight projection (e.g. the projected axis ratio or projected substructure mass fraction), the mean value of $f$ over all ${\mathbf{P}}$ halos is simply $${\left\langle f|{\mathbf{P}} \right\rangle}_{halos} = \int \frac{d^2{{\mathbf{\hat n}}}}{2\pi} f({\mathbf{P}},{{\mathbf{\hat n}}}),$$ whereas the mean value of $f$ over all lenses is given by $${\left\langle f|{\mathbf{P}} \right\rangle}_{lenses} = \frac{1}{{\left\langle \sigma_B|{\mathbf{P}} \right\rangle}}
\int \frac{d^2{{\mathbf{\hat n}}}}{2\pi} \sigma_B({\mathbf{p}}({\mathbf{P}},{{\mathbf{\hat n}}})) f({\mathbf{P}},{{\mathbf{\hat n}}})
\label{eq:losdist}$$ where ${\left\langle \sigma_B \right\rangle}$ is the average value of $\sigma_B$ over all lines of sight, $${\left\langle \sigma_B|{\mathbf{P}} \right\rangle} = \int \frac{d^2{{\mathbf{\hat n}}}}{2\pi} \sigma_B({\mathbf{p}}({\mathbf{P}},{{\mathbf{\hat n}}})).$$ Thus, in general, one expects that the mean value of $f$ over all lenses and over all halos will be different. In the next few sections, we identify a few halo properties that depend on line of sight projection, and determine whether lensing biases induced by triaxiality are likely to be significant.
The Model {#sec:model}
=========
We estimate the impact of halo triaxiality on the properties of lenses by considering a triaxial isothermal profile. The merit of this approach is its simplicity: because of the simple form of the matter distribution in this model, we can compute all of the relevant quantities in a semi-analytic fashion, and the main features of the model can be easily understood, thereby providing an important reference point for investigating more elaborate models. Moreover, by working out in detail a simple analytic model, our results provide an ideal test bed for more involved numerical codes, which would then allow us to investigate how our conclusions are changed as more complicated models are allowed (Chen et al. 2007, in preparation).
Semi-Analytical Modeling
------------------------
Our analytical halo model is that of a simple triaxial isothermal profile of the form $$\rho(\bar{\mathbf{x}}) = N(q_1,q_2)\frac{\sigma_v^2}{2\pi G}
\frac{1}{x^2/q_1^2+y^2+z^2/q_2^2}
\label{eq:3dsie}$$ where $q_1$ and $q_2$ are the axis ratios of the profile and we have chosen a coordinate system that is aligned with the halo’s principal axes, and such that $1\geq q_1 \geq q_2$.[^3] The normalization constant $N(q_1,q_2)$ is chosen to ensure that the mass contained within a sphere of radius $r$ be independent of the axis ratios for fixed velocity dispersion $\sigma_v^2$, the latter being the velocity dispersion of the Singular Isothermal Sphere (SIS) obtained when $q_1=q_2=1$.
Let then $(\theta,\phi)$ denote a line of sight. In appendix \[app:proj\], we show that the corresponding projected surface mass density profile is that of a Singular Isothermal Ellipsoid (SIE) which, following @kormannetal94, we write as $$\Sigma(\tilde x, \tilde y) = \frac{\sqrt{q}\tilde \sigma_v^2}{2G}\frac{1}{\tilde x^2+q^2\tilde y^2}$$ where both $q$ and $\tilde \sigma_v$ are known functions of $q_1,\ q_2$ and, in the case of $\tilde \sigma_v$, of $N(q_1,q_2)\sigma_v^2$ (see Appendix \[app:proj\] for details). In the above expression, $\tilde \sigma$ and $q$ are the effective velocity dispersion and axis ratio respectively of the projected SIE profile. As shown by @kormannetal94, the lensing cross section for an SIE scales trivially with the Einstein radius $b$[^4] $$b= 4\pi\frac{\tilde\sigma_v^2}{c^2}\frac{D_lD_{ls}}{D_s}
\label{eq:erad}$$ of the profile. Consequently, the distribution of halo orientations for a lens sample, $\rho({{\mathbf{\hat n}}})=\sigma_B({{\mathbf{\hat n}}})/{\left\langle \sigma_B \right\rangle}$, is independent of the velocity dispersion $\sigma_v$ of the halo.
There is one last important element of the model that needs to be specified, namely the luminosity function of the sources being lensed. Here, we take the luminosity function to be a power law with slope of $-2$, which, while not exactly correct, is reasonably close to the slope of the luminosity function of CLASS lenses [@chae03; @mckeanetal07]. Moreover, this choice is ideally suited for numerical work since in such a case the biased cross section is simply $\sigma_B={\left\langle \mu \right\rangle}\sigma$, implying that the biased cross section can be easily computed through uniform Monte Carlo sampling of the image plane. Since one of our goals in this work is to provide a test case for more complicated numerical algorithms, we choose $\alpha=-2$.
Having fully specified our model, we can now easily compute the biased lensing cross section for halos of any shape as a function of line of sight. Briefly, we proceed as follows. First, we compute the biased lensing cross section for SIE profiles as a function of the projected axis ratio $q$ for a grid of $q$ values. These data points are then fit using a third order polynomial fit, which we find is accurate to $\lesssim 1\%$. Using this simple fit for $\sigma_b(q)$, and the fact that we can analytically compute the Einstein radius and projected axis ratio for a triaxial halo along any line of sight, we can readily compute the mean lensing cross section of a halo averaged over all lines of sight. For a detailed description of our calculations, we refer the reader to the Appendices.
Before we end, however, it is important to remark here that, despite its simplicity, we expect our model is more than adequate to investigate the qualitative trends that we would expect to observe in the data, and for providing order of magnitude estimates of the impact of triaxiality. Specifically, elliptical isothermal profiles appear to be excellent approximations to the true matter distribution in real lens systems [see e.g. @gerhardetal01; @rusinma01; @rusinetal03; @rusinkochanek05; @treuetal06; @koopmansetal06; @gavazzietal07], so the triaxial isothermal mass distribution considered here should provide a reasonably realistic model for order of magnitude estimates. While more sophisticated models are certainly possible [see e.g. @jiangkochanek07], it is our view that the simplicity of the isothermal model more than justifies our choice of profile for a first pass at the problem.
The Distribution of Halo Orientations for Triaxial Isothermal Profiles
----------------------------------------------------------------------
Before we look at the distribution of halo orientations, it is worth taking a minute to orient ourselves in the coordinate system we have chosen. Consider first Eq. \[eq:3dsie\]. The distance from the center of the halo to the intercept of a constant density contour is maximized for the $y$ axis, and minimized for the $z$ axis, while the $x$ axis is intermediate between the two. If we then parameterize the line of sight using the circular coordinates $\theta$ and $\phi$ where $\theta$ is the angle with the $z$ axis and $\phi$ is the projected angle with the $x$ axis, then our coordinate system is such that it has the following properties.
- The $x, y,$ and $z$ axis of our coordinate system correspond to the middle, long, and short axis of the halo respectively.
- Projections along $\cos(\theta)=1$ are along the short axis of the halo.
- Projections along $\cos(\theta)=0,\ \phi=0$ are along the middle axis of the halo.
- Projections along $\cos(\theta)=0,\ \phi=\pi/2$ are along the long axis of the halo.
The nice thing about this particular choice of coordinates is that in the $\cos(\theta)-\phi$ plane, both the long and the middle axis are represented by a single point, whereas the short axis is represented by an entire line. As we shall see, projections along the middle and long axis maximize the lensing cross section of a halo for quad and double lenses respectively, so having that maximum be a single point in the space of lines of sight is a desirable quality of our chosen coordinate system.
Figure \[fig:orient\] shows the ratio $b({{\mathbf{\hat n}}})/b_0$ where $b_0$ is the Einstein radius of an SIS with velocity dispersion $\sigma_v$, as well as the projected axis ratio $q({{\mathbf{\hat n}}})$, for an isothermal ellipsoid with axis ratios $q_1=0.75,\ q_2=0.5$. We can see the Einstein radius of the projected profile is maximized when projecting along the long axis of the halo, whereas the ellipticity is maximized when projecting along the middle axis of the halo, as it should be. Note we have only considered the range $\theta\in[0,\pi/2]$ and $\phi\in[0,\pi/2]$ rather than the full range of possible lines of sight $\theta\in[0,\pi/2]$ and $\phi\in[0,2\pi]$. This is due to the symmetry of our model; all eight of the octants defined by the symmetry planes of the ellipsoids are identical.
Let us now go back and study the distribution of line of sights for both doubles and quads. Figure \[fig:losdist\] shows these distributions for three types of halos: a prolate halo, an oblate halo, and a halo that is neither strongly oblate nor strongly prolate. As is customary, we parameterize the halo shape in terms of the shape parameter $T$ which is defined as $$T = \frac{1-q_1^2}{1-q_2^2}.
\label{eq:shape_parameter}$$ Note that a perfectly prolate halo ($q_1=q_2$) has $T=1$, whereas a perfectly oblate halo ($q_1=1$) has $T=0$. From top to bottom, the halo shape parameters used to produce Figure \[fig:losdist\] are $T=0.9$ (cigar shape), $T=0.5$ (neither strongly oblate nor strongly prolate), and $T=0.1$ (pancake shape). The axis ratio $q_2$ was held fixed at $q_2=0.5$. Finally, the left column is the distribution of lines of sight for double systems, whereas the right column is the distribution for quads. For ease of comparison, the color scale has been kept fixed in all plots.
Let us begin by looking in detail at the doubles column first. As is to be expected, the distribution of lines of sight is peaked for projections along the long axis of the lens, as this line of sight maximizes the Einstein radius of the projected profile. Moreover, the distribution is very sharply peaked for cigar-like halos (top row), but is rather flat for pancake-like halos (bottom row). The reason that the distribution of lines of sight for pancake-like halos is so flat is simple: for an oblate halo, projecting along either the long or medium axis of the halo results in a large Einstein radius, but also a large ellipticity, so a large part of the multiply imaged region of the source plane actually corresponds to four image configurations, taking away from the cross section for producing doubles. When projecting along the short axis of the lens, the Einstein radius is minimized, but the projected mass distribution is nearly spherical, so the majority of the multiply-imaged region produces only doubles.
The column corresponding to quads has much more interesting structure. First, note that [*the distribution of line of sights for quad lenses peaks for projections along the middle axis of the lens rather than the long axis of the lens.*]{} As noted earlier, projections along the middle axis of the lens maximize the ellipticity of the projected profile, so relative to projections along the long axis of the lens, it is evident that the increase in ellipticity more than offsets the slightly smaller Einstein radii for the purposes of enhancing the lensing cross section for producing quad systems. It is also interesting to note that while the peak of the distribution is always clearly about the middle axis of the lens, the shape of the distribution varies considerably in going from prolate halos to oblate halos. In particular, note that for prolate halos the peak about the middle axis is relatively narrow. What is more, projections along the short axis of the lens are more likely than projections along the long axis because the latter minimizes the ellipticity of the projected profile. For oblate halos, on the other hand, projections along the long axis of the lens are almost as likely as projections along the middle axis. This is simply because for such halos, there is little difference in the ellipticity of the projected profile between projections along the middle and long axis of the halos. Consequently, both axes result in highly effective quad lenses. Note too that for pancake-like halos, projections along the short axis are strongly avoided, since this projection minimizes both the Einstein radius and the projected axis ratio of the lens.
In short, then, prolate halos and oblate halos will have very different orientation distributions: for prolate halos, nearly all doubles will be due to projections along the long axis of the lens, while most quads will be due to projections along the middle axis of the lens, followed by projections along the short axis. For oblate halos, however, all halo orientations are almost equally likely in the case of doubly imaged systems, whereas quads strongly avoid projections along the short axis of the halo.
The remainder of the paper will explore whether these results have a significant impact on the statistical properties of the halo population. Specifically, we will first consider the ellipticity distribution of lensing galaxies compared to that of galaxies as a whole. We will then discuss how these results affect the predicted quad-to-double ratio, and finally, we will investigate whether lensing halos are expected to have a significantly biased projected substructure mass fraction.
The Projected Axis Ratios of Lensing Halos {#sec:axis}
==========================================
As mentioned in the introduction, if one assumes that the ellipticity of the light and that of the mass are monotonically related, then one would naively expect that lensing galaxies that produce quads ought to be more elliptical than the average galaxy because the lensing cross section for quads increases with increasing ellipticity. Similarly, galaxies that produce doubles should tend to be more spherical. In this section, we discuss the impact of halo triaxiality on the distribution of axis ratios for double and quad lenses.
Given a line of sight ${{\mathbf{\hat n}}}$, we can compute the axis ratio $q({{\mathbf{\hat n}}})$ of the projected mass distribution (see Eq. \[eq:qproj\]). Using the distribution of lines of sight $\rho({{\mathbf{\hat n}}})$, one can then easily compute the distribution of projected axis ratios $q$ for a sample of lenses via $$\rho(q|q_1,q_2) = \int \frac{d{{\mathbf{\hat n}}}}{2\pi} \rho({{\mathbf{\hat n}}}) \delta_D(q({{\mathbf{\hat n}}}|q_1,q_2)-q).$$ Figure \[fig:qproj\_dist\] shows the distribution of the projected axis ratio of both quad and double systems for the sample pancake-like (oblate, $T=0.1$) and cigar-like (prolate, $T=0.9$) halos from Figure \[fig:losdist\]. As is to be expected, the distribution for quad systems is considerably skewed towards high ellipticity systems, whereas the distribution for doubles is much flatter. Moreover, the quads distribution is significantly more skewed for prolate (cigar-like) systems than for oblate (pancake-like) halos. Based on Figure \[fig:qproj\_dist\], we have attempted to distill the difference between quads and doubles into a single number. We define the axis ratio $q_{0.75}$ as the axis ratio for which $75\%$ of the lenses have axis ratios $q\leq q_{0.75}$.[^5] The value $q_{0.75}$ for quads and doubles for both sample halos is also shown in Figure \[fig:qproj\_dist\] as lines along the top axis of the plot. It is clear that the projected axis ratio $q_{0.75}$ for doubles and quads is very different, with $\Delta q_{0.75}>0.1$ for both oblate and prolate halos.
Figure \[fig:axis\_ratio\] shows the difference $\Delta q_{0.75}$ between doubles and quads (i.e. $q_{0.75}^{doubles}-
q_{0.75}^{quads}$, solid line) and between doubles and the overall halo populations (i.e. $q_{0.75}^{doubles}-q_{0.75}^{halos}$, dotted line) as a function of the axis ratios $q_1$ and $q_2$. However, rather than using $q_1$ as an axis, we follow standard practice and parameterize the shape of the halo in terms of the shape parameter $T$ defined in Eq. \[eq:shape\_parameter\]. There are several interesting things to be gathered from Figure \[fig:axis\_ratio\]. First, when comparing doubles to quads, note that while $\Delta q_{0.75}$ is indeed large ($q_{0.75}\gtrsim 0.1$) for both prolate and oblate halos, the difference can be larger for oblate halos than for prolate halos. Moreover, note that in going from oblate to prolate halos, the difference $\Delta q_{0.75}$ goes through a minimum when $q_2 \approx q_1^2$ (solid line), in which case values as low as $\Delta q_{0.75} \approx 0.05$ for $q_2\approx 0.5$ are possible. Turning now to the comparison between doubles and random halos, we see that the difference in $q_{0.75}$ for these two halo populations becomes negligible in the case of oblate halos, reflecting the near uniform distribution of lines of sights for doubles for oblate halos (see Figure \[fig:losdist\]). On the other hand, the fact that most prolate doubles are seen along the long axis of the halo implies that $\Delta q_{0.75}$ between doubles and random halos must be significant, and thus doubles tend to be more circular than the typical halo.
In short, then, the quantity $\Delta q_{0.75}$ between doubles and quads and between doubles and random halos can, at least in principle, help determine whether most halos are oblate or prolate. If halos are prolate, the difference $\Delta q_{0.75}$ between doubles and random halos is large. If this difference is small, we can then look at the difference $\Delta q_{0.75}$ between doubles and quads. If this last difference is large, then halos are typically oblate, whereas if the difference is small, then halos are neither strongly oblate nor strongly prolate and $q_2\approx q_1^2$.
In practice, however, the above test is difficult to execute. In particular, while lens modeling can provide some measure of the axis ratio $q$ in quad systems, there remains a fair amount of uncertainty due to the approximate degeneracy between galaxy ellipticity and external shear [see e.g. @keetonetal97]. This degeneracy is even stronger for doubly-imaged systems, and worse, there is no way of determining the axis ratio of the mass for non-lensing galaxies. Fortunately, at the scales relevant for strong lensing ($\lesssim 5\ {\mbox{kpc}}$), baryons dominate the total matter budget in early type galaxies [@rusinetal03], so one expects that the dark matter distribution in these systems will have the same ellipticity and orientation as the baryons. Observationally, @keetonetal98 [see also @keetonetal97] compared the projected ellipticity of the light in lensing galaxies to the ellipticity recovered from explicit lens modeling, and found that the light and the mass tend to be very closely aligned, though the magnitude of the ellipticities is not clearly correlated and the modest quality of the photometry available at the time made their ellipticity measurements difficult. Moreover, the galaxy sample @keetonetal98 included many galaxies that had non-negligible environments that were not incorporated into the model. More recently, a detailed study of the Sloan Lens ACS Survey [SLACS @boltonetal06] with more isolated galaxies supports the hypothesis that the ellipticity of the light is in fact extremely well matched to the ellipticity of the projected mass, at least on scales comparable to the Einstein radii of the galaxies [@koopmansetal06].[^6] Thus, for the purposes of this work, we simply take the isophotal axis ratio of lensing galaxies to be identical to the total matter axis ratio for the purposes of investigating whether lens biasing can be detected in current lensing samples.
Figure \[fig:cumq\] shows the cumulative distribution of isophotal axis ratios for quad lenses (solid) and double lenses (dashed) for all lensing galaxies in the CASTLES[^7] database with isophotal axis ratios measurements.[^8] Of course, the selection function for this sample is impossible to quantify objectively, but our intent is simply to see whether any differences between lensing galaxies and random galaxies can be found. Also shown in the figure are the axis ratio distributions of early type galaxies as reported by two different groups: the dotted line shown is the fit used by @rusintegmark01 to model the distribution of axis ratios in early type galaxies based on measurements by @jorgensenfranx94, and is also quite close to the distribution recovered by @lambasetal92. The dashed-dotted line is the axis ratio distribution obtained by @haoetal06 using the SDSS Data Release 4 photometric catalog, and is a very close match to the distribution recovered by @fasanovio91. @haoetal06 noted that it is unclear why these two distributions differ, though @keetonetal97 note that such a difference can easily arise depending on whether S0 galaxies are included in the galaxy sample or not (with S0 galaxies being more elliptical). Here, we simply consider both distributions.
Given that the axis ratio distribution for both quads and doubles largely fall in between the two model distributions we considered, it is immediately obvious that no robust results can be obtained at this time. Specifically, uncertainties in the details of the selection function of the galaxies used to construct the isophotal axis ratios are a significant systematic. More formally, using a KS-test, we find that the isophotal axis ratio distributions of both quad and double lens galaxies are consistent with that of the early type galaxy population as a whole (irrespective of which model distribution we choose) and with each other as well. Interestingly, whether or not we restrict ourselves to galaxies that are isolated or whether we include all lensing galaxies does not appear to change the result in any way. Naively, then, the consistency of the axis ratio distributions suggests that halos are typically neither strongly oblate nor prolate, but rather somewhere in between, where the quantity $\Delta q_{0.75}$ exhibits a minimum, which occurs at $q_2\approx q_1^2$.
Given that current lens samples are too small for detecting any difference on the ellipticities of quadruply and doubly imaged systems, it is worth asking whether or not a detection is possible in principle. That is, how many lenses must one have in order to detect quad systems as being more elliptical than doubles? To answer this question, we need to first assume a simple model for the distribution of axis ratios $Q(q_1,q_2)$, with which one could then compute the resulting projected axis ratio distributions for doubles, quads, and the galaxy population at large. We should note here, however, that in detail our results will depend on the adopted distribution $Q(q_1,q_2)$, which is not known.
It is not immediately obvious what the most correct model distribution $Q(q_1,q_2)$ should be. While there have been many studies that have investigated the distribution of axis ratios of dark matter halos in simulations [see e.g. @warrenetal92; @jingsuto02; @bailinsteinmetz05], it has become clear that the distribution itself depends on many variables, including halo mass [@kasunevrard05; @bettetal07], radius at which the shape of the halo is measured [@hayashietal07], halo environment [@hahnetal07], and whether the halo under consideration is a parent halo or a subhalo of a larger object [@kuhlenetal07]. Adding to these difficulties is the fact that different authors use different definitions and methods for measuring the shapes of halos, which forces one to go to great lengths in order to ensure a fair comparison of the results from different groups [see for example @allgoodetal06]. Even more problematic that all of these difficulties, however, is the fact that not only can the distributions of baryons have a different shape from the dark matter [@gottloberyepes07], baryons dominate the mass budget in the halo regions where strong lensing occurs, and can therefore dramatically impact halo shapes at those scales [@kazantzidisetal04; @bailinetal05; @gustafssonetal06]. Since our intent here is simply to provide a rough estimate of the number of lenses required to detect a significant difference in the ellipticities of quad and double systems, we simply adopt a fiducial model that is based primarily on the results of @allgoodetal06 and @kazantzidisetal04, and use it to estimate the number of lenses necessary to detect the larger ellipticity of quad systems. Specifically, @allgoodetal06 obtain that for an $M_*$ halo the distribution of the short-to-long axis ratio of dark matter halos is Gaussian with a mean of ${\left\langle q_2 \right\rangle} = 0.54$ and a standard deviation $\sigma_{q_2}=0.1$. As noted by @kazantzidisetal04, baryonic cooling tends to circularize the mass profiles of halos, so we adopt instead a somewhat larger ratio ${\left\langle q_2 \right\rangle}=0.65$, but retain the dispersion $\sigma_{q_2}=0.1$. The adopted value for ${\left\langle q_2 \right\rangle}$ is larger than that obtained from dissipationless simulations, but smaller than that found in the simulations of @kazantzidisetal04, as the latter suffer from the well known over-cooling problem and therefore overestimate the impact of baryons on the profiles. In addition, we truncate the distribution at $q_2=0.4$, as the expressions for the lensing cross sections are no longer valid for systems with projected axis ratios below $0.4$.[^9] Finally, the value of the intermediate axis $q_1$ is obtain following the model of @allgoodetal06 [itself based on the work by @jingsuto02], namely, the quantity $p=q_2/q_1$ is drawn from the distribution $$\rho(p|s) = \frac{3}{2(1-s)}\left[1 - \left(\frac{2p-1-s}{1-s}\right)^2\right]$$ where $s=\mbox{min}(0.55,q_2)$.
Figure \[fig:predict\_qproj\] shows the cumulative distributions of the predicted isophotal axis ratios for all galaxies, as well as for quad and double systems. Also shown for reference are the axis ratio measurements of early type galaxies by @haoetal06 using SDSS DR4 data. Note that, as we expected, the difference in the axis ratio $q_{0.75}$ between doubles and quads is of order $0.05$. The maximum vertical distance between the cumulative distributions functions for quads and doubles is $D\approx 0.15$, which, using a KS-statistic, implies that roughly $300$ lenses ($150$ quads, $150$ doubles) with good isophotal measurements are necessary to detect the difference between the two distributions at the $95\%$ confidence level. A $5\sigma$ detection would require $\approx 1,400$ lenses. Such large number of lenses is larger than the current list of known lensing systems, but is certainly within the realm of what one may expect from future lens searches [see e.g. @koopmansetal04; @marshalletal05].
Triaxiality and Predictions for the Quad-to-Double Ratio {#sec:ratio}
========================================================
We showed above that halo triaxiality can have an important impact on the distribution of axis ratios for lensing galaxies. Since the projected axis ratio of a halo plays a key role in the expected quad-to-double ratio of lensing galaxies, it is easy to see that triaxiality should also affect this statistical observable. This is the problem we wish to consider now: how does triaxiality affect the quad-to-double ratio of lensing galaxies?
Consider first equation \[eq:numlens\]. For our semi-analytic case, the halo parameters ${\mathbf{P}}$ that determine the mass distribution of the halo are simply the halo velocity dispersion and its two axis ratios $q_1$ and $q_2$. What is more, we saw that if we define $b_0(\sigma_v)$ as the Einstein radius of an SIS of velocity dispersion $\sigma_v$, then the ratio $\sigma/b_0^2$ depends only on the axis ratios $q_1$ and $q_2$. If we make the further assumption that the distribution of halo parameters is separable, i.e. that $$\frac{dn_{halos}}{dz_hd\sigma_vdq_1dq_2} = \frac{dn_{halos}}{dz_hd\sigma_v} Q(q_1,q_2),$$ then it is easy to see that the *ratio of the total number of quad systems to double systems depends only on the distribution of axis ratios $Q(q_1,q_2)$ because the overall scaling of the lensing cross sections for both doubles and quads just factors out of the problem. Thus, the ratio of quad-to-doubles is given simply by $$r(q_1,q_2) = \frac{\mbox{No. of quads}}{\mbox{No. of doubles}} =
\frac{ {\left\langle \sigma_B^{(4)}|q_1,q_2 \right\rangle} }{ {\left\langle \sigma_B^{(2)}| q_1,q_2 \right\rangle} }.$$*
The top panel of Figure \[fig:ratio\] shows the dimensionless mean biased lensing cross section ${\left\langle \sigma_B \right\rangle}/b_0^2$ for both doubles and quads averaged over lines of sight for a population of randomly oriented halos. Also shown in the bottom panel is the quad-to-double ratio. As expected, large ($\gtrsim 0.3$) quad-to-doubles ratios require strong deviations from spherical symmetry, so $q_2$ needs to be small. Interestingly, however, all of the contours in both the top and bottom panel of Figure \[fig:ratio\] are nearly vertical: lensing cross sections are nearly independent of halo shape. We can understand this qualitatively as follows. In the case of doubles, there is a tradeoff between two competing effects: for $1\gtrsim q_1\gg q_2$, there are many lines of sight that enhance the Einstein radius of the lens, but only moderately so. For $1\gg q_1\gtrsim q_2$ on the other hand, there are only a few lines of sight that enhance the Einstein radius of the lens (i.e. projections along the long axis of the halo), but the enhancement is much greater. Thus, the overall boost to the Einstein radius is offset by the reduced “volume” of lines of sight available for forming doubles and vice versa. A similar effect occurs for quads: oblate halos make effective lenses when projected along either the long or middle axis of the lens, but strongly avoid the short axis, so the “volume” of lines of sight available to oblate halos is small. Prolate halos, on the other hand, are not quite as effective as oblate halos at making quads, but can produce quads over a larger range of possible lines of sight.
At any rate, one thing that is clear from Figure \[fig:ratio\] is that halo shape does not have a significant impact on the expected quad-to-double ratio. One extremely interesting consequence of this results is that it implies that halo triaxiality can be properly incorporated into lensing statistics studies without greatly increasing the number of degrees of freedom in the problem. More explicitly, traditional lens statistics studies use as input the observed two dimensional ellipticity distribution of early type galaxies, and approximate the effects of triaxiality by multiplying the usual isothermal ellipsoidal profiles with a normalization factor computed assuming halos are either all perfectly oblate, or perfectly prolate [see e.g. @chae03; @chae07; @oguri07]. The main reason this is done, rather than considering triaxial halos and averaging over lines of sight, is that in order to do the latter calculation, one needs to know something about the distribution of axis ratios. We have shown, however, that such a calculation would in fact be nearly independent of assumptions made about the intermediate axis $q_1$. In other words, a proper calculation that weights lines of sight according to their biased lensing cross section rather than uniform weighting (as implicitly done when taking the ellipticity distribution to be that of early type galaxies as a whole) effectively involves no more freedom than the usual approach, the main difference being that the assumptions made will involve not the ellipticity distribution, but rather the distribution of the short-to-long axis ratio $q_2$, which can itself be constrained using the projected ellipticity distribution [e.g. @lambasetal92].
The Substructure Mass Fraction in the Inner Regions of Lensing Halos {#sec:subs}
====================================================================
One of the important predictions of the CDM paradigm of structure formation is that galactic halos contain a large amount of bound substructure within them [see e.g. @whiterees78; @blumenthaletal84]. Observationally, however, both our own galaxy and M31 have an order of magnitude less luminous companions than is predicted if one assumes substructures have a fixed mass to light ratio [@kauffmannetal93; @klypinetal99; @mooreetal99]. Currently, the favored explanation for this discrepancy is that the mass to light ratio of such small structures depends strongly on the history of the objects, and therefore only a select subset of the substructures within the halo become luminous [e.g. @somervilleprimack99; @bensonetal02; @kravtsovetal04; @salesetal07]. While such scenarios appear to be in good agreement with the data, it would still be desirable to provide as direct detection as possible of the remaining dark substructures.
Motivated by the fact that dark substructures can only be discovered via their gravitational signal, @dalalkochanek02a investigated whether the well known flux anomalies problem could be explained as the action of dark substructures embedded within the halo of the lensing galaxy. Using a relatively simple model, they found that in order to explain the observed flux anomalies, one requires a projected substructure mass fraction $f_s$ in the range $7\%>f_s>0.6\%$ at the $90\%$ confidence level. It was then argued by @maoetal04 that such a substructure mass fraction was slightly larger than the mass fraction obtained from simulations $f_s\approx 0.5\%.$
Recently, it has become clear that the distribution of substructures in dark matter halos is not spherically symmetric, but is instead triaxial, and aligned with the major axis of the halo. Since lensing halos are not randomly oriented in space, the mean projected substructure mass fraction for all halos - the $f_s\approx 0.5\%$ value obtained by @maoetal04 - need not be the same as the mean substructure mass fraction for lensing halos, which would in turn affect theoretical predictions [e.g. @rozoetal06; @chenetal07]. Here, we use the results on substructure alignments in numerical simulations to estimate the dependence of the projected substructure mass fraction $f_s$ on the projection axis. More specifically, [*assuming that substructures do not significantly alter the biased lensing cross sections for the halos*]{}, we compute the mean substructure mass fraction for doubles and quad lenses as a function of the axis ratios $q_1$ and $q_2$ of the lensing halos.
We begin by presenting the substructure mass fraction $f_s$, as a function of line-of-sight in simulated dark matter halos. In Figure \[fig:fsublos\], we reproduce the distribution as presented in @zentner06. This figure shows the mass fraction projected within $3\%$ of the virial radius as a function of the projection angle $\cos(\theta_{long})$ for a sample of halos in a dissipationless $N$-body simulation of structure growth. The angle $\theta_{long}$ is defined as the relative angle between the projection axis and the long axis of the halo. The data for the figure come from 26 host dark matter halos with masses in the range $10^{12} h^{-1}{M_\odot}< M < 10^{13} h^{-1}{M_\odot}$, and the error bar shown represents the dispersion in the sample rather than the error on the mean. The halos were drawn from a high-resolution flat, ${\Lambda\mbox{CDM}}$ simulation with $\Omega_m=0.3,$ $\sigma_8=0.9,$ $\Omega_b h^2=0.023,$ and $h=0.7.$ Details on the simulations can be found in @zentner06 or in @gottloberturchaninov06.
Using the fit to $f_s(\cos(\theta_{long}))$ shown in Figure \[fig:fsublos\], we compute the mean projected substructure mass fraction ${\left\langle f_s|q_1,q_2 \right\rangle}$ for a population of double and quad lenses as a function of the halo axis ratios $q_1$ and $q_2$. Our results are shown in Figure \[fig:fsub\]. For reference, the mean substructure mass fraction for randomly oriented halos obtained from the fit shown in figure \[fig:fsublos\] is ${\left\langle f_s \right\rangle}=0.46\%$. As per our expectations, we find that prolate (cigar-like) doubles have substructure mass fractions that are enhanced relative to the average halo, with ${\left\langle f_s \right\rangle}\approx 0.6\%$. Note though that this enhancement is relatively minor, and slowly decreases to the random average as halos become oblate.
More interesting to us is the behavior of quads, for which we find a mild enhancement relative to random for oblate halos, and a decrease in the expected substructure mass fraction for prolate halos. This can be easily understood from Figure \[fig:losdist\]: oblate quads strongly avoid projections along the short axis of the halo, and projections along the middle and long axis of the lens are nearly equally likely. Consequently, one expects an enhancement of the substructure mass fraction because some lines of sight with low $f_s$ are avoided. On the other hand, for prolate halos, projections along the long axis of the lens are the least common, so indeed we expect the mean projected substructure mass fraction for these systems to be reduced.
Overall, though, it is clear that for quad systems - which are the only kind of systems for which ${\left\langle f_s \right\rangle}$ may be estimated using the methods of @dalalkochanek02a - the substructure mass fraction in the inner regions of a halo cannot be significantly enhanced due to lens biasing if the impact of substructures on the lensing cross section of galactic halos can be neglected. Thus, lens biasing does little to soften the slight (and in these authors’ opinion, not terribly significant) discrepancy between the values of $f_s$ recovered by @dalalkochanek02a and those from numerical simulations.
Caveats and Systematics {#sec:caveats}
=======================
Before we finish, we believe it is important to mention two systematics that could significantly affect the conclusions presented in this work. Specifically, throughout we have assumed that the lensing cross section is dominated by the smooth mass distribution of lensing galaxies, and we presented in section \[sec:model\] several studies that suggest that our model for the mass distribution of early type galaxies is a reasonable one. As mentioned in the introduction, the possible discrepancy between theory and observation concerning the quad-to-double ratio of the CLASS lenses has raised the possibility that lensing cross sections are in fact heavily influenced by the environment of the halo or possibly by substructures with in it. We briefly discuss each of these in turn.
We begin by discussing halo environments. In our calculations above, and in most of the lensing statistics literature, the effect of halo environment on lensing statistics is neglected. This is not an entirely ad hoc assumption. Theoretical estimates of the amount of shear that the typical lens experiences are quite small [$\gamma \approx 0.02$, see e.g. @keetonetal97; @dalalwatson04], so its impact should be negligible. Curiously, however, explicit lens modeling of known systems usually requires large external shears ($\gamma\approx 0.1$) in order to provide reasonable fits to observations [see e.g. @keetonetal97]. Moreover, direct estimates of the environment of lensing galaxies also support the idea of a stronger effect from nearby structures [@ogurietal05]. The discrepancy between these observations and the predictions for halo environments are themselves an interesting problem, which ultimately may or may not be related to the usual quad-to-double ratio problem. At any rate, one might hope that even if such large external shears are correct, their impact on the quad-to-double ratio would still be negligible if their orientation is random. This expectation was indeed confirmed by [@rusintegmark01]. Unfortunately, it is known that a significant fraction of lenses are actually member galaxies of intermediate mass groups [@momchevaetal06; @williamsetal06], and that galaxies in groups and clusters tend to be radially aligned [@pereirakuhn05; @donosoetal06; @faltenbacheretal07], implying the randomly oriented shear assumption is likely not justified. Indeed, a careful analysis of the impact of the halo environment for group members shows neglecting to take said environment into account can lead to an underestimate of the ratio of the quad-to-double lensing cross sections for such galaxies as large as a factor of two [@keetonzabludoff04]. At this point, what seems clear is that there is not as of yet a definitive answer as to exactly how important galaxy environments are, and thus, we have opted for making the simplest possible assumption for the purposes of this work, that is, we have ignored the impact of large-scale environments.
The second solution to the quad-to-double ratio problem involves substructures. Specifically, @cohnkochanek04 have shown that the lensing cross section of galaxies is severely affected by substructures. If this is indeed the case, the way in which lensing galaxies are biased relative to the overall galaxy population depend not only on the smooth component of its mass distribution, but also on the spatial distribution of substructures within the galaxy halo. Interestingly, in such a scenario halo triaxiality would impact the orientation of halos relative to the line of sight now only through the biasing due to the smooth matter component, but also because of the previously mentioned alignment between the substructure distributions and the smooth mass distributions. We leave the question of exactly how such a population of halos would be biased to future work (Chen et al., in preparation).
Summary and Conclusions {#sec:summary}
=======================
The triaxial distribution of mass in galactic halos implies that the probability that a galaxy becomes a lens is dependent on the relative orientation of the galaxy’s major axis to the line of sight. Consequently, a subsample of randomly oriented galaxies that act as strong lenses will [*not*]{} be randomly oriented in space. The relative orientation and the strength of the alignment depends on the shape of the matter distribution, and on the type of lens under consideration: prolate doubles have a high probability of being project along their long axis, whereas the distribution of oblate doubles is nearly isotropic. Prolate quads are most often projected along their middle axis, though the degree to which alignment occurs is not as strong as for prolate doubles. Interestingly, highly prolate quads are also more likely to be projected along their short axis than along the long axis, though this very quickly changes as halos become more triaxial and less prolate. Oblate quads strongly avoid projections along the short axis of the lens, but projections along the other two axis are almost equally likely.
An important consequence of the differences in the distribution of halo orientations for quad lenses, double lenses, and the galaxy population as a whole is that the ellipticity distribution of these various samples must be different, even if the distribution of halo shapes is the same. Specifically, we predict that quad lenses are typically more elliptical than random galaxies, and that the ellipticity distribution of doubles is very slightly more circular than that of random galaxies. While current data do not show any indication of these trends, we have shown that $\approx 300\ (1,400)$ lenses are necessary to obtain a $2\sigma\ (5\sigma)$ detection of the effect.
The fact that halo triaxiality affects the ellipticity distribution of lensing galaxies also means that halo triaxiality needs to be properly taken into account in lensing statistics. Consequently, we estimate how the biased lensing cross sections of galaxies depend on halo shape, and find that they are nearly independent of the halo shape parameter $T$. Instead, the mean biased cross section of a lens depends almost exclusive on the distribution on the short-to-long axis ratio $q_2$ (often denoted by $s$).
Finally, given that the distribution of substructures in numerical simulations is observed to be preferentially aligned with the long axis of the host halos, we estimate how the preferred orientation of lensing galaxies affects their predicted substructure mass fraction. We find that biases due to non-isotropic distribution of halos relative to the line of sight have an insignificant impact on the mean substructure mass fraction of lensing galaxies.
[**Acknowledgements:** ]{} ER would like to thank Christopher Kochanek for numerous discussions and valuable comments on the manuscript which have greatly improved both the form and content of this work. The authors would also like to thank to Emilio Falco for kindly providing the isophotal axis ratio data that was needed for producing Figure \[fig:cumq\], and to Charles Keeton for a careful reading of the manuscript. ER was funded by the Center for Cosmology and Astro-Particle Physics (CCAPP) at The Ohio State University. ARZ has been funded by the University of Pittsburgh, the National Science Foundation (NSF) Astronomy and Astrophysics Postdoctoral Fellowship program through grant AST 0602122, and by the Kavli Institute for Cosmological Physics at The University of Chicago. This work made use of the National Aeronautics and Space Administration Astrophysics Data System.
Lensing Cross Sections of Singular Isothermal Ellipsoids
========================================================
The Singular Isothermal Ellipsoid (SIE) is one of the simplest lens models that can produce quadruply imaged sources. @kormannetal94 performed a detailed study of the lensing properties of SIE lenses, and, in particular, derived simple expressions for the total area contained within the tangential and radial caustics of such lenses. Specifically, given an SIE profile $$\Sigma = \frac{\sqrt{q}\sigma_v^2}{2G}\frac{1}{x^2+q^2y^2},$$ @kormannetal94 found that the area $\sigma_r$ and $\sigma_t$ contained inside the radial and tangential caustics is given by $$\sigma_r = \frac{4q}{1-q^2}\int_q^1dx\ \frac{\cos^{-1}(x)}{\sqrt{x^2-q^2}}$$ and $$\sigma_t = \frac{4q}{1-q^2}\int_q^1dx\ \left(\frac{\sqrt{1-x^2}}{x}-\cos^{-1}(x)\right)
\frac{\sqrt{x^2-q^2}}{x^2}$$ respectively. Moreover, they showed that for $q>q_c$ where $q_c\approx 0.394$, the tangential caustic is entirely contained within the radial caustic, and hence the lensing cross section ${\sigma^{(4)}}$ for forming four image lenses is simply ${\sigma^{(4)}}= \sigma_t$. Likewise, the lensing cross section for forming doubles is given by ${\sigma^{(2)}}= \sigma_r-\sigma_t$.
Unfortunately, as derived in section \[sec:cs\], the relevant quantity for lensing statistics of a flux limited sample is not the lensing cross section itself, but the biased cross section $\sigma_B$. Moreover, the latter cross section requires one to compute the magnification distribution $p(\mu)$ for double and quad lenses, for which there are no closed form expressions. In this appendix, we numerically compute the magnification distribution $p(\mu)$, and its first moment ${\left\langle \mu \right\rangle}$ for both doubles and quads, and use them to compute the biased lensing cross section $\sigma_B={\left\langle \mu \right\rangle}\sigma$ appropriate for a source luminosity function $n_s(L)\propto L^{-2}$.
The left panel of Figure \[fig:pmu\] shows the magnification distribution for doubly and quadruply image systems for SIE profiles with axis ratios $q=0.4$ and $q=0.8$. Note that the magnification distribution for doubles is very rich in features. The magnification distribution for quads, on the other hand, is relatively simple, and we can provide a simple fitting formula for it. To do so, first note that we know that in the limit $\mu\rightarrow \infty$, ${p^{(4)}(\mu)}\propto \mu^{-3}$, so we expect that ${p^{(4)}(\mu)}\approx Nx^{-3}f(x)$ where $x=\mu/\mu_{min}$ and $\mu_{min}$ is the minimum magnification for quad lenses, $N$ is a normalization constant, and $f(x)$ is a function which asymptotes to unity and deviates from unity only for $x\approx 1$. Consequently, we expand $f(x)$ in a power series in terms of $x^{-1}$, of which we expect only the first few terms would be necessary to produce a good fit. As it turns out, we found that $f(x)$ needs only one non-constant term to result in excellent fits to ${p^{(4)}(\mu)}$, and our final fitting function for ${p^{(4)}(\mu)}$ is thus $${p^{(4)}(\mu)}\approx \frac{1}{\mu_{min}^{(4)}} \frac{2}{1+a/2} x^{-3}(1+ax^{-2}).
\label{eq:quadfit}$$ A priori, we would expect that the best fit value of the $a$ coefficient in the above expression would be a function of the axis ratio $q$ of the profile. While there does appear to be some such dependence, it is extremely mild, so we have opted for keeping $a$ fixed to the value $a=0.83$. We found that this expression is accurate to better than $5\%$ for $\mu_{min} \lesssim \mu \lesssim 20\mu_{min}$ and $q\geq0.4$.
The right panel of Figure \[fig:pmu\] shows the actual quantities we are interested in, the biased lensing cross sections $\sigma_B = {\left\langle \mu \right\rangle}\sigma$. As is obvious from the figure, the form of these biased cross sections is very simple, so even a simple quadratic fit results in quite good fits (of order a few percent). Since we wish our empirical fit to be accurate, we fit the numerically computed cross sections with a cubic, which is enough to obtain sub-percent level accuracy. Our best fit curves (in a least square sense) are $$\begin{aligned}
\sigma_B^{(2)} & = & -6.902+42.937q-33.240q^2+9.736q^3 \\
\sigma_B^{(4)} & = &\ 11.409-20.833q+13.236q^2-3.816q^3.\end{aligned}$$ Of course, we could have just as easily splined the numerically estimated values to compute the lensing cross section at any axis ratio $q$. We opted to fit the cross sections with a simple form both for simplicity, and in the chance that the fitting formulae provided here will be useful for other works.
Projected Surface Density Profiles of Triaxial Isothermal Halos {#app:proj}
===============================================================
Consider an SIS profile $$\rho_{SIS}(r) = \frac{\sigma_v^2}{2\pi G}\frac{1}{r^2}.$$ Its triaxial generalization takes the form $$\rho_{SIE}(\bar{\mathbf{x}}) = N(q_1,q_2)\frac{\sigma_v^2}{2\pi G}\frac{1}{\bar x^2/q_1^2+\bar y^2+\bar z^2/q_2^2}$$ where $q_1$ and $q_2$ are the halo’s axis ratios, and we have chosen a coordinate system that is aligned the halo’s principal axis and such that $1\geq q_1 \geq q_2$. $p_2$ remains the ratio of the small to large axis. The prefactor $N(q_1,q_2)$ represents a relative normalization for halos of varying axis ratios which we will compute shortly. First however, since @kormannetal94 use the notation where the axis ratios multiply rather than divide the coordinates, we rewrite the mass density as $$\rho_{SIE}(\bar{\mathbf{x}}) = \tilde N(q_1,q_2)\frac{\sigma_v^2}{2\pi G}\frac{1}{p_1^2\bar x^2+p_2^2\bar y^2+\bar z^2}
\label{eq:triaxial}$$ where $p_1=q_2/q_1$, $p_2=q_2$, and $\tilde N(p_1,p_2) = q_2^2N(q_1,q_2)$. Note $p_1$ is the ratio of the small to middle axis, while $p_2$ remains the ratio of the small to large axis. We choose the normalization function $\tilde N(p_1,p_2)$ such that the mass contained within a radius $r$ is independent of the axis ratios $p_1$ and $p_2$, as appropriate if one wishes to investigate the impact of triaxiality on lensing cross sections at fixed mass with the latter defined using spherical overdensities. Integrating the above profiles and setting $M_{SIS}(r)=M_{SIE}(r)$ results in[^10] $$\tilde N(p_1,p_2) = \left\{ \frac{2}{\pi} \int_{0}^{\pi/2} d\phi\
\frac{\tan^{-1}\left[ \sqrt{(1-a)/a} \right]}{\sqrt{a(1-a)}}\right\}^{-1}$$ where we have defined $a(\phi;q_1,q_2)$ via $$a(\phi;q_1,q_2) = p_1^2\cos^2(\phi)+p_2^2\sin^2(\phi).$$
We wish to project $\rho_{SIE}$ along an arbitrary line of sight. Let ${\mathbf{x}}$ be a coordinate system such that the $z$ axis is aligned with the line of sight. We choose the $x$ and $y$ axis to be such that a rotation by an angle $\theta$ along the $y$ axis followed by a rotation along the $z$ axis by an angle $\phi$ recovers the coordinate system $\bar{\mathbf{x}}$ from Eq. \[eq:triaxial\]. The corresponding rotation matrix is given by $$R = \left(\begin{array}{ccc}
\cos\theta\cos\phi & -\sin\phi & \sin\theta\cos\phi \\
\cos\theta\sin\phi & \cos\phi & \sin\theta\sin\phi \\
-\sin\theta & 0 & \cos\theta
\end{array}\right).$$
By construction, the corresponding projected surface density $\Sigma(x,y)$ is given simply by $$\Sigma(x,y) = \int_{-\infty}^\infty dz\ \rho_{SIE}(R{\mathbf{x}})$$ which has the form $$\Sigma(x,y) = \tilde N(p_1,p_2) \frac{\sigma_v^2}{2\pi G} \int_{-\infty}^\infty dz\ \frac{1}{A+Bz+Cz^2}
\label{eq:projint}$$ where $$\begin{aligned}
A & = & A_{xx}x^2+A_{xy}xy+A_{yy}y^2 \\
B & = & B_xx+B_yy \\
C & = & p_1^2\sin^2\theta\cos^2\phi+p_2^2\sin^2\theta\sin^2\phi+\cos^2\theta\end{aligned}$$ and $$\begin{aligned}
A_{xx} & = & p_1^2\cos^2\theta\cos^2\phi+p_2^2\cos^2\theta\sin^2\phi+\sin^2\theta \\
A_{xy} & = & \sin(2\phi)\cos(\theta)(-p_1^2+p_2^2) \\
A_{yy} & = & p_1^2\sin^2\phi+p_2^2\cos^2\phi \\
B_x & = & \sin(2\theta)(p_1^2\cos^2\phi+p_2^2\sin^2\phi-1) \\
B_y & = & \sin(\theta)\sin(2\phi)(-p_1^2+p_2^2).\end{aligned}$$ Note that if $q_1=q_2=1$, then $A_{xx}=A_{yy}=C=1$ and $A_{xy}=B_x=B_y=0$, exactly as it should. Performing the integral in Eq. \[eq:projint\] we find $$\Sigma(x,y) = \tilde N(p_1,p_2) \frac{\sigma_v^2}{2G}\frac{1}{\sqrt{AC-B^2/4}}$$ which has the generic form $$\Sigma(x,y) = \tilde N(p_1,p_2) \frac{\sigma_v^2}{2G}\frac{1}{(\alpha_{xx}x^2+\alpha_{xy}xy+\alpha_{yy}y^2)^{1/2}}
\label{eq:projden}$$ where $$\begin{aligned}
\alpha_{xx} & = & A_{xx}C-B_x^2/4 \\
\alpha_{xy} & = & A_{xy}C-B_xB_y/2 \\
\alpha_{yy} & = & A_{yy}C-B_y^2/4.\end{aligned}$$ For $q_1=q_2=1$, the above expressions reduce to $\alpha_{xx}=\alpha_{yy}=1$ and $\alpha_{xy}=0$ as appropriate for an SIS profile. For the more general case it is evident from equationuation \[eq:projden\] that using an additional rotation of the $x-y$ plane we can diagonalize the projected mass density $\Sigma(x,y)$. We find that the required rotation angle $\psi$ is given by $$\tan2\psi = \frac{\alpha_{xy}}{\alpha_{xx}-\alpha_{yy}}.$$ Using a $\sim$ to denote the new coordinate system, we can thus write $$\Sigma(\tilde x, \tilde y) = \frac{\sqrt{q}\tilde \sigma_v^2}{2G}\frac{1}{(\tilde x^2+q^2\tilde y^2)^{1/2}}
\label{eq:projden1}$$ where $$\begin{aligned}
q^2 & = & \frac{\tilde\alpha_{yy}}{\tilde\alpha_{xx}} \label{eq:qproj} \\
\tilde\sigma_v^2 & = & \frac{\tilde N(p_1,p_2)}{\sqrt{q\tilde\alpha_{xx}}}\sigma_v^2 \label{eq:norm}\end{aligned}$$ and we have defined $$\begin{aligned}
\tilde\alpha_{xx} & = &\alpha_{xx}\cos^2\psi+\alpha_{xy}\sin\psi\cos\psi+\alpha_{yy}\sin^2\psi \\
\tilde\alpha_{yy} & = & \alpha_{xx}\sin^2\psi-\alpha_{xy}\sin\psi\cos\psi+\alpha_{yy}\cos^2\psi.\end{aligned}$$ As expected, the above expression for $q$ reduces to $q=p_2/p_1=q_1$ when we project along the $z$ axis (i.e. the short axis), to $q=p_2$ when projecting along the $y$ axis (i.e. the long axis), and to $q=p_1=q_2/q_1$ when projecting along the $x$ axis (i.e. the middle axis). The particular form of the parameterization of the surface density in Eq. \[eq:projden1\] is meant to match the conventions in @kormannetal94, which was chosen to ensure the mass contained within a given density contour be independent of $q$ for fixed $\tilde\sigma_v$.
and the
[^1]: Throughout this work, we will be using the term galaxy and halo more or less interchangeably. The reason for this is that we are primarily focused on the impact of halo triaxiality on the lensing cross section, and the latter depends only on the [*total*]{} matter density. Consequently, differentiating between halo and galaxy would only obfuscate presentation and introduce unnecessary difficulties. For instance, while modeling the total matter distribution as isothermal is a reasonable approximation, neither the baryons nor the dark matter by itself is isothermally distributed. Thus, it is much simpler to adopt an isothermal model, and refer to the baryons plus dark matter as a single entity, than to try to differentiate between the two. Likewise, when discussing triaxiality, what is important in this work is the triaxiality of the total matter distribution.
[^2]: If ${{\mathbf{\hat n}}}$ denotes the angle between the line of sight and a specified halo axis, and given that ${{\mathbf{\hat n}}}$ and $-{{\mathbf{\hat n}}}$ correspond to the same line of sight, then it is evident that the space of all lines of sight is simply $S^2/Z_2$ - a sphere with its diametrically opposed points identified. The volume of such a space with the usual metric is thus simply $2\pi$.
[^3]: i.e. $q_1$ is the ratio of medium to long axis of the halo, whereas $q_2$ is the ratio of the short to long axis. The motivation behind our particular choice of axis labeling will be made clear momentarily.
[^4]: By trivially, we mean $\sigma\propto b^2$.
[^5]: The $75\%$ number is selected in a somewhat ad hoc manner. Basically, we wanted $q_X$ to fall past the large prominent peak seen in Figure \[fig:qproj\_dist\], and in that sense $X=80\%$ or $X=90\%$ would work just as well. On the other hand, observational estimates of $q_X$ for $X$ close to unity would be quite difficult, so to some extent we wanted $X$ to be as small as possible. We chose $X=75\%$ as a reasonable value.
[^6]: We note, however, that SLAC lenses tend to have Einstein radii that are quite comparable to their optical radii, so the agreement is really expected. In principle, a discrepancy could exist for lenses with larger Einstein radii for which the total mass has a larger dark matter component.
[^7]: http://cfa-www.harvard.edu/castles/
[^8]: This data was kindly provided by Emilio Falco, private communication.
[^9]: For SIE profiles, if the projected axis ratio $q<0.4$, then naked cusp configuration appear. Since the analytical formulae we used to compute $\sigma(q)$ all compute the area contained within the tangential caustic, it follows that for $q<0.4$, our cross section estimates would correspond to the total cross section for producing either quads or naked cusps. To avoid this complication, we simply truncate our axis ratio distribution at $q_2=0.4$. Note however that since $q_2=0.4$ is already $2.5\sigma$ away from the adopted mean we expect the introduced cutoff to have a negligible impact on our results.
[^10]: To obtain the expressions above, we perform first the radial integral and then the $\theta$ integral where $\theta$ is the azimuthal angle.
| 0 |
---
abstract: 'We study the behaviour on rearrangement-invariant (r.i.) spaces of such classical operators of interest in harmonic analysis as the Hardy-Littlewood maximal operator (including the fractional version), the Hilbert and Stieltjes transforms, and the Riesz potential. The focus is on sharpness questions, and we present characterisations of the optimal domain (or range) partner spaces when the range (domain) is fixed. When an r.i. partner space exists at all, a complete characterisation of the situation is given. We illustrate the results with a variety of examples of sharp particular results involving customary function spaces.'
address:
- 'David E. Edmunds, Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, UK'
- 'Zdeněk Mihula, Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic'
- 'Vít Musil, Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic'
- 'Luboš Pick, Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic'
author:
- 'David E. Edmunds, Zdeněk Mihula, Vít Musil and Luboš Pick'
bibliography:
- 'classical-operators-arxiv.bib'
date: '31/10/2019'
title: 'Boundedness of classical operators on rearrangement-invariant spaces'
---
How to cite this paper {#how-to-cite-this-paper .unnumbered}
======================
This paper has been accepted for publication in *Journal of Functional Analysis* and is available on
<https://doi.org/10.1016/j.jfa.2019.108341>.
Should you wish to cite this paper, the authors would like to cordially ask you to cite it appropriately.
Introduction {#sec1}
============
Given function spaces $X,Y$ and an operator $T$ that maps $X$ boundedly into $Y$, it is natural to ask whether there is a space bigger than $X$ that is also mapped boundedly by $T$ into $Y$, or a space smaller than $Y$ into which $T$ maps $X$ boundedly.
Such questions have been attracting a great deal of attention for many years, in particular in connection with embeddings of Sobolev spaces, see, for example [@BMR:03; @CarSo:97; @ClaSo:16a; @ClaSo:16; @CwPu:98; @FiRa:06; @Ker:79; @MaMi:06; @MMP:06; @MaMi:10; @Tal:94; @Tal:16; @Tar:98]. By way of illustration we consider a particularly simple Sobolev embedding. Let $\Omega $ be a bounded open subset of $\mathbb{R}^{n}$, let $p\in [1,n)$ and put $p^{\ast }=np/(n-p)$. It is classical that, in standard notation, the Sobolev space $W_{0}^{1,p}(\Omega )$ is embedded in $L^{p^{\ast }}(\Omega )$. Can $W_{0}^{1,p}(\Omega )$ be embedded in a space smaller than $L^{p^{\ast }}(\Omega )$? Is there a space larger than $W_{0}^{1,p}(\Omega )$ that can be embedded in $L^{p^{\ast }}(
\Omega )?$ To make such questions sensible the class of competing spaces must be specified. If we restrict ourselves to Lebesgue spaces as targets and domain spaces that are Sobolev spaces based on Lebesgue spaces, then the embedding $W_{0}^{1,p}(\Omega )\hookrightarrow $ $L^{p^{\ast }}(\Omega )$ is optimal in the sense that neither the domain nor the target space can be improved. This leaves open the question of optimality in classes of spaces wider than those involving the Lebesgue scale. If the class of admissible target spaces is taken to be that of rearrangement-invariant (r.i.) spaces, then the optimal range space turns out to be the Lorentz space $L^{p^{\ast },p}(\Omega )$ there is a similar improvement of the domain space, involving a Sobolev space based on a Lorentz rather than a Lebesgue space.
The first results in this direction were obtained in [@EKP] in connection with rearrangement-invariant quasinorms. Further extensions concerning r.i. norms were added later in several papers, for instance [@T2; @T3]. A comprehensive treatment of optimal Sobolev embeddings on Euclidean domains equipped with general measures having specific isoperimetric properties was given in [@CPS].
Embeddings are not the only maps for which such questions are of interest and importance. The optimality of r.i. spaces on which the Laplace transform $\mathcal{L}$ acts boundedly was studied in a recent paper [@BEP]. A special case of the results obtained is that if $p\in (1,\infty )$ and $q\in \lbrack 1,\infty ]$, then $\mathcal{L}$ maps the Lorentz space $L^{p,q}(0,\infty )$ boundedly into $L^{p^{\prime },q}(0,\infty )$, a fact which we denote by $
\mathcal{L}\colon L^{p,q}(0,\infty )\rightarrow L^{p^{\prime },q}(0,
\infty )$. Moreover, both the domain and target spaces are optimal: there is no r.i. space smaller than $L^{p^{\prime },q}(0,\infty )$ into which $\mathcal{L}$ maps $L^{p,q}(0,\infty )$, and there is no r.i. space larger than $L^{p,q}(0,\infty )$ mapped by $\mathcal{L}$ into $L^{p^{\prime },q}(0,\infty )$. Thus in particular $\mathcal{L}
\colon L^{p}(0,\infty )\rightarrow L^{p^{\prime },p}(0,\infty )$ and the spaces involved form an optimal pair; if $p>2$, there is no $q$ for which $\mathcal{L}\colon L^{p}(0,\infty )\rightarrow L^{q}(0,\infty )$.
In the present paper we discuss such problems for classical operators of great interest in analysis and its applications, namely the Hilbert and Stieltjes transforms, the Riesz potential and various versions of the maximal operator. The action of these operators on specific classes of function spaces has been extensively studied over several decades. Classical results are available for example in connection with familiar function spaces. The 1970s experienced a real boom of this theory involving weighted Lebesgue spaces and fundamental papers were written ([@Muc:72; @Sa:82] for the Hardy–Littlewood maximal operator, [@MW:71] for singular and fractional integrals, [@CF:74; @HMW; @MW:76] for the Hilbert transform). Later it became apparent that Lebesgue spaces are not sufficient for describing all the important situations and other function spaces were investigated. Classical Lorentz spaces which originated in the 1950s and have been occurring occasionally later (see [@Bag:83; @Boy:67]) became extremely fashionable in the 1990s when the fundamental papers [@AM:90; @Sa:90] appeared. Various important and deep results were obtained, see for example [@ACS:12; @CO:15; @CS:93; @CS:97; @CE:97]. Orlicz spaces which generalize Lebesgue’s scale in a direction essentially different from Lorentz spaces, received much attention too, see for instance [@BK:84; @BP:87; @Cia:97; @Cia:99; @CM:19; @Gal:88; @Mus:19]. The results naturally found their way into important monographs that are considered classic these days, see [@CMP:11; @DHHR:11; @GR:85; @Mabook; @Ruz:00; @Ste:70; @Ste:93; @SW:71]. Let us point out that, in particular, in the monograph [@Mabook], among plenty of other fundamental results, the significance of the connection between embeddings and integral operators is explained in great detail.
On the other hand, surprisingly little attention has been paid to the *sharpness* of the results, perhaps with an exception of results in different direction on optimality obtained e.g. in [@DS:07; @ST:16] and the references therein, where operators related to the Hardy averaging operator are studied, see also [@CR:02]. Optimal range spaces for Calderón operators are studied in the recent paper [@STZ:19].
In this paper we study the behaviour of classical operators on r.i. spaces, a class of function spaces that includes for example all Lebesgue, Lorentz, Orlicz, Lorentz-Zygmund spaces and more. Our focus is mainly on the optimality of function spaces.
We use the Hardy-Littlewood maximal operator $M$ to illustrate the results obtained and serve as an appetiser for the forthcoming attractions. Let $X$ be an r.i. space over $\mathbb{R}^{n}$ with associate space $X^{\prime }$ denote by $X^{\prime }(0,\infty )$ the representation space of $X^{\prime }$ and suppose that the function $\psi $ given by $\psi (t)=\chi _{(0,1)}(t)
\log (1/t)$ belongs to $X^{\prime }(0,\infty )$. Let $Y^{\prime }$ be the set of all $f$ such that $$\begin{aligned}
\varrho (f)=\left \Vert \int \nolimits _{t}^{\infty }f^{\ast }(s)s^{-1}
\,\mathrm{d}s\right \Vert _{X^{\prime }(0,\infty )}<\infty .\end{aligned}$$ Endowed with the norm $\varrho $, $Y^{\prime }$ is an r.i. space with associate space $Y$ that not only has the property that $M\colon X
\rightarrow Y$, but is also the optimal range space corresponding to $X$. If $\psi \notin X^{\prime }(0,\infty )$, there is no r.i. space $Z$ over $\mathbb{R}^{n}$ such that $M\colon X\rightarrow Z$.
The situation turns out to be considerably more complicated in the case of the fractional maximal operator, another classical operator of harmonic analysis. The reason is that the appropriate analogue of the Riesz–Wiener–Herz inequality for the fractional maximal operator leads to an inevitable involvement of a supremum type operator, rather than just an integral mean. Supremum operators are not linear and in general are less manageable than their integral companions. However, using a fine analysis combining known and new techniques and various delicate estimates we are able to characterize the optimal range space for this operator as well. Since the general resulting condition is however naturally not so simple as in the case of the operator $M$, we include another, simpler characterization, available under a rather mild extra assumption. We also include an interesting and perhaps somewhat surprising result describing a vital link between optimality properties of a space and boundedness of a supremum operator on its associate space that leads to a self-explanatory characterization of the above-mentioned extra condition. This part of the paper is one of the most innovative ones.
We finally consider two other classical operators of harmonic analysis, namely the Hilbert transform and the Riesz potential. The importance of these operators is very well known, and their properties have been deeply studied. Our contribution is the characterization of the optimality of the spaces involved. In case of the Hilbert transform we use the Stieltjes transform as the appropriate tool and obtain characterizations for it as well.
For each of the operators considered, we are also able to nail down the optimal domain partner when the range space is fixed, this task being in general slightly simpler than the converse one. To establish all this, a combination of new techniques developed here with those from [@CPS; @EKP] and [@T2] is used.
We illustrate the results obtained with variety of nontrivial examples. For instance, we recover the well-known fact that $$\begin{aligned}
M\colon L(\log L)^{\alpha }(Q)\to L(\log L)^{\alpha -1}(Q)\end{aligned}$$ when $\alpha \geq 1$, $Q\subset \mathbb{R}^{n}$ is a cube of finite measure and $L(\log L)^{\alpha }(Q)$ is the classical Zygmund class defined as the collection of all measurable functions $g$ on $Q$ satisfying $\int _{Q}|g(x)|(\log (1+|g(x)|)^{\alpha }\mathrm{d}x<\infty $, but we add the information that the range space cannot be improved in any way when the competing spaces are rearrangement invariant. Similar examples are even more interesting when the functions act on a set of unbounded measure, say, $\mathbb{R}^{n}$. We will for example prove that if $X$ is the space equipped with the norm $\|f\|_{X}=\int _{0}^{\infty }f^{*}(t)w(t)\,dt$, where $$\begin{aligned}
w(t)=(1-\log t)^{\alpha _{0}}\chi _{(0,1)}+(1+\log t)^{\alpha _{\infty }}
\chi _{[1,\infty )}\end{aligned}$$ and $\alpha _{0}\geq 1$ and $\alpha _{\infty }\in [-1,0]$, then the optimal (smallest possible) r.i. range space $Y$ such that $$\begin{aligned}
M\colon X(\mathbb{R}^{n})\to Y(\mathbb{R}^{n})\end{aligned}$$ is the space whose associate space has norm $$\begin{aligned}
\|f\|=\sup _{0<t<\infty }w(t)\sp{-1}\int _{t}^{\infty }f^{*}(s)\,\frac{
\mathrm{d}s}{s},\quad f\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$ Such results have not been available before, and the latter norm cannot be identified with any customary known one.
We get analogous sets of examples for other operators, too. For example in the case of the fractional maximal operator we essentially improve some results from earlier papers such as [@EO; @EOP; @EOP-broken; @OP].
Preliminaries {#sec2}
=============
In this section we collect all the background material that will be used in the paper. We start with the operation of the nonincreasing rearrangement of a measurable function.
Throughout this section, let $(R,\mu )$ be a $\sigma $-finite nonatomic measure space. We set $$\begin{aligned}
\mathcal{M}(R,\mu )= \{f: f \mbox{ is a } \mu {-}\mbox{measurable function on }R \mbox{ with values in }[-\infty ,\infty ]\},\end{aligned}$$ $$\begin{aligned}
\mathcal{M}_{0}(R,\mu )= \{f \in \mathcal{M}(R,\mu ): f \mbox{ is
finite } \mu \mbox{-a.e. on } R\}\end{aligned}$$ and $$\begin{aligned}
\mathcal{M}_{+}(R,\mu )= \{f \in \mathcal{M}(R,\mu ): f \geq 0\}.\end{aligned}$$ The *nonincreasing rearrangement* $f^{*} \colon [0,\infty )
\to [0, \infty ]$ of a function $f\in \mathcal{M}(R,\mu )$ is defined as $$\begin{aligned}
f^{*}(t)=\inf \{\lambda \in (0,\infty ): \mu(\{s\in R: |f(s)|>
\lambda \})\leq t\},\ t\in [0,\infty ).\end{aligned}$$ The *maximal nonincreasing rearrangement* $f^{**} \colon (0,
\infty ) \to [0, \infty ]$ of a function $f\in \mathcal{M}(R,\mu )$ is defined as $$\begin{aligned}
f^{**}(t)=\frac{1}{t}\int _{0}^{t} f^{*}(s)\,\mathrm{d}s,\quad t\in (0,\infty ).\end{aligned}$$ If $|f|\leq |g|$ $\mu $-a.e. in $R$, then $f^{*}\leq g^{*}$. The operation $f\mapsto f^{*}$ does not preserve sums or products of functions, and is known not to be subadditive. The lack of subadditivity of the operation of taking the nonincreasing rearrangement is, up to some extent, compensated by the following fact [@BS Chapter 2, (3.10)]: for every $t\in (0,\infty )$ and every $f,g\in \mathcal{M}(R,\mu )$, we have $$\begin{aligned}
\label{E:subadditivity-of-doublestar}
\int _{0}^{t}(f +g)^{*}(s)\,\mathrm{d}s
\leq \int _{0}^{t}f^{*}(s)\,\mathrm{d}s + \int _{0}^{t}g
^{*}(s)\,\mathrm{d}s.\end{aligned}$$ This inequality can be also written in the form $$\begin{aligned}
\label{E:subadditivity-of-doublestar-a}
(f+g)^{**}\leq f^{**}+g^{**}.\end{aligned}$$ A fundamental result in the theory of Banach function spaces is the *Hardy lemma* [@BS Chapter 2, Proposition 3.6] which states that if two nonnegative measurable functions $f,g$ on $(0,\infty )$ satisfy $$\begin{aligned}
\int _{0}^{t}f(s)\,\mathrm{d}s\leq \int _{0}^{t}g(s)\,\mathrm{d}s\end{aligned}$$ for all $t\in (0,\infty )$, then, for every nonnegative nonincreasing function $h$ on $(0,\infty )$, one has $$\begin{aligned}
\int _{0}^{\infty }f(s)h(s)\,\mathrm{d}s\leq \int _{0}^{\infty }g(s)h(s)\,\mathrm{d}s.\end{aligned}$$
Another important property of rearrangements is the *Hardy-Littlewood inequality* [@BS Chapter 2, Theorem 2.2], which asserts that, if $f, g \in \mathcal{M}(R,\mu )$, then $$\begin{aligned}
\label{E:HL}
\int _{R} |fg| \,\mathrm{d}\mu \leq \int _{0}^{\infty } f^{*}(t) g^{*}(t)\,\mathrm{d}t.\end{aligned}$$
If $(R,\mu )$ and $(S,\nu )$ are two (possibly different) $\sigma $-finite measure spaces, we say that functions $f\in
\mathcal{M}(R,\mu )$ and $g\in \mathcal{M}(S,\nu )$ are *equimeasurable*, and write $f\sim g$, if $f^{*}=g^{*}$ on $(0,\infty )$.
A functional $\varrho \colon \mathcal{M}_{+} (R,\mu ) \to [0,\infty ]$ is called a *Banach function norm* if, for all $f$, $g$ and $\{f_{j}\}_{j\in \mathbb{N}}$ in $\mathcal{M}_{+}(R,\mu )$, and every $\lambda \geq 0$, the following properties hold:
1. $\varrho (f)=0$ if and only if $f=0$; $\varrho (\lambda f)= \lambda
\varrho (f)$; $\varrho (f+g)\leq \varrho (f)+ \varrho (g)$ (the *norm axiom*);
2. $ f \le g$ a.e. implies $\varrho (f)\le \varrho (g)$ (the *lattice axiom*);
3. $ f_{j} \nearrow f$ a.e. implies $\varrho (f_{j}) \nearrow \varrho (f)$ (the *Fatou axiom*);
4. $\varrho (\chi _{E})<\infty $ for every $E\subset R$ of finite measure (the *nontriviality axiom*);
5. if $E$ is a subset of $R$ of finite measure, then $\int _{E} f\,{\mathrm{d}}\mu \le C_{E} \varrho (f)$ for some positive constant $C_{E}$, depending on $E$ and $\varrho $ but independent of $f$ (the *local embedding in $L^{1}$*).
If, in addition, $\varrho $ satisfies
- $\varrho (f) = \varrho (g)$ whenever $f^{*} = g^{*}$(the *rearrangement-invariance axiom*),
then we say that $\varrho $ is an *r.i. norm*.
If $\varrho $ is an r.i. norm, then the collection $$\begin{aligned}
X=X({\varrho })=\{f\in \mathcal{M}(R,\mu ): \varrho (|f|)<\infty
\}\end{aligned}$$ is called a *rearrangement-invariant space* (*r.i. space* for short), corresponding to the norm $\varrho $. We shall write $\|f\|_{X}$ instead of $\varrho (|f|)$. Note that the quantity $\|f\|_{X}$ is defined for every $f\in \mathcal{M}(R,
\mu )$, and $$\begin{aligned}
f\in X\quad \Leftrightarrow \quad \|f\|_{X}<\infty .\end{aligned}$$
With any r.i. norm $\varrho $ is associated another functional, $\varrho '$, defined for $g \in \mathcal{M}_{+}(R,
\mu )$ as $$\begin{aligned}
\varrho '(g)=\sup \left \{ \int _{R} fg\,{\mathrm{d}}\mu : f\in \mathcal{M}_{+}(R,\mu ),\ \varrho (f)\leq 1\right \}
.\end{aligned}$$ It turns out that $\varrho '$ is also an r.i. norm, which is called the *associate norm* of $\varrho $. Moreover, for every r.i. norm $\varrho $ and every $f\in
\mathcal{M}_{+}(R,\mu )$, we have (see [@BS Chapter 1, Theorem 2.9]) $$\begin{aligned}
\varrho (f)=\sup \left \{ \int _{R}fg\,{\mathrm{d}}\mu : g\in \mathcal{M}_{+}(R,\mu ),\ \varrho '(g)\leq 1\right \}
.\end{aligned}$$ If $\varrho $ is an r.i. norm, $X=X({\varrho })$ is the r.i. space determined by $\varrho $, and $\varrho '$ is the associate norm of $\varrho $, then the function space $X({\varrho '})$ determined by $\varrho '$ is called the *associate space* of $X$ and is denoted by $X'$. We always have $(X')'=X$, and we shall write $X''$ instead of $(X')'$. Furthermore, the *Hölder inequality* $$\begin{aligned}
\int _{R}fg\,{\mathrm{d}}\mu \leq \|f\|_{X}\|g\|_{X'}\end{aligned}$$ holds for every $f,g\in \mathcal{M}(R,\mu )$.
An important consequence of the Hardy lemma, which plays a crucial role in the theory of rearrangement-invariant spaces, is the *Hardy–Littlewood–Pólya principle* [@BS Chapter 2, Theorem 4.6] which asserts that if two functions $f,g$ satisfy the so-called *Hardy–Littlewood–Pólya relation*, defined by $$\begin{aligned}
\int _{0}^{t}f^{*}(s){\mathrm{d}}s\leq \int _{0}^{t}g^{*}(s){\mathrm{d}}s, \quad t\in (0,\infty ),\end{aligned}$$ and sometimes denoted by $f\prec g$ in the literature, then $\|f\|_{X}\leq \|g\|_{X}$ provided that the underlying measure space is resonant. We note that throughout this paper we work solely on nonatomic measure spaces, which are resonant by [@BS Chapter 2, Theorem 2.7].
For every r.i. space $X$ over the measure space $(R,\mu )$ there exists a unique rearrangement-invariant space $X(0,\mu (R))$ over the interval $(0,\mu (R))$ endowed with the one-dimensional Lebesgue measure such that $\|f\|_{X}=\|f^{*}\|_{X(0,
\mu(R))}$. This space is called the *representation space* of $X$. This follows from the Luxemburg representation theorem [@BS Chapter 2, Theorem 4.10]. Throughout this paper, the representation space of an r.i. space $X$ will be denoted by $X(0,\mu (R))$. It will be useful to notice that when $R=(0,\infty )$ and $\mu $ is the Lebesgue measure, then every $X$ over $(R,\mu )$ coincides with its representation space.
If $\varrho $ is an r.i. norm and $X=X({\varrho })$ is the r.i. space determined by $\varrho $, we define its *fundamental function*, $\varphi _{X}$, for every $t\in [0,\mu (R))$ by $\varphi _{X}(t)=\varrho (\chi _{E})$, where $E\subset R$ is such that $\mu (E)=t$. The properties of r.i. norms and the fact that the underlying measure space is nonatomic guarantee that the fundamental function is well defined. Moreover, one has $$\begin{aligned}
\label{E:fundamental-relation}
\varphi _{X}(t)\varphi _{X'}(t)=t, \quad t\in [0,\mu (R)).\end{aligned}$$
Let $X$ and $Y$ be r.i. spaces over $(0,\infty )$ and let $I\colon [0,\infty )\to [0,\infty )$ be a nondecreasing function. Then $$\begin{aligned}
\label{T:Lenka-unrestricted}
\left \| \int _{t}^{\infty }\frac{f(s)}{I(s)}\,{\mathrm{d}}s\right \| _{Y(0,\infty )}
\le C_{1} \|f\|_{X(0,\infty )}
\quad \mbox{for every }f\in \mathcal{M}_{+}(0,\infty )\end{aligned}$$ holds true with some positive constant $C_{1}$ if and only if $$\begin{aligned}
\label{T:Lenka-nonincreasing}
\left \| \int _{t}^{\infty }\frac{g(s)}{I(s)}\,{\mathrm{d}}s\right \| _{Y(0,\infty )}
\leq C_{2} \|g\|_{X(0,\infty )}
\quad
\mbox{for every nonincreasing }g\in \mathcal{M}_{+}(0,\infty )\end{aligned}$$ is valid with some positive constant $C_{2}$. This result originated as a consequence [@CPS Corollary 9.8] of a more general principle established in [@CPS Theorem 9.5] in connection with sharp higher-order Sobolev-type embeddings and its extension to unbounded intervals was given in [@P Theorem 1.10].
An important corollary of the Hardy–Littlewood inequality is the fact that if $f$ is a nonincreasing function on $(0,\infty )$ and $X$ is an r.i. space over $(0,\infty )$, then in fact one has $$\begin{aligned}
\label{E:corollary-of-HL}
\|f\|_{X(0,\infty )}=\sup \left \{ \int _{0}^{\infty }g^{*}(t)f(t)\,
{\mathrm{d}}t: \|g\|_{X'(0,\infty )}\leq 1\right \} .\end{aligned}$$ In other words, for such $f$, the supremum can be reduced to nonincreasing functions only without any loss of information. This fact has deep consequences and will be used in the proofs below.
For each $a\in (0,\infty )$, let $D_{a}$ denote the *dilation operator* defined on every nonnegative measurable function $f$ on $(0,\infty )$ by $$\begin{aligned}
(D_{a}f)(t)=f(at),\quad t\in (0,\infty ).\end{aligned}$$ The operator $D_{a}$ is bounded on every rearrangement-invariant space over $(0,\infty )$ (hence in particular on the representation space of any r.i. space over an arbitrary adequate measure space). More precisely, if $X$ is any given r.i. space over $(0,\infty )$ with respect to the one-dimensional Lebesgue measure, then we have $$\begin{aligned}
\|D_{a}f\|_{X}\leq C\|f\|_{X}, \quad f\in X,\end{aligned}$$ with $C\le \max\{1,\frac{1}{a}\}$. For more details, see [@BS Chapter 3, Proposition 5.11].
Among basic examples of function norms are those associated with the standard Lebesgue spaces $L^{p}$. For $p\in (0,\infty ]$, we define the functional $\varrho _{p}$ by $$\begin{aligned}
\varrho _{p}(f)=\|f\|_{p}=
{\begin{cases}
\left (\int _{R}f^{p}\,{\mathrm{d}}\mu \right )^{\frac{1}{p}}
&\textup{if}\ 0<p<\infty ,
\cr
\operatorname{ess\,\sup }_{R}f
&\textup{if}\ p=\infty
\end{cases}}
$$ for $f \in \mathcal{M}_{+}(R,\mu )$. If $p\in [1,\infty ]$, then $\varrho _{p}$ is an r.i. norm.
If $0< p,q\le \infty $, we define the functional $\varrho _{p,q}$ by $$\begin{aligned}
\varrho _{p,q}(f)=\|f\|_{p,q}=
\left \| s^{\frac{1}{p}-\frac{1}{q}}f
^{*}(s)\right \| _{q}\end{aligned}$$ for $f \in \mathcal{M}_{+}(R,\mu )$. The set $L^{p,q}$, defined as the collection of all $f\in \mathcal{M}(R,\mu )$ satisfying $\varrho _{p,q}(|f|)<
\infty $, is called a *Lorentz space*. If either $1<p<\infty $ and $1\leq q\leq \infty $ or $p=q=1$ or $p=q=\infty $, then $\varrho _{p,q}$ is equivalent to an r.i. norm in the sense that there exists an r.i. norm $\sigma $ and a constant $C$, $0<C<\infty $, depending on $p,q$ but independent of $f$, such that $$\begin{aligned}
C^{-1}\sigma (f)\leq \varrho _{p,q}(f)\leq C\sigma (f).\end{aligned}$$ As a consequence, $L^{p,q}$ is considered to be an r.i. space for these cases of $p,q$, see [@BS Chapter 4]. If either $0<p<1$ or $p=1$ and $q>1$, then $L^{p,q}$ is a quasi-normed space. If $p=\infty $ and $q<\infty $, then $L^{p,q}=\{0\}$. For every $p\in [1,\infty ]$, we have $L^{p,p}=L^{p}$. Furthermore, if $p,q,r\in (0,\infty ]$ and $q\leq r$, then the inclusion $L^{p,q}\subset L^{p,r}$ holds.
If ${\mathbb{A}}=[\alpha _{0},\alpha _{\infty }]\in \mathbb{R}^{2}$ and $t\in \mathbb{R}$, then we shall use the notation ${\mathbb{A}}+t=[
\alpha _{0}+t,\alpha _{\infty }+t]$.
Let $0<p,q\le \infty $, ${\mathbb{A}}=[\alpha _{0},\alpha _{\infty }]
\in \mathbb{R}^{2}$ and ${\mathbb{B}}=[\beta _{0},\beta _{\infty }]
\in \mathbb{R}^{2}$. Then we define the functionals $
\varrho _{p,q;{\mathbb{A}}}$ and $\varrho _{p,q;{\mathbb{A}},{\mathbb{B}}}$ on $\mathcal{M}_{+}(R,
\mu )$ by $$\begin{aligned}
\varrho _{p,q;{\mathbb{A}}}(f)=
\left \| t^{\frac{1}{p}-\frac{1}{q}}
\ell ^{{\mathbb{A}}}(t) f^{*}(t)\right \| _{L^{q}(0,\infty )}\end{aligned}$$ and $$\begin{aligned}
\varrho _{p,q;{\mathbb{A}},{\mathbb{B}}}(f)=
\left \| t^{\frac{1}{p}-
\frac{1}{q}}\ell ^{{\mathbb{A}}}(t)\ell \ell ^{{\mathbb{B}}}(t)
f^{*}(t)\right \| _{L^{q}(0,\infty )},\end{aligned}$$ where $$\begin{aligned}
\ell ^{{\mathbb{A}}}(t)=
{\begin{cases}
(1-\log t)^{\alpha _{0}}
&\textup{if}\ t\in (0,1),
\cr
(1+\log t)^{\alpha _{\infty }}
&\textup{if}\ t\in [1,\infty )
\end{cases}}
$$ and $$\begin{aligned}
\ell \ell ^{{\mathbb{B}}}(t)=
{\begin{cases}
(1+\log (1-\log t))^{\beta _{0}}
&\textup{if}\ t\in (0,1),
\cr
(1+\log (1+\log t))^{\beta _{\infty }}
&\textup{if}\ t\in [1,\infty ).
\end{cases}}
$$ The set $L^{p,q;{\mathbb{A}}}$, defined as the collection of all $f\in \mathcal{M}(R,\mu )$ satisfying $\varrho _{p,q;{\mathbb{A}}}(|f|)<
\infty $, is called a *Lorentz–Zygmund space*, and the set $L^{p,q;{\mathbb{A}},{\mathbb{B}}}$, defined as the collection of all $f\in \mathcal{M}_{+}(R,\mu )$ satisfying $
\varrho _{p,q;{\mathbb{A}},{\mathbb{B}}}(|f|)<\infty $, is called a *generalized Lorentz–Zygmund space*. The functions of the form $\ell ^{{\mathbb{A}}}$, $\ell \ell ^{{\mathbb{B}}}$ are called *broken logarithmic functions*. The spaces of this type proved to be quite useful since they provide a common roof for many customary spaces. These include not only Lebesgue spaces and Lorentz spaces, but also all types of exponential and logarithmic Zygmund classes, and also the spaces discovered independently by Maz’ya (in a somewhat implicit form involving capacitary estimates [@Mabook pp. 105 and 109]), Hansson [@Ha] and Brézis–Wainger [@BW] who used it to describe the sharp target space in a limiting Sobolev embedding (the spaces can be also traced in the works of Brudnyi [@B] and, in a more general setting, Cwikel and Pustylnik [@CP]). One of the benefits of using broken logarithmic functions consists in the fact that the underlying measure space can be considered to have either finite or infinite measure. For the detailed study of generalized Lorentz–Zygmund spaces we refer the reader to [@EOP; @EOP-broken; @OP; @FS].
We further define the spaces $L^{(p,q;{\mathbb{A}})}$ through the functionals $\varrho _{(p,q;{\mathbb{A}})}$ given on $\mathcal{M}_{+}(R,
\mu )$ by $$\begin{aligned}
\varrho _{(p,q;{\mathbb{A}})}(f)=
\left \| t^{\frac{1}{p}-\frac{1}{q}}
\ell ^{{\mathbb{A}}}(t) f^{**}(t)\right \| _{L^{q}(0,\infty )}\end{aligned}$$ and, in an analogous way, all the other spaces involving various levels of logarithms.
Let $X$ and $Y$ be r.i. spaces over possibly different measure spaces $(R,\mu )$ and $(S,\nu )$, respectively, and let $T$ be an operator defined on $X$ with values in $\mathcal{M}(S,
\nu )$. We say that $T$ is *bounded* from $X$ to $Y$, a fact which is denoted by $T\colon X\to Y$, if there exists a positive constant $C$ such that $$\begin{aligned}
\|Tf\|_{Y}\leq C\|f\|_{X} , \quad f\in X.\end{aligned}$$ In an important special case when $T$ is the identity operator, we say that $X$ is *embedded* into $Y$ and write $X\hookrightarrow Y$. If $T'$ is another operator defined at least on $Y'$ with values in $\mathcal{M}(R,\mu )$ and such that $$\begin{aligned}
\label{E:duality-of-operators}
\int _{R}(Tf) g\,{\mathrm{d}}\mu =\int _{S}f(T'g)\,\mathrm{d}\nu\end{aligned}$$ for every $f\in X$ and $g\in Y'$, then $T\colon X\to Y$ is equivalent to $T'\colon Y'\to X'$.
Let $P$ and $Q$ be the integral operators defined by $$\begin{aligned}
(Pf)(t)=\frac{1}{t}\int _{0}^{t} f(s)\,{\mathrm{d}}s,\quad t\in (0,\infty ),\end{aligned}$$ and $$\begin{aligned}
(Qf)(t)=\int _{t}^{\infty }f(s)\frac{{\mathrm{d}}s}{s}, \quad t\in (0,\infty ),\end{aligned}$$ for those functions on $f\in \mathcal{M}_{0}(0,\infty )$ for which the respective integrals have sense. As an interchange of integration shows, $$\begin{aligned}
\int _{0}^{\infty }(Pf)(t)g(t)\,{\mathrm{d}}t=\int _{0}^{\infty }f(t)(Qg)(t)\,{\mathrm{d}}t,\end{aligned}$$ for all $f$ and $g$ for which the integrals make sense. Hence, the operators $P$ and $Q$ are formally adjoint with respect to the $L^{1}$-pairing and therefore satisfy a relation in the spirit of . As a consequence, one has the equivalence $$\begin{aligned}
\label{E:equivalence-of-P-Q}
P\colon X\to Y\quad \Leftrightarrow \quad Q\colon Y'\to X'\end{aligned}$$ for every pair of r.i. spaces $X,Y$ over $(0,\infty )$ (with the same operator norm). Another important example is that when $(R,\mu )$ is arbitrary and both $T$ and $T'$ are identity operators. Then is trivially satisfied and, as a consequence, one gets $$\begin{aligned}
\label{E:equivalence-of-identities}
X\hookrightarrow Y\quad \Leftrightarrow \quad Y'\hookrightarrow X'\end{aligned}$$ for every pair of r.i. spaces $X,Y$, again with the same embedding constant, see [@BS Chapter 1, Proposition 2.10].
We will say that an r.i. space $Y$ over $(S,\nu )$ is a *range partner* for a given r.i. space $X$ over $(R,\mu )$ with respect to a sublinear operator $T$ if $T\colon X\to Y$. We say that $Y$ is the *optimal range partner* for $X$ if one has $Y\hookrightarrow Z$ for every range partner $Z$ for $X$ with respect to $T$. We analogously define a *domain partner* and the *optimal domain partner*, that is, the largest possible domain space.
Throughout the paper the convention that $\frac{1}{\infty }=0$, and $0\cdot \infty =0$ is used without further explicit reference. We write $A\approx B$ when the ratio $A/B$ is bounded from below and from above by positive constants independent of appropriate quantities appearing in expressions $A$ and $B$.
The Hardy-Littlewood maximal operator {#sec3}
=====================================
In this section, the relevant r.i. spaces are considered over $\mathbb{R}^{n}$ endowed with the $n$-dimensional Lebesgue measure. The Lebesgue measure of a measurable set $E\subset
\mathbb{R}^{n}$ will be denoted by $|E|$.
The *Hardy–Littlewood maximal operator*, $M$, is defined for every locally integrable function $f$ on $\mathbb{R}^{n}$ and every $x\in \mathbb{R}^{n}$ by $$\begin{aligned}
Mf(x)=\sup _{Q\owns x}\frac{1}{|Q|}\int _{Q}|f(y)|\,{\mathrm{d}}y,\end{aligned}$$ where the supremum is extended over all cubes $Q\subset \mathbb{R}
^{n}$, whose edges are parallel to the coordinate axes of $\mathbb{R}
^{n}$, that contain $x$.
The operator $M$ is merely sublinear, rather than linear, and it is clearly a contraction on $L^{\infty }$. On the other hand, $Mf$ is never integrable unless $f\equiv 0$. For every locally-integrable function $f$ on $\mathbb{R}^{n}$, one has $|f|\leq Mf$ almost everywhere. The most important information (for our purpose) concerning the operator $M$, now classical, states that there exist positive constants $c,c'$, depending only on $n$, such that $$\begin{aligned}
\label{E:herz}
c(Mf)^{*}(t)\leq f^{**}(t)\leq c'(Mf)^{*}(t), \quad t\in (0,\infty ),\end{aligned}$$ for every locally integrable function $f$ on $\mathbb{R}^{n}$. The first inequality in was established during the 1930s in works of R.M. Gabriel [@G], F. Riesz [@R] and N. Wiener [@W], while the second was added later through the efforts of C. Herz [@H] (for one dimension) and C. Bennett and R. Sharpley [@BS-paper] (for higher dimensions). The result is summarized and proved in [@BS Chapter 3, Theorem 3.8].
We shall now state the first principal result of this section, in which we characterize the optimal range partner to a given space with respect to the operator $M$.
\[T:maximal-operator\] Let $X$ be an r.i. space over $\mathbb{R}^{n}$ such that $$\begin{aligned}
\label{E:psi-condition}
\psi \in X'(0,\infty ),\end{aligned}$$ where $\psi (t)=\chi _{(0,1)}(t)\log \tfrac{1}{t}$, $t\in (0,\infty )$. Define the functional $\sigma $ by $$\begin{aligned}
\sigma (f)=\left \| \int _{t}^{\infty }f^{*}(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,\infty )}, \quad f\in \mathcal{M}_{+}(\mathbb{R}
^{n}).\end{aligned}$$ Then $\sigma $ is an r.i. norm and $$\begin{aligned}
\label{E:M-bounded}
M\colon X\to Y,\end{aligned}$$ where $Y=Y(\sigma ')$. Moreover, $Y$ is the optimal smallest r.i. space for which holds.
Conversely, if is not true, then there does not exist an r.i. space $Y$ for which holds.
We now turn our attention to the question of the optimal domain space when the target space is prescribed. This situation is considerably simpler than the reverse one as no associate norms need to be involved.
\[T:maximal-operator-domain\] Let $Y$ be an r.i. space over $\mathbb{R}^{n}$ such that $$\begin{aligned}
\label{E:psi-condition-domain}
\psi \in Y(0,\infty ),\end{aligned}$$ where $\psi (t)=\min \{1,\frac{1}{t}\}$ for $t\in (0,\infty )$. Define the functional $\varrho $ by $$\begin{aligned}
\varrho (f)=\|f^{**}\|_{Y(0,\infty )}, \quad f\in \mathcal{M}_{+}(
\mathbb{R}^{n}).\end{aligned}$$ Then $\varrho $ is an r.i. norm and is satisfied, where $X=X(\varrho )$. Moreover, $X$ is the optimal largest rearrangement-invariant space for which holds.
Conversely, if is not true, then there does not exist an r.i. space $X$ for which holds.
In our final result of this section we present a collection of nontrivial examples based on Lorentz–Zygmund spaces.
\[T:-maximal-operator-GLZ\] Let $p,q\in [1,\infty ]$, ${\mathbb{A}}\in \mathbb{R}^{2}$. Then $$\begin{aligned}
M\colon L^{p,q; {\mathbb{A}}} \to
\left\{
\begin{array}{l@{\quad }l@{\quad }l}
L^{1, 1, \mathbb{A} - 1}, & p = 1, q = 1, \alpha _{0}\geq 1,
\alpha _{\infty }< -1, &\mathrm{(a)}
\label{E:maximal_p1}
\\
Y,
& p = 1, q = 1, \alpha _{0}\geq 1, -1\leq \alpha _{\infty }
\leq 0, & \mathrm{(b)}
\label{E:maximal_p2}
\\
L^{p,q;{\mathbb{A}}}, & 1< p <\infty ~ or\\
& p=\infty , 1\leq q<\infty , \alpha _{0} + \frac{1}{q} < 0~ or \\
& p=\infty , q=\infty , \alpha _{0}\leq 0,
\end{array}\right.\end{aligned}$$ where $Y$ is the unique rearrangement-invariant space whose associate space $Y'$ satisfies $$\begin{aligned}
\|f\|_{Y'}=\sup _{0<t<\infty }\ell ^{-{\mathbb{A}}}(t)\int _{t}^{\infty
}f^{*}(s)\,\frac{{\mathrm{d}}s}{s}, \quad f\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$ These spaces are the optimal range partners with respect to $M$.
We note that the space $Y'$, given in terms of an operator-induced norm, cannot be expressed in terms of a Lorentz–Zygmund norm.
We shall now proceed to prove the stated results.
The functional $\sigma $ is obviously rearrangement invariant and, thanks to the Monotone Convergence Theorem, it satisfies the lattice axiom and the Fatou axiom. From (P1), only the triangle inequality needs proving. Let $f,g\in \mathcal{M}(\mathbb{R}^{n})$. By the definition of the associate space, one has $$\begin{aligned}
\left \| \int _{t}^{\infty }(f+g)^{*}(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,\infty )}
=
\sup _{\|h\|_{X(0,\infty )}\leq 1}
\int _{0}^{\infty }h(t)\int _{t}^{\infty }(f+g)^{*}(s)\frac{{\mathrm{d}}s}{s}\,{\mathrm{d}}t.\end{aligned}$$ Since the function $$\begin{aligned}
t\mapsto \int _{t}^{\infty }(f+g)^{*}(s)\frac{{\mathrm{d}}s}{s}\end{aligned}$$ is nonincreasing on $(0,\infty )$, we in fact have (cf. ) $$\begin{aligned}
\left \| \int _{t}^{\infty }(f+g)^{*}(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,\infty )}
=
\sup _{\|h\|_{X(0,\infty )}\leq 1}
\int _{0}^{\infty }h^{*}(t)\int _{t}^{\infty }(f+g)^{*}(s)\frac{{\mathrm{d}}s}{s}\,{\mathrm{d}}t.\end{aligned}$$ Thus, by the Fubini theorem, $$\begin{aligned}
\left \| \int _{t}^{\infty }(f+g)^{*}(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,\infty )}
&=
\sup _{\|h\|_{X(0,\infty )}
\leq 1}\int _{0}^{\infty }(f+g)^{*}(s)h^{**}(s)\,{\mathrm{d}}s.\end{aligned}$$ By and the Hardy lemma, one has, for every such $h$, $$\begin{aligned}
\int _{0}^{\infty }(f+g)^{*}(s)h^{**}(s)\,{\mathrm{d}}s\leq \int _{0}^{\infty }f^{*}(s)h^{**}(s)\,{\mathrm{d}}s+\int _{0}^{\infty }g^{*}(s)h^{**}(s)\,{\mathrm{d}}s.\end{aligned}$$ This estimate, combined with the preceding identity and the subadditivity of the supremum, finally yields $$\begin{aligned}
\left \| \int _{t}^{\infty }(f+g)^{*}(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,{\infty })}
\leq \left \| \int _{t}^{\infty
}f^{*}(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,{\infty })}
+
\left \| \int _{t}^{\infty }g
^{*}(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,{\infty })},\end{aligned}$$ establishing the triangle inequality for $\sigma $.
As for (P4), let $E\subset \mathbb{R}^{n}$ be a set of finite measure. We need to prove that $$\begin{aligned}
\left \| \int _{t}^{\infty }\chi _{E}^{*}(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,\infty )}<\infty .\end{aligned}$$ Since $\chi _{E}^{*}=\chi _{(0,|E|)}$, this amounts to showing the finiteness of the quantity $$\begin{aligned}
\left \| \chi _{(0,|E|)}(t)\int _{t}^{|E|}\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,\infty )}=\left \| \chi _{(0,|E|)}(t)\log
\tfrac{|E|}{t}\right \| _{X'(0,\infty )}.\end{aligned}$$ As $D_{|E|}(\chi _{(0,|E|)}(t)\log \tfrac{|E|}{t})=\chi _{(0,1)}(t)
\log \tfrac{1}{t}$, and the dilation operator $D_{|E|}$ is bounded on $X'(0,\infty )$, we obtain that $\left \| \chi _{(0,|E|)}(t)\log
\tfrac{|E|}{t}\right \| _{X'(0,\infty )}$ is finite if and only if holds, which, however, is guaranteed by the assumption. This shows (P4).
Finally, to verify (P5), let $f\in \mathcal{M}_{+}(\mathbb{R}^{n})$ and let $E\subset \mathbb{R}^{n}$ be of finite measure. Then, by the monotonicity of $f^{*}$, we obtain $$\begin{aligned}
\sigma (f)
&
\geq \left \| \int _{t}^{2t}f^{*}(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,\infty )}\geq \left \| f^{*}(2t)\int _{t}
^{2t}\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,\infty )}
=
\left \| f^{*}(2t)\right \|
_{X'(0,\infty )}\log 2.\end{aligned}$$ Since $X'$ itself is an r.i. space, it satisfies (P5). In other words, there is a positive constant $C_{E}$, independent of $f$, such that $$\begin{aligned}
\int _{E}f\,{\mathrm{d}}\mu \leq C_{E}\left \| f\right \| _{X'}.\end{aligned}$$ By the rearrangement invariance of the space $X'$ and the boundedness of the dilation operator on $X'(0,\infty )$, we finally get from the preceding estimates that $$\begin{aligned}
\int _{E}f\,{\mathrm{d}}\mu \leq C_{E}\left \| f^{*}\right \| _{X'(0,\infty )}\leq
\widetilde{C}_{E}\left \| f^{*}(2t)\right \| _{X'(0,\infty )}
\leq \frac{\widetilde{C}_{E}}{\log 2}\sigma (f)\end{aligned}$$ for some positive constant $\widetilde{C}_{E}$, independent of $f$. This shows that $\sigma $ satisfies (P5) and, altogether, that $\sigma $ is an r.i. norm.
We shall now show that $M\colon X\to Y$. Recall that $$\begin{aligned}
\left \| \int _{t}^{\infty }g^{*}(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,{\infty })}=\|g\|_{Y'(0,{\infty })}, \quad
g\in \mathcal{M}_{+}(0,{\infty }).\end{aligned}$$ The next step is getting rid of the star in the last identity, which can be done thanks to the equivalence of and . We conclude that there exists a positive constant $C$ such that, $$\begin{aligned}
\left \| \int _{t}^{\infty }g(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,{\infty })}\leq C\|g\|_{Y'(0,{\infty })}
, \quad g\in \mathcal{M}_{+}(0,{\infty }).\end{aligned}$$ We emphasize that this step (the fall of a star) is quite deep and that it does not follow from the Hardy–Littlewood inequality (as it might deceptively appear) because the integration takes place far away from zero. Once the inequality is unrestricted to monotone functions, we are entitled to apply the standard argument using associate spaces. Using , we get $$\begin{aligned}
\left \| \frac{1}{t}\int _{0}^{t}g(s)\,{\mathrm{d}}s\right \| _{Y(0,{\infty })}\leq C\|g\|_{X(0,{\infty })}, \quad
g\in \mathcal{M}_{+}(0,{\infty }),\end{aligned}$$ with the constant $C$ undamaged. Now we need our star back, but this time that is achieved easily. We just restrict the last inequality to the cone of nonincreasing functions and obtain $$\begin{aligned}
\left \| \frac{1}{t}\int _{0}^{t}g^{*}(s)\,{\mathrm{d}}s\right \| _{Y(0,{\infty })}\leq C\|g^{*}\|_{X(0,{\infty })}
, \quad g\in \mathcal{M}_{+}(0,{\infty }).\end{aligned}$$ Applying the rearrangement invariance of the space $X$ and using the correspondence between an r.i. space and its representation space, we readily see that this can be rewritten as $$\begin{aligned}
\left \| \frac{1}{t}\int _{0}^{t}f^{*}(s)\,{\mathrm{d}}s\right \| _{Y(0,{\infty })}\leq C\|f\|_{X}, \quad
f\in \mathcal{M}(\mathbb{R}^{n}).\end{aligned}$$ By the first inequality in , we obtain that there exists a positive constant $C'$ such that $$\begin{aligned}
\left \| (Mf)^{*}\right \| _{Y(0,{\infty })}\leq C'\|f\|_{X}, \quad
\mathcal{M}(\mathbb{R}^{n}).\end{aligned}$$ Finally, the rearrangement invariance of the space $Y$ yields $$\begin{aligned}
\left \| Mf\right \| _{Y}\leq C'\|f\|_{X}, \quad \mathcal{M}(\mathbb{R}^{n}).\end{aligned}$$ In other words, $M\colon X\to Y$.
We shall now establish the optimality property of $Y$. To this end, assume that, for some r.i. space $Z$ over $\mathbb{R}^{n}$, we have $M\colon X\to Z$. This means that there exists a positive constant $C$ such that for every $f\in L^{1}_{
\operatorname{loc}}(\mathbb{R}^{n})$ the inequality $$\begin{aligned}
\|Mf\|_{Z}\leq C\|f\|_{X}\end{aligned}$$ holds. Translated to the world of rearrangements, this reads $$\begin{aligned}
\|(Mf)^{*}\|_{Z(0,\infty )}\leq C\|f^{*}\|_{X(0,\infty )}.\end{aligned}$$ Using the second inequality in , we get $$\begin{aligned}
\left \| \frac{1}{t}\int _{0}^{t}f^{*}(s)\,{\mathrm{d}}s\right \| _{Z(0,{\infty })}\leq C'\|f^{*}\|_{X(0,{\infty })}
, \quad \mathcal{M}(\mathbb{R}
^{n}).\end{aligned}$$ with some positive constant $C'$. A special case of the Hardy–Littlewood inequality together with (P2) for $Z$ now yields $$\begin{aligned}
\left \| \frac{1}{t}\int _{0}^{t}g(s)\,{\mathrm{d}}s\right \| _{Z(0,{\infty })}
\leq \left \| \frac{1}{t}\int _{0}
^{t}g^{*}(s)\,{\mathrm{d}}s\right \| _{Z(0,{\infty })} , \quad g\in
\mathcal{M}_{+}(0,{\infty }).\end{aligned}$$ Thus, since $\|g\|_{X(0,\infty )}=\|g^{*}\|_{X(0,\infty )}$, we have $$\begin{aligned}
\left \| \frac{1}{t}\int _{0}^{t}g(s)\,{\mathrm{d}}s\right \| _{Z(0,{\infty })}\leq C'\|g\|_{X(0,{\infty })}, \quad
g\in \mathcal{M}_{+}(0,{\infty }).\end{aligned}$$ By , this is nothing else than $$\begin{aligned}
\left \| \int _{t}^{\infty }g(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,{\infty })}\leq C'\|g\|_{Z'(0,{\infty })}
, \quad g\in \mathcal{M}_{+}(0,{\infty }).\end{aligned}$$ Restricting this inequality to nonincreasing functions, we get $$\begin{aligned}
\left \| \int _{t}^{\infty }g^{*}(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,{\infty })}\leq C'\|g^{*}\|_{Z'(0,{\infty })}
, \quad g\in \mathcal{M}_{+}(0,\infty ).\end{aligned}$$ By the definition of $Y'$ and by the rearrangement invariance of $Z'(0,{\infty })$, this can be rewritten as $$\begin{aligned}
\left \| g\right \| _{Y'}\leq C'\|g\|_{Z'},
\quad g\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$ In other words, we have established the embedding $Z'\hookrightarrow
Y'$, which is, due to , equivalent to $Y\hookrightarrow Z$. This shows that $Y$ is indeed the optimal range partner for $X$ with respect to $M$.
Finally, assume that $\psi \notin X'(0,\infty )$ and suppose that $M\colon X\to Y$ for some $Y$. Then, following the same line of argument as above, we obtain that $$\begin{aligned}
\|Qg^{*}\|_{X'(0,\infty )}\leq C\|g\|_{Y'(0,\infty )}, \quad g\in \mathcal{M}_{+}(0,
\infty ),\end{aligned}$$ with some $C$, $0<C<\infty $, independent of $g$. Inserting $g=\chi _{(0,1)}$, we obtain that the right side of the last inequality is finite, since $Y'$ is an r.i. space, and, as such, it must obey the axiom (P4). The left side is however infinite, because we have $$\begin{aligned}
\|Q\chi _{(0,1)}^{*}\|_{X'(0,\infty )}=\|\psi \|_{X'(0,\infty )}=
\infty .\end{aligned}$$ This is absurd, hence there is no such $Y$. The proof is complete.
The functional $\varrho $ obviously obeys (P1), (P2), (P3) and (P6). In particular, the triangle inequality follows immediately from the triangle inequality for $Y(0,\infty )$ and . Thanks to the boundedness of the dilation operator on $Y(0,\infty )$, (P4) is equivalent to $\chi _{(0,1)}^{**}\in Y(0,\infty )$, which is however guaranteed by the assumption of the theorem, since $\chi _{(0,1)}^{**}=\psi $. Finally, (P5) follows easily from the chain $$\begin{aligned}
\varrho (g)\geq \|g^{**}\chi _{(0,|E|)}\|_{Y(0,\infty )}
\geq g^{**}(|E|)
\|\chi _{(0,|E|)}\|_{Y(0,\infty )}
\geq \frac{1}{|E|}\|\chi _{(0,|E|)}\|_{Y(0,
\infty )}\int _{E} g(x)\,{\mathrm{d}}x,\end{aligned}$$ where $E\subset \mathbb{R}^{n}$ is an arbitrary set of finite measure and $g\in \mathcal{M}_{+}(\mathbb{R}^{n})$. We used the monotonicity of $g^{**}$ and the Hardy–Littlewood inequality. The operator $M$ is obviously bounded from $X$ to $Y$ thanks to . The optimality of $X$ follows from the following simple argument. Suppose that $M\colon Z\to Y$ for some r.i. space $Z$. Then $\|Mf\|_{Y}\leq C\|f\|_{Z}$ for some $C>0$ and all $f\in Z$. Therefore, by once again, we have $\|f^{**}\|_{Y}\leq C\|f\|_{Z}$, which, however, is nothing else than the embedding $Z\hookrightarrow
X$. Finally, if $\psi \notin Y(0,\infty )$ then there is no domain partner for $Y$ with respect to $M$, because if there was one, say $X$, then one would have in particular $\|\chi _{(0,1)}^{**}\|_{Y(0,
\infty )}\leq C\|\chi _{(0,1)}\|_{X(0,\infty )}$, but the right-hand side is finite due to (P4) for $X$ and the left-hand side is equal to infinity since $\psi \notin Y(0,\infty )$. The proof is complete.
We first recall that if for an r.i. space $X$ one has $M\colon X\to X$, then automatically $X$ is the optimal range (and domain) partner for itself with respect to $M$. This immediately follows from the inequality $f^{**}\geq f^{*}$ combined with . Now [@OP Theorem 3.8] together with implies that $M\colon L^{p,q;{\mathbb{A}}}\to L^{p,q;{\mathbb{A}}}$ when either $1<p<\infty $ or $p=\infty $, $1\leq q<\infty $ and $\alpha _{0}+
\frac{1}{q}<0$ or $p=\infty $, $q=\infty $ and $\alpha _{0}\leq 0$. This proves the assertion in all cases except (\[E:maximal\_p1\]a) and (\[E:maximal\_p2\]b).
Assume now that $p=1$, $q=1$, $\alpha _{0}\geq 1$ and $\alpha _{\infty
}\leq 0$. By [@OP Theorem 7.1], $L^{1,1;{\mathbb{A}}}$ is equivalent to an r.i. space. Moreover, by [@OP Theorem 6.6], $(L^{1,1;{\mathbb{A}}})'=L^{\infty ,
\infty ;-{\mathbb{A}}}$. Thus, one has $$\begin{aligned}
\|\psi \|_{X'(0,\infty )}
\approx \sup _{0<t\leq 1}(1-\log t)^{1-\alpha
_{0}}(t)<\infty ,\end{aligned}$$ since $\alpha _{0}\geq 1$. In other words, $\psi \in X'(0,\infty )$. Consequently, by [Theorem \[T:maximal-operator\]]{}, the optimal range partner $Y$ for $L^{1,1;{\mathbb{A}}}$ with respect to $M$ satisfies $$\begin{aligned}
\label{E:Y'}
\|f\|_{Y'}=\left \| \int _{t}^{\infty }f^{*}(s)\frac{{\mathrm{d}}s}{s}\right \| _{X'(0,\infty )}
=\sup _{0<t<\infty }\ell ^{-{\mathbb{A}}}(t)
\int _{t}^{\infty }f^{*}(s)\,\frac{{\mathrm{d}}s}{s}, \quad f\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$ This establishes (\[E:maximal\_p2\]b).
It remains to prove (\[E:maximal\_p1\]a). To do this we have to show that, for this choice of parameters, the space $Y$ whose associate space has norm given by coincides with $L^{1,1;{\mathbb{A}}-1}$. We have $$\begin{aligned}
\|f\|_{Y'}
&=&\sup _{0<t<\infty }\ell ^{-{\mathbb{A}}}(t)\int _{t}^{
\infty }f^{*}(s)\,\frac{{\mathrm{d}}s}{s}
\\
&=&\sup _{0<t<\infty }\ell ^{-{\mathbb{A}}}(t)\int _{t}^{\infty }f^{*}(s)
\ell ^{-{\mathbb{A}}+1}(s)\ell ^{{\mathbb{A}}-1}(s)\frac{{\mathrm{d}}s}{s}
\\
&\leq &\left (\sup _{0<s<\infty }f^{*}(s)\ell ^{-{\mathbb{A}}+1}(s)\right )
\left (\sup _{0<t<\infty }\ell ^{-{\mathbb{A}}}(t)\int _{t}^{\infty }
\ell ^{{\mathbb{A}}-1}(s)\frac{{\mathrm{d}}s}{s}\right )
\\
&\approx &\|f\|_{L^{\infty ,\infty ; -{\mathbb{A}}+1}},\end{aligned}$$ and, conversely, $$\begin{aligned}
\|f\|_{Y'}
&\geq &\max \left \{
\sup _{0<t<1}(1-\log t)^{-\alpha _{0}}
\int _{t}^{\sqrt{t}}f^{*}(s)\,\frac{\mathrm{
d}s}{s},
\sup _{1<t<\infty }(1+\log t)^{-\alpha _{\infty }}\int _{t}
^{t^{2}}f^{*}(s)\,\frac{\mathrm{
d}s}{s}
\right \}
\\
&\geq &\max \left \{
\sup _{0<t<1}(1\!-\!\log t)^{-\alpha _{0}}f^{*}(
\sqrt{t})\log (t^{-\frac{1}{2}}),
\sup _{1<t<\infty }(1\!+\!\log t)^{-
\alpha _{\infty }}f^{*}(t^{2})\log t
\right \}
\\
&\approx &\max \left \{
\sup _{0<t<1}(1-\log t)^{1-\alpha _{0}}f^{*}(\sqrt{t}),
\sup _{1<t<\infty }(1+\log t)^{1-
\alpha _{\infty }}f^{*}(t^{2})
\right \}
\\
&\approx &\max \left \{
\sup _{0<t<1}(1-\log t)^{1-\alpha _{0}}f
^{*}(t),
\sup _{1<t<\infty }(1+\log t)^{1-\alpha _{\infty }}f^{*}(t)
\right \}
\\
&\approx &\|f\|_{L^{\infty ,\infty ; -{\mathbb{A}}+1}}.\end{aligned}$$ Therefore, $Y'=L^{\infty ,\infty ; -{\mathbb{A}}+1}$, and, finally, by [@OP Theorem 6.2], we get $Y=L^{1,1; {\mathbb{A}}-1}$, as desired.
The fractional maximal operator {#sec4}
===============================
In this section we shall treat the *fractional maximal operator* $M_{\gamma }$, defined for a fixed $\gamma \in (0,n)$ and for every locally integrable function on $\mathbb{R}^{n}$ by $$\begin{aligned}
M_{\gamma }f(x)=\sup _{Q\owns x}\frac{1}{|Q|^{1-\frac{\gamma }{n}}}\int
_{Q}|f(y)|\,\mathrm{
d}y, \quad x\in \mathbb{R}^{n}.\end{aligned}$$
The operator $M_{\gamma }$ can be defined in the same way also for $\gamma =0$, in which case it coincides with the Hardy–Littlewood maximal operator, and constitutes thereby its natural generalization. The two types of operators nevertheless have to be treated separately because their behaviour in cases $\gamma =0$ and $\gamma >0$ is, rather surprisingly, substantially different, and, in the fractional case, a new approach involving a specific supremum operator is needed for the study of the optimal action of the operator on function spaces. Since the supremum operator is not linear, the use of techniques based on associate norms and spaces is somewhat limited, and a certain care has to be exercised.
The result of [@CKOP Theorem 1.1] shows that there exists a positive constant $C$ depending only on $\gamma $ and $n$ such that, for every $\mathcal{M}(\mathbb{R}^{n})$, one has $$\begin{aligned}
\label{E:upper-bound-for-fractional}
(M_{\gamma }f)^{*}(t)\leq C\sup _{t\leq s<\infty }s^{\frac{\gamma }{n}}
f^{**}(s) , \quad t\in (0,\infty ),\end{aligned}$$ and, conversely, for every nonincreasing function $g$ on $(0,\infty )$ there exists some $f_{0}\in L^{1}_{\operatorname{loc}}(\mathbb{R}^{n})$ such that $f_{0}^{*}=g$ almost everywhere on $(0,\infty )$ and $$\begin{aligned}
\label{E:lower-bound-for-fractional}
(M_{\gamma }f_{0})^{*}(t)\geq c\sup _{t\leq s<\infty }s^{\frac{\gamma
}{n}} g^{**}(s) , \quad t\in (0,\infty ),\end{aligned}$$ where, again, $c$ is some positive constant which depends only on $\gamma $ and $n$. For $\gamma =0$, the combination of and coincides with , since the function $g^{**}$ is nonincreasing on $(0,\infty )$ for any $g$.
\[T:fractional-maximal-operator\] Let $X$ be an r.i. space over $\mathbb{R}^{n}$. Let $\gamma \in (0,n)$ and assume that $$\begin{aligned}
\label{E:fund}
\inf _{1\leq t<\infty } \varphi _{X}(t)t^{-\frac{\gamma }{n}}>0.\end{aligned}$$ Define the functional $\sigma $ by $$\begin{aligned}
\label{E:sigma-frac}
\sigma (f)=\sup _{{\substack{h\sim f \\ h\ge 0}}}
\left \| \int _{t}^{\infty }h(s)s^{\frac{\gamma }{n}-1}\,
\mathrm{
d}s\right \| _{X'(0,\infty )},
\ f\in \mathcal{M}_{+}(\mathbb{R}^{n}),\end{aligned}$$ where the supremum is taken over all $h\in \mathcal{M}_{+}(\mathbb{R}
^{n})$ equimeasurable with $f$. Then $\sigma $ is an r.i. norm and $$\begin{aligned}
\label{E:MG}
M_{\gamma }\colon X\to Y,\end{aligned}$$ where $Y=Y(\sigma ')$. Moreover, $Y$ is the optimal smallest r.i. space for which holds.
Conversely, if is not true, then there does not exist an r.i. space $Y$ for which holds.
The expression for the functional $\sigma $ in [Theorem \[T:fractional-maximal-operator\]]{} is somewhat implicit. Our next result however shows that it can be considerably simplified at a relatively low cost. We shall need a *supremum operator*. For a fixed $\alpha \geq 0$, define the operator $T_{\alpha }$ on $\mathcal{M}(0,\infty )$ by $$\begin{aligned}
T_{\alpha }f(t)= t^{-\alpha }\sup _{t\leq s<\infty }s^{\alpha }f^{*}(s),
\ t\in (0,\infty ).\end{aligned}$$
\[T:fractional-corollary\] Let $0<\gamma <n$ and let $X$ be an r.i. space over $\mathbb{R}^{n}$. Assume that $$\begin{aligned}
\label{E:bundedness-of-T}
T_{\frac{\gamma }{n}}\colon X(0,\infty )\to X(0,\infty ).\end{aligned}$$ Define the functional $\tau $ by $$\begin{aligned}
\tau (f)=\sup _{\|h\|_{X(0,\infty )}\leq 1}\int _{0}^{\infty }f^{*}(s)(PT
_{\frac{\gamma }{n}}h)(s)s^{\frac{\gamma }{n}}\,\mathrm{
d}s.\end{aligned}$$ Then $\tau $ is an r.i. norm such that $$\begin{aligned}
M_{\gamma }\colon X\to Y,\end{aligned}$$ where $Y=Y(\tau ')$, and $Y$ is the optimal smallest r.i. space for which holds. Moreover, $\tau $ is equivalent to the functional $$\begin{aligned}
\label{E:sigma-frac-corollary}
f\mapsto \left \| \int _{t}^{\infty }f^{*}(s)s^{\frac{\gamma }{n}-1}
\,\mathrm{
d}s\right \| _{X'(0,\infty )}, \quad f\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$
\[R:comparison\] The assumption of [Theorem \[T:fractional-corollary\]]{} is natural in view of the fact that the classical endpoint mapping properties for the fractional maximal operator $M_{\frac{\gamma }{n}}$ are of the form $$\begin{aligned}
M_{\frac{\gamma }{n}}\colon L^{1}\to L^{\frac{n}{n-\gamma },\infty }
\qquad
\textup{and}
\qquad
M_{\frac{\gamma }{n}}\colon L^{\frac{n}{\gamma },\infty }\to L^{
\infty },\end{aligned}$$ while those of $T_{{\frac{\gamma }{n}}}$ are (cf. [@T2; @T3; @GOP]) $$\begin{aligned}
T_{{\frac{\gamma }{n}}}\colon L^{1}\to L^{1}
\qquad
\textup{and}
\qquad
T_{{\frac{\gamma }{n}}}\colon L^{\frac{n}{\gamma },\infty }\to L^{\frac{n}{
\gamma },\infty }.\end{aligned}$$ On the other hand, is *strictly stronger* than . Indeed, assume that is satisfied. Then, in particular, there exists a positive constant, $K$, such that for every $a\geq 1$ one has $$\begin{aligned}
\|T_{\frac{\gamma }{n}}\chi _{(0,a)}\|_{X(0,\infty )}
\leq K
\|\chi
_{(0,a)}\|_{X(0,\infty )}.\end{aligned}$$ Since $$\begin{aligned}
T_{\frac{\gamma }{n}}\chi _{(0,a)}(t)
=
\chi _{(0,a)}(t)a^{\frac{\gamma
}{n}}t^{-\frac{\gamma }{n}}\quad \textup{for}\ t\in (0,\infty ),\end{aligned}$$ we in fact have $$\begin{aligned}
\|\chi _{(0,a)}(t)a^{\frac{\gamma }{n}}t^{-\frac{\gamma }{n}}\|_{X(0,
\infty )} \leq K \varphi _{X}(a).\end{aligned}$$ Consequently, $$\begin{aligned}
\varphi _{X}(a)a^{-\frac{\gamma }{n}}
\geq K^{-1}\|\chi _{(0,a)}(t)t
^{-\frac{\gamma }{n}}\|_{X(0,\infty )}
\geq K^{-1}\|\chi _{(0,1)}(t)t
^{-\frac{\gamma }{n}}\|_{X(0,\infty )}.\end{aligned}$$ Hence $$\begin{aligned}
\inf _{1\leq a<\infty }\varphi _{X}(a)a^{-\frac{\gamma }{n}}\geq K^{-1}
\|\chi _{(0,1)}(t)t^{-\frac{\gamma }{n}}\|_{X(0,\infty )}>0,\end{aligned}$$ and follows. This shows the implication $\Rightarrow $. The fact that this implication cannot be reversed follows on considering $X=L^{\frac{n}{
\gamma },q}$ with $q\in [1,\infty )$. Every such space obviously satisfies , but it follows from [@GOP Theorem 3.2] that the operator $T_{{\frac{\gamma }{n}}}$ is not bounded on it, hence does not hold.
For the optimal domain for the fractional maximal operator, we have the following result. Its proof is analogous to that of [Theorem \[T:maximal-operator-domain\]]{} and therefore is omitted.
Let $0<\gamma <n$ and let $Y$ be an r.i. space over $\mathbb{R}^{n}$ such that $$\begin{aligned}
\label{E:fmo-domain-condition}
\psi \in Y(0,\infty ),\end{aligned}$$ where $\psi (t)=(1+t)^{\frac{\gamma }{n}-1}$, $t\in (0,\infty )$. Define the functional $\sigma $ by $$\begin{aligned}
\sigma (f)=\left \| t^{\frac{\gamma }{n}}f^{**}(t) \right \| _{Y(0,
\infty )}, \quad f\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$ Then $\sigma $ is an r.i. norm and $$\begin{aligned}
\label{E:fractional-bounded-domain}
M_{\gamma }\colon X\to Y,\end{aligned}$$ where $X=X(\sigma )$. Moreover, $X$ is the optimal largest r.i. space for which holds.
Conversely, if is not true, then there does not exist an r.i. space $X$ for which holds.
Our next aim is to present an array of results concerning the optimal range partners for Lorentz-Zygmund spaces of the form $L^{p,q;{\mathbb{A}}}$ with respect to $M_{\gamma }$. Mapping properties of $M_{\gamma }$ on Lorentz–Zygmund spaces were studied in [@EO], where the following results were established: $$\begin{aligned}
M_{\gamma }\colon L^{p,q; {\mathbb{A}}} \to
\left\{
\begin{array}{l@{\quad }l}
L^{{\frac{n}{n-\gamma }}, 1;{\mathbb{A}}-1},
& p=1, q=1,
\alpha _{0}\ge 0, \alpha _{\infty }< 0, \\
L^{{\frac{n}{n-\gamma }}, \infty ; {\mathbb{A}}},
& p=1,
q=1, \alpha _{0} \geq 0, \alpha _{\infty }\leq 0, \\
L^{\frac{np}{n-\gamma p},q;{\mathbb{A}}},
& 1<p<\tfrac{n}{
\gamma }, 1\le q \le \infty , \\
L^{\infty ,q;{\mathbb{A}}- \frac{1}{q}},
& p=\tfrac{n}{\gamma
}, 1\le q \leq \infty , \alpha _{0} < 0, \alpha _{\infty }> 0.
\end{array}\right.
$$
Our result concerning optimal range spaces for Lorentz–Zygmund spaces reads as follows.
\[T:fractional-maximal-operator-GLZ\] Let $\gamma \in (0,n)$, $p,q\in [1,\infty ]$, ${\mathbb{A}}\in
\mathbb{R}^{2}$. Then $$\begin{aligned}
M_{\gamma }\colon L^{p,q; {\mathbb{A}}} \to
\left\{
\begin{array}{l@{\quad }l@{\quad }l}
Y_{1},
& p = 1, q = 1, \alpha _{0}\geq 0, \alpha _{\infty }\leq 0, &\mathrm{(a)} \label{E:fractional_p1} \\
L^{\frac{np}{n-\gamma p}, q; {\mathbb{A}}}, & 1<p<\tfrac{n}{\gamma }, & \mathrm{(b)} \label{E:fractional_easy} \\
L^{\infty , \infty ; {\mathbb{A}}}, & p = \tfrac{n}{\gamma }, q=\infty , \alpha _{0}\leq 0, \alpha _{\infty }\geq 0, &\mathrm{(c)}
\label{E:fractional_infty1}
\\
Y_{2},
& p = \tfrac{n}{\gamma }, 1\le q < \infty , \alpha _{\infty
}\ge 0 ~ or & \mathrm{(d)}
\\
& p = \tfrac{n}{\gamma }, q =\infty , \alpha _{0} > 0,
\alpha _{\infty }\ge 0, & \mathrm{(e)}
\label{E:fractional_infty2}
\end{array}\right.\end{aligned}$$ where $Y_{1}$ and $Y_{2}$ are the unique r.i. spaces whose associate spaces, $Y_{1}'$ and $Y_{2}'$, satisfy $$\begin{aligned}
\|f\|_{Y_{1}'}
= \sup _{0<t<\infty } \ell ^{-{\mathbb{A}}}(t)
\int _{t}
^{\infty }f^{*}(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s,
\quad f \in \mathcal{M}_{+}(\mathbb{R}^{n}),\end{aligned}$$ and $$\begin{aligned}
\|f\|_{Y_{2}'}
= \sup _{{\substack{h\sim f \\ h\ge 0}}} \Bigl \|
t^{1-{\frac{\gamma }{n}}-\frac{1}{q'}}
\ell ^{-{\mathbb{A}}}(t)
\int _{t}^{\infty } h(s)\,s^{{\frac{\gamma }{n}}-1}
\,\mathrm{
d}s
\Bigr \|_{L^{q'}(0,\infty )},
\quad f \in \mathcal{M}_{+}(\mathbb{R}
^{n}),\end{aligned}$$ respectively. In particular, in the case ${\mathbb{A}}=[0,0]$, we have $Y_{1}=L^{\frac{n}{n-\gamma },\infty }$ and $Y_{2}=L^{\infty }$.
Moreover, these spaces are the optimal range partners with respect to $M
_{\gamma }$.
Again, there is no simpler way of characterizing the spaces $Y_{1}'$ and $Y_{2}'$.
We note that the range spaces in [Theorem \[T:fractional-maximal-operator-GLZ\]]{} essentially improve those from [@EO] when $p=q=1$, $\alpha _{0}\geq 0$, $\alpha _{
\infty }\leq 0$ and $|\alpha _{0}| + |\alpha _{\infty }| > 0$, and also when $p=\frac{n}{\gamma }$, $1\leq q<\infty $, $\alpha _{0} < 0$ and $\alpha _{\infty }> 0$. It is also worth noting that the spaces $L^{\frac{n}{n-\gamma }, 1;{\mathbb{A}}-1}$ and $L^{\frac{n}{n-\gamma
}, \infty ; {\mathbb{A}}}$ are not comparable in the sense that neither of them is contained in the other (see [@OP] for details).
Our next aim is to describe in more detail the relation between [Theorems \[T:fractional-maximal-operator\] and \[T:fractional-corollary\]]{}. [Theorem \[T:fractional-corollary\]]{} asserts, among other statements, that in the particular cases when is satisfied, the functionals and $\sigma $ from are equivalent. We shall now point out an interesting fact that the converse is also true, namely if is not satisfied, then the functional in is *not equivalent* to $\sigma $ from . That, in fact, means that it is *essentially smaller* than $\sigma $. This is achieved through the following result, which is definitely of independent interest and maybe even a little surprising.
\[T:lenka\] Assume that $X$ is an r.i. space over $\mathbb{R}
^{n}$ and $\gamma \in (0,n)$. Then the following statements are equivalent:
$T_{\frac{\gamma }{n}}\colon X(0,\infty )\to X(0,\infty
)$,
there exists a positive constant $C$ such that, $$\begin{aligned}
\sup _{{\substack{h\sim f \\ h\ge 0}}}\left \| \int _{t}^{\infty }h(s)s^{{\frac{\gamma }{n}}-1}\,
\mathrm{
d}s\right \| _{X'(0,\infty )}\nonumber\\
\qquad \leq C
\left \| \int _{t}^{\infty }f
^{*}(s)s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )} , \quad f\in
\mathcal{M}_{+}(\mathbb{R}^{n}). \label{E:b}\end{aligned}$$
\[R:b\] We note that, since $f\sim f^{*}$, the converse inequality to , namely $$\begin{aligned}
\left \| \int _{t}^{\infty }f^{*}(s)s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}
\leq \sup _{{\substack{h\sim f \\ h\ge 0}}}\left \| \int _{t}^{\infty }h(s)s^{{\frac{\gamma }{n}}-1}\,
\mathrm{
d}s\right \| _{X'(0,\infty )},\end{aligned}$$ is trivial. In other words, if is true, then the two quantities are in fact equivalent.
In the proof of [Theorem \[T:lenka\]]{} we shall need the following auxiliary result of independent interest.
\[L:lenka\] Assume that $I\colon (0,\infty )\to (0,\infty )$ is a nondecreasing function satisfying $$\begin{aligned}
\label{E:I}
\int _{0}^{t} \frac{\mathrm{
d}s}{I(s)}
\approx \int _{t}^{2t} \frac{\mathrm{
d}s}{I(s)}
, \quad t\in (0,\infty ).\end{aligned}$$ Let $N\in \mathbb{N}$, $0<t_{1}<\cdots <t_{N}<\infty $ and $a_{1},
\dots ,a_{N}>0$. Let $$\begin{aligned}
u=\sum _{i=1}^{N}a_{i}\chi _{(0,t_{i})}\end{aligned}$$ and let $X$ be an r.i. space over $(0,\infty)$. Then $$\begin{aligned}
\left \| \int _{t}^{\infty } \frac{u(s)}{I(s)}\,\mathrm{
d}s \right \| _{X(0,\infty )}
\approx \|v\|_{X(0,\infty )},\end{aligned}$$ where $$\begin{aligned}
v=\sum _{i=1}^{N}a_{i}\frac{t_{i}}{I(t_{i})}\chi _{(0,t_{i})}.\end{aligned}$$
First, we have $$\begin{aligned}
\int _{t}^{\infty }\frac{u(s)}{I(s)}\,ds
= \sum _{i=1}^{N}
\int _{t}^{
\infty } \frac{a_{i}\chi _{(0,t_{i})}(s)}{I(s)}\,\mathrm{
d}s
= \sum _{i=1}^{N}
a_{i}\chi _{(0,t_{i})}(t) \int _{t}^{t_{i}} \frac{
\mathrm{
d}s}{I(s)}, \quad t\in (0,\infty ).\end{aligned}$$ By , $$\begin{aligned}
\int _{0}^{t_{i}}\frac{\mathrm{
d}s}{I(s)}
\geq \int _{t}^{t_{i}}\frac{\mathrm{
d}s}{I(s)}
\geq \int _{\frac{t_{i}}{2}}^{t_{i}}\frac{\mathrm{
d}s}{I(s)}
\approx \int _{0}^{t_{i}}\frac{\mathrm{
d}s}{I(s)}, \quad t\in (0,\tfrac{t_{i}}{2}),\end{aligned}$$ whence $$\begin{aligned}
\sum _{i=1}^{N} a_{i}\chi _{(0,t_{i})}(t)\int _{t}^{t_{i}}\frac{\mathrm{
d}s}{I(s)}
& \geq &\sum _{i=1}^{N}
a_{i}\chi _{(0,\frac{t_{i}}{2})}(t)
\int _{t}^{t_{i}}\frac{\mathrm{
d}s}{I(s)}
\approx \sum _{i=1}^{N}
a_{i}\chi _{(0,\frac{t_{i}}{2})}(t)
\int _{0}^{t_{i}}\frac{\mathrm{
d}s}{I(s)}
\\
& \approx & \sum _{i=1}^{N}
a_{i}\frac{t_{i}}{I(t_{i})}
\chi _{(0,\frac{t_{i}}{2})}(t), \quad t\in (0,\infty
).\end{aligned}$$ Therefore, due to the boundedness of the dilation operator on $X(0,\infty )$, we have $$\begin{aligned}
\left \| \int _{t}^{\infty }
\frac{u(s)}{I(s)}\,\mathrm{
d}s
\right \| _{X(0,\infty )}
& \approx & \left \| \sum _{i=1}^{N}
a
_{i}\frac{t_{i}}{I(t_{i})}\chi _{(0,\frac{t_{i}}{2})}
\right \| _{X(0,
\infty )}
\approx \left \| \sum _{i=1}^{N}
a_{i}
\frac{t_{i}}{I(t_{i})} \chi _{(0,t_{i})}
\right \| _{X(0,\infty )}
\\
&= &\|v\|_{X(0,\infty )}
\geq \left \| \sum _{i=1}^{N}
a_{i}
\chi _{(0,t_{i})}(t)\int _{t_{i}}^{2t_{i}}\frac{\mathrm{
d}s}{I(s)}
\right \| _{X(0,\infty )}
\\
&\approx &\left \| \sum _{i=1}^{N}
a_{i}\chi _{(0,t_{i})}(t)\int _{0}
^{t_{i}}\frac{\mathrm{
d}s}{I(s)}
\right \| _{X(0,\infty )}
\geq \left \| \int _{t}^{\infty
}
\frac{u(s)}{I(s)}\,\mathrm{
d}s
\right \| _{X(0,\infty )}.\qedhere\end{aligned}$$
We shall also need a variant of the result obtained in [@T2 Theorem 3.9] and also [@CP-TAMS Lemma 3.3] on the interval $(0,\infty )$. Here we present a more general claim with a shorter and more comprehensive proof.
In the following lemma, we work with the so-called quasiconcave functions instead of power functions. Recall that a nonnegative function $\varphi $ defined on $[0,\infty )$ is said to be *quasiconcave* provided that $\varphi $ is nondecreasing on $[0,\infty )$, $\frac{\varphi (t)}{t}$ is nonincreasing on $(0,\infty )$ and $\varphi (0)=0$. It follows that $\varphi $ is absolutely continuous except perhaps at the origin and $$\begin{aligned}
\label{E:phi_AC}
\varphi (t)-\varphi (s)\le \int _{s}^{t} \frac{\varphi (r)}{r}\,\mathrm{
d}r, \quad t\in (0,\infty), \ s\in(0,t].\end{aligned}$$ See [@Kre Chapter II, Lemma 1.1].
\[L:unsup\] Let $\varphi $ be a quasiconcave function. Then there exists a constant $C>0$ such that $$\begin{aligned}
\label{E:unsup_in}
\int _{0}^{\tau } \sup _{t\leq s<\infty } \varphi (s) f(s)\,\mathrm{
d}t
\le C \int _{0}^{\tau } (\varphi f)^{*}(t) \,\mathrm{
d}t\end{aligned}$$ for every $\tau\in (0,\infty )$ and every nonincreasing $f\in
\mathcal{M}_{+}(0,\infty )$.
Furthermore, if $X$ is an r.i. space over $(0,\infty )$, then $$\begin{aligned}
\label{E:unsup_ri}
\Bigl \| \sup _{t\leq s<\infty } \varphi (s) f(s) \Bigr \|_{X(0,\infty
)}
\le C\, \bigl \| \varphi f \bigr \|_{X(0,\infty )}\end{aligned}$$ for every nonincreasing $f\in \mathcal{M}_{+}(0,\infty )$.
Let $f\in \mathcal{M}_{+}(0,\infty )$ be a nonincreasing function and fix $\tau \in (0,\infty )$. We split the supremum into three parts, namely $$\begin{aligned}
\int _{0}^{\tau } \sup _{t\le s<\infty } \varphi (s) f(s)\,\mathrm{
d}t
& \le & \int _{0}^{\tau } \sup _{t\le s\le \tau } \varphi (s) f(s)\,\mathrm{
d}t
+ \tau \, \sup _{\tau \le s<\infty } \varphi (s) f(s)
\\
& \le &\int _{0}^{\tau } \sup _{t\le s\le \tau }
\bigl [ \varphi (s) - \varphi (t)
\bigr ] f(s)\,\mathrm{
d}t
\\
&&
+ \int _{0}^{\tau } \varphi (t) \sup _{t\le s\le \tau } f(s)\,\mathrm{
d}t
+ \tau \, \sup _{\tau \le s<\infty } \varphi (s) f(s)
\\
& = &\mbox{I} + \mbox{II} + \mbox{III}.\end{aligned}$$ By and the Hardy–Littlewood inequality, we have $$\begin{aligned}
\mbox{I}
& \le &\int _{0}^{\tau } \sup _{t\le s\le \tau } \biggl ( \int _{t}^{s}
\frac{
\varphi (r)}{r}\,\mathrm{
d}r \biggr ) f(s)\,\mathrm{
d}t
\le \int _{0}^{\tau } \sup _{t\le s\le \tau } \int _{t}^{s}
\frac{\varphi (r)}{r}f(r)\,\mathrm{
d}r\, \mathrm{
d}t
\\
& = &\int _{0}^{\tau } \int _{t}^{\tau } \frac{\varphi (r)}{r} f(r)\,\mathrm{
d}r\, \mathrm{
d}t
= \int _{0}^{\tau } \int _{0}^{r} \frac{\varphi (r)}{r} f(r)\,\mathrm{
d}t\, \mathrm{
d}r
\\
& = &\int _{0}^{\tau } \varphi (r) f(r)\, \mathrm{
d}r
\le \int _{0}^{\tau } (\varphi f)^{*}(t)\,\mathrm{
d}t.\end{aligned}$$ The second term is obviously estimated by the right hand side of . Let us consider the third term. Observe that, by , $$\begin{aligned}
\varphi (2t)-\varphi (t)
\le \int _{t}^{2t} \frac{\varphi (r)}{r}\,
\mathrm{
d}r
\le \varphi (t)
, \quad t\in (0,\infty ),\end{aligned}$$ since $\varphi (t)/t$ is nonincreasing, whence $\varphi (2t)\le 2
\varphi (t)$ for $t\in (0,\infty )$. Using this and the fact that $\varphi $ is nondecreasing, we get $$\begin{aligned}
\label{E:phi_int}
\varphi (t)
\le 2 \varphi (t/2)
\le \frac{4}{t} \int _{t/2}^{t} \varphi
(r)\,\mathrm{
d}r
\le \frac{4}{t} \int _{0}^{t} \varphi (r)\,\mathrm{
d}r
, \quad t\in (0,\infty ).\end{aligned}$$ Using we obtain $$\begin{aligned}
\mbox{III}
& = &\tau \, \sup _{\tau \le s<\infty } \varphi (s) f(s)
\le 4\tau
\, \sup _{\tau \le s<\infty }
\Bigl ( { \frac{1}{s} } \int _{0}^{s} \varphi
(r)\,\mathrm{
d}r \Bigr ) f(s)
\\
& \le & 4\tau \, \sup _{\tau \le s<\infty } { \frac{1}{s} } \int _{0}^{s} \varphi
(r) f(r)\,\mathrm{
d}r
\le 4\tau \, \sup _{\tau \le s<\infty } {\frac{1}{s} } \int _{0}^{s} (\varphi
f)^{*}(t)\,\mathrm{
d}t
\\
& =& 4\int _{0}^{\tau } (\varphi f)^{*}(t)\,\mathrm{
d}t,\end{aligned}$$ where in the second inequality we used that $f$ is nonincreasing and the third one is due to the Hardy–Littlewood inequality. Combination of the estimates gives with $C=6$. The inequality (with the same $C$) then follows from by the Hardy-Littlewood-Pólya principle.
Assume first that (a) is true. Then the associate norm of the optimal r.i. range partner space for $X$ with respect to $M_{\gamma }$ is equivalent to owing to [Theorem \[T:fractional-corollary\]]{}. On the other hand, that norm is also equivalent to by [Theorem \[T:fractional-maximal-operator\]]{}. We recall that the assumption of this theorem is satisfied since it follows from (a), as was pointed out in [Remark \[R:comparison\]]{}. Combining these two facts, we immediately obtain (b) (see also [Remark \[R:b\]]{}).
The converse implication is considerably more involved. Suppose that (b) holds. Then the functional $$\begin{aligned}
\label{eq:plenka}
g \mapsto \left \| \int _{t}^{\infty }
g^{*}(s)s^{{\frac{\gamma }{n}}-1}
\,\mathrm{
d}s
\right \| _{X'(0,\infty )}\end{aligned}$$ is equivalent to $\sigma $ from , which in turn is known to be an r.i. norm thanks to [Theorem \[T:fractional-maximal-operator\]]{}. We note that is indeed satisfied because it follows from the proof of [Theorem \[T:fractional-maximal-operator\]]{} that it holds if and only if $\sigma (u) < \infty $ for every nonnegative simple function $u$, which can be readily verified here thanks to (b). Hence the collection=1 $$\begin{aligned}
Y(0,\infty )=\left \{ g\in \mathcal{M}(0,\infty ), \quad \left \| \int
_{t}^{\infty }g^{*}(s)s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}<\infty \right \} ,\end{aligned}$$ endowed with the functional $$\begin{aligned}
\|g\|_{Y(0,\infty )}=\left \| \int _{t}^{\infty }g^{*}(s)s^{{\frac{
\gamma }{n}}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )},\end{aligned}$$ is equivalent to an r.i. space. Define the operator $R$ on $\mathcal{M}(0,\infty )$ by $$\begin{aligned}
Rg(t)=\int _{t}^{\infty }|g(s)|s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s,\quad t\in (0,\infty ).\end{aligned}$$ Then we have $$\begin{aligned}
\|Rg^{*}\|_{X'(0,\infty )}=\|g\|_{Y(0,\infty )}, \quad
g\in \mathcal{M}(0,\infty ).\end{aligned}$$ Therefore, using also the equivalence of and , it clearly follows that $$\begin{aligned}
\label{E:H-bounded}
R\colon Y(0,\infty ) \to X'(0,\infty )\end{aligned}$$ and that $Y(0,\infty )$ is the optimal (largest possible) r.i. space rendering true (in other words, it is the optimal r.i. domain partner space for $X'(0,\infty )$ with respect to the operator $R$).
We however claim a considerably less obvious fact, namely that $X'(0,\infty )$ is also the smallest possible rearrangement-invariant space in , that is, it is the optimal r.i. range partner space for $Y(0,\infty )$ with respect to the operator $R$.
We know that $R$ is bounded from $Y(0,\infty )$ to $X'(0,\infty )$. Therefore we are entitled to denote the optimal rearrangement-invariant range partner for $Y(0,\infty )$ with respect to $R$ by $Y_{R}(0,
\infty )$. Denote further by $Y_{R_{D}}(0,\infty )$ the optimal r.i. domain partner for $Y_{R}(0,\infty )$ with respect to $R$. Then, using the same reasoning as above, we obtain that $$\begin{aligned}
\|g\|_{Y_{R_{D}}(0,\infty )}\approx \left \| \int _{t}^{\infty }g^{*}(s)s
^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s\right \| _{Y_{R}(0,\infty )}, \quad g\in
\mathcal{M}(0,\infty ).\end{aligned}$$
Only an easy observation is needed to realize that once a space is the optimal domain partner of *some* space, then it is necessarily also the optimal domain partner to its own optimal range partner. Indeed, knowing that $Y(0,\infty )$ is optimal in $R\colon Y(0,\infty
)\to X'(0,\infty )$, assume that $R\colon Z(0,\infty )\to Y_{R}(0,
\infty )$. By optimality of $Y_{R}(0,\infty )$ in $R\colon Y(0,\infty
)\to Y_{R}(0,\infty )$, one necessarily has $Y_{R}(0,\infty )\hookrightarrow
X'(0,\infty )$. Thus, $R\colon Z(0,\infty )\to X'(0,\infty )$. But, by optimality of $Y(0,\infty )$ in $R\colon Y(0,\infty )\to X'(0,\infty
)$, it follows that $Z(0,\infty )\hookrightarrow Y(0,\infty )$.
Consequently, $Y(0,\infty )=Y_{R_{D}}(0,\infty )$, that is, $$\begin{aligned}
\left \| \int _{t}^{\infty }g^{*}(s)s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}
\approx \left \| \int _{t}^{\infty }g
^{*}(s)s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s\right \| _{Y_{R}(0,\infty )}, \quad g\in
\mathcal{M}(0,\infty ).\end{aligned}$$ Assume that $u=\sum _{i=1}^{N}b_{i}\chi _{(0,s_{i})}$ for some $N\in \mathbb{N}$, $0 < s_{1} <\dots < s_{N}<\infty $ and $b_{1},
\dots , b_{N} > 0$. Let further $$\begin{aligned}
\label{eq2:lenka}
v=\sum _{i=1}^{N} b_{i}\frac{s_{i}}{I(s_{i})}\chi _{(0,s_{i})},\end{aligned}$$ where $I(t)=t^{1 - {\frac{\gamma }{n}}}$, $t\in (0, \infty )$. Note that the function $I$ satisfies the assumptions of [Lemma \[L:lenka\]]{}. Therefore we are entitled to use the lemma, whence we get $$\begin{aligned}
\|v\|_{X'(0,\infty )}
\approx \left \| \int _{t}^{\infty }
u(s)s^{
{\frac{\gamma }{n}}-1}\,\mathrm{
d}s
\right \| _{X'(0,\infty )}
\approx \left \| \int _{t}^{\infty }
u(s)s
^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s
\right \| _{Y_{R}(0,\infty )}
\approx \|v\|_{Y_{R}(0,\infty )}.\end{aligned}$$ Now, if $f\in \mathcal{M}(\mathbb{R}^{n})$, then there is a sequence $\{v_{n}\}$ of nonnegative simple functions in the form of satisfying $v_{n}\nearrow f^{*}$. By the Fatou property and the computations above, we get $X'(0,\infty )=Y_{R}(0,
\infty )$. This proves that $X'(0,\infty )$ is indeed the optimal range space in .
We next claim that $$\begin{aligned}
\label{E:next-claim}
\|g\|_{X(0,\infty )}\approx \left \|
t^{{\frac{\gamma }{n}}}g^{**}(t)\right \| _{Y'(0,\infty )},\quad
g\in \mathcal{M}(0,\infty ).\end{aligned}$$ Indeed, by the definition of the associate norm, the Fubini theorem and the Hölder inequality, one has, for every $g\in \mathcal{M}(0,
\infty )$, $$\begin{aligned}
\left \| t^{{\frac{\gamma }{n}}}g^{**}(t)\right \| _{Y'(0,\infty )}
&=&
\sup _{\|h\|_{Y(0,\infty )}\leq 1}
\int _{0}^{\infty }|h(t)|t^{\frac{
\gamma }{n}-1}\int _{0}^{t}g^{*}(s)\mathrm{
d}s\,\mathrm{
d}t
\\
&=&
\sup _{\|h\|_{Y(0,\infty )}\leq 1}
\int _{0}^{\infty }g^{*}(s)\int
_{s}^{\infty }|h(t)|t^{\frac{\gamma }{n}-1}\mathrm{
d}t\,\mathrm{
d}s
\\
&\leq &\sup _{\|h\|_{Y(0,\infty )}\leq 1}
\|g\|_{X(0,\infty )}\left \|
\int _{s}^{\infty }|h(t)|t^{\frac{\gamma }{n}-1}\,\mathrm{
d}t\right \| _{X'(0,\infty )}.\end{aligned}$$ Now, the equivalence of and implies that $$\begin{aligned}
\sup _{\|h\|_{Y(0,\infty )}\leq 1}\left \| \int _{s}^{\infty }|h(t)|t
^{\frac{\gamma }{n}-1}\,\mathrm{
d}t\right \| _{X'(0,\infty )}
\leq C
\sup _{\|h\|_{Y(0,\infty )}
\leq 1}\left \| \int _{s}^{\infty }h^{*}(t)t^{\frac{\gamma }{n}-1}\,
\mathrm{
d}t\right \| _{X'(0,\infty )}
= C.\end{aligned}$$ It might be instructive to note that while this estimate, of course, follows from (b), the validity of (b) is in fact not necessary in order to get it. Altogether, combining the estimates, we get $$\begin{aligned}
\label{E:one-inequality}
\left \| t^{{\frac{\gamma }{n}}}g^{**}(t)\right \| _{Y'(0,\infty )}
\leq C \|g\|_{X(0,\infty )} , \quad g\in
\mathcal{M}(0,\infty ).\end{aligned}$$
In order to prove , we now need to show the converse inequality to . Denote $$\begin{aligned}
\|g\|_{Z(0,\infty )}=\left \|
t^{{\frac{\gamma }{n}}}g^{**}(t)\right \| _{Y'(0,\infty )},\quad g
\in \mathcal{M}(0,\infty ).\end{aligned}$$ The functional $g\mapsto \|g\|_{Z(0,\infty )}$ is an r.i. norm. To see this, only (P4) needs proof, since everything else is readily verified. Applying standard techniques, (P4) reduces to $$\begin{aligned}
\label{cond1:lenka}
t^{{\frac{\gamma }{n}}-1}\chi _{[1,\infty )}(t)\in Y'(0,\infty ).\end{aligned}$$ But, using the equivalence of and once again, we get $$\begin{aligned}
\|t^{{\frac{\gamma }{n}}-1}\chi _{[1,\infty )}(t)\|_{Y'(0,\infty )}
&=&
\sup \limits _{\|f\|_{Y(0,\infty )}\leq 1}\int _{0}^{\infty }|f(t)|t^{
{\frac{\gamma }{n}}-1}\chi _{[1,\infty )}(t)\,\mathrm{
d}t
\\
&=&\frac{1}{\|\chi _{(0,1)}\|_{X'(0,\infty )}}\sup
\limits _{\|f\|_{Y(0,\infty )}\leq 1}\left \| \chi _{(0,1)}\int _{1}
^{\infty }|f(t)|t^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}t\right \| _{X'(0,\infty )}
\\
&\leq & \frac{1}{\|\chi _{(0,1)}\|_{X'(0,\infty )}}\sup
\limits _{\|f\|_{Y(0,\infty )}\leq 1}\left \| \int _{s}^{\infty }|f(t)|t
^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}t\right \| _{X'(0,\infty )}
\\
&\le &\frac{C_2}{\|\chi _{(0,1)}\|_{X'(0,\infty )}}\sup
\limits _{\|f\|_{Y(0,\infty )}\leq 1}\left \| \int _{s}^{\infty }f^{*}(t)t
^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}t\right \| _{X'(0,\infty )}
\\
&\le &\frac{C_2}{\|\chi _{(0,1)}\|_{X'(0,\infty )}} < \infty \end{aligned}$$ for some appropriate positive constant $C_2$. We define the operator $R'$ by $$\begin{aligned}
R'g(t)=t^{\frac{\gamma }{n}-1}\int _{0}^{t}|g(s)|\,\mathrm{
d}s,\quad g\in \mathcal{M}(0,\infty ).\end{aligned}$$ Then $$\begin{aligned}
R'\colon Z(0,\infty )\to Y'(0,\infty ),\end{aligned}$$ since, by the Hardy–Littlewood inequality, $$\begin{aligned}
\label{E:H-from-Y'-to-Z'}
\|R'g\|_{Y'(0,\infty )}\leq \|R'g^{*}\|_{Y'(0,\infty )}=\|g\|_{Z(0,
\infty )}, \quad g\in \mathcal{M}(0,\infty ).\end{aligned}$$ We also have $$\begin{aligned}
\label{E:H-from-Z'-to-Y}
R\colon Y(0,\infty )\to Z'(0,\infty ),\end{aligned}$$ since, by the Fubini theorem, the Hölder inequality and , one has $$\begin{aligned}
\|Rg\|_{Z'(0,\infty )}
&=&
\sup _{\|f\|_{Z(0,\infty )}\leq 1}\int _{0}
^{\infty }f(t)Rg(t)\,dt
=
\sup _{\|f\|_{Z(0,\infty )}\leq 1}\int _{0}
^{\infty }|f(t)|Rg(t)\,dt
\\
&=&
\sup _{\|f\|_{Z(0,\infty )}\leq 1}\int _{0}^{\infty }R'f(t)|g(t)|\,dt
\leq \|g\|_{Y(0,\infty )}\sup _{\|f\|_{Z(0,\infty )}\leq 1}\|R'f\|_{Y'(0,
\infty )}
\\
&\leq &\|g\|_{Y(0,\infty )}.\end{aligned}$$ But, as we know, $X'(0,\infty )$ is the optimal (smallest) r.i. target partner for $Y(0,\infty )$ with respect to $R$. Consequently, it must be contained in $Z'(0,\infty )$. By , this means that $Z(0,\infty )$ is continuously embedded into $X(0,\infty )$. In other words, there exists a positive constant, $C'$, such that $$\begin{aligned}
\label{E:converse-inequality}
\|g\|_{X(0,\infty )}\leq C' \|g\|_{Z(0,\infty )}=C'\left \| t^{{\frac{
\gamma }{n}}}g^{**}(t)\right \| _{Y'(0,\infty )},\quad
g\in \mathcal{M}(0,\infty );\end{aligned}$$ hence follows from the combination of and .
Now we know that $X(0,\infty )=Z(0,\infty )$, so in order to prove (a) it suffices to show that $T_{\frac{\gamma }{n}}\colon Z(0,\infty )
\to Z(0,\infty )$. In other words, we claim that there exists a positive constant $C$ such that $$\begin{aligned}
\label{E:last-claim}
\left \|
t^{{\frac{\gamma }{n}}}(T_{\frac{\gamma }{n}}g)^{**}(t)\right \|
_{Y'(0,\infty )}
\leq C
\left \|
t^{{\frac{\gamma }{n}}}g^{**}(t)\right \| _{Y'(0,\infty )},\quad
g\in \mathcal{M}(0,\infty ).\end{aligned}$$
We first recall that there exists a positive constant $K$ depending only on $n$ and $\gamma $ such that $$\begin{aligned}
\label{E:doublestar-in}
(T_{\frac{\gamma }{n}}g)^{**}(t)
\leq K
T_{\frac{\gamma }{n}}(g^{**})(t)
, \quad g\in \mathcal{M}(0,\infty ),\quad t
\in (0,\infty ).\end{aligned}$$ Indeed, this follows from [@Mus:18 Lemma 4.1], where a more general assertion is stated and proved.
Next, it follows from [Lemma \[L:unsup\]]{} that $$\begin{aligned}
\left \|
\sup _{t\leq s<\infty }s^{{\frac{\gamma }{n}}}g^{*}(s)\right \| _{Y'(0,
\infty )}
\leq C
\left \| t^{{\frac{\gamma }{n}}}g^{*}(t)\right \|
_{Y'(0,\infty )}, \quad g\in \mathcal{M}(0,\infty
).\end{aligned}$$ In particular, since $g^{**}$ is also nonincreasing, we have $$\begin{aligned}
\label{E:for-double}
\left \|
\sup _{t\leq s<\infty }s^{{\frac{\gamma }{n}}}g^{**}(s)\right \| _{Y'(0,
\infty )}
\leq C
\left \| t^{{\frac{\gamma }{n}}}g^{**}(t)\right \|
_{Y'(0,\infty )}, \quad g\in \mathcal{M}(0,\infty
).\end{aligned}$$ Thus, combining and , we get $$\begin{aligned}
\left \|
t^{{\frac{\gamma }{n}}}(T_{\frac{\gamma }{n}}g)^{**}(t)\right \|
_{Y'(0,\infty )}
&\leq & K
\left \| t^{{\frac{\gamma }{n}}}T_{\frac{
\gamma }{n}}(g^{**})(t)\right \| _{Y'(0,\infty )}
=K
\left \|
\sup _{t\leq s<\infty }s^{{\frac{\gamma }{n}}}g^{**}(s)\right \| _{Y'(0,
\infty )}
\\
&\leq& KC
\left \| t^{{\frac{\gamma }{n}}}g^{**}(t)\right \| _{Y'(0,
\infty )}, \quad g\in \mathcal{M}(0,\infty ),\end{aligned}$$ proving . Hence (a) holds, as desired. The proof is complete.
Let us now turn our attention to proofs of the main results.
We begin by proving that $\sigma $ is an r.i. norm. As in the proof of [Theorem \[T:maximal-operator\]]{}, only the triangle inequality and axioms (P4) and (P5) have to be verified. The triangle inequality follows by the same argument using measure-preserving transformations as in [@T2 Theorem 3.3].
We shall verify the validity of (P4). Let $E\subset \mathbb{R}$ be a measurable set with $|E|<\infty $ and let $h$ be such that $h\sim \chi
_{E}$. We infer that there is a measurable set $F\subset \mathbb{R}$ such that $h=\chi _{F}$ and $|F|=|E|$. Assume moreover that $|E|\ge 1$. It follows from the regularity of the Lebesgue measure that there exists an open set $G \supseteq F$ such that $|G|\le 2|F|$. Thus there are disjoint intervals $(a_{k},b_{k})$ satisfying $|F| \le a_{k}$, $$\begin{aligned}
F\subseteq (0,|F|) \cup \bigcup _{k} (a_{k},b_{k})\end{aligned}$$ and $$\begin{aligned}
\sum _{k} (b_{k} - a_{k}) \le 2|F|.\end{aligned}$$ Then we have $$\begin{aligned}
\biggl \| \int _{t}^{\infty } h(s)\,s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
& \le &\biggl \| \int _{t}^{\infty }
\biggl (\chi _{(0,|F|)}(s) + \sum _{k} \chi _{(a_{k},b_{k})}(s) \biggr )\,s
^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
\\
& \le &\biggl \| \int _{t}^{\infty } \chi _{(0,|F|)}(s)\,s^{{\frac{
\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
\\
&& + \sum _{k} \biggl \| \int _{t}^{\infty } \chi _{(a_{k},b_{k})}(s)
\,s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
\\
& \le &{\frac{n}{\gamma }}|F|^{\frac{\gamma }{n}}\, \| \chi _{(0,|F|)}
\|_{X'(0,\infty )}
\\
&& + \sum _{k} \biggl \| \chi _{(0,a_{k})}(t) \int _{a_{k}}^{b_{k}}
\,s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
\\
&& + \sum _{k} \biggl \| \chi _{(a_{k},b_{k})}(t) \int _{t}^{b_{k}}
\,s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}.\end{aligned}$$ Let us observe that, due to , is in fact equivalent to the existence of a constant $C$ such that $$\begin{aligned}
\label{cokl}
r^{{\frac{\gamma }{n}}-1} \|\chi _{(0,r)}\|_{X'(0,\infty )} \le C
, \quad r\in[1,\infty).\end{aligned}$$ Next, using the monotonicity of $s^{{\frac{\gamma }{n}}-1}$ and , we get (note that $a_{k}\geq 1$ is satisfied thanks to $a_{k}\geq |F|$) $$\begin{aligned}
\biggl \| \chi _{(0,a_{k})}(t) \int _{a_{k}}^{b_{k}} \,s^{{\frac{\gamma
}{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
\le a_{k}^{{\frac{\gamma }{n}}-1} \|
\chi _{(0,a_{k})} \|_{X'(0,\infty )}\, (b_{k}-a_{k})
\le C (b_{k} - a
_{k}).\end{aligned}$$ Note that $C$ is independent of $k$. Also, $$\begin{aligned}
\biggl \| \chi _{(a_{k},b_{k})}(t) \int _{t}^{b_{k}} \,s^{{\frac{\gamma
}{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
&\le &\biggl \| \chi _{(a_{k},b_{k})}(t)
\int _{a_{k}}^{b_{k}} \,s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
\\
& \le &a_{k}^{{\frac{\gamma }{n}}-1} \|\chi _{(0,b_{k}-a_{k})} \|_{X'(0,
\infty )}\, (b_{k}-a_{k})
\\
& \le & a_{k}^{{\frac{\gamma }{n}}-1} \|\chi _{(0,a_{k})} \|_{X'(0,
\infty )}\, (b_{k}-a_{k})
\le C (b_{k} - a_{k}),\end{aligned}$$ where we, once again, used the monotonicity, , and $$\begin{aligned}
b_{k}-a_{k} \le |F| \le a_{k}.\end{aligned}$$ Therefore $$\begin{aligned}
\biggl \| \int _{t}^{\infty } h(s)\,s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
& \le &{\frac{n}{\gamma }}|F|^{\frac{
\gamma }{n}}\, \| \chi _{(0,|F|)} \|_{X'(0,\infty )}
+ 2C \sum _{k} (b_{k}-a
_{k})
\\
& \le & {\frac{n}{\gamma }}C |F| + 4C |F|
= C_{n,\gamma }|E|.\end{aligned}$$ Taking the supremum over all such $h$, we get $$\begin{aligned}
\label{vlk}
\sigma (\chi _{E}) \le C_{n,\gamma }|E|.\end{aligned}$$ If $E\subset \mathbb{R}^{n}$ has $|E|<1$, we get $\sigma (\chi
_{E})\le C_{n,\gamma }$ by the monotonicity of $\sigma $.
As for (P5), let $E$ be a measurable subset of $\mathbb{R}^{n}$ having finite measure
and assume that $f\in L^{1}(E)$. Denote $r=|E|$ and set $h(s) = f^{*}(s-r)
\chi _{(r,2r)}(s)$. Then $f\sim h$ and $$\begin{aligned}
\sigma (f)
& \ge &\biggl \| \int _{t}^{\infty } h(s)\,s^{{\frac{\gamma
}{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
\nonumber\\
& =& \biggl \| \int _{t}^{\infty } f^{*}(s-r)\chi _{(r,2r)}(s)\,s^{
{\frac{\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
\nonumber\\
& \ge &\biggl \| \chi _{(0,r)}(t) \int _{r}^{2r} f^{*}(s-r)\,s^{{\frac{
\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )} \label{E:sigma-lower-bound} \\
& =& \| \chi _{(0,r)} \|_{X'(0,\infty )}
\int _{r}^{2r} f^{*}(s-r)\,s
^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s
\nonumber\\
& \ge &\| \chi _{(0,r)} \|_{X'(0,\infty )}
(2r)^{{\frac{\gamma }{n}}-1}
\int _{r}^{2r} f^{*}(s-r)\,\mathrm{
d}s
\nonumber\\
& \ge &C_{n,\gamma ,X}
\|f\|_{L^{1}(E)},\nonumber
$$ and (P5) follows.
We now claim that $M_{\gamma }\colon X\to Y$. Assume that $g\in
\mathcal{M}_{+}(0,\infty )$. Define $f(x)=g(\omega _{n}|x|^{n})$ for $x\in \mathbb{R}^{n}\setminus \{0\}$, where $\omega _{n}$ is the volume of the $n$-dimensional unit ball. Then $f$ is defined almost everywhere on $\mathbb{R}^{n}$ and one has $g\sim f$. Thus, by the definitions of $\sigma $ and $Y$, we get $$\begin{aligned}
\left \| \int _{t}^{\infty }g(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}
\leq \sigma (f)=\|f\|_{Y'}=\|g\|_{Y'(0,
\infty )}.\end{aligned}$$ Since $g$ was arbitrary, we obtain by , $$\begin{aligned}
\left \| t^{\frac{\gamma }{n}-1}\int _{0}^{t}g(s)\,\mathrm{
d}s\right \| _{Y(0,\infty )}\leq \|g\|_{X(0,\infty )},\quad
g\in \mathcal{M}_{+}(0,\infty ).\end{aligned}$$ Restricting this inequality to nonincreasing functions, we obtain that $$\begin{aligned}
\left \| t^{\frac{\gamma }{n}}g^{**}(t)\right \| _{Y(0,\infty )}
\leq \|g^{*}\|_{X(0,\infty )}, \quad g\in
\mathcal{M}_{+}(0,\infty ).\end{aligned}$$ Applying [Lemma \[L:unsup\]]{}, we get that there exists a positive constant $C$ such that $$\begin{aligned}
\left \| \sup _{t\leq s<\infty }s^{\frac{\gamma }{n}}g^{**}(s)\right \|
_{Y(0,\infty )}\leq C\|g^{*}\|_{X(0,\infty )},\quad g\in \mathcal{M}_{+}(0,\infty ).\end{aligned}$$ Thus, by , one has $$\begin{aligned}
\|M_{\gamma }f\|_{Y}
&\leq & C\left \| \sup _{t\leq s<\infty }s^{\frac{
\gamma }{n}}f^{**}(s)\right \| _{Y(0,\infty )}
\\
&\leq & C\|f^{*}\|_{X(0,\infty )}=C\|f\|_{X}, \quad f\in X,\end{aligned}$$ whence $M_{\gamma }\colon X\to Y$.
We shall now prove the optimality of the space $Y$ in . Suppose that for some r.i. space $Z$, one has $M_{\gamma }\colon X\to Z$. Let $g$ be a nonincreasing function in $\mathcal{M}_{+}(0,\infty )$. Then there exists a function $f_{0}
\in L^{1}_{\operatorname{loc}}(\mathbb{R}^{n})$ such that $f_{0}
\sim g$ and holds. Since $M_{\gamma }\colon X\to Z$, we have $$\begin{aligned}
\|(M_{\gamma }f_{0})^{*}\|_{Z(0,\infty )}\leq C\|f_{0}^{*}\|_{X(0,
\infty )}=C\|g^{*}\|_{X(0,\infty )}.\end{aligned}$$ By , this yields $$\begin{aligned}
\|\sup _{t\leq s<\infty }s^{\frac{\gamma }{n}}g^{**}(s)\|_{Z(0,\infty
)}\leq C\|g^{*}\|_{X(0,\infty )}.\end{aligned}$$ We emphasize that $C$ does not depend on $g$. The last estimate trivially implies $$\begin{aligned}
\|t^{\frac{\gamma }{n}}g^{**}(t)\|_{Z(0,\infty )}\leq C\|g^{*}\|_{X(0,
\infty )}, \quad g\in \mathcal{M}_{+}(0,\infty ).\end{aligned}$$ Therefore, by the Hardy–Littlewood inequality, we obtain $$\begin{aligned}
\|t^{\frac{\gamma }{n}}Pg(t)\|_{Z(0,\infty )}\leq C\|g\|_{X(0,\infty
)}, \quad g\in \mathcal{M}_{+}(0,\infty ).\end{aligned}$$ By , this yields $$\begin{aligned}
\left \| \int _{t}^{\infty }h(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}\leq C\|h\|_{Z'(0,\infty )},\quad
h\in \mathcal{M}_{+}(0,\infty ).\end{aligned}$$ In particular, for every $f\in \mathcal{M}_{+}(\mathbb{R}^{n})$ and $h\in \mathcal{M}_{+}(0,\infty )$ such that $h\sim f$, one has $$\begin{aligned}
\left \| \int _{t}^{\infty }h(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}
& \leq & C\|h\|_{Z'(0,\infty )}= C\|h^{*}
\|_{Z'(0,\infty )}\nonumber\\
&=& C\|f^{*}\|_{Z'(0,\infty )}=C\|f\|_{Z'}. \label{E:sigma-Z-estimate}\end{aligned}$$ Consequently, $$\begin{aligned}
\sigma (f)=\sup _{{\substack{h\sim f \\ h\ge 0}}}\left \| \int _{t}^{\infty }h(s)s^{\frac{\gamma }{n}-1}\,
\mathrm{
d}s\right \| _{X'(0,\infty )}\leq C\|f\|_{Z'}.\end{aligned}$$ By the definition of $Y$, this means that $Z'\hookrightarrow Y'$, or equivalently $Y\hookrightarrow Z$, proving the optimality of $Y$ in .
Finally, assume that is not true and assume that $M_{\gamma }\colon X\to Y$ for some r.i. space $Y$ over $\mathbb{R}^{n}$. Then it follows from the above that $$\begin{aligned}
\label{E:contradiction}
\sup _{{\substack{h\sim f \\ h\ge 0}}}\left \| \int _{t}^{\infty }h(s)s^{\frac{\gamma }{n}-1}\,
\mathrm{
d}s\right \| _{X'(0,\infty )}\leq C\|f\|_{Y'}, \quad
f\in Y'.\end{aligned}$$ Take any $f\in \mathcal{M}_{+}(\mathbb{R}^{n})$ satisfying $f^{*}=
\chi _{(0,1)}$ and let $h=\chi _{(b,1+b)}$ for some fixed but arbitrary $b\in (1,\infty )$. Then $f\sim h$ and $$\begin{aligned}
\biggl \| \int _{t}^{\infty } h(s)\,s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
&= &\biggl \| \int _{t}^{\infty }
\chi _{(b,1+b)}(s) \,s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
\\
& \ge& \biggl \| \chi _{(0,b)}(t) \int _{b}^{1+b} s^{{\frac{\gamma }{n}}-1}
\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}
\\
&=& \| \chi _{(0,b)} \|_{X'(0,\infty )} \int _{b}^{1+b} s^{{\frac{
\gamma }{n}}-1}\,\mathrm{
d}s
\\
& \ge &\frac{b}{\varphi _{X}(b)} (1+b)^{{\frac{\gamma }{n}}-1}
\ge 2^{
{\frac{\gamma }{n}}-1}\,\frac{b^{\frac{\gamma }{n}}}{\varphi _{X}(b)}.\end{aligned}$$ Since is not satisfied, there exists a sequence $b_{k}\to \infty $ such that $$\begin{aligned}
\lim _{k\to \infty }
\frac{b_{k}^{\frac{\gamma }{n}}}{\varphi _{X}(b_{k})}=\infty .\end{aligned}$$ This implies that $$\begin{aligned}
\biggl \| \int _{t}^{\infty } h(s)\,s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s \biggr \|_{X'(0,\infty )}=\infty .\end{aligned}$$ Since $\|f\|_{Y'}<\infty $ by (P4) for $Y'$, this contradicts . The proof is complete.
We shall first prove that $\tau $ is equivalent to the functional in . By the definition of the associate space, we get $$\begin{aligned}
\left \| \int _{t}^{\infty }f^{*}(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}
=
\sup _{\|h\|_{X(0,\infty )}\leq 1}\int
_{0}^{\infty }h(t)\int _{t}^{\infty }f^{*}(s)s^{\frac{\gamma }{n}-1}\,
\mathrm{
d}s\,\mathrm{
d}t.\end{aligned}$$ Since the function $t\mapsto \int _{t}^{\infty }s^{\frac{\gamma }{n}-1}f
^{*}(s)\,\mathrm{
d}s$ is obviously nonincreasing on $(0,\infty )$ regardless of $f$, we in fact have, by the corollary of the Hardy–Littlewood inequality (see ), $$\begin{aligned}
\left \| \int _{t}^{\infty }f^{*}(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}
=
\sup _{\|h\|_{X(0,\infty )}\leq 1}\int
_{0}^{\infty }h^{*}(t)\int _{t}^{\infty }f^{*}(s)s^{\frac{\gamma }{n}-1}
\,\mathrm{
d}s\,\mathrm{
d}t.\end{aligned}$$ Thus, the Fubini theorem and the definition of $P$ yield $$\begin{aligned}
\left \| \int _{t}^{\infty }f^{*}(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}
=
\sup _{\|h\|_{X(0,\infty )}\leq 1}\int
_{0}^{\infty }f^{*}(s)(Ph^{*})(s)s^{\frac{\gamma }{n}}\,\mathrm{
d}s.\end{aligned}$$ The trivial pointwise estimate $h^{*}\leq T_{\frac{\gamma }{n}}h$ implies that $(Ph^{*})(s)\leq (PT_{\frac{\gamma }{n}}h)(s)$ for every $h$ and every $s$. Hence, we obtain that $$\begin{aligned}
\label{E:tau-lower-estimate}
\left \| \int _{t}^{\infty }f^{*}(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}\leq \tau (f).\end{aligned}$$ To prove the converse inequality, let $K$ be the operator norm of $T_{\frac{\gamma }{n}}$ on $X(0,\infty )$. Then, by the definition of $\tau $, the Fubini theorem, and the Hölder inequality, we have $$\begin{aligned}
\tau (f)
&=&
\sup _{\|h\|_{X(0,\infty )}\leq 1}\int _{0}^{\infty }f^{*}(s)(PT
_{\frac{\gamma }{n}}h)(s)s^{\frac{\gamma }{n}}\,\mathrm{
d}s
=
\sup _{\|h\|_{X(0,\infty )}\leq 1}\int _{0}^{\infty }
(T_{\frac{
\gamma }{n}}h)(t)\int _{t}^{\infty }f^{*}(s)s^{\frac{\gamma }{n}-1}\,
\mathrm{
d}s\,\mathrm{
d}t
\\
&\leq &\sup _{\|h\|_{X(0,\infty )}\leq 1}
\|T_{\frac{\gamma }{n}}h\|
_{X(0,\infty )}\left \| \int _{t}^{\infty }f^{*}(s)s^{
\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}.\end{aligned}$$ By the definition of $K$, we arrive at $$\begin{aligned}
\label{E:upper-bound-for-tau}
\tau (f)
\leq K
\left \| \int _{t}^{\infty }f^{*}(s)s^{
\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )},\end{aligned}$$ and the desired equivalence is established.
Now we shall prove that $\tau $ is an r.i. norm. We first note that the function $s\mapsto s^{\frac{\gamma }{n}}(PT_{\frac{
\gamma }{n}}h)(s)$ is always nonincreasing on $(0,\infty )$, regardless of $h$. This follows from the easily verified fact that the expression $s^{\frac{\gamma }{n}}(PT_{\frac{\gamma }{n}}h)(s)$ is a constant multiple of the integral mean over the interval $(0,s)$ of the obviously nonincreasing function $t\mapsto \sup _{t\leq y<\infty }y^{\frac{
\gamma }{n}}h^{*}(y)$ with respect to the measure $\mathrm{
d}\mu (t)=t^{-\frac{\gamma }{n}}\,\mathrm{
d}t$. Therefore, and Hardy’s lemma yield $$\begin{aligned}
\tau (f+g)
&=&
\sup _{\|h\|_{X(0,\infty )}\leq 1}\int _{0}^{\infty }(f+g)^{*}(s)(PT
_{\frac{\gamma }{n}}h)(s)s^{\frac{\gamma }{n}}\,\mathrm{
d}s
\\
&\leq &\sup _{\|h\|_{X(0,\infty )}\leq 1}\int _{0}^{\infty }f^{*}(s)(PT
_{\frac{\gamma }{n}}h)(s)s^{\frac{\gamma }{n}}\,\mathrm{
d}s
+
\sup _{\|h\|_{X(0,\infty )}\leq 1}\int _{0}^{\infty }g^{*}(s)(PT
_{\frac{\gamma }{n}}h)(s)s^{\frac{\gamma }{n}}\,\mathrm{
d}s
\\
&=&
\tau (f)+\tau (g).\end{aligned}$$ All the other properties in (P1) as well as (P2), (P3) and (P6) are readily verified. We shall show (P4). Let $E\subset (0,\infty )$ be of finite measure and denote $a=|E|$. By , one has $$\begin{aligned}
\tau (\chi _{E})
&\leq & K
\left \| \int _{t}^{\infty }\chi _{E}^{*}(s)s
^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}=
K\left \| \chi _{(0,a)}(t)\int _{t}
^{a}s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}
\\
&\leq & \frac{Kn}{ \gamma }a^{\frac{\gamma }{n}}\left \|
\chi _{(0,a)}\right \| _{X'(0,\infty )},\end{aligned}$$ and so $$\begin{aligned}
\tau (\chi _{E})
\leq \frac{Kn}{ \gamma }a^{\frac{\gamma }{n}}\left \|
\chi _{(0,a)}(t)\right \| _{X'(0,\infty )}<\infty\end{aligned}$$ by the property (P4) for $X'(0,\infty )$. It remains to verify (P5). Let $f\in \mathcal{M}(\mathbb{R}^{n})$ and let $E\subset \mathbb{R}^{n}$ be of finite positive measure. Denote $a=|E|$. Then, by the monotonicity of the function $s\mapsto s^{\frac{\gamma }{n}}(PT_{\frac{\gamma }{n}}h)(s)$ on $(0,\infty )$, we have $$\begin{aligned}
\tau (f)
&=&
\sup _{\|h\|_{X(0,\infty )}\leq 1}\int _{0}^{\infty }f^{*}(s)(PT
_{\frac{\gamma }{n}}h)(s)s^{\frac{\gamma }{n}}\,\mathrm{
d}s
\geq \sup _{\|h\|_{X(0,\infty )}\leq 1}\int _{0}^{a}f^{*}(s)(PT_{\frac{
\gamma }{n}}h)(s)s^{\frac{\gamma }{n}}\,\mathrm{
d}s
\\
&\geq &\sup _{\|h\|_{X(0,\infty )}\leq 1}a^{\frac{\gamma }{n}}(PT_{\frac{
\gamma }{n}}h)(a)\int _{0}^{a}f^{*}(s)\,\mathrm{
d}s.\end{aligned}$$ Now let us take $h_{0}=\frac{\chi _{(0,a)}}{\|\chi _{(0,a)}\|_{X(0,
\infty )}}$. Then $\|h_{0}\|_{X(0,\infty )}=1$, whence $$\begin{aligned}
\sup _{\|h\|_{X(0,\infty )}\leq 1}a^{\frac{\gamma }{n}}(PT_{\frac{
\gamma }{n}}h)(a)
&\geq &a^{\frac{\gamma }{n}}(PT_{\frac{\gamma }{n}}h
_{0})(a)
\\
&=&
\frac{a^{\frac{\gamma }{n}-1}}{\|\chi _{(0,a)}\|_{X(0,\infty )}}a
^{\frac{\gamma }{n}}\int _{0}^{a}s^{-\frac{\gamma }{n}}\,\mathrm{
d}s=
\frac{n}{n-\gamma }\frac{a^{\frac{\gamma }{n}}}{\|\chi _{(0,a)}\|
_{X(0,\infty )}}.\end{aligned}$$ Altogether, $$\begin{aligned}
\int _{E}f(x)\,\mathrm{
d}x\leq \int _{0}^{a}f^{*}(s)\,\mathrm{
d}s\leq \frac{n-\gamma }{n}a^{-\frac{\gamma }{n}}\|\chi _{(0,a)}\|_{X(0,
\infty )}\tau (f),\end{aligned}$$ and (P5) follows. We have shown that $\tau $ is an r.i. norm. This entitles us to take $Y=Y(\tau ')$.
We now claim that $M_{\gamma }\colon X\to Y$. By and since $\tau (f)=\|f\|_{Y'}$ for every $f\in \mathcal{M}_{+}(\mathbb{R}^{n})$, we have $$\begin{aligned}
\label{E:sigma-Y-estimate}
\left \| \int _{t}^{\infty }f^{*}(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}\leq \|f\|_{Y'},\quad f\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$ Let $g\in \mathcal{M}_{+}(0,\infty )$ be nonincreasing. We define $f(x)=g(\omega _{n}|x|^{n})$ for $x\in \mathbb{R}^{n}\setminus \{0\}$, where $\omega _{n}$ is the volume of the $n$-dimensional unit ball. Then $f$ is defined almost everywhere on $\mathbb{R}^{n}$ and one has $g\sim f$. Therefore, implies that $$\begin{aligned}
\left \| \int _{t}^{\infty }g(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}
\leq \|g\|_{Y'(0,\infty )}\end{aligned}$$ for every nonincreasing $g\in \mathcal{M}_{+}(0,\infty )$. Using the equivalence of and with the (nondecreasing) function $I(s)=s^{1-\frac{\gamma }{n}}$, $s\in (0,\infty )$, we obtain that there exists a positive constant $C$ such that $$\begin{aligned}
\left \| \int _{t}^{\infty }g(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}\leq C\|g\|_{Y'(0,\infty )},\quad
g\in \mathcal{M}_{+}(0,\infty ).\end{aligned}$$ By , this in turn gives $$\begin{aligned}
\left \| t^{\frac{\gamma }{n}-1}\int _{0}^{t}g(s)\,\mathrm{
d}s\right \| _{Y(0,\infty )}\leq C\|g\|_{X(0,\infty )},\quad
g\in \mathcal{M}_{+}(0,\infty ).\end{aligned}$$ Restricting this inequality to nonincreasing functions, we obtain that $$\begin{aligned}
\left \| t^{\frac{\gamma }{n}}g^{**}(t)\right \| _{Y(0,\infty )}
\leq C\|g^{*}\|_{X(0,\infty )}, \quad g\in
\mathcal{M}_{+}(0,\infty ).\end{aligned}$$ Applying [Lemma \[L:unsup\]]{}, we get that there exists a (possibly different) positive constant $C$ such that $$\begin{aligned}
\left \| \sup _{t\leq s<\infty }s^{\frac{\gamma }{n}}g^{**}(s)\right \|
_{Y(0,\infty )}\leq C\|g^{*}\|_{X(0,\infty )},\quad g\in \mathcal{M}_{+}(0,\infty ).\end{aligned}$$ Thus, by , one has $$\begin{aligned}
\|M_{\gamma }f\|_{Y}
&\leq C\left \| \sup _{t\leq s<\infty }s^{\frac{
\gamma }{n}}f^{**}(s)\right \| _{Y(0,\infty )}\leq C\|f^{*}\|_{X(0,
\infty )}=C\|f\|_{X}, \quad f\in X,\end{aligned}$$ whence $M_{\gamma }\colon X\to Y$.
It remains to prove the optimality of the space $Y$. Assume that $M_{\gamma }\colon X \to Z$ for some r.i. space $Z$ over $\mathbb{R}^{n}$. Then holds thanks to the same argument as in the proof of [Theorem \[T:fractional-maximal-operator\]]{}, that is, $$\begin{aligned}
\left \| \int _{t}^{\infty }h(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}
\leq C
\|f\|_{Z'},\quad h\sim f.\end{aligned}$$ Since $f^{*}\sim f$, this yields, in particular, $$\begin{aligned}
\left \| \int _{t}^{\infty }f^{*}(s)s^{\frac{\gamma }{n}-1}\,\mathrm{
d}s\right \| _{X'(0,\infty )}
\leq C
\|f\|_{Z'}.\end{aligned}$$ This estimate combined with yields $$\begin{aligned}
\tau (f)\leq KC\|f\|_{Z'},\quad f\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$ As $\tau (f)=\|f\|_{Y'}$, this means that $Z'\hookrightarrow Y'$, or $Y\hookrightarrow Z$, proving the optimality of $Y$. The proof is complete.
Note that ${L^{p,q;\mathbb{A}}}$ is equivalent to an r.i. space under any of the assumptions thanks to [@OP Theorem 7.1].
Let us first treat the cases when $T_{{\frac{\gamma }{n}}}\colon
{L^{p,q;\mathbb{A}}}(0,\infty )\to {L^{p,q;\mathbb{A}}}(0,\infty )$. To this end we have to investigate when there exists a positive constant $C>0$ such that $$\begin{aligned}
\label{E:onestar}
\left \|
t^{-{\frac{\gamma }{n}}}\sup _{t\le s<\infty } s^{\frac{
\gamma }{n}}f^{*}(s)
\right \| _{L^{p,q;\mathbb{A}}}\leq C
\|f\|_{L
^{p,q;\mathbb{A}}}, \quad f\in \mathcal{M}_{+}(
\mathbb{R}^{n}).\end{aligned}$$ We first consider the case when $q=\infty $. Then reads as $$\begin{aligned}
\label{E:twostar}
\sup _{0<t<\infty } t^{\frac{1}{p}-{\frac{\gamma }{n}}} \ell ^{
\mathbb{A}}(t)
\sup _{t\leq s<\infty } s^{\frac{\gamma }{n}}f^{*}(s)
\leq C
\sup _{0<t<\infty } t^{\frac{1}{p}} \ell ^{\mathbb{A}}(t)f^{*}(t).\end{aligned}$$ One has $$\begin{aligned}
\sup _{0<t<\infty } t^{\frac{1}{p}-{\frac{\gamma }{n}}} \ell ^{
\mathbb{A}}(t)
& \sup _{t\leq s<\infty }s^{\frac{\gamma }{n}}f^{*}(s)
=
\sup _{0<t<\infty }t^{\frac{1}{p}-{\frac{\gamma }{n}}} \ell ^{
\mathbb{A}}(t)
\sup _{t\leq s<\infty }s^{\frac{1}{p}}
\ell ^{\mathbb{A}}(s) f^{*}(s)s^{{\frac{\gamma }{n}}-\frac{1}{p}}
\ell ^{-{\mathbb{A}}}(s)
\\
& \leq \left (\sup _{0<t<\infty }
t^{\frac{1}{p}} \ell ^{\mathbb{A}}(t)
f^{*}(t) \right )
\left (\sup _{0<t<\infty }
t^{\frac{1}{p}-{\frac{
\gamma }{n}}} \ell ^{\mathbb{A}}(t)
\sup _{t\leq s<\infty } s^{{\frac{
\gamma }{n}}-\frac{1}{p}}
\ell ^{-{\mathbb{A}}}(s)\right ).\end{aligned}$$ Thus, is obviously satisfied if $s\mapsto s^{{\frac{
\gamma }{n}}-\frac{1}{p}}\ell ^{-{\mathbb{A}}}(s)$ is equivalent to a nonincreasing function. This happens precisely if either $p<{\frac{n}{
\gamma }}$ or $p={\frac{n}{\gamma }}$, $\alpha _{0}\leq 0$ and $\alpha _{\infty }\geq 0$. It is easy to see that in all the remaining cases, that is when either $p>{\frac{n}{\gamma }}$ or $p={\frac{n}{
\gamma }}$ and $\alpha _{0}> 0$, or $p={\frac{n}{\gamma }}$, $\alpha _{0}\leq 0$ and $\alpha _{\infty }<0$, the inequality is false as one can observe by plugging the function $f^{*}=\chi _{(0,a)}$ into the inequality for $a\in (0,1)$ or for $a\in (1,\infty )$, respectively.
Now let us consider the case when $q<\infty $. We recall that then reads as $$\begin{aligned}
\label{E:tristar}
\left ( \int _{0}^{\infty }
t^{-\frac{q\gamma }{n}+\frac{q}{p}-1}
\ell ^{{\mathbb{A}}q}(t)
\sup _{t\leq s<\infty } s^{\frac{q\gamma }{n}}f
^{*}(s)^{q}\,\mathrm{
d}t
\right )^{\frac{1}{q}}
\leq C
\left ( \int _{0}^{\infty }f^{*}(t)^{q}
t^{\frac{q}{p}-1} \ell ^{{\mathbb{A}}q}(t) \,\mathrm{
d}t
\right )^{\frac{1}{q}}\end{aligned}$$ for some $C>0$ and all $f\in \mathcal{M}_{+}(\mathbb{R}^{n})$. By [@GOP Theorem 3.2], holds if and only if there exists a constant $K$ such that, for every $\tau \in (0,\infty
)$, $$\begin{aligned}
\label{E:pes}
\tau ^{{\frac{\gamma }{n}}}
\left ( \int _{0}^{\tau }
t^{-\frac{q\gamma
}{n}+\frac{q}{p} - 1}\ell ^{{\mathbb{A}}q}(t)\,\mathrm{
d}t
\right )^{\frac{1}{q}}
\le K
\left ( \int _{0}^{\tau }
t^{
\frac{q}{p}-1}\ell ^{{\mathbb{A}}q}(t)\,\mathrm{
d}t
\right )^{\frac{1}{q}}.\end{aligned}$$ Elementary calculation shows that holds if and only if $1\leq p<\frac{n}{\gamma }$. Adding all conditions together we infer that $T_{\frac{\gamma }{n}}$ is bounded on the r.i. space ${L^{p,q;
\mathbb{A}}}(0,\infty )$ if and only if one of the conditions (\[E:fractional\_p1\]a), (\[E:fractional\_easy\]b) or (\[E:fractional\_infty1\]c) holds.
We are thus in a position to use [Theorem \[T:fractional-corollary\]]{} in these cases, hence the optimal range $Y$ for the space ${L^{p,q;
\mathbb{A}}}$ with respect to $M_{\gamma }$ satisfies $$\begin{aligned}
\|f\|_{Y'}
= \left \| \int _{t}^{\infty } f^{*}(s)s^{{\frac{\gamma }{n}}-1}
\,\mathrm{
d}s
\right \| _{({L^{p,q;\mathbb{A}}})'}.\end{aligned}$$ Now we have by [@OP Theorems 6.2 and 6.6] that $({L^{p,q;
\mathbb{A}}})'=L^{p',q';-{\mathbb{A}}}$, so we in fact get $$\begin{aligned}
\|f\|_{Y'}
= \left \| \int _{t}^{\infty }f^{*}(s)s^{{\frac{\gamma }{n}}-1}
\,\mathrm{
d}s
\right \| _{L^{p',q';-{\mathbb{A}}}},\end{aligned}$$ that is, $$\begin{aligned}
\|f\|_{Y'}
= \left \|
t^{\frac{1}{p'}-\frac{1}{q'}}\ell ^{-{\mathbb{A}}}(t) \int _{t}^{\infty
}f^{*}(s)s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s
\right \| _{L^{q'}(0,\infty )}.\end{aligned}$$ When $p=1$, $q=1$, $\alpha _{0}\geq 0$ and $\alpha _{\infty }\leq 0$, this establishes the assertion in the case (\[E:fractional\_infty1\]c). In the particular case ${\mathbb{A}}=[0,0]$ we have $$\begin{aligned}
\|f\|_{Y'}
= \sup _{0<t<\infty } \ell ^{-{\mathbb{A}}}(t)
\int _{t}^{
\infty }f^{*}(s)s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s
= \int _{0}^{\infty }f^{*}(s)s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s
= \|f\|_{L^{{\frac{n}{\gamma }},1}},\end{aligned}$$ hence $Y=L^{\frac{n}{n-\gamma },\infty }$. To prove the assertion, our next step will be to simplify the expression for $\|f\|_{Y'}$ if one of the conditions (\[E:fractional\_p1\]a) or (\[E:fractional\_easy\]b) holds. We start with the lower bound. One has, by monotonicity of $f^{*}$, the change of variables and elementary estimates, $$\begin{aligned}
\|f\|_{Y'}
& \geq &\left \| t^{\frac{1}{p'}-\frac{1}{q'}}
\ell ^{-{\mathbb{A}}}(t)
\int _{t}^{2t}f^{*}(s)s^{{\frac{\gamma }{n}}-1}
\,\mathrm{
d}s
\right \| _{L^{q'}(0,\infty )}
\\
& \geq &c
\left \| t^{\frac{1}{p'}-\frac{1}{q'}+{\frac{\gamma }{n}}}
\ell ^{-{\mathbb{A}}}(t)
f^{*}(2t)
\right \| _{L^{q'}(0,\infty )}
\\
& \geq& c'
\left \| t^{\frac{1}{p'}-\frac{1}{q'}+{\frac{\gamma }{n}}}
\ell ^{-{\mathbb{A}}}(t)
f^{*}(t)
\right \| _{L^{q'}(0,\infty )}
\\
& =& c' \|f\|_{L^{r',q';-{\mathbb{A}}}},\end{aligned}$$ where $c,c'$ are positive constants independent of $f$ and $r$ is such that $\tfrac{1}{r'} = \tfrac{1}{p'}+{\frac{\gamma }{n}}$. We shall show however that the converse inequality holds as well. First let $q=1$. Then $$\begin{aligned}
\|f\|_{Y'}
& = &\sup _{0<t<\infty }
t^{\frac{1}{p'}}
\ell ^{-{\mathbb{A}}}(t) \int _{t}^{\infty } f^{*}(s)s^{\frac{1}{p'}+
{\frac{\gamma }{n}}}\ell ^{-{\mathbb{A}}}(s)
s^{-\frac{1}{p'}-1}
\ell ^{{\mathbb{A}}}(s)\,\mathrm{
d}s
\\
& \le &\|f\|_{L^{r',q';-{\mathbb{A}}}}
\sup _{0<t<\infty } t^{
\frac{1}{p'}} \ell ^{-{\mathbb{A}}}(t)
\int _{t}^{\infty } s^{-
\frac{1}{p'}-1} \ell ^{{\mathbb{A}}}(s)\,\mathrm{
d}s
\\
& \approx& \|f\|_{L^{r',q';-{\mathbb{A}}}}.\end{aligned}$$ Now assume that $1<q\leq \infty $. Then, by the classical Hardy inequality (see e.g. [@Mu]), we get that there exists a positive constant $C$ such that $$\begin{aligned}
\left \| t^{\frac{1}{p'}-\frac{1}{q'}} \ell ^{-{\mathbb{A}}}(t)
\int
_{t}^{\infty }g(s)\,\mathrm{
d}s
\right \| _{L^{q'}(0,\infty )} \\
\qquad \leq C
\left \| t^{\frac{1}{p'}-
\frac{1}{q'}+1} \ell ^{-{\mathbb{A}}}(t)
g(t)
\right \| _{L^{q'}(0,
\infty )}, \quad g\in \mathcal{M}_{+}(0,\infty ).\end{aligned}$$ Given $f\in \mathcal{M}$, we set $g(t)=f^{*}(t)t^{{\frac{\gamma }{n}}-1}$, $t\in (0,\infty )$, which leads to $$\begin{aligned}
\|f\|_{Y'}
\leq C
\|f\|_{L^{r',q';-{\mathbb{A}}}},\end{aligned}$$ hence, altogether, $Y'=L^{r',q';-{\mathbb{A}}}$. Since $1<r'<\infty $, we have, by [@OP Theorems 6.2 and 6.6], that $Y=L^{r,q;{\mathbb{A}}}$, establishing the assertion.
We shall now treat the case (\[E:fractional\_infty2\]e). The general formula follows directly by of [Theorem \[T:fractional-maximal-operator\]]{} and the definition of the norm of ${L^{p,q;\mathbb{A}}}$. Note that since $T_{\frac{\gamma }{n}}$ is not bounded on ${L^{p,q;\mathbb{A}}}$ in this case, the supremum in is essential and cannot be avoided by setting $h=f^{*}$ as follows from [Theorem \[T:lenka\]]{}.
Let us now focus on the special case when ${\mathbb{A}}=[0,0]$. We denote the optimal partner for $L^{{\frac{n}{\gamma }},q}$ with respect to $M_{\gamma }$ by $Y$. Our aim is to show that $Y=L^{\infty }$ or, equivalently, that $Y'=L^{1}$. We first notice that $L^{1}$ is (up to equivalence) the only r.i. space whose fundamental function, denoted by $\psi $, satisfies $\psi (t)=t$. Indeed, assume that $X$ has such a fundamental function. Then $$\begin{aligned}
\|f\|_{\Lambda (X)}=\int _{0}^{\infty }f^{*}(t)\mathrm{
d}\psi (t)=\|f\|_{L^{1}}\end{aligned}$$ and $$\begin{aligned}
\|f\|_{M(X)}=\sup _{t\in (0,\infty )}\psi (t)t^{**}(t)=
\sup _{t\in (0,\infty )}\int _{0}^{t}f^{*}(s)\mathrm{
d}s=\|f\|_{L^{1}}.\end{aligned}$$ Consequently, by [@BS Chapter 2, Theorem 5.13], we have $\Lambda (X)=X=M(X)$, hence $X=L^{1}$. Therefore, it is enough to verify that the fundamental function of $Y'$, $\varphi $, say, satisfies $\varphi (t)\approx t$ for $t\in (0,\infty )$. As for the proof of the lower bound, we make use of the same calculation as in with $f=\chi _{E}$ and $|E|=t$. We obtain $$\begin{aligned}
\|\chi _{E}\|_{Y'}
\ge C_{n,\gamma } \,t^{\frac{\gamma }{n}}\|\chi _{(0,t)}
\|_{\bigl (L^{{\frac{n}{\gamma }},q}(0,\infty )\bigr )'},
\quad t\in(0,\infty) ,\end{aligned}$$ which, thanks to , can be rewritten as $$\begin{aligned}
\varphi (t)
\ge C_{n,\gamma } \frac{t^{1+{\frac{\gamma }{n}}}}{\|
\chi _{(0,t)}\|_{L^{{\frac{n}{\gamma }},q}(0,\infty )}},
\quad t\in(0,\infty) ,\end{aligned}$$ and the estimate then follows since the fundamental function of $L^{{\frac{n}{\gamma }},q}$ is $t^{\frac{\gamma }{n}}$. To prove the converse inequality, let us use the same upper bound which appears in the proof of the validity of (P4) in the proof of [Theorem \[T:fractional-maximal-operator\]]{}. Observe that now holds on the whole of $(0,\infty )$ and hence we get also for all sets $E$ with $|E|<1$. That gives the desired relation $\varphi (t)\le C_{n,\gamma} t$, $t\in (0,\infty )$.
The Hilbert transform {#sec5}
=====================
A very important example of a singular integral with odd kernel is the *Hilbert transform*, defined for appropriate functions on $\mathbb{R}$ by $$\begin{aligned}
Hf(x) = \lim _{\varepsilon \rightarrow 0_{+}}\frac{1}{\pi }
\int _{|x-t|\geq \varepsilon }\frac{f(t)}{x-t}\,\mathrm{
d}t.\end{aligned}$$ This operator is defined for every function $f\colon \mathbb{R}\to
\mathbb{R}$ for which the integral converges almost everywhere. The Hilbert transform arises in the study of boundary values of the real and imaginary parts of analytic functions. It is a cornerstone of several important disciplines including real and complex analysis and the theory of PDEs. In this section we shall study its sharp boundedness properties on r.i. spaces over $\mathbb{R}$. A key technical background tool will be the *Stieltjes transform*, $S$, which is defined for every nonnegative measurable function $f$ on $(0,\infty )$ by $$\begin{aligned}
(Sf)(t)
=
\frac{1}{t}\int _{0}^{t}f(s)\,\mathrm{
d}s+\int _{t}^{\infty }f(s)\frac{\mathrm{
d}s}{s}, \quad t\in (0,\infty ).\end{aligned}$$ It might be useful to note that $$\begin{aligned}
\label{E:comparison-of-S-P-and-Q}
S=P+Q=P\circ Q=Q\circ P.\end{aligned}$$
Whenever we say that the Hilbert transform is bounded from a function space $X$ to a function space $Y$, we implicitly assume that $H$ is well defined for every $f\in X$, that is, $f\in L^{1}_{
\operatorname{loc}}(\mathbb{R})$ and the limit in the definition of $Hf$ exists for a.e. $x\in \mathbb{R}$. Let us recall that, by [@BS Chapter 3, Theorem 4.8], a sufficient condition for the existence of this limit, for a given $\mathcal{M}(
\mathbb{R})$, is $$\begin{aligned}
\label{E:2.2}
(Sf^{*})(1) < \infty .\end{aligned}$$
Our main result in this section reads as follows.
\[T:hilbert-transforms\] Let $X$ be an r.i. space over $\mathbb{R}$ such that $$\begin{aligned}
\label{E:eta-satisfied}
\eta \in X'(0,\infty ),\end{aligned}$$ where $$\begin{aligned}
\label{E:definition-of-w}
\eta (t)=\chi _{(0,1]}(t)(1-\log t)+\chi _{(1,\infty )}(t)\frac{1}{t},
\ t\in (0,\infty ).\end{aligned}$$ Define the functional $\sigma $ by $$\begin{aligned}
\sigma (f)=\left \| Sf^{*}\right \| _{X'(0,\infty )}, \quad f\in
\mathcal{M}_{+}(\mathbb{R}).\end{aligned}$$ Then $\sigma $ is an r.i. norm and $$\begin{aligned}
\label{E:boundedness-hilbert}
H\colon X\to Y,\end{aligned}$$ where $Y=Y(\sigma ')$. Moreover, $Y$ is the optimal smallest r.i. space for which holds.
Conversely, if is not true, then there does not exist an r.i. space $Y$ for which holds.
For the optimal domain, we have the following result. Again, the proof is analogous to the appropriate proofs above, and therefore omitted.
Let $Y$ be an r.i. space over $\mathbb{R}$ such that $$\begin{aligned}
\label{E:eta-condition-domain}
\eta \in Y(0,\infty ),\end{aligned}$$ where $\eta $ is the function from . Define the functional $\sigma $ by $$\begin{aligned}
\sigma (f)=\left \| Sf^{*}\right \| _{Y(0,\infty )}, \quad f\in
\mathcal{M}_{+}(\mathbb{R}).\end{aligned}$$ Then $\sigma $ is an r.i. norm and $$\begin{aligned}
\label{E:hilbert-bounded-domain}
H\colon X\to Y,\end{aligned}$$ where $X=X(\sigma )$. Moreover, $X$ is the optimal biggest r.i. space for which holds.
Conversely, if is not true, then there does not exist an r.i. space $X$ for which holds.
We provide several examples of the optimal range partners for Lorentz-Zygmund spaces with respect to the Hilbert transform. The proof is similar to that of [Theorem \[T:-maximal-operator-GLZ\]]{} and therefore omitted.
Assume that $p,q\in [1,\infty ]$, ${\mathbb{A}}\in \mathbb{R}^{2}$. Then $$\begin{aligned}
H\colon L^{p,q; {\mathbb{A}}}\to
\left\{
\begin{array}{l@{\quad }l}
L^{1,1; {\mathbb{A}}-1},
&p=1, q=1, \alpha _{0} \geq 1, \alpha _{
\infty }<0, \\
L^{p,q; {\mathbb{A}}},
&1<p<\infty , \\
Y,
& p=\infty , q=1, \alpha _{0} < -1, \alpha _{\infty }\ge 0~ or \\
&p = \infty , 1 < q <\infty , \alpha _{0} + \frac{1}{q} < 0,
\alpha _{\infty }+ \frac{1}{q'} > 0,\\
L^{\infty ,\infty ; {\mathbb{A}}-1},
&p=\infty , q=\infty , \quad \alpha
_{0} \leq 0, \alpha _{\infty }>1,
\end{array}\right.\end{aligned}$$ where $Y$ is defined by its associate space $Y'$ whose norm is given by $$\begin{aligned}
\|f\|_{Y'}=\left \| \int _{t}^{\infty }f^{**}(s)\frac{\mathrm{
d}s}{s}\right \| _{L^{(1,q';-{\mathbb{A}}-1)}}, \quad f\in \mathcal{M}
_{+}(\mathbb{R}).\end{aligned}$$ These spaces are the optimal range partners with respect to $H$.
At the end of this section, we aim to prove [Theorem \[T:hilbert-transforms\]]{}. We start with a lemma which recalls a well-known fact. We insert a short proof for the sake of completeness.
\[L:comparison-of-hilbert-and-stieltjes\] Let $X$ and $Y$ be r.i. Banach function spaces over $\mathbb{R}$. Assume that is satisfied for every $f\in X$. Then the Hilbert transform $H$ is bounded from $X$ to $Y$ if and only if the Stieltjes transform $S$ is bounded from $X(0,\infty )$ to $Y(0,\infty )$.
Assume first that $H$ is bounded from $X$ to $Y$. Fix a function $f\in \mathcal{M}_{+}(0,\infty )$ such that $(Sf^{*})(1)<\infty $. Then, by a simple modification of [@BS Chapter 3, Proposition 4.10], there exists a function $g\in \mathcal{M}_{+}(\mathbb{R})$, equimeasurable with $f$, such that $$\begin{aligned}
(Sf^{*})(t)\leq 2\pi \left (Hg\right )^{*}(t), \quad
t\in (0,\infty ).\end{aligned}$$ Thus, by the property (P2) of $Y$, we have $$\begin{aligned}
\|(Sf^{*})\|_{Y(0,\infty )}\leq 2\pi \|(Hg)^{*}\|_{Y(0,\infty )}.\end{aligned}$$ By the rearrangement invariance of $Y$, this turns into $$\begin{aligned}
\|(Sf^{*})\|_{Y(0,\infty )}\leq 2\pi \|Hg\|_{Y}.\end{aligned}$$ It follows from the boundedness of $H$ from $X$ to $Y$ that $$\begin{aligned}
\|Hg\|_{Y}\leq C\|g\|_{X}\end{aligned}$$ for some constant $C$, $0<C<\infty $, independent of $g$ (hence of $f$). We thus get, altogether, using also the definition of the representation space and the equimeasurability of $f$ and $g$, that $$\begin{aligned}
\|(Sf^{*})\|_{Y(0,\infty )}\leq 2C\pi \|g\|_{X}=2C\pi \|g^{*}\|_{X(0,
\infty )}=2C\pi \|f^{*}\|_{X(0,\infty )}.\end{aligned}$$ In other words, $S$ is bounded from $X(0,\infty )$ to $Y(0,\infty )$.
Conversely, assume that the Stieltjes transform is bounded from $X(0,\infty )$ to $Y(0,\infty )$. By an appropriate modification of [@BS Chapter 3, Theorem 4.8], there exists a positive constant $C$ independent of $f$ such that $$\begin{aligned}
\left (Hf\right )^{*}(t)\leq C (Sf^{*})(t) , \quad t
\in (0,\infty ).\end{aligned}$$ We then get, similarly as above, $$\begin{aligned}
\|Hf\|_{Y}=\|(Hf)^{*}\|_{Y(0,\infty )}\leq C\|(Sf^{*})\|_{X(0,\infty
)}\leq C'\|f^{*}\|_{X(0,\infty )}=C'\|f\|_{X}\end{aligned}$$ for some suitable constant $C'$, proving that $H\colon X\to Y$. The proof is complete.
Our next step will be a characterization of the optimal range partner with respect to the Stieltjes transform.
\[T:stieltjes-transform\] Let $X$ be an r.i. Banach function space over $(0,\infty )$ such that $$\begin{aligned}
\label{E:eta-satisfied2}
\eta \in X'(0,\infty ),\end{aligned}$$ where $\eta $ is the function from . Define the functional $\sigma $ by $$\begin{aligned}
\sigma (f)=\left \| Sf^{*}\right \| _{X'(0,\infty )}, \quad f\in
\mathcal{M}_{+}(0,\infty ).\end{aligned}$$ Then $\sigma $ is an r.i. norm and $$\begin{aligned}
\label{E:boundedness-stieltjes}
S\colon X\to Y,\end{aligned}$$ where $Y=Y(\sigma ')$. Moreover, $Y$ is the optimal smallest r.i. space for which holds.
Conversely, if is not true, then there does not exist an r.i. space $Y$ for which holds.
Consider the functional $\sigma (f)=\left \| Sf^{*}\right \| _{X'(0,
\infty )}$, $ f\in \mathcal{M}_{+}(0,\infty )$. We shall prove that $\sigma $ is an r.i. norm. As in the proof of [Theorem \[T:maximal-operator\]]{}, the axioms (P2), (P3) and (P6) for $\sigma $ are clearly satisfied. The verification of the triangle inequality is even easier than in the proof of [Theorem \[T:maximal-operator\]]{}. It follows from that $$\begin{aligned}
\label{E:comparison-of-S-and-Q}
Sf^{*}=Qf^{**}, \quad f\in \mathcal{M}(0,\infty ),\end{aligned}$$ which in conjunction with immediately yields the triangle inequality for $\sigma $. As usual, all other properties in (P1) are readily verified. Also the verification of (P5) is easy. In fact, it immediately follows from the analogous property of the functional $\sigma $ from [Theorem \[T:maximal-operator\]]{}, because, by , one has $Sf^{*}\geq Qf^{*}$. It only remains to verify the validity of (P4). To this end, let $E\subset
\mathbb{R}$ be a set of finite measure. We need to prove that $\left \| S\chi _{E}^{*}\right \| _{X'}<\infty $. Calculation shows that this is equivalent to saying that $\eta \in X'$, a fact guaranteed by the assumption. This shows (P4), and, consequently, it completes the proof of the fact that $\sigma $ is an r.i. Banach function norm.
We shall now prove that $S\colon X\to Y$. The operator $S$ is self-adjoint with respect to the $L^{1}$-pairing in the sense that $$\begin{aligned}
\int _{0}^{\infty }(Sf)(t)g(t)\,\mathrm{
d}t=\int _{0}^{\infty }f(t)(Sg)(t)\,\mathrm{
d}t\end{aligned}$$ for every admissible $f$ and $g$. Hence, it suffices to prove that $S\colon Y'\to X'$. That, however, follows trivially from the definition of $Y'$.
The proof of optimality of the space $Y$ as well as that of the nonexistence of an r.i. range partner for $X$ in case $\eta \notin X'$ is completely analogous to its counterpart from the proof of [Theorem \[T:maximal-operator\]]{} and hence is omitted.
Finally, [Theorem \[T:hilbert-transforms\]]{} immediately follows from [Theorem \[T:stieltjes-transform\]]{} and [Lemma \[L:comparison-of-hilbert-and-stieltjes\]]{}.
The Riesz potential {#sec6}
===================
Let $0<\gamma <n$. Then the *Riesz potential* of order $\gamma $, $I_{\gamma }$, of a measurable function $f$ on $\mathbb{R}
^{n}$ is defined by $$\begin{aligned}
(I_{\gamma }f)(x)=\int _{\mathbb{R}^{n}}f(y)\phi (x-y)\,\mathrm{
d}y, \quad x\in \mathbb{R}^{n},\end{aligned}$$ where $$\begin{aligned}
\phi (y)=c(\gamma )|y|^{\gamma -n},
\quad
c(\gamma )=\Gamma \left (\frac{n-\gamma }{2}\right )\left (\pi ^{
\frac{n}{2}}2^{\gamma }\Gamma \left (\frac{\gamma }{2}\right )\right )
^{-1}.\end{aligned}$$
We are going to make use of a special case of the *O’Neil inequality*. In its general form [@ON Lemma 1.5], it states that, for the convolution of two measurable functions $f,g$ on $\mathbb{R}^{n}$, defined by $$\begin{aligned}
(f*g)(x)=\int _{\mathbb{R}^{n}}f(x-y)g(y)\,\mathrm{
d}y, \quad x\in \mathbb{R}^{n},\end{aligned}$$ we have $$\begin{aligned}
(f*g)^{**}(t)\leq tf^{**}(t)+\int _{t}^{\infty }f^{*}(s)g^{*}(s)\,
\mathrm{
d}s , \quad t\in (0,\infty ).\end{aligned}$$ With the particular choice $$\begin{aligned}
g(x)=|x|^{\gamma -n}, \quad x\in \mathbb{R}^{n},\end{aligned}$$ we obtain that $$\begin{aligned}
(I_{\gamma }f)^{*}(t)\leq C\int _{t}^{\infty }f^{**}(s)s^{\frac{\gamma
}{n}-1}\,\mathrm{
d}s , \quad t\in (0,\infty ),\end{aligned}$$ with some positive constant $C$, depending on $\gamma $ and $n$, but independent of $f$ and $t$.
This inequality is known to be sharp, but merely in a broader sense than, for example, the corresponding estimate for the Hardy–Littlewood maximal operator. This was firstly observed by O’Neil in the final remark of the paper [@ON], where it is pointed out that the inequality can be reversed when $f,g$ are radially decreasing positive functions. Furthermore, by an appropriately modified argument from [@EOP Theorem 10.2(iii)]), we get that, for every $f\in \mathcal{M}(\mathbb{R}^{n})$, there exists a function $g\in
\mathcal{M}(0,\infty )$ equimeasurable with $f$ such that $$\begin{aligned}
(I_{\gamma }g)^{*}(t)\geq c\int _{t}^{\infty }f^{**}(s)s^{\frac{\gamma
}{n}-1}\,\mathrm{
d}s , \quad t\in (0,\infty ),\end{aligned}$$ with some constant $c$, $0<c<\infty $, depending on $\gamma $ and $n$, but independent of $f$ and $t$.
We shall now turn our attention to a weighted version of the Stieltjes transform, which plays a key role in the matter of optimal spaces for the Riesz potential.
Let $\alpha \in (1,\infty )$. The *weighted Stieltjes transform*, $S_{\alpha }$, is defined for every nonnegative measurable function $f$ on $(0,\infty )$ by $$\begin{aligned}
(S_{\alpha }f)(t)
=
t^{\frac{1}{\alpha }-1}\int _{0}^{t}f(s)\,\mathrm{
d}s+\int _{t}^{\infty }f(s)s^{\frac{1}{\alpha }-1}\, \mathrm{
d}s, \quad t\in (0,\infty ).\end{aligned}$$
We note that, for every admissible $f$ and $t$, one has $$\begin{aligned}
(S_{\alpha }f)(t)=c_{\alpha }\int _{t}^{\infty }(Pf)(s)s^{\frac{1}{
\alpha }-1}\, \mathrm{
d}s,\end{aligned}$$ where $c_{\alpha }=\frac{\alpha -1}{\alpha }$.
Our main result of this section reads as follows.
\[T:riesz-potential\] Let $\gamma \in (0, n)$ and let $X$ be an r.i. space over $\mathbb{R}^{n}$ such that $$\begin{aligned}
\label{E:xi-satisfied}
\xi _{\frac{n}{\gamma }} \in X'(0,\infty ),\end{aligned}$$ where, for $\alpha >0$, $$\begin{aligned}
\label{E:definition-of-xi}
\xi _{\alpha }(t)=(t+1)^{\frac{1}{\alpha }-1}, \quad t\in (0,\infty ).\end{aligned}$$ Define the functional $\sigma $ by $$\begin{aligned}
\sigma (f)=\left \| S_{\frac{n}{\gamma }}f^{*}\right \| _{X'(0,
\infty )}, \quad f\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$ Then $\sigma $ is an r.i. norm and $$\begin{aligned}
\label{E:boundedness-riesz}
I_{\gamma }\colon X\to Y,\end{aligned}$$ where $Y=Y(\sigma ')$. Moreover, $Y$ is the optimal smallest r.i. space for which holds.
Conversely, if is not true, then there does not exist an r.i. space $Y$ for which holds.
As in the preceding sections, we also characterize optimal domains. We also omit the proof since it is analogous, again, to that of [Theorem \[T:maximal-operator-domain\]]{}.
Let $\gamma \in (0, n)$ and let $Y$ be an r.i. space over $\mathbb{R}^{n}$ such that $$\begin{aligned}
\label{E:xi-condition-domain}
\xi _{\frac{\gamma }{n}} \in Y(0,\infty ),\end{aligned}$$ where $\xi _{\alpha }$ is the function from . Define the functional $\sigma $ by $$\begin{aligned}
\sigma (f)=\left \| S_{\frac{n}{\gamma }}f^{*}\right \| _{Y(0,
\infty )}, \quad f\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$ Then $\sigma $ is an r.i. norm and $$\begin{aligned}
\label{E:riesz-bounded-domain}
I_{\gamma }\colon X\to Y,\end{aligned}$$ where $X=X(\sigma )$. Moreover, $X$ is the optimal biggest r.i. space for which holds.
Conversely, if is not true, then there does not exist an r.i. space $X$ for which holds.
We use [Theorem \[T:riesz-potential\]]{} to provide several examples of the optimal range partners for Lorentz-Zygmund spaces with respect to the Riesz potential.
\[T:riesz-potential-GLZ\] Assume that $\gamma \in (0,n)$, $p,q\in [1,\infty ]$, ${\mathbb{A}}
\in \mathbb{R}^{2}$. Then $$\begin{aligned}
\label{E:riesz_p1}
I_{\gamma }\colon {L^{p,q;\mathbb{A}}}\to
\left\{
\begin{array}{l@{\quad }l}
Y_{1}
& p=1, q=1, \alpha _{0} \geq 0, \alpha _{\infty }\leq 0, \\
L^{\frac{np}{n-\gamma p},q;{\mathbb{A}}}
& 1<p<\frac{n}{\gamma }, \\
L^{\infty ,q;{\mathbb{A}}- 1}
& p={\frac{n}{\gamma }}, 1\leq q
\leq \infty , \alpha _{0} < \frac{1}{q'},
\alpha _{\infty }>
\frac{1}{q'},
\\
L^{\infty ,q;[-\frac{1}{q},\alpha _{\infty }- 1],\left [-1, 0\right ]}
& p={\frac{n}{\gamma }}, 1< q\leq \infty , \alpha _{0} =
\frac{1}{q'},
\alpha _{\infty }>\frac{1}{q'},
\\
Y_{2}
& p={\frac{n}{\gamma }}, q=1, \alpha _{0}<0, \alpha _{
\infty }=0,
\\
L^{\infty ,1;\left [-1,\alpha _{\infty }- 1\right ],\left [-1,0\right ],
\left [-1,0\right ]}
& p={\frac{n}{\gamma }}, q=1, \alpha _{0} = 0,
\alpha _{\infty }> 0,
\\
L^{\infty }
& p={\frac{n}{\gamma }}, q=1, \alpha _{0}\geq 0,
\alpha _{\infty }= 0,
\\
Y_{3}
& p={\frac{n}{\gamma }}, q=1, \alpha _{0} > 0,
\alpha _{\infty }> 0,
\\
Y_{2}
& p={\frac{n}{\gamma }}, 1< q\leq \infty , \alpha _{0} >
\frac{1}{q'},
\alpha _{\infty }>\frac{1}{q'}
,
\end{array}\right.\end{aligned}$$ where $$\begin{aligned}
\|f\|_{Y_{2}}
& = & \|f\|_{L^{\infty }}
+ \| t^{-\frac{1}{q}}
\ell ^{\alpha _{\infty }- 1}(t)f^{*}(t)\|_{L^{q}(1,\infty )},
\\
\|f\|_{Y_{3}}
& = & \| t^{-1}\ell ^{\alpha _{0} - 1}(t)f^{*}(t)\|_{{L
^{1}(0,1)}},\end{aligned}$$ and $Y_{1}$ is defined by its associate space $Y_{1}'$ whose norm is given by $$\begin{aligned}
\|f\|_{Y_{1}'}
= \sup _{0<t<\infty } \ell ^{-{\mathbb{A}}}(t)
\int _{t}
^{\infty }f^{**}(s)s^{{\frac{\gamma }{n}}-1}\,\mathrm{
d}s, \quad f\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$ In particular, if ${\mathbb{A}}=[0,0]$, we have $Y_{1}=L^{\frac{n}{n-
\gamma },\infty }$.
Moreover, these spaces are the optimal range partners with respect to $I
_{\gamma }$.
We note that ${L^{p,q;\mathbb{A}}}$ is equivalent to a rearrangement–invariant Banach function space due to [@OP Theorem 7.1] in all the cases.
Assume that $p\in (1,\infty )$ and $q\in [1,\infty ]$. By [@OP Theorems 6.2 and 6.6], the associate space of $L^{p,q;{\mathbb{A}}}$ is equivalent to $L^{p',q';-{\mathbb{A}}}$. We need to check when $\xi _{\frac{n}{\gamma }}\in X'(0,\infty )$ is satisfied, that is, when $$\int_0^\infty t^{\frac{q'}{p'} - 1}\ell^{-\mathbb{A} q'}(t)(t + 1)^{\frac{\gamma - n}{n}q'}\, \mathrm{d} t < \infty\quad\text{if $q\in(1,\infty]$,}$$ or when $$\sup\limits_{t\in(0,\infty)} t^{\frac1{p'}}\ell^{-\mathbb{A}}(t)(t + 1)^{\frac{\gamma - n}{n}} < \infty\quad\text{if $q = 1$.}$$ It is easy to see that in the former case the integral is finite if and only if either $$p\in(1,\tfrac{n}{\gamma})$$ or $$p=\frac{n}{\gamma}\text{ and }\alpha_\infty > \frac{1}{q'},$$ while in the latter case the supremum is finite if and only if either $$p\in(1,\tfrac{n}{\gamma})$$ or $$p=\frac{n}{\gamma}\text{ and }\alpha_\infty \geq 0.$$ Henceforth, we assume that these conditions are satisfied. By the classical weighted Hardy inequality, there is a positive constant $C$ such that $$\begin{aligned}
\Vert S_{\frac{n}{\gamma }} g^{*}\Vert _{p',q';-{\mathbb{A}}}
&=&
\Vert t^{\frac{1}{p'} - \frac{1}{q'}}\ell ^{-A}(t)\int _{t}^{\infty }g
^{**}(s)s^{\frac{\gamma -n}{n}}\, \mathrm{
d}s\Vert _{q'}
\\
&\le & C\Vert t^{\frac{1}{p'}+\frac{1}{q}}\ell ^{-{\mathbb{A}}}(t)g
^{**}(t)t^{\frac{\gamma }{n} - 1}\Vert _{q'} = C\Vert t^{\frac{1}{p'}+\frac{
\gamma }{n} - \frac{1}{q'}}\ell ^{-{\mathbb{A}}}(t)g^{**}(t)\Vert _{q'}
\\
&= &C\Vert g\Vert _{(r',q';-{\mathbb{A}})},\end{aligned}$$ where $\frac{1}{p'} + \frac{\gamma }{n} = \frac{1}{r'}$, that is, $r'=\frac{np}{(n+\gamma )p - n}$. The converse inequality $$\begin{aligned}
\Vert g\Vert _{(r',q';-{\mathbb{A}})}\le C' \Vert S_{\frac{n}{
\gamma }} g^{*}\Vert _{p',q';-{\mathbb{A}}}\end{aligned}$$ for some positive $C'$ follows immediately from the estimate $$\begin{aligned}
S_{\frac{n}{\gamma }} g^{*}(t)
&= \int _{t}^{\infty }g^{**}(s)s^{\frac{
\gamma }{n}-1}\, \mathrm{
d}s = \int _{t}^{\infty }\frac{1}{s^{2-\frac{\gamma }{n}}}\int _{0}^{s}
g^{*}(u)\, \mathrm{
d}u\, \mathrm{
d}s
\\
&\geq \int _{0}^{t} g^{*}(u)\, \mathrm{
d}u \int _{t}^{\infty }\frac{1}{s^{2-\frac{\gamma }{n}}}\, \mathrm{
d}s = \frac{n}{n-\gamma }t^{\frac{\gamma }{n}}g^{**}(t).\end{aligned}$$ If $p\in (1,\frac{n}{\gamma })$, then $r'\in (1,\frac{n}{\gamma })$. By [@OP Theorem 3.8], $L^{(r',q';-{\mathbb{A}})}$ is equivalent to $L^{r',q';-{\mathbb{A}}}$. Hence $Y$ is equivalent to $L^{r,q;
{\mathbb{A}}}$, where $r=\frac{np}{n-\gamma p}\in (\frac{n}{n-\gamma
},\infty )$, by [@OP Theorem 6.2]. If $p = \frac{n}{\gamma }$, then $r' = 1$. If $q\in (1,\infty )$ (and hence $q'\in (1,\infty )$), we obtain for $q\in (1,\infty )$ by virtue of [@OP Theorem 6.7]. If $q=\infty $ (and hence $q'=1$), we combine [@OP Theorem 3.8] with [@OP Theorem 6.6] in order to prove for $q=\infty $. If $p = \frac{n}{\gamma}$, $q=1$, and, for instance, $\alpha_0=0$ and $\alpha_\infty>0$, then, by the computations above, $\lVert S_{\frac{n}{\gamma}} g^*\rVert_{p',q';-\mathbb{A}}\approx\lVert g\rVert_{(1,\infty;[0,-\alpha_\infty])}$. Hence for this particular case follows from the description of the associate space of $L^{(1,\infty;[0,-\alpha_\infty])}$ provided by [@OP Theorem 6.7]. The other cases when $p = \frac{n}{\gamma}$ and $q=1$ can be proved similarly. In the remaining cases the proof is analogous to that of [Theorem \[T:fractional-maximal-operator-GLZ\]]{}. We omit the details.
We finally note that the result stated in [Theorem \[T:riesz-potential\]]{} follows in the usual way from the corresponding result for the weighted Stieltjes transform. Its proof is analogous to that of [Theorem \[T:stieltjes-transform\]]{}.
Let $\alpha \in (1,\infty )$. Let $X$ be a rearrangement-invariant Banach function space over $(0,\infty )$ such that $$\begin{aligned}
\label{E:xi-satisfied2}
\xi _{\alpha } \in X'(0,\infty ),\end{aligned}$$ where $\xi _{\alpha }$ is defined by . Define the functional $\sigma $ by $$\begin{aligned}
\sigma (f)=\left \| S_{\alpha }f^{*}\right \| _{X'(0,\infty )}, \quad f
\in \mathcal{M}_{+}(0,\infty ).\end{aligned}$$ Then $\sigma $ is an r.i. norm and $$\begin{aligned}
\label{E:boundedness-weighted-stieltjes}
S_{\alpha }\colon X\to Y,\end{aligned}$$ where $Y=Y(\sigma ')$. Moreover, $Y$ is the optimal smallest r.i. space for which holds.
Conversely, if is not true, then there does not exist an r.i. space $Y$ for which holds.
#### Acknowledgment
We wish to thank the referee for valuable comments. We are greatly indebted to Lenka Slavíková for stimulating discussions about the subject.
#### Funding
This research was supported by the grants P201-13-14743S and P201-18-00580S of the Czech Science Foundation, by the grant 8X17028 of the Czech Ministry of Education and by the grant SVV-2017-260455.
| 0 |
---
abstract: |
In a continuing effort to investigate the role of magnetic fields in evolved low and intermediate mass stars (principally regarding the shaping of their envelopes), we present new ALMA high resolution polarization data obtained for the nebula OH 231.8+4.2. We found that the polarized emission likely arises from aligned grains in the presence of magnetic fields rather than radiative alignment and self scattering. The ALMA data show well organized electric field orientations in most of the nebula and the inferred magnetic field vectors (rotated by 90 degrees) trace an hourglass morphology centred on the central system of the nebula. One region in the southern part of OH 231.8+4.2 shows a less organized distribution probably due to the shocked environment. These findings, in conjunction with earlier investigations (maser studies and dust emission analysis at other scales and wavelengths) suggest an overall magnetic hourglass located inside a toroidal field. We propose the idea that the magnetic field structure is closely related to the architecture of a magnetic tower and that the outflows were therefore magnetically launched. While the current dynamical effect of the fields might be weak in the equatorial plane principally due to the evolution of the envelope, it would still be affecting the outflows. In that regard, the measurement of the magnetic field at the stellar surface, which is still missing, combined with a full MHD treatment are required to better understand and constrain the events occurring in OH 231.8+4.2.\
bibliography:
- 'OH231.bib'
date: 'Accepted 2020 May 18. Received 2020 May 16; in original form 2020 April 13 '
title: 'ALMA reveals the coherence of the magnetic field geometry in OH 231.8+4.2'
---
\[firstpage\]
magnetic fields — polarization — stars: AGB and post-AGB — ISM: jets and outflows ISM: individual: OH231.8+4.2
Introduction
============
The detection of magnetic fields in evolved low and intermediate–mass stars ($\sim$0.8–8 M), such as AGB and post-AGB stars (pAGBs), pre-planetary (pPNe) and planetary nebulae (PNe), has generated great interest over the last decade. The main reason lies in our desire to understand the role magnetic fields could play in the dramatic departures from spherical symmetry occurring during the transition from the AGB to the PNe evolutionary phases [@Sahai2007; @Sahai2011].\
Observations, detections and measurements of the field topology and strength have been performed using different methods: (spectro)polarimetry of various maser species (e.g. @Vlemmings2008 [@Ferreira2013; @GOMEZ2016; @Vlemmings2017]), measurement of circular polarization (Stokes [*V*]{}) in stellar atmospheres and photospheres (e.g. @Konstantinova2010 [@Lebre2014; @Sabin2015a]), analysis of the linearly polarized emission of circumstellar dust and molecules (e.g. @Vlemmings2012 [@Vlemmings2017; @Girart2012; @Sabin2014]). Nevertheless, we are still far from getting a full picture of the effects of magnetic fields in evolved stars. Small samples (compared to the total number of stars at each evolutionary stage) and few global analyses (combining the different techniques aforementioned), are likely the main causes for this. For instance, only two pAGBs are known to show a clear Zeeman splitting profile, and hence allow a definite magnetic field detection at their surface [@Sabin2015a]. This number plummets to zero when it comes to the central stars of PNe [@Leone2011; @Jordan2012; @Leone2014; @Steffen2014].\
In this context the post-AGB OH 231.8+4.2 (a.k.a the Calabash nebula) establishes itself as a key object of study based on the various polarimetric observations that have been performed along the years. Indeed, this is one of the few objects for which numerous maser and dust polarization analysis have been undertaken, allowing for a unique view of its magnetic field.\
OH 231.8+4.2 is a bipolar oxygen-rich binary pAGB and has been the subject of maser and dust continuum polarization analyses targeting both the strength and topology of the magnetic field at different scales. @Etoka2009 mapped the distribution of OH masers at 1667 MHz with MERLIN, at an angular resolution of $\sim$0.2 and found a ring-like pattern with a $\sim 4$ diameter centred on the central star. This corresponds to a radius of $\sim$3300 AU considering that OH 231.8+4.2 is part of the open cluster M46 (or NGC 2347, @Jura1985) for which the recent Gaia DR2 release indicates a mean parallax of 0.61 mas [@Cantat2018], leading to a distance to the nebula of 1639 pc. The polarization analysis (circular and linear) indicated a radial magnetic field aligned with the outflow of the pAGB.\
Later on, @Leal2012 observed the H$_{2}$O 6$_{1,6}$–5$_{2,3}$ rotational transition with the Very Long baseline Array (VLBA) and a synthesized beam size of $\sim$1.7$\times$0.9 mas. Of all the 30 detected maser spots, most of them are distributed in the direction of the collimated outflow and at $\sim$40 AU from the central star (assuming a distance of 1540 pc derived by @Choi2012), only a few show detectable level of polarization. Although it was not possible to estimate the direction of the magnetic field via linear polarization analysis, the presence of circular polarization in two masers spots located at opposite sides of the star indicated a strength of the magnetic field along the line of sight of 44$\pm$7 mG and -29$\pm$21 mG. The extrapolation of the field’s strength to the stellar surface, assuming a toroidal distribution, sets a value of $\sim$2.5 G at 1 R$\star$.\
Finally, @Sabin2014 [@Sabin2015b] mapped the polarized thermal emission of the dust continuum at submillimeter (0.87mm/345 GHz) and millimeter (1.3mm/230 GHz) range with the Submillimeter Array (SMA) and Combined Array for Research in Millimeter-wave Astronomy (CARMA), respectively. Based on the radiative torques (RATs) theory regarding the alignment of non-spherical spinning dust grains with respect to the magnetic field [@Lazarian2007; @Lazarian2008; @Hoang2008; @Lazarian2011], the authors were able to infer the presence of an X-shaped magnetic field distribution centred at the base of each optical outflow and encompassing the red- and blue-shifted $^{12}$CO ([*J*]{}=3$\rightarrow$2) molecular outflows. The 230 GHz polarimetric analysis also indicates an organized magnetic field, which not only mirrors part of the X-shaped structure seen at the submillimeter wavelength, but most importantly it also shows the presence of a new structure aligned with the equatorial plane of the nebula. The combined analysis thus revealed the coexistence of an inner dipole/polar magnetic field configuration and an outer toroidal configuration. In addition, the analysis of the variation of “magnetic vectors” (hereafter, $\overrightarrow{B}$-vectors) position angles (PA) led the authors to also consider the possibility of an helical magnetic field geometry. All the detected features seem to point towards a magnetic collimation and launching mechanism in OH 231.8+4.2.\
Thus, in this article we present new high resolution polarimetric observations (at 0.5) obtained with the Atacama Large Millimeter/submillimeter Array (ALMA). These data provide new insights into the inner dusty region of OH 231.8+4.2 by allowing us to probe the magnetic engine (assuming a dynamo mechanism) closer to its launch-site and complete a multi-scale analysis of the magnetic field in OH 231.8+4.2. We present the ALMA observing procedure in §\[obs\] and the results are shown in §\[cont\]. A comparative analysis with all the previously obtained data is presented in §\[comp\] & §\[comp2\]. Our concluding remarks are shown in §\[final\].
![ALMA dust continuum emission map, at 345 GHz, of OH 231.8+4.2. The global hourglass shape is well defined and the peak emission is coincident with the central star location (CS on the map). The origin (0,0) of the map corresponds to the coordinates $\alpha(J2000)$=$07^{h}\,42^{m}\,16^{s}\,947$, $\delta(J2000)=-14\degr\,42\arcmin\,50\farcs199$. North is on top and east is on the left, the synthesized beam is indicated at the lower left corner.[]{data-label="dust"}](StokesIb-eps-converted-to.pdf){height="7cm"}
![ALMA polarization map (at 345 GHz) showing the electric distribution of vectors. The contours are drawn in steps of \[0.01,0.05,0.1,0.3,0.5,0.7\]$\times$73.6 mJy beam$^{-1}$. The letters A,B and C indicate the three main polarized areas. The peak polarized emission, with a value of 1.13$\pm$0.13 mJy beam$^{-1}$, is found in the region B. The central star location is also indicated (CS on the map). North is on top and east is on the left, the synthesized beam is indicated on the left corner.[]{data-label="Evectors"}](ALMA_pol2-eps-converted-to.pdf){height="7.9cm"}
{height="8.5cm"}
Observations and data reduction {#obs}
===============================
The non-standard polarimetric observations were performed with 40$\times$12m ALMA antennae on 2016-12-28. The observations, under the code 2016.1.00196.S (P.I. Sahai), were done using frequency Band 7 with spectral windows centred at 336.495 GHz, 338.432 GHz, 348.495 GHz and 350.495 GHz, each window having 1.875 GHz bandwidth and 64 channels. The C40-3 configuration was used with a maximum baseline of 460m and the maximum recoverable scale (MRS) was 3.9. Two datasets were taken, with a total time of 58.22 min and 91.55 min respectively (including calibrations such as bandpass, flux, phase and polarization) and OH231.8+4.2 was observed for 30.23 min in both cases. The quasars which were used for calibration purposes are J0750+1231 as the flux and polarization calibrator, J0730-1141 as the phase calibrator, and J0522-3627 as the bandpass calibrator. All three quasars were also used for removal of effects of the atmosphere and pointing calibrations.\
The [*uv* ]{} data reduction was performed using the ALMA pipeline and imaging was done with The Common Astronomy Software Applications [@McMullin2007 CASA v 4.7.2]. The two executions were calibrated separately, then combined before imaging and polarization calibration. To achieve a better sensitivity and unveil more details, two rounds of self calibration were performed. We note that dynamical range of the Stokes [*I*]{} image is at best around 300 (despite the self-calibration and careful removal of the channels with line emission), indicating that all the flux is not recovered. With a maximum recoverable scale (MRS) of 3.9we have resolved out all the smooth structures larger than this size. A Briggs weighting with a robust factor of 0.5 was used to generate continuum maps with a resulting synthesized beam of 0.50$\times$0.38with a position angle PA= 85.63$\degr$. The images were also primary-beam corrected and we note that the leakage is expected to be lower than 2% and the 1$\sigma$ continuum instrumental error is around 0.1%.\
The resulting Stokes [*I*]{} (intensity) image has an rms noise level $\sigma_{I}\approx$ 0.260 mJy beam$^{-1}$ and the Stokes [*Q*]{} and [*U*]{} (polarized intensity) images have rms noise levels $\sigma_{Q}\approx\sigma_{U}\approx$ 0.021 mJy beam$^{-1}$. All the maps generated have a 3$\sigma$ cut.
ALMA dust polarization analysis {#cont}
===============================
Fig.\[dust\] presents the distribution of the dust continuum emission (Stokes [*I*]{}) toward the center of OH 231.8+4.2. The hourglass shape (also seen in Fig.\[Evectors\]) is obvious and extends over $\sim$8$\times$3. The eastern side of the distribution appears to be brighter than the western counterpart. We observe two major bright regions noted B and C in Fig.\[dust\] . The first bright section (B) is confined at the waist of the hourglass structure, with the brighter part on its eastern side, and shows a mean flux of 29$\pm$3 mJy beam$^{-1}$ with a peak intensity (P$_{\it{Int}}$ for [Stokes *I*]{} emission) of 74$\pm$8 mJy beam$^{-1}$. The latter, located at the coordinates $\alpha(2000)$=07$^{h}$42$^{m}$16$^{s}$92, $\delta(2000)=-14\degr\,42\arcmin\,50\farcs076$ is coincident with the position of the central star component QX PuP derived by @Dodson2018 and @SC2018 ($\alpha (J2000)$=07$^{h}$42$^{m}$16$^{s}$915, $\delta (J2000)= -14\degr\,42\arcmin\,50\farcs06$). The other region (C) is located in the south-east with a mean flux of 23$\pm$3 mJy beam$^{-1}$ and a peak of 46$\pm$5 mJy beam$^{-1}$.\
The polarization map, shown in Fig.\[Evectors\], indicates the global distribution of the electric vectors ($\overrightarrow{E}$-vectors) and we emphasize that the dusty regions in the nebula where those vectors are not indicated have a polarized intensity below 3$\sigma$. The polarized emission is not uniform but is rather patchy with three main regions (noted A, B and C in Fig.\[Evectors\]). The peak polarization (P$_{\it{Pol}}$), located at the coordinates 07$^{h}$42$^{m}$16$^{s}$976 -14$\degr$42$\arcmin$49$\farcs$357 inside the area B, has a measured flux of 1.13$\pm$0.3 mJy beam$^{-1}$. As observed in many astronomical objects, including OH 231.8+4.2 (see @Sabin2014 [@Sabin2015b]), P$_{\it{Pol}}$ and P$_{\it{Int}}$ are not spatially coincident.\
Most of the $\overrightarrow{E}$-vectors are relatively well organized within each area and this is particularly seen in regions A and B. We found mean position angles for the electric vectors (EVPA) of about -19$\pm$2 and -42$\pm$3 for the regions A and B respectively. However, while the mean PA seems to indicate a global preferential alignment direction of the dust grains in these two regions, locally we observe a curvature in the distribution of the position angle of the $\overrightarrow{E}$-vectors, which is likely to be of interest for the understanding of the magnetic field. In region C, the orientations of the $\overrightarrow{E}$-vectors do not appear to show a coherent pattern, making their interpretation more difficult. Finally, we note that the peak fractional polarization decreases from the outer to the central regions. Its value ranges from 9.7% and 16.4% in the (outer) regions A and C respectively down to 2.2% in the (inner) region B.
Array CARMA SMA ALMA
----------------------- -------------------- -------------------- -------------------
$\nu$ (GHz) 230 345 345
Beam Size ($\arcsec$) 4.6$\times$2.4 2.5$\times$1.9 0.5$\times$0.4
$\Theta{\arcsec}$ 15.3$\times$8.8 7.3$\times$5.7 8.0$\times$3.0
$D_{pc}$ 0.15$\times$0.08 0.07$\times$0.05 0.08$\times$0.03
$D_{AU}$ 30600$\times$17600 14600$\times$11400 16000$\times$6000
: \[tabcomp\] Summary of the available dust polarimetric observations.
$\Theta{\arcsec}$, $D_{pc}$ and $D_{AU}$ indicate the angular extent and approximate size of the thermal emission in arcsec, parsec and AU respectively, assuming a distance of 1639 pc for the nebula. The two last values are corrected from the 35inclination angle of the source.
{height="10cm"}
{height="9cm"}
![ALMA map of the magnetic field distribution of vectors generated with a zoom on the region ”C”. The beam size is shown on the left lower corner. []{data-label="ZoomC"}](ZoomC-eps-converted-to.pdf){height="8.5cm"}
Multi-scale dust polarization analysis {#comp}
======================================
It is generally assumed that the polarization of ISM thermal dust emission observed at long (e.g. submillimeter and millimeter) wavelengths is due to spinning non-spherical dust grains aligned with respect to the magnetic field. However, two other polarization mechanisms have been proposed, mostly in the study of protoplanetary discs.\
First, mm-sized grains may not be aligned by the magnetic field (due to the slow Larmor precession), but rather by the radiative flux, as shown by @Tazaki2017. In this case, the polarization traces the radiation anisotropy (see figure 12 of @Tazaki2017). The application of this mechanism to our bipolar evolved nebula might produce different results due to the different physical and morphological conditions in OH231.8+4.2 compared to protoplanetary disks. However, based on the general distribution of the polarization vectors indicated by the disc model in the case of radiative polarization, we can make a first guess on whether this process could explain the polarization seen in OH 231.8+4.1. In the case of grain alignment caused by a radiative flux the polarization distribution would more likely be centro-symmetric (or spherical symmetric). The millimeter data obtained with CARMA indicates that only the $\overrightarrow{E}$-vectors located in the upper area, corresponding to the regions A and B in our new ALMA dataset, are consistent with radiative grain alignment. However, the high resolution submillimeter ALMA data (Fig.\[Evectors\]) show an arc-like distribution of the polarization vectors in region A (see their position angles) which deviate from the spherical symmetric pattern around the central star. This implies that radiative grain alignment (or alignment with radiation anisotropy) is not the dominant mechanism for grain alignment here. It is possible that the polarization in the inner region of the nebula (area B, see Fig.\[Evectors\]) is due to the radiation field, since many of the $\overrightarrow{E}$-vectors in this area could be related to a spherical symmetric geometry. However, this pattern is incomplete as it is not seen in the southern part of region B, and the mean percentage polarization is relatively low ( $\simeq$1%), both at 230 GHz (CARMA data) and at 345 GHz (ALMA data). So we disregard radiative alignment as the main polarization process operating in OH 231.8+4.2. One would expect higher percentage at this location, or in other words, a more efficient alignment of the dust grains. Detailed modelling, involving different radiation field intensities, observing wavelengths, and dust grain sizes (@Ohashi2018 [@Tazaki2017]), would be needed to assess if and in which conditions this alignment process would be viable.\
The second polarization mechanism unrelated to magnetic fields is due to dust self-scattering. This process becomes important if the dust grain size is comparable to the wavelength, i.e. if a$_{max}$ $\sim$ $\lambda$/$2\pi$ with a$_{max}$ the maximum grain size [@Kataoka2015] and has been mostly studied at millimeter wavelengths. Hence, a low P% is achieved when a$_{max}$ $>$ $\lambda$/$2\pi$ and vice versa. Fig.\[Percentpol\] indicates the variation of the degree of polarization in the nebula at millimeter and sub-millimeter wavelengths. Although there is a clear gradient, which can be associated with a depolarization effect, P% is generally always $\geq$ 2% in most of the nebula except for the region closer to the central star. The relatively high degree of polarization that we find in OH231.8+4.2, compared to the maximum value typically expected from models of self-scattering, about 2.5% at 345 GHz (a$_{max}$=100 $\mu$m) [@Kataoka2015], suggest that self-scattering is not the dominant polarization factor in OH 231.8+4.2. The polarization process occurring in the inner region is less clear though.\
In addition, the good agreement between the 230 GHz and 345 GHz maps in terms of polarization vectors distribution indicates that there is no wavelength dependence. We further convolved the ALMA Stokes [*Q*]{} and [*U*]{} images to the CARMA resolution and confirmed this trend. As scattering polarization is frequency dependent, it is very unlikely that self-scattering is a main element in the polarization seen in the nebula. It is important to note that the different polarization processes are not mutually exclusive. If, as we have argued above, the radiation flux (via radiation anisotropy alignment) and self-scattering are not principal factors operating in the nebula, magnetic fields are likely responsible for the dust alignment and the observed polarization.\
Hence using the ALMA data and bearing in mind the above caveats, it is therefore possible to proceed with the mapping of the “magnetic architecture” of OH 231.8+4.2. Moreover, combining our new findings with the CARMA and SMA data, we now have multi-scale information on the dust distribution, and by extension on the magnetic field geometry. Table \[tabcomp\] compiles the characteristics of each available dataset and Fig.\[area\] shows the extent of the thermal dust emission for the different arrays as well as the HST H$\alpha$ emission. It is therefore possible to pinpoint the distribution of the thermal dust emission with respect to the ionized emission. Viewed on the plane of the sky, the former is centred on the central star of the nebula and extends at most only over a semi-major axis of $\sim$7. This represents $\sim$2.6 times less than the optical northern lobe and 5.1 times less than the southern one. The thermal dust emission observed with our combined dataset, therefore traces the highest dust density regions of the nebula (see also @Balick2017 and @SC2018). Fig.\[CompB\] presents three distinct behaviours (or distributions) of the magnetic fields in the regions above and below the equatorial plane. They are described in the following sections \[reg1\], \[reg2\] and \[reg3\].
Poloidal or hourglass magnetic distribution {#reg1}
-------------------------------------------
The analysis of the ALMA polarization data shows that the $\overrightarrow{B}$-vectors (EVPA rotated by 90) located in the areas A (with a mean PA=63), B1 (with a mean PA=23) and B3 (with mean a PA=-25) align quite well with the edges of the northern outflow within its opening angle of 67 (with a mean PA= 23 based on the optical HST image presented here). The uncertainty on the polarization angles, assuming a 3$\sigma$ threshold, was estimated to be around 10% using the relation derived by @Serkowski74. We can now integrate all the information to assess the distribution of the magnetic field. With the equator of the nebula (at PA $\sim$113) as a demarcation line, the northern section shows an organized “$\overrightarrow{B}$-vectors” V-pattern along the outflows in the 230 GHz CARMA dataset. As it gets closer to the central region, with an increase in resolution and change of wavelength (870 $\mu$m SMA and ALMA dataset), we observe a remarkable consistency in the general distribution of the vectors. This would suggest that the mechanism responsible for aligning the dust grains, that we assume to be the magnetic field, is strong and coherent enough (the vectors showing a noticeable organized pattern) that it is not disturbed along most of the extent of the northern lobe.\
Another significant point is that from the global “V-shaped” structure observed with CARMA (Fig.\[CompB\]-left), a complete hourglass becomes even clearer down to the ALMA resolution. Indeed, the set of $\overrightarrow{B}$-vectors in the region noted “D” in the SMA observations at PA=-20 (Fig.\[CompB\]-middle) and the ones in the region noted “B2” in the ALMA observations at PA= -18 (Fig.\[CompB\]-right) seems to draw the southern section of the magnetic hourglass. In all the maps, the center of this bipolar pattern appears to be coincident with the approximate location of the central star.\
While the Stokes I flux densities of both the SMA and ALMA maps are roughly comparable with values of $\sim$1.65 Jy and $\sim$1.02 Jy respectively, some differences in the polarized emission distribution are worth mentioning. Hence, the strong submillimeter emission in the region “D” (seen with the SMA) is no longer visible in the ALMA data. In addition we observe other discrepancies in the polarization distribution between both maps. Indeed, the strong central polarization in the region A seems to be missing in the ALMA map and conversely the strong polarization in region B1 (seen with ALMA) is not present in the SMA map. We excluded the ”contamination” of our previous SMA maps by some line polarization and conducted independent ALMA data re-reductions. The reason for the absence of structures seen with the SMA (at lower resolution) in the ALMA map (having a higher resolution) is not clear.
A large scale toroidal field component {#reg2}
--------------------------------------
The southern part of the magnetic hourglass structure observed with ALMA, and even the SMA, appears to be surrounded or embedded within a wide toroidal like pattern, with a size of $\sim$6$\times$7, located at larger scale. This can be seen through the orientation of the $\overrightarrow{B}$-vectors in the CARMA image @Sabin2015b (Fig.\[CompB\]-left). This toroidal pattern totally disappears as we are getting closer to the central star perhaps due to the changes in the properties or geometry of the alignment mechanism. At first glance the magnetized torus, seen at 230 GHz, could be considered as an independent secondary element of the magnetic structure of OH 231.8+4.2 (along with the poloidal structure).\
However, @Sabin2015b noted a coherent variation in the PA of the magnetic vectors observed in the millimeter range, between the northern and southern sections (with respect to the equatorial plane), which would be consistent with a single helical structure (composed of a toroidal and a poloidal sections). This will be discussed below.
The case of the region “C” {#reg3}
--------------------------
While the structures described in §§\[reg1\] and \[reg2\] show organized magnetic vectors and preferential directions, this is not the case for the region noted “C” in Figs. \[Evectors\] and \[CompB\]. A closer look at this area (Fig. \[ZoomC\]) indicates that a coherent pattern seems to exist within each internal clump, however this local organization does not hold when the clumps are taken altogether. The changes in the polarization pattern could be linked to the shocked environment at the base of the southern lobe. @Balick2017 presented a detailed hydrodynamical model of OH 231.8+4.2 where they distinguished both lobes. The region coincident with our area “C” is represented with a faster ejection (400 km s$^{-1}$ against 225 km s$^{-1}$ for the northern lobe), a wider opening angle (50/ 30) and a higher injection mass rate (2.3e$^{-2}$ M$_{\odot}$ yr$^{-1}$ / 4.8e$^{-3}$ M$_{\odot}$ yr$^{-1}$). These conditions would therefore contribute to the explanation for the lack of organization in the distribution of vectors.
{height="8cm"}
Global View of Magnetic Fields in OH 231.1+4.2 {#comp2}
==============================================
We can now use the new thermal dust polarization observations with existing OH maser data by @Etoka2009 and @Leal2012, to trace the magnetic field pattern from $\sim$30600 AU down to $\sim$40 AU from the central star (and by extrapolation to its surface).
Overall magnetic topology {#topology}
-------------------------
The OH masers [@Etoka2009] and dust polarization emission data (SMA, ALMA) indicate the presence of a well organized magnetic field in the polar direction and in the “inner” parts of the nebula (i.e. the regions closer to the central star). The most spatially distant magnetic field lines seen with CARMA point towards the presence of an additional toroidal magnetic pattern. Such a dual distribution of the magnetic field is a strong reminiscence of magnetically driven outflows model [@Huarte2012] and is also supported by the possible presence of an helical field, inferred by the distribution of the polarization vectors (i.e. PA variation) in the 230 GHz polarization map (Fig.\[Tower\]).\
Now that we have established the presence of a multi-scale magnetic field in OH 231.8+4.2 we can postulate that it might have a dynamical role in the shaping of the nebula.\
The polarization vectors seen at millimeter and sub-mm wavelengths could indicate the presence of frozen-in toroidal magnetic field lines which would have subsequently been dragged along with the outflows, but they could also trace the remnant of a dynamo-generated magnetic field which has evolved in time. In the later scheme the field is amplified in the rotating disk/torus or within the binary system. The smaller the disk or the closer the binary separation, the higher the magnetic field intensity, and therefore we would expect a magnetic launching mechanism. MHD launching is very likely the only process that would reproduce the kinetic energies observed in such evolved nebulae [@Garcia2020]. Hence, [@Huarte2012] (and references therein) discussed two magnetic models, which could account for the observed magnetic structure in OH 231.8+4.2: magnetic towers and magneto-centrifugal launch. The main distinction between the two models is the distance up to which the magnetic energy density dominates over the kinetic energy. Based on the magneto-hydrodynamics simulations realised by the authors, magneto-centrifugal launch may dominate below $\simeq$100 R$_{star}$ (inside the Alfvén radius). At larger distances the flow is mostly directed by hydrodynamical motions. Magnetic towers on the contrary can have an effective action at much larger scales. We present in Fig.\[Tower\] a model of a magnetic tower, performed with 3D-MHD numerical simulations (see @Huarte2012), with different spatial layers of magnetic field lines which seems to closely mirror the distribution seen in OH 231.8+4.2. The sketch represents the core of the tower only. Knowing the magnetic field intensity of the stellar engine is therefore a key element to determine first, if the field is strong enough to trigger the magnetic launch and then to discern which magnetic launching mechanism applies. The only available data are the extrapolation given by @Leal2012 indicating a surface magnetic field of $\sim$2.5G. But as this value originates from a single H$_{2}$O maser emission located at $\sim$40 AU it cannot be fully established as representative of the stellar surface. @Sabin2014 deduced that for B $\geqslant$ 0.6 mG, the magnetic pressure ($P_{B}=B^{2}/8\pi$) was dominating over the thermal pressure ($P_{th}=n_{H}kT$) in the central region ($\simeq$ 6in size or 12000 AU) defined by @Bujarrabal2002. This corresponds to the area mapped by ALMA. Assuming a temperature of 100 K for a region much closer to the central star ($\simeq$0.5or 1000 AU in size, see @SC2018), the lower limit rises to 1 mG.\
Based on the distances where the fields were observed and the strength derived, it is unlikely that a dynamo-like activity would still be at work in OH231.8+4.2 or at least that it would have any dynamical effect in terms of launching, but magnetic fields can still be involved in shaping of the outflows.\
While the main morphology would be defined by the magnetic tower, there is still the possibility that internal shocks or other localised hydrodynamical motions could locally break the symmetry inside the nebula. For example, @Balick2017 using purely hydrodynamical modeling, find that one way to explain the asymmetry in the lobes is to assume that they are created by the interaction of a fast, bipolar wind with an ambient circumstellar envelope, with the former having different speeds, mass-injection rates and opening angles in the North and South. Presumably, a full MHD simulation that can reproduce the overall large asymmetry in the lobes would have to incorporate similar asymmetries in the magnetically-launched wind, thus implying an underlying asymmetry in the magnetic field at the base of the outflows. Asymmetries in the overall dust distribution, as well as the unexplained structures that we see in the dust distribution (e.g. region C) could presumably represent regions of locally enhanced density and/or temperature, resulting from cooling and associated instabilities (e.g., see Fig. 2 of @Huarte2012).\
The Calabash nebula is somewhat peculiar because of the large striking difference in the size of its North and South lobes. However its dusty nature makes it an ideal target for the type of investigation we present in this article, which addresses fundamental questions regarding the presence of large scale, organized magnetic fields in pre-planetary nebulae in general, and their role in the formation and shaping of these objects. We observed either toroidal or polar magnetic structures in other sources that have been investigated in the past. OH 231.8+4.2 is the only object with a clear X-shaped morphology (but also the best characterised). Although it is too early, and the data too sparse, for us to comment on the commonality of such geometries, the observations presented in this paper can enable testing of theoretical mechanisms involving magnetic fields for shaping PPNe.
As aforementioned, a complete MHD model (i.e. taking into account the detailed hydrodynamics information of the source) would therefore greatly help in determining the field strength needed to trigger the magnetic structures that are observed and if/how it would affect the dynamics of the outflows.\
Summary and Conclusions {#final}
=======================
We present a polarization analysis of OH 231.8+4.2 with ALMA at the high spatial resolution of 0.5”. The new data allowed us to have a clear picture of the magnetic field geometry at smaller scales. In combination with other interferometric polarization data, we now have new insights into the whole magnetic field structure which might govern or be closely related to the object geometry. The main findings are described below.\
$\bullet$ ALMA revealed well organized $\overrightarrow{E}$-vectors in the bright A and B regions corresponding to the northern outflow and the waist of OH 231.8+4.2. The $\overrightarrow{B}$-vectors are consistent with an hourglass morphology whose origin is coincident with the location of the central star of the nebula. In the southern outflow, the polarization vectors (region C) are globally less aligned perhaps due to a shock dominated environment.\
$\bullet$ The absence of a global spherical symmetric pattern, the high percentage polarization ($\geq$2%) in most of the nebula and the absence of a wavelength dependence in terms of the polarization vector distribution argue against radiative alignment and self scattering as the main mechanisms involved in the dust grain alignment pattern, leaving the magnetic field as the most probable cause for the polarization observed. However the polarization process(es) at work in the inner region is less clear and would deserve further study.\
$\bullet$ The combination of all the polarimetric data gathered for OH 231.8+4.2 allows us to perform a multi-scale analysis. The latter tends to confirm the presence (at all scales) of a magnetically collimated outflow most likely originating from the area hosting the central star of the nebula (and where a dynamo would have been at work). This magnetic hourglass, present in the zones of highest dust density regions, is embedded in a toroidal pattern and the whole structure closely mirrors the architecture of a magnetic tower.\
While we probed the presence of a well organized magnetically collimated structure, the precise measurement of the magnetic field in the central engine is still missing. This piece of information is needed to accurately determine the actual (and past) dynamical effects of the field onto the shaping the nebula. While we could only be seeing the remnant of a fossil field dragged by the outflows, the hypothesis of the occurrence of a MHD launching at some early stage in the history of OH 231.8+4.2, which would still have an impact on the outflows geometry, should not be discarded and has been explored in this article.\
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank the referee for the careful review of this paper. LS would like to thank the ALMA/NAASC Staff (in particular Erica Keller and Sarah Wood) for the support provided at the NA-ARC Headquarters at Charlottesville (US). LS also thanks Guillermo García Segura for the useful discussions about MHD launching as well as M.A.Guerrero for commenting on the paper. LS also acknowledges support from the Fundación Marcos Moshinsky. RS’s contribution to the research described here was carried out at the Jet Propulsion Laboratory (JPL), California Institute of Technology, under a contract with NASA, and was funded in part by NASA through an Astrophysics Data Analysis Program award "An X-Ray and UV Study of a New Class of AGB stars with Actively-Accreting Binary Companions: fuv AGB Stars ? (17-ADAP17-0206) and grant number HST-AR-10681.01-A from the Space Telescope Science Institute (operated by AURA, Inc., under NASA contract NAS 5-26555). MHE acknowledges that this work was completed in part with resources provided by the Research Computing Data Core at the University of Houston. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
| 0 |
---
abstract: 'Three empirical lines of evidence, $({\rm \bf P_{QCD}, pQCD, dA)}$, from RHIC have converged and point to the discovery of a strongly coupled Quark Gluon Plasma. The evidence includes (1) bulk collective elliptic flow and (2) jet quenching and mono-jet production, observed in Au+Au collisions at 200 AGeV, and (3) a critical control experiment using D+Au at 200 AGeV.'
author:
- 'M. Gyulassy'
---
————–
The Theoretical QGP
===================
The Standard Model of strong interactions predicts the existence of a new phase of matter, called a Quark Gluon Plasma (QGP), in which the quark and gluon degrees of freedom normally confined within hadrons are mostly liberated. Lattice QCD calculations show that there is a rapid rise of the entropy density, $\sigma(T)$, of matter when the temperature reaches $T\approx T_c\sim 160$ MeV. Beyond $T_c$ the effective number of degrees of freedom, $n(T)$, saturates near the number of quark and gluon helicity states $n_{QCD}=8_c\times 2_s + \frac{7}{8}\times 3_c\times N_f\times 2_s\times 2_{q\bar{q}}\approx
37$. The entropy density $\sigma(T)=dP/dT=(\epsilon+P)/T\propto n(T)T^3$ approaches the Stefan Boltzmann limit $4P_{SB}(T)/3T$. The transition region is a smooth crossover when dynamical quarks are taken into account, but the width of the transition region remains relatively narrow, $\Delta T_c/T_c \sim 0.1$ [@Fodor:2004nz]-[@Rischke:2003mt].
The rapid rise of the entropy was predicted long before QCD by Hagedorn due to the observed exponential rise of the number of hadron resonances [@Karsch:2003zq]. However, the saturation of the number of degrees of freedom near $n_{QCD}$ is a unique feature of QCD. Even though the entropy density approaches the ideal, weakly interacting plasma limit, lattice calculations of correlators show that the QGP is far from ideal below $3T_c$. The nonideal nature of this strongly coupled QGP is also seen from the deviation of the pressure, $P(T)$, and energy density $\epsilon(T)$ from the Stefan Boltzmann limit as shown in Fig.(\[qgpfig1\]) from [@Fodor:2004nz].
The equation of state of the QGP, $P_{QCD}(T)$, is the bulk thermodynamic property that can be investigated experimentally via “barometric” observables. A measure of its stiffness is given by the speed of sound squared, $c_s^2=dP/d\epsilon=d\log T/d\log\sigma=(3+d\log n/d\log T)^{-1}$ shown in Fig.(\[qgpfig2\]). Note that $c_s^2$ drops rapidly below $1/3$ as the effective number of degrees of freedom drops when $T$ approaches $T_c$. This softening of the QGP equation of state near $T_c$ is a key feature can be looked for in the collective hydrodynamic flow patterns produced when the plasma expands.
![A recent Lattice QCD calculation of the pressure, $P(T)/T^4$, and a measure of the deviation from the ideal Stefan-Boltzmann limit $(\epsilon(T)-3 P(T))/T^4$. Note that the scale on both graphs has not been corrected for finite lattice volume effects: see for discussion. []{data-label="qgpfig1"}](FodorKatz_P_e_mu0.eps){height="0.35\textheight" width="100.00000%"}
![Important features of the QGP equation of state. The speed of sound $c_s^2=d\epsilon/dP$ drops below $1/3$ for $T<2T_c \approx300$ MeV. Right panel shows a current estimate of the location of the tricritical point at finite baryon density []{data-label="qgpfig2"}](SoundSpeed_Fig2b.eps "fig:"){height="0.35\textheight" width="45.00000%"} ![Important features of the QGP equation of state. The speed of sound $c_s^2=d\epsilon/dP$ drops below $1/3$ for $T<2T_c \approx300$ MeV. Right panel shows a current estimate of the location of the tricritical point at finite baryon density []{data-label="qgpfig2"}](FodorKatz_Tmu.eps "fig:"){height="0.45\textheight" width="45.00000%"}
Another distinctive feature of the QGP phase diagram is shown in the right panel in Fig.(\[qgpfig2\]). Recent lattice QCD calculations [@Fodor:2004nz] have begun to converge on numerical evidence that the QGP may have a second tricritical point [@Halasz:1998qr; @Rischke:2003mt] at moderate baryon densities with $\mu_B=3\mu_q\sim 360$ MeV and $T\sim T_c$.
QCD predictions of the QGP phase date back thirty years [@Collins:1974ky] and followed immediately after the discovery of asymptotic freedom of QCD. The experimental strategies to search for new forms of dense matter also date back thirty years when T.D. Lee proposed “vacuum engineering” [@Lee:ma]. It was then also realized by W. Greiner and collaborators [@Hofmann:by; @Stocker:bi; @Stocker:ci] that extended regions of dense nuclear matter can be formed in high energy interactions of heavy nuclei, and that the measurement of collective flow patterns will provide the novel barometric probes of the equation of state of ultra-dense matter. The hunt for the QGP and other phases of nuclear matter has been underway since that time using several generations of higher energy accelerators, BEVALAC, AGS, SPS, and now RHIC, and covering an impressive energy range $\sqrt{s}-2m_N=0.2-200$ AGeV. In three years, LHC is expected to start vacuum engineering at $5500$ AGeV.
The first conclusive evidence for (highly dissipative) collective nuclear flow was seen at the BEVALAC in 1984 [@Reisdorf:1997fx], and at AGS and SPS there after. However, the first conclusive evidence for nearly dissipation free collective flow obeying $P_{QCD}$ had to await RHIC. In these lectures, the discovery of the novel low dissipation elliptic flow pattern at RHIC is highlighted as the first of three lines of evidence for QGP production at RHIC. Together with two other convergent lines of evidence, jet quenching and the critical $D+Au$ null control, I conclude that the QGP has not only been discovered but that a few of its remarkable properties have already been established experimentally.
The Empirical QGP
=================
The discovery of the gedanken QGP phase of matter in the laboratory requires an empirical definition of the minimal number of necessary and sufficient conditions in terms of experimentally accessible observables. My empirical definition is summarized by the following symbolic equation $${\rm \bf QGP= P_{QCD} + pQCD + dA}
\;\; .
\label{qgpdef}$$ Why are three independent lines of evidence needed? The first term, $\underline{\rm \bf P_{QCD}}$, stands for a class of observables that provide information about its bulk thermodynamic equation of state. The equation of state characterizes its [*long wavelength*]{} nonperturbative thermodynamic properties briefly reviewed in section 1.
The second term, , stands for class of observables that provide direct evidence about its [*short wavelength*]{} dynamics predicted by perturbative QCD. The QCD plasma differs qualitative from familiar abelian QED plasmas due to its unique non-Abelian color field dynamics. The radiative energy loss of energetic short wavelength partons was predicted to lead to striking quenching patterns [@Gyulassy:2003mc]-[@Wang:1992xy] of moderate and high $p_T$ hadrons. The high RHIC cm energy of $200$ AGeV insures that $p_T\sim 10-20$ GeV jet production rates are large enough to measure via a wide array of inclusive and correlation observables. These hard partons serve as effective “external” tomographic probes of the the QGP and test its -dynamics. Jets play the analogous role of neutrinos that probe the physics of stellar cores, while hadrons play the role of photons that probe the corona of the fireball.
Below RHIC energies, the [***pQCD***]{} line of evidence could not be fully developed because the jet rates decrease too rapidly with energy. However, even more importantly, at the lower $p_T < 4$ GeV available the effects of initial state nuclear dynamics and the final state hadronic dynamics could not be completely deconvoluted from the final spectra. This is the key point that I will repeatedly emphasize which differentiates the observables at the SPS and RHIC energies. The necessity to test for the same complications at RHIC is what gives rise to the third term in Eq.(\[qgpdef\]).
The third term, denoted by $\underline{\rm \bf dA}$, stands for control experiments that can clearly differentiate between alternative nuclear dependences specific to [*initial state*]{} partonic wavefunctions as well as the production mechanisms. The control differential, ${\rm \bf dA}$, is critical at any energy because the QCD plasma must first be created from pure kinetic energy! There has been no hot QGP in the universe (except in cosmic ray collisions) since the last drop condensed into hadrons about 13 billion years ago. Cold crystalline quark matter may lurk in the cores of neutron stars, but the transient hot QGP must be “materialized” in the lab. The “matter” arises from decoherence of [*virtual*]{} quantum chromo fluctuations in the initial wavefuntions of high energy nuclei.
At ultra-relativistic energies, these virtual fluctuations are frozen out due to time dilation into what has been called a Color Glass Condensate (CGC) [@McLerran:2004fg]. The CGC is the high density generalization of the Bjorken-Feynman dilute parton model. At high field strengths, the non-linear interactions of virtual quantum color fluctuations are predicted to limit the very small Bjorken $x_{BJ}\rightarrow 0$ Fourier components. The saturation property of CGC is related to unitarity constraints and determines the maximal entropy that can be produced in $AA$ at a given $\sqrt{s}$ as also pointed out by EKRT [@Eskola:2002qz].
The ${\rm \bf dA}$ control is needed to characterize to what extent these nonlinear initial state physics effects can be differentiated from effects due to final state interactions in the QGP matter that forms from it. At RHIC, the best experimental handle on the ${\rm \bf dA}$ term happens to be the study of $D+A$ reactions. In such light-heavy ion reactions, the initial state CGC physics can be isolated because the produced QGP, if any, is too tenuous.
Why don’t I add more terms in Eq.(\[qgpdef\])? In fact, each term stands for many independent components, as I elaborate below. For the three required terms in Eq.(1) the published experimental evidence is now overwhelming and conclusive. Four independent experiments have converged to complementary very high quality data sets.
The three terms in Eq.(1) are necessary and sufficient for establishing that a discovery has been made of a uniquely different form of strongly interacting QGP. After discussing the three lines of evidence, I will elaborate on why I believe that direct photons, $J/\psi$, HBT, or other interesting observables do not need to be added to Eq.(1). Those observable provide valuable additional constraints on the [*combined and convoluted*]{} properties of the initial state, the QGP, [*AND*]{} the dense hadronic matter into which it condenses. However, the deconvolution of the initial and hadronic final effects has already proven to be very difficult at SPS energies and will continue to be at RHIC.
To avoid misunderstanding, the discovery of the QGP does not mean that its physical properties are now understood. In fact, it only signals the beginning of a long and well focused direction of research. The history of the neutron star discovery offers an instructive analogy. In 1934 Baade and Zwicky proposed the theoretical existence of neutron stars soon after Chadwick discovered free neutrons. Thirty years later in 1967 Hewish and Bell observed the first few pulsars when suitable radio telescopes could finally be constructed. An amusing anecdote is that they actually agonized for a time about whether LGM (little green men) were sending them encrypted messages from the cosmos. T. Gold in 1968 (as $D+Au$ did at RHIC in 2003) put the debate to rest. Gold proposed that radiative energy loss of a magnetized neutron star would cause a predictable spin down. Later precision measurements confirmed this. Seventy years after its proposal, neutron star research still remains a very active experimental and theoretical direction of physics. Current interest has focused on possible [*color field*]{} super-conductivity [@Alford:1999pb] recently predicted in the very high $\mu_B$ sector of the QCD phase diagram, beyond the boundaries [@Rischke:2003mt] of Fig.2b.
The critical $D+Au$ control experiments in 2003 could have found that the $Au+Au$ QGP observables were strongly distorted by the possible initial CGC state that created it. This would have certainly foiled Eq.(1). The search for the bulk QGP phase of matter would then have had to await higher energies and densities at LHC or for a better understanding of how to deconvolute that initial state physics. The large positive signatures in similar $p+Pb$ control experiments at SPS showed in fact initial effects strongly distort key observables. At SPS the physics of high $p_T$ Cronin enhancement and $p+A\rightarrow J/\psi$ suppression remain the important open problems. In contrast, at RHIC energies, the absence of jet quenching at midrapidity and the “return of the jeti” correlations in ${\bf dA=D+Au}$ provided the check-mate completion of Eq.(1).
As emphasized by McLerran[@McLerran:2004fg] in these proceedings, the $D+Au$ control at RHIC at high rapidities does in fact produce a positive signature for new initial state physics. In those kinematic ranges $x_{BJ}<0.001$, ${\rm {\bf dA}}$ fails as a null control for QGP, but may signal the onset of nonlinear CGC initial conditions. In this lecture, I concentrate on the midrapidity region, $x_{BJ}>0.01$, where Eq.(1) was conclusively satisfied.
${\rm \bf P_{QCD}}$ and Bulk Collective Flow
============================================
The identification of a new form of “bulk matter” requires the observation of novel and uniquely different collective properties from ones seen before. This requirement is the first term in Eq.(1). In heavy ion collisions, the primary observables of bulk collectivity are the radial, azimuthal and longitudinal flow patterns of hundreds or now thousands of produced hadrons. Stocker, Greiner, and collaborators were the first to predict [@Hofmann:by; @Stocker:bi; @Stocker:ci; @Stocker:vf; @Stocker:pg] distinctive “side splash and squeeze-out” collective flow patterns in nuclear collisions. The different types of collective flows are conveniently quantified in terms of the first few azimuthal Fourier components [@Ollitrault:bk], $v_n(y,p_T,N_p,h)$, of centrality selected triple differential inclusive distribution of hadrons, $h$. The centrality or impact parameter range is usually specified by a range of associated multiplicities, from which the average number of participating nucleons, $N_p$, can be deduced. The azimuthal angle of the hadrons are measured relative to a globally determined estimate for the collision reaction plane angle $\Phi(M)$. The “directed” $v_1$ and “elliptic” $v_2$ flow components [@Reisdorf:1997fx; @Ollitrault:bk; @Voloshin:1999gs]-[@Back:2002ft] are readily identified from azimuthal dependence $$\begin{aligned}
\frac{dN_h(N_p)}{dydp_T^2d\phi}
=\frac{dN_h(N_p)}{dydp_T^2} \frac{1}{2\pi}(1 &+& 2 v_1(y,p_T,N_p,h)
\cos\phi \nonumber
\\ &+& 2 v_2(y,p_T,N_p,h) \cos 2 \phi + \cdots ) \;\;.
\label{floweq}\end{aligned}$$ The “radial flow” component, “1” , is identified [@Cheng:2003as] from the hadron mass dependence of the blue shifted transverse momentum spectra $$\frac{dN_h(N_p)}{dydp_T^2}\sim \exp[ -m_h\cosh(\rho_\perp-\beta(y))/T_f] \;\;,$$ where $m_h(\sinh(\rho_\perp),\cosh(\rho_\perp))=(p_\perp,\sqrt{m_h^2+p_\perp^2})$ and $\beta(y)$ is the mean collective transverse flow rapidity at $y$.
![First line of evidence: Bulk collective flow is the barometric signature of QGP production. Left figure combines STAR and PHENIX measurements of the azimuthal elliptic flow ($v_2(p_T)$) of $\pi,K,p,\Lambda$ in Au+Au at 200 AGeV. The predicted hydrodynamic flow pattern from agrees well with observations in the bulk $p_T<1$ GeV domain. Right figure from shows $v_2$ scaled to the initial elliptic spatial anisotropy, $\epsilon$, as a function of the charge particle density per unit transverse area. The bulk hydrodynamic limit is only attained at RHIC.[]{data-label="line1fig"}](Line1fig.eps "fig:"){height="0.35\textheight" width="47.00000%"} ![First line of evidence: Bulk collective flow is the barometric signature of QGP production. Left figure combines STAR and PHENIX measurements of the azimuthal elliptic flow ($v_2(p_T)$) of $\pi,K,p,\Lambda$ in Au+Au at 200 AGeV. The predicted hydrodynamic flow pattern from agrees well with observations in the bulk $p_T<1$ GeV domain. Right figure from shows $v_2$ scaled to the initial elliptic spatial anisotropy, $\epsilon$, as a function of the charge particle density per unit transverse area. The bulk hydrodynamic limit is only attained at RHIC.[]{data-label="line1fig"}](NA49STARv2_hydro.eps "fig:"){height="0.35\textheight" width="50.00000%"}
Figure (\[line1fig\]) shows the striking bulk collectivity elliptic flow signature of QGP formation at RHIC. Unlike at SPS and lower energies, the observed large elliptic deformation ($(1+2 v_2)/(1-2 v_2)\sim 1.5$) of the final transverse momentum distribution agrees for the first time with non-viscous hydrodynamic predictions [@Kolb:2000fh]-[@Hirano:2003pw] at least up to about $p_T\sim 1$ GeV/c. However, the right panel shows that when the local rapidity density per unit area [@Voloshin:1999gs; @Alt:2003ab] drops below the values achieved at RHIC $\sim 30/{\rm fm}^2$, then the elliptic flow (scaled by the initial spatial ellipticity, $\epsilon=\langle (y^2-x^2)/(y^2+x^2)\rangle$) falls below the perfect fluid hydrodynamic predictions. We will discuss in more detail the origin of the large discrepancy at SPS energies in the next section.
The most impressive feature in Fig.(\[line1fig\]) is the agreement of the observed hadron mass dependence of the elliptic flow pattern for all hadron species, $\pi, K, p,\Lambda$, with the hydrodynamic predictions below 1 GeV/c. This is the QGP fingerprint that shows that there is a common bulk collective azimuthally asymmetric flow velocity field, $u^\mu(\tau,r,\phi)$.
The flow velocity and temperature fields of a perfect (non-viscous) fluid obeys the hydrodynamic equations: $$\partial_\mu\left\{[\epsilon_{QCD}(T(x))+P_{QCD}(T(x))]u^\mu(x)u^\nu(x)-g^{\mu\nu}
P_{QCD}(T(x))\right\} = 0 \; ,
\label{hydro}$$ where $T(x)$ is the local temperature field, $P_{QCD}(T)$ is the QGP equation of state, and $\epsilon_{QCD}(T)=(TdP/dT -P)_{QCD}$ is the local proper energy density. The above equations apply in the rapidity window $|y|<1$, where the baryon chemical potential can be neglected. Eq.(\[hydro\]) provides the barometric connection between the observed flow velocity and the sought after ${\rm \bf P_{QCD}}$.
![ Left figure combines STAR and NA49 data and shows that the directed sidewards flow, $v_1(y)$, is correlated over 8 units of rapidity at RHIC. At SPS collectivity is dominated by the overlapping fragmentation regions while at RHIC the nearly identical directed flow of in the fragmentation regions is shifted to $|y|>2$. Right figure shows the pseudo rapidity dependence of elliptic from PHOBOS .[]{data-label="line1figb"}](STARFlow_v1.eps "fig:"){height="0.3\textheight" width="45.00000%"} ![ Left figure combines STAR and NA49 data and shows that the directed sidewards flow, $v_1(y)$, is correlated over 8 units of rapidity at RHIC. At SPS collectivity is dominated by the overlapping fragmentation regions while at RHIC the nearly identical directed flow of in the fragmentation regions is shifted to $|y|>2$. Right figure shows the pseudo rapidity dependence of elliptic from PHOBOS .[]{data-label="line1figb"}](PHOBOS_v2_eta.eps "fig:"){height="0.3\textheight" width="45.00000%"}
The long range nature of collective flow has also been conclusively established by STAR and PHOBOS seen in Fig.(\[line1figb\]). The sidewards flow is anti-correlated over 8 units of rapidity! In addition, its azimuthal orientation was shown to coincide with the azimuthal direction of the largest axis of elliptic deformation at $y=0$. This provides an important test of the overall consistency of the hydrodynamic origin of flow. The rapidity dependence of the elliptic flow in Fig.(\[line1figb\]) also shows the long range nature of bulk collectivity.
Why is $v_2$ more emphasized than $v_1$ or radial flow as a signature of QGP formation? The primary reason is that elliptic flow is generated mainly during the highest density phase of the evolution before the initial geometric spatial asymmetry of the plasma disappears. It comes from the azimuthal dependence of the pressure gradients, which can be studied by varying the centrality of the events [@Ollitrault:bk]. Detailed parton transport [@Molnar:2001ux] and hydrodynamic [@Teaney:2001av] calculations show that most of the $v_2$ at RHIC is produced before 3 fm/c and that elliptic flow is relatively insensitive to the late stage dissipative expansion of the hadronic phase. In contrast, radial flow has been observed at all energies [@Cheng:2003as] and has been shown to be very sensitive to late time “pion wind” radial pressure gradients [@Bass:2000ib], which continue to blow after the QGP condenses into hadronic resonances.
The observation of near ideal $v_2$ fluid collectivity as predicted with the $P_{QCD}$ together with $v_1(y)$ and other consistency checks (${\bf \rm\bf c.c.}$) conclusively establish the first term in Eq.(1) : $${\rm\bf P_{QCD}}= v_2(p_T; \pi,K,p,\Lambda) + v_1(y) + \;{ \rm\bf c.c.}
\; \; .
\label{Pdef2}$$ Preliminary Quark Matter 2004 analysis of $\Xi,\Omega$ flow are consistent with the predicted $v_2(p_T,y=0,M,h)$ and this information is lumped into the ${\bf \rm\bf c.c.}$ terms in Eq.(\[Pdef2\]). Other data which provide consistency checks of the hydrodynamic explanation of collective flow include the observed $\pi,K,p$ radial flow data [@Cheng:2003as] for $p_T<2$ GeV. In addition, predicted statistical thermodynamic distributions [@Braun-Munzinger:2003zd] of final hadron yields agree remarkably well with RHIC data. Had hadro-chemistry failed at RHIC, then a large question mark would have remained about bulk equilibration in the QGP phase.
QGP Precursors at SPS and Dissipative Collectivity
==================================================
It is important to point out, that no detailed 3+1D hydrodynamic calculation [@Hirano:2003hq]-[@Hirano:2003pw] has yet been able to reproduce the rapid decrease of $v_2(|\eta|>1)$ observed by PHOBOS in Fig.(\[line1figb\]). This discrepancy is due, in my opinion, to the onset of [*hadronic*]{} dissipation effects as the comoving density decreases with increasing $y$. From the right panel of Fig.(\[line1fig\]), we see that as a decrease of the local transverse density from midrapidity RHIC conditions leads to an increasing deviation from the perfect fluid limit. The initial density was also observed to decrease at RHIC as $|y|$ increases [@Bearden:2001qq]. Therefore, from SPS data alone, we should have expect deviations from the perfect fluid limit away from the midrapidity region. It would be interesting to superpose the PHOBOS data on top of the NA49 systematics.
To elaborate on this point, Fig.\[spsv2\] shows CERES data [@Agakichiev:2003gg] on $v_2(p_T)$ at SPS energy $\sqrt{s}=17$ AGeV. In agreement with the right panel of Fig.(\[line1fig\]), the CERES data falls well below hydrodynamic predictions. At even lower energies, AGS and BEVALAC, the $v_2$ even becomes [*negative*]{} and this “squeeze out” of plane [@Stocker:ci] is now well understood in terms of non-equilibrium BUU nuclear transport theory [@Stoicea:2004kp; @Danielewicz:2002pu].
![Evidence for dissipative collective flow below RHIC energies. Left: Non equilibrium BUU nuclear transport theory [[@Danielewicz:2002pu; @Stoicea:2004kp]]{} can explain the observed elliptic squeeze-out (negative $v_2$) collectivity below 4 AGeV. Right: CERES[[@Agakichiev:2003gg]]{} data on elliptic flow at SPS is well below hydrodynamic predictions with freeze-out $T_f=120$ MeV required to reproduce the single inclusive radial flow. Early freeze-out with $T_f=160$ MeV, simulating effects of dissipation, is needed to reproduce the data.[]{data-label="spsv2"}](Danielv2.eps "fig:"){width="5cm" height="6cm"} ![Evidence for dissipative collective flow below RHIC energies. Left: Non equilibrium BUU nuclear transport theory [[@Danielewicz:2002pu; @Stoicea:2004kp]]{} can explain the observed elliptic squeeze-out (negative $v_2$) collectivity below 4 AGeV. Right: CERES[[@Agakichiev:2003gg]]{} data on elliptic flow at SPS is well below hydrodynamic predictions with freeze-out $T_f=120$ MeV required to reproduce the single inclusive radial flow. Early freeze-out with $T_f=160$ MeV, simulating effects of dissipation, is needed to reproduce the data.[]{data-label="spsv2"}](CERES_v2fig1.eps "fig:"){width="5cm"}
In order to account for the smallness of $v_2$ at SPS, hydrodynamics has to be frozen out at unphysically high $T_f\approx T_c=160$ MeV. However, the observed radial flow rules out this simple fix.
The discrepancy of $v_2$ and hydro at SPS energies can be traced to the important contribution of the dissipative final [*hadronic*]{} state interactions. The hadronic fluid is far from ideal. In approaches [@Bass:2000ib; @Teaney:2001av] that combine perfect fluid QGP hydrodynamics with non-equilibrium hadronic transport dynamics, the importance of dissipative hadron dynamics at SPS was clearly demonstrated. The problem is that the QGP at lower initial densities condenses on a faster time scale cannot take advantage of the of the spatial asymmetry to generate large $v_2$. The subsequent dissipative hadronic fluid is very inefficient in exploiting spatial asymmetry. A factor of two reduction of the initial QGP density, therefore, leads to a significant systematic bias of the $v_2$ barometer, not only at SPS but also at high $|y|$ at RHIC. Current hadronic transport theory is not yet accurate enough to re-calibrate the barometer away from mid-rapdities at RHIC.
In light of the above discussion, the smallness of dissipative corrections in the central regions of RHIC is even more surprising. At mid-rapidities, the lack of substantial dissipation in the QGP phase is in itself remarkable. Calculations based on parton transport theory [@Molnar:2001ux] predict large deviations from the ideal non-viscous hydrodynamic limit. Instead, the data show that the QGP is almost a perfect fluid. A Navier Stokes analysis [@Teaney:2003pb] is consistent with [@Molnar:2001ux] and shows that the viscosity of the QGP must be about ten times less than expected if the QGP were a weakly interactive Debye screened plasma. This unexpected feature of the QGP must be due to nonperturbative and hence strong coupling physics that persists to at least $3T_c$.
The Minimal Viscosity of the QGP
--------------------------------
One intriguing theoretical possibility being explored in the literature [@Policastro:2002tn] is that the shear viscosity, $\eta$, in the strongly coupled QGP may saturate at a universal super-string bound, $\eta/\sigma=1/4\pi$. This conjectured duality between string theory and QCD may help to explain also the $\sim 20\%$ deviation of $P_{QCD}(T)$ from the ideal Stefan Boltzmann limit. The discovery of nearly perfect fluid flow of [*long wavelength*]{} modes with $p_T<1$ GeV at RHIC is certain to fuel more interest in this direction.
I propose that a simpler physical explanation of the lower bound on viscosity follows from the uncertainty principle, as derived in Eq.(3.3) of Ref. [@Danielewicz:ww]. Standard kinetic theory derivation of shear viscosity leads to $\eta=(\rho\langle p\rangle \lambda)/3$ where $\rho$ is the proper density, $\langle p \rangle$ is the average total momentum, and $\lambda$ is the momentum degradation transport mean free path. The uncertainty principle implies that quanta with average momentum components $\langle p \rangle$ cannot be localized to better than $\Delta x\sim
1/\langle p \rangle$. Therefore the momentum degradation mean free path cannot be defined more accurately than $ \lambda>1/\langle p \rangle$. For an ultra-relativistic system, the entropy density is $\sigma\approx 4 \rho$, therefore $$\frac{\eta}{\sigma}\stackrel{~}{>} \frac{1}{12} \;\;,
\label{etabound}$$ which is within 5% of the string theory bound. It is the consequence of the universality of the Heisenberg uncertainty principle. Surprisingly, the QGP found at RHIC saturates this uncertainty bound and the data clearly rule out the order of magnitude larger predictions based on pQCD [@Danielewicz:ww]. See again [@Molnar:2001ux]. The long wavelength modes in the QGP are as maximally coupled as $\hbar=1$ allows.
It is “shear” good luck that the mid rapidity initial conditions at RHIC are dense enough to essentially eliminate the dilution of elliptic flow due to the imperfect hadron fluid formed after the spatial asymmetry vanishes. (Recall that $(\eta/\sigma)_H\sim (T_c/T)^{1/{c_H}^2}>1$ for $T<T_c$ [@Danielewicz:ww]). At lower energies or higher rapidities this good luck runs out and the mixture of near perfect QGP and imperfect hadronic fluid dynamics reduces the elliptic flow.
pQCD and Jet Quenching
=======================
In addition to the breakdown of perfect fluid collectivity at high rapidity seen in Fig.(\[line1figb\]), Fig.(\[line1fig\]) clearly shows that hydrodynamics also breaks down at very short wavelengths and high transverse momenta, $p_T> 2$ GeV. Instead of continuing to rise with $p_T$, the elliptic asymmetry stops growing and the difference between baryon vs meson $v_2$ reverses sign! Between $2<p_T<5$ GeV the baryon $v_2^B(p_T)$ exceeds the meson $v_2^M(p_T)$ by approximately 3/2. For such short wavelength components of the QGP, local equilibrium simply cannot be maintained due the fundamental asymptotic freedom property of QCD. I return to the baryon dominated transition region $1<p_T<5$ GeV in a later section since this involves interesting but as yet uncertain non-equilibrium non-perturbative processes. In this section I concentrate on the $p_T>2$ GeV meson observables that can be readily understood in terms QGP modified [**pQCD**]{} dynamics [@Gyulassy:2003mc; @Baier:2000mf].
The quantitative study of short wavelength partonic [**pQCD**]{} dynamics focuses on the rare high $p_T$ power law tails that extend far beyond the typical (long wavelength) scales $p< 3 T \sim 1$ GeV of the bulk QGP. The second major discovery at RHIC is that the non-equilibrium power law high $p_T$ jet distributions remain power law like but are strongly quenched [@Adcox:2001jp]-[@Adler:2002ct]. Furthermore, the quenching pattern has a distinct centrality, $p_T$, azimuthal angle, and hadron flavor dependence that can be used to test the underlying dynamics in many independent ways.
![Jet Quenching at RHIC. Left shows the jet quenching pattern of $\pi^0$ discovered by PHENIX at RHIC compared to previous observation of high $p_T$ enhancement at ISR and SPS energies. The nuclear modification factor $R_{AA}= dN_{AA}/T_{AA}(b)d \sigma_{pp}$ measures the deviation of $AA$ spectra from factorized pQCD. Right shows predictions of the $\sqrt{s}$ and $p_T$ dependence from SPS, RHIC, LHC based on the GLV theory of radiative energy loss.[]{data-label="line2fig"}](RAA_RHIC_ISR_SPS_EnterriaKemer.eps){height="0.3\textheight" width="47.00000%"}
Below RHIC energies, the initial state Cronin enhancement of moderately high $p_T$ tails was observed in central $Pb+Pb$ reactions at the SPS. At the ISR a reduced Cronin enhancement in $\alpha+\alpha$ reactions was seen. In contrast, at RHIC a large suppression, by a factor of 4-5, was discovered in central $Au+Au$ that extends beyond 10 GeV for $\pi^0$.
Jet quenching in $A+A$ was proposed in [@Gyulassy:1990bh; @Wang:1992xy] as a way to study the dense matter produced at RHIC energies. As noted before, the pQCD jet production rates finally become large enough to measure yields up to high $p_T > 10$ GeV. Order of magnitude suppression effects were predicted based on simple estimates of induced gluon radiative energy loss. Ordinary, elastic energy loss [@Bjorken:1982tu] was known by that time to be too small to lead to significant attenuation.
As reviewed in [@Gyulassy:2003mc; @Baier:2000mf] refinements in the theory since then have opened the possibility of using the observed jet quenching pattern as a tomographic tool [@TOMO] to probe the parton densities in a QGP. The right panel shows a recent jet tomographic analysis [@Vitev:2002pf] of the PHENIX $\pi^0$ data [@Adcox:2001jp; @Adcox:2002pe] based on the GLV opacity formalism [@Gyulassy:2000er]. Vitev and I concluded from Fig.6b that the initial gluon rapidity density required to account for the observed jet quenching pattern must be $dN_g/dy\sim 1000\pm 200$.
This jet tomographic measure of the initial $dN_g/dy$ is in remarkable agreement with three other independent sources: (1) the initial entropy deduced via the Bjorken formula from the measured multiplicity, (2) the initial condition of the QGP required in hydrodynamics to produce the observed elliptic flow, and (3) the estimate of the maximum gluon rapidity density bound from the CGC gluon saturated initial condition [@Eskola:2002qz].
These four independent measures makes it possible to estimate the maximal initial energy density in central collisions $$\epsilon_0 = \epsilon(\tau \sim 1/p_0) \approx \frac{p_0^2}{\pi R^2}\frac{dN_g}{dy}
\approx 20 \frac{{\rm GeV}}{{\rm fm}^3} \sim 100 \times \epsilon_{A}
\label{eps0}$$ where $p_0\approx Q_{sat}\approx 1.0-1.4$ GeV is the mean transverse momentum of the initial produced gluons from the incident saturated virtual nuclear CGC fields[@McLerran:2004fg; @Eskola:2002qz]. This scale controls the formation time $\hbar/p_0\approx 0.2$ fm/c of the initially out-of-equilibrium (mostly gluonic) QGP. The success of the hydrodynamics requires that local equilibrium be achieved on a fast proper time scale $\tau_{eq}\approx (1-3)/p_0 < 0.6$ fm/c. The temperature at that time is $T(\tau_{eq})\approx (\epsilon_0/(1-3)\times 12)^{1/4} \approx 2 T_c$.
In HIJING model[@ToporPop:2002gf], the mini-jet cutoff is $p_0=2-2.2$ GeV limits the number of mini-jets well below 1000. The inferred opacity of the QGP is much higher and consistent with the CGC and EKRT estimates.
and Single Jet Tomography
--------------------------
In order to illustrate the ideas behind jet tomography, I will simplify the discussion here to a schematic form. See [@Gyulassy:2003mc] for details. The fractional radiative energy loss of a high energy parton in an expanding QGP is proportional to the position weighed line integral over color charge density $\rho(\vec{x}_\perp,\tau)$ $$\Delta E_{GLV}/E \approx C_2 \kappa(E) \int_{0}^{L(\phi)} d\tau\;
\tau\rho(\vec{x}_\perp(\tau),\tau)\;\;,
\label{dee1}$$ where $C_2$ is the color Casimir of the jet parton and $\kappa(E)$ is a slowly varying function of the jet energy [@Gyulassy:2003mc]. The azimuthal angle sensitive escape time, $L(\phi)$, depends on the initial production point and direction of propagation relative to the elliptic flow axis of the QGP [@Gyulassy:2000gk]. For a longitudinal expanding QGP, isentropic perfect fluid flow implies that $\tau \rho(\tau) \approx (1/A_\perp) dN_g/dy$ is fixed by the initial gluon rapidity density. In this case $$\frac{\Delta E(\phi)}{E} \propto C_2 \frac{L(\phi)}{A_\perp} \frac{dN_g}{dy}\propto C_2
N_{part}^{2/3} \frac{L(\phi)}{<L>} \;\;.
\label{dee2}$$ Therefore, gluon jets are expected to lose about $9/4$ more energy than quarks. In addition, since the produced gluon density scales as the number of wounded participating nucleons at a given impact parameter, (\[dee2\]) predicts a particular centrality and azimuthal dependence of the energy loss. Detailed numerical studies show that the actual GLV energy loss can account qualitatively for the saturation of $v_2(p_T>2)$ [@Gyulassy:2000gk], the unexpected $p_T$ independence [@Vitev:2002pf] of the quenching pattern, the centrality dependence [@Adler:2003qi] of the suppression factor [@mgcipanp; @agv], and the rapidity dependence [@Arsene:2003yk] of $R_{AA}(\eta,p_T)$ [@Hirano:2003yp; @agv] for pions and high $p_T > 6$ GeV inclusive charged hadrons at RHIC.
To further illustrate qualitatively how (\[dee2\]) influences the quench pattern, consider a simplified initial jet distribution rate, $d^2N/d^2p_0 = c p_0^{-n}$ where $n\sim 7$. In applications these are of course calculated numerically. After passing through the QGP, the final jet $p_T=p_f=p_0(1-\epsilon)$. The average over fluctuations constrains $\langle \epsilon\rangle
=\Delta E(\phi)/E$. The quenched jet distribution is $d^2N/d^2p_T =(d^2 p_0/d^2p_f)
d^2N/d^2p_0 \approx
(1-\epsilon(\phi))^{n-2} \; c p_f^{-n}$. The calculated hadron inclusive distribution is obtained by folding the quench jet distribution over the fragmentation function $D(z=p_h/p_f,Q^2=p_f^2)$. However, in this illustrative example the fragmentation function dependence drops out, and the nuclear modification factor, $R_{AA}(p_h, \phi,N_{part})=dN_h(\epsilon)/{dN_h(0)}$ reduces to $$R_{AA} %(p_h, \phi,N_{part})=\frac{dn(\epsilon)}{dn(0)}
= \langle (1-\epsilon(\phi))^{n-2}\rangle\approx
\left\langle \left(1- \epsilon_c
\frac{L(\phi)}{\langle L\rangle} \left(\frac{N_{part}}{2A} \right)^{2/3} \right)^{n-2}
\right\rangle
\label{raa}$$ The average over $\epsilon$ takes into account fluctuations of the radiative energy loss. The resulting $R_{AA}$ is independent of $p_T$ in this approximation. This was also found in the detailed numerical work in [@Vitev:2002pf]. In central collisions, $N_{part}\approx 2 A$ and $L(\phi)\approx {\langle L\rangle}$, and the magnitude of quenching is fixed by $\epsilon_c\propto
dN_g/dy$. The centrality and azimuthal dependence for non-central collisions follows without additional calculations.
and Di-Jet Tomography
----------------------
Measurements of near side and away side azimuthal angle correlations of di-jet fragments provide the opportunity to probe the evolution of the QGP color charge density in even more detail. Fig.(\[monojet\]) show the discovery [@Jacobs:2003bx; @Adler:2002tq; @Hardtke:2002ph] of mono-jet production [@Gyulassy:1990bh] in central collisions at RHIC.
![Monojets at RHIC from STAR . Strongly correlated back-to-back di-jet production in $pp$ and peripheral $AuAu$ left side is compared to mono-jet production discovered in central $AuAu$.[]{data-label="monojet"}](JacobsCIPANP_Fig4a.eps "fig:"){height="0.33\textheight" width="47.00000%"} ![Monojets at RHIC from STAR . Strongly correlated back-to-back di-jet production in $pp$ and peripheral $AuAu$ left side is compared to mono-jet production discovered in central $AuAu$.[]{data-label="monojet"}](JacobsCIPANP_Fig4b.eps "fig:"){height="0.33\textheight" width="47.00000%"}
In peripheral collisions, the distribution $dN/ d\Delta \phi$ of the azimuthal distribution of $p_T\sim 2$ GeV hadrons relative to a tagged $p_T\sim 4$ GeV leading jet fragment shows the same near side and away side back-to-back jetty correlations as measured in $p+p$. This is a direct proof that the kinematic range studied tests the physics of pQCD binary parton collision processes. For central collisions, on the other hand, away side jet correlations are almost completely suppressed.
The quantitative measure of the nuclear modification of di-jet correlations in $A+B$ reactions at a given $\sqrt{s}$ is given by a formidable multi-variable function $$C_{AB}(y_1,p_{T1},\phi_1, y_2, p_{T2}, \phi_2; b, \Phi_b, h_1, h_2) \;\;,
\label{cab}$$ where $(y_1,p_{T1},\phi_1)$ is the trigger particle of flavor $h_1$ and the $(y_2,p_{T2},\phi_2)$ is an associated particle of flavor $h_2$ for collisions at an impact parameter $b$ with a collective flow axis, the reaction plane, fixing the azimuthal angle $\Phi_b$. Obviously $C_{AB}$ is a very powerful microscope to study the modification of short wavelength correlations in the strongly interacting QGP.
The published data are as yet limited to $y_1\approx y_2\approx 0$, broad $p_T$ cuts: $p_{T1} > 4$ GeV and $p_T\sim 2$ GeV, two bins of $\phi_1-\phi_2$, and of course averaged over $\Phi_b$. The measured modification of di-jet correlations is obtained by subtracting out the correlations due to bulk elliptic flow via the di-jet measure $$\begin{aligned}
I_{AA}&=& \int_{\Delta_{-}}^{\Delta_{+}} d(\phi_1-\phi_2)
\left\{N(\phi_1-\phi_2) \nonumber \right. \\
&& \left. \quad \quad - N_B(1+ 2v_2(p_1)v_2(p_2)\cos(2(\phi_1-\phi_2))
\right\} \;\;,\end{aligned}$$ where the number of triggered pairs $N(\phi_1-\phi_2)$ is normalized relative to the expected number based on $p+p$ measurements in the same $[\Delta_{-},\Delta_{+}]$ relative azimuthal angle range. Wang [@Wang:2003mm; @Wang:2003aw] has analyzed the centrality, $N_{part}$, dependence of $I_{AA}$ as well as $R_{AA}$ and showed that both can be understood from the same pQCD energy loss formalism. This provides another critical consistency check of jet tomography at RHIC.
Additional preliminary data from STAR presented by K. Filimonov at Quark Matter 2004 showed the first direct evidence that back-to-back dijet quenching has a distinctive dependence on the azimuthal orientation of the jets relative to the reaction plane as expected from the obvious generalization of Eq.(\[raa\]) $$I_{AA}(\Delta \phi= \pi, \Phi_b)
\approx
\langle\{(1- k_b L_1(\vec{r}_0,\hat{p}_{T}))(1- k_b L_2(\vec{r}_0,-\hat{p}_{T}))\}^{n-2}
\rangle
\label{iaa}$$ where $\vec{r}_0$ is the initial transverse production point of the approximately back-to-back dijet moving in directions $\hat{p}_{T}$ relative to the reaction plane $\Phi_b$. Here $k_b$ the effective fractional energy loss per unit length in the QGP produced at impact parameter $b$. The high $p_{T1}$ trigger naturally biases $\vec{r}_0$ to be near the surface and $\hat{p}_T$ is biased toward the outward normal direction. This means that on the average, $L_1 \ll L_2$, and the away side fragments should be strongly suppressed in the most central collisions, while the near side fragment correlations should be similar to that seen in $pp$. However, in non-central minimum biased events, Eq.(\[iaa\]) naturally predicts the away side fragments are less quenched when the trigger hadron lies in the reaction plane than perpendicular to it, as observed by STAR.
The Empirical [**pQCD**]{} Line of Evidence
-------------------------------------------
Single and dijet data for pions above 3 GeV and protons above 6 GeV provide conclusive evidence that the QGP matter is partially opaque to short wavelength probes with a quenching pattern as predicted by the [**pQCD**]{} radiative energy loss: $${\rm \bf pQCD}= R_{AuAu}(p_T,\phi, N_{part})
+ I_{AuAu}(\phi_1-\phi_2; b, \Phi_b) \;\;.
\label{pqcd}$$ With the vastly increased statistics from the current RHIC RUN 4, the tests of consistency of the theory will be further extended to $p_T \sim 20$ GeV, and $I_{AA}$ and $C_{AA}$ will attain ever greater resolving power. In addition, heavy quark tomography [@Dokshitzer:2001zm; @Djordjevic:2003zk; @Batsouli:2002qf] will provide new tests of the theory.
The [**dAu**]{} Control
=======================
Only one year ago [@transdyn03] the interpretation of high $p_T$ suppression was under intense debate because it was not yet clear how much of the quenching was due to initial state saturation (shadowing) of the gluon distributions and how much due to jet quenching discussed in the previous section. There was only one way to find out - eliminate the QGP final state interactions by substituting a Deuterium beam for one of the two heavy nuclei. In fact, it was long ago anticipated [@Wang:1992xy] that such a control test would be needed to isolate the unknown nuclear gluon shadowing contribution to the A+A quench pattern. In addition $D+Au$ was required to test predictions of possible initial state Cronin multiple interactions [@Wang:1998ww; @Wang:1996yf; @Vitev:2003xu; @Qiu:2003vd]. In contrast, one model of CGC [@Kharzeev:2002pc] predicted a 30% suppression in central D+Au.
The data [@Adler:2003ii; @Adams:2003im; @Arsene:2003yk; @Back:2003ns] conclusively rule out large initial shadowing as the cause of the $x_{BJ} > 0.01$ quenching in Au+Au and establish the empirical control analog of Eq.(\[pqcd\]) $${\rm \bf dA}= R_{DAu}(p_T,\phi, N_{part}) + I_{DAu}(\phi_1-\phi_2,b) \;\;.
\label{da}$$
![The [**dA**]{} control: PHENIX $\pi^0$ and STAR $h^{\pm}$ data compare $R_{DAu}$ to $R_{AuAu}$. These and BRAHMS and PHOBOS data prove that jet quenching in $Au+Au$ must be due to final state interactions. Curves for $\pi^0$ show predictions from for $AuAu$ and from $DAu$. The curves for $DAu$ show the interplay between different gluon shadow parameterizations (EKS, none, HIJING) and Cronin enhancement and are similar to predictions in . In lower panel, the unquenching of charged hadrons is also seen in $D+Au$ relative to $Au+Au$ at high $p_T$.[]{data-label="rdaudata"}](phen_da_aa.eps "fig:"){height="0.30\textheight" width="100.00000%"} ![The [**dA**]{} control: PHENIX $\pi^0$ and STAR $h^{\pm}$ data compare $R_{DAu}$ to $R_{AuAu}$. These and BRAHMS and PHOBOS data prove that jet quenching in $Au+Au$ must be due to final state interactions. Curves for $\pi^0$ show predictions from for $AuAu$ and from $DAu$. The curves for $DAu$ show the interplay between different gluon shadow parameterizations (EKS, none, HIJING) and Cronin enhancement and are similar to predictions in . In lower panel, the unquenching of charged hadrons is also seen in $D+Au$ relative to $Au+Au$ at high $p_T$.[]{data-label="rdaudata"}](RAA_dAu_AuAu.eps "fig:"){height="0.30\textheight" width="100.00000%"}
The “Return of the Jeti”
------------------------
The $I_{DAu}$ measurement from STAR [@Adams:2003im] is the check mate!
![The [**dA**]{} “Return of the Jeti”: Dijet fragment azimuthal correlations from STAR in $DAu$ are unquenched relative to the mono jet correlation observed in central ${AuAu}$. []{data-label="jeti"}](dAu_ReturnJediFig4.eps){height="0.35\textheight" width="100.00000%"}
The return of back-to-back jet correlation in $D+Au$ to the level observed in $pp$ is seen in Fig.\[jeti\]. The data rule out CGC gluon fusion models that predict mono-jets [@Zoller:2003zs] correlations in the $x_{BJ}>0.01$ region. These $D+Au$ data support the conclusion [@Wang:2003mm; @Wang:2003aw] that the observed jet quenching in $AuAu$ is due to parton energy loss.
Another independent check of the strikingly different nature of the nuclear matter created in $Au+Au$ versus $D+Au$ is provided by the width of the away side correlation function [@Qiu:2003pm]. The transport properties of cold nuclear matter extracted from the Cronin enhancement effect [@Vitev:2003xu] have only a small effect on the measured dijet acoplanarity. Preliminary PHENIX data presented by J. Rak at Quark Matter 2004 confirm the qualitative similarity of $p+p$ and $D+Au$ for $p_T > 2$ GeV, but at the same time demonstrate a strong quantifiable increase in acoplanarity in central $Au+Au$ consistent with multiple semi-hard scattering [@Gyulassy:2002yv] in a dense QGP.
Conclusions
===========
The three lines of evidence have converged from the four RHIC experiments. The empirical QGP that has been found at RHIC via the combination of Eqs.(\[qgpdef\],\[Pdef2\],\[pqcd\],\[da\]). This QGP is, however, not the weakly interacting [@Collins:1974ky], color-dielectric “wQGP”, that we have searched for. Because of its near perfect fluid long wavelength properties, it must be very strongly coupled at least up to several times $T_c$. Symbolically, we should denote [^1] the empirical QGP at RHIC by “[**sQGP**]{}” to emphasize its special properties. The [**sQGP**]{} is not only a near perfect fluid but it also retains part of its QCD asymptotic freedom character through its highly suppressed, but power law, short wavelength spectrum.
In summary, the [**sQGP**]{} found RHIC was seen through the following three convergent lines of evidence $$\begin{aligned}
{\rm \bf sQGP}&=& {\rm \bf P_{QCD} + pQCD + dA} \nonumber \\
&=& \{v_2(p_T; \pi,K,p,\Lambda) + v_1(y) + \;{ \rm\bf c.c.} \}\nonumber \\
&\;& + \{R_{AuAu}(p_T,\phi, N_{part}) + I_{AuAu}(\phi_1-\phi_2; b, \Phi_b)\} \nonumber \\
&\;& + \{ R_{DAu}(p_T,\phi, N_{part}) + I_{DAu}(\phi_1-\phi_2, b)\}
\;\; .
\label{qed}\end{aligned}$$ Other surprising properties are already known. The anomalous $p_T=2-5$ GeV baryon/meson ratios [@Sorensen:2003kp; @Adler:2003cb] , noted in connection with Fig.3 and also seen indirectly in Fig.8, already point to unexpected novel baryon number physics. Current speculations center around possible gluonic baryon junction dynamics [@Kharzeev:1996sq], and possible multi-quark quark coalescence mechanisms [@Csizmadia:1998vp]. The baryon number transport properties in the sQGP will certainly teach us new physics.
The experimental task of mapping out the novel properties of sQGP has only begun. It is important to concentrate, however, on those observables which are least distorted or “polluted” by uninteresting hadronic final interactions. For example, the severity of the HBT puzzle depends on which hadronic transport model is used. Pure hydrodynamics with late freeze-out times fails badly to reproduce final state soft pion correlations. Hybrid hydro+RQMD hadronic cascade does somewhat better [@Soff:2000eh], but there is at least one transport model [@Lin:2002gc] that reproduced the data. Non-equilibrium hadron resonance transport dynamics are unfortunately still not well enough understood at any energy to allow definitive conclusions to be drawn.
Thermal direct photons have yet to be measured, but it is already known that the pre-equilibrium [@Wang:1998ww] and hadronic final state contributions in the few GeV $p_T$ range will produce large backgrounds on top of the thermal component. These must be deconvoluted if thermal photons are to serve as a sQGP thermometer. However, even the theoretical thermal photon rates are still under debate [@Gelis:2002yw].
The $J/\psi$ suppression discovery at SPS was originally attributed to Debye screening of $c\bar{c}$ in a wQGP paradigm. Recent lattice QCD results now indicate that heavy quark correlations persist perhaps up to $~2T_c$. This is another indication for the strongly coupled nature of [**sQCD**]{}. The suppression of $J/\psi$ in $AA$ is also strongly influenced by initial state and final state hadronic (comover) effects. It will be no easier to deconvolute these competing effects at RHIC. These observables of course need to be measured at RHIC, but one should not expect an easy interpretation.
I believe that the most promising direction of future experiments at RHIC will be precision measurements of short wavelength ($p_T>2$ GeV) correlators $C_{AB}$ illustrated in Eq.(\[cab\]). These are very powerful six dimensional microscopes with four discrete $(h_1,h_2, A, B)$ and two continuously adjustable geometric $(b,\Phi_b)$ experimental knobs in addition to the beam energy $\sqrt{s}=20-200$ AGeV. One of the important correlators will be that of direct photon tagged jets [@Wang:1996yh]. Another will be open charm and possibly bottom quark tomographic probes. The available experimental knobs have hardly been varied yet. Much remains to be done to map out to clarify the properties of the [**sQGP**]{} found at RHIC.
This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of Nuclear Physics, of the U.S. Department of Energy under Contract DE-FG-02-93ER-40764. I also gratefully acknowledge partial support from an Alexander von Humboldt-Stiftung Foundation for continuation of collaborative work at the Institut für Theoretische Physik, Frankfurt.
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[^1]: I thank T.D.Lee for discussions and suggesting the sQGP designation of the matter discovered at RHIC to distinguish it from weakly interacting, Debye screened plasmas, wQGP, which may only exist at temperatures $T\gg T_c$.
| 0 |
---
abstract: 'We present the results of our spectroscopic follow-up program of the X-ray sources detected in the 942 ks exposure of the [*C*]{}handra [*D*]{}eep [*F*]{}ield [*S*]{}outh (CDFS). 288 possible counterparts were observed at the VLT with the FORS1/FORS2 spectrographs for 251 of the 349 Chandra sources (including three additional faint X-ray sources). Spectra and R-band images are shown for all the observed sources and R$-$K colours are given for most of them. Spectroscopic redshifts were obtained for 168 X-ray sources, of which 137 have both reliable optical identification and redshift estimate (including 16 external identifications). The R$ < 24$ observed sample comprises 161 X-ray objects (181 optical counterparts) and 126 of them have unambiguous spectroscopic identification. There are two spikes in the redshift distribution, predominantly populated by type-2 AGN but also type-1 AGN and X-ray normal galaxies: that at $z = 0.734$ is fairly narrow (in redshift space) and comprises two clusters/groups of galaxies centered on extended X-ray sources, the second one at $z = 0.674$ is broader and should trace a sheet-like structure. The type-1 and type-2 populations are clearly separated in X-ray/optical diagnostics involving parameters sensitive to absorption/reddening: X-ray hardness ratio ($HR$), optical/near-IR colour, soft X-ray flux and optical brightness. Nevertheless, these two populations cover similar ranges of hard X-ray luminosity and absolute K magnitude, thus trace similar levels of gravitational accretion. Consequently, we introduce a new classification based solely on X-ray properties, $HR$ and X-ray luminosity, consistent with the unified AGN model. This X-ray classification uncovers a large fraction of optically obscured, X-ray luminous AGNs missed by the classical optical classification. We find a similar number of X-ray type-1 and type-2 QSOs ($L_{\rm X}$(0.5-10 keV)$>10^{44}$ erg s$^{-1}$) at $z > 2$ (13 sources with unambiguous spectroscopic identification); most X-ray type-1 QSOs are bright, R$ \lesssim 24$, whereas most X-ray type-2 QSOs have R$ \gtrsim 24$ which may explain the difference with the CDFN results as few spectroscopic redshifts were obtained for R$ > 24$ CDFN X-ray counterparts. There are X-ray type-1 QSOs down to $z\sim0.5$, but a strong decrease at $z<2$ in the fraction of luminous X-ray type-2 QSOs may indicate a cosmic evolution of the X-ray luminosity function of the type-2 population. An X-ray spectral analysis is required to confirm this possible evolution. The red colour of most X-ray type-2 AGN could be due to dust associated with the X-ray absorbing material and/or a substantial contribution of the host galaxy light. The latter can also be important for some redder X-ray type-1 AGN. There is a large population of EROs (R$-$K$>5$) as X-ray counterparts and their fraction strongly increases with decreasing optical flux, up to 25% for the R$ \geq 24$ sample. They cover the whole range of X-ray hardness ratios, comprise objects of various classes (in particular a high fraction of $z\gtrsim 1$ X-ray absorbed AGNs, but also elliptical and starburst galaxies) and more than half of them should be fairly bright X-ray sources ($L_{\rm X}$(0.5-10 keV)$>10^{42}$ erg s$^{-1}$). Photometric redshifts will be necessary to derive the properties and evolution of the X-ray selected EROs.'
author:
- 'G. P. Szokoly, J. Bergeron, G. Hasinger, I. Lehmann. L. Kewley, V. Mainieri, M. Nonino, P. Rosati, R. Giacconi, R. Gilli, R. Gilmozzi, C. Norman, M. Romaniello E. Schreier, P. Tozzi, J. X. Wang, W. Zheng and A. Zirm'
title: |
The Chandra Deep Field South:\
Optical Spectroscopy I. [^1]
---
INTRODUCTION
============
Deep X-ray surveys indicate that the cosmic X-ray background (XRB) is largely due to accretion onto supermassive black holes, integrated over cosmic time. In the soft (0.5–2 keV) band more than 90% of the XRB flux has been resolved using 1.4 Msec observations with [*ROSAT*]{} [@hasinger1998] and 1-2 Msec Chandra observations [@brandt2001a; @rosati2002; @brandt2002] and 100 ksec observations with XMM-Newton [@hasinger2001]. In the harder (2-10 keV) band a similar fraction of the background has been resolved with the above Chandra and XMM-Newton surveys, reaching source densities of about 4000 deg$^{-2}$. Surveys in the very hard (5-10 keV) band have been pioneered using BeppoSAX, which resolved about 30% of the XRB [@fiore1999]. XMM-Newton and Chandra have now also resolved the majority (60-70%) of the very hard X-ray background.
Optical follow-up programs with 8-10m telescopes have been completed for the [*ROSAT*]{} deep surveys and find predominantly Active Galactic Nuclei (AGN) as counterparts of the faint X-ray source population [@schmidt1998; @zamorani1999; @lehmann2001], mainly X-ray and optically unobscured AGN (type-1 Seyferts and QSOs) and a smaller fraction of obscured AGN (type-2 Seyferts). The X-ray observations have so far been about consistent with population synthesis models based on unified AGN schemes [@comastri1995; @gilli2001], which explain the hard spectrum of the X-ray background by a mixture of X-ray absorbed and unabsorbed AGN, folded with the corresponding luminosity function and its cosmological evolution. According to these models, most AGN spectra are heavily absorbed and about 80% of the light produced by accretion will be absorbed by gas and dust which may reside in nuclear starburst regions that feed the AGN [@fabian1998]. However, these models are far from unique and contain a number of often overlooked assumptions, so their predictive power remains limited until complete samples of spectroscopically classified hard X-ray sources are available. In particular, they require a substantial contribution of high-luminosity absorbed X-ray sources (type-2 QSOs), which so far have only scarcely been detected. The cosmic history of obscuration and its potential dependence on intrinsic source luminosity remain completely unknown. @gilli2001 e.g. assumed a strong evolution of the absorbed/obscured fraction (ratio of type-2/type-1 AGN) from 4:1 in the local universe to much larger fractions (10:1) at high redshifts [see also @fabian1998]. The gas-to-dust ratio in high-redshift, high-luminosity AGN could be completely different from the usually assumed Galactic value due to sputtering of the dust particles in the strong radiation field [@granato1997]. There could thus be objects which are heavily absorbed at X-rays and unobscured at optical wavelengths.
After having understood the basic contributions to the X-ray background, the general interest is now focussing on understanding the physical nature of these sources, the cosmological evolution of their properties, and their role in models of galaxy evolution. We know that basically every galaxy with a spheroidal component in the local universe has a supermassive black hole in its centre [@gebhardt2000]. The luminosity function of X-ray selected AGN shows strong cosmological density evolution at redshifts up to 2, which goes hand in hand with the cosmic star formation history [@miyaji2000]. At the redshift peak of optically selected QSOs, around $z$=2.5, the AGN space density is several hundred times higher than locally, which is in line with the assumption that most galaxies have been active in the past and that the feeding of their black holes is reflected in the X-ray background. While the comoving space density of optically and radio-selected QSOs has been shown to decline significantly beyond a redshift of 2.5 [@schmidt1997; @fan2001; @shaver1996], the statistical quality of X-ray selected high-redshift AGN samples still needs to be improved [@miyaji2000]. The new Chandra and XMM-Newton surveys are now providing strong additional constraints.
Optical identifications of the deepest Chandra and XMM-Newton fields are still in progress, however, a mixture of obscured and unobscured AGN with an increasing fraction of obscuration at lower flux levels seems to be the dominant population in these samples [@fiore2000; @barger2001a; @tozzi2001; @rosati2002; @stern2002]. First examples of the long-sought class of high-redshift, radio-quiet, high-luminosity, heavily obscured active galactic nuclei (type-2 QSO) have also been detected in deep Chandra fields [@norman2002; @stern2002] and in the XMM-Newton deep survey in the Lockman Hole field [@hasinger2002].
In this paper we report on our optical identification work in the Chandra Deep Field South, which, thanks to the efficiency of the VLT, has progressed to the faintest magnitudes among the deepest X-ray surveys.
THE CHANDRA DEEP FIELD SOUTH (CDFS)
===================================
The Chandra X-ray Observatory has performed deep X-ray surveys in a number of fields with ever increasing exposure times [@mushotzky2000; @hornschemeier2000; @giacconi2001; @tozzi2001; @brandt2001a] and has completed a 1 Msec exposure in the Chandra Deep Field South, CDFS [@giacconi2002; @rosati2002] and a 2 Msec exposure in the Hubble Deep Field North, HDF-N [@brandt2002]. The Megasecond dataset of the CDFS is the result of the coaddition of 11 individual Chandra ACIS-I exposures with aimpoints only a few arcsec from each other. The nominal aim point of the CDFS is $\alpha=3:32:28.0$, $\delta=-27:48:30$ (J2000). This field was selected in a patch of the southern sky characterized by a low galactic neutral hydrogen column density, $N_{\rm H}=8\times 10^{19} {{\rm cm}}^{-2}$, and a lack of bright stars [@rosati2002].
OPTICAL IDENTIFICATIONS IN THE CDFS\[optid\]
============================================
Our primary optical imaging was obtained using the FORS1 camera on the ANTU (UT-1 at VLT) telescope. The R band mosaics cover $\sim$360 arcmin$^2$ to depths between 26 and 26.7 (Vega magnitudes). This data does not cover the full CDFS area and must be supplemented with other observations (see Figure \[position\]). The ESO Imaging Survey (EIS) has covered this field to moderate depths (5 $\sigma$ limiting AB magnitudes of 26.0, 25.7, 26.4, 25.4, 25.5 and 24.7 in U$^\prime$, U, B, V, R and I, respectively) in several bands [@arnouts2001; @vandame2001]. The EIS data has been obtained using the Wide Field Imager (WFI) on the ESO-MPG 2.2 meter telescope at La Silla. The positioning of the X-ray sources is better than 0.5[@giacconi2002] and we readily identify likely optical counterparts in 85% of the cases.
Figure \[fxR\] shows the classical correlation between the R-band magnitude and the soft X-ray flux of the CDFS sources. The objects are marked according to their classification (see below). By comparison with the deepest ROSAT survey in the Lockman Hole [@lehmann2001], the Chandra data extend the previous ROSAT range by a factor of about 40 in X-ray flux and to substantially fainter optical magnitudes. While the bulk of the type-1 AGN population still follows the general correlation along a constant $f_X/f_{opt}$ line, the type-2 AGNs cluster at higher X-ray-to-optical flux ratios. There is also a population of normal galaxies emerging at low fluxes (thus discovered in the Chandra and XMM era).
To be consistent with the already published deep ROSAT catalogs [@lehmann2001], we used a modified version of the X-ray to optical flux ratios: $$log_{10}(f_x/f_o)\equiv log_{10}(f_{0.5-2{{\rm keV}}}/f_R)\equiv
log_{10}(f_{0.5-2{{\rm keV}}})+0.4 R + 5.71,$$ where the flux is measured in erg cm$^{-2}$ s$^{-1}$ units in the 0.5-2 keV band and R is in Vega magnitudes. The slight change in the normalization [@maccacaro1988] is motivated by the significantly narrower X-ray energy band used (the original definition was based on the [*Einstein*]{} medium sensitivity survey band, 0.3-3.5 keV), which introduces a factor of 1.77 decrease in the flux for objects with a spectral energy index of $-$1 (classical type-1 AGN) and the use of the R-band instead of V (here we assumed a V$-$R color of 0.22, typical value for galaxies).
To use this new X-ray to optical flux ratio definition for source classification, we also had to convert the canonical ranges [@stocke1991] to our new system. The new ranges for different classes of objects are shown in Table \[fxfo\_classes\]. To calculate the new ranges of the X-ray to optical flux ratios, we assumed typical X-ray spectra for each class and calculated the shift in the X-ray flux due to the narrower energy band: a power law with a photon index of $\Gamma$=1-2.7 for AGN, a power law with a photon index of $\Gamma$=1-2 for BL Lac objects, a Raymond-Smith model with $kT=$2-7 keV, abundances of 0.1-0.6 and redshifts of $z=$0-1. For stars and supernova remnants we used Raymond-Smith models with energies of $kT$=0.5-2 keV, for X-ray binaries powerlaw models with photon index $\Gamma$=1-2. For galaxies, we adopted a somewhat ad hoc shift of 0.1-0.3 in the logarithm of the flux due to the different energy bands. This choice was motivated by examining different models for galaxies (warm and hot plasma mixture, powerlaw like emission from X-ray binaries, typical supernova remnant spectra, etc.).
For the shift in the optical flux (using R-band instead of the canonical V-band) we assumed typical values for each class.
The resulting ranges of the X-ray-to-optical flux ratios are shown in Table \[fxfo\_classes\]. As can be seen from the table, the new X-ray-to-optical flux ratio is not significantly different from the canonical one. The typical ranges are a bit wider, but this just a consequence of [*converting*]{} the ranges instead of directly determining it from large surveys. With our new normalization, we can use the original ranges [@stocke1991] to make an educated guess on the galactic/extragalactic nature of objects.
TARGET SELECTION
================
Target selection was primarily based on our deep VLT/FORS imaging data [@giacconi2002], reaching a depth of R $\sim26.5$. In regions not covered by this VLT/FORS deep imaging, we used somewhat shallower VLT/FORS imaging in the R-band obtained as part of the survey.
Possible optical counterparts of X-ray sources were selected based on the estimated astrometry error of the X-ray object (for a relatively bright point source at zero off-axis angle the astrometry rms error is $\sim 0\farcs5$). We used the automatically generated optical catalog, however, [*every*]{} object was visually inspected for deblending problems and artefacts.
The surface density of our X-ray objects is very well suited to MOS spectroscopy with FORS/VLT. We could fill a large part of the masks with program objects and it was quite rare that we had to choose between multiple optical counterpart candidates within the geometrical constraints of the instrument. As a consequence, our target selection is nearly unbiased. The only selection effect that should be considered was related to objects with multiple counterpart candidates. In these cases we usually selected the object in the appropriate magnitude range for the particular mask, but in general we tried to revisit these objects – unless the first one turned out to be clearly the counterpart.
We also took advantage of the extremely high accuracy of the robotic masks: in some cases, we reconfigured [*some*]{} of the slits between read-outs, without changing the telescope pointing to observe many (brighter) optical counterparts. This way, the integration time on bright objects could be shortened and we could use the remaining time on a different object, while maintaining longer integration times for the faint ones.
During our last two runs (in November and December 2001) we were also using the prefabricated masks (MXU mode – only available for FORS2), as opposed to movable robotic slitlets (MOS mode). For our survey, the only important difference between the two modes is more freedom in the placement of slits in MXU mode. This improved our observing efficiency in the later phase of the survey, where we concentrated on fainter objects (with a higher surface density).
*The Reliability of the Target Selection*
-----------------------------------------
The reliability of X-ray follow-up surveys using optical (or near infrared) spectroscopy hinges on matching the X-ray source to the [*right*]{} optical object. This is primarily done through astrometry. Just how reliable are these identifications? Using deep galaxy number counts [@metcalfe2001], we expect roughly 0.02 galaxies in every square arcsec area that are brighter than R $\sim26$. Considering our best $3\sigma$ astrometry error ($1\farcs5$), we expect $\sim0.15$ [*field*]{} galaxies to fall within our error circle – even in the best, zero off axis angle case. In other words we expect one [*false*]{} candidate for every seventh X-ray object at R $<26$. Considering the roughly 250 X-ray sources we observed, we expect that for at least 35 of them, there [*will be*]{} a completely unrelated faint galaxy, even in an error circle of $1\farcs5$. Fortunately, the X-ray counterpart candidates typically have much brighter magnitudes (see Figure \[magdist\]). At these brighter magnitudes the probability of field galaxy contamination is much lower, so we should only worry about contamination for very faint (R=25-27) objects.
As our astrometric accuracy heavily depends on the signal-to-noise ratio of the object (i.e. objects with low photon counts are centered with lower accuracy) and the off-axis angle of the object (there is a significant degradation of the PSF of increasing off-axis angles), the total area covered by the sum of the error circles is quite large, around 3900 arcsec$^2$. In Figure \[magdist\], we show the magnitude distribution of our selected primary optical counterparts and the expected magnitude distribution of random field galaxies over this area, based on galaxy number counts [@metcalfe2001; @jones1991]. Contamination by random field galaxies becomes a serious problem beyond R $\sim24$ and they start to dominate beyond R $\sim26$, the practical limit of our imaging survey.
Therefore, extra caution is required in making sure that the [*right*]{} optical object is identified as the counterpart. This is not always trivial as the optical spectra do not always show clear signatures of active nuclei (AGNs). In some cases we had to observe every object in the error circle. Fortunately this turns out to be feasible. At R $\sim24$ and fainter, deblending is not a serious challenge (using both automated and visual tests). At brighter magnitudes, where deblending would be near impossible (e.g. detecting a R $\sim25$ X-ray object in the halo of a R $\sim19$ galaxy), the probability of field galaxy contamination is negligible. Stellar contamination is negligible at our high galactic latitude.
It is also important to point out that these estimates of field galaxy contamination are for the probability of finding an unrelated object in our X-ray error circle. We can also ask a technical question: what is the probability of finding a field object on a slit? Taking a $20\arcsec\times2\arcsec$ area (the typical slit length in FORS-1 is around 20$\arcsec$), we expect to find a R $<23$ galaxy in 5% of the slits and we expect statistically a field galaxy with magnitude R $<25$ in every second slit. This means that one has to be extremely careful in the data reduction and do a very careful book keeping in the process.
OBSERVATIONS AND DATA REDUCTION
===============================
Data were obtained during 11 nights in 2000 and 2001. A summary of the observations is presented in Table \[tblobs\]. All observations were using the ‘150I’ grism (150I+17 in FORS-1 and 150I+27 in FORS-2). These grisms provide a pixel scale (dispersed) of 280Å/mm, or roughly 5.5Å/pixel. The nominal resolution of the configuration is $R=\lambda/\Delta\lambda$=230, which corresponds to roughly 20[Å]{} at 5600[Å]{}. The pixel scale of these instruments is 0.2$\arcsec$/pixel, so there is no significant degradation of the resolution due to the finite slit width.
In the initial phase of our survey, we exclusively used low resolution multiobject spectroscopy with varying integration time. This strategy maximizes the number of observed objects and provides a (nearly) full spectral coverage for every exposure. This is clearly a trade off, as we then get a significantly lower S/N spectrum for the individual objects, compared to higher resolution long-slit spectroscopy based on photometric redshifts, but the latter technique was deemed to be prohibitevely expensive in observing time in the initial phase of our project.
As our goal was to observe as many objects as possible, we used non standard order separation filters (either no filter, or the GG-375 filter, which cuts out light bluer than $\sim$3750Å). It was thus possible to cover a very wide spectral range in a single exposure (in the standard configuration the order separation filter that cuts the light blueward of 5900Å, thus the whole spectral range can only be covered in two exposures).
*Data Reduction*
----------------
Data were reduced by our own semi-automatic pipeline built on top of IRAF. In general we followed standard procedures, but had to deviate slightly in several cases to accomodate particularities of the FORS instrument and do a very rigorous book-keeping. In the following sections, we enumerate these changes.
*Bias, Overscan and Trim Correction*
------------------------------------
The FORS CCD’s have in principle 4 read-out modes: high and low gain and one and four amplifier modes. To avoid serious complications, we only used the high gain/one amplifier read-out mode for our spectroscopic observations. This decision resulted in a slightly larger overhead, but this was deemed negligible considering our long integration times, compared to the challenges posed by reducing a 4 amplifier read-out mode spectroscopic observations, where we would have to calibrate the gain of each amplifier very accurately (so we do not introduce artifical features in the spectra).
A sufficient number of full frame bias exposures were taken during each run (typically around 20 per run). These were individually overscan corrected and trimmed. The resulting (bias) frames were averaged with suspect pixels (too high or too low values) filtered out to generate the master bias frame. In each case we verified that the bias frame does not change significantly from night to night within a run.
A slight complication was posed by our spectrophotometric standard observations. These frames were also using one amplifier/high gain, but (to save some time) only 500 rows were read out (centered on the standard star). Since ESO does not provide an under/overscan region for windowed frames, we took a sufficient number (typically 10) of bias frames in this configuration. Naturally (lacking under/overscan region) these frames were [*not*]{} overscan corrected, nor trimmed. Instead, they were averaged to create a master bias frame, which [*did*]{} include the artificially introduced bias level. We checked the individual frames and confirmed that the variation of this artifical bias level is negligible for these very high S/N frames.
After creating the full and windowed bias frames, all object and calibration (flat and arc) frames were overscan corrected and trimmed (except the windowed frames) and zero subtracted.
At this point we applied a shift in the dispersion direction, based on the slit position, to bring (very crudely; within 10 pixels or 50Å) the observations on a similar wavelength scale. We also inserted gaps in the spatial direction between the neighbouring slits to reduce the risk of contamination between slits. These two steps are purely practical, but make bookkeeping significantly easier.
*Flatfielding*
--------------
In this processing step, we had to tackle three main issues:
The first one is an inherent complication in the FORS instruments. Due to the mechanical construction of the robotic slit masks and the location of the flat-field lamps, flat-field exposures show higher flux levels in a few rows at the upper or lower edge of the slit. To correct for this effect, there are two sets of flat-field lamps in the instrument. We took a sufficient number of flat-field exposures using both sets of lamps. We generated merged flats independently for each lamp set and generated the final flat-field frame by taking the smaller pixel value in the two frames. As the reflections from the two lamp sets do not overlap, this feature can be fully removed.
The second issue is a consequence of our unusual observing strategy. In some cases (due to geometric constraints imposed by the robotic slit masks) we could not target very faint objects with a particular slit, but we had several bright candidates available. In these cases, to maximize efficiency, we reconfigured these slits between readouts so that all bright candidates were observed, while faint objects targeted with other slits were observed with a longer integration time. Due to the extremely high mechanical stability of the FORS instruments, this strategy is very safe. As the sensitivity variation between pixels is potentially color dependent, we decided to generate flat-field frames for each mask. This may not be the optimal strategy since for the slits that are in the same position in two masks, we could use more exposures, thus to create a more accurate flat-field. This alternative strategy would be too complex and the resulting data quality improvement is very marginal, consequently we decided [*against*]{} it.
The last major issue is due to the extremely wide spectral coverage used. As our intention was to correct [*only*]{} for the pixel-to-pixel sensitivity variations, we had to generate a normalization image (a combination of the flat-field lamp spectrum and the overall quantum efficiency of the system as a function of wavelength and spatial position). For high resolution (and smaller wavelength coverage) observations, this is often achieved by collapsing in the spatial direction and fitting a function in the dispersion direction. Unfortunately, this technique proved to be impractical for us. The main problem was that we were unable to find an ansatz function that could reproduce the very sharp cutoffs at both ends (due to either the order separation filter or the natural cut-off of the CCD detector) without introducing artifical structure on intermediate scale. An additional complication was that the internal flat-field lamps did not illuminate the slits homogenously – there is a slight gradient in the spatial direction. Therefore, after a slight smoothing of the flat-field exposures, we created the normalization image by a linear or (for very long slits) a second order polynomial fit in the spatial direction. Each flat-field exposure was divided by this normalization frame, thus creating a ‘true’ flat-field frame, which only contains pixel-to-pixel sensitivity variations. In regions, where the signal was too low, the flat-field was artificially set to one (to avoid the introduction of too high photon noise).
After these steps, the individual, normalized flat-field frames were merged, eliminating the effect of light reflection on the slit edges. Both science and wavelength calibration frames for a given mask were divided by the resulting master flat-field frame.
*Sky Subtraction*
-----------------
The sky background was estimated in each column by a linear fit (for longer slits) or just calculating the average (shorter slits) in each column of each slit, rejecting too high pixels (i.e. the targeted object) and subtracting the result. It is important to note that we did [*not*]{} correct for the very slight curvature of the dispersed spectra on the CCD in this step. With the FORS instruments, this strategy works quite well (as opposed to LRIS on the Keck telescope). Significant sky residuals are only present around the very bright, narrow sky lines – where sky subtration is doomed anyway due to pixel saturation.
This procedure works only for our typical faint objects. Extremely bright objects can illuminate the whole slit, thus making correct sky subtraction impossible. Fortunately, in those (very few) cases identification was still possible due to the extremely high object signal.
*Fringe Removal*
----------------
In some cases (especially in MXU masks), we could take advantage of our dithering strategy to reduce further the effect of fringing and the sky residuals. As neither the fringe pattern nor the sky residuals are significantly affected by the small (spatial) offsets of the telescope, we could, in some cases (with sufficient number of exposures in a given mask) exclude (most of) the object signal and create a fringe/sky residual template for each slit. Subtracting this from the frames resulted in an improved signal-to-noise ratio for the object spectra. Depending on the seeing conditions and the dithering offsets used, not all object signal was perfectly removed, thus the extracted spectra significantly underestimated the real spectra. As our primary goal was object identification, not spectrophotometry, this was an acceptable trade-off.
*Coadding the Frames*
---------------------
After sky subtraction, all the slits were visually inspected to verify that the object is indeed in the ’good’ region of the slit. This step was necessary since the applied small spatial offsets between the science exposures can result in objects falling too close to the slit edge (MOS blade corners are round, thus the slit is not usable there) or falling completely outside the slit.
After this visual screening, the [*spatial*]{} offset between different exposures of the same object was caculated based on the [*world coordinate system*]{} (WCS) information stored in the frame headers. The individual exposures were coadded (including the rejection of suspicious pixels or cosmic ray hits) after applying these spatial shifts. We only shifted the frames in the spatial direction and only by integer number of pixels. As the objects were sufficiently well sampled (the pixel scale was significantly smaller than the seeing), this step resulted in nearly negligible bluring of the spectra, while preserving the statistical properties of the exposures.
*Extraction*
------------
Even though for [*sky subtraction*]{} we could safely ignore the slight curvature of dispersed spectra on the CCD, for the extraction of the object signal this is no longer possible. Therefore, we estimated the object position on the detector by collapsing at least 30 columns (more for really faint objects) in the dispersion direction and measuring the object center in the resulting profile. The object position was fitted typically with a second order polynomial as a function of column (wavelength).
Then an aperture [*width*]{} was visually determined. Except in special cases (e.g. blended objects), our aim was to include most of the object signal without adding too much sky (to maximize the signal-to-noise ratio). A one dimensional spectrum was obtained using the ‘optimal extraction’ method of IRAF. This procedure calculates a weighted average in each column, based on both the estimated object profile and photon statistics.
*Wavelength Calibration*
------------------------
Wavelength calibration was based on (daytime) arc calibration frames, using four arc lamps (a He, a HgCd and two Ar lamps) which provide a sufficient number of lines over the whole spectral range used (3889–9924Å).
The exact same aperture that was used for the science object was used for the arc frames. The resulting lines were first identified automatically. These identifications were then visually verified, and quite often significantly improved. In most cases, around 20 lines were located in the 3889–9924[Å]{} range, and fitted by a forth order polinomial, with a typical rms accuracy of 1[Å]{} or better. This accuracy is close to that expected from the nominal resolution of the instrument in our configuration. The object spectra were then wavelength calibrated and subsequently rebinned to obtain spectra with a linear wavelength scale.
We also examined the stability of the instrument by repeating the daytime calibrations on different days. No noticeable change was detected. In addition, we verified the wavelength calibration by checking the position of narrow skylines in science exposures – no discrepancy was found within our error estimates.
The wavelength calibration may not be accurate in the range outside the two extreme arc lines identified. As the FORS instruments use a grism, we had to resort to high order polynomial fits, which become unreliable when extrapolating the wavelength solution. In most cases, this is not an important issue, but there were a few unfortunate cases where major object features fell into unreliable regions (typically if the spectra were cut short on the blue side due to the position of the slit).
FLUX CALIBRATION
================
In this step, we [*nearly*]{} followed standard practices. The only significant necessary change arrised due to our choice of a non standard instrument configuration, namely not using the [*right*]{} order separation filters. Consequently, we nearly doubled our efficiency (taking only one exposure per object), but we then had to correct for second order diffracted light.
It is important to point out that we have to correct for this effect both for the science and the spectrophotometric standard observations.
*Second Order Diffraction\[2ndtheory\]*
---------------------------------------
The first step was to determine the nature of the second order contamination. As this contamination affects the red part of the spectrum, where there are typically many lines present already from first order diffraction (in both arc and sky exposures), we used a special set of calibration frames: a 1.3 arcsec wide long slit, the standard arc lamps (He, HgCd, Ar) and a set of (dispersed) exposures through all available broad-band filters (U, B, V, R and I) as well as without any filter. Using the exposure without filter, we established the [*first order*]{} wavelength solution of this configuration. We verified that the use of the Bessel filters does [*not*]{} introduce any noticable shift in this solution. We were then able to identify the second order lines in the exposures taken through the broad-band filters. These identifications are shown in Table \[2ndorder\].
The comparison between the apparent fluxes in first and second order diffracted lines indicates that second order diffraction [*can*]{} be very strong, as much as 30% of the first order strength (especially in the blue part of the spectrum). This effect is made seriously worse by the quantum efficiency of the CCD. The overall quantum efficiency of the system peaks around 6000[Å]{} and declines relatively rapidly (see also Section \[specphotstd\]). Because of this, a relatively weak second order contamination may become the dominant signal beyond 9000Å – due to to the much higher sensitivity of the pixels to these photons. For blue objetcs (for example spectral photometric standards), this problem is even worse: second order contamination can already start at 7000Å(due to the high UV flux of the object) and can contribute over 30% to the observed flux.
To correct for the second order diffraction, we first had to model it. Based on the identified arc lines, we adopted a second order wavelength solution in a linear form: $$\lambda=2.106\Lambda-723\hbox{\AA},$$ where $\Lambda$ is the real wavelength of the feature and $\lambda$ is its apparent wavelength position observed in second order.
It should be noted that, as opposed to [*grating*]{} spectrographs where the coefficient is practically two and the shift is very small, a few times 10Å [@gutierrez1994], FORS, which uses [*grisms*]{}, is significantly different, which makes the detection of second order diffraction harder, unless one takes the appropriate calibration data sets.
We assumed that the measured signal, $d(\lambda)$ is $$d(\lambda)=f(\lambda)s(\lambda)+c(\Lambda)f(\Lambda),$$ where $s(\lambda)$ is the overall quantum efficiency of the system, $\Lambda$ is the real, physical wavelength of features detected at $\lambda$ in second order, $c(\Lambda)$ is the strength of the second order folded into the sensitivity function at $\Lambda$ and $f(\lambda)$ is the real spectrum of the object.
Since we can not derive [*two*]{} functions ($s(\lambda)$ and $c(\Lambda)$) from a single measurement, either we determine the sensitivity, $s(\lambda)$, independently (e.g. observing a standard star through different order separation filters) or we use two independent measurements from observing two different standard stars. As the first option implies the introduction of an additional optical element (the filter), which can affect the strength of the second order diffracted signal (in fact comparing the flux ratios of the 3650.1Å arc line in Table \[2ndorder\] is a strong indication for this to be the case), we selected the latter aproach. We observed two standards with very different spectral shapes, LTT-3218 [@hamuy1992; @hamuy1994 a relatively red DA6 white dwarf] and HD49798 [@turnshek1990 a blue sdO6 subdwarf].
Given two different standards ($f_1(\lambda)$ and $f_2(\lambda)$), but identical instrument setups ($c(\Lambda)$ and $s(\lambda)$), we can write
$$c(\Lambda)={d_1(\lambda)-f_1(\lambda)s(\lambda)\over f_1(\Lambda)}=
{d_2(\lambda)-f_2(\lambda)s(\lambda)\over f_2(\Lambda)},$$
which we can solve for $c(\Lambda)$: $$c(\Lambda)={d_1(\lambda)\over f_1(\Lambda)}\left(1-
{\alpha(\Lambda)-\beta(\lambda)\over\alpha(\Lambda)-\alpha(\lambda)}\right),$$ where $\alpha(\lambda)=f_2(\lambda)/f_1(\lambda)$ (known a’priori) and $\beta(\lambda)=d_2(\lambda)/d_1(\lambda)$ (known from observations).
The derived $c(\Lambda)$ indicates that the contamination is completely negligible up to 6300Å. Up to 7500Å it is somewhat stronger, but typically still negligible as this wavelength corresponds to up to 3800Å in first order, where the CCD is very inefficient. Between 7500 and 9000Å, the effect is strong (12% to 3% of the first order instrumental flux shows up in second order). Depending on the object type (whether it is a blue or red object) and the wavelength of interest (as this result should be folded with the system quantum efficiency, which is increasing with wavelength for second order diffracted photons and decreasing for first order diffracted photons in this range) this may or may not be a strong effect – this decision should be made for each observing program. Beyond that this effect could not be estimated as one of our standards is measured only to 8700Å. As in this range the CCD QE is dropping very sharply, while the first order QE is very high, second order contamination [*must*]{} become the dominant source of signal at some wavelength.
The effect of the second order diffraction is demonstrated in Figure \[secondorderplot\], using a blue spectrophotometric standard star, Feige110 [@hamuy1992; @hamuy1994; @oke1990].
*Overall Quantum Efficiency of the System\[specphotstd\]*
---------------------------------------------------------
The overall throughput of the system was determined using a set of spectro photometric standard stars: LTT-377, LTT-3218, LTT-7379, LTT-7987, LTT-9239 [@hamuy1992; @hamuy1994], HD49798 [@turnshek1990] and Feige110 [@hamuy1992; @hamuy1994; @oke1990]. These objects were observed repeatedly over a wide range of airmasses during each run, using a simulated very wide ($\sim 5\arcsec$) long slit, using the robotic mask facility of the FORS instruments (combining 3 slits).
The observations were reduced nearly the same way as science observations. Second order diffraction was removed as outlined in Section \[2ndtheory\]. The measured spectra (in instrumental units) were compared to the published physical spectra – excluding regions with sharp features in the objects and sharp telluric absorption features in the atmosphere. A smooth sensitivity curve was fitted to the data points. As we were using about the whole wavelength range of the CCD, this fit was done in four parts: 3000-4000[Å]{} (or 3500-4000[Å]{} if the OG375 order separation filter was used during the run), where the throughput raises very sharply and only a 10% accuracy was achieved, between 4000-5000[Å]{}, where the throughput is still rising fast (about 4% accuracy), between 5000-8000[Å]{} (2% accuracy) and finally between 8000-9500[Å]{}, where the accuracy drops again to around 10%. These four data sets were used to construct the sensitivity curve for each observing run and each configuration.
As the Paranal Observatory does not have sufficient data collected to measure an accurate spectroscopic extinction curve, we used the curve published by the Cerro Tololo Inter-American Observatory (CTIO), after verifying, using our standard observations, that the curve is very close to that estimated for Paranal.
This calibration was applied to all measured program spectra. We also corrected for atmospheric extinction using the CTIO extinction curve. The accuracy of the resulting flux calibrated (but not absolute calibrated – see below) spectra is mainly constrained by the signal-to-noise ratio of the object signal in the 4000-8500[Å]{} range (the inaccuracy due to the sensitivity function is negligible in this range). Outside this range the flux calibration can introduce significant structures as the throughput of the whole system drops very rapidly, thus even very small inaccuracies in the wavelength calibration of the object spectra result in significant over or underestimation of the physical spectra. It is also important to point out that no attempt was made to correct for telluric absorption: For the vast majority of our program objects, telluric absorption completely eliminates the signal (between 7600-7630[Å]{}, 7170-7350[Å]{} and 6868-6890[Å]{}), thus a correction is not practical. For the few brighter objects, this correction would have been possible, but was deemed unnecessary for identification of the sources.
*Absolute Calibration – Estimating the Slitloss\[abscal\]*
----------------------------------------------------------
The purpose of our observations was to identify as many X-ray sources as possible with the telescope time available. Therefore, no effort was made to collect data necessary for [*absolute*]{} calibration of the spectra measured. Even though we did [*not*]{} use an elaborate program designed for spectrophotometry, we can still estimate the accuracy of our derived fluxes.
The most important effect to be considered is slit-loss. To maximize the S/N of the data, we tried to match the slit width to the expected seeing of the observations. Therefore, a significant fraction of the light from the object was excluded, but this was more than balanced by the large reduction in the sky background, and thus the increase in the S/N of the source.
We can easily estimate the effect of slit-losses from the high accuracy broad-band photometry. Using the flux calibrated spectra, we can directly calculate the AB-magnitude of the object in any filter: $$m_{AB}=-2.5 \log{\int d(\log \nu) f_\nu S_\nu\over \int d(\log \nu)S_\nu}-48.60,
\label{ABmag}$$ where $f_\nu$ is the energy flux per unit frequency, $S_\nu$ is the overall throughput of the system (telescope and instrument) in arbitrary units.
The first step is to select the filter curve to use, $S_\nu$. In practice, no system can replicate the canonical Cousins-Johnson filter curves [*exactly*]{}. Even a perfect filter response curve would be distorted by the non flatness of the CCD detector. In many cases a slight deviation from this filter response curve is acceptable, assuming that the spectrum of the object is smooth and the slope is not very different from the slope of Vega. Unfortunately, these assumptions do not hold for most of our objects as a significant fraction of the flux is in very sharp features. Therefore, we have to use the effective filter curve of [*our*]{} system used to derive the broad-band magnitudes. Fortunately, the Bessel filter set used by ESO is a sufficiently good approximation of the Cousins-Johnson filters and the quantum efficiency curve of the FORS detectors being relatively flat, this correction would only amount to a fraction of a percent and can be safely ignored. Therefore, we used the published ESO filter response curves folded over the quantum efficiency of the detectors as system throughput, $S_\nu$. To convert to Vega magnitudes, we calculated the AB magnitude of Vega from its spectra [@fukugita1996]. The resulting slitlosses are presented in Table \[tblobs\] for point sources for each mask. We also checked if the slitloss depends on wavelength (by comparing different broad-band magnitudes) but found no significant effect.
*Reddening Correction\[deredden\]*
----------------------------------
To calculate the effect of reddening due to our Galaxy on the spectra, we used the 100$\mu m$ maps [@schlegel1998]. In the direction of the CDFS, $l=223.5\degr$, $b=-54.4\degr$, the color excess is $E$(B$-$V)$\approx0.008$. Assuming the canonical value, $R_{\rm V}=3.1$ for the ratio of extinction in the V-band to the color excess [@cardelli1989], the extintion is $A$(U)$\approx 0.04$ and $A$(I)$\approx 0.01$. As this extinction is heavily dependent on the choice of $R_{\rm V}$, this correction was [*not*]{} applied to the data, introducing an artificial tilt in all spectra on the order of a few percent.
The AGN line strengths were not corrected for absorption lines from the [*host*]{} galaxy [@ho1993] as the S/N of our faint spectroscopic sample is too low. Consequently, the very few optical identifications based on line ratios are possibly affected by this effect.
Correcting for reddening by the AGN host galaxy would require to estimate the extinction using the X-ray spectral information. This correction is deferred to a later paper concentrating on X-ray spectral analysis of the CDFS sources based on the Chandra and XMM-Newton data.
REDSHIFT DETERMINATION AND THE SPECTROSCOPIC SAMPLE\[zestimate\]
================================================================
*Redshift and Luminosity Determination*
---------------------------------------
The first step toward the classification of the spectroscopically observed sources was their [*redshift*]{} determination. In the vast majority of the cases this was done through the identification of prominent features, typically the 4000[Å]{} break and the Ca[ii]{} H and K absorption, Balmer lines or emission lines (e.g. Ly-$\alpha$, C[iv]{}, C[iii]{}\], Mg[ii]{}, \[O[ii]{}\], etc.). In case of prominent emission lines, the wavelength ratio of the line centers was used to identify these features. In cases of single emission line objects with no additional feature, this line was usually identified as either \[O[ii]{}\] or Ly-$\alpha$, depending on the continuum spectral shape. Naturally, these are not secure classifications, the quoted redshifts should only be used as an educated guess to optimize follow-up observations.
The redshift identifications are summarized in Table \[tblspec\]. The ‘No’ column refers to our internal id of the [*X-ray*]{} source – this is the [*unique detection ID (XID)*]{} in the published catalog [@giacconi2002]. In cases of multiple counterparts, a letter is appended to this number to distinguish between the optical candidates. Extended X-ray objects are marked with a star. When an object was observed repeatedly, multiple entries are given in the table. Altogether 249 X-ray sources were observed, of which one point source belongs to the small additional sample given in Table \[newsrc\], and 15 are extended X-ray sources. In 17 cases, the slit was centered on the X-ray position for the search of strong, narrow emission lines although no counterpart was detected in the R-band.
The [*mask*]{} column is our internal name used to identify the set of observations used for individual objects. Multiple mask names indicate that during the observations some slits were reconfigured, but the slit used for the object was identical during the set of observations. The relevant observing conditions and configuration can be found in Table \[tblobs\] (exposure time, slit width, seeing, etc.).
The two position columns (right ascension and declination) give the coordinates of the [*optical*]{} object, [*not*]{} those of the X-ray source. Astrometry is based on the USNO [@usno] reference frame, just like the X-ray positions in @giacconi2002 and the astrometric accuracy is better than 0.2.
Whenever available, we also provide broadband optical information, an R-band magnitude and R$-$K color (both in Vega magnitudes). If no R-band magnitude is given, it implies that our FORS imaging data is not deep enough to measure the magnitude of the object (all program objects are covered by the FORS R-band survey). The lack of R$-$K color can be due to our limited near-infrared coverage ([*NA*]{} entries).
Assuming (throughout this paper) an $\Omega_m=0.3$, $\Omega_\Lambda=0.7$ universe and $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ [@spergel2003] the total X-ray intrinsic luminosity of the object, $L_{\rm X}$, in erg s$^{-1}$, is [@carroll1992] $$L_{\rm X}(f_{\rm X},z)=4\pi f_X
\left(
{c(1+z)\over H_0\sqrt{\vert\Omega_k\vert}}{{\rm sinn}}\left(\sqrt{\vert\Omega_k\vert}
\int\limits_0^z \left(
(1+\zeta)^2(1+\Omega_m\zeta)-\zeta(2+\zeta)\Omega_\Lambda
\right)^{-1/2}d\zeta\right)\right)^2$$ where $f_{X,tot}$ is the [*observed*]{} X-ray flux in the 0.5-10 keV band. The ${{\rm sinn}}(x)$ function is $\sin(x)$ for $\Omega_k<0$, $\sinh(x)$ for $\Omega_k<0$ and simply $x$ for $\Omega_k=0$, where $\Omega_k\equiv 1-\Omega_m-\Omega_\Lambda$. In case of $\Omega_k=0$, the two $\sqrt{\vert\Omega_k\vert}$ terms disappear. This flux being the [*observed*]{} X-ray flux, it should be interpreted as a lower limit for the [*intrinsic*]{} X-ray flux of the object due to potentially strong obscuration of the source.
If we assume that $\Omega_k=0$ (i.e. $\Omega_m+\Omega_\Lambda\equiv1$), we can rewrite this equation as $$L_{\rm X}(f_{\rm X},z)=4\pi f_X
\left(
{c(1+z)\over H_0}\left(
\int\limits_0^z \left(
(1+\zeta)^3\Omega_m+\Omega_\Lambda
\right)^{-1/2}d\zeta\right)\right)^2$$
Introducing $x=(1+\zeta)(\Omega_m/\Omega_\Lambda)^{1/3}$, this simplifies to $$L_{\rm X}(f_{\rm X},z)=
{4\pi f_X c^2(1+z)^2\over H_0^2\Omega_m^{2/3}\Omega_\Lambda^{1/3}}\left(
\int\limits_{(\Omega_m/\Omega_\Lambda)^{1/3}}^{(1+z)(\Omega_m/\Omega_\Lambda)^{1/3}}
{dx\over\sqrt{x^3+1}}
\right)^2$$
The above integral can be evaluated in terms of incomplete elliptical integrals of the first kind. We also give below an analytical fit for flat cosmologies [@pen1999] with $\Omega_m=0.3$: $$d_L={c(1+z)\over H_0}\left(3.308-3.651
\left(0.207 +
0.446(1 + z) +
0.757(1 + z)^2 -
0.204(1 + z)^3 +
(1 + z)^4
\right)^{-1/8}\right)$$
In Figure \[lxcomp\] we show the correction to the calculated X-ray luminosity for slightly different cosmologies. We also show the difference between luminosities calculated using current cosmological parameters and the (now obsolete) $\Omega_m=1$, $\Omega_\Lambda=0$, $H_0=50$ km s$^{-1}$ Mpc$^{-1}$ cosmology, $$L_{\rm X}=4\pi f_{X,tot}\left({2 c \over
H_0}\left(1+z-\sqrt{1+z}\right)\right)^2
\approx
1.72\times 10^{58} {{\rm cm}}^2 f_{X,tot}\left(1+z-\sqrt{1+z}\right)^2,$$
The $HR$ column contains the already published [@giacconi2002] hardness ratios for each object, $HR=(H-S)/(H+S)$, where $H$ and $S$ are the net count rates in the hard (2-10 keV) and soft (0.5-2 keV) band, respectively. It is important to point out that the hardness ratio is defined in [*instrument*]{} counts (for Chandra ACIS-I), thus for different X-ray telescopes or instruments, it should be converted using their specific energy conversion factors.
The $z$ column gives our best redshift estimate. The selected low spectral resolution leads to an uncertainty in the redshift determination of $\pm0.005$. For broad emission line objects the uncertainty is significantly higher. The quoted redshift value [*always*]{} refers to the particular observation of the object, thus, there can be slight discrepancies between observations of the same object or missing redshift values for some masks.
*The Optical and X-ray Classification*
--------------------------------------
The classical/optical and X-ray classifications of the objects are discussed in details in Section \[classes\] and given in Table \[tblspec\].
Based on purely the optical spectra, we define the following [*optical*]{} object classes:
- [*BLAGN*]{}: Objects with emission lines broader than 2000 km s$^{-1}$. This classification implies an optical type-1 AGN or QSO, as discussed in Section \[classes\].
- [*HEX*]{}: Object with unresolved emission lines [*and*]{} exhibiting high ionization lines or emission line ratios indicating AGN activity. These objects are dominanly optical type-2 AGNs or QSOs, but in a few cases the optical type-1/2 distinction is not possible based on the data.
- [*LEX*]{}: Objects with unresolved emission lines consistent with an H[ii]{} region-type spectra. These objects would be classified as normal galaxies based on the optical data alone as the presence of the AGN can not be established.
- [*ABS*]{}: a typical galaxy spectrum showing only absorption lines.
- [*star*]{}: a stellar spectrum.
Our main classification is solely based on the observed X-ray properties ($L_{\rm X}$ and $HR$) of the sources and is summarized below. The type-1 AGN/QSO are soft X-ray sources, while the type-2 AGN/QSO are hard, absorbed X-ray sources [for the relationship between hardness ratio and absorption see e.g. @mainieri2002]. The AGN and QSO classes cover different ranges of X-ray luminosities.
- [*QSO-1*]{}: $L_{\rm X}$(0.5-10 keV)$\geq10^{44}~{{\rm erg}}~{{\rm s}}^{-1}$ and $HR\leq-0.2$.
- [*AGN-1*]{}: $10^{42}\leq L_{\rm X}$(0.5-10 keV)$<10^{44}~{{\rm erg}}~{{\rm s}}^{-1}$ and $HR\leq-0.2$.
- [*QSO-2*]{}: $L_{\rm X}$(0.5-10 keV)$\geq10^{44}~{{\rm erg}}~{{\rm s}}^{-1}$ and $HR>-0.2$.
- [*AGN-2*]{}: $10^{41}\leq L_{\rm X}$(0.5-10 keV)$<10^{44}~{{\rm erg}}~{{\rm s}}^{-1}$ (lower limit smaller than for the AGN-1 population to account for substantial absorption) and $HR>-0.2$.
- [*gal*]{}: $L_{\rm X}$(0.5-10 keV)$<10^{42}~{{\rm erg}}~{{\rm s}}^{-1}$ and $HR<-0.2$.
- [*star*]{}: this class is defined from the optical spectra and/or proper motions.
Throughout the paper we call X-ray type-1/2 AGNs and QSOs together X-ray type-1/2 [*objects*]{}.
*The Spectroscopic Sample*
--------------------------
The [*quality*]{} flag, $Q$, indicates the reliablity of the redshift determination. $Q=2.0$ indicates a reliable redshift determination, a value of 0.0 indicates no success. $Q=1.0$ indicates that we clearly detect [*some*]{} feature (typically a single narrow emission line) in the spectrum that cannot be identified securely. In a few cases, $Q=0.5$ is used when there is a hint of some spectral feature. This quality flag [*only*]{} refers to the reliability of the spectroscopic classification. The identification of the X-ray sources is unambiguous for single counterparts in the X-ray error circles, and for cases with reliable redshift identification we then use a $Q=2.0+$ quality flag. X-ray sources with multiple counterparts are discussed below. The $Q=2.0+$ objects define our spectroscopically identified X-ray sample.
Finally, the [*comments*]{} column contains additional information relevant to the particular observation. The most common ones are a limited wavelength coverage (the full wavelength range is not available due to the positioning of the slit) and the detection of high ionization lines. We also include information necessary to apply the stricy Seyfert definition [@khachikian1974] to optically classify our objects.
Among the X-ray sources with multiple optical counterparts, the identification is considered as highly reliable in the following 13 cases: X-ray type-1 QSO/AGN (XID 30, 101), X-ray type-2 QSO/AGN (XID 56, 201, 263), and interacting galaxy pairs (XID 98, 138, 580). Four of the remaining X-ray sources (XID 553, 567, 582, 620) are the brightest objects in the X-ray error circles, only detected in the soft band and at moderate redshift with H[ii]{} region-type spectra, thus most likely the X-ray counterparts. The last source (XID 189) is detected in the hard band only, has R$-$K $>$ 5 and is the brigthest, best centered counterpart (the additional counterparts are very faint, R $>$ 25). These objects are included in the spectroscopically identified X-ray sample.
As the CDFS has been observed by various teams and covered by wide surveys (2dF and Tycho), we could include in our sample additional spectroscopic redshifts. Four sources (XID 39, 95, 103, 116) were in fact already published in the CDFS 130 ksec paper [@giacconi2001]. Eight objects (XID 33, 38, 149, 171, 204, 526, 563, 600) are covered by the K20 survey [@daddi2003 Cimatti, private communications] and three (XID 90, 92 and 647) by the COMBO-17 survey (Wolf, private communication). Three sources have optically bright, low $z$ counterparts in the 2dFGRS [@colless2001]. One ot them, XID 84 (TGS243Z005), has a soft X-ray spectrum and appears to be a normal X-ray galaxy. The other two sources, XID 247 (TGS243Z011) and XID 514 (TGS243Z010), have hard spectra, luminosities $L_{\rm X}$(0.5-10 keV) $\sim10^{40}~{{\rm erg}}~{{\rm s}}^{-1}$ and are off-centred within the parent galaxy: these properties are similar to those of the brigther, ultraluminous compact X-ray sources (ULXs) detected in nearby spiral galaxies [@makishima2000]. Finally, one source (XID 549), optically very bright, is identified with an object (TYC 6453-888-1) from the Tycho Reference Catalog [@hog1998], that has a clear proper motion (7.6$\pm$2.1 and 15.7$\pm$1.7 mas/yr in right ascension and declination, respectively) and is thus a star in the Milkyway.
The spectroscopically identified X-ray sample comprises 137 sources of which 15 are fainter than R = 24.0. Among the brighter objects, there are seven extended X-ray sources at $ 0.6 < z < 1$ [XID 132, 138, 249, 560, 566, 594, 645: @giacconi2002]. In addition, there are 24 X-ray sources with secure identifications but only tentative redshifts ($Q=0.5$ and 1.0 cases).
FINDING CHARTS AND SPECTRA
==========================
Figure \[spectra\] gives the finding charts and [*all*]{} the spectra obtained for each of our program objects.
Finding charts are $20\arcsec\times20\arcsec$ in size, centered on the X-ray position with its $2\sigma$ position error circle. Individual contrast levels are chosen in each case to give as much information as possible. If multiple optical counterparts are present in (or around) the error circle, they are marked and labeled. The underlying optical images are our $R$-band FORS images.
Next to the finding charts, we show the associated spectra. For cases of repeated observations, all the data are shown. In the plots, we mark the features used for the redshift determination.
In cases of marginal line detections, we also examined the sky-subtracted coadded two-dimensional frames to confirm/infirm the presence of the feature. These images are not included in this paper, but are available through our web-site, <http://www.mpe.mpg.de/CDFS>.
FIELD SAMPLE\[fieldsample\]
===========================
During our survey, we also collected a large number of field object spectra. These were objects either accidentally covered by some of our slits, or observed in slits that could not be placed on X-ray counterpart candidates due to geometrical constraints, or stars used to align the MXU masks. Consequently, this sample is not representative of the field population.
The results of these observations are summarized in Table \[tblfield\]. We give the object position, mask name, $R$-band magnitude (when available), redshift, redshift quality flag, very crude classification and optional comments relevant to the observation. The full dataset (spectra and finding charts) is available on our web site (<http://www.mpe.mpg.de/CDFS>).
X-RAY VERSUS OPTICAL CLASSIFICATION {#classes}
===================================
Seyfert galaxies [@seyfert1943] were originally defined [@khachikian1974] as a recognizable galaxy (on [*Sky Survey*]{} prints) that have broad ($>500$ km s$^{-1}$) emission lines arising in a bright, semi-stellar nucleus. Seyfert galaxies were subdivided into class 1 and class 2, depending on the width of the Balmer lines, compared to that of forbidden lines. For line widths $200<FWHM<500$ km s$^{-1}$, additional criteria were applied, based on emission line ratios [e.g. @osterbrock1989] to establish the Seyfert nature of a galaxy. With the emergence of the unified AGN model [@antonucci1985], it is now widely accepted that these two Seyfert classes are not distinct, but form a continuous distribution between the two extremes and a large number of intermediate classes were introduced since the original definition. Applying these classical definitions poses very serious problems for the study of faint X-ray sources.
At high redshifts, the Balmer lines are no longer in the optical range: the two strongest Balmer lines, H$\alpha$ and H$\beta$, are redwards of 8500[Å]{} for $z>0.3$ and 0.75, respectively. This problem was overcome by extending the original definition to [*permitted*]{} lines in the UV range from Mg[ii]{}$\lambda\lambda$2796,2803 to Ly$\alpha$ [see e.g. @schmidt1998], thus allowing an optical classification of objects up to $z\sim6.5$. Another difficulty stems from the fact that most of the objects associated with faint X-ray sources are at intermediate redshifts and, thus, are comparable in size to the seeing achievable with ground-based optical telescopes. As a consequence, we can only study the [*integrated*]{} emission from these objects, as opposed to [*nuclear*]{} emission from local Seyfert galaxies. Consequently, the nuclear emission can be ‘hidden’ in the stellar light coming from the host galaxy. The study of local Seyfert galaxies confirms that about 60% of the local Seyfert type-2 galaxies would [*not*]{} be classified as Seyfert-2, if only the [*total*]{} emission were available [@moran2002].
Moreover, an obvious challenge in applying the classical Seyfert definition for faint objects is merely to recognize that they are AGNs. The main optical classes introduced in this paper for extragalactic sources are: 1) BLAGN – $FWHM$(permitted lines)$>2000$ km s$^{-1}$, 2) HEX – unresolved emission features but presence of high excitation lines not found in H[ii]{} regions (e.g. \[Ne[v]{}\]$\lambda$3425, He[ii]{}$\lambda$1640), suggesting AGNs of the optical type-2 class, 3) LEX – H[ii]{} region-type spectrum, 4) ABS – typical galaxy absorption line spectrum. For the 130 extragalactic X-ray sources with secure redshift identification, there are 32 BLAGN, 24 HEX, 54 LEX and 21 ABS sources, thus 57% LEX+ABS objects. But among the latter (optically dull), it should be noted that there is a large number of luminous X-ray sources.
To overcome the limitations of the classical/optical definition of AGN, we follow the unified AGN model introduced by @antonucci1985 and classify an object as an AGN [*if it has (nuclear) emission stronger than expected from stellar processes in normal galaxies*]{}. This emission is likely to be produced by strong accretion onto supermassive objects, most probably black holes. A clear signature of the presence of this accretion is a high X-ray luminosity.
An X-ray classification requires first to introduce a conservative lower limit on the (unabsorbed) absolute X-ray luminosity of AGNs. Local, well studied starburst galaxies have X-ray luminosities in the 0.5-10 keV band typically below 10$^{42}$ erg s$^{-1}$ [@rosati2002; @alexander2002]. Thermal haloes of galaxies and intragroup/cluster gas can have higher X-ray luminosities but, in Chandra data, they are spatially resolved and detected only in the soft band thus, at intermediate redshifts, they become fainter than $10^{42}$ erg s$^{-1}$ in the 0.5-10 keV band. Accordingly, objects with $L_{\rm X}$(0.5-10 keV) $\geq 10^{42}$ erg s$^{-1}$, should be classified AGNs. There are 20\[20\] HEX, 31\[53\] LEX and 9\[12\] ABS (excluding XID 645 which only shows extended X-ray emission) high $L_{\rm X}$ sources with secure\[secure+tentative\] redshift identification respectively. Thus the optical classification completely fails to identify as AGN 42% (LEX+ABS fraction) of the luminous X-ray sources (96), or altogether 54% if we include the tentative redshift identifications (120). In these objects, optical extinction of the nuclear component by dust can be very high, and/or the host galaxy can outshine the central AGN [@lehmann2000; @lehmann2001].
The X-ray luminosity is also used to separate the sources of the AGN class, $10^{42} \leq L_{\rm X}$(0.5-10 keV) $< 10^{44}$ erg s$^{-1}$, from those of the QSO class, $L_{\rm X}$(0.5-10 keV) $\geq 10^{44}$ erg s$^{-1}$.
Secondly, following the unified AGN model, we can also define two AGN classes by using the hardness ratio, a parameter sensitive to X-ray absorption which can be measured even for faint objects. In Figure \[HR\_z\_sim\], we give the expected hardness ratios for AGNs with power law X-ray spectra, selecting a photon index $\Gamma$=2 and different absorption levels. Unabsorbed sources have $HR\approx-0.5$, independent of $z$. This is indeed the case for all the BLAGNs; their hardness ratios are in the range $-1.0\leq HR \leq-0.2$, except for one BAL QSO. The scatter is easily explained by introducing different slopes for the X-ray spectra, together with statistical errors associated with low number counts in the X-ray bands. The harder spectra ($HR>-0.2$) are fully consistent with [*absorbed*]{} power law spectra. Significant intrinsic absorption, $10^{21.5} <$ $N_{\rm H}$ $\lesssim 10^{23.5}$ cm$^{-2}$, has indeed already been found for the type-2 AGN population [@mainieri2002; @barger2002]. Figure \[HR\_z\_sim\] shows that, assuming $\Gamma=2$, intrinsic absorption ($HR>-0.2$) can be detected up to $z$ = 0.25, 2.1 and 3.9 for $N_{\rm H}=10^{22}, 10^{23}$ and $3\times10^{23}$ cm$^{-2}$, respectively. Thus the hardness ratio can be used to separate the unabsorbed sources, X-ray type-1: $HR\leq-0.2$, from the absorbed ones, X-ray type-2: $HR>-0.2$. Indeed, in the Chandra and XMM-Newton deep surveys, most of the harder X-ray sources are optical type-2 AGN with an increasing fraction of absorption at decreasing X-ray flux [@barger2001a; @barger2001b; @hasinger2001; @rosati2002; @mainieri2002]. Among this class of objects, there are a few bright type-2 QSOs but the majority of the sources are type-2 AGNs at $z \lesssim 1$ [see e.g. @hasinger2002]. It should be noted that an X-ray classification based on the hardness ratio might be misleading for some high-redshift objects: an increasing absorption makes the sources harder, while a higher redshift makes them softer. Consequently, some high-redshift absorbed/type-2 sources may be mistakenly identified as type-1, but not the other way around.
A consistent X-ray classification should use the intrinsic luminosity. As mentioned above, hard sources ($HR>-0.2$) at $z\sim0.25$ have absorbing column densities $N_{\rm
H}\geq10^{22}$ cm$^{-2}$ and thus their de-absorbed flux in the observed 0.5-10 keV band is at least 5 times larger than the observed flux. Consequently, [*hard*]{} objects ($HR>-0.2$) with lower luminosities (($10^{41} < L_{\rm X}$(0.5-10 keV) $<
10^{42}$ erg s$^{-1}$) can be classified as X-ray type-2 AGN. Four additional objects (XID 55, 525, 538, 598) are thus classified as low $L_{\rm X}$ X-ray type-2 AGNs.
In Figure \[hrlx\], we show the hardness ratio versus the observed X-ray luminosity for all the sources with secure redshift, for both the optical classification (left panel) and the X-ray one (right panel). No source with a very high X-ray luminosity is present in this diagram: this is, at least in part, a selection effect of pencil beam surveys. We now compare the optical and the X-ray classifications.
- Of the 32 BLAGNs in our sample, all are X-ray type-1 objects, except the BAL QSO (XID 62, $HR=-0.07$) which is an X-ray type-2 QSO.
- Among the HEX population (24 objects), there are 16 X-ray type-2 AGNs/QSOs (including one low $L_{\rm X}$ source: XID 55) for which X-ray absorption is indeed associated with optical obscuration. There are eight X-ray type-1 AGNs/QSOs or galaxies, of which four at $z\geq1.6$ (XID 31, 117, 563, 901) with permitted emission lines no broader than $\sim1500$ km s$^{-1}$. These four sources may be partly absorbed ($N_{\rm H}=10^{22}$-$10^{23}$ cm$^{-2}$), thus misclassified as X-ray type-1: the presence of probable X-ray absorption should be confirmed by X-ray spectral analysis, whenever possible. In the spectrum of XID 34a, we do not detect permitted lines. The remaining three HEX objects are X-ray galaxies (XID 98a, 175b, 580a), two being members of interacting pairs, and all have $HR=-1.0$. They could be either low $L_{\rm X}$ type-1 AGN or, in the case of the interacting pairs, shocks might be at the origin of the \[Ne[v]{}\] emission.\
There is a high fraction, 42%, of $z>2$ sources among the HEX class as compared to 25% in the BLAGN class.
- The LEX population comprises 54 sources with secure redhift identification, of which 9 and 24 X-ray type-1 and type-2 (including two low $L_{\rm X}$ sources: XID 525, 538) AGNs/QSOs, respectively. The optical classification thus fails to identify as AGN 61% of this population. Among the remaining sources, there are 21 X-ray galaxies including one ULX (XID 247) at $z=0.038$ with a hard spectrum, $HR=0.31$ (see Section \[zestimate\]). In the LEX class, there are only two high luminosity sources of the X-ray QSO class and no objects at high redshift ($z>1.5$). A few X-ray type-2 AGNs might be of the HEX class but, due to the low S/N ($<5$ per resolution element) of their optical spectra, high excitation lines could be below the detection threshold.\
For most X-ray type-2 AGNs, both the broad (BLR) and narrow (NLR) emission line regions could be obscured by dust absorption. Alternatively, the obscuring region may fully cover the central UV source and the BLR, preventing photo-ionization of external regions thus the existence of a NLR. The AGN nature of all the X-ray type-1 sources ($0.53\leq z\leq 1.03$) is difficult to ascertain from the optical data alone as the H$\alpha$ line is outside the observing range for redshift higher than 0.4. For six of them, the expected Mg[ii]{} emission line is within the observed range (i.e. $0.4<z<2.2$) and away from strong sky lines, but the S/N is not high enough to detect a weak broad line; in one source (XID 138), a broad Mg[ii]{} emission line may be present, although at a low significance level. For comparison, Mg[ii]{} is often seen in absorption or with a P Cygni profile in star-forming galaxies [@kinney1993].
- There are 21 sources in the ABS class of which one and 9 X-ray type-1 and type-2 (including one low $L_{\rm X}$ source: XID 598) AGNs. Thus 48% of the AGN population in the ABS class is missed by the optical classification. All the sources of the X-ray AGN class are at $z<1.2$. The 11 X-ray galaxies are all at $z<0.8$ and have soft spectra ($HR<-0.7$) except one object (XID 514), a ULX at $z=0.103$ with $HR=-0.14$ (see Section \[zestimate\]).
The comparision of the two classification schemes are summarised in Table \[comptab\].
The proposed X-ray classification is more successful than the classical/optical one in revealing the presence of black hole activity, whatever the amount of dust obscuration from the central and/or external parts of the nuclear region. Thus, we use this classification throughout the paper unless otherwise stated. For comparison, we also give the optical classification in Table \[tblspec\]. The latter may be more appropriate in studies that aim to extrapolate the classical Seyfert definition to faint AGNs.
It should be noted that using the X-ray classification is mandatory to properly identify the X-ray normal galaxies among the LEX+ABS optical class. This population provides another means to derive the star formation history of the universe, in addition to the methods using radio or optical data.
CLUSTERS AND EXTENDED SOURCES
=============================
Of the 19 extended sources detected in the CDFS [@giacconi2002], 15 were observed in our survey. In 5 cases (XID 37, 147, 522, 527 and 581) no spectroscopic identification was possible and in one case (XID 132) only a low quality identification was obtained. In two cases (XID 116 and 514) the diffuse X-ray emission could be ascribed to thermal halos of nearby galaxies. In general, most of the remaining extended sources span the regime of galaxy groups (with luminosities of a few$\times 10^{42}$ erg s$^{-1}$) down to X-ray luminosities typical of thermal halos around single early-type galaxies. In some cases, either the hardness ratio or the optical identification suggest the coexistence of a thermal halo with an AGN component (e.g. XID 138). In Figures \[cdf138\]-\[cdf645\] we show K-band images of the identifield clusters/groups with overlaid Chandra contours (2.5,3,4,5,7,10 $\sigma$ above the local background) in the \[0.5-2\] keV band. We also mark objects with concordant redshifts (as listed in Table 5).
Specifically, XID 566,594,645 are ordinary groups showing however a range of surface brightness profiles (see Figs. \[cdf566\],\[cdf594\],\[cdf645\]). XID 566 and 594 belong to the large scale structure at $z\simeq
0.73$ (see below). XID 249, for which we have two concordant redshifts with $<z>=0.964$, is clearly extended with a harder component (Fig. \[cdf249\]). XID 138 was identified as a close pair of AGN at $z=0.97$, surrounded by a soft halo. In two cases, XID 511 and 560, we identified only one galaxy per source, making it difficult to ascertain the existence of a group.
REDSHIFT DISTRIBUTION\[reddis\]
===============================
The spectroscopically identified CDFS sample comprises 135 X-ray sources, including five stars. Reliable redshifts can be obtained typically for objects with R $<25.5$, however, some incompleteness already sets in around R $\sim23$. For the R$<$24 sample (199 objects), 120 (including five stars) of the 159 observed X-ray sources have been spectroscopically identified, thus a success rate of 75% and a completenes of 60%. The sources with inconclusive redshift identification cover a wide range of hardness ratios. In Figure \[position\], we show the spatial distribution of the sources with spectroscopic observations as well as those not observed. The latter lie predominantly in some of the outermost parts of the CDFS.
The histogram of the redshift distribution of the X-ray sources is shown in Figure \[zdist\]. A preliminary version of this diagram was given by @hasinger2002. There is an excess of objects in two redshift bins, revealing large-scale structures of X-ray sources[@gilli2003], similar to that found in the CDFN [@barger2002]. These redshift spikes are populated by X-ray type-1 and type-2 AGNs as well as a few X-ray galaxies. There are 18 X-ray sources within 2000 km s$^{-1}$ of $z=0.674$, of which one (XID 201b) is fainter than R of 24; these objects are distributed loosely across a large fraction of the field and should thus trace a sheet-like structure. The spike centered on $z=0.734$ is narrower and includes 16 X-ray sources within 1000 km s$^{-1}$ of the mean redshift, all brighter than R of 24. In both structures, about 70% of the sources are X-ray type-2 AGNs (+ X-ray galaxies). The brightest X-ray cluster (XID 594) belongs to the $z=0.73$ spike. A few field galaxies, possibly associated with this X-ray cluster and other extended X-ray sources (of which XID 645 at $z = 0.679$), are given at the end of Table \[tblspec\]. The $z=0.67$ and 0.73 structures are also traced by galaxies from the ESO K20 survey which covers $\sim$1/10 of the Chandra field [@cimatti2002a; @cimatti2002b]: they are populated by 24 and 47 galaxies respectively [@gilli2003]. The K20 structure at $z=0.73$ is dominated by a standard cluster with a central cD galaxy (identified with the extended X-ray source XID 566), whereas the K20 galaxies at $z=0.67$ are uniformaly distributed across the field. There is also evidence of higher redshift, narrow spikes in the distribution of the X-ray sources at $z=1.04$, 1.62 and 2.57; that at $z=1.04$ is also present in the K20 sample [@gilli2003].
At $z > 2$, there are similar numbers of X-ray type-1 (5) and type-2 (7) QSOs. The relative paucity of high $z$ X-ray type-2 AGN (1/6) could arise from an observational bias as type-2 sources are optically fainter than the type-1 population. At $z < 1$, the higher number of X-ray type-2 over type-1 sources is mainly due to the large concentration of X-ray type-2 sources within the $z=0.67$ and 0.73 structures.
The redshift distribution of the bright sample with 60% redshift identification completeness can be compared to those predicted by models. The X-ray background population synthesis models [@gilli2001], based on the AGN/QSO X-ray luminosity function and its evolution, predict a maximum in the AGN/QSO redshift distribution at $z\sim1.5$. Contrary to these expectations, accretion onto black holes is still very important at $z<1$: indeed 88 (68%) of the 130 CDFS extragalactic X-ray sources are at $z<1$ and the redshift distribution peaks around $z\sim0.7$, even if the normal starforming galaxies are removed from the sample. Similar results were found for the CDFN [@barger2002]. This clearly demonstrates that the population synthesis models will have to be modified to incorporate different luminosity functions and evolutionary scenarios for intermediate-redshift, lower-luminosity AGNs.
Moreover, the CDFS redshift distribution does not confirm the prediction by @haiman1999, that a large number ($\sim$100) of QSOs at redshifts larger than 5 should be expected in any ultra deep Chandra survey. The highest redshift in the CDFS thus far is 3.70, while there are two confirmed and one uncertain high redshift sources in the CDFN at $z=4.14, 5.19$ and $z=4.42$, respectively [@barger2002; @barger2003a; @brandt2001b], as well as one QSO at $z=4.45$ in the Lockman Hole [@schneider1998]. As our target selection is based primarily on our R-band imaging, we are suffering from a bias against z$>$5 objects (the Ly$_\alpha$ emission is redshifted out of the FORS R-band at z$\sim$5). Therefore, we may have a [*few*]{} QSOs at redshifts larger than 5 in the sample, but we can be certain that the number of these is on the order of a few. Most of the X-ray survey area is covered by near-infrared, where objects well beyond redshift of 15 are detectable. Among the objects covered in the near-IR, we only find 10 that are detected [*only*]{} in the near-IR. Furthermore, 4 of these were still observed spectroscopically, where we can detect Ly$_\alpha$ emission up to a redshift of 6.5. So in the unlikely case that [*all*]{} these objects and five additional objects not detected in optical imaging and without near-IR coverage are all very high redshift QSOs, we are still an order of magnitude below the predicted number of such objects. This suggests a turn-off of the X-ray selected QSO space density beyond $z \sim 4$ [@hasinger2002; @barger2003a].
OPTICAL AND X-RAY DIAGNOSTICS
=============================
*X-ray and Optical Fluxes\[optxflux\]*
--------------------------------------
The soft and hard X-ray fluxes versus redshift diagrams are shown in Figure \[fxz\]. The X-ray type-1 and type-2 populations have similar hard X-ray fluxes, whereas these two populations cover different ranges of soft X-ray fluxes, as can also be seen in Figure \[fxR\]. However, the brighter, rarer objects, $f_{\rm X}$(0.5-2 keV) and $f_{\rm X}$(2-10 keV) larger than (1 and 5)$ \times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$ respectively, are dominated by optically broad-emission line QSOs (at $z < 2$) as already demonstrated by larger samples of luminous X-ray sources detected by ROSAT, Chandra and XMM-Newton [@lehmann2001; @barger2002; @mainieri2002].
The observed R and K magnitudes of the extragalactic sources versus redshift are shown in Figures \[zR\] and \[zK\] respectively. At $z\gtrsim2$, there are seven X-ray type-2 QSOs (XID 27, 54, 57, 62, 112, 202, 263) plus one lower X-ray luminosity type-2 AGN (XID 642); except the BAL QSO, all have narrow Ly$\alpha$ and C[iv]{} emission, $HR>-0.2$, and faint optical magnitudes R $\gtrsim 24.0$. There are also six X-ray type-1 QSOs at $z\gtrsim2$ (XID 11, 15, 21, 24, 68, 117), all but one being otically bright (R $<24$) BLAGN, and five fainter, lower X-ray luminosity type-1 AGN (XID 87, 89, 230, 563, 901). The fraction of high-redshift, X-ray type-2 QSO+AGN sources is thus 42%. Moreover, two of the X-ray type-1 QSO/AGN (XID 117, 901), with narrow Ly$\alpha$ and C[iv]{} emission but $ HR < -0.2$, could be absorbed X-ray sources since the hardness ratio is not a good tracer of intrinsic absorption for high redshift sources. These results differ from those obtained for the CDFN 2 Msec sample [@barger2003b] which comprises 26 objects at $z\gtrsim2.0$ (excluding the sources with tentative redshifts or complex/multiple structure or possible contamination: their Types s and m, respectively) of which 20 are BLAGN, thus an optical type-2 QSO+AGN fraction of 23%. This may arise from an observational selection as 10 (53%) of the 19 CDFS sources at $z\gtrsim2$, with secure redshift identification, have R $> 24$ as compared to only 2 (8%) out of 26 CDFN sources.
The segregation between the X-ray type-1 and type-2 QSOs/AGNs seen in the R versus $z$ diagram (Figure \[zR\]) is far less pronounced in the K versus $z$ diagram (Figure \[zK\]). This is most likely due to the presence of dust in X-ray type-2 QSOs/AGNs associated with the X-ray absorbing material which severely obscures the nuclear component, as well as an increased contribution of the galaxy host light in the K-band relative to that of the AGN. The X-ray and optical versus redshift diagrams (Figure \[fxz\], \[zR\] and \[zK\]) strongly suggest that the X-ray type-1 and type-2 populations cover roughly the same range of intrinsic luminosities [see also @rosati2002; @mainieri2002].
*X-ray and Optical Colours*
---------------------------
A segregation of the X-ray type-1 and type-2 populations is also present in the R$-$K versus $z$ diagram (see Figure \[rkz\]), as first outlined by @lehmann2001 and confirmed by @mainieri2002. The deeper Chandra observations reveal many more X-ray type-2 sources which have optical/near IR colours dominated by the host galaxy and most of them cluster around the SED tracks of elliptical and Sbc galaxies at $0.5 < z <1.0$ [see also @rosati2002]. The X-ray type-1 population usually follows the evolutionary track of an unreddened QSO, except nine AGNs/QSOs at $z \gtrsim 1$, all with R$-$K $\gtrsim$ 4. This may be due to either an important contribution of the galaxy host light in the near IR or obscuration by dust. Among these nine X-ray type-1 sources, the X-ray luminous, red BLAGN at $z = 1.616$ (XID 67) was observed with HST/WFPC2 and is clearly resolved with an elliptical morphology [@koekemoer2002]. A substantial contribution of the host galaxy could also account for the red colour of two BLAGN at $z \simeq 1.62$ (XID 46, 101). Obscuration by dust associated with the X-ray aborbing material is more probable for the remaining six X-ray type-1 sources, of which four belong to the HEX optical class and are at $1.6\lesssim z \lesssim 2.6$ (XID 31, 117, 563, 901) and two belong to the LEX optical class and are at $z \simeq 1.0$ (XID 18, 242).
For a large fraction of the X-ray sources, there is a relationship between the hardness ratio, $HR$, and the R$-$K colour as shown in Figure \[hrrk\]. The bluer objects are X-ray type-1 QSOs, whereas the redder ones are mostly X-ray type-2 AGNs at $z \sim 0.5$ to 1.0. However, the redder objects (R$-$K $>$ 4) cover a wide range of $HR$ values, as already noted by @franceschini2002, and they comprise many X-ray type-1 AGNs, including the nine objects discussed above, while most of the remaining X-ray type-1 are $z < 1$ objects of the optical LEX class. To constrain the nature of the redder, X-ray type-1 AGNs requires to conduct an X-ray spectral analysis (Chandra and XMM-Newton data) of these sources (Streblyanskaya, Mainieri et al. in preparation), primarily those with secure redshifts, and to derive the morphological properties of their host galaxies using the HST observations from the GOODS-ACS Treasury program.
*X-ray Selected Extremely Red Objects*
--------------------------------------
The fraction of extremely red objects (EROs: R$-$K $>$ 5.0) among X-ray sources appears to increase with decreasing optical flux as found for a subset of the CDFN X-ray sources [@alexander2001] and for Lockman Hole (LH) sources detected by XMM-Newton [@mainieri2002]. We use the CDFS sample given in Table \[tblspec\] to confirm this trend. There are 151 X-ray sources with bright, R$<$24, counterparts observed in the R and K bands. Five sources were not detected in the K band (K $>$ 20.3), but the upper limits on their R$-$K colours are smaller than 5. The fraction of EROs in this bright optical sample is 10% (15 objects). The fainter optical sample is limited to $24\leq$ R $<26$ to have meaningful R$-$K upper limits and it comprises 72 X-ray sources. Most of these faint objects do not have spectroscopic redshifts. Thus, in cases of several possible counterparts, the brigthest of the best centred counterparts was selected. To the 14 EROs with measured R$-$K colours, should be added the four objects detected in the K band only (R $>$ 26.3), thus with R$-$K $\gtrsim$ 6.0. For the X-ray counterparts not detected in the K band, all those with $24\leq$ R $<25$ have R$-$K upper limits smaller than 5, but among the 10 objects with $25\leq$ R $<26$ only four have R$-$K $\lesssim$5.0. The remaining six objects have R$-$K upper limits in the range 5.3 to 5.6, but we will consider them as non-EROs in order to get a conservative value of the number of EROs among the fainter optical sample. The ERO fraction in the R$\geq$24 sample is thus 25%, or 2.5 times higher than for the R$<$24 sample.
Six of the optically bright CDFS EROs have redshift estimates (5 secure), all with $z\sim 1$ ($\pm0.3$). One is an X-ray AGN-1 (LEX optical class), and five are X-ray AGN-2 (LEX or ABS class) thus with strong optical obscuration associated with X-ray absorption. The other nine objects do not show any optical emission/absorption feature and all, but one, have $HR<-0.2$. If they were at $z\sim 1$, they would have luminosities $L_{\rm X}$(0.5-10 keV)$>10^{42}$ erg s$^{-1}$. The fraction of hard X-ray sources among the optically bright ERO population is thus 40%. Higher redshift sources are present in the optically faint ERO sample. Among the six objects with redshift estimates (4 secure), five are at $1.6\lesssim z \lesssim 3.7$ of which three are X-ray luminous QSOs. The fraction of hard X-ray, optically faint EROs is 50%, but if there were a majority of high $z$ sources, the bulk of this faint ERO population would be heavily absorbed X-ray sources. A similar result was found for the LH sources [@mainieri2002]. This sample comprises 66 objects with measured R$-$K colour (only 20 are fainter than R=24), of which 18 are EROs. Five EROs (28%) are not detected in the hard band as compared to 27% and 22% for the CDFS optically bright and faint ERO samples, respectively. The X-ray spectral analysis of the LH sources shows that all, but one, of the $HR>-1.0$ sources have high intrinsic absorption: ten have absorbing column densities $N_{\rm H} > 10^{22}$ cm$^{-2}$ and two, without redshift identification, have lower limits (observer frame) of $N_{\rm H,min} > 10^{21.5}$ cm$^{-2}$.
Among the X-ray selected EROs, the dominant population appears to be fairly luminous, absorbed X-ray sources, thus of the X-ray type-2 AGN class, at intermediate and high redshifts. This is consistent with the small fraction (1.5-10%) of near-IR selected EROs detected in X-ray [@cimatti2003]. EROs belonging to other classes, elliptical galaxies or dusty starbursts [@stevens2003], are also most probably present. Indeed, among the optically faint EROs, there are a few sources with optically red and soft X-ray spectra (e.g. XID 579b).
Objects of different classes are also found for EROs in the HDFN and the Lockman hole [@franceschini2002; @stevens2003]: sources at $z \sim 1$ with SEDs typical of elliptical galaxies, dusty starbursts and $z > 1.5$ absorbed AGN.
*Luminosities*
--------------
A trend of increasing hard X-ray luminosity (2-10 keV band) with absolute K magnitude can be seen in Figure \[KLx\]. This trend is not present when the X-ray luminosity in the broad 0.5-10 keV band is considered instead [see also @franceschini2002]. We also show in this figure the effect expected from the correlation found between the bulge luminosity and the black hole mass [@marconi2003]. We used very uncertain assumptions to derive this curve. We assumed that around 40% of the K-band emission originates in the bulge and we assumed that the X-ray luminosity is 0.1% of the Eddington limit luminosity. The observed X-ray luminosity in the hard band is close to the intrinsic one (small K correction for most of the sources) and the reddening correction for the K absolute magnitude is much smaller in the near-IR than in the optical, although it could still be important for the extremely red objects. We thus expect a tighter correlation between the mid-IR luminosity, to be obtained by the Spitzer-GOODS Legacy program, and the hard X-ray luminosity. The trend present in Figure \[KLx\] reinforces the suggestion made above that the X-ray type-1 and type-2 populations cover the same range of luminosities, thus trace similar levels of gravitational accretion. They differ by either the environment close to the AGN and/or the viewing angle to the nucleus, the contribution of the light from an early-type host galaxy, or the dust content and dust-to-gas ratio within the host galaxy, or an associated starburst.
The observed hard X-ray luminosity as a function of redshift is shown in Figure \[zLx\]. There are X-ray luminous X-ray type-1 QSOs down to $z=0.5$, thus no strong evolution, confirming the findings in the CDFN [@barger2002]. However, this only applies to the X-ray type-1 population, as there is only one (8%) X-ray type-2 QSO (XID 51) out of 13 QSOs at $0.5 < z < 2$. Two of the X-ray type-1 QSOs show narrow emission lines only (XID 18, 31) and could be absorbed X-ray sources, as indeed confirmed by X-ray spectral analysis of XID 31 ($z=1.603$) which is an absorbed source with $N_{\rm H} = 1.4\times 10^{22}$ cm$^{-2}$ (V. Mainieri, private communication). Even including these narrow-line QSOs in the type-2 sample would still lead to only 23% X-ray type-2 QSOs at lower redshift compared to 54% at $z > 2$ (see Section \[optxflux\]). This difference (detected at 90% confidence level) would no longer be as significant if sources down to $L_{\rm X}$(2-10 keV) $> 10^{43.5}$ erg s$^{-1}$ were considered instead. Indeed, the ratio of X-ray type-2/type-1 sources at $0.5 < z < 2$ increases with decreasing hard X-ray luminosity, the type-2 population being dominant for $L_{\rm X}$(2-10 keV) $< 10^{43.0}$ erg s$^{-1}$. This trend is confirmed by the analysis of the 2-10 keV luminosity function derived from ASCA, HEAO1 and Chandra surveys [@ueda2003] which shows that, at $z < 1$, the percentage of X-ray type-2 AGN ($N_{\rm H} > 10^{22}$ cm$^{-2}$) decreases with increasing intrinsic luminosity from 49% at $L_{\rm X}$(2-10 keV) $= 10^{43}$ erg s$^{-1}$ to 26% at $L_{\rm X}$(2-10 keV) $= 10^{45}$ erg s$^{-1}$.
The X-ray spectral analysis (absorbing column densities and intrinsic X-ray luminosities) of the QSOs+AGNs of the CDFS and CDFN spectroscopic samples will enable the determination of the cosmic evolution of the X-ray type-1 and the X-ray type-2/absorbed sources, thus of a possible differential cosmic evolution between these two populations. It should be noted that @barger2002 mention the existence of only two type-2 QSOs at $0.5 < z < 2$ (both at $z \approx 1$) while, in their Table 1, there are 12 broad-line QSO+AGN sources in the same redshift range, which is consistent with our results. However, in the CDFN, the fraction of type-2 QSOs is small at both intermediate and high (see Section \[optxflux\]) redshifts.
SUMMARY AND OUTLOOK
====================
We presented a catalog of 137 secure and 24 tentative spectroscopic identifications of the 349 X-ray objects (including 3 new, faint sources) in the CDFS field, based on our survey using the VLT. Our spectroscopic survey is 40% complete considering the whole X-ray catalog, and 70% complete if we consider the subset in the central 8 radius with optical counterparts at R$<$24. This can compared to the somewhat higher spectroscopic completeness achieved in the Chandra Deep Field North identification programme [@barger2002], where the corresponding fractions are 49% and 78%, respectively. Very recently, optical identifications have also been presented for the 2 Msec observation of the HDFN [@barger2003b], which reach a completeness as high as 87% at R$<$24. At fainter optical magnitudes (R$>$24), however, the fraction of reliable spectroscopic identifications is larger for the CDFS compared to the HDFN. This is becoming important in particular, when comparing the fraction of X-ray type-2 QSOs at these faint magnitudes (see below).
We proposed a new, objective and simple scheme, based on X-ray luminosity and hardness ratio, to classify objects into X-ray type-1 (unabsorbed) and X-ray type-2 (absorbed) AGN. Hard ($HR>-0.2$) sources are classified as X-ray type-2 AGN or QSO, depending on their X-ray luminosity. Soft sources ($HR\leq-0.2$) are classified as X-ray galaxies, X-ray type-1 AGN or QSO, depending on their X-ray luminosity. At high optical and X-ray luminosities, this classification scheme is largely coincident with the classical AGN classification purely based on optical spectroscopic diagnostics. However, as soon as the integrated light of the host galaxy becomes larger than the optical emission of the AGN nucleus, the optical classification breaks down. Consequently, we are classifying many more objects as AGN, than would be selected in optical samples. An additional advantage of our proposed classification scheme is that it only relies on X-ray fluxes and redshift (to calculate $L_X$). So far we only used optical spectroscopy to derive the redshift, but our scheme can use photometric redshift techniques, thus going significantly beyond the capabilities of optical spectroscopy. Indeed, using photo-$z$ techniques, more than 95% of the CDFS sources can be identified in our scheme (Mainieri, private communication).
We have spectroscopically identified a sample of 8 secure and 2 tentative high-luminosity X-ray sources with significant absorption, our X-ray type-2 QSO class. Nine ($^{+4.1}_{-3.0}$: 1$\sigma$ errors) of these sources are in the redshift range $2<z<4$ and their optical spectra are dominated by strong, narrow high excitation UV permitted lines, very similar to the prototypical object CDFS-202 [@norman2002]. In contrast, the spectroscopic sample existing in the HDFN so far [@barger2002] only contains 2 ($^{+2.6}_{-1.3}$) similar objects (HDFN $\# 184$ and $\# 287$). This difference may be due to the fact that our spectroscopy is pushing about one magnitude deeper than the HDFN spectroscopy in a part of the field. However, we can not exclude true cosmic field-to-field variations in the number of X-ray type-2 QSOs. The fraction of X-ray type-2 to the total AGN population shows a significant variation with observed X-ray luminosity, consistent with, but even somewhat stronger than the trend found from ASCA surveys in the 2-10 keV band [@ueda2003]: the X-ray type-2 fraction decreases from $75\pm8\%$ (8 type-1 vs. 24 type-2 AGN) in the luminosity range $10^{42-43}$ erg s$^{-1}$, over $44\pm8\%$ (20 vs. 16 AGN) at luminosities $10^{43-44}$ erg s$^{-1}$, to $33\pm10\%$ (16 vs. 8) at $10^{44-45}$ erg s$^{-1}$ (see also Figure \[hrlx\]). This behaviour can probably explain some of the evolutionary trends apparent in Figures \[fxz\] and \[zLx\].
We found spectroscopic evidence for two large-scale structures in the field, predominantly populated by X-ray type-2 AGN but also X-ray type-1 AGN and normal galaxies: one at $z = 0.734$ has a fairly narrow redshift distribution and comprises two clusters/groups of galaxies centered on extended X-ray sources. The redshift distribution of the second one at $z = 0.674$ is broader (velocity space) and traces a sheet-like structure. A detailed comparison with the redshift spikes in a NIR-selected (K20) sample of galaxies in the same field has been performed by [@gilli2003]. Similar, but much less pronounced redshift spikes have also been observed in the HDFN at redshifts around $z = 0.843$ and $z = 1.018$ by [@barger2002]. AGN therefore trace large-scale structures as do normal galaxies. Further studies on larger samples are required to investigate, whether AGN are more strongly clustered than normal galaxies [@gilli2003] and, whether X-ray type-2 AGN are indeed clustering stronger than X-ray type-1 AGN, as indicated by the CDFS results.
However, the objects in these spikes do not dominate the sample. The observed AGN redshift distribution peaks at $z\sim0.7$, even if the objects in the spikes and also the normal, starforming galaxies are removed. Compared to the pre-Chandra and XMM-Newton predictions of population synthesis models of the X-ray background [@gilli2003], there is an excess of $z<1$ AGN, even taking into account the spectroscopic incompleteness of the sample. These models will therefore have to be modified to incorporate different luminosity functions and evolutionary scenarios for intermediate-redshift, lower-luminosity AGNs.
It will be interesting to study the correlation of active galaxies to field galaxies in the sheets and investigate the role that galaxy mergers play in the triggering of the AGN activity. Finally, there may be a relation between the surprisingly low redshift of the bulk of the Chandra sources, the existence of the sheets at the same redshift and the strongly evolving population of dusty starburst galaxies inferred from the ISO mid-infrared surveys [@franceschini2002].
The Chandra Deep Field South has been selected as one of the deep fields in the Spitzer legacy programme Great Observatories Origins Deep Survey (GOODS). GOODS will produce the deepest observations with the Spitzer IRAC instrument at 3.6-8$\mu$m and with the MIPS instrument at 24$\mu$m over a significant fraction of the CDFS [see @fosbury2001]. The same area has already been covered by an extensive set of pointings with the new Advanced Camera for Surveys (ACS) of the Hubble Space Telescope in BVIz to near HDF depth. Following up the deep EIS survey in the CDFS, ESO has undertaken a large program to image the GOODS area with the VLT to obtain deep JHKs images in some 32 ISAAC fields. A small spot inside the CDFS has also been selected as the location of the HST ACS ultradeep field (UDF), aiming at roughly two magnitudes fainter than the Hubble Deep Fields, over a substantially larger area. An even larger field than the CDFS has been surveyed with the HST ACS program GEMS and has also been covered by multiband optical photometry as part of the COMBO-17 survey [@wolf2003]. The next step in the optical identification and classification work is to use the extremely deep HST ACS and VLT ISAAC (or EIS SOFI) data provided by GOODS and the narrow band photometry provided by COMBO-17 to obtain multicolour photometric redshifts for the objects not covered by and/or too faint for our spectroscopic identification programme (Zheng et al., in preparation).
Additional X-ray information in an area wider than the CDFS is existing from a deep XMM-Newton pointing of $\sim$400 ksec exposure time (PI: Bergeron). The already existing Chandra Megasecond coverage will be widened and deepened with four additional 250 ksec ACIS-I pointings (PI: Brandt). The multiwavelength coverage of the field is complemented by deep 20 cm radio data from the VLA and ATCA. The CDFS will therefore ultimately be one of the patches in the sky providing a combination of the widest and deepest coverage at all wavelengths and thus a legacy for the future.
This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. Three of our redshifts have been obtained from the 2dFGRS public dataset.
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.
.
[r@[ ]{}r@r@[ ]{}r@[ ]{}r@[ ]{}r@[ ]{}r@[ ]{}r@[ ]{}r@[ ]{}r@[ ]{}r@[ ]{}r@[ ]{}]{} 901 & J033235.8-274917 & 03 32 35.78 & -27 49 16.82 & 11.0$\pm$5.0 & $<$9.0 & 828.4(857.7) & 6.1e-17 & 2.9e-17 & $<$3.1e-16 & & -1\
902 & J033222.1-275113 & 03 32 22.08 & -27 51 13.05 & $<$7.0 & 12.1$\pm$6.2 & 792.3(804.1) & $<$3.9e-17 & – & 4.5e-16 & 2.0E-16 & +1\
903 & J033226.0-274049 & 03 32 25.97 & -27 40 49.21 & 22.5$\pm$10.4 & $<$3 & 824.3(820.2) & 1.3e-16 & 0.6e-16 & $<1$e-16 & & -1\
[lcccc]{} AGN & $-$1.0…$+$1.2 & 0.25…0.33& 0.0…1.0 & $-$1.4…$+$1.1\
BL Lac & $+$0.3…$+$1.7 & 0.25…0.33& 0.0…1.0 & $-$0.1…$+$1.6\
Clusters & $-$0.5…$+$1.5 & 0.18…0.25& 0.1…0.4 & $-$0.6…$+$1.6\
Galaxies & $-$1.8…$-$0.2 & 0.10…0.30& 0.1…1.0 & $-$2.2…$+$0.0\
M stars & $-$3.1…$-$0.5 & 0.05…0.20& 0.6…1.0 & $-$3.4…$-$0.4\
K stars & $-$4.0…$-$1.5 & 0.05…0.20& 0.4…0.6 & $-$4.1…$-$1.3\
G stars & $-$4.3…$-$2.4 & 0.05…0.20& 0.3…0.5 & $-$4.4…$-$2.2\
B-F stars & $-$4.6…$-$3.0 & 0.05…0.20&-0.5…0.3 & $-$4.6…$-$2.5\
[lrrrrrrc]{} 3888.6 & 3887 & 13847 & 26 & 7462 & 3632 & 30 & U\
3650.1 & 3648 & 90265 & 30 & 6954 & 36364 & 35 & U\
3650.1 & 3649 & 17119 & 27 & 6957 & 8380 & 34 & B\
3888.6 & 3888 & 79904 & 27 & 7465 & 19142 & 30 & B\
4046.6 & 4047 & 346283 & 30 & 7801 & 44653 & 30 & B\
4358.3 & 4359 & 795630 & 28 & 8457 & 51040 & 30 & B\
4471.5 & 4472 & 97692 & 27 & 8694 & 5151 & 30 & B\
5015.7 & 5014 & 172340 & 28 & 9878 & 1946 & 33 & V\
[clcccl]{} 6+7 & 3$\times$1800, 3$\times$1800 & 1.2 & 0.8/0.6 & 40 & 2000 Oct 27-28\
15 & 4$\times$1800+1300 & 1.2 & 0.7/0.8 & 45 & 2000 Oct 27-28\
22+23+24 & 3$\times$1800, 1800, 2$\times$1800 & 1.2 & 1.0/0.9 & 50 & 2000 Oct 28-29\
28+29 & 2$\times$1800, 3$\times$1800 & 1.2 & 0.6/0.9 & 40 & 2000 Oct 28-29\
36+39+40 & 1800, 1800, 4$\times$1800 & 1.2 & 1.3/0.6 & 65 & 2000 Oct 29-30\
46+47 & 4$\times$1800, 1800+900 & 1.2 & 0.5/0.7 & 35 & 2000 Oct 29-30\
78 & 6$\times$1800+945 & 1.2 & 0.5/0.6 & N/A & 2000 Nov 24-25\
82 & 5$\times$1800 & 1.2 & 0.6/0.6 & 35 & 2000 Nov 23-24\
84 & 6$\times$1800 & 1.2 & 0.9/0.8 & 45 & 2000 Nov 23-24\
86 & 3$\times$1800+1535 & 1.2 & 1.1/1.0 & 40 & 2000 Nov 24-25\
88 & 1200 & 1.2 & 0.8/0.6 & 80 & 2000 Nov 23-24\
89 & 1200 & 1.2 & 0.8/0.6 & 75 & 2000 Nov 23-24\
90 & 1200 & 1.2 & 1.3/1.1 & 90 & 2000 Nov 23-24\
99 & 1800 & 1.2 & 0.9/1.3 & 50 & 2000 Nov 24-25\
119+120+121 & 2700, 2$\times$2700, 2$\times$2700+3600 & 1.0 & 0.5/0.7 & 70 & 2001 Sep 20-21\
122 & 4$\times$2700 & 1.0 & 0.6/0.8 & 70 & 2001 Sep 18-19\
134 & 2500+2700 & 1.4 & 0.9/0.8 & 70 & 2001 Sep 17-18, 19-20\
137 & 1800+2700 & 1.4 & 1.0/1.0 & 60 & 2001 Sep 18-19, 19-20\
138+139 & 3$\times1800$, 2$\times$2700 & 1.0 & 0.5/0.6 & 55 & 2001 Sep 17-18\
146 & 3$\times$2700 & 1.4 & 0.7/0.9 & 60 & 2001 Sep 19-20\
MXU2.1 & 9$\times$1800+2400 & 1.0 & 0.7/0.9 & N/A & 2001 Nov 13-14, 14-15\
MXU4.1 & 12$\times$1800+1134 & 1.0 & 0.9/0.8 & N/A & 2001 Nov 12-13\
MXU5.1 & 4$\times$1800 & 1.0 & 0.5/0.7 & 70 & 2001 Nov 14-15\
MXU11.1 & 9$\times$1800+2400 & 1.0 & 0.6/0.8 & sl & 2001 Nov 13-14\
MOS11.1 & 4$\times$1800 & ?.? & 1.2/1.0 & 75? & 2001 Nov 11-12\
MXU1.1 & 2$\times$2100 & ?.? & 0.5/0.7 & sl & 2001 Dec 18-19\
[lrrrr]{} X-ray AGN-1 and QSO-1 & 31 & 5 & 10 & 1\
X-ray AGN-2 and QSO-2 & 1 & 16 & 24 & 8\
X-ray galaxy & 0 & 3 & 20 & 12\
[^1]: Based on observations collected at the European Southern Observatory, Chile (ESO N$^{\rm o}$ 66.A-0270(A) and 67.A-0418(A)).
| 0 |
---
abstract: 'We propose a sequential design method aiming at the estimation of an extreme quantile based on a sample of dichotomic data corresponding to peaks over a given threshold. This study is motivated by an industrial challenge in material reliability and consists in estimating a failure quantile from trials whose outcomes are reduced to indicators of whether the specimen have failed at the tested stress levels. The solution proposed is a sequential design making use of a splitting approach, decomposing the target probability level into a product of probabilities of conditional events of higher order. The method consists in gradually targeting the tail of the distribution and sampling under truncated distributions. The model is GEV or Weibull, and sequential estimation of its parameters involves an improved maximum likelihood procedure for binary data, due to the large uncertainty associated with such a restricted information.'
author:
- |
Michel Broniatowski and Emilie Miranda\
LPSM, CNRS UMR 8001, Sorbonne Universite, Paris
title: 'A sequential design for extreme quantiles estimation under binary sampling.'
---
Consider a non negative random variable $X$ with distribution function $G$.$%
\ $ Let $X_{1},..,X_{n}$ be $n$ independent copies of $X.$ The aim of this paper is to estimate $q_{1-\alpha }$, the $\left( 1-\alpha \right) $-quantile of $G$ when $\alpha $ is much smaller than $1/n.$ We therefore aim at the estimation of so-called extreme quantiles. This question has been handled by various authors, and we will review their results somehow later. The approach which we develop is quite different since we do not assume that the $X_{i}$’s can be observed. For any threshold $x$, we define the r.v. $$Y=\left\{
\begin{array}{l}
1\text{ if }X\leq x \\
0\text{ if }X>x%
\end{array}%
\right.$$which therefore has a Bernoulli distribution with parameter $G(x).$ We may choose $x$, however we do not observe $X$, but merely $Y.$ Therefore any inference on $G$ suffers from a severe loss of information. This kind of setting is common in industrial statistics: When exploring the strength of a material, or of a bundle, we may set a constraint $x$, and observe whether the bundle breaks or not when subjected at this level of constraint.
In the following, we will denote $R$ the resistance of this material, we observe $Y.$ Inference on $G$ can be performed for large $%
n$ making use of many thresholds $x.$ Unfortunately such a procedure will not be of any help for extreme quantiles. To address this issue, we will consider a design of experiment enabling to progressively characterize the tail of the distribution by sampling at each step in a more extreme region of the density. It will thus be assumed in the following that we are able to observe $Y$ not only when $R$ follows $G$ but also when $R$ follows the conditional distribution of $R$ given $\{ R>x\}.$ In such a case we will be able to estimate $q_{1-\alpha }$ even when $\alpha <1/n$ where $n$ designates the total number of trials. In material sciences, this amounts to consider trials based on artificially modified materials; in the case when we aim at estimation of extreme upper quantiles, this amounts to strengthen the material. We would consider a family of increasing thresholds $%
x_{1},..,x_{m}$ and for each of them realize $K_{1},..,K_{m}$ trials, each block of iid realizations $Y$’s being therefore functions of the corresponding unobserved $R$’s with distribution $G$ conditioned upon $%
\{R>x_{l}\}$, $1\leq l\leq m.$ design which allows for the estimation of extreme quantiles.
The present setting is therefore quite different from that usually considered for similar problems under complete information. As sketched above it is specifically suited for industrial statistics and reliability studies in the science of materials.
From a strictly statistical standpoint, the above description may also be considered when the distribution $G$ is of some special form, namely when the conditional distribution of $R$ given $\{R>x\}$ has a functional form which differs from that of $G$ only through some changes of the parameters. In this case, simulation under these conditional distributions can be performed for adaptive choice of the thresholds $x_{l}$’s, substituting the above sequence of trials. This sequential procedure allows to estimate iteratively the initial parameters of $G$ and to obtain $q_{1-\alpha }$ combining corresponding quantiles of the conditional distributions above thresholds, a method named splitting. In this method, we will choose sequentially the $%
x_{l}$’s in a way that $q_{1-\alpha }$ will be obtained easily from the last distribution of $x$ conditioned upon $\{R>x_{m}\}.$
In safety issues or in pharmaceutical control, the focus is usually set on the behavior of a variable of interest (strength, maximum tolerated dose) for small (or even very small) levels. In these settings the above considerations turn to be equivalently stated through a clear change of variable, considering the inverse of the variable of interest. As an example which is indeed at the core of the motivation for this paper, and in order to make this approach more intuitive, we first sketch briefly the industrial situation which motivated this work in Section \[IndContext\]. We look at a safety property, namely thresholds $x$ which specify very rare events, typically failures under very small solicitation.
As stated above, the problem at hand is the estimation of very small quantiles. Classical techniques in risk theory pertain to large quantiles estimation. For example, the Generalized Pareto Distribution, to be referred to later on, is a basic tool in modeling extreme risks and exceedances over thresholds. Denoting $R$ the variable of interest and $\widetilde{R}:=1/R,$ then obviously, for $x>0$, $\left\{ R<x\right\} $ is equivalent to $\left\{
\widetilde{R}>u\right\} $ with $u=1/x$. In this paper we will therefore make use of this simple duality, stating formulas for $R,$ starting with classical results pertaining to $\widetilde{R}$ when necessary. Note that when $q_{\alpha }$ designates the $\alpha -$quantile of $R$ and respectively $\widetilde{q}_{1-\alpha }$ the $\left( 1-\alpha \right)-$quantile of $%
\widetilde{R}$, it holds $q_{\alpha }=1/\widetilde{q}_{1-\alpha }.$The resulting notation may seem a bit cumbersome; however the reader accustomed to industrial statistics will find it more familiar.
This article is organized as follows. Section \[IndContext\] formalizes the problem in the framework of an industrial application to aircraft industry. In Section \[revLit\], a short survey of extreme quantiles estimation and of existing designs of experiment are studied as well as their applicability to extreme quantiles estimation. Then, a new procedure is proposed in Section \[Splitting\] and elaborated for a Generalized Pareto model. An estimation procedure is detailed and evaluated in Section \[EstimationProc\]. Then an alternative Weibull model for the design proposed is presented in Section \[WeibullModel\]. Lastly, Sections \[model\_selection\_missp\] and \[Perspectives\] provide a few ideas discussing model selection and behavior under misspecification as well as hints about extensions of the models studied beforehand.
Industrial challenge {#IndContext}
====================
Estimation of minimal allowable stress in material fatigue
----------------------------------------------------------
In aircraft industry, one major challenge is the characterization of extreme behaviors of materials used to design engine pieces. Especially, we will consider extreme risks associated with fatigue wear, which is a very classical type of damage suffered by engines during flights. It consists in the progressive weakening of a material due to the application of cyclic loadings a large number of times that can lead to its failure. As shown in Figure [cyclefatigue]{}, a loading cycle is defined by several quantities: the minimal and maximal stresses $\sigma _{\min }$ et $\sigma _{\max }$, the stress amplitude $\sigma _{a}=\frac{\sigma _{\max }-\sigma _{\min }}{2}$, and other indicators such as the stress ratio $\frac{\sigma _{\min }}{\sigma
_{\max }}$.
 \[cyclefatigue\]
The fatigue strength of a given material is studied through experimental campaigns designed at fixed environmental covariates to reproduce flight conditions. The trials consist in loading at a given stress level a dimensioned sample of material up to its failure or the date of end of trial. The lifetime of a specimen is measured in terms of number of cycles to failure, usually subject to right censoring.
![S-N curve[]{data-label="wohler"}](wohler.png)
The campaign results are then used to study fatigue resistance and are represented graphically in an S-N scale (see figure \[wohler\]). S-N curves highlight the existence of three fatigue regimes. Firstly, low cycle fatigue corresponds to short lives associated with high levels of stress. Secondly, during high cycle fatigue, the number of cycles to failure decreases log-linearily with respect to the loading. The last regime is the endurance limit, in which failure occurs at a very high number of cycles or doesn’t occur at all. We will focus in the following on the endurance limit, which is also the hardest regime to characterize since there is usually only few and scattered observations.
In this framework, we are focusing on minimal risk. The critical quantities that are used to characterize minimal risk linked to fatigue damage are failure quantiles, called in this framework allowable stresses at a given number of cycles and for a fixed level of probability. Those quantiles are of great importance since they intervene in decisions pertaining engine parts dimensioning, pricing decisions as well as maintenance policies.
Formalization of the industrial problem
---------------------------------------
The aim of this study is to propose a new design method for the characterization of allowable stress in very high cycle fatigue, for a very low risk $\alpha $ of order $10^{-3}$. We are willing to obtain a precise estimation method of the $\alpha -$failure quantile based on a minimal number of trials.
Denote $N$ the lifetime of a material in terms of number of cycles to failure and $S$ the stress amplitude of the loading, in MPa. Let $n_{0}$ be the targeted time span of order $10^{6}-10^{7}$ cycles.
Define the allowable stress $s_{\alpha }$ at $n_{0}$ cycles and level of probability $\alpha $ $=10^{-3}$ the level of stress that guarantee that the risk of failure before $n_{0}$ does not exceed $\alpha $: $$s_{\alpha }=\sup \left\{ s:\mathbb{P}(N\leq n_{0}|S=s)\leq \alpha \right\}
\label{pb1}$$
We will now introduce a positive r.v. $R=R_{n_{0}}$ modeling the resistance of the material at $n_{0}$ cycles and homogeneous to the stress. $R$ is the variable of interest in this study and its distribution $\mathbb{P}$ is defined as: $$\mathbb{P}(R\leq s) = \mathbb{P}(N\leq n_{0}|S=s). \label{loiR}$$
Thus, the allowable stress can be rewritten as the $\alpha-$quantile of the distribution of $R$,
$$s_{\alpha }=q_{\alpha }=\sup \left\{ s:\mathbb{P}(R\leq s)\leq \alpha \right\}.$$
However, $R$ is not directly observed. Indeed, the usable data collected at the end of a test campaign consists in couples of censored fatigue life - stress levels $\left(\min (N,n_{0}),s\right)$ where $s$ is part of the design of the experiment. The relevant information that can be drawn from those observations to characterize $R$ is restricted to indicators of whether or not the specimen tested has failed at $s$ before $n_{0}$. Therefore, the relevant observations corresponding to a campaign of $n$ trials are formed by a sample of variables $Y_{1},...,Y_{n}$ with for $1\leq i\leq n,$ $$Y_{i}=\left\{
\begin{array}{l}
1\text{ if }R_{i}\leq s_{i} \\
0\text{ if }R_{i}>s_{i}%
\end{array}%
\right.$$
where $s_{i}$ is the stress applied on specimen $i.$
Note that the number of observations is constrained by industrial and financial considerations; Thus $\alpha $ is way lower than $1/n$ and we are considering a quantile lying outside the sample range.
While we motivate this paper with the above industrial application, note that this kind of problem is of interest in other domains, such as broader reliability issues or medical trials through the estimation of the maximum tolerated dose of a given drug.
Extreme quantile estimation, a short survey {#revLit}
===========================================
As seen above estimating the minimal admissible constraint raises two issues; on one hand the estimation of an extreme quantile, and on the other hand the need to proceed to inference based on exceedances under thresholds. We present a short exposition of these two areas, keeping in mind that the literature on extreme quantile estimation deals with complete data, or data under right censoring.
Extreme quantiles estimation methods
------------------------------------
Extreme quantile estimation in the univariate setting is widely covered in the literature when the variable of interest $X$ is either completely or partially observed.
The usual framework is the study of the $(1-\alpha)-$quantile of a r.v $X$, with very small $\alpha$.
The most classical case corresponds to the setting where ${x}_{1-\alpha}$ is drawn from a $n$ sample of observations $X_1,\dots X_n$. We can distinguish estimation of high quantile, where $x_{1-\alpha}$ lies inside the sample range, see Weissman 1978 [@Weissman] and Dekkers and al. 1989 [@dekkers1989], and the estimation of an extreme quantile outside the boundary of the sample, see for instance De Haan and Rootzén 1993 [@deHann1993]. It is assumed that $X$ belongs to the domain of attraction of an extreme value distribution. The tail index of the latter is then estimated through maximum likelihood (Weissman 1978 [@Weissman]) or through an extension of Hill’s estimator (see the moment estimator by Dekkers and al. 1989 [@dekkers1989]). Lastly, the estimator of the quantile is deduced from the inverse function of the distribution of the $k$ largest observations. Note that all the above references assume that the distribution has a Pareto tail. An alternative modeling has been proposed by De Valk 2016 [@valk] and De Valk and Cai 2018 [@valk2], and consists in assuming a Weibull type tail, which enables to release some second order hypotheses on the tail. This last work deals with the estimation of extreme quantile lying way outside the sample range and will be used as a benchmark method in the following sections.
Recent studies have also tackled the issue of censoring. For instance, Beirlant and al. 2007 [@beirlant2007] and Einmahl and al. 2008 [@Einmahl2008] proposed a generalization of the peak-over-threshold method when the data are subjected to random right censoring and an estimator for extreme quantiles. The idea is to consider a consistent estimator of the tail index on the censored data and divide it by the proportion of censored observations in the tail. Worms and Worms 2014 [@worms2014] studied estimators of the extremal index based on Kaplan Meier integration and censored regression.
However the literature does not cover the case of complete truncation, i.e when only exceedances over given thresholds are observed. Indeed, all of the above are based on estimations of the tail index over weighed sums of the higher order statistics of the sample, which are not available in the problem of interest in this study. Classical estimation methods of extreme quantiles are thus not suited to the present issue.
In the following, we study designs of experiment at use in industrial contexts and their possible application to extreme quantiles estimation.
Sequential design based on dichotomous data
-------------------------------------------
In this section we review two standard methods in the industry and in biostatistics, which are the closest to our purpose. Up to our knowledge, no technique specifically addresses inference for extreme quantiles.
We address the estimation of small quantiles, hence the events of interest are of the form $\left( R<s\right) $ and the quantile is $q_{\alpha }$ for small $\alpha .$
The first method is the *staircase*, which is the present tool used to characterize a material fatigue strength*.*
The second one is the *Continual Reassessment Method (CRM)* which is adapted for assessing the admissible toxicity level of a drug in Phase 1 clinical trials.
Both methods rely on a parametric model for the distribution of the strength variable $R.$ We have considered two specifications, which allow for simple comparisons of performance, and do not aim at an accurate modelling in safety.
### The Staircase method
Denote $\mathbb{P}(R\leq s)=\phi (s,\theta _{0})$. Invented by Dixon and Mood (1948 [@dixon]), this technique aims at the estimation of the parameter $\theta _{0}$ through sequential search based on data of exceedances under thresholds. The procedure is as follows.
**Procedure**
Fix
- The initial value for the constraint, $S_{ini}$,
- The step $\delta >0$,
- The number of cycles $n_0$ to perform before concluding a trial,
- The total number of items to be tested, $K$.
The first item is tested at level $s_{(1)}=S_{ini}$. The next item is tested at level $s_{(2)}=S_{ini}-\delta$ in case of failure and $s_{(2)}=S_{ini}+\delta$ otherwise. Proceed sequentially on the $K-2$ remaining specimen at a level increased (respectively decreased) by $\delta$ in case of survival (resp. failure). The process is illustrated in figure \[staircase\].
Note that the proper conduct of the Staircase method relies on strong assumptions on the choice of the design parameters. Firstly, $S_{ini}$ has to be sufficiently close to the expectation of $R$ and secondly, $\delta $ has to lay between $0.5\sigma $ and $2\sigma $, where $\sigma$ designates the standard deviation of the distribution of $R$.
Denote $\mathbb{P}(R\leq s)=\phi (s,\theta _{0})$ and $Y_{i}$ the variable associated to the issue of the trial $i$, $1\leq
i\leq K$, where $Y_{i}$ takes value $1$ under failure and $0$ under no failure, $Y_{i}=\mathds{1}_{N_{a}\leq n_{0}}\sim \mathcal{B}(\phi
(s_{i},\theta _{0}))$.
 \[staircase\]
**Estimation**
After the $K$ trials, the parameter $\theta _{0}$ is estimated through maximization of the likelihood, namely
$$\widehat \theta = \underset{\theta}{\text{argmax}}{\prod_{i=1}^K
\phi(s_i,\theta)^{y_i } (1-\phi(s_i,\theta))^{ (1-y_i) }}.$$
**Numerical results**
The accuracy of the procedure has been evaluated on the two models presented below on a batch of 1000 replications, each with $K=100.$
*Exponential case*
Let $R\sim \mathcal{E}(\lambda)$ with $\lambda=0.2$. The input parameters are $S_{\text{ini}}=5
$ and $\delta =15\in \left[ 0.5\times \frac{1}{\lambda ^{2}},2\times \frac{1%
}{\lambda ^{2}}\right] $.
As shown in Table \[stairexp\], the relative error pertaining to the parameter $\lambda $ is roughly $25\%$, although the input parameters are somehow optimal for the method. The resulting relative error on the $10^{-3}$ quantile is $40\%.$ Indeed the parameter $\lambda $ is underestimated, which results in an overestimation of the variance $1/\lambda ^{2}$ , which induces an overestimation of the $10^{-3}$ quantile.
[|c|c|c|c|]{}\
&\
Mean & Std & Mean & Std\
-0.252 & 0.178 & 0.4064874 & 0.304\
*Gaussian case*
We now choose $R\sim \mathcal{N}(\mu,\sigma)$ with $\mu=60$ and $\sigma=10$. The value of $S_{\text{ini}}$ is set to the expectation and $\delta =7$ belongs to the interval $\left[ \frac{\sigma }{2},2\sigma \right] .$ The same procedure as above is performed and yields the results in Table \[stairgaus\].
[|c|c|c|c|c|c|]{}\
& &\
Mean & Std & Mean & Std & Mean & Std\
-0.059 & 0.034 & 1.544 & 0.903 & -1.753 & 0.983\
The expectation of $R$ is recovered rather accurately, whereas the estimation of the standard deviation suffers a loss in accuracy, which in turn yields a relative error of 180 % on the $10^{-3}$ quantile.
**Drawback of the Staircase method**
A major advantage of the Staircase lies in the fact that the number of trials to be performed in order to get a reasonable estimator of the mean is small. However, as shown by the simulations, this method is not adequate for the estimation of extreme quantiles. Indeed, the latter follows from an extrapolation based on estimated parameters, which furthermore may suffer of bias. Also, reparametrization of the distribution making use of the theoretical extreme quantile would not help, since the estimator would inherit of a large lack of accuracy.
### The Continuous Reassesment Method (CRM)
**General principle**
The CRM (O’Quigley, Pepe and Fisher, 1990[@quigley]) has been designed for clinical trials and aims at the estimation of $q_{\alpha }$ among $J$ stress levels $s_{1},...,s_{J}$, when $\alpha$ is of order $20\%$.
Denote $\mathbb{P}(R\leq s)=\psi (s,\beta _{0})$. The estimator of $q_{\alpha }$ is $$s^{\ast }:=\underset{s_j \in \{s_{1},...,s_{J}\}}{\text{arginf}}{|\psi
(s_{j},\beta _{0})-\alpha |}.$$This optimization is performed iteratively and $K$ trials are performed at each iteration.
Start with an initial estimator $\widehat{\beta _{1}}$ of $\beta _{0}$, for example through a Bayesian choice as proposed in [@quigley]. Define $$s_{1}^{\ast }:=\underset{s_j \in \{s_{1},...,s_{J}\}}{\text{arginf}}{|\psi
(s_{j},\widehat{\beta _{1}})-\alpha |}.$$
Every iteration follows a two-step procedure:
**Step 1.** Perform $J$ trials under $\psi (.,\beta _{0})$, say $R_{1,1},..,R_{1,J}$ and observe only their value under threshold, say $Y_{1,j}:={\Large 1}_{R_{1,j}<s_{1}^{\ast }},1\leq j\leq J.$
**Step i.** Iteration $i$ consists in two steps :
- Firstly an estimate $\widehat{\beta _{i}}$ of $\beta _{0}$ is produced on the basis of the information beared by the trials performed in all the preceding iterations through maximum likelihood under $\psi (.,\beta _{0})$ (or by maximizing the posterior distribution of the parameter).
- $$s_{i}^{\ast }:=\underset{s_j\in \{s_{1},...,s_{J}\}}{\text{arginf}}{|\psi
(s,}\widehat{{\beta _{i}}}{)-\alpha |};$$
This stress level $s_{i}^{\ast }$ is the one under which the next $K$ trials $Y_{i,1},\dots,Y_{i,K}$ will be performed in the Bernoulli scheme $\mathcal{B}\left(\psi (s_{i}^{\ast },\beta _{0})\right)$.
The stopping rule depends on the context (maximum number of trials or stabilization of the results).
Note that the bayesian inference is useful in the cases where there is no diversity in the observations at some iterations of the procedure, i.e when, at a given level of test $s_i^*$, only failures or survivals are observed.
**Application to fatigue data**
The application to the estimation of the minimal allowable stress is treated in a bayesian setting. We do not directly put a prior on the parameter $\beta_0$, but rather on the probability of failure. We consider a prior information of the form: *at a given stress level $s$, we can expect $k$ failures out of $n$ trials.* Denote $\pi_s$ the prior indexed on the stress level $s$. $\pi_s$ models the failure probability at level $s$ and has a Beta distribution given by $$\label{priorP}
\pi_{s} \sim \beta(k,n-k+1).$$
Let $R$ follow an exponential distribution: $\forall s \ge 0, \psi(s,\beta_0) = p_s = 1 - \exp(- \beta_0 s)$.
It follows $ \forall s,~ \beta_0 = - \frac{1}{s}\log(1- p_s)$.
Define the random variable $\Lambda_s = - \frac{1}{s}\log(1-\pi_{s})$ which, by definition of $\pi_s$, is distributed as an k-order statistic of a uniform distribution $U_{k,n}$.
The estimation procedure of the CRM is obtained as follows:
**Step 1.** Compute an initial estimator of the parameter $$\Lambda_{s} = \frac{1}{L} \sum_{l=1}^L - \frac{1}{s}\log(1-\pi_{s}^{l} )$$ with $\pi_{s}^l \sim \beta(k,n-k+1), 1\le l\le L$. Define $$s_{1}^{\ast }:=\underset{s_j\in \{s_{1},...,s_{J}\}}{\text{arginf}}{|( 1 - \exp(- \Lambda_s s_j))-\alpha |}.$$ and perform $J$ trials at level $s_1^\ast$. Denote the observations $Y_{1,j}:={\Large 1}_{R_{1,j}<s_{1}^{\ast }},1\leq j\leq J.$
**Step i.** At iteration $i$, compute the posterior distribution of the parameter:
$$\pi^*_{s_i} \sim \beta \left(k + \sum_{l=1}^{i}\sum_{j=1}^{J}Y_{l,j}~,~ n + (J\times i) -(k + \sum_{l=1}^{i}\sum_{j=1}^{J}Y_{l,j}) +1 \right)$$
The above distribution also corresponds an order statistic of the uniform distribution $U_{k + \sum_{l=1}^{i}\sum_{j=1}^{J}Y_{l,j}~,~n + (J\times i) }$. We then obtain an estimate $\Lambda_{s_1^\ast}$.
The next stress level $s_{i+1}^{\ast }$ to be tested in the procedure is then given by $$s_{i+1}^{\ast }:=\underset{s_j\in \{s_{1},...,s_{J}\}}{\text{arginf}}{|( 1 - \exp(- \Lambda_{s_1^\ast} s_j))-\alpha |}.$$
**Numerical simulation for the CRM**
Under the exponential model with parameter $\lambda =0.2$ and through $N=10$ iterations of the procedure, and $J=10$, with equally distributed thresholds $s_{1},..,s_{J}$ , and performing $K=50$ trials at each iteration, the results in Table \[CRMexp\] are obtained.
[|c|c|c|c|]{}\
&\
Mean & Std & Mean & Std\
0.129 & 0.48 & -0.799 & 0.606\
The $10^{-3}-$quantile is poorly estimated on a fairly simple model. Indeed for thresholds close to the expected quantile, nearly no failure is observed. So, for acceptable $K$, the method is not valid; figure \[crm\] shows the increase of accuracy with respect to $K.$
Both the Staircase and the CRM have the same drawback in the context of extreme quantile estimation, since the former targets the central tendency of the variable of interest and the latter aims at the estimation of quantiles of order 0.2 or so, far from the target $\alpha =10^{-3}$. Therefore, we propose an original procedure designed for the estimation of extreme quantiles under binary information.
![Relative error on the $10^{-3}$-quantile with respect to the number of trials for each stress level[]{data-label="crm"}](erreursCRM_m_tronc2.png)
A new design for the estimation of extreme quantiles {#Splitting}
====================================================
Splitting
---------
The design we propose is directly inspired by the general principle of Splitting methods used in the domain of rare events simulation and introduced by Kahn and Harris (1951 [@Kahn1951]).
The idea is to overcome the difficulty of targeting an extreme event by decomposing the initial problem into a sequence of less complex estimation problem. This is enabled by the splitting methodology which decompose a small probability into the product of higher order probabilities.
Denote $\mathbb{P}$ the distribution of the r.v. $R$. The event $\{ R\le s_\alpha \}$ can be expressed as the intersection of inclusive events for $s_{\alpha }=s_{m}<s_{m-1}<...<s_{1}$ it holds: $$\{R\leq s_{\alpha }\}=\{R\leq s_{m}\}\subset \dots \subset \{R\leq s_{1}\}.$$
It follows that
$$\mathbb{P}(R\leq s_{\alpha })=\mathbb{P}(R\leq
s_{1})\prod_{j=1}^{m-1}\mathbb{P}(R\leq s_{j+1}\mid R\leq s_{j})
\label{Prod}$$
\[split\]
The thresholds $(s_{j})_{j=1,\dots ,m}$ should be chosen such that all $\mathbb{P}(R\leq s_{j+1}\mid R\leq s_{j})_{j=1,\dots ,m}$ be of order $p=0.2$ or 0.3, in such a way that $\left\{ R\leq s_{j+1}\right\} $ is observed in experiments performed under the conditional distribution of $R$ given $\left\{ R\leq s_{j}\right\} $, and in a way which makes $\alpha $ recoverable by a rather small number of such probabilities $\mathbb{P}(R\leq s_{j+1}\mid
R\leq s_{j})$ making use of .
From the formal decomposition in , a practical experimental scheme can be deduced. Its form is given in algorithm \[Split\].
\[Split\] **Initialization**
Fix
- the first tested level $s_1$ (ideally the $p-$quantile of the distribution of $R$);
- the number $K$ of trials to be performed at each iteration.
**First step**
- $K$ trials are performed at level $s_1$. The observations are the indicators of failure $Y_{1,1},\dots,Y_{1,K}$, where $Y_{1,i} = \mathds{1}(R_{1,i}<s_1)$ of distribution $\mathcal{B}\left( \mathbb{P}(R \le s_1)\right)$.
- Determination of $s_2$, $p-$quantile of the truncated distribution $R \mid R \le s_1$.
**Iteration $j=2$ to $m$**
- $K$ trials are performed at level $s_j$ under the truncated distribution of $R \mid R \le s_{j-1}$ resulting to observations $Y_{j,1},\dots,Y_{j,k} \sim \mathcal{B}\left( \mathbb{P}(R \le s_j\mid R \le s_{j-1})\right)$.
- Determination of $s_{j+1}$, the $p-$quantile of $R \mid R \le s_{j}$.
The last estimated quantile $s_m$ provides the estimate of $s_\alpha$.
Sampling under the conditional probability {#operational_procedure}
------------------------------------------
In practice batches of specimen are put under trial, each of them with a decreasing strength; this allows to target the tail of the distribution $\mathbb{P}$ iteratively.
 \[fatigue\]
In other words, in the first step, points are sampled in zone (I). Then in the following step, only specimen with strength in zone II are considered, and so on. In the final step, the specimen are sampled in zone IV. At level $s_{m}$, they have a very small probability to fail before $n_{0}$ cycles under $\mathbb{P}$, however under their own law of failure, which is $\mathbb{P}(\mathbf{.}\mid R\leq s_{m-1})$, they have a probability of failure of order 0.2.
In practice, sampling in the tail of the distribution is achieved by introducing flaws in the batches of specimens. The idea is that the strength of the material varies inversely with respect to the size of the incorporated flaws. The flaws are spherical and located inside the specimen (not on its surface). Thus, as the procedure moves on, the trials are performed on samples of materials incorporating flaws of greater diameter. This procedure is based on the hypothesis that there is a correspondence between the strength of the material with flaw of diameter $\theta$ and the truncated strength of this same material without flaw under level of stress $s^*$, i.e. we assume that noting $R_{\theta }$ the strength of the specimen with flaw of size $\theta $, it holds that there exists $s^*$ such that $$\mathcal{L}(R_{\theta })\approx \mathcal{L}(R\mid R\leq s^*).$$
Before launching a validation campaign for this procedure, a batch of 27 specimen has been machined including spherical defects whose sizes vary between 0 and 1.8mm (see Figure \[defautseprouvettes\]). These first trials aim at estimating the decreasing relation between mean allowable stress and defects diameter $\theta$. This preliminary study enabled to draw the abatement fatigue curve as a function of $\theta$, as shown in Figure \[abattement\].
 \[defautseprouvettes\]
 \[abattement\]
Results in Figure \[abattement\] will be used during the splitting procedure to select the diameter $\theta$ to be incorporated in the batch of specimens tested at the current iteration as reflecting the sub-population of material of smaller resistance.
Modeling the distribution of the strength, Pareto model {#GPD_model}
-------------------------------------------------------
The events under consideration have small probability under $\mathbb{P}.$ By (\[Prod\]) we are led to consider the limit behavior of conditional distributions under smaller and smaller thresholds, for which we make use of classical approximations due to Balkema and de Haan (1974[@bal]) which stands as follows, firstly in the commonly known setting of exceedances over increasing thresholds. Denote $\widetilde{R}:=1/R$.
\[Thm de Haan\]For $\widetilde{R}$ of distribution $F$ belonging to the maximum domain of attraction of an extreme value distribution with tail index $c$, i.e. $F\in MDA(c)$, it holds that: There exists $a=a(s)>0$, such that: $$\lim\limits_{s \rightarrow \infty}\sup_{0\le x< \infty}
\left\lvert
\frac{1 - F\left(x + s\right)}{1 - F\left( s \right)} - \left(1 - G_{(c,a}(x)\right)
\right\rvert = 0$$ where $G_{(c,a)}$ is defined through $~$$$G_{(c,a)}(x)=1-\exp \left\{ -\int_{0}^{\frac{x}{a}}\left[ (1+ct)_{+}\right]
^{-1}dt\right\}$$ where $a>0$ and $c\in \mathbb{R}$.
The distribution $G$ is the Generalized Pareto distribution $GPD(c,a)$ is defined explicitly through$$1-G(x)=\left\{
\begin{array}{l}
(1+\frac{c}{a}x)^{-1/c}\text{ when }c\neq 0 \\
\exp (-\frac{x}{a})\text{ when }c=0\end{array}\right.$$where $x\geq 0$ for $c\geq 0$ and $0\leq x\leq -\frac{a}{c}$ if $c<0.$
Generalized Pareto distributions enjoy invariance through threshold conditioning, an important property for our sake. Indeed it holds, for $\widetilde{R}\sim GDP(c,a)$ and $x>s$,
$$\mathbb{P}\left( \widetilde{R}>x\mid \widetilde{R}>s\right) =\left( 1+\frac{c(x-s)}{a+cs}\right) ^{-1/c} \label{r}$$
We therefore state:
\[Prop stability GPD\]When $\widetilde{R}\sim GPD(c,a)$ then, given $\left( \widetilde{R}>s\right) $, the r.v. $\widetilde{R}-s$ follows a $GPD(c,a+cs)$.
The GPD’s are on the one hand stable under thresholding and on the other appear as the limit distribution for thresholding operations. This chain of arguments is quite usual in statistics, motivating the recourse to the ubiquous normal or stable laws for additive models. This plays in favor of GPD’s as modelling the distribution of $\widetilde{R}$ for excess probability inference. Due to the lack of memory property, the exponential distribution which appears as a possible limit distribution for excess probabilities in Theorem \[Thm de Haan\] do not qualify for modelling. Moreover since we handle variables $R$ which can approach $0$ arbitrarily (i.e. unbounded $\widetilde{R}$) the parameter $c$ is assumed positive.
Turning to the context of the minimal admissible constraint, we make use of the r.v. $R=1/\widetilde{R}$ and proceed to the corresponding change of variable.
When $c>0$, the distribution function of the r.v. $R$ writes for nonnegative $x$: $$\label{GPD 1}
F_{c,a}(x)=(1+\frac{c}{ax})^{-1/c}.$$
For $0<x<u$, the conditional distribution of $R$ given $\left\{ R<u\right\} $ is $$\mathbb{P}(R<x\mid R<u)=\left( 1-\frac{c(\frac{1}{x}-\frac{1}{u})}{a+\frac{c}{u}}\right) ^{-1/c}$$ which proves that the distribution of $R$ is stable under threshold conditioning with parameter $\left( a_{u},c\right) $ with $$a_{u}=a+\frac{c}{u}. \label{transition param a}$$In practice at each step $j$ in the procedure the stress level $s_{j}$ equals the corresponding threshold $1/\widetilde{s}_{j}$ , a right quantile of the conditional distribution of $\widetilde{R\text{ }}$ given $\left\{
\widetilde{R}>\widetilde{s}_{j-1}\right\} $. Therefore the observations take the form $Y_{i}=\mathds{1}_{R_{i}<s_{j-1}}=\mathds{1}_{\widetilde{R}_{i}>\widetilde{s}_{j-1}},~~i=1,\dots ,K_{j}$.
A convenient feature of model (\[GPD 1\]) lies in the fact that the conditional distributions are completely determined by the initial distribution of $R$ , therefore by $a\ $ and $c.$ The parameters $a_{j}$ of the conditional distributions are determined from these initial parameters and by the corresponding stress level $s_{j};$ see (\[transition param a\]).
Notations
---------
The distribution function of the r.v. $\widetilde{R}$ is a $GPD(c_{T},a_{T})$ of distrubution function $G_{(c_{T},a_{T})}.$ Note $\overline{G}_{(c_{T},a_{T})}=1-G_{(c_{T},a_{T})}.$
Our proposal relies on iterations. We make use of a set of thresholds $(\widetilde{s}_{1},...,\widetilde{s}_{m})$ and define for any $j\in
\{1,...,m\}$
$$G_{(c_{j},a_{j})}(x - \widetilde{s}_{j})=\mathbb{P(}\left. \widetilde{R}>x\right\vert \widetilde{R}>\widetilde{s}_{j})$$
with $c_{j}=c_{T}$ and $a_{j}=a_{T}+c_{T}\widetilde{s}_{j}$ where we used (\[r\]).
At iteration $j$, denote $(\widehat{c},\widehat{a})_{j}$ the estimators of $(c_{j},a_{j})$.Therefore $1-G_{(\widehat{c},\widehat{a})_{j}}(x - \widetilde{s}_{j})$ estimates $\mathbb{P(}\left. \widetilde{R}>x\right\vert \widetilde{R}>\widetilde{s}_{j})
$. Clearly, estimators of $(c_{T},a_{T})$ can be recovered from $(\widehat{c},\widehat{a})_{j}$ through $\widehat{c}_{T}=\widehat{c}$ and $\widehat{a}_{T}=\widehat{a}-\widehat{c}~\widetilde{s}_{j}.$
Sequential design for the extreme quantile estimation {#proc}
-----------------------------------------------------
Fix $m$ and $p$ , where $m$ denotes the number of stress levels under which the trials will be performed, and $p$ is such that $p^m=\alpha .$
Set a first level of stress, say $s_{1}$ large enough (i.e. $\widetilde{s}_{1}=1/s_{1}$ small enough) so that $p_1 = \mathbb{P}(R<s_{1})$ is large enough and perform trials at this level. The optimal value of $s_{1}$ should satisfy $p_1=p$, which cannot be secured. This choice is based on expert advice.
Turn to $\widetilde{R}:=1/R$. Estimate $c_{T}$ and $a_{T}$, for the GPD $\left( c_{T},a_{T}\right) $ model describing $\widetilde{R}$, say $(\widehat{c},\widehat{a})_1$, based on the observations above $\widetilde{s}_{1}$ (note that under $s_{1}$ the outcomes of $R$ are easy to obtain, since the specimen is tested under medium stress).
Define $$\widetilde{s}_{2}:=\sup \left\{ s:\overline{G}_{(\widehat{c},\widehat{a})_{1}}\left( s - \widetilde s_1\right) <p\right\}$$the $(1-p)-$quantile of $G_{(\widehat{c},\widehat{a})_{1}}.$ $\widetilde s_2$ is the level of stress to be tested at the following iteration.
Iterating from step $j=2$ to $m-1$, perform $K$ trials under $G_{(c_1,a_1)}
$ say $\widetilde{R}_{j,1},..,\widetilde{R}_{j,K}$ and consider the observable variables $Y_{j,i}:={\Large 1}_{\widetilde{R}_{j,i}>\widetilde{s}_{j}}$. Therefore the $K$ iid replications $Y_{j,1},..,Y_{j,K}$ follow a Bernoulli $\mathcal{B}(\overline{G}_{({c}_{j-1},{a}_{j-1})}\left( \widetilde s_{j} - \widetilde s_{j-1}\right) )$, where $\widetilde s_j$ has been determined at the previous step of the procedure. Estimate $(c_{j},a_{j})$ in the resulting Bernoulli scheme, say $(\widehat{c},\widehat{a})_{j}$. Then define $$\begin{split}
\widetilde s_{j+1}&:=\sup \left\{ s:\overline{G}_{(\widehat{c},\widehat{a})_{j}}\left( s - \widetilde s_j\right) <p\right\} \\
&=G_{\left( \widehat{c},\widehat{a}\right)
_{j}}^{-1}(1-p)+\widetilde{s}_{j},
\end{split}$$ which is the $(1-p)-$quantile of the estimated conditional distribution of $\widetilde{R}$ given $\{ \widetilde{R}>\widetilde{s_{j}}\}$, i.e. $G_{(\widehat{c},\widehat{a})_{j}}$, and the next level to be tested.
In practice a conservative choice for $m$ is given by $m=\left\lceil \frac{log\alpha }{logp}\right\rceil $, where $\lceil. \rceil $ denotes the ceiling function. This implies that the attained probability $\widetilde{\alpha}$ is less than or equal to $\alpha.$
The $m$ stress levels $\widetilde{s}_{1}<\widetilde{s}_{1}<\dots <\widetilde{s}_{m}=\widetilde{q}_{1-\alpha}$ satisfy $$\begin{split}
\widetilde{\alpha }& =\overline{G}(\widetilde{s}_{1})\prod_{j=1}^{m-1}\overline{G}_{\left( \widehat{c},\widehat{a}\right)
_{j}}(\widetilde{s}_{j+1} - \widetilde s_j) \\
& ={p}_{1}p^{m-1}
\end{split}$$
Finally by its very definition $\widetilde{s}_{m}$ is a proxy of $\widetilde{q}_{1-\alpha }.$
Although quite simple in its definition, this method bears a number of drawbacks, mainly in the definition of $\left( \widehat{c},\widehat{a}\right) _{j}.$ The next section addresses this question.
Sequential enhanced design in the Pareto model {#EstimationProc}
==============================================
In this section we focus on the estimation of the parameters $\left(
c_{T},a_{T}\right) $ in the $GPD(c_{T},a_{T})$ distribution of $\widetilde{R}.$ One of the main difficulties lies in the fact that the available information does not consist of replications of the r.v. $\widetilde{R}$ under the current conditional distribution $G_{(c_{j},a_{j})}$ of $\widetilde{R}$ given $\left( \widetilde{R}>\widetilde{s_{j}}\right) $ but merely on very downgraded functions of those.
At step $j$ we are given $G_{(\widehat{c},\widehat{a})_{j}}$ and define $\widetilde{s}_{j+1}$ as its $\left( 1-p\right) -$quantile. Simulating $K$ r.v. $\widetilde{R}_{j,i}$ with distribution $G_{(c_{j},a_{j})}$, the observable outcomes are the Bernoulli ($p$) r.v.’s $Y_{j,i}:=1_{\widetilde{R}_{j,i}>\widetilde{s}_{j+1}}.$ This loss of information with respect to the $\widetilde{R}_{j,i}$ ’s makes the estimation step for the coefficients $(\widehat{c},\widehat{a})_{j+1}$ quite complex; indeed $(\widehat{c},\widehat{a})_{j+1}$ is obtained through the $Y_{j,i}$’s, $1\leq i\leq K$.
It is of interest to analyze the results obtained through standard Maximum Likelihood Estimation of $(\widehat{c},\widehat{a})_{j+1}.$ The quantile $\widetilde{q}_{1-\alpha }$ is loosely estimated for small $\alpha $; as measured on 1000 simulation runs, large standard deviation of $\widehat{\widetilde{q}}_{1-\alpha }$ is due to poor estimation of the iterative parameters $(\widehat{c},\widehat{a})_{j+1}.$ We have simulated $n=200$ realizations of r.v.’s $Y_{i}$ with common Bernoulli distribution with parameter $\overline{G}_{\left( c_{T},a_{T}\right) }(\widetilde{s}_{1}).$ Figure \[loglik\] shows the log likelihood function of this sample as the parameter of the Bernoulli $\overline{G}_{\left( c^{\prime },a^{\prime
}\right) }(\widetilde{s_{0}})$ varies according to $\left( c^{\prime
},a^{\prime }\right) .$ As expected this function is nearly flat in a very large range of $\left( c^{\prime },a^{\prime }\right) .$
This explains the poor results in Table \[procQuantm\] obtained through the Splitting procedure when the parameters at each step are estimated by maximum likelihood, especially in terms of dispersion of the estimations. Moreover, the accuracy of the estimator of $\widetilde{q}_{1-\alpha }$ quickly decreases with the number $K$ of replications $Y_{j,i}$, $1\leq i\leq K$.
Changing the estimation criterion by some alternative method does not improve significantly; Figure \[disp\_quant\] shows the distribution of the resulting estimators of $\widetilde{q}_{1-\alpha }$ for various estimation methods (minimum Kullback Leibler, minimum Hellinger and minimum L1 distances - see their definitions in Appendix \[div\]) of $\left( c_{T},a_{T}\right).$
This motivates the need for an enhanced estimation procedure.
Minimum Q25 Q50 Mean Q75 Maximum
--------- -------- -------- -------- -------- -----------
67.07 226.50 327.40 441.60 498.90 10 320.00
: Estimation of the $(1-\alpha)-$quantile, $\widetilde s_{\alpha}=469.103$, through procedure \[proc\] with $K=50$[]{data-label="procQuant"}
----------------------------- ---------- ----------- --------- ---------
**$\widetilde s_{\alpha}$** Mean Std Mean Std
469.103 1 276.00 12 576.98 441.643 562.757
----------------------------- ---------- ----------- --------- ---------
: Estimation of the $(1-\alpha)-$quantile, $\widetilde s_{\alpha}=469.103$, through procedure \[proc\] for different values of $K$[]{data-label="procQuantm"}
![Log-likelihood of the Pareto model with binary data[]{data-label="loglik"}](niv_loglik.png)
![Estimations of the $\protect\alpha-$quantile based on the Kullback-Leibler, L1 distance and Hellinger distance criterion[]{data-label="disp_quant"}](disp_quantiles.png)
An enhanced sequential criterion for estimation {#procImpML}
-----------------------------------------------
We consider an additional criterion which makes a peculiar use of the iterative nature of the procedure. We will impose some control on the stability of the estimators of the conditional quantiles through the sequential procedure.
At iteration $j-1$, the sample $Y_{j-1,i}$ , $1\leq i\leq K$ has been generated under $G_{(\widehat{c},\widehat{a})_{j-2}\text{ }}$and provides an estimate of $p$ through $$\widehat{p}_{j-1}:=\frac{1}{K}\sum_{i=1}^{n}Y_{j-1,i}. \label{p_ji}$$The above $\widehat{p}_{j-1}$ estimates $\mathbb{P}\left( \widetilde{R}>\widetilde{s}_{j-1}\mid \widetilde{R}>\widetilde{s}_{j-2}\right) $ conditionally on $\widetilde{s}_{j-1}$ and $\widetilde{s}_{j-2}.$ We write this latter expression $\mathbb{P}\left( \widetilde{R}>\widetilde{s}_{j-1}\mid \widetilde{R}>\widetilde{s}_{j-2}\right) $ as a function of the parameters obtained at iteration $j$ , namely $(\widehat{c},\widehat{a})_{j}.
$The above r.v’s $\ Y_{j-1,i}$ stem from variables $\widetilde{R}_{j-1,i}$ greater than $\ \widetilde{s}_{j-2}.$ At step $j,$ estimate then $\mathbb{P}\left( \widetilde{R}>\widetilde{s}_{j-1}\mid \widetilde{R}>\widetilde{s}_{j-2}\right) $ making use of $G_{(\widehat{c},\widehat{a})_{j}}.$ This backward estimator writes $$\frac{\overline{G}_{(\widehat{c},\widehat{a})_{j}}(\widetilde{s}_{j-1})}{\overline{G}_{(\widehat{c},\widehat{a})_{j}}(\widetilde{s}_{j-2})}=1-G_{(\widehat{c},\widehat{a})_{j}}(\widetilde{s}_{j-1}-\widetilde{s}_{j-2}).$$The distance $$\left\vert \left( \overline{G}_{(\widehat{c},\widehat{a})_{j}}(\widetilde{s}_{j-1}-\widetilde{s}_{j-2})\right) -\widehat{p_{j-1}}\right\vert \label{A}$$should be small, since both $ \overline{G}_{(\widehat{c},\widehat{a})_{j}}(\widetilde{s}_{j-1}-\widetilde{s}_{j-2}) $ and $\ \widehat{p}_{j-1}$ should approximate $p.$
Consider the distance between quantiles $$\left\vert (\widetilde{s}_{j-1}-\widetilde{s}_{j-2})-G_{(\widehat{c},\widehat{a})_{j}}^{-1}(1-\widehat{p}_{j-1})\right\vert . \label{B}$$
An estimate $(\widehat{c},\widehat{a})_{j}$ can be proposed as the minimizer of the above expression for $(\widetilde{s}_{j-1}-\widetilde{s}_{j-2})$ for all $j$. This backward estimation provides coherence with respect to the unknown initial distribution $G_{\left( c_{T},a_{T}\right) }$. Would we have started with a good guess $(\widehat{c},\widehat{a})=\left(
c_{T},a_{T}\right) $ then the successive $(\widehat{c},\widehat{a})_{j},\
\widetilde{s}_{j-1}$ etc would make (\[B\]) small, since $\widetilde{s}_{j-1}$ (resp. $\widetilde{s}_{j-2}$) would estimate the $p-$conditional quantile of $\mathbb{P}\left( \left. .\right\vert \widetilde{R}>\widetilde{s}_{j-2}\right) $ (resp. $\mathbb{P}\left( \left. .\right\vert \widetilde{R}>\widetilde{s}_{j-3}\right) $).
It remains to argue on the set of plausible values where the quantity in (\[B\]) should be minimized.
We suggest to consider a confidence region for the parameter $\left(
c_{T},a_{T}\right) .$ With $\widehat{p}_{j}$ defined in (\[p\_ji\]) and $\gamma \in \left( 0,1\right) $ define the $\gamma -$confidence region for $p$ by
$$I_{\gamma }=\left[ \widehat{p}_{j}-z_{1-\gamma /2}\sqrt{\frac{\widehat{p}_{j}(1-\widehat{p}_{j})}{K-1}};\widehat{p}_{j}+z_{1-\gamma /2}\sqrt{\frac{\widehat{p}_{j}(1-\widehat{p}_{j})}{K-1}}\right]$$where $z_{\tau }$ is the $\tau -$quantile of the standard normal distribution. Define $$\mathcal{S}_{j}=\left\{ (c,a):\left( 1-G_{(c,a)}(\widetilde{s}_{j}-\widetilde{s}_{j-1})\right) \in I_{\gamma }\right\} .$$Therefore $\mathcal{S}_{j}$ is a plausible set for $(\widehat{c}_{T},\widehat{a}_{T}).$
We summarize this discussion:
At iteration $j,$ the estimator of $\left( c_{T},a_{T}\right) $ is a solution of the minimization problem $$\min_{(c,a)\in \mathcal{S}_{j}}\left\vert (\widetilde{s}_{j-1}-\widetilde{s}_{j-2})-G_{(c,a+c\widetilde{s}_{j-2})}^{-1}(1-\widehat{p}_{j-1})\right\vert .$$ The optimization method used is the Safip algorithm (Biret and Broniatowski, 2016 [@Biret]) As seen hereunder, this heuristics provides good performance.
Simulation based numerical results\[Subsection numGPD\]
-------------------------------------------------------
This procedure has been applied in three cases. A case considered as reference is $(c_{T},a_{T})=(1.5,1.5)$; secondly the case when $(c_{T},a_{T})=(0.8,1.5)$ describes a light tail with respect to the reference. Thirdly, a case $(c_{T},a_{T})=(1.5,3)$ defines a distribution with same tail index as the reference, but with a larger dispersion index.
Table \[parIC\] shows that the estimation of $\widetilde{q}_{1-\alpha }$ deteriorates as the tail of the distribution gets heavier; also the procedure underestimates $\widetilde{q}_{1-\alpha }.$
----------------------------------------------------------- -------- -------
**Parameters**
Mean Std
$c=0.8$, $a_0=1.5$ and $\widetilde s_{\alpha}= 469.103$ -0.222 0.554
$c=1.5$, $a_0=1.5$ and $\widetilde s_{\alpha}=31621.777 $ -0.504 0.720
$c=1.5$, $a_0=3$ and $\widetilde s_{\alpha}=63243.550 $ 0.310 0.590
----------------------------------------------------------- -------- -------
: Mean and std of relative errors on the $(1-\alpha)-$quantile of GPD calculated through 400 replicas of procedure \[procImpML\].[]{data-label="parIC"}
Despite these drawbacks, we observe an improvement with respect to the simple Maximum Likelihood estimation; this is even more clear, when the tail of the distribution is heavy. Also, in contrast with the ML estimation, the sensitivity with respect to the number $K$ of replications at each of the iterations plays in favor of this new method: As $K$ decreases, the gain with respect to Maximum Likelihood estimation increases notably, see Figure \[compMVQm\].
\[compMVQ\]   [The red line stands stands for the real value of $s_\alpha$ ]{}
![Estimations of the $(1-\alpha)-$quantile of a $GPD(0.8,1.5)$ obtained by Maximum Likelihood and by the improved Maximum Likelihood method for different values of $K$.[]{data-label="compMVQm"}](compMV_ICm50.png "fig:") ![Estimations of the $(1-\alpha)-$quantile of a $GPD(0.8,1.5)$ obtained by Maximum Likelihood and by the improved Maximum Likelihood method for different values of $K$.[]{data-label="compMVQm"}](compMV_ICm30.png "fig:") ![Estimations of the $(1-\alpha)-$quantile of a $GPD(0.8,1.5)$ obtained by Maximum Likelihood and by the improved Maximum Likelihood method for different values of $K$.[]{data-label="compMVQm"}](compMV_ICm15.png "fig:")
[The red line stands stands for the real value of $s_\alpha$ ]{}
Performance of the sequential estimation\[subsection comparaison de Valk GPD\]
------------------------------------------------------------------------------
As stated in chapter \[revLit\], there is to our knowledge no method dealing with similar question available in the literature. Therefore we compare the results of our method, based on observed exceedances over thresholds, with the results that could be obtained by classical extreme quantiles estimation methods assuming we have complete data at our disposal; those may be seen as benchmarks for an upper bound of the performance of our method.
### Estimation of an extreme quantile based on complete data, de Valk’s estimator
In order to provide an upper bound for the performance of the estimator, we make use of the estimator proposed by De Valk and Cai (2016). This work aims at the estimation of a quantile of order $p_{n}\in \lbrack n^{-\tau
_{1}};n^{-\tau _{2}}]$, with $\tau _{2}>\tau _{1}>1$ , where $n$ is the sample size. This question is in accordance with the industrial context which motivated the present paper. De Valk’s proposal is a modified Hill estimator adapted to log-Weibull tailed models. De Valk’s estimator is consistent, asymptotically normally distributed, but is biased for finite sample size.We briefly recall some of the hypotheses which set the context of de Valk’s approach.
Let $X_{1},\dots ,X_{n}$ be $n$ iid r.v’s with distribution $F$, and denote $X_{k:n}$ the $k-$ order statistics. A tail regularity assumption is needed in order to estimate a quantile with order greater than $1-$ $1/n.$
Denote $U(t)=F^{-1}\left( 1-1/t\right) $, and let the function $q$ be defined by
$$q(y)=U(e^{y})=F^{-1}\left( 1-e^{-y}\right)$$
for $y>0$.
Assume that $$\lim\limits_{y\rightarrow \infty }~\frac{\log q(y\lambda )-\log q(y)}{g(y)}=h_{\theta }(\lambda )~~~\lambda >0 \label{logweibulltail}$$where $g$ is a regularly varying function and $$h_{\theta }(\lambda )=\left\{
\begin{array}{l}
\frac{\lambda ^{\theta }-1}{\theta }\text{ if }\theta \neq 0 \\
\log \lambda \text{ if }\theta =0\end{array}\right.$$
de Valk writes condition \[logweibulltail\] as $\log q\in ERV_{\theta }(g)$.
*Remark :* Despite its naming of log-Generalized tails, this condition also holds for Pareto tailed distributions, as can be checked, providing $\theta =1.$
We now introduce de Valk’s extreme quantile estimator.
Let $$\vartheta _{k,n}:=\sum_{j=k}^{n}\frac{1}{j}.$$
Let $q(z)$ be the quantile of order $e^{-z}=p_{n}$ of the distribution $F$. The estimator makes use of $X_{n-l_{n}:n}$, an intermediate order statistics of $X_{1},..,X_{n}$, where $l_{n}$ tends to infinity as $n\rightarrow
\infty $ and $l_{n}$ $\ /n\rightarrow 0.$
de Valk’s estimator writes $$\widehat{q}(z)=X_{n-l_{n}:n}\exp \left\{ g(\vartheta _{l_{n},n})h_{\theta
}\left( \frac{z}{\vartheta _{l_{n+1},n}}\right) \right\} .$$
When the support of $F$ overlaps $\mathbb{R}^{-}$ then the sample size $n$ should be large; see de Valk ([@valk2]) for details.
Note that, in the case of a $GPD(c,a)$, parameter $\theta$ is known and equal to 1 and the normalizing function $g$ is defined by $g(x)=cx$ for $x>0$.
### Loss in accurracy due to binary sampling
In Table \[ValkIt\] we compare the performance of de Valk’s method with ours on the model, making use of complete data in de Valk’s estimation, and of dichotomous ones in our approach. Clearly de Valk’s results cannot be attained by the present sequential method, due to the loss of information induced by thresholding and dichotomy. Despite this, the results can be compared, since even if the bias of the estimator clearly exceeds the corresponding bias of de Valk’s, its dispersion is of the same order of magnitude, when handling heavy tailed GPD models. Note also that given the binary nature of the data considered, the average relative error is quite honorable. We can assess that a large part of the volatility of the estimator produced by our sequential methodology is due to the nature of the GPD model as well as to the sample size.
---------------------------------------------- ------- ------- -------- -------
**Parameters**
Mean Std Mean Std
$c=0.8$, $a_0=1.5$ and $s_\alpha= 469.103$ 0.052 0.257 -0.222 0.554
$c=1.5$, $a_0=1.5$ and $s_\alpha=31621.777 $ 0.086 0.530 -0.504 0.720
$c=1.5$, $a_0=3$ and $s_\alpha=63243.550 $ 0.116 0.625 0.310 0.590
---------------------------------------------- ------- ------- -------- -------
: Mean and std of the relative errors on the $1-\alpha-$quantile of GPD on complete and binary data for samples of size $n=250$ computed through $400$ replicas of both estimation procedures.Estimations on complete data are obtained with de Valk’s method; estimations on binary data are provided by the sequential design.[]{data-label="ValkIt"}
Sequential design for the Weibull model {#WeibullModel}
=======================================
The main property which led to the GPD model is the stability through threshold conditioning. However the conditional distribution of $\widetilde{R}$ given $\left\{ \widetilde{R}>s\right\}$ takes a rather simple form which allows for some variation of the sequential design method.
The Weibull model
-----------------
Denote $\widetilde{R}\sim W(\alpha ,\beta )$, with $\alpha ,\beta >0$ a Weibull r.v. with scale parameter $\alpha $ and shape parameter $\beta .$ let $G$ denote the distribution function of $\widetilde{R}$ , $g$ its density function and $G^{-1}$ its quantile function. We thus write for non negative $x$ $$\begin{split}
~G(x)& =1-\exp \left( -\left( \frac{x}{\alpha }\right) ^{\beta }\right) \\
\text{ for }0<u<1,~~G^{-1}(u)& =\alpha (-\log (1-u))^{1/\beta }
\end{split}$$
The conditional distribution of $\widetilde{R}$ is a truncated Weibull distribution
$$\begin{split}
\text{ for }\widetilde{s}_{2}>\widetilde{s}_{1},~~\mathbb{P}(\widetilde{R}>\widetilde{s}_{2}\mid \widetilde{R}>\widetilde{s}_{1})& =\frac{\mathbb{P}(\widetilde{R}>\widetilde{s}_{2})}{\mathbb{P}(\widetilde{R}>\widetilde{s}_{1})} \\
& =\exp \left\{ \left( -\left( \frac{s_{2}}{\alpha }\right) ^{\beta }+\left(
\frac{s_{1}}{\alpha }\right) ^{\beta }\right) \right\}
\end{split}$$
Denote $G_{s_{2}}$ the distribution function of $\widetilde{R}$ given $\left( \widetilde{R}>\widetilde{s}_{2}\right) $.
The following result helps. For $\widetilde{s}_{2}>\widetilde{s}_{1}$,
$$\log \mathbb{P}(\widetilde{R}>\widetilde{s}_{2}\mid \widetilde{R}>\widetilde{s}_{1})=\left[ \left( \frac{\widetilde{s}_{2}}{\widetilde{s}_{1}}\right)
^{\beta }-1\right] \log \mathbb{P}(\widetilde{R}>\widetilde{s}_{1})$$
Assuming $\mathbb{P}(\widetilde{R}>\widetilde{s}_{1})=p$, and given $\widetilde{s}_{1}$ we may find $\widetilde{s}_{2}$ the conditional quantile of order $1-p$ of the distribution of $\widetilde{R}$ given $\left\{ \widetilde{R}>\widetilde{s}_{1}\right\} $. This solves the first iteration of the sequential estimation procedure through $$\log p=\left[ \left( \frac{\widetilde{s}_{2}}{\widetilde{s}_{1}}\right)
^{\beta }-1\right] \log p$$
where the parameter $\beta $ has to be estimated on the first run of trials.
The same type of transitions holds for the iterative procedure; indeed for $\widetilde{s}_{j+1}>\widetilde{s}_{j}>\widetilde{s}_{j-1}$
$$\begin{split}
\log \mathbb{P}(\widetilde{R}>\widetilde{s}_{j+1}\mid \widetilde{R}>\widetilde{s}_{j})& =\left[ \frac{\log \mathbb{P}(\widetilde{R}>\widetilde{s}_{j+1}\mid \widetilde{R}>\widetilde{s}_{j-1})}{\log \mathbb{P}(\widetilde{R}>\widetilde{s}_{j}\mid \widetilde{R}>\widetilde{s}_{j-1})}-1\right] \log
\mathbb{P}(\widetilde{R}>\widetilde{s}_{j}\mid \widetilde{R}>\widetilde{s}_{j-1}) \\
& =\left[ \frac{\widetilde{s}_{j-1}^{\beta }-\widetilde{s}_{j+1}^{\beta }}{\widetilde{s}_{j-1}^{\beta }-\widetilde{s}_{j}^{\beta }}-1\right] \log
\mathbb{P}(\widetilde{R}>\widetilde{s}_{j}\mid \widetilde{R}>\widetilde{s}_{j-1})
\end{split}$$
At iteration $j$ the thresholds $\widetilde{s}_{j}$ and $\widetilde{s}_{j-1}$ are known; the threshold $\widetilde{s}_{j+1}$ is the $(1-p)-$ quantile of the conditional distribution, $\mathbb{P}(\widetilde{R}>\widetilde{s}_{j+1}\mid
\widetilde{R}>\widetilde{s}_{j})=p$, hence solving $$\log p=\left[ \frac{\widetilde{s}_{j-1}^{\beta }-\widetilde{s}_{j+1}^{\beta }}{\widetilde{s}_{j-1}^{\beta }-\widetilde{s}_{j}^{\beta }}-1\right] \log p$$where the estimate of $\beta $ is updated from the data collected at iteration $j.$
Numerical results
-----------------
Similarly as in Sections \[Subsection numGPD\] and \[subsection comparaison de Valk GPD\] we explore the performance of the sequential design estimation on the Weibull model. We estimate the $(1-\alpha)-$ quantile of the Weibull distribution in three cases. In the first one, the scale parameter $a$ and the shape parameter $b$ satisfy $\left( a,b\right)
=\left( 3,0.9\right)$. This corresponds to a strictly decreasing density function, with heavy tail. In the second case, the distribution is skewed since $\left( a,b\right) =\left( 3,1.5\right) $ and the third case is $\left( a,b\right) =\left( 2,1.5\right) $ and describes a less dispersed distribution with lighter tail.
Table \[errWeibulldeValk\] shows that the performance of our procedure here again depends on the shape of the distribution. The estimators are less accurate in case 1, corresponding to a heavier tail. Those results are compared to the estimation errors on complete data through de Valk’s methodology. As expected, the loss of accuracy linked to data deterioration is similar to what was observed under the Pareto model, although a little more important. This can be explained by the fact that the Weibull distribution is less adapted to the splitting structure than the GPD.
------------------------------------------ -------- ------- ------- -------
**Parameters**
Mean Std Mean Std
$a_0=3$, $b_0=0.9$ et $s_\alpha= 25.69 $ 0.282 0.520 0.127 0.197
$a_0=3$, $b_0=1.5$ et $s_\alpha=10.88 $ -0.260 0.490 0.084 0.122
$a_0=2$, $b_0=1.5$ et $s_\alpha=7.25$ -0.241 0.450 0.088 0.140
------------------------------------------ -------- ------- ------- -------
: Mean and std of relative errors on the $(1-\alpha)-$quantile of Weibull distributions on complete and binary data for samples of size $n=250$ computed through $400$ replicas.Estimations on complete data are obtained with de Valk’s method; estimations on binary data are provided by the sequential design.[]{data-label="errWeibulldeValk"}
Model selection and misspecification {#model_selection_missp}
====================================
In the above sections, we considered two models whose presentation was mainly motivated by theoretical properties. As it has already been stated in paragraph \[GPD\_model\], the modeling of $\widetilde R$ by a GPD with $c$ strictly positive is justified by the assumption that the support of the original variable $R$ may be bounded by 0. However, note that the GPD model can be easily extended to the case where $c=0$. It then becomes the trivial case of the estimation of an exponential distribution.
Though we did exclude the exponential case while modeling the excess probabilities of $\widetilde R$ by a GPD, we still considered the Weibull model in section \[WeibullModel\], which belongs to the max domain of attraction for $c=0$. On top of being exploitable in the splitting structure, the Weibull distribution is a classical tool when modeling reliability issues, it thus seemed natural to propose an adaptation of the sequential method for it.
In this section, we discuss the modeling decisions and give some hints on how to deal with misspecification.
Model selection
---------------
The decision between the Pareto model with tail index strictly positive and the Weibull model has been covered in the literature. There exists a variety of tests on the domain of attraction of a distribution.
Dietrich and al. (2002 [@Dietrich2002]) Drees and al. (2006 [@Drees2006]) both propose a test for extreme value conditions related to Cramer-von Mises tests. Let $X$ of distribution function $G$. The null hypothesis is $$H_{O}: G \in MDA(c_0).$$ In our case, the theoretical value for the tail index is $c_0=0$. The former test provides a testing procedure based on the tail empirical quantile function, while the latter uses a weighted approximation of the tail empirical distribution. Choulakian and Stephens (2001 [@Choulakian2001]) proposes a goodness of fit test in the fashion of Cramer-von Mises tests in which the unknown parameters are replaced by maximum likelihood estimators. The test consists in two steps: firstly the estimation of the unknown parameters, and secondly the computation of the Cramer-von Mises $W^2$ or Anderson-Darling $A^2$ statistics. Let $X_1,\dots,X_n$ be a random sample of distribution $G$. The hypothesis to be tested is: $H_O$: The sample is coming from a $GPD(c_0, \widehat{a})$. The associated test statistics are given by: $$\begin{split}
&W^2 = \sum_{i=1}^n \left(\widehat{G}(x_{(i)}) - \frac{2i-1}{2n}\right)^2 + \frac{1}{12n};\\
&A^2 = -n-\frac{1}{n} \sum_{i=1}^n (2i-1)\left\{ \log(\widehat{G}(x_{(i)})) + \log(1-\widehat{G}(x_(n+1-i)) \right\},
\end{split}$$ where $x_{(i)}$ denotes the $i-$th order statistic of the sample. The authors provide the corresponding tables of critical points.
Jurečková and Picek (2001 [@Jureckova2001]) designed a non-parametric test for determining whether a distribution $G$ is light or heavy tailed. The null hypothesis is defined by : $$H_{c_O}: x^{1/c_0} (1 - G(x)) \le 1 ~~ \forall x>x_0 \text{ for some } x_0>0$$ with fixed hypothetical $c_0$. The test procedure consists in splitting the data set in $N$ samples and computing the empirical distribution of the extrema of each sample.
The evaluation of the suitability of each model for fatigue data is precarious. The main difficulty here is that it is not possible to perform goodness-of-fit type tests, since firstly, we collect the data sequentially during the procedure and do not have a sample of available observations beforehand, and secondly, we do not observe the variable of interest $R$ but only peaks over chosen thresholds. The existing tests procedures are not compatible with the reliability problem we are dealing with. On the first hand, they assume that the variable of interest is fully observed and are mainly semi-parametric or non-parametric tests based on order statistics. On the other hand, their performances rely on the availability of a large volume of data. This is not possible in the design we consider since fatigue trial are both time consuming and extremely expensive.
Another option consists of validating the model *a posteriori*, once the procedure is completed using expert advices to confirm or not the results. For that matter, a procedure following the design presented in \[operational\_procedure\] is currently being carried out. Its results should be available in a few months and will give hints on the most relevant model.
Misspecification
----------------
In paragraph \[GPD\_model\], we assumed that $\widetilde{R}$ initially follows a GPD. In practice, the distribution may have its excess probabilities converge towards it as the thresholds increase but differ from a GPD. In the following, let us assume that $\widetilde{R}$ does not follow a GPD (of distribution function $F$) but another distribution $G$ whose tail gets closer and closer to a GPD.
In this case, the issue is to control the distance between $G$ and the theoretical GPD and to determine from which thresholding level it becomes negligible. One way to deal with this problem is to restrict the model to a class of distributions that are not so distant from $F$: Assume that the distribution function $G$ of the variable of interest $\widetilde{R}$ belongs to a neighborhood of the $GPD(c,a)$ of distribution function $F$, defined by: $$\label{GPD_neighborhood}
V_\epsilon(F) = \left\{G: \sup_x |\bar F(x) - \bar G(x)|w(x) \le \epsilon\right\},$$ where $\epsilon \ge 0$ and $w$ an increasing weight function such that $\lim_{x\rightarrow\infty} w(x) = \infty$.
$V_\epsilon(F)$ defines a neighborhood which does not tolerate large departures from $F$ in the right tail of the distribution.
Let $x \ge s$, it follows from a bound for the conditional probability of $x$ given $R>s$: $$\label{cond_ineq}
\frac{\bar F(x) - \epsilon/w(x)}{\bar F(s) + \epsilon/w(s)}
\le
\frac{\bar G(x)}{\bar G(s)}
\le
\frac{\bar F(x+) + \epsilon/w(x)}{\bar F(s) - \epsilon/w(s)}.$$ When $\epsilon=0$, the bounds of match the conditional probabilities of the theoretical Pareto distribution.
In order to control the distance between $F$ and $G$, the bound above may be rewritten in terms of relative error with respect to the Pareto distribution. Using a Taylor expansion of the right and left bounds when $\epsilon$ is close to 0, it becomes: $$1 - u(s,x).\epsilon
\le
\frac{\frac{\bar G(x)}{\bar G(s)} }{\frac{\bar F(x)}{\bar F(s)}}
\le
1 + u(x,s).\epsilon,$$ where $$u(s,x) = \frac{\left(1+\frac{cs}{a}\right)^{1/c}}{w(s)} + \frac{\left(1+\frac{cx}{a}\right)^{1/c}}{w(x)}.$$
For a given $\epsilon$ close to 0, the relative error on the conditional probabilities can be controlled upon $s$. Indeed, then the relative error is bounded by a fixed level $\delta>0$ whenever: $$\frac{\left(1+\frac{cs}{a}\right)^{1/c}}{w(s)} \le \frac{\delta}{\epsilon}
\frac{\left(1+\frac{cx}{a}\right)^{1/c}}{w(x)}.$$
Perspectives, generalization of the two models {#Perspectives}
==============================================
In this work, we have considered two models for $\widetilde R$ that exploits the thresholding operations used in the splitting method. This is a limit of this procedure as the lack of relevant information provided by the trials do not enable a flexible modeling of the distribution of the resistance. In the following, we present ideas of extensions and generalizations of those models, based on common properties of the GPD and Weibull models.
Variations around mixture forms
-------------------------------
When the tail index is positive, the GPD is completely monotone, and thus can be written as the Laplace transform of a probability distribution. Thyrion (1964[@thyrion]) and Thorin (1977[@thorin]) established that a $GPD(a_T,c_T)$, with $c_T>0$, can be written as the Laplace transform of a Gamma r.v $V$ whose parameters are functions of $a_T$ and $c_T$: $V ~ \sim~ \Gamma\left(\frac{1}{c_T},\frac{a_T}{c_T}\right)$. Denote $v$ the density of $V$,
$$\label{lapla}
\begin{split}
\forall x\ge 0, ~~ \bar{G}(x) = &\int_{0}^{\infty}\exp(-xy) v(y)dy \\
& \text{ where } ~~ v(y) = \frac{(a_T/c_T)^{1/c}}{\Gamma(1/c_T)}y^{1/c_T -1}\exp\left(-\frac{a_Ty}{c_T}\right).
\end{split}$$
It follows that the conditional survival function of $\widetilde R$, $\bar{G}_{s_j}$, is given by: $$\begin{aligned}
{2}
\mathbb{P}(\widetilde{R}>\widetilde s_{j+1} \mid \widetilde{R}_j > \widetilde s_j) & = \bar{G}_{\widetilde s_j}(\widetilde s_{j+1}-\widetilde s_j) \\
& = \int_{0}^{\infty}\exp \left\{-(\widetilde s_{j+1} - \widetilde s_j)y \right\} v_j(y)dy, &&\\
&\text{ ~~where } V_j \text{ is a r.v of distribution } \Gamma\left(\frac{1}{c_j},\frac{a_j}{c_j}\right). &&\end{aligned}$$
with $c_j=c_T$ and $a_j=a_{j-1}+c_T (\widetilde s_j - \widetilde s_{j-1})$.
Expression gives room to an extension of the Pareto model. Indeed, we could consider distributions of $\widetilde R$ that share the same mixture form with a mixing variable $W$ that possesses some common characteristics with the Gamma distributed r.v. $V.$
Similarly, the Weibull distribution $W(\alpha, \beta)$ can also be written as the Laplace transform of a stable law of density $g$ whenever $\beta\le1$. Indeed, it holds from Feller 1971[@feller]) (p. 450, Theorem 1) that: $$\label{fellerTh1}
\forall x\ge 0, ~~ \exp\left\{-x^{\beta} \right\}= \int_{0}^{\infty}\exp(-xy) g(y)dy$$ where $g$ is the density of an infinitely divisible probability distribution.
It follows, for $s_j< s_{j+1}$ $$\label{condWeibullLaplace}
\begin{split}
\mathbb{P}(\widetilde{R}>\widetilde s_{j+1} \mid \widetilde{R}_j > \widetilde s_j) &= \frac{\exp\left\{-(\widetilde s_{j+1}/\alpha)^{\beta} \right\}}{\exp\left\{-(\widetilde s_{j}/\alpha)^{\beta} \right\}} \\
& = \frac{ \int_{0}^{\infty}\exp\left\{(-(\widetilde s_{j+1}/\alpha)y\right\} g(y)dy }{ \int_{0}^{\infty}\exp\left\{-(\widetilde s_{j}/\alpha)y\right\} g(y)dy }
= \frac{ \int_{0}^{\infty}\exp\left\{-(\widetilde s_{j+1}/\alpha)y\right\} g(y)dy }{K(s_j)} \\
&= \frac{1}{K(s_j)} \int_{0}^{\infty}\exp\left\{-\widetilde s_{j+1}u \right\} g_{\alpha}(u)
)du \\
& \quad \quad\text{ with } u=y/\alpha \text{ and } g_{\alpha}(u)=\alpha g(\alpha u)
\end{split}$$
Thus an alternative modeling of $\widetilde R$ could consist in any distribution that can be written as a Laplace transform of a stable law of density $w_{\alpha,\beta}$ defined on $\mathbb{R}_+$ and parametrized by $(\alpha,\beta)$, that complies to the following condition: For any $s>0$, the distribution function of the conditional distribution of $\widetilde R$ given $\widetilde R>s$ can be written as the Laplace transform of $w_{\alpha,\beta}^{(\alpha,s)}( . )$ where $$x>s, w_{\alpha,\beta}^{(\alpha,s)}(x) = \frac{\alpha w_{\alpha,\beta}(\alpha x)}{K(s)},$$ where $K( . )$ is defined in .
Variation around the GPD
------------------------
Another approach, inspired by Naveau et al. (2016[@naveau2016]), consists in modifying the model so that the distribution of $\widetilde R$ tends to a GPD as $x$ tends to infinity and it takes a more flexible form near 0.
$\widetilde R$ is generated through $G_{(c_T,a_T)}^{-1}(U)$ with $U\sim\mathcal{U}[0,1]$. Let us consider now a deformation of the uniform variable $V=L^{-1}(U)$ defined on $[0,1]$, and the transform $W$ of the GPD: $W^{-1}(U)=G_{(c_T,a_T)}^{-1}(L^{-1} (U))$.
The survival function of the GPD being completely monotone, we can choose $W$ so that the distribution of $\widetilde R$ keeps this property.
If $\phi : [0,\infty[ \rightarrow \mathbb{R}$ is completely monotone and let $\psi$ be a positive function, such that its derivative is completely monotone, then $\phi(\psi)$ est completely monotone.
The transformation of the GPD has cumulative distribution function $W=L(G_(c_T,a_T))$ and survival function $\bar W= \bar L(G_(c_T,a_T))$. $G(c_T,a_T)$ is a Berstein function, thus $\bar W$ is completely monotone if $\bar L$ is also.
### Examples of admissible functions:
*(1) Exponential form :*
$$\begin{split}
&L(0) = 0 \\
& L(x) = \frac{1-\exp(-\lambda x^\alpha)}{1-\exp(-\lambda)} ~~~ \text{avec } 0\le \alpha \le 1 \text{ et } \lambda>0 \\
& L(1) = 1
\end{split}$$
The obtained transformation is: $\forall x>0$,
$$\bar W_{(\lambda,c_T,a_T)}(x) = \bar L ( G(x)) = \frac{\exp\left(-\lambda \left[1-(1+\frac{c_T}{a_T})^{-1/c_T}\right]^\alpha \right) - \exp(-\lambda) }{1-\exp(-\lambda)}$$
with $\bar W_{(\lambda,c_T,a_T)}(x) $ completely monotone.
*(2) Logarithmic form:* $$\begin{split}
&L(0) = 0 \\
& L(x) = \frac{\log(x+1)}{\log 2 } ~~~~ \text{ \big( or more generally } \frac{\log(\alpha x+1)}{\log 2 }, ~\alpha >0 \text{\big )}\\
& L(1) = 1
\end{split}$$
and $\forall x>0$, $$\bar W_{(c_T,a_T)}(x) = 1-\frac{\log\left(2-(1+\frac{c_T}{a_T})^{-1/c_T}\right)}{\log 2}$$ *(3) Root form:*
$$\begin{split}
&L(0) = 0 \\
& L(x) = \frac{\sqrt{x+1}-1 }{\sqrt{2} -1}\\
& L(1) = 1
\end{split}$$ and $$\bar W_{(c_T,a_T)}x) = 1-\frac{\sqrt{2-(1+\frac{c_T x}{a_T})^{-1/c_T}} -1}{\sqrt{2}}$$ *(4) Fraction form:*
$$\begin{split}
&L(0) = 0 \\
& L(x) = \frac{(\alpha + 1)x}{x+ \alpha}, ~~ \alpha >0\\
& L(1) = 1
\end{split}$$ and $$\bar W_{(\alpha,c_T,a_T)}(x) = 1-\frac{ (\alpha+1)\left(1-(1+\frac{c_T x}{a_T})^{-1/c}\right) }{1-(1+\frac{c_T x}{a_T})^{-1/c_T} + \alpha}$$ The shapes of the above transformations of the GPD are shown in Figure \[transf\_gpd\].
![Survival functions associated with transformations of the GPD$(0.8,1.5)$[]{data-label="transf_gpd"}](transf_gpd.png)
However those transformations do not conserve the stability through thresholding of the Pareto distribution. Thus, their implementation does not give stable results. Still they give some insight on a simple generalization of the proposed models usable under additional information on the variable of interest.
Conclusion
==========
The splitting induced procedure presented in this article proposes an innovative experimental plan to estimate an extreme quantile. Its development has been motivated by on the one hand major industrial stakes, and on the other hand the lack of relevance of existing methodologies. The main difficulty in this setting is the nature of the information at hand, since the variable of interest is latent, therefore only peaks over thresholds may be observed. Indeed, this study is directly driven from an application in material fatigue strength: when performing a fatigue trial, the strength of the specimen obviously can not be observed; only the indicator of whether or not the strength was greater than the tested level is available.
Among the methodologies dealing with such a framework, none is adapted to the estimation of extreme quantiles. We therefore proposed a plan based on splitting methods in order to decompose the initial problem into less complex ones. The splitting formula introduces a formal decomposition which has been adapted into a practical sampling strategy targeting progressively the tail of the distribution of interest.
The structure of the splitting equation has motivated the parametric hypothesis on the distribution of the variable of interest. Two models exploiting a stability property have been presented: one assuming a Generalized Pareto Distribution and the other a Weibull distribution.
The associated estimation procedure has been designed to use the iterative and stable structure of the model by combining a classical maximum likelihood criterion with a consistency criterion on the sequentially estimated quantiles. The quality of the estimates obtained through this procedure have been evaluated numerically. Though constrained by the quantity and quality of information, those results can still be compared to what would be obtained ideally if the variable of interest was observed.
On a practical note, while the GPD is the most adapted to the splitting structure, the Weibull distribution has the benefit of being particularly suitable for reliability issues. The experimental campaign launched to validate the method will contribute to select a model.
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abstract: |
We obtained thorough photometric observations of two binary near-Earth asteroids (66391) Moshup = 1999 KW4 and (88710) 2001 SL9 from June 2000 to June 2019. We modeled the data and derived physical and dynamical properties of the binary systems. For (66391) 1999 KW4, we derived its mutual orbit’s pole, semimajor axis and eccentricity that are in agreement with radar-derived values (Ostro et al. \[2006\]. Science, 314, 1276–1280). However, we found that the data are inconsistent with a constant orbital period and we obtained unique solution with a quadratic drift of the mean anomaly of the satellite of $-0.65 \pm 0.16$ deg/$\mbox{yr}^2$ (all quoted uncertainties correspond to 3$\sigma$). This means that the semimajor axis of the mutual orbit of the components of this binary system increases in time with a mean rate of $1.2 \pm 0.3$ cm/yr.
For (88710) 2001 SL9, we determined that the mutual orbit has a pole within ${\sim}10^{\circ}$ of $(L, B) = (302^{\circ},-73^{\circ})$ (ecliptic coordinates), and is close to circular (eccentricity $< 0.07$). The data for this system are also inconsistent with a constant orbital period and we obtained two solutions for the quadratic drift of the mean anomaly: $2.8 \pm 0.2$ and $5.2 \pm 0.2$ deg/$\mbox{yr}^2$, implying that the semimajor axis of the mutual orbit of the components decreases in time with a mean rate of $-2.8 \pm 0.2$ or $-5.1 \pm 0.2$ cm/yr for the two solutions, respectively.
The expanding orbit of (66391) 1999 KW4 may be explained by mutual tides interplaying with binary YORP (BYORP) effect (McMahon, J., Scheeres, D. \[2010\]. Icarus 209, 494-–509). However, a modeling of the BYORP drift using radar-derived shapes of the binary components predicted a much higher value of the orbital drift than the observed one. It suggests that either the radar-derived shape model of the secondary is inadequate for computing the BYORP effect, or the present theory of BYORP overestimates it. It is possible that the BYORP coefficient has instead an opposite sign than predicted; in that case, the system may be moving into an equilibrium between the BYORP and the tides.
In the case of (88710) 2001 SL9, the BYORP effect is the only known physical mechanism that can cause the inward drift of its mutual orbit.
Together with the binary (175706) 1996 FG3 which has a mean anomaly drift consistent with zero, implying a stable equilibrium between the BYORP effect and mutual body tides (Scheirich et al. \[2015\]. Icarus 245, 56-63), we now have three distinct cases of well observed binary asteroid systems with their long-term dynamical models inferred. That indicates a presence of all the three states of the mutual orbit evolution – equilibrium, expanding and contracting – in the population of near-Earth binary asteroids.
address:
- 'Astronomical Institute, Academy of Sciences of the Czech Republic, Fričova 1, CZ-25165 Ondřejov, Czech Republic'
- 'Department of Aerospace Engineering Sciences, The University of Colorado at Boulder, Boulder, CO, USA'
- 'Sugarloaf Mountain Observatory, South Deerfield, MA, USA'
- 'Modra Observatory, Department of Astronomy, Physics of the Earth, and Meteorology, FMPI UK, Bratislava SK-84248, Slovakia'
- 'Institute of Astronomy of Kharkiv National University, Sumska Str. 35, Kharkiv 61022, Ukraine'
- 'Lowell Observatory, 1400 W Mars Hill Road, Flagstaff, AZ 86001, USA'
- 'Lunar and Planetary Laboratory, University of Arizona, 1629 East University Boulevard, Tucson, AZ 85721, USA'
- 'U.S. Naval Academy, Annapolis, MD, USA'
- 'Sonoita Research Observatory, 77 Paint Trail, Sonoita, AZ 85637, USA'
- 'Deptartment of Space Studies, Southwest Research Institute, Boulder, CO 80302, USA'
- 'Ulugh Beg Astronomical Institute, Astronomicheskaya Street33, 100052 Tashkent, Uzbekistan'
- 'Shed of Science South Observatory, Pontotoc, TX, USA'
- 'Kharadze Abastumani Astrophysical Observatory, Ilya State University, K. Cholokashvili Avenue 3/5, Tbilisi 0162, Georgia'
- 'Samtskhe-Javakheti State University, Rustaveli Street 113, Akhaltsikhe 0080, Georgia'
- 'Crimean Astrophysical Observatory of Russian Academy of Sciences, 298409 Nauchny, Ukraine'
- 'Planetary Science Institute, 1700 E. Fort Lowell Road, Tucson, AZ 85719, USA'
- 'Northern Arizona University, Flagstaff, AZ, USA'
- 'Planetary Defense Coordination Office, NASA Headquarters, 300 E Street SW, Washington, DC 20546, USA'
- 'Astronomical Institute, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 180 00, Prague 8, Czech Republic'
- 'Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic'
- 'Keldysh Institute of Applied Mathematics, RAS, Miusskaya sq. 4, Moscow 125047, Russia'
author:
- 'P. Scheirich'
- 'P. Pravec'
- 'P. Kušnirák'
- 'K. Hornoch'
- 'J. McMahon'
- 'D. J. Scheeres'
- 'D. Čapek'
- 'D. P. Pray'
- 'H. Kučáková'
- 'A. Galád'
- 'J. Vraštil'
- 'Yu. N. Krugly'
- 'N. Moskovitz'
- 'L. D. Avner'
- 'B. Skiff'
- 'R. S. McMillan'
- 'J. A. Larsen'
- 'M. J. Brucker'
- 'A. F. Tubbiolo'
- 'W. R. Cooney'
- 'J. Gross'
- 'D. Terrell'
- 'O. Burkhonov'
- 'K. E. Ergashev'
- 'Sh. A. Ehgamberdiev'
- 'P. Fatka'
- 'R. Durkee'
- 'E. Lilly Schunova'
- 'R. Ya. Inasaridze'
- 'V. R. Ayvazian'
- 'G. Kapanadze'
- 'N. M. Gaftonyuk'
- 'J. A. Sanchez'
- 'V. Reddy'
- 'L. McGraw'
- 'M. S. Kelley'
- 'I. E. Molotov'
title: 'A satellite orbit drift in binary near-Earth asteroids (66391) 1999 KW4 and (88710) 2001 SL9 — Indication of the BYORP effect'
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Proposed running head: BYORP effect on 1999 KW4 and 2001 SL9
Editorial correspondence to:\
Peter Scheirich, Ph.D.\
Astronomical Institute AS CR\
Fričova 1\
Ondřejov\
CZ-25165\
Czech Republic\
Phone: 00420-323-620115\
Fax: 00420-323-620263\
E-mail address: [email protected]\
Asteroids, dynamics; Near-Earth objects; Photometry
Introduction {#Introduction}
============
Binary asteroids exhibit interesting internal dynamics driven by thermal emission from irregularly shaped components, but there is only one study constraining its limits based on direct measurements so far. Scheirich et al. (2015) found an upper limit on drift of the mutual orbit of binary near-Earth asteroid (175706) 1996 FG3, that is consistent with the theory of Jacobson and Scheeres (2011a) of that synchronous binary asteroids are in a state of stable equilibrium between binary YORP (BYORP) effect (which is a secular change of the mutual orbit of the components of a binary asteroid due to the emission of thermal radiation from asymmetric shapes of the components) and mutual body tides. In this paper, we present a comprehensive analysis of mutual orbit drifts in two well-observed binary near-Earth asteroids (NEAs).
The NEA (66391) Moshup = 1999 KW4 was discovered by Lincoln Near-Earth Asteroid Research in Socorro, New Mexico, on 1999 May 20. Its binary nature was revealed by Benner et al. (2001). We obtained thorough photometric observations for it in six apparitions from 2000 to 2019. Since the asteroid was named only recently and its original designation 1999 KW4 is well-known to the asteroid science community, we use it throughout this paper.
The NEA (88710) 2001 SL9 was discovered by Near-Earth Asteroid Tracking at Palomar on 2001 September 18. Its binary nature was revealed by Pravec et al. (2001). We obtained thorough photometric observations for it in five apparitions from 2001 to 2015.
Among binary NEAs observed so far, our photometric datasets for these three sytems (together with 1996 FG3) are the longest coverages obtained, providing a unique opportunity to study an evolution of the mutual orbits of components of small binary asteroids.
The structure of this paper is as follows. In Section \[Model\], we present a model of the mutual orbit of the components of 1999 KW4 and 2001 SL9 constructed from our complete photometric datasets. Then in Sections \[ParamsKW4\] and \[ParamsSL9\], we summarize our results with already known parameters of the two binaries. In Section \[byorp\], we then discuss implications of the observed characteristics, especially on the BYORP theory, from the derived drifts of the mutual orbits.
Mutual orbit models of 1999 KW4 and and 2001 SL9 {#Model}
================================================
Observational data {#KW4ObsData}
------------------
Time span No. of nights Telescope References
------------------------------ --------------- ----------------------------- ------------
2000-06-19.0 to 2000-06-29.0 5 0.65-m Ondřejov P06
2001-06-03.2 to 2001-06-20.9 7 0.41-m River Oaks P06
4 0.65-m Ondřejov P06
2016-06-07.9 to 2016-06-22.3 6 0.65-m Ondřejov This work
6 0.5-m Sugarloaf Mountain This work
2017-06-01.8 to 2017-06-27.0 8 0.65-m Ondřejov This work
6 0.5-m Sugarloaf Mountain This work
2018-06-05.9 to 2018-06-18.9 9 0.65-m Ondřejov This work
2019-05-31.1 to 2019-06-09.2 6 1.8-m Spacewatch II This work
6 0.65-m Ondřejov This work
5 0.5-m Sonoita This work
3 0.5-m Sugarloaf Mountain This work
3 0.5-m Shed of Science South This work
: Observations of (66391) 1999 KW4[]{data-label="TableDataKW4"}
Reference: P06 (Pravec et al., 2006)\
Time span No. of nights Telescope References
------------------------------ --------------- --------------------- ------------
2001-10-10.9 to 2001-10-21.3 7 0.65-m Ondřejov P06
2 0.5-m Palmer Divide P06
2012-09-11.9 to 2012-11-15.4 4 1.54-m La Silla This work
4 1.5-m Maidanak This work
2013-10-12.0 to 2013-12-05.1 7 1.54-m La Silla This work
2 0.7-m Abastumani This work
1 1.0-m Simeiz This work
2014-10-18.0 to 2014-10-26.1 4 1.54-m La Silla This work
2015-07-09.2 to 2015-08-17.3 6 1.8-m Lowell This work
3 2.2-m U. Hawaii This work
: Observations of (88710) 2001 SL9 []{data-label="TableDataSL9"}
Reference: P06 (Pravec et al., 2006)
----------------------------- ------------------------------------------------ ----------------------
Telescope Observatory References for
observational and
reduction procedures
2.2-m U. Hawaii Mauna Kea, Hawaii 1
1.8-m Lowell Lowell Observatory, Arizona 2
1.8-m Spacewatch II Spacewatch, Arizona M07, L20
1.54-m La Silla La Silla, European Southern Observatory, Chile P14
1.5-m Maidanak Maidanak Astronomical Observatory, Uzbekistan P19
1.0-m Simeiz Simeiz, Crimea 3
0.7-m Abastumani Abastumani, Georgia K16, P19
0.65-m Ondřejov Ondřejov, Czech Republic P06
0.5-m Sugarloaf Mountain Sugarloaf Mountain Observatory, Massachusetts V17
0.5-m Sonoita Sonoita Research Observatory, Arizona C15
0.5-m Palmer Divide Palmer Divide Observatory, Colorado P06
0.5-m Shed of Science South Shed of Science South Observatory, Texas 4
0.41-m River Oaks River Oaks Observatory, Texas P06
----------------------------- ------------------------------------------------ ----------------------
: Observational stations[]{data-label="TableObs"}
References: 1: The observations were made in the Cousins R filter. Standard procedure of image reduction included dark removal and flatfield correction. 2: The observations were reduced using the same procedure as the observations from the 1.54-m La Silla, see Pravec et al. (2014) for details. 3: The observations were carried with a 1-m Ritchey-Chrétien telescope at Simeiz Department of the Crimean Astrophysical Observatory using camera FLI PL09000. The observations were made in the Johnson-Cousins photometric system. Standard procedure of image reduction included dark removal and flatfield correction. The aperture photometry was done with the AstPhot package described in Mottola et al. (1995). The differential lightcurves were calculated with respect to an ensemble of comparison stars by the method described in Erikson et al. (2000) and Krugly (2004). 4: The Shed of Science South utilizes a 0.5m Corrected Dall Kirkham telescope operating at a focal ratio of f4.5 and a pixel scale of 1.24 arc seconds per pixel using an SBIG ST10XME. Flat, dark, and bias images were applied using MaximDL and photometry was done using MPO Canopus. All images were unfiltered. C15 (Cooney et al., 2015), K16 (Krugly et al., 2016), L20 (Larsen, J. A., et al. 2020. In preparation.), M07 (McMillan et al., 2007), P06 (Pravec et al., 2006), P14 (Pravec et al., 2014), P19 (Pravec et al., 2019), V17 (Vokrouhlický et al., 2017).
The data used in our analysis, obtained during six and five apparitions for 1999 KW4 and 2001 SL9, respectively, are summarized in Tables \[TableDataKW4\] and \[TableDataSL9\]. The references and descriptions of observational procedures of the individual observatories are summarized in Table \[TableObs\].
The data were reduced using the standard technique described in Pravec et al. (2006). By fitting a two-period Fourier series to data points outside mutual (occultation or eclipse) events, the rotational lightcurves of the primary (short-period) and the secondary (long-period), which are additive in intensities, were separated. The long-period component containing the mutual events and the secondary rotation lightcurve is then used for subsequent numerical modeling.
Numerical model {#NumModel}
---------------
We constructed models of the two binary asteroids using the technique of Scheirich and Pravec (2009) that was further developed in Scheirich et al. (2015). In following, we outline the basic points of the method, but we refer the reader to the 2009 and 2015 papers for details of the technique.
The shapes of the binary asteroid components were represented with ellipsoids, orbiting each other on a Keplerian orbit with apsidal precession and allowing for a quadratic drift in mean anomaly. The primary was modeled as an oblate spheroid, with its spin axis assumed to be normal to the mutual orbital plane of the components (i.e., assuming zero inclination of the mutual orbit). The shape of the secondary was modeled as a prolate spheroid in synchronous rotation, with its long axis aligned with the centers of the two bodies (i.e., assuming zero libration). The shapes were approximated with 1016 and 252 triangular facets for the primary and the secondary, respectively. The components were assumed to have the same albedo. The brightness of the system as seen by the observer was computed as a sum of contributions from all visible facets using a ray-tracing code that checks which facets are occulted by or are in shadow from the other body. A combination of Lommel-Seeliger and Lambert scattering laws was used (see, e.g., Kaasalainen et al., 2002).
The quadratic drift in mean anomaly, $\Delta M_d$, was fitted as an independent parameter. It is the coefficient in the second term of the expansion of the time-variable mean anomaly: $$M (t) = M (t_0) + n (t-t_0) + \Delta M_d (t - t_0)^2, \label{dMd1}$$ where $$\Delta M_d = \frac{1}{2} \dot{n}, \label{dMd2}$$ where $n$ is the mean motion, $t$ is the time, and $t_0$ is the epoch. $\Delta M_d$ was stepped from $-15$ to $+15$ deg/yr$^2$ in the case of 1999 KW4 and from $-9$ to $+39$ deg/yr$^2$ in the case of 2001 SL9 and all other parameters were fitted at each step.[^1]
To reduce the complexity of the model, we estimated upper limits on the eccentricity of the mutual orbits by fitting the data from the best covered apparitions: the 2001 apparition for 1999 KW4 and the 2013 apparition for 2001 SL9. The model includes a precession of the line of apsides. The pericenter drift rate depends on the polar flattening of the primary (see Murray and Dermott, 1999, Eq. (6.249)), but as the polar flattenings are poorly constrained from the data (see Tables \[tablePropKW4\] and \[tablePropSL9\]), we instead fit the drift rate as an independent parameter. Its initial values were stepped in a range from zero to $25^{\circ}/{\rm day}$. This range encompassed all plausible values for the flattening of the primaries and other parameters of the systems.
Since we found that the upper limits on eccentricity were low, in further modeling of the data from all apparitions together, we set the eccentricity equal to zero for simplicity and efficiency. This assumption had a negligible effect on the accuracy of other derived parameters of the models.
Across all observations, we found a unique solution for the system parameters except for an ambiguity in the quadratic drift in mean anomaly and the orbital period of 2001 SL9, see Tables \[tablePropKW4\] and \[tablePropSL9\]. We describe and discuss these parameters in Sections \[ParamsKW4\] and \[ParamsSL9\]. Plots of the RMS residuals (root mean square of observed magnitudes minus the values calculated from the model) vs $\Delta M_d$ are shown in Figs. \[RMS\_vs\_DMdKW4\] and \[RMS\_vs\_DMdSL9\]. In order to save computing time, the plots were constructed using spherical shapes of both components. However, neighborhoods of local minima were then revisited using elipsoidal shapes in order to improve the fit.
For 1999 KW4, the RMS residuals of the two best local minima obtained using the spherical shapes (with $\Delta M_d$ of $-0.65$ and $-1.3$ deg/yr$^2$) were 0.0307 and 0.0320 mag, respectively. The fits improved to 0.0251 and 0.0266 mag using the elipsoidal shapes. The fit is significanly poorer for the latter solution. The former solution provides a satisfactory fit to the data and it is accepted as real solution for the binary asteroid parameters.
For 2001 SL9, the RMS residuals of the five best local minima obtained using the spherical shapes (with $\Delta M_d$ of 2.8, 5.2, 7.6, 4.0 and 0.5 deg/yr$^2$) were 0.0238. 0.0238, 0.0245, 0.0246 and 0.0248 mag, respectively. The fits improved to 0.0236, 0.0236, 0.0243, 0.0245 and 0.0245 mag using the elipsoidal shapes; the marginal improvement is due to that the secondary of 2001 SL9 is not prominently elongated. The first two solutions provide satisfactory fit to the data; one of them is a real solution for the binary asteroid parameters, but we cannot resolve this ambiguity with the available data. The other three solutions with the higher RMS residuals provide significantly poorer fits to the data and they do not appear real. Figures \[additionalM0KW4\] and \[additionalM0SL9\] show the quadratic drift in the mean anomaly with respect to a solution with constant orbital period. Examples of the long-period component data together with the synthetic lightcurves of the best-fit solutions are presented in Figs. \[66391\_00-19\_synth\] and \[2001sl9\_01-15\_synth\]. Uncertainty areas of the orbital poles are shown in Figs. \[66391\_LB\_polar\] and \[2001sl9\_01-15\_LB\_polar\].
We estimated realistic uncertainties of the fitted parameters using the procedure described in Scheirich and Pravec (2009). For each parameter, we obtained its admissible range that corresponds to a 3-$\sigma$ uncertainty.
![The RMS residuals vs. $\Delta M_d$ for solutions of the model of (66391) 1999 KW4 presented in Section \[NumModel\]. Each dot represents the best-fit result with $\Delta M_d$ fixed and other parameters varied. The plots were constructed using spherical shapes of both components; see text for details. []{data-label="RMS_vs_DMdKW4"}](66391_RMS_vs_DMd_png){width="\textwidth"}
![The RMS residuals vs. $\Delta M_d$ for solutions of the model of (88710) 2001 SL9 presented in Section \[NumModel\]. Each dot represents the best-fit result with $\Delta M_d$ fixed and other parameters varied. The plots were constructed using spherical shapes of both components; see text for details.[]{data-label="RMS_vs_DMdSL9"}](2001sl9_RMS_vs_DMd_png){width="\textwidth"}
Parameters of (66391) 1999 KW4 {#ParamsKW4}
==============================
In this section, we summarize the best-fit model parameters of the binary system (66391) 1999 KW4 and overview previous publications. The parameters are listed in Table \[tablePropKW4\].
[cccc]{} Parameter & Value & Unc. & Reference\
\
$H_V$ & $16.74 \pm 0.22$ & 1$\sigma$ & This work\
$G$ & $(0.24 \pm 0.11)^a$ & 1$\sigma$ & This work\
$p_{\rm V}$ & $0.162 \pm 0.034$ & 1$\sigma$ & This work\
Taxon. class & Q & & This work\
\
$D_{\rm 1,C}$ (km) & $1.367 \pm 0.041^b$ & 1$\sigma$ & From O06\
$D_{\rm 1,V}$ (km) & $1.317 \pm 0.040$ & 1$\sigma$ & O06\
$P_1$ (h) & $2.7645 \pm 0.0003$ & 1$\sigma$ & O06\
$(A_1 B_1)^{1/2}/C_1$ & $\le 1.6^c$ / $1.17 \pm 0.15$ & 3$\sigma$ & This work / O06\
$A_{\rm 1}/B_{\rm 1}$ & $1.04 \pm 0.04$ & 1$\sigma$ & O06\
$\rho_1$ (g cm$^{-3}$) & $1.3^{+0.7}_{-0.4}$ / $1.97 \pm 0.72$ & 3$\sigma$ & This work / O06\
\
$D_{\rm 2,C}/D_{\rm 1,C}$ & $0.42 \pm 0.03^d$ & 3$\sigma$ & This work\
$D_{\rm 2,C}$ (km) & $0.574 \pm 0.066$ & 3$\sigma$ & This work\
$D_{\rm 2,V}$ (km) & ($0.59 \pm 0.04)^e$ & 1$\sigma$ & This work\
$P_2$ (h) & (17.46)$^f$ & & This work\
$A_{\rm 2}/B_{\rm 2}$ & $ 1.3^{+0.3}_{-0.1}$ & 3$\sigma$ & This work\
\
$a/(A_1 B_1)^{1/2}$ & $1.7\pm 0.2$ & 3$\sigma$ & This work\
$a$ (km) & $2.548 \pm 0.015$ & 1$\sigma$ & O06\
$(L_{\rm P}, B_{\rm P})$ (deg.) & $(329.6, -62.3) \pm (12 \times 4)^g$ & 3$\sigma$ & This work\
$P_{\rm orb}$ (h) & $17.45763 \pm 0.00004^h$ & 3$\sigma$ & This work\
$L_0$ (deg.) & $40 \pm 5^h$ & 3$\sigma$ & This work\
$e$ & $\le 0.006$ & 3$\sigma$ & O06\
$\Delta M_d$ (deg/yr$^2$) & $-0.65 \pm 0.16$ & 3$\sigma$ & This work\
$\dot{P}_{\rm orb}$ (h/yr) & $0.00013 \pm 0.00003 $ & 3$\sigma$ & This work\
$\dot{a}$ (cm/yr) & $1.2 \pm 0.3$ & 3$\sigma$ & This work\
References: O06 (Ostro et al., 2006)\
$^a$ The range of high solar phase angles covered by the observations did not allow to determine the $G$ parameter. We assumed the mean $G$ value for S-complex asteroids (Warner et al., 2009).\
$^b$ Derived from the primary shape model by O06 and for the average observed aspect. See text for details.\
$^c$ The formal best-fit value is 1.1.\
$^d$ This is a ratio of the cross-section equivalent diameters for the average observed aspect of 27 deg. See text for details.\
$^e$ Derived using the shape model of the secondary from O06 rescaled by 130%. See text for details.\
$^f$ The secondary appears to be in synchronous rotation. See text for details.\
$^g$ These are the semiaxes of the uncertainty area; see its actual shape in Fig. \[66391\_LB\_polar\].\
$^h$ The $P_{\rm orb}$ and $L_0$ values are for epoch JD 2455305.0, for which $P_{\rm orb}$ and $\Delta M_d$ do not correlate.
In the first part of the table, we present data derived from optical and spectroscopic observations of the system. $H_V$ and $G$ are the mean absolute magnitude and the phase parameter of the $H$–$G$ phase relation (Bowell et al., 1989). Using $H_V$ and effective diameter of the whole system ($D_{\rm eff} \equiv (D_{1,{\rm C}}^2 + D_{2,{\rm C}}^2)^{1/2}$) at the mean observed aspect of 27 deg. (see below), we derived the visual geometric albedo $p_{\rm V}$. We note that our value is in agreement with the $0.19 \pm 0.05$ value derived by Devogèle et al. (2019) from their polarimetric observations. We also observed 1999 KW4 in near-infrared spectral range and classified it as a Q type asteroid (see Appendix A.).
In the next two parts of Table \[tablePropKW4\], we give parameters for the components of the binary. The indices 1 and 2 refer to the primary and the secondary, respectively.
$D_{i,{\rm C}}$ is the cross-section equivalent diameter, i.e., the diameter of a sphere with the same cross section, of the $i$-th component at the observed aspect. Since the aspect is changing over time, the given value is an average over all lightcurve sessions. To quantify the mean aspect we used an asterocentric latitude of a Phase Angle Bisector (PAB), which is the mean direction between the heliocentric and geocentric directions to the asteroid. As discussed in Harris et al. (1984), this is an approximation for the effective viewing direction of an asteroid observed at non-zero solar phase. The average absolute value of the asterocentric latitude of the PAB (computed using the nominal pole of the mutual orbit, assumed to be the rotational pole of both components) was 27 deg.
$D_{i,{\rm V}}$ is the volume equivalent diameter, i.e., the diameter of a sphere with the same volume, of the $i$-th component. $D_{2,{\rm C}}/D_{1,{\rm C}}$ is the ratio between the cross-section equivalent diameters of the components. $P_i$ is the rotational period of the $i$-th component.
An analysis of the best subset of data for the secondary rotation taken from 2018-06-07.9 to -11.0 gave a formal best-fit estimate for the secondary rotation period of $17.53 \pm 0.12$ h ($3\sigma$; this includes also a synodic-sidereal difference uncertainty). This agrees with the mutual orbit period, within the error bar. Considering that all the observed secondary lightcurve minima coincide with or lie close to the mutual events —small differences may be due to a phase effect or secondary libration—, it is very likely that the secondary is in synchronous rotation. We therefore assume that $P_2$ is equal to the orbital period (see Table \[tablePropKW4\]).
$(A_1 B_1)^{1/2}/C_1$ is a ratio between the mean equatorial and the polar axes of the primary. $A_i/B_i$ is a ratio between the equatorial axes of the $i$-th component (equatorial elongation). $\rho_1 = \rho_2$ are the bulk densities of the two components, which we assumed to be the same in our modeling.
Most of the quantities were parameters of our model given in Section \[NumModel\] and we derived them from our observations.
The cross-section and volume equivalent diameters of the primary were derived using the shape model of the primary from Ostro et al. (2006). Assuming its rotational pole is the same as the mutual orbital pole (see below), we computed its rotationally averaged cross-section for each lightcurve session and presented the mean value over all sessions. Its 1$\sigma$ uncertainty was computed using the uncertainties of the dimensions of the primary from Ostro et al.
$D_{1,{\rm V}}$ was taken from Table 2 of Ostro et al.
$D_{2,{\rm V}}$ was derived using the shape model of the secondary by Ostro et al. (2006), rescaled to 130% of its original size to match mutual events’ depths from our data (see below). Its 1$\sigma$ uncertainty is a formal value taken from Table 2 of Ostro et al. (2006), but the real uncertainty may be higher because of uncertainties of the secondary radar shape model (Lance Benner, personal communication).
In the last part of Table \[tablePropKW4\], we summarize the parameters of the mutual orbit of the binary components. $a$ is the semimajor axis, $L_{\rm P}, B_{\rm P}$ are the ecliptic coordinates of the orbital pole in the equinox J2000, $L_0$ is the mean length of the secondary (i.e., the sum of angular distance from the ascending node and the length of the ascending node) for epoch JD 2455305.0, $e$ is the orbit eccentricity (only the upper limit was derived), and $\Delta M_d$ is the quadratic drift in mean anomaly. Since the orbital period $P_{\rm orb}$ is changing in time, the value presented in Table \[tablePropKW4\] is valid for epoch JD 2455305.0. For this epoch, which is approximately the mean time of all observed events, a correlation between $P_{\rm orb}$ and $\Delta M_d$ is zero. We also give the time derivatives of the orbital period and the semimajor axis, derived from $\Delta M_d$.
Although the orbit of 1999 KW4 crosses those of Earth, Venus and Mercury, according to JPL HORIZONS system the asteroid experienced only four close approaches to Earth between 2000 and 2019. The approaches took place in May 2001, May 2002, May 2018 and May 2019 at distances of 0.032, 0.089, 0.078 and 0.035 AU, respectively. Since the observed mutual orbital period increase is based on the observations at six effective epochs (apparitions), we can rule out planetary-tug effects as a potential mechanism for the increase.
The uncertainty area of the orbital pole is shown in Fig. \[66391\_LB\_polar\]. The size of the area shrinks with increasing the flattening of the primary $(A_1 B_1)^{1/2}/C_1$. To demonstrate the effect, we constrained the orbital pole uncertainties using three fixed values of the flattening (1.0, 1.2 and 1.4) and plotted the respective areas in the figure.
The uncertainties of the mutual semimajor axis and flattening of the primary are the main sources of the uncertainty of the bulk density of the system. In addition to that, the uncertainties of the two parameters are not independent. We therefore stepped $a$ and $(A_1 B_1)^{1/2}/C_1$ on a grid (while all other parameters were fitted at each step) to obtain an uncertainty area of both parameters together. The area is shown in Fig. \[66391\_20Ba\_a\_PrimA\_Rho\] with values of the bulk density for each combination of the parameters indicated.
The mutual orbit and shapes of the binary asteroid components of 1999 KW4 were modeled by Ostro et al. (2006) with radar observations taken in 2001. They report the size of the primary to be close to a tri-axial ellipsoid with axes 1417 $\times$ 1361 $\times$ 1183 m ($1\sigma$ uncertainties of $\pm$ 3%), and the secondary to be a tri-axial ellipsoid with axes 595 $\times$ 450 $\times$ 343 m ($1\sigma$ uncertainties of $\pm$ 5%). The dimensions given are extents of dynamically equivalent equal-volume ellipsoid (DEEVE; a homogeneous ellipsoid having the same moment-of-inertia ratios and volume as the shape model).
They also found the parameters of the mutual orbit to be as follows: orbital period $P_{\rm orb} = 17.422 \pm 0.036$ h, semimajor axis $a = 2548 \pm 15$ m, eccentricity $e = 0.0004 \pm 0.0019$, pole direction in ecliptic coordinates: $L_{\rm P} = 325.8 \pm 3.5$ deg, $B_{\rm P} = -61.8 \pm 1.2$ deg (uncertainties correspond to $1 \sigma$).
To compare our results with the values from Ostro et al. (2006), we computed $(A_1 B_1)^{1/2}/C_1$ and $a/(A_1 B_1)^{1/2}$ using their DEEVE for the primary and their semimajor axis of the mutual orbit. The result is plotted as a solid point in Fig. \[66391\_20Ba\_a\_PrimA\_Rho\] with $1 \sigma$ error bars.
There is one significant discrepancy between our results and those by Ostro et al. (2006): We obtained a significantly larger secondary-to-primary size ratio. To compare their result with ours, we computed a mean (rotationally averaged) cross-section ratio from the component shapes by Ostro et al. (2006): $(D_{\rm 2,C}/D_{\rm 1,C})_{\rm radar} = 0.34 \pm 0.02$ ($1 \sigma$) at the same mean aspect as our observations (asterocentric latitude of the Phase Angle Bisector, $B_{\rm PAB} = 27$ deg). The value is significantly lower than our $D_{\rm 2,C}/D_{\rm 1,C} = 0.42 \pm 0.03$ ($3 \sigma$).
To look more into the discrepancy between the secondary-to-primary size ratios by Ostro et al. (2006) and by us, we performed following test. Using the shape models of both components from Ostro et al. and the orbital parameters from Table \[tablePropKW4\], we generated a synthetic long-period component of the lightcurve. We then increased the size of the secondary until the depths of the secondary events (occultations and eclipses of the secondary) matched the observed event depths. We obtained a match when we increased the secondary axes by Ostro et al. (2006) to 130% of their original values. This is even slightly greater than $0.42/0.34 \doteq 124 \%$ because in this test the actual light scattering model was used for calculating the synthetic lightcurve, which models the scattering from non-spherical component shapes at the high solar phases and it is more precise than simply comparing the estimated mean cross-sections above. We note that replacing the parameters of the mutual orbit with those derived by Ostro et al. did not change the result.
We discussed this issue with Lance Benner and we received following information: “The dimensions of the secondary might be underestimated by Ostro et al. (2006) because the radar images were obtained at relatively coarse range and Doppler resolutions and at modest signal-to-noise ratios. Consequently, it is plausible that the trailing edge of the secondary in the radar images were less than would be detected if the SNRs were substantially higher.” (Lance Benner, personal communication.)
![Selected data of the long-period lightcurve component of (66391) 1999 KW4. The observed data are marked as points. The solid curve represents the synthetic lightcurve for the best-fit solution with $\Delta M_d = -0.65$ deg/yr$^2$. For comparison, the dashed curve is for the best-fit model with $\Delta M_d$ fixed at 0.0 deg/yr$^2$. []{data-label="66391_00-19_synth"}](66391_00-19_synth){width="\textwidth"}
![A time evolution of the mean anomaly difference $\Delta M$ between the best-fit solution with a constant orbital period (i.e., with $\Delta M_d$ fixed at zero) and the best-fit solution with $\Delta M_d$ fitted for (66391) 1999 KW4. Each point corresponds to the middle of one of the six apparitions from 2000 to 2019. Vertical error bars represent estimated $3\sigma$ uncertainties of the event times, expressed in mean anomaly. The solid curve is a quadratic fit to the data points. []{data-label="additionalM0KW4"}](66391_additionalM0){width="\textwidth"}
![Area of admissible poles for the mutual orbit of (66391) 1999 KW4 in ecliptic coordinates (grey area) for $(A_1 B_1)^{1/2}/C_1 = 1.$ The dot is the nominal solution given in Table \[tablePropKW4\]. This area corresponds to $3\sigma$ confidence level. To demonstrate the effect of a flattening of the primary on the estimated pole, the areas confined by solid lines shows the admissible poles constrained using $(A_1 B_1)^{1/2}/C_1 = 1.2$ (middle area) and 1.4 (the smallest area). The open circle with error bars represents a solution for the orbital pole from Ostro et al. (2006) with $1\sigma$ uncertainties. The south pole of the current asteroid’s heliocentric orbit is marked with the cross.[]{data-label="66391_LB_polar"}](66391_LB_polar){width="\textwidth"}
![Area of admissible combinations of the ratio between the mean equatorial and the polar axes of the primary ($(A_1 B_1)^{1/2}/C_1$) and the semimajor axis of the mutual orbit $a$ of (66391) 1999 KW4. This area corresponds to $3\sigma$ confidence level. Values of the bulk density of the system ($\rho$) in g cm$^{-3}$ are indicated. The dot with the error bars is the result from Ostro et al. (2006) and its $1 \sigma$ uncertainties (see text for details). []{data-label="66391_20Ba_a_PrimA_Rho"}](66391_20Ba_a_PrimA_Rho){width="\textwidth"}
Parameters of (88710) 2001 SL9 {#ParamsSL9}
==============================
In this section, we summarize the best-fit model parameters of the binary system (88710) 2001 SL9 and overview previous publications. The parameters are listed in Table \[tablePropSL9\].
[ccccc]{} Parameter & & Value & Unc. & Reference\
\
$H_V$ & & $17.98 \pm 0.02$ & 1$\sigma$ & This work\
$G$ & & $0.34 \pm 0.03$ & 1$\sigma$ & This work\
$V-R$ & & $0.457 \pm 0.010$ & 1$\sigma$ & This work\
$D_{\rm eff}$ (km) & & $0.75 \pm 0.10^a$ & 1$\sigma$ & This work\
Taxon. class & & Sr, Q & & P18, L05\
\
$D_{\rm 1,C}$ (km) & & $0.73 \pm 0.32$ & 3$\sigma$ & This work\
$D_{\rm 1,V}$ (km) & & $0.77 \pm 0.34$ & 3$\sigma$ & This work\
$P_1$ (h) & & $2.4004 \pm 0.0002$ & 1$\sigma$ & P06\
$(A_1 B_1)^{1/2}/C_1$ & & $\le 2.2^b$ & 3$\sigma$ & This work\
$A_{\rm 1}/B_{\rm 1}$ & & $1.07 \pm 0.01$ & 1$\sigma$ & PH07\
$\rho_1 = \rho_2$ (g cm$^{-3}$) & & $1.8^{+2.5}_{-0.5} $ & 3$\sigma$ & This work\
\
$D_{\rm 2,C}/D_{\rm 1,C}$ & & $0.24 \pm 0.02$ & 3$\sigma$ & This work\
$D_{\rm 2,C}$ (km) & & $0.18 \pm 0.08$ & 3$\sigma$ & This work\
$D_{\rm 2,V}$ (km) & & $(0.18 \pm 0.08)^c$ & 3$\sigma$ & This work\
$P_2$ (h) & & $(16.40)^d$ & &\
$A_{\rm 2}/B_{\rm 2}$ & & $\le 1.2$ & 3$\sigma$ & This work\
Mutual orbit: &\
$a/(A_1 B_1)^{1/2}$ & & $1.75\pm 0.3$ & 3$\sigma$ & This work\
$(L_{\rm P}, B_{\rm P})$ (deg.) & & $(302, -73) \pm (10 \times 4)^e$ & 3$\sigma$ & This work\
$P_{\rm orb}$ (h) &1. & $16.4022 \pm 0.0002^f$ & 3$\sigma$ & This work\
&2. & $16.4027 \pm 0.0002^f$ & &\
$L_0$ (deg.) &1. & $51 \pm 5^f$ & 3$\sigma$ & This work\
&2. & $56 \pm 5^f$ & &\
$e$ & & $\le 0.07$ & 3$\sigma$ & This work\
$\Delta M_d$ (deg/yr$^2$) &1. & $2.8 \pm 0.2$ & 3$\sigma$ & This work\
&2. & $5.2 \pm 0.2$ & &\
$\dot{P}_{\rm orb}$ (h/yr) &1. & $-0.00048 \pm 0.00003$ & 3$\sigma$ & This work\
&2. & $-0.00089 \pm 0.00004$ & &\
$\dot{a}$ (cm/yr) &1. & $-2.8 \pm 0.2$ & 3$\sigma$ & This work\
&2. & $-5.1 \pm 0.2$ & &\
References: L05 (Lazzarin et al., 2005), P18 (Pajuelo et al., 2018), P06 (Pravec et al., 2006), PH07 (Pravec and Harris, 2007).\
$^a$ From the derived $H_V$ and assumed $p_V = 0.20 \pm 0.05$ that is the mean albedo for S-complex asteroids (Pravec et al., 2012).\
$^b$ The formal best-fit value is 1.7.\
$^c$ Assuming a spherical shape of the secondary.\
$^d$ The secondary is assumed to in synchronous rotation. See text for details.\
$^e$ These are the semiaxes of the uncertainty area; see its actual shape in Fig. \[2001sl9\_01-15\_LB\_polar\].\
$^f$ These are the periods and $L_0$ for epoch JD 2456182.39026.
The notation of the values in the table and their uncertainties are the same as in Table \[tablePropKW4\] (see Section \[ParamsKW4\]).
The average absolute value of the asterocentric latitude of the PAB (computed using the nominal pole of the mutual orbit, assumed to be the rotation pole of both components) was 11 deg; we observed the asteroid close to equator-on.
Three works were published reporting spectroscopic observations of 2001 SL9 in the visual and near-infrared spectral range: Lazzarin et al. (2004, 2005) and Pajuelo et al. (2018). Based on moderate slope and broad $1{\mu}\rm m$ and $2{\mu}\rm m$ absorbtion bands, Lazzarin et al. (2004) and (2005) classified 2001 SL9 as an Sr and Q type, respectively. Pajuelo et al. found that the taxonomic types that fit their NIR spectrum are Sr, S and Sq, with Sr being the best fit.
From the measured $H_V$ and assuming the mean albedo for S-complex asteroids (Pravec et al., 2012), we estimated the effective diameter of the system $D_{\rm eff}$ at the observed (near equator-on) aspect.
A rotational state of the secondary is particularly important for the interpretations we present in Section \[byorp\]. However, as the amplitude of the secondary rotation lightcurve is very low, we could not derive its rotation period from the available data. It appears that the secondary is nearly spheroidal with low equatorial elongation.
Pravec et al. (2016) showed that asynchronous secondaries are absent among observed binary systems with close orbits ($a/D_1 \lesssim 2.2$, $P_{\rm orb} \lesssim 20$ h). They also pointed out that asynchronous secondaries are typically observed on eccentric orbits. Based on that, the parameters of the mutual orbit of 2001 SL9 (a close orbit with low or zero eccentricity) and the fact that the secondary spin relaxation is typically faster than the orbit circularization (Goldreich and Sari, 2009), we assume that the secondary of 2001 SL9 is in synchronous rotation, i.e., its rotation period is the same as the orbit period.
Earlier work where some of the binary parameters were derived is Pravec et al. (2006). Their values are generally in agreement with our current best estimated parameters, but they did not perform a modeling in order to get parameters of the mutual orbit.
\(88710) 2001 SL9 appears to be a typical near-Earth binary asteroid according to its basic parameters. Its bulk density of $\sim 1.8$ g cm$^{-3}$ is in good agreement with its rocky taxonomical class. The normalized total angular momentum content of 2001 SL9 is $\alpha_L = 1.1 \pm 0.2$ (1-$\sigma$ uncertainty), i.e., in the range 0.9–1.3 for small near-Earth and main belt asteroid binaries and exactly as expected for the proposed formation of small binary asteroids by fission of critically spinning rubble-pile progenitors (Pravec and Harris, 2007).
According to JPL HORIZONS system, the closest Earth, Venus and Mars approaches of 2001 SL9 from 2001 to 2015 were 0.22, 0.13 and 0.36 AU, respectively. We can therefore rule out planetary-tug effects as a potential mechanism for the observed mutual orbital period decrease.
![Selected data of the long-period lightcurve component of 2001 SL9. The observed data are marked as points. The solid and dashed curves represent the synthetic lightcurves of the two best-fit solutions with $\Delta M_d = 2.8$ and 5.2 deg/yr$^2$, respectively. For comparison, the dotted curve is for the best-fit model with $\Delta M_d$ fixed at 0.0 deg/yr$^2$. []{data-label="2001sl9_01-15_synth"}](2001sl9_01-15_synth){width="\textwidth"}
![ Time evolutions of the mean anomaly difference $\Delta M$ between the best-fit solution with a constant orbital period (i.e., with $\Delta M_d$ fixed at zero) and the two best-fit solutions with $\Delta M_d$ fitted for (88710) 2001 SL9. Each point corresponds to the middle of one of the five apparitions from 2001 to 2015. The open and solid circles stand for the two solutions with $\Delta M_d = 2.8$ and 5.2 deg/yr$^2$, respectively. The sizes of the symbols in vertical direction represent estimated $3\sigma$ uncertainties in the timing of events ($\pm 5^{\circ}$ in mean anomaly). The curves are quadratic fits to the data points. []{data-label="additionalM0SL9"}](2001sl9_additionalM0){width="\textwidth"}
![Area of admissible poles for the mutual orbit of (88710) 2001 SL9 in ecliptic coordinates (grey area). The dot is the nominal solution given in Table \[tablePropSL9\]. This area corresponds to $3\sigma$ confidence level. The south pole of the current asteroid’s heliocentric orbit is marked with the cross.[]{data-label="2001sl9_01-15_LB_polar"}](2001sl9_01-15_LB_polar){width="\textwidth"}
![Area of admissible combinations of the ratio between the mean equatorial and the polar axes of the primary ($(A_1 B_1)^{1/2}/C_1$) and the semimajor axis of the mutual orbit $a$ of (88710) 2001 SL9. This area corresponds to $3\sigma$ confidence level. Values of the bulk density of the system ($\rho$) in g cm$^{-3}$ are indicated.[]{data-label="2001sl9_a_PrimA_Rho"}](2001sl9_a_PrimA_Rho){width="\textwidth"}
Implications for the BYORP effect {#byorp}
=================================
(66391) 1999 KW4 BYORP Modeling
-------------------------------
McMahon and Scheeres (2010b) computed a BYORP coefficient, $B$, for the secondary shape of 1999 KW4 based on the model published by Ostro et al. (2006). The nominal coefficient was found to be $B_{nom} = 2.082 \times 10^{-2}$, and based on the other parameters of the system this produced a semimajor axis drift rate of approximately 7 cm/yr, according to the relationship $$\label{eq:adotBYORP}
\dot{a}_B = \frac{2 P_\Phi}{ a_h^2 \sqrt{1 - e_h^2}} \frac{a^{3/2} R_{mean}^2}{m_{s} \sqrt{\mu}} B$$ where $P_\Phi$ is the solar radiation pressure constant, whose value is taken to be $10^{14}$ kg km/s$^2$; $a_h$ is the heliocentric orbit semimajor axis of 0.642 AU, and $e_h$ = 0.688 is the heliocentric orbit eccentricity. The other values – binary orbit semimajor axis, $a$, secondary mass, $m_s$, binary gravitational parameter, $\mu$ can be obtained from Table \[tablePropKW4\]. The secondary mean radius, $R_{mean}$ was computed as average of vertices of the shape model of the secondary from Ostro et al., scaled up by 130%. The secondary mass can be expressed in terms of the estimated secondary volume, $V_s$ and bulk density, $\rho_s$. The values are: $V_s$ = 0.108 km$^3$, $\mu = $ 131.5 m$^3$/s$^2$, $a = 2.361$ km, $R_{mean} = 0.284$.
For $\rho_s$ we used a value of 1.97 g/cm$^3$ – the density of the primary from Ostro et al., assuming that both components have the same density. The BYORP modeling with these newly estimated parameters gives $\dot{a} =$ 7.46 cm/year, which is significantly larger than the observed value of 1.2 cm/year.
Given the previous discussion of the uncertainty in the secondary shape from Ostro et al. (2006), and the fact that we find an increase in size of approximately 30%, it is reasonable to assume that many details of the shape may not be accurately known. If the topography changes, the predicted BYORP coefficient will also change. To investigate this, we modeled the predicted BYORP effect for a suite of shapes similar to the KW4 secondary radar shape model from Ostro et al. (2006), to compute the likely range of values for the BYORP effect, using the computational model of McMahon and Scheeres (2010a), which incorporates self-shadowing and secondary intersections of re-radiated energy. The shapes were changed by perturbing the vertices vertically using the random Gaussian spheroid method (Muinonen 2010). The vertical perturbations were set to approximate the size estimate accuracy given in Ostro et al. (2006) of 6% of the long axis, which comes out to 17.1 m for a 1$\sigma$ radial dispersion. The correlation distances were set as 50 m (making small scale, “€œspiky" topography features), 150 m, and 300 m (smoother global variations in topography). The BYORP coefficients were computed for 90 such randomly perturbed shapes for each correlation distance.
The results of this process can be seen in Fig. \[fig:kw4\_byorp\]. As can be seen, these drift rates are all still higher than the measured value. The associated BYORP coefficients range from $B_{min} = 7.701 \times 10^{-3}$ to $B_{max} = 3.323 \times 10^{-2}$.
![Histogram of the resulting BYORP induced semimajor axis drift rates for the 270 perturbed secondary shape models of 1999 KW4.[]{data-label="fig:kw4_byorp"}](KW4_coeffs_rho1p97){width=".75\textwidth"}
One other parameter that is poorly constrained in Eq. \[eq:adotBYORP\] is the secondary density. In fact, while the previous computations assumed an equal density across both components, Ostro et al (2006) reported its large uncertainty. The effect of a variation in secondary density (with total system mass $\mu$ being held constant) can be seen in Fig. \[fig:density\]. It can be seen here that in order for the secondary density alone to modify the semimajor axis drift rate to match the measured value – even with the minimum BYORP coefficient seen – the secondary density would have to be approximately 3.6 g/cm$^3$ – significantly higher than Ostro’s estimate, and requiring a significantly more dense secondary than primary, but not impossible in terms of bulk density alone.
![Variation of $\dot{a}_B$ with secondary density, for the nominal and minimum BYORP coefficients.[]{data-label="fig:density"}](KW4_adot_vs_beta_density){width=".75\textwidth"}
The total semimajor axis drift rate for a binary asteroid is governed by the interplay between BYORP and tides, however tides are always expansive for 1999 KW4. The tide induced semimajor axis drift rates can be computed (Jacobson and Scheeres, 2011) as $$\dot{a}_{T}=3 \frac{k_{p}}{Q}\left(\frac{\omega_{d}}{a_{rp}^{11 / 2}}\right) q \sqrt{1+q}$$ where the surface disruption spin limit for a sphere is given by $$\omega_{d}=(4 \pi G \rho / 3)^{1 / 2}$$ and $q$ is the mass ratio, $k_p$ is the tidal Love number of the primary, $Q$ is the tidal dissipation number, and $G$ is the gravitational constant, and $a_{rp} = a/(D_{\rm 1,V}/2)$ is the binary semimajor axis in units for primary radii. For the current estimate of 1999 KW4, $q = m_{s}/m_p$ = 0.090, $\omega_d = 6.501 \times 10^{-4}$ rad/s, and the primary radius is taken to be $R_p$ = 0.6585 km. $Q/k_p$ is a relatively unknown parameter for rubble pile asteroids, but two values have emerged from the literature: $2.7 \times 10^{7}$ (Taylor and Margot, 2010) and $2.4 \times 10^{5}$ (Scheirich et al. 2015). Using these two values as bounds, we find that the tide induced semimajor axis drift rate to range from 0.0126 - 1.413 cm/yr.
The semimajor axis drift rate from BYORP is also lowered if the secondary is librating significantly, however the lightcurve observations show little evidence of this, implying that if there is any libration it is small and the degradation in the drift rate would be minimal. Thus, the overall BYORP coefficient may be significantly lower than predicted from our direct geometric theory or have an opposite sign, implying that the system may be moving into an equilibrium.
(88710) 2001 SL9 BYORP Modeling
-------------------------------
Unlike with 1999 KW4, there is no shape model available for the secondary of 2001 SL9, so that no informed forward modeling for the BYORP coefficient can be carried out. Instead, we compute the value of the BYORP coefficient that would produce the measured semimajor axis drift rates.
Given that the secondary is assumed to be in synchronous rotation while the primary is spinning much faster than the orbit period, the tides work to expand the semimajor axis. Thus, inward BYORP must overcome tides to achieve the measured semimajor axis rates. Due to the uncertainty in the $Q/k$, we report four possible BYORP coefficients for 2001 SL9 in Table \[tab:sl9\] - one for each combination of drift rate and tidal parameters.
------------------------ ------------------------- -------------------------
$Q/k = 2.4 \times 10^5$ $Q/k = 2.7 \times 10^7$
$\dot{a}$ = -2.8 cm/yr $-6.057 \times 10^{-3}$ $-6.173 \times 10^{-3}$
$\dot{a}$ = -5.1 cm/yr $-1.109 \times 10^{-2}$ $-1.120 \times 10^{-2}$
------------------------ ------------------------- -------------------------
: Computed BYORP coefficient, $B$, for SL9 based on measured semimajor axis drift rates and possible $Q/k$ values.
\[tab:sl9\]
Note that for 2001 SL9, $q = 0.0128$, $\omega_d = 7.094 \times 10^{-4}$ rad/s, and given the primary radius of 0.385 km ($= D_{1,V}/2$), we get $a_{rp}=4.177$. It is important to point out that BYORP is the only known physical mechanism that can cause an inward semimajor axis drift rate, as measured here for 2001 SL9. The computed $B$ coefficient magnitudes are in line with the modeled values for 1999 KW4, providing some confidence that the results are reasonable.
The results shown here, combined with the BYORP-tide equilibrium state detected for FG3 (Scheirich et al., 2015) does imply that BYORP effect seems to be real, but that we cannot adequately compute it as of yet. This inadequacy could either be from error in the shape models or a deficit in the theory.
Differential Yarkovsky force in binary asteroid system
------------------------------------------------------
Another effect affecting the magnitude of the mutual semimajor axis drift is the Yarkovsky force, which affects not only the motion of the center of mass of the whole binary system but also the relative motion of components. We computed the effect by a method described by Vokrouhlický et al. (2005). The shapes of the components were approximated by spheres represented by regular polyhedrons with 504 surface elements. The Yarkovsky accelerations $\bf{f}_1$ and $\bf{f}_2$ of both components were determined by numerical solution of the heat diffusion problem. The accelerations for the two components differ because of different sizes and spin rates. Moreover, they are affected by mutual shadowing of the components. Assuming zero eccentricity, the drift of the semimajor axis of the mutual orbit is $\dot{a}=2/n\, \langle f_\tau \rangle $, where $n$ represents the mean motion and $\langle f_\tau \rangle$ is a heliocentric-orbit averaged value of $f_\tau$ – a projection of the difference between the two Yarkovsky accelerations to the transversal direction of the relative motion $\bf{e}_\tau$, $$f_\tau=\bf{e}_\tau \cdot (\bf{f}_2-\bf{f}_1).$$
Without the mutual shadowing of the components the value of $\langle f_\tau \rangle$ would be zero. Therefore, the resulting mutual semimajor axis drift depends also on the orientation of the heliocentric and mutual orbits.
The Yarkovsky acceleration is less sensitive to body’s shape than to its thermophysical parameters. The results for the semimajor axis drift are shown in Fig. \[yark\]. For the nominal solution of (66391) 1999 KW4 and the thermal inertia range 100–1000 Jm$^{-2}$s$^{-1/2}$K$^{-1}$ (Delbo et al., 2015), the semimajor axis drift is between $-4$ mm and $-8$ mm per year. With the pole of mutual orbit inside the admissible area (see Fig. \[66391\_LB\_polar\]), the drift of mutual semimajor axis can differ by a factor of $\sim$2 from the value for nominal solution.
![Semimajor axis drift of the mutual orbit due to differential Yarkovsky effect as a function of thermal inetria $\Gamma$. Solid curve corresponds to (66391) 1999 KW4 and the dashed one corresponds to (88710) 2001 SL9.[]{data-label="yark"}](y.pdf){width="\textwidth"}
In the case of (88710) 2001 SL9, the Yarkovsky force has only negligible effect on the mutual semimajor axis drift. For the nominal parameters the drift is $\sim -1$ mm/yr (see Fig. \[yark\]). Depending on the orientation of the mutual orbit within its admissible area, the value can differ by a factor of $\sim$2.
Conclusions
===========
The near-Earth asteroids (66391) 1999 KW4 and (88710) 2001 SL9 are among the best characterized small binary asteroid systems. They are typical members of the population of near-Earth asteroid binaries for most of its parameters. With the data from our photometric observations taken during six apparitions over the time interval of 18 years, and during five apparitions over almost 14 years, for (66391) and (88710), respectively, we constrained the long-term evolution of their binary orbits.
For (66391), we found that the semimajor axis of its mutual orbit is expanding with a rate of $1.2 \pm 0.3$ cm/yr ($3\sigma$). The observed drift is on an order of the theoretical drift rate caused by mutual tides (0.0126 – 1.413 cm/yr). However the predicted drift caused by the BYORP effect (7.46 cm/yr) is much higher than the observed value. Thus, the BYORP coefficient may be significantly lower than predicted from a direct geometric theory by McMahon and Scheeres (2010a) or have an opposite sign, implying that the system may be moving into an equilibrium.
For (88710), we found that the semimajor axis of its mutual orbit is shrinking with a rate of $-2.8 \pm 0.2$ or $-5.1 \pm 0.2$ cm/yr ($3\sigma$). The BYORP effect is the only known physical mechanism (except the differential Yarkovsky effect, which is much slower than the observed value) that can cause an inward drift. Since there is no shape model available for the secondary, no forward modeling for the BYORP coefficient is possible. Instead, the BYORP coefficient can be computed from the measured drift rates. The computed coefficient magnitudes are similar to the modeled values for (66391) 1999 KW4, providing some confidence that the results are reasonable.
**Acknowledgements**
The work at Ondřejov Observatory and observations with the Danish 1.54-m telescope on the ESO La Silla station were supported by the Grant Agency of the Czech Republic, Grant 17-00774S. Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum provided under the program “Projects of Large Research, Development, and Innovations Infrastructures” (CESNET LM2015042), and the CERIT Scientific Cloud LM2015085, is greatly appreciated. The observations at Maidanak Observatory were supported by grant VA-FA-F-2-010 of the Ministry of Innovative Development of Uzbekistan. The work at Abastumani was supported by the Shota Rustaveli National Science Foundation, Grant RF-18-1193. The authors acknowledge the sacred nature of Mauna Kea, and appreciate the opportunity to observe from the mountain. The authors would like to thank the University of Hawaii for using the 2.2 m telescope.
**Appendix A. Taxonomic classification of (66391) 1999 KW4**
Near-infrared (NIR) spectra (0.7-2.5 $\mu$m) of (66391) 1999 KW4 were obtained in low-resolution prism mode on May 28, 2019 UTC with the SpeX instrument (Rayner et al., 2003) on NASA Infrared Telescope Facility (IRTF). The asteroid was 12.6 visual magnitude and was observed at a phase angle of $81^{\circ}$, and an airmass of $\sim$1.2-1.7. Weather conditions were stable during the observing run, with a seeing of 0.9“ and a humidity of $\sim$25%. During the observations, the 0.8”-slit was oriented along the parallactic angle in order to minimize the effects of differential atmospheric refraction. To avoid saturation, the integration time was limited to 60 seconds, and a total of 37 spectra of (66391) were obtained. A G-type local extinction star was observed before and after the asteroid in order to correct the telluric bands. Solar analog SAO 120107 was also observed to correct for possible spectral slope variations. All spectra were reduced using the IDL-based software Spextool (Cushing et al., 2004). A detailed description of the steps involved in the data reduction process can be found in Sanchez et al. (2013).
The NIR spectrum of (66391) 1999 KW4 is shown in Fig. \[66391\_spectrum\]. The spectrum exhibits two very deep absorption bands at 0.94 and 1.94 $\mu$m, due to the presence of olivine and pyroxene. Using the online Bus-DeMeo taxonomy calculator (http://smass.mit.edu/busdemeoclass.html) we found that (66391) is classified as either O- or Q-type in this taxonomic system (DeMeo et al., 2009). A visual inspection shows that the overall spectral characteristics of (66391) are more similar to a Q-type asteroid. However, we noticed that the absorption bands in the NIR spectrum of (66391) are much deeper than those of a typical Q-type. Band depths are measured from the continuum to the band centers and are given as percentage depths (Clark and Roush, 1984). For (66391), we found that the Band I depth is $34.4 \pm 0.2\%$, and the Band II depth is $15.9 \pm 0.2\%$, while the mean spectrum of a Q-type asteroid (DeMeo et al. 2009) has Band I and II depths of $23.8 \pm 0.1\%$, and $6.0 \pm 0.2\%$, respectively. This difference could be attributed to several factors, including mineral abundance, grain size, and the high phase angle at which (66391) was observed (e.g., Sanchez et al. 2012).
![NIR spectrum of (66391) 1999 KW4 normalized to unity at 1.5 $\mu$m. []{data-label="66391_spectrum"}](1999KW4_NIR){width="\textwidth"}
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[^1]: $\Delta M_d$ of 2001 SL9 was sampled on the larger interval because in our initial modeling runs, there appeared possible solutions at high positive $\Delta M_d$ values. Therefore, we expanded the interval in order to examine them; there turned out to be no significant solution at high $\Delta M_d$ finally.
| 0 |
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abstract: 'This study investigates the capacity region of a three-user cognitive radio network with two primary users and one cognitive user. A three-user Cognitive Interference Channel (C-IFC) is proposed by considering a three-user Interference Channel (IFC) where one of the transmitters has cognitive capabilities and knows the messages of the other two transmitters in a non-causal manner. First, two inner bounds on the capacity region of the three-user C-IFC are obtained based on using the schemes which allow all receivers to decode all messages with two different orders. Next, two sets of conditions are derived, under which the capacity region of the proposed model coincides with the capacity region of a three-user C-IFC in which all three messages are required at all receivers. Under these conditions, referred to as strong interference conditions, the capacity regions for the proposed three-user C-IFC are characterized. Moreover, the Gaussian three-user C-IFC is considered and the capacity results are derived for the Gaussian case. Some numerical examples are also provided.'
author:
- |
Mahtab Mirmohseni, Bahareh Akhbari, and Mohammad Reza Aref\
Information Systems and Security Lab (ISSL)\
Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran\
Email: [email protected], b\[email protected], and [email protected] [^1]
title: 'Three-User Cognitive Interference Channel: Capacity Region with Strong Interference'
---
Cognitive interference channel, three-user interference channel, strong interference, capacity region.
Introduction
============
Interference avoidance techniques have traditionally been used in wireless networks wherein multiple source-destination pairs share the same medium. However, the broadcasting nature of wireless networks may enable cooperation among entities, which ensures higher rates with more reliable communication. On the other hand, due to the increasing number of wireless systems, spectrum resources have become scarce and expensive. The exponentially growing demand for wireless services along with the rapid advancements in wireless technology have lead to cognitive radio technology which aims to overcome the spectrum inefficiency problem by developing communication systems that have the capability to sense the environment and adapt to it [@GolJafMarSri09; @Mit91].
In overlay cognitive networks, the cognitive user can transmit simultaneously with the non-cognitive users and compensate for the interference by cooperation in sending, i.e., relaying, the non-cognitive users’ messages [@GolJafMarSri09]. In order to obtain the fundamental limits of these networks by information theoretical techniques, researchers have to consider the models with idealized assumptions. The assumption of *full non-causal* knowledge of the primary messages (as side information) at the cognitive users is a standard one, which is still very useful in practical applications if one considers a phase for obtaining this side information. From an information theoretic point of view, Cognitive Interference Channel (C-IFC) was first introduced in [@DevrMitTar06] to model an overlay cognitive radio and refers to a two-user Interference Channel (IFC) in which the cognitive user (secondary user) has the ability to obtain the message being transmitted by the other user (primary user), either in a non-causal or causal manner. For the non-causal C-IFC, where the cognitive user has non-causal full or partial knowledge of the primary user’s transmitted message, an achievable rate region was first derived in [@DevrMitTar06], by combining the Gel’fand-Pinsker (GP) binning [@GelfPin80] with a well known simultaneous superposition coding scheme (rate splitting) applied to IFC [@HanKob81]. Subsequently, several achievable rate regions and capacity results in some special cases for the C-IFC have been established [@JoviVis09]-[@JiangXin08]. Yet, capacity results have been known only in special cases. C-IFC with strong interference conditions is one of these cases, where interference is such that both messages can be decoded at both receivers with no rate penalty. Strong interference conditions for C-IFC and the capacity regions under these conditions have been derived in [@MariYatKra07; @MirAkhArefITW10; @MariYatKra06]. For an overview on the capacity results of C-IFC, see [@RiniIT11].
The *$k$-user* IFC consists of $k$ independent transmitters sending messages to $k$ independent receivers. Extending the results of the classic two-user IFC to the IFCs with more than two user pairs is non-trivial; because each receiver is affected by the joint interference from the all other transmitters rather by each transmitter’s signal separately [@ElgKim11 P. 157]. Recently, the capacity region of a three-user Gaussian IFC under mixed strong-very strong interference conditions has been characterized in [@ChaSez10]. In C-IFC, asymmetric nature of the transmitters’ cooperation makes this extension even more challenging, since there are several ways for applying the cognition capabilities and also the obtained setups may involve different aspects of IFCs such as independent channel inputs at the transmitters which makes difficult to apply the results of C-IFC to these setups. An achievable rate region for a three-user Multiple Access Channel (MAC) [@ElgKim11 Chapter 4] with three transmitters and *one receiver* has been derived in [@NagKis_Gl10]. By increasing the number of receivers, a *three-user* C-IFC with *one* primary user and *two* cognitive users has been studied in [@NagMur_ITW09; @NagMur_Gl09], where an achievable rate region is derived for this setup. The authors in [@NagMohMurKis_11], proposed the achievable rate regions for the different non-causal message-sharing mechanism in the three-user C-IFC and also derived an outer bound in the Gaussian case.
In this paper, we consider a *three-user* C-IFC with *two* primary users and *one* cognitive user, where the cognitive transmitter non-causally knows the messages of both primary transmitters. Up to our best knowledge, in all of the previous works on *three-user* C-IFC in the general discrete memoryless setup, only achievable rate regions have been obtained and the capacity result in all setups of *three-user* C-IFC is an open problem. In this paper, we consider the strong interference regime and derive *capacity regions* in this case. First, we obtain two inner bounds on the capacity region (achievable rate regions) based on using superposition coding and allowing all receivers to decode all messages. In the achievablity scheme of the first region, we utilize simultaneous joint decoding in the decoding part at all receivers. However, in the second scheme, each primary receiver first decodes the other primary user’s message, while treating the remaining signals as noise, i.e., the combination of its intended transmitter’s signal, the cognitive transmitter’s signal and additive noise. This strategy is useful for the channels where the other primary user’s signal (as seen by each primary user) is strong enough and it is possible to decode this primary interference first. Then, the primary receiver decodes the message of the cognitive user and its own message by a joint typicality decoding. The receiver of the cognitive user pair uses joint typicality decoding. Next, deriving two sets of strong interference conditions, we show that the obtained inner bounds achieve capacity under these conditions by proving converse proofs. In these cases, decoding the unintended messages causes no additional constraint on the rate region. Therefore, the channel model is equal to the one in which all three messages are required at all receivers and the capacity region coincides with the capacity region of a three-user C-IFC in which each receiver should decode all three messages. In fact, we determine the conditions, referred to as *Set1*, under which the three-user C-IFC can be seen as a compound three-user MAC with common information. Under the second set of conditions, referred to as *Set2*, the considered channel can be seen as a compound of three channels: two *two-user MAC*s with common information at the primary receivers and a three-user MAC with common information at the cognitive receiver. Further, we compare these two sets of conditions and show that *Set1* is weaker than *Set2*. Moreover, we consider the Gaussian three-user C-IFC and find capacity results for the Gaussian case based on *Set2*. We also provide some numerical examples.
The rest of the paper is organized as follows. Section \[sec:definition\] introduces the three-user C-IFC model and the notations. In Section \[sec:Ach\], we obtain the achievable rate regions; while in Section \[sec:Cap\], we state the capacity results for the discrete memoryless three-user C-IFC. In Section \[sec:Gaussian\], Gaussian three-user C-IFC is investigated. Finally, Section \[sec:conclusion\] concludes the paper.
Channel Models and Preliminaries {#sec:definition}
================================
Throughout the paper, upper case letters (e.g. $X$) are used to denote RVs and lower case letters (e.g. $x$) show their realizations. The probability mass function (p.m.f) of a RV $X$ with alphabet set ${\mathcal{X}}$, is denoted by $p_X(x)$, where subscript $X$ is occasionally omitted. $A_\epsilon^n(X,Y)$ specifies the set of $\epsilon$-strongly, jointly typical sequences of length $n$. The notation $X^j_i$ indicates a sequence of RVs $(X_i,X_{i+1},...,X_j)$, where $X^j$ is used instead of $X^j_1$, for brevity. ${\mathcal{N}}(0,\sigma^2)$ denotes a zero mean normal distribution with variance $\sigma^2$.
![Three-user Cognitive Interference Channel (C-IFC)[]{data-label="fig:channelmodel"}](fig_block_diag.eps){width="11cm"}
Consider the three-user C-IFC in Fig.\[fig:channelmodel\], which is denoted by (${\mathcal{X}}_1\times{\mathcal{X}}_2\times{\mathcal{X}}_3,p(y_1^n,y_2^n,y_3^n|x_1^n,x_2^n,x_3^n),{\mathcal{Y}}_1\times{\mathcal{Y}}_2\times{\mathcal{Y}}_3$), where $X_u\in{\mathcal{X}}_u$ is the channel input of Transmitter $u$ (Tx$u$) and $Y_u\in{\mathcal{Y}}_u$ is the channel output at Receiver $u$ (Rx$u$) for $u\in\{1,2,3\}$. Also, $p(y_1^n,y_2^n,y_3^n|x_1^n,x_2^n,x_3^n)$ is the channel transition probability distribution. In $n$ channel uses, each Tx$u$ desires to send a message $m_u$ to Rx$u$ where $u\in\{1,2,3\}$.
\[def:code\] A $(2^{nR_1},2^{nR_2},2^{nR_3},n)$ code for the three-user C-IFC consists of (i) three independent message sets ${\mathcal{M}}_u=\{1,...,2^{nR_u}\}$, where $u\in\{1,2,3\}$, (ii) two encoding functions at the primary transmitters, $f_1:{\mathcal{M}}_1\mapsto{\mathcal{X}}_1^n$ at Tx1 and $f_2:{\mathcal{M}}_2\mapsto{\mathcal{X}}_2^n$ at Tx2, (iii) an encoding function at the cognitive transmitter, $f_3:{\mathcal{M}}_1\times{\mathcal{M}}_2\times{\mathcal{M}}_3\mapsto{\mathcal{X}}_3^n$, and (iv) three decoding functions, $g_u:{\mathcal{Y}}_u^n\mapsto{\mathcal{M}}_u$ at Rx$u$ where $u\in\{1,2,3\}$. We assume that the channel is memoryless. Thus, the channel transition probability distribution is given by $$\label{eqn:pmf}
p(y_1^n,y_2^n,\,y_3^n|x_1^n,x_2^n,x_3^n)=\prod\limits_{i=1}^np(y_{1,i},y_{2,i},y_{3,i}|x_{1,i},x_{2,i},x_{3,i}).$$
The probability of error for this code is defined as $P_e=max\{P_{e,1},P_{e,2},P_{e,3}\}$, where we have
[l]{} P\_[e,u]{}=\_[m\_1,m\_2,m\_3]{}[P(g\_u(Y\^n\_[u]{})m\_u | (m\_1,m\_2,m\_3))]{}
for $u\in\{1,2,3\}$.
\[def:rate\] A rate triple $(R_1,R_2,R_3)$ is achievable if there exists a sequence of $(2^{nR_1},2^{nR_2},2^{nR_3},n)$ codes with $P_e\rightarrow 0$ as $n\rightarrow \infty$. The capacity region ${\mathcal{C}}$, is the closure of the set of all achievable rates.
Achievable Rate Regions for Discrete Memoryless three-user C-IFC {#sec:Ach}
================================================================
In this section, we consider the discrete memoryless three-user C-IFC and present two achievable rate regions for this setup. The coding schemes contain superposition coding in the encoding part. In the decoding part, all messages are common to all receivers, i.e., all three receivers decode $m_1$, $m_2$ and $m_3$. In the scheme of the first achievable rate region, the simultaneous joint decoding is utilized at all receivers. However, in the second scheme, Rx1 first decodes the other primary user’s message $m_2$, while treating the remaining signals as noise, i.e., the signals of $m_1$ and $m_3$ plus additive noise. This strategy is useful for the channels where the signal of $m_2$ at Rx1 is strong enough and it is possible to decode this primary interference first. Then, Rx1 decodes the message of the cognitive user $m_3$ and its own message $m_1$ by a joint typicality decoding. Rx2 proceeds similarly, while, Rx3 uses joint typicality decoding. Detailed proofs are provided in Appendix \[app:ach\_proof\].
Let ${\mathcal{P}}$ denotes the set of all joint p.m.fs $p(.)$, that factor as
[c]{} p(x\_1,x\_2,x\_3)=p(x\_1)p(x\_2)p(x\_3|x\_1,x\_2).\[eqn:pmf\_ach\]
\[thm:ach1\] The union of rate regions given by
[rcl]{} R\_[3]{}&& I(X\_3;Y\_3|X\_1,X\_2) \[eqn:ach1\_I\]\
R\_[1]{}+R\_[3]{}&& {I(X\_1,X\_3;Y\_1|X\_2),I(X\_1,X\_3;Y\_3|X\_2)} \[eqn:ach1\_II\]\
R\_[2]{}+R\_[3]{}&& {I(X\_2,X\_3;Y\_2|X\_1),I(X\_2,X\_3;Y\_3|X\_1)} \[eqn:ach1\_III\]\
R\_[1]{}+R\_[2]{}+R\_[3]{}&&{I(X\_1,X\_2,X\_3;Y\_1),I(X\_1,X\_2,X\_3;Y\_2),I(X\_1,X\_2,X\_3;Y\_3)} \[eqn:ach1\_IV\]
is achievable for the three-user C-IFC (denoted as ${\mathcal{R}}_1(p)$), where the union is over $p(.)\in{\mathcal{P}}$ (defined in (\[eqn:pmf\_ach\])).
\[thm:ach2\] The union of rate regions given by (\[eqn:ach1\_I\])-(\[eqn:ach1\_III\]) and
[rcl]{} R\_[1]{}&& I(X\_1;Y\_2) \[eqn:ach2\_I\]\
R\_[2]{}&& I(X\_2;Y\_1) \[eqn:ach2\_II\]\
R\_[1]{}+R\_[2]{}+R\_[3]{}&& I(X\_1,X\_2,X\_3;Y\_3) \[eqn:ach2\_III\]
is achievable for the three-user C-IFC (denoted as ${\mathcal{R}}_2(p)$), where the union is over $p(.)\in{\mathcal{P}}$ (defined in (\[eqn:pmf\_ach\])).
Strong Interference Conditions and Capacity Results {#sec:Cap}
===================================================
In this section, we derive two sets of strong interference conditions (*Set1* and *Set2*), under which the regions of Theorem \[thm:ach1\] and Theorem \[thm:ach2\] achieve capacity. First, we give an intuition about deriving the conditions at each receiver:
- **Strong interference at the cognitive receiver (Rx3):** In *both schemes*, Rx3 jointly decodes $m_1$, $m_2$ and $m_3$. Therefore, it is assumed that $m_1$ and $m_2$ jointly cause strong interference. These conditions are shown in (\[eqn:set1\_IV\]) and the second terms of (\[eqn:set1\_II\]) and (\[eqn:set1\_III\]) for the first scheme. In other words, assuming the above conditions, the joint received signal from Tx1 and Tx2 at Rx3 is strong enough to decode without imposing any rate constraint on $R_{1}$ and $R_{2}$. Similar conditions are also provided for the second scheme in (\[eqn:set2\_IV\]) and the second terms of (\[eqn:set2\_II\]) and (\[eqn:set2\_III\]). Therefore, there is no difference between two schemes about the strong interference condition at the cognitive receiver (Rx3).
- **Strong interference at the primary users (Rx1 and Rx2):** We illustrate the condition for Rx1 and the one for Rx2 follows due to the symmetry. In the first scheme, condition at Rx1 is similar to Rx3 and it is assumed that $m_2$ and $m_3$ jointly cause strong interference, which is shown in the first terms of (\[eqn:set1\_I\]) and (\[eqn:set1\_III\]). Note that, the asymmetric nature of the conditions, compared to the one for Rx3, is due to the cognition capability of Tx3, i.e., $x_3$ depends on $m_1$ and $m_2$ in addition to $m_3$. However, in the second scheme, it is assumed that the interference caused by $m_2$ at Rx1 is stronger than the joint received signals of $m_1$ and $m_3$ (first term of (\[eqn:set2\_III\])). Therefore, it is possible to decode $m_2$ first. The second level for the strong interference condition at Rx1, assumes that after decoding $m_2$, the cognitive message ($m_3$) causes strong interference in comparison to the desired message ($m_1$) (first term of (\[eqn:set1\_III\])).
The above intuitions are summarized in Table \[tbl:str\_cond\].
Tx3 $\rightarrow$ Rx1 Tx3 $\rightarrow$ Rx2 Tx1 $\rightarrow$ Rx2 Tx2 $\rightarrow$ Rx1 Tx1 and Tx2 $\rightarrow$ Rx3
-------- --------------------------------- ---------------------------------- ---------------------------------- ----------------------------------- ---------------------------------------------------------------------------------
*Set1* first term of (\[eqn:set1\_I\]) second term of (\[eqn:set1\_I\]) first term of (\[eqn:set1\_II\]) first term of (\[eqn:set1\_III\]) (\[eqn:set1\_IV\]) + second terms of (\[eqn:set1\_II\]) and (\[eqn:set1\_III\])
*Set2* first term of (\[eqn:set1\_I\]) second term of (\[eqn:set1\_I\]) first term of (\[eqn:set2\_II\]) first term of (\[eqn:set2\_III\]) (\[eqn:set2\_IV\]) + second terms of (\[eqn:set2\_II\]) and (\[eqn:set2\_III\])
Theorem \[thm:ach2\] includes (\[eqn:ach2\_I\])-(\[eqn:ach2\_III\]) instead of (\[eqn:ach1\_IV\]) in Theorem \[thm:ach1\]. In fact, in the *Gaussian case*, the converse proof can not be established for the two first terms in (\[eqn:ach1\_IV\]). Therefore, we propose Theorem \[thm:ach2\] and find the stronger conditions than *Set1*, i.e., *Set2*, which makes the bounds in (\[eqn:ach2\_I\])-(\[eqn:ach2\_III\]) redundant. Hence, we intend to use *Set2* to derive the capacity results for the Gaussian case in Section \[sec:Gaussian\].
In *Set1*, (\[eqn:set1\_IV\]) and the second terms of (\[eqn:set1\_II\]) and (\[eqn:set1\_III\]) are used to make the second terms of (\[eqn:ach1\_II\]) and (\[eqn:ach1\_III\]), and the third term of (\[eqn:ach1\_IV\]) redundant. However, (\[eqn:set1\_I\]) and the first terms of (\[eqn:set1\_II\]) and (\[eqn:set1\_III\]) are used to prove the converse part for the rates in (\[eqn:Cap1\_II\])-(\[eqn:Cap1\_IV\]).
In *Set2*, the second terms of (\[eqn:set2\_II\]) and (\[eqn:set2\_III\]) are used to make the second terms of (\[eqn:ach1\_II\]) and (\[eqn:ach1\_III\]) redundant. The first terms of (\[eqn:set2\_II\]) and (\[eqn:set2\_III\]) make the (\[eqn:ach2\_I\]) and (\[eqn:ach2\_II\]) redundant and (\[eqn:set2\_IV\]) is used to make the (\[eqn:ach2\_III\]) redundant. However, (\[eqn:set1\_I\]) is used to prove the converse part for the rates in (\[eqn:Cap2\_II\]) and (\[eqn:Cap2\_III\]).
These results are summarized in Table \[tbl:str\_cond2\].
Achievability Converse
-------- ------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------
*Set1* (\[eqn:set1\_IV\]) + second terms of (\[eqn:set1\_II\]),(\[eqn:set1\_III\]) (\[eqn:set1\_I\]) $\rightarrow$ (\[eqn:Cap1\_II\]),(\[eqn:Cap1\_III\])
$\rightarrow$ second terms of (\[eqn:ach1\_II\]),(\[eqn:ach1\_III\]) + third term of (\[eqn:ach1\_IV\]) (\[eqn:set1\_I\]) + first terms of (\[eqn:set1\_II\]),(\[eqn:set1\_III\]) $\rightarrow$ (\[eqn:Cap1\_IV\])
*Set2* second terms of (\[eqn:set2\_II\]),(\[eqn:set2\_III\]) $\rightarrow$ (\[eqn:ach1\_II\]),(\[eqn:ach1\_III\]) (\[eqn:set1\_I\]) $\rightarrow$ (\[eqn:Cap2\_II\]),(\[eqn:Cap2\_III\])
first terms of (\[eqn:set2\_II\]),(\[eqn:set2\_III\]) $\rightarrow$ (\[eqn:ach2\_I\]),(\[eqn:ach2\_II\])
(\[eqn:set2\_IV\]) $\rightarrow$ (\[eqn:ach2\_III\])
Assume that the following set of strong interference conditions (*Set1*) holds for every $p(.)\in{\mathcal{P}}$:
[rcl]{} I(X\_3;Y\_3|X\_1,X\_2)&& {I(X\_3;Y\_1|X\_1,X\_2),I(X\_3;Y\_2|X\_1,X\_2)}\[eqn:set1\_I\]\
I(X\_1,X\_3;Y\_1|X\_2)&& {I(X\_1;Y\_2|X\_2),I(X\_1,X\_3;Y\_3|X\_2)}\[eqn:set1\_II\]\
I(X\_2,X\_3;Y\_2|X\_1)&& {I(X\_2;Y\_1|X\_1),I(X\_2,X\_3;Y\_3|X\_1)}\[eqn:set1\_III\]\
{I(X\_1,X\_2,X\_3;Y\_1),I(X\_1,X\_2,X\_3;Y\_2)}&&I(X\_1,X\_2,X\_3;Y\_3) \[eqn:set1\_IV\]
In fact, under these conditions interfering signals at the receivers are strong enough that all messages can be jointly decoded by all the receivers.
\[thm:cap\_set1\] The capacity region of the three-user C-IFC, satisfying (\[eqn:set1\_I\])-(\[eqn:set1\_IV\]), is given by
[rl]{} \_1 =\_[p(.)]{} {(R\_1,R\_2,R\_3): &R\_1 0, R\_2 0, R\_3 0\
&R\_[3]{}I(X\_3;Y\_3|X\_1,X\_2) \[eqn:Cap1\_I\]\
&R\_[1]{}+R\_[3]{}I(X\_1,X\_3;Y\_1|X\_2) \[eqn:Cap1\_II\]\
&R\_[2]{}+R\_[3]{}I(X\_2,X\_3;Y\_2|X\_1) \[eqn:Cap1\_III\]\
&R\_[1]{}+R\_[2]{}+R\_[3]{} {I(X\_1,X\_2,X\_3;Y\_1),I(X\_1,X\_2,X\_3;Y\_2)}}. \[eqn:Cap1\_IV\]
\[remark:cap\_set1\_I\] The message of the cognitive user ($M_3$) can be decoded at Rx1 and Rx2, under condition (\[eqn:set1\_I\]). Rx1 can decode $M_2$ considering the condition of the first term in the RHS of (\[eqn:set1\_III\]). Note that, $X_3$ is required in this condition due to the dependance on $M_2$. Similarly, the condition of the first term in the RHS of (\[eqn:set1\_II\]) enables Rx2 to decode $M_1$. Moreover, $(M_1,M_2)$ can be decoded at Rx3 under (\[eqn:set1\_IV\]) and the second terms in the RHS of (\[eqn:set1\_II\]) and (\[eqn:set1\_III\]). Therefore, ${\mathcal{C}}_1$ gives the capacity region for a compound three-user MAC with common information, where $R_1$ and $R_2$ are the common rates of Tx1-Tx3 and Tx2-Tx3, respectively, $R_3$ is the private rate for Tx3, and the private rates for Tx1 and Tx2 are zero.
\[remark:cap\_set1\_II\] If we omit the second pair, i.e., $X_2=Y_2=\emptyset$ and $R_2=0$, the model reduces to a two-user C-IFC and ${\mathcal{C}}_1$ coincides with the capacity region of the strong interference channel with unidirectional cooperation, which was characterized in [@MariYatKra07].
First, we provide a useful lemma which we need in the proof of the converse part for Theorem \[thm:cap\_set1\].
\[lemma:cond\_str\^n\] If (\[eqn:set1\_I\])-(\[eqn:set1\_III\]) hold for all distribution $p(.)\in{\mathcal{P}}$, then we have:
[rcl]{} I(X\_3\^n;Y\_3\^n|X\_1\^n,X\_2\^n,U)&&I(X\_3\^n;Y\_1\^n|X\_1\^n,X\_2\^n,U)\[eqn:set1\_I\^n1\]\
I(X\_3\^n;Y\_3\^n|X\_1\^n,X\_2\^n,U)&&I(X\_3\^n;Y\_2\^n|X\_1\^n,X\_2\^n,U)\[eqn:set1\_I\^n2\]\
I(X\_1\^n,X\_3\^n;Y\_1\^n|X\_2\^n,U)&&I(X\_1\^n;Y\_2\^n|X\_2\^n,U)\[eqn:set1\_II\^n\]\
I(X\_2\^n,X\_3\^n;Y\_2\^n|X\_1\^n,U)&&I(X\_2\^n;Y\_1\^n|X\_1\^n,U).\[eqn:set1\_III\^n\]
Proof relies on the result in [@KorMar77 Proposition 1] and follows the same lines as in [@MariYatKra07 Lemma 5] and [@CosElg79 Lemma].
Considering (\[eqn:set1\_II\])-(\[eqn:set1\_IV\]), the proof follows from Theorem \[thm:ach1\].
Consider a $(2^{nR_1},2^{nR_2},2^{nR_3},n)$ code with average error probability $P_e^n\rightarrow 0$, which implies that $P_{e,u}^{(n)}\rightarrow 0$ for $u\in\{1,2,3\}$. Applying Fano’s inequality [@CovTho06], [@ElgKim11 P. 19] results in
[rcl]{} H(M\_u|Y\_u\^n)P\_[e,u]{}\^[(n)]{} log(2\^[nR\_u]{} - 1) + h(P\_[e,u]{}\^[(n)]{})n\_[un]{}\[eqn:Fano\_delta\]
for $u\in\{1,2,3\}$, where $\delta_{un}\rightarrow 0$ as $P_{e,u}^{(n)}\rightarrow 0$. Note that, due to the encoding functions $f_1$, $f_2$ and $f_3$, defined in Definition \[def:code\] and the independence of the messages, we have $p(.)\in{\mathcal{P}}$. Now, we derive the bounds in Theorem \[thm:cap\_set1\]. For the first bound, we obtain
[ll]{} nR\_3=H(M\_3)&H(M\_3|M\_1,M\_2)\
&=I(M\_3;Y\_3\^n|M\_1,M\_2)+H(M\_3|Y\_3\^n,M\_1,M\_2)\
&I(M\_3;Y\_3\^n|M\_1,M\_2)+H(M\_3|Y\_3\^n)\[eqn:cap\_set1\_fanoI\_before\]\
&I(M\_3;Y\_3\^n|M\_1,M\_2)+n\_[3n]{}
where (a) follows since $M_1$, $M_2$ and $M_3$ are independent, (b) is due to the fact that conditioning does not increase the entropy and (c) follows from (\[eqn:Fano\_delta\]) for $u=3$. Hence,
[ll]{} nR\_3- n\_[3n]{}&I(M\_3;Y\_3\^n|M\_1,M\_2)\
& I(M\_3,X\_3\^n;Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n)\
&=H(Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n)-H(Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n,M\_3,X\_3\^n)\
& H(Y\_3\^n|X\_1\^n,X\_2\^n)-H(Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n,M\_3,X\_3\^n)\
& H(Y\_3\^n|X\_1\^n,X\_2\^n)-H(Y\_3\^n|X\_1\^n,X\_2\^n,X\_3\^n)=I(X\_3\^n;Y\_3\^n|X\_1\^n,X\_2\^n)\
&\_[i=1]{}\^[n]{}I(X\_3\^n;Y\_[3,i]{}|X\_1\^n,X\_2\^n,Y\_3\^[i-1]{})\[eqn:cap\_set1\_fanoI\]\
&=\_[i=1]{}\^[n]{}H(Y\_[3,i]{}|X\_1\^n,X\_2\^n,Y\_3\^[i-1]{})-I(Y\_[3,i]{}|X\_1\^n,X\_2\^n,Y\_3\^[i-1]{},X\_3\^n)\
&\_[i=1]{}\^[n]{}H(Y\_[3,i]{}|X\_[1,i]{},X\_[2,i]{})-I(Y\_[3,i]{}|X\_1\^n,X\_2\^n,Y\_3\^[i-1]{},X\_3\^n)\
&\_[i=1]{}\^[n]{}H(Y\_[3,i]{}|X\_[1,i]{},X\_[2,i]{})-I(Y\_[3,i]{}|X\_[1,i]{},X\_[2,i]{},X\_[3,i]{})=\_[i=1]{}\^[n]{}I(X\_[3,i]{};Y\_[3,i]{}|X\_[1,i]{},X\_[2,i]{})
where (a) is due to the encoding functions $f_1$, $f_2$ and $f_3$, defined in Definition \[def:code\], (b) and (e) are due to the fact that conditioning does not increase the entropy, (c) follows from the fact that $(M_1,M_2,M_3)\rightarrow (X_1^n,X_2^n,X_3^n)\rightarrow Y_3^n$ forms a Markov chain, (d) is obtained from the chain rule, and (f) follows from the memoryless property of the channel.
Now, applying (\[eqn:Fano\_delta\]) for $u\in\{1,3\}$ and the independence of the messages, we can bound $R_1+R_3$ as
[ll]{} n(R\_1+R\_3)- n(\_[1n]{}+\_[3n]{})&I(M\_1;Y\_1\^n|M\_2)+I(M\_3;Y\_3\^n|M\_1,M\_2)\
& I(M\_1,X\_1\^n;Y\_1\^n|M\_2,X\_2\^n)+I(M\_3,X\_3\^n;Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n)\
& I(M\_1,X\_1\^n;Y\_1\^n|M\_2,X\_2\^n)+I(X\_3\^n;Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n)\
& I(M\_1,X\_1\^n;Y\_1\^n|M\_2,X\_2\^n)+I(X\_3\^n;Y\_1\^n|M\_1,M\_2,X\_1\^n,X\_2\^n)\
&= I(M\_1,X\_1\^n,X\_3\^n;Y\_1\^n|M\_2,X\_2\^n)\[eqn:cap\_set1\_fanoII\]\
&\_[i=1]{}\^[n]{}I(M\_1,X\_1\^n,X\_3\^n;Y\_[1,i]{}|M\_2,X\_2\^n,Y\_1\^[i-1]{})\
&\_[i=1]{}\^[n]{}I(X\_[1,i]{},X\_[3,i]{};Y\_[1,i]{}|X\_[2,i]{})
where (a) follows from the encoding functions $f_1$, $f_2$ and $f_3$, defined in Definition \[def:code\], (b) follows from the fact that $M_3\rightarrow (X_1^n,X_2^n,X_3^n)\rightarrow Y_3^n$ forms a Markov chain, (c) is obtained from (\[eqn:set1\_I\^n1\]), (d) follows from the chain rule, and (e) follows from the memoryless property of the channel and the fact that conditioning does not increase the entropy (like parts (d)-(f) in (\[eqn:cap\_set1\_fanoI\])). Applying similar steps using (\[eqn:Fano\_delta\]) for $u\in\{2,3\}$ and (\[eqn:set1\_I\^n2\]), we can show that,
[l]{} n(R\_2+R\_3)-n(\_[2n]{}+\_[3n]{})\_[i=1]{}\^[n]{}I(X\_[2,i]{},X\_[3,i]{};Y\_[2,i]{}|X\_[1,i]{}).\[eqn:cap\_set1\_fanoIII\]
Finally, using (\[eqn:Fano\_delta\]) for $u\in\{1,2,3\}$ and the independence of the messages, the sum-rate bounds can be obtained as
[rl]{} n(R\_1+R\_2+R\_3)- &n(\_[1n]{}+\_[2n]{}+\_[3n]{}) I(M\_1;Y\_1\^n)+I(M\_2;Y\_2\^n|M\_1,M\_3)+I(M\_3;Y\_3\^n|M\_1,M\_2)\
&I(M\_1,X\_1\^n;Y\_1\^n)+I(M\_2,M\_3,X\_2\^n,X\_3\^n;Y\_2\^n|M\_1,X\_1\^n)\
&+I(M\_3,X\_3\^n;Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n)\
=&I(M\_1,X\_1\^n;Y\_1\^n)+I(M\_2,M\_3,X\_2\^n,X\_3\^n;Y\_2\^n|M\_1,X\_1\^n)\
&+H(Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n)-H(Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n,M\_3,X\_3\^n)\
&I(M\_1,X\_1\^n;Y\_1\^n)+I(M\_2,M\_3,X\_2\^n,X\_3\^n;Y\_2\^n|M\_1,X\_1\^n)\
&+H(Y\_3\^n|M\_1,X\_1\^n,X\_2\^n)-H(Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n,M\_3,X\_3\^n)\
&I(M\_1,X\_1\^n;Y\_1\^n)+I(X\_2\^n,X\_3\^n;Y\_2\^n|M\_1,X\_1\^n)+I(X\_3\^n;Y\_3\^n|M\_1,X\_1\^n,X\_2\^n)\
&I(M\_1,X\_1\^n;Y\_1\^n)+I(X\_2\^n;Y\_1\^n|M\_1,X\_1\^n)+I(X\_3\^n;Y\_1\^n|M\_1,X\_1\^n,X\_2\^n)\
= &I(M\_1,X\_1\^n,X\_2\^n,X\_3\^n;Y\_1\^n)\
&\_[i=1]{}\^[n]{}I(M\_1,X\_1\^n,X\_2\^n,X\_3\^n;Y\_[1,i]{}|Y\_1\^[i-1]{})\
&\_[i=1]{}\^[n]{}I(X\_[1,i]{},X\_[2,i]{},X\_[3,i]{};Y\_[1,i]{})\[eqn:cap\_set1\_fanoIV\]
where (a) follows from the encoding functions $f_1$, $f_2$ and $f_3$, defined in Definition \[def:code\], and the fact that conditioning does not increase the entropy, (b) is due to the fact that conditioning does not increase the entropy, (c) follows since $(M_2,M_3)\rightarrow (X_1^n,X_2^n,X_3^n)\rightarrow (Y_2^n,Y_3^n)$ forms a Markov chain, (d) is obtained from (\[eqn:set1\_I\^n1\]) and (\[eqn:set1\_III\^n\]), (e) follows from the chain rule, and (f) is due to the memoryless property of the channel and the fact that conditioning does not increase the entropy (like parts (d)-(f) in (\[eqn:cap\_set1\_fanoI\])).
By applying a similar technique based on (\[eqn:set1\_I\^n2\]) and (\[eqn:set1\_II\^n\]), we obtain:
[rcl]{} n(R\_1+R\_2+R\_3)&-& n(\_[1n]{}+\_[2n]{}+\_[3n]{})\_[i=1]{}\^[n]{}I(X\_[1,i]{},X\_[2,i]{},X\_[3,i]{};Y\_[2,i]{}).\[eqn:cap\_set1\_fanoV\]
Using a standard time-sharing argument [@ElgKim11 P. 85] for (\[eqn:cap\_set1\_fanoI\])-(\[eqn:cap\_set1\_fanoV\]) completes the proof.
Next, we derive the second set of strong interference conditions (*Set2*), under which the region of Theorem \[thm:ach2\] is the capacity region. For every $p(.)\in{\mathcal{P}}$, *Set2* includes (\[eqn:set1\_I\]) and the following conditions:
[rcl]{} I(X\_1,X\_3;Y\_1|X\_2)&& {I(X\_1;Y\_2),I(X\_1,X\_3;Y\_3|X\_2)}\[eqn:set2\_II\]\
I(X\_2,X\_3;Y\_2|X\_1)&& {I(X\_2;Y\_1),I(X\_2,X\_3;Y\_3|X\_1)}\[eqn:set2\_III\]\
{I(X\_1;Y\_2)I(X\_1;Y\_3)}&&{I(X\_2;Y\_1)I(X\_2;Y\_3)} \[eqn:set2\_IV\]
\[thm:cap\_set2\] The capacity region of the three-user C-IFC, satisfying (\[eqn:set1\_I\]) and (\[eqn:set2\_II\])-(\[eqn:set2\_IV\]), is given by
[rl]{} \_2 =\_[p(.)]{} {(R\_1,R\_2,R\_3): &R\_1 0, R\_2 0, R\_3 0\
&R\_[3]{}I(X\_3;Y\_3|X\_1,X\_2) \[eqn:Cap2\_I\]\
&R\_[1]{}+R\_[3]{}I(X\_1,X\_3;Y\_1|X\_2) \[eqn:Cap2\_II\]\
&R\_[2]{}+R\_[3]{}I(X\_2,X\_3;Y\_2|X\_1) \[eqn:Cap2\_III\]}}.
Consider the region of Theorem \[thm:ach2\]. Using the second terms of conditions (\[eqn:set2\_II\]) and (\[eqn:set2\_III\]), the bounds in (\[eqn:ach1\_I\])-(\[eqn:ach1\_III\]) reduce to ${\mathcal{C}}_2$. Based on the first term of condition (\[eqn:set2\_II\]), the bound in (\[eqn:ach2\_I\]) is redundant due to (\[eqn:Cap1\_II\]). Similarly, (\[eqn:Cap1\_III\]) and (\[eqn:set2\_III\]) make the bound in (\[eqn:ach2\_II\]) redundant. Moreover, considering (\[eqn:ach2\_I\]) (or (\[eqn:ach2\_II\])), (\[eqn:set2\_IV\]), and the second bound in (\[eqn:ach1\_III\]) (or (\[eqn:ach1\_II\])), the bound in (\[eqn:ach2\_III\]) becomes redundant.
The bounds in ${\mathcal{C}}_2$ are same as the bounds (\[eqn:Cap1\_I\])-(\[eqn:Cap1\_III\]) in ${\mathcal{C}}_1$, which are shown in the converse proof of Theorem \[thm:cap\_set1\]. This completes the proof.
We compare the different terms in *Set1* and *Set2*. Since $X_1$ and $X_2$ are independent, we obtain
[rl]{} I(X\_2;Y\_1|X\_1)=H(X\_2|X\_1)-H(X\_2|X\_1,Y\_1)&H(X\_2)-H(X\_2|X\_1,Y\_1)\
&H(X\_2)-H(X\_2|Y\_1)= I(X\_2;Y\_1)
where (a) follows from the independence of $X_1$ and $X_2$, and (b) is a consequence of the fact that conditioning does not increase the entropy. Hence, condition (\[eqn:set2\_III\]) implies condition (\[eqn:set1\_III\]). Similarly, condition (\[eqn:set2\_II\]) implies condition (\[eqn:set1\_II\]). Moreover, the second terms of conditions (\[eqn:set2\_II\]) and (\[eqn:set2\_IV\]) give the first term in condition (\[eqn:set1\_IV\]). Also, the second term of condition (\[eqn:set2\_III\]) along with the first term of (\[eqn:set2\_IV\]) give the second term in condition (\[eqn:set1\_IV\]). Therefore, *Set2* implies *Set1*, and the conditions of *Set1* are weaker compared to thoes of *Set2*. In fact, we use *Set2* and ${\mathcal{C}}_2$ to derive the capacity results for the Gaussian case in the next section.
Gaussian three-user C-IFC {#sec:Gaussian}
=========================
In this section, we consider the Gaussian three-user C-IFC and characterize capacity results for the Gaussian case. Moreover, we present some numerical examples. The Gaussian three-user C-IFC, as depicted in Fig. \[fig:Gauschannelmodel\], at time $i=1,\ldots,n$ and at each Rx$r$, for $r\in\{1,2,3\}$, can be mathematically modeled as
[rcl]{} Y\_[r,i]{}&=&\_[t=1]{}\^[3]{}h\_[tr]{}X\_[t,i]{}+Z\_[r,i]{}\[eqn:Gaussian\_model\]
where $h_{tr}$, for $t,r\in\{1,2,3\}$, are known channel gains. $X_{t,i}$ is the input signal with average power constraint:
[rcl]{} \_[i=1]{}\^n(x\_[t,i]{})\^2P\_t\[eqn:power\_cons\]
for $t\in\{1,2,3\}$. $Z_{r,i},\: r\in\{1,2,3\}$ is an independent and identically distributed (i.i.d) zero mean Gaussian noise component with unit power, i.e., $Z_{r,i}\sim{\mathcal{N}}(0,1)$.
![Gaussian three-user C-IFC.[]{data-label="fig:Gauschannelmodel"}](fig_gaus.eps){width="9cm"}
Now, we extend the results of Theorem \[thm:cap\_set2\], i.e., ${\mathcal{C}}_2$ and *Set2*, to the Gaussian case. The strong interference conditions of *Set2*, i.e., (\[eqn:set1\_I\]), (\[eqn:set2\_II\])-(\[eqn:set2\_IV\]) for the above Gaussian model, respectively, become ($Set_G$):
[rcl]{} h\_[33]{}\^2&&{h\_[31]{}\^2,h\_[32]{}\^2}\[eqn:set1\_I\_Gaus\]\
h\_[11]{}\^2P\_1+h\_[31]{}\^2P\_3(1-\_2\^2)+2h\_[11]{}h\_[31]{}\_1&&{A\_[12]{},h\_[13]{}\^2P\_1+h\_[33]{}\^2P\_3(1-\_2\^2)+2h\_[13]{}h\_[33]{}\_1}\
&&\[eqn:set1\_II\_Gaus\]\
h\_[22]{}\^2P\_2+h\_[32]{}\^2P\_3(1-\_1\^2)+2h\_[22]{}h\_[32]{}\_2&&{A\_[21]{},h\_[23]{}\^2P\_1+h\_[33]{}\^2P\_3(1-\_1\^2)+2h\_[23]{}h\_[33]{}\_2}\
&&\[eqn:set1\_III\_Gaus\]\
{A\_[12]{}B\_[12]{}}&&{A\_[21]{}B\_[21]{}}\[eqn:set1\_IV\_Gaus\]
where $-1\leq\rho_u\leq 1$ is the correlation coefficient between $X_u$ and $X_3$, i.e., $E(X_uX_3)=\rho_u\sqrt{P_uP_3}$ for $u\in\{1,2\}$, and $A_{ij}$ and $B_{ij}$ are defined as,
[rcl]{} A\_[ij]{}&=&\
B\_[ij]{}&=&
for $i,j\in \{1,2\}$.
\[thm:Gaus\_cap\_set1\] For the Gaussian three-user C-IFC, satisfying conditions (\[eqn:set1\_I\_Gaus\])-(\[eqn:set1\_IV\_Gaus\]), the capacity region is given by
[rl]{} \_1\^G =\_[-1\_1,\_21:\_1\^2+\_2\^21 ]{} {&(R\_1,R\_2,R\_3): R\_1,R\_2,R\_3 0\
&R\_3(h\_[33]{}\^2P\_3(1-\_1\^2-\_2\^2))\[eqn:Gaus\_Cap1\_I\]\
&R\_[1]{}+R\_[3]{}(h\_[11]{}\^2P\_1+h\_[31]{}\^2P\_3(1-\_2\^2)+2h\_[11]{}h\_[31]{}\_1)\[eqn:Gaus\_Cap1\_II\]\
&R\_[2]{}+R\_[3]{}(h\_[22]{}\^2P\_2+h\_[32]{}\^2P\_3(1-\_1\^2)+2h\_[22]{}h\_[32]{}\_2)}\[eqn:Gaus\_Cap1\_III\]
where to simplify notation we define $$\label{eqn:theta}
\theta(x)\doteq \frac{1}{2}\log(1+x).$$
\[remark:Gaus\_cap\_set1\_I\] Condition (\[eqn:set1\_I\_Gaus\]) implies that Tx3 causes strong interference at Rx1 and Rx2. This fact enables Rx1 and Rx2 to decode $m_3$. Moreover, due to the first terms in the RHS of (\[eqn:set1\_II\_Gaus\]) and (\[eqn:set1\_III\_Gaus\]), $m_1$ and $m_2$ can be decoded at Rx2 and Rx1, respectively. Also, (\[eqn:set1\_II\_Gaus\])-(\[eqn:set1\_IV\_Gaus\]) provides strong interference conditions at Rx3, under which all messages can be decoded by Rx3.
The achievablity follows from ${\mathcal{C}}_2$ in Theorem \[thm:cap\_set2\] by evaluating *Set2* and ${\mathcal{C}}_2$ with zero mean jointly Gaussian channel inputs $X_1$, $X_2$ and $X_3$. In fact, $X_1\sim{\mathcal{N}}(0,P_1)$, $X_2\sim{\mathcal{N}}(0,P_2)$ and $X_3\sim{\mathcal{N}}(0,P_3)$, where $E(X_1X_2)=0$, $E(X_1X_3)=\rho_1\sqrt{P_1P_3}$, and $E(X_2X_3)=\rho_2\sqrt{P_2P_3}$. The converse proof is based on the similar reasoning as in [@Sato81] and is provided in Appendix \[app:thm:Gaus\_converse\].
Note that, the channel parameters, i.e., $h_{tr}$, $P_t$ for $t,r\in\{1,2,3\}$, must satisfy (\[eqn:set1\_I\_Gaus\])-(\[eqn:set1\_IV\_Gaus\]) for all $-1\leq\rho_1,\rho_2\leq 1:\rho_1^2+\rho_2^2\leq 1$, to numerically evaluate the ${\mathcal{C}}_1^G$ using (\[eqn:Gaus\_Cap1\_I\])-(\[eqn:Gaus\_Cap1\_III\]). Here, we choose $P_1=P_3=3$, $P_2=6$, $h_{11}=h_{22}=h_{33}=1$, $h_{31}=h_{32}=\sqrt{1.5}$, $h_{12}=7$, $h_{13}=3$, $h_{21}=5$, and $h_{23}=15$, which satisfy (\[eqn:set1\_I\_Gaus\])-(\[eqn:set1\_IV\_Gaus\]); hence, the regions are derived under strong interference conditions $Set_G$.
![Capacity region for the Gaussian three-user C-IFC for fixed $\rho_1=\rho_2$.[]{data-label="fig:Gaus1"}](Case1.eps){width="11cm"}
Fig. \[fig:Gaus1\] shows the capacity region for the Gaussian three-user C-IFC of Theorem \[thm:Gaus\_cap\_set1\], for the above parameter selection, where $\rho_1=\rho_2$ is fixed in each surface. $\rho_1=\rho_2=0$ region corresponds to the *no cooperation* case, where the channel inputs are independent. It can be seen that as $\rho_1=\rho_2$ increases, the bound on $R_3$ becomes more restrictive while the sum-rate bounds become looser; because Tx3 dedicates parts of its power for cooperation. The capacity for this channel is the union of all the regions obtained for different values of $\rho_1$ and $\rho_2$ satisfying $\rho_1^2+\rho_2^2\leq 1$. This union is shown in Fig. \[fig:Gaus2\].
Conclusion {#sec:conclusion}
==========
We considered a three-user cognitive radio network with two primary users and one cognitive user and investigated its capacity region in the strong interference regime. For this purpose, we introduced the three-user Cognitive Interference Channel (C-IFC) by providing cognition capabilities for one of the transmitters in the three-user IFC. We derived two sets of strong interference conditions under which we established the capacity regions. Under these conditions, all three messages are required at all receivers. We also found capacity results for the Gaussian case.
![Capacity region for the Gaussian three-user C-IFC under strong interference conditions $Set_G$.[]{data-label="fig:Gaus2"}](Case2.eps){width="11cm"}
Proofs of Theorem \[thm:ach1\] and Theorem \[thm:ach2\] {#app:ach_proof}
=======================================================
We propose the following random coding scheme, which contains superposition coding in the encoding part and simultaneous joint decoding in the decoding part. All messages are common to all receivers, i.e., all three receivers decode $m_1$, $m_2$ and $m_3$.
***Codebook Generation**:* Fix $p(.)\in{\mathcal{P}}$. For $u\in\{1,2\}$, generate $2^{nR_{u}}$ i.i.d $x_{u}^n$ sequences, each with probability $\prod\limits_{i=1}^np(x_{u,i})$. Index them as $x_{u}^n(m_{u})$ where $m_{u}\in[1,2^{nR_{u}}]$. For each $(x_{1}^n(m_{1}),x_{2}^n(m_{2}))$, generate $2^{nR_{3}}$ i.i.d $x_{3}^n$ sequences, each with probability $\prod\limits_{i=1}^np(x_{3,i}|x_{1,i},x_{2,i})$. Index them as $x_{3}^n(m_{3},m_2,m_1)$ where $m_{3}\in[1,2^{nR_{3}}]$.
***Encoding**:* In order to transmit the message $(m_{1},m_{2},m_3)$, Txu sends $x_{u}^n(m_{u})$ for $u\in\{1,2\}$ and Tx3 sends $x_{3}^n(m_{3},m_2,m_1)$.
***Decoding**:*
*Rx1:* After receiving $y_1^n$, Rx1 looks for a unique index $\hat{m}_{1}$ and some $(\hat{m}_{2},\hat{m}_{3})$ such that,
[c]{} (y\_1\^n,x\_[1]{}\^n(\_[1]{}),x\_[2]{}\^n(\_[2]{}),x\_[3]{}\^n(\_[3]{},\_2,\_1))A\_\^n(Y\_1,X\_1,X\_2,X\_3).
Using the packing lemma [@ElgKim11 P. 45] (or [@CovTho06 Theorem 15.2.3]), for large enough $n$, with arbitrarily high probability $\hat{m}_{1}=m_{1}$ if
[rcl]{} R\_[1]{}+R\_[3]{}&& I(X\_1,X\_3;Y\_1|X\_2)\[eqn:ach1\_rec1\_I\]\
R\_[1]{}+R\_[2]{}+R\_[3]{}&& I(X\_1,X\_2,X\_3;Y\_1).\[eqn:ach1\_rec1\_II\]
*Rx2:* The decoding process at Rx2 is similar to Rx1. Therefore, based on packing lemma [@ElgKim11 P. 45], the decoding error at Rx2 can be made sufficiently small if
[rcl]{} R\_[2]{}+R\_[3]{}&& I(X\_2,X\_3;Y\_2|X\_1)\[eqn:ach1\_rec2\_I\]\
R\_[1]{}+R\_[2]{}+R\_[3]{}&& I(X\_1,X\_2,X\_3;Y\_2).\[eqn:ach1\_rec2\_II\]
*Rx3:* After receiving $y_3^n$, Rx3 finds a unique index $\hat{\hat{m}}_{3}$ and some pair $(\hat{\hat{m}}_{1},\hat{\hat{m}}_{2})$ such that,
[c]{} (y\_3\^n,x\_[1]{}\^n(\_[1]{}),x\_[2]{}\^n(\_[2]{}),x\_[3]{}\^n(\_[3]{},\_2,\_1))A\_\^n(Y\_3,X\_1,X\_2,X\_3).
Using packing lemma [@ElgKim11 P. 45], With arbitrary high probability $\hat{\hat{m}}_{3}=m_3$, if $n$ is large enough and
[rcl]{} R\_[3]{}&& I(X\_3;Y\_3|X\_1,X\_2) \[eqn:ach1\_rec3\_I\]\
R\_[1]{}+R\_[3]{}&& I(X\_1,X\_3;Y\_3|X\_2)\[eqn:ach1\_rec3\_II\]\
R\_[2]{}+R\_[3]{}&& I(X\_2,X\_3;Y\_3 |X\_1)\[eqn:ach1\_rec3\_III\]\
R\_[1]{}+R\_[2]{}+R\_[3]{}&& I(X\_1,X\_2,X\_3;Y\_1).\[eqn:ach1\_rec3\_IV\]
This completes the proof.
The *codebook generation* and *encoding* parts of the proof follow the same lines as in Theorem \[thm:ach1\]. Therefore, we only describe the decoding part. Similar to Theorem \[thm:ach1\], all three receivers decode $m_1$, $m_2$ and $m_3$. However, here Rx1 (or Rx2) first decodes $m_2$ (or $m_1$). Then, it jointly decodes $m_1$ (or $m_2$) and $m_3$.
***Decoding**:*
*Rx1:* After receiving $y_1^n$, Rx1 first finds a unique index $\hat{m}_{2}$ such that,
[c]{} (y\_1\^n,x\_[2]{}\^n(\_[2]{}))A\_\^n(Y\_1,X\_2).
Applying packing lemma [@ElgKim11 P. 45], with arbitrary high probability $\hat{m}_{2}=m_{2}$, if $n$ is large enough and
[rcl]{} R\_[2]{}&& I(X\_2;Y\_1). \[eqn:ach2\_rec1\_I\]
Then, it looks for a unique index $\hat{m}_{1}$ and some $\hat{m}_{3}$ such that,
[c]{} (y\_1\^n,x\_[1]{}\^n(\_[1]{}),x\_[2]{}\^n(m\_[2]{}),x\_[3]{}\^n(\_[3]{},m\_2,\_1))A\_\^n(Y\_1,X\_1,X\_2,X\_3).
For large enough $n$, with arbitrarily high probability $\hat{m}_{1}=m_{1}$ if (\[eqn:ach1\_rec1\_I\]) holds.
*Rx2:* Rx2 proceeds similarly. This step can be accomplished with sufficiently small probability of error for large enough $n$, if (\[eqn:ach1\_rec2\_I\]) holds and
[rcl]{} R\_[1]{}&& I(X\_1;Y\_2).\[eqn:ach2\_rec2\_I\]
The decoding procedure at Rx3 is similar to Theorem \[thm:ach1\] and the error in this receiver can be bounded, if (\[eqn:ach1\_rec3\_I\])-(\[eqn:ach1\_rec3\_IV\]) hold. This completes the proof.
Proof of the Converse Part for Theorem \[thm:Gaus\_cap\_set1\] {#app:thm:Gaus_converse}
==============================================================
For any rate triple $(R_1,R_2,R_3)\in{\mathcal{C}}$, Rx1 is able to decode $m_1$ reliably. Assume that Rx1 knows $X_2$ by a genie. Obviously, the genie aided channel has a larger capacity region than ${\mathcal{C}}$. Now, Rx1 knows $X_1$ from $m_1$ and $X_2$ from genie. Then, Rx1 is able to construct
[rcl]{} \_3&=&(Y\_1-h\_[11]{}X\_1-h\_[12]{}X\_2)+h\_[13]{}X\_1+h\_[23]{}X\_2\
&=&h\_[13]{}X\_1+h\_[23]{}X\_2+h\_[33]{}X\_3+Z\_1
If condition (\[eqn:set1\_I\_Gaus\]) holds, then $\tilde{Y}_3$ is a less noisy version of $Y_3$. Since Rx3 has to decode $m_3$, Rx1 can decode $m_3$ via $\tilde{Y}_3$. Therefore, $(R_1,R_2,R_3)$ is contained in the capacity region of a MAC with common information from Tx1 and Tx3 to Rx1 with $X_2$ as a receiver side information, where $R_1$ is the common rate, $R_3$ is the private rate for Tx3, and the private rates for Tx1 is zero. Therefore, the sum-rate $R_1+R_3$ is bounded as (\[eqn:Cap2\_II\]). From the maximum-entropy theorem [@CovTho06] (or [@ElgKim11 P. 21]), this bound is largest for the Gaussian inputs and is evaluated to (\[eqn:Gaus\_Cap1\_II\]). In a similar manner, we can obtain (\[eqn:Gaus\_Cap1\_III\]) at Rx2. The bound in (\[eqn:Gaus\_Cap1\_I\]) follows by applying the standard methods as in (\[eqn:cap\_set1\_fanoI\]).
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Costa M. H. M. and El Gamal A., “The capacity region of the discrete memoryless interference channel with strong interference,” *IEEE Trans. Inf. Theory*, Sept. 1987, **33**, (5), pp. 710-–711
Sato H., “The capacity of the Gaussian interference channel under strong interference,” *IEEE Trans. Inf. Theory*, Nov. 1981, **27**, (6), pp. 786–-788
[^1]: This work was partially supported by Iran National Science Foundation (INSF) under contract No. 88114.46-2010 and by Iran Telecom Research Center (ITRC) under contract No. T500/17865.
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abstract: 'Let $E$ be an elliptic curve defined over a number field $K$, let $\alpha \in E(K)$ be a point of infinite order, and let $N^{-1}\alpha$ be the set of $N$-division points of $\alpha$ in $E(\overline{K})$. We prove strong effective and uniform results for the degrees of the Kummer extensions $[K(E[N],N^{-1}\alpha) : K(E[N])]$. When $K={\mathbb{Q}}$, and under a minimal assumption on $\alpha$, we show that the inequality $[{\mathbb{Q}}(E[N],N^{-1}\alpha) : {\mathbb{Q}}(E[N])] {\geqslant}cN^2$ holds with a constant $c$ independent of both $E$ and $\alpha$.'
address:
- 'Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy'
- 'Mathematics Research Unit, University of Luxembourg, 6 av. de la Fonte, 4364 Esch-sur-Alzette, Luxembourg'
author:
- Davide Lombardo
- Sebastiano Tronto
bibliography:
- 'biblio.bib'
title: Explicit Kummer Theory for Elliptic Curves
---
Introduction
============
Setting
-------
Let $E$ be an elliptic curve defined over a number field $K$ (for which we fix an algebraic closure ${\overline{K}}$) and let $\alpha\in E(K)$ be a point of infinite order. The purpose of this paper is to study the extensions of $K$ generated by the division points of $\alpha$; in order to formally introduce these extensions we need to set some notation.
Given a positive integer $M$, we denote by $E[M]$ the group of $M$-torsion points of $E$, that is, the set $\{P \in E({\overline{K}}) : MP=0\}$ equipped with the group law inherited from $E$. Moreover, we denote by $K_M$ the *$M$-th torsion field* $K(E[M])$ of $E$, namely, the finite extension of $K$ obtained by adjoining the coordinates of all the $M$-torsion points of $E$. For each positive integer $N$ dividing $M$, we let $N^{-1}\alpha:={\left\{ \beta\in E({\overline{K}})\mid N\beta=\alpha \right\}}$ denote the set of $N$-division points of $\alpha$ and set $$\begin{aligned}
K_{M,N}:=K(E[M],N^{-1}\alpha).\end{aligned}$$ The field $K_{M,N}$ is called the *$(M,N)$-Kummer extension* of $K$ (related to $\alpha$), and both $K_M$ and $K_{M,N}$ are finite Galois extensions of $K$.
It is a classical question to study the degree of $K_{M,N}$ over $K_M$ as $M,N$ vary, see for example [@Durham Théorème 5.2], [@Hindry Lemme 14], or Ribet’s foundational paper [@MR552524]. In particular, it is known that there exists an integer $C=C(E/K, \alpha)$, depending only on $E/K$ and $\alpha$, such that $$\begin{aligned}
\frac{N^2}{\left[K_{M,N}:K_M\right]}\quad\text{divides}\quad C\end{aligned}$$ for every pair of positive integers $(M,N)$ with $N\mid M$.
The aim of this paper is to give an explicit version of this result, and to show that it can even be made uniform when the base field is $K={\mathbb{Q}}$. Our first result is that, under the assumption $\operatorname{End}_K(E)={\mathbb{Z}}$, the integer $C$ can be bounded (explicitly) in terms of the $\ell$-adic Galois representations attached to $E$ and of divisibility properties of the point $\alpha$, and that this statement becomes false if we remove the hypothesis $\operatorname{End}_K(E)={\mathbb{Z}}$.
On the other hand, the assumption $\operatorname{End}_{\mathbb{Q}}(E)={\mathbb{Z}}$ is always satisfied when $K={\mathbb{Q}}$, and we show that in this case $C$ can be taken to be independent of $E$ and $\alpha$, provided that $\alpha$ and all its translates by torsion points are not divisible by any $n>1$ in the group $E({\mathbb{Q}})$. This is a rather surprising statement, especially given that such a strong uniformity result is not known for the closely connected problem of studying the degrees of the torsion fields $K_M$ over $K$.
Main results
------------
Our main results are the following.
\[thm:Main\] Assume that $\operatorname{End}_K(E)={\mathbb{Z}}$. There is an explicit constant $C$, depending only on $\alpha$ and on the $\ell$-adic torsion representations associated to $E$ for all primes $\ell$, such that $$\begin{aligned}
\frac{N^2}{\left[K_{M,N}:K_M\right]}\quad\text{divides}\quad C\end{aligned}$$ for all pairs of positive integers $(M,N)$ with $N$ dividing $M$.
The proof gives an explicit expression for $C$ that depends on computable parameters associated with $E$ and $\alpha$. We also show that all these quantities can be bounded effectively in terms of standard invariants of the elliptic curve and of the height of $\alpha$, see Remark \[rmk:MainTheoremIsEffective\].
\[thm:UniformIntroduction\] There is a universal constant $C$ with the following property. Let $E/{\mathbb{Q}}$ be an elliptic curve, and let $\alpha \in E({\mathbb{Q}})$ be a point such that the class of $\alpha$ in the free abelian group $E({\mathbb{Q}})/E({\mathbb{Q}})_{\operatorname{tors}}$ is not divisible by any $n>1$. Then $$\frac{N^2}{\left[ {\mathbb{Q}}_{M,N} : {\mathbb{Q}}_M \right]} \quad \text{ divides } \quad C$$ for all pairs of positive integers $(M,N)$ with $N$ dividing $M$.
Structure of the paper
----------------------
We start with some necessary general preliminaries in Section \[sec:Preliminaries\], leading up to a factorisation of the constant $C$ of Theorem \[thm:Main\] as a product of certain contributions which we dub the *$\ell$-adic* and *adelic* failures (corresponding to $E$, $\alpha$, and a fixed prime $\ell$). In the same section we also introduce some of the main actors of this paper, in the form of several Galois representations associated with the torsion and Kummer extensions. In Section \[sec:PropertiesTorsionRepresentation\] we then recall some important properties of the torsion representations that will be needed in the rest of the paper. In Sections \[sec:lAdicFailure\] and \[sec:AdelicFailure\] we study the $\ell$-adic and adelic failures respectively. In Section \[sec:CMCounterexample\] we show that one cannot hope to naïvely generalise some of the results in section \[sec:lAdicFailure\] to CM curves. Finally, in Section \[sec:UniformBounds\] we prove Theorem \[thm:UniformIntroduction\] by establishing several auxiliary results about the Galois cohomology of the torsion modules $E[M]$ that might have an independent interest.
Acknowledgements
----------------
It is a pleasure to thank Antonella Perucca for suggesting the problem that led to this paper, for her constant support, and for her useful comments. We are grateful to Peter Bruin for many interesting discussions, and to Peter Stevenhagen and Francesco Campagna for useful correspondence about the results of section \[sec:IntersectionTorsionFields\].
Preliminaries {#sec:Preliminaries}
=============
Notation and definitions
------------------------
The letter $K$ will always denote a number field, $E$ an elliptic curve defined over $K$, and $\alpha$ a point of infinite order in $E(K)$.
For $n$ a positive integer, we denote by $\zeta_n$ a primitive root of unity of order $n$.
Given a prime $\ell$, we denote by $v_\ell$ the usual $\ell$-adic valuation on ${\mathbb{Q}}$ and on ${\mathbb{Q}}_\ell$. If $X$ is a vector in ${\mathbb{Z}}_\ell^n$ or a matrix in $\operatorname{Mat}_{m \times n}({\mathbb{Z}}_\ell)$, we call *valuation* of $X$, denoted by $v_\ell(X)$, the minimum of the $\ell$-adic valuations of its coefficients.
We shall often use divisibility conditions involving the symbols $\ell^\infty$ (where $\ell$ is a prime) and $\infty$. Our convention is that every power of $\ell$ divides $\ell^\infty$, every positive integer divides $\infty$, and $\ell^\infty$ divides $\infty$.
Recall from the Introduction that we denote by $K_M$ the field $K(E[M])$ generated by the coordinates of the $M$-torsion points of $E$, and by $K_{M,N}$ (for $N \mid M$) the field $K(E[M], N^{-1} \alpha)$. We extend this notation by setting $K_{\ell^\infty} = \bigcup_n K_{\ell^n}$, $K_\infty = \bigcup_M K_M$, and more generally, for $M, N \in \mathbb{N}_{>0} \cup \{\ell^\infty, \infty\}$ with $N \mid M$, $$K_M = \bigcup_{d \mid M} K_d, \quad K_{M,N} = \bigcup_{d \mid M} \bigcup_{\substack{e \mid d \\ e \mid N}} K_{d,e}$$
If $H$ is a subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$, we denote by ${\mathbb{Z}}_\ell[H]$ the sub-${\mathbb{Z}}_\ell$-algebra of $\operatorname{Mat}_2({\mathbb{Z}}_\ell)$ generated by the elements of $H$.
Let $G$ be a (profinite) group. We write $G'$ for its derived subgroup, namely, the subgroup of $G$ (topologically) generated by commutators, and $G^{\operatorname{ab}}=G/G'$ for its abelianisation, namely, its largest abelian (profinite) quotient. We say that a finite simple group $S$ *occurs* in a profinite group $G$ if there are closed subgroups $H_1, H_2$ of $G$, with $H_1 \triangleleft H_2$, such that $H_2/H_1$ is isomorphic to $S$. Finally, we denote by $\exp G$ the exponent of a finite group $G$, namely, the smallest integer $e {\geqslant}1$ such that $g^e=1$ for every $g \in G$.
The $\ell$-adic and adelic failures
-----------------------------------
We start by observing that it is enough to restrict our attention to the case $N=M$:
\[rem-NM\] Suppose that there is a constant $C{\geqslant}1$ satisfying $$\begin{aligned}
\frac{M^2}{\left[K_{M,M}:K_M\right]}\quad\text{divides}\quad C\end{aligned}$$ for all positive integers $M$. Then for any $N\mid M$, since $[K_{M,M}:K_{M,N}]$ divides $(M/N)^2$, we have that $$\begin{aligned}
\frac{N^2}{\left[K_{M,N}:K_M\right]}=\frac{N^2[K_{M,M}:K_{M,N}]}{[K_{M,M}:K_M]}\quad \text{divides} \quad \frac{M^2}{\left[K_{M,M}:K_M\right]},\end{aligned}$$ which in turn divides $C$.
Elementary field theory gives $$\begin{aligned}
\frac{N^2}{[K_{N,N}:K_{N}]}=&\prod_{\overset{\ell|N}{\ell\text{ prime}}}\frac{\ell^{2n_\ell}}{[K_{N,\ell^{n_\ell}}:K_N]}=\\
=&\prod_{\overset{\ell|N}{\ell\text{ prime}}}\frac{\ell^{2n_\ell}} {[K_{\ell^{n_\ell},\ell^{n_\ell}}:K_{\ell^{n_\ell}}]}\cdot\frac{[K_{\ell^{n_\ell},\ell^{n_\ell}}:K_{\ell^{n_\ell}}]} {[K_{N,\ell^{n_\ell}}:K_N]}=\\
=&\prod_{\overset{\ell|N}{\ell\text{ prime}}} \frac{\ell^{2n_\ell}} {[K_{\ell^{n_\ell},\ell^{n_\ell}}:K_{\ell^{n_\ell}}]} \cdot [K_{\ell^{n_\ell},\ell^{n_\ell}}\cap K_N:K_{\ell^{n_\ell}}]\end{aligned}$$ where $n_\ell={v_\ell}(N)$. To see why the first equality holds, recall that the degree $[K_{N, \ell^{n_\ell}} : K_N]$ is a power of $\ell$, so the fields $K_{N, \ell^{n_\ell}}$ are linearly disjoint over $K_N$, and clearly they generate all of $K_{N,N}$.
Let $\ell$ be a prime and $N$ a positive integer. Let $n:={v_\ell}(N)$. We call $$\begin{aligned}
A_\ell(N):=\frac{\ell^{2n}} {[K_{\ell^{n},\ell^{n}}:K_{\ell^{n}}]}\end{aligned}$$ the *$\ell$-adic failure* at $N$ and $$\begin{aligned}
B_\ell(N):=\frac{[K_{\ell^{n},\ell^{n}}:K_{\ell^{n}}]} {[K_{N,\ell^{n}}:K_N]}=[K_{\ell^{n},\ell^{n}}\cap K_N:K_{\ell^{n}}]\end{aligned}$$ the *adelic failure* at $N$ (related to $\ell$). Notice that both $A_\ell(N)$ and $B_\ell(N)$ are powers of $\ell$.
It is clear that the $\ell$-adic failure $A_\ell(N)$ can be nontrivial, that is, different from $1$. Suppose for example that $\alpha=\ell\beta$ for some $\beta\in E(K)$: then we have $$\begin{aligned}
K_{\ell^n,\ell^n}=K_{\ell^n}(\ell^{-n}\alpha)=K_{\ell^n}(\ell^{-n+1}\beta),\end{aligned}$$ and the degree of this field over $K_{\ell^n}$ is at most $\ell^{2(n-1)}$, so $\ell^2\mid A_\ell(N)$. In Example \[exa-17739g1\] we will show that the $\ell$-adic failure can be non-trivial also when $\alpha$ is strongly $\ell$-indivisible (see Definition \[def-divisib\]).
We now show that the adelic failure $B_\ell(N)$ can be non-trivial as well. Consider the elliptic curve $E$ over ${\mathbb{Q}}$ given by the equation $$\begin{aligned}
y^2 = x^{3} + x^{2} - 44 x - 84\end{aligned}$$ and with Cremona label 624f2 (see [@lmfdb [label 624f2](http://www.lmfdb.org/EllipticCurve/Q/624f2/)]). One can show that $E({\mathbb{Q}})\cong {\mathbb{Z}}\oplus ({\mathbb{Z}}/2{\mathbb{Z}})^2$, so that the curve has full rational $2$-torsion, and that a generator of the free part of $E({\mathbb{Q}})$ is given by $P=(-5,6)$. The $2$-division points of $P$ are given by $(1 + \sqrt{-3}, -3 + 7\sqrt{-3})$, $(-11 + 3\sqrt{-3},27 + 15\sqrt{-3})$, and their Galois conjugates, so they are defined over ${\mathbb{Q}}(\zeta_3)\subseteq {\mathbb{Q}}_3$, and we have $B_2(6):=[{\mathbb{Q}}_{2,2}\cap {\mathbb{Q}}_6:{\mathbb{Q}}_2]=[{\mathbb{Q}}(\zeta_3):{\mathbb{Q}}]=2$.
These computations have been checked with SageMath [@sagemath].
The torsion, Kummer and arboreal representations {#subsec:Representations}
------------------------------------------------
In this section we introduce three representations of the absolute Galois group of $K$ that will be our main tool for studying the extensions $K_{M,N}$. For further information about these representations see for example [@JonesRouse Section 3], [@2018arXiv180208527B], and [@2016arXiv161202847L].
### The torsion representation
Let $N$ be a positive integer. The group $E[N]$ of $N$-torsion points of $E$ is a free ${\mathbb{Z}}/N{\mathbb{Z}}$-module of rank $2$. Since the multiplication-by-$N$ map is defined over $K$, the absolute Galois group of $K$ acts ${\mathbb{Z}}/N{\mathbb{Z}}$-linearly on $E[N]$, and we get a homomorphism $$\begin{aligned}
\tau_N:\operatorname{Gal}({\overline{K}}\mid K)\to \operatorname{Aut}(E[N]).\end{aligned}$$
The field fixed by the kernel of $\tau_N$ is exactly the $N$-th torsion field $K_N$. Thus, after fixing a ${\mathbb{Z}}/N{\mathbb{Z}}$-basis of $E[N]$, the Galois group $\operatorname{Gal}(K_N\mid K)$ is identified with a subgroup of $\operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ which we denote by $H_N$.
As $N$ varies, and provided that we have made compatible choices of bases, these representations form a compatible projective system, so we can pass to the limit over powers of a fixed prime $\ell$ to obtain the *$\ell$-adic torsion representation* $\tau_{\ell^\infty} : \operatorname{Gal}(\overline{K} \mid K) \to \operatorname{GL}_2({\mathbb{Z}}_\ell)$. We can also take the limit over all integers $N$ (ordered by divisibility) to obtain the *adelic torsion representation* $\tau_\infty : \operatorname{Gal}(\overline{K} \mid K) \to \operatorname{GL}_2(\hat{{\mathbb{Z}}})$. We denote by $H_{\ell^\infty}$ (resp. $H_\infty$) the image of $\tau_{\ell^\infty}$ (resp. $\tau_\infty$). The group $H_{\ell^\infty}$ (resp. $H_{\infty}$) is isomorphic to $\operatorname{Gal}(K_{\ell^\infty} \mid K)$ (resp. $\operatorname{Gal}(K_\infty \mid K)$).
One can also pass to the limit on the torsion subgroups themselves, obtaining the *$\ell$-adic Tate module* $T_\ell E=\varprojlim_n E[\ell^n] \cong {\mathbb{Z}}_\ell^2$ and the *adelic Tate module* $TE = \varprojlim_M E[M] \cong \hat{{\mathbb{Z}}}^2 \cong \prod_{\ell} {\mathbb{Z}}_\ell^2$.
### The Kummer representation
Let $M$ and $N$ be positive integers with $N\mid M$. Let $\beta\in E({\overline{K}})$ be a point such that $N\beta=\alpha$. For any $\sigma\in\operatorname{Gal}({\overline{K}}\mid K_M)$ we have that $\sigma(\beta)-\beta$ is an $N$-torsion point, so the following map is well-defined: $$\begin{array}{cccc}
\kappa_N: & \operatorname{Gal}({\overline{K}}\mid K_M) &\to & E[N]\\
& \sigma &\mapsto & \sigma(\beta)-\beta.
\end{array}$$ Since any other $N$-division point $\beta'$ of $\alpha$ satisfies $\beta'=\beta+T$ for some $T\in E[N]$, and the coordinates of $T$ belong to $K_N \subseteq K_M$, the map $\kappa_N$ does not depend on the choice of $\beta$. It is also immediate to check that $\kappa_N$ is a group homomorphism, and that the field fixed by its kernel is exactly the $(M,N)$-Kummer extension of $K$. Fixing a basis of $E[N]$ we can identify the Galois group $\operatorname{Gal}(K_{M,N}\mid K_M)$ with a subgroup of $({\mathbb{Z}}/N{\mathbb{Z}})^2$. It is then clear that $K_{M,N}$ is an abelian extension of $K_M$ of degree dividing $N^2$, and the Galois group of this extension has exponent dividing $N$.
In the special case $M=N$ we denote by $V_N$ the image of $\operatorname{Gal}\left( K_{N,N} \mid K_N \right)$ in $({\mathbb{Z}}/N{\mathbb{Z}})^2$.
By passing to the limit in the previous constructions we also obtain the following:
- There is an $\ell$-adic Kummer representation $\kappa_{\ell^\infty} : \operatorname{Gal}(\overline{K} \mid K_{\ell^\infty}) \to T_\ell E$ which factors via a map $\operatorname{Gal}(K_{\ell^\infty, \ell^\infty} \mid K_{\ell^\infty}) \to T_\ell E$ (still denoted by $\kappa_{\ell^\infty}$).
- The image $V_{\ell^\infty}$ of $\kappa_{\ell^\infty}$ is a sub-${\mathbb{Z}}_\ell$-module of $T_\ell E \cong {\mathbb{Z}}_\ell^2$, isomorphic to $\operatorname{Gal}(K_{\ell^\infty, \ell^\infty} \mid K_{\ell^\infty})$ as a profinite group. We therefore identify $\operatorname{Gal}(K_{\ell^\infty, \ell^\infty} \mid K_{\ell^\infty})$ with $V_{\ell^\infty}$.
- We can identify the Galois group $\operatorname{Gal}(K_{\infty, \ell^\infty} \mid K_\infty)$ with a ${\mathbb{Z}}_\ell$-submodule $W_{\ell^\infty}$ of $V_{\ell^\infty}$ (hence also of $T_\ell E$) via the representation $\kappa_{\ell^\infty}$.
- We can identify the Galois group $\operatorname{Gal}(K_{\infty, \infty} \mid K_\infty)$ with a sub-$\hat{{\mathbb{Z}}}$-module $W_\infty$ of $TE \cong \hat{{\mathbb{Z}}}^2$.
Notice that $W_{\ell^\infty}$ is the projection of $W_{\infty}$ in ${\mathbb{Z}}_\ell^2$, and since $W_{\ell^\infty}$ is a pro-$\ell$ group and there are no nontrivial continuous morphisms from a pro-$\ell$ group to a pro-$\ell'$ group for $\ell \neq \ell'$ we have $W_\infty = \prod_\ell W_{\ell^\infty}$.
### The arboreal representation
Fix a sequence $\{\beta_i\}_{i\in{\mathbb{N}}}$ of points in $E(\overline{K})$ such that $\beta_1=\alpha$ and $N \beta_M=\beta_{M/N}$ for all pairs of positive integers $(N,M)$ with $N \mid M$. For every $N{\geqslant}1$ fix furthermore a ${\mathbb{Z}}/N{\mathbb{Z}}$-basis $\{T_1^N,T_2^N\}$ of $E[N]$ in such a way that $N T_1^M=T_1^{M/N}$ and $N T_2^M=T_2^{M/N}$ for every pair of positive integers $(N,M)$ with $N \mid M$. For every $N {\geqslant}1$, the map $$\begin{aligned}
\omega_N:\operatorname{Gal}(K_{N,N}\mid K)&\to \left({\mathbb{Z}}/N{\mathbb{Z}}\right)^2\rtimes\operatorname{GL}_2\left({\mathbb{Z}}/N{\mathbb{Z}}\right)\\
\sigma&\mapsto \left(\sigma(\beta_N)-\beta_N,\tau_{N}(\sigma)\right)\end{aligned}$$ is an injective homomorphism (similarly to [@JonesRouse Proposition 3.1]) and thus identifies the group $\operatorname{Gal}(K_{N,N}\mid K)$ with a subgroup of $\left({\mathbb{Z}}/N{\mathbb{Z}}\right)^2\rtimes\operatorname{GL}_2\left({\mathbb{Z}}/N{\mathbb{Z}}\right)$.
It will be important for our applications to notice that $V_N$ comes equipped with an action of $H_N$ coming from the fact that $V_N$ is the (abelian) kernel of the natural map $\operatorname{Gal}(K_{N,N}\mid K) \to H_N$. More precisely, the action of $h \in H_N$ on $v \in V_N$ is given by conjugating the element $(v, \operatorname{Id}) \in ({\mathbb{Z}}/N{\mathbb{Z}})^2 \rtimes \operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ by $(0,h)$. Explicitly, we have $$(0,h) (v, \operatorname{Id}) (0,h)^{-1} = (hv,h)(0,h^{-1}) = (hv, \operatorname{Id}),$$ so that the action of $H_N$ on $V_N$ is induced by the natural action of $\operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ on $\left({\mathbb{Z}}/N{\mathbb{Z}}\right)^2$. We obtain similar statements by suitably passing to the limit in $N$:
\[lemma:HnActionOnVn\] For every positive integer $N$, the group $V_N$ is an $H_N$-submodule of $({\mathbb{Z}}/N{\mathbb{Z}})^2$ for the natural action of $H_N {\leqslant}\operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ on $V_N {\leqslant}({\mathbb{Z}}/N{\mathbb{Z}})^2$. Similarly, both $V_{\ell^\infty}$ and $W_{\ell^\infty}$ are $H_{\ell^\infty}$-modules.
\[rem:Vsubgroup\] Let $N \in \mathbb{N} \cup \{\ell^\infty\}$ and $M \in \mathbb{N} \cup \{ \ell^\infty, \infty \}$ with $N \mid M$. Then $\operatorname{Gal}(K_{M,N} \mid K_M)$ can be identified with a subgroup of $V_N$: this follows from inspection of the diagram $$\xymatrix{
& K_{M,N} \ar@{-}[dr] \ar@{-}[dl] \\
K_M \ar@{-}[dr] && K_{N,N} \ar@{-}[dl] \\
& K_M \cap K_{N,N} \ar@{-}[d] \\
& K_N
}$$ which shows that $\operatorname{Gal}(K_{M,N} \mid K_M)$ is isomorphic to $\operatorname{Gal}( K_{N,N} \mid K_M \cap K_{N,N})$, which in turn is clearly a subgroup of $\operatorname{Gal}(K_{N,N} \mid K_N) \cong V_N$.
Curves with complex multiplication {#subsect:CMCurves}
----------------------------------
If $\operatorname{End}_{{\overline{K}}}(E)\neq \mathbb{Z}$ we say that *$E$ has complex multiplication*, or CM for short. In this case $\operatorname{End}_{{\overline{K}}}(E)$ is an order in an imaginary quadratic field, called *the CM-field of $E$*. The torsion representations in the CM case have been studied for example in [@Deur1] and [@Deur2].
In this case, the image of the torsion representation $\tau_{\ell^\infty}$ is closely related to the *Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ corresponding to $\operatorname{End}_{\overline{K}}(E)$*, defined as follows:
Let $F$ be a reduced ${\mathbb{Q}}_\ell$-algebra of degree $2$ and let ${\mathcal{A}}_\ell$ be a ${\mathbb{Z}}_\ell$-order in $F$. The *Cartan subgroup* corresponding to ${\mathcal{A}}_\ell$ is the group of units of ${\mathcal{A}}_\ell$, which we embed in $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ by fixing a ${\mathbb{Z}}_\ell$-basis of ${\mathcal{A}}_\ell$ and considering the left multiplication action of ${\mathcal{A}}_\ell^\times$. If ${\mathcal{A}}$ is an order in an imaginary quadratic number field, the Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ corresponding to ${\mathcal{A}}$ is defined by taking ${\mathcal{A}}_\ell={\mathcal{A}}\otimes {\mathbb{Z}}_\ell$ in the above.
More precisely, when $E/K$ is an elliptic curve with CM, the image of the $\ell$-adic torsion representation $\tau_{\ell^\infty}$ is always contained (up to conjugacy in $\operatorname{GL}_2({\mathbb{Z}}_\ell)$) in the normaliser of the Cartan subgroup corresponding to $\operatorname{End}_{\overline{K}}(E)$, and is contained in the Cartan subgroup itself if and only if the complex multiplication is defined over the base field $K$.
In order to have a practical representation of Cartan subgroups, we recall the following definition from [@MR3690236]:
\[def:CartanParameters\] Let $C$ be a Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$. We say that $(\gamma, \delta) \in {\mathbb{Z}}_\ell^2$ are *parameters for $C$* if $C$ is conjugated in $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ to the subgroup $$\label{nfaa}
\left\{ \begin{pmatrix}
x & \delta y \\ y & x+\gamma y
\end{pmatrix} : x,y \in {\mathbb{Z}}_{\ell},\; v_{\ell}(x(x+\gamma y)-\delta y^2)=0 \right\}\,.$$ Parameters for $C$ always exist, see [@MR3690236 §2.3].
\[rem:Parameters\] One may always assume that $\gamma,\delta$ are integers. Furthermore, one can always take $\gamma \in \{0,1\}$, and $\gamma=0$ if $\ell \neq 2$.
We also recall the following explicit description of the normaliser of a Cartan subgroup [@MR3690236 Lemma 14]:
\[lem-Norm\] A Cartan subgroup has index 2 in its normaliser. If $C$ is as in , its normaliser $N$ in $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ is the disjoint union of $C$ and $\displaystyle C':=\begin{pmatrix}
1 & \gamma \\ 0 & -1
\end{pmatrix} \cdot C\,.$
Properties of the torsion representation {#sec:PropertiesTorsionRepresentation}
========================================
Torsion representations are studied extensively in the literature; we have in particular the following fundamental theorem of Serre [@Serre], which applies to all elliptic curves (defined over number fields) without complex multiplication:
\[thm:Serre\] If $\operatorname{End}_{{\overline{K}}}(E)={\mathbb{Z}}$, then $H_\infty$ is open in $\operatorname{GL}_2(\hat{{\mathbb{Z}}})$. Equivalently, the index of $H_N$ in $\operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ is bounded independently of $N$.
There is also a CM analogue of Theorem \[thm:Serre\], which is more easily stated by introducing the following definition:
\[def:MaximalRepresentation\] Let $E/K$ be an elliptic curve and $\ell$ be a prime number. We say that the image of the $\ell$-adic representation is *maximal* if one of the following holds:
- $E$ does not have CM over $\overline K$ and $H_{\ell^\infty}=\operatorname{GL}_2({\mathbb{Z}}_\ell)$.
- $E$ has CM over $K$ by an order ${\mathcal{A}}$ in the imaginary quadratic field $F$, the prime $\ell$ is unramified in $F$ and does not divide $[\mathcal{O}_F : {\mathcal{A}}]$, and $H_{\ell^\infty}$ is conjugated to the Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ corresponding to ${\mathcal{A}}$.
- $E$ has CM over $\overline K$ (but not over $K$) by an order ${\mathcal{A}}$ in the imaginary quadratic field $F$, the prime $\ell$ is unramified in $F$ and does not divide $[\mathcal{O}_F : {\mathcal{A}}]$, and $H_{\ell^\infty}$ is conjugated to the normaliser of the Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ corresponding to ${\mathcal{A}}$.
\[thm:CMSerre\] Let $E/K$ be an elliptic curve admitting CM over $\overline{K}$. Then the $\ell$-adic representation attached to $E/K$ is maximal for all but finitely many primes $\ell$.
In the rest of this section we recall various important properties of the torsion representations: we shall need results that describe both the asymptotic behaviour of the $\bmod \,\ell^n$ torsion representation as $n \to \infty$ (§\[subsect:MaximalGrowth\] and \[subsect:UniformGrowth\]) and the possible images of the $\bmod\,\ell$ representations attached to elliptic curves defined over the rationals (§\[subsec:PossibleImagesModl\]).
Maximal growth {#subsect:MaximalGrowth}
--------------
We recall some results on the growth of the torsion extensions from [@2016arXiv161202847L §2.3].
\[prop-max-growth\] Let $\ell$ be a prime number. Let $\delta=2$ if $E$ has complex multiplication and $\delta=4$ otherwise. There exists a positive integer $n_\ell$ such that $$\begin{aligned}
\#H_{\ell^{n+1}}/\#H_{\ell^n}=\ell^\delta \qquad \text{for every }n{\geqslant}n_\ell.\end{aligned}$$
This follows from Theorem \[thm:Serre\] in the non-CM case and from classical results in the CM case. See also [@2016arXiv161202847L Lemma 10 and Remark 13] for a more general result.
We call an integer $n_\ell$ as in Proposition \[prop-max-growth\] a *parameter of maximal growth for the $\ell$-adic torsion representation*. We say that it is *minimal* if $n_\ell-1$ is not a parameter of maximal growth; when $\ell=2$, we require that the minimal parameter be at least 2.
The assumption $n_\ell {\geqslant}2$ when $\ell=2$ will be needed to apply [@2016arXiv161202847L Theorem 12].
\[rmk:nlIsEffective\] Given an explicit elliptic curve $E/K$ and a prime $\ell$, the problem of determining the optimal value of $n_\ell$ can be solved effectively (see [@2016arXiv161202847L Remark 13]). However, computing $n_\ell$ can be challenging in practice, because the naïve algorithm requires the determination of the Galois groups of the splitting fields of several large-degree polynomials. The situation is usually better for smaller primes $\ell$, and especially for $\ell=2$, for which the 2-torsion tower is known essentially explicitly (see [@MR3500996] for a complete classification result when $K=\mathbb{Q}$, and [@MR3426175] for a description of the 2-torsion tower of a given elliptic curve over a number field).
In Section \[sec:AdelicFailure\] we will need to bound the minimal parameter of maximal growth for the $\ell$-adic torsion representation defined over certain extensions of the base field. We will do so with the help of the following Lemma:
\[lem-param-tilde-bound\] Let $\tilde K$ be a finite extension of $K$. Let $n_\ell$ (resp. $\tilde n_\ell$) be the minimal parameter of maximal growth for the $\ell$-adic torsion representation attached to $E/K$ (resp. $E/\tilde K$). Then $\tilde n_\ell{\leqslant}n_\ell+v_\ell([\tilde K:K])$.
Let $n_0:=n_\ell+v_\ell([\tilde K:K])+1$ and consider the following diagram:
(L2tilde) at (0,1) [$\tilde K_{\ell^{n_0}}$]{}; (L1tilde) at (-2,0) [$\tilde K_{\ell^{n_\ell}}$]{}; (L2) at (2,0) [$K_{\ell^{n_0}}$]{}; (L1tildecapL2) at (0,-1) [$\tilde K_{\ell^{n_\ell}}\cap K_{\ell^{n_0}}$]{}; (Ktilde) at (-4,-1) [$\tilde K$]{}; (fantasma) at (4,-1) ; (L1) at (2,-2) [$K_{\ell^{n_\ell}}$]{}; (KtildecapL1) at (0,-3) [$\tilde K \cap K_{\ell^{n_\ell}}$]{}; (K) at (0,-4) [$K$]{}; (L2)–(L2tilde)–(L1tilde)–(Ktilde); (L1)–(L1tildecapL2)–(L1tilde); (K)–(KtildecapL1)–(Ktilde); (L2)–(L1tildecapL2); (L1)–(KtildecapL1);
Since clearly $[\tilde K_{\ell^{n_\ell}}\cap K_{\ell^{n_0}}:K_{\ell^{n_\ell}}]$ divides $[\tilde K_{\ell^{n_\ell}}:K_{\ell^{n_\ell}}]$, which in turn divides $[\tilde K:K]$, and since $[\tilde K_{\ell^{n_0}}:\tilde K_{\ell^{n_\ell}}]=[K_{\ell^{n_0}}:\tilde K_{\ell^{n_\ell}}\cap K_{\ell^{n_0}}]$, we have $$\begin{aligned}
v_\ell\left([K_{\ell^{n_0}}:K_{\ell^{n_\ell}}]\right) & = v_\ell\left([K_{\ell^{n_0}}:\tilde K_{\ell^{n_\ell}} \cap K_{\ell^{n_0}}] \right) + v_\ell\left( [\tilde K_{\ell^{n_\ell}}\cap K_{\ell^{n_0}}:K_{\ell^{n_\ell}}]\right) \\ &{\leqslant}v_\ell\left([\tilde K_{\ell^{n_0}}:\tilde K_{\ell^{n_\ell}}]\right)+ v_\ell\left( [\tilde K:K]\right).
\end{aligned}$$ By [@2016arXiv161202847L Theorem 12] we have $v_\ell\left([K_{\ell^{n_0}}:K_{\ell^{n_\ell}}]\right)=\delta(n_0-n_\ell)=\delta\left( v_\ell\left([\tilde K:K]\right)+1\right)$, where $\delta$ is as in Proposition \[prop-max-growth\], and we get $$\begin{aligned}
v_\ell\left([\tilde K_{\ell^{n_0}}:\tilde K_{\ell^{n_\ell}}]\right){\geqslant}\delta+ (\delta-1)v_\ell\left([\tilde K:K]\right)> (\delta-1)(n_0-n_\ell).\end{aligned}$$ Consider now the tower of extensions $$\begin{aligned}
\tilde K_{\ell^{n_\ell}}\subseteq \tilde K_{\ell^{n_\ell+1}}\subseteq \cdots \subseteq \tilde K_{\ell^{n_0}}\end{aligned}$$ and notice that by the pigeonhole principle for at least one $n\in {\left\{ n_\ell, n_\ell+1,\dots, n_0-1 \right\}}$ we must have $[\tilde K_{\ell^{n+1}}:\tilde K_{\ell^{n}}]{\geqslant}\delta$. But then by [@2016arXiv161202847L Theorem 12] we have maximal growth over $\tilde K$ from $n< n_0$. Thus we get $\tilde n_\ell{\leqslant}n_\ell+v_\ell\left([\tilde K:K]\right)$ as claimed.
Uniform growth of $\ell$-adic representations {#subsect:UniformGrowth}
---------------------------------------------
The results in this subsection and the next will be needed in Section \[sec:UniformBounds\]. We start by recalling the following result, due to Arai:
\[thm:Arai\] Let $K$ be a number field and let $\ell$ be a prime. Then there exists an integer $n {\geqslant}0$, depending only on $K$ and $\ell$, such that for any elliptic curve $E$ over $K$ with no complex multiplication over $\overline{K}$ we have $$\tau_{\ell^\infty}(\operatorname{Gal}(\overline{K} \mid K)) \supseteq \{M \in \operatorname{GL}_2({\mathbb{Z}}_\ell) : M \equiv \operatorname{Id}\pmod{\ell^n}\}.$$
For the next result we shall need a well-known Lemma about twists of elliptic curves:
\[lemma:Untwisting\] Let $E_1, E_2$ be elliptic curves over $K$ such that $(E_1)_{\overline{{\mathbb{Q}}}}$ is isomorphic to $(E_2)_{\overline{{\mathbb{Q}}}}$. There is an extension $F$ of $K$, of degree dividing $12$, such that $E_1$ and $E_2$ become isomorphic over $F$.
Fixing a $\overline{{\mathbb{Q}}}$-isomorphism between $E_1$ and $E_2$ allows us to attach to $E_2$ a class in $H^1\left( \operatorname{Gal}(\overline{K} \mid K), \operatorname{Aut}(E_1) \right)$. Since $H^1\left( \operatorname{Gal}(\overline{K} \mid K), \operatorname{Aut}(E_1) \right) \cong K^\times/K^{\times n}$ for some $n \in \{2, 4, 6\}$ (see [@SilvermanEC Proposition X.5.4]), the class of $E_2$ corresponds to the class of a certain $[\alpha] \in K^\times/K^{\times n}$. Letting $F=K(\sqrt[n]{\alpha})$, whose degree over $K$ divides $12$, it is clear that $[\alpha] \in F^\times/F^{\times n}$ is the trivial class, so the same is true for $[E_2] \in H^1(\operatorname{Gal}(\overline{F} \mid F), \operatorname{Aut}(E_1))$, which means that $E_2$ is isomorphic to $E_1$ over $F$ as desired.
\[cor:UniversalBoundGrowthParameter\] Let $K$ be a number field and $\ell$ be a prime number. There exists an integer $n_\ell$ with the following property: for every elliptic curve $E/K$, the minimal parameter of maximal growth for the $\ell$-adic representation attached to $E$ is at most $n_\ell$.
Let $n$ be the integer whose existence is guaranteed by Theorem \[thm:Arai\]. By the general theory of CM elliptic curves, we know that there are finitely many values $j_1,\ldots,j_k \in \overline{{\mathbb{Q}}}$ such that for every CM elliptic curve $E/K$ we have $j(E) \in \{j_1,\ldots,j_k\}$. For each such $j_i$, fix an elliptic curve $E_i/K$ with $j(E_i)=j_i$. To every $E_i/K$ corresponds a minimal parameter of maximal growth for the $\ell$-adic representation that we call $m_i$. Let $n_\ell=\max\{n, m_i+2 \bigm\vert i =1,\ldots,k\}$: we claim that this value of $n_\ell$ satisfies the conclusion of the Corollary. Indeed, let $E/K$ be any elliptic curve. If $E$ does not have CM, the minimal parameter of maximal growth for its $\ell$-adic representation is at most $n {\leqslant}n_\ell$. If $E$ has CM, then there exists $i$ such that $j(E)=j_i=j(E_i)$, so $E$ is a twist of $E_i$. By Lemma \[lemma:Untwisting\] the curves $E$ and $E_i$ become isomorphic over an extension $F/K$ of degree dividing $12$, so if $m$ (resp. $\tilde{m}$, resp. $\tilde{m}_i$) denotes the minimal parameter of maximal growth for $E/K$ (resp. for $E/F$, resp. for $E_i/F$) we have $$m {\leqslant}\tilde{m} = \tilde{m}_i {\leqslant}m_i + 2 {\leqslant}n_\ell,$$ where the equality follows from the fact that $E$ and $E_i$ are isomorphic over $F$, while the inequality $\tilde{m}_i {\leqslant}m_i + 2$ follows from Lemma \[lem-param-tilde-bound\] and the fact that $v_\ell([F:K]) {\leqslant}v_\ell(12) {\leqslant}2$ for every prime $\ell$.
Possible images of $\bmod\,\ell$ representations {#subsec:PossibleImagesModl}
------------------------------------------------
We recall several results concerning the images of the $\bmod\,\ell$ representations attached to elliptic curves over ${\mathbb{Q}}$.
We begin with a famous Theorem of Mazur. Let $\mathcal{T}_0:={\left\{ p\text{ prime }\mid p{\leqslant}17 \right\}}\cup\{37\}$.
\[thm:Mazur\] Let $E/{\mathbb{Q}}$ be an elliptic curve and assume that $E$ has a $\mathbb{Q}$-rational subgroup of order $p$. Then $p\in \mathcal T_0\cup {\left\{ 19,43,67,163 \right\}}$. If $E$ does not have CM over $\overline {\mathbb{Q}}$, then $p\in\mathcal{T}_0$.
\[thm:Zywyna\] Let $E/{\mathbb{Q}}$ be a non-CM elliptic curve and $p\not \in \mathcal{T}_0$ be a prime. Let $C_{\operatorname{ns}}(p)$ be the subgroup of $\operatorname{GL}_2({\mathbb{F}}_p)$ consisting of all matrices of the form $\begin{pmatrix}
a & b\epsilon\\
b & a
\end{pmatrix}$ with $(a,b)\in{\mathbb{F}}_p^2\setminus \{(0,0)\}$ and $\epsilon$ a fixed element of ${\mathbb{F}}_p^\times\setminus {\mathbb{F}}_p^{\times 2}$. Then $H_p$ is conjugate to one of the following:
(i) $\operatorname{GL}_2({\mathbb{F}}_p)$;
(ii) the normaliser $N_{\operatorname{ns}}(p)$ of $C_{\operatorname{ns}}(p)$;
(iii) the index $3$ subgroup $$D(p):={\left\{ a^3\mid a\in C_{\operatorname{ns}}(p) \right\}}\cup {\left\{ \begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}\cdot a^3\mid a\in C_{\operatorname{ns}}(p) \right\}}$$ of $N_{\operatorname{ns}}(p)$.
Moreover, the last case can only occur if $p\equiv 2\pmod 3$.
\[cor:ContainsScalarsAndConjugation\] Let $E/{\mathbb{Q}}$ be a non-CM elliptic curve and $p\not \in \mathcal{T}_0$ be a prime. The following hold:
(1) The image $H_p$ of the modulo-$p$ representation attached to $E$ contains $\{\lambda\operatorname{Id}\mid \lambda\in{\mathbb{F}}_p^\times\}$.
(2) Suppose $H_p\neq \operatorname{GL}_2({\mathbb{F}}_p)$ and let $g_p\in \operatorname{GL}_2({\mathbb{F}}_p)$ be an element that normalises $H_p$. Then there is $h\in \operatorname{GL}_2({\mathbb{F}}_p)$ such that $h^{-1}g_ph\in N_{\operatorname{ns}}(p)$ and $h^{-1}H_ph\subseteq N_{\operatorname{ns}}(p)$.
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(1) We apply Theorem \[thm:Zywyna\]. If $H_p$ is either $\operatorname{GL}_2({\mathbb{F}}_p)$ or conjugate to $N_{\operatorname{ns}}(p)$, the conclusion follows trivially, since $C_{\operatorname{ns}}(p)$ contains all scalars. In case (iii) of Theorem \[thm:Zywyna\], $H_p$ contains the cubes of the scalars, hence all scalars since $p\equiv 2\pmod 3$.
(2) We only have to consider cases (ii) and (iii) of Theorem \[thm:Zywyna\]. Up to conjugation, we may assume that $H_p\subseteq N_{\operatorname{ns}}(p)$ and the claim becomes $g_p\in N_{\operatorname{ns}}(p)$.
In case (ii) it suffices to check that the normaliser of $N_{\operatorname{ns}}(p)$ is $N_{\operatorname{ns}}(p)$ itself. This holds because $C_{\operatorname{ns}}(p)$, being the only cyclic subgroup of index $2$ of $N_{\operatorname{ns}}(p)$, is characteristic in $N_{\operatorname{ns}}(p)$; hence any element that normalises $N_{\operatorname{ns}}(p)$ normalises $C_{\operatorname{ns}}(p)$ as well, so it must be in $N_{\operatorname{ns}}(p)$. In case (iii), one similarly sees that $\{a^3\mid a\in C_{\operatorname{ns}}(p)\}$ is characteristic in $D(p)$ and that its normaliser is $N_{\operatorname{ns}}(p)$, and the conclusion follows as above.
\[lem:LiftHomothety\] Let $\ell$ be a prime number and let $H$ be a closed subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$. Denote by $H_\ell$ the reduction of $H$ modulo $\ell$ and suppose that $H_\ell$ contains a scalar matrix $\overline{\lambda} \operatorname{Id}$. Then $H$ contains a scalar matrix $\lambda \operatorname{Id}$ for some $\lambda \in {\mathbb{Z}}_\ell^\times$ with $\lambda \equiv \overline{\lambda} \pmod{\ell}$.
Let $h \in H$ be any element that is congruent modulo $\ell$ to $\overline{\lambda} \operatorname{Id}$. Let $\lambda \in {\mathbb{Z}}_\ell^\times$ be the Teichmüller lift of $\overline{\lambda}$ (that is, $\lambda^\ell=\ell$ and $\lambda \equiv \overline{\lambda} \pmod{\ell}$) and write $h=\lambda h_1$, where $h_1 =\operatorname{Id}+\ell A$ for some $A\in\operatorname{Mat}_2({\mathbb{Z}}_\ell)$. The sequence $h^{\ell^n} = \lambda^{\ell^n} h_1^{\ell^n}= \lambda h_1^{\ell^n}$ converges to $\lambda \operatorname{Id}$, because for every $n$ we have $h_1^{\ell^n} = \left( \operatorname{Id} + \ell A \right)^{\ell^n} \equiv \operatorname{Id} \pmod{\ell^n}$. As $H$ is closed, the limit of this sequence, namely $\lambda \operatorname{Id}$, also belongs to $H$ as claimed.
The $\ell$-adic failure {#sec:lAdicFailure}
=======================
The aim of this section is to study the $\ell$-adic failure $A_\ell(N)$ for a fixed prime $\ell$. The divisibility properties of $\alpha$ in the group $E(K)$ play a crucial role in the study of this quantity, so we begin with the following definition:
\[def-divisib\] Let $\alpha \in E(K)$ and let $n$ be a positive integer. We say that $\alpha$ is *$n$-indivisible over $K$* if there is no $\beta \in E(K)$ such that $n \beta=\alpha$; otherwise we say that $\alpha$ is *$n$-divisible* or *divisible by $n$ over $K$*. Let $\ell$ be a prime number. We say that $\alpha$ is *strongly $\ell$-indivisible over $K$* if the point $\alpha+T$ is $\ell$-indivisible over $K$ for every torsion point $T\in E(K)$ of $\ell$-power order. Finally, we say that $\alpha$ is *strongly indivisible over $K$* if its image in the free abelian group $E(K)/E(K)_{\operatorname{tors}}$ is not divisible by any $n>1$, or equivalently if $\alpha$ is strongly $\ell$-indivisible over $K$ for every prime $\ell$.
Our aim is to give an analogue of the following result, which bounds the index of the image of the Kummer representation, in those cases when the torsion representation is not surjective.
\[thm:JonesRouseSurjectivity\] Assume that the $\ell$-adic torsion representation $\tau_{\ell^\infty}:\operatorname{Gal}(K_{\ell^\infty}\mid K)\to \operatorname{GL}_2({\mathbb{Z}}_\ell)$ is surjective. Assume that $\alpha$ is $\ell$-indivisible in $E(K)$ and, if $\ell=2$, that $K_{2,2}\not\subseteq K_4$. Then the $\ell$-adic Kummer representation $\kappa_{\ell^\infty}:\operatorname{Gal}(K_{\ell^\infty,\ell^\infty}\mid K_{\ell^\infty})\to {\mathbb{Z}}_\ell^2$ is surjective.
An exact sequence
-----------------
We shall need to understand the divisibility properties of $\alpha$ not only over the base field $K$, but also over the division fields of $E$. Thus we turn to studying how the divisibility of the point $\alpha$ by powers of $\ell$ changes when passing to a field extension. Our main tool will be the following Lemma.
\[lem-divibility-1\] Let $L$ be a finite Galois extension of $K$ with Galois group $G$. For every $m{\geqslant}1$ there is an exact sequence of abelian groups $$\begin{aligned}
0 \to m E(K)\to E(K)\cap m E(L)\to H^1(G,E[m](L)),\end{aligned}$$ where the injective map on the left is the natural inclusion.
Consider the short exact sequence of $G$-modules $$\begin{aligned}
0\to E[m](L)\to E(L) \xrightarrow{[m]} m E(L)\to 0\end{aligned}$$ and the beginning of the long exact sequence in cohomology, $$\begin{aligned}
0\to (E[m](L))^G\to (E(L))^G\to (m E(L))^G\to H^1(G,E[m](L)).\end{aligned}$$ Noticing that $$\begin{aligned}
(E[m](L))^G= E[m](K), && (E(L))^G=E(K), && (m E(L))^G = E(K)\cap m E(L)\end{aligned}$$ and that $$\begin{aligned}
E(K)/E[m](K) \cong m E(K)\end{aligned}$$ the lemma follows.
The quotient $\left(E(K)\cap m E(L)\right)/m E(K)$ gives a measure of “how many” $K$-points of $E$ are $m$-divisible in $E(L)$ but not $m$-divisible in $E(K)$. We shall often use this Lemma in the special case of $m=\ell^n$ being a power of $\ell$: in this context, the quotient $\left(E(K)\cap \ell^n E(L)\right)/\ell^n E(K)$ is a subgroup of $E(K)/\ell^n E(K)$, so it has exponent dividing $\ell^n$. We conclude that if $\ell\nmid \#H^1(G,E[\ell^n](L))$ then no $\ell$-indivisible $K$-point of $E$ can become $\ell$-divisible in $E(L)$. This applies in particular when $\ell\nmid \#G$, see [@MR2392026 Proposition 1.6.2].
Divisibility in the $\ell$-torsion field
----------------------------------------
As an example, we investigate the situation of Lemma \[lem-divibility-1\] with $m=\ell$ and $L=K_\ell$. In this case the exact sequence becomes $$\begin{aligned}
0 \to \ell E(K)\to E(K)\cap\ell E(K_{\ell})\to H^1(H_\ell,E[\ell]).\end{aligned}$$
The following Lemma can also be found in [@LawsonWutrich Section 3].
The cohomology group $H^1(H_\ell,E[\ell])$ is either trivial or cyclic of order $\ell$. When $\ell=2$ it is always trivial.
Since $\ell E[\ell]=0$, we have $\ell H^1(H_\ell,E[\ell])=0$. It follows from [@Serre-LocalFields Theorem IX.4] that we have an injective map $H^1(H_\ell,E[\ell])\to H^1(S_\ell, E[\ell])$, where $S_\ell$ is an $\ell$-Sylow subgroup of $H_\ell$. This is either trivial, in which case $H^1(H_\ell,E[\ell])=0$, or cyclic of order $\ell$. In the latter case, up to a change of basis for $E[\ell]$ we can assume that $S_\ell$ is generated by $\sigma=\left(\begin{array}{cc}1&1\\ 0&1\end{array}\right)$. One can conclude the proof by explicitly computing the cohomology of the cyclic group $\langle \sigma\rangle$ as in [@LawsonWutrich Lemma 7].
In [@LawsonWutrich] the authors classify the cases when $H^1(H_\ell,E[\ell]) \neq 0$ for $K={\mathbb{Q}}$ and they give rather complete results in case $K$ is a number field with $K\cap{\mathbb{Q}}(\zeta_\ell)={\mathbb{Q}}$. In particular, it turns out that, for $K={\mathbb{Q}}$, the group $H^1(H_\ell,E[\ell])$ can be non-trivial only when $\ell=3,5,11$, and only when additional conditions are satisfied (see [@LawsonWutrich Theorem 1]).
The next Example shows that for $K={\mathbb{Q}}$ a point in $E({\mathbb{Q}})$ that is strongly $3$-indivisible may become $3$-divisible over the $3$-torsion field.
\[exa-17739g1\] Consider the elliptic curve $E$ over ${\mathbb{Q}}$ given by the equation $$\begin{aligned}
y^2 + y = x^{3} - 216 x - 1861 \end{aligned}$$ with Cremona label 17739g1 (see [@lmfdb [label 17739g1](http://www.lmfdb.org/EllipticCurve/Q/17739g1/)]). We have $E({\mathbb{Q}})\cong {\mathbb{Z}}\oplus {\mathbb{Z}}/3{\mathbb{Z}}$, with a generator of the free part given by $P=\left(\frac{23769}{400}, \frac{3529853}{8000}\right)$. One can show that $P$ is strongly $3$-indivisible.
Since the $\mathbb{Q}$-isogeny class of $E$ consists of exactly two curves, by [@LawsonWutrich Theorem 1] we have $H^1(H_3,E[3])={\mathbb{Z}}/3{\mathbb{Z}}$. The $3$-torsion field is given by ${\mathbb{Q}}(z)$, where $z$ is any root of $x^6+3$. Over this field the point $$\begin{aligned}
Q=\left(\frac{803}{400}z^4 - \frac{416}{400}z^2 + \frac{507}{400}, \frac{89133}{8000}z^4 - \frac{199071}{8000}z^2 - \frac{95323}{8000}\right)\in E({\mathbb{Q}}(z))\end{aligned}$$ is such that $3Q=P$.
A computer search performed with the help of the LMFDB [@lmfdb] and of Pari/GP [@PARI2] shows that there are only $20$ elliptic curves with conductor less than $4\times 10^{5}$ satisfying this property for $\ell=3$, none of which has conductor less than $17739$.
Divisibility in the $\ell$-adic torsion tower
---------------------------------------------
As we have seen in the previous Section, the $\ell$-divisibility of a point can increase when we move along the $\ell$-adic torsion field tower. We would now like to give a bound on the extent of this phenomenon.
Our purpose in this section is to prove Proposition \[prop-not-divisible\] (essentially an application of Sah’s lemma, see [@MR0229713 Proposition 2.7 (b)] and [@MR2018998 Lemma A.2]), which will allow us to give such a bound in terms of the image of the torsion representation.
\[lem-not-divisible\] Let $L$ be a finite Galois extension of $K$ containing $K_{\ell^{n}}$ and let $G:=\operatorname{Gal}(L|K)$. Assume that $\ell^{k}H^1(G,E[\ell^n])=0$. If $\alpha\in E(K)$ is strongly $\ell$-indivisible in $E(K)$, then $\alpha$ is not $\ell^{k+1}$-divisible in $E(L)$.
Applying Lemma \[lem-divibility-1\] with $M=\ell^{k+1}$ we have that the quotient $\frac{E(K)\cap \ell^{k+1}E(L)}{\ell^{k+1} E(K)}$ embeds in $H^1(G,E[\ell^n])$, so it is killed by $\ell^k$. Therefore $\ell^k\left(E(K)\cap \ell^{k+1}E(L)\right)\subseteq\ell^{k+1} E(K)$. Assuming by contradiction that $\alpha\in \ell^{k+1} E(L)$ we get $\ell^{k}\alpha=\ell^{k+1}\beta$ for some $\beta\in E(K)$. But then $T=\ell\beta-\alpha\in E[\ell^{k}](K)$ is such that $\alpha+T\in \ell E(K)$, contradicting our assumption that $\alpha$ is strongly $\ell$-indivisible.
\[lemma:ExponentOfH1\] Assume that for some $n_0{\geqslant}1$ we have $(1+\ell^{n_0})\operatorname{Id}\in H_{\ell^n}$ (if $n {\leqslant}n_0$ the condition is automatically satisfied). Then the exponent of $H^1(H_{\ell^n},E[\ell^k])$ divides $\ell^{n_0}$ for every $k {\leqslant}n$.
Let $\lambda=(1+\ell^{n_0})\operatorname{Id}$ and let $\varphi:H_{\ell^n}\to E[\ell^{k}]$ be a cocycle. Using that $\lambda$ is central in $H_{\ell^n}$ and that $\varphi$ is a cocycle, for any $g\in H_{\ell^n}$ we have $$\begin{aligned}
g\varphi(\lambda)+\varphi(g)=\varphi(g\lambda)=\varphi(\lambda g)= \lambda\varphi(g)+\varphi(\lambda),\end{aligned}$$ so $$\begin{aligned}
\ell^{n_0}\varphi(g)=(\lambda-1)\varphi(g)=g\varphi(\lambda)-\varphi(\lambda),\end{aligned}$$ that is, $\ell^{n_0}\varphi$ is a coboundary. This proves that $\ell^{n_0}H^1(H_{\ell^n},E[\ell^{k}])=0$ as claimed.
\[lem-all-mat\] Assume that $E$ does not have complex multiplication and let $n_\ell{\geqslant}1$ be a parameter of maximal growth for the $\ell$-adic torsion representation. Then for every $n{\geqslant}n_\ell$ and for every $g\in \operatorname{Mat}_{2}({\mathbb{Z}}_\ell)$ we have that $(\operatorname{Id}+ \ell^{n_\ell}g) \bmod \ell^n$ is an element of $H_{\ell^n}$.
We prove this by induction. For $n=n_\ell$ the statement is trivial, so suppose $(\operatorname{Id}+\ell^{n_\ell}g)\mod \ell^n$ belongs to $H_{\ell^n}$ for some $n> n_\ell$. Since the map $H_{\ell^{n+1}}\to H_{\ell^n}$ is surjective we can lift this element to an element of the form $\operatorname{Id}+ \ell^{n_\ell}g + \ell^n g' \in H_{\ell^{n+1}}$, where $g'\in\operatorname{Mat}_{2}({\mathbb{F}}_\ell)$. Since $$\ker(H_{\ell^{n+1}}\to H_{\ell^n})={\left\{ \operatorname{Id}+ \ell^n h \mid h\in \operatorname{Mat}_{2}({\mathbb{F}}_\ell) \right\}}$$ we have that $\operatorname{Id}- \ell^n g'$ is in $H_{\ell^{n+1}}$, hence $H_{\ell^{n+1}}$ contains the product $$(\operatorname{Id}-\ell^{n}g')(\operatorname{Id}+ \ell^{n_\ell} g + \ell^n g') \equiv (\operatorname{Id}+ \ell^{n_\ell}g)\pmod{\ell^{n+1}},$$ where we use the fact that $\ell^{2n}(g')^2 = \ell^{n+n_\ell} g'g = 0 $ since we are working modulo $\ell^{n+1}$.
In the special case $g=\operatorname{Id}$, the same result also holds for elliptic curves with complex multiplication:
\[lemma:ImageAlwaysContainsScalars\] Let $E$ be an arbitrary elliptic curve and let $n_\ell {\geqslant}1$ be a parameter of maximal growth (in particular, $n_\ell {\geqslant}2$ if $\ell=2$). For every $n {\geqslant}n_\ell$ we have $(1+\ell^{n_\ell}) \operatorname{Id} \in H_{\ell^n}$.
In the light of the previous lemma we may assume that $E$ has complex multiplication, so that the image of the torsion representation is contained in the normaliser of a Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$. The equality $\#H_{\ell^{n+1}} = \ell^2 \# H_{\ell^n}$ for $n {\geqslant}n_\ell$ is equivalent to the fact that $$\ker \left( H_{\ell^{n+1}} \to H_{\ell^n} \right) = \operatorname{Id} + \ell^n \mathbb{T} \subseteq \{ M \in \operatorname{Mat}_2({\mathbb{Z}}/\ell^{n+1}{\mathbb{Z}}) : M \equiv \operatorname{Id} \pmod{\ell^n} \},$$ where $\mathbb{T}$ is the tangent space to the image of the Galois representation as introduced in [@2016arXiv161202847L Definition 9] and further studied in [@MR3690236 Definition 18]. We proceed by induction, the base case $n=n_\ell$ being trivial. By surjectivity of $H_{\ell^{n+1}} \to H_{\ell^n}$ and the inductive hypothesis, we know that $H_{\ell^{n+1}}$ contains an element reducing to $(1+\ell^{n_\ell}) \operatorname{Id}$ modulo $\ell^n$, that is, an element of the form $M_{n+1} := (1 + \ell^{n_\ell})\operatorname{Id} + \ell^n t$. Here $t$ is an element of $\mathbb{T}$: to see this, notice that $M_{n+1}$ is congruent to the identity modulo $\ell^{n_\ell}$, so it cannot lie in the non-trivial coset of the normaliser of a Cartan subgroup ([@MR3690236 Theorem 40]), and therefore belongs to the Cartan subgroup itself. But then $M_{n+1}$ is of the form $\begin{pmatrix}
x & \delta y \\
y & x+\gamma y
\end{pmatrix}$ for appropriate parameters $(\gamma,\delta)$, hence $$t=\frac{1}{\ell^n}\begin{pmatrix}
x-1-\ell^{n_\ell} & \delta y \\
y & (x-1-\ell^{n_\ell})+\gamma y
\end{pmatrix} \in \operatorname{Mat}_2(\mathbb{F}_\ell)$$ belongs to $\mathbb{T}$ by the explicit description given in [@MR3690236 Definition 18]. Using the equality $\ker \left( H_{\ell^{n+1}} \to H_{\ell^n} \right) = \operatorname{Id}+ \ell^n \mathbb{T}$ we see that $H_{\ell^{n+1}}$ also contains $\operatorname{Id}-\ell^n t$, so it contains $$((1+\ell^{n_\ell})\operatorname{Id} + \ell^n t)(\operatorname{Id}-\ell^n t) \equiv \operatorname{Id}-\ell^{2n} t^2 + \ell^{n_\ell} \operatorname{Id} - \ell^{n+n_\ell} t \equiv (1 + \ell^{n_\ell}) \operatorname{Id}\pmod{\ell^{n+1}}$$ as claimed.
\[prop-not-divisible\] Assume that $\alpha$ is strongly $\ell$-indivisible in $E(K)$. Let $n_\ell$ be a parameter of maximal growth for the $\ell$-adic torsion representation. Then for every $n$ the point $\alpha$ is not $\ell^{n_\ell+1}$-divisible in $K_{\ell^n}$; equivalently, $\alpha$ is not $\ell^{n_\ell+1}$-divisible in $K_{\ell^{\infty}}$.
By Lemma \[lemma:ImageAlwaysContainsScalars\] the group $H_{\ell^n}$ contains $(1+\ell^{n_\ell})\operatorname{Id}$, so by Lemma \[lemma:ExponentOfH1\] the exponent of $H^1(H_{\ell^n}, E[\ell^{n}])$ divides $\ell^{n_\ell}$. We conclude by Lemma \[lem-not-divisible\].
The $\ell$-adic failure is bounded
----------------------------------
In this section we establish some general results that will form the basis of all subsequent arguments (in particular Lemma \[lem-order-n-d\] and Proposition \[prop:GroupTheory\]) and use them to show that the $\ell$-adic failure $A_\ell(N)$ can be effectively bounded (Theorem \[thm:UpperBoundAl\]).
\[lem-order-n-d\] Assume that for some $d{\geqslant}0$ the point $\alpha\in E(K)$ is not $\ell^{d+1}$-divisible over $K_{\ell^\infty}$. Then $V_{\ell^\infty}$ contains a vector of valuation at most $d$.
Similarly, if $\alpha\in E(K)$ is not $\ell^{d+1}$-divisible over $K_{\infty}$ then $W_{\ell^\infty}$ contains a vector of valuation at most $d$.
Assume by contradiction that every element of $V_{\ell^\infty}$ has valuation at least $d+1$. Then the image of $V_{\ell^\infty}$ in $E[\ell^{d+1}]=T_\ell(E)/\ell^{d+1}T_\ell(E)$ is zero. As this image is exactly $\operatorname{Gal}(K_{\ell^\infty,\ell^{d+1}}\mid K_{\ell^\infty})$, we obtain $K_{\ell^\infty,\ell^{d+1}}=K_{\ell^\infty}$, so $\alpha$ is $\ell^{d+1}$-divisible in $K_{\ell^\infty}$, a contradiction.
The second part can be proved in exactly the same way.
The following group-theoretic Proposition will be applied in this section and in Section \[sec:UniformBounds\]. In all of our applications the group $H$ will be the image of the $\ell$-adic torsion representation associated with some elliptic curve.
\[prop:GroupTheory\] Let $\ell$ be a prime number, $d$ be a positive integer, $H$ be a closed subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$, and $A={\mathbb{Z}}_\ell[H]$ be the sub-${\mathbb{Z}}_\ell$-algebra of $\operatorname{Mat}_2({\mathbb{Z}}_\ell)$ generated by the elements of $H$. Let $V \subseteq {\mathbb{Z}}_\ell^2$ be an $A$-submodule of ${\mathbb{Z}}_\ell^2$, and suppose that $V$ contains at least one vector of $\ell$-adic valuation at most $d$.
1. Suppose that $H$ contains $\{ M \in \operatorname{Mat}_2({\mathbb{Z}}_\ell) : M \equiv \operatorname{Id} \pmod{\ell^n} \}$ for some $n {\geqslant}1$. Then $V$ contains $\ell^{d+n} {\mathbb{Z}}_\ell^2$.
2. Suppose that the reduction of $H$ modulo $\ell$ acts irreducibly on ${\mathbb{F}}_\ell^2$. Then $V$ contains $\ell^d {\mathbb{Z}}_\ell^2$.
3. Let $C$ be a Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ with parameters $(\gamma,\delta)$ and let $N$ be its normaliser. Suppose that $H$ is an open subgroup of $N$ not contained in $C$, and that $H$ contains $\{ M \in C : M \equiv \operatorname{Id} \pmod{\ell^n} \}$ for some $n {\geqslant}1$. Then $V$ contains $\ell^{3n+d+v_\ell(4\delta)}{\mathbb{Z}}_\ell^2$.
The assumptions and the conclusions of the Proposition are invariant under changes of basis in ${\mathbb{Z}}_\ell^2$, so we may assume that $v=\ell^d e_1$ is in $V$, where $e_1=\begin{pmatrix}
1 \\ 0
\end{pmatrix}$.
1. It is clear that $A$ contains $\ell^n \operatorname{Mat}_2({\mathbb{Z}}_\ell)$, so we have $$V \supseteq A \cdot v \supseteq \ell^n \operatorname{Mat}_2({\mathbb{Z}}_\ell) \cdot v = \ell^{n+d} \operatorname{Mat}_2({\mathbb{Z}}_\ell) \cdot e_1=\ell^{n+d}{\mathbb{Z}}_\ell^2.$$
Let $H_\ell$ denote the reduction of $H$ modulo $\ell$. The condition that $H_\ell$ acts irreducibly on $\mathbb{F}_\ell^2$ implies that there exists $\overline{M} \in {\mathbb{F}}_\ell[H_\ell]$ such that $\overline{M}e_1 \equiv \begin{pmatrix}
0 \\ 1
\end{pmatrix} \pmod{\ell}$. Fix a lift $M \in A$ of $\overline{M}$, which exists because the natural reduction map $A={\mathbb{Z}}_\ell[H] \to {\mathbb{F}}_\ell[H_\ell]$ is clearly surjective. Then $M v=\ell^d M e_1$ is a vector whose second coordinate has valuation exactly $d$ and whose first coordinate has valuation strictly larger than $d$. It is then immediate to see that $v$ and $Mv$, that are contained in $V$, generate $\ell^d {\mathbb{Z}}_\ell^2$.
2. It is enough to show that $A$ contains $\ell^{3n+v_\ell(4\delta)}\operatorname{Mat}_2({\mathbb{Z}}_\ell)$, and the conclusion follows as in (1) above.
Suppose first that $\gamma=0$, and let $M_0=\begin{pmatrix}
x_0 & -\delta y_0\\
y_0 & -x_0
\end{pmatrix}\in H\setminus C$ and $M_1=\begin{pmatrix}
1+\ell^nx_0 & \delta\ell^ny_0\\
\ell^n y_0 & 1+\ell^nx_0
\end{pmatrix}\in H$. The existence and the form of such matrices follow from the assumptions and from the description of Cartan subgroups and their normaliser given in Definition \[def:CartanParameters\] and Lemma \[lem-Norm\]. Then $A$ contains $M_2=M_1-\operatorname{Id}+\ell^nM_0=2\ell^n\begin{pmatrix}
x_0 & 0 \\ y_0 & 0
\end{pmatrix}$. Let moreover $M_3=\ell^n\begin{pmatrix}
0 & \delta \\ 1 & 0
\end{pmatrix}$, which is in $A$ since it can be written as $\begin{pmatrix}
1 & \ell^n\delta \\ \ell^n & 1
\end{pmatrix}-\operatorname{Id}$, where both matrices are in $H$ by assumption. Then we have $$4\ell^{2n}\begin{pmatrix}
x_0^2-\delta y_0^2 & 0\\ 0 & 0
\end{pmatrix} = \left(M_2-2y_0M_3\right)\cdot M_2 \in A$$ and $x_0^2-\delta y_0^2=-\det M_0\in {\mathbb{Z}}_\ell^\times$. It follows that $A$ contains $4\ell^{2n}\begin{pmatrix}
1 & 0\\ 0 & 0
\end{pmatrix}$, and since $\operatorname{Id}\in A$ we have that all diagonal matrices of valuation at least $2n+v_\ell(4)$ are in $A$, which therefore also contains $\displaystyle
\begin{pmatrix}
0 & 0 \\ \ell^{3n+v_\ell(4)} & 0
\end{pmatrix}= M_3\begin{pmatrix}
\ell^{2n+v_\ell(4)} & 0 \\ 0 & 0
\end{pmatrix}$ and $\displaystyle \begin{pmatrix}
0 & \ell^{3n+v_\ell(4)}\delta \\ 0 & 0
\end{pmatrix}= M_3\begin{pmatrix}
0 & 0 \\ 0 & \ell^{2n+v_\ell(4)}
\end{pmatrix}$. Together with the diagonal matrices found above, these elements clearly generate $\ell^{3n+v_\ell(4\delta)}\operatorname{Mat}_2({\mathbb{Z}}_\ell)$, and we are done.
If $\gamma\neq 0$, by Remark \[rem:Parameters\] we may assume $\gamma =1$ and $\ell=2$. In this case let $M_0=\begin{pmatrix}
x_0+y_0& \delta y_0+x_0+y_0\\
-y_0 & -x_0-y_0
\end{pmatrix}\in H\setminus C$ and $M_1=\operatorname{Id}+\ell^n\begin{pmatrix}
x_0 & \delta y_0\\
y_0 & x_0+y_0
\end{pmatrix}\in H$. Then $A$ contains $M_2=M_1-\operatorname{Id}+\ell^nM_0=\ell^n\begin{pmatrix}
2x_0+y_0 & 2\delta y_0 + x_0 + y_0\\
0 & 0
\end{pmatrix}$. Let moreover $M_3=\ell^n\begin{pmatrix}
-1 & \delta \\
1 & 0
\end{pmatrix}\in A$. Then we have $$M_2(\delta M_2 - (2\delta y_0+x_0+y_0) M_3)=
-\ell^{2n}\det(M_0)(1+4\delta)\begin{pmatrix}
1&0\\0&0
\end{pmatrix}\in A,$$ and using the fact that $\det(M_0) \in {\mathbb{Z}}_\ell^\times$ (since $M_0 \in H \subseteq \operatorname{GL}_2({\mathbb{Z}}_\ell)$) we obtain that $A$ contains all diagonal matrices of valuation at least $2n$. We can then conclude as before.
\[prop-max-growth-kummer\] Assume that $\alpha$ is strongly $\ell$-indivisible in $E(K)$ and let $n_\ell$ be a parameter of maximal growth for the $\ell$-adic torsion representation.
(1) Assume that $E$ does not have complex multiplication. Then for every $k{\geqslant}1$ we have $E[\ell^k]\subseteq V_{\ell^{k+2n_\ell}}$.
(2) Assume that $E$ has complex multiplication by ${\mathcal{A}}:=\operatorname{End}_{\overline K}(E)$, and that $K$ does not contain the imaginary quadratic field ${\mathcal{A}}\otimes_{\mathbb{Z}} \mathbb{Q}$. Let $(\gamma,\delta)$ be parameters for the Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ corresponding to ${\mathcal{A}}$. Then for all $k{\geqslant}1$ we have $E[\ell^k]\subseteq V_{\ell^{k+4n_\ell+v_\ell(4\delta)}}$.
By Remark \[rem:Vsubgroup\], in order to show part (1) it is enough to prove $\ell^{2n_\ell} T_\ell(E)\subseteq V_{\ell^\infty}$. To see that this holds, notice that by Lemma \[lem-order-n-d\] and Proposition \[prop-not-divisible\] there is an element of valuation at most $n_\ell$ in $V_{\ell^\infty}$. Now we just need to apply Proposition \[prop:GroupTheory\](1) with $H=H_{\ell^\infty}$, $V=V_{\ell^\infty}$ and $d=n=n_\ell$. Part (2) can be proved in the same way using Proposition \[prop:GroupTheory\](3).
In §\[sec:CMCounterexample\] we will show that a naïve analogue of Proposition \[prop-max-growth-kummer\] does not hold in case $E$ has complex multiplication defined over $K$.
\[rem-parameters\] Write $\alpha=\ell^d\beta+T_h$, where $\beta\in E(K)$ is strongly $\ell$-indivisible and $T_h\in E[\ell^h](K)$ is a point of order $\ell^h$, for some $h,d{\geqslant}0$. Notice that it is always possible to do so: first, let $\beta\in E(K)$ and $d$ be such that $\alpha=\ell^{d}\beta+T$ for some $T\in E(K)$ of order a power of $\ell$, with $d$ maximal. Assume then by contradiction that $\beta$ is not strongly $\ell$-indivisible. This means that there are $\gamma,S\in E(K)$ with $S$ of order a power of $\ell$ such that $\beta =\ell\gamma+S$. But then $\alpha=\ell^d(\ell\gamma+S)+T=\ell^{d+1}\gamma +(\ell^dS+T)$, contradicting the maximality of $d$.
\[rmk:dIsEffective\] Let $\hat{h}$ be the canonical (Néron-Tate) height on $E$, as described in [@SilvermanEC Section VIII.9]. Following [@Petsche], it is possible to bound the divisibility parameters $d$ and $h$ in terms of $\hat h(\alpha)$, the degree of $K$ over ${\mathbb{Q}}$, the discriminant $\Delta_E$ of $E$ over $K$ and the Szpiro ratio $$\begin{aligned}
\sigma=\begin{cases}
1 & \text{if $E$ has everywhere good reduction}\\
\frac{\log|N_{K/{\mathbb{Q}}}(\Delta_{E})|}{\log|N_{K/{\mathbb{Q}}}(N_{E})|} & \text{otherwise}
\end{cases}\end{aligned}$$ where $N_E$ denotes the conductor of $E$ over $K$. In fact, [@Petsche Theorem 1] gives the bound $$\begin{aligned}
h{\leqslant}v_\ell \left\lfloor c_1[K:{\mathbb{Q}}]\sigma^2\log\left(c_2[K:{\mathbb{Q}}]\sigma^2\right)\right\rfloor\end{aligned}$$ where $c_1=134861$ and $c_2=104613$.
For the parameter $d$ we can reason as follows. For $\alpha=\ell^d \beta+T_h$, by [@SilvermanEC Theorem 9.3] we have $$\begin{aligned}
\hat h(\alpha) = \hat h(\ell^d\beta+T_h)=\hat h(\ell^d\beta)=\ell^{2d}\hat h(\beta)\end{aligned}$$ so we get $\displaystyle d{\leqslant}\frac{1}{2 \log \ell}\log\left(\frac{\hat h(\alpha)}{\hat h(\beta)}\right).$ Now in view of [@Petsche Theorem 2] for any non-torsion point $\beta\in E(K)$ we have $$\begin{aligned}
\hat h(\beta){\geqslant}B:=\frac{\log|N_{K/{\mathbb{Q}}}(\Delta_{E})|}{10^{15}[K:{\mathbb{Q}}]^3\sigma^6\log^2(c_2[K:{\mathbb{Q}}]\sigma^2)},\end{aligned}$$ where again $c_2=104613$. We thus obtain the effective bound $d{\leqslant}\frac{1}{2 \log \ell}\log\left(\frac{\hat h(\alpha)}{B}\right)$.
\[thm:UpperBoundAl\] Let $\ell$ be a prime and assume that $\operatorname{End}_K(E)={\mathbb{Z}}$ (i.e. either $E$ does not have CM, or it has CM but the complex multiplication is not defined over $K$). There is an effectively computable constant $a_\ell$, depending only on $\alpha$ and on the $\ell$-adic torsion representation associated to $E$, such that $A_\ell(N)$ divides $\ell^{a_\ell}$ for all positive integers $N$.
Moreover, $a_\ell$ is zero for every odd prime $\ell$ such that $\alpha$ is $\ell$-indivisible and for which the $\ell$-adic torsion representation associated with $E$ is maximal (see Definition \[def:MaximalRepresentation\]). For the finitely many remaining primes $\ell$ we can take $a_\ell$ as follows: let $n_\ell$ be a parameter of maximal growth for the $\ell$-adic torsion representation and let $d$ be as in Remark \[rem-parameters\]. If $E$ has CM over $\overline K$, let $(\gamma,\delta)$ be parameters for the Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ corresponding to $\operatorname{End}_{\overline K}(E)$. Then:
- $a_\ell=4n_\ell+2d$ if $E$ does not have CM over $\overline K$;
- $a_\ell=8n_\ell+2v_\ell(4\delta)+2d$ if $E$ has CM over $\overline K$.
Let $\alpha=\ell^d\beta+T_h$ as described above. Notice that if $\alpha$ is strongly $\ell$-indivisible we have $d=0$, and the conclusion follows from Proposition \[prop-max-growth-kummer\]. If the $\ell$-adic torsion representation is maximal, the fact that $a_\ell$ is zero in the cases stated follows from [@JonesRouse Theorem 5.2 and Theorem 5.8].
We now study the $\ell$-adic failure $A_\ell(N)$ in the general case. Let $n=v_\ell(N)$ and notice that the claim is trivial for $n{\leqslant}d$, so we may assume $n> d$. Since $$\begin{aligned}
[K_{\ell^{n+h}}(\ell^{-n}\alpha):K_{\ell^{n+h}}]=[K_{\ell^{n}}(\ell^{-n}\alpha) K_{\ell^{n+h}}:K_{\ell^n}K_{\ell^{n+h}}]\quad \text{divides}\quad [K_{\ell^{n}}(\ell^{-n}\alpha):K_{\ell^n}]\end{aligned}$$ we have that $\displaystyle \frac{\ell^{2n}}{[K_{\ell^{n}}(\ell^{-n}\alpha):K_{\ell^n}]}$ divides $\displaystyle \frac{\ell^{2n}}{[K_{\ell^{n+h}}(\ell^{-n}\alpha):K_{\ell^{n+h}}]}
$, and since we have $$\begin{aligned}
K_{\ell^{n+h}}(\ell^{-n}\alpha)=K_{\ell^{n+h}}(\ell^{-(n-d)}\beta)\end{aligned}$$ we get $$\begin{aligned}
\frac{\ell^{2n}}{[K_{\ell^{n+h}}(\ell^{-n}\alpha):K_{\ell^{n+h}}]}=\ell^{2d}\frac{\ell^{2(n-d)}}{[K_{\ell^{n+h}}(\ell^{-(n-d)}\beta):K_{\ell^{n+h}}]}\end{aligned}$$ so in view of Remark \[rem-NM\] we are reduced to proving the statement for $\beta$ instead of $\alpha$. Since $\beta$ is strongly $\ell$-indivisible, we can conclude as stated at the beginning of the proof.
The fact that $a_\ell$ is effective follows from the fact that one can effectively compute a parameter of maximal growth for the $\ell$-adic torsion representation (Remark \[rmk:nlIsEffective\]), an upper bound for the value of $d$ (Remark \[rmk:dIsEffective\]), and the endomorphism ring $\operatorname{End}_{\overline K}(E)$ ([@Achter], [@CMSV], [@LombardoEndo]).
The adelic failure {#sec:AdelicFailure}
==================
In this section we study the adelic failure $B_\ell(N)$, that is, the degree of the intersection $K_{\ell^{n},\ell^{n}}\cap K_N$ over $K_{\ell^{n}}$. Notice that this intersection is a finite Galois extension of $K_{\ell^n}$.
Intersection of torsion fields in the non-CM case {#sec:IntersectionTorsionFields}
-------------------------------------------------
We first aim to establish certain properties of the intersections of different torsion fields of $E$, assuming for this subsection that $E$ does not have complex multiplication over $\overline{K}$. Our main tool (Theorem \[th-intersection\]) is a refinement of [@Campagna Theorem 3.3.1], and will appear in an upcoming paper of F. Campagna and P. Stevenhagen. The proof of the stronger version we need requires only minor changes with respect to that of [@Campagna Theorem 3.3.1], and can be easily derived from it using the following well-known lemmas (see [@Serre] and [@MR1484415]).
\[lemma-solvable\] Let $p$ be a prime and let $H$ be a subgroup of $\operatorname{GL}_2({\mathbb{F}}_p)$. Let $S$ be a non-abelian simple group that occurs in $H$. Then $S$ is isomorphic either to $A_5$ or to $\operatorname{PSL}_2({\mathbb{F}}_p)$; the latter case is only possible if $H$ contains $\operatorname{SL}_2({\mathbb{F}}_p)$.
\[lemma:SerreLifting\] Let $\ell {\geqslant}5$ be a prime and let $G \subseteq \operatorname{SL}_2({\mathbb{Z}}/\ell^k{\mathbb{Z}})$ be a subgroup. Let $\pi : \operatorname{SL}_2({\mathbb{Z}}/\ell^k{\mathbb{Z}}) \to \operatorname{SL}_2({\mathbb{Z}}/\ell{\mathbb{Z}})$ be the reduction homomorphism and suppose that $\pi(G) = \operatorname{SL}_2({\mathbb{Z}}/\ell{\mathbb{Z}})$: then $G=\operatorname{SL}_2({\mathbb{Z}}/\ell^k{\mathbb{Z}})$.
\[th-intersection\] Assume that $E$ does not have complex multiplication. Let $S$ be the set consisting of the primes $\ell$ satisfying one or more of the following three conditions:
(i) $\ell\mid 30\operatorname{disc}(K\mid \mathbb{Q})$;
(ii) $E$ has bad reduction at some prime of $K$ above $\ell$;
(iii) the modulo $\ell$ torsion representation is not surjective.
For every $\ell\not \in S$ we have $K_{\ell^n}\cap K_M=K$ for all $M,n{\geqslant}1$ with $\ell\nmid M$.
The finite set $S$ appearing in Theorem \[th-intersection\] can be computed explicitly. In fact, it is well known that one can compute the discriminant of $K$ and the set of primes of bad reduction of $E$. An algorithm to compute the set of primes for which the $\bmod\, \ell$ representation is not surjective is described in [@Zywina].
As a corollary, we give a slightly more precise version of [@MR1484415 §3.4, Lemma 6].
\[cor-direct-prod\] Assume that $E$ does not have complex multiplication and let $S$ be as in Theorem \[th-intersection\]. Let $M$ be a positive integer and write $M=M_1 M_2$, where $$\begin{aligned}
M_1&=\prod_{p\not \in S}p^{e_p}&& p\text{ prime, }e_p{\geqslant}0,\\
M_2&=\prod_{q\in S}q^{e_q} && q\text{ prime, }e_q{\geqslant}0.\end{aligned}$$ Then we have $$\begin{aligned}
\operatorname{Gal}(K_M\mid K)\cong\operatorname{GL}_2\left( {\mathbb{Z}}/M_1{\mathbb{Z}}\right)\times \operatorname{Gal}\left(K_{M_2}\mid K\right).\end{aligned}$$
By Theorem \[th-intersection\] we have that, for any $p\not \in S$ and any $e{\geqslant}0$, the field $K_{p^e}$ is linearly disjoint over $K$ from $K_{M_2}$ and from $K_{q^f}$ for every $q \neq p$ and every $f{\geqslant}1$. Moreover we have $$\begin{aligned}
\operatorname{GL}_2\left( {\mathbb{Z}}/M_1{\mathbb{Z}}\right)\cong \prod_{p\not\in S}\operatorname{GL}_2({\mathbb{Z}}/p^{e_p}{\mathbb{Z}})\cong \prod_{p\not\in S}\operatorname{Gal}(K_{p^{e_p}}\mid K),\end{aligned}$$ and the Corollary follows by standard Galois theory.
\[rem-S-stable\] Let $\tilde K$ be the compositum of the fields $K_p$ for all $p\in S$, where $S$ is as in Theorem \[th-intersection\]. In the following section it will be important to notice that $S$ is stable under base change to $\tilde K$. More precisely, let $\tilde S$ be the set of all primes $\ell$ that satisfy one of the following:
1. $\ell\mid 30\operatorname{disc}(\tilde K\mid \mathbb{Q})$;
2. $E$ has bad reduction at some prime of $\tilde K$ above $\ell$;
3. the modulo $\ell$ torsion representation attached to $E/\tilde{K}$ is not surjective.
Then $\tilde S=S$.
Indeed, the inclusion $\tilde S\supseteq S$ is easy to see: clearly conditions (i) and (iii) imply (i’) and (iii’) respectively, so we only need to discuss (ii). Let $\mathfrak{p}$ be a prime of $K$ (of characteristic $\ell$) at which $E$ has bad reduction, and let $\mathfrak{q}$ be a prime of $\tilde{K}$ lying over $\mathfrak{p}$. We need to show that $\ell \in \tilde{S}$. If $E$ has bad reduction at $\mathfrak{q}$ we have $\ell \in \tilde{S}$ by (ii’), while if $E$ has good reduction at $\mathfrak{q}$ then $\mathfrak{p}$ ramifies in $\tilde{K}$ by [@SilvermanEC Proposition VII.5.4 (a)], so we have $\ell \mid \operatorname{disc}(\tilde{K} \mid {\mathbb{Q}})$ and $\ell$ is in $\tilde{S}$ by (i’).
Conversely, let $\ell \in \tilde{S}$. If (ii’) holds, then clearly also (ii) holds, and $\ell$ is in $S$. Suppose that (i’) holds. If $\ell$ divides $30$, then it is in $S$ by (1). Otherwise $\ell$ divides $\operatorname{disc}(\tilde K\mid {\mathbb{Q}})$, which by [@Serre-LocalFields III.§4, Proposition 8] is equal to $\operatorname{disc}(K \mid {\mathbb{Q}})^{[\tilde{K}:K]} N_{K/{\mathbb{Q}}} \operatorname{disc}(\tilde{K} \mid K)$; if $\ell$ divides $\operatorname{disc}(K \mid {\mathbb{Q}})$, then it is in $S$ by (1), while if it divides $\operatorname{disc}(\tilde{K} \mid K)$ then we have $\ell\in S$ by [@SilvermanEC Proposition VIII.1.5(b)]. We may therefore assume that (i’) and (ii’) do not hold. Since $\ell$ is in $\tilde{S}$, (iii’) must hold, that is, the modulo-$\ell$ torsion representation attached to $E/\tilde{K}$ is not surjective. We claim that the same is true for $E/K$. Indeed, if $\ell$ is in $S$ this is true by definition, while if $\ell \not \in S$ the previous corollary shows that $K_{\ell}$ is linearly disjoint from $\tilde{K}$, so the images of the modulo-$\ell$ representations over $K$ and over $\tilde{K}$ coincide.
The adelic failure is bounded
-----------------------------
We now go back to the general case of $E$ possibly admitting complex multiplication.
Fix an integer $N>1$ and a prime number $\ell$ dividing $N$. Write $N=\ell^nR$ with $\ell\nmid R$ and recall that the adelic failure $B_\ell(N)$ is defined to be the degree $[K_{\ell^{n},\ell^{n}}\cap K_N:K_{\ell^{n}}]$. In this section we study this failure for $N=\ell^nR$, starting with a simple Lemma in Galois theory.
\[lem-gal\] Let $L_1$, $L_2$ and $L_3$ be Galois extensions of $K$ with $L_1\subseteq L_2$. Then the compositum $L_1(L_2\cap L_3)$ is equal to the intersection $L_2\cap (L_1L_3)$.
For $i=1,2,3$ let $G_i:=\operatorname{Gal}(\overline{K}\mid L_i)$. The claim is equivalent to $G_1\cap(G_2\cdot G_3)=G_2\cdot(G_1\cap G_3)$, where the inclusion “$\supseteq$” is obvious. Let then $g\in G_1\cap(G_2\cdot G_3)$, so that there are $g_1\in G_1$, $g_2\in G_2$ and $g_3\in G_3$ such that $g=g_1=g_2g_3$. But then $g_2^{-1}g_1=g_3\in G_3$ and, since $G_2\subseteq G_1$, also $g_2^{-1}g_1\in G_1$, so that $g=g_2(g_2^{-1}g_1)\in G_2\cdot(G_1\cap G_3)$.
We now establish some properties of certain subfields of $K_{\ell^nR,\ell^n}$.
\[nice-lemma\] Setting $$\begin{aligned}
L:=K_{\ell^n,\ell^n}\cap K_{N}, && F:= L\cap K_R=K_{\ell^n,\ell^n}\cap K_R, && T:= F\cap K_{\ell^n}=K_{\ell^n}\cap K_R\end{aligned}$$ we have:
1. The compositum $F K_{\ell^n}$ is $L$.
2. $\operatorname{Gal}(F\mid T)\cong \operatorname{Gal}(L\mid K_{\ell^n})$; in particular, $\operatorname{Gal}(F\mid T)$ is an abelian $\ell$-group.
3. $F$ is the intersection of the maximal abelian extension of $T$ contained in $K_{\ell^n,\ell^n}$ and the maximal abelian extension of $T$ contained in $K_R$.
\(a) By Lemma \[lem-gal\] we have $FK_{\ell^n}=K_{\ell^n}(K_{\ell^n,\ell^n}\cap K_R)=K_{\ell^n,\ell^n}\cap K_{\ell^nR}=L$. (b) Follows from (a) and standard Galois theory. For (c), notice that $F$ is abelian over $T$ by (b), so it must be contained in the maximal abelian extension of $T$ contained in $K_{\ell^n,\ell^n}$ and in the maximal abelian extension of $T$ contained in $K_R$. On the other hand, $F$ cannot be smaller than the intersection of these abelian extensions, because by definition it is the intersection of $K_{\ell^n,\ell^n}$ and $K_R$.
$$\xymatrix{
& K_{\ell^nR,\ell^n} \ar@{-}[dl]\ar@{-}[dr]\\
K_{\ell^n,\ell^n}\ar@{-}[dd]\ar@{-}[dr] & & K_{\ell^nR}\ar@{-}[dl]\ar@{-}[dd]\\
& L:=K_{\ell^n,\ell^n}\cap K_{\ell^nR} \ar@{-}[dl]\ar@{-}[dd]\\
K_{\ell^n}\ar@{-}[dddr] & & K_R\ar@{-}[dl]\ar@{-}[dddl]\\
& F:=K_{\ell^n,\ell^n}\cap K_R\ar@{-}[dd]\\
\\
&T:=K_{\ell^n}\cap K_R\ar@{-}[d]\\
&K
}$$
\[prop-1\] The adelic failure $B_\ell(N)$ is equal to $[F:T]$, where $F=K_{\ell^n,\ell^n}\cap K_{R}$ and $T=K_{\ell^n}\cap K_R$.
Let as above $L=K_{\ell^n,\ell^n}\cap K_{\ell^nR}$. We have $\operatorname{Gal}(K_{\ell^n,\ell^n}\,|\,L)\cong \operatorname{Gal}(K_{\ell^nR,\ell^n}\,|\,K_{\ell^nR})$, so we get $$\begin{aligned}
[K_{\ell^n,\ell^n}:K_{\ell^n}]=[K_{\ell^n,\ell^n}:L][L:K_{\ell^n}]=[K_{\ell^nR,\ell^n}:K_{\ell^nR}][L:K_{\ell^n}]\end{aligned}$$ and we conclude by Lemma \[nice-lemma\](b).
In what follows we will need to work over a certain extension $\tilde K$ of $K$; this extension will depend on the prime $\ell$. More precisely, we give the following definition.
\[def-tildeK\] Let $\tilde K$ be the finite extension of $K$ defined as follows:
- If $E$ has complex multiplication, we take $\tilde K$ to be the compositum of $K$ with the CM field of $E$. This is an at most quadratic extension of $K$. Notice that in this case by [@2018arXiv180902584L Lemma 2.2] we have $\tilde K_n=K_n$ for every $n{\geqslant}3$.
- If $E$ does not have CM and $\ell$ is not one of the primes in the set $S$ of Theorem \[th-intersection\], we just let $\tilde K = K$. Notice that this happens for all but finitely many primes $\ell$.
- If $E$ does not have CM and $\ell$ is one of the primes in the set $S$ of Theorem \[th-intersection\], we let $\tilde K$ be the compositum of all the $K_p$ for $p\in S$. Notice that in this case $\tilde{K}_\ell=\tilde{K}$.
We shall use the notation $\tilde{K}_M$ (respectively $\tilde{K}_{M,N}$) for the torsion (resp. Kummer) extensions of $\tilde{K}$. We shall also write $$\begin{aligned}
\tilde H_{\ell^n}&:=\operatorname{Im}\left(\tau_{\ell^n}:\operatorname{Gal}({\overline{K}}\mid\tilde K)\to \operatorname{Aut}(E[\ell^n]) \right)\cong \operatorname{Gal}\left(\tilde K_{\ell^n}\mid\tilde K\right),\\
\tilde V_{\ell^n}&:=\operatorname{Im}\left(\kappa_{\ell^n}:\operatorname{Gal}({\overline{K}}\mid\tilde K_{\ell^n})\to E[\ell^n]\right)\cong \operatorname{Gal}\left(\tilde K_{\ell^n,\ell^n}\mid\tilde K_{\ell^n}\right)\end{aligned}$$ for the images of the $\ell^n$-torsion representation and of the $(\ell^n,\ell^n)$-Kummer map attached to $E/\tilde{K}$. Finally, we let $\tilde n_\ell$ be the minimal parameter of maximal growth for the $\ell$-adic torsion representation over $\tilde K$. Notice that, thanks to Lemma \[lem-param-tilde-bound\], we have $\tilde n_\ell {\leqslant}n_\ell+v_\ell([\tilde K:K])$.
\[prop-F-abelian\] The extension $F':=\tilde K_{\ell^n,\ell^n}\cap \tilde K_R$ is abelian over $\tilde K$.
This is well known if $E$ has complex multiplication because then $\tilde{K}_R$ is itself abelian over $\tilde K$, see for example [@MR1312368 Theorem II.2.3]. In case $E$ does not have complex multiplication and $\ell$ is not in the set $S$ of Theorem \[th-intersection\], this follows easily by considering the diagram $$\begin{aligned}
\xymatrix{
& \tilde K_{\ell^n,\ell^n} \ar@{-}[d]\\
& \tilde K_{\ell^n}F'\ar@{-}[dl]\ar@{-}[dr]\\
\tilde K_{\ell^n} \ar@{-}[dr]& & F'\ar@{-}[dl]\\
&\tilde K
}\end{aligned}$$ In fact, since $\tilde K_{\ell^n}\cap F'=\tilde K$ by Theorem \[th-intersection\] (notice that in this case $\tilde{K}=K$), we have that $\operatorname{Gal}(F'\mid \tilde K)\cong \operatorname{Gal}(\tilde K_{\ell^n} F' \mid \tilde K_{\ell^n})$ is a quotient of $\tilde V_{\ell^n}$, hence abelian. Thus we can assume that $E$ does not have CM and that $\ell$ is in the set $S$ of Theorem \[th-intersection\].
Notice that $F'$ is a Galois extension of $\tilde K$ with degree a power of $\ell$, since the same is true for $\tilde K_{\ell^n,\ell^n} \mid \tilde{K}$ and $F' \subseteq \tilde K_{\ell^n,\ell^n}$. Letting $r$ denote the radical of $R$, the degree of $[F' : F' \cap \tilde{K}_r]$, which is still a power of $\ell$, divides $[\tilde{K}_R : \tilde{K}_r]$, which is a product of primes dividing $R$. So since $\ell \nmid R$ we obtain $[F' : F' \cap \tilde{K}_r]=1$, that is $\tilde{K}_{\ell^n, \ell^n} \cap \tilde{K}_R =\tilde{K}_{\ell^n, \ell^n} \cap \tilde{K}_r$, and we may assume that $R$ is squarefree. Write now $R=R_1R_2$, where $R_1$ is the product of the prime factors of $R$ that are *not* in $S$ and $R_2$ is the product of the prime factors of $R$ that belong to $S$. By definition of $\tilde K$ we have $\tilde{K}_{R} = \tilde{K}_{R_1}$, so we may further assume that no prime $p\in S$ divides $R$. By Corollary \[cor-direct-prod\] we then have $\operatorname{Gal}(\tilde K_{R}\,|\,\tilde K)\cong \operatorname{GL}_2({\mathbb{Z}}/R{\mathbb{Z}})$.
Since $F'\subseteq \tilde K_{R}$, there must be a normal subgroup $D=\operatorname{Gal}(\tilde{K}_R\mid F')\trianglelefteq \operatorname{GL}_2({\mathbb{Z}}/R{\mathbb{Z}})$ of index a power of $\ell$. In order to conclude we just need to show that $D$ contains $\operatorname{SL}_2({\mathbb{Z}}/R{\mathbb{Z}})$, for then $\operatorname{Gal}( F'\mid \tilde{K})\cong \operatorname{GL}_2({\mathbb{Z}}/R{\mathbb{Z}})/D$ is abelian.
Since $\operatorname{SL}_2({\mathbb{Z}}/R{\mathbb{Z}})\cong\prod_{p\mid R}\operatorname{SL}_2({\mathbb{F}}_p)$, we can consider the intersection $D_p:=D\cap \operatorname{SL}_2({\mathbb{F}}_p)$, which is a normal subgroup of $\operatorname{SL}_2({\mathbb{F}}_p)$. Here we identify $\operatorname{SL}_2({\mathbb{F}}_p)$ with the corresponding direct factor of $\operatorname{SL}_2({\mathbb{Z}}/R{\mathbb{Z}})$. The quotient $\operatorname{SL}_2({\mathbb{F}}_p)/D_p$ cannot have order a power of $\ell$ unless it is trivial (recall that in our case $p{\geqslant}5$), so we deduce that $D\supseteq \operatorname{SL}_2({\mathbb{F}}_p)$. As this is true for every $p\mid R$, we have $D\supseteq \operatorname{SL}_2({\mathbb{Z}}/R{\mathbb{Z}})$, and we are done.
In what follows, whenever $A$ is an abelian group and $Q$ is a group acting on $A$, we denote by $[A,Q]$ the subgroup of $A$ generated by elements of the form $gv-v$ for $v\in A$ and $g\in Q$. For example, we will consider the case $A=\tilde V_{\ell^n}$ and $Q=\tilde H_{\ell^n}$.
\[lemma-ab\] Let $$\begin{aligned}
1\to A\to G\to Q \to 1\end{aligned}$$ be a short exact sequence of groups, with $A$ abelian, so that $Q$ acts naturally on $A$. Let $G^{\operatorname{ab}}$ and $Q^{\operatorname{ab}}$ be the maximal abelian quotients of $G$ and $Q$ respectively. Then $A/[A,Q]$ surjects onto $\ker(G^{\operatorname{ab}}\to Q^{\operatorname{ab}})$.
We have an injective map of short exact sequences
1& AG’& G’& Q’ & 1\
1& A & G & Q&1
from which we get the exact sequence $$\begin{aligned}
1\to \frac{A}{A\cap G'}\to G^{\operatorname{ab}}\to Q^{\operatorname{ab}}\to 1\end{aligned}$$ and since $[A,Q]\subseteq A\cap G'$ we get that $A/[A,Q]$ surjects onto $A/A\cap G'=\ker(G^{\operatorname{ab}}\to Q^{\operatorname{ab}})$.
\[bound-F-over-K\] The adelic failure $B_\ell(N)$ divides $\displaystyle [\tilde{K}:K]\cdot \#\frac{\tilde V_{\ell^n}}{[\tilde V_{\ell^n},\tilde H_{\ell^n}]}$.
Let ${J}_1$ and ${J}_2$ be the maximal abelian extensions of $\tilde K$ contained in $\tilde K_{\ell^n}$ and $\tilde K_{\ell^n,\ell^n}$ respectively. Then we have $\operatorname{Gal}({J}_1\,|\,\tilde K)=\tilde H_{\ell^n}^{\operatorname{ab}}$ and $\operatorname{Gal}({J}_2\,|\,\tilde K)=\tilde G_{\ell^n}^{\operatorname{ab}}$, where $\tilde G_{\ell^n}=\operatorname{Gal}(\tilde K_{\ell^n,\ell^n}\,|\,\tilde K)$. Notice that $[{J}_2:{J}_1]=\#W$, where $W=\ker(\tilde G_{\ell^n}^{\operatorname{ab}}\to \tilde H_{\ell^n}^{\operatorname{ab}})$ is a quotient of $\tilde V_{\ell^n}/[\tilde V_{\ell^n},\tilde H_{\ell^n}]$ by Lemma \[lemma-ab\]. Let moreover $F':=\tilde K_{\ell^n,\ell^n}\cap \tilde K_R$ and $T':= \tilde K_{\ell^n}\cap \tilde K_R$. By Proposition \[prop-F-abelian\] we have $F'\subseteq {J}_2$ and clearly also $T'\subseteq {J}_1$ (indeed $T'$ is abelian over $\tilde{K}$ since it is a sub-extension of $F'$). Consider the compositum $ {J}_1F'$ inside ${J}_2$. $$\begin{aligned}
\xymatrix{
& {J}_2 \ar@{-}[d]\\
& {J}_1F'\ar@{-}[dl]\ar@{-}[dr]\\
{J}_1 \ar@{-}[dr]& & F'\ar@{-}[dl]\\
&\tilde K_\ell
}\end{aligned}$$ It is easy to check that $F'\cap {J}_1=T'$, so we have that $[F':T']=[{J}_1F':{J}_1]$ divides $[{J}_2:{J}_1]$, which in turn divides $\tilde V_{\ell^n}/[\tilde V_{\ell^n},\tilde H_{\ell^n}]$.
Now applying Proposition \[prop-1\] with $\tilde K$ in place of $K$ we get that $$\begin{aligned}
\frac{[\tilde K_{\ell^n,\ell^n}:\tilde K_{\ell^n}]}{[\tilde K_{\ell^nR,\ell^n}:\tilde K_{\ell^nR}]}\qquad \text{divides}\qquad [F':T'],\end{aligned}$$ and using that $[\tilde K_{\ell^nR,\ell^n}:\tilde K_{\ell^nR}]$ divides $[K_{\ell^nR,\ell^n}:K_{\ell^nR}]$ it is easy to see that $$\begin{aligned}
\frac{[K_{\ell^n,\ell^n}:K_{\ell^n}]}{[K_{\ell^nR,\ell^n}:K_{\ell^nR}]}\qquad \text{divides}\qquad [\tilde K:K]\cdot\frac{[\tilde K_{\ell^n,\ell^n}:\tilde K_{\ell^n}]}{[\tilde K_{\ell^nR,\ell^n}:\tilde K_{\ell^nR}]}.\end{aligned}$$ We conclude that $$\begin{aligned}
B_\ell(N)=\frac{[K_{\ell^n,\ell^n}:K_{\ell^n}]}{[K_{\ell^nR,\ell^n}:K_{\ell^nR}]} \qquad \text{divides} \qquad [\tilde{K}:K]\cdot \#\frac{\tilde V_{\ell^n}}{[\tilde V_{\ell^n},\tilde H_{\ell^n}]}.\end{aligned}$$
So we are left with giving an upper bound on the ratio $\#\tilde V_{\ell^n}/\#[\tilde V_{\ell^n},\tilde H_{\ell^n}]$: this is achieved in the following Proposition.
\[prop-quotient-bounded\] For every $n$, the order of $\tilde V_{\ell^n}/[\tilde V_{\ell^n},\tilde H_{\ell^n}]$ divides $\ell^{2\tilde n_\ell}$, where $\tilde n_\ell$ is the minimal parameter of maximal growth for the $\ell$-adic torsion representation of $E/\tilde K$.
By Lemma \[lemma:ImageAlwaysContainsScalars\], the group $\tilde H_{\ell^n}$ contains $(1+\ell^{\tilde n_\ell}) \operatorname{Id}$. This implies that for every $v \in \tilde V_{\ell^n}$ the group $[\tilde V_{\ell^n}, \tilde H_{\ell^n}]$ contains $$\left[v, (1+\ell^{\tilde n_\ell}) \operatorname{Id}\right] = (1+\ell^{\tilde{n}_\ell}) \operatorname{Id} \cdot v - v = \ell^{\tilde n_\ell} v,$$ that is, $[\tilde V_{\ell^n}, \tilde H_{\ell^n}]$ contains $\ell^{\tilde n_\ell} \tilde V_{\ell^n}$. The claim now follows from the fact that $\tilde{V}_{\ell^n}$ is generated over ${\mathbb{Z}}/\ell^n{\mathbb{Z}}$ by at most two elements.
\[lem-adelic-good-ell\] Assume that $\ell{\geqslant}5$ is unramified in $K\mid {\mathbb{Q}}$ and that the image of the $\bmod\,\ell$ torsion representation is $\operatorname{GL}_2({\mathbb{F}}_\ell)$ (so in particular $E$ does not have CM over ${\overline{K}}$). Assume moreover that $\alpha$ is $\ell$-indivisible. Then $V_{\ell^n}=[V_{\ell^n},H_{\ell^n}]$.
Since $H_{\ell^\infty}'$ is a closed subgroup of $\operatorname{SL}_2({\mathbb{Z}}_\ell)$ whose reduction modulo $\ell$ contains $H_\ell'=\operatorname{GL}_2({\mathbb{F}}_\ell)'=\operatorname{SL}_2({\mathbb{F}}_\ell)$, by Lemma \[lemma:SerreLifting\] the group $H_{\ell^\infty}$ contains $\operatorname{SL}_2({\mathbb{Z}}_\ell)$. The assumption that $\ell$ is unramified in $K$ implies that $\det(H_{\ell^\infty})={\mathbb{Z}}_\ell^\times$, which together with the inclusion $\operatorname{SL}_2({\mathbb{Z}}_\ell) \subseteq H_{\ell^\infty}$ implies $H_{\ell^\infty}=\operatorname{GL}_2({\mathbb{Z}}_\ell)$, and in particular $H_{\ell^n} = \operatorname{GL}_2({\mathbb{Z}}/\ell^n{\mathbb{Z}})$. By [@JonesRouse Theorem 5.2] we have $V_{\ell^n} = ({\mathbb{Z}}/\ell^n{\mathbb{Z}})^2$, so it is enough to consider $$\begin{aligned}
\begin{array}{ccc}
g_1:=\left(\begin{array}{cc}
1 & 1 \\
0 & 1
\end{array}
\right)\in H_{\ell^n}, &
g_2:=\left(\begin{array}{cc}
1 & 0 \\
1 & 1
\end{array}
\right) \in H_{\ell^n},&
v:=\left(\begin{array}{c}
1\\
1
\end{array}\right)\in V_{\ell^n}
\end{array}\end{aligned}$$ to conclude that $$\begin{aligned}
\begin{array}{ccc}
\left(\begin{array}{c}
1\\
0
\end{array}\right)=g_1v-v\in [V_{\ell^n},H_{\ell^n}] &
\text{and} &
\left(\begin{array}{c}
0\\
1
\end{array}\right)=g_2v-v\in [V_{\ell^n},H_{\ell^n}],
\end{array}\end{aligned}$$ so that $V_{\ell^n}\subseteq[V_{\ell^n},H_{\ell^n}]$.
\[lemma:CMVanishing\] Let $E/K$ be an elliptic curve such that $\operatorname{End}_{\overline{K}}(E)$ is an order ${\mathcal{A}}$ in the imaginary quadratic field ${\mathbb{Q}}(\sqrt{-d})$. Let $\ell {\geqslant}3$ be a prime unramified both in $K$ and in $\mathbb{Q}(\sqrt{-d})$, and suppose that $E$ has good reduction at all places of $K$ of characteristic $\ell$. Then $V_{\ell^n}=[V_{\ell^n},H_{\ell^n}]$ and $\tilde{V}_{\ell^n}=[\tilde{V}_{\ell^n},\tilde{H}_{\ell^n}]$.
By [@MR3766118 Theorem 1.5], the image of the $\ell$-adic representations attached to both $E/K$ and $E/\tilde{K}$ contains $({\mathcal{A}}\otimes {\mathbb{Z}}_\ell)^\times$, hence in particular it contains an operator that acts as multiplication by $2$ on $E[\ell^n]$ for every $n$. Let $\lambda$ be such an operator: then $[V_{\ell^n},H_{\ell^n}]$ contains $[V_{\ell^n}, \lambda] = \{\lambda v - v \bigm\vert v \in V_{\ell^n} \} = V_{\ell^n}$ as claimed. The case of $\tilde{V}_{\ell^n}$ is similar.
\[thm:UpperBoundBl\] Let $\ell$ be a prime. There is a constant $b_\ell$, depending only on the $p$-adic torsion representations associated with $E$ for all the primes $p$, such that $B_\ell(N)$ divides $\ell^{b_\ell}$ for all positive integers $N$. Moreover,
- Suppose that $E$ does not have complex multiplication over $\overline{\mathbb{Q}}$. Then $b_\ell$ is zero whenever the following conditions all hold: $\alpha$ is $\ell$-indivisible, $\ell > 5$ is unramified in $K\mid {\mathbb{Q}}$, the $\bmod \, \ell$ torsion representation is surjective, and $E$ has good reduction at all places of $K$ of characteristic $\ell$.
- Suppose that $\operatorname{End}_{\overline{K}}(E)$ is an order in the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Then $b_\ell$ is zero whenever the following conditions all hold: $\ell {\geqslant}3$ is a prime unramified both in $K$ and in $\mathbb{Q}(\sqrt{-d})$, and $E$ has good reduction at all places of $K$ of characteristic $\ell$.
Both in the CM and non-CM cases, for the finitely many remaining primes $\ell$ we can take $b_\ell=2n_\ell+3v_\ell\left([\tilde K:K]\right)$, where $\tilde K$ is as in Definition \[def-tildeK\] and $n_\ell$ is a parameter of maximal growth for the $\ell$-adic torsion part.
Let $n$ be the $\ell$-adic valuation of $N$. By Proposition \[bound-F-over-K\], the adelic failure $B_\ell(N)$ divides $\displaystyle [\tilde{K}:K] \cdot \# \frac{\tilde{V}_{\ell^n}}{[\tilde{V}_{\ell^n}, \tilde{H}_{\ell^n}]}$.
- Suppose that $E$ does not have CM over $\overline{{\mathbb{Q}}}$, that $\alpha$ is $\ell$-indivisible, that $\ell > 5$ is unramified in $K\mid {\mathbb{Q}}$, that the $\bmod\,\ell$ torsion representation is surjective, and that $E$ has good reduction at all places of $K$ of characteristic $\ell$. Under these assumptions, the prime $\ell$ does not belong to the set $S$ of Theorem \[th-intersection\], so we have $\tilde{K}=K$ and $\displaystyle [\tilde{K}:K] \cdot \# \frac{\tilde{V}_{\ell^n}}{[\tilde{V}_{\ell^n}, \tilde{H}_{\ell^n}]}$ is simply $\displaystyle \# \frac{V_{\ell^n}}{[V_{\ell^n}, H_{\ell^n}]}$. We conclude because this quotient is trivial by Lemma \[lem-adelic-good-ell\].
- In the CM case, the conclusion follows from Lemma \[lemma:CMVanishing\] since $\ell \nmid [\tilde{K}:K]{\leqslant}2$.
For all other primes, combining Proposition \[bound-F-over-K\] and Proposition \[prop-quotient-bounded\] we get that $B_\ell(N)$ divides $[\tilde{K}:K]\cdot \ell^{2\tilde n_\ell}$ and we conclude using Lemma \[lem-param-tilde-bound\].
The proof shows that the inequality $v_\ell(B_\ell(N)) {\leqslant}2n_\ell + 3v_\ell \left([\tilde{K}:K] \right)$ holds for every prime $\ell$ and for every rational point $\alpha \in E(K)$. In other words, for a fixed prime $\ell$ the adelic failure can be bounded independently of the rational point $\alpha$.
We can finally prove our first Theorem from the introduction:
By Remark \[rem-NM\], Theorem \[thm:Main\] follows from Theorems \[thm:UpperBoundAl\] and \[thm:UpperBoundBl\] by taking $C:=\prod_\ell \ell^{a_\ell+b_\ell}$.
\[rmk:MainTheoremIsEffective\] Theorem \[thm:Main\] is completely effective, in the following sense: the quantities $a_\ell$ and $b_\ell$ can be computed in terms of $[\tilde{K}:K]$, $n_\ell$, and the divisibility parameter $d$. The integer $d$ can be bounded effectively in terms of the height of $\alpha$ and of standard invariants of the elliptic curve, as showed in Remark \[rmk:dIsEffective\]. The remaining quantities $[\tilde{K}:K]$ and $n_\ell$ can be bounded effectively in terms of $[K:{\mathbb{Q}}]$ and of the height of $E$, as shown in [@MR3437765].
A counterexample in the CM case {#sec:CMCounterexample}
===============================
We give an example showing that Proposition \[prop-max-growth-kummer\] does not hold in the CM case when $\ell$ is split in the field of complex multiplication, and that in fact in this case there can be no uniform lower bound on the image of the Kummer representation depending only on the image of the torsion representation, even when $\alpha$ is strongly $\ell$-indivisible.
Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication over $\overline {\mathbb{Q}}$ by the imaginary quadratic field $F$. Let $\alpha \in E(\mathbb{Q})$ be such that the $\ell^n$-arboreal representation attached to $(E,\alpha)$ maps onto $\left( {\mathbb{Z}}/\ell^n{\mathbb{Z}}\right)^2 \rtimes N_{\ell^n}$ for every $n {\geqslant}1$, where $N_{\ell^n}$ is the normaliser of a Cartan subgroup $C_{\ell^n}$ of $\operatorname{GL}_2({\mathbb{Z}}/\ell^n{\mathbb{Z}})$. Suppose furthermore that $\ell$ is split in $F$ and does not divide the conductor of the order $\operatorname{End}_{\overline{{\mathbb{Q}}}}{E} \subseteq \mathcal{O}_F$. Such triples $(E, \alpha, \ell)$ exist: by [@JonesRouse Example 5.11] we can take $E : y^2 = x^3 + 3x$ (which has CM by $\mathbb{Z}[i]$), $\alpha = (1, -2)$ and $\ell = 5$ (which splits in $\mathbb{Z}[i]$). Notice that for this elliptic curve and this $\alpha$ the same property holds for every $\ell \equiv 1 \pmod 4$: [@MR3766118 Theorem 1.5 (2)] implies that for all $\ell {\geqslant}5$ the image of the Galois representation is the full normaliser of a Cartan subgroup, at which point surjectivity of the Kummer representation follows from [@JonesRouse Theorem 5.8].
Consider now the image of the arboreal representation associated with the triple $(E / F, \alpha, \ell)$. Base-changing $E$ to $F$ has the effect of replacing the normaliser of the Cartan subgroup with Cartan itself: more precisely we have $\omega_{\ell^n}\left( \operatorname{Gal}( F_{\ell^n, \ell^n} \mid F ) \right) = \left( {\mathbb{Z}}/\ell^n{\mathbb{Z}}\right)^2 \rtimes C_{\ell^n}$ for every $n {\geqslant}1$. As $\ell$ is split in the quadratic ring $\operatorname{End}_{\overline{{\mathbb{Q}}}}(E)$, so is the Cartan subgroup $C_{\ell^n}$, and therefore we can assume – choosing a different basis for $E[\ell^n]$ if necessary – that $C_{\ell^n}$ is the subgroup of diagonal matrices in $\operatorname{GL}_2({\mathbb{Z}}/\ell^n{\mathbb{Z}})$. Fix now a large $n$ and let $$B_{\ell^n} = \left\{ (t,M) \in \left( {\mathbb{Z}}/\ell^n{\mathbb{Z}}\right)^2 \rtimes C_{\ell^n} : t \equiv (*,0) \pmod{\ell^{n-1}} \right\}.$$ Using the explicit group law on $({\mathbb{Z}}/\ell^n{\mathbb{Z}})^2 \rtimes C_{\ell^n}$ one checks without difficulty that $B_{\ell^n}$ is a subgroup of $\left( {\mathbb{Z}}/\ell^n{\mathbb{Z}}\right)^2 \rtimes C_{\ell^n}$: indeed, given two elements $g_1=(t_1,M_1)$ and $g_2=(t_2,M_2)$ in $B_{\ell^n}$, we have $$g_1 \cdot g_2 = (t_1, M_1) \cdot (t_2, M_2) = (t_1 + M_1 t_2, M_1M_2),$$ and (since $M_1$ is diagonal) the second coordinate of $t_1 + M_1 t_2$ is a linear combination (with ${\mathbb{Z}}/\ell^n{\mathbb{Z}}$-coefficients) of the second coordinates of $t_1, t_2$, hence is zero modulo $\ell^{n-1}$. Finally, let $K \subset F_{\ell^n, \ell^n}$ be the field corresponding by Galois theory to the subgroup $B_{\ell^n}$ of $\left( {\mathbb{Z}}/\ell^n{\mathbb{Z}}\right)^2 \rtimes C_{\ell^n} \cong \operatorname{Gal}(F_{\ell^n,\ell^n} \mid F)$.
We now study the situation of Proposition \[prop-max-growth-kummer\] for the elliptic curve $E/K$ and the point $\alpha$. By construction, the image of the $\ell^{n-1}$-torsion representation attached to $(E/K, \ell)$ is $C_{\ell^{n-1}}$, so the parameter of maximal growth can be taken to be $n_\ell=1$. We claim that $\alpha \in E(K)$ is strongly $\ell$-indivisible. The modulo-$\ell$ torsion representation is surjective onto $C_\ell$, so that in particular no $\ell$-torsion point of $E$ is defined over $K$, and strongly $\ell$-indivisible is equivalent to $\ell$-indivisible. To see that this last condition holds, notice that if $\alpha$ were $\ell$-divisible then we would have $K_{\ell, \ell}=K_\ell$. However this is not the case, because by construction $\operatorname{Gal}(K_{\ell,\ell} \mid K_\ell) = \{ t \in ({\mathbb{Z}}/\ell{\mathbb{Z}})^2 : t \equiv (*,0) \pmod{\ell} \}$ has order $\ell$. Finally, for $k=n-3$ we have $$V_{\ell^{k+2n_\ell}} = V_{\ell^{n-1}} = \{ t \in ({\mathbb{Z}}/\ell^{n-1}{\mathbb{Z}})^2 : t \equiv (*,0) \pmod{\ell^{n-1}} \},$$ which is very far from containing $E[\ell^k]$ – in fact, the index of $V_{\ell^{k+2n_\ell}}$ in $E[\ell^{k+2n_\ell}]$ can be made arbitrarily large by choosing larger and larger values of $n$. Notice that in any such example the $\ell$-adic representation will be surjective onto a split Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$.
Uniform Bounds for the Adelic Kummer Representation {#sec:UniformBounds}
===================================================
Our aim in this section is to show:
\[thm:QUniformity\] There is a positive integer $C$ with the following property: for every elliptic curve $E/{\mathbb{Q}}$ and every strongly indivisible point $\alpha \in E({\mathbb{Q}})$, the image $W_\infty \cong \operatorname{Gal}(K_{\infty, \infty} \mid K_\infty)$ of the Kummer map associated with $(E/{\mathbb{Q}},\alpha)$ has index dividing $C$ in $\prod_\ell T_\ell(E)$.
This result immediately implies Theorem \[thm:UniformIntroduction\]:
By Remark \[rem:Vsubgroup\], for every $N\mid M$ the ratio $
\displaystyle \frac{N^2}{[{\mathbb{Q}}_{M,N}: {\mathbb{Q}}_M]}$ divides $$\displaystyle \frac{N^2}{[{\mathbb{Q}}_{\infty,N}: {\mathbb{Q}}_\infty]}=
\left[(\hat {\mathbb{Z}}/N\hat{\mathbb{Z}})^2:W_\infty/N W_\infty\right],$$ which in turn divides $[\hat{\mathbb{Z}}^2:W_\infty]$.
As in Subsection \[subsec:PossibleImagesModl\], we will denote by $\mathcal{T}_0$ the finite set of primes $$\mathcal{T}_0:={\left\{ p\text{ prime }\mid p{\leqslant}17 \right\}}\cup\{37\}.$$
Bounds on Cohomology Groups
---------------------------
Let $E/{\mathbb{Q}}$ be an elliptic curve and $N_1,N_2$ be positive integers with $N_1\mid N_2$. The first step in the proof of Theorem \[thm:QUniformity\] is to bound the exponent of the cohomology group $H^1(H_{N_2},E[N_1])$. In the course of the proof we shall need the following technical result, which will be proved in Section \[section:ProofOf\].
\[prop:NewBoundCohomology\] There is a universal constant $e$ satisfying the following property. Let $E/{\mathbb{Q}}$ be a non-CM elliptic curve, $N$ a positive integer and $\ell$ a prime factor of $N$. Let $\ell^k$ be the largest power of $\ell$ dividing $N$ and ${J}=\operatorname{Gal}({\mathbb{Q}}_N \mid {\mathbb{Q}}_{\ell^k})\triangleleft H_N$. Consider the action of $H_N$ on $\operatorname{Hom}(J,E[\ell^k])$ defined by $(h\psi)(x)=h\psi(h^{-1}xh)$ for all $h\in H_N$, $\psi:{J}\to E[\ell^k]$ and $x\in {J}$. Then the exponent of $\operatorname{Hom}\left({J}, E[\ell^k] \right)^{H_N}$ divides $e$.
\[prop:UniformBoundOnCohomology\] There is a positive integer $C_1$ with the following property. Let $E/\mathbb{Q}$ be an elliptic curve, $N_1$ and $N_2$ be positive integers with $N_1\mid N_2$. Then the exponent of $H^1(H_{N_2}, E[N_1])$ divides $C_1$.
We can prove the statement separately for CM and non-CM curves, and then conclude by taking the least common multiple of the two constants obtained in the two cases.
Assume first that $E/{\mathbb{Q}}$ has CM over $\overline {\mathbb{Q}}$. Let $F$ be the CM field of $E$, $\mathcal{O}_F$ the ring of integers of $F$ and $\mathcal{O}_\ell:=\mathcal{O}_F \otimes_{{\mathbb{Z}}}{\mathbb{Z}}_\ell$. By [@MR3766118 Theorem 1.5] we have $d:=\left[\prod_\ell \mathcal{O}_\ell^\times:H_\infty\cap \prod_\ell \mathcal{O}_\ell^\times\right]{\leqslant}6$. In particular all the $d$-th powers of elements in $\prod_\ell \mathcal{O}_\ell^\times$ are in $H_\infty$, hence we have $\hat {\mathbb{Z}}^{\times d}\subseteq H_\infty\subseteq \prod_\ell \operatorname{GL}_2({\mathbb{Z}}_\ell)$ and $H_\infty$ contains the nontrivial homothety $\lambda=(\lambda_\ell)$, where $\lambda_2=3^d$ and $\lambda_\ell=2^d$ for $\ell\neq 2$. By Sah’s Lemma [@MR2018998 Lemma A.2] we have $(\lambda-1)H^1(H_{N_2}, E[N_1])=0$. Notice that the image of $\lambda-1$ in ${\mathbb{Z}}_\ell$ is nonzero for all $\ell$, and that it is invertible for almost all $\ell$. The claim follows from the fact that $d$ is bounded.
Assume now that $E$ does not have complex multiplication over $\overline {\mathbb{Q}}$. As cohomology commutes with finite direct products we have $$\begin{aligned}
H^1(H_{N_2}, E[N_1]) & \cong H^1 \left( H_{N_2}, \bigoplus_{\ell^v \mid {N_1}} E[\ell^v] \right) \cong \bigoplus_{\ell^v \mid {N_1}} H^1 \left( H_{N_2}, E[\ell^v] \right).
\end{aligned}$$ Fix an $\ell$ in this sum and let ${J}= \operatorname{Gal}({\mathbb{Q}}_{N_2} \mid {\mathbb{Q}}_{\ell^k}) \triangleleft H_{N_2}$, where $\ell^k$ is the largest power of $\ell$ dividing ${N_2}$. By the inflation-restriction sequence we get $$0 \to H^1(H_{N_2}/{J}, E[\ell^v]^{{J}}) \to H^1(H_{N_2}, E[\ell^v]) \to H^1({J}, E[\ell^v])^{H_{N_2}};$$ since by definition ${J}$ fixes $E[\ell^v]$, this is the same as $$0 \to H^1( H_{\ell^k} , E[\ell^v]) \to H^1(H_{N_2}, E[\ell^v]) \to \operatorname{Hom}({J}, E[\ell^v])^{H_{N_2}}.$$ It is clear that the exponent of $H^1(H_{N_2}, E[{N_1}])$ is the least common multiple of the exponents of the direct summands $H^1 \left(H_{N_2}, E[\ell^v] \right)$ for $\ell \mid {N_1}$, so we can focus on one such summand at a time. Furthermore, the above inflation-restriction exact sequence shows that the exponent of $H^1(H_{N_2},E[\ell^v])$ divides the product of the exponents of $H^1( H_{\ell^k} , E[\ell^v])$ and of $\operatorname{Hom}({J}, E[\ell^v])^{H_{N_2}}$. It is enough to give a uniform bound for the exponents of these two cohomology groups.
- $\boxed{H^1( H_{\ell^k}, E[\ell^v])}$ Assume first that $\ell\not\in\mathcal T_0$. By Theorem \[thm:Mazur\], $H_\ell$ is not contained in a Borel subgroup of $\operatorname{GL}_2({\mathbb{F}}_\ell)$, so by [@LawsonWutrich Lemma 4] it contains a nontrivial homothety. By Lemma \[lem:LiftHomothety\] the image $H_{\ell^\infty}$ of the $\ell$-adic representation contains a homothety that is non-trivial modulo $\ell$, so by Sah’s Lemma [@MR2018998 Lemma A.2] we have $H^1( H_{\ell^k}, E[\ell^v])=0$. For $\ell \in \mathcal{T}_0$ let $n_\ell$ be a universal bound on the parameter of maximal growth of the $\ell$-adic representation, as in Corollary \[cor:UniversalBoundGrowthParameter\]. By Lemma \[lemma:ImageAlwaysContainsScalars\] we have $(1+\ell^{n_\ell})\operatorname{Id}\in H_{\ell^k}$, and from Lemma \[lemma:ExponentOfH1\] we obtain that the exponent of $H^1( H_{\ell^k}, E[\ell^v])$ divides $\ell^{n_\ell}$.
- $\boxed{\operatorname{Hom}({J}, E[\ell^v])^{H_{N_2}}}$ As $v {\leqslant}k$, this group is contained in ${\operatorname{Hom}({J}, E[\ell^k])^{H_{N_2}}}$, whose exponent is uniformly bounded by Proposition \[prop:NewBoundCohomology\]. Notice that the action of $H_{N_2}$ on $\operatorname{Hom}(J, E[\ell^k])$ is precisely the one considered in Proposition \[prop:NewBoundCohomology\] by well-known properties of the inflation-restriction exact sequence (see for example [@MR1282290 Theorem 4.1.20]).
\[cor:UniformBoundDivisibility\] Let $C_1$ be as in Proposition \[prop:UniformBoundOnCohomology\]. Let $E/\mathbb{Q}$ be an elliptic curve and let $\alpha \in E({\mathbb{Q}})$ be a strongly indivisible point. If $\alpha$ is divisible by $n{\geqslant}1$ over ${\mathbb{Q}}_\infty$, then $n \mid C_1$.
Without loss of generality we can assume that $n=\ell^e$ is a power of a prime $\ell$. Since ${\mathbb{Q}}_\infty$ is the union of the torsion fields ${\mathbb{Q}}_N$, there exists $N$ such that $\alpha$ is divisible by $\ell^e$ over ${\mathbb{Q}}_N$, and we may assume that $\ell^e$ divides $N$. The claim then follows from Lemma \[lem-not-divisible\], since by Proposition \[prop:UniformBoundOnCohomology\] the exponent of $H^1(\operatorname{Gal}({\mathbb{Q}}_N\mid{\mathbb{Q}}),E[\ell^e])$ is a power of $\ell$ that divides $C_1$.
\[cor:BoundInTermsOfC1\] Let $C_1$ be as in Proposition \[prop:UniformBoundOnCohomology\]. The following hold for every prime $\ell$:
1. The ${\mathbb{Z}}_\ell$-module $W_{\ell^\infty}$, considered as a submodule of ${\mathbb{Z}}_\ell^2$, contains a vector of valuation at most $v_\ell(C_1)$.
2. If $E$ does not have CM over $\overline{\mathbb{Q}}$ and $n_\ell$ is a parameter of maximal growth for the $\ell$-adic torsion representation, then $W_{\ell^\infty}$ contains $\ell^{n_\ell + v_\ell(C_1)} T_\ell(E)$.
3. If $E[\ell]$ is an irreducible $H_\ell$-module, then $W_{\ell^\infty}$ contains $\ell^{v_\ell(C_1)} T_\ell(E)$.
4. If $E$ has CM over $\overline {\mathbb{Q}}$, let $(\gamma,\delta)$ be parameters for the Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ corresponding to $\operatorname{End}_{\overline {\mathbb{Q}}}(E)$. If $n_\ell$ is a parameter of maximal growth for the $\ell$-adic torsion representation, then $W_{\ell^\infty}$ contains $\ell^{3n_\ell + v_\ell(4\delta C_1)} T_\ell(E)$.
Part (1) follows from Lemma \[lem-order-n-d\], since by Corollary \[cor:UniformBoundDivisibility\] the point $\alpha$ is not divisible by $\ell^{v_\ell(C_1)+1}$ over ${\mathbb{Q}}_\infty$. Parts (2), (3) and (4) then follow from Proposition \[prop:GroupTheory\] (for part (4) observe that no elliptic curve over ${\mathbb{Q}}$ has CM defined over ${\mathbb{Q}}$).
We can now prove the main Theorem of this section.
As already explained, we have $W_\infty = \prod_\ell W_{\ell^\infty}$, so we obtain $$\left[\prod_\ell T_\ell(E) : W_\infty\right] = \prod_\ell [T_\ell(E) : W_{\ell^\infty}].$$
Let $$\mathcal T_1= \mathcal T_0\cup\{\ell\text{ prime}\mid\ell \text{ divides }C_1\}\cup {\left\{ 19,43,67,163 \right\}}.$$ Notice that by Theorem \[thm:Mazur\] for $\ell\not\in \mathcal T_1$ there is no elliptic curve over ${\mathbb{Q}}$ with a rational subgroup of order $\ell$. By Lemma \[cor:BoundInTermsOfC1\] (3), for $\ell\not\in \mathcal T_1$ we have $W_{\ell^{\infty}}=T_\ell(E)$, so $$\label{eqn:boundIndexKummerUniform}
\left[\prod_\ell T_\ell(E) : W_\infty\right] = \prod_{\ell \in \mathcal T_1} [T_\ell(E) : W_{\ell^\infty}].$$
Now it is enough to prove the Theorem separately in the CM and in the non-CM case, and then take the least common multiple of the two constants obtained.
Suppose first that $E$ does not have CM over $\overline{{\mathbb{Q}}}$. Applying Lemma \[cor:BoundInTermsOfC1\](2) we see that $[T_\ell(E) : W_{\ell^\infty}]$ divides $\ell^{2(n_\ell+v_\ell(C_1))}$, where $n_\ell$ is a parameter of maximal growth for the $\ell$-adic torsion for $E$. By Theorem \[thm:Arai\] this can be bounded uniformly in $E$. Since $C_1$ does not depend on $E$, each factor of the right hand side of is uniformly bounded.
Assume now that $E$ has complex multiplication over $\overline {\mathbb{Q}}$ and let $(\gamma,\delta)$ be parameters for the Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ corresponding to $\operatorname{End}_{\overline {\mathbb{Q}}}(E)$. Applying Lemma \[cor:BoundInTermsOfC1\](4), we see that $[T_\ell(E) : W_{\ell^\infty}]$ divides $\ell^{2(3n_\ell+v_\ell(4\delta C_1))}$, where $n_\ell$ is a parameter of maximal growth for the $\ell$-adic torsion representation for $E$, which is uniformly bounded by Corollary \[cor:UniversalBoundGrowthParameter\]. It remains to show that $v_\ell(\delta)$ can be bounded uniformly as well. This follows from the fact that $\delta$ only depends on the $\overline {\mathbb{Q}}$-isomorphism class of $E$, and that there are only finitely many rational $j$-invariants corresponding to CM elliptic curves.
Proof of Proposition \[prop:NewBoundCohomology\] {#section:ProofOf}
------------------------------------------------
Recall the setting of Proposition \[prop:NewBoundCohomology\]: $E/{\mathbb{Q}}$ is a non-CM elliptic curve, $N$ is a positive integer, and $\ell$ is a prime factor of $N$. Let $\ell^k$ be the largest power of $\ell$ dividing $N$ and ${J}=\operatorname{Gal}({\mathbb{Q}}_N \mid {\mathbb{Q}}_{\ell^k})\triangleleft H_N$. The question is to study the exponent of the group $\operatorname{Hom}\left({J}, E[\ell^k] \right)^{H_N}$. In order to do this, we shall study the conjugation action of $g \in H_N$ on the abelianisation of ${J}$. More generally, we shall also consider the conjugation action of elements in $\operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ that normalise ${J}$.
It will be useful to work with a certain subgroup ${J(2)}$ of ${J}$. More generally, we introduce the following notation.
Let $G$ be a group and $M$ a positive integer. We denote by $G(M)$ the subgroup of $G$ generated by ${\left\{ g^M\mid g\in G \right\}}$.
\[lem:SubgroupL\] The subgroup ${J(2)}$ is normal in ${J}$, the quotient group ${J}/{J(2)}$ has exponent at most 2, ${J(2)}$ is stable under the conjugation action of $H_N$, and $$\operatorname{exp} \operatorname{Hom}\left({J}, E[\ell^k] \right)^{H_N} \mid 2 \operatorname{exp} \operatorname{Hom}\left({J(2)}, E[\ell^k] \right)^{H_N}.$$
Clearly ${J(2)}$ is a characteristic subgroup of ${J}$, so it is normal in ${J}$ and stable under the conjugation action of $H_N$ on ${J}$. Given a coset $h{J(2)}\in {J}/{J(2)}$ we have $(h{J(2)})^2 = h^2{J(2)}= {J(2)}$ since $h^2 \in {J(2)}$ by definition, so the quotient ${J}/{J(2)}$ is killed by $2$. Finally, take a homomorphism $\psi : {J}\to E[\ell^k]$ stable under the conjugation action of $H_N$ and denote by $d$ the exponent of the abelian group $\operatorname{Hom}\left({J(2)}, E[\ell^k] \right)^{H_N}$. The restriction of $\psi$ to ${J(2)}$ is an element of $\operatorname{Hom}\left({J(2)}, E[\ell^k] \right)^{H_N}$, so it satisfies $d\psi|_{{J(2)}} = 0$, and thus given any $h \in {J}$ we have $d\psi|_{{J(2)}}(h^2)=0$. This implies that for every $h \in J$ we have $2d \psi(h) = 0$, hence $\psi$ is killed by $2d$. Since this is true for all $\psi$, the claim follows.
We will also need the following two simple lemmas:
\[lem:NontrivialHomothety\] Let $E/{\mathbb{Q}}$ be an elliptic curve and let $M{\geqslant}37$ be an integer. If $\ell > M+1$ is a prime number, then $H_{\ell^\infty}{(M)}$ contains a homothety $\lambda \operatorname{Id}$ with $\lambda \not \equiv 1 \pmod{\ell}$.
By Corollary \[cor:ContainsScalarsAndConjugation\], since $\ell>M+1>37$, the image of the modulo-$\ell$ representation contains all the homotheties. In particular, if $\overline \mu \in {\mathbb{F}}_\ell^\times$ is a generator of the multiplicative group ${\mathbb{F}}_\ell^\times$, then $H_\ell$ contains $\overline \mu \operatorname{Id}$, so by Lemma \[lem:LiftHomothety\] $H_{\ell^\infty}$ contains $\mu\operatorname{Id}$, where $\mu\in{\mathbb{Z}}_\ell^\times$ is congruent to $\overline{\mu}$ modulo $\ell$. So $H_{\ell^\infty}{(M)}$ contains $\mu^{M} \operatorname{Id}$, which is nontrivial modulo $\ell$ since $\overline \mu$ has order $\ell-1 > M$.
\[lem:Cohen\] Let $p$ be a prime and let $n$ be a positive integer (with $n{\geqslant}2 $ if $p=2$). For every positive integer $k$ let $U_k={\left\{ x\in{\mathbb{Z}}_p\mid x\equiv 1 \pmod {p^k} \right\}}$. Let $M$ be a positive integer. Then ${\left\{ x^M\mid x\in U_n \right\}}\supseteq U_{n+v_p(M)}$.
Let $y\in U_{n+v_p(M)}$ and let $a=y-1$. By [@MR2312337 Corollary 4.2.17 and Corollary 4.2.18(1)], the $p$-adic integer $x=\exp(M^{-1}\log y)$ is well defined and satisfies $v_p(x-1){\geqslant}v_p(M^{-1}a){\geqslant}n$. Therefore $x\in U_n$ and clearly $x^M=y$.
We will derive Proposition \[prop:NewBoundCohomology\] from the following statement:
\[prop:BoundExponentg\] There is a universal constant $M$ with the following property. For every elliptic curve $E/{\mathbb{Q}}$, every positive integer $N$, every prime power $\ell^k$ dividing $N$, and every $g \in H_N$, the conjugation action of $g^M$ on the abelianisation of ${J(2)}$ is trivial.
By Lemma \[lem:SubgroupL\] it suffices to prove Proposition \[prop:NewBoundCohomology\] with ${J}$ replaced by ${J(2)}$. Let $\psi \in \operatorname{Hom}\left({J(2)}, E[\ell^k] \right)$: then as $E[\ell^k]$ is abelian $\psi$ factors through ${J(2)}^{\operatorname{ab}}$.
For every $g \in H_N$, every $\psi \in \operatorname{Hom}\left({J(2)}, E[\ell^k] \right)^{H_N}$ and every $h\in {J(2)}$ we have $$\psi(h)=g^M \cdot \psi(g^{-M} h g^M ) =g^M \cdot \psi(h),$$ where the first equality holds because $\psi$ is $H_N$-invariant and the second because the automorphism induced by $g^M$ on ${J(2)}^{\operatorname{ab}}$ is trivial by Proposition \[prop:BoundExponentg\]. This means that the image of $\psi$ is contained in $E[\ell^k]^{H_N{(M)}}$. Since the action of $H_N$ on $E[\ell^k]$ factors via the canonical projection $H_N \to \operatorname{GL}_2({\mathbb{Z}}/\ell^k{\mathbb{Z}})$, this is the same as saying that the image of $\psi$ is contained in the subgroup of $E[\ell^k]$ fixed under $H_{\ell^k}(M)$. It remains to show that the exponent of $E[\ell^k]^{H_{\ell^k}(M)}$ is uniformly bounded, and trivial for $\ell$ sufficiently large.
To see this, recall that by Theorem \[thm:Arai\] there exists an integer $n{\geqslant}1$, independent of $E$, such that $H_{\ell^k}$ contains $\operatorname{Id} + \ell^{n} \operatorname{Mat}_2({\mathbb{Z}}/\ell^k{\mathbb{Z}})$ (and we have $n {\geqslant}2$ if $\ell=2$). By Lemma \[lem:Cohen\], for every $E/{\mathbb{Q}}$ the group $H_{\ell^k}(M)$ contains all scalar matrices in $\operatorname{Mat}_2({\mathbb{Z}}/\ell^k{\mathbb{Z}})$ that are congruent to the identity modulo $\ell^{n+v_\ell(M)}$. We claim that the exponent of $E[\ell^k]^{H_{\ell^k}(M)}$ divides $\ell^{n+v_\ell(M)}$. In fact, by what we have seen $H_{\ell^k}(M)$ contains $(1+\ell^{n+v_\ell(M)})\operatorname{Id}$, so $E[\ell^k]^{H_{\ell^k}(M)}$ is in particular fixed by $(1+\ell^{n+v_\ell(M)})\operatorname{Id}$, hence it is contained $E[\ell^{n+v_\ell(M)}]$.
Finally, we show that $\operatorname{Hom}({J},E[\ell^k])^{H_N}$ is trivial for $\ell>M+1$. Since $\ell>2$, by Lemma \[lem:SubgroupL\] it is enough to show that $\operatorname{Hom}({J(2)},E[\ell^k])^{H_N}$ is trivial. As above, the image of any $H_N$-stable homomorphism from ${J(2)}$ to $E[\ell^k]$ is contained in the $H_{\ell^k}(M)$-fixed points of $E[\ell^k]$. By Lemma \[lem:NontrivialHomothety\], $H_{\ell^k}(M)$ contains a homothety which is nontrivial modulo $\ell$, so we are done since the only fixed point of this homothety is $0$.
We now turn to the proof of Proposition \[prop:BoundExponentg\]. We start by showing that we may assume $N$ to be of the form $\ell^k \cdot \prod_{p \mid N, p \neq \ell} p$. To see this, let $N=\ell^k \prod_{p \mid N, p \neq \ell} p^{e_p}$ be arbitrary and let $N':=\ell^k \prod_{p \mid N, p \neq \ell} p$. There is an obvious reduction map ${J}\to \operatorname{Gal}({\mathbb{Q}}_{N'} \mid {\mathbb{Q}}_{\ell^k})$. The kernel $\mathcal K$ of this map is a subgroup of ${J}$ whose order is divisible only by primes $p \mid N, p \neq \ell$. Recall that we will be considering $\operatorname{Hom}({J}, E[\ell^k])^{H_N}$. Let $\psi : {J}\to E[\ell^k]$ be a homomorphism: we claim that $\psi$ factors via the quotient $\operatorname{Gal}({\mathbb{Q}}_{N'} \mid {\mathbb{Q}}_{\ell^k})$. Indeed, all the elements in $\mathcal K$ have order prime to $\ell$, hence they must go to zero in $E[\ell^k]$. Therefore we may assume $N=N'$, that is, $N=\ell^k \cdot \prod_{p \mid N, p \neq \ell} p$.
We identify $H_N$ with a subgroup of $\operatorname{GL}_2({\mathbb{Z}}/\ell^k{\mathbb{Z}}) \times \prod_{p \mid N, p \neq \ell} \operatorname{GL}_2({\mathbb{Z}}/p{\mathbb{Z}})$ and ${J}$ with the subgroup of $H_N$ consisting of elements having trivial first coordinate, and for $g\in H_N$ we write $g=(g_\ell, g_{p_1}, \ldots, g_{p_r})$ with $g_\ell \in \operatorname{GL}_2({\mathbb{Z}}/\ell^k{\mathbb{Z}})$ and $g_{p_i} \in \operatorname{GL}_2({\mathbb{Z}}/p_i{\mathbb{Z}})$. Finally, for $p\mid N$, $p\neq \ell$ we denote by $\pi_{p_i}:H_N\to \operatorname{GL}_2({\mathbb{Z}}/p_i{\mathbb{Z}})$ the projection on the factor corresponding to $p_i$, and we denote by $\pi_\ell:H_N\to \operatorname{GL}_2({\mathbb{Z}}/\ell^k{\mathbb{Z}})$ the projection on the factor corresponding to $\ell$.
\[lem:SL2Factor\] Let $p$ be a prime factor of $N$ with $p {\geqslant}7$, $p \neq \ell$. Suppose that the modulo-$p$ representation attached to $E/{\mathbb{Q}}$ is surjective. Then ${J(2)}$ contains $\{1\} \times \cdots \times \{1\} \times \operatorname{SL}_2({\mathbb{Z}}/p{\mathbb{Z}}) \times \{1\} \times \cdots \times \{1\}$.
Clearly $\operatorname{PSL}_2({\mathbb{F}}_p)$ occurs in $H_N$. Hence it must occur either in ${J}$ or in $H_N/{J}$, but the latter is isomorphic to a subgroup of $\operatorname{GL}_2({\mathbb{Z}}/\ell^k{\mathbb{Z}})$ with $\ell \neq p$, so it must occur in ${J}$. Consider the kernel of the projection ${J}\to \prod_{q \mid N, q \neq p} \operatorname{GL}_2({\mathbb{Z}}/q{\mathbb{Z}})$: then $\operatorname{PSL}_2({\mathbb{F}}_p)$ must occur either in this kernel or in $\prod_{q \mid N, q \neq p} \operatorname{GL}_2({\mathbb{Z}}/q{\mathbb{Z}})$, but the latter case is impossible. Using Lemma \[lemma-solvable\], it follows immediately that ${J}$ contains $\{1\} \times \cdots \times \{1\} \times \operatorname{SL}_2({\mathbb{Z}}/p{\mathbb{Z}}) \times \{1\} \times \cdots \times \{1\}$. We conclude by noting that $\operatorname{SL}_2({\mathbb{F}}_p)$ is generated by its squares.
\[lem:SameAutomorphism\] Let $g \in H_N$ and $h \in {J(2)}$. Then $gh \in H_N$, and the automorphisms of ${J(2)}^{\operatorname{ab}}$ induced by $g$ and by $gh$ coincide.
As ${J(2)}$ is a subgroup of $H_N$, the fact that $gh \in H_N$ is obvious. For the second statement, notice that for every $x \in {J(2)}$ the element $(gh)^{-1} x (gh)$ differs from $g^{-1}xg$ by multiplication by $h^{-1}(g^{-1}x^{-1}g)^{-1}h(g^{-1}x^{-1}g)$, which is a commutator in ${J(2)}$. Hence the classes of $(gh)^{-1} x (gh)$ and $g^{-1}xg$ are equal in ${J(2)}^{\operatorname{ab}}$.
\[lem:gpNormalisesKp\] For each $p \mid N, p \neq \ell$, the component $g_p$ of $g$ along $\operatorname{GL}_2({\mathbb{Z}}/p{\mathbb{Z}})$ normalises $\pi_{p}({J(2)})$ in $\operatorname{GL}_2({\mathbb{Z}}/p{\mathbb{Z}})$.
Since $H_N$ normalises ${J(2)}$ by Lemma \[lem:SubgroupL\], we have $\pi_p(g^{-1}{J(2)}g)=\pi_p({J(2)})$. On the other hand $\pi_p(g^{-1}{J(2)}g) = \pi_p(g)^{-1} \pi_p({J(2)}) \pi_p(g)$, so $g_p^{-1} \pi_p({J(2)}) g_p=\pi_p({J(2)})$ as desired.
\[cor:gpId\] Let $p_1,\dots,p_s {\geqslant}7$ be primes all different from $\ell$ and such that the mod-$p_i$ representation attached to $E/{\mathbb{Q}}$ is surjective for each $p_i$. Let $g\in H_N$ and let $\hat g$ be the element of $\operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ obtained by replacing every $p_i$-component (for $i=1,\ldots,s$) of $g$ by $\operatorname{Id}$. Then $\hat g^2$ normalises ${J(2)}$, and it induces on ${J(2)}^{\operatorname{ab}}$ the same conjugation action as $g^2$.
By Lemma \[lem:SameAutomorphism\], if we multiply $g^2$ by any element of ${J(2)}$ the conjugation action on ${J(2)}^{\operatorname{ab}}$ does not change. By construction, the determinant of $\pi_{p_i}(g^2)=g_{p_i}^2$ is a square in ${\mathbb{F}}_{p_i}^\times$, say $\lambda_i^2$. It follows that the determinant of $g_{p_i}^2/\lambda_i$ is 1, so $g_{p_i}^2/\lambda_i \in \operatorname{SL}_2({\mathbb{Z}}/p_i{\mathbb{Z}})$. By Lemma \[lem:SL2Factor\] we have that ${J(2)}$ contains $h_i=(1,1,\ldots,1,g_{p_i}^2/\lambda_i,1,\ldots,1)$. Letting $h=h_1\cdots h_s$, we obtain that the action of $g^2h^{-1}$ is the same as that of $g^2$. But the element $$\mu=(1,\dots,1,\lambda_1,1,\dots,1)\cdots (1,\dots,1,\lambda_s,1,\dots,1)$$ is central in $\operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$, so $\hat g^2=g^2h^{-1}\mu^{-1}$ normalises ${J(2)}$ and it induces the same action as $g^2$ on $J(2)^{\operatorname{ab}}$.
Let $M=\operatorname{lcm} \{\exp \operatorname{PGL}_2({\mathbb{F}}_p) : p \in \mathcal T_0\}$, where $\exp \operatorname{PGL}_2({\mathbb{F}}_p)$ denotes the exponent of the group $\operatorname{PGL}_2({\mathbb{F}}_p)$.
\[rem:evenScalar\] Notice that $M$ is even. Moreover, for any $g\in \operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ and any $p\in \mathcal T_0$ with $p\mid N$ and $p\neq \ell$ we have that $\pi_p(g^M)$ is a scalar in $\operatorname{GL}_2({\mathbb{F}}_p)$, since it is trivial in $\operatorname{PGL}_2({\mathbb{F}}_p)$.
We now prove Proposition \[prop:BoundExponentg\], using the constant $M$ just introduced.
Write as before $g=(g_p)$. We divide the prime factors of $N$ different from $\ell$ into three sets as follows: $$\begin{aligned}
\mathcal{P}_0&={\left\{ p\mid N\text{ such that } p\in\mathcal{T}_0,\,p\neq \ell \right\}},\\
\mathcal{P}_1&={\left\{ p\mid N\text{ such that } H_p=\operatorname{GL}_2({\mathbb{F}}_p),\,p\neq \ell \right\}},\\
\mathcal{P}_2&={\left\{ p\mid N\text{ such that } H_p\text{ is conjugate to a subgroup of }N_{\operatorname{ns}}(p),\,p\neq \ell \right\}}.\end{aligned}$$ Notice that by Theorem \[thm:Zywyna\] each prime factor of $N$ different from $\ell$ belongs to one of these three sets.
We now apply Corollary \[cor:gpId\] with $\{p_1,\dots,p_s\}=\mathcal{P}_1$ to obtain an element $\hat g\in \operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ such that $\pi_{p}(\hat g)=\operatorname{Id}$ for every $p\in\mathcal{P}_1$ and such that $\hat{g}^2$ induces on ${J(2)}^{\operatorname{ab}}$ the same conjugation action as $g^2$. In particular, $\hat g^M$ induces on ${J(2)}^{\operatorname{ab}}$ the same conjugation as $g^M$ (recall that $M$ is even).
We now prove that this conjugation action is trivial by showing that $\hat g^M$ commutes with every element of ${J(2)}$. It suffices to show that for each $p\mid N$ the projection $\pi_p(\hat g^M)$ commutes with every element of $\pi_p({J(2)})$.
- *Case $p\in \mathcal{P}_0$:* by Remark \[rem:evenScalar\], $\pi_p(\hat g^M)$ is a scalar, thus it commutes with all of $\operatorname{GL}_2({\mathbb{F}}_p)$.
- *Case $p\in \mathcal{P}_1$:* by construction $\pi_p(\hat g^M)$ is trivial.
- *Case $p\in \mathcal{P}_2$:* by Corollary \[cor:ContainsScalarsAndConjugation\] applied to $\pi_p(\hat g)$, there is $h\in \operatorname{GL}_2({\mathbb{F}}_p)$ such that $\pi_p(\hat g)\in hN_{\operatorname{ns}}(p)h^{-1}$ and $H_p\subseteq hN_{\operatorname{ns}}(p)h^{-1}$. Since $M$ is even and $C_{\operatorname{ns}}(p)$ has index $2$ in $N_{\operatorname{ns}}(p)$, $\pi_p(\hat g^M)\in hC_{\operatorname{ns}}(p)h^{-1}$ and $\pi_p({J(2)})\subseteq \langle a^2\mid a\in H_p\rangle\subseteq hC_{\operatorname{ns}}(p)h^{-1}$. Since $C_{\operatorname{ns}}(p)$ is abelian, $\pi_p(\hat g^M)$ commutes with every element of $\pi_p({J(2)})$.
- *Case $p=\ell$:* by construction $\pi_p(J(2))$ is trivial.
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abstract: 'In this paper we develop an abstract theory for the Codazzi equation on surfaces, and use it as an analytic tool to derive new global results for surfaces in the space forms $\r^3$, $\s^3$ and $\h^3$. We give essentially sharp generalizations of some classical theorems of surface theory that mainly depend on the Codazzi equation, and we apply them to the study of Weingarten surfaces in space forms. In particular, we study existence of holomorphic quadratic differentials, uniqueness of immersed spheres in geometric problems, height estimates, and the geometry and uniqueness of complete or properly embedded Weingarten surfaces.'
---
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[**The Codazzi Equation for Surfaces**]{}\
[**Juan A. Aledo$^a$, José M. Espinar${}^{b}$ and José A. Gálvez${}^{c}$**]{}
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[${}^a$Departamento de Matemáticas, Universidad de Castilla-La Mancha, EPSA, 02071 Albacete, Spain; e-mail: [email protected]\
${}^b$Institut de Mathématiques, Université Paris VII, 175 Rue du Chevaleret, 75013 Paris, France; e-mail: [email protected]\
${}^c$Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain; e-mail: [email protected]]{}
Introduction
============
The *Codazzi equation* for an immersed surface ${\Sigma}$ in $\r^3$ yields $$\label{ecod}
\nabla_XSY-\nabla_YSX-S[X,Y]=0,\qquad X,Y\in{\mathfrak{X}}({\Sigma}).$$ Here $\nabla$ is the Levi-Civita connection of the first fundamental form $I$ of ${\Sigma}$ and $S$ is the shape operator, defined by $II(X,Y)=I(S(X),Y)$, where $II$ is the second fundamental form of the surface. This Codazzi equation is, together with the Gauss equation, one of the two classical integrability conditions for surfaces in $\r^3$, and it remains invariant if we substitute the ambient space $\r^3$ by other space form $\s^3$ or $\h^3$.
It is remarkable that some crucial results of surface theory in $\r^3$ only depend, in essence, of the Codazzi equation. This is the case, for instance, of Hopf’s theorem (resp. Liebmann’s theorem) on the uniqueness of round spheres among immersed constant mean curvature spheres (resp. among complete surfaces of constant positive curvature). This suggests the possibility of adapting these results to an abstract setting of *Codazzi pairs* (i.e. pairs of real quadratic forms $(I,II)$ on a surface verifying ), and to explore its possible consequences in surface theory. The basic idea in this sense is to use the Codazzi pair $(I,II)$ as a geometric object in a *non-standard* way, i.e. so that $(I,II)$ are no longer the first and second fundamental forms of a surface in a space form.
Motivated by this, our objective here is to develop an abstract theory for the Codazzi equation on surfaces, and use it subsequently as an analytic tool to derive new global results for surfaces in the space forms $\r^3$, $\s^3$ and $\h^3$.
Our results on Codazzi pairs here provide an extremely general extension of some classical theorems of surface theory that mainly depend on the Codazzi equation. But, moreover, this abstract approach has some very definite applications to the study of complete or properly embedded Weingarten surfaces in $\r^3$ or $\h^3$:
1. It reveals the existence of holomorphic quadratic differentials for some classes of surfaces in space forms (and also in product spaces $\s^2\times \r$, $\h^2\times \r$, see [@AEG1]).
2. It unifies the proof of apparently non related theorems. For example, it shows that uniqueness in the Christoffel problem in $\r^3$ is basically equivalent to the Bonnet theorem on uniqueness of immersed spheres with prescribed mean curvature.
3. It gives an analytic tool to prove uniqueness results for complete or compact Weingarten surfaces in space forms.
These applications show the flexibility of the use of Codazzi pairs in surface theory, and suggest the possibility of obtaining further global results with the techniques employed here.
We have organized this paper as follows. We shall start by reminding in Section \[s2\] the definitions of fundamental pair, Codazzi pair, and some of their associated invariants such as the mean curvature $H$, the extrinsic curvature $K$ and the Hopf differential. We prove in Theorem \[elteorema\] that a topological sphere $\Sigma$ endowed with a Codazzi pair satisfying a general Weingarten relationship $W(H,K)=0$ must be totally umbilical, if some necessary conditions are fulfilled by the functional $W$. This generalizes the previous Hopf theorem and Liebmann theorem.
This abstract treatment lets us apply Theorem \[elteorema\] to some seemingly unrelated situations. More specifically, as a consequence of that result, we obtain generalizations of the Bonnet Theorem and the theorem of uniqueness in the Christoffel problem. We point out that our proof to the Christoffel problem is different from the classical approach, which uses integration theory on surfaces (see [@H; @Sp]).
We will finish Section \[s2\] proving that two Codazzi pairs $(I_i,II_i)$, $i = 1,2$, on a topological sphere $\Sigma$, such that $II_1=II_2$ and with the same positive extrinsic curvature must be isometric, that is, $I_1=I_2$. This result is a wide generalization of a classical result by Grove [@Gr] about rigidity of ovaloids in $\r^3$.
In Section \[s3\] we study when a real quadratic form $II$ on a Riemannian surface is conformal to the metric, even if the Codazzi equation is not satisfied. For that, we will define the Codazzi function on a surface associated to its induced metric $I$ and a real quadratic form $II$. This function will measure how far is the pair $(I,II)$ from satisfying the Codazzi equation.
We devote Section \[s4\] to the fundamental relation between the Codazzi equation and the existence of holomorphic quadratic differentials. Thus, given a Codazzi pair on a surface $\Sigma$, we find, under certain conditions, the existence of a new Codazzi pair on $\Sigma$ whose Hopf differential is holomorphic. This new pair will provide geometric information about the initial one.
We particularize this result to the study of Codazzi pairs of special Weingarten type, that is, pairs satisfying $H=f(H^2-K)$ for a certain function $f$. The corresponding problem for surfaces in $\r^3$ and $\h^3$ was studied by Bryant in [@Br]. We will also prove that every Codazzi pair on a surface $\Sigma$ satisfying $H=f(H^2-K)$ can be recovered in terms of a metric on $\Sigma$ and a holomorphic quadratic form.
Finally, in Section \[s5\], we give some applications of our abstract approach to surfaces in space forms. We begin by obtaining height estimates for a wide family of surfaces of elliptic type. Although these estimates are not optimal, the existence of such height estimates with respect to planes constitute a fundamental tool for studying the behaviour of complete embedded surfaces.
Following the ideas developed by Rosenberg and Sa Earp in [@RoS], we show that the theory developed by Korevaar, Kusner, Meeks and Solomon [@KKMS; @KKS; @M] for constant mean curvature surfaces in $\r^3$ and $\h^3$ remains valid for some families of surfaces satisfying the maximum principle (Theorem \[finales\]).
In particular, when Theorem \[finales\] is applied to a properly embedded Weingarten surface $\Sigma$ of elliptic type satisfying $H=f(H^2-K)$ in $\r^3$ or $\h^3$, we obtain: If $\Sigma$ has finite topology and $k$ ends, then $k\geq 2$, $\Sigma$ is rotational if $k=2$, and $\Sigma$ is contained in a slab if $k=3$.
To finish the paper, we study the problem of classifying Weingarten surfaces of elliptic type satisfying $H=f(H^2-K)$ in $\r^3$ such that $K$ does not change signs [@ST3]. We show that, in the above conditions, if $\Sigma$ is a complete surface with $K\geq0$ then it must be a totally umbilical sphere, a plane or a right circular cylinder, and if $\Sigma$ is properly embedded and $K\leq 0$, then it is a right circular cylinder or a surface of minimal type (i.e. $f(0)=0$).
Fundamental pairs and Codazzi pairs {#s2}
===================================
Let us start this section by recalling some classical results about fundamental pairs. A classical reference about this topic is [@Mi]. Besides we point out that, although throughout this paper we will assume that the differentiability used is always $\cal{C}^{\infty}$, the differentiability requirements are much lower.
We will denote by ${\Sigma}$ an orientable (and oriented) differentiable surface. Otherwise we would work with its oriented two-sheeted covering.
A fundamental pair on ${\Sigma}$ is a pair of real quadratic forms $(I,II)$ on ${\Sigma}$, where $I$ is a Riemannian metric.
Associated to a fundamental pair $(I,II)$ we define the shape operator $S$ of the pair as $$\label{ii}
II(X,Y)=I(S(X),Y)$$ for any vector fields $X,Y$ on ${\Sigma}$.
Conversely, from (\[ii\]) it becomes clear that the quadratic form $II$ is totally determined by $I$ and $S$. In other words, giving a fundamental pair on ${\Sigma}$ is equivalent to giving a Riemannian metric on ${\Sigma}$ and a self-adjoint endomorphism $S$.
We define the [*mean curvature*]{}, the [*extrinsic curvature*]{} and the [*principal curvatures*]{} of $(I,II)$ as half the trace, the determinant and the eigenvalues of the endomorphism $S$, respectively.
In particular, given local parameters $(x,y)$ on ${\Sigma}$ such that $$I=E\,dx^2+2F\,dxdy+G\,dy^2,\qquad II=e\,dx^2+2f\,dxdy+g\,dy^2,$$ the mean curvature and the extrinsic curvature of the pair are given, respectively, by $$H=H(I,II)=\frac{E g+G e-2F f}{2(EG-F^2)},\qquad K=K(I,II)=\frac{eg-f^2}{EG-F^2}.$$ Moreover, the principal curvatures of the pair are $H\pm\sqrt{H^2-K}$.
We will say that the pair $(I,II)$ is [*umbilical*]{} at $p\in {\Sigma}$ if $II$ is proportional to $I$ at $p$, or equivalently:
- if both principal curvatures coincide at $p$, or
- if $S$ is proportional to the identity map on the tangent plane at $p$, or
- if $H^2-K=0$ at $p$.
We define the [*Hopf differential*]{} of the fundamental pair $(I,II)$ as the (2,0) part of $II$ for the Riemannian metric $I$. In other words, if we consider $\Sigma$ as a Riemann surface with respect to the metric $I$ and take $z$ a local conformal parameter, then $$\label{parholomorfo}
\begin{array}
{c} I=2\lambda\,|dz|^2\\[2mm]
II=Q\,dz^2+2\lambda\,H\,|dz|^2+\overline{Q}\,d{\bar{z}}^2.
\end{array}$$ The quadratic form $Q\,dz^2$, which does not depend on the chosen parameter, is known as the Hopf differential of the pair $(I,II)$. We note that $(I,II)$ is umbilical at $p\in\Sigma$ if, and only if, $Q(p)=0$.
All the above definitions can be understood as natural extensions of the corresponding ones for isometric immersions of a Riemann surface in a 3-dimensional ambient space, where $I$ plays the role of the induced metric and $II$ the role of its second fundamental form.
A specially interesting case happens when the fundamental pair satisfies, in an abstract way, the Codazzi equation for surfaces in $\r^3$,
We say that a fundamental pair $(I,II)$, with shape operator $S$, is a Codazzi pair if $$\label{ecuacioncodazzi}
\nabla_XSY-\nabla_YSX-S[X,Y]=0,\qquad X,Y\in{\mathfrak{X}}({\Sigma}),$$ where $\nabla$ stands for the Levi-Civita connection associated to the Riemannian metric $I$ and ${\mathfrak{X}}({\Sigma})$ is the set of differentiable vector fields on ${\Sigma}$.
Many Codazzi pairs appear in a natural way in the study of surfaces. For instance, the first and second fundamental forms of a surface isometrically immersed in a 3-dimensional space form is a Codazzi pair. The same occurs for spacelike surfaces in a 3-dimensional Lorentzian space form. More generally, if the surface is immersed in an $n$-dimensional (semi-Riemannian) space form and has a parallel unit normal vector field $N$, then its induced metric and its second fundamental form associated to $N$ make up a Codazzi pair.
Classically, Codazzi pairs also arise in the study of harmonic maps. Many others examples of Codazzi pairs also appear in [@AEG1; @Bi; @Mi; @Ol]. All of this shows that the results that we present in this work can be used in many different contexts.
Many classical results in surface theory depend on the Codazzi equation of the immersion. This fact allows to generalize such results to the Codazzi pairs theory. Some examples of that, are Hopf’s results proving that the only surfaces immersed in $\r^3$ with constant mean curvature are totally umbilical. Analogously, Liebmann proved that the only complete surfaces with positive constant Gaussian curvature in $\r^3$ are totally umbilical spheres. Now, we obtain a generalization of both results to the wider family of Weingarten pairs, which we define next
We say that a fundamental pair $(I,II)$ on a surface ${\Sigma}$ is a Weingarten pair if its mean and extrinsic curvatures, $H$ and $K$ respectively, satisfy a non trivial relationship $$W(H,K)=0,$$ where $W$ is a differentiable function defined on an open set of $\r^2$ containing the set of points $\{(H(p),K(p)):\ p\in{\Sigma}\}$.
\[elteorema\] Let $(I,II)$ be a Codazzi pair on a surface ${\Sigma}$. If $(I,II)$ is a Weingarten pair for a functional $W(x,y)$ such that $$\label{hipotesis} W_x(t,t^2)+2t\,W_y(t,t^2)\neq0\qquad\mbox{for all
} t,$$ then either the umbilical points of $(I,II)$ are isolated and of negative index, or the pair is totally umbilical.
In particular, if ${\Sigma}$ is a topological sphere then $(I,II)$ is totally umbilical.
Several proofs of this result when the ambient space is $\r^3$ or $\h^3$ have been given by Hopf [@Ho], Chern [@Ch], Hartman and Wintner [@HW], Bryant [@Br] or Alencar, do Carmo and Tribuzy [@AdCT].
Let us consider ${\Sigma}$ as a Riemann surface with the conformal structure induced by $I$. Given a local conformal parameter $z$, we can write the fundamental pair $(I,II)$ as in (\[parholomorfo\]). Hence we have $$\label{levicivita}
\nabla_{\frac{\partial\
}{\partial z}}\frac{\partial\ }{\partial z}=\frac{\lambda_z}{\lambda}\
\frac{\partial\ }{\partial z},\qquad \nabla_{\frac{\partial\ }{\partial
z}}\frac{\partial\ }{\partial {\bar{z}}}=0$$ and the shape operator $S$ becomes $$\label{endomorfismoweingarten}
S\frac{\partial\ }{\partial z}=H\,\frac{\partial\ }{\partial z}+\frac{Q}{\lambda}\
\frac{\partial\ }{\partial {\bar{z}}}.$$ Consequently, if we take $X=\frac{\partial\ }{\partial z}$ and $Y=\frac{\partial\ }{\partial {\bar{z}}}$ in the Codazzi equation (\[ecuacioncodazzi\]) we get $$\label{qzbarra}
Q_{{\bar{z}}}=\lambda\,H_z.$$ In addition, from (\[endomorfismoweingarten\]) we obtain that the extrinsic curvature is given by $$\label{modulodeq}
K=H^2-\frac{|Q|^2}{\lambda^2}.$$ Thus, differentiating the equality $W(H,K)=0$ with respect to $z$ $$\begin{aligned}
0&=&H_{z}\,W_x(H,K)+K_{z}\,W_y(H,K)\\
&=&H_{z}\,W_x(H,K)+\left(2H\,H_{z}-|Q|^2\left(\frac{1}{\lambda^2}\right)_{z}-
\frac{Q_{z}\overline{Q}+Q\,\overline{Q}_{z}}{\lambda^2}\right)\,W_y(H,K),\end{aligned}$$ and using (\[qzbarra\]) $$(W_x(H,K)+2H\,W_y(H,K))\,Q_{{\bar{z}}}=\lambda\,W_y(H,K)\,\left(|Q|^2\left(\frac{1}{\lambda^2}\right)_{z}+
\frac{Q_{z}\overline{Q}+Q\,\overline{Q}_{z}}{\lambda^2}\right).$$
Therefore, from (\[hipotesis\]), if $p\in{\Sigma}$ is an umbilical point (i.e. $Q(p)=0$ or equivalently $H^2=K$), there exists a continuous function $h$ in a neighborhood $U$ of $p$ such that $|Q_{{\bar{z}}}|\leq h\,|Q|$ on $U$.
Hence, from [@AdCT Main Lemma] (see also [@Jost Lemma 2.7.1]), either $Q$ vanishes identically on $U$ or $p$ is an isolated zero of negative index of $Q$.
In particular, if ${\Sigma}$ is a topological sphere, from the Poincaré index Theorem we get that the Hopf differential $Q\,dz^2$ must vanish identically on ${\Sigma}$, as we wanted to prove.
The above result can be globally used not only for topological spheres. Indeed, if ${\Sigma}$ is a topological torus under the assumptions of Theorem \[elteorema\], then we deduce that the pair $(I,II)$ is either totally umbilical or umbilically free. Analogously, if ${\Sigma}$ is a closed topological disk and its boundary $\partial{\Sigma}$ is a line of curvature for $(I,II)$, then the pair is totally umbilical.
It is well-known that the hypothesis (\[hipotesis\]) cannot be removed. Examples of this are the non totally umbilical rotational spheres in any space form, since every rotational sphere is a Weingarten surface.
The abstract use of Codazzi pairs allows us to see some classical results, apparently non related, as immediate consequences of Theorem \[elteorema\]. Two good examples are Bonnet Theorem and the uniqueness to the Christoffel problem in $\r^3$, as we show next
\[bonnet\] [**(Abstract Bonnet Theorem)**]{} Let ${\Sigma}$ be a topological sphere and $(I,II_1)$, $(I,II_2)$ two Codazzi pairs with the same Riemanian metric $I$. If both pairs have the same mean curvature, then $II_1=II_2$.
In $\r^3$, this result says that two isometric immersions from a Riemannian sphere in $\r^3$ with the same mean curvature must coincide, up to an isometry of the ambient space.
Since $(I,II_1)$ and $(I,II_2)$ are Codazzi pairs, so is the new pair $(I,II_1-II_2)$. Besides, as $H(I,II_1)=H(I,II_2)$ the mean curvature of $(I,II_1-II_2)$ vanishes identically. In particular, by taking a local conformal parameter $z$, we can put (see (\[parholomorfo\])) $$I=2\lambda\,|dz|^2,\qquad II_1-II_2=Q\,dz^2+\overline{Q}\,d{\bar{z}}^2.$$ Thus, using Theorem \[elteorema\] for the pair $(I,II_1-II_2)$ and the functional $W(H,K)=H=0$, we get that $Q\equiv 0$, which finishes the proof.
For a fundamental pair $(I,II)$ with mean and extrinsic curvatures $H$ and $K$ respectively, the *third fundamental form* is given by $III=-K\,I+2H\,II$ (see, for instance, [@Mi]). In particular, given a surface isometrically immersed in a 3-dimensional manifold with first and second fundamental forms $I$ and $II$ respectively, $III$ is nothing but its classical third fundamental form. In other words, $III={\langle dN,dN \rangle }$ where $N$ is a unit normal vector field on the surface and ${\langle , \rangle }$ is the metric of the ambient space.
\[chris\] Let ${\Sigma}$ be a topological sphere and $(I_i,II_i)$, $i=1,2$, two Codazzi pairs with mean curvature $H_i$ and extrinsic curvature $K_i$. If both pairs have the same third fundamental form with $K_i(p)\neq 0$ for all $p\in{\Sigma}$ and $\frac{H_1}{K_1}=\frac{H_2}{K_2}$, then $(I_1,II_1)=(I_2,II_2)$.
When we particularize this result to the case of two isometric immersions from a Riemannian sphere in $\r^3$ satisfying the assumptions above, we get an easy proof of the uniqueness to the Christoffel problem. Besides this proof is original in the sense that the classical approaches to this problem use integration theory on surfaces (see [@H; @Sp]).
It is known [@Mi] that if $(I_i,II_i)$ is a Codazzi pair with non vanishing extrinsic curvature, then $(III_i,II_i)$ is also a Codazzi pair with mean curvature $\frac{H_i}{K_i}$. Consequently, from Corollary \[bonnet\] we deduce that $II_1=II_2$.
Thus, since $K(III_i,II_i)=\frac{1}{K_i}$ we have that $K_1=K_2$. Finally, using that $I_i=-\frac{1}{K_i}\,III_i+2\frac{H_i}{K_i}\,II_i$, it follows that $I_1=I_2$.
We observe that the previous proof is based in the simple fact that $(III,II)$ is a Codazzi pair. That is, the Bonnet theorem in $\r^3$ and the theorem of uniqueness of the Christoffel problem are a direct consequence of the Abstract Bonnet Theorem, when it is applied to the Codazzi pair $(I,II)$ or the Codazzi pair $(III,II)$, respectively.
In [@Gr] Grove proved that two ovaloids in $\r^3$ with the same second fundamental form and extrinsic curvature are congruent. We give a different proof, generalizing that result to Codazzi pairs. The original proof by Grove involves techniques from integration theory on surfaces.
\[gengr\] Let ${\Sigma}$ be a topological sphere and $(I_i,II)$, $i=1,2$, two Codazzi pairs on $\Sigma$ with the same extrinsic curvature $K>0$. Then $I_1=I_2$.
Since $K>0$, we can assume (changing $II$ by $-II$ if necessary) that $II$ is a Riemannian metric on ${\Sigma}$. Taking a local isothermal parameter $z$ for $II$, we can write $$\begin{aligned}
&I_i=P_i\,dz^2+2\lambda_i\,|dz|^2+\overline{P_i}\,d{\bar{z}}^2&\\
&II=2\rho\,|dz|^2,\end{aligned}$$ with $\rho>0$.
Hence, the mean and extrinsic curvatures of the pair $(I_i,II)$, $i=1,2$ can be written as $$H_i=\frac{\lambda_i\,\rho}{\lambda_i^2-|P_i|^2} \label{HK},\qquad
K=\frac{\rho^2}{\lambda_i^2-|P_i|^2},$$ and the shape operator is given by $$S_i\frac{\partial\ }{\partial z}=\frac{K}{\rho}\,\left(\lambda_i\,\frac{\partial\
}{\partial z}-P_i\,\frac{\partial\ }{\partial {\bar{z}}}\right).$$ Let us denote by $\nabla^i$ the Levi-Civita connection associated to the metric $I_i$ and put $$\nabla^i_{\frac{\partial\ }{\partial z}}\frac{\partial\ }{\partial
z}=\Gamma_{11}^{1,i}\,\frac{\partial\ }{\partial z}+
\Gamma_{11}^{2,i}\,\frac{\partial\ }{\partial {\bar{z}}},\qquad \nabla^i_{\frac{\partial\
}{\partial z}}\frac{\partial\ }{\partial {\bar{z}}}=\Gamma_{12}^{1,i}\,\frac{\partial\
}{\partial z}+ \overline{\Gamma_{12}^{1,i}}\,\frac{\partial\ }{\partial {\bar{z}}}.$$ Since $(I_i,II)$ is a Codazzi pair we have $$\begin{aligned}
0&=&I_i\left(\nabla^i_{\frac{\partial\ }{\partial
z}}S_i{\frac{\partial\ }{\partial {\bar{z}}}},{\frac{\partial\ }{\partial
z}}\right)-I_i\left(\nabla^i_{\frac{\partial\ }{\partial
{\bar{z}}}}S_i{\frac{\partial\ }{\partial z}},{\frac{\partial\ }{\partial z}}\right)\\
&=&\rho_z-I_i\left(S_i{\frac{\partial\ }{\partial
{\bar{z}}}},\nabla^i_{\frac{\partial\ }{\partial z}}\frac{\partial\
}{\partial z}\right)+I_i\left(S_i{\frac{\partial\ }{\partial
z}},\nabla^i_{\frac{\partial\ }{\partial z}}\frac{\partial\
}{\partial {\bar{z}}}\right)\end{aligned}$$ and so $$\label{formula1}
\rho_z=\rho\left(\Gamma_{11}^{1,i}-\overline{\Gamma_{12}^{1,i}}\right).$$
On the other hand, a direct calculation gives $$\left(\lambda_i^2-|P_i|^2\right)_z=2\,\left(\lambda_i^2-|P_i|^2\right)\,\left(\Gamma_{11}^{1,i}+
\overline{\Gamma_{12}^{1,i}}\right).$$
With all of this, we obtain from (\[HK\]) by differentiating $K$ $$\label{formula2}
\frac{K_z}{K}=2\frac{\rho_z}{\rho}-2\left(\Gamma_{11}^{1,i}+
\overline{\Gamma_{12}^{1,i}}\right).$$ Therefore, from (\[formula1\]) and (\[formula2\]), $$\frac{K_z}{K}=-4\,\overline{\Gamma_{12}^{1,i}}=\frac{2K}{\rho^2}\left(P_i\,\overline{P_i}_z-\lambda_i\,{P_i}_{{\bar{z}}}\right),$$ or equivalently, $$\label{f1}
{P_i}_{{\bar{z}}}=-\frac{1}{2K}\left(\lambda_i\,K_z+P_i\,K_{{\bar{z}}}\right).$$
Moreover, from (\[HK\]) we get $$\begin{aligned}
|\lambda_1-\lambda_2|&=&\frac{\rho}{K}\,|H_1-H_2|=\frac{\rho}{K}\,\sqrt{(H_1-H_2)^2} \label{f2} \\
&\leq& \frac{\rho}{K}\,\left|\sqrt{H_1^2-K}-\sqrt{H_2^2-K}\right|=
\left||P_1|-|P_2|\right| \leq |P_1-P_2|. \nonumber\end{aligned}$$
Now, we can derive from (\[f1\]) and (\[f2\]) that $$|{P_1}_{{\bar{z}}}-{P_2}_{{\bar{z}}}|\leq\frac{|K_z|+|K_{{\bar{z}}}|}{2K}\,|P_1-P_2|.$$ Finally, using [@AdCT Main Lemma] or [@Jost Lemma 2.7.1], we conclude that the quadratic form $(P_1-P_2)dz^2$ vanishes identically on the topological sphere ${\Sigma}$. Thus, $P_1\equiv P_2$ and, from (\[HK\]), we get $\lambda_1=\lambda_2$. Or equivalently, $I_1=I_2$.
The Codazzi function {#s3}
====================
Under certain natural conditions, it is possible to obtain important consequences about a surface endowed with a fundamental pair although the Codazzi equation is not satisfied. In order to study these conditions, next we define the Codazzi tensor and the Codazzi function, which will play an essential role in our study.
\[d4\] Given a fundamental pair $(I,II)$ on a surface ${\Sigma}$ with associated shape operator $S$, we will call Codazzi tensor of $(I,II)$ to the map $T_S:{\mathfrak{X}}({\Sigma})\times {\mathfrak{X}}({\Sigma})\fl {\mathfrak{X}}({\Sigma})$ defined by $$T_S(X,Y)=\nabla_XSY-\nabla_YSX-S[X,Y],\qquad X,Y\in {\mathfrak{X}}({\Sigma}).$$
Although the definition above has been made in an abstract context, the Codazzi tensor appears naturally in the study of isometric immersions of surfaces. To be more precise, the Codazzi equation of a surface isometrically immersed in a 3-dimensional manifold $M^3$ is $$\nabla_XSY-\nabla_YSX-S[X,Y]=-\overline{R}(X,Y)N,\qquad X,Y\in{\mathfrak{X}}({\Sigma}),$$ where $N$ is the unit normal vector field of the immersion, $S$ the associated shape operator and $\overline{R}$ the curvature tensor of $M^3$ $$\overline{R}(X,Y)Z=\overline{\nabla}_X\overline{\nabla}_YZ-
\overline{\nabla}_Y\overline{\nabla}_XZ-\overline{\nabla}_{[X,Y]}Z,$$ $\overline{\nabla}$ being the Levi-Civita connection of $M^3$.
A straightforward computation shows that the Codazzi tensor of a fundamental pair on a surface satisfies the following properties:
\[l1\] Let $(I,II)$ be a fundamental pair on a surface ${\Sigma}$ with associated shape operator $S$ and Codazzi tensor $T_S$. Then
1. $T_S$ is skew-symmetric, i.e. $T_S(X,Y)=-T_S(Y,X)$ for all $X,Y\in {\mathfrak{X}}({\Sigma})$.
2. $T_S$ is ${\cal C}^{\infty}(\Sigma)$-bilineal, that is, $$T_S(f_1\,X_1+f_2\,X_2,Y)=f_1\,T_S(X_1,Y)+f_2\,T_S(X_2,Y)$$ for all vector fields $X_1,X_2,Y\in {\mathfrak{X}}({\Sigma})$ and differentiable real functions $f_1,f_2$.
3. Moreover, given vector fields $X,Y\in {\mathfrak{X}}({\Sigma})$ and a differentiable real function $f$ on ${\Sigma}$ $$T_{fS}(X,Y)=f\,T_S(X,Y)+X(f)\,SY-Y(f)\,SX.$$
Associated to the Codazzi tensor of a fundamental pair we define the Codazzi function, thanks to which we will [*measure how distant*]{} the pair is from satisfying the Codazzi equation.
\[d5\] Let $(I,II)$ be a fundamental pair on a surface ${\Sigma}$ with associated shape operator $S$. We will call Codazzi function of $(I,II)$ to the map ${\cal T}_S:{\Sigma}\fl\r$ given by $$\label{funcioncodazzi}
I\left(T_S(v_1,v_2),T_S(v_1,v_2)\right)={\cal T}_S(p)\,
\left(I(v_1,v_1)\,I(v_2,v_2)-I(v_1,v_2)^2\right),$$ where $v_1,v_2\in T_p{\Sigma}$, $p\in{\Sigma}$.
Observe that ${\cal T}_S$ is a well-defined differentiable function since the Codazzi tensor is skew-symmetric. Besides, ${\cal T}_S$ vanishes identically if, and only if, $(I,II)$ is a Codazzi pair.
\[anterior\] Let $(I,II)$ be a fundamental pair on a surface ${\Sigma}$ with associated shape operator $S$, mean curvature $H$ and extrinsic curvature $K$. Let $z$ be a local conformal parameter for $I$ such that $$I=2\lambda\,|dz|^2,\qquad II=Q\,dz^2+2H\,\lambda\,|dz|^2+\overline{Q}\,d{\bar{z}}^2.$$ Then $$|Q_{{\bar{z}}}|^2=\frac{\lambda\,{\cal T}_{\widetilde{S}}}{2(H^2-K)}\,|Q|^2,$$ where $\widetilde{S}$ is the traceless operator $S-H\,Id$, $Id_p$ being the identity map on the tangent plane at $p\in {\Sigma}$.
Since the Levi-Civita connection of $I$ is given by (\[levicivita\]) and the shape operator $S$ by (\[endomorfismoweingarten\]), we have $$\begin{aligned}
T_{\widetilde{S}}\left(\frac{\partial\ }{\partial z},\frac{\partial\ }{\partial
{\bar{z}}}\right)&=&\nabla_{\frac{\partial\ }{\partial z}}\widetilde{S}\frac{\partial\
}{\partial {\bar{z}}}-\nabla_{\frac{\partial\ }{\partial {\bar{z}}}}\widetilde{S}\frac{\partial\
}{\partial z}\ =\ \nabla_{\frac{\partial\ }{\partial
z}}\frac{\overline{Q}}{\lambda}\frac{\partial\ }{\partial z}-\nabla_{\frac{\partial\
}{\partial {\bar{z}}}}\frac{Q}{\lambda}\frac{\partial\ }{\partial {\bar{z}}}\nonumber\\
&=&\frac{1}{\lambda}\left(\overline{Q}_z\,\frac{\partial\ }{\partial z}-
Q_{{\bar{z}}}\,\frac{\partial\ }{\partial {\bar{z}}}\right).\label{primero1}\end{aligned}$$
Hence, using (\[funcioncodazzi\]) one gets $\frac{2}{\lambda}|Q_{{\bar{z}}}|^2={\cal T}_{\widetilde{S}}\,\lambda^2$. The proof finishes using (\[modulodeq\]).
Given a fundamental pair $(I,II)$ on a surface ${\Sigma}$, we will denote by ${\Sigma}_U\subseteq{\Sigma}$ the set of umbilical points of the pair. Then we have
\[thglover\] Let $(I,II)$ be a fundamental pair on a surface ${\Sigma}$ with associated shape operator $S$, mean curvature $H$ and extrinsic curvature $K$. Let us suppose that every point $p\in\partial\Sigma_U$ has a neighborhood $V_p$ such that $$\frac{{\cal T}_{\widetilde{S}}}{H^2-K}\quad \mbox{is bounded in}\,
V_p\cap({\Sigma}-{\Sigma}_U),$$ where $\widetilde{S}=S-H\, Id$. Then either the Hopf differential of $(I,II)$ vanishes identically or its zeroes are isolated and of negative index.
In particular, if ${\Sigma}$ is a topological sphere then the pair is totally umbilical.
If $p\in\partial\Sigma_U$, then there exists an open neighborhood $V_p$ and a constant $m_0$ such that $\frac{{\cal T}_{\widetilde{S}}}{H^2-K}\leq m_0$ in $V_p\cap({\Sigma}-{\Sigma}_U)$. Thus, using Lemma \[anterior\] $$\label{noseyo}
|Q_{{\bar{z}}}|^2\ \leq\ m_0\,\frac{\lambda}{2}\,|Q|^2$$ in $V_p\cap({\Sigma}-{\Sigma}_U)$. Besides, since this inequality is also valid in the interior of $\Sigma_U$, we conclude that (\[noseyo\]) holds in $V_p$.
Therefore, using again [@AdCT] or [@Jost], we have that $p$ is an isolated zero of negative index of the Hopf differential, as we wanted to prove.
To end up, if ${\Sigma}$ is a topological sphere the result follows from the Poincaré index Theorem as in Theorem \[elteorema\].
Again, we point out that if ${\Sigma}$ is a topological torus under the assumptions above, then the pair is either totally umbilical or umbilically free. Analogously, if ${\Sigma}$ is a closed topological disk and $\partial{\Sigma}$ is a line of curvature of $(I,II)$, then the pair is totally umbilical.
If $(I,II)$ is a Codazzi pair we have $$T_{\widetilde{S}}\left(\frac{\partial\ }{\partial z},\frac{\partial\ }{\partial
{\bar{z}}}\right)=-\nabla_{\frac{\partial\ }{\partial z}}H\frac{\partial\
}{\partial {\bar{z}}}+\nabla_{\frac{\partial\ }{\partial {\bar{z}}}}H\frac{\partial\
}{\partial z}=H_{{\bar{z}}}\frac{\partial\
}{\partial z}-H_{z}\frac{\partial\
}{\partial {\bar{z}}}$$ and, therefore, its Codazzi function is $${\cal T}_{\widetilde{S}}\,=\,\frac{2}{\lambda}\,|H_{z}|^2\,=\,\|\nabla H\|^2,$$ where $\|\nabla H\|$ stands for the modulus of the gradient of $H$ for the Riemannian metric $I$. Thus, Theorem \[thglover\] can be applied for Codazzi pairs whenever the quotient $\|\nabla
H\|^2/(H^2-K)$ is bounded.
However, this result can also be applied to fundamental pairs which are not Codazzi pairs, as was implicitly made in [@EGR] in order to classify the complete surfaces with constant extrinsic curvature in the product spaces $\h^2\times\r$ and $\s^2\times\r$.
Holomorphic quadratic differentials. {#s4}
====================================
In this section we will see that, under certain assumptions on a Codazzi pair, it is possible to obtain a new Codazzi pair with vanishing constant mean curvature which is geometrically related to the first one. Thanks to this second Codazzi pair, we will show the existence of a holomorphic quadratic differential which will provide important information on the geometric behavior of the initial pair.
If $(u,v)$ are doubly orthogonal parameters for a fundamental pair $(I,II)$, then we can write $$I=E\,du^2+G\,dv^2,\qquad II=k_1\,E\,du^2+k_2\,G\,dv^2.$$ Hence, the Codazzi tensor acting on the vector fields $\frac{\partial\ }{\partial u},\frac{\partial\ }{\partial v}$ can be expressed as $$\begin{aligned}
T_S\left(\frac{\partial\ }{\partial u},\frac{\partial\ }{\partial
v}\right)&=&\nabla_{\frac{\partial\ }{\partial u}}S\frac{\partial\ }{\partial
v}-\nabla_{\frac{\partial\ }{\partial v}}S\frac{\partial\ }{\partial
u}\,=\,\nabla_{\frac{\partial\ }{\partial u}}k_2\frac{\partial\ }{\partial
v}-\nabla_{\frac{\partial\ }{\partial v}}k_1\frac{\partial\ }{\partial u}\nonumber\\
&=&(k_2)_u\,\frac{\partial\ }{\partial v}-(k_1)_v\,\frac{\partial\ }{\partial
u}+(k_2-k_1)\left(\frac{E_v}{2E}\frac{\partial\ }{\partial
u}+\frac{G_u}{2G}\frac{\partial\ }{\partial v}\right)\label{Tuv}\\
&=&-\frac{1}{E}\left((k_1E)_v-H\,E_v\right)\,\frac{\partial\ }{\partial u}+
\frac{1}{G}\left((k_2G)_u-H\,G_u\right)\,\frac{\partial\ }{\partial v}\nonumber\,,\end{aligned}$$ where $H$ is the mean curvature of $(I,II)$.
We observe that we can take doubly orthogonal parameters in a neighborhood of every non umbilical point as well as in a neighborhood of every point in the interior of ${\Sigma}_U=\{\mbox{umbilical points of }(I,II)\}$. Thus, the set of points where there exist doubly orthogonal parameters is dense in ${\Sigma}$. Consequently, all the properties that we prove by using this kind of parameters, will be extended to the whole surface by continuity.
Throughout this section we will use the new quadratic form $II'$ associated to the fundamental pair $(I,II)$ given by $$II'=II-H\,I.$$
\[lemita\] Let $(I,II)$ be a Codazzi pair on a surface ${\Sigma}$ with mean and extrinsic curvatures $H$ and $K$ respectively. Let $\varphi$ be a differentiable function on ${\Sigma}$ such that the function $\sinh\varphi/\sqrt{H^2-K}$ can be differentiably extended to ${\Sigma}$. Then $$\begin{array}
{l} {\displaystyle A=\cosh\varphi\,I+\frac{\sinh\varphi}{\sqrt{H^2-K}}\,II'}\\[6mm]
{\displaystyle B=\sqrt{H^2-K}\sinh\varphi\,I+\cosh\varphi\,II'},
\end{array}$$ is a fundamental pair with mean curvature $H(A,B)=0$, extrinsic curvature $K(A,B)=-(H^2-K)$ and such that its Codazzi tensor $T_{\widetilde{S}}$ satisfies $$\label{TStilde}
T_{\widetilde{S}}(X,Y)=\omega(Y)\,X-\omega(X)\,Y,\qquad\omega=\frac{1}{2}(dH-\sqrt{H^2-K}d\varphi),$$ for all $X,Y\in{\mathfrak{X}}({\Sigma})$.
Let $(u,v)$ be doubly orthogonal parameters for the Codazzi pair $(I,II)$ such that $$I=E\,du^2+G\,dv^2,\qquad II=k_1\,E\,du^2+k_2\,G\,dv^2,$$ being $k_1\geq k_2$.
Then we can write $(A,B)$ as $$A=e^\varphi\,E\,du^2+e^{-\varphi}\,G\,dv^2,\qquad
B=\frac{k_1-k_2}{2}\,\left(e^\varphi\,E\,du^2-e^{-\varphi}\,G\,dv^2\right).$$
Hence, $A$ is a Riemannian metric on ${\Sigma}$ and the mean and extrinsic curvatures of the pair are given by $H(A,B)=0$ and $K(A,B)=-(H^2-K)$.
In addition, from (\[Tuv\]) and taking into account that $(I,II)$ is a Codazzi pair, we get $$\begin{aligned}
T_{\widetilde{S}}\left(\frac{\partial\ }{\partial u},\frac{\partial\ }{\partial
v}\right)&=&-\frac{1}{e^\varphi\,E}\left(\frac{k_1-k_2}{2}\,\left(e^\varphi\,E\right)\right)_v
\frac{\partial\ }{\partial
u}-\frac{1}{e^{-\varphi}\,G}\left(\frac{k_1-k_2}{2}\,\left(e^{-\varphi}\,G\right)\right)_u
\frac{\partial\ }{\partial v}\\
&=&-\frac{1}{2}\left((k_1)_v-(k_2)_v+(k_1-k_2)\varphi_v-2(k_1)_v)\right)\frac{\partial\
}{\partial u}\\
&&-\frac{1}{2}\left((k_1)_u-(k_2)_u-(k_1-k_2)\varphi_u+2(k_2)_u)\right)\frac{\partial\
}{\partial v}\\
&=&\omega\left(\frac{\partial\ }{\partial v}\right)\frac{\partial\ }{\partial
u}-\omega\left(\frac{\partial\ }{\partial u}\right)\frac{\partial\ }{\partial v}.\end{aligned}$$ Finally, by linearity we obtain (\[TStilde\]).
Under the assumptions of Lemma \[lemita\], if we take a conformal parameter $z$ for $A$ and put $$\label{elnuevopar}
A=2\lambda\,|dz|^2,\qquad B=Q\,dz^2+\overline{Q}\,d{\bar{z}}^2,$$ then from (\[primero1\]) and (\[TStilde\]) we get $$Q_{{\bar{z}}}=\lambda\,\omega\left(\frac{\partial\ }{\partial
z}\right)=\frac{\lambda}{2}\left(H_z-\sqrt{H^2-K}\,\varphi_z\right).$$
Moreover, we obtain from (\[TStilde\]) that the pair $(A,B)$ given by (\[elnuevopar\]) is a Codazzi pair if and only if $dH-\sqrt{H^2-K}d\varphi=0$, or equivalently $Q_{{\bar{z}}}=0$.
With all of this we have
\[corolario3\] Let $(I,II)$ be a Codazzi pair on a surface ${\Sigma}$ with mean and extrinsic curvatures $H$ and $K$ respectively. Let $\varphi$ be a differentiable function on ${\Sigma}$ such that the function $\sinh\varphi/\sqrt{H^2-K}$ can be differentiably extended to ${\Sigma}$. Then the fundamental pair $$\label{AB}
\begin{array}
{l}
{\displaystyle A=\cosh\varphi\,I+\frac{\sinh\varphi}{\sqrt{H^2-K}}\,II'}\\[6mm]
{\displaystyle B=\sqrt{H^2-K}\sinh\varphi\,I+\cosh\varphi\,II'},
\end{array}$$ has mean curvature $H(A,B)=0$ and extrinsic curvature $K(A,B)=-(H^2-K)$. In addition, the following conditions are equivalent:
- $(A,B)$ is a Codazzi pair,
- the Hopf differential of $(A,B)$ is holomorphic for the conformal structure induced by $A$,
- $
dH-\sqrt{H^2-K}d\varphi=0.$
Next we see some situations where the corollary above can be used. In order to do that and following the classical notation, we give the following definition
\[sw\] We say that a Codazzi pair $(I,II)$ is a special Weingarten pair if there exists a differentiable function $f$ defined on an interval ${\cal
J}\subseteq [0,\infty)$ such that its mean curvature $H$ and extrinsic curvature $K$ satisfy $$H=f(H^2-K).$$
Now, let us suppose that the mean and extrinsic curvatures of a Codazzi pair $(I,II)$ satisfy a general Weingarten relationship $W(H,K)=0$. Let us parametrize by taking $H=H(t)$, $K=K(t)$, for $t$ varying in a certain interval. Then, if we look for a solution of the type $\varphi=\varphi(t)$ for the previous Equation $dH-\sqrt{H^2-K}d\varphi=0$, we have $$\sqrt{H(t)^2-K(t)}\,\varphi'(t)=H'(t).$$
Therefore, if there exists a primitive $\varphi(t)$ of the function $$\frac{H'(t)}{\sqrt{H(t)^2-K(t)}}$$ with $\sinh\varphi(t)/\sqrt{H(t)^2-K(t)}$ well-defined even at the umbilical points, then the Codazzi pair $(A,B)$, given as (\[AB\]), will exist on the whole surface $\Sigma$.
In addition, if $(I,II)$ is a special Weingarten pair satisfying $H=f(H^2-K)$, then we can parametrize in the way $H^2-K=t^2$ and $H=f(t^2)$. This allows us to take $$\label{varfi}
\varphi(t)=\int_0^t2f'(s^2)\,ds$$ whenever there exist umbilical points (i.e. $t=0$ has sense), or any primitive of $2f'(t^2)$ otherwise.
Thus, for special Weingarten pairs, the metric $A$ defined as in Corollary \[corolario3\] is always well-defined, because so is the function $\sinh\varphi(t)/t$.
The metric $A$ for special Weingarten surfaces in $\r^3$ and $\h^3$ was first defined by R.L. Bryant in [@Br]. In that work, he also found a holomorphic quadratic form for the metric $A$ which agrees with the Hopf differential of the pair $(A,B)$. Thanks to it, Bryant provided an easy proof of the fact that every topological sphere in $\r^3$ or $\h^3$ satisfying a special Weingarten relationship must be totally umbilical.
The abstract formulation which we have adopted in this work, allows us to extend this result to general special Weingarten surfaces.
Another remarkable fact is that, in our abstract context, we are able to recover every special Weingarten pair as follows
\[corolario4\] Let ${\Sigma}$ be a surface and $f$ a differentiable function defined on an interval ${\cal J}\subseteq [0,\infty)$. Let us take a primitive $\varphi(t)$ of $2f'(t^2)$ on that interval such that the function $\sinh\varphi(t)/t$ is well-defined. Then every special Weingarten pair $(I,II)$ on ${\Sigma}$ satisfying $H=f(H^2-K)$ is given by $$\label{III}
\begin{array}
{l}
{\displaystyle I=-\frac{\sinh\varphi(t)}{t}\,Q+\cosh\varphi(t)\,A-
\frac{\sinh\varphi(t)}{t}\,\overline{Q},}\\[5mm]
{\displaystyle
II-f(t^2)\,I=-\cosh\varphi(t)\,Q+t\,\sinh\varphi(t)\,A-\cosh\varphi(t)\,\overline{Q},
}
\end{array}$$ where $A$ is a Riemannian metric on ${\Sigma}$ and $Q$ a holomorphic 2-form for $A$ such that the image of the function $t:{\Sigma}\fl[0,\infty)$ defined as $2\,|Q|=t\,A$ is contained in ${\cal
J}$. In particular, $t^2=H^2-K$.
It suffices to observe that given a special Weingarten pair $(I,II)$, if we take $H^2-K=t^2$ (and therefore $H=f(t^2)$), we have already proved that there exists $\varphi=\varphi(t)$, primitive of $2f'(t^2)$, in the conditions of Corollary \[corolario3\]. Thus, there exists a pair $(A,B)$ made up of a Riemannian metric $A$ and a quadratic form $B$ which can be written as $B=Q+\overline{Q}$, since $H(A,B)=0$. Besides, the Hopf differential $Q$ of $(A,B)$ is a holomorphic 2-form for the metric $A$ and $$t^2\,=\,H^2-K\,=\,-K(A,B)\,=\,4\,\frac{|Q|^2}{|A|^2}.$$
Summing up, using (\[AB\]) it is possible to recover $(I,II)$ from $(A,B)$ as (\[III\]). Finally, it is a straightforward computation to check that any pair $(A,B)$ as above, gives a Codazzi pair $(I,II)$ which is special Weingarten.
Applications in Space Forms {#s5}
===========================
In this section we focus our attention on surfaces in space forms. We will obtain several results as a consequence of the abstract study developed previously.
We start giving a geometrical argument in order to obtain height bounds for a large amount of families of surfaces which satisfy a maximum principle. The proof is based on some ideas used in [@EGR].
\[d7\] We say that a family ${\cal A}$ of oriented surfaces in $\r^3$ satisfies the Hopf maximum principle if the following properties are satisfied:
1. ${\cal A}$ is invariant under isometries of $\r^3$. In other words, if ${\Sigma}\in{\cal A}$ and $\varphi$ is an isometry of $\r^3$, then $\varphi({\Sigma})\in{\cal
A}$.
2. If ${\Sigma}\in{\cal A}$ and $\widetilde{{\Sigma}}$ is another surface contained in ${\Sigma}$, then $\widetilde{{\Sigma}}\in{\cal A}$.
3. There is an embedded compact surface without boundary in ${\cal A}$.
4. Whichever two surfaces in ${\cal A}$ satisfy the maximum principle (interior and boundary).
Note that a large amount of families of surfaces verify the Hopf maximum principle. Classical examples of this fact are the family of surfaces with constant mean curvature $H\neq 0$ and the family of surfaces with positive constant extrinsic curvature $K$. And, more generally, the family of special Weingarten surfaces in $\r^3$ satisfying a relation of the type $H=f(H^2-K)$, where $f$ is a differentiable function defined on an interval ${\cal J}\subseteq
[0,\infty)$ with $0\in {\cal J}$, such that $f(0)\neq 0$ and $4tf'(t)^2<1$ for all $t\in{\cal J}$ (see [@RoS]).
We also point out that if a family of surfaces ${\cal A}$ satisfy the Hopf maximum principle, then there exists, up to isometries of $\r^3$, a unique embedded compact surface ${\Sigma}$ without boundary in ${\cal A}$. Such surface is, necessarily, a totally umbilical sphere.
To see this, it suffices to observe that the Alexandrov reflection principle works for surfaces in ${\cal A}$. Thus, for every plane $P\subseteq\r^3$ there exists a plane, parallel to $P$, which is a symmetry plane of ${\Sigma}$. Therefore, ${\Sigma}$ is a round sphere.
In addition, there cannot be two totally umbilical spheres ${\Sigma}_1,{\Sigma}_2$ in ${\cal A}$ which are non isometric. Otherwise, up to isometries, we can suppose that one of them, let us say ${\Sigma}_1$, is contained in the bounded region determined by ${\Sigma}_2$. If we move ${\Sigma}_1$ until it meets first ${\Sigma}_2$ and at this contact point the normal vectors to ${\Sigma}_1,{\Sigma}_2$ coincide, we can conclude that ${\Sigma}_1={\Sigma}_2$ from the maximum principle. If the normal vectors at that point do not coincide, we keep on moving ${\Sigma}_1$ until it meets ${\Sigma}_2$ at a last contact point, where necessarily the normal vectors do coincide, which allows us, as before, to assert that ${\Sigma}_1={\Sigma}_2$.
Now, let us see that there exists a constant $c_{\cal A}$ such that for all compact surface ${\Sigma}\in{\cal A}$ whose boundary is contained in a plane $P$, the maximum distance from a point $p\in{\Sigma}$ to $P$ is bounded by $c_{\cal A}$. This bound only depends on the radius of the unique totally umbilical sphere contained in the family ${\cal
A}$.
Although we will not provide optimal estimates here, only the existence of such height estimates respect to planes will allow us to get interesting consequences regarding several aspects of embedded surfaces in $\r^3$ (see [@KKMS; @KKS; @M; @RoS]).
We will start studying graphs $\Sigma$ with boundary contained in a plane $P$ of $\r^3$. Up to an isometry, we can assume that $P$ is the $xy-$plane, and so $$\Sigma = {\left\{ (x,y , u(x,y)) \in \r ^3 : \, \, (x,y)\in \Omega \subseteq \r^2 \right\}}.$$
\[teoremaco\] Let ${\cal A}$ be a family of surfaces in $\r^3$ satisfying the Hopf maximum principle, and $\Sigma\in{\cal A}$ a compact graph on a domain $\Omega$ in the $xy-$plane with $\partial\Sigma$ contained in this plane. Then for all $p\in{\Sigma}$, the distance in $\r ^3$ from $p$ to the $xy-$plane is less or equal to $4R_{\cal A}$. Here, $R_{\cal
A}$ stands for the radius of the unique totally umbilical sphere in the family ${\cal A}$.
Let $\Sigma\in{\cal A}$ be a graph on a domain $\Omega$ in the $xy-$plane and $\Sigma _0$ the unique totally umbilical sphere of ${\cal A}$. Let $P(t)$ be the foliation of $\r^3$ by horizontal planes, $P(t)$ being the plane at height $t$.
Let us see that for every $t > 2R_{\cal A}$, the diameter of any open connected component bounded by $\Sigma (t) = P(t)\cap \Sigma $ is less than or equal to $2R_{\cal A}$.
Indeed, let us suppose that this assertion is not true. Then, for some connected component $C(t)$ of $\Sigma (t)$, there are points $p,q$ in the interior of the domain $\Omega (t)$ in $P(t)$ bounded by $C(t)$ such that $\mbox{dist}(p,q)> 2R_{\cal A}$. Let $Q$ be the domain in $\r^3$ bounded by $\Sigma \cup \Omega $. Let $\beta$ be a curve in $\Omega (t)$ joining $p$ and $q$, $\beta$ and $C(t)$ being disjoint. Let $\Pi$ be the *rectangle* given by $$\Pi = {\left\{\alpha _s (r) \, : \, \, s \in \mathcal{I} , \, r \in [0, t]\right\}}$$ where $\mathcal{I}$ is the interval where $\beta$ is defined, and $\alpha _s$ is the geodesic with initial data $\alpha _s (0) = \beta
(s)$ and $\alpha _s ' (0) = -e_3$, $r$ being the length arc parameter along $\alpha _{s}$ and $e_3=(0,0,1)$.
Since $\Sigma $ is a graph and $\beta $ is contained in the interior of the domain determined by $C(t)$, then $\Pi\subset Q$. Let $\widetilde{p}\in \Pi$ be a point whose distance to $\partial \Pi $ is greater than $R_{\cal A}$. Note that, according to our construction of $\Pi$, the point $\widetilde{p}$ necessarily exists.
Let $\eta(r)$ be a horizontal geodesic passing through $\widetilde{p}$ and such that every point in $\eta(r)$ is far from $\partial\Pi$ a distance greater than $R_{\cal A}$. Observe that such a geodesic can be chosen as the horizontal line in the plane $P(t_1)$ containing the point $\widetilde{p}$ and being orthogonal to the vector joining $p$ and $q$. Let $\widetilde{q_1}$ be the first point where $\eta$ meets $Q$, and $\widetilde{q_2}$ the last one.
Now, let us consider the spheres ${\Sigma}_0(r)\in{\cal A}$ centered at every point $\eta(r)$. Note that these spheres can be obtained from the rotational sphere $\Sigma _0$ by means of a translation of $\r
^3$.
There exists a first sphere in this family (coming from $\widetilde{q_1}$) which meets ${\Sigma}$. If the normal vectors of both surfaces coincide at this point, we conclude that both surfaces agree by the maximum principle. On the other hand, if the normal vectors are opposite, we reason as follows.
Let us consider the first sphere ${\Sigma}_0(r_0)$ in the family above (coming from $\widetilde{q_1}$) which meets $\Pi$ at an interior point of $\Pi$.
For every $r>r_0$ we consider the piece $\widetilde{{\Sigma}}_0(r)$ of the sphere ${\Sigma}_0(r)$ which has gone through $\Pi$. Since these spheres leave $Q$ at $\widetilde{q_2}$ and none of them meets $\partial\Pi$, there exists a first value $r_1$ such that $\widetilde{{\Sigma}}_0(r_1)$ meets first $\partial Q
\cap \Sigma $ at a point $\widetilde{q_0}$. Thus, applying the maximum principle to ${\Sigma}_0(r_1)$ and ${\Sigma}$ at $\widetilde{q}_0$, we conclude that both surfaces agree, which is a contradiction.
Therefore we obtain that, for height $t=2R_{\cal A}$, the diameter of every open connected component bounded by $\Sigma (t) = P(t)\cap
\Sigma $ is less than or equal to $2R_{\cal A}$.
To finish, we will see that $P(t)\cap \Sigma$ is empty for $t>
4R_{\cal A}$. To do that, it suffices to prove the following assertion
> Let $\Omega_1$ be a connected component bounded by $\Sigma (2R_{\cal
> A})$ in $P(2R_{\cal A})$. The distance from any point in $\Sigma$ (which is a graph on $\Omega_1$) to the plane $P(2R_{\cal A})$ is less than or equal to the diameter of $\Omega_1$.
Let $\sigma$ be a support line of $\partial\Omega_1$ in $P(2R_{\cal
A})$ with exterior unitary normal vector $v$, and let us take $\eta(r)$ a geodesic such that $\eta(0)\in\sigma$ and $\eta'(0)=\frac{1}{\sqrt{2}}(v+e_3)$.
Now, let us consider for every $r$ the plane $\Pi(r)$ in $\r^3$ passing through $\eta(r)$ which is orthogonal to $\eta'(r)=\eta'(0)$. Such planes intersect every horizontal plane in a line parallel to $\sigma$, being $\pi/2$ the angle between them.
If the assertion above was not true, there would exist a point $p\in{\Sigma}$ over $\Omega_1$ such that its height on the plane $P(2R_{\cal A})$ would be greater than the diameter of $\Omega_1$.
Let ${\Sigma}_1$ be the compact piece of ${\Sigma}$ which is a graph on $\Omega_1$. Observe that, for $r$ big enough, $\Pi(r)$ does not meet ${\Sigma}_1$. In addition, for $r=0$ the plane $\Pi(0)$ contains the line $\sigma$, and the reflection of $p$ with respect to $\Pi(0)$ is a point whose vertical projection on $P(2R_{\cal A})$ is not in $\Omega_1$. Therefore, using the Alexandrov reflection principle for the planes $\Pi(r)$ with $r$ coming from infinity, there exists a first value $r_0>0$ such that either the reflection of the piece of ${\Sigma}_1$ which is over $\Pi(r)$ meets first ${\Sigma}_1$ at an interior point or both surfaces are tangent at a point in the boundary. But this is a contradiction, by the maximum principle.
This finishes the proof.
As a consequence of this result, we are able to bound the maximum distance attained by an embedded compact surface whose boundary is contained in a plane.
\[corolariaco\] Let ${\cal A}$ be a family of surfaces in $\r^3$ satisfying the Hopf maximum principle. Then every embedded compact surface $\Sigma\in{\cal A}$ whose boundary is contained in a plane $P$ verifies that for every $p\in{\Sigma}$ the distance in $\r ^3 $ from $p$ to the plane $P$ is less than or equal to $8R_{\cal A}$. Here, $R_{\cal A}$ denotes the radius of the unique totally umbilical sphere contained in ${\cal A}$.
This result follows from Theorem \[teoremaco\] as a standard consequence of the Alexandrov reflection principle for planes parallel to $P$.
The techniques used in Theorem \[teoremaco\] and Corollary \[corolariaco\] are valid not only in $\r^3$, but also more generally for hypersurfaces in $\r ^n$. Even more, they can easily be adapted to study hypersurfaces in $\h ^n$.
The existence of a maximum principle and height estimates with respect to planes for a family of surfaces ${\cal A}$, allow us to extend the theory developed by Korevaar, Kusner, Meeks and Solomon [@KKMS; @KKS; @M] for constant mean curvature surfaces in $\r^3$ and $\h^3$ to our family ${\cal A}$. On the other hand, in [@RoS] Rosenberg and Sa Earp showed that those techniques are also suitable to study some families of surfaces satisfying a relationship of the type $H=f(H^2-K)$. However, they do not use that the surfaces satisfy $H=f(H^2-K)$ actually, but only that they satisfy the Hopf maximum principle and there exist height estimates for them. Thus, following [@RoS] we get
\[cilindrico\] Let ${\cal A}$ be a family of surfaces in $\r^3$ satisfying the Hopf maximum principle. Let us take $\Sigma\in{\cal A}$ an annulus, i.e. ${\Sigma}$ homeomorphic to a punctured closed disc of $\r^2$. If ${\Sigma}$ is properly embedded, then it is contained in a half-cylinder of $\r^3$.
A unitary vector $v\in\s^2$ is said to be an axial vector for ${\Sigma}\subseteq\r^3$ if there exists a sequence of points $p_n\in{\Sigma}$ such that $|p_n|\rightarrow\infty$ and $p_n/|p_n|\rightarrow v$. In particular, the theorem above asserts that for any properly embedded annulus there exists a unique axial vector. In addition, this vector is the generator of the rulings of the cylinder.
Finally, following [@RoS] for properly embedded complete surfaces, we have
\[finales\] Let ${\cal A}$ be a family of surfaces in $\r^3$ satisfying the Hopf maximum principle. If $\Sigma\in{\cal A}$ is a properly embedded surface with finite topology in $\r ^3$, then every end of ${\Sigma}$ is cylindrically bounded. Moreover, if $a_1 , \ldots , a_k$ are the $k$ axial vectors corresponding to the ends, then these vectors cannot be contained in an open hemisphere of $\s ^2$. In particular,
- $k =1$ is impossible.
- If $k =2$, then $\Sigma$ is contained in a cylinder and is a rotational surface with respect to a line parallel to the axis of the cylinder.
- If $k=3$, then $\Sigma$ is contained in a slab.
\[d8\] Let $(I,II)$ be a Codazzi pair on a surface $\Sigma$. We will say that the pair is special Weingarten of elliptic type if its mean and extrinsic curvatures $H$ and $K$ satisfy that $H=f(H^2-K)$, where $f$ is a differentiable function defined on $[0,a)$, $0<a\leq\infty$, such that $$4tf'(t)^2<1$$ for all $t\in[0,a)$.
It was proved by Rosenberg and Sa Earp [@RoS] (see also [@BSE]) that the set of Weingarten surfaces of elliptic type in $\r^3$ and $\h^3$ with $f(0)\neq 0$ is a family satisfying the Hopf maximum principle. Thus, the above theorems are true for this kind of surfaces. Actually these results were also proved in [@RoS], although under the additional hypothesis $f'(t)(1-2f(t)f'(t))\geq 0$.
The special Weingarten surfaces in $\r^3$ and $\h^3$ satisfying $H=f(H^2-K)$ have been widely studied. In particular, an exhaustive study of the rotational surfaces was developed by Sa Earp and Toubiana [@ST5; @ST2; @ST3; @ST4].
In [@ST3] was posed the question of classifying the surfaces satisfying $H=f(H^2-K)$ whose extrinsic curvature does not change signs. More specifically, it is asked if such surfaces are totally umbilical spheres, cylinders or surfaces of minimal type (i.e. with $f(0)=0$). Observe that this fact is known for surfaces with constant mean curvature. In fact, a minimal surface has non-positive extrinsic curvature at every point and a complete surface with non zero constant mean curvature and whose extrinsic curvature does not change signs, must be a sphere or a cylinder [@Hof; @KO].
Next, and as a consequence of the study developed for Codazzi pairs, we study that problem for the general case of special Weingarten surfaces of elliptic type.
\[otroth\] Let $\Sigma$ be a special Weingarten surface of elliptic type in $\r^3$ satisfying that $H=f(H^2-K)$. Let us suppose that its extrinsic curvature does not change signs:
1. If $\Sigma$ is complete and $K\geq 0$ at every point, then $\Sigma$ is a totally umbilical sphere, a plane or a right circular cylinder.
2. If $\Sigma$ is properly embedded and $K\leq 0$ at every point, then $\Sigma$ is either a right circular cylinder or a surface of minimal type (i.e. $f(0)=0$).
In order to prove this theorem, we will first establish the following general Lemma for Codazzi pairs.
\[lematecnico\] Let $(I,II)$ be a special Weingarten pair of elliptic type on a surface $\Sigma$, with mean and extrinsic curvatures $H$ and $K$ respectively. If $H^2-K\neq0$ on $\Sigma$, then the new metric $$g_0=\sqrt{H^2-K}\ A$$ is a flat metric on $\Sigma$. Here, $A$ is the metric given by (\[AB\]) for the function $\varphi$ defined in (\[varfi\]).
Moreover, if $I$ is complete and $H^2-K\geq c_0>0$ then the metric $g_0$ is complete. In particular, $\Sigma$ with the conformal structure given by $A$ (or by $g_0$) is the complex plane, the once punctured complex plane or a torus.
From Corollary \[corolario4\] we get that $2|Q|=t\,A$, where $t=\sqrt{H^2-K}$ and $Q$ is a holomorphic quadratic form for $A$. Thus, since $H^2-K>0$, $g_0=2|Q|$ is a well-defined flat metric on $\Sigma$.
Let us see that $g_0$ is complete if $I$ is complete. In such a case $g_0$ would be a complete flat metric and, so, the universal Riemannian covering of $\Sigma$ for the metric $g_0$ would be the Euclidean plane. Hence, $\Sigma$ would be conformally equivalent to the complex plane, to the once punctured complex plane or to a torus.
Observe that, from (\[III\]), we get that $$\label{ladesigualdad}
I\leq 2\cosh\varphi(t)\,A.$$ On the other hand, since $(I,II)$ is a special Weingarten pair of elliptic type it follows that $4\,s^2\,f'(s^2)^2<1$ and so, from (\[varfi\]), $$s^2\,\varphi'(s)^2<1,\quad {\rm or\ equivalently}\quad
-\frac{1}{s}<\varphi'(s)<\frac{1}{s}.$$ Hence, by integrating between a fixed point $s_0>0$ and $s$ one gets that there exists a constant $c_1>0$ such that $|\varphi(s)|\leq
|\log s|+c_1$. Therefore, since $$\lim_{s\rightarrow\infty}\frac{\cosh\log(s)}{s}=\frac{1}{2}$$ and $t\geq\sqrt{c_0}$, we deduce the existence of a constant $c_2>0$ such that $\cosh\varphi(t)\leq c_2\,t$.
Finally, from (\[ladesigualdad\]) it follows that $$I\,\leq \,2c_2\,t\,A\,=\,2c_2\,g_0,$$ that is, $g_0$ is complete.
[*Proof of Theorem \[otroth\]:*]{} Firstly, let us suppose that $\Sigma$ is a complete surface in $\r^3$ with $K\geq0$.
If $K$ vanishes identically, then it is easy to conclude that $\Sigma$ is either a plane or a right circular cylinder (see [@ST5]).
If there exists a point where the extrinsic curvature is positive, then either $\Sigma$ is homeomorphic to a sphere or it is properly embedded and homeomorphic to the plane [@Wu]. In addition, we have $f(0)\neq 0$ (see [@ST5]).
In the first case $\Sigma$ must be a totally umbilical sphere from Theorem \[elteorema\]. The second case is not possible from Theorem \[finales\] applied to our family of special Weingarten surfaces with $H=f(H^2-K)$.
Now, let us suppose that $\Sigma$ is properly embedded and $K\leq0$. Then we have that $$0\geq K=H^2-(H^2-K)=f(H^2-K)^2-(H^2-K).$$ Hence, if $f(0)\neq0$, since the function $f(s)^2-s$ is continuous for $s\geq0$ and takes a positive value at $s=0$, then there exists $s_0>0$ such that $f(s)^2-s>0$ for $s\in[0,s_0]$. Consequently $H^2-K\geq s_0>0$ on $\Sigma$ since $K\leq 0$.
From Lemma \[lematecnico\], $\Sigma$ is homeomorphic to the plane, to the once punctured plane or to a torus. Using once again Theorem \[finales\], $\Sigma$ cannot be homeomorphic to a plane. In addition, every compact surface in $\r^3$ must have a point with positive extrinsic curvature, and so $\Sigma$ cannot be homeomorphic to a torus. With all of this, $\Sigma$ must be homeomorphic to the once punctured plane and so, from Theorem \[finales\], it must be a rotational surface and must be contained in a cylinder $C$ of $\r^3$.
To finish, let us see that $\Sigma$ is a right circular cylinder. In fact, up to an isometry of $\r^3$, we can suppose that $\Sigma$ is a rotational surface with respect to the $z$-axis. Let us denote by $$\alpha=\Sigma\cap\{(x,y,z)\in\r^3:\ x>0,\ y=0\}$$ a generatrix curve of $\Sigma$. It is clear that $\alpha$ is a line of curvature of $\Sigma$ and its signed-curvature on the plane $y=0$ changes signs if and only if $K$ changes signs.
Since $K\leq 0$, the sign of the curvature of $\alpha$ on the plane $y=0$ does not change, and so $\alpha$ is a convex curve. But, since $\alpha$ is contained in the strip determined by the $z$-axis and the line parallel to $C\cap\{(x,y,z)\in\r^3:\ x>0,\ y=0\}$, we conclude that $\alpha$ must be a line parallel to the $z$-axis, as we wanted to prove.
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The first author is partially supported by Junta de Comunidades de Castilla-La Mancha, Grant No PCI-08-0023. The second and third authors are partially supported by Grupo de Excelencia P06-FQM-01642 Junta de Andalucía. The authors are partially supported by MCYT-FEDER, Grant No MTM2007-65249
| 0 |
---
abstract: |
The Parker or field line tangling model of coronal heating is studied comprehensively via long-time high-resolution simulations of the dynamics of a coronal loop in cartesian geometry within the framework of reduced magnetohydrodynamics (RMHD). Slow photospheric motions induce a Poynting flux which saturates by driving an anisotropic turbulent cascade dominated by magnetic energy. In physical space this corresponds to a magnetic topology where magnetic field lines are barely entangled, nevertheless current sheets (corresponding to the original tangential discontinuities hypothesized by Parker) are continuously formed and dissipated.
Current sheets are the result of the nonlinear cascade that transfers energy from the scale of convective motions ($\sim 1,000\, km$) down to the dissipative scales, where it is finally converted to heat and/or particle acceleration. Current sheets constitute the dissipative structure of the system, and the associated magnetic reconnection gives rise to impulsive “bursty” heating events at the small scales. This picture is consistent with the slender loops observed by state-of-the-art (E)UV and X-ray imagers which, although apparently quiescent, shine bright in these wavelengths with little evidence of entangled features.
The different regimes of weak and strong MHD turbulence that develop, and their influence on coronal heating scalings, are shown to depend on the loop parameters, and this dependence is quantitatively characterized: weak turbulence regimes and steeper spectra occur in [*stronger loop fields*]{} and lead to [*larger heating rates*]{} than in weak field regions.
author:
- 'A.F. Rappazzo, M. Velli'
- 'G. Einaudi'
- 'R.B. Dahlburg'
title: Nonlinear Dynamics of the Parker Scenario for Coronal Heating
---
INTRODUCTION
============
In a previous letter [@rap07] we described simulations, within the framework of RMHD in cartesian geometry, aimed at solving the Parker field-line tangling (coronal heating) problem [@park72; @park88]. We also developed a phenomenological model for nonlinear interactions, taking into account the inertial photospheric line-tying effect, which explained how the average coronal heating rate would depend on the only free parameter present in the simulations, namely the ratio of the coronal loop Alfvén crossing time and the photospheric eddy turnover time. This paper is devoted to a more detailed discussion of the numerical simulations and of the relationship between this work, the original Parker conjecture, and the nanoflare scenario of coronal heating.
Parker’s book [@park94] is devoted to an examination of the basic theorem of magnetostatics, namely that the lowest available energy state of a magnetic field in an infinitely conducting fluid contains surfaces of tangential discontinuity, or current sheets. It is Parker’s conjecture that the continuous footpoint displacement of coronal magnetic field lines must lead to the development of such discontinuities as the field continuously tries to relax to its equilibrium state, and it is the dynamical interplay of energy accumulation via footpoint motion and the bursty dissipation in the forming current sheets which gives rise to the phenomenon of the high temperature solar corona, heated by the individual bursts of reconnection, or nanoflares.
What then does turbulence have to do with the nanoflare heating scenario? Parker himself strongly criticizes the use of the “t” word, the formation of the current sheets being due in his opinion to the “requirement for ultimate static balance of the Maxwell stresses”. But what better way is there to describe the nonlinear global dynamics of a magnetically dominated plasma in which the formation of an equilibrium state containing current sheets is the inevitable asymptotic state (once the photospheric driver is turned off)?
The striving of the global magnetic field toward a state containing current sheets must occur through local violations of the force-free condition, the induction of local flows, the collapse of the currents into ever thinner layers: a nonlinear process generating ever smaller scales. From the spectral point of view, a power law distribution of energy as a function of scale is expected, even though the kinetic energy is much smaller than the magnetic energy. The last two statements are clear indications that the word turbulence provides a correct description of the dynamical process.
A final important issue is whether the overall dissipated power tends to a finite value as the resistivity and viscosity of the coronal plasma become arbitrarily small. That this must be the case is easy to understand (see § \[sec:eod\]). For suppose that for an arbitrary, continuous, foot-point displacement the coronal field were only to map the foot-point motion, and that there were no non-linear interactions, i.e. the Lorentz force and convective derivatives were negligible everywhere. In this case, the magnetic field and the currents in the corona would then grow linearly in time, until the coronal dissipation at the scale of photospheric motions balanced the forcing. The amplitudes of the coronal fields and currents would then be inversely proportional to resistivity (eqs. (\[eq:bdiff\])-(\[eq:jdiff\])), and the dissipated power, product of resistivity and square of the current, would also scale as the inverse power of the resistivity (eq. (\[eq:hrdiff\])). In other words, the smaller the resistivity in the corona, the higher the power dissipated would be. But the amplitudes can not become arbitrarily large, because non-linear effects intervene to stop the increase in field amplitudes, increasing the effective dissipation at a given resistivity. Since the power can not continue to increase monotonically as the resistivity is decreased, it is clear that at some point non-linear interactions must limit the dissipated power to a finite value, regardless of the value of the resistivity. Finite dissipation at arbitrarily small values of dissipative coefficients is another definition of a turbulent system.
All this assuming that a statistically stationary state may be reached in a finite time, a question closely related to the presence of finite time singularities in 3D magnetohydrodynamics. It now appears that magnetic field relaxation in an unforced situation does not lead to the development of infinitely thin current sheets in a finite time, but rather the current development appears to be only exponential in time [@GRM00]. In forced numerical simulations, as the ones we will describe in detail here, this is a moot point: for all intents and purposes a statistically stationary state is achieved at a finite time independent of resistivity for sufficiently high resolution. In fact, even if the growth is exponential, we can estimate that the width of the current sheets reaches the meter-scale in a few tens Alfvén crossing times $\tau_{\mathcal{A}}$. A typical value is $\tau_{\mathcal{A}} = 40\, s$, so that this initial time is not only finite, but also short compared with an active region timescale. Once the steady state has been reached this phenomenon is no longer important. The nonlinear regime is in fact characterized by the presence of numerous current sheets, so that while some of them are being dissipated others are being formed, and a [*statistical*]{} steady state is maintained.
It therefore seems that the Parker field-line tangling scenario of coronal heating may be described as a particular instance of magnetically dominated MHD turbulence. Numerous analytical and numerical models of this process have been presented in the past, each discussing in some detail aspects of the general problem as presented above [@park72; @park88; @HP92; @vb86; @ber91; @StUch81; @GomFF92; @mik89; @hen96; @long94; @dmi99; @ein96; @georg98; @dmi98; @ein99].
The numerical simulations presented here bring closure to the [*original*]{} question as posed in cartesian geometry by Parker, starting from a uniform axial magnetic field straddling from one boundary plane to another, subject to continuous independent footpoint motions in either photosphere. This does not imply that we have fully solved the coronal heating problem as due to footpoint dragging by the photospheric velocity field.
A number of relevant effects have been neglected: first, the field line expansion between the photosphere and corona, which, if the photospheric flux is confined to bundles in granular and supergranular network lanes, would allow the mapping of the photospheric velocity field to the coronal volume to contain discontinuities. We are presently carrying out a dedicated set of simulations to capture this effect. Second, the projection of the 3D photospheric velocity to 2D coronal base motions parallel to the photosphere also introduces compressibility in the forcing flow, again neglected here. Third, we have considered stationary photospheric flows. The effect of a finite eddy-turnover time in the flow was considered in [@ein96; @georg98] in 2 dimensions, and in the “3 dimensional” shell model calculations of [@buc07]. These showed that time-dependence does not change things substantially provided the flow pattern does not contain degenerate symmetries, a fact confirmed by shorter simulations we defer to a future paper. Finally, we do not address the more realistic case of a single photosphere with curved coronal loops, such as the simulations presented recently by @gud05. While this approach has advantages when investigating the coronal loop dynamics within its coronal neighborhood, modeling a larger part of the solar corona numerically drastically reduces the number of points occupied by the coronal loops. At the moment the very low resolution attainable with this kind of simulations does not allow the development of turbulence with a well-developed inertial range. The transfer of energy from the scale of convection cells $\sim 1000\, km$ toward smaller scales is inhibited, because the smaller scales are not resolved (their linear resolution is $\sim 500\, km$). Thus, these simulations have not been able to shed light on the detailed coronal statistical response nor on the different regimes which may develop and how they depend on the coronal magnetic field crossing time and the photospheric eddy turnover time.
In § \[sec:model\] we introduce the coronal loop model, whose properties are qualitatively analyzed in § \[sec:an\]. The results of our simulations are described in § \[sec:ns\], and their turbulence properties are analyze in more detail in § \[sec:wts\]. Finally in § \[sec:disc\] we summarize and discuss our results.
PHYSICAL MODEL {#sec:model}
==============
A coronal loop is a closed magnetic structure threaded by a strong axial field, with the footpoints rooted in the photosphere. This makes it a strongly anisotropic system, as measured by the relative magnitude of the Alfvén velocity associated with the axial magnetic field $v_{\mathcal A} \sim 2000\ \textrm{km}\, \textrm{s}^{-1}$ compared to the typical photospheric velocity $u_{ph} \sim 1\ \textrm{km}\, \textrm{s}^{-1}$.
We study the loop dynamics in a simplified Cartesian geometry, neglecting field line curvature, i.e. the toroidality of loops. Our loop is a “straightened out” box, with an orthogonal square cross section of size $\ell$ (along which the x-y directions lie), and an axial length $L$ (along the z direction) embedded in an axial homogeneous uniform magnetic field $\mathbf{B}_0 = B_0\ \mathbf{e}_z$. This simplified geometry allows us to perform simulations with both high numerical resolution and long-time duration.
In § \[sec:ge\] we introduce the equations used to model the dynamics, while in § \[sec:bc\] we give the boundary and initial conditions used in our numerical simulations.
Governing Equations {#sec:ge}
-------------------
The dynamics of a plasma embedded in a strong axial magnetic field are well described by the equations of reduced MHD (RMHD) [@kp74; @stra76; @mont82].
These equations are valid for a plasma with small ratio of kinetic to magnetic pressures, in the limit of a large loop-aspect ratio ($\epsilon = l/L \ll 1$, $L$ being the length of the loop and $l$ being the minor radius of the loop) and of a small ratio of poloidal to axial magnetic field ($b_\perp /B_0 \le \epsilon$). In dimensionless form they can be written as: $$\begin{gathered}
\frac{\partial \boldsymbol{u}_{\perp}}{\partial t}
+ \left( \boldsymbol{u}_{\perp} \cdot \boldsymbol{\nabla}_{\perp} \right) \boldsymbol{u}_{\perp}
= - \boldsymbol{\nabla}_{\perp} \left( p + \frac{\boldsymbol{b}_{\perp}^2}{2} \right) \\
+ \left( \boldsymbol{b}_{\perp} \cdot \boldsymbol{\nabla}_{\perp} \right) \boldsymbol{b_{\perp}}
+ c_{\mathcal{A}} \, \frac{\partial \boldsymbol{b}_{\perp}}{\partial z}
+ \frac{(-1)^{n+1}}{Re_n} \boldsymbol{\nabla}^{2n}_{\perp} \boldsymbol{u}_{\perp}, \label{eq:adim1}\end{gathered}$$ $$\begin{gathered}
\frac{\partial \boldsymbol{b}_{\perp}}{\partial t} = \left( \boldsymbol{b}_{\perp} \cdot \boldsymbol{\nabla}_{\perp} \right) \boldsymbol{u_{\perp}} - \left( \boldsymbol{u}_{\perp} \cdot \boldsymbol{\nabla}_{\perp} \right) \boldsymbol{b _{\perp}} \\
+ c_{\mathcal{A}} \, \frac{\partial \boldsymbol{u}_{\perp}}{\partial z}
+ \frac{(-1)^{n+1}}{Re_n} \boldsymbol{\nabla}^{2n}_{\perp}
\boldsymbol{b}_{\perp}, \label{eq:adim2}\end{gathered}$$ $$\boldsymbol{\nabla}_{\perp} \cdot \boldsymbol{u}_{\perp} = 0, \qquad
\boldsymbol{\nabla}_{\perp} \cdot \boldsymbol{b}_{\perp} = 0, \qquad \qquad \qquad \quad
\label{eq:adim3}$$ where $\mathbf{u}_{\perp}$ and $\mathbf{b}_{\perp}$ are the components of the velocity and magnetic fields perpendicular to the mean field, and $p$ is the kinetic pressure. The gradient operator likewise has only components in the $x$-$y$ plane perpendicular to the axial direction $z$, i.e.$$\mathbf{\nabla}_{\perp} = \mathbf{e}_{x}\, \frac{\partial}{\partial x} +
\mathbf{e}_{y}\, \frac{\partial}{\partial y},$$ while the dynamics in the planes is coupled to the axial direction through the linear terms $\propto \partial_z$.
To render the equation non dimensional magnetic fields have first been expressed in velocity units by dividing by $\sqrt{4\pi \rho_0}$ (where $\rho_0$ is a density supposed homogeneous and constant), i.e. considering the associated Alfvén velocities ($b \rightarrow b/\sqrt{4\pi \rho_0}$), and then both velocity and magnetic fields have been normalized to a typical photospheric velocity $u_{ph}$; lengths and times have been expressed in units of the perpendicular length of the computational box $\ell$ and its related “eddy turnover time” $t_{\perp} = \ell / u_{ph}$.
As a result, in equations (\[eq:adim1\])-(\[eq:adim2\]), the linear terms $\propto \partial_z$ are multiplied by the *dimensionless* Alfvén velocity $c_{\mathcal{A}} = v_{\mathcal A} / u_{ph} $, i.e the ratio between the Alfvén velocity associated with the axial magnetic field $v_{\mathcal A} = B_0 / \sqrt{4 \pi \rho_0}$, and the photospheric velocity $u_{ph}$.
The index $n$ is called *dissipativity*: the diffusive terms adopted in equations (\[eq:adim1\])-(\[eq:adim2\]) correspond to ordinary diffusion for $n=1$ and to so-called hyperdiffusion for $n > 1$. When $n=1$ the $\mathbf{\nabla}_{\perp}^2 / Re$ diffusive operator is recovered, so that $Re_1 = Re = Re_m$ corresponds to the kinetic and magnetic Reynolds number (considered of equal and uniform value): $$\label{r1}
Re = \frac{\rho_0\, \ell u_{ph}}{\nu}, \qquad
Re_m = \frac{4\pi \rho_0 \, \ell u_{ph} }{\eta c^2},$$ where viscosity $\nu$ and resistivity $\eta$ are taken to be constant and uniform ($c$ is the speed of light).
We have performed numerical simulations with both $n=1$ and $n=4$. Hyperdiffusion is used because with a limited resolution the diffusive timescales associated with ordinary diffusion are small enough to affect the large scale dynamics and render very difficult the resolution of an inertial range, even with a grid with 512x512 points in the x-y plane (the highest resolution grid we used for the plane). The diffusive time $\tau_n$ at the scale $\lambda$ associated with the dissipative terms used in (\[eq:adim1\])-(\[eq:adim2\]) is given by: $$\label{eq:taud}
\tau_n \sim Re_n\, \lambda^{2n}$$ While for $n=1$ the diffusive time decreases relatively slowly towards smaller scales, for $n=4$ it decreases far more rapidly. This allows to have longer diffusive timescales at large spatial scales and similar diffusive timescales at the resolution scale. Numerically we require that the diffusion time at the resolution scale $\lambda_{min} = 1 / N$, where N is the number of grid points, to be of the same order of magnitude for both normal and hyperdiffusion, i.e.$$\frac{Re_1}{N^2} \sim \frac{Re_n}{N^{2n}} \quad \longrightarrow \quad Re_n \sim Re_1\, N^{ 2(n-1) }$$ For instance a numerical grid with $N=512$ points which requires a Reynolds number $Re_1 = 800$ with ordinary diffusion, can implement $Re_4 \sim 10^{19}$, removing diffusive effects at the large scales, and allowing (if present) the resolution of an inertial range.
The numerical integration of the RMHD equations (\[eq:adim1\])-(\[eq:adim3\]) is substantially simplified by using the potentials of the velocity ($\varphi$) and magnetic field ($\psi$), $$\label{eq:potfi}
\mathbf{u}_\perp = \mathbf{\nabla} \times
\left( \varphi\, \mathbf{e}_z \right), \qquad
\mathbf{b}_\perp = \mathbf{\nabla} \times
\left( \psi\, \mathbf{e}_z \right),$$ linked to vorticity and current by $\omega =- \mathbf{\nabla}^2_\perp \varphi$ and $j = - \mathbf{\nabla}^2_\perp \psi$.
We solve numerically equations (\[eq:adim1\])-(\[eq:adim3\]) written in terms of the potentials (see [@rap07]) in Fourier space, i.e. we advance the Fourier components in the $x$-$y$ directions of the scalar potentials $\varphi$ and $\psi$. Along the $z$ direction no Fourier transform is performed so that we can impose non-periodic boundary conditions (specified in § \[sec:bc\]), and a central second-order finite difference scheme is used. In the $x$-$y$ plane a Fourier pseudospectral method is implemented. Time is discretized with a third-order Runge-Kutta method.
We use a computational box with an aspect ratio of 10, which spans $$0 \le x, y \le 1, \qquad 0 \le z \le 10.$$
Boundary and Initial Conditions {#sec:bc}
-------------------------------
As boundary conditions at the photospheric surfaces ($z=0,\ L$) we impose two independent velocity patterns, intended to mimic photospheric motions, made up of large spatial scale projected convection cell flow patterns constant in time. The velocity potential at each boundary is given by: $$\begin{gathered}
\label{eq:bc}
\varphi (x,y) = \frac{1}{\sqrt{ \sum_{mn} \alpha_{mn}^2}} \ \sum_{k,l}
\frac{\ell\, \alpha_{kl}}{2\pi \sqrt{k^2+l^2}} \ \\
\sin \left[ \frac{2 \pi}{\ell} \left( kx+ly \right) + 2\pi \, \xi_{kl} \right]\end{gathered}$$ We excite all the wave number values $(k,l) \in \mathbb{Z}^2$ included in the range $3 \le \left( k^2 + l^2 \right)^{1/2} \le 4$, so that the resulting average injection wavenumber is $k_{c} \sim 3.4$, and the average injection scale $\ell_{c}$, the convection cell scale, is given by $\ell_c = \ell / k_{c}$. $\alpha_{kl}$ and $\xi_{kl}$ are two sets of random numbers whose values range between $0$ and $1$, and are independently chosen for the two boundary surfaces. The normalization adopted in eq. (\[eq:bc\]) sets the value of the corresponding velocity rms (see eq. (\[eq:potfi\])) to $1/\sqrt{2}$, i.e. $$\label{eq:rmscon}
\int_0^{\ell} \! \! \! \int_0^{\ell} \mathrm{d}x\, \mathrm{d}y \
\left( u_x^2 + u_y^2 \right) = \frac{1}{2}$$
At time $t=0$ no perturbation is imposed inside the computational box, i.e. $\mathbf{b}_{\perp} = \mathbf{u}_{\perp} = 0$, and only the axial magnetic field $B_0$ is present: the subsequent dynamics are then the effect of the photospheric forcing (\[eq:bc\]) on the system, as described in the following sections.
ANALYSIS {#sec:an}
========
In order to clarify aspects of the linear and nonlinear properties of the RMHD system, we provide an equivalent form of the equations (\[eq:adim1\])-(\[eq:adim3\]). In terms of the Elsässer variables $\mathbf{z}^{\pm} = \mathbf{u}_{\perp} \pm \mathbf{b}_{\perp}$, which bring out the basic symmetry of the equations in terms of parallel and anti-parallel propagating Alfvén waves, they can be written as $$\begin{gathered}
\frac{\partial \boldsymbol z^+}{\partial t} =
- \left( \boldsymbol z^- \cdot \boldsymbol{\nabla}_{\perp} \right) \boldsymbol{z}^+
+ c_\mathcal{A} \, \frac{\partial \boldsymbol z^+}{\partial z} \\
+ \frac{(-1)^{n+1}}{Re_n} \boldsymbol{\nabla}^{2n}_{\perp} \boldsymbol z^+
- \boldsymbol{\nabla}_{\perp} P, \label{eq:els1}\end{gathered}$$ $$\begin{gathered}
\frac{\partial \boldsymbol z^-}{\partial t} =
- \left( \boldsymbol z^+ \cdot \boldsymbol{\nabla}_{\perp} \right) \boldsymbol{z}^-
- c_\mathcal{A} \, \frac{\partial \boldsymbol z^-}{\partial z} \\
+ \frac{(-1)^{n+1}}{Re_n} \boldsymbol{\nabla}^{2n}_{\perp} \boldsymbol z^-
- \boldsymbol{\nabla}_{\perp} P, \label{eq:els2}\end{gathered}$$ $$\boldsymbol{\nabla}_{\perp} \cdot \boldsymbol{z}^{\pm} = 0, \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \ \ \, \label{eq:els3}$$ where $P = p + \mathbf{b}_{\perp}^2/2 $ is the total pressure, and is linked to the nonlinear terms by incompressibility (\[eq:els3\]): $$\label{eq:els4}
\mathbf{\nabla}_{\perp}^2 P =
- \sum_{i,j=1}^2 \Big( \partial_i z_j^- \Big) \Big( \partial_j z_i^+ \Big).$$
In terms of the Elsässer variables $\mathbf{z}^{\pm} = \mathbf{u}_{\perp} \pm \mathbf{b}_{\perp}$, a velocity pattern $\mathbf{u}_{\perp}^{0,L}$ at upper or lower boundary surface becomes the constraint $\mathbf{z}^{+} + \mathbf{z}^{-} = 2 \mathbf{u}_{\perp}^{0,L}$ at that boundary. Since, in terms of characteristics (which in this case are simply $\mathbf{z}^{\pm}$ themselves), we can specify only the incoming wave (while the outgoing wave is determined by the dynamics inside the computational box), this velocity pattern implies a reflecting condition at the top ($z=L$) and bottom ($z=0$) planes: $$\label{eq:bc0}
\mathbf{z^{-}} = - \mathbf{z^{+}} + 2 \, \mathbf{u}^{0}_{\perp}
\quad \textrm{at} \ z=0,$$ $$\label{eq:bcL}
\mathbf{z^{+}} = - \mathbf{z^{-}} + 2 \, \mathbf{u}^{L}_{\perp}
\quad \textrm{at} \ z=L.$$
The linear terms ($\propto \partial_z$) in equations (\[eq:els1\])-(\[eq:els2\]) give rise to two distinct wave equations for the $\mathbf{z}^{\pm}$ fields, which describe Alfvén waves propagating along the axial direction $z$. This wave propagation, which is present during both the linear and nonlinear regimes, is responsible for the continuous energy influx on large perpendicular scales (see eq. (\[eq:bc\])) from the boundaries into the loop. The nonlinear terms $\left( \mathbf z^{\mp} \cdot \mathbf{\nabla}_{\perp} \right) \mathbf{z}^{\pm}$ are then responsible for the transport of this energy from the large scales toward the small scales, where energy is finally dissipated, i.e. converted to heat and/or particle acceleration.
A well-known important feature of the nonlinear terms in equations (\[eq:els1\])-(\[eq:els3\]) is the absence of self-coupling, i.e. only counterpropagating waves interact non-linearly, and if one of the two fields $\mathbf{z}^{\pm}$ is zero, there are no nonlinear interactions at all. This fact, i.e. that counter-propagating wave-packets may interact only while they are crossing each other, lies at the basis of the so-called Alfvén effect [@iro64; @kra65], which ultimately renders the nonlinear timescales longer and slows down the dynamics.
From this description three different timescales arise naturally: $\tau_{\mathcal A}$, $\tau_{ph}$ and $\tau_{nl}$. $\tau_{\mathcal A} = L/v_{\mathcal A}$ is the crossing time of the Alfvén waves along the axial direction $z$, i.e.the time it takes for an Alfvén wave to cover the loop length $L$. $\tau_{ph} \sim 5~m$ is the characteristic time associated with photospheric motions, while $\tau_{nl}$ is the nonlinear timescale.
For a typical coronal loop $\tau_{\mathcal A} \ll \tau_{ph}$. For instance for a coronal loop long $L = 40,000~km$ and with an Alfvén velocity $v_{\mathcal A} = 2,000~km\, s^{-1}$ we obtain $\tau_{\mathcal A} = 20~s$, which is small compared to $\tau_{ph} \sim 5~m = 300~s$. This is the reason we carried out simulations with a photospheric forcing constant in time (see eq. (\[eq:bc\])), i.e. for which formally $\tau_{ph} = \infty$.
In the RMHD ordering the nonlinear timescale $\tau_{nl}$ is bigger than the Alfvén crossing time $\tau_{\mathcal A}$. As we shall see this ordering is maintained during our simulations and we will give analytical estimates of the value of $\tau_{nl}$ as a function of the characteristic parameters of the system.
An important feature of equations (\[eq:els1\])-(\[eq:els3\]) that we will use to generalize our results is that, apart from the Reynolds numbers, there is only one fundamental non-dimensional parameter: $$\label{eq:fsc}
f = \frac{\ell_c\, v_{\mathcal A}}{L\, u_{ph}}.$$ Hence all the physical quantities which result from the dynamical evolution, e.g. energy, Poynting flux, heating rate, timescales, etc., must depend on this single parameter $f$.
Energy Equation {#sec:ee}
---------------
From equations (\[eq:adim1\])-(\[eq:adim3\]), with $n=1$, and considering the Reynolds numbers equal, the following energy equation can be derived: $$\label{eq:poy}
\frac{\partial }{\partial t} \left( \frac{1}{2} \mathbf u_{\perp}^2
+ \frac{1}{2} \mathbf b_{\perp}^2 \right)
= - \mathbf{\nabla} \cdot \mathbf{S}
- \frac{1}{Re} \left( \mathbf{ j }^2 + \mathbf{ \omega }^2 \right),$$ where $\mathbf{S} = \mathbf{B} \times (\mathbf{u} \times \mathbf{B})$ is the Poynting vector. As expected the energy balance of the system described by eq. (\[eq:poy\]) is due to the competition between the energy (Poynting) flux flowing into the computational box and the ohmic and viscous dissipation. Integrating eq (\[eq:poy\]) over the whole box the only relevant component of the Poynting vector is the component along the axial direction $z$, because in the $x$-$y$ plane periodic boundary conditions are used and their contribution to the Poynting flux is null. As $\mathbf{B} = c_{\mathcal{A}}\, \mathbf{e}_z + \mathbf{b}_{\perp}$ and $\mathbf{u} = \mathbf{u}_{\perp}$, this is given by $$\label{eq:sz}
S_z = \mathbf{S} \cdot \mathbf{e}_z = - c_{\mathcal{A}}
\left( \mathbf{u}_{\perp} \cdot \mathbf{b}_{\perp} \right).$$ Considering that the velocity fields at the photospheric boundaries are given by $\mathbf{u}_{\perp}^0$ and $\mathbf{u}_{\perp}^L$, for the integrated energy flux we obtain $$\label{eq:tsz}
S =
c_{\mathcal A} \int_{z=L} \! \mathrm{d} a\, \left( \mathbf{u}_{\perp}^L
\cdot \mathbf{b}_{\perp} \right)
- c_{\mathcal A} \int_{z=0} \! \mathrm{d} a\, \left( \mathbf{u}_{\perp}^0
\cdot \mathbf{b}_{\perp} \right).$$ The injected energy flux therefore depends not only on the photospheric forcing and the axial Alfvén velocity (which have fixed values), but also on the value of the magnetic fields at the boundaries, which is determined by the dynamics of the system inside the computational box: *the injection of energy depends on the nonlinear dynamics which develops, and viceversa*.
The simplified topology investigated in this paper, i.e. a strong axial magnetic field whose footpoints are dragged by 2D orthogonal motions applies to regions where emerging flux may be neglected. Consider the axial component of the velocity $u_z$ field carrying new magnetic field ($\mathbf{b}_{\perp}^{ef}$) into the corona. The associated Poynting flux is $$\label{eq:ef}
S_z^{ef} = \left( \mathbf{b}_{\perp}^{ef} \right)^2 u_z.$$ This flux component is negligible when $S_z^{ef} < S_z$, i.e., since all the components of the photospheric velocity fields are of the same order, $u_z \sim u_{ph}$, when
$$\label{eq:efc}
\left( b_{\perp}^{ef} \right)^2 < B_0\, b_{\perp}^{turb}.$$
In § \[sec:sca\] we give an estimate of the value of the field $b_{\perp}^{turb}$ generated by the field-line dragging, and will be able to quantify for which value of $b_{\perp}^{ef}$ the emerging flux can be neglected.
Linear Stage {#sec:ls}
------------
For $t < \tau_{nl}$ nonlinear terms can be neglected. Neglecting also the diffusion terms, which play no role on large scales, equations (\[eq:adim1\])-(\[eq:adim3\]) reduce to two simple wave equations. Coupled with the boundary conditions (\[eq:bc\]) the solution for times longer than the crossing time $\tau_{\mathcal{A}}$ reads:[ $$\begin{aligned}
& &\mathbf{b}_{\perp} (x,y,z,t) =
\left[ \mathbf{u}^{L} (x,y) - \mathbf{u}^{0} (x,y) \right]\
\frac{t}{\tau_{\mathcal A}}, \label{eq:blin}\\
& &\mathbf{u}_{\perp} (x,y,z,t) = \mathbf{u}^L (x,y) \frac{z}{L}
+ \mathbf{u}^0 (x,y) \left( 1 - \frac{z}{L} \right). \label{eq:vlin}\end{aligned}$$ ]{} This shows that A) the loop velocity field is bounded by the imposed photospheric fields and b) the magnetic field grows linearly in time, uniform along the loop, while *mapping* the photospheric velocity field in the perpendicular planes. Therefore, for a generic set of velocities $\mathbf{u}^L$ and $\mathbf{u}^0$, the resulting magnetic fields (\[eq:blin\])-(\[eq:vlin\]) give rise to non-vanishing forces in the perpendicular planes which grow quadratically in time, becoming dynamically important after a certain interval [@buc07].
There exists a (singular) set of velocity forcing patterns, for which the generated coronal field has a vanishing Lorentz force. For simplicity consider $\mathbf{u}^L = 0$: in terms of potentials it follows that $\psi = -\varphi^0 \, t/\tau_{\mathcal A}$ and $\varphi = \varphi^0 \, (1-z/L)$ (where $\mathbf{u}^0_\perp = \mathbf{\nabla} \times
\left( \varphi^0\, \mathbf{e}_z \right)$). In this case both $\mathbf{b}_{\perp}$ and $\mathbf{u}_{\perp}$ are proportional to $\mathbf{\nabla}_{\perp} \times ( \varphi^0\, \mathbf{e}_z )$. The condition for the vanishing of nonlinear terms then becomes $$\mathbf{\nabla} \left( \mathbf{\nabla}^2 \, \varphi^0 \right) \times
\mathbf{\nabla} \varphi^0 = 0, \quad \textrm{with} \quad \varphi^0 = \varphi^0 \left(x, y\right).$$ This condition is then satisfied by those fields for which the laplacian is constant along the streamlines of the field. As $\omega = - \mathbf{\nabla}^2 \, \varphi$ this is equivalent to the statement that the *vorticity is constant along the streamlines*. This condition is in general not verified, unless very symmetric functions are chosen, e.g. in cartesian geometry by any 1D function like $\varphi^0 = f(x)$, and in polar coordinates by any radial function $\varphi^0 = g(r)$.
Generally speaking even in such peculiar configurations non-linear interactions will arise due to the onset of instabilities. We defer discussion of these extreme examples to a subsequent paper, the random photospheric fields (\[eq:bc\]) discussed here always giving rise to non-vanishing forces .
Inserting the linear evolution fields (\[eq:blin\]) in the expression for the integrated energy flux (\[eq:tsz\]), we find $$\label{eq:slin}
S = c_{\mathcal A} \int\! \mathrm{d} a\, | \mathbf{u}^L - \mathbf{u}^0 |^2
\cdot \frac{t}{\tau_{\mathcal A}},$$ i.e. the Poynting flux $S$ grows linearly in time until such a time that non-linear interactions set in.
A similar linear analysis was already performed by [@park88], who noted that if this is the mechanism responsible for coronal heating, then the energy flux $S_z \sim S / \ell^2$ must approach the value $S_z \sim 10^7\, erg\, cm^2\, s^{-1}$ necessary to sustain an active region before a saturating mechanism, magnetic reconnection of singular current sheets in Parker’s language, takes over.
In fact however the value reached by $S_z$ depends on the nonlinear dynamics, its value *self-consistently* determined by solving the *nonlinear problem*. An $S_z$ too small compared with observational constraints would then rule out the Parker model.
Effects of Diffusion {#sec:eod}
--------------------
The linear solution (\[eq:blin\])-(\[eq:vlin\]) has been obtained without taking into account the diffusive terms. This is justified, given the large value of the Reynolds numbers for the solar corona. But numerically it can be important. At very low resolution diffusion is so important that little or no nonlinear dynamics develop and the system reaches a balance between the photospheric forcing and diffusion of the large scale fields.
![Streamlines of the velocity field $\mathbf{u}_{\perp}^0$, the boundary forcing at the *bottom* plane $z=0$ for run A. In lighter vortices the velocity field is directed anti-clockwise while in darker vortices it is directed clockwise. The cross-section shown in the figure is roughly $4000 \times 4000\, km^2$, where the typical scale of a convective cell is $1000\, km$.\
\[fig:v0\]](f1.eps){width="47.00000%"}
One can attempt to bypass the non-linear problem by adopting a much smaller “turbulent” value of the Reynolds number [@HP92]. For this “ad hoc” value of the Reynolds number the average dissipation would be the same as in the high Reynolds number active turbulence limit. Linearizing equation (\[eq:adim2\]) (with $n=1$ and $Re_1 = Re$), we obtain $$\label{eq:lind2}
\frac{\partial \mathbf{b}_{\perp} }{\partial t}
= c_\mathcal{A} \frac{\partial \mathbf{u}_{\perp}}{\partial z}
+ \frac{1}{Re} \mathbf{ \nabla}^2_{\perp} \mathbf{b}_{\perp}.$$ Taking into account that the forcing velocities are dominated by components at the injection scale $\ell_{c}$ (see eq. (\[eq:bc\])), the relation $\mathbf{\nabla}^2_{\perp} \varphi = - \left( 2 \pi / \ell_{c} \right)^2
\varphi$, where $\ell_{c} = \ell / k_{c}$ with the average wavenumber $k_{c} \sim 3.4$, is approximately valid. Integrating then eq. (\[eq:lind2\]) over $z$ and dividing by the length $L$, we obtain for $\mathbf{b}_{\perp}$ averaged along $z$: $$\label{eq:eqdiff}
\frac{\partial \mathbf{b}_{\perp} }{\partial t}
= \frac{c_\mathcal{A}}{L}
\left[ \mathbf{u}^L \left( x, y \right) - \mathbf{u}^0 \left( x, y \right) \right]
- \frac{\left( 2 \pi \right)^2}{\ell_{c}^2 Re} \mathbf{b}_{\perp}.$$ Indicating with $\mathbf{u}_{ph} = \mathbf{u}^L - \mathbf{u}^0$, with $\tau_{\mathcal{R}} = {\ell_{c}^2 Re}/ \left( 2 \pi \right)^2$ the diffusive time-scale and with $\tau_{\mathcal A} = L / c_{\mathcal A}$ the Alfvén crossing time, the solution is given by: $$\label{eq:bdiff}
\boldsymbol{b}_{\perp} \left( x, y, t \right) =
\boldsymbol{u}_{ph} \left( x, y \right)
\frac{\tau_{\mathcal R}}{\tau_{\mathcal A}}
\left[ 1 - \exp \left( - \frac{t}{\tau_{\mathcal R}} \right) \right],$$ $$\begin{gathered}
\label{eq:jdiff}
\left| j \left( x, y, t \right) \right| =
\left| \boldsymbol{u}_{ph} \left( x, y \right) \right|
\left( \frac{2 \pi}{\ell_{c}} \right)
\frac{\tau_{\mathcal R}}{\tau_{\mathcal A}} \times \\
\times \left[ 1 - \exp \left( - \frac{t}{\tau_{\mathcal R}} \right) \right].\end{gathered}$$ So that the magnetic energy $E_M$
![Streamlines of the velocity field $\mathbf{u}_{\perp}^L$, the boundary forcing at the *top* plane $z=L$ for run A. The numerical grid has 512x512 points in the x-y planes, with a linear resolution of $\sim 8\, km$.\
\[fig:vL\]](f2.eps){width="47.00000%"}
and the ohmic dissipation rate $J$ are given by $$\begin{gathered}
\label{eq:emdiff}
E_M = \frac{1}{2} \int_V \mathrm{d}^3 \boldsymbol{x} \ \boldsymbol{b}_{\perp}^2 = \\
= \frac{1}{2} \ell^2\, L\, u^2_{ph}\,
\left( \frac{\tau_{\mathcal R}}{\tau_{\mathcal A}} \right)^2
\left[ 1 - \exp \left( - \frac{t}{\tau_{\mathcal R}} \right) \right]^2,\end{gathered}$$ $$\begin{gathered}
\label{eq:hrdiff}
J = \frac{1}{Re} \int_V \mathrm{d}^3 \boldsymbol{x} \ j^2 = \\
= \ell^2\, L\, u^2_{ph}\,
\frac{\tau_{\mathcal R}}{\tau_{\mathcal A}^2}
\left[ 1 - \exp \left( - \frac{t}{\tau_{\mathcal R}} \right) \right]^2,\end{gathered}$$ where $u_{ph}$ is the rms of $\mathbf{u}_{ph}$, and with the rms of the boundary velocities $\mathbf{u}^0$ and $\mathbf{u}^L$ fixed to $1/2$ (\[eq:rmscon\]) we have $u_{ph} \sim 1$. Both total magnetic energy (\[eq:emdiff\]) and ohmic dissipation (\[eq:hrdiff\]) grow quadratically in time for time smaller than the resistive time $\tau_{\mathcal R}$, while on the diffusive time scale they saturate to the values $$\label{eq:satdiff}
E_M^{sat} = \frac{\ell^6\, c_{\mathcal{A}}^2\, u^2_{ph}\, Re^2}{
L\, \left( 2\pi k_{c} \right)^4}, \qquad
J^{sat} = \frac{\ell^4\, c_{\mathcal{A}}^2\, u^2_{ph}\, Re}{
L\, \left( 2\pi k_{c} \right)^2},$$ written explicitly in terms of the loop parameters and Reynolds number.
Magnetic energy saturates to a value proportional to the square of both the Reynolds number and the Alfvén velocity, while the heating rate saturates to a value that is proportional to the Reynolds number and the square of the axial Alfvén velocity. We have also used equations (\[eq:emdiff\])-(\[eq:hrdiff\]) as a check in our numerical simulations, and during the linear stage, before nonlinearity sets in they are well satisfied.
From equation (\[eq:emdiff\])-(\[eq:hrdiff\]) we can estimate the saturation time as the time at which the functions (\[eq:emdiff\])-(\[eq:hrdiff\]) reach 2/3 of the saturation values. It is approximately given by $$\label{eq:tdiff}
\tau^{sat} \sim 2\, \tau_{\mathcal{R}} = \frac{2\,\ell^2 Re}{\left( 2\pi k_{c} \right)^2}$$
In the next section we describe the results of our simulations, which investigate the linear and nonlinear dynamics. The average values may be used in conjunction with \[eq:satdiff\] to define the equivalent turbulence Reynolds number.
NUMERICAL SIMULATIONS {#sec:ns}
=====================
Run $c_{\mathcal A}$ $n_x \times n_y \times n_z$ n $Re$, $Re_4$ $t_{max}/\tau_{\mathcal A}$
----- ------------------ ----------------------------- --- ----------------- -----------------------------
A 200 $512 \times 512 \times 200$ 1 $8\cdot10^{2}$ 548
B 200 $256 \times 256 \times 100$ 1 $4\cdot10^{2}$ 1061
C 200 $128 \times 128 \times 100$ 1 $2\cdot10^{2}$ 2172
D 200 $128 \times 128 \times 100$ 1 $1\cdot10^{2}$ 658
E 200 $128 \times 128 \times 100$ 1 $1\cdot10^{1}$ 1272
F 50 $512 \times 512 \times 200$ 4 $3\cdot10^{20}$ 196
G 200 $512 \times 512 \times 200$ 4 $10^{19}$ 453
H 400 $512 \times 512 \times 200$ 4 $10^{20}$ 77
I 1000 $512 \times 512 \times 200$ 4 $10^{19}$ 502
: Summary of the simulations. $c_{\mathcal A}$ is the axial Alfvén velocity and $n_x \times n_y \times n_z$ is number of points for the numerical grid. n is the *dissipativity*, $n=1$ indicates normal diffusion, $n=4$ hyperdiffusion. $Re$ ($=Re_1$) or $Re_4$ indicates respectively the value of the Reynolds number or of the hyperdiffusion coefficient (see eq.(\[eq:els1\])-(\[eq:els2\])). The duration of the simulation $t_{max}/\tau_{\mathcal A}$ is given in Alfvén crossing time unit $\tau_{\mathcal A} = L / v_{\mathcal A}$. \[tab:r\]
In this section we present a series of numerical simulations, summarized in Table \[tab:r\], modeling a coronal layer driven by a forcing velocity pattern *constant in time*. On the bottom and top planes we impose two independent velocity forcings as described in § \[sec:bc\], which result from the linear combination of large-scale eddies with random amplitudes, normalized so that the rms of the photospheric velocity is $u_{ph} \sim 1\, km\, s^{-1}$. For each simulation a different set of random amplitudes is chosen, corresponding to different patterns of the forcing velocities. A realization of this forcing with a specific choice (run A) of the random amplitudes is shown in Figures \[fig:v0\]-\[fig:vL\].
The length of a coronal section is taken as the unitary length. As we excite all the wavenumbers between 3 and 4, and the typical convection cell scale is $\sim 1,000\, km$, this implies that each side of our section is roughly $4,000\, km$ long. Our typical grid for the cross-sections has 512x512 grid points, corresponding to $\sim 128^2$ points per convective cell, and hence a linear resolution of $\sim 8~km$.
Between the top and bottom plates a uniform magnetic field $\mathbf{B} = B_0\, \mathbf{e}_z$ is present. The subsequent evolution is due to the shuffling of the footpoints of the magnetic field lines by the photospheric forcing.
In the different numerical simulations, keeping fixed the cross-section length ($\sim 4,000\, km$) and axial length ($\sim 40,000\, km$), we explore the behavior of the system for different values of $c_{\mathcal A}$, i.e. the ratio between the Alfvén velocity associated with the axial magnetic field and the rms of the photospheric motions (density is supposed uniform and constant).
Nevertheless, as shown in (\[eq:fsc\]) the fundamental parameter is $f = \ell_c v_{\mathcal{A}} / L u_{ph}$, so that changing $c_{\mathcal A} = v_{\mathcal{A}} / u_{ph}$ is equivalent to explore the behaviour of the system for different values of $f$, where the same value of $f$ can be realized with a different choice of the quantities, provided that the RMHD approximation is valid, i.e. we are describing a slender loop threaded by a strong magnetic field.
We also perform simulations with different numerical resolutions, i.e. different Reynolds numbers, and both normal ($n=1$) and hyper-diffusion ($n=4$).
The qualitative behaviour of the system is the same for all the simulations performed. In the next section we describe these qualitative features in detail for run A, and then describe the quantitative differences found in the other simulations.
Run A {#sec:runA}
-----
In this section we present the results of a simulation performed with a numerical grid with 512x512x200 points, normal ($n=1$) diffusion with a Reynolds number $Re=800$, and the Alfvén velocity $v_{\mathcal A} = 200\, km\, s^{-1}$ corresponding to a ratio $c_{\mathcal A} = v_{\mathcal A}/u_{ph} = 200$. The streamlines of the forcing velocities applied in the top ($z=L$) and bottom ($z=0$) planes are shown in Figures \[fig:v0\]-\[fig:vL\]. The total duration is roughly 550 axial Alfvén crossing times ($\tau_{\mathcal A} = L / v_{\mathcal A}$).
Plots of the total magnetic and kinetic energies $$\label{eq:em}
E_M = \frac{1}{2} \int\! \mathrm{d} V\, \mathbf{b}_{\perp}^2,
\qquad
E_K = \frac{1}{2} \int\! \mathrm{d} V\, \mathbf{u}_{\perp}^2,$$ and of the total ohmic and viscous dissipation rates $$\label{eq:tdiss}
J = \frac{1}{Re} \int\! \mathrm{d} V\, \mathbf{j}^2,
\qquad
\Omega = \frac{1}{Re} \int\! \mathrm{d} V\, \mathbf{\omega}^2,$$ along with the incoming energy rate (integrated Poynting flux) $S$ (see eq. (\[eq:tsz\])), are shown in Figures \[fig:fig3\]-\[fig:diss\]. At the beginning the system has a linear behavior (see eqs. (\[eq:blin\])-(\[eq:vlin\]), and (\[eq:slin\])), characterized by a linear growth in time for the magnetic energy, the Poynting flux and the electric current, which implies a quadratic growth for the ohmic dissipation $\propto (t/\tau_{\mathcal A})^2$, until time $t \sim 6\, \tau_{\mathcal A}$, when nonlinearity sets in. We can identify this time as the nonlinear timescale, i.e. $\tau_{nl} \sim 6\, \tau_{\mathcal A}$. The timescales of the system will be analyzed in more details in §\[sec:tmsc\].
After this time, in the fully nonlinear stage, a *statistically steady state* is reached, in which the Poynting flux, i.e. the energy that is entering the system for unitary time, balances on time average the total dissipation rate ($J+\Omega$). As a result there is no average accumulation of energy in the box, beyond what has been accumulated during the linear stage, while a detailed examination of the dissipation time series (see inset in Figure \[fig:diss\]) shows that the Poynting flux and total dissipations are decorrelated around dissipation peaks.
![*Run A*: High-resolution simulation with $v_{\mathcal A}/u_{ph} = 200$, 512x512x200 grid points and $Re=800$. Magnetic ($E_M$) and kinetic ($E_K$) energies as a function of time ($\tau_{\mathcal A}=L / v_{\mathcal A}$ is the axial Alfvén crossing time).\
\[fig:fig3\]](f3.eps){width="47.00000%"}
In the diffusive case from eqs. (\[eq:emdiff\])-(\[eq:tdiff\]), with the values of this simulation we would obtain $\tau^{sat} \sim 50\, \tau_{\mathcal A}$, $E_M^{sat} \sim 6100$ and $J^{sat} \sim 7100$; all values well beyond those of the simulation. A value of $Re=85$ would fit the simulated average dissipation, while $Re=140$ would approximately fit the average magnetic energy. In any case this would only fit the curves, but *the physical phenomena would be completely different*, as we describe in the following sections.
*An important characteristic of the system is the magnetic predominance for both energy and dissipation* (Figures \[fig:fig3\] and \[fig:diss\]). In the linear stage (§ \[sec:ls\]) while the magnetic field grows linearly in time, the velocity field does not, and its value is roughly the sum of the boundary forcing fields. The physical interpretation is that because we are bending the axial magnetic field with a constant forcing, as a result the perpendicular magnetic field grows linearly in time, while the velocity remains limited. More formally this is a consequence of the fact that, while on the perpendicular magnetic field no boundary condition is imposed, the velocity field must approach the imposed boundary values at the photosphere both during the linear and nonlinear stages.
In Figure \[fig:avz\] the 2D averages in the x-y planes of the magnetic and velocity fields and of the ohmic dissipation $j^2/Re$, are plotted as a function of z at different times. These macroscopic quantities are smooth and present almost no structure along the axial direction. The reason is that every disturbance or gradient along the axial direction, at least considering the large perpendicular scales (for the small scales behavior see § \[sec:wts\] ), is smoothed out by the fast propagation of Alfvén waves along the axial direction, their propagation time $\tau_{\mathcal A}$ is in fact the fastest timescale present (in particular $\tau_{\mathcal A} < \tau_{nl}$), and then the system tends to be homogeneous along this direction.
The predominance of the ohmic over the viscous dissipation is due to the fact that, as we show in the next sections, the dissipative structures are current sheets, where magnetic reconnection takes place.
![*Run A*: The integrated Poynting flux $S$ dynamically balances the Ohmic ($J$) and viscous ($\Omega$) dissipation. Inset shows a magnification of total dissipation and $S$ for $150 \le t/\tau_{\mathcal{A}} \le 250$.\
\[fig:diss\]](f4.eps){width="47.00000%"}
The phenomenology described in this section is general and we have found it in all the simulations that we have performed, in particular we have always found that in the nonlinear stage a statistical steady state is reached where energies fluctuate around a mean value and total dissipation and Poynting flux on the average balance while on small timescales decorrelate. In particular, to check the temporal stability of these features, which are fully confirmed, we have performed a numerical simulation (run C) with the same parameters as run A, but with a lower resolution (128x128x100), a Reynolds number $Re = 200$ and a longer duration ($t \sim 2,000\, \tau_{\mathcal A}$). On the opposite the average levels of the energies and of total dissipation depend on the parameters used as we describe in the next sections.
Before describing these features, in the next section we describe the current sheets formation, their temporal evolution and other properties.
### Current Sheets, Magnetic Reconnection, Global Magnetic Field Topology and Self-Organization {#sec:csf}
The nonlinear stage is characterized by the presence of current sheets elongated along the axial direction (Figures (18a)-(18b)), which exhibit temporal dynamics and are the dissipative structures of the system. We now show that they are the result of a nonlinear cascade. Figure \[fig:emmod\] shows the time evolution of the first 11 modes of magnetic energy for the first 20 crossing times $\tau_{\mathcal A}$ for run A. During the linear stage the magnetic field is given by eq. (\[eq:blin\]) and is the mapping of the difference between the top ($z=10$) and bottom ($z=0$) photospheric velocities $\mathbf{u}^{L} (x,y) - \mathbf{u}^{0} (x,y)$, whose streamlines are shown in Figure 17a. The field lines of the orthogonal magnetic field in the midplane ($z=5$) at time $t=0.63\, \tau_{\mathcal A}$ are shown in Figure 17b, and as expected they map the velocity field. The same figure shows in colour the axial current $j$. As shown by eq. (\[eq:blin\]) (taking the curl) the large scale motions that we have imposed at the photosphere induce large scale currents in all the volume and, as described in the previous section, if there was not a nonlinear dynamics a balance between diffusion and forcing would be reached, where no small scale would be formed and the magnetic field would always map the photospheric velocities.
![*Run A*: 2D averages in the x-y planes of the ohmic dissipation $j^2/Re$, the magnetic and velocity fields $\mathbf{b}_{\perp}^2$, and $\mathbf{u}_{\perp}^2$, as a function of z. The different colours represent 10 different times separated by $\Delta t = 50\, \tau_{\mathcal A}$ in the interval $30\, \tau_{\mathcal A} \le t \le 480\, \tau_{\mathcal A}$.\
\[fig:avz\]](f5rgb.eps){width="47.00000%"}
As time proceeds the magnetic field grows and a cascade transfers energy from the large scales, where the photospheric forcing (\[eq:bc\]) injects energy at the wavenumbers $n=3$ and $4$, to the small scales (Figure \[fig:emmod\]). In physical space this cascade corresponds to the collapse of the large scale currents which lead to the formation of current sheets, as shown in Figures 17c and 17d. In Figures 17e and 17f we show the magnetic field lines at time $t = 18.47\, \tau_{\mathcal A}$, in the fully nonlinear stage, with respectively the axial component of the current $j$ and of the vorticity $\omega$. The resulting magnetic topology is quiet complex, X and Y-points are not in fact easily distinguished. They are distorted and very often a component of the magnetic field orthogonal to the current sheet length is present, so that the sites of reconnection are more easily identified by the corresponding vorticity quadrupoles. As shown in Figures 17e and 17f, the more or less distorted current sheets are always embedded in quadrupolar structures for the vorticity, a characteristic maintained throughout the whole simulation, and a clear indication that *nonlinear magnetic reconnection* is taking place.
Figures 18a and 18b show a view from the side and the top of the 3D current sheets at time $t = 18.47\, \tau_{\mathcal A}$. When looked from the side the current sheets, which are elongated along the axial direction, look space filling, but the view from the top shows that the filling factor is actually small (see also Figure 17).
![*Run A*: First 11 magnetic energy modes as a function of time covering the first 20 Alfvén crossing times $\tau_{\mathcal{A}}$. Photospheric motions inject energy at $n=3$ and $4$.\
\[fig:emmod\]](f6rgb.eps){width="47.00000%"}
Another aspect of the dynamics is *self-organization*: while until time $t = 4.79\, \tau_{\mathcal A}$ the magnetic field lines are still approximately a mapping of the photospheric velocities, in the fully nonlinear stage they *depart* from it and have an independent topology that evolves dynamically in time (see the associated movie for the time evolution covering $40$ crossing times from $\sim 508\, \tau_{\mathcal{A}}$ up to $\sim 548\, \tau_{\mathcal{A}}$; notations and simulation are the same used in Figure 17). The reason for which the photospheric forcing does not determine the spatial shape of the magnetic field lines is due to the bigger value of the rms of the magnetic field $b_{\perp} = < \mathbf{b}_{\perp}^2>^{1/2}$ in the volume respect to the rms of the photospheric forcings $u_{ph} = < (\mathbf{u}_{\perp}^0 - \mathbf{u}_{\perp}^L)^2 >^{1/2} \sim 1$ (eqs. (\[eq:bc0\])-(\[eq:bcL\])).
This means that the contribution to the dynamics of the Alfvénic perturbations propagating from the boundary are much smaller, over short periods of time, than the self-consistent non-linear evolution due to the magnetic fields inside the domain, and therefore can not determine the topology of the magnetic field. For run A and G, both with $c_{\mathcal{A}} = 200$, the ratio is $b_{\perp} / u_{ph} \sim 6$ and it increases up to $b_{\perp}/u_{ph} \sim 27$ in run I with $c_{\mathcal{A}} = 1000$. On the other hand these waves continuously transport from the boundaries the energy that sustain the system in a magnetically dominated statistically steady state.
All the facts presented in this section, and the properties of the cascade and of the resulting current sheets in presence of a magnetic guide field outlined in § \[sec:wts\], lead to the conclusion that the current sheets do not generally result directly from a ÒgeometricalÓ misalignment of neighboring magnetic field lines stirred by their footpoints motions, but that *they are the result of a nonlinear cascade in a self-organized system*.
Although the magnetic energy dominates over the kinetic energy, the ratio of the rms of the orthogonal magnetic field over the axial dominant field $B_0$ is quite small. For $c_{\mathcal{A}} = 200, 400$ and $1000$ it is $\sim 3\%$, so that the average inclination of the magnetic fieldlines respect to the axial direction is just $\sim 2^{\circ}$, it is only for the lower value $c_{\mathcal{A}} = 50$ that $b_{\perp}/B_0 \sim 4\%$ and the angle is $\sim 4^{\circ}$. The field lines of the total magnetic field at time $t=18.47\tau_{\mathcal A}$ are shown in Figures 18c and 18d. The computational box has been rescaled for an improved viewing, and to attain the original aspect ratio the box should be stretched 10 times along the axial direction. The magnetic topology for the total field is quiete simple, as the line appear slightly bended. It is only in correspondence of the small scale current-sheets that field lines on the opposite side may show a relative inclination. But as the current sheets are very tiny (and their width decreases at higher Reynolds numbers), they occupy only a very small fraction of the volume, so that the bulk of the magnetic field lines appears only slightly bended.
It is often suggested, or implicitly assumed, that current sheets are formed because the magnetic field line footpoints are subject to a *random walk*. The complexity of the footpoint trajectory would then be a necessary ingredient. In fact it would give rise to a complex topology for the coronal magnetic field, leading either to tangled field lines which would then release energy via fast magnetic reconnection, or to turbulence. So that the “complexity” of the footpoint motions would be responsible for the “complex” dynamics in the corona.
On the opposite our simulations show that *this system in inherently turbulent*, and that “simple” footpoint motions give rise to turbulent dynamics characterize by the presence of an inertial range (§ \[sec:wts\]) and dynamical current sheets. In fact our photospheric forcing velocities (Figures \[fig:v0\]-\[fig:vL\]) are constant in time and have only large-scale components (eq. (\[eq:bc\])), so that the *footpoint motions* are “ordered” and *do not follow any random walk*. During the linear stage this gives rise to a magnetic field that grows linearly in time (eq. (\[eq:blin\])) and that is a mapping of the velocity fields (see eq. (\[eq:blin\]) and Figures 17a and 17b), i.e. both the magnetic field and the current have only large-scale components. The footpoint motions of our photospheric velocities never bring two magnetic field lines close to one another, i.e.they never geometrically produce a current sheet. Current sheets are produced on an ideal timescale, the nonlinear timescale, by the cascade. Furthermore, as we show in the next section, the statistically steady state that characterizes the nonlinear stage results from the *balance at the large-scales between the injection of energy and the flow of this energy from the large scales toward the small scales*, where it is finally dissipated.
As the system is self-organized and the magnetic energy increases at higher values of the axial magnetic fields, very likely different static or time-dependent (with the characteristic photospheric time $\sim 300\, s$) forcing functions, will not be able to determine the spatial shape of the orthogonal magnetic field. In our more realistic simulation with $c_{A} =1000$ the ratio $b_{\perp} / u_{ph}$ is in fact $\sim 27$. Other forcing functions are currently being investigated, and time-dependent forcing functions are likely to modulate with their associated timescale the rms of the system, like total energy and dissipation.
TURBULENCE {#sec:wts}
==========
Before analyzing in detail further aspects of our simulations, namely inertial spectra, anisotropies and scaling laws, let us briefly justify the statement that the time-dependent Parker problem, i.e. the dynamics of a magnetofluid threaded by a strong axial field whose footpoints are stirred by a velocity field, is an MHD turbulence problem.
The fact that at the large orthogonal scales the Alfvén crossing time $\tau_{\mathcal{A}}$ is the fastest timescale so that during the linear stage the fields evolves as (\[eq:blin\])-(\[eq:vlin\]), means that the photosphere’s role is to contribute an anisotropic magnetic forcing function that stirs the fluid, with an orthogonal length typical of the convective cells ($\sim 1000\, km$) and an axial length is given by the loop length $L$.
Typically, forced MHD turbulence simulations (e.g. see @bisk03 and references therein) are performed using a 3-periodic numerical cube with a volumetric forcing function which mimics some physical process injecting energy at the large scales.
Solutions (\[eq:blin\])-(\[eq:vlin\]) can be approximately obtained introducing the*magnetic* forcing function $\mathbf{F}_m$ in equation (\[eq:adim2\]) $$\label{eq:mforc}
\mathbf{F}_m =
\frac{ \mathbf{u}^{L} (x,y) - \mathbf{u}^{0} (x,y) }{
\tau_{\mathcal A}},$$ and implementing 3-periodic boundary conditions in our elongated ($0 \le x, y \le 1$, $0 \le z \le L$) computational box. During the linear stage this forcing would give rise, apart from the small velocity field (\[eq:vlin\]), to the same magnetic field. During the nonlinear stage, as $\tau_A < \tau_{nl}$, it would still give rise to a similar injection of energy. This property was the basis for the body of previous 2D calculations [@ein96; @dmi98; @georg98]
In particular the photospheric motions imposed at the boundaries for the Parker problem take the place of, and represent a different physical realizations of the forcing function generally used for the 3-periodic MHD turbulence box. In the Parker model, the equivalent forcing stirs the magnetic field, whiile in standard simulations the forcing stirs both velocity and magnetic fields or mostly the velocity field. The main differences between “standard” MHD turbulence simulations and the problem at hand are that a) the peculiarity of the low-frequency photospheric forcing leads to magnetic energy largely dominating over the kinetic energy in the system b) the forcing involves line-tying of the magnetic field with 3-periodic boundary conditions. Line-tying inhibit the inverse cascade for the magnetic field, as described later in this section (§ \[sec:lt\]). Equivalently, one may say that line-tying hinders magnetic reconnection by rendering it less energetically favorable due to the increased field line-curvature it requires compared to the unbound system. This property is fundamental to the anomalous scaling laws and enhanced overall heating rates that will be found below.
In MHD, the cascade takes place preferentially in planes orthogonal to the local mean magnetic field ([@sheb83]). The small scales formed are not uniformly distributed in this plane, rather they are organized in dynamical current-vortex sheets extended along the direction of the local main field. These current sheets with associated quadrupolar vorticity filaments *form the dissipative structures of MHD turbulence* (e.g. [@bisk00], [@bisk03] and references therein). In our case, because the axial field is strong, the current sheets are elongated along the axial direction to the point of being quasi-uniform along the loop axis (Figure 18).
Spectral Properties {#sec:sp}
-------------------
In order to investigate inertial range spectra, we have carried out four simulations (runs F, G, H and I in Table \[tab:r\]) with a resolution of 512x512x200 grid points using a mild power ($n=4$) for hyperdiffusion (\[eq:els1\])-(\[eq:els2\]).
![*Run G*: Ratio between cross-helicity $H^C$ and total energy $E$ as a function of time. $H^C \ll E$ shows that the system is in a regime of balanced turbulence.\
\[fig:hc\]](f7.eps){width="47.00000%"}
In turbulence the fundamental physical fields are the Elsässer variables $\mathbf{z}^{\pm} = \mathbf{u}_{\perp} \pm \mathbf{b}_{\perp}$. Their associated energies $$E^{\pm} = \frac{1}{2} \int\! \mathrm{d} V\, \left( \mathbf{z}^{\pm} \right)^2,$$ are linked to kinetic and magnetic energies $E_K$, $E_M$ and to the cross helicity $H^C$ $$H^{C} = \frac{1}{2} \int\! \mathrm{d} V\, \mathbf{u}_{\perp} \cdot \mathbf{b}_{\perp}$$ by $$E^{\pm} = E_K + E_M \pm H^{C}$$ Nonlinear terms in equations (\[eq:els1\])-(\[eq:els4\]) are symmetric under the exchange $\mathbf{z}^{+} \leftrightarrow \mathbf{z}^{-}$, so as substantially are also boundary conditions (\[eq:bc0\])-(\[eq:bcL\]), given that the two forcing velocities are different but have the same rms values ($=1/\sqrt{2}$). It is then expected that $H^C \ll E$ so that none of the two energies prevails $E^{+} \sim E^{-} \sim E$, where $E = E_K + E_M$ is total energy. In Figure \[fig:hc\] the ratio $H^C / E$ is shown as a function of time for run G. Cross helicity has a maximum value of 5% of total energy, and its time average is $\sim 1\%$, and similar values are found for all the simulations. Furthermore perpendicular spectra of $E$ and $E^{\pm}$ in simulations F, G, H and I, overlap each other, so that as expected we can also assume that $$\delta z^+_{\lambda} \sim \delta z^-_{\lambda} \sim \delta z_{\lambda},$$ where $\delta z_{\lambda}$ is the rms value of the Elsässer fields $\mathbf{z}^{\pm}$ at the perpendicular scale $\lambda$.
![Total energy spectra as a function of the wavenumber $n$ for simulations F, G, H and I. To higher values of $c_{\mathcal{A}} = v_{\mathcal{A}} / u_{ph}$, the ratio between the Alfvén and photospheric velocities, correspond steeper spectra, with spectral index respectively $1.8$, $2$, $2.3$ and $2.7$.\
\[fig:multisp\]](f8.eps){width="47.00000%"}
In the following we always consider the spectra in the orthogonal plane $x$-$y$ integrated along the axial direction $z$, unless otherwise noted. Furthermore as they are isotropic in the Fourier $k_x$-$k_y$ plane, we will consider the integrated 1D spectra, so that for total energy $$\begin{gathered}
E =
\frac{1}{2}\, \int_0^L \! \mathrm{d} z\,\iint_{0\ \ \ }^{\ell\ \, }
\displaylimits \! \mathrm{d} x\, \mathrm{d} y\,
\left( \boldsymbol{u}^2 + \boldsymbol{b}^2 \right) = \\
= \frac{1}{2}\, \int_0^L \! \mathrm{d} z\, \ell^2 \sum_{\boldsymbol{k}}
\left( | \boldsymbol{\hat{u}} |^2 + | \boldsymbol{\hat{b}} |^2
\right) \left( \boldsymbol{k}, z \right)
= \sum_n E_n, \\
n = 1, 2, \dots \label{eq:esp}\end{gathered}$$ where, similarly to eq. (\[eq:bc\]), $n$ indicates “rings” in $k$-space. Figure \[fig:multisp\] shows the total energy spectra $E_n$ averaged in time, obtained from the hyperdiffusive simulations F, G, H and I with dissipativity $n=4$ (eqs. (\[eq:els1\])-(\[eq:els2\])) and respectively $c_{\mathcal{A}}=50$, $200$, $400$ and $1000$. An inertial range displaying a power law behaviour is clearly resolved. The spectra visibly steepens increasing the value of $c_{\mathcal{A}}$, with spectral index ranging from $1.8$ for $c_{\mathcal{A}}=50$ up to $\sim 2.7$ for $c_{\mathcal{A}} = 1000$. The spectra are clearly always steeper than the well known (strong) MHD inertial range turbulence spectra $k_{\perp}^{-5/3}$ or $k_{\perp}^{-3/2}$ .
This steepening is certainly not a numerical artifact: the use of hyperdiffusion gives rise to a hump at high wave-number values, known as the bottleneck effect [@falk94], which when present *flattens* the spectra. Furthermore we use the same value of dissipativity ($n=4$) used by [@mg01], who find the same IK spectral slope ($-3/2$), also confirmed in recent higher-resolution simulations performed by [@mg05] with standard $n=1$ diffusion.
In our simulations, a hump or flattening at high wavenumbers is best visible in run H with $c_{\mathcal{A}} = 400$, which might be due to the bottleneck effect, but a more probable interpretation involves a transition from weak to strong turbulence at the smaller scales within the inertial range, which requires a preliminary discussion of strong vs. weal turbulence in MHD.
![*Run I*: Snapshot of the 2D spectrum $E(n_{\perp}, n_z)$ in bilogarithmic scale at time $t \sim 145\, \tau_{\mathcal{A}}$. $n_{\perp}$ and $n_z$ are respectively the orthogonal and axial wavenumbers. The 2D spectrum is shown as a function of $n_{\perp}$ and $n_z+1$, to allow the display of the $n_z=0$ component.\
\[fig:spzp\]](f9rgb.eps){width="47.00000%"}
Recently a lot of progress has been made in the understanding MHD turbulence both in the condition of so-called strong [@gs95; @gs97; @cv00; @bisk00; @mbg03; @mg05; @bd05; @bd06; @mcb06] and weak turbulence [@ng97; @gs97; @gal00; @gc06]. Weak turbulence has been investigated mainly through analytical methods. The total energy spectrum can be characterized by a $k_{\perp}^{-2}$ power law, which is easily found phenomenologically by considering that the Alfvén effect occurs along the field while the cascade proceeds in the orthogonal direction [@ng97]. While our MHD simulations, even with our line-tying boundary conditions and anomalous energetic regime ($b$ dominating over $u$ except at the smallest scales), confirm the presence of the $k_{\perp}^{-2}$ spectrum for a range of loop parameters, steeper spectra are also found nearly reaching $k_{\perp}^{-3}$, clearly linked to the strength of the axial field $B_0$ an effect we discuss more in detail in the following subsection.
The formation of an inertial range is crucially related to the anisotropy of the cascade, where a relationship between spectral extent in the perpendicular and parallel directions known as “critical balance” may be derived. To understand this feature, consider the timescale $T_{\lambda}$, the energy-transfer time at the corresponding scale $\lambda$, characterizing the nonlinear dynamics at that scale. $T_{\lambda}$ does not necessarily coincide with the eddy turnover time $\tau_{\lambda} = \lambda / \delta z_{\lambda}$ because of the Alfvén effect. Spatial structures along the axial direction result from wave propagation (at the Alfvén speed $c_{\mathcal{A}}$) of the orthogonal fluctuations. In other words, the cascading of turbulence in two different planes separated by a distance $\ell_{\parallel}$ leads to formation of scales in the parallel direction whose smallest size can be [@gs95; @cho02; @ou94] $$\ell_{\parallel} (\lambda) \sim c_{\mathcal{A}} T_{\lambda},$$ the critical balance condition. $T_{\lambda}$ will be smaller at smaller scales, so that smaller perpendicular scales create smaller axial scales.
Figure \[fig:spzp\] shows a snapshot at time $t \sim 145\, \tau_{\mathcal{A}}$ of the 2D spectrum $E(n_{\perp}, n_z)$ for run I in bilogarithmic scale, where $n_z$ and $n_{\perp}$ are respectively the axial and orthogonal wavenumbers. Consider vertical cuts at $n_{\perp} = const$: it is clearly visible that from $n_{\perp} =1$ up to $n_{\perp} \sim 20$ the wavenumbers with $n_z > 1$ are scarcely populated compared to the respective wavenumbers with $n_z \le 1$ (the parallel spectrum has also the $n_z=0$ component, in Figure \[fig:spzp\] the vertical coordinate is $n_z+1$). However note also how the loci of maximum parallel wave-number do not precisely follow the critical balance line, rather they are offset at larger $n_{\perp}$: in our case, the hypothetical length of the axial structures (from critical balance) can be longer than the characteristic length of the system, in our case the length of the coronal loop $L$. But in the range of perpendicular wavenumbers for which $$\ell_{\parallel} (\lambda) > L,$$ boundary conditions, i.e. line-tying, intervenes and the cascade along the axial direction is strongly inhibited. In our simulations this occurs roughly at $n_{\perp} \sim 20$. Beyond $n_{\perp} \sim 20$ the spectrum is roughly constant along $n_{\perp} = const$ up to a critical value where it drops.
Interestingly enough, the slope of the 1D spectrum for run I (Figure \[fig:multisp\]) diminishes its value around $n_{\perp} \sim 20$. The reason is that the condition $\ell_{\parallel} (\lambda) > L$ with $\ell_{\parallel} (\lambda)$ defined by critical balance, turns out to play a major role in the “strength” or “weakness” of the cascade: *for $n_{\perp} \lesssim 20$ the system is in a weak turbulent regime, while for $n_{\perp} \gtrsim 20$ a transition to strong turbulence is observed*.
![Total energy at the injection scale (modes 3 and 4), time-averaged for the four simulations F, G, H and I with different Alfvén velocities. The dashed line shows the curve $E_{in} \propto c_{\mathcal{A}}^2$, while the continuous line shows $E_{in}$ as a function of $c_{\mathcal{A}}$ as obtained from equation (\[eq:amp\]) for $\alpha = 0$ corresponding to a Kolmogorov spectrum. The actual growth of $E_{in}$, both simulated and derived from (\[eq:amp\]), show that the growth is less than quadratical but higher than in the simple Kolmogorov case.\
\[fig:ein\]](f10.eps){width="47.00000%"}
In our runs, larger values of $c_{\mathcal{A}}$, i.e. of the parameter $f$ (\[eq:fsc\]), lead to larger magnetic energy and total energies, while the kinetic energy remains smaller than magnetic energy and increases much less (increasing its value by a factor of $6$ from $c_{\mathcal{A}}=50$ to $c_{\mathcal{A}}=1000$). In particular Figure \[fig:ein\] shows total energy at the injection scales (see § \[sec:bc\]), i.e. the sum of the modes $n=3$ and $n=4$ (see eq. \[eq:esp\]) of total energy, $$E_{in} = E_3 + E_4$$ as a function of the non-dimensional Alfvén velocity $c_{\mathcal{A}}$. Their growth is less than quadratical in $c_{\mathcal{A}}$, which implies that the rms of the velocity and magnetic fields at the injection scale (or equivalently the Elsässer fields $\delta z_{in}$) grow less than linearly. Hence as $c_{\mathcal{A}}$ is increased, the ratio $\chi$, a measure of the relative strength of the nonlinear interactions at the injection scale, decreases: at different values of $c_{\mathcal{A}}$ different regimes of weak turbulence are therefore realized at the larger scales of the inertial range, as the different spectra in Figure \[fig:multisp\] confirm.
The presence of a “double” inertial range, with a [*weak-type*]{} power-law index at larger scales, and a flatter [*strong-type*]{} power-law index at smaller scales would not affect the overall cascade rate, and therefore the scalings of loop heating with loop parameters. These are set at the larger scales, and are therefore dependent on the cascade rate determined by the [*weak-type*]{} scaling law, for which a physically motivated phenomenological derivation is presented below. We stress that the possible existence of a “double” inertial range, surmised here with scaling laws and somewhat tenuous numerical evidence, does not appear to have been predicted before and requires substantiating evidence from higher numerical resolution simulations which are planned for the near future.
Phenomenology of the Inertial Range and Coronal Heating Scalings {#sec:sca}
----------------------------------------------------------------
We now introduce a phenomenological model to determine the energy transfer time-scale $T_{\lambda}$ and as a consequence the properties of the cascade. This time-scale, and therefore the different spectra which result, can only depend on the single non-dimensional quantity defining our system, namely $f ={\ell_c\, v_{\mathcal A}} / {L\, u_{ph}}$ (\[eq:fsc\]). The simulations show that as this parameter is increased the spectra steepen leading to a weakened cascade. We revert here to *dimensional quantities* for the scalings, so that we can quantify the resulting coronal heating rates.
The Alfvén effect is based on the idea that two counterpropagating Alfvén waves interact only for the time $\tau_{\parallel} = \ell_{\parallel} / v_{\mathcal{A}}$, leading to a transfer energy time longer that the “generalized” eddy turnover time $$\tau_{\lambda} = \frac{\lambda}{\delta z_{\lambda}},$$ The ratio between these two timescales $$\chi = \frac{\tau_{\mathcal{A}}}{\tau_{\lambda}}=
\frac{\ell_{\parallel}\, \delta z_{\lambda}}{\lambda\, c_{\mathcal{A}}}$$ gives a measure of their relative strength. [@iro64] and [@kra65] proposed that the energy transfer time $T_{\lambda}$, because of the Alfvén effect, is longer than the eddy turnover time, and is given by $$\label{eq:ik}
T_{\lambda} \sim \tau_{\lambda} \, \frac{\tau_{\lambda}}{\tau_{A}},$$ where however they considered an isotropic situation, so that the Alfvén time was given by the propagation time over the scale of the Alfvénic packet. For weak turbulence however $\ell_{\parallel} > L$, so that the Alfvén time must be based on the scale $L$: $\tau_{\mathcal{A}} = L / v_{\mathcal{A}}$.
In addition, we must allow line-tying which acts to slow the destruction of eddies on a given scale ${\lambda}$ more effectively than the standard random encounter effect $ {\tau_{\lambda}}/{\tau_{A}}$ [@DMV80]. We can therefore assume a sub-diffusive behaviour for ${\bf z^+} --- {\bf z^-}$ non-linear encounters leading to $$\label{eq:gik}
T_{\lambda} \sim \tau_{\lambda} \,
\left( \frac{\tau_{\lambda}}{\tau_{A}} \right)^{\alpha},$$ with values $\alpha >1$ and depending in some way on the parameter $f$ \[recall that $\alpha=0,1$ correspond respectively to anisotropic Kolmogorov and Kraichnan cases (the latter leading to a $k^{-2}$ inertial range spectrum)\].
Our simulations then close this ansatz by determining how $alpha$ depends on $f$: integrating over the whole volume ($\ell \times \ell \times L$), the energy cascade rate may now be written as $$\label{eq:etr}
\epsilon \sim \ell^2\, L\, \rho\,
\frac{{\delta z_{\lambda}}^2}{T_{\lambda}},$$ Using (\[eq:gik\]) the energy transfer rate is given by $$\label{eq:sbe1}
\epsilon \sim \ell^2 L\cdot \rho\, \frac{\delta z_{\lambda}^2}{T_{\lambda}}
\sim \ell^2 L\cdot \rho\, \left( \frac{L}{v_{A}} \right)^{\alpha} \,
\frac{\delta z_{\lambda}^{\alpha + 3}}{\lambda^{\alpha + 1}}.$$ Identifying, as usual, the eddy energy with the band-integrated Fourier spectrum $\delta z^2_{\lambda} \sim k_{\perp} E_{k_{\perp}}$, where $k_{\perp} \sim \ell / \lambda$, from eq. (\[eq:sbe1\]) we obtain the spectrum $$\label{eq:gsp}
E_{k_{\perp}} \propto k_{\perp}^{ - \frac{3 \alpha + 5}{\alpha + 3} },$$ where for $\alpha = 0,1$ the $-5/3,-3/2$ slope for the anisotropic Kolmogorov, Kraichnan spectra are is recovered, but steeper spectral slopes up to an asymptotic value of $-3$ are obtained with *higher values of $\alpha$*.
Correspondingly, from eqs. (\[eq:etr\])-(\[eq:sbe1\]), the following scaling relations for $\delta z_{\lambda}$ and $T_{\lambda}$ follow: $$\label{eq:zsc}
\delta z_{\lambda} \sim
\left( \frac{\epsilon}{\ell^2 L \rho} \right)^{\frac{1}{\alpha + 3}}
\left( \frac{v_{\mathcal{A}}}{ L} \right)^{\frac{\alpha}{\alpha + 3}} \
\lambda^{\frac{\alpha + 1}{\alpha + 3}}$$ $$\label{eq:tsc}
T_{\lambda} \sim
\left( \frac{\ell^2 L \rho}{\epsilon} \right)^{\frac{\alpha + 1}{\alpha + 3}}
\left( \frac{v_{\mathcal{A}}}{ L} \right)^{\frac{2 \alpha}{\alpha + 3}} \
\lambda^{2\, \frac{\alpha + 1}{\alpha + 3}}$$
Recently [@bd05] has proposed a similar model, which aims to overcome some discrepancies between previous models and numerical simulations, and that self-consistently accounts for the formation of current sheets, for the cascade of strong turbulence. His energy transfer time is given by $$\label{eq:bd}
T_{\lambda} = \frac{\lambda}{\delta z_{\lambda}}
\left( \frac{v_{\mathcal{A}}}{\delta z_{\lambda}} \right)^{\alpha},$$ but he suggests the interval $ 0 \le \alpha \le 1$ as appropriate to strong turbulence.
As pointed out above of § \[sec:an\], the solutions of equations (\[eq:els1\])-(\[eq:els3\]) depend only on the non-dimensional parameter $f=\ell_c\, v_{\mathcal{A}} / L\, u_{ph}$ (eq. (\[eq:fsc\])) and so $\alpha$ (\[eq:gik\]) is only a function of $f$ $$\label{eq:af}
\alpha = \alpha \left( \frac{\ell_c\, v_{\mathcal{A}}}{L\, u_{ph}} \right).$$
We estimate the value of $\alpha$ from the slope of the total energy spectra (\[eq:gsp\]), as described in [@rap07]. As shown in Figure \[fig:multisp\] to different values of $c_{\mathcal{A}} = v_{\mathcal{A}} / u_{ph}$, (i.e. $f$) ranging from $50$ up to $1000$ correspond spectral slopes from $\sim -1.8$ up to $\sim -2.7$. Thes in turn correspond (through eq. (\[eq:gsp\])) to values of $\alpha$ ranging from $\sim 0.33$ up to $\sim 10.33$.
How do the above results affect coronal heating scalings? The energy that is injected at the large scales by photospheric motions, and whose energy rate ($\epsilon_{in}$) is given quantitatively by the Poynting flux (\[eq:tsz\]), is transported (without being dissipated) along the inertial range at the rate $\epsilon$ (\[eq:sbe1\]), to be finally dissipated at the rate $\epsilon_d$. In a stationary state all these fluxes must be equal $$\label{eq:bal}
\epsilon_{in} = \epsilon = \epsilon_{d}$$
The injection energy rate (\[eq:tsz\]) is given by S, the Poynting flux integrated over the photospheric surfaces: $$\begin{gathered}
\label{eq:dtsz}
\epsilon_{in} = S = \\
= \rho v_{\mathcal A}
\left[ \int_{z=L} \! \mathrm{d} a\, \left( \boldsymbol{u}_{\perp}^L
\cdot \boldsymbol{b}_{\perp} \right)
- \int_{z=0} \! \mathrm{d} a\, \left( \boldsymbol{u}_{\perp}^0
\cdot \boldsymbol{b}_{\perp} \right) \right].\end{gathered}$$ 2D spatial periodicity in the orthogonal planes allows us to expand the velocity and magnetic fields in Fourier series, e.g.$$\label{eq:2dfe}
\mathbf{u}_{\perp} \left( x, y \right) = \sum_{r,s} \mathbf{u}_{r,s}\, e^{i \mathbf{k}_{r,s} \cdot \mathbf{x}},$$ where $$\mathbf{k}_{r,s} = \frac{2\pi}{\ell} \left( r, s, 0 \right)
\qquad r, s \in \mathbb{Z}$$ The surface integrated scalar product of $\mathbf{u}_{\perp}$ and $\mathbf{b}_{\perp}$ at the boundary is then given by $$\begin{gathered}
\label{eq:par}
\int \! \mathrm{d} a\, \boldsymbol{u}_{\perp}
\cdot \boldsymbol{b}_{\perp}
= \sum_{r,s} \boldsymbol{u}_{r,s} \cdot
\int_0^{\ell} \! \! \! \int_0^{\ell} \mathrm{d}x \mathrm{d}y \
\boldsymbol{b}_{\perp} e^{i \boldsymbol{k}_{r,s} \cdot \boldsymbol{x}} = \\
= \ell^2\, \sum_{r,s} \boldsymbol{u}_{r,s} \cdot \boldsymbol{b}_{-r,-s},
\qquad r, s \in \mathbb{Z}\end{gathered}$$ This integral is clearly dominated by large scales consistent with observations of photospheric motions. In our case (eq. (\[eq:bc\])) boundary velocities only have components for wave numbers $(r, s) \in \mathbb{Z}^2$ with absolute values between $3$ and $4$, $3 \le \left( r^2 + s^2 \right)^{1/2} \le 4$. Then in (\[eq:par\]) only the corresponding components of $\mathbf{b}_{\perp}$ are selected.
At the injection scale, which is the scale of convective motions $\ell_c \sim 1,000\, km$, a weak turbulence regime develops, so that the cascade along the axial direction $z$ is limited and in particular the magnetic field $\mathbf{b}_{\perp}$ can be considered approximately uniform along $z$ at the large orthogonal scales. Then from eq. (\[eq:dtsz\]) we obtain $$\label{eq:dein}
\epsilon_{in} = S \sim \rho v_{\mathcal A}
\int \! \mathrm{d} a\, \left( \mathbf{u}_{\perp}^L - \mathbf{u}_{\perp}^0 \right)
\cdot \mathbf{b}_{\perp}$$ Introducing $\mathbf{u}_{ph} = \mathbf{u}_{\perp}^L - \mathbf{u}_{\perp}^0$, using relation (\[eq:par\]), and integrating over the surface, we can now write $$\label{eq:dein2}
\epsilon_{in} = S \sim \ell^2 \rho v_{\mathcal A}
u_{ph} \delta z_{\ell_c},$$ where we have approximated the value of $\delta b_{\ell_c}$, the rms of the magnetic field at the injection scale $\ell_c$, with the rms of the Elsässer variable because the system is magnetically dominated, i.e. $\delta z_{\ell_c} = \left( \delta u^2_{\ell_c} + \delta b^2_{\ell_c} \right)^{1/2} \sim \delta b_{\ell_c}$.
We now have an expression for $\epsilon_{in}$, where the only unknown variable is $\delta z_{\ell_c}$, as $\ell_c$, $\rho$, $v_{\mathcal{A}}$ and $u_{ph}$ are the parameters characterizing our model of a coronal loop.
The transfer energy rate $\epsilon$ does not depend on $\lambda$. Considering then $\lambda=\ell_c$ in equation (\[eq:sbe1\]), we have $$\label{eq:sbe2}
\epsilon
\sim \frac{\rho\, \ell^2 L^{\alpha+1}}{\ell_c^{\alpha+1} \, v_{A}^{\alpha}} \,
\delta z_{\ell_c}^{\alpha + 3}.$$
Equations (\[eq:dein2\]) and (\[eq:sbe2\]) show another aspect of self-organization. Both $\epsilon_{in}$ and $\epsilon$, respectively the rate of the energy flowing in the system at the large scales, and the rate of the energy flowing from the large scales toward the small scales depend on $\delta z_{\ell_c}$, the rms of the fields at the large scale. This shows that the energetic balance of the system is determined by the balance of the energy fluxes $\epsilon$ and $\epsilon_{in}$ at the large scales. The small scales will then dissipate the energy that is transported along the inertial range (see eq. (\[eq:bal\])). This implies that *beyond a numerical threshold total dissipation (dissipation integrated over the whole volume) is independent of the Reynolds number*. In fact beyond a value of the Reynolds number for which the diffusive time at the large scale is negligible, i.e. when the resolution is high enough to resolve an inertial range, the large-scale balance between $\epsilon$ and $\epsilon_{in}$ is no longer influenced by diffusive processes. Of course this threshold is quite low respect to the high values of the Reynolds numbers for the solar corona, but it is still computationally very demanding.
An analytical expression for the coronal heating scalings may be obtained from (\[eq:dein2\]) and (\[eq:sbe2\]) yielding the value of $\delta z_{\ell_c}^{\ast}$ for which the balance $\epsilon_{in} = \epsilon$ is realized: $$\label{eq:amp}
\frac{\delta z_{\ell_c}^{\ast}}{u_{ph}}
\sim \left( \frac{\ell_c v_{A}}{L u_{ph}} \right)^{\frac{\alpha + 1}{\alpha + 2}}$$ Substituting this value in (\[eq:sbe2\]) or equivalently in (\[eq:dein2\]) we obtain the energy flux $$\label{eq:chs}
S^{\ast}
\sim \ell^2 \rho \, v_{A} u_{ph}^2
\left( \frac{\ell_c v_{A}}{L u_{ph}} \right)^{\frac{\alpha+1}{\alpha+2}}.$$ As stated in (\[eq:bal\]) in a stationary cascade all energy fluxes are equal on the average. $S^{\ast}$ is then the energy that for unitary time flows through the boundaries in the coronal loop at the convection cell scale, and that from these scales flows towards the small scales. This is also the dissipation rate, and hence the *coronal heating scaling, i.e. the energy which is dissipated in the whole volume for unitary time.* As shown in equation (\[eq:af\]) the power $\alpha$ depends on the parameters of the coronal loop, and its value is determined numerically with the aforementioned technique.
The observational constraint with which to compare our results is the energy flux sustaining an active region. The energy flux at the boundary is the axial component of the Poynting vector $S_z$ (see § \[sec:ee\]). This is obtained dividing $S^{\ast}$ (\[eq:chs\]), the Poynting flux integrated over the surface, by the surface $\ell^2$: $$\label{eq:fchs}
S_z = \frac{S^{\ast}}{\ell^2} \sim \rho \, v_{A} u_{ph}^2
\left( \frac{\ell_c v_{A}}{L u_{ph}} \right)^{\frac{\alpha+1}{\alpha+2}},$$ where $\alpha$ is not a constant, but a function of the loop parameters (\[eq:af\]). The exponent in (\[eq:fchs\]) goes from $0.5$ for $\alpha = 0$ up to the asymptotic value $1$ for larger $\alpha$. We determine $\alpha$ numerically, measuring the slope of the inertial range (Figure \[fig:multisp\]), and inverting the spectral power index (\[eq:gsp\]). We have used simulations F, G, H and I to compute the values of $\alpha$, because they implement hyperdiffusion, resolve the inertial range, and then are beyond the numerical threshold below which total dissipation does not depend on the Reynolds number. These simulations implement $v_{\mathcal{A}} = 50$, $200$, $400$ and $1,000$, and the corresponding $\alpha$ are $\sim 0.33$, $1$, $3$, $10.33$. The corresponding values for the power $(\alpha+1)/(\alpha+2)$ (\[eq:fchs\]) are $\sim 0.58$, $0.67$, $0.8$ and $0.91$, close to the asymptotic value $1$. $S_z$ is shown in Figure \[fig:dissA\] (diamond points) as a function of the axial Alfvén velocity $v_{\mathcal{A}}$. To compute the value of $S_z$ for $v_{\mathcal{A}} = 2,000\, km\, s^{-1}$ we have estimated $\alpha \sim .95$, although for values close to $1$ $S_z$ does not have a critical dependence on the value of the exponent.
In Figure \[fig:dissA\] we compare the analytical function $S_z$ (\[eq:fchs\]) with the respective value determined from our numerical simulations (star points), i.e. with the total dissipation rate by the surface and converted to dimensional units ($(J+\Omega)/\ell^2$, see (\[eq:tdiss\])). For the numerical simulation values, the error-bar is defined as 1 standard deviation of the temporal signal. The analytical and computational values are in good agreement for all the 4 simulations considered, and for the more realistical value $v_{\mathcal{A}} = 2,000\, km\, s^{-1}$ the dissipated flux is $\sim 1.6\times10^6\, erg\, cm^{-2}\, s^{-1}$. This value is in the lower range of the observed constraint $10^7\, erg\, cm^{-2}\, s^{-1}$.
![Analytical (\[eq:fchs\]) and numerically computed dissipated flux as a function of the axial Alfvén velocity $v_{\mathcal{A}}$. The continuous line shows the Poynting flux (\[eq:fchs\]) as a function of $v_{\mathcal{A}}$ in the case $\alpha = 0$, corresponding to a Kolmogorov-like cascade. To higher values of $v_{\mathcal{A}}$ correspond a higher dissipation rate because a weak turbulence regime develops.\
\[fig:dissA\]](f11.eps){width="47.00000%"}
The continuous line in Figure \[fig:dissA\] corresponds to the function $S_z$ for $\alpha = 0$ (which is approximately realized for $v_{\mathcal{A}} \lesssim 50\, km\, s^{-1}$), in correspondence of which a Kolmogorov spectrum would be present, and $S_z \propto v_{\mathcal{A}}^{3/2}$. The computed and analytical values of $S_z$ for higher $v_{\mathcal{A}}$ are always beyond this curve, because $\alpha$ increases its values, and a more efficient dissipation takes place. This is due to the fact that to higher values of $\alpha$ correspond higher values of the energy transfer time, and consequently a longer linear stage, higher values of the fields at the large scales (\[eq:amp\]), and hence a higher value of the energy rates (see (\[eq:dein2\]), (\[eq:sbe2\]) and (\[eq:chs\])). So that it is realized the only apparently paradox that to a weaker turbulent regime, to which corresponds less efficiency in the nonlinear terms, corresponds a higher total dissipation.
In the last paragraph of § \[sec:ee\] we have shown that when the condition (\[eq:efc\]) is satisfied the emerging flux can be neglected. But in eq. (\[eq:efc\]) we have to specify the value of the magnetic field $b_{\perp}^{turb}$ self-consistently generated by the non-linear dynamics. This value is given by (\[eq:amp\]) as the magnetic field dominates ($\delta z^{\ast}_{\ell_c} \sim b_{\perp}^{turb}$). By substitution we can now estimate that the emerging flux is negligible when the emerging component of the magnetic field satisfies $$\label{eq:efcc}
b_{\perp}^{ef} < B_0\, \sqrt{
\left( \frac{\ell_c}{L} \right)^{\frac{\alpha+1}{\alpha+2}}
\left( \frac{u_{ph}}{v_{\mathcal{A}}} \right)^{\frac{1}{\alpha+2}}
}$$ In the asymptotic state $\alpha \gg 1$ the condition reduces to $b_{\perp}^{ef} / B_0 < \sqrt{ \ell_c / L }$. For a coronal loop with $L \sim 40,000\, km$, as $\ell_c \sim 1,000\, km$ this implies that emerging flux does not play a role if $b_{\perp}^{ef} / B_0 < 1/ 6$.
![*Transition to turbulence*: Total ohmic and viscous dissipation as a function of time for simulations A, B, C, G (displayed on the same scales). All the simulations implement $c_{\mathcal{A}} = 200$, but different Reynolds numbers, from $Re=200$ up to $800$. Run G implements hyperdiffusion. For Reynolds numbers lower than $100$ the signal is completely flat and displays no dynamics, at higher Reynolds smaller temporal structures are present.\
\[fig:turbod\]](f12.eps){width="47.00000%"}
Transition to Turbulence and Dissipation vs. Reynolds Number {#sec:trans}
------------------------------------------------------------
Turbulence is a characteristic of high Reynolds number systems (e.g. [@fr95]). For a sufficiently high viscosity nonlinear dynamics is strongly suppressed, and our system relaxes to a diffusive equilibrium (§\[sec:eod\]), and no significant small scale is formed. Increasing the Reynolds number, the diffusive time at the injection scale (\[eq:taud\]) $\tau_d \sim Re\, \ell_c^2$ increases. At a certain point it will be big enough not to influence the dynamics as the large scales, an inertial range will then be resolved and total dissipation will not depend any longer from the Reynolds numbers. In fact for higher values of $Re$ the intertial range will extend to higher wave-numbers, but the energy flux will remain the same.
At higher Reynolds numbers smaller scales are resolved, and each scale will contribute with its characteristic time $T_{\lambda}$ to the temporal structure of the rms of the system. Figure \[fig:turbod\] shows total dissipation as a function of time for simulations A, B, C and G, on the same time interval, and on the same scale. At increasingly higher values of the Reynolds numbers smaller and smaller temporal structures are added to the signal. Ideally the temporal structure of total dissipation at higher Reynolds numbers is well described by shell-model simulations. For smaller values of Re the signal is completely flat (see Figure \[fig:multid\]). This behaviour identifies a *transition to turbulence*.
![Total ohmic and viscous dissipation as a function of time for simulations A, B, C, D, E and G, all of them implement $c_{\mathcal{A}} = 200$ but different Reynolds numbers. The threshold beyond which dissipation is independent of the Reynolds number can be identified around $Re = 800$, corresponding to a numerical resolution of 512x512 points in the orthogonal planes.\
\[fig:multid\]](f13rgb.eps){width="47.00000%"}
Figure \[fig:multid\] shows total dissipation as a function of time for the same 4 simulations shown in Figure \[fig:turbod\], plus other 2 simulations at lower Reynolds number, respectively $Re =100$ and $Re=10$ for the complete time interval. For the lowest value of $Re$ no dynamics is present, so that the threshold value for the transition to turbulence can be set to $Re \sim 100$. To higher values of $Re$ dissipation grows. An inertial range is barely solved with a resolution of 512x512 grid points in the $x$-$y$ planes, so that simulation with $Re = 800$ can be considered at the threshold. On the other hand simulation G implements hyperdiffusion, so that an inertial range is solved, and the dynamics is not affected by diffusion. The presence of a sufficiently extended inertial range implies in fact that we are beyond the numerical threshold where dissipation does not depend on the Reynolds number (§ \[sec:sca\]). The threshold value can be identified to a sufficient extent at $Re =800$, i.e. for a numerical grid of 512x512 points. The number of points to use along the axial direction should be enough to allow the formation of all the small scales due to the “critical balance” (Figure \[fig:spzp\]), but a larger number of points would only result in a waste of computational time.
Inverse Cascade and Line-tying {#sec:lt}
------------------------------
Two dimensional simulations [@ein96; @georg98] have shown an inverse cascade for the magnetic energy, corresponding in physical space to the coalescence of magnetic islands. In the 3D case the DC magnetic field along the axial direction is present, giving rise to a field line tension that tends to inhibit an inverse cascade, as motions linked to the coalescence would bend the field lines of the total magnetic field, which are mostly elongated along the axial direction (Figure 18). On the other hand field line tension depends on the strength of the axial field, becoming stronger for a stronger field.
In Figures \[fig:fig14\] and \[fig:fig15\] the first 4 modes of magnetic energy for simulations F and I, respectively with $c_{\mathcal{A}} = 50$ and $1000$, are plotted as a function of time. Energy is injected at wave-numbers $n=3$ and $4$. Modes associated to wave-numbers 1 and 2 grow to higher values than at the injection scale in run F, while in run I they are always limited to lower values. In runs G and H, with respectively $c_{\mathcal{A}} = 200$ and $400$ an intermediate behaviour is found, but none of the modes $n=1$ or $2$ never becomes bigger than the injection energy modes.
![*Run F:* In this simulation with $c_{\mathcal{A}} = 50$ an inverse cascade at the wavenumbers $n=1$ and $2$ is realized. Energy is injected at wavenumbers $n=3$ and $4$.\
\[fig:fig14\]](f14rgb.eps){width="47.00000%"}
Timescales {#sec:tmsc}
----------
In the previous sections we have always affirmed that the Alfvén crossing time $\tau_{\mathcal{A}} = L / v_{\mathcal{A}}$ is the fastest timescale in the system, and that in particular it is smaller than the nonlinear timescale $\tau_{nl}$, which we can identify with the energy transfer time (\[eq:tsc\]) at the injection scale $\tau_{nl} = T_{\ell_c}$.
In Figure \[fig:fig3\] it is already clear that the nonlinear timescale is longer that $\tau_{\mathcal{A}}$, in fact it shows that the timescale over which energy has substantial variations is bigger than the Alfvén crossing time.
The same behaviour is identified in Figures \[fig:fig14\]-\[fig:fig15\], which show the time evolution of the magnetic energy modes for runs F and I. These are more relevant quantities, because to realize a weak MHD turbulence regime it is required that the energy transfer time $T_{\lambda}$ is bigger than the crossing time $\tau_{\mathcal{A}}$ at the injection scale $\lambda = \ell_c$ and for a limited range of smaller scales down to some lower bound $\lambda^{\ast}$: $\lambda^{\ast} \le \lambda \le \ell_c$. The magnetic energy modes at the injection scale ($n=3$ and $4$) change their values on scales bigger than $\tau_{\mathcal{A}}$, and for a larger value of the Alfvén velocity the nonlinear timescale is longer respect to the crossing time (Figures \[fig:fig14\]-\[fig:fig15\]). We can roughly estimate $\tau_{nl} \sim 5 \tau_{\mathcal{A}}$ for run F with $c_{\mathcal{A}} = 50$ and $\tau_{nl} \sim 20 \tau_{\mathcal{A}}$ for run I with $c_{\mathcal{A}} = 1,000$.
![*Run I:* Simulation performed with $c_{\mathcal{A}} = 1000$. The increased magnetic field line tension inhibits an inverse cascade for the orthogonal magnetic field.\
\[fig:fig15\]](f15rgb.eps){width="47.00000%"}
Using our scaling relations we can derive an analytical estimate for the energy transfer time $T_{\lambda}$. Substituting the energy rate (\[eq:chs\]) in equation (\[eq:tsc\]) we obtain: $$\label{eq:tsc1}
T_{\lambda} \sim
\left( \tau_{\mathcal{A}} \tau_c^{\alpha+1} \right)^{\frac{1}{\alpha+2}}
\left( \frac{\lambda}{\ell_c} \right)^{2 \frac{\alpha+1}{\alpha+3}},$$ where $\tau_c = \ell_c / u_{ph}$. In particular the ratio over the Alfvén crossing time is: $$\label{eq:tsc2}
\frac{T_{\lambda}}{\tau_{\mathcal{A}}} \sim
\left( \frac{\tau_c}{\tau_{\mathcal{A}}} \right)^{\frac{\alpha+1}{\alpha+2}}
\left( \frac{\lambda}{\ell_c} \right)^{2 \frac{\alpha+1}{\alpha+3}},$$ and as $\tau_c > \tau_{\mathcal{A}}$ then self-consistently $T_{\lambda} > \tau_{\mathcal{A}}$. For our loop $\ell_c \sim 1,000\, km$ and $u_{ph} \sim 1\, km\, s^{-1}$, so that $\tau_c \sim 1,000\, s$. For runs F and I shown in Figures \[fig:fig14\] and \[fig:fig15\], the loop length is always $L = 40,000\, km$, while the Alfvén velocity is respectively $v_{\mathcal{A}} = 50$ and $1,000\, km\, s^{-1}$, and the corresponding crossing times $\tau_{\mathcal{A}} = 800$ and $40\, s$. Using the values of $\alpha$ computed in §\[sec:sca\] (respectively $\alpha = 0.33$ and $10.33$) we can then roughly estimate from (\[eq:tsc2\]), the nonlinear timescale $\tau_{nl}=T_{\lambda = \ell_c}$ and its ratio with the Alfvén crossing time: $$\label{eq:tsc3}
\frac{\tau_{nl}}{\tau_{\mathcal{A}}} =
\frac{T_{\ell_c}}{\tau_{\mathcal{A}}} \sim
\left( \frac{\tau_c}{\tau_{\mathcal{A}}} \right)^{\frac{\alpha+1}{\alpha+2}}.$$ For runs F and I we find $\tau_{nl} / \tau_{\mathcal{A}} = 1.2$ and $22.3$ in agreement with the simulations.
Equation (\[eq:tsc2\]) can also be used to estimate the extension of the weak turbulence inertial range. The region for which the weak turbulence condition $T_{\lambda} > \tau_{\mathcal{A}}$ is satisfied is: $$\lambda > \lambda^{\ast} =
\ell_c \left( \frac{\tau_{\mathcal{A}}}{\tau_c} \right)^{\frac{\alpha+3}{2(\alpha+2)}}$$
![Temporal spectrum of magnetic energy for run G. $\nu_{\mathcal{A}} = 1 / \tau_{\mathcal{A}}$ is the frequency corresponding to the Alfvén crossing time. The intermediate part of the spectrum exhibits a $\nu^{-2}$ power law.\
\[fig:fig16\]](f16.eps){width="47.00000%"}
Figure \[fig:fig16\] shows the temporal spectrum of magnetic energy for run G with $c_{\mathcal{A}} = 200$, i.e. we perform the Fourier transform of the magnetic energy as a function of time, and then plot its squared modulus. We use run G because it is the one for which we have saved more frequently the rms quantity and then the plot covers a wider range at high frequencies. The power spectrum is roughly constant up to $\nu / \nu_{\mathcal{A}} \sim 0.2$, which corresponds to $t / \tau_{\mathcal{A}} \sim 5$ in agreement with our scaling (\[eq:tsc3\]) which for this case gives $\tau_{nl} / \tau_{\mathcal{A}} \sim 3.3$. Beyond this critical point the power spectrum exhibits a power law which fits $\nu^{-2}$, in agreement with shell-model simulations [@buc07].
DISCUSSION AND CONCLUSIONS {#sec:disc}
==========================
We would like first to clarify a few concepts that might otherwise result in misunderstandings of the work that we have presented. The concept of turbulence is used to describe different processes in different research fields, so that its use, without specifications, can result vague and misleading. It is in fact very often used to describe chaotic behaviors at the small scales, often linked to the intermittent dissipation of energy. Although this aspect is present in our simulations, when we say that the Parker problem is an MHD turbulence problem, we refer mainly to the property of turbulence to transfer energy from large to small scales. Namely to its ability to transport the energy from the scale of photospheric motions ($\sim 1000\, km$), where it is injected, down to the small dissipative scales (meters?), without dissipating it at the intermediate scales. This property is clearly identified by the presence of an inertial range with a power law spectrum, which extends from the injection scale to the dissipative scale.
Furthermore turbulence, magnetic reconnection and ohmic heating associated to currents are sometime presented as alternative and/or mutually exclusive coronal heating models. This contraposition is artificial. Current sheets are in fact [**the dissipative structures**]{} of MHD turbulence, and magnetic reconnection at the loci of current sheets is observed in virtually every MHD turbulence simulation in both 2D and 3D (see e.g., [@bisk03] and references therein ). Nanoflares are then naturally associated with the time and space intermittency of the small scale deposition of energy (as shown in the 2D case by [@georg98]), which is due to the cascade which leads to the formation and dissipation of current sheets, and to which we refer collectively with the term MHD turbulence.
In summary, the main results presented in this paper are the following:
- The time-dependent Parker problem may be seen as an MHD turbulence problem, where the large scale forcing function is realized by the photospheric motions.
- This system is genuinely turbulent, in the sense that small scale formation is not driven passively by the random walk of the footpoints, rather it is a property of the maxwell stresses developing in the coronal volume. Current sheets therefore do not generally result *directly* from a “geometrical” misalignment of neighboring magnetic field lines stirred by their footpoint (random) motions, *they are the result of a nonlinear cascade in a self-organized system*.
- Nanoflares are naturally associated with the intermittent dissipation of the energy that, injected at the large scales by photospheric motions, is transported to the dissipative scales through a cascade, and is finally dissipated through nonlinear magnetic reconnection.
- Beyond a threshold, that is low compared to the coronal Reynolds numbers, but still computationally very demanding, total dissipation is independent of the Reynolds numbers. This threshold corresponds to a numerical resolution of $\sim 512\times512$ grid points in the planes orthogonal to the dominant DC magnetic field.
- As the loop parameters vary, different regimes of turbulence develop: strong turbulence is found for weak axial magnetic fields and long loops, leading to Kolmogorov-like spectra in the perpendicular direction, while weaker and weaker regimes (steeper spectral slopes of total energy) are found for strong axial magnetic fields and short loops. There is no single universal scaling law (see (\[eq:fchs\])), as a consequence the scaling of the heating rate with axial magnetic field intensity, which depends on the spectral index of total energy for given loop parameters, must vary from $B_0^{3/2}$ for weak fields to $B_0^2$ for strong fields at a given aspect ratio.
- For a loop $40,000\, km$ long , with an Alfvén velocity $v_{\mathcal{A}} = 2,000\, km\, s^{-1}$ and a numerical density of $10^{10}\, cm^{-3}$, whose footpoints are subject to photospheric motions of $u_{ph} \sim 1\, km\, s^{-1}$ on a scale of $\ell_c \sim 1,000\, km$, the energy flux entering the system and being dissipated is $S_z \sim 1.6\times10^6\, erg\, cm^{-2}\, s^{-1}$. On the other hand, for a coronal loop typical of a quiet-Sun region, that has the same parameter of the previous case but with a length of $100,000\, km$ and $v_{\mathcal{A}} = 500\, km\, s^{-1}$, the resulting Poynting flux is $S_z \sim 7\times10^4\, erg\, cm^{-2}\, s^{-1}$.
The most advanced EUV and X-RAY imagers (e.g. those onboard SOHO, TRACE, STEREO and HINODE) have space resolutions ($\sim 800\, km$) of the order of the granulation cells. Hence they do not resolve the small-scales where current sheets, magnetic reconnection and all the dynamical features of the system take place. Their resolution is roughly $1/5$ the length of the perpendicular cross-section of our numerical box ($\sim 4000\, km$). Hence, even if the system is highly dynamical on small-scales (see Figure 17 and the associated movie), integrating over these scales has the effect to “averaging” the small scale dynamics. In particular small scale reconnection cannot be detected, magnetic fieldlines will appear only slightly bended (Figure 18), and their dynamics will appear slower (a modulation of the nonlinear timescale with the thermodinamical timescales).
The topological and dynamical effects associated with magnetic reconnection should be taken into account when modeling the thermodynamical and observational properties of coronal loops [@schr07], recalling that most of the dynamics take place at sub-resolution scales while we observe the integrated emission.
Two density current fields that have the same “steady” integrated ohmic dissipation, balanced by a corresponding Poynting Flux (see § \[sec:eod\], equation (\[eq:tdiss\]) and Figure \[fig:diss\]), but with different spatial distributions will have different emissions. Consider the first with only large scale components, as the one that would result from a diffusive process (§ \[sec:eod\]), while in the second the current has only small scale components, as in the simulations that we have presented. In the second case the filling factor is small (Figures 17 and 18) so that the density of current has a far larger value, and this would correspond to two very different thermodynamical and observational outcomes. But the highly dynamical effects associated with the second case will be averaged and result less dynamical when integrated. Still the integrated observables should be very distinct between the two cases.
Finally, while our simulations give an accurate description of the time-dependent Parker problem, with the limitations on the photospheric forcing field described in the introduction, the use of the reduced MHD equation is justified only for slender loops threaded by a strong axial magnetic field. For short loops, or loops that have orthogonal component of the magnetic field comparable to the axial component, the full set of MHD equation should be implemented. For the slender loops that we have simulated we observe a modest accumulation of energy, which subsequently is released via nanoflares. On the other hand shorter loops, or loops in a more complicated geometry, or subject to loop-loop interactions, and more generally loops affected by the neighboring coronal environment, might exhibit the ability to accumulate more energy (e.g. [@low06]) and then release it in larger flares, possibly via a “secondary instability” [@dahl05] or fast magnetic reconnection [@cas06].
Magnetohydrodynamics (MHD) has proved to be a useful tool to investigate the properties of the turbulent cascade [@bisk03]. MHD is very well known to give an approximate description of the plasma dynamics at *large scales* and *low frequencies*. In MHD turbulence it is generally supposed that at the small scales a “dissipative mechanism” is present. Most of the properties of the turbulent cascade do not depend on the details of the dissipative mechanism, whether it is described by the diffusive operator present in equations (\[eq:adim1\])-(\[eq:adim2\]), or more properly by a kinetic mechanism.
In particular in our case, the timescales associated at the scale $\lambda$ ((\[eq:tsc1\]) for weak turbulence and (\[eq:bd\]) for the strong case) decrease for smaller scales. In this way the small-scale dynamics is characterized by high-frequency phenomena, and then it is not well described by MHD, but rather a kinetic model would be more appropriate. It is then possible that (self-consistently) at the small scales *particle acceleration* plays an important role in the dissipation of energy, a physical process that should be investigated through kinetic models. Nevertheless the coronal heating rates (\[eq:fchs\]), like the cascade properties over an extended range of scales, are independent of the details of the dissipation mechanism. They are determined by the balance, at the *large scales* (see § \[sec:sca\]), between the rate of the energy flowing into the loop from the boundaries due to the work done by photospheric motions on the magnetic field line footpoints at the scale of the convective cells, and the rate at which the energy flows along the inertial range from the large scales towards the small scales.
The authors would like to thank Bill Matthaeus and the anonymous referee for very useful comments. A.F.R. is supported by the NASA Postdoctoral Program, M.V. is supported by NASA LWS-TR&T and SR&T, and R.B.D. is supported by NASA SPTP. A.F.R. and M.V. thank the IPAM program “Grand Challenge Problems in Computational Astrophysics” at UCLA. Simulations were carried out on JPL supercomputers.\
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| 0 |
---
author:
- 'R. Schulze[^1]'
- 'M. Bluhm'
- 'B. Kämpfer'
title: 'Plasmons, plasminos and Landau damping in a quasiparticle model of the quark-gluon plasma'
---
Introduction {#intro}
============
Intense experimental and theoretical investigations [@KMR03] suggest the existence of a new, deconfined phase of strongly interacting matter, where quarks and gluons form a fluid or gas, the quark-gluon plasma (QGP). If confirmed, the QGP would have existed during the Big Bang prior to hadronization and might be found inside of massive neutron stars. Indeed, recent results from the Relativistic Heavy Ion Collider (RHIC) experiments point to the formation of a quark-gluon medium of low viscosity [@BRA05; @PHO05; @STA05; @PHE05].
However, on the theoretical side, much work remains to be done. Perturbative solutions of QCD [@AZ95; @ZK95; @Kaj03; @Vuo03a; @Vuo03b; @IRV04] are limited to the region of asymptotic freedom and fail for the strong coupling regime (e.g. in the vicinity of the pseudocritical temperature [@BI02] of deconfinement $T_{c}$; at somewhat higher temperatures, say above $2T_c$ resummation improves the convergence of perturbation theory noticeably [@KPP97; @ABS99; @BIR01]). Numerical evaluations of the full theory, on the other hand, are still restricted to small chemical potential as being useful for the Big Bang or heavy-ion collisions at present RHIC top energies or future LHC energies. However, at RHIC bottom energies, at SPS energies and, in particular, at FAIR energies, baryon density effects become significant and require different approaches.
Quasiparticle models (QPM), describing the quark-gluon plasma as assembly of essentially non-interacting excitations emerging from the strong interaction, have proven to represent useful phenomenological parametrizations of QCD thermodynamics above $T_{c}$ [@Pes94; @LH98; @Pes02; @LR03; @TSW04]. At zero chemical potential, lattice results are described with surprising accuracy allowing the adjustment of model parameters. Thermodynamic self-consistency, supplemented by the stationarity of the thermodynamic potential, can then be used to extrapolate thermodynamic properties of systems to nonzero net baryon density.
Our quasiparticle model [@KBS06; @BKS06; @BKS07a; @BKS07b] is based on the HTL approximation [@BP90a] to the 1-loop self-energies. This gives rise to four quasiparticle families. While quasiquarks and transversal gluons within the model represent excitations with quantum numbers of actual quarks and gluons with modified masses, quark holes (plasminos) and longitudinal gluons (plasmons) are quanta of collective excitations. The residues of the poles in the spectral density of the collective modes vanish exponentially for momenta $k\sim T$, from which the main contributions to thermodynamic phase space integrals originate. Therefore, they were neglected in the previous simple form of the model (dubbed eQP in [@Pes05]). Additionally, damping contributions were neglected as they are small at zero chemical potential, $\mu=0$.
The procedure of mapping the eQP results from $\mu=0$ into the $T$-$\mu$ plane is plagued by some ambiguities leading to non-unique solutions close to the presumed phase transition. Therefore, the model is restricted to sufficiently large temperatures. First attempts to include collective excitations into a two-flavor quasiparticle model at nonzero chemical potential have been made [@RR03] and suggest that these ambiguities might vanish. In this work we show that both collective excitations and damping effects are necessary to preserve the self-consistency of the model and ensure unique solutions when extrapolating towards large baryon densities at moderate temperatures. In the present work, the 2+1 flavor case is considered, allowing the use and extrapolation of recent lattice data [@Kar07].
Our paper is organized as follows. Section \[sec:derivation\] will comprise the derivation of the full HTL-based QPM, including collective modes and damping, as a series of approximations from QCD. The necessary further approximations leading to the eQP and its problems are discussed in section \[sec:eQP\]. The results for both models are then contrasted in section \[sec:fullHTL\] and investigated in some detail. Finally, a conclusion is given in section \[sec:conclusion\].
Derivation of the full HTL model\[sec:derivation\]
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The effective action
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A connection of the fundamental theory of QCD and the thermodynamic potential of the QGP is provided by the Luttinger-Ward formalism [@LW60; @Bay62] as shown in [@BKS07a]. Alternatively, the Cornwall-Jackiw-Tomboulis (CJT) formalism [@CJT74] may be used, as a translationally invariant QGP in equilibrium and without spontaneously broken symmetries is considered. In this case, both formalisms are equivalent [@Ris03].
The CJT formalism requires the stationarity of the effective action$$\begin{aligned}
\Gamma[D,S]=I & -\frac{1}{2}\left\{ \text{Tr}\left[\ln D^{-1}\right]+\text{Tr}\left[D_{0}^{-1}D-1\right]\right\} \nonumber \\
& +\quad\left\{ \text{Tr}\left[\ln S^{-1}\right]+\text{Tr}\left[S_{0}^{-1}S-1\right]\right\} \,\,+\,\,\Gamma_{2}[D,S],\label{eq:cjt effective action}\end{aligned}$$ where $I$ is the classical action and $D$ and $S$ are the full gluon and quark propagators while the subscript $0$ denotes the respective free equivalents. The functional $\Gamma_{2}$ represents the sum over all two-particle irreducible skeleton graphs of the theory. The traces $\text{Tr}$ contain the integration over the four-dimensional phase space as well as a trace $\text{tr}$ over discrete indices. The integration is performed using the imaginary time formalism [@LeB96; @Kap89; @YHM95]. For the grand canonical potential $\Omega=-T\Gamma$ [@Bro92; @Riv88] this yields$$\begin{aligned}
\frac{\Omega}{V} & = & \mbox{tr}\!\!\int\!\!\frac{{\mathrm{d}^{4}k}}{(2\pi)^{4}}{n_\text{B}}(\omega)\,\mbox{Im}\!\left(\ln D^{-1}-\Pi D\right)+2\,\mbox{tr}\!\!\int\!\!\frac{{\mathrm{d}^{4}k}}{(2\pi)^{4}}{n_\text{F}}(\omega)\,\mbox{Im}\!\left(\ln S^{-1}-\Sigma S\right)-\frac{T}{V}\Gamma_{2},\label{eq:Omegafinal}\end{aligned}$$ where ${n_\text{B}}=(\exp\,(\beta\omega)-1)^{-1}$ with $\beta=1/T$ is the Bose-Einstein, and ${n_\text{F}}=(\exp(\beta(\omega-\mu))+1)^{-1}$ the Fermi-Dirac distribution function.
Application to QCD
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In our present approach, the infinite sum $\Gamma_{2}$ is truncated at 2-loop order leaving the contributions exhibited e.g. in equation (25) in [@BKS07a]. Within the CJT formalism, the self-energies then follow from a functional derivative of $\Gamma_{2}$, giving the well-known 1-loop self-energies (e.g. equations (26) and (27) ibid.). In order to achieve a gauge invariant formulation of the model, we apply an additional approximation of hard thermal loops (HTL). Although originally being derived for soft external momenta $\omega,k\sim gT\ll T$, HTL results coincide with the complete one-loop results on the lightcone [@BKS07a; @Pes98b] and thus provide the correct limiting behaviour.
The resulting HTL self-energies can be found in textbooks. Here we follow the conventions of Blaizot et al. [@BIR01], where essential features of our model have been worked out, and use$$\begin{aligned}
\Pi_{\mu\nu} & = & {\Pi_\text{T}}(\omega,k)\left(\Lambda_{\text{T}}(\vec{k})\right)_{\mu\nu}-\,\,\,\,{\Pi_\text{L}}(\omega,k)\left(\Lambda_{\text{L}}(\vec{k})\right)_{\mu\nu},\\
\gamma_{0}\Sigma & = & \Sigma_{+}(\omega,k)\,\,\,\,\,\Lambda_{+}(\vec{k})\quad-\,\,\,\,\Sigma_{-}(\omega,k)\,\,\,\,\Lambda_{-}(\vec{k})\end{aligned}$$ with the scalar self-energies$$\begin{aligned}
{\Pi_\text{T}}(\omega,k)=\frac{m_{D}^{2}}{2}\left(1+\frac{\omega^{2}-k^{2}}{k^{2}}{\Pi_\text{L}}(\omega,k)\right), & & {\Pi_\text{L}}(\omega,k)=m_{D}^{2}\left(1-\frac{\omega}{2k}\ln\frac{\omega+k}{\omega-k}\right),\label{eq:HTL PiT}\\
\text{and}\quad\quad\Sigma_{\pm}(\omega,k) & = & \frac{\hat{M}^{2}}{k}\left(1-\frac{\omega\mp k}{2k}\ln\frac{\omega+k}{\omega-k}\right),\label{eq:HTL Sigmapm}\end{aligned}$$ where $\hat{M}(T,\,\mu,\, g^{2})$ is the thermal fermion mass or plasma frequency and $m_{D}(T,\,\mu,\, g^{2})$ denotes the Debye screening mass$$\begin{aligned}
m_{D}^{2}=\!\left(\left[2N_{c}\!+\! N_{q}\!+\! N_{s}\right]T^{2}+\frac{N_{c}}{\pi^{2}}\sum_{i}\mu_{i}\right)\frac{g^{2}}{6} & \quad\text{and}\quad & \hat{M}^{2}=\frac{N_{c}^{2}-1}{16N_{c}}\left(T^{2}+\frac{\mu^{2}}{\pi^{2}}\right)g^{2}.\label{eq:mDebye and plasma freq}\end{aligned}$$ The number of colors $N_{c}$ is fixed at $3$. The numbers of light quarks $N_{q}=2$ and one strange quark, $N_{s}=1$, are chosen as in the lattice calculations [@Kar07].
Properties of HTL self-energies and dispersion relations
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The real and the imaginary parts of the HTL self-energies (\[eq:HTL PiT\]) and (\[eq:HTL Sigmapm\]) are [@LeB96]$$\begin{aligned}
{\text{Re}}\Pi_{T}=\frac{m_{D}^{2}}{2}\left(1+\frac{\omega^{2}\!-\! k^{2}}{k^{2}}{\text{Re}}{\Pi_\text{L}}(\omega,k)\right)\!, & & {\text{Im}}\Pi_{T}=\frac{1}{2}m_{D}^{2}\frac{\omega^{2}\!-\! k^{2}}{k^{2}}\frac{\omega}{2k}\pi\Theta\left(k^{2}\!-\!\omega^{2}\right)\varepsilon(k),\label{eq:Im PiT}\\
{\text{Re}}\Pi_{L}=m_{D}^{2}\left(1-\frac{\omega}{2k}\ln\left|\frac{\omega+k}{\omega-k}\right|\right)\!, & & {\text{Im}}\Pi_{L}=m_{D}^{2}\frac{\omega}{2k}\pi\Theta\left(k^{2}-\omega^{2}\right)\varepsilon(k),\label{eq:Im PiL}\\
{\text{Re}}\Sigma_{\pm}=\frac{\hat{M}^{2}}{k}\left(1-\frac{\omega\mp k}{2k}\ln\left|\frac{\omega+k}{\omega-k}\right|\right)\!, & & {\text{Im}}\Sigma_{\pm}=\frac{\hat{M}^{2}}{k}\frac{\omega\mp k}{2k}\pi\Theta\left(k^{2}-\omega^{2}\right)\varepsilon(k),\label{eq:Im Sigmapm}\end{aligned}$$ where ${\varepsilon}(k)$ is the sign function. The real parts are symmetric with respect to $\omega$, while the imaginary parts are antisymmetric and differ from zero only for $|\omega|<k$, i.e. below the light cone. The real and imaginary parts for $k=0.5T$ are shown in Figure \[fig:gluon se\]. Analogously, the quark self-energies fulfill the parity relations ${\text{Re}}\Sigma_{+}(-\omega)={\text{Re}}\Sigma_{-}(\omega)$ and ${\text{Im}}\Sigma_{+}(-\omega)=-{\text{Im}}\Sigma_{-}(\omega)$ as shown for $k=0.5T$ in Figure \[fig:quark se\].
![The real and imaginary parts of the retarded transverse (left) and longitudinal (right) gluon self-energies scaled by the Debye mass squared are shown as functions of the energy $\omega$ scaled by the momentum $k$ which is fixed at $k=0.5T$. \[fig:gluon se\]](PiTGraph2.eps "fig:") ![The real and imaginary parts of the retarded transverse (left) and longitudinal (right) gluon self-energies scaled by the Debye mass squared are shown as functions of the energy $\omega$ scaled by the momentum $k$ which is fixed at $k=0.5T$. \[fig:gluon se\]](PiLGraph2.eps "fig:")
![The real and imaginary parts of the retarded quark self-energies for the normal (left) and abnormal branch (right) scaled by the plasma frequency squared are shown as functions of the energy $\omega$ scaled by the momentum $k$ which is fixed at $k=0.5T$. \[fig:quark se\]](SigmaPlusGraph2.eps "fig:") ![The real and imaginary parts of the retarded quark self-energies for the normal (left) and abnormal branch (right) scaled by the plasma frequency squared are shown as functions of the energy $\omega$ scaled by the momentum $k$ which is fixed at $k=0.5T$. \[fig:quark se\]](SigmaMinusGraph2.eps "fig:")
It follows directly from eqs. (\[eq:Im PiT\])-(\[eq:Im Sigmapm\]) that the HTL self-energies do not account for quasiparticle widths since the imaginary parts are zero at the poles of the quasiparticle propagators, i.e. above the light cone. The nonzero imaginary parts of the self-energies below the light cone are due to *Landau damping* (LD). LD is a collective effect caused by energy transfer between the gauge field and plasma particles with velocities close to the phase velocity (“resonant particles”).
Even though the imaginary parts are formally nonzero only below the light cone, retardation leads to an infinitely small contribution even above the light cone, giving a definite sign to the self-energies for all energies: ${\varepsilon}({\text{Im}}{\Pi_\text{T}}(\omega))=-{\varepsilon}(\omega)$, ${\varepsilon}({\text{Im}}{\Pi_\text{L}}(\omega))=+{\varepsilon}(\omega)$ and ${\varepsilon}({\text{Im}}\Sigma_{\pm}(\omega))=\mp1$. Note that this is not related to Landau damping which is found below the light cone only.
The HTL propagators follow from Dyson’s equations as ${D_\text{T}}^{-1}=-\omega^{2}+k^{2}+{\Pi_\text{T}}$, ${D_\text{L}}^{-1}=-k^{2}-{\Pi_\text{L}}$ and $S_{\pm}^{-1}=-\omega\pm(k+\Sigma_{\pm})$. On-shell (quasi)particles satisfy a dispersion relation determined by ${\text{Re}}D_{\text{T},\text{L}}^{-1}=0$ and ${\text{Re}}S_{\pm}^{-1}=0$ respectively. It is, therefore, useful to first investigate the real part of the inverse retarded HTL propagators.
![The real parts of the inverse gluon propagators $D_{\text{T,L}}^{-1}$ scaled by the Debye screening mass squared are shown as functions of the energy $\omega$ scaled by the momentum $k$ which is fixed at $k=0.5T$. \[fig:ReGluPropm1\]](ReDTm1xGraph2.eps "fig:") ![The real parts of the inverse gluon propagators $D_{\text{T,L}}^{-1}$ scaled by the Debye screening mass squared are shown as functions of the energy $\omega$ scaled by the momentum $k$ which is fixed at $k=0.5T$. \[fig:ReGluPropm1\]](ReDLm1xGraph2.eps "fig:")
![The real parts of the inverse quark propagators $S_{\pm}^{-1}$ scaled by the fermionic mass parameter squared are shown as functions of the energy $\omega$ scaled by the momentum $k$ which is fixed at $k=0.5T$. \[fig:ReQuarkPropm1\]](ReSplusm1xGraph2.eps "fig:") ![The real parts of the inverse quark propagators $S_{\pm}^{-1}$ scaled by the fermionic mass parameter squared are shown as functions of the energy $\omega$ scaled by the momentum $k$ which is fixed at $k=0.5T$. \[fig:ReQuarkPropm1\]](ReSminusm1xGraph2.eps "fig:")
Due to symmetry properties of $D_{\text{T,L}}^{-1}$ there is just one positive-energy dispersion relation above the light cone: $\omega_{\text{T},k}$ and $\omega_{\text{L},k}$, respectively. This means that - up to the sign - transverse and longitudinal gluons have the same dispersion relations as their anti(quasi)particle counterparts. The additional tachyonic dispersion relation for longitudinal gluons is related to Landau damping. Figure \[fig:ReGluPropm1\] explicitly shows the real parts for fixed momentum $k=0.5T$. Both $D_{\text{T},\text{L}}^{-1}$ are symmetric with respect to $\omega$. The zero of ${\text{Re}}{D_\text{T}}^{-1}$ determines the dispersion relation $\omega_{\text{T},k}$ for transverse gluons. The zero of ${\text{Re}}{D_\text{L}}^{-1}$ above the light cone indicates the dispersion relation $\omega_{\text{L},k}$ of longitudinal gluons, while the tachyonic dispersion relation $\omega_{\text{L},k}^{t}$ (below the light cone) is due to Landau damping.
The inverse quark propagators are not symmetric but, as a consequence of the symmetry of the self-energy, satisfy the parity property ${\text{Re}}S_{+}^{-1}(-\omega)=-{\text{Re}}S_{-}^{-1}(\omega)$ (cf. Figure \[fig:ReQuarkPropm1\]). Hence, quarks are described by the positive energy dispersion relation related to $S_{+}$, while the dispersion relation of antiquarks is found from the negative energy solution of ${\text{Re}}S_{-}^{-1}=0$. The remaining two dispersion relations represent collective quark excitations: the positive energy dispersion relation related to $S_{-}$ describes the plasminos, while the negative energy solution of ${\text{Re}}S_{+}^{-1}=0$ represents antiplasminos. Again, a tachyonic solution appears within the regime of Landau damping.
The evolution of the zeros of the real part of the inverse retarded propagators as a function of the momentum $k$ gives the dispersion relations $\omega_{i,k}$. These dispersion relations cannot be expressed as analytic functions $\omega(k)$ in closed form. ${\text{Re}}D_{i}^{-1}(\omega,\, k,\,\Pi_{i}(\omega,k))=0$ and ${\text{Re}}S_{i}^{-1}(\omega,\, k,\,\Sigma_{i}(\omega,k))=0$ lead to transcendental equations and have to be solved numerically. The results are shown in Figures \[fig:disp gluons\] and \[fig:disp quarks\]. Due to the above parity property quarks and antiquarks obey identical dispersion relations up to the sign, as do plasminos and antiplasminos.
![The dispersion relations $\omega_{\text{T},k}$ of transverse and $\omega_{\text{L},k}$ of longitudinal gluon modes scaled by the Debye screening mass are shown as functions of the momentum $k$ scaled by the Debye screening mass in linear (left) and quadratic (right) scales.\[fig:disp gluons\]](GluonsLinear2.eps "fig:") ![The dispersion relations $\omega_{\text{T},k}$ of transverse and $\omega_{\text{L},k}$ of longitudinal gluon modes scaled by the Debye screening mass are shown as functions of the momentum $k$ scaled by the Debye screening mass in linear (left) and quadratic (right) scales.\[fig:disp gluons\]](GluonsSquared2.eps "fig:")
![The dispersion relations $\omega_{i,k}$ of quarks (solid black), antiquarks (dashed grey), plasminos (black dashes) and antiplasminos (black points) scaled by the fermionic mass parameter are shown as functions of the momentum $k$ scaled by the fermionic mass parameter in linear (left) and quadratic (right) scales. \[fig:disp quarks\]](QuarksLinearBW2.eps "fig:") ![The dispersion relations $\omega_{i,k}$ of quarks (solid black), antiquarks (dashed grey), plasminos (black dashes) and antiplasminos (black points) scaled by the fermionic mass parameter are shown as functions of the momentum $k$ scaled by the fermionic mass parameter in linear (left) and quadratic (right) scales. \[fig:disp quarks\]](QuarksSquaredBW2.eps "fig:")
2-loop QCD entropy
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Given the explicit form of the HTL self-energies and the respective propagators, we evaluate the remaining traces $\text{tr}$ in eq. (\[eq:Omegafinal\]). Assuming equal masses for $u$ and $d$ quarks and zero chemical potential of strange quarks the isospin chemical potential $\mu_{I}=(\mu_{u}-\mu_{d})/2$ is supposed to vanish at zero net charge. Therefore, there is only one independent chemical potential $\mu=\mu_{q}=\mu_{B}/3$, where $\mu_{B}$ is the baryo-chemical potential. As a consequence the flavor trace gives equal contributions of light quarks to the thermodynamic potential (i.e. a factor $N_{q}$). The contribution of the strange quark flavor is here supposed to be equal to the light quark contribution up to a substitution of $\mu\rightarrow0$ and $N_{q}\rightarrow N_{s}$ in the following formulae.
Taking the trace in Minkowski space, the gluonic part decomposes into three contributions for one longitudinal and two (equivalent) transverse polarizations, while the quark contribution becomes the sum of the normal and the abnormal quark branch (positive and negative chirality over helicity ratio, respectively) when taking the Dirac trace. The remaining traces only give overall factors: the color trace $(N_{c}^{2}-1)$ for the gluons and $N_{c}$ for the quarks, and the spin traces for quarks an additional $2$. Defining the prefactors $d_{g}=N_{c}^{2}-1$, $d_{q}=2N_{c}N_{q}$ and $d_{s}=2N_{c}N_{s}$ and introducing the abbreviation $\int_{{\mathrm{d}^{n}k}}=\int{\mathrm{d}^{n}k}/(2\pi)^{4}$ the HTL grand canonical potential then reads$$\begin{aligned}
\frac{\Omega}{V} & = & d_{g}\int_{{\mathrm{d}^{4}k}}\!\!{n_\text{B}}\,\Big\{2{\text{Im}}\!\left(\ln{D_\text{T}}^{-1}-{D_\text{T}}{\Pi_\text{T}}\right)+{\text{Im}}\!\left(\ln\left(-{D_\text{L}}^{-1}\right)+{D_\text{L}}{\Pi_\text{L}}\right)\!\Big\}\\
& & +2\sum_{i=q,s}d_{i}\int_{{\mathrm{d}^{4}k}}\!\!{n_\text{F}}\,\Big\{{\text{Im}}\!\left(\ln S_{+}^{-1}-S_{+}\Sigma_{+}\right)+{\text{Im}}\!\left(\ln\left(-S_{-}^{-1}\right)+S_{-}\Sigma_{-}\right)\!\Big\}-\frac{T}{V}\Gamma_{2}.\nonumber \end{aligned}$$
Differentiating the thermodynamic potential with respect to the temperature at constant chemical potential gives the entropy. In contrast to the pressure, which is influenced by vacuum fluctuations, the entropy is sensitive to thermal excitations and therefore manifestly ultraviolet (UV) finite. As such, it is ideally suited to investigate the properties of the QGP [@BIR01].
Due to the stationarity of the thermodynamic potential with respect to the full propagators, $\delta\Omega/\delta D=0$, only the derivatives of the statistical distribution functions contribute. Using ${\text{Im}}({D_\text{T}}{\Pi_\text{T}})={\text{Re}}{D_\text{T}}{\text{Im}}{\Pi_\text{T}}+{\text{Im}}{D_\text{T}}{\text{Re}}{\Pi_\text{T}}$, the entropy density can be written as $s:=-V^{-1}\left.\partial\Omega/\partial T\right|_{\mu}=s_{g,\text{T}}+s_{g,\text{L}}+\sum_{q,s}(s_{i,+}+s_{i,-})$ with$$\begin{aligned}
s_{g,\text{T}} & = & -2d_{g}\int_{{\mathrm{d}^{4}k}}\frac{\partial{n_\text{B}}(\omega)}{\partial T}\Big\{\mbox{Im}\ln\left(+{D_\text{T}}^{-1}\right)-\mbox{Re}{D_\text{T}}\mbox{Im}{\Pi_\text{T}}\Big\},\label{eq:sgT ImLn}\\
s_{g,\text{L}} & = & -\,\,\, d_{g}\int_{{\mathrm{d}^{4}k}}\frac{\partial{n_\text{B}}(\omega)}{\partial T}\Big\{\mbox{Im}\ln\left(-{D_\text{L}}^{-1}\right)+\mbox{Re}{D_\text{L}}\mbox{Im}{\Pi_\text{L}}\Big\},\label{eq:sgL ImLn}\\
s_{q/s,\pm} & = & -2d_{q/s}\!\int_{{\mathrm{d}^{4}k}}\!\!\frac{\partial{n_\text{F}}(\omega)}{\partial T}\Big\{\mbox{Im}\ln\left(\pm S_{\pm}^{-1}\right)\mp\mbox{Re}S_{\pm}\mbox{Im}\Sigma_{\pm}\Big\},\label{eq:sq ImLn}\end{aligned}$$ each describing the entropy density of one quasiparticle species in the absence of the others. An interaction (correlation) entropy density contribution would contain terms of the form ${\text{Im}}{D_\text{T}}{\text{Re}}{\Pi_\text{T}}$ and the derivative of $\Gamma_{2}T$ with respect to the temperature. However, at 2-loop order these terms exactly cancel each other [@BIR01]. In fact, this seems to be a generic, topological feature [@CP75] which has explicitly been proven for QED [@VB98] and $\Phi^{4}$ theory [@Pes01] too.
We now focus on the terms ${\text{Im}}\ln(\pm D_{\text{T,L}}^{-1})$ and ${\text{Im}}\ln(\pm S_{\pm}^{-1})$, which can be written as$$\begin{aligned}
{\text{Im}}\ln{D_\text{T}}^{-1} & = & \arctan\left(\frac{{\text{Im}}{D_\text{T}}^{-1}}{{\text{Re}}{D_\text{T}}^{-1}}\right)+\pi\varepsilon({\text{Im}}{D_\text{T}}^{-1})\Theta\!\left(-{\text{Re}}{D_\text{T}}^{-1}\right),\label{eq:ImLn+DTm1}\\
{\text{Im}}\ln\left(-{D_\text{L}}^{-1}\right) & = & \arctan\left(\frac{{\text{Im}}{D_\text{L}}^{-1}}{{\text{Re}}{D_\text{L}}^{-1}}\right)-\pi\varepsilon({\text{Im}}{D_\text{L}}^{-1})\Theta\!\left(+{\text{Re}}{D_\text{L}}^{-1}\right).\label{eq:ImLn-DLm1}\end{aligned}$$ Similar expressions apply for the two quark propagators: one has to substitute $S_{+}^{-1}$ for ${D_\text{T}}^{-1}$ in (\[eq:ImLn+DTm1\]) and $S_{-}^{-1}$ for ${D_\text{L}}^{-1}$ in (\[eq:ImLn-DLm1\]).
From the properties of the imaginary parts of the self-energies discussed above, we find ${\varepsilon}({\text{Im}}D_{i}^{-1}(\omega))=-{\varepsilon}(\omega)$ for the gluons and ${\varepsilon}({\text{Im}}S_{\pm}(\omega))\equiv-1$ for the normal and abnormal quark branches. We end up with$$\begin{aligned}
s_{g,\text{T}} & = & +2d_{g}\int_{{\mathrm{d}^{4}k}}\frac{\partial{n_\text{B}}}{\partial T}\Big\{\pi\varepsilon(\omega)\Theta\!\left(-{\text{Re}}{D_\text{T}}^{-1}\right)-\arctan\frac{{\text{Im}}{\Pi_\text{T}}}{{\text{Re}}{D_\text{T}}^{\text{-}1}}+\mbox{Re}{D_\text{T}}\mbox{Im}{\Pi_\text{T}}\Big\},\label{eq:si eq theta arctan reim gT}\\
s_{g,\text{L}} & = & -\,\,\, d_{g}\int_{{\mathrm{d}^{4}k}}\frac{\partial{n_\text{B}}}{\partial T}\Big\{\pi\varepsilon(\omega)\Theta\!\left(+{\text{Re}}{D_\text{L}}^{-1}\right)-\arctan\frac{{\text{Im}}{\Pi_\text{L}}}{{\text{Re}}{D_\text{L}}^{\text{-}1}}+\mbox{Re}{D_\text{L}}\mbox{Im}{\Pi_\text{L}}\Big\},\label{eq:si eq theta arctan reim gL}\\
s_{q/s,\pm} & = & \pm2d_{q/s}\!\int_{{\mathrm{d}^{4}k}}\frac{\partial{n_\text{F}}}{\partial T}\Big\{\quad\,\,\,\pi\Theta\!\left(\mp{\text{Re}}S_{\pm}^{-1}\right)-\arctan\frac{{\text{Im}}\Sigma_{\pm}}{{\text{Re}}S_{\pm}^{\text{-}1}}\,+\mbox{Re}S_{\pm}\mbox{Im}\Sigma_{\pm}\Big\}.\label{eq:si eq theta arctan reim q}\end{aligned}$$ The partial entropy densities (\[eq:si eq theta arctan reim gT\])-(\[eq:si eq theta arctan reim q\]) and, therefore the whole entropy density expression, are independent of possible renormalization factors. As required, the expression is also explicitly UV finite, as the derivatives of the distribution functions soften the UV behavior. The terms $\pi\Theta(\ldots)$ represent the quasiparticle contributions to the entropy, while the terms containing the imaginary parts of the self-energies are related to damping effects and quasiparticle widths. In the case of HTL self-energies, Landau damping is contained within the latter terms.
The quark entropy density $s_{q/s}=s_{q/s,+}+s_{q/s,-}$ can be simplified by utilizing the parity properties for quark propagators and self-energies. Introducing the distribution function of antiparticles ${n_\text{F}}^{A}=(e^{\beta(\omega+\mu)}+1)^{-1}$ with $\partial{n_\text{F}}(-\omega)/\partial\omega=-\partial{n_\text{F}}^{A}(\omega)/\partial\omega$ and substituting $\omega\rightarrow-\omega$ within $s_{q,-}$, we find$$s_{q/s}=2d_{q/s}\int_{{\mathrm{d}^{4}k}}\left(\frac{\partial{n_\text{F}}}{\partial T}\!+\!\frac{\partial{n_\text{F}}^{A}}{\partial T}\right)\left\{ \pi\Theta\!\left(\text{-}{\text{Re}}S_{+}^{-1}\right)-\arctan\!\left(\frac{{\text{Im}}\Sigma_{+}}{{\text{Re}}S_{+}^{\text{-}1}}\!\right)+\mbox{Re}S_{+}\mbox{Im}\Sigma_{+}\right\} .\label{eq:sq combined}$$ Regarding the quasiparticle pole term $\pi\Theta(\text{-}{\text{Re}}S_{+}^{-1})$, the energy integration from $-\infty$ to $0$ gives the (anti)plasmino contribution, while the integration from $0$ to $+\infty$ delivers the contributions of the (anti)particles to the entropy density. Isolating both parts of the spectrum by applying the parity properties once more gives the explicit expressions$$\begin{aligned}
s_{q/s,\text{TL}} & \!\!=\!\! & \,\,\,\,\,2d_{q/s}\int_{{\mathrm{d}^{3}k}}\int\limits _{0}^{\infty}\!\frac{{\mathrm{d}\omega}}{2\pi}\,(.)\left\{ \pi\Theta\!\left(\text{-}{\text{Re}}S_{+}^{-1}\right)-\arctan\!\left(\frac{{\text{Im}}\Sigma_{+}}{{\text{Re}}S_{+}^{\text{-}1}}\!\right)+\mbox{Re}S_{+}\mbox{Im}\Sigma_{+}\!\right\} ,\label{eq:sq TL-Pl split}\\
s_{q/s,\text{Pl}} & \!\!=\!\! & -2d_{q/s}\int_{{\mathrm{d}^{3}k}}\int\limits _{0}^{\infty}\!\frac{{\mathrm{d}\omega}}{2\pi}\,(.)\left\{ \pi\Theta\!\left(\,\,{\text{Re}}S_{-}^{-1}\right)-\arctan\!\left(\frac{{\text{Im}}\Sigma_{-}}{{\text{Re}}S_{-}^{\text{-}1}}\!\right)+\mbox{Re}S_{-}\mbox{Im}\Sigma_{-}\!\right\} ,\label{eq:sq TL-Pl split Pl}\end{aligned}$$ where the sum of the derivatives of the distribution functions is abbreviated by the parentheses $(.)$. While this separation seems straightforward, it has to be handled with care as the Landau damping term within the quark self-energies $\Sigma_{\pm}$ (see the imaginary parts in Figure \[fig:quark se\]) can, in general, not be separated into quark and plasmino contributions in this simple way.
The full HTL QPM
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Since the entropy density of the quark-gluon plasma for 2-loop QCD is the sum of the single quasiparticle entropy density contributions, it can be considered as mixture of non-interacting ideal quasiparticle gases. It is natural to assume that the pressure, which follows from the entropy density by integration, consists of single partial pressures, too. Therefore, we use the ansatz $p=p_{g,\text{T}}+p_{g,\text{L}}+\sum_{i=q,s}p_{i}-B({\Pi_\text{T}},{\Pi_\text{L}},\Sigma_{\pm})$ for the pressure, where $B$ is chosen appropriately to ensure thermodynamic consistency. The ansatz has to satisfy $s_{i}=\partial p_{i}/\partial T|_{\mu}$ which leads to$$\begin{aligned}
p_{g,\text{T}} & = & +2d_{g}\int_{{\mathrm{d}^{4}k}}{n_\text{B}}\Big\{\pi\varepsilon(\omega)\Theta\!\left(-{\text{Re}}{D_\text{T}}^{-1}\right)-\arctan\frac{{\text{Im}}{\Pi_\text{T}}}{{\text{Re}}{D_\text{T}}^{\text{-}1}}+\mbox{Re}{D_\text{T}}\mbox{Im}{\Pi_\text{T}}\Big\},\\
p_{g,\text{L}} & = & -\,\,\, d_{g}\int_{{\mathrm{d}^{4}k}}{n_\text{B}}\Big\{\pi\varepsilon(\omega)\Theta\!\left(+{\text{Re}}{D_\text{L}}^{-1}\right)-\arctan\frac{{\text{Im}}{\Pi_\text{L}}}{{\text{Re}}{D_\text{L}}^{\text{-}1}}+\mbox{Re}{D_\text{L}}\mbox{Im}{\Pi_\text{L}}\,\Big\},\\
p_{q/\! s} & = & 2d_{q/\! s}\!\int_{{\mathrm{d}^{4}k}}\!\!\left({n_\text{F}}\!+\!{n_\text{F}}^{A}\right)\!\Big\{\pi\Theta\!\left(-{\text{Re}}S_{+}^{-1}\right)-\arctan\frac{{\text{Im}}\Sigma_{+}}{{\text{Re}}S_{+}^{\text{-}1}}\,+\mbox{Re}S_{+}\mbox{Im}\Sigma_{+}\Big\},\end{aligned}$$ where the integrability condition $\partial B/\partial\Pi_{i}=\partial p/\partial\Pi_{i}$ has to be fulfilled for every quasiparticle species $i$. Thus $B$ ensures the stationarity of the thermodynamic potential under functional variation with respect to the self-energies [@GY95]. Note that the plasma frequency within the $s$-quark pressure differs from the plasma frequency within $p_{q}$ as $\mu_{s}=0$.
The pressure fully defines the model. The particle density follows by differentiation of the pressure with respect to the chemical potential at constant temperature. The Bose-Einstein distribution function ${n_\text{B}}$ does not depend on $\mu_{g}$ and strange quarks are included into the model with manifest zero net particle density, therefore $n_{g,\text{T}}=n_{g,\text{L}}=n_{s}=0$ . Due to the integrability condition, the terms containing the derivatives of the self-energies with respect to $\mu$ vanish, so that$$n\!=\! n_{q}\!\!=\!2d_{q}\!\!\int_{{\mathrm{d}^{4}k}}\!\!\left(\frac{\partial{n_\text{F}}}{\partial\mu}\!+\!\frac{\partial{n_\text{F}}^{A}}{\partial\mu}\right)\!\!\left\{ \pi\Theta\!\left(\text{-}{\text{Re}}S_{+}^{-1}\right)\!-\!\arctan\frac{{\text{Im}}\Sigma_{+}}{{\text{Re}}S_{+}^{\text{-}1}}\!+\!\mbox{Re}S_{+}\mbox{Im}\Sigma_{+}\right\} ,\label{eq:fullHTL nq}$$ thus $n_{q}(\mu\rightarrow0)\rightarrow0$.
Effective coupling
------------------
Obviously, 2-loop QCD is only a crude approximation of the full theory. In order to accommodate non-perturbative effects in the quasiparticle model, we introduce some flexibility by parameterizing the QCD coupling constant $g^{2}$ in a phenomenologically motivated way. The truncated 2-loop running QCD coupling $g^{2}$ is given by $g^{2}(x)=16\pi^{2}(\beta_{0}\ln(x))^{-1}\,(1-2\beta_{1}\ln[\ln(x)]\,(\beta_{0}^{2}\ln(x))^{-1})$ [@PDG06], where $\beta_{0}=11N_{c}/3-2N_{f}/3$, $\beta_{1}=51-19N_{f}/3$ and $N_{f}=N_{q}+N_{s}$. It depends on the ratio $x=\bar{\mu}^{2}/\Lambda^{2}$ of the renormalization scale $\bar{\mu}$ and the QCD scale parameter $\Lambda$. A term involving $\ln^{-2}(\bar{\mu}^{2}/\Lambda^{2})$ which is only a small correction for $\bar{\mu}^{2}\approx\Lambda^{2}$ was neglected.
The renormalization scale is usually taken to be the first Matsubara frequency $2\pi T$, while the latter one is just a parameter to be adjusted using experimental data. Introducing the pseudocritical temperature of QCD matter at vanishing net baryon density $T_{c}$ and substituting $\Lambda\rightarrow2\pi T_{c}/\lambda$, the ratio $\bar{\mu}/\Lambda$ becomes $\lambda T/T_{c}$. In order to avoid the Landau pole of $g^{2}(T/T_{c})$ at $T_{c}$ a temperature shift with parameter $T_{s}$ is introduced, replacing $x$ by $\xi=\lambda(T-T_{s})/T_{c}$. The result $$G^{2}(T\geq T_{c},\mu=0)=\frac{16\pi^{2}}{\beta_{0}\ln\xi^{2}}\left(1-\frac{2\beta_{1}}{\beta_{0}^{2}}\frac{\ln\left[\ln\xi^{2}\right]}{\ln\xi^{2}}\right)\label{eq:eff coupling}$$ is our *effective coupling*. Within the plasma phase for temperatures $T>T_{c}/\lambda+T_{s}$ it is well-behaved; however, at some point within the hadronic phase, i.e. below $T_{c}$, an infrared (IR) divergence does occur. In order to prevent this divergence a phenomenological infrared cutoff for $G^{2}$ can be applied. See [@Blu04a] for details.
Adjustment to lattice calculations
----------------------------------
The two QPM parameters $\lambda$ and $T_{s}$ have to be adjusted to results of numerical first-principle QCD calculations dubbed lattice data. Most of the past work on the QPM has been tested against lattice data from [@KLP00] for rather large and temperature dependent lattice restmasses of $m_{q}=0.4T$ and $m_{s}=1.0T$ compared to the physical quark masses $m_{u,d}\sim10\,{\text{M}\hspace{-.5px}e\hspace{-.8px}\text{V}}$ and $m_{s}\sim90-150\,{\text{M}\hspace{-.5px}e\hspace{-.8px}\text{V}}$ [@PDG06]. Recently, new lattice data has become available [@Kar07], which relies on lattice restmasses much closer to the physical quark masses and which is used in this work.
Also, lattice calculations are performed on a finite lattice, while our quasiparticle model is formulated in the thermodynamic limit, i.e. aimed at describing a spatially infinite plasma. In order to compare our model with lattice data, the proper continuum extrapolation of the latter one is required. A safe continuum extrapolation on the lattice is a fairly demanding work. Therefore, various estimates have been applied, e.g. simply scaling the lattice results by a factor being strictly valid only for asymptotically high temperatures or for the non-interacting limit. To account for a possible deficit of such rough continuum estimates of the lattice data we introduce an ad hoc scaling factor $d_{\text{lat}}$ which turns out to be nearly unity.
Nonzero chemical potential
--------------------------
The parametrization of $G^{2}$ in eq. (\[eq:eff coupling\]) is valid for $\mu=0$ only. However, it is possible to use the thermodynamic consistency of the QPM to map the results at zero chemical potential into the $T$-$\mu$ plane [@Pes00]. Specifically, this means to impose the Maxwell relation $\partial s/\partial\mu|_{T}=\partial n/\partial T|_{\mu}$ on the thermodynamic quantities. Ordering the terms with respect to the partial derivatives of the effective coupling gives an elliptic quasilinear partial differential equation$$a_{T}\frac{\partial G^{2}}{\partial T}+a_{\mu}\frac{\partial G^{2}}{\partial\mu}=b,\label{eq:floweq}$$ named hereafter flow equation, with the coefficients $a_{T}$, $a_{\mu}$ and $b$ depending on $T$ and $\mu$ explicitly and, via the self-energies, implicitly. It is solved by the method of characteristics by introducing a curve parameter $x$, assuming that $T=T(x)$, $\mu=\mu(x)$ and $G^{2}=G^{2}(x)$. Subsequently, the comparison of $G_{,x}^{2}=G_{,T}^{2}T_{,x}+G_{,\mu}^{2}\mu_{,x}$ with the flow equation gives a system of three linear, coupled ordinary differential equations: $G_{,x}^{2}=-b$, $T_{,x}=-a_{T}$ and $\mu_{,x}=-a_{\mu}$ which can be solved using standard numerical methods. The initial condition for the flow equation is the effective coupling at $\mu=0$, with model parameters fixed by comparison of the entropy density with lattice results.
The effective QPM {#sec:eQP}
=================
Assuming that transversal gluons and quark particle excitations propagate predominantly on mass shells the full HTL QPM can be significantly simplified as it implies explicit (asymptotic) dispersion relations $\omega_{i}(k)$ of the form $\omega_{i}^{2}(k)=k^{2}+m_{i,\infty}^{2}$ as approximations to the full, implicit ones [@Pes02]. The $m_{i,\infty}$ terms depend neither on energy nor momentum and are therefore called asymptotic (thermal) masses. In order to adjust the eQP to lattice data, they can be modified to accomodate lattice restmasses $m_{i}$ using a prescription from [@Pis93]: $m_{i,\infty}^{2}\rightarrow m_{i}^{2}+2m_{i}m_{i,\infty}+2m_{i,\infty}^{2}$. Additionally neglecting Landau damping, i.e. assuming vanishing imaginary parts of the self-energies, and contributions from collective excitations, i.e. plasmons and (anti)plasminos, leads to the eQP. As collective excitations are exponentially suppressed[^2] and the effect of Landau damping is small at vanishing chemical potential the eQP seemed to be sufficient.
However, as previous studies of the flow equation [@Pes02; @Blu04a] have shown the characteristic curves emerging at $T\approx T_{c}$ cross each other in some region of finite values of $\mu$ for parameters adjusted to lattice QCD results (cf. dashed lines in Figure \[fig:HTL characteristics\] below). This unfortunate feature prevents an unambiguous extrapolation of thermodynamic quantities into the full $T$-$\mu$-plane. Romatschke [@RR03] has shown for the $2$ flavor case that these crossings can be avoided by using the full HTL model. We are going to extend his line of arguing to the physically interesting case of $2+1$ flavors.
Investigation of the full HTL QPM\[sec:fullHTL\]
================================================
Adjustment to lattice data at $\mu=0$
-------------------------------------
While the eQP is able to accommodate arbitrary lattice restmasses by means of a modified asymptotic dispersion relation, the full HTL model relies on the HTL dispersion relations and thus massless particles. We assume here that the employed lattice restmasses in [@Kar07] are sufficiently small to be absorbed in suitably adjusted parameters.
For $T>T_{c}$ and $N_{f}=2$ the full HTL model has been shown to give a description of lattice data being equally well as the eQP [@Rom04]. For $N_{f}=2+1$ flavors we meet a similar situation: The full HTL QPM describes the lattice QCD data [@Kar07] as good as the eQP model (see Figure \[fig:fullHTL fit quad\], left). The extension to $T<T_{c}$, on the other hand, is not straightforward. Instead of a linear IR regulator [@Blu04b], it is necessary to use a quadratic parametrization of the effective coupling in order to achieve agreement with lattice data also below the pseudocritical temperature, which we consider here as mere parametrisation of the lattice data.
![The scaled entropy densities $s/T^{3}$ of the full HTL QPM with quadratic IR regulator (solid black lines; $T_{s}=0.728T_{c}$ and $\lambda=6.10$) and the eQP (grey dashed lines; $T_{s}=0.752T_{c}$ and $\lambda=6.26$) adjusted to lattice data for $N_{f}=2+1$ from [@Kar07] with $d_{\text{lat}}=0.96$ are shown as functions of the scaled temperature $T/T_{c}$. The adjustment quality of the full HTL QPM to lattice data is indistinguishable from the eQP. The single contributions to $s^{HTL}$, including their respective LD contributions, are given in the right figure (dashed black: transversal gluons+(anti)quarks, dash-dotted: longitudinal gluons, dotted grey: (anti)plasminos). \[fig:fullHTL fit quad\]\[fig:sHTL contributions\]](fullHTLquadFitSimplified.eps "fig:") ![The scaled entropy densities $s/T^{3}$ of the full HTL QPM with quadratic IR regulator (solid black lines; $T_{s}=0.728T_{c}$ and $\lambda=6.10$) and the eQP (grey dashed lines; $T_{s}=0.752T_{c}$ and $\lambda=6.26$) adjusted to lattice data for $N_{f}=2+1$ from [@Kar07] with $d_{\text{lat}}=0.96$ are shown as functions of the scaled temperature $T/T_{c}$. The adjustment quality of the full HTL QPM to lattice data is indistinguishable from the eQP. The single contributions to $s^{HTL}$, including their respective LD contributions, are given in the right figure (dashed black: transversal gluons+(anti)quarks, dash-dotted: longitudinal gluons, dotted grey: (anti)plasminos). \[fig:fullHTL fit quad\]\[fig:sHTL contributions\]](fullHTLcontributions2.eps "fig:")
When evaluating the individual contributions to the entropy density of the full HTL QPM we find the entropy density contributions of longitudinal gluon $s_{g,\text{L}}$ (eq. (\[eq:si eq theta arctan reim gL\])) and (anti)plasminos $s_{q,\text{Pl}}$ (eq. (\[eq:sq TL-Pl split Pl\])) to be negative. This is due to the fact that both represent collective phenomena of the QGP resulting in correlations not present in a noninteracting medium. As a consequence, the transverse gluon and (anti)quark entropy density contributions have to increase in comparison to the eQP in order for the sum of the partial entropy densities to describe the same lattice data as the eQP (see Figure \[fig:sHTL contributions\], right). This not only allows for a pure quasiparticle entropy contribution much closer to the Stefan-Boltzmann limit than in the eQP but also proves to have a positive impact on the extension to nonzero chemical potential.
Solution of the flow equation
-----------------------------
Solving the flow equation (\[eq:floweq\]) with coefficients listed in Appendix \[app:flow eq coeff\] the characteristics are found to be well-behaving, as can be seen in Figure \[fig:HTL characteristics\]. Also a stronger curvature of the full HTL characteristics compared to the eQP characteristics (shown as dashed lines in the right panel) is observable.
To explain the disappearance of the ambiguities caused by crossing characteristics in the eQP we mention that the crossings appear due to the effective coupling $G^{2}$ being too large near the pseudocritical temperature [@Blu04]. Since the entropy density increases with decreasing mass parameters $m_{D}^{2}$ and $\hat{M}^{2}$ (which are proportional to $G^{2}T^{2}$ at $\mu=0$) the crossings would therefore disappear for a larger eQP entropy density. One way to allow for a larger eQP entropy density is to take into account collective modes. As medium effects indicate correlations between the constituents of the eQP plasma, including them causes a decrease of overall entropy density. Consequently, the eQP parameters have to change in order to still describe the same lattice data, causing the entropy density to increase. With the resulting decrease of the effective coupling $G^{2}$ the crossings partially disappear.
However, the different parametrization at $\mu=0$ alone cannot account for the complete absence of ambiguities for the full HTL model. Instead, the influence of collective modes and Landau damping on the flow equation has to be examined. We therefore calculate the characteristics of the full HTL model respectively disregarding terms stemming from these contributions. While neglecting plasmon/(anti)plasminos terms from the coefficients $a_{T}$, $a_{\mu}$ and $b$ (see Appendix \[app:flow eq coeff\]) but keeping the Landau damping contributions leads to deformed characteristics meeting $T=0$ at smaller $\mu$ and no crossings appear. Hence, it is neither the plasmon nor the plasmino term which accounts for the vanishing crossings. However, neglecting the Landau damping terms immediately leads to crossing characteristics. Therefore, both collective excitations (in order to obtain a reasonably small coupling $G^{2}$) and Landau damping (in order to ultimately remove the crossings) are necessary to obtain a flow equation with unique solutions. Using this flow equation, it is possible to extrapolate the equation of state from lattice QCD at $\mu=0$ towards $T=0$.
![The solid curves in both graphs are several characteristics of the full HTL flow equation for $2+1$ quark flavors using parameters from the adjustment of the full HTL QPM to lattice data from [@Kar07] shown in Figure \[fig:fullHTL fit quad\]. The characteristic curve emerging from $T_{c}$ is depicted as bold solid line. All crossings have disappeared (left panel). For a comparison, the right panel shows the characteristics of the eQP flow equation using the parameters of the eQP adjustment to the same lattice QCD data (dashed curves).[]{data-label="fig:HTL characteristics"}](HTLcloseup2.eps "fig:") ![The solid curves in both graphs are several characteristics of the full HTL flow equation for $2+1$ quark flavors using parameters from the adjustment of the full HTL QPM to lattice data from [@Kar07] shown in Figure \[fig:fullHTL fit quad\]. The characteristic curve emerging from $T_{c}$ is depicted as bold solid line. All crossings have disappeared (left panel). For a comparison, the right panel shows the characteristics of the eQP flow equation using the parameters of the eQP adjustment to the same lattice QCD data (dashed curves).[]{data-label="fig:HTL characteristics"}](FlowEqMod2.eps "fig:")
Conclusion {#sec:conclusion}
==========
The mapping of a previous quasiparticle model (eQP) into the $T$-$\mu$ plane was plagued by crossing characteristics. It is shown here for the 2+1 flavor case that, if using the full HTL model, these crossings disappear. Collective modes (longitudinal gluon and (anti)quark hole excitations, i.e. plasmons and (anti)plasminos respectively) as well as Landau damping of the collisionless quasiparticle plasma, both neglected hitherto in the eQP, need to be taken into account to avoid the ambiguities.
With the problem of crossing characteristics solved, one can proceed to derive an equation of state, following from the full HTL QPM, especially for the cold and dense quark-gluon plasma of interest in future heavy ion collision experiments or for the simulation of possible quark stars.
*Acknowledgment:* R.S. would like to thank the organizers of the *Zimányi 75 Memorial Workshop* for the invitation to present his results at this very inspiring workshop.
Coefficients of the flow equation\[app:flow eq coeff\]
======================================================
For the reader’s convenience and to extend the results in [@Rom04] to $N_{f}=2+1$ flavors the full HTL flow equation is presented. To calculate the Maxwell relation, the derivatives $\partial s_{g}/\partial\mu=A_{g}\partial m_{D}^{2}/\partial\mu$, $(\partial s_{q}/\partial\mu)_{impl.}=A_{q}\partial\hat{M}^{2}/\partial\mu$, $(\partial s_{s}/\partial\mu)_{impl.}=A_{s}\partial\hat{M}^{2}/\partial\mu$ and $(\partial n_{q}/\partial T)_{impl.}=A_{n}\partial\hat{M}^{2}/\partial T$ are necessary. The explicit derivatives cancel within the Maxwell relation due to Schwarz’s Theorem. We find$$\begin{aligned}
A_{g} & = & \frac{d_{g}}{2\pi^{3}m_{D}^{2}}\int\limits _{0}^{\infty}{\mathrm{d}\hspace{-.5pt}k}\, k^{2}\Bigg(\int\limits _{0}^{k}{\mathrm{d}\omega}\left[\frac{\partial{n_\text{B}}}{\partial T}\frac{4(\omega^{2}-k^{2}){\text{Im}}^{3}{\Pi_\text{T}}}{({\text{Re}}^{2}{D_\text{T}}^{-1}\!+\!{\text{Im}}^{2}{\Pi_\text{T}})^{2}}\!-\!\frac{\partial{n_\text{B}}}{\partial T}\frac{2k^{2}{\text{Im}}^{3}{\Pi_\text{L}}}{({\text{Re}}^{2}{D_\text{L}}^{-1}\!+\!{\text{Im}}^{2}{\Pi_\text{L}})^{2}}\right]\label{eq:dsgdmu HTL}\\
& & \quad\quad\quad\quad\quad\quad\quad\quad-\,\,\pi\,\frac{\omega_{\text{T},k}(\omega_{\text{T},k}^{2}-k^{2})^{2}}{|(\omega_{\text{T},k}^{2}\!-\! k^{2})^{2}\!-\! m_{D}^{2}\omega_{\text{T},k}^{2}|}\left.\frac{\partial{n_\text{B}}}{\partial T}\right|_{\omega_{\text{T},k}}\!\!\!\!\!\!\!\!-\,\pi\,\frac{\omega_{\text{L},k}(\omega_{\text{L},k}^{2}-k^{2})}{|\omega_{\text{L},k}^{2}\!-\! k^{2}\!-\! m_{D}^{2}|}\left.\frac{\partial{n_\text{B}}}{\partial T}\right|_{\omega_{\text{L},k}}\Bigg)\nonumber \end{aligned}$$ $$\begin{aligned}
A_{q} & = & \frac{d_{q}}{2\pi^{3}\hat{M}^{2}}\int\limits _{0}^{\infty}{\mathrm{d}\hspace{-.5pt}k}\, k^{2}\Bigg(\int\limits _{-k}^{k}{\mathrm{d}\omega}\left[{\cal N}_{T}\frac{2(\omega-k){\text{Im}}^{3}\Sigma_{+}}{({\text{Re}}^{2}S_{+}^{-1}\!+\!{\text{Im}}^{2}\Sigma_{+})^{2}}\right]\label{eq:dsqdmu HTL}\\
& & \quad\quad\quad\quad\quad\quad\quad\,\,\,\,-\pi\frac{\omega_{\text{TL},k}^{2}\!\!-k^{2}}{2\hat{M}^{2}}(\omega_{\text{TL},k}-k)\left.{\cal N}_{T}\right|_{\omega_{\text{TL},k}}-\pi\frac{\omega_{\text{Pl},k}^{2}\!\!-k^{2}}{2\hat{M}^{2}}(\omega_{\text{Pl},k}+k)\left.{\cal N}_{T}\right|_{\omega_{\text{Pl},k}}\Bigg)\nonumber \end{aligned}$$ $$\begin{aligned}
A_{n} & = & \frac{d_{q}}{2\pi^{3}\hat{M}^{2}}\int\limits _{0}^{\infty}{\mathrm{d}\hspace{-.5pt}k}\, k^{2}\Bigg(\int\limits _{-k}^{k}{\mathrm{d}\omega}\left[{\cal N}_{\mu}\frac{2(\omega-k){\text{Im}}^{3}\Sigma_{+}}{({\text{Re}}^{2}S_{+}^{-1}\!+\!{\text{Im}}^{2}\Sigma_{+})^{2}}\right]\label{eq:dnqdT HTL}\\
\, & \, & \quad\quad\quad\quad\quad\quad\quad\,\,\,\,-\pi\frac{\omega_{\text{TL},k}^{2}\!\!-k^{2}}{2\hat{M}^{2}}(\omega_{\text{TL},k}-k)\left.{\cal N}_{\mu}\right|_{\omega_{\text{TL},k}}-\pi\frac{\omega_{\text{Pl},k}^{2}\!\!-k^{2}}{2\hat{M}^{2}}(\omega_{\text{Pl},k}+k)\left.{\cal N}_{\mu}\right|_{\omega_{\text{Pl},k}}\Bigg)\nonumber \end{aligned}$$ with abbreviations ${\cal N}_{T}:=\partial/\partial T({n_\text{F}}+{n_\text{F}}^{A})$ and ${\cal N}_{\mu}:=\partial/\partial\mu({n_\text{F}}+{n_\text{F}}^{A})$. The derivative of the strange quark entropy density with respect to the temperature equals the light quark expression at vanishing chemical potential with $A_{s}=A_{q}(\mu=0)$ and $\partial\hat{M}_{s}^{2}/\partial\mu=(\partial\hat{M}^{2}/\partial\mu)|_{\mu=0}$. Imposing the Maxwell relation $\partial s/\partial\mu|_{T}=\partial n/\partial T|_{\mu}$ and employing the prefactors in eq. (\[eq:mDebye and plasma freq\]) the coefficients of the flow equation (\[eq:floweq\]) are given by$$\begin{aligned}
a_{T} & = & -\frac{N_{c}^{2}-1}{16N_{c}}\left(T^{2}+\frac{\mu^{2}}{\pi^{2}}\right)A_{n},\label{eq:full HTL floweq aT}\\
a_{\mu} & = & \frac{1}{6}\left(\left[2N_{c}+N_{q}+N_{s}\right]T^{2}+\frac{N_{c}N_{q}}{\pi^{2}}\mu^{2}\right)A_{g}\label{eq:full HTL floweq amu}\\
& & \quad\quad\quad\quad\quad\quad\,\,\,+\frac{N_{c}^{2}-1}{16N_{c}}\left(T^{2}+\frac{\mu^{2}}{\pi^{2}}\right)A_{q}+\frac{N_{c}^{2}-1}{16N_{c}}T^{2}A_{s},\nonumber \\
b & = & \frac{N_{c}^{2}-1}{8N_{c}}TG^{2}A_{n}-\frac{N_{c}N_{q}}{3\pi^{2}}\mu\, G^{2}A_{g}-\frac{N_{c}^{2}-1}{8N_{c}\pi^{2}}\mu\, G^{2}A_{q}.\label{eq:full HTL floweq b}\end{aligned}$$ These expressions correct a few typos in [@Rom04] (see eqs. (B.1)-(B.5) therein).
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M. Bluhm, Master’s thesis, Technical University Dresden (2004)
[^1]:
[^2]: That is after calculating the propagators using Dyson’s relation from the HTL self-energies, the residues of the poles in the spectral density of both plasmon and (anti)plasmino propagators vanish exponentially for momenta $k\sim T,\mu$, which give the dominant main contributions to thermodynamic integrals.
| 0 |
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abstract: 'Multiple imputation (MI) has become popular for analyses with missing data in medical research. The standard implementation of MI is based on the assumption of data being missing at random (MAR). However, for missing data generated by missing not at random (MNAR) mechanisms, MI performed assuming MAR might not be satisfactory. For an incomplete variable in a given dataset, its corresponding population marginal distribution might also be available in an external data source. We show how this information can be readily utilised in the imputation model to calibrate inference to the population, by incorporating an appropriately calculated offset termed the ‘calibrated-$\delta$ adjustment’. We describe the derivation of this offset from the population distribution of the incomplete variable and show how in applications it can be used to closely (and often exactly) match the post-imputation distribution to the population level. Through analytic and simulation studies, we show that our proposed calibrated-$\delta$ adjustment MI method can give the same inference as standard MI when data are MAR, and can produce more accurate inference under two general MNAR missingness mechanisms. The method is used to impute missing ethnicity data in a type 2 diabetes prevalence case study using UK primary care electronic health records, where it results in scientifically relevant changes in inference for non-White ethnic groups compared to standard MI. Calibrated-$\delta$ adjustment MI represents a pragmatic approach for utilising available population-level information in a sensitivity analysis to explore potential departure from the MAR assumption.'
author:
- 'Tra My Pham^$\star$^'
- James R Carpenter
- Tim P Morris
- Angela M Wood
- Irene Petersen
bibliography:
- 'ama.bib'
title: 'Population-calibrated multiple imputation for a binary/categorical covariate in categorical regression models'
---
Introduction {#sec1}
============
Multiple imputation (MI) [@Rubin1987] has increasingly become a popular tool for analyses with missing data in medical research [@Sterne2009; @Klebanoff2008]; the method is now incorporated in many standard statistical software packages.[@StataCorp2015a; @VanBuuren2011; @Yuan2011] In MI, several completed datasets are created and in each, missing data are replaced with values drawn from an imputation model which is the Bayesian posterior predictive distribution of the missing data, given the observed data. Each completed dataset is then analysed using the substantive analysis model that would have been used had there been no missing data. This process generates several sets of parameter estimates, which are then combined into a single set of results using Rubin’s rules.[@Rubin1987; @Barnard1999] Given congenial specification of the imputation model, Rubin’s rules provide estimates of standard errors and confidence intervals that correctly reflect the uncertainty introduced by missing data.
The standard implementation of MI in widely available software packages provides valid inference under the assumption that missing values are missing completely at random (MCAR) or missing at random (MAR). However, in many applied settings, it is possible that the unseen data are missing not at random (MNAR). For example, in primary care, individuals with more frequent blood pressure readings may, on average, have higher blood pressure compared to the rest of the primary care population. Although MI can be used when data are MNAR, imputation becomes more difficult because a model for the missing data mechanism needs to be specified, which describes how missingness depends on both observed and unobserved quantities. This implies that in practice, it is necessary to define a model for either the association between the probability of observing a variable and its unseen values (selection models) [@Little-Rubin2002]; or the difference in the distribution of subjects with and without missing data (pattern-mixture models).[@Little1993; @Little1994] Due to the potential complexity of modelling the missingness mechanism under MNAR, analyses assuming MNAR are relatively infrequently performed and reported in the applied literature. Instead, in practice, researchers more often try to enhance the plausibility of the MAR assumption as much as possible by including many variables in the imputation model.[@White2011; @Collins2001]
The extra model specification requirement in MI for MNAR data raises several issues. Firstly, the underlying MAR and MNAR mechanisms are not verifiable from the observed data alone. Secondly, there can be an infinite number of possible MNAR models for any dataset, and it is very rare to know which of these models is appropriate for the missingness mechanism. However, for an incomplete variable in a given dataset, its corresponding population marginal distribution might be available from an external data source, such as a population census or survey. If our study sample in truth comes from such a population, it is sensible to feed this population information into the imputation model, in order to calibrate inference to the population.
In this paper, we propose a version of MI for an incomplete binary/categorical variable, termed *calibrated-$\delta$ adjustment MI*, which exploits such external information. In this approach, the population distribution of the incomplete variable can be used to calculate an adjustment in the imputation model’s intercept, which is used in MI such that the post-imputation distribution much more closely (and often exactly) matches the population distribution. The idea of the calibrated-$\delta$ adjustment is motivated by van Buuren et al.’s $\delta$ adjustment (offset) approach in MI.[@VanBuuren1999] However, while values of $\delta$ are often chosen arbitrarily (and independently of covariates in the imputation model) in van Buuren et al.’s approach, the incomplete variable’s population distribution is used to derive the value of $\delta$ in calibrated-$\delta$ adjustment MI. We show that our proposed method gives equivalent inference to standard MI when data are MAR, and can produce unbiased inference under two general MNAR mechanisms.
From a practical point of view, the development of calibrated-$\delta$ adjustment MI is motivated by the issue incomplete recording of ethnicity data in UK primary care electronic health records. Routine recording of ethnicity has been incorporated at the general practice level in the UK, and the variable is therefore available in many large primary care databases. However, research addressing ethnicity has been constrained by the low level of recording.[@Kumarapeli2006; @Aspinall2007; @Mathur2013b] Studies often handle missing data in ethnicity by either dropping ethnicity from the analysis [@Osborn2015], performing a complete record analysis (i.e. excluding individuals with missing data), or single imputation of missing values with the White ethnic group [@Hippisley-Cox2008]; these methods will generally lead to biased estimates of association and standard errors.[@Sterne2009] In addition, the probability that ethnicity is recorded in primary care may well vary systematically by ethnic groups, even after adjusting for other variables.[@Mathur2013b] This implies a potential MNAR mechanism for ethnicity, and as a result, standard MI might fail to give valid inference for the underlying population. Since the population marginal distribution of ethnicity is available in the UK census data, the plausibility of the MAR assumption for ethnicity in UK primary care data can be assessed by using standard MI to handle missing data, and comparing the resulting ethnicity distribution to that in the census. In earlier work, we explored departures from the MAR assumption for other incomplete heath indicators by comparing the results with external nationally representative datasets.[@Marston2010; @Marston2014] As an example of this, Marston et al. (2014) reported that if smoking status is missing for a patient then he or she is typically either an ex-smoker or non-smoker, and accordingly proposed only allowing imputed data to take one of these two values [@Marston2014]. The method we describe here supersedes this ad-hoc approach, providing a way to incorporate population distribution information into MI.
This paper focuses on missing data in an incomplete binary/categorical covariate in an analysis model, where the outcome variable and other covariates are all binary/categorical and fully observed. The remainder of this paper is structured as follows. works through a simple example analytically to describe the derivation of the calibrated-$\delta$ adjustment. In \[sec3\], we formally introduce the procedure of calibrated-$\delta$ adjustment MI and evaluate the performance of the method in simulation studies. illustrates the use of this MI method in a case study which uses electronic health records to examine the association between ethnicity and the prevalence of type 2 diabetes diagnoses in UK primary care. We conclude the paper with a discussion in \[sec5\].
Analytic study – bias in a $2\times 2$ contingency table {#sec2}
========================================================
In this section, we present the development of calibrated-$\delta$ adjustment MI in a simple setting of a $2 \times 2$ contingency table and describe the derivation of the calibrated-$\delta$ adjustment.
Suppose it is of interest to study the association between a binary variable $x$ taking values $j=0,1$ and a binary outcome $y$ taking values $k=0,1$, whose full-data distribution is given in Table \[tab:analytic\_study\_full\_data\]. The full-data distribution is assumed to be identical to the population distribution, such that the population marginal distribution of $x$ is given by $p_{j}^{\text{pop}} = \frac{n_{j+}}{n_{++}}$. The data generating model is $$\text{logit}\left[p\left(y=1\mid x\right)\right] = \beta_{0} + \beta_{x}x,$$ whose parameters can be written in terms of cell counts, $\beta_{0} = \text{ln}\left(\frac{n_{01}}{n_{00}}\right)$ and $\beta_{x} = \text{ln}\left(\frac{n_{11}n_{00}}{n_{01}n_{10}}\right)$.
\[tab:analytic\_study\_full\_data\]
$y=0$ $y=1$ $\sum_{j=0}^{1}x$
-------------------- ---------- ---------- -------------------
$x=0$ $n_{00}$ $n_{01}$ $n_{0+}$
$x=1$ $n_{10}$ $n_{11}$ $n_{1+}$
$\sum_{k=0}^{1} y$ $n_{+0}$ $n_{+1}$ $n_{++}$
: Note: $r$: response indicator of $x$; $j$ and $k$: index categories of $x$ and $y$, respectively; $j, k$ take values $0/1$.
\[tab:analytic\_study\_r=1\]
$y=0 \mid r=1$ $y=1 \mid r=1$ $\sum_{j=0}^{1} x\mid r=1$ Population
--------------------------- ----------------------- ----------------------- ---------------------------- ------------
$x=0 \mid r=1$ $n_{00}^{\text{obs}}$ $n_{01}^{\text{obs}}$ $n_{0+}^{\text{obs}}$ $n_{0+}$
$x=1 \mid r=1$ $n_{10}^{\text{obs}}$ $n_{11}^{\text{obs}}$ $n_{1+}^{\text{obs}}$ $n_{1+}$
$\sum_{k=0}^{1}y\mid r=1$ $n_{+0}^{\text{obs}}$ $n_{+1}^{\text{obs}}$ $n_{++}^{\text{obs}}$
$\sum_{k=0}^{1}y\mid r=0$ $n_{+0}^{\text{mis}}$ $n_{+1}^{\text{mis}}$ $n_{++}^{\text{mis}}$
: Note: $r$: response indicator of $x$; $j$ and $k$: index categories of $x$ and $y$, respectively; $j, k$ take values $0/1$.
\[tab:analytic\_study\_selections\]
[lcc]{}
---------------------------------------------------------
Linear predictor of selection model
$\text{logit}\left[p\left[(r=1 \mid x, y\right)\right]$
---------------------------------------------------------
: Note: $r$: response indicator of $x$; $j$ and $k$: index categories of $x$ and $y$, respectively; $j, k$ take values $0/1$.
&
----------------------------
Selection probability
$p\left(r_{jk} = 1\right)$
----------------------------
: Note: $r$: response indicator of $x$; $j$ and $k$: index categories of $x$ and $y$, respectively; $j, k$ take values $0/1$.
&
-------
Label
-------
: Note: $r$: response indicator of $x$; $j$ and $k$: index categories of $x$ and $y$, respectively; $j, k$ take values $0/1$.
\
$\alpha_{0}$ & $p_{r}$ & M1\
$\alpha_{0} + \alpha_{y}y$ & $p_{r_{k}}$ & M2\
$\alpha_{0} + \alpha_{x}x$ & $p_{r_{j}}$ & M3\
$\alpha_{0} + \alpha_{x}x + \alpha_{y}y$ & $p_{r_{jk}}$ & M4\
\
In addition, suppose that $y$ is fully observed, while some data in $x$ are set to missing (i.e. the sample contains no individuals with missing $y$ and observed $x$, Table \[tab:analytic\_study\_r=1\]). Let $r$ be the response indicator taking values $1$ if $x$ is observed and $0$ if $x$ is missing. Four different missingness mechanisms considered for $x$ and the corresponding selection models are presented in Table \[tab:analytic\_study\_selections\]. Observed cell counts, $n_{jk}^{\text{obs}}$, can be written as a product of the full-data cell counts, $n_{jk}$, and the cell-wise probability of observing $x$, $p_{r_{jk}}$, such that $n_{jk}^{\text{obs}} = n_{jk}p_{r_{jk}}$.
To perform standard MI of missing values in $x$, an imputation model $$\label{eq:standard_imp_model}
\text{logit}\left[p\left(x=1\mid y\right)\right] = \theta_{0} + \theta_{y}y,$$ is fitted to the $n_{++}^{\text{obs}}$ complete records (Table \[tab:analytic\_study\_r=1\]) to obtain the $\theta$ parameter estimates, where $$\theta_{0}^{\text{obs}} = \text{ln}\left(\frac{n_{10}^{\text{obs}}}{n_{00}^{\text{obs}}}\right); \quad \theta_{y}^{\text{obs}} = \text{ln}\left(\frac{n_{11}^{\text{obs}}n_{00}^{\text{obs}}}{n_{01}^{\text{obs}}n_{10}^{\text{obs}}}\right).$$ When $x$ is MCAR or MAR conditional on $y$, we can obtain an unbiased estimate of the association between $x$ and $y$ in the missing data by fitting the above logistic regression imputation model to the complete records. No adjustment is needed in the intercept of the imputation model, and standard MI provides unbiased estimates of the marginal distribution of $x$ as well as the association between $x$ and $y$. We focus on two general MNAR mechanisms described below.
$x$ is MNAR dependent on $x$
----------------------------
Under this missingness mechanism, the posited model for the response indicator $r$ of $x$ is given by $$\label{eq:selection_mnar_x}
\text{logit}\left[p\left(r=1\mid x\right)\right] = \alpha_{0} + \alpha_{x}x,$$ and the corresponding probabilities of observing $x$ are $$p\left(r=1 \mid x=j\right) = p_{r_{j}} = \text{expit}\left(\alpha_{0} + \alpha_{x}x\right); \quad j=0,1.$$ For imputation model , the log odds ratios of $x=1$ for $y=1$ compared to $y=0$ in the observed and missing data are $$\begin{aligned}
\left[\theta_{y} \mid r=1\right] &= \theta_{y}^{\text{obs}} = \text{ln}\left(\frac{n_{00}p_{r_{0}}n_{11}p_{r_{1}}}{n_{01}p_{r_{0}}n_{10}p_{r_{1}}}\right) = \text{ln}\left(\frac{n_{00}n_{11}}{n_{01}n_{10}}\right); \\
\left[\theta_{y} \mid r=0\right] &= \theta_{y}^{\text{mis}} = \text{ln}\left(\frac{n_{00}\left(1-p_{r_{0}}\right)n_{11}\left(1-p_{r_{1}}\right)}{n_{01}\left(1-p_{r_{0}}\right)n_{10}\left(1-p_{r_{1}}\right)}\right) = \text{ln}\left(\frac{n_{00}n_{11}}{n_{01}n_{10}}\right),
\end{aligned}$$ respectively. Hence, $\theta_{y}^{\text{obs}} = \theta_{y}^{\text{mis}}$, which are also the same as the log odds ratio $\theta_{y}$ in the full data (i.e. before values in $x$ are set to missing). The log odds of $x=1$ for $y=0$ in the observed and missing data are given by $$\begin{aligned}
\left[\theta_{0} \mid r=1\right] &= \theta_{0}^{\text{obs}} = \text{ln}\left(\frac{n_{10}p_{r_{1}}}{n_{00}p_{r_{0}}}\right); \\
\left[\theta_{0} \mid r=0\right] &= \theta_{0}^{\text{mis}} = \text{ln}\left(\frac{n_{10}\left(1-p_{r_{1}}\right)}{n_{00}\left(1-p_{r_{0}}\right)}\right),
\end{aligned}$$ respectively. This implies that the correct adjustment in the imputation model’s intercept should be $$\begin{aligned}
\theta_{0}^{\text{mis}} - \theta_{0}^{\text{obs}} &= \text{ln}\left(\frac{\left(1-p_{r_{1}}\right)p_{r_{0}}}{\left(1-p_{r_{0}}\right)p_{r_{1}}}\right) \\
&= \text{ln}\left(\frac{\text{exp}\left(\alpha_{0}\right)}{\text{exp}\left(\alpha_{0}+\alpha_{x}\right)}\right) \\
&= -\alpha_{x},
\end{aligned}$$ which is minus the log odds ratio of observing $x$ for $x = 1$ compared to $x = 0$ in .
$x$ is MNAR dependent on $x$ and $y$
------------------------------------
Under this missingness mechanism, the posited model for the response indicator $r$ of $x$ is given by $$\label{eq:selection_mnar_x_y}
\text{logit}\left[p\left(r=1 \mid x,y\right)\right] = \alpha_{0} + \alpha_{x}x + \alpha_{y}y,$$ and the corresponding probabilities of observing $x$ are $$p\left(r=1 \mid x=j, y=k\right) = p_{r_{jk}} = \text{expit}\left(\alpha_{0} + \alpha_{x}x + \alpha_{y}y\right); \quad j,k = 0,1.$$ For imputation model , the log odds ratios of $x=1$ for $y=1$ compared to $y=0$ in the observed and missing data are $$\begin{aligned}
\theta_{y}^{\text{obs}} &= \text{ln}\left(\frac{n_{00}p_{r_{00}}n_{11}p_{r_{11}}}{n_{01}p_{r_{01}}n_{10}p_{r_{10}}}\right); \label{eq:thetay_obs_mnar_x_y}\\
\theta_{y}^{\text{mis}} &= \text{ln}\left(\frac{n_{00}\left(1-p_{r_{00}}\right)n_{11}\left(1-p_{r_{11}}\right)}{n_{01}\left(1-p_{r_{01}}\right)n_{10}\left(1-p_{r_{10}}\right)}\right). \label{eq:thetay_mis_mnar_x_y}
\end{aligned}$$ Again, it can be shown from and that $\theta_{y}^{\text{obs}} = \theta_{y}^{\text{mis}}$, since $$\begin{aligned}
\theta_{y}^{\text{mis}} - \theta_{y}^{\text{obs}} &= \text{ln}\left(\frac{\left(1-p_{r_{00}}\right)\left(1-p_{r_{11}}\right)p_{r_{01}}p_{r_{10}}}{\left(1-p_{r_{01}}\right)\left(1-p_{r_{10}}\right)p_{r_{00}}p_{r_{11}}}\right)\\
&= \text{ln}\left(\frac{\text{exp}\left(\alpha_{0} + \alpha_{x}\right)\text{exp}\left(\alpha_{0} + \alpha_{y}\right)}{\text{exp}\left(\alpha_{0}\right)\text{exp}\left(\alpha_{0} + \alpha_{x} + \alpha_{y}\right)}\right) \\
&= 0.
\end{aligned}$$ The log odds of $x=1$ for $y=0$ in the observed and missing data are given by $$\begin{aligned}
\theta_{0}^{\text{obs}} = \text{ln}\left(\frac{n_{10}p_{r_{10}}}{n_{00}p_{r_{00}}}\right); \\
\theta_{0}^{\text{mis}} = \text{ln}\left(\frac{n_{10}\left(1-p_{r_{10}}\right)}{n_{00}\left(1-p_{r_{00}}\right)}\right),
\end{aligned}$$ which implies that the correct adjustment in the imputation model’s intercept should be $$\begin{aligned}
\theta_{0}^{\text{mis}} - \theta_{0}^{\text{obs}} &= \text{ln}\left(\frac{\left(1-p_{r_{10}}\right)p_{r_{00}}}{\left(1-p_{r_{00}}\right)p_{r_{10}}}\right) \\
&= \text{ln}\left(\frac{\text{exp}\left(\alpha_{0}\right)}{\text{exp}\left(\alpha_{0}+\alpha_{x}\right)}\right) \\
&= -\alpha_{x},
\end{aligned}$$ which is again minus the log odds ratio of observing $x$ in .
Derivation of the calibrated-$\delta$ adjustment
------------------------------------------------
The analytic calculations above confirm that in a $2\times 2$ contingency table setting, appropriately adjusting the intercept of the imputation model for the covariate $x$ can sufficiently correct bias introduced by MNAR mechanisms under which missingness in $x$ depends on either its values or both its values and the outcome (M3 and M4). The population distribution of $x$ can be used to calculate the correct adjustment in the imputation model’s intercept. This adjustment is referred to as the *calibrated-$\delta$ adjustment* to clarify its relationship to van Buuren et al.’s $\delta$ adjustment.[@VanBuuren1999]
The probability of $x=1$ can be written in terms of the conditional probabilities among subjects with observed and missing $x$ $$ p\left(x=1\right) = p\left(x=1 \mid r=1\right)p\left(r=1\right) + p\left(x=1 \mid r=0\right)p\left(r=0\right),$$ where $p\left(x=1\right)$ is the population proportion; $p\left(x=1 \mid r=1\right)$ , $p\left(r=1\right)$, and $p\left(r=0\right)$ can be obtained from the observed data. Thus, $p\left(x=1 \mid r=0\right)$ can be solved for as $$\label{eq:p_x=1_partitioned2}
p\left(x=1 \mid r=0\right) = \frac{p\left(x=1\right) - p\left(x=1 \mid r=1\right)p\left(r=1\right)}{p\left(r=0\right)}.$$ Note that $p\left(x=1 \mid r=0\right)$ can be further written as $$\begin{aligned}
p\left(x=1 \mid r=0\right) &= \sum_{k=0}^{1}p\left(x=1 \mid y=k, r=0\right)p\left(y=k \mid r=0\right) \nonumber\\
&= \sum_{k=0}^{1}\text{expit}\left(\theta_{0}^{\text{mis}} + \theta_{y}^{\text{mis}} I\left[y=k\right]\right)\frac{n_{+k}^{\text{mis}}}{n_{++}^{\text{mis}}} \nonumber\\
&= \frac{1}{n_{++}^{\text{mis}}}\text{expit}\left(\theta_{0}^{\text{mis}} + \theta_{y}^{\text{mis}} I\left[y=k\right]\right)n_{+k}^{\text{mis}}, \label{eq:p_x=1_r=0}\end{aligned}$$ where $I\left[A\right]$ is an indicator function taking values 1 if $A$ is true and 0 otherwise. It is shown earlier that when $x$ is MNAR dependent on either the values of $x$ or both $x$ and $y$, $\theta_{y}^{\text{obs}}=\theta_{y}^{\text{mis}}$; is therefore equal to $$\begin{aligned}
p\left(x=1 \mid r=0\right) &= \frac{1}{n_{++}^{\text{mis}}}\text{expit}\left(\theta_{0}^{\text{mis}} + \theta_{y}^{\text{obs}} I\left[y=k\right]\right)n_{+k}^{\text{mis}} \\
&= \frac{1}{n_{++}^{\text{mis}}}\text{expit}\left(\left(\theta_{0}^{\text{obs}} + \delta\right) + \theta_{y}^{\text{obs}} I\left[y=k\right]\right)n_{+k}^{\text{mis}} \\
&= \frac{1}{n^{\text{mis}}}\sum_{i=1}^{n^{\text{mis}}}\text{expit}\left(\left(\theta_{0}^{\text{obs}} + \delta \right) + \theta_{y}^{\text{obs}}y_{i}\right),\end{aligned}$$ where $\delta$ is the adjustment factor in the intercept of the imputation model for $x$. The value of the calibrated-$\delta$ adjustment can be obtained numerically from and using interval bisection [@Russ1980; @Burden2011] (or any other root-finding method).
When the population marginal distribution of the incomplete covariate $x$ is available, a natural alternative to adjusting the intercept of the imputation model based on this information is to weight the complete records in the imputation model (which we term ‘weighted multiple imputation’), in order to match the post-imputation distribution of $x$ to the population. In the supporting information section we explore two such weighting approaches, marginal and conditional weighted MI; we show analytically that while these methods can provide more accurate results compared to standard MI under certain MNAR mechanisms, they do not provide a general solution as does calibrated-$\delta$ adjustment MI.
Simulation studies {#sec3}
==================
This section presents univariate simulation studies to evaluate performance measures of the calibrated-$\delta$ adjustment MI method for an incomplete binary covariate $x$, when the fully observed outcome variable $y$ is also binary. The term ‘univariate’ is used here to refer to the setting where missingness occurs in a single covariate. The aims of these simulation studies are (i) to examine finite-sample properties of calibrated-$\delta$ adjustment MI including bias in parameter estimates, efficiency in terms of the empirical and average model standard errors (SE), and coverage of $95\%$ confidence intervals (CI); and (ii) to compare the method with standard MI and complete record analysis (CRA) under various missingness mechanisms for $x$.
When the population distribution is ‘known’ {#subsec3.1}
-------------------------------------------
Below we consider the setting where the population distribution of the incomplete variable is obtained from a population census or equivalent, i.e. it is ‘known’. The uncertainty associated with having to estimate the population distribution is explored in \[subsec3.2\].
### Method {#subsubsec3.1.1}
Similar to the analytic study presented in \[sec2\], the analysis model in this simulation study is a logistic regression model for a fully observed binary outcome $y$ on an incomplete binary covariate $x$. Calibrated-$\delta$ adjustment MI is compared to standard MI and CRA under four missingness mechanisms of increase complexity. The data generating mechanism and analysis procedures are as follows.
1. Simulate $n=5\,000$ complete values of the binary $0/1$ covariate $x$ and binary $0/1$ outcome $y$ from the following models $$\begin{aligned}
&x \sim \text{Bernoulli}\left(p_{x}^{\text{pop}} = 0.7\right); \nonumber\\
& \text{logit}\left[p\left(y=1 \mid x\right)\right] = \beta_{0} + \beta_{x}x, \label{eq:moi_sim1}
\end{aligned}$$ where $\beta_{0}$ and $\beta_{x}$ are arbitrarily set to $\text{ln}\left(0.5\right)$ and $\text{ln}\left(1.5\right)$, respectively. The same values of the $\beta$ parameters are used throughout to make bias comparable across all simulation settings. This sample size is chosen to minimise the issue of small-sample bias associated with the logistic regression [@Nemes2009];
2. Simulate a binary indicator of response $r$ of $x$ from each of the selection models M1–M4 (Table \[tab:analytic\_study\_selections\]). Values of $1.5$ and $-1.5$ are chosen for $\alpha_{y}$ and $\alpha_{x}$ in M2 and M3, respectively, to reflect strong odds ratios (OR) of observing $x$ (OR $= 4.5$ and $0.2$, respectively). For M4, $\alpha_{y} = 1.5$ and $\alpha_{x} = -1.5$ are chosen as bias in the three MI methods under evaluation is likely to be apparent with these coefficients predicting missingness in $x$. For all selection models, $\alpha_{0}$ is altered to achieve approximately $45\%$ missing $x$. For M1, $\alpha_{0}$ is calculated directly as $\text{ln}\left(\frac{0.55}{0.45}\right)$; for M2–M4, $\alpha_{0} = -0.2; 1.35$; and $0.75$ appear to work well;
3. For $i=1 \ldots 5\,000$, set $x_{i}$ to missing if $r_{i}=0$;
4. Impute missing values in $x$ $M=50$ times using standard MI and calibrated-$\delta$ adjustment MI in turn;
5. In each MI method, fit the analysis model to each completed dataset and combine the results using Rubin’s rules.[@Rubin1987; @Barnard1999]
Steps 1–5 are repeated $S=2\,000$ times under each of the four selection models M1–M4, so the same set of simulated independent datasets is used to compare the three MI methods under the same missingness scenario, but a different set of datasets is generated for each missingness scenario.[@Burton2006] The parameters of interest are $\beta_{0}$ and $\beta_{x}$, although in practice $\beta_{x}$ is usually of more interest. Bias, efficiency of $\hat{\beta}_{0}$ and $\hat{\beta}_{x}$ in terms of the empirical standard errors, and coverage of 95% CIs are calculated over $2\,000$ repetitions for each combination of simulation settings,[@White2010a] with analyses of full data (i.e. before any values in $x$ are set to missing) and complete records also provided for comparison.
All simulations are performed in Stata 14 [@StataCorp2015b]; `mi impute logit` is used for standard MI, the community-contributed command `uvis logit` [@Royston2004] for calibrated-$\delta$ adjustment MI, and `mi estimate: logit` for fitting the analysis model to the completed datasets and combining the results using Rubin’s rules.[@Rubin1987; @Barnard1999] Simulated datasets are analysed using the community-contributed command `simsum`.[@White2010a]
Based on the analytic calculations presented in \[sec2\], we propose the following procedure for imputing missing values in the covariate $x$ using calibrated-$\delta$ adjustment MI.
1. Fit a logistic regression imputation model for $x$ conditional on $y$ to the complete records to obtain the maximum likelihood estimates of the imputation models’ parameters $\hat{\theta}$ and their asymptotic sampling variance $\widehat{\boldsymbol{U}}$;
2. Draw new parameters $\tilde{\boldsymbol{\theta}}$ from the large-sample normal approximation $N(\widehat{\boldsymbol{\theta}}, \widehat{\boldsymbol{U}})$ of their posterior distribution, assuming non-informative priors;
3. Draw a new probability of observing $x$, $\tilde{p}_{r}$, from the normal approximation $N\left(\hat{p}_{r}, \frac{\hat{p}_{r}\left(1-\hat{p}_{r}\right)}{n}\right)$, where $\hat{p}_{r}$ is the sample proportion of the response indicator of $x$, $\hat{p}_{r} = \frac{n_{++}^{\text{obs}}}{n_{++}}$;
4. Draw a new probability of observed $x=1$, $\tilde{p}_{x}$, from the normal approximation $N\left(\hat{p}_{x}, \frac{\hat{p}_{x}\left(1-\hat{p}_{x}\right)}{n}\right)$, where $\hat{p}_{x}$ is the observed proportion of $x=1$, $\hat{p}_{x} = \frac{n_{1+}^{\text{obs}}}{n_{++}^{\text{obs}}}$;
5. Derive the value of the calibrated-$\delta$ adjustment from the equation $$\frac{1}{n^{\text{mis}}}\sum_{i=1}^{n^{\text{mis}}}\text{expit}\left(\left(\tilde{\theta}_{0} + \delta\right) + \tilde{\theta}_{y}y_{i}\right) = \frac{p_{x}^{\text{pop}} - \tilde{p}_{x}}{\tilde{p}_{r}},$$ where $p_{x}^{\text{pop}}$ is the probability of $x=1$ in the population;
6. Fit the logistic regression imputation model for $x$ conditional on $y$ (in step 1) to the complete records with the intercept adjustment fixed to $\delta$ to obtain the maximum likelihood estimates of the imputation models’ parameters $\hat{\boldsymbol{\theta}}$ and their asymptotic sampling variance $\widehat{\boldsymbol{U}}$;
7. Draw new parameters $\dot{\boldsymbol{\theta}}$ from the large-sample normal approximation $N(\widehat{\boldsymbol{\theta}}, \widehat{\boldsymbol{U}})$ of their posterior distribution, assuming non-informative priors;
8. Draw imputed values for $x$ from the above logistic regression imputation model, using the newly drawn parameters $\dot{\boldsymbol{\theta}}$ and calibrated-$\delta$ adjustment.
### Results {#subsubsec3.1.2}
Results of the simulation study are summarised graphically in Figure \[fig:sim\_base\]. Full data and CRA both give the results that the theory predicts. Analysis of full data is always unbiased with coverage close to the $95\%$ level and the smallest standard errors of all methods. CRA is unbiased under M1 and M3 as expected,[@White2010b] but bias is observed under the other two missingness mechanisms. Coverage is correspondingly low when bias is present, and efficiency is lower than that in full data.
Under M1, when $x$ is MCAR, all methods appear unbiased, with comparable empirical and average model standard errors and correct coverage. This is as expected.
Under M2, when $x$ is MAR conditional on $y$, CRA is severely biased in the estimate of $\beta_{0}$ and the corresponding coverage of 95% CIs falls to 0. However, the method provides an unbiased estimate of $\beta_{x}$ with correct coverage. This result is specific to this simulation set-up, where the probability of being a complete record depends on the outcome, and the analysis model is a logistic regression. This mimics case-control sampling, where the log odds of the logistic regression is biased in case-control studies but the log odds ratio is not.[@White2010b; @Bartlett2015] The outcome–covariate association can therefore be estimated consistently among the complete records. Standard MI and calibrated-$\delta$ adjustment MI are unbiased for both parameter estimates. Standard MI yields comparable empirical and average model standard errors and coverage attains the nominal level. In calibrated-$\delta$ adjustment MI, empirical standard errors are slightly smaller than the average model counterparts, leading to a minimal increase in coverage.
[0.69]{} ![Simulation study: bias in point estimates, empirical and average model SE, and coverage of 95% CIs under different missingness mechanism for $x$.[]{data-label="fig:sim_base"}](figures/bias_base.pdf "fig:"){width="\textwidth"}
[0.69]{} ![Simulation study: bias in point estimates, empirical and average model SE, and coverage of 95% CIs under different missingness mechanism for $x$.[]{data-label="fig:sim_base"}](figures/se_base.pdf "fig:"){width="\textwidth"}
[0.69]{} ![Simulation study: bias in point estimates, empirical and average model SE, and coverage of 95% CIs under different missingness mechanism for $x$.[]{data-label="fig:sim_base"}](figures/cv_base.pdf "fig:"){width="\textwidth"}
Under M3, when $x$ is MNAR dependent on $x$, CRA yields unbiased estimates of both parameters. Standard MI is biased in the estimate of $\beta_{0}$ but provides an unbiased estimate of $\beta_{x}$ due to the symmetry property of the odds ratios. Generally, in logistic regression with an incomplete covariate $x$, when the missingness mechanism is such that both standard MI and CRA are unbiased, standard MI tends not to be more efficient than CRA in estimating $\beta_{x}$.[@White2010b] This is because without auxiliary variables in the imputation model, standard MI does not carry any extra information on the odds ratio compared to CRA. This is seen in the simulation results for $\beta_{x}$ under models M1–M3. Under M3, calibrated-$\delta$ adjustment MI is also unbiased in both parameter estimates. Given that all three methods are unbiased for $\beta_{x}$ under M3, there is a small gain in efficiency in the estimate of $\beta_{x}$ in calibrated-$\delta$ adjustment MI, as the empirical standard error for this parameter is slightly smaller than that in CRA. Under this missingness mechanism, empirical and average model standard errors are comparable across methods; for methods that are unbiased, their corresponding coverage of $95\%$ CIs generally attains the nominal level.
Under M4, when $x$ is MNAR dependent on $x$ and $y$, standard MI and CRA are again biased in both parameter estimates, leading to coverage close or equal to 0. In contrast, calibrated-$\delta$ adjustment MI produces unbiased estimates of both parameters. In this method, empirical standard errors are again slightly smaller than the average model counterparts (as seen previously under M2), which leads to coverage slightly exceeding the $95\%$ level.
When the population distribution is estimated with uncertainty {#subsec3.2}
--------------------------------------------------------------
So far, the population distribution of the incomplete covariate that is used to derive the calibrated-$\delta$ adjustment is assumed to be obtained from a population census or equivalent. In other words, it is assumed that there is no uncertainty associated with estimating the reference distribution, and hence, the adjustment. In calibrated-$\delta$ adjustment MI, we believe that the extra uncertainty in estimating the calibrated-$\delta$ adjustment should be ignored when the population distribution of the incomplete covariate is assumed to be invariant, unless the reference population is not a census or equivalent. Since MI is a Bayesian procedure in which all sources of uncertainty are modelled, this explains why, if there is uncertainty about the population distribution of the incomplete covariate, this uncertainty needs to be accounted for in the derivation of the calibrated-$\delta$ adjustment across imputations.
When the population distribution of the incomplete covariate is not ‘known’ and is estimated, a natural approach for incorporating this extra uncertainty would be to draw values of the population proportions from their distribution and calculate the calibrated-$\delta$ adjustment using these draws, so that this uncertainty is reflected in the MI variance estimation. This additional step is expected to have an effect on the between-imputation variance of Rubin’s variance estimator.
An extension of the simulation study presented in section \[subsec3.1\] is conducted to explore this setting.
### Method {#subsubsec3.2.1}
This extended simulation study of a fully observed binary outcome y and a partially observed binary covariate $x$ follows the same method described in \[subsubsec3.1.1\], except that two variations of the population proportions of $x$ are evaluated in the imputation step of calibrated-$\delta$ adjustment MI. The reference distribution is assumed to either come from a census or equivalent (case 1), or be estimated in an external dataset of larger size (case 2) or smaller size (case 3) than the study sample.
Suppose that in an external dataset of size $n^{\text{ex}}$ which comes from the same population as the study sample, the sample proportion $\hat{p}_{x}^{\text{pop}}$ provides an unbiased estimate of the population proportion $p_{x}^{\text{pop}}$. Assuming that the sampling distribution of the sample proportions is approximately normal, its standard error is given by $$\text{SE}\left(\hat{p}_{x}^{\text{pop}}\right) = \sqrt{\frac{\hat{p}_{x}^{\text{pop}}\left(1-\hat{p}_{x}^{\text{pop}}\right)}{n^{\text{ex}}}}.$$ The data generating mechanism and analysis procedures are as follows.
1. For cases 2 and 3, the following two steps are performed to incorporate the sampling behaviour of $\hat{p}_{x}^{\text{pop}}$, which is estimated in an external dataset of size $n^{\text{ex}}$, into the data generating mechanism in repeated simulations.
1. Simulate $n^{\text{ex}} = 10\,000$ (case 2) or $1\,000$ (case 3) complete values of the binary $0/1$ covariate $x$ from the model $$x \sim \text{Bernoulli}\left(p_{x}^{\text{pop}} = 0.7\right);$$
2. Obtain the sample proportion $\hat{p}_{x}^{\text{pop}}$ of $x$, which is an unbiased estimate of the population proportion $p_{x}^{\text{pop}}$;
2. Simulate $n=5\,000$ complete values of the binary $0/1$ covariate $x$ and binary $0/1$ covariate $y$ from the models $$\begin{aligned}
&x \sim \text{Bernoulli}\left(p_{x}^{\text{pop}} = 0.7\right); \nonumber \\
&\text{logit}\left[p\left(y=1 \mid x\right)\right] = \beta_{0} + \beta_{x}x \label{eq:moi_sim2},
\end{aligned}$$ where $\beta_{0}$ and $\beta_{x}$ are arbitrarily set to $\text{ln}\left(0.5\right)$ and $\text{ln}\left(1.5\right)$, respectively. The same values of the $\beta$ coefficients are used throughout to make bias comparable across all simulation settings;
3. Simulate a binary indicator of response $r$ of $x$ from each of the selection models M1–M4 (Table \[tab:analytic\_study\_selections\]). Values of $1.5$ and $-1.5$ are chosen for $\alpha_{y}$ and $\alpha_{x}$ in M2 and M3, respectively. For M4, $\alpha_{y} = 1.5$ and $\alpha_{x} = -1.5$ are used. In all selection models, $\alpha_{0}$ is altered to achieve approximately $45\%$ missing $x$. For M1, $\alpha_{0}$ is calculated directly as $\text{ln}\left(\frac{0.55}{0.45}\right)$; for M2–M4, $\alpha_{0} = -0.2; 1.35$; and $0.75$ are used;
4. For $i=1, \ldots, 5\,000$, set $x_{i}$ to missing if $r_{i} = 0$;
5. Impute missing values in $x$ $M=10$ times using standard MI and calibrated-$\delta$ adjustment MI in turn. For cases 2 and 3, calibrated-$\delta$ adjustment MI is performed as follows.
1. Draw a value $\tilde{p}_{x}^{\text{pop}}$ from the normal approximation $N\left(\hat{p}_{x}^{\text{pop}}, \frac{\hat{p}_{x}^{\text{pop}}\left(1-\hat{p}_{x}^{\text{pop}}\right)}{n^{\text{ex}}}\right)$, with values of $n^{\text{ex}} = 10\,000$ (case 2) and $1\,000$ (case 3). This is done by first taking a draw $\tilde{z}$ from the standard normal distribution, $z \sim N\left(0,1\right)$, followed by drawing $\tilde{p}_{x}^{\text{pop}} = \hat{p}_{x}^{\text{pop}} + \tilde{z}\sqrt{\frac{\hat{p}_{x}^{\text{pop}}\left(\hat{p}_{x}^{\text{pop}}\right)}{n^{\text{ex}}}}$;
2. Derive the calibrated-$\delta$ adjustment and perform MI according to the algorithm set out in \[subsubsec3.1.1\], using $\tilde{p}_{x}^{\text{pop}}$ as the reference proportion;
6. For each MI method, fit the analysis model to each completed dataset and combine the results using Rubin’s rules.[@Rubin1987; @Barnard1999]
Step 5 is designed to mimic the full Bayesian sampling process, which is always the aim in proper (or Rubin’s) MI. Again, steps 1–6 are repeated $S=2\,000$ times under each of the four selection models M1–M4, so the same set of simulated independent datasets is used to compare the two MI methods under the same missingness scenario, but a different set of datasets is generated for each missingness scenario.[@Burton2006] The parameters of interest are $\beta_{0}$ and $\beta_{x}$ . Bias in $\hat{\beta}_{0}$ and $\hat{\beta}_{x}$, efficiency in terms of the empirical and average model standard errors, and coverage of $95\%$ CIs are calculated over $2\,000$ repetitions for each combination of simulation settings,[@White2010a] with analyses of full data and complete records also provided for comparison.
All simulations are performed in Stata 14 [@StataCorp2015b] with `mi impute logit` for standard MI, the community-contributed command `uvis logit` [@Royston2004] for calibrated-$\delta$ adjustment MI, and `mi estimate: logit` for fitting the analysis model to the completed datasets and combining the results using Rubin’s rules [@Rubin1987; @Barnard1999]; simulated datasets are analysed using the community-contributed command `simsum`.[@White2010a]
### Results {#subsubsec3.2.2}
Results of the extended simulation study are presented in Figure \[fig:sim\_var\]. Bias in point estimates is similar when $p_{x}^{\text{pop}}$ is invariant or estimated in a large external dataset (cases 1 and 2, respectively). Bias slightly increases, particularly under M2 and M4, when $p_{x}^{\text{pop}}$ is estimated in a small external dataset with higher variance (case 3).
[0.65]{} ![Extended simulation study: bias in point estimates, empirical and average model SE, and coverage of 95% CIs under different missingness mechanism for $x$; the population distribution of $x$ is assumed to be invariant (case 1) or estimated in an external dataset of size 10000 (case 2) or 1000 (case 3).[]{data-label="fig:sim_var"}](figures/bias_var.pdf "fig:"){width="\textwidth"}
[0.65]{} ![Extended simulation study: bias in point estimates, empirical and average model SE, and coverage of 95% CIs under different missingness mechanism for $x$; the population distribution of $x$ is assumed to be invariant (case 1) or estimated in an external dataset of size 10000 (case 2) or 1000 (case 3).[]{data-label="fig:sim_var"}](figures/se_var.pdf "fig:"){width="\textwidth"}
[0.65]{} ![Extended simulation study: bias in point estimates, empirical and average model SE, and coverage of 95% CIs under different missingness mechanism for $x$; the population distribution of $x$ is assumed to be invariant (case 1) or estimated in an external dataset of size 10000 (case 2) or 1000 (case 3).[]{data-label="fig:sim_var"}](figures/cv_var.pdf "fig:"){width="\textwidth"}
Empirical and average model standard errors are comparable and remain stable for calibrated-$\delta$ adjustment MI across the three cases under M1 and M3. Under M2 and M4, the discrepancy previously seen between the empirical and average model standard errors in calibrated-$\delta$ adjustment MI (\[subsubsec3.1.2\]) decreases in case 3 compared to cases 1 and 2. When there is increased uncertainty in estimating the population proportions of $x$ (case 3 compared to case 1), there is also a marked increase in both the empirical and average model standard errors in calibrated-$\delta$ adjustment MI. This extra uncertainty is reflected in the variation of the point estimates across the simulation repetitions according to how the simulation is set up, and is also acknowledged by an increase in the between-imputation variance component of Rubin’s variance estimator (results for between-imputation variances not shown).
In line with results seen for the standard errors, coverage attains the nominal level for calibrated-$\delta$ adjustment MI under M1 and M3. Under M2 and M4, since the empirical standard errors are closer to the average model standard errors in case 3 compared to case 1, the slight over-coverage of 95% CIs seen in case 1 seems to disappear in case 3.
Case study – ethnicity and the prevalence of type 2 diabetes diagnoses in The Health Improvement Network primary care database {#sec4}
==============================================================================================================================
This case study is conducted to illustrate the use of calibrated-$\delta$ adjustment MI for handling missing data in ethnicity in UK primary care electronic health records, when ethnicity is included as a covariate in the analysis model. In particular, this is a cross-sectional study which examines the association between ethnicity and the prevalence of type 2 diabetes diagnoses in a large UK primary care database in 2013. Prevalence of type 2 diabetes is chosen as the outcome variable to illustrate the application of the calibrated-$\delta$ adjustment MI method as developed and evaluated in \[sec2,sec3\].
The Health Improvement Network database {#subsec4.1}
---------------------------------------
The Health Improvement Network (THIN) [@IMSHealth2015] is one of the largest databases in the UK to collect information on patient demographics, disease symptoms and diagnoses, and prescribed medications in primary care. THIN contains anonymised electronic health records from over 550 general practices across the UK, with more than 12 million patients contributing data. The database is broadly generalisable to the UK population in terms of demographics and crude prevalences of major health conditions.[@Blak2011; @Bourke2004]
Information is recorded during routine patient consultations with General Practitioners (GP) from when the patients register to general practices contributing data to THIN to when they die or transfer out. Symptoms and diagnoses of disease are recorded using Read codes, a hierarchical coding system.[@Chisholm1990; @Dave2009] THIN also provides information on referrals made to secondary care and anonymised free text information. Patient demographics include information on year of birth, sex, and social deprivation status measured in quintiles of the Townsend deprivation score.[@Townsend1988]
The acceptable mortality reporting (AMR) [@Maguire2009] and the acceptable computer usage (ACU) [@Horsfall2013] dates are jointly used for data quality assurance in THIN. The AMR date is the date after which the practice is deemed to be reporting a rate of all-cause mortality sufficiently similar to that expected for a practice with the same demographics, based on data from the Office for National Statistics (ONS).[@Maguire2009] The ACU date is designed to exclude the transition period between the practice switching from paper-based records to complete computerisation; it is defined as the date from which the practice is consistently recording on average at least two drug prescriptions, one medical record and one additional health record per patient per year.[@Horsfall2013]
Use of THIN for scientific research was approved by the NHS South-East Multi-Centre Research Ethics in 2003. Scientific approval to undertake this study was obtained from IQVIA World Publications Scientific Review Committee in September 2017 (SRC Reference Number: 17THIN083).
Study sample {#subsec4.2}
------------
All individuals who are permanently registered with general practices in London contributing data to THIN are considered for inclusion in the study sample. This sample is chosen since it is not only more practical to perform MI on a smaller dataset, but also because London is the most ethnically diverse region in the UK, and hence incorrect assignment of ethnicity from imputing missing data with the White ethnic group is expected to be more apparent compared to other regions.
For each individual, a start date is defined as the latest of: date of birth, ACU and AMR dates,[@Maguire2009; @Horsfall2013] and registration date. Similarly, an end date is defined as the earliest of: date of death, date of transfer out of practice, and date of last data collection from the practice. Point prevalence of type 2 diabetes on 01 January 2013 is calculated, since THIN is a dynamic database in which individuals register with and leave their general practices at different times. Individuals are selected into the study sample if they are actively registered to THIN practices in London on 01 January 2013, and in addition they need to have been registered with the same general practices for at least 12 months by this date. This criterion is introduced to ensure that there is enough time for the individuals to have their type 2 diabetes diagnoses recorded in their electronic health data, after registration with their general practices.
Outcome variable and main covariate {#subsec4.3}
-----------------------------------
The recording of diabetes diagnoses and management in THIN is comprehensive and therefore there are several ways an individual may be identified as diabetic. For this study, an algorithm developed by Sharma et al. [@Sharma2016a] is used to identify individuals with diabetes mellitus, as well as to distinguish between type 1 and type 2 diabetes. According to this algorithm, individuals are identified as having diabetes if they have at least two of the following records: a diagnostic code for diabetes, supporting evidence of diabetes (e.g. screening for diabetic retinophany), or prescribed treatment for diabetes. In this study, the first record of any of these three is considered as the date of diagnosis. In addition to identifying individuals with diabetes, the algorithm also distinguishes between type 1 and type 2 diabetes based on individuals’ age at diagnosis, types of treatment and timing of the diabetes diagnosis. [@Sharma2016a; @Sharma2016b] After the study sample is selected using the method described in \[subsec4.2\], prevalent cases of type 2 diabetes are defined as individuals who have a diagnosis of type 2 diabetes on or before 01 January 2013.
Ethnicity is typically recorded in THIN using the Read code system [@Chisholm1990]; it can also be recorded using free text entries. A list containing Read codes related to ethnicity is developed using a published method.[@Dave2009] The majority of ethnicity records are identified by searching both the medical and additional health data files for Read codes in the ethnicity code list. Minimal additional information is found by searching the pre-anonymised free text as well as other free text linked to ethnicity-related Read codes. Ethnicity is then coded into the five-level ONS classification as White, Mixed, Asian, Black, and Other ethnic groups.[@OfficeforNationalStatistics2012] Subsequently, the Mixed and Other ethnic groups are combined due to the small counts and heterogeneity in these two groups. Searching for ethnicity-related Read codes reveals that there is a small number of individuals with multiple inconsistent records of ethnicity. For these individuals, it can not be determined with certainty whether their ethnicity is in fact one of the recorded categories or if all the recorded categories are incorrect. Therefore, their ethnicity is set to missing for simplicity, since the issue of inconsistency in ethnicity recording is not the focus of this study.
Statistical analysis {#subsec4.4}
--------------------
The analysis model in this study is a logistic regression model for a binary indicator of whether an individual has a diagnosis of type 2 diabetes on or before 01 January 2013, conditional on the individual’s age in 2013, sex, Townsend deprivation score (five quintiles, from the least to the most deprived), and ethnic group (White, Asian, Black, Mixed/Other). Age is analysed in $10$-year age groups for individuals aged 0–79 years, and all individuals aged 80 years and above are grouped into the $80+$ category. Ethnicity information is extracted and categorised as described in \[subsec4.3\]. Since this study is conducted to illustrate the application of calibrated-$\delta$ adjustment MI in a univariate missing data setting where missing data occurs in a single covariate (ethnicity), individuals with incomplete information on age, sex, and deprivation status were excluded from the analysis.
Missing values in ethnicity are handled by (i) a CRA, (ii) single imputation with the White ethnic group, (iii) standard MI, and (iv) calibrated-$\delta$ adjustment MI using the 2011 ONS census distribution of ethnicity in London [@OfficeforNationalStatistics2012] as the reference distribution. For MI of ethnicity, a multinomial logistic regression imputation model is constructed for ethnicity using all variables in the analysis model, including individuals’ age group in 2013, sex, and quintiles of the Townsend score. In MI, the outcome variable must be explicitly included in the imputation model for the incomplete covariate. [@Sterne2009] Since the analysis model is a logistic regression model, the type 2 diabetes indicator is also included as a covariate in the imputation model for ethnicity.
In this study, ethnicity is analysed as a four-level categorical variable. Therefore, the calibrated-$\delta$ adjustment MI method for handling missing data in an incomplete binary covariate discussed in \[sec2,sec3\] can be generalised for handling missing values in ethnicity as a categorical covariate. The overall proportion of the $j$th level of ethnicity, $j=1, \ldots, 4$ can be written as $$\label{eq:eth_partition}
p\left(\text{eth} = j\right) = p\left(\text{eth} = j \mid r = 1\right) p\left(r=1\right) + p\left(\text{eth} = j \mid r = 0\right) p\left(r=0\right),$$ where $p\left(\text{eth} = j\right)$ is available in the census; $p\left(\text{eth} = j \mid r = 1\right)$, $p\left(r=1\right)$, and $p\left(r=0\right)$ can be obtained in the observed data.
A multinomial logistic regression imputation model for ethnicity conditional on age group (40–49 years old as the base level), sex (male as the base level), Townsend score (quintile 1 as the base level), and the binary indicator of type 2 diabetes (no diagnosis as the base level) is fitted to the observed data. Setting the first level of ethnicity (White, $j = 1$) as the base level to identify the model, the probability of the level $j$th of ethnicity in the observed data, $j=2, \ldots, 4$ can be written in terms of the observed-data linear predictors, $\text{linpred}_{j}^{\text{obs}}$, which is estimated from the multinomial logistic regression model for ethnicity as $$\label{eq:pethj_r1}
p\left(\text{eth}=j \mid r=1\right) = \frac{1}{n^{\text{obs}}} \sum_{i=1}^{n^{\text{obs}}}\frac{1}{1+\sum_{j=2}^{4}\left(\text{linpred}_{ij}^{\text{obs}}\right)},$$ where $i$ indexes individuals in the dataset, and $$\begin{aligned}
\label{eq:linpred_obs}
\text{linpred}_{ij}^{\text{obs}} &= \theta_{j0}^{\text{obs}} + \sum_{a=10}^{30} \theta_{j\text{age}_{a}}^{\text{obs}}I\left[\text{age}_{ij}=a\right] + \sum_{a=50}^{80} \theta_{j\text{age}_{a}}^{\text{obs}}I\left[\text{age}_{ij}=a\right] + \theta_{j\text{sex}}^{\text{obs}}I\left[\text{sex}_{ij} = \text{female}\right] \nonumber \\
& + \sum_{t=2}^{5}\theta_{j\text{town}_{t}}^{\text{obs}}I\left[\text{Townsend}_{ij} = t\right] + \theta_{j\text{t2d}}^{\text{obs}}I\left[\text{type 2 diabetes}_{ij} = \text{yes}\right].\end{aligned}$$
Following the methods outlined in \[sec3\], since covariates in the imputation model for ethnicity are all binary or categorical, the relative risk ratios are the same among those with ethnicity observed and missing. The linear predictors in the missing data, $\text{linpred}_{j}^{\text{mis}}$, can therefore be written as $$\begin{aligned}
\label{eq:linpred_mis}
\text{linpred}_{ij}^{\text{mis}} &= \left(\theta_{j0}^{\text{obs}} + \delta_{j0}\right)+ \sum_{a=10}^{30} \theta_{j\text{age}_{a}}^{\text{obs}}I\left[\text{age}_{ij}=a\right] + \sum_{a=50}^{80} \theta_{j\text{age}_{a}}^{\text{obs}}I\left[\text{age}_{ij}=a\right] + \theta_{j\text{sex}}^{\text{obs}}I\left[\text{sex}_{ij} = \text{female}\right] \nonumber \\
& + \sum_{t=2}^{5}\theta_{j\text{town}_{t}}^{\text{obs}}I\left[\text{Townsend}_{ij} = t\right] + \theta_{j\text{t2d}}^{\text{obs}}I\left[\text{type 2 diabetes}_{ij} = \text{yes}\right],\end{aligned}$$ where $\delta_{j0}$ is the level-$j$ intercept adjustment in the multinomial logistic regression imputation model for ethnicity. Hence, the probability of the $j$th level of ethnicity in the missing data, $j=2, \ldots, 4$, is given by $$\label{eq:pethj_r0}
p\left(\text{eth}=j \mid r=0\right) = \frac{1}{n^{\text{mis}}} \sum_{i=1}^{n^{\text{mis}}}\frac{1}{1+\sum_{j=2}^{4}\left(\text{linpred}_{ij}^{\text{mis}}\right)}.$$
From \[eq:eth\_partition,eq:pethj\_r1,eq:linpred\_obs,eq:pethj\_r0,eq:linpred\_mis\], to implement calibrated-$\delta$ adjustment MI, we need to find the solutions $\delta_{j0}$, $j=2, \ldots, 4$, of a system of three non-linear equations for the three categories of ethnicity. The solutions of this system of equations can be obtained simultaneously using the Stata base command `nl` [@StataCorp2015b] and defining a function evaluator program. Once the values of the calibrated-$\delta$ adjustments are obtained, the imputation is performed using the same procedure as outlined in \[subsec3.1\].
Both MI methods are performed using $M=30$ imputations, and Rubin’s rules [@Rubin1987; @Barnard1999] are used to obtain estimates of association and standard errors. All analyses are conducted using Stata 14, [@StataCorp2015b] where `mi impute mlogit` is used for standard MI, the community-contributed command `uvis mlogit` [@Royston2004] for calibrated-$\delta$ adjustment MI, and `mi estimate: logit` for performing the main analysis in the completed datasets and obtaining the final results using Rubin’s rules.[@Rubin1987; @Barnard1999]
Results {#subsec4.5}
-------
Figure \[fig:flowchart\] depicts a flowchart of the selection criteria used to obtain the relevant sample for this study. In total, data from 13532630 individuals are extracted from THIN, of which 2137874 (15.8%) individuals are not permanently registered, 293 (less than 0.1%) individuals do not have their year of birth recorded, 1308 (less than 0.1%) individuals have missing sex, 1376098 (10.2%) individuals have invalid or missing Townsend score, and 2160435 (16.0%) have their start date after their end date. Applying the selection criteria results in 9065617 (70.0%) individuals who are eligible for inclusion in this study. In this eligible sample, there are 1090248 (8.1%) individuals who are registered to THIN general practices in London, of whom 470863 (3.5%) individuals are actively registered on 01 January 2013. Finally, $n=404\,318 \left(3.0\%\right)$ individuals have at least 12 months of follow-up by 01 January 2013 and make up the sample for this study. Table \[tab:example2\_vars\] presents a summary of variables considered in this study. The sample comprises $51\%$ women; the majority of individuals in the sample (approximately $80\%$) are below 60 years of age; there are slightly more than $70\%$ of the individuals with quintiles of the Townsend score of 3 and above; and $5.5\%$ of the individuals have a diagnosis of type 2 diabetes on or before 01 January 2013.
Ethnicity is recorded for $309\,684 \left(76.6\%\right)$ and missing for $94\,634 \left(23.4\%\right)$ individuals (Table \[tab:dist\_etht2d\]). Among individuals with ethnicity recorded, the estimated proportion of the White ethnic group is higher, and the non-White ethnic groups lower compared to the corresponding ethnic breakdown in the 2011 ONS census data for London (Table \[tab:dist\_etht2d\]). Single imputation with the White ethnic group further overestimates the White group and underestimates the other non-White groups, under the assumption that the ethnicity distribution in THIN should match that in the census (Table \[tab:dist\_etht2d\]).
\[tab:example2\_vars\]
Variable Frequency %
----------------------------- ----------- -------
*Age group (years)*
0–9 41601 10.29
10–19 45664 11.29
20–29 50065 12.38
30–39 65695 16.25
40–49 64837 16.04
50–59 53272 13.18
60–69 39427 9.75
70–79 25348 6.27
80+ 18409 4.55
*Sex*
Male 198301 49.05
Female 206017 50.95
*Townsend score*
Quintile 1 (least deprived) 48934 12.10
Quintile 2 64788 16.02
Quintile 3 101305 25.06
Quintile 4 102626 25.38
Quintile 5 (most deprived) 86665 21.43
*Type 2 diabetes* 22100 5.47
: Case study: summary of variables in the analysis; $n = 404\,318$.
\[tab:dist\_etht2d\]
[lcccccc]{} Ethnicity & Frequency &
-----------
%
including
missing
-----------
: Case study: summary of variables in the analysis; $n = 404\,318$.
&
-----------
%
excluding
missing
-----------
: Case study: summary of variables in the analysis; $n = 404\,318$.
&
------------
Frequency
missing
imputed
with White
------------
: Case study: summary of variables in the analysis; $n = 404\,318$.
&
------------
%
missing
imputed
with White
------------
: Case study: summary of variables in the analysis; $n = 404\,318$.
&
----------
%
2011 ONS
census
London
----------
: Case study: summary of variables in the analysis; $n = 404\,318$.
\
White & 224403 & 55.50 & 72.46 & 319037 & 78.91 & 59.8\
Asian & 35027 & 8.66 & 11.31 & 35027 & 8.66 & 18.8\
Black & 30771 & 7.61 & 9.94 & 30771 & 7.61 & 13.3\
Other & 19483 & 4.82 & 6.29 & 19483 & 4.82 & 8.4\
Missing & 94634 & 23.41 & & & &\
$\sum$ including missing & 404318 & & & & &\
$\sum$ excluding missing & 309684 &\
Figure \[fig:dist\_eth\] shows the distribution of four-level ethnicity after missing values in ethnicity are handled by the various methods for missing data. CRA, single imputation with the White ethnic group, and standard MI overestimate the White group while underestimating the other non-White ethnic proportions, compared to the corresponding census statistics. In calibrated-$\delta$ adjustment MI, the majority of missing values in ethnicity are imputed with the Asian and Black groups. This method recovers the ethnic breakdown in the census as expected, since the census distribution is used as the reference.
![Case study: distribution of four-level ethnicity in different methods for handling missing ethnicity data, compared to the 2011 ONS census distribution for London (horizontal black lines).[]{data-label="fig:dist_eth"}](figures/disteth_methodf.pdf)
Figure \[fig:combine\_methodf\] and Table \[tab:case\_study\_or\] present estimated odds ratios of type 2 diabetes diagnosis and $95\%$ CIs for age group, sex, Townsend score, and ethnicity in the analysis model. Age 40–49 years, male, quintile 1, and the White ethnic group are selected as base levels for age group, sex, Townsend score, and ethnicity, respectively. $M = 30$ imputations produce Monte Carlo errors for point estimates of less than 10% of the estimated standard errors for all parameters. The relative efficiency versus an infinite number of imputations is above 0.988 for all parameter estimates and MI methods. Overall, the odds of being diagnosed with type 2 diabetes increase relatively smoothly with older age groups and higher quintiles of the Townsend score; are lower in women compared to men; and are higher in the Asian, Black, and Mixed/Other ethnic groups compared to the White group in all methods for handling missing data in ethnicity.
Compared to the other three methods under consideration, calibrated-$\delta$ adjustment MI produces comparable estimated odds ratios for the younger age groups, and smaller estimated odds ratios for the older ($60+$) age groups. Calibrated-$\delta$ adjustment MI leads to slightly higher estimated odds ratio for women compared to CRA, single imputation with the White ethnic group, and standard MI; this increase is towards the null. All missing data methods produce odds ratios that increase with more deprived quintiles of the Townsend score. Calibrated-$\delta$ adjustment MI yields similar estimated odds ratios compared to the other methods for the first three quintiles of the Townsend score, and higher estimates for the top two quintiles.
The most noticeable differences in point estimates associated with the prevalence of type 2 diabetes diagnoses are seen in the estimated odds ratios for ethnicity. CRA, single imputation, and standard MI again return similar results, in which the odds of having a diagnosis of type 2 diabetes are around 3.6 times higher in the Asian ethnic group compared to the White group, and individuals in the Black ethnic group are about 2.3 times more likely to receive a diagnosis of type 2 diabetes compared to those of White ethnic background. Single imputation with the White ethnic group slightly increases the estimated odds ratios for the non-White groups. This is because explanatory analyses conducted to examine predictors of both ethnicity and missingness in ethnicity suggest that individuals with missing ethnicity are, on average, less likely to have a diagnosis of type 2 diabetes (OR of observing ethnicity for type 2 diabetes (adjusted for age group, sex, Townsend score) = 1.39, $95\%$ CI 1.34 to 1.44, full results not shown). Replacing missing values with the White ethnic group means that this group will contain a lower percentage of type 2 diabetes diagnoses, which implies that the estimated odds ratios for the non-White ethnic groups will increase. Compared to CRA, single imputation with the White ethnic group, and standard MI, calibrated-$\delta$ adjustment MI leads to a reduction in the estimated odds ratios for the non-White ethnic groups (Figure \[fig:combine\_methodf\] and Table \[tab:case\_study\_or\]). For these groups, the $95\%$ CIs of the ethnicity point estimates in calibrated-$\delta$ adjustment MI do not cross that of the other methods.
Fraction of missing information (FMI) [@White2011] for the estimates of association between ethnicity and the prevalence of type 2 diabetes diagnoses was 0.132 (Monte Carlo standard error (MCSE) $=0.003$); 0.193 (MCSE $= 0.05$); 0.230 (MCSE $= 0.066$) for Asian, Black, and Mixed/Other ethnic group, respectively in standard MI. The corresponding quantities for these three groups in calibrated-$\delta$ adjustment MI are 0.283 (MCSE $=0.052$); 0.245 (MCSE $=0.045$); 0.327 (MCSE $=0.051$). Calibrated-$\delta$ adjustment MI appears to have higher FMI compared to standard MI. This could be explained by the fact that non-White ethnic groups, which are under-represented in the observed data, are imputed more often in calibrated-$\delta$ adjustment MI than in standard MI. Therefore, the between-imputation variance relies on more imputed values in the non-White ethnic groups and less frequently imputed values in the White group, which leads to the non-White proportion estimates being more variable across the completed datasets.
--------------------------------------------------------------------------------------------------------- ------- ---------------- ------- ---------------- ------- ---------------- ------- ----------------
(l[2pt]{}r[2pt]{})[2-3]{} (l[2pt]{}r[2pt]{})[4-5]{} (l[2pt]{}r[2pt]{})[6-7]{} (l[2pt]{}r[2pt]{})[8-9]{} OR 95% CI OR 95% CI OR 95% CI OR 95% CI
*Age group (years)*
0-9 0.010 0.006 to 0.016 0.010 0.006 to 0.016 0.010 0.006 to 0.016 0.010 0.006 to 0.017
10-19 0.022 0.016 to 0.032 0.026 0.020 to 0.035 0.025 0.019 to 0.033 0.025 0.019 to 0.033
20-29 0.120 0.103 to 0.139 0.122 0.107 to 0.139 0.120 0.106 to 0.137 0.122 0.107 to 0.139
30-39 0.308 0.283 to 0.336 0.316 0.292 to 0.342 0.320 0.296 to 0.347 0.330 0.305 to 0.357
40-49 1 1 1
50-59 2.641 2.495 to 2.796 2.605 2.474 to 2.743 2.604 2.473 to 2.742 2.516 2.390 to 2.649
60-69 5.255 4.968 to 5.559 5.190 4.933 to 5.46 5.309 5.044 to 5.587 4.928 4.685 to 5.184
70-79 7.662 7.230 to 8.120 7.748 7.352 to 8.166 7.984 7.573 to 8.417 7.484 7.102 to 7.886
80+ 8.154 7.655 to 8.685 8.003 7.560 to 8.472 8.379 7.910 to 8.876 7.596 7.175 to 8.043
*Sex*
Male 1 1 1 1
Female 0.727 0.704 to 0.751 0.752 0.731 to 0.774 0.760 0.738 to 0.782 0.773 0.751 to 0.796
*Townsend score*
Quintile 1 (least deprived) 1 1 1 1
Quintile 2 1.125 1.057 to 1.196 1.121 1.060 to 1.185 1.115 1.054 to 1.179 1.119 1.058 to 1.183
Quintile 3 1.217 1.149 to 1.288 1.242 1.180 to 1.307 1.208 1.147 to 1.272 1.249 1.187 to 1.316
Quintile 4 1.376 1.300 to 1.457 1.420 1.349 to 1.496 1.381 1.312 to 1.455 1.474 1.400 to 1.553
Quintile 5 (most deprived) 1.693 1.596 to 1.796 1.783 1.691 to 1.879 1.708 1.619 to 1.802 1.864 1.768 to 1.966
*Ethnic group*
White 1 1 1 1
Asian 3.588 3.431 to 3.753 3.629 3.474 to 3.789 3.577 3.425 to 3.735 2.355 2.259 to 2.456
Black 2.253 2.135 to 2.378 2.257 2.142 to 2.379 2.254 2.136 to 2.379 1.638 1.555 to 1.725
Mixed/Other 1.606 1.486 to 1.736 1.617 1.497 to 1.746 1.615 1.491 to 1.749 1.174 1.085 to 1.270
--------------------------------------------------------------------------------------------------------- ------- ---------------- ------- ---------------- ------- ---------------- ------- ----------------
: Case study: adjusted ORs and $95\%$ CIs from a multivariable logistic regression model for the prevalence of type 2 diabetes diagnoses, conditional on age group in 2013, sex, Townsend deprivation score, and ethnic group in different methods for handling missing data in ethnicity, $n=404\,138$.[]{data-label="tab:case_study_or"}
Discussion {#sec5}
==========
Our proposed calibrated-$\delta$ adjustment MI method for missing data in a binary/categorical covariate involves utilising population-level information about the incomplete covariate to generate a calibrated-$\delta$ adjustment, which is then used in the intercept of the imputation model in order to improve the analysis of data suspected to be MNAR. The development of this method was motivated by van Buuren et al.’s [@VanBuuren1999] $\delta$ (offset) approach in MI, but where $\delta$ is derived based on external information instead of chosen arbitrarily or based on expert’s belief (which is arguably not arbitrary, but can be subjective). Direct linkage to external data has also increasingly been used for the analysis of missing data generated by a MNAR mechanism.[@Cornish2015] However, external linked data might not always be available, or the linkage might not be possible, whereas our proposed calibrated-$\delta$ adjustment MI method does not require records from the same individuals to be directly linked between the datasets.
Under the MNAR assumption of missing data, MI results rely on subtle, untestable assumptions, and may depend heavily on the particular way the missingness mechanism is modelled. This issue emphasises the central role of sensitivity analysis, which explores how inference may vary under different missingness mechanisms.[@Kenward2007] MI offers flexibility for sensitivity analysis, since the imputation model can be tuned to incorporate possible departures from the MAR assumption.[@Kenward2007; @White2011] Unfortunately, a sensitivity analysis is often not performed or reported sufficiently in practice,[@Wood2004; @HayatiRezvan2015] a tendency abetted by the practical constraints of many applied projects. When the population-level information about the incomplete covariate is available, our proposed calibrated-$\delta$ adjustment MI method provides a useful tool for performing a single, calibrated sensitivity analysis to assess the impact of potential departures from the MAR assumption.
The analytic study of a $2\times2$ contingency table with a binary outcome variable $y$ and a binary covariate $x$ gave insights into how the method works, and will work for more general contingency table settings with one incomplete variable. The analytic study explored the appropriate derivation of the calibrated-$\delta$ adjustment under increasingly complex missingness mechanisms. We showed that when data in $x$ were MNAR dependent on $x$ or both $x$ and $y$, appropriately adjusting the intercept of the imputation model sufficiently corrected bias in the analysis model’s parameter estimates. Based on this setting, simulation studies were conducted to explore scenarios when the population distribution of $x$ was either invariant (i.e. ‘known’) or estimated in an external dataset with uncertainty. Calibrated-$\delta$ adjustment MI was shown to perform as well as standard MI in terms of bias when data were MAR. Further, calibrated-$\delta$ adjustment MI also produced unbiased parameter estimates with good coverage, and was preferred to standard MI under the two general MNAR mechanisms being evaluated.
In the analytic and simulation studies, we did not consider the MNAR selection model where the probability of observing $x$ depends on both $x$, $y$, and their interaction. We suspect that calibrated-$\delta$ adjustment MI with a single intercept adjustment calculated based on the marginal distribution of $x$ alone will not fully correct bias introduced by this missingness mechanism; and that an additional sensitivity parameter for the $x$–$y$ association is present. Information about the population distribution of $x$ conditional on $y$ might be required to produce unbiased estimates when the probability of observing $x$ given $x$ differs across the levels of $y$. However, such information might not always be available in practice. Similarly, when the outcome variable $y$ is continuous, a second sensitivity parameter for the covariate–outcome association in the imputation model is needed; we will explore this setting in another paper.
In the case study which examined the association between ethnicity and the prevalence of type 2 diabetes diagnoses in THIN, calibrated-$\delta$ adjustment MI using information from census data yielded a more plausible estimate of the ethnicity distribution compared to CRA, single imputation of missing values with the White ethnic group, and standard MI. Subsequently, estimates of association for the non-White ethnic groups produced by calibrated-$\delta$ adjustment MI were lower than that in the other methods. Previously, it was found that ethnicity was more likely to be recorded for individuals with a diagnosis of type 2 diabetes. By imputing missing values with the non-White ethnic groups more frequently, calibrated-$\delta$ adjustment MI led to a decrease in the percentage of prevalent type 2 diabetes cases among these groups, which we thought was the primary reason explaining the lower odds ratios compared to the other methods. In addition, it was also possible that the explanatory power of ethnicity for type 2 diabetes was partially diluted by the stronger effect of deprivation status, which compensated for the reduction in the odds ratios for ethnicity. The odds ratios for Townsend deprivation score were higher in calibrated-$\delta$ adjustment MI compared to CRA for the top two quintiles. These findings seemed to suggest that some effect of ethnicity was absorbed in Townsend score in calibrated-$\delta$ adjustment MI, where deprivation status explained some of the effect which might otherwise have been explained by ethnicity. This could be attributed to a possibility that individuals of Asian and Black ethnic groups, whose ethnicity was not recorded, were more likely to belong to the more deprived quintiles of the Townsend score.
Given the missingness mechanisms considered thus far for the development of calibrated-$\delta$ adjustment MI in \[sec2,sec3\], results in the case study suggested a potential departure from the MAR assumption for missingness in ethnicity. This was because, conditional on the outcome variable type 2 diabetes and other fully observed variables included in the analysis model, standard MI did not yield a distribution of ethnicity that was comparable to the census ethnic breakdown. Ethnicity was also not likely to be MNAR dependent only on the values of ethnicity, since the point estimates in CRA and standard MI were broadly comparable. Results from the exploratory analyses examining the associations between covariates in the imputation model for ethnicity and missingness in ethnicity among the complete records suggested that age group, sex, Townsend score, and type 2 diabetes were factors likely to be associated with whether ethnicity was recorded. This finding indicated that ethnicity was likely to be MNAR depending on the ethnic groups, fully observed outcome variable (type 2 diabetes diagnoses), as well as other fully observed covariates in the analysis model (age group, sex, and deprivation status).
The major strength of calibrated-$\delta$ adjustment MI is its flexibility to be adapted to impute variables in a given dataset whose distributions might be available in some external data. Here we used census data for ethnicity in primary care electronic health records, but information obtained from other nationally representative datasets (such as the Health Survey for England [@UKDataService]) could similarly be used to impute missing data in other health indicators routinely recorded in primary care such as smoking status or alcohol consumption. In such instances, the variability associated with estimating the reference distribution used for calibration needs to be accounted for in calibrated-$\delta$ adjustment MI as illustrated in \[subsec3.2\], although this source of uncertainty might be negligible depending on the size of the external dataset.
Throughout this paper, we restricted our development of calibrated-$\delta$ adjustment MI to the case of a single partially observed covariate. However, we believe this approach can be extended for handling missing data in more than one variable. Multivariate imputation by chained equations (MICE) [@VanBuuren1999; @VanBuuren2011] is a popular procedure for performing MI of multivariate missing data, and is commonly implemented under the MAR assumption.[@Marston2010; @Marston2014] MICE is an iterative procedure which requires the specification of an imputation model for each incomplete variable, conditional on all other variables. Our proposed univariate calibrated-$\delta$ adjustment MI method can, in principle, be embedded into MICE to impute certain MNAR variables whose distributions are available externally, while the standard MI method can be used for the imputation of other variables assuming data are MAR. Under the MICE framework, when there are several MNAR variables to be imputed, information from more than one external data source can potentially be drawn on and utilised in calibrated-$\delta$ adjustment MI for these variables.
Finally, returning to the analytic and simulation studies, we did not consider the setting where both the outcome variable $y$ and the covariate $x$ are incomplete. When $y$ is MNAR dependent on its values and in addition to the population information on $x$ we can obtain the marginal distribution of $y$ from an external dataset, then this information can be used in calibrated-$\delta$ adjustment MI for $y$ when $y$ is imputed in the MICE algorithm. If $y$ is MAR then there must be some artificial mechanism whereby the dataset is divided into two subsets; one where $y$ is MAR dependent on the observed values of $x$ and another one where $x$ is MNAR dependent on its values. In this setting, our proposed MI method should work for $x$ when it is imputed in the MICE algorithm. The more complex missingness settings involving several incomplete covariates are subjected to on-going work and will be reported in the future.
Acknowledgements {#acknowledgements .unnumbered}
================
Tra My Pham was supported by awards to establish the Farr Institute of Health Informatics Research, London, from the Medical Research Council, Arthritis Research UK, British Heart Foundation, Cancer Research UK, Chief Scientist Office, Economic and Social Research Council, Engineering and Physical Sciences Research Council, NIHR, National Institute for Social Care and Health Research, and Wellcome Trust (grant MR/K006584/1); and the NIHR School for Primary Care Research (project number 379). James Carpenter and Tim Morris were supported by the Medical Research Council (grant numbers MC\_UU\_12023/21 and MC\_UU\_12023/29).
Conflict of interest {#conflict-of-interest .unnumbered}
====================
The authors declare no potential conflict of interests.
Supporting information {#supporting-information .unnumbered}
======================
Weighted multiple imputation for a binary/categorical covariate
===============================================================
The procedure of the weighted multiple imputation is as follows. In the imputation step, weights derived from the population marginal distribution of the incomplete variable are attached to the complete records, and a weighted (multinomial) logistic regression model is fitted to the complete records to obtain the maximum likelihood estimates of the imputation model’s parameters $\widehat{\boldsymbol{\theta}}$ and their asymptotic sampling variance $\widehat{\boldsymbol{U}}$. New parameters are then drawn from the large-sample normal approximation $N(\widehat{\boldsymbol{\theta}}, \widehat{\boldsymbol{U}})$ of its posterior distribution, assuming non-informative priors. Finally, imputed values are drawn from the (multinomial) logistic regression using these new parameters. Note that *no weights* are used when fitting the substantive scientific model to the imputed data.
Derivation of the marginal weights
----------------------------------
The idea of augmenting the standard MI method with weights is related to the technique of post-stratification weighting, which is commonly used in survey non-responses when the population distributions are known.[@Raghunathan2015] To post-stratify the sample, weights are calculated to bring the sample distribution in line with the population. Suppose that in a survey, one of the variables measured is ethnicity, which is categorised into four groups (White, Black, Asian, and Other). If the population distribution of ethnicity is available, the distribution of ethnicity among survey respondents can be compared with the population distribution. Suppose that a proportion $p^{\text{obs}} = 0.8$ of the survey respondents give their ethnicity as White, whereas the population has $p^{\text{pop}} = 0.6$ in this category. The White category is over-represented in the survey respondents, but can be made representative of the population by assigning to the responses a post-stratification weight $w^{\text{ps}} < 1$, such that $$w^{\text{ps}} = 1/(p^{\text{obs}}/p^{\text{pop}}) = 1/(0.8/0.6)=0.75.$$ In adapting this idea to MI, we need to address the complication arising because the *completed* data obtained after MI consist of both observed and imputed (missing) data. Naive use of post-stratification weights in MI will recover the correct population distribution in the imputed data. However, since the observed data remain the same, the distribution in the completed data will not be matched to that in the population. Therefore, some *compensation* for the lack of representativeness in the observed data is needed in the imputed data so that the correct population distribution can be recovered after imputation. Continuing with the survey example, suppose that we survey $200$ individuals, $100$ of whom respond with their ethnicity. A proportion $p^{\text{obs}} = 0.8$ of these $100$ responses are in the White group. If the population proportion of this group is $p^{\text{pop}} = 0.6$, we would expect to have $120$ White individuals in the survey sample. This implies that among the $100$ individuals with missing ethnicity, we need to impute ethnicity of $40$ individuals as White, i.e. the proportion of the White category required in the missing data, $p^{\text{req}}$, is equal to $0.4$. To make the completed (observed and imputed) data of this category representative of the population, we need to weight respondents of this category in the imputation model by $$1/(p^{\text{obs}}/p^{\text{req}}) = 1/(0.8/0.4) = 0.5,$$ which is smaller than the corresponding naive post-stratification weight above, since it compensates for the over-representation among the survey respondents of White ethnicity.
More generally, suppose that we seek to collect a $J$-level variable $x$ in a sample of size $n$, resulting in $x$ being observed for $n^{\text{obs}}$ subjects and missing for $n^{\text{mis}}$ subjects, $n^{\text{obs}}+n^{\text{mis}}=n$. Let $p_{j}^{\text{obs}}$ and $p_{j}^{\text{req}}$ denote the level-$j$ proportions of $x$ in the observed and imputed data respectively, such that $p_{j}^{\text{obs}}n^{\text{obs}}= n_{j}^{\text{obs}}$, and $p_{j}^{\text{req}}n^{\text{mis}}= n_{j}^{\text{req}}$, where $j = 1, \ldots, J$. Let $p_{j}^{\text{pop}}$ denote the level-$j$ proportion of $x$ in the population, which is assumed to be known. The aim here is to find $p_{j}^{\text{req}}$ for each level of $x$ such that the number of subjects in the completed data after imputation is equal to the expected number implied by the corresponding population proportion, i.e. $n_{j}^{\text{obs}}+n_{j}^{\text{req}}=p_{j}^{\text{pop}}n$. The level-$j$ proportion of $x$ required in the imputed data, $p_{j}^{\text{req}}$, is given by $$p_{j}^{\text{req}}=\frac{p_{j}^{\text{pop}}n - p_{j}^{\text{obs}} n^{\text{obs}}}{n^{\text{mis}}}.$$ Therefore, the weight for group $j$, which we refer to as the ‘marginal weight’ and denote by $w_{j}^{\text{m}}$, is $$w_{j}^{\text{m}}=1/(p_{j}^{\text{obs}}/p_{j}^{\text{req}}).$$
Derivation of the conditional weights
-------------------------------------
The marginal weights introduced above only depend on the population distribution of the incomplete variable. However, if there are (fully observed) covariates in the imputation model, the associations between these variables and the incomplete variable distribution are not reflected in such weights. We therefore adjust the marginal weights to obtain another set of weights, termed the ‘conditional weights’, which account for covariates in the imputation model. These weights are derived using the marginal distribution of the incomplete variable obtained after having estimated the parameters of an imputation model assuming MAR in the complete records. Suppose that an imputation model is fitted to the complete records, and the corresponding predicted probabilities of the incomplete variable (averaged over the covariates) are obtained and applied to the missing data. Let ${p}_{j}^{\text{pred}}$ denote the resulting predicted level-$j$ proportion of $x$ in the completed data, then the level-$j$ proportion required in the imputed data is given by $$p_{j}^{\text{req}} = \frac{p_{j}^{\text{pop}} n - {p}_{j}^{\text{pred}} n^{\text{obs}}}{n^{\text{mis}}},$$ and the conditional weight for group $j$, denoted by $w_{j}^{\text{c}}$ , is $$w_{j}^{\text{c}}=1/({p}_{j}^{\text{pred}}/p_{j}^{\text{req}}).$$
In this approach, the effects of covariates in the imputation model are reflected in the predicted probabilities ${p}_{j}^{\text{pred}}$, which are then used to derive the conditional weights for weighted MI.
Analytic study – bias in a $2 \times 2$ contingency table {#app2}
=========================================================
In the $2\times 2$ contingency table of a complete binary outcome variable $y$ and an incomplete binary covariate $x$ (\[sec2\]), we calculate analytic bias in the analysis model’s parameter estimates (defined as $\hat{\beta} - \beta$) after missing values in $x$ are handled by (i) a CRA, (ii) standard MI, (iii) marginal weighted MI, and (iv) conditional weighted MI. The analytic calculations are then verified by simulating a full-data sample with $n=10\,000$ observations of $x$ and $y$ from the following model $$\begin{aligned}
& x \sim \text{Bernoulli}\left(p_{x}^{\text{pop}} = 0.7\right);\\
&\text{logit}\left[p\left(y=1 \mid x\right)\right] = \beta_{0} +\beta_{x}x,\end{aligned}$$ where $\beta_{0}=\text{ln}\left(0.5\right)$ and $\beta_{x}=\text{ln}\left(1.5\right)$. Missing values in $x$ are generated using selection models M1–M4 with a range of values for the selection parameters $\alpha$ (Table \[tab:anstudy\_selectionparam\]).
----------------------------------------------------------------------------- ------------------------------------------ --------------------- --------------------- --------------------- ----------
(l[2pt]{}r[2pt]{})[3-3]{}(l[2pt]{}r[2pt]{})[4-4]{}(l[2pt]{}r[2pt]{})[5-5]{} $\alpha_{0}$ $\alpha_{x}$ $\alpha_{y}$
M1 $\alpha_{0}$ $\left[-3,3\right]$ $5$–$95$
M2 $\alpha_{0} + \alpha_{y}y$ $\left[-3,3\right]$ $\left[-3,3\right]$ $3$–$97$
M3 $\alpha_{0} + \alpha_{x}x$ $\left[-3,3\right]$ $\left[-3,3\right]$ $2$–$98$
M4 $\alpha_{0} + \alpha_{x}x + \alpha_{y}y$ $0.5$ $\left[-3,3\right]$ $\left[-3,3\right]$ $9$–$84$
----------------------------------------------------------------------------- ------------------------------------------ --------------------- --------------------- --------------------- ----------
: Note: $r$: response indicator of $x$.
Figure \[fig:biascal\_mary,fig:biascal\_mnarx,fig:biascal\_mnarxy\] present the analytic bias in CRA, standard MI, marginal and conditional weighted MI under MAR and MNAR mechanisms with the various values of the selection parameters. When $x$ is MCAR (M1), all methods provide unbiased parameter estimates, as suggested by the calculations (results not shown).
When $x$ is MAR conditional on $y$ (M2, Figure \[fig:biascal\_mary\]), standard MI and conditional weighted MI are unbiased, while bias is observed for CRA in $\beta_{0}$, and for marginal weighted MI in both parameter estimates. This bias is due to the marginal weights not accounting for the association between $x$ and $y$ in the imputation model for $x$. As a result, marginal weights do not successfully recover the correct distribution of $x$ after MI.
Both parameter estimates are unbiased in marginal weighted MI when $x$ is MNAR dependent on $x$ (M3, Figure \[fig:biascal\_mnarx\]), while standard MI leads to noticeable bias in the estimate of $\beta_{0}$. Bias in conditional weighted MI is small and occurs for extreme values of the selection parameters. Since missingness in $x$ does not depend on $y$ under M3, CRA is unbiased in both parameter estimates as the theory predicts.
Under the last missingness mechanism when $x$ is MNAR dependent on both $x$ and $y$ (M4, Figure \[fig:biascal\_mnarxy\]), none of the methods result in unbiased parameter estimates. However, bias appears to be the smallest in conditional weighted MI. Although bias is present in both standard MI and marginal weighted MI, the magnitude of bias is smaller in marginal weighted MI compared to standard MI. Under this missingness mechanism, conditional weighted MI can be regarded as a hybrid of marginal weighted MI and standard MI. The conditional weights correct for some bias introduced by $x$ in the selection model in a similar manner to the marginal weights under M3; the method also alleviates some residual bias similarly to standard MI under M2.
Overall, these results suggest that under the missingness mechanisms considered in this paper, calibrated-$\delta$ adjustment MI provides a more general solution for accommodating missing data in $x$ and is therefore the preferred method compared to standard MI and marginal and conditional weighted MI.
![Note: selection parameters $\alpha_{0} \in [-3,3]$, $\alpha_{y} \in [-3,3]$; corresponding percentages of missing $x$ are presented for extreme values ($\pm 3$) of the $\alpha$ parameters.](figures/biascal_mary)
![Note: selection parameters $\alpha_{0} \in [-3,3]$, $\alpha_{x} \in [-3,3]$; corresponding percentages of missing $x$ are presented for extreme values ($\pm 3$) of the $\alpha$ parameters.](figures/biascal_mnarx)
![Note: selection parameters $\alpha_{0} = 0.5, \alpha_{x} \in [-3,3]$, $\alpha_{y} \in [-3,3]$; corresponding percentages of missing $x$ are presented for extreme values ($\pm 3$) of the $\alpha$ parameters.](figures/biascal_mnarxy)
| 0 |
---
abstract: 'Iron chalcogenides display a rich variety of electronic orders in their phase diagram. A particularly enigmatic case is FeTe, a metal which possesses co-existing hole and electron Fermi surfaces as in the iron pnictides but has a distinct ($\pi$/2,$\pi/2$) bicollinear antiferromagnetic order in the Fe square lattice. While local-moment physics has been recognized as essential for understanding the electronic order, it has been a long-standing challenge to understand how the bicollinear antiferromagnetic ground state emerges in a proper quantum spin model. We show here that a bilinear-biquadratic spin-$1$ model on a square lattice with nonzero ring-exchange interactions exhibits the bicollinear antiferromagnetic order over an extended parameter space in its phase diagram. Our work shows that frustrated magnetism in the quantum spin model provides a unified description of the electronic orders in the iron chalcogenides and iron pnictides.'
author:
- 'Hsin-Hua Lai'
- 'Shou-Shu Gong'
- 'Wen-Jun Hu'
- Qimiao Si
bibliography:
- 'bib4FeTeBC.bib'
title: Frustrated magnetism and bicollinear antiferromagnetic order in FeTe
---
*Introduction.*— Iron-based superconductors (FeSCs) have been of extensive interest during the past eight years [@Kamihara2008; @Johnston2010; @PCDai2015; @Si2016]. Early work in the field focused on the iron pncitides, such as BaFe$_2$As$_2$ with various chemical substitutions. More recently, the iron chalcogenides have occupied the center stage, in part because they have provided a new record of the superconducting transition temperature ($T_c$) [@QYWang2012; @JJLee_Nature; @SLHe2013; @YWang2015] and a renewed hope of reaching even higher $T_c$. Because superconductivity in these materials occurs at the border of correlation-induced electronic orders [@Johnston2010; @Si2016], it is vitally important to understand the origin and nature of the ordered states.
One of the outstanding puzzles in the field arises in the structurally simplest iron chalcogenide FeTe. Compared to the iron pnictides, it has a similar Fermi surface with hole and electron pockets [@Subedi2008; @Xia2009]. Yet, instead of having a ($\pi$, $0$) collinear antiferromagnetic (AFM) order as in the latter case, FeTe has a ($\pi/2$,$\pi/2$) bicollinear AFM order (BC) illustrated in Fig. \[Fig:bicollinear\] [@Bao2009; @ShiliangLi2009; @JWen_FeTe]. This order is entirely unexpected in the weak-coupling Fermi-surface nesting picture. Thus, it is widely believed that the origin lies in the frustrated magnetism of correlation-induced local moments [@SiAbrahams08; @Ma2009; @Fang2009]. A natural starting point would be to consider bilinear exchange interactions between the local moments with nearest-neighbor ($J_1$) and further neighbor (second-neighbor $J_2$ and third-neighbor $J_3$) interactions on the square lattice. However, a classical spin model with bilinear $J_1-J_2-J_3$ interactions yields incommensurate spiral magnetic order, and the ordering wavevector can reach $(\pi/2, \pi/2)$ only for an infinitesimal $J_1$ [@Moreo1990; @Ferrer1993]. The multi-band nature of these systems makes it natural to consider the role of the biquadratic interactions, which have recently been discussed [@YuSi_AFQ; @FaWang2015; @Glasbrenner2015; @Wangzhentao2016; @Lai2016; @Wenjun_nematic2016; @Ergueta2016] as providing a route towards understanding the intriguing phenomenologies of the iron chalcogenides. Indeed, it is also known in classical spin models that the presence of the biquadratic $K$ interactions can make the ordering wavevector to be $(\pi/2,\pi/2)$ [@Hu2012]. The problem is that the BC state is degenerate with the plaquette AFM order (PL) shown in Fig. \[Fig:plaquette\]. How the BC order emerges as the true ground state in quantum spin models remains a long-standing puzzle.
In this *Letter*, we propose that ring exchange ($\mathfrak{R}$) interactions provide a robust mechanism to stabilize the BC order. Strongly correlated bad metals such as FeTe possess significant charge fluctuations [@Si2016]. We therefore expect that the ring exchange interactions involving cyclic permuting spin degrees of freedoms on more-than-two lattice sites can be significant, in addition to the biquadratic terms that only capture charge fluctuations between two lattice sites.
More specifically, we study the frustrated quantum spin model on a square lattice that contains bilinear and biquadratic interactions. For spin $S=1$, using semi-classical site-factorized wavefunction analysis and fully-quantum density matrix renormalization group (DMRG) studies, we provide evidence for a phase regime in which the BC order is degenerate with the PL order. The degeneracy is robust, persisting even for the higher $S=3/2, 2$ systems as found by the DMRG calculations. We then determine the fluctuation spectra of the BC and PL phases using a flavor-wave theory analysis for the spin-$1$ case. This allows us to identify a regime where quantum fluctuations select the BC order, by destabilizing the PL order. For the much more extended parameter regime where both the BC and PL states are stable and degenerate, we show that a nonzero $4$-site ring-exchange interaction selects the BC order.
\
*Model.*— We study the $J$-$K$-$\mathfrak{R}$ model whose Hamiltonian is defined as $$\begin{aligned}
H&=& \sum_{i, j} \left[ J_{ij} {\bf S}_i \cdot {\bf S}_j + K_{ij} \left( {\bf S}_i \cdot {\bf S}_j \right)^2 \right] \nonumber \\
&-& \sum_{\triangpic} \mathfrak{R}_3 \left( P_{123} + \Hc \right) + \sum_{\rhombpic} \mathfrak{R}_4 \left( P_{1234} + \Hc \right),
\label{ham_ring}~~~~~\end{aligned}$$ where ${\bf S}_{i}$ is the local moment at site $i$ of the Fe square lattice, and the bilinear and biquadratic interactions are chosen up to the third neighbors. The $P_{123}$ and $P_{1234}$ stand for the $3$-site and $4$-site ring exchanges with $\mathfrak{R}_3, \mathfrak{R}_4 >0$ [@Lai_SU(3)_gapless], which rotate the states such as $P_{123} | \alpha \beta \gamma\ra = |\gamma \alpha \beta\ra,~P_{1234} | \alpha \beta \gamma \xi \ra = | \xi \alpha \beta \gamma \ra$. The ring exchange operators can be re-expressed via physical spin operators [@Itoi97] as detailed in Supplemental Material [@supplemental]. The biquadratic terms also introduce the quadrupolar operator ${\bf Q}_i$, which has five components: $Q^{x^2 - y^2}_i = (S^x_i)^2 - (S^y_i)^2$, $Q^{3z^2 - r^2}_i = [ 2 (S^z_i)^2 - (S^x_i)^2 - (S^y_i)^2]/\sqrt{3}$, $Q^{xy} = S^x_i S^y_i + S^y_i S^x_i$, $Q^{yz} = S^y_i,S^z_i +S^z_i S^y_i$, and, $Q^{zx} = S^z_i S^x_i + S^x_i S^z_i$. The biquadratic term can be re-expressed as $({\bf S}_i \cdot {\bf S}_j )^2 = ({\bf Q}_i \cdot {\bf Q}_j) /2 - ({\bf S}_i \cdot {\bf S}_j) /2 + ({\bf S}^2_i {\bf S}^2_j)/3$.
*Semi-classical phase diagram for spin-$1$ model.*— We first use the site-factorized wavefunction analysis to study the semi-classical phase diagram for $S = 1$. Based on the time-reversal invariant basis of the SU(3) fundamental representation, $
|x\ra = \left[ i |1\ra - i |\bar{1}\ra \right]/\sqrt{2}, ~ |y\ra = \left[ |1\ra + |\bar{1}\ra \right]/\sqrt{2},~~ |z\ra = -i |0\ra,$ where we abbreviate $|S^z = \pm1\ra \equiv |\pm1\ra$, $|S^z = 0\ra \equiv |0\ra$, and $|\bar{1}\ra \equiv |-1\ra$, we can introduce the complex site-factorized wavefunction vectors at each site to characterize any possible ordered state with short-ranged correlations as $ {\bf d}_i = ( d^x_i, d^y_i, d^z_i )$ in the $\{|x\ra, |y\ra, |z\ra\}$ basis (${\bf d}_i = {\bf u}_i + i {\bf v}_i$). The normalization of the wavefunction leads to the constraint ${\bf d}_i \cdot \bar{\bf d}_i = 1$, or equivalently, ${\bf u}^2_i + {\bf v}^2_i =1$, and the overall phase can be fixed by requiring ${\bf d}_i^2 = \bar{\bf d}_i^2$, i.e., ${\bf u}_i \cdot {\bf v}_i = 0$. In terms of ${\bf d}$, the Hamiltonian is expressed as $$\begin{aligned}
&&H = \sum_{i, \delta_n} \left[J_n\left| {\bf d}_i \cdot \bar{\bf d}_j\right|^2
+ \left(K_n - J_n\right) \left|{\bf d}_i \cdot {\bf d}_j \right|^2 + K_n\right]~~~~ \nonumber \\
&& - \sum_{\triangpic} \mathfrak{R}_3 (\bf{d}_1 \cdot \bar{\bf{d}}_3) ( \bf{d}_2 \cdot \bar{\bf{d}}_1) ( \bf{d}_3 \cdot \bar{\bf{d}}_2) + \Hc \nonumber \\
&& + \sum_{\rhombpic} \mathfrak{R}_4 (\bf{d}_1 \cdot \bar{\bf{d}}_4) (\bf{d}_2 \cdot \bar{\bf{d}}_1)(\bf{d}_3 \cdot \bar{\bf{d}}_2) (\bf{d}_4 \cdot \bar{\bf{d}}_3) + \Hc. \label{Hd_4ring}~~~~~\end{aligned}$$ In the following, we will drop the irrelevant constant terms.
To illustrate the energetic degeneracy of the BC and PL orders, we first set ring exchanges $\mathfrak{R}_3, \mathfrak{R}_4=0$ and without loss of generality we study the $J$-$K$ model with $(J_1,J_2, J_3) = (1, 0.8, 1)$. We vary $K_1 = K_3\equiv -K <0$ and $K_2 \in [-0.5,0.5]$ on an $L\times L$ square lattice with $L$ up to $8$ to obtain the site-factorized wavefunction phase diagram, Fig. \[Fig:PD\_noring\]. Besides the intriguing doubly degenerate BC/PL orders, we find another doubly degenerate orders dubbed $(\pi/2, \pi/2)$ AFM$^*_{a/b}$ and abbreviated as AFM$^*_{a/b}$ below, whose spin patterns are illustrated in Figs. \[Fig:pi/2pi/2\_phase\](c)-(d). [@AFMs_note] The semi-classical energies for these two orders in the absence of the ring exchange interactions are $ \mathcal{E}_{BC/PL}= K_1 + K_2 + 2(K_3 - J_3)$ and $ \mathcal{E}_{AFM_{a/b}^*} = \frac{3}{4}K_1 + \frac{1}{2} K_2 + 2(K_3 - J_3)$. This gives the phase boundary, $K_1 + 2K_2 =0$, consistent with the numerical result in Fig. \[Fig:PD\_noring\].
The BC/PL orders have different “spin dipolar nematicity" along the off-diagonal directions of the square lattice, defined as $\sigma^s_{2} \equiv (1/N_s) \sum_j {\bf S}_j \cdot \left[{\bf S}_{j + \hat{x} + \hat{y}} - {\bf S}_{j - \hat{x} + \hat{y}}\right]$ (where $N_s$ is the total number of lattice site). This nematic order parameter is nonzero for the BC phase, but vanishes for the PL phase. The two orders, AFM$^*_{a/b}$, do not possess the spin dipolar nematicity. Still, they have broken $C_4$ symmetry due to the nonzero “spin quadrupolar nematicity", defined in terms of $\sigma^Q_1 = 1/N_s \sum_j {\bf Q}_j \cdot \left[ {\bf Q}_{j + \hat{x}} - {\bf Q}_{j + \hat{y}}\right]$. The spin quadrupolar nematicity along off-diagonal direction are also nonzero but are much smaller.
Since our main focus is to examine the mechanism for stabilizing the BC order relevant to FeTe, below we focus on the BC/PL regime.
*DMRG calculations for the $J$-$K$ model.*— To analyze the robustness of the degeneracy in the BC and PL orders, we next turn to analyzing the quantum fluctuations in an unbiased way using the DMRG [@White1992] method with spin rotational SU(2) symmetry [@Mcchlloch2002; @gong2014square]. We study cylinder system on two different geometries–the rectangular cylinder (RC) and the $\pi/4$-rotated tilt cylinder (TC) [@DMRG]. We show the spin correlations for spin-$1$ and spin-$2$ models at the parameter $(J_1,J_2,J_3,K_1,K_2,K_3)=(1,0.8,1,-0.5,0.1,-0.5)$ in Fig. \[Fig:DMRG\_main\] (see Supplemental Material for spin-$3/2$ [@supplemental]).
For the models with different spin-$S$, we find the PL order on the RC cylinder and the BC order on the TC cylinder. On a finite-size system, different states being stabilized on different geometries suggest energetically degenerate competing states [@gong2015honeycomb]. This is corroborated by a comparison of the bulk energies of the two states on different system sizes [@supplemental]. On the large cylinders, we find the energies of the two states to be quite close to each other. For the spin-$1$ model, the energy difference between the RC ($L_y = 8$) and TC ($L_y = 6$) cylinders is only $0.2\%$. Thus, our DMRG results strongly suggest the (quasi-) degeneracy of the BC and PL states in the $J$-$K$ model, even after the quantum fluctuations effects are considered, which is also consistent with the flavor-wave theory results to be presented below.
\
*Stiffness of the BC/PL orders in the $J$-$K$ model.*— To address the mechanisms for breaking the BC/PL degeneracy in the BC and PL phases of the $J$-$K$ model, we perform the flavor-wave theory analysis [@Bauer2012; @Lai_SU(3)_gapless], which considers partial quantum fluctuations going beyond the semi-classical picture. Within the flavor wave calculations for BC and PL, we consider a square lattice consisting of $16$ sublattice per unit cell illustrated in Fig. \[Fig:FLWT\_lattice\]. In addition, we associate $3$ Schwinger-bosons at each site $i$, $b_{i\alpha=x,y,z}$, to the states under SU(3) time-reversal invariant basis, $|x\ra,~|y\ra,~|z\ra$, where $b^\dagger_{i\alpha} |vac\ra = |\alpha\ra$ with $|vac\ra$ being the vacuum state of the Schwinger bosons. The bosons satisfy a local constraint, $\sum_{\alpha} b^\dagger_{i \alpha}b_{i \alpha} = 1.$ The model Hamiltonian in terms of spin operators can be re-expressed in terms of the bosons, $$\begin{aligned}
H \simeq \sum_{i, \delta_n, \alpha, \beta} \left[ J_nb^\dagger_{i \alpha} b_{i \alpha} b^\dagger_{j\beta} b_{i\beta} + \left(K_n - J_n\right) b_{i\alpha}^\dagger b^\dagger_{j \alpha} b_{i\beta} b_{i\beta}\right],~~~\end{aligned}$$ where $j = i + \delta_n,$ and $\delta_n$ (with $n = 1,~2,~3$) connects site $i$ to its $n$th nearest neighbor sites, and we have ignored the constant terms $K_n - J_n$ in the above equation. We also introduce different local rotations for different site $j$. For site $j \in | S^z = \pm1\ra$, we introduce $$\begin{aligned}
\begin{pmatrix}
b_{jx} \\
b_{jy} \\
b_{jz}
\end{pmatrix} =
\begin{pmatrix}
\mp \frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2} }& 0\\
\frac{1}{\sqrt{2}} & \mp \frac{i}{\sqrt{2}} & 0 \\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
d_{jx}\\
d_{jy}\\
d_{jz}
\end{pmatrix},\end{aligned}$$ where for each site there is only one flavor of bosons $d_{ix}$ condensing, and we replace $d^\dagger_{ix}$ and $d_{ix}$ by $(M - d_{iy}^\dagger d_{iy} - d^\dagger_{iz} d_{iz})^{1/2}$, with $M=1$ in the present case. A $1/M$ expansion up to the quadratic order of the Holstein-Primakoff bosons $d_y$ and $d_z$ followed by a Bogoliuobov transformation allows us to extract the ground state energy [@Xiao2009]. The detailed derivations are presented in the Supplemental Material [@supplemental]. Numerically diagonalizing the quadratic boson Hamiltonians on a square network consisting of $100\times 100$ cluster unit cells gives the energies of BC and PL which can be used to extract the regimes of BC and PL with finite stiffness.
The results are summarized in the shaded regimes in Fig. \[Fig:PD\_noring\]. The blue/green shaded regions in Fig. \[Fig:PD\_noring\] represent the stable BC/PL regimes in which the stiffness of BC/PL are finite. Surprisingly, we find that the BC and PL have *different* stiffness. In particular, there is a parameter range over which the BC order is stable, while the PL order is not due to negative stiffness (see supplementa material [@supplemental]). This represents a fluctuation mechanism to select the BC AFM order.
We find a much wider parameter regime where both BC and PL have positive stiffness. The flavor-wave theory also suggests that the energy splitting between these two orders are negligible (for a point deep inside this regime, the energy splitting is $\leq 0.1\%$; near the boundary, the energy splitting is $\sim 1\%$). Because this common regime, where both orders are stable, covers a large parameter space, we need to explore additional inputs for another mechanism that breaks the degeneracy and stabilizes the BC. This leads us to discuss the effect of ring exchanges in the next subsection.
We close this subsection with a remark on the limit of vanishing $K,~K_2$ (and in the absence of the ring exchange interactions); further details are given in the Supplemental Material [@supplemental]. For $(J_1, J_2, J_3) = (1, 0.8, 1)$, which we have so far focused on, Fig. \[Fig:PD\_noring\] shows that the BC/PL phases are unstable at $K, K_2 \rightarrow 0$. In this limit, the flavor-wave theory suggests that neither of the BC/PL phases can be stable except for infinitesimally small ratios of $J_1/J_3$ and $J_1/J_2$ in the $J_1$-$J_2$-$J_3$-only model. Consider, for example, $(J_2, J_3) =( 0.8,1.0)$, $K,K_2 = 0$: we find that the BC and PL can only be stable for a very small range of $J_1 \leq 0.03$. (Within this regime, the energies between the two orders are degenerate, with their difference being $\sim 0.001\%$.) This implies that the previous conclusion suggesting that $1/S$ quantum fluctuations stabilize the PL phase in an extended parameter range cannot apply to the $J_1$-$J_2$-$J_3$ model [@Chubukove_PL2012]. Going “slightly” beyond the $J$-only model by considering the effects of $K_1 = K_3 = -K$ with $K_2=0$, we find that increasing $K$ significantly enhances the threshold values of $J_1$ for stable BC and PL orders. Taking $K=0.1, K_2 =0$, we find threshold values $J^{BC}_1 \simeq 0.42$ for BC and $J^{PL}_1 \simeq 0.5$ for PL (see Supplemental Material [@supplemental]). On the other hand, increasing $K_2 > 0$ reduces the stable BC and PL regimes.
*Ring exchanges stabilizing BC.*— We are now in position to address the role of the ring exchange interactions in the wide parameter regime where the BC and PL phases are degenerate, with both having positive stiffness. In the presence of the ring exchanges, the energy corrections, within the site-factorized wavefunction, to each order are $$\begin{aligned}
&\mathcal{E}^{BC}_\mathfrak{R} = -2 \mathfrak{R}_3,~~~&\mathcal{E}^{PL}_\mathfrak{R} = -2 \mathfrak{R}_3 + \frac{\mathfrak{R}_4}{2},\\
&\mathcal{E}^{AFM^{*}_a}_\mathfrak{R} = - \mathfrak{R}_3, ~~~& \mathcal{E}^{AFM^{*}_b}_\mathfrak{R} = - \mathfrak{R}_3 + \frac{\mathfrak{R}_4}{4}.\end{aligned}$$ The $3$-site ring exchange interaction, $\mathfrak{R}_3$, does not split the degeneracy of BC/PL and that of AFM$^*_{a/b}$, although it does lower the energies of BC/PL and AFM$^*_{a/b}$ by different amounts and make those of BC/PL lower. By contrast, the $4$-site ring exchange, $\mathfrak{R}_4$, *does* lift the degeneracy between BC/PL (as well as that between AFM$^*_{a/b}$).
To illustrate such effects of the ring exchanges, we take relatively small values $\mathfrak{R}_3 = \mathfrak{R}_4 = 0.1 \ll J_n, | K_n |$. The phase diagram of the $J$-$K$-$\mathfrak{R}$ model for this case is shown in Fig. \[Fig:PD\_wring\]. The red line is the phase boundary with $K - 2K_2 = -0.4$. We note that, in Fig. \[Fig:PD\_wring\], the BC order will remain stable over an extended regime in the presence of quantum fluctuations, based on what we have established in Fig. \[Fig:PD\_noring\]. Moreover, the energy splitting between BC and PL due to the presence of ring exchanges can already be observed at semi-classical level and we expect the splitting will be further enhanced if additional quantum fluctuation effects are incorporated by, *e.g.*, the DMRG method.
![(color online) The site-factorized wavefunction phase diagram with small ring exchange terms. Here we fix $(J_1, J_2, J_3, \mathfrak{R}_3,\mathfrak{R}_4) = (1, 0.8,1, 0.1, 0.1)$ and vary $K_1 = K_3=-K$ and $K_2$. The $4$-site ring exchange interaction is responsible for lifting the degeneracy between BC/PL and that betweenAFM$^*_{a/b}$.[]{data-label="Fig:PD_wring"}](SFPD_wring_J3_1_K1=K3.eps){width="2"}
*Discussions.*— We close by remarking on several points. The small ring exchanges do not affect dramatically the energetics of $(\pi,0)$ collinear AFM phase and the $(\pi,0)$ antiferro-quadrupolar order (AFQ) that are suggested to play major roles in the normal state of the iron chalcogenide FeSe [@YuSi_AFQ; @Lai2016; @Wenjun_nematic2016]. Indeed, at the semi-classical level, there are no corrections to the energies of these states from the ring exchanges.
From the above, we can conclude that the frustrated magnetism encoded in the Hamiltonian, Eq. , is sufficiently rich to understand the magnetism and nematic properties of FeTe on the same footing with those of FeSe and the iron pnictides. Given the strong indication for the importance of the spin physics to the iron-based superconductivity [@PCDai2015; @Si2016], the unification we have achieved here represents an important virtue of the mechanism we have advanced in the present work.
Finally, it is worth noting that alternative mechanisms for the BC in FeTe have assumed that the spin physics itself does not yield the ($\pi/2$,$\pi/2$) bicollinear antiferromagnetic state, but such an order arises under additional FeTe-specific driving forces beyond the spin physics. The additional driving forces that have been discussed include an orbital order [@Turner2009; @Singh2009] or spin-lattice interactions [@Fang2009; @EDagotto_FeTe].
*Conclusion.*— We have studied the quantum bilinear-biquadratic model in the presence of ring exchange interactions, and identified two mechanisms that stabilize the $(\pi/2,\pi/2)$ bicollinear antiferromagnetic order experimentally observed in FeTe. In the absence of ring exchanges, we demonstrate a relatively narrow parameter range where quantum fluctuations select the bicollinear order, by destabilizing the classically-degenerate $(\pi/2,\pi/2)$ plaquette antiferomagnetic order. We also identify a larger parameter range where the bicollinear and plquettes orders are degenerate and both are stable. In this regime, the presence of a $4$-site ring exchange interaction breaks the degeneracy and selects the bicollinear order. Because the second mechanism operates over a considerably more extended parameter regime, it represents a more robust mechanism to understand the magnetic and nematic orders of FeTe. Our work unifies the electronic order of FeTe with those of the other iron chalcogenides and iron pnictides within a single Hamiltonian, and highlights the importance of spin frustration to the magnetism and superconductivity of the iron-based systems.
*Acknowledgement.*— The work was supported in part by the NSF Grant No. DMR-1611392 and the Robert A. Welch Foundation Grant No. C-1411 (W.-J.H., H.-H.L. and Q.S.), the NSF Grant No. DMR-1350237 (W.-J.H. and H.-H.L.), a Smalley Postdoctoral Fellowship of the Rice Center for Quantum Materials (H-H. L.), the National High Magnetic Field Laboratory through the NSF Grant No. DMR-1157490 and the State of Florida (S.-S.G.). The majority of the computational calculations have been performed on the Shared University Grid at Rice funded by NSF under Grant EIA-0216467, a partnership between Rice University, Sun Microsystems, and Sigma Solutions, Inc., the Big-Data Private-Cloud Research Cyberinfrastructure MRI-award funded by NSF under Grant No. CNS-1338099 and by Rice University, the Extreme Science and Engineering Discovery Environment (XSEDE) by NSF under Grants No. DMR160003 and DMR160057. Computational support has also been provided by XSEDE from the NSF under Grant No. DMR160004 (S.-S.G.).
Supplemental Material
=====================
Ring exchanges and spin operators {#App:ring}
=================================
The $3$-site and $4$-site ring exchanges $P_{123}$ and $P_{1234}$ are equivalently expressed in terms of the physical spin operators. In general ground, the ring exchange $P_{ijk\cdots \ell}$ can be expressed in terms of the products of two-site exchange operator $P_{ijk\cdots \ell m} = P_{ij} P_{jk} \cdots P_{\ell m}$. The two-site ring exchange operator can be written in terms of physical spin operators. Defining $X\equiv {\bf S}_i \cdot {\bf S}_j$, we can specify the results as follows [@Itoi97] $$\begin{aligned}
&S=\frac{1}{2}: ~~~& ~~~ P^{1/2}_{ij} = 2X + \frac{1}{2},\\
&S=1:~~~& ~~~ P^1_{ij} = X^2 + X -1,\\
&S=\frac{3}{2}:~~~& ~~~ P^{3/2}_{ij} = \frac{2}{9} X^3 + \frac{11}{18}X^2 - \frac{9}{8}X - \frac{67}{32},\\
&S = 2:~~~& ~~~ P^2_{ij} = \frac{1}{36} X^4 + \frac{1}{6} X^3 - \frac{7}{12}X^2 - \frac{5}{2} X -1.\end{aligned}$$
For studying the $S=1$ system in the current work, we introduce a site-factorized wavefunction in the time-reversal-invariant SU(3) basis, $\{ |x\ra, |y\ra, |z\ra \}$ as a complex vector ${\bf d}_j = {\bf u}_j + i {\bf v}_j$ at site $j$. Within the site-factorized wavefunction analysis, the wavefunction of $n$-sites is simply the tensor product of the wavefunction at each site. The ring exchange operator cyclically rotates the states between these sites. $$\begin{aligned}
&& P_{12\cdots n} |\Psi\ra_{12\cdots n} = P_{12\cdots n} \left[ {\bf d}_1 \otimes {\bf d}_2\otimes \cdots \otimes {\bf d}_n \right] = {\bf d}_n \otimes {\bf d}_1 \otimes {\bf d}_2 \otimes \cdots {\bf d}_{n-1}.\end{aligned}$$ Therefore, within the site-factorized wavefunction, the $n$-site ring exchange terms contribute to the Hamiltonian as $$\begin{aligned}
&& _{12\cdots n}\la \Psi| P_{12\cdots n} | \Psi \ra_{12\cdots n} = \left[ \bar{\bf d}_1 \otimes \bar{\bf d}_2 \otimes \cdots \otimes \bar{\bf d}_n \right] \cdot \left[ {\bf d}_n \otimes {\bf d}_1 \otimes \cdots \otimes \bar{\bf d}_{n-1} \right] = (\bar{\bf d}_1 \cdot {\bf d}_n )(\bar{\bf d}_2 \cdot {\bf d}_1) \cdots (\bar{\bf d}_n \cdot {\bf d}_{n-1}) .\end{aligned}$$ The cases of $n=3,4$ correspond to those shown in the main text.
Site-factorized wavefunction energies of competing orders {#App:SFW}
=========================================================
In the present work, within the site-factorized wavefunction studies we find four competing phases, $(\pi/2,\pi/2)$ bicollinear order(BC), $(\pi/2,\pi/2)$ plaquette order(PL), $(\pi/2,\pi/2)$ AFM$_a^*$, and $(\pi/2,\pi/2)$ AFM$_b^*$, in the $J$-$K$-$\mathfrak{R}$ model as $$\begin{aligned}
&& \mathcal{E}_{BC}= K_1 + K_2 + 2(K_3 - J_3)- 2\mathfrak{R}_3,\\
&& \mathcal{E}_{PL}= K_1 + K_2 + 2(K_3 - J_3) - 2\mathfrak{R}_3+ \frac{1}{2}\mathfrak{R}_4,\\
&& \mathcal{E}_{(\pi/2, \pi/2)AFM_a^*} = \frac{3}{4}K_1 + \frac{1}{2} K_2 + 2(K_3 - J_3) -\mathfrak{R}_3, \\
&& \mathcal{E}_{(\pi/2, \pi/2)AFM_b^*} = \frac{3}{4}K_1 + \frac{1}{2} K_2 + 2(K_3 - J_3) -\mathfrak{R}_3+ \frac{1}{4}\mathfrak{R}_4.\end{aligned}$$ In the absence of the ring exchanges, $\mathfrak{R}_{3/4}$, BC and PL are degenerate and so are $(\pi/2,\pi/2)$ AFM$_a^*$ and $(\pi/2,\pi/2)$ AFM$^*_b$. The boundary between degenerate BC/PL and degenerate $(\pi/2,\pi/2)$ AFM$_{a/b}^*$ in the absence of ring exchange terms is determined by $K_1 = -2 K_2 \Rightarrow K = 2 K_2$, where we explicitly use the definition $K_1 = -K$. We remark that the site-factorized wavefunction analysis in the present spin-$1$ model suggests that in the regime of $(\pi/2,\pi/2)$ AFM$^*_{a/b}$ the nearest-neighbor spins be tilted on the plane from the spin patterns shown in Figs. 1(c)-(d) by a small angle $\theta$ in such a way that each second-neighbor spins remain orthogonal, while third-neighbor spins are always anti-parallel to each other. Despite of the angle $\theta$, the energy of AFM$_{a/b}^*$ is independent of $\theta$.
We note that in numerics, there are indeed some fluctuations in the quadrupolar degrees of freedom and the dipolar magnetization can be slightly smaller than one, $|\la{\bf S}\ra| \leq 1$. The energy difference between the phases obtained in the exact numerical result and the AFM$^*_{a/b}$ are within $O(10^{-2})$, which may be due to the numerical errors in searching for a global minimum in the $4\times L^2$ parameter space. On other other hand, this may suggests there are many competing local minima in the AFM$^*_{a/b}$ regime in the parameter space in the $S=1$ model. Enlarging the spin size $S >1$ sufficiently suppresses the fluctuations in the quadrupolar degrees of freedom and the spin pattern can be unbiasedly confirmed in the density matrix renormalization group analysis. [@FeTe-long]
In the presence of ring exchanges, the $3$-site ring exchanges within the site-factorized wavefunction analysis can not split the degeneracies, but they lower the energies of BC/PL more than those of $(\pi/2,\pi/2)$ AFM$_{a/b}^*$. The $4$-site ring exchanges split the double degeneracy in both BC/PL and $(\pi/2,\pi/2)$ AFM$^*_{a/b}$. The competing phases become BC and $(\pi/2,\pi/2)$ AFM$_a^*$. The boundary can be determined by the equation, $K_1 + 2K_2 = 4 \mathfrak{R}_3 \Rightarrow K = 2K_2 - 4 \mathfrak{R}_3.$
Flavor wave theory for $(\pi/2,\pi/2)$ bicollinear order and $(\pi/2,\pi/2)$ plaquette order {#App:FLWT}
============================================================================================
Within the flavor wave calculation for BC and PL with $16$ sublattice per unit cell illustrated in Fig. 2(b) in the main text, we associate $3$ Schwinger-bosons at each site $i$, $b_{i\alpha=x,y,z}$, to the states under SU(3) time-reversal invariant basis, $|x\ra,~|y\ra,~|z\ra$, where $b^\dagger_{i\alpha} |vac\ra = |\alpha\ra$ with $|vac\ra$ being the vacuum state of the Schwinger bosons. The bosons satisfy a local constraint $$\begin{aligned}
\label{Eq:constraint}
\sum_{\alpha} b^\dagger_{i \alpha}b_{i \alpha} = 1.\end{aligned}$$ The model Hamiltonian in terms of spin operators can be re-expressed in terms of the bosons, $$\begin{aligned}
H = \sum_{i, \delta_n} \left[ J_n {\bf S}_i \cdot {\bf S}_j + K_n \left( {\bf S}_i \cdot {\bf S}_j \right)^2 \right] = \sum_{i, \delta_n, \alpha, \beta} \left[ J_nb^\dagger_{i \alpha} b_{i \alpha} b^\dagger_{j\beta} b_{i\beta} + \left(K_n - J_n\right) b_{i\alpha}^\dagger b^\dagger_{j \alpha} b_{i\beta} b_{i\beta}\right],\end{aligned}$$ where $j = i + \delta_n,$ and $\delta_n$ (with $n = 1,~2,~3$) connects site $i$ to its $n$th nearest neighbor sites, and we ignore the constant terms $K_n - J_n$ in the above equation. For performing flavor wave theory calculation, we introduce different local rotations for different site $j$. For site $j \in | S^z = \pm1\ra$, we introduce $$\begin{aligned}
\begin{pmatrix}
b_{jx} \\
b_{jy} \\
b_{jz}
\end{pmatrix} =
\begin{pmatrix}
\mp \frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2} }& 0\\
\frac{1}{\sqrt{2}} & \mp \frac{i}{\sqrt{2}} & 0 \\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
d_{jx}\\
d_{jy}\\
d_{jz}
\end{pmatrix},\end{aligned}$$ which still preserves the local constraint of Eq. (\[Eq:constraint\]) with $b_{i\alpha} \rightarrow d_{i\alpha}$. At each site only one flavor of bosons $d_{ix}$ condenses, and we replace $d^\dagger_{ix}$ and $d_{ix}$ by $(M - d_{iy}^\dagger d_{iy} - d^\dagger_{iz} d_{iz})^{1/2}$, with $M=1$ in the present case. A $1/M$ expansion up to the quadratic order of the Holstein-Primakoff bosons $d_y$ and $d_z$ followed by an appropriate transformation allows us to extract the ground state energy. We will replace the labeling $d_{i\alpha} = d_{\alpha}({\bf r},a)$, where ${\bf r}$ runs over the Bravais lattice of unit cells of the square network and $a=1,2,3,4,\cdots, 15,16$ runs over the sub lattices. The different unit cells are connected by ${\bf e}_1 \equiv \hat{x}$ and ${\bf e}_2 \equiv \hat{y}$.
For clarity, we introduce $ D^T_{\alpha = y,z} ({\bf k}) = \{d_\alpha({\bf k},1), d_\alpha({\bf k},2),\cdots,d_\alpha({\bf k},15),d_\alpha({\bf k},16)\}$, and $A_\alpha({\bf k}) = \left\{ D^T_\alpha({\bf k}), D^\dagger_\alpha(-{\bf k})\right\}$ for clarity below. Within the linear flavor wave theory calculation, we find that the Hamiltonian is $H_\mu = H_c + H^\mu_B$, where $\mu =$ BC or PL, and $ H_c = 32\sum_{\bf k} \left[ J_1 + J_2 - \frac{13}{4} J_3 + K_1 + K_2 + 3 K_3 \right]$, and can be determined straightforwardly independent of the boson fields and independent of BC or PL order. The $H^\mu_B \equiv \sum_{n=1,2,3}H^\mu_n$ represent the boson Hamiltonian related to the $n$th neighbor couplings that are different for $\mu=BC,PL$. After Fourier transform, we find that $H^\mu_n = \sum_{{\bf k}, \eta = y,z} (A^\mu_\eta)^\dagger \mathcal{H}^\mu_{n,\eta} A^\mu_\eta $, with $$\begin{aligned}
\mathcal{H}^\mu_{n,\eta} = \begin{pmatrix}
\alpha^\mu_{n,\eta} & \gamma^\mu_{n,\eta} \\
(\gamma^\mu_{n,\eta})^\dagger & \alpha^\mu_{n,\eta}
\end{pmatrix},\end{aligned}$$ where $\alpha^\mu_{n,\eta}$ and $\gamma^\mu_{n,\eta}$ are $16\times16$ Hermitian matrices. The $H^\mu_n$ can be straightforwardly diagonalized to get the energies of BC and PL, which can be used to estimate the finite stiffness regimes of BC and PL shown in the main texts. Below we list the matrix elements for $\alpha^\mu$ and $\gamma^\mu$ in BC and PL respectively combining the contributions from nearest-neighbor, second-neighbor, and third-neighbor terms (i.e., $\alpha=\sum_n \alpha_n$, $\gamma = \sum_n \gamma_n$): $$\begin{aligned}
&& \alpha^{BC}_{y,a~a=1\sim16} = \frac{9}{2}J_3 - 2 K_3, \\
&& \alpha^{BC}_{y,1~4} = \alpha^{BC}_{y,9~12}= \frac{K_1}{2} e^{-i k_y},\\
\nonumber && \alpha^{BC}_{y,1~5} = \alpha^{BC}_{y,2~3}= \alpha^{BC}_{y,3~7}= \alpha^{BC}_{y,5~6}= \alpha^{BC}_{y,6~10}=\alpha^{BC}_{y,7~8}= \alpha^{BC}_{y,8~12}=\alpha^{BC}_{y,9~13}= \alpha^{BC}_{y,10~11}= \alpha^{BC}_{y,11~15}= \alpha^{BC}_{y,13~14}=\alpha^{BC}_{y,15~16}= \frac{K_1}{2},\\ \\
&& \alpha^{BC}_{y,1~6} = \alpha^{BC}_{y,2~7}=\alpha^{BC}_{y,3~8}= \alpha^{BC}_{y,5~10}= \alpha^{BC}_{y,6~11}= \alpha^{BC}_{y,7~12}= \alpha^{BC}_{y,9~14}= \alpha^{BC}_{y,10~15}= \alpha^{BC}_{y,11~16}= \frac{K_2}{2},\\
&& \alpha^{BC}_{y,1~16} = \frac{1}{2}K_2 e^{-i (k_x + k_y)}, \\
&& \alpha^{BC}_{y,2~14}= \alpha^{BC}_{y,4~16}= \frac{K_1}{2} e^{-i k_x},\\
&& \alpha^{BC}_{y,2~13}= \alpha^{BC}_{y,3~14}= \alpha^{BC}_{y,4~15}= \frac{K_2}{2} e^{-i k_x},\\
&& \alpha^{BC}_{y,4~5}= \alpha^{BC}_{y,8~9}= \alpha^{BC}_{y,12~13}= \frac{K_2}{2} e^{i k_y}, \end{aligned}$$ where $\alpha^{BC}_{y, ab} = (\alpha^{BC}_{y, ba})^*$ and rest of the elements are zero. Similarly, $$\begin{aligned}
&& \gamma^{BC}_{y,1~2} = \gamma^{BC}_{y,2~6}= \gamma^{BC}_{y,3~4} = \gamma^{BC}_{y,4~8} = \gamma^{BC}_{y,5~9}= \gamma^{BC}_{y, 6~7}= \gamma^{BC}_{y,7~11} = \gamma^{BC}_{y, 9~10} = \gamma^{BC}_{y,10~14}= \gamma^{BC}_{y, 11~12} = \gamma^{BC}_{y, 12~16}= \gamma^{BC}_{y,14~15}=\frac{K_1}{2},~~~\\
&& \gamma^{BC}_{y, 1~3} = \gamma^{BC}_{y,2~4}=\gamma^{BC}_{y,5~7} = \gamma^{BC}_{y, 6~8} = \gamma^{BC}_{y,9~11} = \gamma^{BC}_{y,10~12}=\gamma^{BC}_{y,13~15} = \gamma^{BC}_{y,14~16} = \frac{K_3}{2} \left( 1 + e^{-i k_y}\right),\\
&& \gamma^{BC}_{y,1~8} = \gamma^{BC}_{y,5~12}= \gamma^{BC}_{y,9~16}=\frac{K_2}{2} e^{-i k_y},\\
&& \gamma^{BC}_{y,1~9} = \gamma^{BC}_{y,2~10} = \gamma^{BC}_{y,3~11} = \gamma^{BC}_{y,4~12} = \gamma^{BC}_{y,5~13} = \gamma^{BC}_{y,6~14} = \gamma^{BC}_{y,7~15} = \gamma^{BC}_{y,8~16} = \frac{K_3}{2}\left(1 + e^{-i k_x} \right),\\
&& \gamma^{BC}_{y,1~13} = \gamma^{BC}_{y,3~15} = \frac{K_1}{2} e^{-i k_x},\\
&& \gamma^{BC}_{y,1~14} = \gamma^{BC}_{y,2~15} = \gamma^{BC}_{y,3~16} = \frac{K_2}{2} e^{-i k_x},\\
&& \gamma^{BC}_{y,2~5} = \gamma^{BC}_{y,3~6} = \gamma^{BC}_{y,4~7} = \gamma^{BC}_{y,6~9} = \gamma^{BC}_{y,7~10} = \gamma^{BC}_{y,8~11}=\gamma^{BC}_{y,10~13} = \gamma^{BC}_{y,11~14}=\gamma^{BC}_{y,12~15}=\frac{K_2}{2},\\
&& \gamma^{BC}_{y,4~13} = \frac{K_2}{2}e^{-i (k_x - k_y)},\\
&& \gamma^{BC}_{y, 5~8} = \gamma^{BC}_{y,13~16} = \frac{K_1}{2}e^{-i k_y},\end{aligned}$$ where $\gamma^{BC}_{y, ab} = (\alpha^{BC}_{y, ba})^*$ and rest of the elements are zero.
The diagonal matrix elements of $\alpha^{BC}_z$ for the $z$-flavored bosons in the BC are $\alpha^{BC}_{z,a~a=1\sim16} = 2J_3 - K_1 - K_2 - 2 K_3$, while the off-diagonal matrix elements can be obtained from those of $\alpha^{BC}_y$ with $K_n \rightarrow J_n$. In addition, the matrix elements of $\gamma^{BC}_z$ can be obtained from those of $\gamma^{BC}_y$ by replacing $K_n \rightarrow J_n$.
The matrix elements for the PL are, $$\begin{aligned}
&& \alpha^{PL}_{y,a~a=1\sim16} = \frac{9}{2}J_3 - 2 K_3, \\
\nonumber && \alpha^{PL}_{y,1~2} = \alpha^{PL}_{y,1~5} = \alpha^{PL}_{y,2~6}=\alpha^{PL}_{y,3~4} =\alpha^{PL}_{y,3~7} = \alpha^{PL}_{y,4~8} = \alpha^{PL}_{y,5~6} =\alpha^{PL}_{y,7~8} = \alpha^{PL}_{y,9~10} = \alpha^{PL}_{y,9~13} = \alpha^{PL}_{y,10~14}=\alpha^{PL}_{y,11~12} =\\
&&\hspace{9cm} =\alpha^{PL}_{y,11~15} = \alpha^{PL}_{y,12~16} = \alpha^{PL}_{y,13~14} = \alpha^{PL}_{y,15~16} =\frac{K_1}{2},\\
&& \alpha^{PL}_{y,1~6} = \alpha^{PL}_{y,2~5} =\alpha^{PL}_{y,3~8} =\alpha^{PL}_{y,4~7} = \alpha^{PL}_{y,6~11} = \alpha^{PL}_{y,7~10} = \alpha^{PL}_{y,9~14} =\alpha^{PL}_{y,10~13} = \alpha^{PL}_{y,11~16} = \alpha^{PL}_{y,12~15} =\frac{K_2}{2},\\
&& \alpha^{PL}_{y,1~6} = \frac{K_2}{2}e^{-i(k_x + k_y)},\\
&& \alpha^{PL}_{y,2~15} = \alpha^{PL}_{y,3~14} = \frac{K_2}{2} e^{-i k_x}, \\
&& \alpha^{PL}_{y,4~13} = \frac{K_2}{2} e^{-i(k_x - k_y)},\\
&& \alpha^{PL}_{y,5~12} = \alpha^{PL}_{y,8~9}=\frac{K_2}{2} e^{-i k_y},\end{aligned}$$ where $\alpha^{PL}_{y, ab} = (\alpha^{PL}_{y, ba})^*$ and rest of the elements are zero. Similarly, $$\begin{aligned}
&& \gamma^{PL}_{y,1~3} = \gamma^{PL}_{y,2~4} = \gamma^{PL}_{y,5~7} = \gamma^{PL}_{y,6~8} = \gamma^{PL}_{y,9~11} = \gamma^{PL}_{y,10~12} = \gamma^{PL}_{y,13~15} = \gamma^{PL}_{y,14~16} = \frac{K_3}{2} \left( 1 + e^{-i k_y} \right),\\
&& \gamma^{PL}_{y,1~4}= \gamma^{PL}_{y,5~8} = \gamma^{PL}_{y,9~12} = \gamma^{PL}_{y,13~16} = \frac{K_1}{2} e^{-i k_y},\\
&& \gamma^{PL}_{y,1~8} = \gamma^{PL}_{y,5~4} = \gamma^{PL}_{y,9~16} = \gamma^{PL}_{y,13~12} = \frac{K_2}{2} e^{-i k_y}, \\
&& \gamma^{PL}_{y,1~9} = \gamma^{PL}_{y,2~10} = \gamma^{PL}_{y,3~11} = \gamma^{PL}_{y,4~12} = \gamma^{PL}_{y,5~13}= \gamma^{PL}_{y,6~14} = \gamma^{PL}_{y,7~15}= \gamma^{PL}_{y,8~16} = \frac{K_3}{2} \left( 1 + e^{-i k_x} \right), \\
&& \gamma^{PL}_{y,1~13} = \gamma^{PL}_{y,2~14} = \gamma^{PL}_{y,3~15}= \gamma^{PL}_{y,4~16} = \frac{K_1}{2} e^{-i k_x},\\
&& \gamma^{PL}_{y,1~14} = \gamma^{PL}_{y,2~13} = \gamma^{PL}_{y,3~16}= \gamma^{PL}_{y,4~15} = \frac{K_2}{2} e^{-i k_x},\\
&& \gamma^{PL}_{y,2~3} = \gamma^{PL}_{y,5~9}= \gamma^{PL}_{y,6~7} = \gamma^{PL}_{y,6~10} = \gamma^{PL}_{y,7~11}= \gamma^{PL}_{y,8~12} = \gamma^{PL}_{y,10~11} = \gamma^{PL}_{y,14~15}=\frac{K_1}{2},\\
&& \gamma^{PL}_{y,2~7} = \gamma^{PL}_{y,3~6} = \gamma^{PL}_{y,5~10}= \gamma^{PL}_{y,6~9} = \gamma^{PL}_{y,7~12}= \gamma^{PL}_{y,8~11}= \gamma^{PL}_{y,10~15} = \gamma^{PL}_{y,11~14}=\frac{K_2}{2},\end{aligned}$$ where $\gamma^{PL}_{y, ab} = (\alpha^{PL}_{y, ba})^*$ and rest of the elements are zero.
The diagonal matrix elements for the $z$-flavored bosons in the PL are $ \alpha^{PL}_{z,a~a=1\sim16} = 2J_3 -K_1-K_2- 2 K_3$, and the off-diagonal matrix elements of $\alpha^{PL}_z$ can be obtained from those of $y$-flavored bosons with $K_n \rightarrow J_n$. The matrix elements of $\gamma^{PL}_z$ can be obtained from $\alpha^{PL}_y$ with $K_n \rightarrow K_n - J_n$.
Diagonalizing the Hamiltonian to extract the $y$ or $z$ bosons dispersions, we can then determine the regimes where the BC or PL become unstable due to the negative stiffness (i.e. complex eigenvalues near the gapless point in the momentum space). For example, we take a line cut in the Fig. 2(a) parallel to $K_2$-axis. Deep inside the BC/PL regime, we see a single linear-$k$ gapless dispersion at ${\bf k} = {\bf 0}$ representing the spin-dipolar- and quadrupolar-wave modes in the present lattice setup. Approaching the boundary, we observe other local minima near the original linear-$k$ gapless point. At the boundary, we observe that the there are multiple minima in the boson dispersions, which implies the original starting point assuming the BC/PL order is not appropriate. Across the boundary, the original minimum point of gapless spin-dipolar- and quandrupolar-wave modes is no longer a global minimum, and the dispersions at certain momenta ${\bf k}$ become complex, indicating the assumption of the stable BC/PL orders is no longer appropriate, and this suggest the negative stiffness of BC/PL orders.
We then arrive at the conclusions that the BC and PL have different stiffness, and BC and PL can be stable in different parameter regime. In the regime where both BC and PL have positive stiffness, the energies of BC and PL are very close to each other, the difference being $0.1\%$ to $1 \%$, which implies that quantum fluctuations in the $J$-$K$ only model *cannot* efficiently split the degeneracy of BC and PL, consistent with unbiased density matrix renormalization group analysis, and additional inputs such as ring exchanges are needed.
For testing the stable regime of BC/PL order in the $J$-only model, we focus on the Fig. 2(a) in the main texts at fixed $J_2=0.8, J_3=1, K_2=0$, and we vary $J_1$ and $K$. The result is illustrated in Fig. \[Fig:app\_J1-K\_boundary\]. We find that at $K=0$, the BC and PL are both stable only up to a very small threshold, $J_1 = 0.03$. Increasing the value of $K$ substantially increase the stable regimes of BC and PL, although the stable regimes of BC and PL are different in size indicating the difference stiffness of BC and PL.
![(Color online)The stable regimes of BC and PL near the purely Heisneberg$J_{n=1,2,3}$ model suggested by flavor-wave theory calculations. At $K=0$, both BC and PL are only stable at very small $J_1 \simeq 0.03$, in which their energies are highly degenerate, the difference being $\sim 0.001\%$. Increasing $K$ substantially increases the stable regime of BC and PL.[]{data-label="Fig:app_J1-K_boundary"}](app_J1-K_boundary.eps){width="1.5"}
Spin correlation for spin-$3/2$ and the ground-state energy {#App:DMRG}
===========================================================
In the main texts, we have shown the spin correlation functions for the spin-$1$ and spin-$2$ models in the (nearly) degenerate BC/PL phase regime obtained from DMRG, where we find the BC state on the TC cylinder and the PL state on the RC cylinder. Here, we also present our DMRG results for the spin-$3/2$ model at the same parameters $(J_1, J_2, J_3, K_1, K_2, K_3)
= (1.0, 0.8, 1.0, -0.5, 0.1, -0.5)$ in Fig. \[Fig:spin3/2\]. The results are consistent those for the spin-$1$ and spin-$2$ systems.
As an additional test of the near degeneracy of the BC and PL states, we compare the ground-state bulk energies of the two states on the different system sizes for the spin-$1$ and spin-$3/2$ models. In Fig. \[Fig:energy\], we demonstrate the energies on the RC cylinder with $L_y = 8$ and the TC cylinder with $L_y = 4, 6$ versus cylinder width $L$. We note that RC cylinders with $L_y = 4, 6$ are strongly affected by finite-size effects and RC6 does not match the $(\pi/2, \pi/2)$ magnetic ordering. For both models, we find that the energies of the two states on large cylinders are quite close. For the spin-$1$ and spin-$3/2$ models, the energy differences on the RC ($L_y = 8$) and TC ($L_y = 6$) cylinders are only about $0.2\%$ and $0.1\%$, respectively.
| 0 |
---
abstract: |
We report on optical spectroscopic observations of the X-ray Transient XTE J1118+480 covering the period from April 7, 2000 to July 4, 2000. The spectrum is characterized by weak, broad, double-peaked Balmer and He lines on top of a blue continuum of slope $p \approx
1/3$, as expected for an optically-thick accretion disk. The weak Bowen blend seen in our spectra may indicate a low metallicity for the source. The presence of a partial S-wave pattern in the He $\lambda 4686$ line appears consistent with the reported photometric period $P_{\rm orb} = 4.1$ hr for XTE J1118+480. By using a combination of Doppler mapping and various theoretical arguments, we constrain plausible orbital parameters for the system: a mass ratio $0.02 \lsim q \lsim 0.1$, an inclination $i
\gsim 70^o$ for a neutron star primary, or $30^o \lsim i \lsim 50^o$ for a black hole primary with a mass between $4$ and $10~M_\odot$. Ca[ii]{} absorption features observed at very high resolution constrain the interstellar hydrogen absorption column $\log [N_{\rm
H{\sc i}}~({\rm ~cm^{-2}})] \approx 20.45 \pm 0.2$ and the identification of three absorbing clouds indicate a distance to the source $\lsim 1$ kpc, assuming the line-of-sight to XTE J1118+480 has average high-latitude properties. These results are discussed in the context of previous multiwavelength observations of this unusual system.
author:
- |
Guillaume Dubus, Rita S.J. Kim, Kristen Menou, Paula Szkody\
and David V. Bowen
title: 'Optical Spectroscopy Of The X-Ray Transient XTE J1118+480 In Outburst'
---
Introduction
============
Soft X-ray Transients (SXTs) are compact binary systems in which a low-mass secondary (either a main-sequence star or a subgiant) transfers mass via Roche-lobe overflow onto a black hole (BH) or a neutron star (NS) primary (see reviews by Tanaka & Lewin 1995; van Paradijs & McClintock 1995; White, Nagase & Parmar 1995). SXTs have highly variable luminosities. They spend most of their lifetime in a low luminosity quiescent state, but occasionally undergo dramatic outbursts during which both their optical and X-ray emission increase by several orders of magnitude (see, e.g., Tanaka & Shibazaki 1996; Chen, Shrader & Livio 1997).
The X-ray emission during outburst of an SXT is typically dominated by relatively soft, thermal emission from the accretion disk surrounding the compact object, while the optical emission is usually interpreted as reprocessed X-rays from the disk and/or the companion star. The new source XTE J1118+480 belongs to the class of SXTs, but it also possesses some rather unusual characteristics.
XTE J1118+480 was discovered on March 29 2000 with the RXTE All-Sky Monitor as a brightening X-ray source. Pointed RXTE observations confirmed the presence of this high Galactic latitude source ($l=157.7^o$, $b=+62.3^o$), with a rather hard power law emission spectrum of photon index $\Gamma \simeq 1.8$ up to 30 keV. It was subject to rapid X-ray flares, but no pulsation was detected. Retrospective ASM analysis revealed that the source experienced another modest outburst in January 2000 (Remillard et al. 2000; see Fig. \[fig:lightcurve\]). BATSE observations showed that the source is visible up to 120 keV (Wilson & McCollough 2000), and a 6.2 mJy [ (at 15 GHz)]{} variable radio counterpart was later discovered (Pooley & Waldram 2000).
The $V \sim 13$ optical counterpart of XTE J1118+480 was found to correspond to an $18.8$ mag star in the USNO catalog by Uemura, Kato & Yamaoka (2000; see also Uemura et al. 2000). A photometric modulation on a 4.1 hr period was observed by Cook et al. (2000) and was later confirmed as a plausible orbital period by Patterson (2000) and Uemura et al. (2000; see Stull, Ioannou & Webb in Haswell et al. 2000b for a discrepant claim at twice this value). The photometric modulation was later reported changing shape and period, possibly showing the development of superhumps in this source (Uemura 2000). [ If these are normal superhumps, then the orbital period would be smaller than the photometric superhump modulation by a few percent.]{}
Garcia et al. (2000) reported the first optical spectroscopic results on XTE J1118+480. They found an optical spectrum typical of SXTs in outburst, with very broad H$\alpha$, H$\beta$ and He[ii]{} lines ( FWHM $\gsim 2000$ km s$^{-1}$). These observations also indicated the presence of absorption features and a very low interstellar absorption ($E(B-V) \lsim 0.024$) to the source. These authors suggested that the surprisingly low X-ray ($\sim 40$ mCrab, 2-12 keV) to optical ($V \sim
13$) flux ratio of XTE J1118+480 could be due to a nearly edge-on viewing angle.
Additional X-ray observations revealed the presence of a strong Quasi-Periodic Oscillation (QPO) at 0.085 Hz in the X-ray lightcurve of XTE J1118+480. The shape of the power density spectrum and the hard emission spectrum of XTE J1118+480 prompted Revnivtsev, Sunyaev & Borozdin (2000) to propose that the source is a BH transient, by analogy with other such systems. The QPO was confirmed by ASCA observations, which also suggest the presence of a soft component in the spectrum (below 2 keV), possibly due to emission from the accretion disk in the system (Yamaoka et al. 2000). [ Soft X-ray observations by [*Chandra*]{} did not confirm this (McClintock et al. 2000)]{}. The QPO frequency was reported to have shifted from 0.085 Hz to $\gsim 0.1$ Hz in subsequent observations (Yamaoka et al. 2000; Wood et al. 2000). Pointed XTE observations at the end of May do not reveal significant changes in the emission spectrum (P. Jonker, private communication).
XTE J1118+480 has also been the subject of an extensive multiwavelength observation campaign with HST, EUVE, UKIRT and RXTE. Thanks to the very low interstellar absorption to the source, the first EUVE spectrum of a BH-candidate X-ray Transient was obtained (Mauche et al. 2000). No periodic modulation was found in the EUVE data (Hynes et al. 2000b). An HST spectrum revealed a very broad ($>
10,000$ km s$^{-1}$) Ly$\alpha$ absorption feature, suggestive of a massive accretor (Haswell et al. 2000b). Haswell et al. (2000c) obtained a near-UV power density spectrum of XTE J1118+480 with a QPO and an overall shape in agreement with previous RXTE timing data. In addition, the near-UV variability was found to lag by 1-2 seconds behind the X-ray variations, as would be expected from light echoes in a system with $P_{\rm orb} =4.1$ hr. Hynes et al. (2000b), combining data from HST, EUVE, UKIRT and RXTE, find that the IR to UV data suggests emission from both an optically-thick disk and another, flat spectrum component (possibly synchrotron), while the EUVE and X-ray data suggest a power law emission typical of a Galactic X-ray binary in a low-hard state. They conclude that XTE J1118+480, rather than experiencing a full outburst [ approaching the Eddington luminosity]{}, seems to be in a mini-outburst state [ (but see Kuulkers 2000)]{}.
In this paper, we describe the results of our optical spectroscopic campaign to monitor the evolution of XTE J1118+480 during its recent outburst (and early decline). In §2, we describe our observations and the data reduction techniques used. Our results concerning the continuum and line emission, a partial S-wave pattern and the interstellar absorption to the source are presented in §3 and discussed in the context of the current knowledge on XTE J1118+480 in §4. Our main conclusions are summarized in §5.
Observations and Reduction
==========================
We obtained optical spectra of XTE J1118+480 from April 7, 2000 to July 4, 2000 with the ARC 3.5 m telescope at Apache Point Observatory. [ We mostly used the Double Imaging Spectrograph (DIS),]{} but spectra with the Echelle spectrograph were also obtained on April 7, 2000. Table \[tab:obs\] summarizes the dates and other characteristics of our observational campaign.
Most of our observations with DIS were carried out with the high resolution gratings [ (hereafter [*hires*]{})]{} with a 1.5” slit (dispersion 1.6Å pixel$^{-1}$ in the blue, 1.1Å pixel$^{-1}$ in the red, and a resolution of 2 pixels), but we also obtained spectra with the low-resolution gratings (same slit size, dispersion 6.2Å pixel$^{-1}$ in the blue, 7Å pixel$^{-1}$ in the red, and a resolution of 2 pixels;[ hereafter [*lowres*]{})]{} on April 7, 2000. For the DIS high resolution observations, the blue and red gratings were centered on slightly different wavelengths during our various nights of observations, but we generally centered the blue side to cover the H$\beta$ and He[ii]{} $\lambda 4686$ lines, and the red side to cover the H$\alpha$ line. The complete list of spectral coverage for our observations can be found in Table \[tab:obs\]. The Echelle spectrograph covers the (fixed) wavelength range 3500-10000Å with $R\sim 30000$ (10 km s$^{-1}$ at 5000Å) and a resolution element of $\sim$ 2.5 pixels. All the exposure times can be found in Table \[tab:obs\].
The DIS observations were reduced in the standard way using [*IRAF*]{} and the spectra were optimally-extracted and dispersion corrected without any particular difficulty.
The Echelle spectra were reduced using the IRAF [*ecspec*]{} package. Direct extraction of the object spectra proved difficult for the highest (bluest) orders where the trace was hard to follow. We decided to use the flat field as a guide and validated the method with a standard star observed on the same night. This resulted in significantly higher S/N in the blue part of the spectrum. A total of 115 orders were extracted covering the spectral range 3500Å-10000Å, in which the source is detected, albeit with varying sensitivity. The dispersion varied between $\sim$ 0.05Å/pixel to 0.1Å/pixel and the usable spectral range in each order between $\sim$ 70Å to 150Å from the blue to the red end.
The emission lines in the spectrum of XTE J1118+480 are both broad and weak (see below, Fig. \[fig:lowres\]) and their identification in the Echelle spectra turned out to be challenging. The broad lines can cover almost half of the spectral range in one order making a continuum fit unreliable. We therefore proceeded by normalizing to an interpolated continuum determined from the two closest orders where no lines are expected. The normalised summed spectra clearly show H$\alpha$ (in one order, left panel of Fig. \[fig:echha\]) and He[ii]{} $\lambda$4686 (spread over two orders, right panel of Fig. \[fig:echha\]). These lines are also detectable in the individual continuum-subtracted [ Echelle]{} spectra. [ The different profile of the He[ii]{} line in the [*lowres*]{} and Echelle spectra is most probably due to the S-wave modulation discussed below. This S-wave is much less prominent in H$\alpha$]{}.
Results
=======
The RXTE ASM lightcurve of XTE J1118+480 is shown in Fig. \[fig:lightcurve\] with the dates of our spectroscopic observations indicated by dashed lines. The source was in outburst during the entire period covered by our optical observations, with a flux about 40 mCrab in the 2-12 keV band (Fig. \[fig:lightcurve\]). Optical photometry [^1] also shows the source in outburst with a mean magnitude that decreased at most by $0.5$ mag (from its peak value $\lsim 13$) at the end of our observational program. Since then, the optical and X-ray flux have decreased significantly, indicating that the source may soon enter quiescence.
Continuum
---------
Figure \[fig:lowres\] shows the sum of the two DIS [*lowres*]{} spectra obtained on April 7, 2000, which is also representative of other spectra of the source obtained later on during our campaign. The strong, blue power-law continuum and the weak emission lines are typical of an SXT in outburst (see also Garcia et al. 2000). A power-law fit to the blue part of the spectrum yields a slope $p
\simeq +0.4 \pm 0.2$ (where the flux F$_\nu \propto \nu^p$), while a similar fit to the red part of the spectrum yields $p \simeq +0.33 \pm
0.15$. These values are consistent with the expectation $p=1/3$ for an optically-thick accretion disk (see, e.g., Frank, King & Raine 1992). Hynes et al. (2000b) find that the spectrum is flat using a wider wavelength range.
The broad, double-peaked H$\alpha$ emission line is the only one easily identified on the spectrum of Fig. \[fig:lowres\], but a closer examination reveals the presence of He[ii]{} $\lambda4686$, and of additional Balmer and He lines in the other spectra that we collected (see §3.2 below). We find only marginal evidence for continuum variability in XTE J1118+480, both during the same night and from night to night. The lack of obvious variability is consistent with the long-term variations in the optical VSNET lightcurve ($<$ 0.5 mag), the short timescale ($<$10 s) of the large amplitude ($>$0.3 mag) flickering and the low amplitude (0.052 mag) of the 4 hr modulation (Patterson 2000).
Line Profiles and their Evolution during Outburst
-------------------------------------------------
We identify three Balmer lines, three He lines and the Bowen blend in our set of spectra. These are: H$\gamma~\lambda 4340.5$, He[i]{} $\lambda 4471.5$, the Bowen blend at $\sim~\lambda4638$, He[ii]{} $\lambda 4686.7$, H$\beta~\lambda 4861.3$, He[i]{} $\lambda
4921.9$, H$\alpha~\lambda 6562.8$ and He[i]{} $\lambda 6678.1$ The lines are weak, broad and double-peaked. H$\beta$, H$\gamma$ and also the weaker He[i]{} $\lambda$4471.5 have double-peaked emission clearly embedded in a large absorption trough. There is no clear evidence of such absorption around H$\alpha$.
Fig. \[fig:echha\] shows the H$\alpha$ and He[ii]{} line profiles in the summed Echelle spectrum with the low resolution spectrum taken on the same night overplotted. We find no evidence for structures at high spectral resolution in the lines, except for an excess at rest wavelength which would not be expected in a pure double-peaked disk profile. This component also seems to be present in the high resolution DIS spectra (see Fig. \[fig:sumlines\]). The Bowen blend, which is very weak in the DIS [*hires*]{} spectra, was not detected on April 7 in either the DIS [*lowres*]{} or the Echelle spectra.
Measuring the lines proved difficult, not because of a lack of counts in the spectra but because of their intrinsic broadness and weakness. As a rule, we performed two-component Gaussian fits to the emission lines. For H$\gamma$ and H$\beta$ the two emission components were subtracted to measure the equivalent width (EW) and FWHM of the absorption trough. The EW of the lines varies (with large associated errors) within a night but we could find no periodic behaviour linked to the [ photometric]{} period. We also tried to estimate the semi-amplitude velocity of the primary $K_1$ by fitting the wings of the He[ii]{} line (expected to trace the motion of the compact star) and folding around the suspected $P_{\rm orb}$ of 0.17 day. This did not lead to any result as could be expected from the usually low $K_1$ of SXTs ($K_1\lsim 50$ km s$^{-1}$).
The evolution of these lines during the 3 months of observations is shown in Fig. \[fig:sumlines\], after the underlying continuum was normalized to unity. When available, multiple spectra were summed up to increase the S/N of the spectra. A comparison between the most comprehensive datasets (Apr. 17 and Jul. 4), where measurement errors can be minimized, shows that the two strongest lines, H$\alpha$ and He[ii]{}, did not change significantly. We conclude that there do not seem to be any long term variations. The average properties of the lines are summed up in Table. \[tab:lines\] (the quoted errors are the rms of the measurements).
In all our spectra the blue side of the absorption troughs in H$\gamma$ and H$\beta$ is less [ strong]{} than the red side. The measured central wavelengths of the absorptions are redshifted compared to the rest wavelength (which could be an artefact of the method we used to remove the double peaked emission). Fitting a gaussian to the red part of the absorption while ignoring (but not subtracting) the emission lines gave higher equivalent widths (2.4Åfor H$\gamma$ and 2.0Å for H$\beta$) and better agreement with the rest wavelength (7Å redshift for both instead of 15Å). Similar absorption redshifts were reported by Callanan et al. (1995) for GRO J0422+32. A plausible explanation would be that the shifts are due to distorted line profiles. Such asymetric lines are observed in dwarf novae when the disk becomes eccentric and the system shows superhumps (Warner 1995). There is evidence for such superhumps in XTE J1118+480 (see discussion in §4.3).
Constraint on the EUV flux
--------------------------
The He[ii]{} emission may be used to obtain a crude estimate of the extreme ultraviolet (EUV) flux from the source (e.g. Patterson & Raymond 1985) [ if one assumes]{} the $\lambda$4686 line stems from the recombination of He[i]{} photoionized in the disk by photons with energies between 55-280 eV. [ On the other hand, the doppler map discussed below in §3.5 shows most of the He[ii]{} emission is localized and in all likelihood associated with stream-disk interaction. In this case the $\lambda$4686 line would be pumped by collisional excitation rather than photoionization. The following estimate therefore only yields an upper limit on the EUV flux. This is still of interest since the upper limit is independent of the column density to the source.]{}
We derive from our observations $F_{\lambda4686}\approx
7\cdot10^{-14}$ erg s$^{-1}$ cm$^{-2}$ and, from the values given by Haswell et al. (2000b), $F_{\lambda1640}\approx 7\cdot10^{-13}$ erg s$^{-1}$ cm$^{-2}$. This favours Case B recombination for which one would expect $F_{\lambda1640}/F_{\lambda4686}\sim7$ (Seaton 1978). Therefore, the region is optically-thick to the ionizing flux but thin to the line and a fraction $\epsilon\approx0.2$ of the photoionized He[i]{} recombinations lead to $\lambda$4686 emission (Hummer & Storey 1987).
Following previous applications to soft X-ray transients (Marsh, Robinson & Wood 1994; Hynes et al. 1998), the EUV flux $F_{\rm EUV}$ and the He[ii]{} line flux $F_{\lambda4686}$ are related through $$\epsilon\alpha\frac{F_{\rm EUV}}{E_{\rm EUV}}=\frac{F_{\lambda4686}}
{E_{\lambda4686}}$$ where $\alpha$ is the fraction of EUV photons intercepted by the disc while $E_{\rm EUV}\approx100$ eV and $E_{\lambda4686}\approx2.6$ eV refer to the mean energy of the photons. We have assumed $L_{\rm
EUV}/L_{\lambda4686}=F_{\rm EUV}/F_{\lambda4686}$, which may be incorrect if the EUV emission is not isotropic or if we see only a fraction of the He[ii]{} emission. In both cases this assumption leads to an underestimate of the true EUV luminosity. Observations suggest that a fraction $\sim 10^{-3}$ of the soft X-ray flux is intercepted and reprocessed in the optical by the accretion disk in SXTs (e.g. Dubus et al. 1999). By analogy, we take $\alpha=10^{-3}$ from which we finally derive [ an upper limit to the]{} 55-280 eV flux of $10^{-8}$ erg s$^{-1}$ cm$^{-2}$. This is obviously a very crude estimate.
[ An EUVE spectrum was obtained for this source, the first time ever for a SXT. The source flux at these wavelengths is heavily dependent on the extinction, but the upper limit on $F_{\rm EUV}$ derived above appears consistent with both a simple extrapolation of the optical–UV flux observed and the EUVE flux inferred if a value of $N_{\rm H} \approx 10^{20}$ cm$^{-2}$ is assumed when dereddening the EUVE spectrum (Mauche et al. 2000; Hynes et al. 2000). ]{}
Peak-to-peak velocities
-----------------------
The mean peak-to-peak separation for all the observations is 18Å(1240 km s$^{-1}$) for H$\gamma$, 23Å (1470 km s$^{-1}$) for He[ii]{}, 21Å (1300 km s$^{-1}$) for H$\beta$ and 28Å (1280 km s$^{-1}$) for H$\alpha$. The S-wave is much less prominent in the Balmer lines [ and]{} we assume this represents the projected Keplerian velocity of the outer disk. [ The peak-to-peak separations have been found in well-known systems to overestimate the outer disk radius by about 20% because of sub-Keplerian motion or local broadening (Marsh 1998). This uncertainty is acceptable considering other assumptions made below.]{} For a Keplerian disk with an emissivity $\propto R^{-n}$, $\Delta v$ can be related to the outer (emitting) disk radius $R_{\rm d}$ (Smak 1981): $$\Delta v = 2 R_{\rm d}\Omega_{\rm K} \sin i = 2 \sin i (GM_1/R_{\rm
d})^{1/2},
\label{eq:vrout}$$ where $M_1$ is the mass of the primary and $i$ the system inclination. We can also express $R_{\rm d}/a$ as a function of the velocity semi-amplitude of the secondary $K_2$ and the mass ratio $q=M_2/M_1$ by combining the above equation with the mass function (see below, Eq. \[eq:fm\]) and Kepler’s third law[^2], resulting in: $$\frac{R_{\rm d}}{a}=(1+q)\left(\frac{2 K_2}{\Delta v}\right)^2
\label{eq:rova}$$ The disk outer radius is expected to reach the tidal truncation radius for $q\gsim0.2$ and to be limited to $R_{\rm d}/a\approx0.48$ by the 3:1 resonance for $q\lsim 0.2$ (e.g. Warner 1995). Papaloizou & Pringle (1977; their Tab. 1) give a theoretical estimate of the outer disk radius that we recklessly use for $R_{\rm d}/a$. Eq. \[eq:rova\] can therefore be used to give the semi-amplitudes $K_2(q)$ for which the disk radius is consistent with the theoretical expectations. For $0.01<q<1$ the disk radius varies between 0.5 and 0.3 in units of the binary separation $a$. The expected $K_2$ as a function of $q$ is shown as a solid line in the left panel of Fig. \[fig:cons1\]. We find $440>K_2$ (km s$^{-1}$) $>260$ for $0.01<q<1$.
The observed velocity FWHM ($v_{\rm fwhm}$) of the lines provides an upper limit to the minimum emitting Keplerian disk radius: $$R_{\rm in}\lsim 1.3\cdot 10^{10}\ (M_1/M_\odot)\left(\sin i/v_{\rm
fwhm}\right)^2\ {\rm cm},$$ where $v_{\rm fwhm}$ is expressed in units of 1000 km s$^{-1}$. The absorption features from the optically-thick disk (see §4.1 and Table. \[tab:lines\]) have FWHM $\gsim 3000$ km s$^{-1}$ implying $R_{\rm in}\lsim 10^9$ cm. Haswell et al. (2000b) detect broad Ly$\alpha$ absorption with $v_{\rm fwhm} \sim 10^4$ km s$^{-1}$ which gives a stricter $R_{\rm in}\lsim 10^8$ cm or $R_{\rm in} \lsim 500$ in Schwarzschild units. Spectral models of the X-ray low/hard state of BH candidates tend to predict larger values of $R_{\rm in} (\gsim
1000$ in Schwarzschild units) at which the transition from a thin disk to a hot advection-dominated flow occurs (see, e.g., Esin et al. 1998). The EUV flux of XTE J1118+480 could also imply a smaller transition radius (see §4.2).
Doppler mapping
---------------
Figure \[fig:swave\] shows the evolution of the He[ii]{} $\lambda
4686$ line profile over an approximately continuous 2.9 hr period on April 20, 2000. A partial S-wave pattern moves from the red to the blue side of the He[ii]{} $\lambda 4686$ line rest wavelength and appears consistent with the claimed photometric period of $4.1$ hr (Cook et al. 2000; Patterson 2000). The modulated component is particularly strong towards the middle of the observation at HJD 51654.876 where it is blueshifted by about 900 km s$^{-1}$.
The trailed spectrum clearly shows that the Bowen blend, He[i]{} $\lambda4921.9$, H$\gamma$ and H$\beta$ follow the same S-wave pattern (see Fig. \[fig:swave\] ). We note that identical behaviour is seen in the smaller continuous set (1.8 hr) of July 4 although with poorer S/N (not shown here). However, the presence of the S-wave in the H$\alpha$ emission profile is not clear on either dates.
Despite the incomplete phase coverage and the absence of a reference for superior conjunction of the secondary, we attempted to locate the emission site of the S-wave in the binary velocity plane using the Doppler tomography technique (Marsh & Horne 1988). Maps were reconstructed for He[ii]{} and H$\alpha$ for the April 20 observations using the software developed by Spruit (1998), assuming an orbital period of 0.1708 day and a null systemic velocity. It was not possible to combine this dataset with our other observations as this would have required $P_{\rm orb}$ to be known much more accurately ($\Delta P_{\rm orb}/P_{\rm orb}= \Delta \phi P_{\rm orb}/
T \sim 10^{-5}$ day with $T\approx90$ days and an error on the phase set by the exposure time of $\Delta \phi=0.01$). [ Furthermore, the 0.1708 day modulation is probably related to superhumps so that the orbital period can be expected to be close to this value with an uncertainty of a few percent. We tried several different orbital periods around 0.17 day but this had no significant impact on the maps. ]{} The reconstructed trailed profiles and corresponding velocity maps are shown in Fig. \[fig:prof\] and Fig. \[fig:map\] respectively.
Each pixel on the projected Doppler map corresponds to an S-wave in the trailed spectra. The binary components are located on the y-axis of the map at positions corresponding to their semi-amplitude velocities $(v_x,v_y)=(0,-K_1)$ for the primary and $(0,K_2)$ for the companion (indicated by crosses). However, since the orbital reference phase is arbitrary (here we chose $t_0=$51654.420 in MJD), the map can be rotated in any fashion around the velocity origin. The secondary position, the ballistic gas stream trajectory and the corresponding Keplerian velocity at the location of the stream shown on Fig. \[fig:map\] result from particular choices of $K_2$ and $q=M_2/M_1=K_1/K_2$ discussed further below.
The trailed emission of He[ii]{} shows a nice S-wave pattern which is consistent with the reported 0.1708 day photometric period (Cook et al. 2000; Patterson 2000). Most of the He[ii]{} emission originates from a localized region but there is also an underlying (double-peaked) disk component which is visible in the original data. On the other hand, the disk dominates the H$\alpha$ profile producing a ring of emission in the velocity map. We also note the regions of enhanced H$\alpha$ emission at about (-910,0) km s$^{-1}$ (consistent with the He[ii]{} S-wave) and $(-210,-730)$ km s$^{-1}$. The peak at the velocity origin (the center of mass) is due to the H$\alpha$ emission component at rest wavelength discussed in §3.2.
The bright emission region is typically associated with emission arising from the stream-disk interaction region or from the X-ray heated surface of the secondary (e.g. Smak 1985; Marsh, Robinson & Wood 1994; Casares et al. 1995ab; Harlaftis et al. 1996, 1997ab; Hynes et al. 2000a). Some SXTs also have low-velocity emission regions in their doppler maps, possibly associated with a magnetic propeller (Hynes et al. 2000a). Here, the velocity of $\sim 900$ km s$^{-1}$ makes it doubtful that we observe such a phenomenon.
Assuming the He[ii]{} emission arises from the heated hemisphere of the secondary, the semi-amplitude $900$ km s$^{-1}$ [ (underestimating the real $K_2$)]{} gives a rough constraint on the mass function through : $$f(M)=\frac{M_1^3\sin^3 i}{(M_1+M_2)^2}=\frac{K_2^3 P_{\rm orb}}{2 \pi G}
\label{eq:fm}$$ This gives a [*minimum*]{} mass for the primary of 12 M$_\odot$, which [ would make XTE J1118+480 at least as massive as the black hole in V404 Cyg. Lower inclinations would imply an even greater mass making this system rather unusual compared to the dynamical masses inferred in other SXTs (see, e.g., van Paradijs & McClintock 1995). Although we cannot formally reject this possibility, the He[ii]{} emission is unlikely to originate from the heated surface of the secondary.]{}
The He[ii]{} emission is more likely to originate from the stream-disk interaction. [ Typically, the hotspot is located (see above references) somewhere between the ballistic gas stream trajectory and its corresponding Keplerian velocity (which defines a circularization radius)]{}. We derive a [ lower limit on $K_2(q)$ (hence a lower limit on the primary mass)]{} by assuming that the emission is located at the intersection of the two trajectories. The values of $K_2$ and $q$ for which the intersection matches the He[ii]{} emission are plotted as a dashed line in the left panel of Fig. \[fig:cons1\].
A comparison between the two estimates of $K_2(q)$ shows that values of $0.02\lsim q\lsim 0.1$ best fit the data in the sense that they minimize the discrepancies. Formal agreement requires $q\approx0.045$ ($K_2\approx430$ km s$^{-1}$) but this is, of course, model-dependent. Knowing $q$ and $K_2$, the mass function (Eq. \[eq:fm\]) gives the inclination $i$ as a function of $M_1$. This relation is plotted in Fig. \[fig:cons1\] (right panel) for the two extreme cases ($q=0.02$,$K_2=500$ km s$^{-1}$) and ($q=0.1$,$K_2=350$ km s$^{-1}$). Clearly, a NS would require large inclinations while a BH gives intermediate inclinations. [ Given the assumptions made for the interpretation of the peak-to-peak velocity and the He[ii]{} emission, we are likely to underestimate $K_2$ in both cases. This would only strengthen the case for a black hole primary.]{}
The stream-disk interaction apparently extends some way along the Keplerian velocity trace as previously seen in other SXTs (e.g. Harlaftis et al. 1996). This could also be a numerical artifact [ possibly due to an inaccurate orbital period]{}. The second weaker emission region in the H$\alpha$ velocity map could be due to continued disturbance of the accretion disk further along in azimuth. In this picture the stream-disk interaction takes place inside the disk as emphasized by the larger velocity amplitude of the S-wave with respect to the peak-to-peak amplitude of the Balmer emission lines. This interpretation suggests significant stream overflow above the outer disk and an interaction region close to the circularization radius. [ Alternatively, systematic effects leading to lower observed peak-to-peak velocities (e.g. local line broadening, Marsh 1998) would place the hot spot closer to the disk outer edge]{}.
Interstellar Absorption
-----------------------
The Echelle spectra of XTE J1118+480 obtained on April 7, 2000 reveal the presence of weak Ca[ii]{} $\lambda 3933$ absorption features, which can be used to estimate the neutral hydrogen absorption column to the source. The spectra also show Ca[ii]{} $\lambda 3968$ in absorption, with an equivalent width of order half that of Ca[ii]{} $\lambda 3933$ features, as expected for this doublet (see, e.g., Cohen 1975). Both features are seen on two overlapping orders with similar structure. The Ca[ii]{} lines are detected in some of the individual Echelle exposures with identical profiles. Figure \[fig:ca2\] shows the Ca[ii]{} $\lambda\lambda 3933,3968$ absorption features in the summed Echelle spectra. The two orders have also been summed for this plot with weighting corresponding to the different sensitivities at the locations of the Ca[ii]{} lines.
A Gaussian fit to each of the three absorption components yields heliocentric velocities $v_0 \simeq -44$, $-26$ and $-5$ km s$^{-1}$. These velocities are consistent with an absorption component clearly seen at $\sim -44$ km s$^{-1}$ and the hint of another component at $-26$ km s$^{-1}$ in NaI $\lambda\lambda 5889, 5895$. The presence of strong Na emission at the rest wavelength (sky emission) precludes the detection of the presumed third absorption component at $-5$ km s$^{-1}$ and the use of Na absorption for the determination of the H[i]{} absorption column, N$_{\rm H{\sc i}}$, to XTE J1118+480.
We proceed as follows to determine N$_{\rm H{\sc i}}$.[^3] We assume that the gas in each absorbing cloud has a Maxwellian velocity distribution centered on its heliocentric velocity $v_0$, so that the Ca[ii]{} optical depth of the cloud can be described by (Spitzer 1978): $$\tau [v-v_0] = {\rm N_{Ca{\sc ii}}} \frac{\sqrt{\pi} e^2 f \lambda_0}{m_e c b}
e^{-[(v-v_0)/b]^2},$$ where ${\rm N_{Ca{\sc ii}}}$ is the total cloud absorption column, $\lambda_0 = 3933.6~\AA$ is the rest wavelength of the Ca[ii]{} K line, $f=0.688$ is the oscillator strength of this transition (Cardelli & Wallerstein 1986), $b$ is the velocity dispersion of the cloud, $e$ is the charge of the electron, $m_e$ is the mass of the electron and $c$ is the speed of light. Relative to a unity continuum, the line profile of a given absorbing cloud component has a depth given by: $$e^{- \tau[v-v_0]}$$ at any speed $v$ (or equivalently wavelength) around the central value $v_0$.
To deduce precise values of $b$ and ${\rm N_{Ca{\sc ii}}}$, we have compared the observed Ca${\sc ii}$ line profiles with theoretical, multi-component absorption line profiles. Initial profiles were calculated using the values of $b$ and ${\rm N_{Ca{\sc ii}}}$ derived from Eqs. (6) & (7), which were then convolved with a Gaussian instrumental Line Spread Function whose FHWM was measured from the arc lines used to wavelength calibrate the data. The best values of $b$ and ${\rm N_{Ca{\sc ii}}}$ were found by minimizing $\chi^2$ between the data and the theoretical fits. The resulting values for the Ca${\sc ii}$ lines are: $b\:=\:9.1, 2.4 $ and 6.2 km s$^{-1}$, $\log
{\rm N_{Ca{\sc ii}}} = 11.85, 11.34$ and 11.67 for the components at $v_0\:=\:-5, -26$ and $-44$ km s$^{-1}$ respectively.
The translation of the Ca[ii]{} absorption column to an hydrogen absorption column is somewhat delicate, especially in view of the possibly efficient deposition of gaseous Ca[ii]{} onto dust grains (see, e.g., Vallerga et al. 1993). However, there is a substantial body of observational data on Ca[ii]{} absorption which allows us to use an average value for the ratio ${\rm N_{Ca{\sc ii}}/N_{H{\sc
i}}}$. In their independent studies, both Cohen (1975) and Sembach & Danks (1994) find an average average value of $\log[{\rm N_{Ca{\sc
ii}}/N_{H{\sc i}}}] \simeq -8.3$ for high latitude line-of-sights, with approximate errorbars of $\pm 0.2$ in log in the two cases. Assuming this average is representative of the line of sight to XTE J1118+480, this translates in a value $\log[{\rm N_{H{\sc
i}}~(cm^{-2})}] = 20.45 \pm 0.2$, where the errorbars come from the conversion of ${\rm Ca{\sc ii}}$ to ${\rm H{\sc i}}$ absorption.
[ We note that this estimate is somewhat larger than the values $\log[{\rm N_{H{\sc i}}~(cm^{-2})}] \approx 20$ considered as reasonable by Hynes et al. (2000; see also McClintock et al. 2000; Esin et al. 2000) when dereddening the low-energy spectrum of XTE J1118+480. This discrepancy raises the possibility that the value obtained from the Ca${\sc ii}$ measurement overestimates the true value of N$_{H{\sc i}}$ on the line of sight to the source. Most likely, this would come from the conversion from Ca${\sc ii}$ to H${\sc i}$ which may not be “average” as assumed above for XTE J1118+480. It is also possible to measure the total $N$(H I) along the sightline towards XTE J1118+480 from 21 cm emission observations. Using data from the Leiden/Dwingeloo Survey (Hartmann & Burton 1997), the H I column at ($l,b$)=(157.5,62.0) is $1.32\times10^{20}$ cm$^{-2}$, a factor of two lower than we derive. This value represents an [*upper*]{} limit of course, since the 21 cm emission is integrated along the line of sight through the entire Milky Way. This therefore also suggests our value of $N$(H I) is over-estimated, although the Dwingeloo beam size of 36$'$ is too large to measure higher $N$(H I) over smaller angular scales, and it remains possible that a small, dense knot of H I with $N$(H I)$>>
1.3\times10^{20}$ cm$^{-2}$ could be lying directly along the line of sight. ]{}
It is encouraging to see that the properties derived above for the clouds are consistent with the average properties of high latitude line-of-sights described in Sembach & Danks (1994): $\langle{\rm
N_{Ca{\sc ii}}}\rangle =4.3 \pm 1.3 \times
10^{11}$ cm$^{-2}$. Furthermore, these authors find that, along extended sight lines at high latitude, the average number of absorbing Ca[ii]{} clouds is 3.6 kpc$^{-1}$. This indicates that XTE J1118+480 is located at a distance $\approx 0.83$ kpc if the line of sight to the source has average properties, an estimate which is consistent with those for other SXTs. The actual distance to XTE J1118+480 could be somewhat different if this line-of-sight has properties deviating significantly from the mean. Note that our estimate of the distance is in rough agreement with that of Uemura et al. (2000) who use the 18.8 quiescent magnitude to derive $d\sim0.5$ kpc (M-type secondary) or $d\sim1.5$ kpc (K secondary), and is consistent with the 0.8 kpc distance estimate of McClintock et al. (2000), assuming that the primary is a massive BH and that a third of the light in quiescence is provided by the M-dwarf secondary. [ It is also of interest to note that the 21 cm emission from the Leiden/Dwingeloo Survey mentioned above shows two peaks in brightness temperature which correspond closely to the Ca II components at $-5$ and $-44$ km s$^{-1}$. Although the distance of the H I is not known, the scale height of H I in the Milky Way is $\sim 1$ kpc, and since the 21 cm emission arises from [*all*]{} the H I in the Galaxy along the sightline, XTE J1118+480 must be sufficiently far away so as to show the Ca II absorption from both H I components. This again suggests a significant distance between us and XTE J1118+480.]{}
Discussion
==========
The absorption troughs
----------------------
An interesting feature of the spectra are the absorption troughs in $H\gamma$ and $H\beta$. Although rarely discussed (see, however, Soria et al. 2000), these have been observed in several other transients with similar FWHM, [ including]{} GRO J1655-40, A0620-00, GS1124-68, Nova Mus 91, XTE J2123-058, XTE J1859+226, and especially GRO J0422+32 which bears many similarities to XTE J1118+480 (see conclusion).
This type of line profile is also seen in dwarf novae (DN) and [ nova-like]{} (NL), systems which are analogous to low-mass X-ray binaries (LMXBs), except for a primary which is a white dwarf. Typically, the spectrum of quiescent DN display strong emission lines which are gradually replaced by absorption features during the rise to outburst and vice-versa during the decay (e.g. Szkody et al. 1990).
Schematically, the absorption is thought to arise from the optically-thick accretion disk whilst the emission is thought to result from photoionization in a chromosphere-like optically thin region above the disk. A key feature is that the expected spectrum is inclination dependent since the absorption should disappear as the system gets closer to being edge-on due to strong line limb-darkening in the 2D disk (see e.g. Marsh & Horne 1990; la Dous 1989; Wade & Hubeny 1998). There is indeed such an observed dependence in both DN and NL with the highest inclination systems showing only emission lines (la Dous 1991).
Although the detected absorption makes it unlikely that the inclination of XTE J1118+4800 is very high, this constraint is weak. For instance, GRO J1655-40 also showed Balmer absorption (Soria et al. 2000) but has a well-determined inclination of 70. In contrast with DN and NL, irradiation heating is much stronger in SXT where it usually dominates the heat balance in the outer disk regions (hence the optical emission, van Paradijs & McClintock 1994). Energy deposited by soft X-rays easily causes a thermal inversion in the top layers where the emission lines originate (e.g. Ko & Kallman 1994). If the optically thin atmospheres of SXTs are indeed more extended than in DN, one would expect Balmer absorption not to be as common an occurence as in the non-magnetic cataclysmic variables.
The spectral models of X-ray irradiated disks computed by Sakhibullin et al. (1998) show that softer irradiation leads to the disappearance of the absorption troughs. As the irradiation spectrum hardens, the X-ray photons deposit their energy in deeper disk layers, which do not contribute to the line emission. Inversely, any parameter change resulting in additional heating of the disk atmosphere leads to more emission. Any combination of a low inclination, low X-ray luminosity, hard X-ray spectrum or low fraction of reprocessed X-ray photons in the disk could explain the absorption troughs.
Those SXTs in which Balmer absorption has been observed have probably been guilty of one or several of the above. Interestingly, in GRO J0422+32, the H$\alpha$ and H$\beta$ lines evolved from absorption to emission within 3 days during the rise to one of the mini-outbursts (Callanan et al. 1995). Although the authors could not find any evidence for increased X-ray flux between 0.5-10 keV, the simultaneous rise in He[ii]{} emission does suggest a larger flux of soft 0.05-0.3 keV photons which would have a strong effect on the chromosphere. However, a hard X-ray upturn in GRO J1655-40 was accompanied by a dramatic increase of H$\alpha$ emission but also by a [*decrease*]{} of the He[ii]{} $\lambda 4686$ flux (Shrader et al. 1996). It seems difficult to identify the dominant culprit but we propose the low X-ray flux (see next section) and the hard X-ray spectrum of XTE J1118+480 as prime suspects responsible for the absorption troughs observed.
The optical magnitude
---------------------
A unique characteristic of this system is the large optical to X-ray ratio, as compared to other SXTs. The ratio for XTE J1118+480 is not dissimilar (Garcia et al. 2000) to that found in Accretion Disk Corona (ADC) sources where the X-ray flux is partly hidden by the disk seen edge-on. Yet, XTE J1118+4800 has not shown any of the typical behaviour associated with ADC sources, namely X-ray dips or eclipses. In addition, our crude ‘fitting’ of the peak-to-peak velocity of the Balmer lines and of the velocity map rather hints at intermediate inclinations.
This high ratio might simply be due to the low absorption and proximity of the source. At the $\sim$ 0.8 kpc distance implied by the Ca[ii]{} absorption, the 2-10 keV X-ray flux ($\approx
8\cdot10^{-10}$erg s$^{-1}$ cm$^{-2}$, Yamaoka et al. 2000) is such that $L_{\sc x}\approx 6\cdot 10^{34}$ erg s$^{-1}$. By analogy with the low/hard state of Cyg X-1, Fender et al. (2000) find the broad band X-ray luminosity is $\sim 10^{36}$erg s$^{-1}$ cm$^{-2}$ at 1 kpc, i.e. this outburst was weak for a SXT.
We compare the X-ray flux and optical magnitude $M_V$ by computing the quantity $\Sigma=(L_{\sc x}/L_{\rm Edd})^{1/2}(P_{\rm orb}/1{\rm
~hr})^{(2/3)}$, where $L_{\rm Edd}= 1.3 \times 10^{38}
(M_1/M_\odot)$ erg s$^{-1}$ is the Eddington luminosity. With $P_{\rm
orb}=0.1708$ day, we find $\log\Sigma\lsim -1.2$ with $M_1=1$ M$_\odot$ while $M_V\approx 3.5$ for $V=13$ and $E(B-V)=0$. Within the error bars, this is consistent with the correlation found for other SXTs between $\log\Sigma$ and $M_V$ by van Paradijs & McClintock (1994). In other words, the optical flux of XTE J1118+480 is roughly in agreement with what would be expected if the disk is irradiation-dominated.
The agreement is less good for higher $M_1$, the optical flux being larger than expected from the correlation. The X-ray heated companion could contribute to the optical as in, e.g., the persistent LMXB Cyg X-2. [ In this case we might expect the He[ii]{} emission to originate from the X-ray heated surface of the secondary. We have argued this is unlikely. A better candidate for this extra optical emission is]{} synchrotron emission (Hynes et al. 2000b) originating in a magnetic corona (Merloni et al. 2000),[^4] an advection-dominated accretion flow (ADAF; Esin et al. 2000) and/or a jet ($\sim$ equivalent to the first model with the base of the jet acting as the magnetic corona; [ Markoff et al. 2000]{}). XTE J1118+480 [ probably has]{} a powerful compact jet, with emission extending at least to the near-infrared (Fender et al. 2000). Both ADAF and magnetic corona models can explain the high optical/X-ray ratio but the EUV predictions may differ. The model presented by Merloni et al. (2000) has a large blackbody component peaking at $\nu\approx10^{16.5}$ Hz due to hard X-ray reprocessing in the disk. [ The soft X-ray observations of [*Chandra*]{} do not seem to support this model (McClintock et al. 2000; Esin et al. 2000).]{}
Superhumps ?
------------
The detection of superhumps (Uemura 2000) could add several constraints to the system parameters. According to current models, superhumps appear when the disk can reach the 3:1 resonance radius, requiring small mass ratio $q\lsim0.3$ (Warner 1995). In principle, the relative difference between the superhump period and the orbital period can give an even better constraint on $q$ (e.g. O’Donoghue & Charles 1996). Such a small mass ratio is compatible with the constraint $0.02\lsim q\lsim 0.1$ that we derived in §3.5.
In DN and NL, the origin of the optical variation could lie in the enhanced viscous dissipation modulated on the superhump period. However, these variations should be swamped in SXTs where irradiation heating rather than viscous heating dominates the optical output. Haswell et al. (2000a) recently argued that the small change in the accretion disk area on the superhump cycle, rather than the enhanced viscous dissipation, would give the desired effect for SXTs. The amplitude of the superhump would however decrease with larger inclinations.
The detection of superhumps in XTE J1118+4800, if confirmed, would therefore either mean the disk was not irradiation-dominated (which is unlikely considering, e.g., the correlated X-ray/near-UV behaviour reported by Haswell et al. 2000c) or that the inclination is low enough for the variations of the area of the disk to be observable.
Is XTE J1118+480 a halo object ?
--------------------------------
The high Galactic latitude of XTE J1118+480 ($b=+62^o$) is in sharp contrast with the latitudes of other LMXBs which cluster next to the Galactic plane. Whether or not XTE J1118+480 is a halo object obviously depends on its actual distance. At our inferred $d\sim
0.8$ kpc, XTE J1118+480 lies $\sim 0.7$ kpc above the Galactic plane.
The distribution of BH LMXBs around the Galactic plane has a dispersion of $\sim 0.5$ kpc. The larger dispersion of NS LMXBs ($\sim
1$ kpc) is probably due to larger kick velocities at birth (White & van Paradijs 1996). Whether XTE J1118+480 has a BH or a NS primary, its height above the Galactic plane is not much larger than for other LMXBs. For instance, White & van Paradijs (1996) find $z\approx
0.9$ kpc for the BH Nova Oph 1977.
A low metallicity environment would be supported by the low Bowen to He[ii]{} flux ratio $\approx 0.3$ (Table. \[tab:lines\]) that we find (Motch & Pakull 1989). But the detection of N and the non-detection of C and O lines in the UV spectra has been used to argue that the matter has been CNO processed in the companion (Haswell et al. 2000b) and that, from an evolutionary point of view, $M_1<3$ M$_\odot$ is preferred to allow for an evolved secondary. Considering that XTE J1118+480 is not extremely far from the distribution of other LMXBs above the Galactic plane and that CNO processing might explain the low Bowen/He[ii]{} flux ratio, we conclude that there is no compelling evidence for a halo origin of XTE J1118+480.
Conclusion
==========
Let us first summarize the main conclusions of this work.
$\bullet$ The optical spectrum of XTE J1118+480 in outburst shows several variable broad and weak emission lines superposed on a $F_\nu\propto \nu^{1/3}$ continuum typical of an optically-thick accretion disk. The He[ii]{} line shows a strong S-wave pattern which is consistent with the claimed 0.1708 day photometric period. We see no [ other]{} obvious periodic or long-term behaviour in the lines.
$\bullet$ We find from the Ca[ii]{} lines that the interstellar absorption to the source is low ($\log[{\rm N_{H{\sc i}}~(cm^{-2})}] =
20.45 \pm 0.2$) and that the presence of three intervening clouds suggests a distance of $\sim 0.8$ kpc to the source. Since the low Bowen to He[ii]{} ratio could indicate CNO-processed material (Haswell et al. 2000b) rather than an intrinsically low metallicity, we conclude there is no strong evidence to support a halo origin for XTE J1118+480.
$\bullet$
We estimate an upper limit for the 0.05-0.3 keV source flux of $\approx 10^{-8}$ erg s$^{-1}$ cm$^{-2}$. The dereddened EUVE spectrum (using a column absorption $\log[{\rm N_{H{\sc
i}}~(cm^{-2})}] \approx 20$; Hynes et al. 2000; Mcclintock et al. 2000) appears compatible with this value. This suggests that a significant fraction of the source flux is emitted in the EUV/Soft X-ray band. Given the very strong dependence of the slope of the dereddened EUVE spectrum on ${\rm N_{H_{\sc i}}}$ , it may be difficult to accomodate a higher value for ${\rm N_{H_{\sc i}}}$ as derived from Ca[ii]{} absorption in §3.6.
$\bullet$ The absorption in H$\gamma$, H$\beta$ and the Balmer jump (Hynes et al. 2000b), originating from the optically-thick disk, suggests that X-ray heating of the atmosphere is not as strong as in other SXTs, which may be consistent with the (very) low X-ray flux and hard X-ray spectrum. This also disfavors very high inclinations for which limb-darkening would remove the absorption features.
$\bullet$ The Balmer lines have peak-to-peak velocities of $\approx
1250$ km s$^{-1}$. Assuming this corresponds to the Keplerian velocity of the outer disk, we use theoretical arguments on the disk size to estimate $K_2(q)$. We find $250 \lsim K_2$ (km s$^{-1}$)$\lsim 450$ for $0.01<q<1$.
$\bullet$ Using the reconstructed Doppler tomogram, we tentatively identify the He[ii]{} S-wave with the stream-disk interaction. This gives us a second estimate of $K_2(q)$ which, combined with the first one, restricts possible values for $q$ to $0.02\lsim q\lsim0.1$. This further implies that if the primary is a NS, the system should have a high inclination ($i\gsim70$), while for a BH primary, an intermediate inclination is preferred (30$\lsim i
\lsim$50 for $10\gsim M_1/M_\odot\gsim4$).
Superhumps (Uemura 2000), if they require $q<0.3$ and low to intermediate inclinations (Haswell et al. 2000a), the absence of dips or eclipses in the X-ray lightcurve, the lack of high frequency QPOs (Revnivtsev et al. 2000) and the broad Ly$\alpha$ line (Haswell et al. 2000b) all favor an intermediate inclination system containing a BH. All of the arguments above are, however, rather uncertain and model-dependent. [ The assumptions made to interpret the peak-to-peak velocity and He[ii]{} emission probably underestimate $K_2$. A larger $K_2$ would strengthen the case for a black hole primary]{}.
We conclude by emphasizing the similarities between XTE J1118+480 and the black hole LMXB GRO J0422+32 in mini-outburst: both have orbital periods of $\sim 4$ hours, stayed in the low/hard X-ray state during outburst, had low X-ray luminosities, showed superhumps and had identical optical spectra (including the absorption troughs). Scaling the $V=19.5$ secondary of GRO J0422+32 from $d\approx2.5$ kpc to the distance of XTE J1118+480 ($\approx 0.8$ kpc) gives $V=18.5$, consistent with the reported 18.8 quiescent magnitude. The system parameters for GRO J0422+32 (taken from Chen et al. 1997) are also within our allowed range for a coherent explanation of both the Balmer line peak-to-peak velocity and the location of the He[ii]{} emission in XTE J1118+480. Quiescent studies, which can constrain the spectral type of the secondary and $K_2$, will shed more light on this interesting object.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are grateful to Mike Garcia and Jeff McClintock for very useful discussions, to Todd Tripp for sharing his knowledge of the APO Echelle spectrograph and to [ Rob Hynes for a very useful referee report]{}. We are indebted to Ed Turner for granting us some of his Director’s discretionary time at APO. Support for this work was provided by NASA through Chandra Postdoctoral Fellowship grant number PF9-10006 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-39073. GD acknowledges support from the European Commission through the TMR network ‘Accretion on to Black Holes, Compact Stars and Protostars’ (contract number ERBFMRX-CT98-0195) and from the Leids Kerkhoven-Bosscha Fonds. RSJK acknowledges support from NSF grant AST96-16901 and the Princeton University Research Board. PS acknowledges support from NASA grant NAG 5-7278.
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[ccccc]{}\
Date (UT) & Epoch & Instrument & Wavelength coverage & Exposures\
& (HJD-2400000) & & (Å) &\
\
\
April 7, 2000 & 51641.7 & DIS lowres & 4000-9000& $2 \times 60$s\
& & Echelle & 3500-9800 & $2 \times 600$s + $12 \times 1200$s\
April 20, 2000 & 51654.9 & DIS hires & 4200-5000 / 5800-6800 & $26 \times 300$s\
April 23, 2000 & 51657.8 & DIS hires & 4200-5000 / 6200-7200 & $1 \times 300$s\
May 6, 2000 & 51670.8 & DIS hires & 4200-5000 / 6300-7300 & $1 \times 480$s\
May 12, 2000 & 51676.8 & DIS hires & 4200-5000 / 6300-7300 & $1 \times 300$s\
May 15, 2000 & 51679.7 & DIS hires & 4200-5000 / 6300-7300 & $1 \times 300$s\
May 28, 2000 & 51692.8 & DIS hires & 4200-5000 / 6300-7300 & $1 \times 300$s\
July 3, 2000 & 51728.7 & DIS hires & 4200-5000 / 6300-7300 & $1 \times 300$s\
July 4, 2000 & 51729.7 & DIS hires & 4350-5150 / 6050-7050 & $9 \times 300$s + $ 2 \times 600$s\
\
\[tab:obs\]
[lcccc]{}\
Line & $\lambda_0$ (Å) & $\lambda_{\rm obs}-\lambda_0$ (Å) & EW (Å) & FWHM (1000 km s$^{-1}$)\
\
\
H$\gamma$ & 4340.5 & -2$\pm$4 & -1.1$\pm$0.4 & 2.1$\pm$0.5\
& 4340.5 & 15$\pm$5 & 1.8$\pm$0.5 & 3.5$\pm$0.5\
He[i]{} & 4471.5 & 1$\pm$5 & -0.3$\pm$0.2 & 1.9$\pm$0.6\
& 4471.5 & 4$\pm$5 & 0.7$\pm$0.4 & 4.0$\pm$0.8\
Bowen blend & 4638 & 8$\pm$10 & -0.5$\pm$0.3 & 2.0$\pm$1.1\
He[ii]{} & 4686.7 & 0$\pm$3 & -1.5$\pm$0.4 & 2.3$\pm$0.4\
H$\beta$ & 4861.3 & -1$\pm$5 & -1.0$\pm$0.3 & 1.8$\pm$0.3\
& 4861.3 & 13$\pm$6 & 1.5$\pm$0.5 & 3.1$\pm$0.4\
He[i]{} & 4921.9 & 2$\pm$5 & -0.2$\pm$0.2 & 1.4$\pm$0.6\
H$\alpha$ & 6562.8 & 0$\pm$2 & -1.9$\pm$0.4 & 1.8$\pm$0.4\
He[i]{} & 6678.1 & -1$\pm$5 & -0.6$\pm$0.4 & 1.8$\pm$0.8\
\[tab:lines\]
NOTE. – Negative equivalent widths indicate emission lines, while positive values correspond to absorption troughs. $\lambda_0$ is the rest wavelength and $\lambda_{\rm obs}$ is the central wavelength at which each feature is observed.
[^1]: VSNET observations at http://www.kusastro.kyoto-u.ac.jp/vsnet
[^2]: From $P_{\rm orb}=$0.17 day, Kepler’s third law gives a binary separation $a \approx 10^{11}$ cm with a weak dependence on $M_1$ and $q$.
[^3]: Note that we do not have to worry about a stellar Ca[ii]{} absorption component from the companion star because the emission of XTE J1118+480 in outburst is presumably dominated by accretion
[^4]: Note that the corona model of Merloni et al. 2000 may be challenged by the observations of Haswell et al. (2000c) which show that the near-UV variability lags behind the X-ray variability by 1 to 2 seconds.
| 0 |
---
abstract: |
Based on microscopic Hartree-Fock + random phase approximation calculations with Skyrme interactions, we study the correlations between the nuclear breathing mode energy $E_{\mathrm{ISGMR}}$ and properties of asymmetric nuclear matter with a recently developed analysis method. Our results indicate that the $E_{\mathrm{ISGMR}}$ of $^{208}$Pb exhibits moderate correlations with the density slope $L$ of the symmetry energy and the isoscalar nucleon effective mass $m_{s,0}^{\ast }$ besides a strong dependence on the incompressibility $K_{0}$ of symmetric nuclear matter. Using the present empirical values of $L=60\pm 30$ MeV and $m_{s,0}^{\ast
}=(0.8\pm 0.1)m$, we obtain a theoretical uncertainty of about $\pm 16$ MeV for the extraction of $K_{0}$ from the $E_{\mathrm{ISGMR}}$ of $^{208}$Pb. Furthermore, we find the $E_{\mathrm{ISGMR}}$ difference between $^{100}$Sn and $^{132}$Sn strongly correlates with $L$ and thus provides a potentially useful probe of the symmetry energy.
author:
- 'Lie-Wen Chen'
- 'Jian-Zhong Gu'
title: Correlations between the nuclear breathing mode energy and properties of asymmetric nuclear matter
---
Introduction
============
Determination of the equation of state (EOS) for isospin asymmetric nuclear matter (ANM) is among fundamental questions in both nuclear physics and astrophysics. Knowledge on the nuclear EOS is important for understanding not only the structure of finite nuclei, the nuclear reaction dynamics, and the liquid-gas phase transition in nuclear matter, but also many critical issues such as properties of neutron stars and supernova explosion mechanism in astrophysics [@LiBA98; @Dan02; @Lat04; @Ste05; @Bar05; @LCK08]. In the past more than $30$ years, significant progress has been made in determining the EOS of symmetric nuclear matter from subsaturation density to about $5$ times normal nuclear matter density $\rho _{0}$ by studying the nuclear isoscalar giant monopole resonances (ISGMR) [@You99], collective flows [@Dan02] and subthreshold kaon production [@Aic85; @Fuc06a] in nucleus-nucleus collisions. On the other hand, the isospin dependent part of the nuclear EOS, characterized essentially by the nuclear symmetry energy $E_{\text{\textrm{sym}}}({\rho })$, is still largely uncertain [Bar05,LCK08]{}. Lack of knowledge on the symmetry energy actually hinders us to extract more accurately the EOS of symmetric nuclear matter. Therefore, to explore and narrow down the uncertainties of both the theoretical methods and the experimental data is of crucial importance for extracting more stringently information on the nuclear EOS.
During the past more than 30 years, it has been established that the nuclear ISGMR provides a good tool to probe the nuclear EOS around the nuclear normal density. In particular, it is generally believed that the incompressibility $K_{0}$ of symmetric nuclear matter can be extracted from a self-consistent microscopic theoretical model that successfully reproduces the experimental ISGMR energies as well as the ground state binding energies and charge radii of a variety of nuclei [Bla80]{}. Experimentally, thanks to new and improved experimental facilities and techniques, the ISGMR centroid energy $E_{\mathrm{ISGMR}}$, i.e., the so-called nuclear breathing mode energy, of $^{208}$Pb (a heavy, doubly-magic nucleus with a well-developed monopole peak) has been measured with a very high precision (less than $2\%$). Indeed, a value of $E_{\mathrm{ISGMR}}=14.17\pm 0.28$ MeV was extracted from the giant monopole resonance in $^{208}$Pb based on an improved $\alpha $-scattering experiment [You99]{} (another value of $E_{\mathrm{ISGMR}}=13.96\pm 0.20$ MeV was extracted in Ref. [@You04]). The $E_{\mathrm{ISGMR}}$ of $^{208}$Pb has been extensively used to constrain the $K_{0}$ parameter in the literature [@You99; @You04; @Lui04; @Ma02; @Vre03; @Col04; @Shl06; @LiT07; @Gar07; @Paa07; @Col09; @Pie10]. It is thus important to estimate and eventually narrow down the theoretical uncertainty of extracting $K_{0}$ from the nuclear ISGMR. Theoretically, in fact, it has been found that the uncertainty of the density dependence of the symmetry energy has significantly influenced the precise extraction of the $K_{0}$ parameter from ISGMR in $^{208}$Pb and it also provides an explanation for the observed model dependence of the $K_{0}$ extraction from the ISGMR in $^{208}$Pb based on non-relativistic and relativistic models [@Pie02; @Agr03; @Vre03; @Col04; @Pie04].
In the present work, we estimate the theoretical uncertainty when one extracts the $K_{0}$ parameter from the nuclear ISGMR based on microscopic Hartree-Fock (HF) + random phase approximation (RPA) calculations with Skyrme interactions. In particular, we study the correlations between the ISGMR centroid energy and properties of ANM with a recently developed analysis method [@Che10] in which instead of varying directly the $9$ parameters in the Skyrme interaction, we express them explicitly in terms of $9$ macroscopic quantities that are either experimentally well constrained or empirically well known. Then, by varying individually these macroscopic quantities within their known ranges, we can examine more transparently the correlation of the ISGMR centroid energy with each individual macroscopic quantity and thus estimate the theoretical uncertainty of the ISGMR centroid energy based on the empirical uncertainties of the macroscopic quantities. Our results indicate that the density slope $L$ of the symmetry energy and the isoscalar nucleon effective mass $m_{s,0}^{\ast }$ can significantly change the $E_{\mathrm{ISGMR}}$ of $^{208}$Pb and the present uncertainties of $L$ and $m_{s,0}^{\ast }$ can lead to a theoretical uncertainty of about $\pm 16$ MeV for the extraction of $K_{0}$. We further find the $E_{\mathrm{ISGMR}}$ difference between $^{100}$Sn and $^{132}$Sn displays a strong correlation with $L$ and thus provides a potential probe of the symmetry energy.
Methods {#Theory}
=======
Skyrme-Hartree-Fock approach and macroscopic properties of asymmetric nuclear matter
------------------------------------------------------------------------------------
The EOS of isospin asymmetric nuclear matter, given by its binding energy per nucleon, can be expanded to $2$nd-order in isospin asymmetry $\delta $ as $$E(\rho ,\delta )=E_{0}(\rho )+E_{\mathrm{sym}}(\rho )\delta ^{2}+O(\delta
^{4}), \label{EOSANM}$$where $\rho =\rho _{n}+\rho _{p}$ is the baryon density with $\rho _{n}$ and $\rho _{p}$ denoting the neutron and proton densities, respectively; $\delta
=(\rho _{n}-\rho _{p})/(\rho _{p}+\rho _{n})$ is the isospin asymmetry; $E_{0}(\rho )=E(\rho ,\delta =0)$ is the binding energy per nucleon in symmetric nuclear matter, and the nuclear symmetry energy is expressed as $$E_{\mathrm{sym}}(\rho )=\frac{1}{2!}\frac{\partial ^{2}E(\rho ,\delta )}{\partial \delta ^{2}}|_{\delta =0}. \label{Esym}$$Around $\rho _{0}$, the symmetry energy can be characterized by using the value of $E_{\text{\textrm{sym}}}({\rho _{0}})$ and the density slope parameter $L=3{\rho _{0}}\frac{\partial E_{\mathrm{sym}}(\rho )}{\partial
\rho }|_{\rho ={\rho _{0}}}$, i.e.,
$$E_{\mathrm{sym}}(\rho )=E_{\text{\textrm{sym}}}({\rho _{0}})+\frac{L}{3}(\frac{\rho -{\rho _{0}}}{{\rho _{0}}})+O((\frac{\rho -{\rho _{0}}}{{\rho _{0}}})^{2}).$$
In the standard Skyrme Hartree-Fock approach, the nuclear effective interaction is taken to have a zero-range, density- and momentum-dependent form [@Cha97], i.e., $$\begin{aligned}
V_{12}(\mathbf{R},\mathbf{r}) &=&t_{0}(1+x_{0}P_{\sigma })\delta (\mathbf{r})
\notag \\
&+&\frac{1}{6}t_{3}(1+x_{3}P_{\sigma })\rho ^{\sigma }(\mathbf{R})\delta (\mathbf{r}) \notag \\
&+&\frac{1}{2}t_{1}(1+x_{1}P_{\sigma })(\mathbf{K}^{^{\prime }2}\delta (\mathbf{r})+\delta (\mathbf{r})\mathbf{K}^{2}) \notag \\
&+&t_{2}(1+x_{2}P_{\sigma })\mathbf{K}^{^{\prime }}\cdot \delta (\mathbf{r})\mathbf{K} \notag \\
&\mathbf{+}&iW_{0}(\mathbf{\sigma }_{1}+\mathbf{\sigma }_{2})\cdot \lbrack
\mathbf{K}^{^{\prime }}\times \delta (\mathbf{r})\mathbf{K]}, \label{V12Sky}\end{aligned}$$with $\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$ and $\mathbf{R}=(\mathbf{r}_{1}+\mathbf{r}_{2})/2$. In the above expression, the relative momentum operators $\mathbf{K}=(\mathbf{\nabla }_{1}-\mathbf{\nabla }_{2})/2i$ and $\mathbf{K}^{\prime }=-(\mathbf{\nabla }_{1}-\mathbf{\nabla }_{2})/2i$ act on the wave function on the right and left, respectively. The quantities $P_{\sigma }$ and $\sigma _{i}$ denote, respectively, the spin exchange operator and Pauli spin matrices. The $\sigma $, $t_{0}-t_{3}$, $x_{0}-x_{3}$ are the $9$ Skyrme interaction parameters which can be expressed analytically in terms of $9$ macroscopic quantities $\rho _{0}$, $E_{0}(\rho
_{0})$, the incompressibility $K_{0}$, the isoscalar effective mass $m_{s,0}^{\ast }$, the isovector effective mass $m_{v,0}^{\ast }$, $E_{\text{\textrm{sym}}}({\rho _{0}})$, $L$, the gradient coefficient $G_{S}$, and the symmetry-gradient coefficient $G_{V}$ [@Che10; @Che11b], i.e., $$\begin{aligned}
t{_{0}} &=&4\alpha /(3{\rho _{0}}) \\
x{_{0}} &=&3(y-1)E_{\text{\textrm{sym}}}^{\mathrm{loc}}({\rho _{0}})/\alpha
-1/2 \\
t{_{3}} &=&16\beta /\left[ {\rho _{0}}^{\gamma }(\gamma +1)\right] \\
x{_{3}} &=&-3y(\gamma +1)E_{\text{\textrm{sym}}}^{\mathrm{loc}}({\rho _{0}})/(2\beta )-1/2 \\
t_{1} &=&20C/\left[ 9{\rho _{0}(}k_{\mathrm{F}}^{0})^{2}\right] +8G_{S}/3 \\
t_{2} &=&\frac{4(25C-18D)}{9{\rho _{0}(}k_{\mathrm{F}}^{0})^{2}}-\frac{8(G_{S}+2G_{V})}{3} \\
x_{1} &=&\left[ 12G_{V}-4G_{S}-\frac{6D}{{\rho _{0}(}k_{\mathrm{F}}^{0})^{2}}\right] /(3t_{1}) \\
x_{2} &=&\left[ 20G_{V}+4G_{S}-\frac{5(16C-18D)}{3{\rho _{0}(}k_{\mathrm{F}}^{0})^{2}}\right] /(3t_{2}) \\
\text{\ }\sigma &=&\gamma -1 \label{SkySigma}\end{aligned}$$where $k_{\mathrm{F}}^{0}=(1.5\pi ^{2}{\rho _{0}})^{1/3}$, $E_{\text{\textrm{sym}}}^{\mathrm{loc}}({\rho _{0}})=E_{\text{\textrm{sym}}}({\rho _{0}})-E_{\text{\textrm{sym}}}^{\mathrm{kin}}({\rho _{0}})-D$, and the parameters $C$, $D$, $\alpha $, $\beta $, $\gamma $, and $y$ are defined as [@Che09] $$\begin{aligned}
C &=&\frac{m-m_{s,0}^{\ast }}{m_{s,0}^{\ast }}E_{\mathrm{kin}}^{0} \\
D &=&\frac{5}{9}E_{\mathrm{kin}}^{0}\left( 4\frac{m}{m_{s,0}^{\ast }}-3\frac{m}{m_{v,0}^{\ast }}-1\right) \\
\alpha &=&-\frac{4}{3}E_{\mathrm{kin}}^{0}-\frac{10}{3}C-\frac{2}{3}(E_{\mathrm{kin}}^{0}-3E_{0}(\rho _{0})-2C) \notag \\
&&\times \frac{K_{0}+2E_{\mathrm{kin}}^{0}-10C}{K_{0}+9E_{0}(\rho _{0})-E_{\mathrm{kin}}^{0}-4C} \\
\beta &=&(\frac{E_{\mathrm{kin}}^{0}}{3}-E_{0}(\rho _{0})-\frac{2}{3}C)
\notag \\
&&\times \frac{K_{0}-9E_{0}(\rho _{0})+5E_{\mathrm{kin}}^{0}-16C}{K_{0}+9E_{0}(\rho _{0})-E_{\mathrm{kin}}^{0}-4C} \\
\gamma &=&\frac{K_{0}+2E_{\mathrm{kin}}^{0}-10C}{3E_{\mathrm{kin}}^{0}-9E_{0}(\rho _{0})-6C}. \\
y &=&\frac{L-3E_{\text{\textrm{sym}}}({\rho _{0}})+E_{\text{\textrm{\ sym}}}^{\mathrm{kin}}({\rho _{0}})-2D}{3(\gamma -1)E_{\text{ \textrm{sym}}}^{\mathrm{loc}}({\rho _{0}})}\end{aligned}$$with $E_{\mathrm{kin}}^{0}=\frac{3\hbar ^{2}}{10m}\left( \frac{3\pi ^{2}}{2}\right) ^{2/3}\rho _{0}^{2/3}$ and $E_{\text{\textrm{sym}}}^{\mathrm{kin}}({\rho _{0}})=\frac{\hbar ^{2}}{6m}\left( \frac{3\pi ^{2}}{2}{\rho _{0}}\right) ^{2/3}$. In the above, the isoscalar effective mass $m_{s,0}^{\ast }$ is the nucleon effective mass in symmetric nuclear matter at its saturation density ${\rho _{0}}$ while the isovector effective mass $m_{v,0}^{\ast }$ corresponds to the proton (neutron) effective mass in pure neutron (proton) matter at baryon number density ${\rho _{0}}$ (See, e.g., Refs. [Cha97,Che09]{}). Furthermore, $G_{S}$ and $G_{V}$ are respectively the gradient and symmetry-gradient coefficients in the interaction part of the binding energies for finite nuclei defined as $$E_{\mathrm{grad}}=G_{S}(\nabla \rho )^{2}/(2{\rho )}-G_{V}\left[ \nabla
(\rho _{n}-\rho _{p})\right] ^{2}/(2{\rho )}.$$
HF + continuum-RPA calculations
-------------------------------
Since the energy of the giant monopole resonance is above the single particle continuum threshold, a proper calculation should, in principle, involve a complete treatment of the particle continuum. In the present work, we study the ISGMR of nuclei by using a microscopic HF + continuum-RPA calculations with Skyrme interactions [@Ham96]. The RPA response function is solved in the coordinate space with the proton-neutron formalism including simultaneously both the isoscalar and the isovector correlation. In this way, we can take properly into account the coupling to the continuum and the effect of neutron (proton) excess on the structure of the giant resonances in nuclei near the neutron (proton) drip lines [@Ham96].
The RPA strength distribution of ISGMR of nuclei $$S(E_{x})=\sum_{n}|<n|Q|0>|^{2}\delta (E_{x}-E_{n}) \label{RPAStrength}$$can be obtained by using the isoscalar monopole operator $$Q^{\lambda =0,\tau =0}=\frac{1}{\sqrt{4\pi }}\sum_{i}r_{i}^{2}.
\label{ISGMROperator}$$Furthermore, one can define the *k*-th energy moment of the transition strength by $$m_{k}=\int dE_{x}E_{x}^{k}S(E_{x}). \label{RPAmoment}$$The average energy of ISGMR can be defined by the ratio between the moments $m_{1}$ and $m_{0}$, i.e., $$E_{ave}=m_{1}/m_{0}. \label{Eave}$$In addition, the so-called scaling energy of ISGMR can be expressed as $$E_{sca}=\sqrt{m_{3}/m_{1}}, \label{Esca}$$while the ISGMR energy obtained from the constrained HF approach can be written as $$E_{con}=\sqrt{m_{1}/m_{-1}}. \label{Econ}$$The ISGMR energies defined by Eqs. (\[Eave\])-(\[Econ\]) will become identical in the case of a sharp single peak exhausting $100\%$ of the sum rule. In practice, it is found that both the experimental data and the theoretical calculations show a large width of a few MeV even in the most well-established ISGMR in $^{208}$Pb. However, it is interesting to note that $E_{ave}$ and $E_{con}$ are rather close within a $0.1\sim 0.2$ MeV difference even when the ISGMR peak has a large width although the scaling energy $E_{sca}$ has a large uncertainty due to the high energy tail of monopole strength, which is always the case in experimental data (and see the theoretical results in the following). Furthermore, from the relation of the energy moments $m_{k+1}m_{k-1}\geq m_{k}^{2}$, one can obtain $E_{sca}\geq E_{ave}\geq E_{con}$. Therefore, the average energy $E_{ave}$ is usually defined as the ISGMR centroid energy and compared between the experimental data and the theoretical calculations. It should be noted that the situation of light nuclei may be quite different from that of medium and heavy nuclei considered in the present work since the strength distribution of ISGMR for light nuclei is usually very fragmented [@Lui01; @Joh03; @You09]. It was suggested in a recent work [@Fur10] that the fragmentation of the strength distribution for the light nuclei might be explained by the clustering effects which are not considered in the present work.
Results {#Result}
=======
In the present work, as a reference for the correlation analyses performed below with the standard Skyrme interactions, we use the MSL0 parameter set [@Che10], which is obtained by using the following empirical values for the $9$ macroscopic quantities: $\rho _{0}=0.16$ fm$^{-3}$, $E_{0}(\rho
_{0})=-16$ MeV, $K_{0}=230$ MeV, $m_{s,0}^{\ast }=0.8m$, $m_{v,0}^{\ast
}=0.7m$, $E_{\text{\textrm{sym}}}({\rho _{0}})=30$ MeV, $L=60$ MeV, $G_{V}=5$ MeV$\cdot $fm$^{5}$, and $G_{S}=132$ MeV$\cdot $fm$^{5}$. And the spin-orbit coupling constant $W_{0}=133.3$ MeV $\cdot $fm$^{5}$ is used to fit the neutron $p_{1/2}-p_{3/2}$ splitting in $^{16}$O. It has been shown [Che10]{} that the MSL0 interaction can describe reasonably the binding energies and charge rms radii for a number of closed-shell or semi-closed-shell nuclei. We further find that the MSL0 parameter set predicts a value of $1.06$ MeV for the splitting of the neutron $3p$ shell in $^{208}$Pb, which reasonably describes the experimental value of $0.9$ MeV. It should be pointed out that the MSL0 is only used here as a reference for the correlation analyses. Using other Skyrme interactions obtained from fitting measured binding energies and charge rms radii of finite nuclei as in usual Skyrme parametrization will not change our conclusion.
As numerical examples, in the present work, we choose the spherical closed-shell doubly-magic nuclei $^{208}$Pb, $^{100}$Sn, and $^{132}$Sn. Thus, we do not include the pairing interaction since it has negligible effects on these spherical closed-shell doubly-magic nuclei considered in this work [@Kha10]. In addition, the two-body spin-orbit and the two-body Coulomb interactions are not taken into account in the present continuum-RPA calculations although the HF calculations include both of the interactions. As pointed out in Ref. [@Sil06], the net effect of the two interactions in RPA decreases the centroid energy of ISGMR in $^{208}$Pb by about $300$ keV. It should be stressed that, in the present work, we do not intend to extract accurately the value of the $K_{0}$ parameter by comparing the measured ISGMR centroid energy with that from HF + continuum-RPA calculations, and our main motivation is to explore the theoretical uncertainty for extracting $K_{0}$. Meanwhile, we are mainly interested in the ISGMR centroid energy difference between $^{100}$Sn and $^{132}$Sn rather than their respective absolute value. Therefore, we do not expect that the two-body spin-orbit and the two-body Coulomb interactions in RPA will significantly change our conclusion and further work is needed to see how exactly the two interactions in continuum-RPA calculations will affect our results. Furthermore, in the following calculations, the sum rules $m_{k} $ are obtained by integrating the RPA strength from excitation energy $E_{x}=5$ MeV to $E_{x}=35$ MeV in Eq. (\[RPAmoment\]).
Isospin scalar giant monopole resonance in $^{208}$Pb
-----------------------------------------------------
![(Color online) The ISGMR energies of $^{208}$Pb obtained from SHF + continuum-RPA calculations with MSL0 by varying individually $L$ (a), $G_{V}$ (b), $G_{S}$ (c), $E_{0}(\protect\rho _{0})$ (d), $E_{\text{\textrm{sym}}}(\protect\rho _{0})$ (e), $K_{0}$ (f), $m_{s,0}^{\ast }$ (g), $m_{v,0}^{\ast
} $ (h), $\protect\rho _{0}$ (i), and $W_{0}$ (j). The three lines from upper to lower in each panel correspond to $E_{sca}$, $E_{ave}$, and $E_{con} $, respectively.[]{data-label="XEaPb208"}](XEaPb208.eps)
Shown in Fig. \[XEaPb208\] are the ISGMR energies, i.e., $E_{sca}$, $E_{ave}$, and $E_{con}$ of $^{208}$Pb obtained from SHF + continuum-RPA calculations with MSL0 by varying individually $L$, $G_{V}$, $G_{S}$, $E_{0}(\rho _{0})$, $E_{\text{\textrm{sym}}}(\rho _{0})$, $K_{0}$, $m_{s,0}^{\ast }$, $m_{v,0}^{\ast }$, $\rho _{0}$, and $W_{0}$, namely, varying one quantity at a time while keeping all the others at their default values in MSL0. Firstly, one can see clearly the ordering of $E_{sca}\geq
E_{ave}\geq E_{con}$ as expected. In particular, for the default parameters in MSL0, we obtain $E_{sca}=14.962$ MeV, $E_{ave}=14.453$ MeV, and $E_{con}=14.338$ MeV. We note that the centroid energy of ISGMR $E_{ave}=14.453$ MeV is essentially in agreement with the measured value of $14.17\pm 0.28$ MeV for the ISGMR in $^{208}$Pb [@You99] (a more recent experimental value of $13.96\pm 0.20$ MeV was extracted in Ref. [@You04]). The agreement will become much better if the two-body spin-orbit and the two-body Coulomb interactions are taken into account in the continuum-RPA calculations since the net effect of the two interactions in RPA reduces the centroid energy of ISGMR in $^{208}$Pb by about $300$ keV [@Sil06]. These features imply that the default Skyrme parameter set MSL0 can give a good description for the ISGMR in $^{208}$Pb. Furthermore, one can see from Fig. \[XEaPb208\] that, within the uncertain ranges considered here for the macroscopic quantities, the ISGMR energies display a very strong positive correlation with $K_{0}$ as expected. On the other hand, however, the ISGMR energies also exhibit moderate negative correlations with both $L$ and $m_{s,0}^{\ast }$ while weak dependence on the other macroscopic quantities. These results indicate that the uncertainties of $L$ and $m_{s,0}^{\ast }$ may significantly influence the extraction of $K_{0}$ by comparing the theoretical value of the ISGMR energies of $^{208}$Pb from SHF + RPA calculations with the experimental measurements.
![(Color online) SHF + continuum-RPA response functions of $^{208}$Pb with Skyrme interaction MSL0 by varying individually $K_{0}$ (a), $L$ (b), and $m_{s,0}^{\ast }$ (c).[]{data-label="SRPAPb208"}](SRPAPb208.eps)
As for the correlation analysis method in Fig. \[XEaPb208\], we would like to stress that, when the macroscopic quantities (except $E_{0}(\rho _{0})$ and $\rho _{0}$) change individually from their base values in MSL0 within the empirical uncertain ranges considered here, the values of the binding energy or the charge rms radii of finite nuclei will vary by only about $2\%$ at most (See, e.g., Figs. 4 and 5 in Ref. [@Che10]). Therefore, the original agreement of MSL0 with the experimental data of binding energies or charge radii of finite nuclei will essentially still hold with the individual change of the macroscopic quantities. In this way, changing the macroscopic quantities individually in the present correlation analysis approach is equivalent to constructing a number of different Skyrme interaction sets which can give reasonable description on the ground state binding energy and charge rms radii of finite nuclei. The key point and the most important advantage of the present analysis approach is that in the present correlation analysis, one knows exactly what is the difference among different Skyrme interaction sets constructed by using different macroscopic quantities. Furthermore, it should be mentioned that the centroid energy of ISGMR in heavy nuclei are not sensitive to the values of $E_{0}(\rho _{0})$ and $\rho _{0}$ as shown in Fig. \[XEaPb208\], and thus in principle we can adjust $E_{0}(\rho _{0})$ and $\rho
_{0}$ to give better description for the ground state binding energy and charge rms radii of finite nuclei without changing significantly the results of ISGMR and thus the conclusions in the present work.
In order to see the dependence of the detailed structure of ISGMR in $^{208}$Pb on the values of $K_{0}$, $L$ and $m_{s,0}^{\ast }$, we show in Fig. \[SRPAPb208\] the SHF + continuum-RPA response functions of $^{208}$Pb with MSL0 by varying individually $K_{0}$, $L$, and $m_{s,0}^{\ast }$, i.e., $K_{0}=200$ and $270$ MeV, $L=30$ and $90$ MeV, and $m_{s,0}^{\ast }=0.6m$ and $0.9m$. As can be seen in Fig. \[SRPAPb208\], the RPA result displays a single collective peak in each case, consistent with the experimental data [@You04; @Sag07b]. Furthermore, it is seen that varying the value of $K_{0} $ from $200$ MeV to $270$ MeV strongly shifts the single collective peak from about $13.3$ MeV to $15.4$ MeV while varying the value of $L$ ($m_{s,0}^{\ast }$) from $30$ MeV ($0.6m$) to $90$ MeV ($0.9m$) shifts the single collective peak from about $14.6$ ($15.0$) MeV to $13.9$ ($13.9$) MeV. These results are consistent with the results shown in Fig. [XEaPb208]{}. In addition, the calculated width with MSL0 by varying individually $K_{0}$, $L$, and $m_{s,0}^{\ast }$ shows roughly the same value as that of experimental data [@You04; @Sag07b]. This agreement implies that the width of ISGMR is essentially determined by the Landau damping and the coupling to the continuum, which are properly taken into account in the present calculations.
The ISGMR energy $E_{\mathrm{ISGMR}}$ is conventionally related to a finite nucleus incompressibility $K_{A}(N,Z)$ for a nucleus with $N$ neutrons and $Z
$ protons ($A=N+Z$) by the following definition$$E_{\mathrm{ISGMR}}=\sqrt{\frac{\hbar ^{2}K_{A}(N,Z)}{m\left\langle
r^{2}\right\rangle }}, \label{EGMRKa}$$where $m$ is the nucleon mass and $\left\langle r^{2}\right\rangle $ is the mean square mass radius of the nucleus in the ground state. Similarly to the semi-empirical mass formula, the finite nucleus incompressibility $K_{A}(N,Z)
$ is usually expanded as [@Bla80] $$\begin{aligned}
K_{A}(N,Z) &=&K_{0}+K_{\mathrm{surf}}A^{-1/3}+K_{\mathrm{curv}}A^{-2/3}
\notag \\
&&+(K_{\tau }+K_{\mathrm{ss}}A^{-1/3})\left( \frac{N-Z}{A}\right) ^{2}
\notag \\
&&+K_{\mathrm{Coul}}\frac{Z^{2}}{A^{4/3}}+\cdot \cdot \cdot . \label{KA1}\end{aligned}$$Neglecting the curvature term $K_{\mathrm{curv}}$, the surface symmetry term $K_{\mathrm{ss}}$ and the other higher-order terms of the finite nucleus incompressibility $K_{A}(N,Z)$ in Eq. (\[KA1\]), one can express $K_{A}(N,Z)$ as $$\begin{aligned}
K_{A}(N,Z) &=&K_{0}+K_{\mathrm{surf}}A^{-1/3}+K_{\tau }\left( \frac{N-Z}{A}\right) ^{2} \notag \\
&&+K_{\mathrm{Coul}}\frac{Z^{2}}{A^{4/3}}, \label{KA2}\end{aligned}$$where $K_{0}$, $K_{\mathrm{surf}}$, $K_{\tau }$, and $K_{\mathrm{coul}}$ represent the volume, surface, symmetry, and Coulomb terms, respectively. The $K_{\tau }$ parameter is usually thought to be equivalent to the isospin dependent part, i.e., the $K_{\mathrm{sat,2}}$ parameter, of the isobaric incompressibility coefficient of ANM (incompressibility evaluated at the saturation density of ANM) defined as$$K_{\mathrm{sat}}(\delta )=K_{0}+K_{\mathrm{sat,2}}\delta ^{2}+O(\delta ^{4}).
\label{Ksat}$$We would like to point out that the $K_{\mathrm{sat,2}}$ parameter is a theoretically well-defined physical property of ANM [@Pie09; @Che09] while the value of the $K_{\tau }$ parameter may depend on the details of the truncation scheme in Eq. (\[KA1\]) [@Bla81; @Sha88; @Pea91; @Shl93; @Pea10]. Here, we assume $K_{\mathrm{sat,2}}$ has similar influences on $K_{A}(N,Z)$ as $K_{\tau }$ and thus $K_{\mathrm{sat,2}}$ will affect the $E_{\mathrm{ISGMR}}$ through Eq. (\[EGMRKa\]), and then we can analyze the $L$ and $m_{s,0}^{\ast }$ dependences of $E_{\mathrm{ISGMR}}$ from the correlations of $K_{\mathrm{sat,2}}$ parameter with $L$ and $m_{s,0}^{\ast }$.
![The $K_{\mathrm{sat,2}}$ parameter obtained from SHF with MSL0 by varying individually $L$ (a) and $m_{s,0}^{\ast }$ (b).[]{data-label="XKsat2"}](XKsat2.eps)
The effects of the density dependence of the symmetry energy on the ISGMR centroid energy $E_{ave}$ of $^{208}$Pb has been extensively investigated in the literature [@Pie02; @Vre03; @Agr03; @Pie04; @Col04]. It was firstly proposed by Piekarewicz [@Pie02] that the different symmetry energies used in the non-relativistic models and the relativistic models may be responsible for the puzzle that the former predicted an incompressibility in the range of $K_{0}=210-230$ MeV while the latter predicted a significantly larger value of $K_{0}\approx 270$ MeV from the analysis of the ISGMR centroid energy. It is seen from Fig. \[XEaPb208\] that a larger $L$ value (as in usual relativistic models) leads to a smaller $E_{ave}$ value and thus a larger $K_{0}$ value is necessary to counteract the deceasing of $E_{ave}$ due to a larger $L$ value. Furthermore, Fig. \[XEaPb208\] shows that $E_{ave}$ displays a very weak dependence on $E_{\text{\textrm{sym}}}(\rho _{0})$, which is in contrast to the results in Ref. [@Col04] where $E_{ave}$ is shown to be sensitive to $E_{\text{\textrm{sym}}}(\rho _{0})$. This is due to the fact that a constrain on the value of $E_{\text{\textrm{sym}}}({\rho
=0.1}$ [fm]{}$^{{-3}})$ was imposed in Ref. [@Col04], which leads to a strong linear correlation between $E_{\text{\textrm{sym}}}({\rho _{0}})$ and $L$ as shown recently in Ref. [@Che11a].
The symmetry energy dependence of the ISGMR centroid energy of $^{208}$Pb can be understood from the fact that the ISGMR in $^{208}$Pb does not constrain the compression modulus of symmetric nuclear matter but rather the one of neutron-rich matter, i.e., the isobaric incompressibility coefficient in Eq. (\[Ksat\]). From Eq. (\[Ksat\]) it is clear that the ISGMR in $^{208}$Pb (with an isospin asymmetry of $\delta =0.21$) should be sensitive to a linear combination of $K_{0}$ and $K_{\mathrm{sat,2}}$. The $K_{\mathrm{sat,2}}$ parameter is completely determined by the slope and curvature of the symmetry energy at saturation density as well as the third derivative of the EOS of symmetric nuclear matter (see, e.g., Ref. [@Che09]). Fig. [XKsat2]{} shows the $K_{\mathrm{sat,2}}$ parameter from SHF with MSL0 by varying individually $L$ and $m_{s,0}^{\ast }$. As can be seen in Fig. [XKsat2]{}, the $K_{\mathrm{sat,2}}$ parameter decreases with both $L$ and $m_{s,0}^{\ast }$, and thus $K_{A}(N,Z)$ for $^{208}$Pb will decrease correspondingly if the $K_{\mathrm{sat,2}}$ parameter has similar effects on $K_{A}(N,Z)$ as the $K_{\tau }$ parameter and the $K_{\mathrm{ss}}$ term as well as the other higher-order terms in Eq. (\[KA1\]) are not important for $K_{A}(N,Z)$. These results provide an explanation on the behavior that the ISGMR energies decrease with $L$ and $m_{s,0}^{\ast }$ observed in Fig. \[XEaPb208\].
![The $\protect\sigma $ parameter obtained from SHF with MSL0 by varying individually $K_{0}$ (a), $E_{0}(\protect\rho _{0})$ (b), $m_{s,0}^{\ast }$ (c), and $\protect\rho _{0}$ (d).[]{data-label="XSigmaPara"}](XSigmaPara.eps)
To understand more clearly why the ISGMR energies decrease with $m_{s,0}^{\ast }$ observed in Fig. \[XEaPb208\], it is useful to note the fact that, with the standard Skyrme interaction, the $K_{0}$ and $m_{s,0}^{\ast }$ cannot be chosen independently if the Skyrme interaction parameter $\sigma $ in Eq. (\[V12Sky\]), $E_{0}(\rho _{0})$ and $\rho _{0}$ are fixed [@Boh79]. It should be stressed here that, instead of assuming a fixed value of $\sigma $ as in the usual parametrization and correlation analysis [@Cha97; @Col04], in the present work, the $\sigma $ parameter is determined by four macroscopic quantities, i.e., $K_{0}$, $E_{0}(\rho _{0})$, $m_{s,0}^{\ast }$ and $\rho _{0}$ as shown in Eq. (\[SkySigma\]), and thus $K_{0}$ and $m_{s,0}^{\ast }$ can be chosen independently. Neglecting the isospin dependence (assuming $N\approx Z)$, the nuclear breathing mode energy for medium and heavy nuclei can be approximated by [@Boh79] $$E_{\mathrm{ISGMR}}\approx \sqrt{\frac{\hbar ^{2}(K_{0}-63\sigma )}{m\left\langle r^{2}\right\rangle }}\text{ (}K_{0}\text{ in MeV).}
\label{EGMRBohigas}$$Eq. (\[EGMRBohigas\]) implies that the nuclear breathing mode energy can be closely related to both $K_{0}$ and $m_{s,0}^{\ast }$ if the parameter $\sigma $ is free and the values of $E_{0}(\rho _{0})$ and $\rho _{0}$ are fixed. In Fig. \[XSigmaPara\], we show the $\sigma $ parameter obtained from SHF with MSL0 by varying individually $K_{0}$, $E_{0}(\rho _{0})$, $m_{s,0}^{\ast }$, and $\rho _{0}$. One can see clearly that the $\sigma $ parameter indeed exhibits a strong correlation with $K_{0}$ as expected. However, it also displays a moderate dependence on $m_{s,0}^{\ast }$, a small dependence on $E_{0}(\rho _{0})$, and a very weak correlation with $\rho _{0}$. As can be seen in Fig. \[XSigmaPara\], the $\sigma $ parameter increases with $m_{s,0}^{\ast }$, leading to smaller ISGMR energies according to Eq. (\[EGMRBohigas\]), which is consistent with the results shown in Fig. \[XEaPb208\]. In addition, the fact that $K_{\mathrm{sat,2}}$ parameter decreases with $m_{s,0}^{\ast }$ observed in Fig. \[XKsat2\] will also be partially responsible for the behavior of ISGMR energies decreasing with $m_{s,0}^{\ast }$ as seen in Fig. \[XEaPb208\] since a smaller $K_{\mathrm{sat,2}}$ value will lead to a smaller $E_{\mathrm{ISGMR}} $ as discussed previously.
The above results indicate that the ISGMR centroid energy of $^{208}$Pb exhibits moderate correlations with both $L$ and $m_{s,0}^{\ast }$ besides a strong dependence on $K_{0}$. The accurate knowledge on $L$ and $m_{s,0}^{\ast }$ is thus important for a precise determination of the $K_{0}$ parameter from the ISGMR centroid energy of $^{208}$Pb. In recent years, significant progress has been made in determining $L$ and its value is essentially consistent with $L=60\pm 30$ MeV depending on the observables and methods used in the studies [Mye96,Che05a,She07,Kli07,Tri08,Tsa09,Dan09,Cen09,Car10,XuC10,Liu10,Che11a]{}. Using $L=60\pm 30$ MeV, we can estimate an uncertainty of about $\pm 0.281$ MeV for the ISGMR centroid energy in $^{208}$Pb from Fig. \[XEaPb208\]. On the other hand, for the isoscalar effective mass, the empirical value of $m_{s,0}^{\ast }=(0.8\pm 0.1)m$ has been obtained from the analysis of both isoscalar quadrupole giant resonances data in doubly closed-shell nuclei and single-particle spectra [@Liu76; @Boh79; @Far97; @Rei99; @Les06]. From Fig. [XEaPb208]{}, we can obtain an uncertainty of about $\pm 0.382$ MeV for the ISGMR centroid energy in $^{208}$Pb using the empirical value of $m_{s,0}^{\ast }=(0.8\pm 0.1)m$. Assuming the two uncertainties due to the present uncertainties of $L$ and $m_{s,0}^{\ast }$ on the ISGMR centroid energy in $^{208}$Pb are independent, we thus can add them quadratically to obtain an uncertainty of about $\pm 0.474$ MeV for the ISGMR centroid energy in $^{208}$Pb. Then, using the approximate relation $(\delta K_{0}/K_{0})=2$($\delta E_{\mathrm{ISGMR}}$/$E_{\mathrm{ISGMR}}$) from Eq. (\[EGMRKa\]), we can obtain an uncertainty of $\pm 7\%$ for $K_{0}$ with $E_{\mathrm{ISGMR}}\approx 14$ MeV, namely, about $\pm 16$ MeV for $K_{0}=230$ MeV.
Furthermore, including other uncertainties due to $G_{V}$, $G_{S}$, $E_{0}(\rho _{0})$, $E_{\text{\textrm{sym}}}(\rho _{0})$, $m_{v,0}^{\ast }$, $\rho _{0}$ and $W_{0}$ with empirical values of $G_{V}=0\pm 40$ MeV, $G_{S}=130\pm 10$ MeV, $E_{0}(\rho _{0})=-16\pm 1$ MeV, $E_{\text{\textrm{sym}}}(\rho _{0})=30\pm 5$ MeV, $m_{v,0}^{\ast }=(0.7\pm 0.1)m$, $\rho
_{0}=0.16\pm 0.01$ fm$^{-3}$ and $W_{0}=130\pm 20$ MeV, and assuming all the uncertainties are independent, we can obtain from Fig. \[XEaPb208\] a total uncertainty of about $\pm 0.647$ MeV for the ISGMR centroid energy in $^{208}$Pb, which gives an uncertainty of about $\pm 9\%$ for $K_{0}$, namely, about $\pm 21$ MeV for $K_{0}=230$ MeV.
Isospin scalar giant monopole resonances in $^{100}$Sn and $^{132}$Sn
---------------------------------------------------------------------
To see the isotopic dependence of the ISGMR centroid energy, we study here the spherical closed-shell doubly-magic nuclei $^{100}$Sn and $^{132}$Sn. Shown in Fig. \[XEcSn100132\] are the ISGMR centroid energy $E_{ave}$ of $^{100}$Sn and $^{132}$Sn obtained from SHF + RPA calculations with MSL0 by varying individually $L$, $G_{V}$, $G_{S}$, $E_{0}(\rho _{0})$, $E_{\text{\textrm{sym}}}(\rho _{0})$, $K_{0}$, $m_{s,0}^{\ast }$, $m_{v,0}^{\ast }$, $\rho _{0}$, and $W_{0}$. One can see that the results for neutron-rich nucleus $^{132}$Sn are quite similar to those for $^{208}$Pb as shown in Fig. \[XEaPb208\]. On the other hand, for the symmetric nucleus $^{100}$Sn, it is interesting to see that the dependence of $E_{ave}$ on the isospin relevant macroscopic quantities, namely, $L$, $G_{V}$, $E_{\text{\textrm{sym}}}(\rho _{0})$, $m_{v,0}^{\ast }$ is very weak. We have also checked the case of the stable nucleus $^{90}$Zr for the correlation analysis as in Fig. \[XEcSn100132\], and we find the results are very similar to the case of $^{100}$Sn, namely, displaying a much weak correlation with the $L$ parameter while a stronger correlation with the $G_{S}$ parameter compared with the case of $^{208}$Pb. This may be understandable from the fact that the $^{90}$Zr has a smaller isospin asymmetry, i.e., $(N-Z)/A=0.11$ compared with $^{208}$Pb where we have ($N-Z)/A=0.21$. In addition, the surface coefficient $G_{S}$ may become more important for lighter nuclei as expected, leading to a stronger correlation with the $G_{S}$ parameter. From these results, it seems that the ISGMR of a heavier and more symmetric nucleus, where the symmetry energy effects will be reduced significantly, may be more suitable for extracting the $K_{0}$ parameter. In addition, the different $E_{ave}$-$m_{s,0}^{\ast }$ correlations between $^{100}$Sn and $^{132}$Sn observed in Fig. \[XEcSn100132\] can be understood from the fact that $K_{\mathrm{sat,2}}$ parameter decreases with $m_{s,0}^{\ast }$ as shown in Fig. \[XKsat2\], leading additional decrement of $E_{ave}$ with $m_{s,0}^{\ast }$ for the neutron-rich nucleus $^{132}$Sn.
![(Color online) Same as Fig. \[XEaPb208\] but for the ISGMR centroid energy $E_{ave}$ of $^{100}$Sn and $^{132}$Sn. The results of $^{100}$Sn shift down by $1.5$ MeV for a more clear comparison with those of $^{132}$Sn.[]{data-label="XEcSn100132"}](XEcSn100132.eps)
![Same as Fig. \[XEaPb208\] but for the ISGMR centroid energy difference between $^{100}$Sn and $^{132}$Sn.[]{data-label="dEcSn100132"}](dEcSn100132.eps)
It is instructive to see the ISGMR centroid energy difference between $^{100} $Sn and $^{132}$Sn, which is shown in Fig. \[dEcSn100132\] with MSL0 by varying individually $L$, $G_{V}$, $G_{S}$, $E_{0}(\rho _{0})$, $E_{\text{\textrm{sym}}}(\rho _{0})$, $K_{0}$, $m_{s,0}^{\ast }$, $m_{v,0}^{\ast
}$, $\rho _{0}$, and $W_{0}$. It is very interesting to see from Fig. [dEcSn100132]{} that, within the uncertain ranges considered here for the macroscopic quantities, the ISGMR centroid energy difference displays a very strong correlation with $L$. However, on the other hand, the ISGMR centroid energy difference exhibits only moderate correlations with $m_{s,0}^{\ast }$ and $m_{v,0}^{\ast }$ while weak dependence on the other macroscopic quantities. These features imply that the ISGMR centroid energy difference between $^{100}$Sn and $^{132}$Sn provides a potential probe of the $L$ parameter. Furthermore, it is seen that the ISGMR centroid energy difference displays opposite correlation with $m_{s,0}^{\ast }$ and $m_{v,0}^{\ast }$, namely, increases with $m_{s,0}^{\ast }$ while decreases with $m_{v,0}^{\ast
}$. Recently, a constraint of $m_{s,0}^{\ast }-m_{v,0}^{\ast }=(0.126\pm
0.051)m$ has been extracted from global nucleon optical potentials constrained by world data on nucleon-nucleus and (p, n) charge-exchange reactions [@XuC10]. Imposing the constraint $m_{s,0}^{\ast
}-m_{v,0}^{\ast }=(0.126\pm 0.051)m$, we can expect from Fig. [dEcSn100132]{} that the correlation of the ISGMR centroid energy difference with $m_{s,0}^{\ast }$ and $m_{v,0}^{\ast }$ will become significantly weak, making the ISGMR centroid energy difference really a good probe of the $L$ parameter. Our results indicate that a precise determination of the ISGMR centroid energy difference between $^{100}$Sn and $^{132}$Sn will be potentially useful to constraint accurately the symmetry energy, especially the $L$ parameter. This provides strong motivation for measuring the ISGMR strength in unstable nuclei, which can be investigated at the new/planning rare isotope beam facilities at CSR/HIRFL and BRIF-II/CIAE in China, RIBF/RIKEN in Japan, SPIRAL2/GANIL in France, FAIR/GSI in Germany, and FRIB/NSCL in USA.
Summary {#Summary}
=======
The isoscalar giant monopole resonances of finite nuclei have been investigated based on microscopic Hartree-Fock + random phase approximation calculations with Skyrme interactions. In particular, we have studied the correlations between the ISGMR centroid energy, i.e., the so-called nuclear breathing mode energy, and properties of asymmetric nuclear matter within a recently developed correlation analysis method. Our results indicate that the ISGMR centroid energy of $^{208}$Pb displays a very strong correlation with $K_{0}$ as expected. On the other hand, however, the ISGMR centroid energy also exhibits moderate correlation with both $L$ and $m_{s,0}^{\ast }$ while weak dependence on the other macroscopic quantities. Using the present empirical values of $L=60\pm 30$ MeV and $m_{s,0}^{\ast }=(0.8\pm 0.1)m$, we have obtained an uncertainty of about $0.474$ MeV for the ISGMR centroid energy in $^{208}$Pb, leading to a theoretical uncertainty of about $\pm 16$ MeV for the extraction of $K_{0}$ from the $E_{\mathrm{ISGMR}}$ of $^{208}$Pb. Including additionally other uncertainties due to $G_{V}$, $G_{S}$, $E_{0}(\rho _{0})$, $E_{\text{\textrm{sym}}}(\rho _{0})$, $m_{v,0}^{\ast }$, $\rho _{0}$ and $W_{0}$ with empirical values of $G_{V}=0\pm 40$ MeV, $G_{S}=130\pm 10$ MeV, $E_{0}(\rho _{0})=-16\pm 1$ MeV, $E_{\text{\textrm{sym}}}(\rho _{0})=30\pm 5$ MeV, $m_{v,0}^{\ast }=(0.7\pm 0.1)m$, $\rho
_{0}=0.16\pm 0.01$ fm$^{-3}$ and $W_{0}=130\pm 20$ MeV, we have estimated a total uncertainty of about $\pm 21$ MeV for the extraction of $K_{0}$ by assuming all the uncertainties are independent. These results show that the accurate knowledge on $L$ and $m_{s,0}^{\ast }$ is important for a precise determination of the $K_{0}$ parameter by comparing the measured ISGMR centroid energy of $^{208}$Pb with that from Hartree-Fock + random phase approximation calculations.
Furthermore, we have investigated how the ISGMR centroid energy difference between $^{100}$Sn and $^{132}$Sn correlates with properties of asymmetric nuclear matter. We have found that the ISGMR centroid energy difference between $^{100}$Sn and $^{132}$Sn displays a strong correlation with the $L$ parameter while weak dependence on the other macroscopic quantities. This feature implies that the ISGMR centroid energy difference between $^{100}$Sn and $^{132}$Sn provides a potentially useful probe of the nuclear symmetry energy. Our results also provide strong motivation for measuring the ISGMR strength in unstable nuclei, which can be investigated at the new/planing rare isotope beam facilities around the world.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This work was supported in part by the NNSF of China under Grant Nos. 10975097, 10975190 and 11135011, Shanghai Rising-Star Program under Grant No. 11QH1401100, “Shu Guang" project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation, the National Basic Research Program of China (973 Program) under Contract Nos. 2007CB815003 and 2007CB815004, and the Funds for Creative Research Groups of China under Grant No. 11021504.
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| 0 |
---
abstract: 'In this work we analyse the Parisi’s $\infty$-replica symmetry breaking solution of the Sherrington - Kirkpatrick model without external field using high order perturbative expansions. The predictions are compared with those obtained from the numerical solution of the $\infty$-replica symmetry breaking equations which are solved using a new pseudo-spectral code which allows for very accurate results. With this methods we are able to get more insight into the analytical properties of the solutions. We are also able to determine numerically the end-point $x_{\rm max}$ of the plateau of $q(x)$ and find that $\lim_{T\to 0} x_{\rm max}(T) > 0.5$.'
author:
- 'A. Crisanti'
- 'T. Rizzo'
date: 'V2.1, '
title: 'Analysis of the $\infty$-replica symmetry breaking solution of the Sherrington-Kirkpatrick model'
---
Introduction
============
Since its proposal in the 80’s the behaviour of the Parisi $\infty$-replica symmetry breaking ($\infty$-RSB) solution of the Sherrington-Kirkpatrick model has been extensively investigated both qualitatively and quantitatively [@MPV; @FH]. Despite this enormous amount of work, which has revealed many of the properties of the solutions, a complete control of the solution is still missing. One of the reasons can be traced back to the fact that till now only low order expansions were used, moreover applied often to reduced forms $\infty$-replica symmetry breaking equations valid only near the critical temperature. From the numerical point of view there are only few works which confirm the general properties of the solution but do not allow for high accuracy. On the other hand $\infty$-replica symmetry breaking solutions of the type encountered in the SK model have been found in other models of interest in different fields, e.g., in computer science with solvability problems [@Luca] or in the study of the structural glass transition [@G85; @KT87; @SNA97].
Motivated by these problems we believe that it would be quite useful to have some reliable and efficient tool to find good approximations of the full solution also far from the critical points. In this work we reconsider two approaches. The first one is based on expansions for temperatures near the critical temperature $T_c$. As said above previous works considered only low order expansions [@P7980; @K83; @S85]. Here, by using algebraic manipulators, we push the expansion to rather high orders and resumming it via Padè resummation technique we are able to a have good estimate of the solution for a wide range of temperature below $T_c$.
The second approach is numerical. Previous numerical studies of the $\infty$-replica symmetry breaking solution used a naive integration scheme based on the direct discretization of the Parisi’s equation [@VTP81; @SDJPC84; @topomoto; @B90]. The main disadvantages of this approach are the large amount of memory needed for a good resolution of the solution and the numerical problems arising when $\dot{q}(x)$ is small. To overcome these problems we developed a new numerical scheme based on a pseudo-spectral algorithm which allows for rather accurate results for all temperatures with a reasonable amount of memory. Moreover the use of pseudo-spectral methods makes the whole code rather fast.
We stress that while the methods we are going to discuss are applied here to the Sherrington - Kirkpatrick model, they have a wider range of application. In principle can be applied to any model with $\infty$-replica symmetry breaking type solution [@Luca].
We find that for the Sherrington - Kirkpatrick model the Parisi solution $q(x)$ is not an odd function as one may expect from its physical meaning. At any $T < T_c$, the Taylor expansion of $q(x)$ in powers of $x$ contains both odd as well as even powers of $x$. The only term which is missing is $x^2$. The presence of the fourth oder derivative was first noted by Temesvari [@Temes]. Often, instead of $q(x)$, it is more useful to consider the overlap probability distribution function $P(q)$, which gives the probability of finding two states with mutual overlap $q$ according to the Gibbs measure. The two quantities are related by [@DY83; @P83]: $$\label{eq:pdq}
P(q)=\frac{dx}{dq}$$ where $x(q)$ is the inverse function of $q(x)$. In the absence of external magnetic fields the function $P(q)$ must be an even function of $q$. The computed function $q(x)$ is however defined only for positive values, therefore it determines only the right branch of the function $P(q)$. If we define $\tilde{P}(q)=dx/dq$ for $q>0$ then full $P(q)$ is given by the symmetrized expression $$P(q)=\frac{1}{2}\tilde{P}(-q)+\frac{1}{2}\tilde{P}(q)$$ It is easy to see that the presence of non-zero even derivatives of $q(x)$ at $x=0$ makes the function $P(q)$ non analytical at $q=0$: $$P(q)=c_{0}+c_{2}q^{2}+c_{3}|q|^{3}+\ldots$$ so that $P(q)$ has discontinuous derivatives at $q=0$.
We shall discuss two different methods of computing the expansions. The first, discussed in Section \[SecExp\], performs expansion before imposing stationarity of the free energy functional. The two steps however can be inverted, i.e., the expansion can be done after stationarity is imposed, Section \[SecDeriv\]. The two approaches are obviously equivalent and the advantage of using one or the other only depends on which quantity one is interested in. Since the expansions are likely to be asymptotic some resummation scheme, such as Padè discussed in Section \[SecPad\], are needed. Finally in Section \[SecNum\] we present a new integration procedure and compare the analytical results with those obtained from a direct numerical solution of the $\infty$-replica symmetry breaking equations.
Expansion of the free energy functional {#SecExp}
=======================================
The Parisi’s free energy $f$ for a the SK model in an external field $h$ at temperature $T$ is [@P80]:
$$-f = \frac{\beta}{4}\,\Bigl(
1 - 2\,q(1) + \int_0^1dx\, q^2(x)
\Bigr)
+ \int_{-\infty}^{+\infty} \frac{d y}{\sqrt{2 \pi q(0)}}
\exp\left(-\frac{(y-h)^2}{2\,q(0)}\right)\phi(0,y)
\label{eqfree}$$
where $\phi(0,y)$ is the solution evaluated at $x=0$ of the the Parisi’s equation $$\dot\phi(x,y)=-\frac{\dot{q}(x)}{2}\,
\Bigl[
\phi''(x,y)+\beta\,x\,\phi'(x,y)^2
\Bigr]
\label{eqPhi}$$ with the boundary condition $$\phi(1,y)= \beta^{-1}\log\left(2\cosh \beta y\right)
\label{Phi1}$$ where we have used the standard notation and denote derivatives with respect to $x$ by a dot and derivatives with respect to $y$ by a prime. The order parameter $q(x)$ at temperature $T$ is obtained by the stationarity condition of (\[eqfree\]) with respect to variations of $q(x)$, while the value of (\[eqfree\]) at the stationarity point gives the free energy $f(T)$.
To expand the free energy functional (\[eqfree\]) in powers of $\tau = T_c - T = 1 - T$ we observe that in the absence of external fields $q(x)$ is different from $q(1)$ only in a region $[0,x_{\rm max}]$ with $x_{max} = O(\tau)$ [@P7980], so that an expansion in power of $\tau$ must correspond to an expansion of the same order in $x$. Therefore to compute the free energy to order $n$ we insert into eq. (\[eqfree\]) the following expansions: $$\label{eqq1}
q(1) = \sum_{i=1}^{n-2}\, a_i\,\tau^i$$ and $$\label{eqxq}
x(q) = \sum_{i=1}^{n-3}\,\sum_{j=0}^{n-3-i} b_{ij}\,q^i\,\tau^j.$$ The coefficients of the expansion of the function $\phi(q,y)$ about $q=q(1)$ and $y=0$ can be obtained by repeated differentiation with respect to $q$ of the equation $$\frac{\partial \phi}{\partial q} = -\frac{1}{2}\left[
\frac{\partial^{2} \phi}{\partial y^{2}}
+ x(q)\left( \frac{\partial \phi}{\partial y}\right)^{2}
\right].$$ Differentiating $j$ times with respect to $y$ this equation, mixed derivatives $\phi^{(1,j)}(q,y)$ can be eliminated in favor of derivatives with respect to $y$ only. In the absence of an external magnetic field the last term in equation (\[eqfree\]) reduces to $\phi(0,0)$ greatly simplifying the calculation since at each step we can eliminate all terms containing odd derivatives of $\phi$ with respect to $y$, as for example $(\partial \phi / \partial y)^{2}$ in the previous equation, since all these vanish if evaluated at $y=0$ being $\phi(q,y)$ and even function of $y$.
Collecting all terms with the same power of $\tau$ the free energy functional (\[eqfree\]) is written as $$f = \sum_{i=0}^{n}\, c_i[\{a\},\{b\}]\, \tau^i.$$ This expression must be stationary with respect to variations of $a$’s and $b$’s for any $\tau$. Imposing stationarity of each $c_i$ we can find the value of the parameters $a$ and $b$. For example to order $\tau^6$ we have: $$\begin{aligned}
\label{eq:qx6}
q(x) = \left(
\frac{1}{2} + \frac{3}{2}\,\tau + 2\,\tau^3
- 9\, \tau^4 + \frac{336}{5}\,\tau^5
\right)\, x
&+& \left(
-\frac{1}{8} + \frac{25}{8}\,\tau + 3\,\tau^2
+ 38\,\tau^3
\right)\, x^3
\nonumber\\
&+& \left(
-1 - 9\,\tau - 30\,\tau^2
\right)\, x^4
+ \left(
\frac{351}{320} + \frac{9189}{320}\,\tau
\right)\, x^5
- \frac{27}{5}\, x^6\end{aligned}$$ and $$\label{eq:xmax6}
x_{\rm max} = 2\,\tau - 4\,\tau^2 + 12\,\tau^3 - 69\,\tau^4
+\frac{2493}{5}\,\tau^5 - \frac{20544}{5}\,\tau^6.$$ By using this procedure we have obtained the free energy up to order $30$, $q(x)$ to order $13$ and $q(1)$ to order $14$ because despite the fact that the free energy is evaluated to order $n$, the variational relations allow to determine $x(q)$ only to order $[(n-3)/2]$ and $q(1)$ only to order $[(n-1)/2]$.
From eq. (\[eq:qx6\]) we clearly see that $q(x)$ contains even powers of $x$, with the exclusion of $x^2$. In the next Section we shall derive exact relations among the derivatives of $q(x)$ at $x=0$ from which follow that $q^{(2)}(x=0) = 0$ but $q^{(4)}(x=0) \not= 0$.
Expansion of the order parameter $q(x)$ {#SecDeriv}
=======================================
To evaluate the derivatives of the order parameter $q(x)$ at $x=0$ we use a variational approach developed by Sommers and Dupont[@SDJPC84]. This method as also the advantage of leading to exact relations among derivatives of different order, so can be used to test the findings of the previous Section in a non-perturbative way. The starting point is the variational form of the Parisi’s free energy $f$: $$\begin{aligned}
-f &=& \frac{\beta}{4}\,\Bigl(
1 - 2\,q(1) + \int_0^1dx\, q^2(x)
\Bigr)
+ \int_{-\infty}^{+\infty} \frac{d y}{\sqrt{2 \pi q(0)}}
\exp\left(-\frac{(y-h)^2}{2\,q(0)}\right)\phi(0,y)
\nonumber
\\
&-&\int_{-\infty}^{+\infty} dy\ P(1,y)\,
\bigl[\phi(1,y)-T \log\left(2\cosh \beta y\right)\bigr]
\nonumber
\\
&+&\int_0^1dx\int_{-\infty}^{+\infty} dy\ P(x,y)
\left[\dot\phi(x,y)+
\frac{\dot{q}(x)}{2}\,
\Bigl[
\phi''(x,y)+\beta\,x\,\phi'(x,y)^2
\Bigr]
\right].
\label{eqfrev}\end{aligned}$$ Imposing stationarity with respect to variations of $P(x,y)$, $P(1,y)$, $\phi(x,y)$, $\phi(0,y)$ and $q(x)$, one obtains the variational equations: $$\label{SP1}
q(x)=\int dy\, P(x,y)\,m^{2}(x,y)$$ $$\dot m(x,y)=-\frac{\dot{q}(x)}{2}\,
\Bigl[
m''(x,y) + 2\,\beta\,x\, m(x,y)\ m'(x,y)
\Bigr]
\label{SP2}$$
$$\label{SP3}
\dot{P}(x,y) = \frac{\dot{q}(x)}{2}
\Bigl[
P''(x,y) - 2\,\beta\,x\,[m(x,y)\,P(x,y)]'
\Bigr]$$
with initial conditions (in the absence of a magnetic field) $$\begin{aligned}
m(1,y) & = & \tanh (y/T)\label{condm} \\
P(0,y) & = & \delta (y)\label{condP}\end{aligned}$$ These equations are the starting point of both the expansion discussed in this Section and the numerical solution.
A time scale $\tau_x$ can be associated to the order parameter $q(x)$ such that for times of order $\tau_x$ states with an overlap equal to $q(x)$ or greater can be reached by the system. In this picture the $P(x,y)$ and $m(x,y)$ become respectively the probability distribution of frozen local fields $y$ and the local magnetization in a local field $y$ at the time scale labeled by $x$ [@SDJPC84; @MPV].
The derivatives of $q(x)$ can be obtained by successive $x$-derivation of eq. (\[SP1\]). The procedure is simplified by the use of the following identity [@S85]: $$\label{int1}
\frac{d}{dx}\int dy\,P(x,y)\,f(x,y)=\int dy\,P(x,y)\,\Omega(x,y)\, f(x,y)$$ where $$\Omega(x,y) =\frac{\partial }{\partial x}+\frac{\dot{q}}{2}
\left( \frac{\partial ^{2}}{\partial y^{2}} + 2\,\beta\,x\,m(x,y)
\frac{\partial }{\partial y}\right)$$ The application of the operator $\Omega(x,y)$ generates derivatives of the function $m(x,y)$ with respect to $x$ and $y$. Mixed derivatives such as $m^{(1,j)}(x,y)$ can be eliminated in favor of derivatives of $m(x,y)$ respect only to $y$ by deriving equation (\[SP2\]) $j$ times with respect to $y$.
The first application of this procedure yields $$\dot{q}(x)=\dot{q}(x)\,\int dyP(x,y)(m')^{2}$$ which for $\dot{q}(x)\neq 0$ simply reads $$\label{d1}
1=\int dy\,P(x,y)\, m'(x,y)^{2}.$$ The procedure can be iterated infinitely. For example, the next three applications leads respectively to $$\label{d2}
0=-\frac{2x}{T}\int dyP(m')^{3}+\int dyP(m'')^{2}$$ $$\label{d3}
\frac{2}{T}\int dyP(m')^{3}=\dot{q}\int Pdy\left( (m''')^{2}-\frac{12x}{T}m'(m'')^{2}+6\left( \frac{x}{T}\right) ^{2}(m')^{4}\right)$$ and $$\int Pdy\left( \frac{(18x\dot{q}+6x^{2}\ddot{q})(m')^{4}}{T^{2}}-\frac{(18\dot{q}T+12x\ddot{q}T-120m'x^{2}\dot{q}^{2})m'(m'')^{2}}{T^{2}}\right.$$ $$\label{dev2x}
\left. +\frac{-30x\dot{q}^{2}(m'')^{2}+\ddot{q}m'''T-20x\dot{q}^{2}m'(m''')}{T}m'''-\frac{24x^{3}\dot{q}^{2}(m')^{5}}{T^{3}}+\dot{q}^{2}(m'''')^{2}\right) =0$$ We are interested into the derivatives of $q(x)$ at $x=0$, so we take the limit $x\rightarrow 0$ of the above equations. The limit can be done in trivial way and, since the function $P(0,y)$ reduce to a $\delta$-function \[see eq. (\[condP\])\], the equations are greatly simplified. Moreover since $m(x,y)$ is an odd function of $y$ for any $x$ clearly $m^{(0,j)}(0,0)=0$ for any even $j$. In this limit equations (\[d1\]), (\[d3\]) and (\[dev2x\]) reduce respectively to: $$\begin{aligned}
\label{dev10}
1 & = & m'(0,0)\label{limz2} \\
\label{dev20}
\frac{2}{T}m'(0,0)^{3} & = & \dot{q}(0)m'''(0,0)^{2}\label{limz} \\
\label{eq:d40}
\ddot{q}(0)m'''(0,0)^{2} &=& 0 \end{aligned}$$ while equations (\[SP1\]) and (\[d2\]) become trivial identities.
From equations (\[limz2\]) and (\[limz\]) we have $$\label{m30}
m'''(0,0)=-\sqrt{\frac{2}{T\dot{q}(0)}}\neq 0$$ therefore (\[eq:d40\]) implies that $\ddot{q}(0)=0$ as already found in Ref. [@S85].
To obtain information on the fourth derivative of $q(x)$ the above procedure must be iterated two more times. Since successive derivatives yields expressions with a rapidly growing number of terms we only report the limit $x\rightarrow 0$ result: $$\label{dev30}
\frac{18\dot{q}(0)}{T^{2}}+q^{(3)}(0)m'''(0,0)^{2}-\frac{38\dot{q}(0)^{2}m'''(0,0)^{2}}{T}+\dot{q}(0)^{3}m^{(0,5)}(0,0)=0$$ $$\label{eq:dev40}
q^{(4)}(0)m'''(0,0)-\frac{96\dot{q}(0)m'''(0,0)^{3}}{T}=0$$ where equation (\[limz2\]) and the exact result $ \ddot{q}(0)=0 $ have been used. Note that equation (\[eq:dev40\]), with equation (\[m30\]), gives a complete determination of the quartic derivative of $q(x)$ at $x=0$ as a function of the temperature $T$ and of the first derivative $\dot{q}(x=0)$: $$q^{(4)}(0)=-\frac{96\sqrt{2}\dot{q}(0)^{5/2}}{T^{3/2}}$$ This relation shows that the function $q(x)$ does not have a well defined parity [@Temes].
Going to higher orders one can show that all the even derivatives can be expressed in terms of the odd ones; for instance we have $$q^{(6)}(0)=-\frac{34272\sqrt{2}\dot{q}(0)^{7/2}}{T^{5/2}}
-\frac{1056\sqrt{2}\dot{q}(0)^{3/2}q^{(3)}(0)}{T^{3/2}}$$ and so on.
In the limit $T\to 0$ we have $T\,\dot{q}(0) = 0.743\pm 0.002$. Note that if we take $\dot{q}(0)\sim 1 /T$ for $T\to 0$ the previous equations implies that all the derivatives diverge with the temperature as $q^{(n)}(0)\sim 1/T^{n}$, in agreement with the Parisi-Toulouse scaling $q(x,T)= q(\beta x)$ [@VTP81; @PT80]. Note that we have derived this scaling under strong hypothesis that it must be valid asymptotically for $T\to 0$ and $\beta x\to 0$.
This approach also provides an alternative method to compute the expansion of $q(x)$ in powers of $x$ and $\tau$: starting from $q(x)$ evaluated at a given order in $x$ and $\tau$ we can compute $ m^{(0,j)}(0,0) $ through (\[SP2\]) and then $q(x)$ at the next order through the set of equations (\[limz2\]),(\[eq:d40\]), (\[dev30\]),(\[eq:dev40\]) and so on. The set of equations can be solved iteratively. By this method we were able to compute the series expansion of $q(x)$ up to order $20$, improving the results of previous section.
Resummation of the expansions {#SecPad}
=============================
Unfortunately all the expansions derived in the previous Sections are likely to be asymptotic and to obtain sensible estimates of the various quantities of interest some form of resummation must be done. Here we shall consider the standard Padè approximants which for a series of degree $N+M$ reads [@BO78]: $$P^N_M(x) = \frac{\sum_{i=0}^{N}\, a_i\, x^i}
{1 + \sum_{i=1}^{M}\, b_i\, x^i}$$ where the coefficients are chosen so that the first $(N+M+1)$ terms of the Taylor expansion of $P^N_M(x)$ match the the first $(N+M+1)$ terms of the of the original series. In the following we shall call this the Padè approximant $(N,M)$.
The first problem we faced is that despite the fact that the series have alternate signs, they are not Stijlties integral and therefore we cannot obtain in a systematic way a sequence of lower and upper bounds [@BO78]. This difficulty can be overcome by noticing that most of the quantities we are interested in, such as for example free energy or entropy or $q(1)$, do have a null derivative at $T=0$. Therefore an indication on the quality of the approximants can be obtained by analyzing the behaviour near $T=0$. For example, the free energy as a function of $T$ is reproduced quite well by many Padè approximants, even at very low orders, however some of these have a positive derivative at $T=0$ while others negative, see Fig. \[fig:freepad\]. By inspecting the figure we can safely assume that approximants with positive derivative give an upper bound, while those with negative derivative a lower bound, for the true free energy [@Note].
As a general fact we obtain that the best Padè approximants at a given order in $\tau$ are those with nearly the same degree of the numerator and the denominator. We stress, however, that as usual with asymptotic expansion an increase of the order in $\tau$ does not necessarily correspond to an improvement of the precision. With this procedure we obtain for the free energy an estimate with at least $16$ digits precision at $T=0.9$ and $8$ digits at $T=0.5$, and for the ground state energy $E_{0}=-.76321\pm .00003$ in agreement with Parisi’s estimate $E_{0}=-.7633\pm .0001$ [@P80]. A similar analysis can be used to determine the value of $x_{\rm max}$ as a function of temperature, the result is shown in Fig. \[fig:xmaxpad\]. The value of the breaking point is finite in the limit $T\rightarrow 0$ $$\label{xmax}
x_{max}(0)=.548\pm .005,$$ see inset Figure \[fig:xmaxpad\], and slightly larger than the value $1/2$ predicted by the Parisi-Toulouse scaling, in agreement with the approximate nature of this relation [@VTP81; @PT80].
The analysis of the function $q(x,T)$ is more complex, because not only the Taylor expansion of $q(x)$ in powers of $x$ around any $0<x<x_{max}$ is likely to be asymptotic for any fixed temperature, but the expansion in $\tau$ of the coefficients of the $x$-expansion are themselves non convergent. Therefore one should use a double Padè expansion, one for the coefficients and one for $q(x)$. The procedure however is quite difficult because we do not have a systematic way of choosing the best approximant and, moreover, coefficients of higher order are known with less precision in $\tau$. A better approach is to construct the function $q(x)$ directly point by point by computing $q(mx_{\rm max})$ where $m=i/n$, $(i=0,1,\ldots \, n)$ for fixed $n$. For any $m$ and $T$ the quantity $q(mx_{\rm max})$ is itself a power series in $\tau$ which can be summed up using Padè approximants. With this procedure the function $q(x)$ can be determined for different $x$-resolution just changing the value of $n$, e.g., $n=50,100,1000$, and using the value of $x_{\rm max}$ previously found, see Fig. \[fig:xmaxpad\]. In Figure \[fig:qxpad\] the function $q(x)$ is shown for various temperatures $T$.
This method can be extended to any function of $x$ or $q$, for example, we computed the overlap probability function $P(mq_{\rm max})$ in a wide range of temperature $T>0.3$, see Figure \[fig:pdqpad\]. We found that the best Padè approximant is given by the $(12,7)$. By using the relation (\[eq:pdq\]) we can have an independent estimation of $q(x)$ with which to test the precision of our results. By using a norm $d_\infty(q,q') = \max_{0\leq x \leq 1} |q(x) - q'(x)|$ and expansions up to order $20$ we find, for example, that $d_\infty(q,q') = O(10^{-5})$ for $T=0.6$ and $d_\infty(q,q') = O(10^{-4})$ for $T=0.4$.
The form of the function $q(x)$ confirms the prediction of Ref. [@VTP81] obtained from interpolation of the $11$-RSB solution. In particular it confirms the approximate scaling $q(x,T)\sim q(x/T)$ at low temperatures, see Figure \[fig:rqxpad\]. Note that the scaling fails when $\beta x\sim O(1)$, in agreement with the findings of previous section.
Finally we mention that an alternative resummation technique based on the Borel transform give results consistent with those obtained with the Padè approximants.
Numerical Integration of the $\infty$-RSB equations {#SecNum}
===================================================
To check the analytical results of the previous sections we have solved numerically the $\infty$-RSB equations (\[SP1\]) - (\[condP\]) on a discrete set of points in the infinite strip $[0\leq x\leq 1$; $-\infty< y <\infty]$ and determined $q(x)$, $P(x,y)$ and $m(x,y)$. The numerical method is based on the iterative procedure of Ref. [@topomoto]: from an initial guess of $q(x)$ the fields $m(x,y)$, $P(x,y)$ and the associated $q(x)$ are computed in order as:
1. Compute $m(x,y)$ integrating from $x=x_0$ to $x=0$ eqs. (\[SP2\]) with initial condition (\[condm\]).
2. Compute $P(x,y)$ integrating from $x=0$ to $x=x_0$ eqs. (\[SP3\]) with initial condition (\[condP\]).
3. Compute $q(x)$ using eq. (\[SP1\]).
where $x_0\leq 1$ (See later). The steps $1.\,\to\, 2.\,\to\, 3.$ are repeated until a reasonable convergence is reached, typically mean square error on $q$, $P$ and $m$ of the order $O(10^{-6})$.
The core of the numerical scheme is the integration of the partial differential equations (\[SP2\]) and (\[SP3\]) along the $x$ direction which, at difference with previous numerical studies [@topomoto; @B90], is done in the Fourier Space of the $y$ variables where the equations take the form: $$\label{eq:mk}
\frac{\partial}{\partial x}\,m(x,k) =
\frac{k^2\dot{q}(x)}{2}\, m(x,k)
-\frac{\beta \dot{q}(x)}{2}\, \mbox{i}k\, {\cal FT}\left[
m^2\right](x,k)$$ and $$\label{eq:mP}
\frac{\partial}{\partial x}\,P(x,k) =
-\frac{k^2\dot{q}(x)}{2}\, P(x,k)
-\beta \dot{q}(x)\, \mbox{i}k\, {\cal FT}\left[
P\,m\right](x,k)$$ For each wave-vector $k$ these are ordinary differential equations which can be integrated using standard methods. To avoid the time consuming calculation of the convolutions in the non-linear term we use a pseudo-spectral[@Orsz] code on a grid mesh of $N_x\,\times\, N_y$ points, which covers the $x$-interval $[0,x_0]$ and the $y$-interval $[-y_{\rm max},y_{\rm max}]$. The truncation of wave-number may introduce anisotropic effects for large $k$, therefore to ensure a better isotropy of numerical treatment we perform de-aliasing via a $N_y/2$ truncation [@deal]. Finally the $x$ integration has been performed using an third-order Adam-Bashfort scheme which has the advantage of reducing the number of Fast Fourier calls [@AB]. Typical values used are $N_x=100 \div 5000$, $N_y = 512 \div 4096$ and $y_{\rm max} = 12 \div 48$. The difference between the values used for $N_x$ and $N_y$ follows from the observation that if the solution in the $y$-direction is smooth enough, then only few low wave-vectors are exited. The value of the parameter $y_{\rm max}$ fixes the $y$-range where the solution is assumed different form zero, since in the numerical algorithm is implicitly assumed that $$P(x,y)\equiv m(x,y) = 0 \qquad |y|> y_{\rm max}.$$ This explain the rather large value used. The number of iterations necessary to reach a mean square error on $q$, $P$ and $m$ of order $O(10^{-6})$ depends on the initial guess of $q(x)$ but it is typically of few hundreds.
In Figure \[fig:qx-pdq.T0.6-1.000\] are shown the order parameter $q(x)$ and the overlap probability distribution function $P(q)$ at $T=0.6$ computed for increasing $x$-resolution and $x_0=1$. As expected the agreement between the numerical and the perturbative solutions increases with the number of $N_x$ of $x$-grid points. However, the convergence is not uniform: it is rather fast far from $x_{\rm max}$ and much slower for $x\simeq x_{\rm max}$, see the inset of Figure \[fig:qx-pdq.T0.6-1.000\] panel (a). This is not unexpected because for $x=x_{\rm max}$ the derivative of the order parameter $q^{(1)}(x)$ has a cusp: $$\label{eq:cusp}
\lim_{x\to x_{\rm max}^-} q^{(1)}(x) > 0, \qquad
\lim_{x\to x_{\rm max}^+} q^{(1)}(x) = 0$$ making the convergence more difficult. We recall that in deriving equations (\[SP2\]) and (\[SP3\]) differentiability of $q(x)$ was assumed. The use of lower order integration schemes, as second-order Adam-Bashfort or Euler schemes, does not give sensible improvements.
Larger values of $N_x$ requires larger needs of computer memory therefore to increase the precision we adopted a different approach. Since $\dot{q}(x)=0$ for $x>x_{\rm max}$ equations (\[SP2\]) and (\[SP3\]) are trivial in this range and we can reduce the upper bound of the $x$-integration from $x=1$ to $x=x_0=x_{\rm max}$. This obviously requires the knowledge of $x_{\rm max}$ for the given temperature. However if we assume [*no a priori*]{} knowledge of $x_{\rm max}$ we must proceeds for successive approximations: we start from $x_0=1$ an then reducing it until we ‘hit’ the value of $x_{\rm max}$. This procedure is simplified by the fact that if $x_0<x_{\rm max}$ the shape of $q(x)$ near $x_0$ changes dramatically with the concavity passing from negative values for $x_0>x_{\rm max}$ to positive values for $x_0<x_{\rm max}$. In Figure \[fig:qx-pdq.T0.6-x0\] panel (a) are shown $q(x)$ (panel a) and $P(q)$ (panel b) at $T=0.6$ for different values of $x_0$, the improvement in rather evident. As additional check we have considered the equality $$1 - \int_{0}^{1} dx\, q(x) = T$$ which is satisfied by our numerical solution for all studied temperatures with at least four digits. For example we for $T=0.8$ we get $0.79999(4)$, while for $T=0.5$ the value is $0.49999(3)$.
Note that not only by fine tuning of $x_0$ we can have a good solution for $q(x)$ at the given temperature, but we also have [*the value*]{} of $x_0$. This is best seen by analyzing the concavity of $q(x)$ near $x_0$. In Figure \[fig:q2x.T0.4-x0-500\] we show the second derivative of $q(x)$ near $x_0$ for $T=0.4$ and $N_x=500$, from which one may conclude that $0.505 < x_{\rm max} < 0.510$.
A careful analysis of the stability of this results as function of $N_x$, see Figure \[fig:q2x.T0.4-x0.510-0.515\], reveals, however, that the correct estimation is $0.510 < x_{\rm max} < 0.515$, in rather good agreement with the analytical result $x_{\rm max} = 0.5111\pm 0.0002$. The same analysis for $T=0.6$ leads to $0.438 < x_{\rm max} < 0.440$.
We are now in the position of checking the results of previous section about the derivative of the order parameter at $x=0$, and in particular the conclusion $$\label{eq:q30}
\lim_{x\to 0} q^{(3)}(x) > 0.$$ In Figure \[fig:q2-3x.T0.6-1.000\] we show the second and third derivative of $q(x)$ obtained from numerical differentiation of $q(x)$. The agreement with the perturbative result is sufficiently good, moreover from the right panel of Figure \[fig:q2-3x.T0.6-1.000\] we clearly see that the prediction (\[eq:q30\]) is verified.
We conclude this Section with a short discussion on the entropy which, using the stationarity of the free energy functional (\[eqfrev\]), can be written as: $$s = -\frac{\beta^2}{4}\bigl[1-q(1)\bigr]^2
+ \int_{-\infty}^{\infty} dy \ P(1,y)
\bigl[\log 2 \cosh \beta y - y \ \tanh(\beta y)\bigr].
\label{s}$$ For other equivalent forms see, e.g., Ref. [@Luca]. The entropy as function of temperature is shown in the left panel of Figure \[fig:s-q1\]. The entropy must vanish quadratically with the temperature as $T\to 0$ [@SDJPC84]. From our numerical data we find $$\lim_{T\to 0} \frac{s(T)}{T^2} = a \simeq 0.72$$ to be compared with $0.718\pm 0.004$ of the analytic expansions.
In the limit $T\to 0$ the quantity $1-q(1)$ must also vanish as $T^2$ [@SDJPC84]. The behaviour of $q(1)$ as function of $T$ is shown in the right panel of Figure \[fig:s-q1\]. Using this data we obtain $$\lim_{T\to 0} \frac{1 - q(1)}{T^2} \simeq 1.60$$ in very good agreement the value $1.60\pm 0.01$ obtained with the expansions of previous sections.
Conclusions {#SecCon}
===========
In this paper we have studied the properties of the $\infty$-replica symmetry breaking solution of the Sherrington - Kirkpatrick model without external fields. Using high order expansions in $\tau = T_c - T$ we are able to compute the order parameter $q(x)$ and other relevant quantities for a large range of temperatures with high precision. In particular we found that $q(x)$ [*is not*]{} an odd function of $x$, confirming the prediction of Ref. [@Temes]. Direct consequence of this is that the overlap probability distribution function $P(q)$ has discontinuous derivatives at $q=0$. Another consequence of our findings is that the Parisi-Toulouse scaling becomes exact asymptotically for $T\to 0$ [*and*]{} $\beta x\to 0$, while for $T\to 0$ is a fairly good approximation. This is also consistent with the $T=0$ limit of the breaking point which we found to be $x_{max}(0)=.548\pm .005$.
Having reached very high orders we can reasonably speculate on the analytical properties of the function $ q(x) $. In particular we believe that
- All the expansions in power of $ \tau $ are asymptotic expansions.
- At any temperature, the function $ q(x) $ is infinitely differentiable but not analytical for any $ x $, in particular the Taylor expansion of the function $ q(x) $ around any $ 0<x<x_{max} $ does not converge but is asymptotic.
This singular behaviour is not connected neither with the replica limit nor with the Parisi Ansatz, actually it originates from the singularities in the complex plain of the initial condition of the Parisi equation: $ f(1,y)=\ln 2\cosh \beta y $. This is clearly seen for the replica-symmetric solution$$q=\int ^{+\infty }_{-\infty }\frac{dz}{\sqrt{2\pi }}e^{\frac{z^{2}}{2}}\tanh ^{2}(\beta \sqrt{q}z)$$ In this case it is easy to prove that the expansion of $ (1-T^{2}) $ in powers of $ p=\beta ^{2}q $ is asymptotic because it corresponds to substitute $ \tanh ^{2}z $ in the integrand with its Taylor expansion which is not convergent on the whole real axes. Then one can prove that the expansion of $ q $ in powers of $ \tau =1-T $ is asymptotic recalling that standard manipulation (e.g. multiplication, division, inversion...) on an asymptotic expansion in power series do not change its character. A detailed treatment of the RSB solution is much more complex, but the origin of the asymptotic character is likely to be the same. Indeed an expansion in small $ \tau $ (and therefore in small $ q $) corresponds to an expansion in small $ y $ of all the quantities like $ f(x,y) $ and $ m(x,y) $; the appearance of integrals of the form $ \int Pfdy $where $ P(x,y)\sim \exp (y^{2}/x) $ generates asymptotic expansions since the Taylor expansions of $ f(x,y) $ and $ m(x,y) $ in powers of $ y $ do not converge on the whole real axes. These arguments can be very useful in practice to guess the position of the singularities of the Borel transform if one want to sum the expansions through a conformal mapping [@Pbook]. For instance in the expression of the free energy appear integrals of the following form:$$\label{as2}
\int ^{+\infty }_{-\infty }\frac{dz}{\sqrt{2\pi \tau }}e^{-\frac{z^{2}}{2\tau }}\ln \cosh (z)$$ The singularities of the Borel transform of the previous integral are located on a cut running from $ -\infty $ to $ -\pi ^{2}/8 $ and a possible guess is that this be also the singularity structure of the Borel transform of the free energy. This guess is supported by the analysis of the series expansions.
The analytical results have been compared with numerical solutions of the $\infty$-replica symmetry breaking equations. We have developed a new numerical approach based on a pseudo-spectral code which leads to a strong enhancement of the quality of the numerical results. We have also shown how, for example, to determine the value of $x_{\rm max}$ numerically. In all cases the agreement between the numerical and the analytical results is rather good.
We conclude by stressing that our results go beyond the interest on the Sherrington - Kirkpatrick model, since the method we used here are far more general and can be employed to a wider class of models with generalized $\infty$-replica symmetry breaking equations such as those introduced in Ref. [@Luca]. In particular in this reference the numerical method was applied to the 3-SAT model, and the extension to other relevant models is under development.
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---
abstract: 'We develop a formalism to describe the formation of bound states in quantum field theory using an exact renormalization group flow equation. As a concrete example we investigate a nonrelativistic field theory with instantaneous interaction where the flow equations can be solved exactly. However, the formalism is more general and can be applied to relativistic field theories, as well. We also discuss expansion schemes that can be used to find approximate solutions of the flow equations including the essential momentum dependence.'
author:
- 'S. Floerchinger'
title: Exact flow equation for bound states
---
Introduction
============
The formation of bound states was one of the first problems discussed in quantum mechanics. Although the quantum mechanical formalism for nonrelativistic particles with instantaneous interaction is simple and easy to apply, it is much more difficult to treat the problem in quantum field theory as needed for example for relativistic particles. The Bethe-Salpeter [@BetheSalpeter] equation can be used to sum Ladder diagrams but it is difficult to go beyond [@Weinberg]. For that reason it is sensible to look for alternative approaches.
In this paper we show how a recently derived exact renormalization group flow equation [@FW09] can be used to treat the formation of bound states. As a simple example and to develop the formalism, we consider well known problem of nonrelativistic particles with instantaneous interaction. In this case we can solve the flow equations exactly and we show that our formalism is equivalent to the standard quantum mechanical description although it is based on a field theoretic approach.
In principle, since the flow equation is exact, it can describe also more complicated situations for example with relativistic particles, retarded interactions or at nonzero density. However, one cannot expect to find exact solutions to the flow equations, there. The merit of our formalism is to provide a convenient starting point for approximations. These do not have to rely on the existence of a small coupling constant as in usual perturbation theory and have the potential to capture nonperturbative effects. We discuss a generic scheme for the construction of such approximations. The resulting formalism is quite general and can be seen as a method to resolve the momentum dependence of vertices in renormalization group flow equations.
We mention that flow equations for bound states have been investigated previously. A setup based on Wegner’s flow equation for Hamiltonians [@Wegner] and the similarity renormalization group [@SimilarityRG] has been employed to investigate for example a two-dimensional particle with contact potential [@GlazekWilson]. Note that the flow equation of Wegner and the similarity renormalization group break space-time symmetry explicitly. This is in contrast to Wetterich’s formulation of renormalization in terms of the effective average action or flowing action $\Gamma_k$ [@CWFloweq]. Ellwanger demonstrated that this formulation can be used to investigate bound state formation for relativistic theories [@Ellwanger]. By tracing the flow of the four-point vertex, Ellwanger calculates for example binding energies for the Wick-Cutkosky model and finds good agreement with known results in various limits. As a drawback of his approach one might see that properties of the bound states are somewhat hidden in the analytic structure of the four-point vertex. This makes is it hard to refine approximations or to investigate interactions between bound states or spontaneous symmetry breaking.
These difficulties can be overcome by introducing auxiliary fields for composite operators as will be discussed in more detail below. It is important to adapt the composite fields while solving the flow equation. The reason is that one wants to absorb the essential parts of the four-point vertex by means of an Hubbard-Stratonovich transformation at each step in the flow. A first proposal, how this can be realized with help of a scale-dependent nonlinear field transformation was formulated in ref. [@GiesWetterich] and used to study bound state formation in the NJL model. The analysis in the present paper is based on a recently derived, simple, but nevertheless exact flow equation [@FW09]. Although the spirit is similar to the formalism proposed in ref. [@GiesWetterich], there are some differences in the implementation.
This paper is organized as follows. In chapter \[sec:nonrelativisticmodel\] we introduce our notation and the microscopic model for nonrelativistic particles with an instantaneous interaction. In the subsequent section \[sec:Partialbosonization\] we show how this model can be treated using partial bosonization and an exact flow equation where the Hubbard-Stratonovich transformation is kept fixed, i.e. independent of the renormalization scale $k$. In this approach we can integrate the flow equation exactly. However, this solution relies on some particular features of the nonrelativistic theory and is difficult to generalize to more complicated cases. Therefore we discuss an alternative approach to the problem in section \[sec:scaledeppartialbos\]. Here we work with a scale-dependent Hubbard-Stratonovich transformation. Again, it is possible to integrate the resulting flow equations exactly in the nonrelativistic case. We argue at the end of the section that this formalism is suited much better for the generalization away from the nonrelativistic and instantaneous approximation. In section \[sec:approximationschemes\] we suggest approximation schemes and apply our formalism to a nonrelativistic Yukawa potential. Finally, section \[sec:generalformalismandapproxschemes\] gives a discussion of a somewhat generalized formalism that can be applied for example in relativistic quantum field theory or at nonzero density and we draw some conclusions in section \[sec:conclusions\].
Microscopic model for nonrelativistic particles {#sec:nonrelativisticmodel}
===============================================
Let us consider the problem of two nonrelativistic particles interacting via a potential $V(\vec x-\vec y)$. We will use a quantum field theoretic description using the functional integral. The microscopic action we employ is $$\begin{aligned}
\nonumber
S_\psi&=&\int_x \psi_1^*(x)\left(i\partial_t+\frac{1}{2M_1}\vec \nabla^2\right) \psi_1(x) \\
\nonumber
&&+ \int_x \psi_2^*(x)\left(i\partial_t+\frac{1}{2M_2}\vec \nabla^2\right) \psi_2(x) \\
\nonumber
&&- \int_{x_1,x_2} \psi_2^*(x_2)\psi_1^*(x_1) \; V(\vec x_1-\vec x_2) \\
&&\times\; \delta((x_1)_0-(x_2)_0)\; \psi_1(x_1)\psi_2(x_2).
\label{eq:microscopicaction2}\end{aligned}$$ We use here $\int_x=\int_{x_0}\int_{\vec x}=\int dx_0 \int d^3x$. For the Coulomb problem the interaction is of the form $$V(\vec x_1-\vec x_2) = \frac{-e^2}{4\pi |\vec x_1-\vec x_2|}.
\label{eq:Coulombpot}$$ It is useful to change variables of integration according to $x=\eta_1 x_1+\eta_2 x_2$, $y=x_1-x_2$ with $\eta_1=M_1/(M_1+M_2)$, $\eta_2=M_2/(M_1+M_2)$. The interaction term becomes then $$\begin{aligned}
\nonumber
&&-\int_{x,y} \psi_2^*(x-\eta_1 y)\psi_1^*(x+\eta_2 y) V(\vec y)\\
&& \delta((x_1)_0-(x_2)_0) \psi_1(x+\eta_2 y)\psi_2(x-\eta_1 y).\end{aligned}$$ It is also useful to introduce the Fourier transformed fields $$\begin{aligned}
\nonumber
\psi_i(x) &=& \int_p \psi_i(p)\, e^{-ipx}, \quad i=1,2,\quad p x=p_0 x_0-\vec p\vec x,\end{aligned}$$ with $$\int_p = \int_{p_0}\int_{\vec p}, \quad \int_{p_0}=\int_{-\infty}^\infty \frac{d p_0}{2\pi}, \quad \int_{\vec p}=\int \frac{d^3p}{(2\pi)^3}.$$ We also introduce an abbreviation for a composition of fields $(\psi_1\psi_2)$. Its position and momentum space representation reads $$\begin{aligned}
\nonumber
(\psi_1\psi_2)(x,\vec y) &=& \psi_1(x_0,\vec x+\eta_2 \vec y) \psi_2(x_0, \vec x-\eta_1 \vec y)\\
&=& \int_{p,q} e^{-ipx} e^{i\vec q\vec y} (\psi_1\psi_2)(p,\vec q),\end{aligned}$$ with $(\psi_1\psi_2)(p,\vec q)=\int_{q_0}\psi_1(\eta_1 p+q)\psi_2(\eta_2p-q)$. The conjugate field is $$(\psi_1\psi_2)^*(p,\vec q) = \int_{q_0} \psi_2^*(\eta_2 p-q) \psi^*(\eta_1 p+q),
\label{eq:}$$ where the reversed order of $\psi_1$ and $\psi_2$ is convenient when these are fermionic fields. Obviously, the first variable ($x=(x_0,\vec x)$ or $p=(p_0,\vec p)$) describes the center of mass motion, while the second variable ($\vec y$ or $\vec q$) describes the relative motion of the two particles. The fact that no relative time or energy appears is due to the instantaneous approximation for the interaction. For relativistic systems this point may be different and the generalization of the formalism to that case will be discussed in section \[sec:generalformalismandapproxschemes\].
Using the momentum representation, the interaction term in the microscopic action becomes $$\begin{aligned}
\nonumber
&&-\int_{x,\vec y} (\psi_1\psi_2)^*(x,\vec y)\, V(\vec y)\, (\psi_1\psi_2)(x,\vec y)\\
&=& -\int_{p,\vec q,\vec q^\prime} (\psi_1\psi_2)^*(p,\vec q)\, V(\vec q-\vec q^\prime)\, (\psi_1\psi_2)(p,\vec q^\prime).\end{aligned}$$ In the last line we used $$V(\vec q-\vec q^\prime) = \int_{\vec y} e^{i(\vec q-\vec q^\prime)\vec y} \,V(\vec y).$$
In this work we investigate the microscopic action in Eq. using the functional integral formalism. The partition function as a functional of the source field $j$ is given by $$Z[j] = \int D \psi \; e^{iS_\psi[\psi]-i\int_p\sum_{i=1,2} \{j_i^*(p) \psi_i(p)+c.c.\}}.
\label{eq:partitionfunctnoRk}$$ In order to derive an exact flow equation we modify this expression by introducing an infrared cutoff term $$Z_k[j] = e^{-i W_k[j]} = \int D \psi \, e^{i S_\psi[\psi]+i\Delta S_k[\psi]-i\int \{j^*\psi+c.c.\}},$$ where we choose $$\begin{aligned}
\nonumber
\Delta S_k & = \int_p & \psi_1^*(p)\left[-\frac{1}{2M_1}R_k(\vec p^2)\right]\psi_1(p) \\
&& + \psi_2^*(p)\left[-\frac{1}{2M_2}R_k(\vec p^2)\right]\psi_2(p),
\label{eq:cutoffaction}\end{aligned}$$ with $R_k\to\infty$ for $k\to0$ and $R_k\to0$ for $k\to0$. The flowing action is now defined as the modified Legendre transform $$\Gamma_k[\psi] = \int_p \sum_{i=1,2}\{j_i^*(p)\psi_i(p)+c.c.\} -W_k[j]-\Delta S_k[\psi],$$ where $\psi(p)=\frac{\delta W_k}{\delta j^*(p)}$ is now the expectation value. The flowing action satisfies the exact flow equation [@CWFloweq] $$\partial_k \Gamma_k[\psi] = \frac{1}{2} \text{STr} (\Gamma^{(2)}[\psi]+{\cal R}_k)^{-1} \partial_k {\cal R}_k.
\label{eq:exactfloweqCW}$$ The operation $STr$ is a trace over both continuous and discrete degrees of freedom such as momentum or spin. It also includes a minus sign for fermions. In addition we use in Eq. the matrix of second functional derivatives $$\begin{aligned}
&&\Gamma^{(2)}_k(p,q)=\\
\nonumber
&&\begin{pmatrix} \overset{\rightharpoonup}{\delta}_{\psi_1^*(p)} \\ \overset{\rightharpoonup}{\delta}_{\psi_2^*(p)}\\
\overset{\rightharpoonup}{\delta}_{\psi_1(-p)}\\
\overset{\rightharpoonup}{\delta}_{\psi_2(-p)} \end{pmatrix} \Gamma_k[\psi] \begin{pmatrix}
\overset{\leftharpoonup}{\delta}_{\psi_1(q)} && \overset{\leftharpoonup}{\delta}_{\psi_2(q)} && \overset{\leftharpoonup}{\delta}_{\psi_1^*(-q)} &&
\overset{\leftharpoonup}{\delta}_{\psi_2^*(-q)} \end{pmatrix}\end{aligned}$$ and the cutoff matrix $${\cal R}_k(p,q) = \begin{pmatrix} \frac{-R_k(\vec p^2)}{2M_1} && 0 && 0 && 0 \\ 0 && \frac{-R_k(\vec p^2)}{2M_2} && 0 && 0 \\ 0 && 0 && \frac{R_k(\vec p^2)}{2M_1} && 0 \\ 0 && 0 && 0 && \frac{R_k(\vec p^2)}{2M_2} \end{pmatrix} \delta(p-q),
\label{eq:secondfunctder}$$ with $$\delta(p) = (2\pi)^4 \delta(p_0) \delta^{(3)}(\vec p).
\label{eq:}$$
For large values of $k$ the flowing action approaches the microscopic action $$\lim_{k\to\infty} \Gamma_k[\psi] = S[\psi],$$ while for $k\to0$ the flowing action equals the quantum effective action $\Gamma$ which is the generating functional of one-particle irreducible Feynman diagrams $$\lim_{k\to0}\Gamma_k[\psi]=\Gamma[\psi].$$ We note that the flow equation can be written as $$\partial_k \Gamma_k[\psi] = \tilde \partial_k \frac{1}{2} \text{STr} \ln (\Gamma^{(2)}[\psi]+{\cal R}_k)$$ with the formal derivative $\tilde \partial_k$ that applies to the cutoff term ${\cal R}_k$, only. For reviews of the flow equation method see [@ReviewRG; @Pawlowski; @Metzner; @SalmhoferHonerkamp].
Fixed Hubbard-Stratonovich transformation {#sec:Partialbosonization}
=========================================
In this section we use a Hubbard-Stratonovich transformation [@HS] to calculate the effective bound state propagator. We modify the functional integral over the field $\psi$ in Eq. by including another integral over the (composite) field $\Phi$ $$\begin{aligned}
\nonumber
Z[j,J] &=& \int D\psi \; e^{i S_\psi[\psi]+i\int_p \sum_i \{j_i^*(p) \psi_i(p)+c.c.\}}\\
& \times & \int D \Phi \; e^{i S_\text{pb}[\psi,\Phi]+i \int_{p,\vec q}\{J^*(p,\vec q)\Phi(p,\vec q)+c.c.\}}\end{aligned}$$ with $$\begin{aligned}
\nonumber
S_\text{pb}[\psi,\Phi] &=& \int_{p,\vec q,\vec q^\prime} \left[\Phi^*(p,\vec q) -(\psi_1\psi_2)^*(p,\vec q)\right]\\
&& V(\vec q-\vec q^\prime) \left[\Phi(p,\vec q^{\prime})-(\psi_1\psi_2)(p,\vec q^\prime)\right].
\label{eq:spb1}\end{aligned}$$ For $J=0$ the functional integral over $\Phi$ contributes only a multiplicative constant to $Z$ which is irrelevant for most purposes. In the combined action $S=S_\psi+S_\text{pb}$ the term quadratic in $\psi$ cancels and we arrive at $$\begin{aligned}
\nonumber
S &=& \int_p \psi_1^*(p_0-\frac{1}{2M_1}\vec p^2) \psi_1 + \psi_2^*(p_0-\frac{1}{2M_2}\vec p^2)\psi_2\\
\nonumber
&&- \int_{p,\vec q, \vec q^\prime} \left\{\Phi^*(p,\vec q)V(\vec q-\vec q^\prime)(\psi_1\psi_2)(p,\vec q)+c.c.\right\}\\
&&+ \int_{p,\vec q,\vec q^\prime} \Phi^*(p,\vec q) V(\vec q-\vec q^\prime) \Phi(p,\vec q^\prime).
\label{eq:HStransformedaction}\end{aligned}$$ Note that there is a certain ambiguity in the precise from of the Hubbard-Stratonovich transformation. Linear transformations of the field $\Phi$ are always possible. For the Hubbard-Stratonovich transformed action in Eq. it is now possible to solve some parts of the functional integral.
Let us consider the functional renormalization group equations for this theory when an infrared cutoff of the form in Eq. is added for the field $\psi$. For the truncation of the effective action we use a vertex expansion. Due to the nonrelativistic dispersion relation and the instantaneous interaction, the propagator of the field $\psi$ and the Yukawa-like coupling $\sim \Phi^* \psi_1\psi_2+c.c.$ are independent of the scale $k$. Also, no additional interaction vertex involving the field $\psi$ is generated by the renormalization group flow. Indeed, all one-loop Feynman diagrams that contribute to the flow equations of the above quantities consist of a closed tour of particles and vanish in the nonrelativistic few-body limit. This feature is a manifestation of the fact that the nonrelativistic few-body problems decouple in the sence that for example the three-body problem can be solved independent of the four-body problem. More general, the flow equations for correlation functions that determine the $n$-body problem are independent of the correlation functions determining the $n+1$-body problem. For a more detailed discussion of these matters we refer to ref. [@SFdiss]. As another consequence of the decoupling feature, the flow equation for the propagator of the field $\Phi$ depends only on the propagator of the field $\psi$ and the Yukawa-like vertex but is independent of the higher vertex functions. This implies that the flow equation for the propagator of $\Phi$ can be solved exactly. More concrete, we write the term of $\Gamma_k$ that is quadratic in $\Phi$ as $$\Gamma_k^{(\Phi,2)} = \int_{p,\vec q,\vec q^\prime} \Phi^*(p,\vec q) P_\Phi(p,\vec q,\vec q^\prime) \Phi(p,\vec q^\prime)$$ with a $k$-dependent function $P_\Phi(p,\vec q,\vec q^\prime)$ denoting the inverse propagator. Its flow equation reads $$\begin{split}
& \partial_k P_\Phi(p,\vec q^\prime,\vec q^{\prime\prime}) = \tilde \partial_k (-i) \int_{q} V(\vec q^{\prime},\vec q) V(\vec q, \vec q^{\prime\prime}) \\
& \times \frac{1}{(\eta_1 p+q)_0-\frac{1}{2M_1}[(\eta_1 \vec p+\vec q)^2+R_k((\eta_1 \vec p+\vec q)^2)]+i\epsilon}\\
& \times \frac{1}{(\eta_2 p-q)_0-\frac{1}{2M_2}[(\eta_1 \vec p-\vec q)^2+R_k((\eta_1 \vec p-\vec q)^2)]+i\epsilon}.
\end{split}$$ We can write this in a symbolic notation as $$\partial_k P_\Phi(p) = - \tilde \partial_k (V A_k^{-1}(p) V)$$ where the “matrix indices” $\vec q$ etc. and the corresponding integrations are implicit. We use the abbreviation $$\begin{split}
& A_k^{-1}(p,\vec q,\vec q^\prime) = i \int_{q_0} \delta^{(3)}(\vec q-\vec q^\prime) \\
& \times \frac{1}{(\eta_1 p+q)_0-\frac{1}{2M_1}[(\eta_1 \vec p+\vec q)^2+R_k((\eta_1 \vec p+\vec q)^2)]+i\epsilon}\\
& \times \frac{1}{(\eta_2 p-q)_0-\frac{1}{2M_2}[(\eta_1 \vec p-\vec q)^2+R_k((\eta_1 \vec p-\vec q)^2)]+i\epsilon}\\
& = \delta^{(3)}(\vec q-\vec q^\prime) {\bigg ( }p_0-\frac{1}{2M_1}[(\eta_1 \vec p+\vec q)^2+R_k((\eta_1\vec p+\vec q)^2)]\\
& \quad \quad -\frac{1}{2M_2}[(\eta_1 \vec p-\vec q)^2+R_k((\eta_1\vec p-\vec q)^2)]+i\epsilon {\bigg )}^{-1}.
\end{split}
\label{eq:akinverse}$$ Since the only $k$-dependence in this expression comes from $R_k$, the flow equation for $P_\Phi$ can be integrated directly. For $\Lambda\to\infty$ we obtain $$\begin{aligned}
P_{\Phi,k}(p)-P_{\Phi,\Lambda}(p) &=&-V A_k^{-1} V.\end{aligned}$$ For $P_{\Phi,\Lambda}$ we can use $$P_{\Phi,\Lambda}(p) = V.$$ The propagator $$G_k(p) = P_{\Phi,k}^{-1}(p) = \left(-V A_k^{-1} V + V\right)^{-1}
\label{eq:propagatorformal}$$ describes correlations of two particles with relative momentum $\vec q^\prime$ to two particles with relative momentum $\vec q$. We use here a matrix notation where $\vec q$ and $\vec q^\prime$ are indices and $P_{\Phi,k}(p,\vec q,\vec q^\prime)$ is inverted as a matrix for fixed value of the center of mass momentum $p$ which is conserved.
To understand the physical meaning of $G_k(p)$ we consider field configurations $\Phi$ for which $G_k$ has a pole, or for which $$\int_{\vec q^\prime} P_{\Phi,k}(p,\vec q, \vec q^\prime) \Phi(\vec q^\prime) = 0$$ holds. In the center of mass frame where $p=(p_0,0,0,0)$ and using the reduced mass $\mu=M_1 M_2/(M_1+M_2)$, this condition can be written as $$\int_{\vec q^\prime} V(\vec q,\vec q^\prime) \Phi(\vec q^\prime) = \left[p_0-\frac{1}{2\mu}(\vec q^2+R_k(\vec q^2))\right] \Phi(\vec q).$$ For $R_k=0$ this is, of course, just the Schrödinger equation for the two-body problem. This becomes clear in position space where one obtains $$\left[p_0-\frac{1}{2\mu}(-\vec \nabla^2+R_k(-\vec \nabla^2))-V(\vec x)\right] \Phi(\vec x)=0.$$
In principle, one could obtain the bound state propagator $G_k$ for arbitrary values of the center of mass momentum $p$ by inverting the expression for $P_{\Phi,k}$ according to Eq. . To do that one would first diagonalize the matrix $P_{\Phi,k}$ by finding its eigenvalues $\lambda_i(p)$ and eigenfunctions $\varphi_i(p,\vec q)$. This is a problem of similar complexity as solving the Schrödinger equation for the two-particle problem. It is clear that for $p=(E,0,0,0)$ and $k=0$ the spectrum of eigenvalues of $P_\Phi$ contains as many vanishing eigenvalues as bound states exist with energy $E$.
Note, that the on-shell information obtained from solving Schrödingers equation (i.e. the binding energies and corresponding eigenfunctions) is not sufficient to determine the bound state propagator $G_k$ uniquely. In particular one can add to $G_k$ a term that is regular as a function of the center of mass momentum $p$ without changing the poles and therefore the on-shell information. It is especially useful to consider the combination $$\tilde G_k(p) = G_k(p)-G_\Lambda(p) = G_k(p)-V^{-1}.$$ This is just the part of $G_k$ that is generated from the flow equation with the “classical” part subtracted. Using Eq. this can be written as $$\tilde G_k(p) = (A(p)-V)^{-1}.$$ In this representation it is particularly clear that the poles of $\tilde G_k$ correspond to the solutions of Schrödingers equation for two particles.
Moreover, it is clear how to diagonalize this propagator, at least for $R_k=0$. To that end we note that the inverse propagator has the position space representation $$\tilde G_{k=0}^{-1}(p) = p_0-\frac{1}{2(M_1+M_2)}\vec p^2-\frac{1}{2\mu}(-\vec \nabla_y^2)-V(\vec y)$$ which is diagonalized, of course, by the solutions of the stationary Schrödinger equation $$\tilde G_{k=0}^{-1}(p) g_n(\vec y)=(p_0-\frac{1}{2(M_1+M_2)}\vec p^2-E_n) g_n(\vec y).$$ Here, $n$ is a combined index that labels all quantum numbers. For a spherical symmetric potential this includes the radial quantum number as well as those for angular momentum. In the basis $$\Phi(p,\vec y) = \sum_n \phi_n(p) g_n(\vec y)$$ we can write $$(\tilde G_{k=0}(p))_{n m} = (p_0-\frac{1}{2(M_1+M_2)}\vec p^2-E_n) \delta_{n m}.$$ For completeness we note the explicit form of the Yukawa coupling term in the effective action in this basis $$\Gamma_k^{\Phi\psi\psi} = \sum_n \int_{p,\vec y} \left\{\phi^*_n(p) g^*_n(\vec y) V(\vec y) (\psi_1\psi_2)(p,\vec y)+c.c.\right\}.$$ For a constant cutoff function $R_k=k^2$ the basis of functions $g_n$ diagonalizes also $\tilde G_k$ for nonzero $k$. However, the situation becomes more complicated for other choices of $R_k$ where the basis would depend on the center of mass momentum $p$ and the flow parameter $k$.
At this point we could in principle undo the Hubbard-Stratonovich transformation and thus go back to our original formulation of the theory in terms of the fields $\psi$. This can be done by solving the field equation $$\frac{\delta \Gamma_k}{\delta \Phi}=0$$ for the field $\Phi$ as a functional of $\psi$. Plugging this solution $\Phi[\psi]$ into the action leads to $$\Gamma_k^{(\psi)} = \Gamma_k[\psi,\Phi[\psi]].$$ In general, the dependence of $\Gamma_k$ on $\Phi$ is complicated and undoing the Hubbard-Stratonovich transformation leads to complicated interaction vertices for the field $\psi$. However, it is quite easy to calculate the effective vertex with two incoming and two outgoing $\psi$-particles. We note that the only contribution to this interaction is given by a tree level diagram involving the propagator $G_k$. The corresponding term in $\Gamma_{k=0}^{(\psi)}$ is $$\begin{split}
& \Gamma_{k=0}^{(\psi,4)} = -\int_{p,\vec y_1,\vec y_2} (\psi_1\psi_2)(p,\vec y_1)^* g_n(\vec y_1) V(\vec y_1) \\
& \times \frac{1}{p_0-\frac{1}{2(M_1+M_2)}\vec p^2-E_n} V(\vec y_2) g^*_n(\vec y_2) (\psi_1\psi_2)(p,\vec y_2)\\
& - \int_{p,\vec y} (\psi_1\psi_2)^*(p,\vec y) V(\vec y) (\psi_1\psi_2)(p,\vec y).
\end{split}
\label{eq:efffourfermioninteraction}$$ This expression has a simple interpretation. The first term is the contribution from bound state exchange processes, while the second term is just the classical term. For an electromagnetic Coulomb interaction, the second term can be seen as a contribution from photon exchange processes.
Let us summarize what we have done in this section. Starting from the microscopic model in Eq. we performed a Hubbard-Stratonovich transformation and introduced the bilocal field $\Phi$. In this “partially bosonized” language it was possible to find a solution of the flow equation for the propagator of the field $\Phi$ in a closed form. Note however, that this solution relies on some features particular to the nonrelativistic few-body problem. First, the description of the interaction between particles in terms of a instantaneous interaction potential $V(\vec x_1-\vec x_2)$ is usually not possible for relativistic problems. In relativistic quantum field theory, interactions are mediated by exchange particles such as the photon. In momentum space the resulting interaction term has nontrivial frequency- and momentum dependence and is subject to renormalization group modifications for example due to a “running” fine-structure constant $\alpha$ or charge $e$. These features make it difficult to absorb the interaction term by an appropriate Hubbard-Stratonovich transformation. Even if this was possible, solving the flow equations would be more difficult than in the nonrelativistic case presented above. Both the propagator for the field $\psi$ and the Yukawa interaction $\sim \Phi^* \psi_1\psi_2+c.c.$ might have non-vanishing flow equations. These equations as well as the flow equations for the propagator of $\Phi$ would depend on higher-order vertex functions such that no solution in a closed form can be expected.
Although one cannot expect to find analytic solutions by transferring the above calculation to a relativistic field theory, one might ask whether it can be helpful for finding approximate solutions. One could make a truncation of the flowing action $\Gamma_k$ for example in terms of a derivative expansion and consider the flow equations in the theory-subspace spanned by the finite number of operators included in this truncation. Indeed, we will argue below that such truncations can lead to good approximate solutions. The formulation in the present section has one drawback for approximate solutions, however. As already discussed above, interactions in relatvistic field theories are usually mediated by exchange fields such as the photon. The renormalization of the corresponding couplings and propagators can be calculated most efficient in a formulation of the theory which directly takes the exchange fields as propagating fields into account. Other formulations where these fields (as for example the photon) are “integrated out” might be equivalent in principle, but are usually much harder to treat by approximate methods. The reason is that the essential momentum- and frequency dependence is often hidden in such formulations.
Transfered to the example of a nonrelativistic theory it would be useful to have a formulation where the two terms in Eq. are treated in different ways. While the term in the first line (the contribution from bound state exchange processes) is treated most efficient in terms of the Hubbard-Stratonovich field $\Phi$, this is different for the second term. For a relativistic field theory this corresponds to the contribution from photon or other exchange processes and it is therefore most efficient to write it in terms of this exchange field. In other words, for the nonrelativistic theory, the Hubbard-Stratonovich transformation should be constructed such that the effective action $\Gamma_{k=0}$ contains as an explicit contribution only the classical term (the second term in Eq. ). All additional terms should be described by bound state exchange processes. For the flowing action one can use the same prescription. To that end it is necessary to work with a scale-dependent Hubbard-Stratonovich transformation, however. An exact flow equation that can be used for this purpose was derived in [@FW09] and will be discussed in the next section.
Scale dependent Hubbard-Stratonovich transformation {#sec:scaledeppartialbos}
===================================================
In this section we investigate the bound state problem using a $k$-dependent version of the Hubbard Stratonovich transformation. Instead of $S_\text{pb}$ in Eq. we employ $$\begin{split}
& S_\text{pb}=\int_p\left[\Phi^*(p)-(\psi_1\psi_2)^*(p) V Q_\Lambda^{-1}(p)\right]\; Q(p)\\
& \left[\Phi(p)-Q_\Lambda^{-1}(p) V (\psi_1\psi_2)(p)\right].
\end{split}
\label{eq:Spb2}$$ Again we use a matrix notation where the summation over the index $\vec q$ etc. is left implicit. In Eq. the matrix $Q(p)$ is $k$-dependent, while $Q_\Lambda(p)$ equals $Q(p)$ for $k=\Lambda$ but is independent of $k$. We choose $$\begin{split}
& \lim_{\Lambda\to\infty} Q_\Lambda(p,\vec q,\vec q^\prime) = \infty \;\delta^{(3)}(\vec q-\vec q^\prime),\\
& \lim_{\Lambda\to\infty} Q_\Lambda^{-1}(p,\vec q,\vec q^\prime)=0.
\end{split}$$ In this limit the combined action $S=S_\psi+S_\text{pb}$ becomes $$\begin{aligned}
\nonumber
S &=& \int_p \, \psi_1^*\left(p_0-\frac{1}{2M_1}\vec p^2\right)\psi_1 + \psi_2^*\left(p_0-\frac{1}{2M_2}\vec p^2\right)\psi_2\\
\nonumber
&& - \int_{p,\vec q,\vec q^\prime} (\psi_1\psi_2)^*(p,\vec q) V(\vec q-\vec q^\prime) (\psi_1\psi_2)(p,\vec q^\prime)\\
\nonumber
&& - \int_{p,\vec q,\vec q^\prime} \left\{ \Phi^*(p,\vec q) V(\vec q-\vec q^\prime) (\psi_1\psi_2)(p,\vec q^\prime)+c.c.\right\}\\
&& + \int_{p,\vec q, \vec q^\prime} \Phi^*(p,\vec q) Q(p,\vec q,\vec q^\prime) \Phi(p,\vec q^\prime).
\label{eq:combinedaction2}\end{aligned}$$ The field $\Phi$ is very “massive” for large $\Lambda$. The action in Eq. is the starting point for the flow of the functional $\Gamma_k[\psi,\Phi]$ for $k=\Lambda$. In ref. [@FW09] we showed that the flowing action for a scale dependent Hubbard-Stratonovich transformation of the form in Eq. satisfies the exact flow equation $$\begin{aligned}
\nonumber
\partial_k \Gamma_k &=& \frac{1}{2} \text{STr} (\Gamma_k^{(2)}+{\cal R}_k)^{-1} (\partial_k {\cal R}_k - {\cal R}_k (\partial_k Q^{-1}){\cal R}_k)\\
&&-\frac{1}{2} \Gamma_k^{(1)} (\partial_k Q^{-1}) \Gamma_k^{(1)}+\gamma_k
\label{eq:exactflow}\end{aligned}$$ where $\gamma_k$ is a field independent constant that is irrelevant for most purposes and will be dropped from here on. The flow equation holds for fixed ($k$-independent) field $\Phi$. For our purpose it is useful to perform a $k$-dependent linear transformation on the field $$\begin{aligned}
\nonumber
\hat \Phi(p,\vec q) &=& \int_{\vec q^\prime} \left(e^{M_k(p)}\right)(\vec q,\vec q^\prime) \Phi(p,\vec q^\prime),\\
\hat \Phi^*(p,\vec q) &=& \int_{\vec q^\prime} \Phi^*(p,\vec q^\prime) \left(e^{M_k^\dagger(p)}\right)(\vec q^\prime,\vec q),\end{aligned}$$ with an exponentiated matrix $e^{M_k(p)}$ with “indices” $\vec q,\vec q^\prime$. This gives the flow equation $$\begin{aligned}
\nonumber
\partial_k \Gamma_k {\big |}_{\hat\Phi} &=& \partial_k \Gamma_k {\big |}_\Phi - \int_{p,\vec q}\frac{\delta \Gamma_k}{\delta \hat \Phi(p,\vec q)} \partial_k \hat \Phi(p,\vec q){\big |}_\Phi\\
\nonumber
&& -\int_{p,\vec q}(\partial_k \hat \Phi^*(p,\vec q)){\big |}_\Phi\frac{\delta \Gamma_k}{\delta \hat \Phi^*(p,\vec q)}\\
\nonumber
&=& \frac{1}{2} \text{STr} (\Gamma_k^{(2)}+\hat {\cal R}_k)^{-1} (\widehat{\partial_k {\cal R}_k} - \hat {\cal R}_k (\widehat{\partial_k Q^{-1}}) \hat {\cal R}_k)\\
\nonumber
&& -\frac{1}{2} \Gamma_k^{(1)} (\widehat{\partial_k Q^{-1}}) \Gamma_k^{(1)} \\
&& - \frac{\delta\Gamma_k}{\delta \hat \Phi}(\partial_k M_k) \hat \Phi- \hat \Phi^* (\partial_k M_k^\dagger) \frac{\delta\Gamma_k}{\delta \hat \Phi^*} .
\label{eq:floweqwithvariablechange}\end{aligned}$$ In the last equation we used an obvious matrix notation and the abbreviations $$\begin{split}
& \widehat{\partial_k Q^{-1}} = e^{M_k^\dagger} (\partial_k Q^{-1}) e^{M_k}, \quad \hat {\cal R}_k = e^{-M_k^\dagger} {\cal R}_k e^{-M_k}, \\
& \widehat{\partial_k {\cal R}_k} = e^{-M_k^\dagger} \partial_k {\cal R}_k e^{-M_k}
\end{split}
\label{eq:abbreviations}$$ and functional derivatives are now taken with respect to $\hat \Phi$. It is important to note that the matrices $\partial_k Q$, $\widehat{\partial_k Q}$ and $M_k$ have entries only in the $\Phi$-$\Phi$-block. For example, Eq. implies a transformation of the cutoff term for the composite bosons but not for the fundamental fields $\psi_1, \psi_2$. For simplicity we drop the hats at most places below.
To investigate the implications of the flow equation we use again a truncation of the flowing action in terms of a vertex expansion. Due to the nonrelativistic dispersion relation and the instantaneous interaction, this leads to exact flow equations for the considered $n$-point functions since their flow equations decouple from higher vertex functions. More concrete, we choose as our truncation $$\begin{aligned}
\nonumber
\Gamma_k &=& \int_p \left\{\psi_1^*\left(p_0-\frac{1}{2M_1}\vec p^2\right)\psi_1+\psi_2^*\left(p_0-\frac{1}{2M_2}\vec p^2\right)\psi_2\right\}\\
\nonumber
&& - \int_{p,\vec q,\vec q^\prime} (\psi_1\psi_2)^*(p,\vec q) V(\vec q- \vec q^\prime) (\psi_1\psi_2)(p,\vec q^\prime)\\
\nonumber
&& - \int_{p,\vec q,\vec q^\prime} (\psi_1\psi_2)^*(p,\vec q) \lambda_\psi(p,\vec q, \vec q^\prime) (\psi_1\psi_2)(p,\vec q^\prime)\\
\nonumber
&& - \int_{p,\vec q,\vec q^\prime} \left\{ \Phi^*(p,\vec q) h_\Phi(p,\vec q, \vec q^\prime) (\psi_1\psi_2)(p,\vec q^\prime)+c.c. \right\}\\
&& + \int_{p,\vec q, \vec q^\prime} \Phi^*(p,\vec q) P_\Phi(p,\vec q,\vec q^\prime) \Phi(p,\vec q^\prime).
\label{eq:truncationrebosonization}\end{aligned}$$ Once again, the propagators for the fields $\psi_1, \psi_2$ remain unmodified by the renormalization flow, while the functions $\lambda_\psi$, $h_\Phi$ and $P_\Phi$ are $k$-dependent objects. For $k=\Lambda$ they have the initial values $$\begin{aligned}
\nonumber
\lambda_{\psi,\Lambda}(p,\vec q,\vec q^\prime) &=& 0,\\
\nonumber
h_{\Phi,\Lambda} (p,\vec q,\vec q^\prime) &=& V(\vec q-\vec q^\prime),\\
P_{\Phi,\Lambda} (p,\vec q,\vec q^\prime) &=& Q_\Lambda(\vec q,\vec q^\prime) = \infty\; \delta^{(3)}(\vec q-\vec q^\prime).\end{aligned}$$ Projecting the flow equation onto the truncation in Eq. we find for $\lambda_\psi=0$ the flow equations (again in matrix notation and suppressing the argument $p$ at several places) $$\begin{aligned}
\nonumber
\partial_k \lambda_\psi &=& \tilde \partial_k (V A_k^{-1} V) + h_\Phi (\partial_k Q^{-1}) h_\Phi,\\
\nonumber
\partial_k h_\Phi &=& \tilde \partial_k (h_\Phi A_k^{-1} V) - P_\Phi (\partial_k Q^{-1}) h_\Phi\\
\nonumber
&&- (\partial_k M_k^\dagger) h_\Phi,\\
\nonumber
\partial_k P_\Phi &=& -\tilde \partial_k (h_\Phi A_k^{-1} h_\Phi) - P_\Phi (\partial_k Q^{-1}) P_\Phi\\
&& - (\partial_k M_k^\dagger) P_\Phi - P_\Phi (\partial_k M_k).
\label{eq:floweq234}\end{aligned}$$ As discussed in ref. [@FW09] we can now use our freedom in the choice of $\partial_k Q^{-1}$ to enforce $\partial_k \lambda_\psi=0$, i.e. $$\lambda_\psi(p,\vec q, \vec q^\prime)=0 \quad \text{for all } k.$$ This fixes $\partial_k Q^{-1}=-h_\Phi^{-1} V (\partial_k A_k^{-1}) V h_\Phi^{-1}$. In addition, the $k$-dependent field rescaling determined by the matrix $M_k(p)$ can be chosen arbitrary as well such that we can use it to enforce $\partial_k h_\Phi(p)=0$, i.e. $$h_\Phi(p,\vec q, \vec q^\prime)=V(\vec q-\vec q^\prime) \quad \text{for all } k.$$ In summary, we use $$\begin{aligned}
\nonumber
\partial_k Q^{-1} &=& -\partial_k A_k^{-1},\\
\nonumber
\partial_k M_k &=& (\partial_k A_k^{-1})^\dagger(P_\Phi+V)^\dagger,\\
\partial_k M_k^\dagger &=& (P_\Phi+V) (\partial_k A_k^{-1}).
\label{eq:dkQdkM}\end{aligned}$$ For $P_\Phi$ this leads to the flow equation $$\begin{aligned}
\nonumber
\partial_k P_\Phi &=& - V (\partial_k A_k^{-1}) V + P_\Phi (\partial_k A_k^{-1}) P_\Phi\\
\nonumber
&& - (P_\Phi + V)(\partial_k A_k^{-1}) P_\Phi - P_\Phi (\partial_k A_k^{-1})(P_\Phi+V)\\
&=& - (P_\Phi+V)(\partial_k A_k^{-1})(P_\Phi+V)\end{aligned}$$ which is solved by $$P_\Phi = A_k - V.$$ Using the definition of $A_k$ in Eq. , we see that, as expected, the zero crossings of $P_\Phi$ at $k=0$ correspond to the solution of Schrödingers equation for the two particle problem. Note that we can now also determine via Eqs. and the $k$-dependent matrix $Q$ in the Hubbard-Stratonovich transformation .
To summarize, we found the solution to the flow equation by first adjusting the scale-dependent Hubbard-Stratonovich transformation $\partial_k Q^{-1}$ such that no additional term $\lambda_\psi$ is generated by the flow. All contributions to an interaction of this form are dynamically expressed by bound state exchange processes. As a second step we choose a $k$-dependent linear field redefinition (encoded by the matrix $M_k$) such that also the Yukawa interaction $h_\Phi$ remains $k$-independent. While the first step (to enforce vanishing $\lambda_\psi$) is rather natural, the second is more arbitrary. In our case the choice of $M_k$ was guided by the insight we obtained in section \[sec:Partialbosonization\]. More general it is useful to redefine the composite fields such that the Yukawa-like interaction $h_\Phi$ remains independent of the center of mass momentum $p$, which is always possible. For some problems and for suitable choices of the cutoff function $R_k$ it may also be possible to choose the basis for the composite fields $\Phi$ such that the inverse propagator $P_\Phi$ is a diagonal matrix. As discussed in section \[sec:Partialbosonization\] this is possible for the problem considered here for very simple cutoff functions such as $R_k=k^2$. For more complicated $R_k$ one can still diagonalize $P_\Phi$ but the drawback is then that $h_\Phi$ might depend on $p$.
Approximative solutions {#sec:approximationschemes}
=======================
Up to this point we discussed the application of the functional RG to the nonrelativistic two-body problem in a rather formal way. No approximations were needed, but we had to introduce an additional functional integral over a bilocal composite fields $\Phi(x,\vec y)$ where $\vec y$ labels the relative coordinate of the two particles. For more complicated problems it is difficult to follow the RG flow for bilocal fields. It is therefore necessary to find useful approximation schemes. One possibility is to expand the field in terms of an complete orthonormal set of functions $f_n(\vec y)$ like $$\Phi(x,\vec y) = \sum_{n\in {\cal I}} \phi_n(x) f_n(\vec y).$$ One can then consider the flow of the propagator and coupling constants in that basis. A sensible approximation would now be to take only a finite subset ${\cal J}$ of the infinite index set ${\cal I}$ into account and to neglect all couplings between the fields $\phi_n$, $n\in {\cal J}$ and the fields $\phi_n$, $n\notin {\cal J}$. The approximation becomes good if the influence of the neglected couplings onto the flow of the considered quantities is small. This will, of course, only work for some particular choices of the set $f_n(\vec y)$.
In the following we will consider as an example a Yukawa potential of the form $$V(\vec x-\vec y) = \frac{-e^2}{4\pi |\vec x_1-\vec x_2|} e^{-m|\vec x_1-\vec x_2|}.
\label{eq:Yukawapot}$$ Note that for $m\to \infty$ this potential approaches a contact interaction $$V(\vec x-\vec y) \to -\frac{e^2}{m^2} \delta^{(3)}(\vec x-\vec y).
\label{eq:Yukawatocontact}$$ We choose a particular simple set of orthonormal functions $$\Phi(x,\vec y) = \sum_{n,l,m} \phi_{nlm} (x)\; Y_{lm}(\Omega_{\vec y})\; R_{nl}(|\vec y|).
\label{eq:expansion}$$ Here we use the spherical harmonics $Y_{lm}$ and some functions $R_{nl}$ for the radial direction. A possible choice would be the associate Laguerre Polynomials or any other suitable normalized set of orthogonal functions. Depending on the cutoff function, it may be possible to find analytic solutions for the flow equations in a particular basis.
As a test of the robustness of our formalism, we will try a very crude approximation in the following and include only the term with $l=m=0$ in Eq. . Also the radial dependence is truncated to a single function $R(|\vec y|)$ with support only for $|\vec y|=0$. In other words, we use $$\Phi(x,\vec y) = \phi(x) \delta^{(3)}(\vec y).
\label{eq:compfieldpointbos}$$
The microscopic action corresponding to Eq. reads in momentum space $$\begin{aligned}
\nonumber
S_\psi &=& \int_p{\bigg \{} \psi_1^*(p) \left(p_0-\frac{1}{2M_1}\vec p^2\right)\psi_1(p) \\
\nonumber
&& + \psi_2^*(p) \left(p_0-\frac{1}{2M_2}\vec p^2\right) \psi_2(p) {\bigg \}} \\
\nonumber
&& + \int_{q_1..q_4} \psi_2^*(q_4) \psi_1^*(q_3) \frac{e^2}{m^2+(\vec q_1-\vec q_3)^2} \\
&& \times \psi_1(q_1) \psi_2(q_2) \delta(q_1+q_2-q_3-q_4).
\label{eq:Coulombmomentumspace}\end{aligned}$$ It is useful to introduce the real auxiliary field $\sigma$ which can be seen as the remnant of a massive photon in the nonrelativistic limit. Similar to the Hubbard-Stratonovich transformation performed in section \[sec:Partialbosonization\], we multiply to the partition function a functional integral over the fields $\sigma_1$ and $\sigma_2$ weighted by the quadratic action $$\begin{split}
S_\sigma = & -\int_p \left(\sigma_1(-p)-\frac{e}{\vec p^2}(\psi_1^* \psi_1)(-p)\right) (m^2+\vec p^2) \\
& \left(\sigma_2(p)-\frac{e}{\vec p^2}(\psi_2^* \psi_2)(p)\right)
\end{split}
\label{eq:sigmaHST}$$ with $$(\psi_1^*\psi_1)(p) = \int_q \psi_1^*(q) \psi_1(q+p)$$ and similar for $(\psi_2^*\psi_2)$. Adding this to Eq. we arrive at $$\begin{aligned}
\nonumber
S_{\psi\sigma} &=& \int_p {\bigg \{}\psi_1^*(p)\left(p_0-\frac{1}{2M_1}\vec p^2\right)\psi_1(p)\\
\nonumber
&& +\psi_2^*(p)\left(p_0-\frac{1}{2M_2}\vec p^2\right)\psi_2(p)\\
\nonumber
&&+ \sigma_1(-p) (-m^2-\vec p^2) \sigma_2(p) {\bigg \}}\\
&&+e\int_{p_1,p_2} {\big \{} \sigma_1(p_1-p_2) \psi_2^*(p_1) \psi_2(p_2)\\
&&+\sigma_2(p_1-p_2) \psi_1^*(p_1) \psi_1(p_2){\big \}}.\end{aligned}$$ For the flowing action $\Gamma_k$ we make the following truncation $$\begin{split}
& \Gamma_k[\psi,\phi,\sigma] = S_{\psi\sigma} + \int_p \phi^*(p) P_\phi(p) \phi(p)\\
& - \int_{p,q} \left\{\phi^*(p) h_\phi(p)\psi_1(\eta_1 p+q)\psi_2(\eta_2 p-q)+c.c.\right\}\\
&- \int_{p,q_1,q_2} \psi_2^*(\eta_2 p-q_1) \psi_1^*(\eta_1 p+q_1)\\
& \times \lambda_\psi(p)\, \psi_1(\eta_1 p+ q_2) \psi_2 (\eta_2 p-q_2).
\end{split}
\label{eq:truncationsingleboson}$$ We will use our freedom in the choice of the scale-dependent Hubbard-Stratonovich transformation to ensure that $\lambda_\psi(p)=0$ at all scales. It is a consequence of the nonrelativistic dispersion relation that the couplings in the $\sigma$-sector of the theory (the charge $e$ and the propagator of the $\sigma$-boson) do not receive any modifications from the renormalization group flow. This can also be checked explicitly by looking at the flow equations for these quantities. One advantage of introducing the field $\sigma$ instead of working with $V(\vec q-\vec q^\prime)$ as before is that one can also introduce a cutoff function for $\sigma$. In principle, one could do this also for the composite boson $\phi$, but we will not do this for simplicity, here. We use $$\begin{aligned}
\nonumber
\Delta S_k &=& \int_p {\bigg \{} \psi_1^*(p) \left[-\frac{1}{2M_1} R_k(\vec p^2)\right]\psi_1(p) \\
\nonumber
&& + \psi_2^*(p) \left[-\frac{1}{2M_1} R_k(\vec p^2)\right] \psi_2(p) {\bigg \}}\\
&& + \sigma_1(-p) [-R_k(\vec p^2)] \sigma_2(p)
\label{eq:cutoff}\end{aligned}$$ with $R_k(z)=(k^2-z)\theta(k^2-z)$.
Now that we have fixed the truncation and the cutoff function, we can determine the flow equations. By setting the ansatz in Eq. into the flow equation we find in the center of mass frame ($\vec p=0$) $$\begin{aligned}
\nonumber
\partial_k \lambda_\psi(p) &=& \tilde \partial_k \int_{\vec q} \frac{e^4}{(m^2+\vec q^2+R_k(\vec p^2))^2}\\
&&\times \frac{1}{p_0-\frac{1}{2\mu}(\vec q^2+R_k(\vec q^2))}\\
\nonumber
&& + h_\phi(p)^2 \partial_k Q^{-1}(p),\\
\nonumber
\partial_k h_\phi(p) &=& -\tilde \partial_k \int_{\vec q} h_\phi(p) \frac{e^2}{m^2+\vec q^2+R_k(\vec p^2)} \\
&& \times \frac{1}{p_0-\frac{1}{2\mu}(\vec q^2+R_k(\vec q^2))}\\
\nonumber
&&-P_\phi(p) h_\phi(p) \partial_k Q^{-1}(p) - h_\phi(p) \partial_k M(p),\\
\nonumber
\partial_k P_\phi(p) &=& -\tilde \partial_k \int_{\vec q} h_\phi(p) \frac{1}{p_0-\frac{1}{2\mu}(\vec q^2+R_k(\vec q^2))} h_\phi(p)\\
&&- P_\phi(p)^2 \partial_k Q^{-1}(p) - 2 h_\phi(p) \partial_k M(p).\end{aligned}$$ Here, the derivative $\tilde \partial_k$ hits only the explicit $k$-dependence of the cutoff function $R_k$. Before we solve the flow equations it remains to perform the integration over the spatial momentum $\vec q$. For the cutoff function in Eq. the integration is very simple and gives $$\begin{aligned}
\nonumber
\partial_k \lambda_\psi(p) &=& \frac{e^4 k^3}{6\pi^2} \partial_k \frac{1}{(m^2+k^2)^2(p_0-\frac{1}{2\mu}k^2)}\\
&& + h_\phi(p)^2 \partial_k Q^{-1}(p),\\
\nonumber
\partial_k h_\phi(p) &=& -\frac{e^2 h_\phi(p) k^3}{6\pi^2} \partial_k \frac{1}{(m^2+k^2)(p_0-\frac{1}{2\mu}k^2)}\\
&&-P_\phi(p) h_\phi(p) \partial_k Q^{-1}(p) - h_\phi(p) \partial_k M(p),\\
\nonumber
\partial_k P_\phi(p) &=& -\frac{h_\phi(p)^2 k^3}{6\pi^2}\partial_k \frac{1}{p_0-\frac{1}{2\mu}k^2}\\
&&- P_\phi(p)^2 \partial_k Q^{-1}(p) - 2 P_\phi(p) \partial_k M(p).\end{aligned}$$ We choose now the scale-dependent Hubbard-Stratonovich transformation $\partial_k Q^{-1}(p)$ such that $\partial_k \lambda_\psi(p)=0$. This gives $$\partial_k Q^{-1}(p) = -\frac{e^4 k^3}{6\pi^2 h_\phi(p)^2} \partial_k \frac{1}{(m^2+k^2)^2(p_0-\frac{1}{2\mu}k^2)}.$$ In addition, we choose the rescaling of the field $\phi(p)$ encoded by $\partial_k M(p)$ such that $\partial_k h_\phi(p)=\partial_k h_\phi(0)$. What remains then is the freedom to choose $\partial_k M(0)$. This is just the freedom to make a $p$-independent rescaling of the field $\phi$. Usually, one chooses this wavefunction renormalization such that the propagator for the field $\phi$ has a residue of value unity at a pole that corresponds to a propagating particle. However, since $\partial_k M(0)$ can be chosen arbitrary in principle, we choose it here for simplicity such that $\partial_k h_\phi(0)=0$. This results in $$\begin{aligned}
\partial_k M(p) &=& -\frac{e^2 k^3}{6\pi^2} \partial_k \frac{1}{(m^2+k^2)(p_0-\frac{1}{2\mu}k^2)} \\
\nonumber
&&+ \frac{e^4 k^3}{h_\phi^2 6\pi^2} P_\phi(p) \partial_k \frac{1}{(m^2+k^2)^2(p_0-\frac{1}{2\mu}k^2)}.\end{aligned}$$ We keep in mind, that with this choice the poles in the propagator for the field $\phi$ may have residues with values different from one. To summarize, we have obtained the flow equations $$\begin{aligned}
\partial_k \lambda_\psi(p) &=& 0,\\
\partial_k h_\phi(p) &=& 0,\\
\nonumber
\partial_k P_\phi(p) &=& -\frac{h_\phi^2 k^3}{6\pi^2} \partial_k \frac{1}{p_0-\frac{1}{2\mu} k^2}\\
\nonumber
&+& 2 P_\phi(p) \frac{e^2 k^3}{6\pi^2} \partial_k \frac{1}{(m^2+k^2)(p_0-\frac{1}{2\mu}k^2)}\\
&-& P_\phi(p)^2 \frac{e^4 k^3}{h_\phi^2 6\pi^2} \partial_k \frac{1}{(m^2+k^2)^2(p_0-\frac{1}{2\mu}k^2)}.
\label{eq:flowbder}\end{aligned}$$
Before we proceed with the solution of the flow equation let us discuss the initial conditions for large values of the scale $k\to\Lambda$. In the limit $k\to\Lambda\to\infty$ the contribution of the $\phi$-particle exchange to the effective interaction between the $\psi$-particles should vanish which implies $$\lim_{\Lambda\to\infty} \frac{h_{\phi,\Lambda}^2}{P_{\phi,\Lambda}}=0.$$ For large but finite $\Lambda$ this value gets modified by one-loop contributions which can be obtained from the flow equation for $\lambda_\psi$ $$\begin{aligned}
\nonumber
\frac{h_{\phi,\Lambda}^2}{P_{\phi,\Lambda}} &=& \int_{\vec q} \frac{e^4}{(m^2+\vec q^2+R_k(\vec q^2))^2} \frac{1}{p_0-\frac{1}{2\mu}(\vec q^2+R_\Lambda(\vec q^2))}\\
&\to& -\frac{2\mu e^4}{3\pi^2 \Lambda^3}.\end{aligned}$$ In the last line we assumed $\Lambda^2 \gg m^2$, $\Lambda^2\gg 2\mu p_0$. Indeed we find that this vanishes in the limit $\Lambda\to\infty$. Since only the above ratio is fixed, there is still some ambiguity in the choice of the initial conditions. Indeed it is always possible to rescale the field $\phi$ which changes $h_{\phi,\Lambda}^2$ and $P_{\phi,\Lambda}$ but keeps the ratio fixed. Physical observables are of course not affected by such a rescaling. For simplicity we choose $$\begin{aligned}
\nonumber
h_{\phi,\Lambda} &=& h_\phi = e,\\
P_{\phi,\Lambda} &=& -\frac{3\pi^2}{2\mu e^2} \Lambda^3 = -\Lambda^3/c.
\label{eq:initialcond}\end{aligned}$$ The last equation defines the abbreviation $c$. We have now all ingredients to solve the flow equation. From Eq. we obtain $$\begin{split}
\partial_k P_\phi = & -c \frac{1}{(1-2\mu p_0/k^2)^2}\\
& +\frac{2 c P_\phi}{k^2} {\bigg [}\frac{1}{(1-2\mu p_0/k^2)^2(1+m^2/k^2)}\\
& +\frac{1}{(1-2\mu p_0/k^2)(1+m^2/k^2)^2} {\bigg ]}\\
& -\frac{c P_\phi^2}{k^4} {\bigg [ } \frac{1}{(1-2\mu p_0/k^2)^2(1+m^2/k^2)^2}\\
& +\frac{2}{(1-2\mu p_0/k^2)(1+m^2/k^2)^3}{\bigg ]}.
\end{split}
\label{eq:floweqader}$$ We start our investigation with $p_0=0$ and $m^2=0$. The flow equation simplifies to $$\partial_k P_\phi = -c + \frac{4c P_\phi}{k^2} - \frac{3c P_\phi}{k^4}.$$ The solution of this flow equation with the initial condition in Eq. is quite simple. It is given for $\Lambda\to\infty$ by $$P_\phi=-k^3/c+k^2.$$ We note that the propagator for the composite field $\phi$ changes its sign at the scale $k=c$. The zero crossing of the iinverse propagator corresponds to a bound state. Would we solve the flow equation for a negative frequency $p_0<0$, would the terms involving $p_0$ in Eq. act as an infrared cutoff and effectively stop the flow at the scale where $k^2\approx 2\mu |p_0|$. Choosing $|p_0|\approx c^2/(2\mu)$ leads to $P_\phi=0$ at the macroscopic scale $k=0$ which corresponds to a pole in the propagator for the composite field $\phi$. We note that $c^2/(2\mu)$ is proportional to the Rydberg energy $\sim e^4 \mu$ and the bound state can therefore be interpreted as the lowest bound state of the Coulomb potential. That this comes out qualitatively correct is encouraging, although quantitative agreement with the expectations from quantum mechanics can not be expected in this case due to the simple approximation made in Eq. .
Let us now come to the limit of very large $m^2$. The potential becomes now of the singular form in Eq. . For that reason, the theory must be regularized by introducing a finite UV cutoff $\Lambda_\text{UV}$. However, we will see that many results do not depend on the precise choice of $\Lambda_\text{UV}$. The initial values of couplings are fixed by imposing $h_{\phi,\Lambda}^2/P_{\phi,\Lambda}=0$ at $\Lambda=\Lambda_\text{UV}$. In addition we choose $h_\phi=e$ as before. The flow equation becomes for large $m^2$ $$\partial_k P_\phi = \left(-c+\frac{2cP_\phi}{m^2}-\frac{cP_\phi^2}{m^4}\right)\frac{1}{(1+w/k^2)^2}$$ with the abbreviation $w=-2\mu p_0$. It has the solution $$\begin{split}
& P_\phi = m^2 \\
&- 2m^4(k^2+w) {\bigg (}-ck(2k^2+3w)\\
& +(k^2+w)\left[3c\sqrt{w}\;\text{arctan}(k/\sqrt{w})+D\right]{\bigg )}^{-1},
\end{split}$$ with an arbitrary constant $D$. The condition that $e^2/P_{\phi,\Lambda}=0$ fixes $$D=\frac{c\Lambda(2\Lambda^2+3w)}{\Lambda^2+w}-3c\sqrt{w} \; \text{arctan}(\Lambda/\sqrt{w}).$$ We note that for large $k$ but $k<\Lambda$ the inverse propagator $P_\phi$ is negative. Depending on the choice of the parameters $m^2$ and $e^2$ (or $c$) $P_\phi$ may change its sign during the flow towards small values of $k$. This corresponds then again to a bound state. To see this we consider $P_\phi$ for $k=0$ which becomes for $\Lambda^2\gg w$ $$P_\phi(k=0)=m^2-\frac{m^4}{c\left(\Lambda-\frac{3\pi}{4}\sqrt{w}\right)}.
\label{eq:sollargemk0}$$ To interpret this expression let us assume in the following that the fields $\psi_1$ and $\psi_2$ describe fermions with equal mass $M_1=M_2=M$, $\mu=M/2$. We can obtain the scattering length $a$ from Eq. as $$a=\frac{M}{4\pi}\lambda_{\psi,\text{eff}},
\label{eq:alambda}$$ with $$\lambda_{\psi,\text{eff}} = -\frac{e^2}{m^2}+\frac{e^2}{P_\phi(k=0,w=0)}=\frac{e^2}{c\Lambda-m^2}.$$ For a more detailed explanation of Eq. we refer to refs. [@DKS; @MFSW]. Inserting the abbreviation $c=2\mu e^2/(3\pi^2)$ this gives $$a=\frac{1}{\frac{4\Lambda}{3\pi}-\frac{4\pi m^2}{e^2 M}}.$$ Depending on the choice of the initial parameters $e^2$ and $m^2$ for a given value of $\Lambda$, the scattering length can be both negative and positive. For large positive scattering length one expects the presence of a shallow dimer state in the spectrum [@BraatenHammer]. Its binding energy can be calculated from the on-shell condition of the $\phi$-particle which implies $P_\phi(k=0)=0$. From this we find $$c\left(\Lambda-\frac{3\pi}{4}\sqrt{w}\right)=m^2$$ or after reinserting $w=-2\mu p_0 = 2\mu E$ $$E=\frac{1}{M}\frac{1}{a^2}.$$ This is precisely the well known relation between the scattering length $a$ and the binding energy $E$ of the shallow dimer that exists for $a>0$ [@BraatenHammer].
It is interesting that our treatment yields exact results in the limit of large $m$. On the other side this could have been expected, since a quite similar formalism using a $k$-independent bosonization yields exact results for the contact potential, as well [@DKS; @MFSW; @Birse]. In any case it is interesting that the relatively simple approximation including only one boson already yields qualitatively correct results for the Coulomb potential and quantitatively precise results for the contact potential. More elaborate approximations will allow quantitative investigations for a large class of nonrelativistic and instantaneous interactions. Since this is not the purpose of the present paper we will discuss in the last chapter some generalizations of the formalism to problems that are in a certain sense more complicated than the nonrelativistic few-body problem.
Generalized formalism {#sec:generalformalismandapproxschemes}
=====================
While most parts of this paper where devoted to the treatment of nonrelativistic particles with instantaneous interactions, we will use the present section to present some generalizations of the formalism to a wider class of problems. For this purpose we first somewhat generalize the notion of a composite field. For some set of functions $f_n(y)$ where $y=(y_0,\vec y)$ is a space-time coordinate and for two real numbers $\eta_1,\eta_2\in(0,1)$ with $\eta_1+\eta_2=1$ we define $$\begin{aligned}
\nonumber
(\psi_1\psi_2)(x,y) &=& \psi_1(x+\eta_2 y) \psi_2(x-\eta_1 y),\\
(\psi_1\psi_2)_n(x) &=& \int_y f_n^*(y) (\psi_1\psi_2)(x,y).\end{aligned}$$ We assume that the functions $f_n(y)$, $n\in{\cal N}$ constitute a complete orthonormal set such that the following relations hold $$\begin{aligned}
\nonumber
\int_y f_n^*(y) f_m(y) &=& \delta_{nm},\\
\sum_{n\in{\cal N}} f_n^*(x) f_n(y) &=& \delta(x-y).\end{aligned}$$ In momentum space with $f_n(y)=\int_p e^{-ipx} \tilde f_n(p)$, $f_n^*(y)=\int_p e^{ipx} \tilde f_n^*(p)$ these relations become $$\begin{aligned}
\nonumber
\int_p f_n^*(p) f_m(p) &=& \delta_{nm},\\
\sum_{n\in N} f_n^*(p) f_n(q) &=& \delta(p-q),\end{aligned}$$ and the composite field reads in momentum space $$\begin{aligned}
\nonumber
(\psi_1\psi_2)_n(p) &=& \int_q \tilde f_n^*(q) \psi_1(\eta_1p+q) \eta_2(\eta_2p-q),\\
(\psi_1\psi_2)_n^*(p) &=& \int_q f_n(q) \psi_2^*(\eta_2p-q) \psi_1^*(\eta_1p + q).
\label{eq:compositefieldsmomspace}\end{aligned}$$ The index $n$ we use to label the functions $f_n$ is an abstract index that can be either continuous or discrete. Note that the bilocal fields used for the investigation of nonrelativistic particles with instantaneous interactions can be embedded into the above description by choosing $f_n(y)$ to have support only for $y_0=0$, i.e. $$f_n(y) = \delta(y_0)\; g_n(\vec y).$$
Similar to previous chapters we introduce bosonic fields for every composite combination. For example, the term in the action corresponding to the Yukawa-type coupling is $$\Gamma_k^{(\phi\psi\psi)} = -\sum_{n,m} \int_p \left\{\phi_n^*(p) h_\phi(p)_{nm} (\psi_1\psi_2)_m(p)+c.c.\right\}.$$ It will often be possible and convenient to work with a Yukawa coupling that is independent of $p$ and diagonal with respect to the indices $n,m$. The action involves also a term quadratic in the fields $\phi_n^*$, $\phi_n$. We write it as $$\Gamma_k^{(\phi,2)} = \sum_{n,m} \int_p \phi_m^*(p) P_\phi(p)_{nm} \phi_m(p).$$ Similarly, we formally introduce also a quartic interaction term for the fields $\psi$ $$\Gamma_k^{(\psi,4)} = -\sum_{n,m} (\psi_1\psi_2)^*_n(p) \lambda_\psi(p)_{nm} (\psi_1\psi_2)_m(p).
\label{eq:quanricintexpnm}$$
As an important side remark, we note that a very large class of quartic interactions can be writte in the form . A completely general interaction between the fields $\psi_1$ and $\psi_2$ can be written in a homogeneous situation as $$\begin{split}
& \Gamma_k^{(\psi,4)} = -\int_{k_1\dots k_4} \psi_2^*(k_2)\psi_1^*(k_1) \;\lambda_\psi(k_1,k_2,k_3,k_4)\\
& \psi_1(k_3) \psi_2(k_4)\;\delta(k_1+k_2-k_3-k_4).
\end{split}
\label{eq:lambdapsigeneral}$$ Due to momentum conservation, the vertex function $\lambda_\psi$ depends actually only on three independent variables. One possibility to choose them is $$\begin{aligned}
\nonumber
p &=& k_1+k_2=k_3+k_4,\\
\nonumber
q_1 &=& \eta_2 k_1-\eta_1 k_2,\\
q_2 &=& \eta_2 k_3-\eta_1 k_4.\end{aligned}$$ Eq. can then be written as $$\Gamma_k^{(\psi,4)} = -\int_{p,q_1,q_2} (\psi_1\psi_2)^*(p,q_1)\;\lambda_\psi(p,q_1,q_2) \; (\psi_1\psi_2)(p,q_2),
\label{eq:explambdapsiqq}$$ with $$\begin{aligned}
\nonumber
(\psi_1\psi_2)(p,q) &=& \psi_1(\eta_1p+q) \psi_2(\eta_2p-q)\\
(\psi_1\psi_2)^*(p,q) &=& \psi_2^*(\eta_2p-q) \psi_1^*(\eta_1p+q).\end{aligned}$$ Assuming now that $\lambda_\psi(p,q_1,q_2)$ is regular with respect to the arguments $q_1$ and $q_2$ we can expand $$\lambda_\psi(p,q_1,q_2) = \sum_{n,m} \tilde f_n(q_1) \;\lambda_\psi(p)_{nm} \tilde f_m^*(q_2),
\label{eq:expansionlambdapsi}$$ with coefficients $$\lambda_\psi(p)_{nm} = \int_{q_1,q_2} \tilde f_n^*(q_1) \; \lambda_\psi(p,q_1,q_2) \; f_m(q_2).$$ Plugging Eq. into Eq. and using we find that this is precisely of the form in Eq. . Note that we did not assume that $\lambda_\psi(p,q_1,q_2)$ is regular with respect to the argument $p$. Possible singularities are transfered to the coefficients $\lambda_\psi(p)_{nm}$. In fact, it is a big advantage of our formalism that poles or branch cuts in $\lambda_\psi(p)_{nm}$ can be described by relatively simple parameterizations of the inverse propagator $P_\phi(p)_{nm}$ for a composite particle.
Let us assume that for a fixed Hubbard-Stratonovich transformation a term of the form in Eq. is generated by the renormalization group flow, i.e. $\partial_k \lambda_\psi(p)_{nm}{\big |}_\text{HS}\neq0$. We can then follow the calculation in Sect. \[sec:scaledeppartialbos\] and use a scale-dependent Hubbard-Stratonovich transformation to realize $\partial_k \lambda_\psi(p)_{nm}=0$. This will lead to additional contributions in the flow equations for $h_\phi(p)_{nm}$ and $P_\phi(p)_{nm}$. In addition, a $k$-dependent linear transformation of the fields $\phi_n(p)$ can be employed, for example to keep $h_\phi(p)_{nm}$ independent of $p$. More concrete, the flow equations can be derived similar to Eq. and read in symbolic notation $$\begin{aligned}
\nonumber
\partial_k \lambda_\psi(p)_{nm} &=& \partial_k \lambda_\psi(p)_{nm}{\big |}_\text{HS} \\
\nonumber
&&+ \sum_{r,s} h_\phi(p)_{nr} \partial_k Q^{-1}(p)_{rs} h_\phi(p)_{sm},\\
\nonumber
\partial_k h_\phi(p)_{nm} &=& \partial_k h_\phi(p)_{nm}{\big |}_\text{HS} \\
\nonumber
&&- \sum_{r,s} P_\phi(p)_{nr} \partial_k Q^{-1}(p)_{rs} h_\phi(p)_{sm}\\
\nonumber
&&-\sum_r (\partial_k M_k^\dagger)(p)_{nr} h_\phi(p)_{rm},\\
\nonumber
\partial_k P_\phi(p)_{nm} &=& \partial_k P_\phi(p)_{nm} {\big |}_\text{HS} \\
\nonumber
&&- \sum_{r,s} P_\phi(p)_{nr} \partial_k Q^{-1}(p)_{rs} P_\phi(p)_{rm}\\
\nonumber
&&-\sum_r (\partial_k M_k^\dagger)(p)_{nr} P_\phi(p)_{rm}\\
&& - \sum_r P_\phi(p)_{nr} (\partial_k M_k)(p)_{rm}.\end{aligned}$$
We see that by choosing $\partial_k Q^{-1}(p)_{nm}$ conveniently, we can absorb a quartic interaction term $\lambda_\psi$ as in Eq. that is regular with respect to the relative momenta $q_1$ and $q_2$ into the exchange of a composite boson of the type $\psi_1\psi_2$. Besides this particle-particle pair also other combinations are possible, such as the combination $\psi_1^*\psi_1$ or $\psi_2^*\psi_2$ corresponding to particle-hole pairs in nonrelativistic physics or a particle-antiparticle pair in relativistic quantum field theory. With a construction similar to the one presented above, one can show that a contribution to the general interaction term $\lambda_\psi$ in Eq. that is regular as a function of $k_1+k_3$ and $k_2+k_4$ (but might have singularities with respect to $k_1-k_3=k_4-k_2$) can be absorbed into the exchange of a $\psi_1^*\psi_1$ pair. The construction would be similar to introducing the $\sigma$-field in Eq. but now with a $k$-dependent Hubbard-Stratonovich transformation. In a relativistic field theory (or in a nonrelativistic field theory at nonzero density) the propagator of this field might get a nontrivial frequency dependence and thus become a dynamical field. Finally, the exchange of a $\psi_1^*\psi_2$-pair can describe contributions that are regular as functions of the combinations $k_1+k_4$ and $k_2+k_3$ but might have singularities with respect to $k_1-k_4=k_3-k_2$.
Conclusions {#sec:conclusions}
===========
We have shown in this paper how a recently derived exact flow equation can be employed for the treatment of bound state formation in quantum field theory. For a nonrelativistic field theory, the relevant flow equations can be integrated exactly and our approach is equivalent to the standard quantum mechanical treatment. The presented formalism can also be employed to find approximate solutions to bound state problems. This can be useful under circumstances where no exact solution from quantum mechanics is available. Examples for such situations arise in relativistic quantum field theory, due to non-instantaneous interactions or in nonrelativistic quantum field theory at nonzero density and temperature.
One big advantage of our formalism is that fundamental and composite particles are described in a very similar way by fluctuating quantum fields. A scale dependent Hubbard-Stratonovich transformation allows to absorb complicated interaction vertices into relatively simple tree diagrams of a Yukawa-type theory. Depending on the situation, a particular composite field might become a propagating degree of freedom, end as a gapped excitation or remain an auxiliary field without own dynamics when the infrared cutoff scale is lowered. The formalism is particularly useful in situations where the relevant degrees of freedom change during the renormalization group flow. This is often the case when interaction effects are strong. Prominent examples are QCD where the ultraviolet physics is dominated by quarks and gluons while mesons and baryons dominate the physics in the infrared or the Hubbard model, where fermionic degrees of freedom dominate the UV, while various competing pairing instabilities dominate the infrared physics. For the example of QCD the flow from UV to IR degrees of freedom was discussed in a setup closely related to the one discussed here [@GiesWetterich; @GiesWetterich2; @Braun].
Functional renormalization group equations have been applied with quite some success to many problems in quantum field theory. For reviews see ref. [@ReviewRG; @Pawlowski; @SalmhoferHonerkamp; @Metzner]. It is a general feature that attempts to increase the precision of these calculations face the difficult problem to find an efficient parameterization for the momentum dependence of vertex functions. The formalism presented in this paper does not only provide a general framework for such parameterizations (similar to the one proposed in ref. [@HusemannSalmhofer]) but also allows to take the essential composite degrees of freedom as dynamical fields into account. This includes also possible interactions between composite particles which typically correspond to complicated and nonlocal higher-order interactions in terms of the fundamental fields. Another advantage of the description involving the composite fields is that spontaneous symmetry breaking, for example in a superconducting phase, is straight-forward to describe.
Many of the phenomena discussed above – including the formation of bound states – can also be described with other functional methods of QFT such as Schwinger-Dyson and Bethe-Salpeter equations. Renormalization group equations have the advantage that they allow a straightforward discussion of interesting features such as fixed point behavior and universality. Close to renormalization group fixed points one can distinguish between relevant and irrelevant (and marginal) operators. This implies that the flow equations determine which terms in a given expansion become important and which can be neglected! Often, a large class of microscopic theories are attracted towards the same fixed point and the system shows universal features that are independent of the concrete microscopical realization. Using a scale-dependent Hubbard-Stratonovich transformation, one can study for a theory with bound states the different fixed points as well as the crossovers between them [@GiesWetterich2].
In summary, we have developed a formalism to describe bound states in quantum field theory using an exact flow equation and look forward to applications to many interesting problems in the future.
The author thanks C. Wetterich for useful discussions. This work has been supported by the DFG research group FOR 723 and the Helmholtz Alliance HA216/EMMI.
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---
abstract: 'In this work, we investigate what role the redshift drift data of Square Kilometre Array (SKA) will play in the cosmological parameter estimation in the future. To test the constraint power of the redshift drift data of SKA-only, the $\Lambda$CDM model is chosen as a reference model. We find that using the SKA1 mock data, the $\Lambda$CDM model can be loosely constrained, while the model can be well constrained when the SKA2 mock data are used. When the mock data of SKA are combined with the data of the European Extremely Large Telescope (E-ELT), the constraints can be significantly improved almost as good as the data combination of the type Ia supernovae observation (SN), the cosmic microwave background observation (CMB), and the baryon acoustic oscillations observation (BAO). Furthermore, we explore the impact of the redshift drift data of SKA on the basis of SN+CMB+BAO+E-ELT in the $\Lambda$CDM model, the $w$CDM model, the CPL model, and the HDE model. We find that the redshift drift measurement of SKA could help to significantly improve the constraints on dark energy and could break the degeneracy existing between the cosmological parameters. Therefore, we conclude that redshift-drift observation of SKA would provide a good improvement in the cosmological parameter estimation in the future and have the enormous potential to be one of the most competitive cosmological probes in constraining dark energy.'
author:
- Yan Liu
- 'Jing-Fei Zhang'
- 'Xin Zhang[^1]'
title: 'Real-time cosmology with SKA'
---
Introduction {#sec:intro}
============
The accelerated expansion of the universe has been discovered and confirmed by cosmological observations for about twenty years, which is undoubtedly one of the greatest scientific discoveries in the modern cosmology. However, the science behind the cosmic acceleration, i.e., the nature of dark energy, still remains mysterious for us. To measure the physical property of dark energy, one should precisely measure the expansion history of the universe. Currently, the mainstream way is to measure the cosmic distances (luminosity distance or angular diameter distance) and the corresponding redshifts, and to establish a distance-redshift relation, by which constraints on the parameters of dark energy (and other cosmological parameters) can be made. However, a more straightforward way is to directly measure the expansion rate of the universe at different redshifts, although this measurement is more difficult in the observational cosmology.
With the fast advancement in technology over the past several decades, the possibility of measuring the temporal variation of astrophysical observable quantities over a few decades is becoming more and more realistic. This kind of real-time observations can be called the “[*real-time cosmology*]{}”. The most typical real-time observable is the [*redshift drift*]{}, which can give a direct measurement for the expansion rate (namely, the Hubble parameter) of the universe in a specific range of redshift.
The approach of measuring the redshift drift was first proposed by Sandage, who suggested a direct measurement of the redshift variation for the extra–galactic sources [@sandage]. At that time, obviously, such a measurement was out of reach with the technological limitation of the day. Then, the method was further improved by Loeb, who suggested a more realistic way of measuring the redshift drift using Lyman-$\alpha$ absorption lines of the distant quasars (QSOs) to detect the redshift variation [@Loeb:1998bu]. Loeb concluded that the signal would be detectable when 100 quasars can be observed over 10 years with a 10-meter class telescope. Thus, the method of redshift drift measurement is also referred to as the “Sandage-Loeb" (SL) test.
Based on the SL test, the scheduled European Extremely Large Telescope (E-ELT), a giant 40-meter class optical telescope, is equiped with a high-resolution spectrograph to perform the COsmic Dynamics EXperiment (CODEX). The experiment is designed to detect the SL-test signals by observing the Lyman–$\alpha$ absorption lines within the redshift range of $2\lesssim z\lesssim 5$. The forecast of using the redshift drift from the E-ELT to constrain dark energy models has been extensively discussed; see, e.g., Refs. [@Liske:2008ph; @Geng:2014hoa; @Geng:2014ypa; @Guo:2015gpa; @He:2016rvp; @Liu:2018kjv; @Lazkoz:2017fvx; @Zhang:2013mja; @Martinelli; @Corasaniti:2007bg; @Balbi:2007fx; @Zhang:2007zga; @Zhang:2010im; @Geng:2018pxk; @Geng:2015ara; @Yuan:2013wpa]. It has been shown that the redshift drift in the redshift range of $2< z< 5$ is rather useful to break the parameter degeneracies generated by other observations and thus can play an important role in the cosmological estimation in the future.
The Square Kilometre Array (SKA) has recently started construction for the stage of Phase one. Actually, SKA can also perform the research of real-time cosmology. Instead of detecting the Lyman-$\alpha$ absorption lines of quasar, SKA will measure the spectral drift in the neutral hydrogen (HI) emission signals of galaxies to implement the measurement of redshift drift in the redshift range of $0 < z < 1$. Obviously, the redshift drift data of SKA provide an important supplement to those of E-ELT.
In this work, we will study the real-time cosmology with the redshift drift observation from SKA. We will simulate the redshift drift data of SKA and use these data to constrain cosmological parameters. We have the following aims in this work: (i) We wish to learn what extent the cosmological parameters can be constrained to by using the redshift drift data of SKA-only. (ii) What will happen when the redshift drift data of SKA and E-ELT are combined to perform constraints on cosmological parameters. (iii) What role the redshift drift data of SKA will play in the cosmological estimation in the future.
We will employ several typical and simple dark energy models to perform the analysis of this work. We will consider the $\Lambda$ cold dark matter ($\Lambda$CDM) model in this work, which is the simplest cosmological model and is able to explain the various current cosmological observations quite well. The $w$CDM model is the simplest extension to the $\Lambda$CDM model, in which the equation-of-state (EoS) parameter $w$ of dark energy is assumed to be a constant. The Chevalliear-Polarski-Linder (CPL) [@Chevallier:2000qy; @Linder:2002et] model of dark energy is a further extension to the $\Lambda$CDM model, in which the form of $w(a) = w_{0} + w_{a}(1-a)$ with two free parameters $w_0$ and $w_a$ is proposed to describe the cosmological evolution of the EoS of dark energy. We will also consider the holographic dark energy (HDE) model in this work, which is a dynamical dark energy model based on the consideration of quantum effective field theory and holographic principle of quantum gravity [@Li:2004rb; @Cohen:1998zx]. In the HDE model, the type (quintessence or quintom) and the cosmological evolution of dark energy are solely determined by a dimensionless constant $c$ (note that this is not the speed of light). For more detailed studies on the HDE model, see e.g. Refs. [@Li:2009jx; @Li:2009bn; @Li:2004rb; @Zhang:2005hs; @Zhang:2007sh; @Gao:2007ep; @Cui:2014sma; @Zhang:2014sqa; @Zhang:2019ple; @Zhang:2006av; @Zhang:2006qu; @Zhang:2007es; @Zhang:2007an; @Zhang:2008mb; @Zhang:2009un; @Cui:2009ns; @Feng:2009hr; @Wang:2012uf; @Zhang:2014ija; @Zhang:2015rha; @Zhao:2017urm; @Feng:2018yew; @Zhang:2015uhk; @Wang:2016tsz; @Geng:2014hoa; @Geng:2015ara]. In this work, we use these four typical, simple dark energy models, namely, the $\Lambda$CDM, $w$CDM, CPL, and HDE models, as examples to make an analysis for the real-time cosmology.
The structure of this paper is arranged as follows. In Sect. \[Method and data\], we present the analysis method and the observational data used in this work. In Sect. \[sec:Results and Discussions\], we report the constraint results of cosmological parameters and make some relevant discussions. In Sect. \[sec:Conclusion\], the conclusion of this work is given.
Method and data {#Method and data}
===============
We will simulate the redshift drift data of SKA, and use these mock data to constrain the cosmological models. We will also simulate the redshift drift data of E-ELT, and make comparison and combination with the data of SKA. In order to check how the redshift drift data of SKA will break the parameter degeneracies generated by other cosmological observations, we will also consider the current mainstream observations in this work.
A brief description of the dark energy models
---------------------------------------------
In this subsection, we will briefly describe the dark energy models employed in the analysis of this work. In a spatially flat universe with a dark energy having an EoS $w(z)$, the form of the Hubble expansion rate is given by the Friedmann equation, $$\begin{aligned}
E^2(z)&\equiv\frac{H^{2}(z)}{H^{2}_{0}}=\Omega_{\rm m}(1+z)^{3}+\Omega_{\rm r}(1+z)^{4}\\
& +(1-\Omega_{\rm m}-\Omega_{\rm r})\exp(3 \int^{z}_{0} \frac{{1+w(z')}}{{1+z'}}dz'),
\end{aligned}$$ where $\Omega_{\rm m}$ and $\Omega_{\rm r}$ correspond to the present-day fractional densities of matter and radiation, respectively. Next, we will directly give the expressions of $E(z)$ for the $\Lambda$CDM, $w$CDM, CPL, and HDE models. Note that since we mainly focus on the evolution of the late universe, in the following we shall neglect the radiation component.
- $\Lambda$CDM model: Since the cosmological constant $\Lambda$ can explain the various cosmological observations quite well, it has nowadays become the preferred and simplest candidate for dark energy, although it has been suffering the severe theoretical puzzles. The EoS of the cosmological constant is $w=-1$, and thus we have $$E^2(z)=\Omega_{\rm{m}}(1+z)^{3}+(1-\Omega_{\rm{m}}).
$$
- $w$CDM model: In this model, the EoS of dark energy is assumed to be a constant, i.e., $w={\rm constant}$, and thus it is the simplest case for the dynamical dark energy. For this model, the expression of $E(z)$ is given by $$E^2(z)=\Omega_{\rm{m}}(1+z)^{3}+(1-\Omega_{\rm{m}})(1+z)^{3(1+w)}.$$
- CPL model: In this model, the form of the EoS of dark energy $w(a)$ is parameterized as $w(a) = w_{0}+ w_{a}(1 - a)$ with two free parameters $w_{0}$ and $w_{a}$. Thus, we have $$\begin{aligned}
E^{2}(z)&=\Omega_{\rm{m}}(1+z)^{3}+(1-\Omega_{\rm{m}})\\
&\times(1+z)^{3(1+w_{\rm{0}}+w_{\rm{a}})}\exp\left(-\frac{3w_{\rm{a}}z}{1+z}\right).
\end{aligned}$$
- HDE model: In this model, the dark energy density is assumed to be of the form $\rho_{{\rm de}}=3c^{2}M^{2}_{\rm{pl}}R_{\rm{eh}}^{-2}$ [@Li:2004rb], where $c$ is a dimensionless parameter, $M_{\rm pl}$ is the reduced Planck mass, and $R_{\rm eh}$ is the future event horizon defined as $R_{\rm{eh}}(t)=ar_{\rm{max}}(t)=a(t)\int_t^\infty{dt'}/{a(t')}$. The evolution of the universe in this model is determined by the following two differential equations, $$\begin{aligned}
\frac{1}{E(z)}\frac{dE(z)}{dz}=-\frac{\Omega_{\rm{de}}(z)}{1+z}\left(\frac{1}{2}+\frac{\sqrt{\Omega_{\rm{de}}(z)}}{c}-\frac{3}{2\Omega_{\rm{de}}(z)}\right),
\end{aligned}$$ $$\begin{aligned}
\frac{d\Omega_{\rm{de}}(z)}{dz}=-\frac{2\Omega_{\rm{de}}(z)(1-\Omega_{\rm{de}}(z))}{1+z}\left(\frac{1}{2}+\frac{\sqrt{\Omega_{\rm{de}}(z)}}{c}\right).
\end{aligned}$$ Numerically solving the two differential equations with the initial conditions $E(0)=1$ and $\Omega_{\rm de}(0)=1-\Omega_{\rm m}$ will directly give the evolutions of $E(z)$ and $\Omega_{\rm de}(z)$.
Current mainstream cosmological observations
--------------------------------------------
**SN data:** We use the largest compilation of type Ia supernovae (SN) data in this work, which is named the Pantheon compilation [@Scolnic:2017caz]. The Pantheon compilation consists of 1048 SN data, which is composed of the subset of 279 SN data from the Pan-STARRS1 Medium Deep Survey in the redshift range of $0.03 < z < 0.65$ and useful distance estimates of SN from SDSS, SNLS, various low-redshift and HST samples in the redshift range of $0.01 < z < 2.3$. According to the observational point of view, using a modified version of the Tripp formula [@Tripp], in the SALT2 spectral model [@Guy], the distance modulus can be expressed as [@Scolnic:2017caz] $$\label{SNu}
{\mu}=m_{\rm{B}}-M+\alpha \times x_{1}-\beta \times c +\Delta_{M}+\Delta_{B},$$ where $m_{\rm{B}}$, $x_{1}$, and $c$ represent the log of the overall flux normalization, the light-curve shape parameter, and the color in the light-curve fit of SN, respectively, $M$ repersents the absolute B-band magnitude with $x_{1} = 0$ and $c = 0$ for a fiducial SN, $\alpha$ and $\beta$ are the coefficients of the relation between luminosity and stretch and of the relation between luminosity and color, respectively, $\Delta_{M}$ is the distance correction from the host-galaxy mass of the SN, and $\Delta_{B}$ is the distance correction from predicted biases of simulations.
The luminosity distance $d_{\rm L}$ to a supernova can be given by $$d_{{\rm L}}(z)=\frac{1+z}{H_{0}} \int_{0}^{z} \frac{dz'}{E(z')},$$ where $E(z)= H(z)/H_{0}$. Note that we consider a flat universe throughout this work. The $\chi^{2}$ function for SN observation is expressed as $$\chi^{2}_{\rm{SN}}=({\mu}-\mu_{\rm{th}})^{\dagger}C_{\rm SN}^{-1}({\mu}-\mu_{\rm{th}}),$$ where $C_{\rm SN}$ is the covariance matrix of the SN observation [@Scolnic:2017caz], and the theoretical distance modulus $\mu_{\rm{th}}$ is given by $$\mu_{\rm{th}}=5\log_{10}\frac{d_{\rm{L}}}{10\rm{pc}}.$$
**CMB data:** For the cosmic microwave background (CMB) anisotropies data, we use the “Planck distance priors” from the Planck 2015 data [@Ade:2015rim]. The distance priors include the shift parameter $R$, the “acoustic scale” $\ell_{\rm{A}}$, and the baryon density $\omega_{b}$, defined by $$R\equiv\sqrt{\Omega_{\rm{m}}H^{2}_{0}}(1+z_{\ast})D_{\rm{A}}(z_{\ast}),$$ $$\ell_{\rm{A}}\equiv(1+z_{\ast})\frac{\pi D_{\rm{A}}(z_{\ast})}{r_{\rm{s}}(z_{\ast})},\label{la}$$ $$\omega_{b}\equiv \Omega_{b}h^{2},$$ where $\Omega_{\rm{m}}$ is the present-day fractional matter density, and $D_{\rm{A}}(z_{\ast})$ denotes the angular diameter distance at $z_{\ast}$ with $z_{\ast}$ being the redshift of the decoupling epoch of photons. In a flat universe, $D_{\rm{A}}$ can be expressed as $$D_{\rm{A}}(z)=\frac{1}{H_{0}(1+z)}\int_{0}^{z}\frac{dz'}{E(z')},\label{DA}$$ and $r_{\rm{s}}(a)$ can be given by $$r_{\rm{s}}(a)=\frac{1}{\sqrt{3}}\int_{0}^{a}\frac{da'}{a'H(a')\sqrt{1+(3\Omega_{{\rm b}}/4\Omega_{{\rm \gamma}})a'}},\label{rs}$$ where $\Omega_{{\rm b}}$ and $\Omega_{{\gamma}}$ are the present-day energy densities of baryons and photons, respectively. In this work, we adopt $3\Omega_{{\rm b}}/4\Omega_{{\rm \gamma}}=31500\Omega_{\rm{b}}h^{2}(T_{\rm{cmb}}/2.7{\rm K})^{-4}$ and $T_{\rm{cmb}}=2.7255$ K. $z_{\ast}$ can be calculated by the fitting formula [@Hu:1995en], $$z_{\ast}=1048[1+0.00124(\Omega_{{\rm b}}h^{2})^{-0.738}][1+g_{1}(\Omega_{{\rm m}}h^{2})^{g_{2}}],$$ where $$g_{1}=\frac{0.0783(\Omega_{\rm{b}}h^{2})^{-0.238}}{1+39.5(\Omega_{\rm{b}}h^{2})^{-0.76}}, \; g_{2}=\frac{0.560}{1+21.1(\Omega_{\rm{b}}h^{2})^{1.81}}.$$
The three values can be obtained from the Planck TT+LowP data [@Ade:2015rim]: $R=1.7488\pm0.0074$, $\ell_{\rm{A}}=301.76\pm0.14$, and $\Omega_{\rm{b}}h^{2}=0.02228\pm0.00023$. The $\chi^{2}$ function for CMB is $$\chi^{2}_{\rm{CMB}}=\Delta p_{i}[{\rm Cov}^{-1}_{\rm{CMB}}(p_{i},p_{j})]\Delta p_{j}, \quad \Delta p_{i}=p_{i}^{\rm{th}}-p_{i}^{\rm{obs}},$$ where $p_{1}=\ell_{\rm{A}}$, $p_{2}=R$, $p_{3}=\omega_{b}$, and ${\rm Cov}^{-1}_{\rm CMB}$ is the inverse covariance matrix and can be found in Ref. [@Ade:2015rim].
**BAO data:** From the baryon acoustic oscillations (BAO) measurements, we can obtain the distance ratio $D_{{\rm V}}(z)/r_{\rm s}(z_{\rm d})$ or $D_{{\rm M}}(z)/r_{\rm s}(z_{\rm d})$. The spherical average gives the expression of $D_{{\rm V}}(z)$, $$D_{\rm{V}}(z)\equiv\left[D^{2}_{\rm{M}}(z)\frac{z}{H(z)}\right]^{1/3},$$ where $D_{\rm{M}}(z)=(1+z)D_{\rm A}(z)$ is the the comoving angular diameter distance [@Alam:2016hwk]. $r_{\rm s}(z_{\rm d})$ is the comoving sound horizon size at the redshift $z_{\rm d}$ of the drag epoch and its calculated value can be given by Eq. (\[rs\]). $z_{\rm{d}}$ is given by the fitting formula [@Hu:1995en], $$z_{\rm{d}}=\frac{1291(\Omega_{\rm{m}}h^2)^{0.251}}{1+0.659(\Omega_{\rm{m}}h^2)^{0.828}}[1+b_1(\Omega_{\rm{b}}h^2)^{b_2}],$$ with $$\begin{gathered}
b_1=0.313(\Omega_{\rm{m}}h^2)^{-0.419}[1+0.607(\Omega_{\rm{m}}h^2)^{0.674}],\\
b_2=0.238(\Omega_{\rm{m}}h^2)^{0.223}.
\end{gathered}$$
We use five BAO data points form the 6dF Galaxy Survey at $z_{\rm eff} = 0.106$ [@Beutler], the SDSS-DR7 at $z_{\rm eff} = 0.15$ [@Ross:2014qpa], and the BOSS-DR12 at $z_{\rm eff} = 0.38$, $z_{\rm eff} = 0.51$, and at $z_{\rm eff} = 0.61$ [@Alam:2016hwk]. The distance ratio $D_{\rm V}(z)/r_{\rm s}(z_{\rm d})$ or $D_{{\rm M}}(z)/r_{\rm s}(z_{\rm d})$ for the BAO data are shown in Table \[bao\].
$z$ $\xi(z)$ Experiment Reference
------- ------------------------------------------------------------- ------------ ------------------
0.106 $D_{\rm{V}}(z)/r_{\rm s}(z_{\rm d})=2.976\pm0.133$ 6dFGS [@Beutler]
0.15 $D_{\rm{V}}(z)/r_{\rm s}(z_{\rm d})=4.466\pm0.168$ SDSS-DR7 [@Ross:2014qpa]
0.38 $D_{\rm{M}}(z)/r_{\rm s}(z_{\rm d})=10.231\pm0.149\pm0.074$ BOSS-DR12 [@Alam:2016hwk]
0.51 $D_{\rm{M}}(z)/r_{\rm s}(z_{\rm d})=13.364\pm0.183\pm0.095$ BOSS-DR12 [@Alam:2016hwk]
0.61 $D_{\rm{M}}(z)/r_{\rm s}(z_{\rm d})=15.611\pm0.223\pm0.115$ BOSS-DR12 [@Alam:2016hwk]
{width="8cm"}
The $\chi^{2}$ function for BAO measurements is $$\chi^2_{\rm BAO}=\sum\limits_{ i=1}^5 \frac{(\xi^{\rm obs}_{ i}-\xi^{\rm th}_{ i})^2}{\sigma_{ i}^2},$$ where $\xi_{\rm th}$ and $\xi_{\rm obs}$ represent the theoretically predicted value and the experimentally measured value of the $i$-th data point for the BAO observations, respectively, and $\sigma_{i}$ is the standard deviation of the $i$-th data point.
Redshift drift observations from E-ELT and SKA
----------------------------------------------
The actual measurement for the SL-test signal is the shift in the spectroscopic velocity ($\Delta v$) for a source in a given time interval ($\Delta t_o$). The spectroscopic velocity shift is usually expressed as [@Loeb:1998bu] $$\label{3}
\Delta v=\frac{\Delta z}{1+z} =H_{\rm{0}} \Delta t_{o} \left[1-\frac{E(z)}{1+z}\right],$$ where $E(z)$ is determined by a specific cosmological model.
The measurement of velocity shift will be achieved by the upcoming experiments such as the E-ELT and SKA through two different means. The E-ELT will be able to observe the Lyman-$\alpha$ absorption lines of distant quasar systems to achieve the measurement of $\Delta v$ in the redshift range of $z\in[2,5]$ [@Loeb:1998bu; @Liske]. The SKA will measure the spectroscopic velocity shift $\Delta v$ by observing the neutral hydrogen emission signals of galaxies at the precision of one percent in the redshift range of $z\in[0,1]$. Obviously, the E-ELT and SKA experiments will be the ideal complements with each other, because of the explorations of different periods for the cosmic evolution.
**E-ELT mock data:** For the E-ELT data, as discussed in Ref. [@Liske:2008ph], the standard deviation on $\Delta v$ can be estimated as $$\label{4}
\sigma_{\Delta v} = 1.35\left(\frac{2370}{S/N}\right)\left(\frac{N_{\rm QSO}}{30}\right)^{-1/2}\left(\frac{1+z_{\rm QSO}}{5}\right)^{x} ~\mathrm{cm}~\mathrm{s}^{-1},$$ where $S/N$ is the signal-to-noise ratio of the Lyman-$\alpha$ spectrum, $N_{\rm QSO}$ is the number of observed quasars at the effective redshift $z_{\rm QSO}$, and $x$ is $1.7$ for $2\leq z\leq4$ and $0.9$ for $z\geq4$. In this work, we assume $S/N=3000$ and $N_{\rm QSO}=30$. We generate 30 mock data with a uniform distribution for the E-ELT’s redshift drift observation in six redshift bins (the redshift interval $\Delta z=0.5$ for each bin), and we assume the observation time of $\Delta t_o =10$ years.
**SKA mock data:** For the case of SKA, we follow the prescription given in Refs. [@Klockner:2015rqa; @Martins:2016bbi] to produce the mock data of redshift drift. It is shown in Refs. [@Klockner:2015rqa; @Martins:2016bbi] that if SKA could have the full sensitivity and detect a billion galaxies, the evolution of the frequency shift in redshift space would be estimated to a precision of one percent. Thus, we consider the following two scenarios:
1. For SKA Phase 1, in our simulation, we produce 3 mock data of the drift $\Delta v$ in redshift bin centered on $z_{i}=[0.1,0.2,0.3]$ with velocity uncertainties $\sigma_{\Delta v}$ respectively of $3\%$ in the first bin, $5\%$ in the second bin and $10\%$ in the third bin. The redshift interval $\Delta z$ is 0.1 for each bin and the timespan $\Delta t_o$ is 40 years. Note that although a timespan of 40 years is long integration time, it can be as a benchmark scenario to improve sensitivity and redshift coverage in the full SKA configuration.
2. For SKA Phase 2, we generate 10 mock data of the drift $\Delta v$ in ten redshift bins. The mock data are covering from $z=0.1$ to $z=1.0$ with the velocity uncertainties $\sigma_{\Delta v}$ ranging from 1% to 10%. This could be reached in the timespan $\Delta t_o = 0.5$ years, which leads to an extremely competitive and ideal scenario. Note that the requirement of this scenario is $10^7$ galaxies observed in each bin [@Martins:2016bbi].
In addition, in the mock data simulation, we adopt the scheme accordant with our previous papers [@Geng:2018pxk; @Geng:2014ypa; @Guo:2015gpa; @He:2016rvp; @Geng:2014hoa; @Geng:2015ara; @Zhang:2010im]. In other words, the fiducial cosmology for the SL simulated data from the E-ELT or the SKA is chosen to be the best-fit cosmology according to the analysis of the data combination of SN+CMB+BAO in the $\Lambda$CDM model, the $w$CDM model, the CPL model, and the HDE model, respectively.
Results and discussion {#sec:Results and Discussions}
======================
Since the $\Lambda$CDM model is widely regarded as a prototype of the standard cosmology, we take this model as a reference model to test the constraining power of the SKA-only mock data and make an analysis of constraints on cosmological parameters when the redshift drift data of SKA and E-ELT are combined. In Fig. \[mockDV\], we show the simulated redshift-drift data for E-ELT, SKA1, and SKA2, using the $\Lambda$CDM model as the fiducial model. In this figure, the curve of $\Delta v(z)$ is plotted according to Eq. (\[3\]), with the fiducial values of parameters given by the best fit to the SN+CMB+BAO data; the error bars on $\Delta v$, i.e., $\sigma_{\Delta v}$, for each redshift bin, are plotted according to Eq. (\[4\]) for E-ELT, and according to the detailed prescriptions described in the above section (the part entitled “SKA mock data”) for SKA1 and SKA2. We find that in the E-ELT case the error of $\Delta v$ decreases with the increase of redshift, and vice versa in the SKA1 case or the SKA2 case. In Fig. \[ELTvsSKA\], we plot the two-dimensional posterior contours at $68\%$ and $95\%$ confidence level (CL) in the $\Lambda$CDM model. We clearly see that using the SKA1-only mock data, the $\Lambda$CDM model can only be loosely constrained, while the model can be well constrained using the SKA2-only mock data.
In addition, form Fig. \[ELTvsSKA\], we clearly see that in the $\Lambda$CDM model, from the E-ELT, $\Omega_{\rm m}$ and $h$ are in strong anti-correlation while constraints from SKA1 or SKA2 provide a positive correlation for $\Omega_{\rm m}$ and $h$, and thus the orthogonality of the two degeneracy orientations leads to a complete breaking for the parameter degeneracy. Thus, the constraints from the combination of E-ELT and SKA (SKA1 or SKA2) would have a tremendous improvement, as shown by the gray and red contours in Fig. \[ELTvsSKA\]. This may be due to the fact that the experiments of the E-ELT and the SKA are complementary in mapping the expansion history of the universe with a model-independent way. That is to say, these two experiments will be able to directly perform reconstruction of the expansion history of the universe in the dark matter- or dark energy-dominated epochs by using different observational techniques. Particularly, the result from the combination of E-ELT+SKA2 is almost as good as the constraint from the combination of SN+CMB+BAO, which implies that the redshift drift observation would have chance to be one of the most competitive cosmological probes.
Meanwhile, we find that the degeneracy orientation of E-ELT+SKA1 or E-ELT+SKA2 in the parameter plane is evidently different from result for the combination of SN+CMB+BAO. This phenomenon would result in an effective breaking of the parameter degeneracy and a significant improvement of the constraints on dark energy. It is of extreme interest to know what role the redshift drift data of SKA will play in constraining dark energy in the future. Next we will explore this issue in detail.
Paramerer Prior
----------------------- -- -- -- -- -- -------------------- -- -- --
$\Omega_{\rm b}h^{2}$ $[0.005,0.100]$
$\Omega_{\rm c}h^{2}$ $[0.001,0.990]$
$w_{0}$ $[-3.000,-0.900]$
$w_{a}$ $[-14.000,-0.700]$
$c$ $[0.200,1.200]$
------------------ ------------------------------ ------------------------------- ------------------------------- ------------------------------ -- ------------------------------ ------------------------------- ------------------------------- ------------------------------
Parameter $\Lambda$CDM $w$CDM CPL HDE $\Lambda$CDM $w$CDM CPL HDE
$w_{0}$ $-$ $-1.0401^{+0.0477}_{-0.0452}$ $-1.1777^{+0.0614}_{-0.0683}$ $-$ $-$ $-1.0385^{+0.0442}_{-0.0447}$ $-1.1792^{+0.0644}_{-0.0663}$ $-$
$w_{a}$ $-$ $-$ $0.6028^{+0.1989}_{-0.2096}$ $-$ $-$ $-$ $0.6012^{+0.2036}_{-0.2107}$ $-$
$c$ $-$ $-$ $-$ $0.6350^{+0.0391}_{-0.0374}$ $-$ $-$ $-$ $0.6318^{+0.0358}_{-0.0316}$
$\Omega_{\rm m}$ $0.3277^{+0.0039}_{-0.0039}$ $0.3248^{+0.0054}_{-0.0049}$ $0.3135^{+0.0050}_{-0.0053}$ $0.3086^{+0.0052}_{-0.0054}$ $0.3277^{+0.0035}_{-0.0034}$ $0.3250^{+0.0043}_{-0.0044}$ $0.3136^{+0.0049}_{-0.0055}$ $0.3082^{+0.0039}_{-0.0041}$
$\emph{h}$ $0.6645^{+0.0029}_{-0.0029}$ $0.6702^{+0.0069}_{-0.0074}$ $0.6751^{+0.0067}_{-0.0060}$ $0.6770^{+0.0077}_{-0.0072}$ $0.6646^{+0.0026}_{-0.0026}$ $0.6700^{+0.0065}_{-0.0065}$ $0.6751^{+0.0065}_{-0.0062}$ $0.6776^{+0.0058}_{-0.0058}$
Parameter $\Lambda$CDM $w$CDM CPL HDE $\Lambda$CDM $w$CDM CPL HDE
$w_{0}$ $-$ $-1.0382^{+0.0373}_{-0.0392}$ $-1.1805^{+0.0565}_{-0.0541}$ $-$ $-$ $-1.0383^{+0.0288}_{-0.0310}$ $-1.1764^{+0.0395}_{-0.0443}$ $-$
$w_{a}$ $-$ $-$ $0.6103^{+0.1753}_{-0.2093}$ $-$ $-$ $-$ $0.6016^{+0.1615}_{-0.1871}$ $-$
$c$ $-$ $-$ $-$ $0.6345^{+0.0134}_{-0.0125}$ $-$ $-$ $-$ $0.6354^{+0.0075}_{-0.0072}$
$\Omega_{\rm m}$ $0.3277^{+0.0029}_{-0.0029}$ $0.3248^{+0.0032}_{-0.0033}$ $0.3132^{+0.0040}_{-0.0037}$ $0.3082^{+0.0032}_{-0.0028}$ $0.3277^{+0.0011}_{-0.0011}$ $0.3248^{+0.0013}_{-0.0012}$ $0.3135^{+0.0017}_{-0.0016}$ $0.3081^{+0.0024}_{-0.0025}$
$\emph{h}$ $0.6645^{+0.0023}_{-0.0023}$ $0.6701^{+0.0050}_{-0.0049}$ $0.6753^{+0.0050}_{-0.0052}$ $0.6773^{+0.0024}_{-0.0026}$ $0.6645^{+0.0011}_{-0.0013}$ $0.6701^{+0.0026}_{-0.0024}$ $0.6750^{+0.0034}_{-0.0030}$ $0.6774^{+0.0017}_{-0.0016}$
------------------ ------------------------------ ------------------------------- ------------------------------- ------------------------------ -- ------------------------------ ------------------------------- ------------------------------- ------------------------------
-------------------------- -------------- ---------- ---------- ---------- -- -------------- ------------ ---------- ----------
Error $\Lambda$CDM $w$CDM CPL HDE $\Lambda$CDM $w$CDM CPL HDE
$\sigma(w_{0})$ $-$ $0.0465$ $0.0649$ $-$ $-$ $0.0445$ $0.0654$ $-$
$\sigma(w_{a})$ $-$ $-$ $0.2043$ $-$ $-$ $-$ $0.2072$ $-$
$\sigma(c)$ $-$ $-$ $-$ $0.0383$ $-$ $-$ $-$ $0.0337$
$\sigma(\Omega_{\rm m})$ $0.0039$ $0.0052$ $0.0052$ $0.0053$ $0.0035$ $0.0044$ $0.0052$ $0.0040$
$\sigma(\emph{h})$ $0.0029$ $0.0072$ $0.0064$ $0.0075$ $0.0026$ $0.0065$ $0.0064$ $0.0058$
Error $\Lambda$CDM $w$CDM CPL HDE $\Lambda$CDM $w$CDM CPL HDE
$\sigma(w_{0})$ $-$ $0.0383$ $0.0553$ $-$ $-$ $0.0299$ $0.0419$ $-$
$\sigma(w_{a})$ $-$ $-$ $0.1923$ $-$ $-$ $-$ $01743$ $-$
$\sigma(c)$ $-$ $-$ $-$ $0.0130$ $-$ $-$ $-$ $0.0074$
$\sigma(\Omega_{\rm m})$ $0.0029$ $0.0033$ $0.0039$ $0.0030$ $0.0011$ $0.0013$ $0.0017$ $0.0025$
$\sigma(\emph{h})$ $0.0023$ $0.0050$ $0.0051$ $0.0025$ $0.0012$ $0.0.0025$ $0.0032$ $0.0017$
-------------------------- -------------- ---------- ---------- ---------- -- -------------- ------------ ---------- ----------
------------------------------- -------------- ---------- ---------- ---------- -- -------------- ---------- ---------- ----------
Precision $\Lambda$CDM $w$CDM CPL HDE $\Lambda$CDM $w$CDM CPL HDE
$\varepsilon(w_{0})$ $-$ $0.0447$ $0.0551$ $-$ $-$ $0.0438$ $0.0555$ $-$
$\varepsilon(w_{a})$ $-$ $-$ $0.3389$ $-$ $-$ $-$ $0.3446$ $-$
$\varepsilon(c)$ $-$ $-$ $-$ $0.0603$ $-$ $-$ $-$ $0.0533$
$\varepsilon(\Omega_{\rm m})$ $0.0119$ $0.0160$ $0.0166$ $0.0172$ $0.0107$ $0.0135$ $0.0166$ $0.0130$
$\varepsilon(\emph{h})$ $0.0044$ $0.0107$ $0.0095$ $0.0111$ $0.0039$ $0.0097$ $0.0094$ $0.0086$
Precision $\Lambda$CDM $w$CDM CPL HDE $\Lambda$CDM $w$CDM CPL HDE
$\varepsilon(w_{0})$ $-$ $0.0398$ $0.0468$ $-$ $-$ $0.0288$ $0.0356$ $-$
$\varepsilon(w_{a})$ $-$ $-$ $0.3151$ $-$ $-$ $-$ $0.2897$ $-$
$\varepsilon(c)$ $-$ $-$ $-$ $0.0205$ $-$ $-$ $-$ $0.0116$
$\varepsilon(\Omega_{\rm m})$ $0.0088$ $0.0102$ $0.0125$ $0.0097$ $0.0034$ $0.0040$ $0.0054$ $0.0081$
$\varepsilon(\emph{h})$ $0.0035$ $0.0075$ $0.0076$ $0.0037$ $0.0018$ $0.0037$ $0.0047$ $0.0025$
------------------------------- -------------- ---------- ---------- ---------- -- -------------- ---------- ---------- ----------
We constrain the $\Lambda$CDM, $w$CDM, CPL and HDE models by using the data combinations of SN+CMB+BAO, SN+CMB+BAO+E-ELT, SN+CMB+BAO+E-ELT+SKA1, and SN+CMB+BAO+E-ELT+SKA2 to complete our analysis. The priors of the free parameters are given in Table \[prior\]. Here, $\Omega_{\rm b}h^{2}$ and $\Omega_{\rm c}h^{2}$ respectively stand for the physical baryon and cold dark matter densities. The constraint results are presented in Tables \[Parameter\]–\[Precision\] and Figs. \[h\]–\[parameters\]. In Table \[Parameter\], we show the best-fit results with the 1$\sigma$ errors quoted. The constraint errors and precisions of the cosmological parameters are given in Tables \[Error\]–\[Precision\], respectively. Here, for a parameter $\xi$, we use $\sigma(\xi)$ to denote its 1$\sigma$ error. For the cases that its distribution slightly deviates from the gaussian distribution, we adopt the value of averaging the upper-limit and lower-limit errors. We use $\varepsilon(\xi)=\sigma(\xi)/\xi_{\rm bf}$ to denote the relative error of the parameter $\xi$, where $\xi_{\rm bf}$ is its best-fit value. In this paper, for convenience, we also informally call $\varepsilon(\xi)$ the “constraint precision” of the parameter $\xi$. In Figs. \[h\]–\[parameters\], we show the two-dimensional posterior distribution contours of constraint results in the $\Lambda$CDM, $w$CDM, CPL and HDE models at the 68$\%$ and 95$\%$ CL.
From these figures, we clearly see that when the E-ELT mock data are combined with SN+CMB+BAO, the parameter spaces can be significantly reduced in the $\Lambda$CDM, $w$CDM, and HDE models, while there is little significant improvement in the parameter space for the CPL model. Adding the SKA1 mock data to the data combination of SN+CMB+BAO+E-ELT, the parameter spaces are sharply reduced. In particular, when the SKA2 mock data are combined with SN+CMB+BAO+E-ELT, the improvement is actually much more significant than the case of SN+CMB+BAO+E-ELT+SKA1. Meanwhile, from Fig. \[parameters\], we can easily find that the E-ELT and SKA mock data can help to break the parameter degeneracies, in particular between the parameters $\Omega_{\rm m}$ and $c$ in the HDE model.
From Table \[Precision\], we can easily find that the E-ELT, SKA1, and SKA2 can significantly improve the constraints on almost all the parameters to different extent, in particular for SKA2. Concretely, when the E-ELT mock data are combined with SN+CMB+BAO, the precision of $\Omega_{m}$ is improved from $1.19\%$ to $1.07\%$ in the $\Lambda$CDM model, from $1.60\%$ to $1.35\%$ in the $w$CDM model, from $1.72\%$ to $1.30\%$ in the HDE model. The precision of $h$, $w_{0}$, $w_{a}$ and $c$ are also enhanced in the $\Lambda$CDM, $w$CDM, and HDE models; for details, see Table \[Precision\]. Adding the SKA1 mock data to the data combination of SN+CMB+BAO+E-ELT, the improvement of the constraint on parameter $\Omega_{\rm m}$ is from $1.07\%$ to $0.88\%$ in the $\Lambda$CDM model, from $1.35\%$ to $1.02\%$ in the $w$CDM model, from $1.66\%$ to $1.25\%$ in the CPL model, and from $1.30\%$ to $0.97\%$ in the HDE model. For the parameter $h$, the constraint is improved from $0.39\%$ to $0.35\%$ in the $\Lambda$CDM model, from $0.97\%$ to $0.75\%$ in the $w$CDM model, from $0.94\%$ to $0.76\%$ in the CPL model, and from $0.86\%$ to $0.37\%$. For the parameters of dark energy, the improvement is from $4.38\%$ to $3.98\%$ for the parameter $w$ in the $w$CDM model, from $5.55\%$ to $4.68\%$ for the parameter $w_{0}$ in the CPL model, from $34.46\%$ to $31.51\%$ for the parameter $w_{a}$ in the CPL model, and from $5.33\%$ to $2.05\%$ for the parameter $c$ in the HDE model.
Furthermore, when the SKA2 mock data are combined with SN+CMB+BAO+E-ELT, the improvement of the constraint on parameter $\Omega_{\rm m}$ is from $1.07\%$ to $0.34\%$ in the $\Lambda$CDM model, from $1.35\%$ to $0.40\%$ in the $w$CDM model, from $1.66\%$ to $0.54\%$ in the CPL model, and from $1.30\%$ to $0.81\%$ in the HDE model. For the parameter $h$, the constraint is improved from $0.39\%$ to $0.18\%$ in the $\Lambda$CDM model, from $0.97\%$ to $0.37\%$ in the $w$CDM model, from $0.94\%$ to $0.47\%$ in the CPL model, and from $0.86\%$ to $0.25\%$ in the HDE model. For the parameters of dark energy, the improvement is from $4.38\%$ to $2.88\%$ for the parameter $w$ in the $w$CDM model, from $5.55\%$ to $3.56\%$ for the parameter $w_{0}$ in the CPL model, from $34.46\%$ to $28.97\%$ for the parameter $w_{a}$ in the CPL model, and from $5.49\%$ to $1.16\%$ for the parameter $c$ in the HDE model. Therefore, we conclude that the redshift drift data of SKA will help to significantly improve the constraints of parameters and break the degeneracy between the parameters in constraining dark energy in the future.
Conclusion {#sec:Conclusion}
==========
In this work, we wish to investigate what extent the cosmological parameters can be constrained to when the redshift drift data of SKA are used and what will happen when the combination of SKA and E-ELT mock data is considered. We use the five data sets, i.e., SKA1, SKA2, E-ELT, E-ELT+SKA1, E-ELT+SKA2, and SN+CMB+BAO to reach our aims in the $\Lambda$CDM model. We find that using the SKA2 mock data alone, the $\Lambda$CDM model can be constrained well, while the constraint is weak from the mock data of SKA1-only. When the redshift drift mock data of SKA and E-ELT are combined, the results show that the parameter space is dramatically reduced almost as good as SN+CMB+BAO. Thus, the last aim of this work is to investigate what role the redshift drift data of SKA will play in constraining dark energy in the future. To fulfill the task, we employ several concrete dark energy models, including the $\Lambda$CDM, $w$CDM, CPL, and HDE models, which are still consistent with the current observations at least to some extent.
We first use the data combination of SN+CMB+BAO to constrain the four dark energy models, and then we consider the addition of the E-ELT mock data in the data combination, i.e., we use the data combination of SN+CMB+BAO+E-ELT to constrain the models. The constraints on cosmological parameters are tremendously improved for the $\Lambda$CDM, $w$CDM, and HDE models, while E-ELT mock data do not help improve constraints in the CPL model. When adding the SKA1 mock data to the SN+CMB+BAO+E-ELT, the constraint results are significantly improved in all the four dark energy models. For example, with the help of the SKA1 mock data, the constraints on $\Omega_{\rm m}$ are improved by 10$\%$–25$\%$, and the constraints on $h$ are improved by 10$\%$–70$\%$. Furthermore, when the SKA2 mock data are combined with the dataset of SN+CMB+BAO+E-ELT, the constraint results are tremendously improved in all the four dark energy models. Concretely, the constraints on $\Omega_{\rm m}$ are improved by 40$\%$–70$\%$, and the constraints on $h$ are improved by 50$\%$–75$\%$. We also find that the degeneracy between cosmological parameters could be effectively broken by the combination of the E-ELT and SKA mock data. Therefore, we can conclude that in the future the redshift-drift observation of SKA would help to improve the constraints in constraining dark energy and have a good potential to be one of the most competitive cosmological probes in constraining dark energy.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11875102, 11975072, 11835009, 11522540, and 11690021) and the Top-Notch Young Talents Program of China.
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[^1]: Corresponding author
| 0 |
---
abstract: 'We propose adjustable phase shift pilots (APSPs) for channel acquisition in wideband massive multiple-input multiple-output (MIMO) systems employing orthogonal frequency division multiplexing (OFDM) to reduce the pilot overhead. Based on a physically motivated channel model, we first establish a relationship between channel space-frequency correlations and the channel power angle-delay spectrum in the massive antenna array regime, which reveals the channel sparsity in massive MIMO-OFDM. With this channel model, we then investigate channel acquisition, including channel estimation and channel prediction, for massive MIMO-OFDM with APSPs. We show that channel acquisition performance in terms of sum mean square error can be minimized if the user terminals’ channel power distributions in the angle-delay domain can be made non-overlapping with proper phase shift scheduling. A simplified pilot phase shift scheduling algorithm is developed based on this optimal channel acquisition condition. The performance of APSPs is investigated for both one symbol and multiple symbol data models. Simulations demonstrate that the proposed APSP approach can provide substantial performance gains in terms of achievable spectral efficiency over the conventional phase shift orthogonal pilot approach in typical mobility scenarios.'
author:
- 'Li You, Xiqi Gao, A. Lee Swindlehurst, and Wen Zhong [^1][^2][^3][^4]'
title: 'Channel Acquisition for Massive MIMO-OFDM with Adjustable Phase Shift Pilots'
---
Adjustable phase shift pilots, massive MIMO-OFDM, channel estimation, channel prediction, channel acquisition, pilot phase shift scheduling.
Introduction
============
5G cellular wireless systems are expected to support 1000 times faster data rates than the currently deployed 4G long-term evolution (LTE) system. To achieve the high data rates required by 5G, many technologies have been proposed [@Andrews14What; @Boccardi14Five; @Wang14Cellular]. Among them, massive multiple-input multiple-output (MIMO) systems, which deploy unprecedented numbers of antennas at the base stations (BSs) to simultaneously serve a relatively large number of user terminals (UTs), are believed to be one of the key candidate technologies for 5G [@Marzetta10Noncooperative; @Larsson14Massive; @Lu14overview].
Orthogonal frequency division multiplexing (OFDM) is a multi-carrier modulation technology suited for high data rate wideband wireless transmission [@Cimini85Analysis; @Stuber04Broadband]. Due to its robustness to channel frequency selectivity and relatively efficient implementation, OFDM combined with massive MIMO is a promising technique for wideband massive MIMO transmission [@Marzetta10Noncooperative]. As in conventional MIMO-OFDM, the performance of massive MIMO-OFDM is highly dependant on the quality of the channel acquisition. Pilot design and channel acquisition for massive MIMO-OFDM is of great practical importance.
Optimal pilot design and channel acquisition for conventional MIMO-OFDM has been extensively investigated in the literature. The most common approach is to estimate the channel response in the delay domain, and optimal pilots sent from different transmit antennas are typically assumed to satisfy the phase shift orthogonality condition in both the single-user case [@Li02Simplified; @Barhumi03Optimal; @Minn06Optimal] and the multi-user case [@Chi11Training]. Note that such phase shift orthogonal pilots (PSOPs) have been adopted in LTE [@Dahlman11LTE]. When channel spatial correlations are taken into account, optimal pilot design has been investigated for both the single-user case [@Tuan10Optimized] and multi-user case [@Tran12Training]. Although these orthogonal pilot approaches can eliminate pilot interference in the same cell, they do not take into account the pilot overhead issue, which is thought to be one of the limiting factors for throughput in massive MIMO-OFDM \[4\]. When such approaches are directly adopted in time-division duplex (TDD) massive MIMO-OFDM, the corresponding pilot overhead is proportional to the sum of the number of UT antennas, and would be prohibitively large as the number of UTs becomes large. This becomes the system bottleneck, especially in high mobility scenarios where pilots must be transmitted more frequently. Therefore, a pilot approach that takes the pilot overhead issue into account is of importance for massive MIMO-OFDM systems.
In this paper, we propose adjustable phase shift pilots (APSPs) for massive MIMO-OFDM to reduce the pilot overhead. For APSPs, one sequence along with different adjustable phase shifted versions of itself in the frequency domain are adopted as pilots for different UTs. The proposed APSPs are different from conventional PSOPs [@Li02Simplified; @Barhumi03Optimal; @Chi11Training], in which phase shifts for different pilots are fixed, and phase shift differences between different pilots are no less than the maximum channel delay (divided by the system sampling duration) of all the UTs. Since in our approach the phase shifts for different pilots are adjustable, more pilots are available compared with conventional PSOPs, which leads to significantly reduced pilot overhead.
The proposed APSPs exploit the following two channel properties: First, wireless channels are sparse in many typical propagation scenarios; most channel power is concentrated in a finite region of delays and/or angles due to limited scattering [@Tse05Fundamentals; @Clerckx13MIMO; @win2chanmod; @Bajwa10Compressed]. Such channel sparsity can be resolved in the angle domain in massive MIMO due to the relatively large antenna array apertures, which has been observed in recent massive MIMO channel measurement results [@Payami12Channel; @Gao13Massive]. Second, channel sparsity patterns, i.e., channel power distributions in the angle-delay domain, for different UTs are usually different.[^5] For APSPs, when the phase shifts for pilots employed by different UTs are properly scheduled according to the above channel properties, channel acquisition can be achieved simultaneously in an almost interference-free manner as with conventional PSOPs. There has recently been increased research interest on utilizing channel sparsity for channel acquisition in massive MIMO. For instance, a time-frequency training scheme [@Dai13Spectrally] and a distributed Bayesian channel estimation scheme [@Masood15Efficient] were proposed for massive MIMO-OFDM by exploiting the channel sparsity. As the approaches in [@Masood15Efficient] and [@Dai13Spectrally] focus on channel acquisition for a single UT, the corresponding pilot overhead would still grow linearly with the number of UTs. Channel sparsity has also been exploited to mitigate pilot contamination in multi-cell massive MIMO [@Chen14Pilot; @Wen15Channel]. Note that compressive sensing has been applied to sparse channel acquisition in some recent works (see, e.g., [@Bajwa10Compressed; @Barbotin12Estimation; @Berger10Application; @Rao14Distributed] and references therein), in which the corresponding pilot signals are usually assumed to be randomly generated. However, it is usually quite difficult to implement random pilot signals in practical systems [@Candes07Sparsity]. For example, adopting large dimensional random pilot signals in the massive MIMO-OFDM systems considered here requires huge storage space and high complexity channel acquisition algorithms. In addition, a low peak-to-average power ratio (PAPR) for randomly generated pilot signals usually cannot be guaranteed. These drawbacks can be mitigated via proper design of the deterministic sensing matrices (see, e.g., [@Calderbank10Construction; @Strohmer12Measure] and references therein).
The main contributions of this paper are summarized as follows:
- Based on a physically motivated channel model, we establish a relationship between the space-frequency domain channel covariance matrix (SFCCM) and the channel power angle-delay spectrum for massive MIMO-OFDM. We show that when the number of BS antennas is sufficiently large, the eigenvectors of the SFCCMs for different UTs tend to be equal, while the eigenvalues depend on the respective channel power angle-delay spectra, which reveals the channel sparsity in the angle-delay domain. Then we propose the angle-delay domain channel response matrix (ADCRM) and the corresponding angle-delay domain channel power matrix (ADCPM), which can model the massive MIMO-OFDM channel sparsity in the angle-delay domain, and are convenient for further analyses.
- With the presented channel model, we propose APSP-based channel acquisition (APSP-CA) for massive MIMO-OFDM in TDD mode. For APSPs, equivalent channels for different UTs will experience corresponding cyclic shifts in the delay domain. Using this property, we show that the sum mean square error (MSE) of channel estimation (MSE-CE) can be minimized if the UTs’ channel power distributions in the angle-delay domain can be made non-overlapping with proper pilot phase shift scheduling. Taking the time-varying nature of the channel into account, we further investigate channel prediction during the data segment using the received pilot signals. We show that the sum MSE of channel prediction (MSE-CP) can also be minimized if the UTs’ channel power distributions in the angle-delay domain can be made non-overlapping with proper pilot phase shift scheduling, which coincides with the optimal channel estimation condition. A simplified pilot phase shift scheduling algorithm is developed based on this optimal channel acquisition condition. The proposed APSP-CA approach is investigated for cases involving both one symbol and multiple consecutive symbols.
- The proposed APSP-CA is evaluated in several typical propagation scenarios, and significant performance gains in terms of achievable spectral efficiency over the conventional PSOP-based channel acquisition (PSOP-CA) are demonstrated, especially in high mobility scenarios.
Portions of this work previously appeared in the conference paper [@You15Adjustable].
Notations {#sec:prwb_not}
---------
We adopt the following notation throughout the paper. We use ${\bar{\jmath}}=\sqrt{-1}$ to denote the imaginary unit. $\left\lfloor x\right\rfloor$ ($\left\lceil x\right\rceil$) denotes the largest (smallest) integer not greater (smaller) than $x$. ${\left\langle\cdot\right\rangle_{N}}$ denotes the modulo-$N$ operation. $\delta(\cdot)$ denotes the delta function. Upper (lower) case boldface letters denote matrices (column vectors). The notation $\triangleq$ is used for definitions. Notations $\sim$ and $\propto$ represent “distributed as” and “proportional to”, respectively. We adopt ${{\mathbf{I}}}_{N}$ to denote the $N\times N$ dimensional identity matrix, and ${\mathbf{I}}_{N\times G}$ to denote the matrix composed of the first $G\left(\leq N\right)$ columns of ${\mathbf{I}}_{N}$. We adopt ${\mathbf{0}}$ to denote the all-zero vector or matrix. The superscripts $(\cdot)^{H}$, $(\cdot)^{T}$, and $(\cdot)^{*}$ denote the conjugate-transpose, transpose, and conjugate operations, respectively. The operator ${\mathsf{diag}\left\{{\mathbf{x}}\right\}}$ denotes the diagonal matrix with ${\mathbf{x}}$ along its main diagonal. We employ ${\left[{\mathbf{a}}\right]_{i}}$, ${\left[{\mathbf{A}}\right]_{i,j}}$ and ${\left[{\mathbf{A}}\right]_{i,:}}$ to denote the [$i$th]{} element of the vector ${\mathbf{a}}$, the $(i,j)$th element of the matrix ${\mathbf{A}}$ and the [$i$th]{} row of the matrix ${\mathbf{A}}$, respectively, where the element indices start with 0. ${\mathbb{C}}^{M\times N}$ (${\mathbb{R}}^{M\times N}$) denotes the $M\times N$ dimensional complex (real) vector space. ${{{\mathsf{E}}}\left\{\cdot\right\}}$ denotes the expectation operation. ${\mathcal{CN}\left( {\mathbf{a}}, {\mathbf{B}}\right) }$ denotes the circular symmetric complex Gaussian distribution with mean ${\mathbf{a}}$ and covariance ${\mathbf{B}}$. $\otimes$ and $\odot$ denote the Kronecker product and Hadamard product, respectively. ${\mathsf{vec}\left\{\cdot\right\}}$ represents the vectorization operation. ${\mathbf{F}}_{N}$ denotes the $N$-dimensional unitary discrete Fourier transform (DFT) matrix. ${\mathbf{F}}_{N\times G}$ denotes the matrix composed of the first $G\left(\leq N\right)$ columns of ${\mathbf{F}}_{N}$. ${\mathbf{f}}_{N,q}$ denotes the [$q$th]{} column of the matrix $\sqrt{N}{\mathbf{F}}_{N}$. We further define the permutation matrix ${{\boldsymbol\Pi}}_{N}^{n} \triangleq \left[\begin{IEEEeqnarraybox*}[][c]{,c/c,}
{\mathbf{0}}&{\mathbf{I}}_{N-{\left\langlen\right\rangle_{N}}}\\
{\mathbf{I}}_{{\left\langlen\right\rangle_{N}}}&{\mathbf{0}}\end{IEEEeqnarraybox*}\right]$. The notation $\backslash$ denotes the set subtraction operation.
Outline
-------
The rest of the paper is organized as follows. In [Section \[sec:massive\_channel\]]{}, we investigate the sparse nature of the massive MIMO-OFDM channel model. In [Section \[sec:one\_tr\]]{}, we propose APSP-CA over one OFDM symbol in massive MIMO-OFDM, including channel estimation and prediction. We investigate the multiple consecutive pilot symbol case in [Section \[sec:mul\_tr\]]{}. Simulation results are presented in [Section \[sec:sim\_res\]]{}, and conclusions are given in [Section \[sec:conc\_pw\]]{}.
Massive MIMO-OFDM Channel Model {#sec:massive_channel}
===============================
In this section, we propose a physically motivated massive MIMO-OFDM channel model, and investigate the inherent channel sparsity property. We consider a single-cell TDD wideband massive MIMO wireless system which consists of one BS equipped with $M$ antennas and $K$ single-antenna UTs. We denote the UT set as ${\mathcal{K}}=\left\{{0,1,\ldots,K-1}\right\}$ where $k\in{\mathcal{K}}$ represents the UT index. We assume that the channels of different UTs are statistically independent. We assume that the BS is equipped with a one-dimensional uniform linear array (ULA),[^6] with antennas separated by one-half wavelength. Then the BS array response vector corresponding to the incidence angle $\theta$ with respect to the perpendicular to the array is given by [@Clerckx13MIMO] $$\begin{aligned}
\label{eq:ula steer vec}
{{\mathbf{v}}_{M,\theta}}&=\bigg[1\quad{\exp\left(-{\bar{\jmath}}\pi{\sin\left(\theta\right)}\right)}\quad\ldots{\notag\\}&\qquad\ldots\quad{\exp\left(-{\bar{\jmath}}\pi(M-1){\sin\left(\theta\right)}\right)}\bigg]^{T}\in{\mathbb{C}}^{M\times 1}.\end{aligned}$$ We assume that the signals seen at the BS are constrained to lie in the angle interval ${\mathcal{A}}=[-\pi/2,\pi/2]$, which can be achieved through the use of directional antennas at the BS, and thus no signal is received at the BS for incidence angles $\theta\notin{\mathcal{A}}$ [@You15Pilot].
We consider OFDM modulation with ${N_{\mathrm{c}}}$ subcarriers, performed via the ${N_{\mathrm{c}}}$-point inverse DFT operation, appended with a guard interval (a.k.a. cyclic prefix) of length ${N_{\mathrm{g}}}\left(\leq{N_{\mathrm{c}}}\right)$ samples. We employ ${T_{\mathrm{sym}}}=\left({N_{\mathrm{c}}}+{N_{\mathrm{g}}}\right){T_{\mathrm{s}}}$ and ${T_{\mathrm{c}}}={N_{\mathrm{c}}}{T_{\mathrm{s}}}$ to denote the OFDM symbol duration with and without the guard interval, respectively, where ${T_{\mathrm{s}}}$ is the system sampling duration [@Dahlman11LTE]. We assume that the guard interval length ${T_{\mathrm{g}}}={N_{\mathrm{g}}}{T_{\mathrm{s}}}$ is longer than the maximum channel delay of all the UTs [@Edfors98OFDM; @Li98Robust].
We assume that the channels remain constant during one OFDM symbol, and evolve from symbol to symbol. We denote the uplink (UL) channel gain between the antenna of the [$k$th]{} UT and the [$m$th]{} antenna of the BS over OFDM symbol $\ell$ and subcarrier $n$ as ${\left[{\mathbf{g}}_{k,\ell,n}\right]_{m}}$. Using a physical channel modeling approach (see, e.g., [@Clerckx13MIMO; @Liu03Capacity; @Barriac04Characterizing; @Auer12MIMO; @Fleury00First]), the channel response vector ${\mathbf{g}}_{k,\ell,n}\in{\mathbb{C}}^{M\times1}$ can be described as $$\begin{aligned}
\label{eq:wb_cha_mod}
{\mathbf{g}}_{k,\ell,n}
&=\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\infty}^{\infty}\int\limits_{-\pi/2}^{\pi/2}\!{{\mathbf{v}}_{M,\theta}}\cdot{\exp\left(-{\bar{\jmath}}2\pi \frac{n}{{T_{\mathrm{c}}}}\tau\right)}{\notag\\}&\quad\cdot{\exp\left({\bar{\jmath}}2\pi \nu\ell{T_{\mathrm{sym}}}\right)}\cdot
g_{k}\left(\theta,\tau,\nu\right)\cdot{\delta\left(\tau-q{T_{\mathrm{s}}}\right)} {{\mathrm{d}}\theta}{{\mathrm{d}}\nu}{\notag\\}&=\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\infty}^{\infty}\int\limits_{-\pi/2}^{\pi/2}\!{{\mathbf{v}}_{M,\theta}}\cdot{\exp\left(-{\bar{\jmath}}2\pi \frac{n}{{N_{\mathrm{c}}}}q\right)}{\notag\\}&\quad\cdot{\exp\left({\bar{\jmath}}2\pi \nu\ell{T_{\mathrm{sym}}}\right)}\cdot
g_{k}\left(\theta,q{T_{\mathrm{s}}},\nu\right){{\mathrm{d}}\theta}{{\mathrm{d}}\nu}\end{aligned}$$ where ${{\mathbf{v}}_{M,\theta}}$ is given in , $g_{k}\left(\theta,\tau,\nu\right)$ is the complex-valued joint angle-delay-Doppler channel gain function of UT $k$ corresponding to the incidence angle $\theta$, delay $\tau$, and Doppler frequency $\nu$. Note that the number of significant channel taps in the delay domain is usually limited, and smaller than ${N_{\mathrm{g}}}$; i.e., ${\left|g_{k}\left(\theta,q{T_{\mathrm{s}}},\nu\right)\right|}$ is approximately $0$ for most $q$. Since the locations of the significant channel taps in the delay domain are usually different for different UTs, we adopt in this paper to obtain a general channel representation applicable for all the UTs.
We write the [$k$th]{} UT’s channel at OFDM symbol $\ell$ over all subcarriers as $$\label{eq:chgkl}
{\mathbf{G}}_{k,\ell}=
\left[{\mathbf{g}}_{k,\ell,0}\quad{\mathbf{g}}_{k,\ell,1}\quad\ldots\quad{\mathbf{g}}_{k,\ell,{N_{\mathrm{c}}}-1}\right]
\in{\mathbb{C}}^{M\times{N_{\mathrm{c}}}}$$ which will be referred to as the space-frequency domain channel response matrix (SFCRM). From , it is not hard to show that $$\begin{aligned}
\label{eq:phych}
{\mathsf{vec}\left\{{\mathbf{G}}_{k,\ell}\right\}}&=\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\infty}^{\infty}\int\limits_{-\pi/2}^{\pi/2}\!
\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{{\mathbf{v}}_{M,\theta}}\right]\cdot{\exp\left({\bar{\jmath}}2\pi \nu\ell{T_{\mathrm{sym}}}\right)}{\notag\\}&\quad\cdot g_{k}\left(\theta,q{T_{\mathrm{s}}},\nu\right){{\mathrm{d}}\theta}{{\mathrm{d}}\nu}.\end{aligned}$$
We assume that channels with different incidence angles, delays, and/or Doppler frequencies are uncorrelated [@Clerckx13MIMO; @Auer12MIMO; @Fleury00First]. We also assume that the temporal correlations and joint space-frequency domain correlations of the channels can be separated [@Li98Robust; @Auer12MIMO], i.e., $$\begin{aligned}
\label{eq:uncorcha}
&{{{\mathsf{E}}}\left\{g_{k}\left(\theta,\tau,\nu\right)g_{k}^{*}\left(\theta',\tau',\nu'\right)\right\}}{\notag\\}&={{\mathtt{S}}_{k}^{\mathrm{ADD}}}\left(\theta,\tau,\nu\right)\cdot{\delta\left(\theta-\theta'\right)}{\delta\left(\tau-\tau'\right)}{\delta\left(\nu-\nu'\right)}{\notag\\}&={{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau\right)\cdot{{\mathtt{S}}_{k}^{\mathrm{Dop}}}\left(\nu\right)\cdot{\delta\left(\theta-\theta'\right)}{\delta\left(\tau-\tau'\right)}{\delta\left(\nu-\nu'\right)}\end{aligned}$$ where ${{\mathtt{S}}_{k}^{\mathrm{ADD}}}\left(\theta,\tau,\nu\right)$, ${{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau\right)$, and ${{\mathtt{S}}_{k}^{\mathrm{Dop}}}\left(\nu\right)$ represent the power angle-delay-Doppler spectrum, power angle-delay spectrum, and power Doppler spectrum of UT $k$, respectively [@Patzold12Mobile; @Clerckx13MIMO].
From and , we can obtain the following channel statistical property (see [Appendix \[app:der\_sta\]]{} for the derivations) $$\begin{aligned}
\label{eq:expbgkldel}
{{{\mathsf{E}}}\left\{{\mathsf{vec}\left\{{\mathbf{G}}_{k,\ell+\Delta_{\ell}}\right\}}{\mathsf{vec}^{H}\left\{{\mathbf{G}}_{k,\ell}\right\}}\right\}}=\varrho_{k}\left(\Delta_{\ell}\right)\cdot{\mathbf{R}}_{k}\end{aligned}$$ where $\varrho_{k}\left(\Delta_{\ell}\right)$ is the channel temporal correlation function (TCF) given by $$\varrho_{k}\left(\Delta_{\ell}\right)\triangleq\int\limits_{-\infty}^{\infty}\!{\exp\left({\bar{\jmath}}2\pi\nu\Delta_{\ell}{T_{\mathrm{sym}}}\right)}\cdot{{\mathtt{S}}_{k}^{\mathrm{Dop}}}\left(\nu\right){{\mathrm{d}}\nu}$$ and ${\mathbf{R}}_{k}$ is the space-frequency domain channel covariance matrix (SFCCM) given by $$\begin{aligned}
\label{eq:spa_fre_cov}
{\mathbf{R}}_{k}&\triangleq\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\pi/2}^{\pi/2}\!
\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{{\mathbf{v}}_{M,\theta}}\right]
\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{{\mathbf{v}}_{M,\theta}}\right]^{H}{\notag\\}&\qquad\cdot{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,q{T_{\mathrm{s}}}\right)
{{\mathrm{d}}\theta}\in{\mathbb{C}}^{M{N_{\mathrm{c}}}\times M{N_{\mathrm{c}}}}.\end{aligned}$$
In this work, we consider the widely accepted Clarke-Jakes channel power Doppler spectrum,[^7] with the corresponding channel TCF given by [@Jakes94Microwave; @Patzold12Mobile] $$\label{eq:timcor}
\varrho_{k}\left(\Delta_{\ell}\right)=\mathrm{J}_{0}\left(2\pi\nu_{k}{T_{\mathrm{sym}}}\Delta_{\ell}\right)$$ where $\mathrm{J}_{0}\left(\cdot\right)$ is the zeroth-order Bessel function of the first kind, and $\nu_{k}$ is the Doppler frequency of UT $k$. Note that the Clarke-Jakes power Doppler spectrum is an even function, i.e., $\varrho_{k}(\Delta_{\ell})=\varrho_{k}(-\Delta_{\ell})$, and satisfies $\varrho_{k}\left(0\right)=1$. Also, we assume that according to the law of large numbers, the channel elements exhibit a joint Gaussian distribution, i.e., ${\mathsf{vec}\left\{{\mathbf{G}}_{k,\ell}\right\}}\sim{\mathcal{CN}\left( {\mathbf{0}}, {\mathbf{R}}_{k}\right) }$.
Before proceeding, we investigate in the following proposition a property of the large dimensional SFCCM, and present a relationship between the SFCCM and the power angle-delay spectrum for massive MIMO-OFDM channels.
\[prop:Decomp\_cov\] Define ${\mathbf{V}}_{M}\in{\mathbb{C}}^{M\times M}$ as ${\left[{\mathbf{V}}_{M}\right]_{i,j}}\triangleq\frac{1}{\sqrt{M}}\cdot{\exp\left(-{\bar{\jmath}}2\pi\frac{i\left(j-M/2\right)}{M}\right)}$, and ${{\boldsymbol\Omega}}_{k}\in{\mathbb{R}}^{M\times{N_{\mathrm{g}}}}$ as $$\begin{aligned}
\label{eq:cov_eigen_ele}
{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}\triangleq M{N_{\mathrm{c}}}\left(\theta_{i+1}-\theta_{i}\right)\cdot {{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta_{i},\tau_{j}\right)\end{aligned}$$ where $\theta_{m}\triangleq\arcsin\left(2m/M-1\right)$, and $\tau_{n}\triangleq n{T_{\mathrm{s}}}$. Then when the number of antennas $M\to\infty$, the SFCCM ${\mathbf{R}}_{k}$ tends to $\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right){\mathsf{diag}\left\{{\mathsf{vec}\left\{{{\boldsymbol\Omega}}_{k}\right\}}\right\}}\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)^{H}$ in the sense that, for fixed non-negative integers $i$ and $j$, $$\begin{aligned}
\label{eq:Decomp_cov}
\lim_{\substack{{M \to\infty}}}&\Big[{\mathbf{R}}_{k}
-\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)
{\mathsf{diag}\left\{{\mathsf{vec}\left\{{{\boldsymbol\Omega}}_{k}\right\}}\right\}}{\notag\\}&\quad\cdot\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)^{H}\Big]_{i,j}=0.\end{aligned}$$
See [Appendix \[app:prop\_Decomp\_cov\]]{}.
The relationship between the space-frequency domain channel joint correlation property and the channel power distribution in the angle-delay domain for massive MIMO-OFDM is established in [Proposition \[prop:Decomp\_cov\]]{}. Specifically, for massive MIMO-OFDM channels in the asymptotically large array regime, the eigenvectors of the SFCCMs for different UTs tend to be the same, which shows that massive MIMO-OFDM channels can be asymptotically decorrelated by the fixed space-frequency domain statistical eigendirections, while the eigenvalues depend on the corresponding channel power angle-delay spectra.
[Proposition \[prop:Decomp\_cov\]]{} indicates that, for massive MIMO-OFDM channels, when the number of BS antennas $M$ is sufficiently large, the SFCCM can be well approximated by $$\begin{aligned}
\label{eq:cov_appr}
{\mathbf{R}}_{k}&\approx\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right){\mathsf{diag}\left\{{\mathsf{vec}\left\{{{\boldsymbol\Omega}}_{k}\right\}}\right\}}{\notag\\}&\qquad\cdot\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)^{H}.\end{aligned}$$ It is worth noting that the approximation in is consistent with existing results in the literature. For frequency-selective single-input single-output channels, agrees with the results in [@Li98Robust; @van95channel]. For frequency-flat massive MIMO channels, the approximation given in has been shown to be accurate enough for a practical number of antennas, which usually ranges from 64 to 512 [@Wen15Channel; @Yin13coordinated; @Adhikary13Joint; @You15Pilot], and a detailed numerical example can be found in [@Wen15Channel]. Since the SFCCM model given in is a good approximation to the more complex physical channel model in when the number of BS antennas is sufficiently large, we will thus exclusively use the simplified SFCCM model in in the rest of the paper.
Realistic wireless channels are usually not wide-sense stationary [@Clerckx13MIMO], i.e., ${\mathbf{R}}_{k}$ varies as time evolves, although with a relatively large time scale.[^8] In practice, acquisition of the large dimensional ${\mathbf{R}}_{k}$ is rather difficult and resource-intensive for massive MIMO-OFDM. However, when we shift our focus from the space-frequency domain to the angle-delay domain, the problem can be significantly simplified. Motivated by the eigenvalue decomposition of the SFCCM given in , we decompose the SFCRM as follows $$\label{eq:gkl}
{\mathbf{G}}_{k,\ell}={\mathbf{V}}_{M}{\mathbf{H}}_{k,\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}$$ where $$\label{eq:adcr}
{\mathbf{H}}_{k,\ell}={\mathbf{V}}_{M}^{H}{\mathbf{G}}_{k,\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}\in{\mathbb{C}}^{M\times{N_{\mathrm{g}}}}$$ is referred to as the angle-delay domain channel response matrix (ADCRM) of UT $k$ at OFDM symbol $\ell$. In the following proposition, we derive a statistical property of the ADCRM.
\[prop:adcmsts\] For massive MIMO-OFDM channels, when the number of antennas $M\to\infty$, elements of the ADCRM ${\mathbf{H}}_{k,\ell}$ satisfy $$\begin{aligned}
\label{eq:exbhk}
&{{{\mathsf{E}}}\left\{{\left[{\mathbf{H}}_{k,\ell+\Delta_{\ell}}\right]_{i,j}}{\left[{\mathbf{H}}_{k,\ell}\right]_{i',j'}}^{*}\right\}}{\notag\\}&\qquad=\varrho_{k}\left(\Delta_{\ell}\right){\delta\left(i-i'\right)}{\delta\left(j-j'\right)}\cdot{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}\end{aligned}$$ where ${{\boldsymbol\Omega}}_{k}$ is given in .
See [Appendix \[app:prop\_adcmsts\]]{}.
[Proposition \[prop:adcmsts\]]{} shows that, for massive MIMO-OFDM channels, different elements of the ADCRM ${\mathbf{H}}_{k,\ell}$ are approximately mutually statistically uncorrelated, which lends the ADCRM in its physical interpretation. Specifically, different elements of the ADCRM correspond to the channel gains for different incidence angles and delays, which can be resolved in massive MIMO-OFDM with a sufficiently large antenna array aperture. Note that ${\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}$ corresponds to the average power of ${\left[{\mathbf{H}}_{k}\right]_{i,j}}$, and can describe the sparsity of the wireless channels in the angle-delay domain. Hereafter we will refer to ${{\boldsymbol\Omega}}_{k}$ as the angle-delay domain channel power matrix (ADCPM) of UT $k$. The dimension of the ADCPM ${{\boldsymbol\Omega}}_{k}$ is much smaller than that of the SFCCM ${\mathbf{R}}_{k}$, and most elements in ${{\boldsymbol\Omega}}_{k}$ are approximately zero due to the channel sparsity. In addition, ${{\boldsymbol\Omega}}_{k}$ is composed of the variances of independent angle-delay domain channel elements, and thus can be estimated in an element-wise manner. Therefore, in practice there will be enough resources for one to obtain an estimate of ${{\boldsymbol\Omega}}_{k}$ with guaranteed accuracy. In the rest of the paper, we will assume that the ADCPMs of all the UTs are known by the BS.
Before we conclude this section, we define the extended ADCRM as follows $$\begin{aligned}
\label{eq:extadcrm}
{\bar{{\mathbf{H}}}}_{k,\ell,{\left({{N_{\mathrm{c}}}}\right)}}
&\triangleq{\mathbf{H}}_{k,\ell}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\notag\\}&=\left[{\mathbf{H}}_{k,\ell}
\quad{\mathbf{0}}_{M\times\left({N_{\mathrm{c}}}-{N_{\mathrm{g}}}\right)}\right]
\in{\mathbb{C}}^{M\times{N_{\mathrm{c}}}}.\end{aligned}$$ Similarly, the extended ADCPM, which corresponds to the power distribution of the extended ADCRM ${\bar{{\mathbf{H}}}}_{k,\ell,{\left({{N_{\mathrm{c}}}}\right)}}$, is defined as $$\begin{aligned}
\label{eq:extadcpm}
{\bar{{{\boldsymbol\Omega}}}}_{k,{\left({{N_{\mathrm{c}}}}\right)}}
&\triangleq{{\boldsymbol\Omega}}_{k}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\notag\\}&=\left[{{\boldsymbol\Omega}}_{k}
\quad{\mathbf{0}}_{M\times\left({N_{\mathrm{c}}}-{N_{\mathrm{g}}}\right)}\right]
\in{\mathbb{R}}^{M\times{N_{\mathrm{c}}}}.\end{aligned}$$ Such definitions will be employed to simplify the analyses in the following sections.
Channel Acquisition with APSPs over One Symbol {#sec:one_tr}
==============================================
Based on the sparse massive MIMO-OFDM channel model presented in the previous section, we propose APSP-CA for massive MIMO-OFDM, including channel estimation and prediction. In this section, we first investigate the case where the APSPs are sent over one OFDM symbol, while the multiple symbol case will be investigated in the next section.
APSPs over One Symbol
---------------------
We assume that all the UTs are synchronized. During the UL pilot segment, namely, the [$\ell$th]{} OFDM symbol of each frame, all the UTs transmit the scheduled pilots simultaneously, and the space-frequency domain signal received at the BS can be represented as $$\begin{aligned}
\label{eq:recsigult}
{\mathbf{Y}}_{\ell}
=\sum_{k'=0}^{K-1}{\mathbf{G}}_{k',\ell}{\mathbf{X}}_{k'}+{\mathbf{Z}}_{\ell}\in{\mathbb{C}}^{M\times{N_{\mathrm{c}}}}\end{aligned}$$ where ${\left[{\mathbf{Y}}_{\ell}\right]_{i,j}}$ denotes the received pilot signal at the [$i$th]{} antenna over the [$j$th]{} subcarrier, ${\mathbf{G}}_{k,\ell}$ is the SFCRM defined in , ${\mathbf{X}}_{k}={\mathsf{diag}\left\{{\mathbf{x}}_{k}\right\}}\in{\mathbb{C}}^{{N_{\mathrm{c}}}\times{N_{\mathrm{c}}}}$ denotes the frequency domain pilot signal sent from the [$k$th]{} UT, ${\mathbf{Z}}_{\ell}$ is the additive white Gaussian noise (AWGN) matrix during the UL pilot segment with elements identically and independently distributed (i.i.d.) as ${\mathcal{CN}\left( 0 , {\sigma_{\mathrm{ztr}}}\right) }$, and ${\sigma_{\mathrm{ztr}}}$ is the noise power.
The proposed APSP over one OFDM symbol for a given UT $k$ is given by $$\begin{aligned}
\label{eq:pls}
{\mathbf{X}}_{k}\triangleq\sqrt{{\sigma_{\mathrm{xtr}}}}
\underbrace{{\mathsf{diag}\left\{{\mathbf{f}}_{{N_{\mathrm{c}}},\phi_{k}}\right\}}}_{\triangleq{\mathbf{D}}_{\phi_{k}}}
{\mathbf{X}},\quad\phi_{k}={0,1,\ldots,{N_{\mathrm{c}}}-1}\end{aligned}$$ where ${\mathbf{X}}={\mathsf{diag}\left\{{\mathbf{x}}\right\}}\in{\mathbb{C}}^{{N_{\mathrm{c}}}\times{N_{\mathrm{c}}}}$ satisfying ${\mathbf{X}}{\mathbf{X}}^{H}={\mathbf{I}}_{{N_{\mathrm{c}}}}$ is the basic pilot matrix shared by all UTs in the same cell, and ${\sigma_{\mathrm{xtr}}}$ is the pilot signal transmit power. The APSP signal given in can be seen as a phase shifted version of $\sqrt{{\sigma_{\mathrm{xtr}}}}{\mathbf{X}}$ with phase shift $\phi_{k}$ in the frequency domain. Note that the proposed APSP has the same PAPR as that of ${\mathbf{X}}$ in the time domain, thus existing low PAPR sequence designs can be easily incorporated into our approach. In addition, as the basic pilot matrix ${\mathbf{X}}$ can be predetermined, only ${\mathbf{X}}$ and the pilot phase shift indices rather than the entire pilot matrices are required to be stored, and the required storage space can be significantly reduced.
From , it can be readily obtained that, for $\forall k,k'\in{\mathcal{K}}$, $$\label{eq:ijcrocor}
{\mathbf{X}}_{k'}{\mathbf{X}}_{k}^{H}
={\sigma_{\mathrm{xtr}}}{\mathbf{D}}_{\phi_{k'}-\phi_{k}}$$ which indicates that cross correlations of the proposed APSPs for different UTs depend only on the associated phase shift difference. It is worth noting that, for conventional PSOPs, the phase shift differences for different pilots are set to satisfy the orthogonality condition ${\left|\phi_{k'}-\phi_{k}\right|}\geq{N_{\mathrm{g}}}\ \forall k'\neq k$. However, for our APSPs, the phase shifts for different pilots are adjustable, and pilots for different UTs can even share the same phase shift, which leads to more available pilots, and thus pilot overhead can be significantly reduced.
Channel Estimation with APSPs
-----------------------------
In this section we investigate channel estimation during the pilot segment under the minimum MSE (MMSE) criterion using the proposed APSPs. Direct MMSE estimation of the SFCRM ${\mathbf{G}}_{k,\ell}$ requires information about the large dimensional SFCCM ${\mathbf{R}}_{k}$ and a large dimensional matrix inversion, which is difficult to implement in practice. However, with the sparse massive MIMO-OFDM channel model presented above, when we shift our focus from the space-frequency domain to the angle-delay domain, channel estimation can be greatly simplified. The BS can first estimate the ADCRM to obtain ${\hat{{\mathbf{H}}}}_{k,\ell}$, then the SFCRM estimates can be readily obtained as ${\hat{{\mathbf{G}}}}_{k,\ell}={\mathbf{V}}_{M}{\hat{{\mathbf{H}}}}_{k,\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}$ via exploiting the unitary equivalence between the angle-delay domain channels and the space-frequency domain channels given in , while the same MSE-CE performance can be maintained. In the following, we focus on estimation of the ADCRM ${\mathbf{H}}_{k,\ell}$ under the MMSE criterion.
Recalling , the received pilot signal at the BS in can be rewritten as $$\begin{aligned}
\label{eq:recsigad}
{\mathbf{Y}}_{\ell}
=\sum_{k'=0}^{K-1}{\mathbf{V}}_{M}{\mathbf{H}}_{k',\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}
{\mathbf{X}}_{k'}+{\mathbf{Z}}_{\ell}.\end{aligned}$$ After decorrelation and power normalization of ${\mathbf{Y}}_{\ell}$, the BS can obtain an observation of the UL channel ${\mathbf{H}}_{k,\ell}$, given by shown at the top of the next page,
$$\begin{aligned}
\label{eq:ykltr}
{\mathbf{Y}}_{k,\ell}
&={\frac{1}{{\sigma_{\mathrm{xtr}}}}}{\mathbf{V}}_{M}^{H}{\mathbf{Y}}_{\ell}{\mathbf{X}}_{k}^{H}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}{\notag\\}&={\frac{1}{{\sigma_{\mathrm{xtr}}}}}\sum_{k'=0}^{K-1}{\mathbf{H}}_{k',\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\mathbf{X}}_{k'}{\mathbf{X}}_{k}^{H}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}
+{\frac{1}{{\sigma_{\mathrm{xtr}}}}}{\mathbf{V}}_{M}^{H}{\mathbf{Z}}_{\ell}{\mathbf{X}}_{k}^{H}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}{\notag\\}&{\mathop{=}^{(\mathrm{a})}}{\mathbf{H}}_{k,\ell}+\underbrace{\sum_{k'\neq k}{\mathbf{H}}_{k',\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\mathbf{D}}_{\phi_{k'}-\phi_{k}}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}}_{\mathrm{pilot\ interference}\triangleq\sum_{k'\neq k}{\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}}
+\underbrace{{\frac{1}{{\sigma_{\mathrm{xtr}}}}}{\mathbf{V}}_{M}^{H}{\mathbf{Z}}_{\ell}{\mathbf{X}}_{k}^{H}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}}_{\mathrm{pilot\ noise}}\end{aligned}$$
where (a) follows from . Using the unitary transformation property, it can be readily shown that the pilot noise term in exhibits a Gaussian distribution with i.i.d. elements distributed as ${\mathcal{CN}\left( 0 , {\sigma_{\mathrm{ztr}}}/{\sigma_{\mathrm{xtr}}}\right) }$, and can be simplified as $$\begin{aligned}
\label{eq:simykltr}
{\mathbf{Y}}_{k,\ell}
={\mathbf{H}}_{k,\ell}+\sum_{k'\neq k}{\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}
+{\frac{1}{\sqrt{{\rho_{\mathrm{tr}}}}}}{{\mathbf{Z}}_{\mathrm{iid}}}\end{aligned}$$ where ${\rho_{\mathrm{tr}}}\triangleq{\sigma_{\mathrm{xtr}}}/{\sigma_{\mathrm{ztr}}}$ is the signal-to-noise ratio (SNR) during the pilot segment, and ${{\mathbf{Z}}_{\mathrm{iid}}}\in{\mathbb{C}}^{M\times{N_{\mathrm{g}}}}$ is the normalized AWGN matrix with i.i.d. elements distributed as ${\mathcal{CN}\left( 0 , 1\right) }$.
Note that the pilot interference term ${\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}$ defined in satisfies $$\begin{aligned}
\label{eq:hulphi}
{\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}
&={\mathbf{H}}_{k',\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\mathbf{D}}_{\phi_{k'}-\phi_{k}}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}{\notag\\}&={\mathbf{H}}_{k',\ell}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\mathbf{F}}_{{N_{\mathrm{c}}}}^{T}{\mathbf{D}}_{\phi_{k'}-\phi_{k}}{\mathbf{F}}_{{N_{\mathrm{c}}}}^{*}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}{\notag\\}&{\mathop{=}^{(\mathrm{a})}}{\bar{{\mathbf{H}}}}_{k',\ell,{\left({{N_{\mathrm{c}}}}\right)}}{\mathbf{F}}_{{N_{\mathrm{c}}}}^{T}{\mathbf{D}}_{\phi_{k'}-\phi_{k}}{\mathbf{F}}_{{N_{\mathrm{c}}}}^{*}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}{\notag\\}&{\mathop{=}^{(\mathrm{b})}}{\bar{{\mathbf{H}}}}_{k',\ell,{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k'}-\phi_{k}}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\end{aligned}$$ where (a) follows from , and (b) follows from the permutation matrix definition given in [Section \[sec:prwb\_not\]]{}. Thus, the pilot interference term ${\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}$ in is a column truncated version of the extended ADCRM ${\bar{{\mathbf{H}}}}_{k',\ell,{\left({{N_{\mathrm{c}}}}\right)}}$ with a cyclic column shift, where the shift factor depends on the corresponding pilot phase shift difference $\phi_{k'}-\phi_{k}$. Thus elements of ${\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}$ can be readily obtained as $$\begin{aligned}
{\left[{\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}\right]_{i,j}}
=\left\{ {\begin{array}{l}
{\left[{\mathbf{H}}_{k',\ell}\right]_{i,{\left\langlej-\left(\phi_{k'}-\phi_{k}\right)\right\rangle_{{N_{\mathrm{c}}}}}}}, \\
\quad {\left\langlej-\left(\phi_{k'}-\phi_{k}\right)\right\rangle_{{N_{\mathrm{c}}}}}\leq{N_{\mathrm{g}}}-1 \\
0, \quad\textrm{else}.
\end{array}} \right.\end{aligned}$$
Recalling [Proposition \[prop:adcmsts\]]{}, elements of the ADCRM ${\mathbf{H}}_{k',\ell}$ are statistically uncorrelated. Consequently, elements of the pilot interference term ${\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}$, a column truncated copy of ${\mathbf{H}}_{k',\ell}$ with cyclic column shift, are also statistically uncorrelated. Thus, using the same methodology as in the previous section, the corresponding power matrix of the pilot interference term ${\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}$ can be defined as $$\begin{aligned}
\label{eq:omeuphuk}
{{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}
&\triangleq{{{\mathsf{E}}}\left\{{\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}\odot\left({\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}\right)^{*}\right\}}{\notag\\}&={\bar{{{\boldsymbol\Omega}}}}_{k',{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k'}-\phi_{k}}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\end{aligned}$$ which is a column truncated version of the extended ADCPM ${\bar{{{\boldsymbol\Omega}}}}_{k',{\left({{N_{\mathrm{c}}}}\right)}}$ defined in with cyclic column shift $\phi_{k'}-\phi_{k}$.
With the channel observation ${\mathbf{Y}}_{k,\ell}$ in , and the fact that the angle-delay domain channel elements are uncorrelated as derived in [Proposition \[prop:adcmsts\]]{}, the MMSE estimate ${\hat{{\mathbf{H}}}}_{k,\ell}$ can be obtained in an element-wise manner as follows [@Kailath00Linear] $$\begin{aligned}
\label{eq:hathkl}
{\left[{\hat{{\mathbf{H}}}}_{k,\ell}\right]_{i,j}}=\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}}
{\sum_{k'=0}^{K-1}{\left[{{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}
{\left[{\mathbf{Y}}_{k,\ell}\right]_{i,j}}.\end{aligned}$$ Let ${\tilde{{\mathbf{H}}}}_{k,\ell}={\mathbf{H}}_{k,\ell}-{\hat{{\mathbf{H}}}}_{k,\ell}$ be the angle-delay domain channel estimation error of the [$k$th]{} UT, then the corresponding MSE-CE can be obtained as $$\begin{aligned}
\label{eq:epsilonkl}
{\epsilon_{k}^{\mathrm{CE}}}
&\triangleq\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}
{{{\mathsf{E}}}\left\{{\left|{\left[{\tilde{{\mathbf{H}}}}_{k,\ell}\right]_{i,j}}\right|^{2}}\right\}}{\notag\\}&{\mathop{=}^{(\mathrm{a})}}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}
{{{\mathsf{E}}}\left\{{\left|{\left[{\mathbf{H}}_{k,\ell}\right]_{i,j}}\right|^{2}}-{\left|{\left[{\hat{{\mathbf{H}}}}_{k,\ell}\right]_{i,j}}\right|^{2}}\right\}}{\notag\\}&=\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}\left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}
-\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}{\sum_{k'=0}^{K-1}{\left[{{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}\right\}\end{aligned}$$ where (a) follows from the orthogonality principle of MMSE estimation [@Kailath00Linear].
Before we proceed, we define the sum MSE-CE of all the UTs as $$\begin{aligned}
\label{eq:epsicesum}
{{\epsilon}^{\mathrm{CE}}}\triangleq&\sum_{k=0}^{K-1}{\epsilon_{k}^{\mathrm{CE}}}.\end{aligned}$$ Due to the incurred pilot interference, performance of the APSP-based channel estimation might deteriorate. However, we will show in the following proposition that such effects can be eliminated with proper phase shift scheduling for different pilots.
\[prop:mmsetrseqcon\] The sum MSE-CE ${\epsilon^{\mathrm{CE}}}$ is lower bounded by $$\label{eq:varepsilonl}
{\epsilon^{\mathrm{CE}}}\geq{\varepsilon^{\mathrm{CE}}}=\sum_{k=0}^{K-1}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}
\left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}-\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}\right\}$$ and the lower bound can be achieved under the condition that, for $\forall k,k'\in{\mathcal{K}}$ and $k\neq k'$, $$\label{eq:condokou}
\left({\bar{{{\boldsymbol\Omega}}}}_{k,{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k}}\right)
\odot\left({\bar{{{\boldsymbol\Omega}}}}_{k',{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k'}}\right)
={\mathbf{0}}.$$
See [Appendix \[app:prop\_mmsetrseqcon\]]{}.
[Proposition \[prop:mmsetrseqcon\]]{} shows that with the proposed APSPs, the sum MSE-CE can be minimized when phase shifts for different pilots are properly scheduled according to the condition given in . The interpretation is very intuitive. With frequency domain phase shifted pilots, equivalent channels will exhibit corresponding cyclic shifts in the delay domain, as seen from . If the equivalent channel power distributions in the angle-delay domain for different UTs can be made non-overlapping after pilot phase shift scheduling, the pilot interference effect can be eliminated, and the sum MSE-CE can be minimized.
Wireless channels are approximately sparse in the angle-delay domain in many practical propagation scenarios, and typically only a few elements of the ADCPM ${{\boldsymbol\Omega}}_{k}$ are dominant in massive MIMO-OFDM. When such channel sparsity is properly taken into account, the equivalent angle-delay domain channels for different UTs are almost non-overlapping with high probability, assuming proper pilot phase shifts. This suggests the feasibility of the proposed APSPs for massive MIMO-OFDM.
Note that performance of the proposed APSP approach is related to the channel sparsity level. For the case where channels of different UTs have a sparse common support with $s\left(\leq{N_{\mathrm{g}}}\right)$ representing the number of the columns containing non-zero elements in the ADCPM [@Barbotin12Estimation; @Rao14Distributed], the maximum number of UTs that can be served without pilot interference is $\left\lfloor{N_{\mathrm{c}}}/s\right\rfloor$. However, for practical wireless channels, most of the channel elements in the angle-delay domain are close to zero, and the condition in usually cannot be satisfied exactly, which will lead to degradation of the channel acquisition performance. In such cases, it is clear that the more sparse the channels are, the better performance can be achieved by the proposed APSP approach.
Before we conclude this subsection, we remark here that several existing pilot approaches satisfy the optimal condition given in [Proposition \[prop:mmsetrseqcon\]]{}. For the case where channel sparsity property is not known, it is reasonable to assume that all the angle-delay domain channel elements are identically distributed, i.e., all the ADCPM elements are equal, in which case the optimal condition in can be achieved when ${\left|\phi_{k}-\phi_{k'}\right|}\geq{N_{\mathrm{g}}}$ for $\forall k\neq k'$, i.e., the extended channels in the delay domain for different UTs are totally separated, which coincides with the conventional PSOPs [@Chi11Training]. For frequency-flat massive MIMO channels, i.e., ${N_{\mathrm{c}}}=1$, the condition in can be achieved when ${{\boldsymbol\Omega}}_{k}\odot{{\boldsymbol\Omega}}_{k'}={\mathbf{0}}$ for $\forall k\neq k'$, i.e., different UTs can share the same pilot when the respective channels have non-overlapping support in the angle domain, which coincides with previous works such as [@You15Pilot; @Yin13coordinated]. In our work, the proposed APSPs exploit the joint angle-delay domain channel sparsity in massive MIMO-OFDM, and are more efficient and general from the pilot overhead point of view.
Channel Prediction with APSPs
-----------------------------
In the previous subsection, we investigated channel estimation during the pilot segment. Directly employing the pilot segment channel estimates in the data segment might not always be appropriate [@Truong13Effects], especially in high mobility scenarios, which are the main focus of the APSPs. In this subsection, we investigate channel prediction during the data segment based on the received pilot signals, using the proposed APSPs.
For frame-based massive MIMO-OFDM transmission, the BS utilizes the received signals during the pilot segment to acquire the channels in the current frame. If the pilot segment channel estimate ${\hat{{\mathbf{H}}}}_{k,\ell}$ is directly employed as the estimate of the channel ${\mathbf{H}}_{k,\ell+\Delta_{\ell}}$ during the data segment, the corresponding sum MSE-CE for a given delay $\Delta_{\ell}$ between the pilot symbol and data symbol can be written as $$\begin{aligned}
\label{eq:epscek}
{\epsilon^{\mathrm{CE}}}\left(\Delta_{\ell}\right)
&=\sum_{k=0}^{K-1}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}{{{\mathsf{E}}}\left\{{\left|{\left[{\mathbf{H}}_{k,\ell+\Delta_{\ell}}-{\hat{{\mathbf{H}}}}_{k,\ell}\right]_{i,j}}\right|^{2}}\right\}}{\notag\\}&=\sum_{k=0}^{K-1}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}
\Bigg\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+\left[1-2\varrho_{k}\left(\Delta_{\ell}\right)\right]{\notag\\}&\qquad\cdot\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}
{\sum_{k'=0}^{K-1}{\left[{{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}\Bigg\}.\end{aligned}$$ In high mobility scenarios, the channel TCF satisfies $\varrho_{k}\left(\Delta_{\ell}\right)\to0$ for relatively large delay ${\left|\Delta_{\ell}\right|}$. When $\varrho_{k}\left(\Delta_{\ell}\right)<1/2$, i.e., $1-2\varrho_{k}\left(\Delta_{\ell}\right)>0$, it can be observed from that the sum MSE-CE expression ${\epsilon^{\mathrm{CE}}}\left(\Delta_{\ell}\right)$ is even larger than the sum channel power $\sum_{k=0}^{K-1}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}$, and channel estimation performance cannot be guaranteed, which motivates the need for channel prediction.
For channel prediction, the BS utilizes the received pilot signals as well as the channel TCF to get estimates of the channels during the data segment. Under the MMSE criterion, with the angle-delay domain channel property of massive MIMO-OFDM given in [Proposition \[prop:adcmsts\]]{}, it is not hard to show that an estimate of the ADCRM ${\mathbf{H}}_{k,\ell+\Delta_{\ell}}$ based on ${\mathbf{Y}}_{k,\ell}$ can be obtained in an element-wise manner as follows $$\begin{aligned}
&{\left[{\hat{{\mathbf{H}}}}_{k,\ell+\Delta_{\ell}}\right]_{i,j}}{\notag\\}&\quad=\varrho_{k}\left(\Delta_{\ell}\right)\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}}
{\sum_{k'=0}^{K-1}{\left[{{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}{\left[{\mathbf{Y}}_{k,\ell}\right]_{i,j}}.\end{aligned}$$ Recalling the pilot segment channel estimate in , it can be seen that $$\begin{aligned}
\label{eq:hathkldell}
{\hat{{\mathbf{H}}}}_{k,\ell+\Delta_{\ell}}=\varrho_{k}\left(\Delta_{\ell}\right){\hat{{\mathbf{H}}}}_{k,\ell}\end{aligned}$$ which indicates that optimal channel estimates during the data segment can be easily obtained via prediction with initial channel estimates obtained during the pilot segment, and the complexity of channel prediction in massive MIMO-OFDM can be further reduced. Similar to , the sum MSE-CP for a given delay $\Delta_{\ell}$ between the data symbol and pilot symbol can be defined as $$\begin{aligned}
\label{eq:epsilonklcp}
{\epsilon^{\mathrm{CP}}}\left(\Delta_{\ell}\right)
&\triangleq\sum_{k=0}^{K-1}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}
{{{\mathsf{E}}}\left\{{\left|{\left[{\mathbf{H}}_{k,\ell+\Delta_{\ell}}-{\hat{{\mathbf{H}}}}_{k,\ell+\Delta_{\ell}}\right]_{i,j}}\right|^{2}}\right\}}{\notag\\}&=\sum_{k=0}^{K-1}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}
\Bigg\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}
-\varrho_{k}^{2}\left(\Delta_{\ell}\right){\notag\\}&\qquad\cdot\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}
{\sum_{k'=0}^{K-1}{\left[{{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}\right]_{i,j}}
+{\frac{1}{{\rho_{\mathrm{tr}}}}}}\Bigg\}.\end{aligned}$$ From , it can be seen that pilot interference will affect channel prediction performance similar to the channel estimation case. However, we will show in the following proposition that such effects can still be eliminated with proper pilot phase shift scheduling.
\[prop:mmsetrscp\] The sum MSE-CP ${\epsilon^{\mathrm{CP}}}\left(\Delta_{\ell}\right)\ \forall\Delta_{\ell}$ is lower bounded by $$\begin{aligned}
\label{eq:varepsiloncp}
&{\epsilon^{\mathrm{CP}}}\left(\Delta_{\ell}\right)\geq{\varepsilon^{\mathrm{CP}}}\left(\Delta_{\ell}\right){\notag\\}&=\sum_{k=0}^{K-1}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}
\left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}-\varrho_{k}^{2}\left(\Delta_{\ell}\right)\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}\right\}\end{aligned}$$ and the lower bound can be achieved under the condition that, for $\forall k,k'\in{\mathcal{K}}$ and $k\neq k'$, $$\label{eq:condokoudcp}
\left({\bar{{{\boldsymbol\Omega}}}}_{k,{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k}}\right)
\odot\left({\bar{{{\boldsymbol\Omega}}}}_{k',{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k'}}\right)
={\mathbf{0}}.$$
The proof is similar to that of [Proposition \[prop:mmsetrseqcon\]]{}, and is omitted for brevity.
Frame Structure {#subsec:fra_str}
---------------
There exist two typical frame structures for TDD massive MIMO transmission [@Bjornson15Massive]. One type of frame structure (which will be referred to as type-A) begins with the UL pilot segment, followed by the UL and downlink (DL) data segments, as shown in [Fig. \[fig:frame\]]{}. In the second type (which will be referred to as type-B), the UL pilot segment is placed between the UL data segment and DL data segment, as shown in [Fig. \[fig:frame\]]{}. For the proposed APSP approach, the delay between the tail-end symbols of the data segment and the pilot segment will be longer than the PSOP approach due to the reduced pilot segment length. In addition, the APSP approach focuses on high mobility scenarios where channels vary relatively quickly. Thus the type-B frame structure is well-suited for the proposed APSP approach.
Pilot Phase Shift Scheduling
----------------------------
In the previous subsections, we investigated channel estimation and prediction for massive MIMO-OFDM with APSPs, and obtained the optimal pilot phase shift scheduling condition applicable to both channel estimation and prediction. However, such an optimal condition cannot always be met in practice, but pilot phase shift scheduling can still be beneficial. Several scheduling criteria can be adopted. For example, if we schedule the pilot phase shifts based on the MMSE-CE criterion, the problem can be formulated as $$\begin{aligned}
\label{eq:pspsch}
{\mathop{\arg\min}\limits_{\left\{\phi_{k}:k\in{\mathcal{K}}\right\}}}&\qquad{{\epsilon}^{\mathrm{CE}}}\end{aligned}$$ where ${{\epsilon}^{\mathrm{CE}}}$ is defined in . Such a scheduling problem is combinatorial, and optimal solutions must be found through an exhaustive search. Note that the optimal phase shift scheduling conditions for channel estimation and prediction are the same, thus solution of the problem can also be expected to perform well under the MMSE-CP criterion.
Motivated by the optimal condition for channel estimation and prediction obtained in previous subsections, a simplified pilot phase shift scheduling algorithm can be developed. We first define the following function that measures the degree of overlap between two real matrices ${\mathbf{A}},{\mathbf{B}}\in{\mathbb{R}}^{M\times N}$ as follows $$\label{eq:xiab}
\xi\left({\mathbf{A}},{\mathbf{B}}\right)\triangleq
\frac{\sum_{i,j}{\left[{\mathbf{A}}\odot{\mathbf{B}}\right]_{i,j}}}
{\sqrt{\sum_{i,j}{\left[{\mathbf{A}}\right]_{i,j}}^{2}}\cdot
\sqrt{\sum_{i,j}{\left[{\mathbf{B}}\right]_{i,j}}^{2}}}.$$ From the Cauchy-Schwarz inequality, it is obvious that the overlapping degree function in satisfies $0\leq\xi\left({\mathbf{A}},{\mathbf{B}}\right)\leq1$. When ${\mathbf{A}}$ is a scaled version of ${\mathbf{B}}$, $\xi\left({\mathbf{A}},{\mathbf{B}}\right)=1.$ When the locations of non-zero elements in ${\mathbf{A}}$ and ${\mathbf{B}}$ are non-overlapping, $\xi\left({\mathbf{A}},{\mathbf{B}}\right)=0$. In our algorithm, we preset a threshold to balance the tradeoff between the algorithm complexity and channel acquisition performance. Specifically, we schedule the pilot phase shifts for different UTs to make the overlap function between the ADCPMs for different UTs smaller than the preset threshold $\gamma$. Intuitively, the smaller the threshold $\gamma$, the better the channel acquisition performance will be, although with a higher algorithm complexity. The description of the proposed algorithm is summarized in [Algorithm \[alg:GPPSSA\]]{}.
The UT set ${\mathcal{K}}$ and the corresponding ADCPMs $\left\{{{\boldsymbol\Omega}}_{k}:k\in{\mathcal{K}}\right\}$; the preset threshold $\gamma$ Pilot phase shift pattern $\left\{\phi_{k}:k\in{\mathcal{K}}\right\}$ Initialization: $\phi_{0}=0$, scheduled UT set ${{\mathcal{K}}^{\mathrm{sch}}}=\left\{0\right\}$, unscheduled UT set ${{\mathcal{K}}^{\mathrm{un}}}={{\mathcal{K}}\backslash {{\mathcal{K}}^{\mathrm{sch}}}}$ Search for a phase shift $\phi$ that satisfies $\xi\left({\bar{{{\boldsymbol\Omega}}}}_{k,\left({N_{\mathrm{c}}}\right)}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi},
\sum_{k'\in{{\mathcal{K}}^{\mathrm{sch}}}}{\bar{{{\boldsymbol\Omega}}}}_{k',\left({N_{\mathrm{c}}}\right)}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k'}}\right)\leq\gamma$ If $\phi$ cannot be found in step 3, then $\phi={\mathop{\arg\min}\limits_{x}}\ \xi\left({\bar{{{\boldsymbol\Omega}}}}_{k,\left({N_{\mathrm{c}}}\right)}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{x},
\sum_{k'\in{{\mathcal{K}}^{\mathrm{sch}}}}{\bar{{{\boldsymbol\Omega}}}}_{k',\left({N_{\mathrm{c}}}\right)}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k'}}\right)$ Update $\phi_{k}=\phi$, ${{\mathcal{K}}^{\mathrm{sch}}}\leftarrow{{\mathcal{K}}^{\mathrm{sch}}}\cup\left\{k\right\}$, ${{\mathcal{K}}^{\mathrm{un}}}\leftarrow{{{\mathcal{K}}^{\mathrm{un}}}\backslash \left\{k\right\}}$
Channel Acquisition with APSPs over Multiple Symbols {#sec:mul_tr}
====================================================
In the previous section, we investigated channel acquisition for massive MIMO-OFDM with the proposed APSPs over one OFDM symbol. Sometimes pilots over one symbol might be not sufficient to accommodate a large number of UTs. In this section, we extend the use of APSPs to the case of multiple consecutive OFDM symbols.
We assume that the pilots are sent over $Q$ consecutive OFDM symbols starting with the [$\ell$th]{} symbol in each frame. In practice, the pilot segment length $Q$ is usually short, and we adopt the widely accepted assumption that the channels remain constant during the pilot segment [@Barhumi03Optimal; @Minn06Optimal; @Chi11Training]. Then the received signals by the BS during the pilot segment can be written as $$\begin{aligned}
\label{eq:ytrlbq}
{\mathbf{Y}}_{\ell,{\left({Q}\right)}}
&=\sum_{k'=0}^{K-1}{\mathbf{G}}_{k',\ell}{\mathbf{X}}_{k',{\left({Q}\right)}}+{\mathbf{Z}}_{\ell,{\left({Q}\right)}}{\notag\\}&=\sum_{k'=0}^{K-1}{\mathbf{V}}_{M}{\mathbf{H}}_{k',\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\mathbf{X}}_{k',{\left({Q}\right)}}{\notag\\}&\qquad+{\mathbf{Z}}_{\ell,{\left({Q}\right)}}\in{\mathbb{C}}^{M\times{N_{\mathrm{c}}}Q}\end{aligned}$$ where ${\mathbf{Y}}_{\ell,{\left({Q}\right)}}\triangleq\left[{\mathbf{Y}}_{\ell}\quad{\mathbf{Y}}_{\ell+1}\quad\ldots\quad{\mathbf{Y}}_{\ell+Q-1}\right]$, ${\mathbf{Y}}_{\ell}\in{\mathbb{C}}^{M\times{N_{\mathrm{c}}}}$ represents the received pilot signal at the BS during the [$\ell$th]{} symbol, ${\mathbf{X}}_{k,{\left({Q}\right)}}\triangleq\left[{\mathbf{X}}_{k,0}\quad{\mathbf{X}}_{k,1}\quad\ldots\quad{\mathbf{X}}_{k,Q-1}\right]$ represents the pilot signals and ${\mathbf{X}}_{k,q}={\mathsf{diag}\left\{{\mathbf{x}}_{k,q}\right\}}\in{\mathbb{C}}^{{N_{\mathrm{c}}}\times{N_{\mathrm{c}}}}$ represents the signal sent from the [$k$th]{} UT during the [$q$th]{} symbol of the pilot segment, ${\mathbf{Z}}_{\ell,{\left({Q}\right)}}$ is AWGN with i.i.d. elements distributed as ${\mathcal{CN}\left( 0 , {\sigma_{\mathrm{ztr}}}\right) }$ and ${\sigma_{\mathrm{ztr}}}$ is the noise power.
Recalling , the maximum adjustable phase shift for different pilots over one OFDM symbol is ${N_{\mathrm{c}}}-1$. For the $Q$ pilot symbol case, the maximum adjustable pilot phase shift can be extended to $Q{N_{\mathrm{c}}}-1$. By exploiting the modulo operation, we construct the APSPs over multiple OFDM symbols as follows $$\begin{aligned}
\label{eq:xtruq}
{\mathbf{X}}_{k,{\left({Q}\right)}}
\triangleq\sqrt{Q}{\left[{\mathbf{U}}\right]_{{\left\langle\phi_{k}\right\rangle_{Q}},:}}
&\otimes{\mathbf{X}}_{\left\lfloor\phi_{k}/Q\right\rfloor},{\notag\\}&\qquad\phi_{k}={0,1,\ldots,Q{N_{\mathrm{c}}}-1}\end{aligned}$$ where ${\mathbf{U}}$ is an arbitrary $Q\times Q$ dimensional unitary matrix, and ${\mathbf{X}}_{\left\lfloor\phi_{k}/Q\right\rfloor}$ is the APSP signal over one symbol defined in . Then it can be obtained that, for $\forall k,k'\in{\mathcal{K}}$, $$\begin{aligned}
\label{eq:xiqxjq}
&{\mathbf{X}}_{k',{\left({Q}\right)}}\left({\mathbf{X}}_{k,{\left({Q}\right)}}\right)^{H}{\notag\\}&\qquad=Q\left({\left[{\mathbf{U}}\right]_{{\left\langle\phi_{k'}\right\rangle_{Q}},:}}
\otimes{\mathbf{X}}_{\left\lfloor\phi_{k'}/Q\right\rfloor}\right){\notag\\}&\qquad\quad\cdot\left({\left[{\mathbf{U}}\right]_{{\left\langle\phi_{k}\right\rangle_{Q}},:}}
\otimes{\mathbf{X}}_{\left\lfloor\phi_{k}/Q\right\rfloor}\right)^{H}{\notag\\}&\qquad{\mathop{=}^{(\mathrm{a})}}Q\left({\left[{\mathbf{U}}\right]_{{\left\langle\phi_{k'}\right\rangle_{Q}},:}}{\left[{\mathbf{U}}\right]_{{\left\langle\phi_{k}\right\rangle_{Q}},:}}^{H}\right)\otimes
\left({\mathbf{X}}_{\left\lfloor\phi_{k'}/Q\right\rfloor}{\mathbf{X}}_{\left\lfloor\phi_{k}/Q\right\rfloor}^{H}\right){\notag\\}&\qquad{\mathop{=}^{(\mathrm{b})}}{\sigma_{\mathrm{xtr}}}Q{\delta\left({\left\langle\phi_{k'}\right\rangle_{Q}}-{\left\langle\phi_{k}\right\rangle_{Q}}\right)}
\cdot{\mathbf{D}}_{\left\lfloor\phi_{k'}/Q\right\rfloor-\left\lfloor\phi_{k}/Q\right\rfloor}\end{aligned}$$ where (a) follows from the Kronecker product identities $\left({\mathbf{A}}\otimes{\mathbf{B}}\right)\left({\mathbf{C}}\otimes{\mathbf{D}}\right)=\left({\mathbf{A}}{\mathbf{C}}\right)\otimes\left({\mathbf{B}}{\mathbf{D}}\right)$ and $\left({\mathbf{A}}\otimes{\mathbf{B}}\right)^{H}={\mathbf{A}}^{H}\otimes{\mathbf{B}}^{H}$ [@Seber08Matrix], and (b) follows from . This shows that the available phase shifts for the $Q$ symbol case are divided into $Q$ groups for the proposed APSPs in , and the group index depends on the residue of the pilot phase shift $\phi$ with respect to the pilot segment length $Q$. Pilot interference can only affect the UTs using APSPs with phase shifts in the same group. For example, if ${\left\langle\phi_{k'}\right\rangle_{Q}}={\left\langle\phi_{k}\right\rangle_{Q}}$, then phase shifts $\phi_{k'}$ and $\phi_{k}$ are within the same group, and the corresponding channel acquisition of UTs $k'$ and $k$ might be mutually affected.
Given the APSP correlation property over multiple symbols in , the channel estimation and prediction operations can be performed similarly to the single-symbol case investigated in the previous section, and we will briefly discuss such issues below.
After decorrelation and power normalization with ${\mathbf{Y}}_{\ell,{\left({Q}\right)}}$ given in , the BS can obtain an observation of the pilot segment ADCRM ${\mathbf{H}}_{k,\ell}$ as $$\begin{aligned}
\label{eq:yklqtr}
&{\mathbf{Y}}_{k,\ell,{\left({Q}\right)}}{\notag\\}&\quad=
{\frac{1}{{\sigma_{\mathrm{xtr}}}Q}}{\mathbf{V}}_{M}^{H}{\mathbf{Y}}_{\ell,{\left({Q}\right)}}{\mathbf{X}}_{k,{\left({Q}\right)}}^{H}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}{\notag\\}&\quad={\frac{1}{{\sigma_{\mathrm{xtr}}}Q}}\sum_{k'=0}^{K-1}{\mathbf{H}}_{k',\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\mathbf{X}}_{k',{\left({Q}\right)}}
{\mathbf{X}}_{k,{\left({Q}\right)}}^{H}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}{\notag\\}&\qquad+{\frac{1}{{\sigma_{\mathrm{xtr}}}Q}}{\mathbf{V}}_{M}^{H}{\mathbf{Z}}_{\ell,{\left({Q}\right)}}{\mathbf{X}}_{k,{\left({Q}\right)}}^{H}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}{\notag\\}&\quad{\mathop{=}^{(\mathrm{a})}}\sum_{k'=0}^{K-1}{\delta\left({\left\langle\phi_{k'}\right\rangle_{Q}}-{\left\langle\phi_{k}\right\rangle_{Q}}\right)}\cdot{\mathbf{H}}_{k',\ell}
{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\notag\\}&\qquad\cdot{\mathbf{D}}_{\left\lfloor\phi_{k'}/Q\right\rfloor-\left\lfloor\phi_{k}/Q\right\rfloor}
{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}
+{\frac{1}{\sqrt{{\rho_{\mathrm{tr}}}Q}}}{{\mathbf{Z}}_{\mathrm{iid}}}{\notag\\}&\quad{\mathop{=}^{(\mathrm{b})}}{\mathbf{H}}_{k,\ell}+\underbrace{\sum_{k'\neq k}{\delta\left({\left\langle\phi_{k'}\right\rangle_{Q}}-{\left\langle\phi_{k}\right\rangle_{Q}}\right)}\cdot{\mathbf{H}}_{k',\ell}^{\left\lfloor\phi_{k'}/Q\right\rfloor-\left\lfloor\phi_{k}/Q\right\rfloor}}_{\mathrm{pilot\ interference}}{\notag\\}&\qquad+\underbrace{{\frac{1}{\sqrt{{\rho_{\mathrm{tr}}}Q}}}{{\mathbf{Z}}_{\mathrm{iid}}}}_{\mathrm{pilot\ noise}}\end{aligned}$$ where (a) follows from , ${\rho_{\mathrm{tr}}}\triangleq{\sigma_{\mathrm{xtr}}}/{\sigma_{\mathrm{ztr}}}$ is the pilot segment SNR, ${{\mathbf{Z}}_{\mathrm{iid}}}$ is the normalized AWGN matrix with i.i.d. elements distributed as ${\mathcal{CN}\left( 0 , 1\right) }$, and (b) follows from .
With the channel observation ${\mathbf{Y}}_{k,\ell,{\left({Q}\right)}}$ in , the MMSE estimate of the ADCRM ${\mathbf{H}}_{k,\ell}$ can be readily obtained in an element-wise manner as shown at the top of the next page,
$$\begin{aligned}
\label{eq:hathklmul}
{\left[{\hat{{\mathbf{H}}}}_{k,\ell}\right]_{i,j}}=
\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}}
{\sum_{k'=0}^{K-1}{\delta\left({\left\langle\phi_{k'}\right\rangle_{Q}}-{\left\langle\phi_{k}\right\rangle_{Q}}\right)}
{\left[{{\boldsymbol\Omega}}_{k'}^{\left\lfloor\phi_{k'}/Q\right\rfloor-\left\lfloor\phi_{k}/Q\right\rfloor}\right]_{i,j}}
+{\frac{1}{{\rho_{\mathrm{tr}}}Q}}}{\left[{\mathbf{Y}}_{k,\ell,{\left({Q}\right)}}\right]_{i,j}}\end{aligned}$$
------------------------------------------------------------------------
and the corresponding sum MSE-CE is given by shown at the top of the next page,
$$\begin{aligned}
\label{eq:epsilonklq}
{\epsilon_{{\left({Q}\right)}}^{\mathrm{CE}}}
=\sum_{k=0}^{K-1}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}\left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}
-\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}
{\sum_{k'=0}^{K-1}{\delta\left({\left\langle\phi_{k'}\right\rangle_{Q}}-{\left\langle\phi_{k}\right\rangle_{Q}}\right)}
{\left[{{\boldsymbol\Omega}}_{k'}^{\left\lfloor\phi_{k'}/Q\right\rfloor-\left\lfloor\phi_{k}/Q\right\rfloor}\right]_{i,j}}
+{\frac{1}{{\rho_{\mathrm{tr}}}Q}}}\right\}\end{aligned}$$
------------------------------------------------------------------------
In addition, prediction of the ADCRM ${\mathbf{H}}_{k,\ell+\Delta_{\ell}}$ based on ${\mathbf{Y}}_{k,\ell,{\left({Q}\right)}}$ can be performed as shown at the top of the next page,
$$\begin{aligned}
\label{eq:hathkldellmul}
{\left[{\hat{{\mathbf{H}}}}_{k,\ell+\Delta_{\ell}}\right]_{i,j}}=
\frac{\varrho_{k}\left(\Delta_{\ell}\right){\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}}
{\sum_{k'=0}^{K-1}{\delta\left({\left\langle\phi_{k'}\right\rangle_{Q}}-{\left\langle\phi_{k}\right\rangle_{Q}}\right)}
{\left[{{\boldsymbol\Omega}}_{k'}^{\left\lfloor\phi_{k'}/Q\right\rfloor-\left\lfloor\phi_{k}/Q\right\rfloor}\right]_{i,j}}
+{\frac{1}{{\rho_{\mathrm{tr}}}Q}}}{\left[{\mathbf{Y}}_{k,\ell,{\left({Q}\right)}}\right]_{i,j}}\end{aligned}$$
------------------------------------------------------------------------
and the corresponding sum MSE-CP with a given delay $\Delta_{\ell}$ is given by shown at the top of the next page.
$$\begin{aligned}
\label{eq:sumcpmul}
{\epsilon_{{\left({Q}\right)}}^{\mathrm{CP}}}\left(\Delta_{\ell}\right)
=\sum_{k=0}^{K-1}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}\left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}
-\frac{\varrho_{k}^{2}\left(\Delta_{\ell}\right){\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}
{\sum_{k'=0}^{K-1}{\delta\left({\left\langle\phi_{k'}\right\rangle_{Q}}-{\left\langle\phi_{k}\right\rangle_{Q}}\right)}
{\left[{{\boldsymbol\Omega}}_{k'}^{\left\lfloor\phi_{k'}/Q\right\rfloor-\left\lfloor\phi_{k}/Q\right\rfloor}\right]_{i,j}}
+{\frac{1}{{\rho_{\mathrm{tr}}}Q}}}\right\}\end{aligned}$$
------------------------------------------------------------------------
Based on the above sum MSE-CE and MSE-CP expressions for the multiple symbol APSP case, we can readily obtain the following proposition.
\[prop:mmsetrscpmul\] The sum MSE-CE ${\epsilon_{{\left({Q}\right)}}^{\mathrm{CE}}}$ is lower bounded by $$\label{eq:varepsiloncpmul}
{\epsilon_{{\left({Q}\right)}}^{\mathrm{CE}}}\geq
{\varepsilon_{{\left({Q}\right)}}^{\mathrm{CE}}}=\sum_{k=0}^{K-1}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}
\left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}-\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}Q}}}\right\}$$ and the sum MSE-CP ${\epsilon_{{\left({Q}\right)}}^{\mathrm{CP}}}\left(\Delta_{\ell}\right)$ for $\forall\Delta_{\ell}$ is lower bounded by $$\begin{aligned}
\label{eq:cpminmul}
&{\epsilon_{{\left({Q}\right)}}^{\mathrm{CP}}}\left(\Delta_{\ell}\right)\geq
{\varepsilon_{{\left({Q}\right)}}^{\mathrm{CP}}}\left(\Delta_{\ell}\right){\notag\\}&\qquad=
\sum_{k=0}^{K-1}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}
\left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}-\frac{\varrho_{k}^{2}\left(\Delta_{\ell}\right){\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}Q}}}\right\}.\end{aligned}$$ Both the lower bounds in and can be achieved under the condition that, for $\forall k,k'\in{\mathcal{K}}$ and $k\neq k'$, $$\begin{aligned}
\label{eq:condokoudcpmul}
\left({\bar{{{\boldsymbol\Omega}}}}_{k,{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\left\lfloor\phi_{k}/Q\right\rfloor}\right)
&\odot\left({\bar{{{\boldsymbol\Omega}}}}_{k',{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\left\lfloor\phi_{k'}/Q\right\rfloor}\right)={\mathbf{0}},{\notag\\}&\qquad\qquad\textrm{when} \quad {\left\langle\phi_{k}\right\rangle_{Q}}={\left\langle\phi_{k'}\right\rangle_{Q}}.\end{aligned}$$
The proof is similar to that of [Proposition \[prop:mmsetrseqcon\]]{}, and is omitted for brevity.
[Proposition \[prop:mmsetrscpmul\]]{} extends the single-symbol APSP case in the previous section to the multiple symbol case. Actually, when $Q=1$, [Proposition \[prop:mmsetrscpmul\]]{} reduces to the results in [Proposition \[prop:mmsetrseqcon\]]{} and [Proposition \[prop:mmsetrscp\]]{}. The interpretation of [Proposition \[prop:mmsetrscpmul\]]{} is straightforward. For multiple symbol APSPs, different pilot phase shifts are divided into several groups, and pilot interference only affects the UTs using the phase shifts within the same group. If pilot interference can be eliminated through proper phase shift scheduling in all the groups, then optimal channel estimation and prediction performance can be achieved. When the optimal pilot phase shift scheduling condition in [Proposition \[prop:mmsetrscpmul\]]{} cannot be met, a straightforward extension of the pilot phase shift scheduling algorithm in the previous section can be applied. Specifically, the UT set can be divided into $Q$ groups, and pilot phase shift scheduling can be performed within each UT group using [Algorithm \[alg:GPPSSA\]]{}. The tradeoff between channel acquisition performance and algorithm complexity can still be balanced with the preset threshold to determine the degree of allowable channel overlap.
Numerical Results {#sec:sim_res}
=================
In this section, we present numerical simulations to evaluate the performance of the proposed APSP-CA in massive MIMO-OFDM. The major OFDM parameters, which are based on 3GPP LTE [@3gpp.36.211], are summarized in [Table \[tb:ofdm\_para\]]{}. The massive MIMO-OFDM system considered is assumed to be equipped with a 128-antenna ULA at the BS with half wavelength antenna spacing. The number of UTs is set to $K=42$ as in [@Marzetta10Noncooperative].
We consider channels with 20 taps in the delay domain, which exhibit an exponential power delay profile [@Pedersen00stochastic; @win2chanmod] $$\begin{aligned}
\label{eq:exp_pdp}
{{\mathtt{S}}_{k}^{\mathrm{del}}}\left(\tau\right)
\propto{\exp\left(-\tau/\varsigma_{k}\right)},\ {\textrm{for}}\ \tau\in \left[0,{N_{\mathrm{g}}}{T_{\mathrm{s}}}\right]\end{aligned}$$ where $\varsigma_{k}$ denotes the channel delay spread of UT $k$. We assume that transmissions from all the UTs are synchronized [@Dahlman11LTE; @win2chanmod]. The [$q$th]{} channel tap of UT $k$ is assumed to exhibit a Laplacian power angle spectrum [@Pedersen00stochastic; @win2chanmod; @You15Pilot] $$\begin{aligned}
\label{eq:lap_pas}
{{\mathtt{S}}_{k,q}^{\mathrm{ang}}}\left(\theta\right)&\propto {\exp\left(-\sqrt{2}{\left|\theta-\theta_{k,q}\right|}/\varphi_{k,q}\right)},{\notag\\}&\qquad\qquad {\textrm{for}}\ \theta\in{\mathcal{A}}=[-\pi/2,\pi/2]\end{aligned}$$ where $\theta_{k,q}$ and $\varphi_{k,q}$ represent the corresponding mean angle of arrival (AoA) and angle spread for the given channel tap, respectively. We assume that the UTs are uniformly distributed in a $120^{\circ}$ sector, and the mean AoA $\theta_{k,q}$ is uniformly distributed in the angle interval $[-\pi/3,\pi/3]$ in radians. We do not consider large scale fading in the simulations, and channels are normalized as $\sum_{i,j}{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}=M{N_{\mathrm{c}}}$ for $\forall k$. We consider channel propagation under several typical mobility scenarios including suburban (SU), urban macro (UMa), and urban micro (UMi). The primary statistical channel parameters under these scenarios are based on the WINNER II channel model [@win2chanmod; @Auer12MIMO], and are summarized in [Table \[tb:cha\_sta\_par\]]{}. We assume that all UTs exhibit the same Doppler, delay, and angle spread in the simulations.
[LcR]{} Parameter & & Value\
System bandwidth && 20 MHz\
Sampling duration ${T_{\mathrm{s}}}$ && 32.6 ns\
Subcarrier spacing && 15 kHz\
Subcarrier number ${N_{\mathrm{c}}}$ && 2048\
Guard interval ${N_{\mathrm{g}}}$ && 144\
Symbol length ${T_{\mathrm{sym}}}$ && 71.4 $\mu$s\
[LcccR]{} Scenario &&
---------------------------------
Doppler $\nu{T_{\mathrm{sym}}}$
(Velocity)
---------------------------------
: Statistical Channel Parameters in Typical Scenarios[]{data-label="tb:cha_sta_par"}
&
--------------------
Delay
spread $\varsigma$
--------------------
: Statistical Channel Parameters in Typical Scenarios[]{data-label="tb:cha_sta_par"}
&
------------------
Angle
spread $\varphi$
------------------
: Statistical Channel Parameters in Typical Scenarios[]{data-label="tb:cha_sta_par"}
\
Suburban && $31\times10^{-3}$ & 0.77 $\mu$s & $2^{\circ}$\
(SU) && ([250 km/h]{}) & &\
Urban macro && $14\times10^{-3}$ & 1.85 $\mu$s & $2^{\circ}$\
(UMa) && ([100 km/h]{}) &&\
Urban micro && $6.6\times10^{-3}$ & 0.62 $\mu$s & $10^{\circ}$\
(UMi) && ([50 km/h]{}) & &\
With the above settings, we compare the performance of the proposed APSP-CA approach with that of the conventional PSOP-CA approach, which serves as the benchmark for comparison of channel acquisition performance. For the conventional PSOP-CA, the required pilot segment length is $Q=\left\lceil K/\left({N_{\mathrm{c}}}/{N_{\mathrm{g}}}\right)\right\rceil=3$ OFDM symbols [@Marzetta10Noncooperative]. For the proposed APSP-CA, the pilot segment length can be set to $Q=1$ or $2$. We adopt [Algorithm \[alg:GPPSSA\]]{} to schedule the pilot phase shifts in the simulations, and the overlap threshold in the algorithm is set as $\gamma=10^{-4}$. Although this algorithm is suboptimal in general compared with exhaustive search, substantial performance gains over the conventional PSOP-CA in terms of achievable spectral efficiency can still be achieved with relatively little computational cost.
In [Fig. \[fig:psmsecom\]]{}, the pilot segment MSE-CE performance[^9] obtained by the proposed APSPs (with $Q=1$ and $2$) are compared with those for conventional PSOPs ($Q=3$) under several typical propagation scenarios. It can be observed that, in all the considered scenarios, the MSE-CE performance with APSPs approaches the performance obtained with PSOPs, while the pilot overhead is reduced by 66.7% ($Q=1$) and 33.3% ($Q=2$), respectively.
In [Fig. \[fig:msecpcomp\]]{}, we compare the channel acquisition performance during the data segment in terms of MSE versus the delay $\Delta_{\ell}$ between the data symbol and pilot segment. Both the APSP-CA ($Q=1$) and PSOP-CA ($Q=3$) are evaluated. Also, for APSPs, both the channel estimation and prediction MSE performance are calculated. It can be observed that the MSE-CP performance obtained with APSPs approaches that for PSOPs, with the pilot overhead reduced by 66.7%. In addition, with APSPs, channel prediction outperforms channel estimation in all the evaluated scenarios. Note that the channel acquisition performance in terms of both MSE-CE and MSE-CP grows almost linearly with delay, and thus the channel acquisition performance can be improved when combined with the type-B frame structure, as shown in the following simulation results.
At the end of this section, we compare the achievable spectral efficiency of the proposed APSP and the conventional PSOP approaches.[^10] We assume that the frame length equals 500 $\mu$s as in [@Marzetta10Noncooperative], which is equal to the length of 7 OFDM symbols [@3gpp.36.211], and that UL and DL data transmission each occupies half of the data segment length. For the conventional PSOP-CA approach, channel estimation and the type-A frame structure in [Fig. \[fig:frame\]]{} are adopted. For the proposed APSP-CA approach, both APSPs ($Q=1$) and channel prediction are adopted, and both type-A and type-B frame structures are considered. A MMSE receiver and precoder are employed for both UL and DL data transmissions, and the SNR is assumed to be equal to the pilot SNR. In [Fig. \[fig:ratecomp\]]{}, the achieved spectral efficiency[^11] of the APSP-CA and PSOP-CA approaches are depicted. It can be observed that the proposed APSP-CA approach shows substantial performance gain in terms of the achievable spectral efficiency over the conventional PSOP-CA approach, especially in the high mobility regime where pilot overhead dominates and the high SNR regime where pilot interference dominates. Specifically, in the high mobility SU scenario (250 km/h) with an SNR of 10 dB, the proposed APSPs can provide about 69% in average spectral efficiency gains over the conventional PSOPs. In addition, the type-B frame structure can provide a gain of about 64% over the type-A frame structure when APSPs are adopted.
Conclusion {#sec:conc_pw}
==========
In this paper, we proposed a channel acquisition approach with adjustable phase shift pilots (APSPs) for massive MIMO-OFDM to reduce the pilot overhead. We first investigated the channel sparsity in massive MIMO-OFDM based on a physically motivated channel model. With this channel model, we investigated channel estimation and prediction for massive MIMO-OFDM with APSPs, and provided an optimal pilot phase shift scheduling condition applicable to both channel estimation and prediction. We further developed a simplified pilot phase shift scheduling algorithm based on this optimal channel acquisition condition with APSPs. The proposed APSP-CA implemented over both one and multiple symbols were investigated. Significant performance gains in terms of achievable spectral efficiency were observed for the proposed APSP-CA approach over the conventional PSOP-CA approach in several typical mobility scenarios.
Derivation of {#app:der_sta}
==============
The derivation of is detailed in , shown at the top of the next page, where (a) follows from , and (b) follows from the definition of the delta function.
$$\begin{aligned}
\label{eq:derofsta}
&{{{\mathsf{E}}}\left\{{\mathsf{vec}\left\{{\mathbf{G}}_{k,\ell+\Delta_{\ell}}\right\}}{\mathsf{vec}^{H}\left\{{\mathbf{G}}_{k,\ell}\right\}}\right\}} {\notag\\}&\quad=\mathsf{E}\left\{\left[\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\infty}^{\infty}\int\limits_{-\pi/2}^{\pi/2}\!
\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{{\mathbf{v}}_{M,\theta}}\right]\cdot{\exp\left({\bar{\jmath}}2\pi \nu\left(\ell+\Delta_{\ell}\right){T_{\mathrm{sym}}}\right)}
\cdot g_{k}\left(\theta,q{T_{\mathrm{s}}},\nu\right){{\mathrm{d}}\theta}{{\mathrm{d}}\nu}\right]\right.{\notag\\}&\qquad\quad\cdot\left.\left[\sum_{q'=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\infty}^{\infty}\int\limits_{-\pi/2}^{\pi/2}\!
\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q'}\otimes{\mathbf{v}}_{M,\theta'}\right]^{H}\cdot{\exp\left(-{\bar{\jmath}}2\pi \nu'\ell{T_{\mathrm{sym}}}\right)}
\cdot g_{k}\left(\theta',q'{T_{\mathrm{s}}},\nu'\right){{\mathrm{d}}\theta'}{{\mathrm{d}}\nu'}\right]\right\}{\notag\\}&\quad=\sum_{q=0}^{{N_{\mathrm{g}}}-1}\sum_{q'=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\infty}^{\infty}\int\limits_{-\pi/2}^{\pi/2}\int\limits_{-\infty}^{\infty}\int\limits_{-\pi/2}^{\pi/2}\!
\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{{\mathbf{v}}_{M,\theta}}\right]\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q'}\otimes{\mathbf{v}}_{M,\theta'}\right]^{H}
\cdot{\exp\left({\bar{\jmath}}2\pi \nu\left(\ell+\Delta_{\ell}\right){T_{\mathrm{sym}}}\right)}
\cdot{\exp\left(-{\bar{\jmath}}2\pi \nu'\ell{T_{\mathrm{sym}}}\right)}
{\notag\\}&\qquad
\cdot {{{\mathsf{E}}}\left\{g_{k}\left(\theta,q{T_{\mathrm{s}}},\nu\right)
g_{k}\left(\theta',q'{T_{\mathrm{s}}},\nu'\right)\right\}}{{\mathrm{d}}\theta}{{\mathrm{d}}\nu}{{\mathrm{d}}\theta'}{{\mathrm{d}}\nu'}{\notag\\}&\quad{\mathop{=}^{(\mathrm{a})}}\sum_{q=0}^{{N_{\mathrm{g}}}-1}\sum_{q'=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\infty}^{\infty}\int\limits_{-\pi/2}^{\pi/2}\int\limits_{-\infty}^{\infty}\int\limits_{-\pi/2}^{\pi/2}\!
\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{{\mathbf{v}}_{M,\theta}}\right]\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q'}\otimes{\mathbf{v}}_{M,\theta'}\right]^{H}
\cdot{\exp\left({\bar{\jmath}}2\pi \nu\left(\ell+\Delta_{\ell}\right){T_{\mathrm{sym}}}\right)}
\cdot{\exp\left(-{\bar{\jmath}}2\pi \nu'\ell{T_{\mathrm{sym}}}\right)}
{\notag\\}&\qquad
\cdot {{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,q{T_{\mathrm{s}}}\right)\cdot{{\mathtt{S}}_{k}^{\mathrm{Dop}}}\left(\nu\right)\cdot{\delta\left(\theta-\theta'\right)}{\delta\left(q-q'\right)}{\delta\left(\nu-\nu'\right)}{{\mathrm{d}}\theta}{{\mathrm{d}}\nu}{{\mathrm{d}}\theta'}{{\mathrm{d}}\nu'}{\notag\\}&\quad{\mathop{=}^{(\mathrm{b})}}\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\infty}^{\infty}\int\limits_{-\pi/2}^{\pi/2}\!
\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{{\mathbf{v}}_{M,\theta}}\right]\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{\mathbf{v}}_{M,\theta}\right]^{H}
\cdot{\exp\left({\bar{\jmath}}2\pi\nu\Delta_{\ell}{T_{\mathrm{sym}}}\right)}
\cdot {{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,q{T_{\mathrm{s}}}\right)\cdot{{\mathtt{S}}_{k}^{\mathrm{Dop}}}\left(\nu\right){{\mathrm{d}}\theta}{{\mathrm{d}}\nu}{\notag\\}&\quad=\underbrace{\int\limits_{-\infty}^{\infty}\!{\exp\left({\bar{\jmath}}2\pi\nu\Delta_{\ell}{T_{\mathrm{sym}}}\right)} \cdot{{\mathtt{S}}_{k}^{\mathrm{Dop}}}\left(\nu\right){{\mathrm{d}}\nu}}_{\varrho_{k}\left(\Delta_{\ell}\right)}
\cdot
\underbrace{\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\pi/2}^{\pi/2}\!
\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{{\mathbf{v}}_{M,\theta}}\right]\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{\mathbf{v}}_{M,\theta}\right]^{H}
\cdot{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,q{T_{\mathrm{s}}}\right){{\mathrm{d}}\theta}}_{{\mathbf{R}}_{k}}\end{aligned}$$
------------------------------------------------------------------------
Proof of [[Proposition \[prop:Decomp\_cov\]]{}]{} {#app:prop_Decomp_cov}
=================================================
We start by defining some auxiliary variables to simplify the derivations. We define $n_{d}\triangleq\left\lfloor d/M\right\rfloor$ and $m_{d}\triangleq{\left\langled\right\rangle_{M}}$ for an arbitrary non-negative integer $d$. Note that the element indices start from $0$ in this paper. Then we can readily obtain that for a matrix ${{\boldsymbol\Omega}}_{k}\in{\mathbb{R}}^{M\times{N_{\mathrm{g}}}}$, the $d$th element of ${\mathsf{vec}\left\{{{\boldsymbol\Omega}}_{k}\right\}}$ equals the $\left(m_{d},n_{d}\right)$th element of ${{\boldsymbol\Omega}}_{k}$, i.e., ${\left[{\mathsf{vec}\left\{{{\boldsymbol\Omega}}_{k}\right\}}\right]_{d}}={\left[{{\boldsymbol\Omega}}_{k}\right]_{m_{d},n_{d}}}$. We can also obtain that for matrices ${\mathbf{F}}\in{\mathbb{C}}^{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}$ and ${\mathbf{V}}\in{\mathbb{C}}^{M\times M}$, ${\left[{\mathbf{F}}\otimes{\mathbf{V}}\right]_{i,j}}={\left[{\mathbf{F}}\right]_{n_i,n_j}}{\left[{\mathbf{V}}\right]_{m_i,m_j}}$ from the definition of the Kronecker product. With the above definitions and related properties, the proof can be obtained as follows:
$$\begin{aligned}
\label{eq:pradlm}
&\lim_{\substack{{M \to\infty}}}
\Big[{\mathbf{R}}_{k}-\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)
{\mathsf{diag}\left\{{\mathsf{vec}\left\{{{\boldsymbol\Omega}}_{k}\right\}}\right\}}{\notag\\}&\qquad\cdot\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)^{H}\Big]_{i,j}{\notag\\}&\quad=\lim_{\substack{{M \to\infty}}}
{\left[{\mathbf{R}}_{k}\right]_{i,j}}-\lim_{\substack{{M \to\infty}}}\sum_{d=0}^{M{N_{\mathrm{g}}}-1}{\left[{\mathsf{vec}\left\{{{\boldsymbol\Omega}}_{k}\right\}}\right]_{d}}{\notag\\}&\qquad\cdot{\left[{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right]_{i,d}}{\left[{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right]_{j,d}}^{*}{\notag\\}&\quad{\mathop{=}^{(\mathrm{a})}}\lim_{\substack{{M \to\infty}}}{\left[{\mathbf{R}}_{k}\right]_{i,j}}-\lim_{\substack{{M \to\infty}}}
\sum_{n_{d}=0}^{{N_{\mathrm{g}}}-1}\sum_{m_{d}=0}^{M-1}{\left[{{\boldsymbol\Omega}}_{k}\right]_{m_{d},n_{d}}}{\notag\\}&\qquad
\cdot{\left[{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\right]_{n_{i},n_{d}}}{\left[{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\right]_{n_{j},n_{d}}}^{*}
{\left[{\mathbf{V}}_{M}\right]_{m_{i},m_{d}}}{\left[{\mathbf{V}}_{M}\right]_{m_{j},m_{d}}}^{*}{\notag\\}&\quad{\mathop{=}^{(\mathrm{b})}}\lim_{\substack{{M \to\infty}}}{\left[{\mathbf{R}}_{k}\right]_{i,j}}
-\lim_{\substack{{M \to\infty}}}{\frac{1}{M{N_{\mathrm{c}}}}}\sum_{n_{d}=0}^{{N_{\mathrm{g}}}-1}\sum_{m_{d}=0}^{M-1}
M{N_{\mathrm{c}}}{\notag\\}&\qquad\cdot\left(\theta_{m_{d}+1}-\theta_{m_{d}}\right)
{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta_{m_{d}},\tau_{n_{d}}\right){\notag\\}&\qquad\cdot{\exp\left(-{\bar{\jmath}}2\pi\frac{\left(n_{i}-n_{j}\right)n_{d}}{{N_{\mathrm{c}}}}\right)}{\notag\\}&\qquad\cdot{\exp\left(-{\bar{\jmath}}2\pi\frac{\left(m_{i}-m_{j}\right)\left(m_{d}-M/2\right)}{M}\right)}{\notag\\}&\quad{\mathop{=}^{(\mathrm{c})}}\sum_{q=0}^{{N_{\mathrm{g}}}-1}
\int\limits_{-\pi/2}^{\pi/2}\!\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{{\mathbf{v}}_{M,\theta}}\right]_{i}
\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\left(q\right)\otimes{{\mathbf{v}}_{M,\theta}}\right]_{j}^{*}{\notag\\}&\qquad\cdot{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,q{T_{\mathrm{s}}}\right){{\mathrm{d}}\theta}
-\lim_{\substack{{M \to\infty}}}\sum_{n_{d}=0}^{{N_{\mathrm{g}}}-1}\sum_{m_{d}=0}^{M-1}
\left(\theta_{m_{d}+1}-\theta_{m_{d}}\right){\notag\\}&\qquad\cdot{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta_{m_{d}},\tau_{n_{d}}\right)
{\exp\left(-{\bar{\jmath}}2\pi\frac{\left(n_{i}-n_{j}\right)n_{d}}{{N_{\mathrm{c}}}}\right)}{\notag\\}&\qquad\cdot
{\exp\left(-{\bar{\jmath}}\pi\left(m_{i}-m_{j}\right)\sin\left(\theta_{m_{d}}\right)\right)}{\notag\\}&\quad{\mathop{=}^{(\mathrm{d})}}\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\pi/2}^{\pi/2}\!
{\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\right]_{n_{i}}}{\left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\right]_{n_{j}}}^{*}
{\left[{{\mathbf{v}}_{M,\theta}}\right]_{m_{i}}}{\left[{{\mathbf{v}}_{M,\theta}}\right]_{m_{j}}}^{*}{\notag\\}&\qquad\cdot{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau_{q}\right){{\mathrm{d}}\theta}
-\sum_{r=0}^{{N_{\mathrm{g}}}-1}\int\limits_{\theta_{0}}^{\theta_{M}}\!
{\exp\left(-{\bar{\jmath}}2\pi\frac{\left(n_{i}-n_{j}\right)}{{N_{\mathrm{c}}}}r\right)}{\notag\\}&\qquad\cdot
{\exp\left(-{\bar{\jmath}}\pi\left(m_{i}-m_{j}\right)\sin\left(\theta\right)\right)}\cdot
{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau_{r}\right){{\mathrm{d}}\theta}{\notag\\}&\quad{\mathop{=}^{(\mathrm{e})}}\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\pi/2}^{\pi/2}\!
{\exp\left(-{\bar{\jmath}}2\pi\frac{\left(n_{i}-n_{j}\right)}{{N_{\mathrm{c}}}}q\right)}{\notag\\}&\qquad\cdot
{\exp\left(-{\bar{\jmath}}\pi\left(m_{i}-m_{j}\right)\sin\left(\theta\right)\right)}\cdot
{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau_{q}\right){{\mathrm{d}}\theta}{\notag\\}&\qquad-\sum_{r=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\pi/2}^{\pi/2}\!
{\exp\left(-{\bar{\jmath}}2\pi\frac{\left(n_{i}-n_{j}\right)}{{N_{\mathrm{c}}}}r\right)}{\notag\\}&\qquad\cdot
{\exp\left(-{\bar{\jmath}}\pi\left(m_{i}-m_{j}\right)\sin\left(\theta\right)\right)}\cdot
{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau_{r}\right){{\mathrm{d}}\theta}{\notag\\}&\quad=0\end{aligned}$$
where (a) follows from the definition of Kronecker product and the definitions of $m_{d}$ and $n_{d}$, (b) follows from and the definitions of ${\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}$ and ${\mathbf{V}}_{M}$, (c) follows from and the definitions of $\tau_{n}$ and $\theta_{m}$, (d) follows from the definition of the Kronecker product, and (e) follows from and the fact that $\theta_{0}=-\pi/2$ and $\theta_{M}=\pi/2$.
Before concluding the proof, we also have to show that both of the limits in the first equation of exist and are finite. For this purpose, as can be seen from (e) of , we only need to show that $$\begin{aligned}
&\Bigg|\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\pi/2}^{\pi/2}\!
{\exp\left(-{\bar{\jmath}}2\pi\frac{\left(n_{i}-n_{j}\right)}{{N_{\mathrm{c}}}}q\right)}{\notag\\}&\qquad\cdot{\exp\left(-{\bar{\jmath}}\pi\left(m_{i}-m_{j}\right)\sin\left(\theta\right)\right)}\cdot
{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau_{q}\right){{\mathrm{d}}\theta}\Bigg| {\notag\\}&\quad\mathop{\leq}^{(\mathrm{a})}\sum_{q=0}^{{N_{\mathrm{g}}}-1}\Bigg|\int\limits_{-\pi/2}^{\pi/2}\!
{\exp\left(-{\bar{\jmath}}2\pi\frac{\left(n_{i}-n_{j}\right)}{{N_{\mathrm{c}}}}q\right)}{\notag\\}&\quad\qquad\cdot{\exp\left(-{\bar{\jmath}}\pi\left(m_{i}-m_{j}\right)\sin\left(\theta\right)\right)}\cdot
{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau_{q}\right){{\mathrm{d}}\theta}\Bigg| {\notag\\}&\quad\mathop{\leq}^{(\mathrm{b})}\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\pi/2}^{\pi/2}\!
\Bigg|{\exp\left(-{\bar{\jmath}}2\pi\frac{\left(n_{i}-n_{j}\right)}{{N_{\mathrm{c}}}}q\right)}{\notag\\}&\quad\qquad\cdot{\exp\left(-{\bar{\jmath}}\pi\left(m_{i}-m_{j}\right)\sin\left(\theta\right)\right)}\cdot
{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau_{q}\right)\Bigg|{{\mathrm{d}}\theta} {\notag\\}&\quad=\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\pi/2}^{\pi/2}\!
{\left|{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau_{q}\right)\right|}{{\mathrm{d}}\theta}{\notag\\}&\quad\mathop{<}^{(\mathrm{c})}\infty\end{aligned}$$ where (a) follows from the triangle inequality ${\left|\sum_{q=0}^{N-1}a_{q}\right|}\leq\sum_{q=0}^{N-1}{\left|a_{q}\right|}$, (b) follows from the integral property ${\left|\int_{a}^{b}\!f\left(x\right){{\mathrm{d}}x}\right|}\leq\int_{a}^{b}\!{\left|f\left(x\right)\right|}{{\mathrm{d}}x}$, and (c) follows from the fact that the power angle-delay spectrum function ${{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau\right)$, which represents the channel power in the angle-delay domain, is bounded. This concludes the proof.
Proof of [[Proposition \[prop:adcmsts\]]{}]{} {#app:prop_adcmsts}
=============================================
To show , it suffices to show that $$\begin{aligned}
&{{{\mathsf{E}}}\left\{{\mathsf{vec}\left\{{\mathbf{H}}_{k,\ell+\Delta_{\ell}}\right\}}{\mathsf{vec}^{H}\left\{{\mathbf{H}}_{k,\ell}\right\}}\right\}}{\notag\\}&\qquad=\varrho_{k}\left(\Delta_{\ell}\right)\cdot{\mathsf{diag}\left\{{\mathsf{vec}\left\{{{\boldsymbol\Omega}}_{k}\right\}}\right\}}.\end{aligned}$$ From the definition of ${\mathbf{H}}_{k,\ell}$ given in , we can obtain $$\begin{aligned}
{\mathsf{vec}\left\{{\mathbf{H}}_{k,\ell}\right\}}&=\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{H}\otimes{\mathbf{V}}_{M}^{H}\right){\mathsf{vec}\left\{{\mathbf{G}}_{k,\ell}\right\}}{\notag\\}&=\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)^{H}{\mathsf{vec}\left\{{\mathbf{G}}_{k,\ell}\right\}}\end{aligned}$$ via employing the Kronecker product identities ${\mathsf{vec}\left\{{\mathbf{A}}{\mathbf{B}}{\mathbf{C}}\right\}}=\left({\mathbf{C}}^{T}\otimes{\mathbf{A}}\right){\mathsf{vec}\left\{{\mathbf{B}}\right\}}$ and ${\mathbf{A}}^{H}\otimes{\mathbf{B}}^{H}=\left({\mathbf{A}}\otimes{\mathbf{B}}\right)^{H}$ [@Seber08Matrix].
Then it can be shown that $$\begin{aligned}
&{{{\mathsf{E}}}\left\{{\mathsf{vec}\left\{{\mathbf{H}}_{k,\ell+\Delta_{\ell}}\right\}}{\mathsf{vec}^{H}\left\{{\mathbf{H}}_{k,\ell}\right\}}\right\}}{\notag\\}&\qquad{\mathop{=}^{(\mathrm{a})}}\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)^{H}
{{{\mathsf{E}}}\left\{{\mathsf{vec}\left\{{\mathbf{G}}_{k,\ell+\Delta_{\ell}}\right\}}{\mathsf{vec}^{H}\left\{{\mathbf{G}}_{k,\ell}\right\}}\right\}}{\notag\\}&\qquad\quad\cdot\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right){\notag\\}&\qquad{\mathop{=}^{(\mathrm{b})}}\varrho_{k}\left(\Delta_{\ell}\right)\cdot\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)^{H}
{\mathbf{R}}_{k}\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right){\notag\\}&\qquad{\mathop{=}^{(\mathrm{c})}}\varrho_{k}\left(\Delta_{\ell}\right)\cdot{\mathsf{diag}\left\{{\mathsf{vec}\left\{{{\boldsymbol\Omega}}_{k}\right\}}\right\}}\end{aligned}$$ where (a) follows from the fact that ${\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}$ and ${\mathbf{V}}_{M}$ are both deterministic matrices, (b) follows from , and (c) follows from [Proposition \[prop:Decomp\_cov\]]{}. This concludes the proof.
Proof of [[Proposition \[prop:mmsetrseqcon\]]{}]{} {#app:prop_mmsetrseqcon}
==================================================
Due to the fact that the elements of ${{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}$ are non-negative, we can obtain $$\begin{aligned}
{{\epsilon}^{\mathrm{CE}}}=&\sum_{k=0}^{K-1}
\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}\Bigg\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}
{\notag\\}&\qquad-\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+\sum_{k'\neq k}{\left[{{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}\Bigg\}{\notag\\}\geq&\sum_{k=0}^{K-1}
\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}\left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}
-\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}\right\}
={\varepsilon^{\mathrm{CE}}}.\end{aligned}$$ Furthermore, when the condition $\left({\bar{{{\boldsymbol\Omega}}}}_{k,{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k}}\right)
\odot\left({\bar{{{\boldsymbol\Omega}}}}_{k',{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k'}}\right)={\mathbf{0}}$ is satisfied, then with the same column permutation and column truncation, multiplications of the corresponding elements still equal zero, i.e., $$\begin{aligned}
\label{eq:condomeg1}
&\left({\bar{{{\boldsymbol\Omega}}}}_{k,{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k}}
{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{-\phi_{k}}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\right){\notag\\}&\qquad\odot\left({\bar{{{\boldsymbol\Omega}}}}_{k',{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k'}}
{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{-\phi_{k}}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\right)={\mathbf{0}}.\end{aligned}$$
Recalling the definition in and exploiting the permutation matrix property that ${{\boldsymbol\Pi}}_{N}^{a}{{\boldsymbol\Pi}}_{N}^{b}={{\boldsymbol\Pi}}_{N}^{a+b}$, the condition in is equivalent to $${{\boldsymbol\Omega}}_{k}\odot{{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}={\mathbf{0}}.$$ Then for $\forall i,j$, $$\begin{aligned}
&{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}\left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+\sum_{k'\neq k}{\left[{{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}\right\}{\notag\\}&\qquad={\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}\left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}\right\}\end{aligned}$$ which leads to $$\label{eq:okijom}
\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+\sum_{k'\neq k}{\left[{{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}
=\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}.$$ Substituting into , the MSE-CE expression ${\epsilon_{k}^{\mathrm{CE}}}$ reduces to $$\begin{aligned}
{\varepsilon_{k}^{\mathrm{CE}}}
=\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}\left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}
-\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}\right\}.\end{aligned}$$ Then the minimum in can be achieved. This concludes the proof.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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[^1]: Copyright © 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].
[^2]: This work was supported in part by the National Natural Science Foundation of China under Grants 61471113, 61320106003, and 61201171, the China High-Tech 863 Plan under Grants 2015AA01A701 and 2014AA01A704, the National Science and Technology Major Project of China under Grant 2014ZX03003006-003, and the Program for Jiangsu Innovation Team. The work of L. You was supported in part by the China Scholarship Council (CSC). This work was presented in part at the IEEE 16th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, 2015.
[^3]: L. You, X. Q. Gao, and W. Zhong are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China (e-mail: [email protected]; [email protected]; [email protected]).
[^4]: A. L. Swindlehurst is with the Center for Pervasive Communications and Computing (CPCC), University of California, Irvine, CA 92697 USA (e-mail: [email protected]).
[^5]: There has been recent work that considers channels with a sparse common support [@Barbotin12Estimation; @Rao14Distributed]. However, for massive MIMO channels, the common support assumption might not hold due to the increased angle resolution [@Masood15Efficient; @Barbotin12Estimation]. Thus, in this work we assume that the channel sparsity patterns of different UTs are different (but not necessarily totally different), although the proposed APSP approach can also be applied to the common support cases.
[^6]: We adopt the ULA model in this paper for clarity, although our work can be readily extended to more general antenna array models using the techniques in [@You15Pilot].
[^7]: Although the waves impinging on the BS are assumed to be sparsely distributed in the angle domain due to limited scattering around the BS (typically mounted at an elevated position), the waves departing the mobile UTs are usually uniformly distributed in angle of departure. Thus the Clarke-Jakes spectrum is suitable to model the time variation of the channel [@Jakes94Microwave; @Patzold12Mobile].
[^8]: The degree of channel stationarity depends on the propagation scenarios. In typical scenarios, the channel statistics vary on the order of seconds [@Liu15Two], while the OFDM symbol length is usually on the order of millisecond [@3gpp.36.211].
[^9]: All the simulated MSE results are normalized by the number of subcarriers ${N_{\mathrm{c}}}$ and the number of UTs $K$.
[^10]: Note that the achievable spectral efficiency can reflect the tradeoff between the transmission performance and pilot overhead. Intuitively, reducing the pilot overhead decreases the channel acquisition quality (which leads to degradation of the achievable spectral efficiency), but also increases the length of the data segments (which leads to increased achievable spectral efficiency).
[^11]: The achievable UL rate is evaluated using the classical worst case approach as in [@Hassibi03How], and the achievable DL rate is evaluated using the approach in [@Jose11Pilot]. The OFDM guard interval overhead is taken into account.
| 0 |
---
abstract: 'The Aw-Rascle-Zhang (ARZ) model can be interpreted as a generalization of the Lighthill-Whitham-Richards (LWR) model, possessing a family of fundamental diagram curves, each of which represents a class of drivers with a different empty road velocity. A weakness of this approach is that different drivers possess vastly different densities at which traffic flow stagnates. This drawback can be overcome by modifying the pressure relation in the ARZ model, leading to the generalized Aw-Rascle-Zhang (GARZ) model. We present an approach to determine the parameter functions of the GARZ model from fundamental diagram measurement data. The predictive accuracy of the resulting data-fitted GARZ model is compared to other traffic models by means of a three-detector test setup, employing two types of data: vehicle trajectory data, and sensor data. This work also considers the extension of the ARZ and the GARZ models to models with a relaxation term, and conducts an investigation of the optimal relaxation time.'
address:
- |
Department of Civil and Environmental Engineering\
University of Illinois at Urbana Champaign\
205 N. Mathews Ave\
Urbana, IL 61801
- |
Department of Mathematics\
RWTH Aachen University\
Templergraben 55\
D-52056 Aachen
- |
Department of Mathematics\
Temple University\
1805 North Broad Street\
Philadelphia, PA 19122
author:
- Shimao Fan
- Michael Herty
- Benjamin Seibold
bibliography:
- 'references\_complete.bib'
title: 'Comparative Model Accuracy of a Data-Fitted Generalized Aw-Rascle-Zhang Model'
---
.9ex
Introduction
============
The mathematical modeling of vehicular traffic flow knows a variety of types of descriptions (see [@Helbing2001; @BellomoDogbe2011] for review papers): *microscopic* (e.g., [@Pipes1953; @Newell1961; @BandoHesebeNakayama1995; @Helbing1995]), which model the individual vehicles and their interactions by ODE; *cellular* (e.g., [@NagelSchreckenberg1992; @FukuiIshibashi1996; @Daganzo2006; @SakaiNishinariIIda2006; @AlperovichSopasakis2008]), which divide the road into cells and prescribe stochastic rules how vehicles advance through cells; and *continuum*. This latter class divides into *mesoscopic/gas-kinetic* (e.g., [@HermanPrigogine1971; @Phillips1979; @KlarWegener2000; @IllnerKlarMaterne2003; @HertyPareschi2010; @HertyIllner2010]) and *macroscopic* (e.g., [@LighthillWhitham1955; @Richards1956; @Underwood1961; @Payne1971; @Payne1979; @Lebacque1993; @KernerKonhauser1993; @KernerKonhauser1994; @Daganzo1995; @AwRascle2000; @Daganzo2006; @GaravelloPiccoli2006; @Goatin2006; @BerthelinDegondDelitalaRascle2008; @BayenClaudel2011]), i.e., fluid-dynamical, models. Among the (inviscid) macroscopic models one distinguishes between first-order models based on scalar hyperbolic equations and second-order models comprised of systems of hyperbolic equations. Specific examples for the latter are the Payne-Whitham model [@Payne1971; @Whitham1974], two-phase models [@ColomboGoatin2006], and the Aw-Rascle-Zhang model [@AwRascle2000; @Greenberg2001; @Zhang2002; @BerthelinDegondDelitalaRascle2008].
All of these types of traffic models are of practical importance, however—due to their different mathematical structure—for different purposes. For instance, microscopic models are well-suited for traffic simulation, i.e., the “in silico” study of a specific scenario; cellular models reproduce jamming behavior while being simple to implement and easy to parallelize; mesoscopic models provide a high modeling flexibility; and macroscopic models provide a suitable framework for the incorporation of on-line data. Moreover, there are mathematical relations between these types of models. For example, microscopic models as well as cellular models converge to mesoscopic or macroscopic models in the limit of vanishing cell size or vehicle spacing, respectively [@Daganzo1994; @Daganzo2006; @AlperovichSopasakis2008; @BorscheKimathiKlar2012]. Similarly, macroscopic models arise as suitable limits of mesoscopic models [@NelsonSopasakis1999; @KlarWegener2000].
Microscopic and cellular models are nowadays widely used in traffic engineering, and their combination with data is ubiquitous. In contrast, certain types of continuum models have been studied mathematically, but very little work has been conducted on their validation with traffic data. Examples of macroscopic first-order models used in traffic engineering practice are the Mobile Century project [@Aminetal2008] and the Mobile Millennium project [@MobileMillennium], including approaches based on the reformulation of the Lighthill-Whitham-Richards model in terms of Hamilton-Jacobi equations [@BayenClaudel2010; @BayenClaudel2011], or in terms of the velocity variable [@WorkBlandinTossavainenPiccoliBayen2010]. Further examples are [@BlandinBrettiCutoloPiccoli2009; @BlandinCoqueBayen2012] and the references therein. Those projects focus on the assimilation of data, the reconstruction of traffic states (e.g., from cell phone data), and the combination of macroscopic models with filtering techniques.
In contrast, here different types of traffic models are generated via historic data, and then their predictive accuracy is investigated using time-dependent data. Moreover, the main focus lies on second-order models; and first-order models are considered mainly for comparison purposes. The contributions of this paper are: (i) the design of a generalized Aw-Rascle-Zhang model and the analysis of its mathematical properties; (ii) a systematic methodology to construct data-fitted first-order and second-order traffic models, using historic fundamental diagram data; (iii) the validation of first- and second-order macroscopic traffic models via time-dependent trajectory and sensor data, and the comparison of the predictive accuracy of different models; and (iv) the investigation of the optimal relaxation time in second-order models with a relaxation term.
This paper is organized as follows. In §\[sec:models\] an overview over existing macroscopic traffic models is provided, including a discussion of some of the modeling shortcomings of the Aw-Rascle-Zhang model. We then introduce the generalized Aw-Rascle-Zhang model as an approach that addresses the shortcomings, and discuss its mathematical properties. The fitting of the model parameters and functions is then described in §\[sec:data-fitted\_models\]. Given historic fundamental diagram data in the flow rate vs. density plane, we systematically construct data-fitted first- and second-order macroscopic models. In §\[sec:numerical\_methods\] the numerical methods used to conduct the model validation and comparison are presented. Unlike studies of cell-transmission models [@Daganzo1994], in this paper all studies are carried out in a macroscopic sense; in particular the governing PDE are numerically solved with high enough accuracy such that the numerical approximation errors are negligibly small relative to the model errors. The comparison of the models on a three-detector test setup [@Daganzo1997] is then carried out in §\[sec:validation\]. In addition to the macroscopic traffic models, we also consider a predictor that simply interpolates the traffic state from the boundaries. For vehicle trajectory data, and for sensor data, the predictive accuracies of the models are compared, and their reproduction of features in the traffic states are studied. In §\[sec:inhomogeneous\_models\] we then extend the studies to data-fitted second-order models with relaxation terms. In particular, we study the dependence of the model accuracy on the relaxation time at which drivers adjust their driving behavior. Finally, in §\[sec:conclusions\] we present the conclusions from our studies.
Existing and New Macroscopic Traffic Models {#sec:models}
===========================================
Common to all macroscopic traffic models is the continuity equation $$\label{eq:continuity_equation}
\rho_t+(\rho u)_x = 0\;,$$ which gives the conservation of vehicles. In , the vehicle density is $\rho(x,t)$, and the vehicle velocity field is $u(x,t)$, where $x$ is the position along the road, and $t$ is time. If the road has multiple lanes (in a given direction), we consider these aggregated into the scalar field quantities $\rho$ and $u$.
The Lighthill-Whitham-Richards Model and Flow Rate Functions
------------------------------------------------------------
The simplest macroscopic traffic model, the Lighthill-Whitham-Richards (LWR) model [@LighthillWhitham1955; @Richards1956], is obtained by assuming a functional relationship between $\rho$ and $u$, i.e., $u = U(\rho)$. This turns equation into a scalar hyperbolic conservation law $$\label{eq:lighthill_whitham_richards_model}
\rho_t+(Q(\rho))_x = 0\;,$$ where the flux $Q$ is given by the flow rate function $Q(\rho) = \rho U(\rho)$. Because the LWR model is a closed model consisting of a single equation, it is denoted a *first order model*. The velocity function $U(\rho)$ is commonly assumed to be decreasing in $\rho$ with $U(\rho_\text{max}) = 0$ for some maximal vehicle density $\rho_\text{max}>0$.
Popular examples of flow rate functions are the Greenshields flux [@Greenshields1935], in which $Q(\rho)$ is a quadratic function, and the Newell-Daganzo flux [@Newell1993; @Daganzo1994], in which $Q(\rho)$ is a piecewise linear function. While these different choices of functions $Q(\rho)$ lead to well-posed first-order models, the second-order models derived below call for further properties that the function $Q(\rho)$ must satisfy. In particular, the velocity function $U(\rho) = Q(\rho)/\rho$ must nowhere be constant, because otherwise hyperbolicity would be lost (see the analysis in §\[subsubsec:characteristics\]). This rules out the Newell-Daganzo flux; and as a consequence, in this paper we consider flow rate functions that resemble the shape of the Newell-Daganzo flux, but that are strictly concave.
The Aw-Rascle-Zhang Model
-------------------------
The strict functional relationship between $\rho$ and $u$ is loosened in *second order models*, which augment by an evolution equation for the velocity field. Payne and Whitham proposed a model [@Payne1971; @Whitham1974] in which the vehicle velocity relaxes towards a velocity function, while also being affected by a “traffic pressure” that is analogous to a pressure in fluid dynamics. Because this pressure can lead to unrealistic solutions (vehicles going backwards on the road, shocks that overtake vehicles, etc., see [@Daganzo1995]), Aw and Rascle [@AwRascle2000], and independently Zhang [@Zhang2002], proposed a different form of “pressure” that remedies the shortcomings of the Payne-Whitham model. The homogeneous Aw-Rascle-Zhang (ARZ) model reads as $$\label{eq:aw_rascle_zhang_model_homogeneous}
\begin{split}
\rho_t+(\rho u)_x &= 0\;, \\
(u+h(\rho))_t+u(u+h(\rho))_x &= 0\;,
\end{split}$$ where we call $h(\rho)$ the *hesitation function*.[^1] We assume that $h'(\rho)>0$ and use the gauge $h(0) = 0$. The addition of a relaxation term (analogous to the Payne-Whitham model) yields the inhomogeneous ARZ model [@Greenberg2001; @Rascle2002] $$\label{eq:aw_rascle_zhang_model_inhomogeneous}
\begin{split}
\rho_t+(\rho u)_x &= 0\;, \\
(u+h(\rho))_t+u(u+h(\rho))_x &= \tfrac{1}{\tau}{\left(U_\text{eq}(\rho)-u\right)}\;.
\end{split}$$ We call $U_\text{eq}(\rho)$ the *desired velocity function* or the *equilibrium velocity function*, and $\tau$ the *relaxation time scale*.
As one can easily verify, the homogeneous ARZ model possesses no mechanism to make drivers move when starting with all vehicles at rest, i.e., $u(x,0) = 0$. In turn, the inhomogeneous ARZ model does. We therefore expect the homogeneous ARZ model to yields reasonable results only when the traffic flow is close to its equilibrium state, i.e., $u\approx U_\text{eq}(\rho)$. Yet, in general the inhomogeneous ARZ model has the potential to yield more realistic predictions.
The conservative form of is given by $$\label{eq:aw_rascle_zhang_model_conservative}
\begin{split}
\rho_t+{\left(q-\rho h(\rho)\right)}_x &= 0\;, \\
q_t+{\left(\tfrac{q^2}{\rho}-h(\rho)q\right)}_x &= \tfrac{1}{\tau}{\left(Q_\text{eq}(\rho)+\rho h(\rho)-q\right)}\;,
\end{split}$$ where the two conserved variables are $\rho$ and $q = \rho (u+h(\rho))$, and $Q_\text{eq}(\rho) = \rho U_\text{eq}(\rho)$ is called the *equilibrium curve* that the momentum density $\rho u$ relaxes to.
\[rem:dependent\_h\_U\] Various authors (e.g., [@Greenberg2001; @SiebelMauser2006]) have proposed to choose the functions $h(\rho)$ and $U_\text{eq}(\rho)$ dependent on each other, namely $h(\rho) = U_\text{eq}(0)-U_\text{eq}(\rho)$. In this case, the solution relaxes towards an equilibrium state $u = U_\text{eq}(\rho)$ (see §\[subsubsec:relaxtion\_GARZ\_LWR\]). In this paper, we follow this philosophy, since it generates both functions from the same data-fitting procedure. However, it should be noted that it is in principle perfectly reasonable to choose $h(\rho)$ and $U_\text{eq}(\rho)$ independently of each other. As shown in [@Greenberg2004; @FlynnKasimovNaveRosalesSeibold2009; @SeiboldFlynnKasimovRosales2013; @KasimovRosalesSeiboldFlynn2013], such a choice can generate (whenever $h'(\rho)+U_\text{eq}'(\rho)<0$) instabilities and self-sustaining traveling wave solutions that model phantom traffic jams and traffic waves, respectively.
Interpretation of ARZ as Generalization of LWR {#subsec:ARZ_generalization_of_LWR}
----------------------------------------------
As pointed out in [@Lebacque1993; @AwRascle2000; @BerthelinDegondDelitalaRascle2008; @FanSeibold2013], the homogeneous ARZ model can be interpreted as a generalization of the LWR model, by introducing the variable $w = u+h(\rho)$. Thus, system takes the form $$\label{eq:aw_rascle_zhang_model_w}
\begin{split}
\rho_t+(\rho u)_x &= 0\;, \\
w_t+uw_x &= 0\;, \\
\text{where~}u &= w-h(\rho)\;.
\end{split}$$ The interpretation of is that $w$ is advected with the flow $u$, i.e., it moves with the vehicles. One can therefore interpret $w$ as a quantity that is associated with each vehicle, and that is influencing the velocity. Since for $\rho = 0$, we have $h(0) = 0$ and thus $w = u$, we call $w$ the *empty road velocity*. The actual vehicle velocity is given by its empty road velocity, reduced by the hesitation $h(\rho)$.
Using this interpretation, the homogeneous ARZ model generalizes the LWR model , as follows. Given a (decreasing) LWR velocity function $U(\rho)$, we define $h(\rho) = U(0)-U(\rho)$ (clearly, $h'(\rho)>0$ and $h(0) = 0$). Then, model possesses a one-parameter family of velocity curves, namely $u_w(\rho) = w-h(\rho) = U(\rho)+(w-U(0))$, and the LWR velocity curve $u(\rho) = U(\rho)$ is one of them, namely the one corresponding to $w = U(0)$. The same behavior translates to the flow rate curves that live in the fundamental diagram ($Q$ vs. $\rho$). The ARZ model possesses a one-parameter family of flow rate curves, namely $Q_w(\rho) = Q(\rho)+\rho(w-U(0))$, and the LWR flow rate curve $Q(\rho)$ is one of them, namely the one corresponding to $w = U(0)$. This has been observed by Lebacque [@Lebacque1993].
The aforementioned relationship between the LWR and the ARZ model is shown in Fig. \[fig:fd\_lwr\_arz\]. The single velocity curve (left panel) and flow rate curve (right panel), shown in red, is one representative of the family of curves, shown in black, that the ARZ model possesses. By construction, each velocity curve in the ARZ model is merely a vertical translation of the LWR velocity curve. Hence, the homogeneous ARZ model possesses the same number of parameter functions (namely: a single one) as the LWR model.
In line with Remark \[rem:dependent\_h\_U\], i.e., by choosing $U_\text{eq}(\rho) = U(\rho) = U(0)-h(\rho)$, we can extend model to the inhomogeneous case, yielding $$\label{eq:aw_rascle_zhang_model_w_inhomogeneous}
\begin{split}
\rho_t+(\rho u)_x &= 0\;, \\
w_t+uw_x &= \tfrac{1}{\tau}{\left(U(0)-w\right)}\;, \\
\text{where~}u &= w-h(\rho)\;,
\end{split}$$ which adds a temporal relaxation of each vehicle’s empty road velocity $w$ towards a uniform value $U(0)$. In other words: the dynamics, that can be on any velocity curve $u_w(\rho)$, are driven towards the LWR velocity function $U(\rho)$, i.e., the red curves in Fig. \[fig:fd\_lwr\_arz\]—unless the system is driven away from equilibrium by another effect, such as boundary conditions (see §\[subsubsec:relaxtion\_GARZ\_LWR\]).
Generalized ARZ Model {#subsec:garz}
---------------------
The interpretation of the ARZ model as possessing a family of velocity curves (see ) reveals a fundamental shortcoming of the model: due to the additive relationship between velocity, empty road velocity, and hesitation, there is not a unique maximum density $\rho_\text{max}$, at which the flow stagnates. On the contrary, as the plots in Fig. \[fig:fd\_lwr\_arz\] indicate, variations in $w$ can lead to significant variations in the density at which $u_w(\rho) = 0$. However, since in reality the maximum density is largely a property of the road, it should not depend (at least not strongly) on the velocity that drivers assume when alone on the road. In order to remedy this shortcoming, the relationship between $u$, $w$, and $\rho$ must be generalized.
To that end, we consider a generalized Aw-Rascle-Zhang (GARZ) model, which is a representative of the class of generic second order models (GSOM), proposed by Lebacque, Mammar, and Haj-Salem [@LebacqueMammarHajSalem2007]. Specifically, the homogeneous ARZ model generalizes to $$\label{eq:GARZ_model}
\begin{split}
\rho_t+(\rho u)_x &= 0\;, \\
w_t+uw_x &= 0\;, \\
\text{where~}u &= V(\rho,w)\;,
\end{split}$$ where we impose the following requirements on the velocity function $V(\rho,w)$, and the associated generalized flow rate function $Q(\rho,w) = \rho V(\rho,w)$:
- $V(\rho,w)\ge 0$, i.e., vehicles never go backwards on the road.
- $V(0,w) = w$, i.e., we gauge the convected quantity $w$ to play the role of the empty road velocity, as in the ARZ model .
- $\frac{\partial^2 Q}{\partial \rho^2}(\rho,w) < 0$ for $w > 0$, i.e., each flow-rate curve $Q_w(\rho) = Q(\rho,w)$ is strictly concave. This condition implies (see Lemma \[lem:concave\_Q\_decreasing\_U\]) in particular that ${\frac{\partialV}{\partial\rho}}(\rho,w) < 0$, i.e., each velocity curve $u_w(\rho) = V(\rho,w)$ is strictly decreasing w.r.t. the density.
- ${\frac{\partialV}{\partialw}}(\rho,w) > 0$, i.e., a faster empty road velocity results in a faster velocity for all possible densities.
- $V(\rho,0) = 0$, i.e., for $w = 0$, the concavity of $Q$ and the slope of $V$ hold with an equality sign.
\[lem:concave\_Q\_decreasing\_U\] Consider a $C^2$ function $U(\rho)$, and let $Q(\rho) = \rho U(\rho)$. If $Q''(\rho)<0$ everywhere, then $U'(\rho)<0$ everywhere.
The function $a(\rho) = Q(\rho)-Q'(\rho)\rho$ satisfies: (i) $a(0) = 0$, and (ii) $a'(\rho) = -Q''(\rho)\rho > 0$ everywhere. Hence, $U'(\rho) = \frac{-a(\rho)}{\rho^2} < 0$ everywhere.
In order to define an inhomogeneous GARZ model, an equilibrium velocity curve must be specified. We assume that it is a member of the family of velocity curves defined by $V$, i.e. $$U_\text{eq}(\rho) = V(\rho,w_\text{eq})\;,$$ for some equilibrium empty road velocity $w_\text{eq}$. We choose to generalize to the GARZ case as follows: $$\label{eq:GARZ_model_inhomogeneous}
\begin{split}
\rho_t+(\rho u)_x &= 0\;, \\
w_t+uw_x &= \tfrac{1}{\tau}{\left(U_\text{eq}(\rho)-u\right)}\;, \\
\text{where~}u &= V(\rho,w)\;.
\end{split}$$ Note that it would alternatively be conceivable to propose a relaxation in of the form $$\label{eq:alternative_relaxation}
w_t+uw_x = \tfrac{1}{\tau}{\left(w_\text{eq}-w\right)}\;.$$ While for the ARZ model, the forms and are equivalent, for the GARZ model the relaxations and are not. Specifically, if ${\frac{\partialV}{\partialw}}(\rho,w_\text{eq})>0$, then both forms relax to the same limit, but at different rates. A simple Taylor expansion yields that for $w$ nearby $w_\text{eq}$, the relaxation in happens ${\frac{\partialV}{\partialw}}(\rho,w_\text{eq})$ times as rapidly as the relaxation in .
Finally, in line with , the GARZ model is meant to be interpreted in the conservative form $$\label{eq:GARZ_conservative}
\begin{split}
\rho_t+{\left(V(\rho,q/\rho)\rho\right)}_x &= 0\;, \\
q_t+{\left(V(\rho,q/\rho)q\right)}_x &= \tfrac{1}{\tau}{\left(Q_\text{eq}(\rho)-Q(\rho,q/\rho)\right)}\;,
\end{split}$$ where the two conserved variables are $\rho$ and $q = \rho w$. Moreover, $Q_\text{eq}(\rho) = \rho U_\text{eq}(\rho)$ and $Q(\rho,w) = \rho V(\rho,w)$.
Properties of the GARZ Model
----------------------------
Most properties of the classical ARZ model transfer over to its generalization, the GARZ model. Here we only collect relevant results, many of which have been presented in [@LebacqueMammarHajSalem2007], or that are relatively straightforward generalizations of the results given in [@AwRascle2000; @Greenberg2001; @Rascle2002]. The theoretical results presented below are in particular important for the data-fitting methodologies conducted in §\[sec:data-fitted\_models\], and for the interpretation of the results obtained in §\[sec:inhomogeneous\_models\].
### Regions of GARZ Variables and Inverse Velocity Functions {#subsubsec:regions}
Because of their relations and because of their physical meaning, the quantities $\rho$, $w$, and $u$ cannot assume any arbitrary values. In this paper, we assume that there is a unique stagnation density $\rho_\text{max}$, at which vehicles come to a stop, independent of their empty road velocity, i.e., $V(\rho_\text{max},w) = 0$ for all $w$. We therefore have $\rho\in [0,\rho_\text{max})$ and $0 < u\le w$, where the latter inequality follows from ${\frac{\partialV}{\partial\rho}}(\rho,w) < 0$. Moreover, we assume that there is a minimum and maximum empty road velocity, i.e., $0<w_\text{min}\le w\le w_\text{max}$. Because ${\frac{\partialV}{\partial\rho}} < 0$ and ${\frac{\partialV}{\partialw}} > 0$, the function $V(\rho,w)$ can be “inverted” to define the functions (and their domains): $$\begin{aligned}
V:\mathcal{D}_V \longrightarrow [0,w_\text{max}]
&\;\,\text{where~~}
\mathcal{D}_V = \{(\rho,w)\;|\; 0\le\rho<\rho_\text{max},\;
w_\text{min}\le w\le w_\text{max}\}\;, \\
R:\mathcal{D}_R \longrightarrow [0,\rho_\text{max})
&\;\,\text{where~~}
\mathcal{D}_R = \{(u,w)\;|\; 0<u\le w,\; w_\text{min}\le w\le w_\text{max}\}\;, \\
W:\mathcal{D}_W\! \longrightarrow [w_\text{min},w_\text{max}]
&\;\,\text{where~~}
\mathcal{D}_W\! = \{(\rho,u)\;|\; 0\le\rho<\rho_\text{max},\,
V(\rho,w_\text{min})\le u\le V(\rho,w_\text{max})\}\;.\end{aligned}$$ Here the functions $R$ and $W$ are defined as the unique solutions to the problems:
1. given $u$ and $w$, find $\rho = R(u,w)$, s.t. $V(\rho,w) = u$;
2. given $\rho$ and $u$, find $w = W(\rho,u)$, s.t. $V(\rho,w) = u$.
From the fact that the quantity $w$ is transported with the flow (while possibly relaxing to some $w_\text{eq} \in (w_\text{min},w_\text{max})$), and from the solution of the Riemann problems of the GARZ model (see below), it follows that the dynamics of the model never generate values $w\notin [w_\text{min},w_\text{max}]$ or $\rho\notin [0,\rho_\text{max})$. Hence, analogous to the ARZ model (cf. [@AwRascle2000]), the domain $\mathcal{D}_V$ is an invariant region.
### Characteristics and Associated Fields {#subsubsec:characteristics}
The homogeneous part of the GARZ model is a conservation law of the form $${\mathbf{U}}_t + {\mathbf{F}}({\mathbf{U}})_x = 0\;,$$ where $${\mathbf{U}} = \begin{pmatrix} \rho \\ q \end{pmatrix}
\quad\text{and}\quad
{\mathbf{F}}({\mathbf{U}}) = \begin{pmatrix} u\rho \\ uq \end{pmatrix}\;,
\quad\text{where}\quad
u = V(\rho,q/\rho)\;.$$ The Jacobian of the flux function ${\mathbf{F}}({\mathbf{U}})$ is $$\nabla{\mathbf{F}}({\mathbf{U}})
= \begin{pmatrix} u+\rho{\frac{\partialu}{\partial\rho}} & \rho{\frac{\partialu}{\partialq}} \\
q{\frac{\partialu}{\partial\rho}} & u+q{\frac{\partialu}{\partialq}} \end{pmatrix}\;,$$ and its eigenvalues and associated eigenvectors are $$\begin{aligned}
\lambda^{(1)} &= u+\rho{\frac{\partialu}{\partial\rho}}+q{\frac{\partialu}{\partialq}} = u+\rho{\frac{\partialV}{\partial\rho}}
\quad\text{with}\quad
{\mathbf{\gamma}}^{(1)} = \begin{pmatrix} \rho \\ q \end{pmatrix}
\quad\text{and thus}\quad
\nabla\lambda^{(1)}\cdot {\mathbf{\gamma}}^{(1)} \neq 0\;,
\intertext{and}
\lambda^{(2)} &= u \hspace{10em}
\quad\text{with}\quad
{\mathbf{\gamma}}^{(2)}
= \begin{pmatrix} -{\frac{\partialu}{\partialq}} \\ \phantom{-}{\frac{\partialu}{\partial\rho}} \end{pmatrix}
\quad\text{and thus}\quad
\nabla\lambda^{(2)}\cdot {\mathbf{\gamma}}^{(2)} = 0\;.\end{aligned}$$ Hence, like the ARZ model , the GARZ model is strictly hyperbolic for $\rho>0$. One of its characteristic velocities, $\lambda^{(1)}$, is slower than the vehicles (i.e., $\lambda^{(1)}<u$) and its associated field is genuinely nonlinear (i.e., it corresponds to shocks and rarefaction waves, see below). Its other characteristic velocity $\lambda^{(2)}$ equals the vehicle velocity and its associated characteristic field is linearly degenerate (i.e., its associated waves are contact discontinuities that are transported with the flow).
\[lem:lambdas\_decreasing\] Both characteristic velocities are strictly decreasing w.r.t. $\rho$, i.e., ${\frac{\partial\lambda^{(1)}}{\partial\rho}} < 0$ and ${\frac{\partial\lambda^{(2)}}{\partial\rho}} < 0$.
We have that $\lambda^{(1)} = V+\rho{\frac{\partialV}{\partial\rho}} = {\frac{\partialQ}{\partial\rho}}$. Since $Q$ is assumed concave w.r.t. $\rho$, it follows that ${\frac{\partial\lambda^{(1)}}{\partial\rho}} = \frac{\partial^2 Q}{\partial \rho^2} < 0$. Moreover, ${\frac{\partial\lambda^{(2)}}{\partial\rho}} = {\frac{\partialV}{\partial\rho}} < 0$ by Lemma \[lem:concave\_Q\_decreasing\_U\].
We continue with the discussion of the characteristic fields. The scalar function $I^{(1)} = q/\rho = w$ satisfies $\nabla I^{(1)}\cdot {\mathbf{\gamma}}^{(1)} = 0$, and it is a Riemann invariant to $\lambda^{(1)}$. Hence, across waves of the first family, the empty road velocity $w$ is constant. The field associated with the second eigenvalue $\lambda^{(2)}$ is linearly degenerate. It is given by $I^{(1)} = \lambda^{(2)} = u$ and across waves of the second family, the velocity $u$ is constant. The solution to a Riemann problem, i.e., the Cauchy problem to system on the real line with discontinuous piecewise constant initial data ${\mathbf{U}}(x,0) = (1-H(x)){\mathbf{U}}_\text{L}+H(x){\mathbf{U}}_\text{R}$, where $H$ is the Heaviside function, generalizes naturally from the ARZ model as well. In general, the solution is obtained by superposition of simple waves connecting different constant states: from a given left state ${\mathbf{U}}_\text{L}$ to a given right state ${\mathbf{U}}_\text{R}$ via an intermediate state ${\mathbf{U}}_\text{M}$ that is connected to ${\mathbf{U}}_\text{L}$ by a 1-wave (i.e., a Lax-shock or rarefaction associated to the first characteristic field), and to ${\mathbf{U}}_\text{R}$ by a 2-wave (i.e., a contact discontinuity). Since the GARZ system is—as the ARZ model—of Temple class [@Temple1983], shocks and rarefaction wave curves in phase space coincide. Moreover, due to Lemma \[lem:lambdas\_decreasing\], a simple wave of the first family is either a shock or a rarefaction wave.
In the phase space $(\rho,w)$-plane we discuss the shape of the characteristic fields. The second fields are parallel to the $\rho$-axis, and the first fields are the contours $V(\rho,w) = \text{const}$. Since by assumption ${\frac{\partialV}{\partial\rho}}<0$ and ${\frac{\partialV}{\partialw}}>0$, the contours of $V(\rho,w)$ always have a finite and truly positive slope in the $(\rho,w)$-plane. Thus, for any two states $(\rho_\text{L},w_\text{L})$ and $(\rho_\text{R},w_\text{R})$ that satisfy $w_\text{L}\ge u_\text{R}$, where $u_\text{R} = V(\rho_\text{R},w_\text{R})$, there is a unique intermediate state $(\rho_\text{M},w_\text{M}) = (R(u_\text{R},w_\text{L}), w_\text{L})$, defined via the inverse function given in §\[subsubsec:regions\]. Moreover, because $\lambda^{(1)}$ is decreasing with $\rho$ (see Lemma \[lem:lambdas\_decreasing\]), the Lax entropy conditions [@Evans1998] imply that for $\rho_\text{L}<\rho_\text{M}$ the 1-wave is a shock wave (moving with speed $s = \frac{\rho_\text{M}V(\rho_\text{M},w_\text{M}) - \rho_\text{L}V(\rho_\text{L},w_\text{L})}{\rho_\text{M}-\rho_\text{L}}$, given by the Rankine-Hugoniot conditions [@Evans1998]), while for $\rho_\text{L}>\rho_\text{M}$ it is a rarefaction wave. The condition $w_\text{L}\ge u_\text{R}$ means that drivers on the left wish to drive at least as fast as the vehicles on the right are driving. If this is not the case, i.e., if $w_\text{L} < u_\text{R}$, then there is no non–negative density at which the 1-wave and the 2-wave intersect. Here, a vacuum state will be generated, analogously to the construction for the ARZ model [@AwRascle2000; @Rascle2002]. The left state $(\rho_\text{L},w_\text{L})$ is connected by a rarefaction wave to a left vacuum state $(0,w_\text{L})$; this state is connected to a right vacuum state $(0,u_\text{R})$ via another rarefaction (which is feasible because $w_\text{L} < u_\text{R}$); and this then connects to the right state $(\rho_\text{R},w_\text{R})$ via a 2-contact discontinuity.
### Relaxation of GARZ to LWR {#subsubsec:relaxtion_GARZ_LWR}
Smooth solutions to the Cauchy problem of the *inhomogeneous* GARZ model relax in time towards solutions of the LWR model , because $w$ tends to $w_\text{eq}$ along characteristic curves. Note that for general relaxation systems, convergence to a first-order equation is only warranted if a sub-characteristic condition is satisfied, cf. [@Liu1987; @ChenLevermoreLiu1994; @SeiboldFlynnKasimovRosales2013]. Here, we are in the characteristic case. Moreover, shocks of the $2\times 2$ hyperbolic system are also shocks of the LWR model . Therefore, for the Cauchy problem, GARZ solutions converge to LWR solutions as $t\to\infty$, and/or as $\tau\to 0$.
![Riemann problem at right domain boundary. Consider a constant initial state $(\rho_\text{R},Q_\text{eq}(\rho_\text{R}))$ in the domain, and prescribed boundary data $(\rho_\text{R},\rho_\text{R}u_\text{R})$ with $u_\text{R} > U_\text{eq}(\rho_\text{R})$. The LWR model preserves the constant state (the boundary data is projected onto the equilibrium curve vertically along the dashed line). In contrast, the GARZ model generates a new state $(\rho_\text{M},Q_\text{eq}(\rho_\text{M}))$ at the boundary (projected onto the equilibrium curve along a ray through the origin), from which a shock moves into the domain, thus changing the initial state to the new boundary state.[]{data-label="fig:Riemann_problem_bc"}](fig_riemann_problem_bc){width=".75\textwidth"}
In contrast, for initial-boundary-value problems on a bounded domain $x\in [x_\text{L},x_\text{R}]$, this last property is in general not true. To highlight this fact we consider a simplified setting depicted in Fig. \[fig:Riemann\_problem\_bc\]. Let constant boundary data $\rho(x_\text{L})$, $u(x_\text{L})$, $\rho(x_\text{R})$, $u(x_\text{R})$ be given. With this data, we can solve the LWR model and the inhomogeneous GARZ model . Then, in general solutions to the latter problem do not converge to solutions of the former, even in the limit $\tau\to 0$. Consider a constant state $(\rho_\text{R},Q_\text{eq}(\rho_\text{R}))$ inside the domain. At the outflow boundary $x_\text{R}$, let a state $(\rho_\text{R},\rho_\text{R}u_\text{R})$ be given where $u_\text{R} > U_\text{eq}(\rho_\text{R})$. The LWR model only uses the density information, and thus the constant state is preserved. In contrast, the GARZ model uses the full state in the Riemann problem. Its solution yields an intermediate state $(\rho_\text{M},Q_\text{eq}(\rho_\text{M}))$, whose density is determined via the relation $\rho_\text{M}u_\text{R} = Q_\text{eq}(\rho_\text{M})$. This intermediate state connects to the boundary state via a contact discontinuity (a 2-wave) moving with speed $u_\text{R}$, and to the interior state via a shock (a 1-wave) that moves with speed $$s = \frac{Q_\text{eq}(\rho_\text{R})-Q_\text{eq}(\rho_\text{M})}
{\rho_\text{R}-\rho_\text{M}}\;,$$ which in the situation depicted in Fig. \[fig:Riemann\_problem\_bc\] moves *into* the domain, since $s<0$. Thus, after some time in the GARZ model the intermediate state $(\rho_\text{M},\rho_\text{M}u_\text{R})$ is observed within the domain. Note that this argument holds independent of the value of the relaxation time $\tau$ in the model .
Data-Fitted Traffic Models {#sec:data-fitted_models}
==========================
In this section we describe how the parameter functions of the traffic models presented in §\[sec:models\] can be fitted to historic fundamental diagram data. We assume that flow rate vs. density pairs $(\rho_j,Q_j),\;j=1,\dots n$ are given from long-term measurements (commonly obtained via stationary sensors). As visible in the right panel of Fig. \[fig:fd\_lwr\_arz\], these data (gray dots) tend to exhibit a relatively clear functional relationship between $\rho$ and $Q$ for low densities. In turn, for medium densities, a significant spread is visible, i.e., a single $\rho$-value corresponds to many different flow rates $Q$. Finally, for large densities, very few data points are available at all.
Data-Fitting for the LWR and ARZ Models
---------------------------------------
The first-order LWR model must represent these data via a single function $Q(\rho)$. As the spread of the data cannot be captured, it is reasonable to find a function that lies “in the middle” of the cloud of data points. Specifically, we employ the approach presented in [@FanSeibold2013]. First, since the stagnation density $\rho_\text{max}$ is not represented well via data, we prescribe it as a fixed constant, given by a typical vehicle length of 5 meters, plus 50% of additional safety distance, $$\rho_\text{max} = \frac{\text{number of lanes}}
{\text{typical vehicle length}\times\text{safety distance factor}}
= \frac{\#\text{lanes}}{7.5\text{m}}\;.$$
Second, a three-parameter family of smooth and strictly concave flow rate curves is selected as $$\label{eq:flow_rate_curve}
Q_{\alpha,\lambda,p}(\rho) = \alpha{\left(a+(b-a)\rho/\rho_\text{max}-\sqrt{1+y^2}\right)}\;,$$ where $$\begin{aligned}
a = \sqrt{1+\left(\lambda p\right)^2}\;, \quad
b = \sqrt{1+\left(\lambda(1-p)\right)^2}\;,\text{~and}\quad
y = \lambda \left(\rho/\rho_\text{max}-p\right).\end{aligned}$$ Each flow rate function $Q_{\alpha,\lambda,p}(\rho)$ in this family vanishes for $\rho = 0$ and $\rho = \rho_\text{max}$. The three free parameters allow for controlling three important features of $Q_{\alpha,\lambda,p}(\rho)$: the value of maximum flow rate $Q_\text{max}$ (mainly determined by $\alpha$), the critical density $\rho_\text{c}$ (mainly controlled by $p$), and the “roundness" of the curve, i.e., how rapidly the slope transitions from positive to negative near $\rho_\text{c}$ (dominated by $\lambda$).
Third, from this three-parameter family of flow rate curves, the one is selected that is the closest to the data points $(\rho_j,Q_j),\;j=1,\dots n$ in a least-squares sense, i.e. we solve $$\label{eq:LSQ}
\min_{\alpha,\lambda,p}\; \sum_{j=1}^n
(Q_{\alpha,\lambda,p}(\rho_{j})-Q_{j})^2\;.$$ In the right panel of Fig. \[fig:fd\_lwr\_arz\], the resulting least-squares fit to the given gray data points, called $Q_\text{eq}(\rho)$, is depicted by the red curve. The red curve in the left panel represents the resulting velocity function $U_\text{eq}(\rho) = Q_\text{eq}(\rho)/\rho$.
As described in §\[subsec:ARZ\_generalization\_of\_LWR\], the ARZ model generalizes the LWR model to a one-parameter family of velocity curves $u_w(\rho) = U_\text{eq}(\rho)+(w-U_\text{eq}(0))$ (black curves in the left panel of Fig. \[fig:fd\_lwr\_arz\]) and flow rate curves $Q_w(\rho) = Q_\text{eq}(\rho)+(w-U_\text{eq}(0))\rho$ (right panel of Fig. \[fig:fd\_lwr\_arz\]). Due to this property, the ARZ model possesses the same amount data-fitted parameters as the LWR model.
An interpretation of the ARZ family of velocity curves is that different $w$-values represent different types of drivers; the larger $w$, the faster the corresponding drivers tend to drive. As motivated in §\[subsec:garz\], this captures the spread in the fundamental diagram (which is desirable), but it also results in vastly varying stagnation densities for different types of drivers. This last property is unrealistic, as the maximum density is a property of the road, rather than of the behavior of drivers [@FanSeibold2013]. We therefore need to construct a family of curves that are not simple shifts of each other, as done below.
![Velocity vs. density (left panel) and flow rate vs. density (right panel) curves of the smooth three-parameter model , fitted with historic fundamental diagram data (gray dots), for the ARZ model.[]{data-label="fig:fd_lwr_arz"}](fig_curves_arz_rho_u){width="\textwidth"}
![Velocity vs. density (left panel) and flow rate vs. density (right panel) curves of the smooth three-parameter model , fitted with historic fundamental diagram data (gray dots), for the ARZ model.[]{data-label="fig:fd_lwr_arz"}](fig_curves_arz_rho_q){width="\textwidth"}
Data-Fitting for the GARZ Model {#sec:data-fitting_GARZ}
-------------------------------
The GARZ model is based on a generalized velocity function $V(\rho,w)$, where—as for the ARZ model—$w$ represents different types of drivers, and thus parameterizes a families of velocity and flow rate curves, respectively. We construct these families of curves by generalizing the least-squares fit to a weighted least-squares fit, as follows.
Given a weight parameter $0<\beta<1$, we consider the minimization problem $$\label{eq:WLSQ}
\min_{\alpha,\lambda,p}\left\{
(1-\beta) \sum^{n}_{j=1}{\left((Q_{\alpha,\lambda,p}(\rho_{j})-Q_{j})_{+}\right)^2}+
\beta \sum^{n}_{j=1}{\left((Q_{\alpha,\lambda,p}(\rho_{j})-Q_{j})_{-}\right)^2}
\right\}\;,$$ where $$\begin{aligned}
\left(Q_{\alpha,\lambda,p}(\rho_{j})-Q_{j}\right)_{+}\
&= \max\left\{ Q_{\alpha,\lambda,p}(\rho_{j})-Q_{j},0\right\}\;, \\
\left(Q_{\alpha,\lambda,p}(\rho_{j})-Q_{j}\right)_{-}\
&= \max\left\{-Q_{\alpha,\lambda,p}(\rho_{j})+Q_{j},0\right\}\;.\end{aligned}$$ For $\beta=\frac{1}{2}$, problem reduces to , i.e., the LWR equilibrium curve is recovered. For $\beta<\frac{1}{2}$, data below the curve is penalized more, and consequently the resulting curve moves downwards. In turn, if $\beta>\frac{1}{2}$, curves above the equilibrium curve are obtained.
The weighted least-squares problem generates a one-parameter family of curves $Q_\beta(\rho) = Q_{\alpha(\beta),\lambda(\beta),p(\beta)}(\rho)$, parameterized by $\beta$; and consequently it also generates a family of velocity functions $$V_\beta(\rho) = \begin{cases}
Q_\beta(\rho)/\rho &\text{if~}\rho>0 \\
\frac{\partial Q_\beta}{\partial\rho}(0) &\text{if~}\rho=0 \end{cases}\;.$$ In this paper, we restrict to the case that the velocity curves in the family are non-intersecting, i.e., $$\label{eq:non_intersecting}
\text{If~}\beta_1<\beta_2\;,\text{~~then~~}
V_{\beta_1}(\rho)<V_{\beta_2}(\rho)
\text{~~for~~}\rho\in [0,\rho_{\text{max}})\;.$$ Note that in general (i.e., for a general family of flow rate functions, and for general data points), property is not necessarily satisfied; and furthermore, problem need not have a unique solution. However, in all cases studied in this paper, the minimization problem does have a unique solution; and property is in fact satisfied.
![Family of velocity vs. density (left panel) and flow rate vs. density (right panel) curves generated from the weighted least square (WLSQ) algorithm in constructing the velocity function $u = V(\rho,w)$ in the GARZ model.[]{data-label="fig:curves_garz"}](fig_curves_garz_rho_u){width="\textwidth"}
![Family of velocity vs. density (left panel) and flow rate vs. density (right panel) curves generated from the weighted least square (WLSQ) algorithm in constructing the velocity function $u = V(\rho,w)$ in the GARZ model.[]{data-label="fig:curves_garz"}](fig_curves_garz_rho_q){width="\textwidth"}
While problem is defined for all $0<\beta<1$, values of $\beta$ that are extremely close to 0 or 1 tend to lead to unreasonable curves. The reason is that in the limit $\beta\nearrow 1$, the resulting curve is the lowest curve that has no data points above it, and as a result it adjusts to outliers in the data (similar arguments hold for $\beta\searrow 0$). We therefore define a lower/upper flow rate curve, such that 99.9% of all data points lie above/below it. Consequently, together with the equilibrium curve, we have the following three special flow rate curves $$Q_\text{min}(\rho) = Q_{\beta_\text{min}}(\rho)\;, \quad
Q_\text{eq }(\rho) = Q_{\frac{1}{2}}(\rho)\;,\;\text{and} \quad
Q_\text{max}(\rho) = Q_{\beta_\text{max}}(\rho)\;,$$ where here we use $\beta_\text{min} = 10^{-4}$ and $\beta_\text{max} = 1-10^{-4}$. In the right panel of Fig. \[fig:curves\_garz\] these three curves are depicted by the lowest black curve, the red curve, and the uppermost black curve, respectively.
Even though the parameter $\beta$ defines a family of flow rate curves as desired, it has the shortcoming that it does not have an immediate interpretation as a property of traffic flow. We therefore re-parameterize the family in terms of the empty road velocity $w$, as follows. For any $\beta\in [\beta_\text{min},\beta_\text{max}]$, we define $w$ as the resulting slope of the curve $Q_\beta$ at $\rho = 0$, i.e., $$w = w(\beta) = V_\beta(0)\;.$$ Due to property , the relationship $w = w(\beta)$ is strictly increasing, and thus can be inverted into $\beta = \beta(w)$, defined on the interval $w\in [w_\text{min},w_\text{max}]$, where $w_\text{min} = Q_\text{min}'(0)$ and $w_\text{max} = Q_\text{max}'(0)$. Using this re-parameterization, we obtain a generalized flow rate function $$Q(\rho,w) = Q_{\beta(w)}(\rho)\;,$$ and a generalized velocity function $$V(\rho,w) = V_{\beta(w)}(\rho)\;,$$ as used in the GARZ model . The properties of $V(\rho,w)$ assumed in §\[subsec:garz\] are satisfied by construction.
Domain Extension for the GARZ Model
-----------------------------------
The systematical construction of a generalized velocity function $V(\rho,w)$, presented in §\[sec:data-fitting\_GARZ\], is in line with the regions of the GARZ variables defined in §\[subsubsec:regions\]. In particular, the function $W(\rho,u)$ is defined only for $V(\rho,w_\text{min})\le u\le V(\rho,w_\text{max})$. However, when applying the GARZ model in a forward computation, velocity data may be provided through initial and boundary conditions that lie outside of the domain of $W(\rho,u)$. In order to make sense of the model for such data, we effectively extend the domain of the function $W(\rho,u)$ via a projection of such data, as follows.
Given a density–velocity pair $(\rho,u)$, where $0<\rho<\rho_\text{max}$, we define a projected velocity as $$\tilde{U}(\rho,u) = \min\{\max\{u,V(\rho,w_\text{min})\},V(\rho,w_\text{max})\}\;,$$ and thus obtain the extended function $$\tilde{W}(\rho,u) = W(\rho,\tilde{U}(\rho,u))$$ that is defined for arbitrary velocity values. This simple projection (densities are left unchanged, and velocities are moved onto the lowest or highest curve, respectively) provides a constant extension of the function $W$ beyond its domain $\mathcal{D}_W$. Note that the range of $W$ remains unaffected as being $[w_\text{min},w_\text{max}]$, and consequently the function $\tilde{W}$ is not invertible outside of $\mathcal{D}_W$.
Numerical Methods {#sec:numerical_methods}
=================
All models are approximated numerically using a finite volume method on a regular grid of cell size $\Delta x$ and time step $\Delta t$, chosen so that the CFL condition [@CourantFriedrichsLewy1928] $$s_\text{max} \Delta t \leq \Delta x\;,$$ is satisfied, where $s_\text{max} = \max_k |\lambda_k|$ is the largest wave speed (see §\[subsubsec:characteristics\] for the characteristic velocities of the models). In all examples throughout this paper, the grid resolution is chosen small enough ($\Delta x\le 50\text{cm}$) so that the numerical approximation errors are much smaller than the model errors. Hence, the studies are conducted truly on the continuum level.
The first order model is solved using Godunov’s method [@Godunov1959]. For the second order models, we have to account for the fact that the inhomogeneous GARZ model becomes stiff if $\tau$ is small. Hence, we employ a semi-implicit finite volume scheme that treats the nonlinear hyperbolic terms explicitly and the relaxation terms implicitly (to prevent a time step restriction $\Delta t = O(\tau)$). The update rule of a state $(\rho_j^n,q_j^n)$ in cell $j$ from time $t_n$ to the state $(\rho_j^{n+1},q_j^{n+1})$ at time $t_{n+1} = t_n+\Delta t$ reads as $$\begin{aligned}
\label{eq:update_rho}
\rho_j^{n+1} &= \rho_j^n - \frac{\Delta t}{\Delta x}\!
{\left((\mathcal{F}_\rho)_{j+\frac{1}{2}}^n-(\mathcal{F}_\rho)_{j-\frac{1}{2}}^n\right)}\;, \\
\label{eq:update_q}
q_j^{n+1} &= q_j^n - \frac{\Delta t}{\Delta x}\!
{\left((\mathcal{F}_q)_{j+\frac{1}{2}}^n-(\mathcal{F}_q)_{j-\frac{1}{2}}^n\right)}
+\frac{\Delta t}{\tau}\!
{\left(Q_\text{eq}(\rho_j^{n+1})-Q(\rho_j^{n+1},q_j^{n+1}/\rho_j^{n+1})\right)}\;.\end{aligned}$$ Here $(\mathcal{F}_\rho)_{j+\frac{1}{2}}^n = \mathcal{F}_\rho(\rho_j^n,q_j^n,\rho_{j+1}^n,q_{j+1}^n)$ denotes the numerical flux for the quantity $\rho$ through the boundary between cells $j$ and $j+1$; the other fluxes are defined accordingly. Moreover, $Q(\rho,w) = \rho V(\rho,w)$ is the model’s two-parameter flow rate function, and $Q_\text{eq}(\rho) = \rho U_\text{eq}(\rho) = Q(\rho,w_\text{eq})$ is the equilibrium flow rate function.
As in the Godunov method [@Godunov1959] one could use the exact solution to the Riemann problem (cf. [@AwRascle2000; @Fan2013]) to define the numerical fluxes. However, this would require the inversion of the velocity function $u = V(\rho,w)$, which is costly for the GARZ model. A less expensive approach, employed here, is to define the numerical fluxes via the HLL approximate Riemann solver [@HartenLaxVanLeer1983], which approximates the true Riemann problem by a single constant intermediate region. Note that due to the fine grid resolution, and due to the fact that initial and boundary conditions are continuous, spurious overshoots that may occur in the velocity (cf. [@ChalonsGoatin2007]) are negligibly small.
Since the inhomogeneous model possesses a relaxation only in the momentum equation, the time update of the density variable, given by , is fully explicit. Therefore, in the time update of the generalized momentum, given by , the quantity $\rho_j^{n+1}$ is known. Specifically, the numerical scheme is implemented in three steps:
1. Based on the data $(\rho_j^n,q_j^n)\,\forall j$ at time $t_n$, the fluxes $((\mathcal{F}_\rho)_{j+\frac{1}{2}}^n, (\mathcal{F}_q)_{j+\frac{1}{2}}^n)\,\forall j$ are computed.
2. The new density states $\rho_j^{n+1}\,\forall j$ are computed via .
3. The new generalized momenta $q_j^{n+1}\,\forall j$ are computed according to . On the cell $j$, the new state $q_j^{n+1}$ is obtained as the solution of the scalar nonlinear equation $G(q) = 0$, where $$\label{eq:nonlinear_equation}
G(q) = q + \frac{\Delta t}{\tau} Q(\rho_j^{n+1},q/\rho_j^{n+1})
- q_j^n + \frac{\Delta t}{\Delta x}\!
{\left((\mathcal{F}_q)_{j+\frac{1}{2}}^n-(\mathcal{F}_q)_{j-\frac{1}{2}}^n\right)}
-\frac{\Delta t}{\tau}Q_\text{eq}(\rho_j^{n+1})\;.$$ The root of is found up to machine accuracy within a few Newton steps, using $q_j^n$ as the starting guess.
In the special case of the ARZ model , the update is given by the explicit formula $$q_j^{n+1}
= \frac{ q_j^n - \frac{\Delta t}{\Delta x}\!
{\left((\mathcal{F}_q)_{j+\frac{1}{2}}^n-(\mathcal{F}_q)_{j-\frac{1}{2}}^n\right)}
+\frac{\Delta t}{\tau} \rho_j^{n+1} w_\text{eq} }{1+\frac{\Delta t}{\tau}}\;.$$ It should further be remarked that the semi-implicit scheme and is equivalent to the fractional step approach that first approximates the homogeneous part of via a forward Euler step, and then approximates the relaxation part via a backward Euler step. Hence, in the limit $\tau\to 0$ (while $\Delta t$ fixed), the scheme amounts to simply projecting $q$ onto the equilibrium curve in the relaxation step, i.e., equation turns into $q_j^{n+1} = Q_\text{eq}(\rho_j^{n+1})$. In turn, in the homogeneous case, i.e., $\tau\to\infty$, the relaxation terms are simply omitted.
The boundary data are provided by introducing a ghost cell adjacent to the outermost grid cell (on either side of the domain), in which the boundary state $(\rho,q)$ is assumed. The numerical fluxes in and then by construction pick up the information corresponding to waves that enter the computational domain.
Validation and Comparison of Models via Real Data {#sec:validation}
=================================================
In the following, we validate the presented models by studying how well they reproduce the evolution of real traffic data, and in that process we compare the predictive accuracy of the models. A particular focus lies on the investigation of the extent to which the GARZ model, that addresses various shortcoming of traditional models, improves the actual model agreement with real traffic data. We conduct the validations using the NGSIM trajectory data set [@TrafficNGSIM] and the RTMC sensor data set [@TrafficMnDOT].
The test framework considered here is based on the methodology presented in [@FanSeibold2013] and further developed in [@FanSeibold2014]: on a segment of highway, a three-detector test problem [@Daganzo1997] is formulated. At each end of the segment, the traffic state is (at all times) provided to the traffic model, which is advanced forward in time (using the numerical methods described in §\[sec:numerical\_methods\]) inside the segment. The predictions that the traffic model produces in time are then compared to real data inside the segment, and the deviation between predicted and real traffic states is used to quantify the model error.
Treatment of Data {#subsec:data_treatment}
-----------------
As described in [@FanSeibold2013], continuous field quantities $\rho(x,t)$ and $u(x,t)$ are constructed from the NGSIM vehicle trajectory data [@TrafficNGSIM_I80], using kernel density estimation [@Parzen1962; @Rosenblatt1956]. In this approach, given vehicle locations $x_j(t)$ (including “ghost vehicle” positions, obtained via reflection at the boundaries, see [@KarunamuniAlberts2005]), density and flow rate functions are obtained as superpositions of Gaussian profiles, $$\rho(x,t) = \sum_{j=1}^n K(x-x_j(t))
\quad\text{and}\quad
Q(x,t) = \sum_{j=1}^n u_j K(x-x_j(t))\;,
\text{~where~}
K(x) = \tfrac{1}{\sqrt{2\pi}h}e^{-\frac{x^2}{2h^2}}\;,$$ and the velocity field is then given by $u(x,t) = Q(x,t)/\rho(x,t)$. The kernel width is chosen $h = 25$meters. These field quantities then define initial conditions ($t=0$) and boundary conditions (when evaluated at the segment boundary positions) for the traffic models, and reference states for the validation (inside the segment for $t>0$). Before the boundary data can be provided to the traffic model, one additional processing step must be applied to address spurious fast oscillations in the reconstructed boundary data (due to variations in the starting and end position of each vehicle trajectory in the data set): the time domain is divided into intervals of length 15seconds, and on each interval the boundary data is replaced by a cubic polynomial that is a least-squares fit to the data, under the constraint that the resulting evolution is globally $C^1$.
For the RTMC sensor data, vehicles densities and flow rates are given at three sensor positions, aggregated in intervals of length 30seconds. Temporally continuous quantities $\rho(x_s,t)$ and $u(x_s,t)$ at a sensor position $x_s$ are generated via cubic spline interpolation (in time) of the aggregated information. One shortcoming of the RTMC data is the absence of a reliable initial state (because information is given only at the sensor positions). This problem is circumvented by running the models forward through an initialization phase (5 minutes), before the actual model comparison is started. During this phase, the boundary data has time to move into the domain and create a reasonable initial state for the actual validation.
Quantification of Model Errors {#sec:error}
------------------------------
The quantification of the deviation of the model predictions from the actual data requires two aspects to be specified: first, which field quantities to consider and how to combine them into a single quantity; and second, if data is available at multiple positions and/or times, how to combine these multiple pieces of information into a single quantity?
Regarding the choice of field quantities, in this study we are interested in models that predict traffic densities (as required for instance for ramp metering) and velocities (as required for instance for travel time estimates) accurately. Since densities and velocities have different physical units, suitable normalization constants must be found, so that the deviations in each quantity contribute with equal influence to the total model error.
Given model predictions $\rho^\text{model}(x,t)$ and $u^\text{model}(x,t)$, and real data $\rho^\text{data}(x,t)$ and $u^\text{data}(x,t)$, we define a space-and-time-dependent error measure as $$\label{eq:error_measure}
E(x,t) = \frac{|\rho^\text{model}(x,t)-\rho^\text{data}(x,t)|}{\Delta\rho}
+\frac{|u^\text{model}(x,t)-u^\text{data}(x,t)|}{\Delta u}\;,$$ where the normalization constants $\Delta\rho$ and $\Delta u$ represent the ranges of the fundamental diagram data, as defined below. Note that various choices of normalization constants have been proposed in the literature. For instance, in [@BlandinWorkGoatinPiccoliBayen2011] the absolute errors in density and velocity are scaled with $\|\rho^\text{data}(x,t)\|_{L^1}$ and $\|u^\text{data}(x,t)\|_{L^1}$, respectively. A shortcoming of this choice is that for traffic flow at low densities, errors in density get divided by a very small number and thus significantly amplified. An alternative choice is employed in [@FanSeibold2013] by using $\rho_\text{max}$ and $u_\text{max}$ as normalization constants. However, these tend to give too much influence to velocity errors, because even in moving congested traffic flow, $\rho/\rho_\text{max}$ tends to be significantly smaller than $u/u_\text{max}$.
![Construction of ranges in density and velocity in historic fundamental diagram data, using an upper density (red line) and lower and upper velocity boundaries (blue lines). Data points with densities below a threshold (black line), as well as outliers, are systematically excluded.[]{data-label="fig:normalization"}](fig_error_normalization){width=".90\textwidth"}
In line with [@FanSeibold2014], we argue that balanced weights are given when the error in each field quantity is related to the maximum variation that the respective quantity exhibits in the historic fundamental diagram. In order to exclude the influence of outliers in the data, we conduct the following four-step approach. First, all data points $(\rho_j,u_j)$ with $\rho_j<5\,\text{veh}/\text{km}/\text{lane}$ are neglected. The rationale is that these data do not contribute any meaningful information about the spread in the traffic states, and moreover such low density values are not meaningful in the context of a macroscopic description of traffic flow. In Fig. \[fig:normalization\], this boundary is depicted by the black line. Second, similar to the method presented in [@BlandinBrettiCutoloPiccoli2009], the upper density boundary $\rho^\text{up}$ is defined such that 99.9% of the remaining data points lie below it (red line in Fig. \[fig:normalization\]). Third, the lower (upper) velocity boundary $u^\text{low}$ ($u^\text{up}$) is defined such that 99.9% of the remaining data points lie above (below) it (blue lines in Fig. \[fig:normalization\]). Fourth, we define the data ranges $$\Delta\rho = \rho^\text{up}
\quad\text{and}\quad
\Delta u = u^\text{up}-u^\text{low}\;.$$
Regarding the norms and averages to measure the model errors, we use the following expressions. On a segment $x\in [0,L]$ and time interval $t\in [0,T]$, spatial and spatio-temporal averages are considered $$\begin{aligned}
E^{x}(t) &= \frac{1}{L}\int_0^L E(x,t){\,\mathrm{d}x}\;, \label{eq:error_x} \\
E &= \frac{1}{TL}\int_0^T\int_0^L E(x,t){\,\mathrm{d}x}{\,\mathrm{d}t}\;. \label{eq:error_xt}\end{aligned}$$ Moreover, for the RTMC data set, the temporal error at a sensor position $x_s$ inside the road segment on a given day is considered, as well averages over multiple days $$\begin{aligned}
E_\text{day} &= \frac{1}{T}\int_0^T E(x_s,t){\,\mathrm{d}t}\;, \label{eq:error_t} \\
E &= \frac{1}{\#\text{days}}\sum_{\text{day}=1}^{\#\text{days}} E_\text{day}\;. \label{eq:error_day}\end{aligned}$$
![Flow rate vs. density curves for the models LWR (left column), ARZ (middle column), and GARZ (right column), together with the traffic states observed in the test cases (gray dots). The results show the NGSIM data (top row) and the RTMC data (bottom row).[]{data-label="fig:models_flow_rate_curves"}](fig_models_flow_rate_data_ngsim_lwr){width="\textwidth"}
![Flow rate vs. density curves for the models LWR (left column), ARZ (middle column), and GARZ (right column), together with the traffic states observed in the test cases (gray dots). The results show the NGSIM data (top row) and the RTMC data (bottom row).[]{data-label="fig:models_flow_rate_curves"}](fig_models_flow_rate_data_ngsim_arz){width="\textwidth"}
![Flow rate vs. density curves for the models LWR (left column), ARZ (middle column), and GARZ (right column), together with the traffic states observed in the test cases (gray dots). The results show the NGSIM data (top row) and the RTMC data (bottom row).[]{data-label="fig:models_flow_rate_curves"}](fig_models_flow_rate_data_ngsim_garz){width="\textwidth"}
![Flow rate vs. density curves for the models LWR (left column), ARZ (middle column), and GARZ (right column), together with the traffic states observed in the test cases (gray dots). The results show the NGSIM data (top row) and the RTMC data (bottom row).[]{data-label="fig:models_flow_rate_curves"}](fig_models_flow_rate_data_rtmc_lwr){width="\textwidth"}
![Flow rate vs. density curves for the models LWR (left column), ARZ (middle column), and GARZ (right column), together with the traffic states observed in the test cases (gray dots). The results show the NGSIM data (top row) and the RTMC data (bottom row).[]{data-label="fig:models_flow_rate_curves"}](fig_models_flow_rate_data_rtmc_arz){width="\textwidth"}
![Flow rate vs. density curves for the models LWR (left column), ARZ (middle column), and GARZ (right column), together with the traffic states observed in the test cases (gray dots). The results show the NGSIM data (top row) and the RTMC data (bottom row).[]{data-label="fig:models_flow_rate_curves"}](fig_models_flow_rate_data_rtmc_garz){width="\textwidth"}
List of Models {#subsec:list_of_models}
--------------
We compare the following four models in terms of their predictive accuracy of the real data:
1. **Interpolation:** A predictor that, at any instance in time, constructs the traffic density and velocity via direct linear interpolation of the boundary conditions, i.e., on the road segment $x\in [0,L]$, the predicted state is $\rho(x,t) = \rho(0,t)(1-x/L)+\rho(L,t)x/L$ and $u(x,t) = u(0,t)(1-x/L)+u(L,t)x/L$. Of course, this predictor is not an actual traffic model. However, due to its simplistic nature, it represents an important means of comparison.
2. **LWR:** The first-order model , in which only the density state $\rho(x,t)$ is evolved, based on the data-fitted equilibrium velocity curve $V(\rho,w_\text{eq})$, resulting from the least-squares fit of the family to the fundamental diagram data.
3. **ARZ:** The second-order ARZ model that generalizes the least-squares fitted flow rate curve of the LWR model to a family of curves $V(\rho,w) = V(\rho,w_\text{eq})+(w-w_\text{eq})$.
4. **GARZ:** The second-order generalized ARZ model , whose generalized velocity function $V(\rho,w)$ is obtained via a weighted least-squares fit of the family to the fundamental diagram data.
The fundamental diagram curves $Q_w(\rho) = \rho V(\rho,w)$ of the three traffic models are shown in Fig. \[fig:models\_flow\_rate\_curves\], overlayed on top of the traffic state data that are actually observed in the test cases (gray dots). The top row of figures corresponds to the NGSIM data, and the bottom row represents the RTMC data. The LWR model is shown on the left, the ARZ model in the middle, and the GARZ model on the right.
Regarding the reproduction of real traffic data, it should be stressed that the models/predictors differ in the way they use data. In the model generation step, the Interpolation predictor uses no historic data; the LWR and ARZ model are based on the same least-squares fit; and the GARZ model employs more information from the fundamental diagram data due to the weighted least-squares fit. In turn, during the advance forward in time, the Interpolation predictor uses two pieces of information ($\rho$ and $u$) at each boundary. In contrast, the LWR model uses only one piece of information ($\rho$) at one of the two boundaries (assuming the traffic states at the two boundaries are either both in free flow or both congested). Finally, the ARZ and GARZ model use a total of two pieces of information through the boundary conditions.
![Model comparison on the NGSIM 5:15pm–5:30pm data set. In each panel, the time-evolutions of the model predictions (colored dashed curve) and measured data (solid gray curve) at the middle position of the study area are shown (top box: $\rho$, bottom box: $u$). The four panels correspond to: Interpolation (top left, green), LWR (top right, blue), ARZ (bottom left, red), GARZ (bottom right, black).[]{data-label="fig:NGSIM_evolution"}](fig_prediction_ngsim_interp){width="\textwidth"}
![Model comparison on the NGSIM 5:15pm–5:30pm data set. In each panel, the time-evolutions of the model predictions (colored dashed curve) and measured data (solid gray curve) at the middle position of the study area are shown (top box: $\rho$, bottom box: $u$). The four panels correspond to: Interpolation (top left, green), LWR (top right, blue), ARZ (bottom left, red), GARZ (bottom right, black).[]{data-label="fig:NGSIM_evolution"}](fig_prediction_ngsim_lwr){width="\textwidth"}
![Model comparison on the NGSIM 5:15pm–5:30pm data set. In each panel, the time-evolutions of the model predictions (colored dashed curve) and measured data (solid gray curve) at the middle position of the study area are shown (top box: $\rho$, bottom box: $u$). The four panels correspond to: Interpolation (top left, green), LWR (top right, blue), ARZ (bottom left, red), GARZ (bottom right, black).[]{data-label="fig:NGSIM_evolution"}](fig_prediction_ngsim_arz){width="\textwidth"}
![Model comparison on the NGSIM 5:15pm–5:30pm data set. In each panel, the time-evolutions of the model predictions (colored dashed curve) and measured data (solid gray curve) at the middle position of the study area are shown (top box: $\rho$, bottom box: $u$). The four panels correspond to: Interpolation (top left, green), LWR (top right, blue), ARZ (bottom left, red), GARZ (bottom right, black).[]{data-label="fig:NGSIM_evolution"}](fig_prediction_ngsim_garz){width="\textwidth"}
![Comparison of traffic models for the data sets NGSIM 4:00pm–4:15pm (at the top), NGSIM 5:00pm–5:15pm (in the middle), and NGSIM 5:15pm–5:30pm (at the bottom). In each panel, the time-evolution of spatially averaged errors (left top box), measurement traffic density data (left bottom box), and space-time average errors (right box) are shown in log-scale for Interpolation, LWR, ARZ, and GARZ.[]{data-label="fig:NGSIM_errors"}](fig_results_ngsim_1){width=".97\textwidth"}
![Comparison of traffic models for the data sets NGSIM 4:00pm–4:15pm (at the top), NGSIM 5:00pm–5:15pm (in the middle), and NGSIM 5:15pm–5:30pm (at the bottom). In each panel, the time-evolution of spatially averaged errors (left top box), measurement traffic density data (left bottom box), and space-time average errors (right box) are shown in log-scale for Interpolation, LWR, ARZ, and GARZ.[]{data-label="fig:NGSIM_errors"}](fig_results_ngsim_2){width=".97\textwidth"}
![Comparison of traffic models for the data sets NGSIM 4:00pm–4:15pm (at the top), NGSIM 5:00pm–5:15pm (in the middle), and NGSIM 5:15pm–5:30pm (at the bottom). In each panel, the time-evolution of spatially averaged errors (left top box), measurement traffic density data (left bottom box), and space-time average errors (right box) are shown in log-scale for Interpolation, LWR, ARZ, and GARZ.[]{data-label="fig:NGSIM_errors"}](fig_results_ngsim_3){width=".97\textwidth"}
Model Validation with the NGSIM Trajectory Data {#subsec:validation_ngsim}
-----------------------------------------------
As in [@FanSeibold2013], we consider a segment of 450 meters in length in the domain of the NSGIM vehicle trajectories. Data for three time intervals is available: 4:00pm–4:15pm, 5:00pm–5:15pm, and 5:15pm–5:30pm. However, because our model validation requires a traffic state be defined on the complete study segment, slightly shorter study time intervals must be chosen that guarantee that recorded vehicles are present everywhere on the road. We choose: 4:00:30pm–4:14:00pm, 5:00:30pm–5:13:30pm, and 5:15:30pm–5:28:00pm. The parameters of the traffic models, obtained by fitting to the fundamental diagram data provided with NSGIM, are given in the NGSIM row of Table \[tab:model\_parameters\]. In line with the temporal resolution of the data, we generate density and velocity functions (real data and model predictions) in intervals of 0.1 seconds.
Figure \[fig:NGSIM\_evolution\] displays the time-evolution (5:15pm–5:30pm) of the traffic density and velocity that are predicted by the selected models, in comparison with the real evolution of these quantities, at the mid-point of the study area. Both the LWR and the ARZ model do reproduce the general trends present in the true density evolution, albeit with a systematic delay of 30–60 seconds (see [@FanSeibold2014] for a detailed discussion on this delay). One difference between LWR and ARZ is that the latter reproduces, modulo the delay, the velocity evolution better than the former. This is particularly visible in the predicted velocity in the plateau between 5:19:00pm and 5:22:30pm. In turn, the GARZ model captures the evolution of density and velocity significantly better than the other two traffic models. There is still a systematic delay in the model predictions, but this delay is very small. Finally, the Interpolation predictor captures the large–scale features of the real data as well. However, it also exhibits an oscillatory behavior of a larger frequency. This is due to the fact that the linear interpolations transmit temporal oscillations from both boundaries immediately to the observation site. In contrast, the traffic models pick up fewer boundary data and furthermore oscillations can turn into N-waves and thus reduce in magnitude as they move into the domain.
Data set $\alpha$ (veh/h/lane) $\lambda$ $p$ $w_\text{min}$ (km/h) $w_{\text{max}}$ (km/h)
---------- ----------------------- ----------- ------ ----------------------- -------------------------
NGSIM 247.38 23.41 0.16 39.49 82.01
RTMC 316.46 23.91 0.16 75.81 102.82
: Model parameters for the data-fitted models for the two data sets. Here, $\alpha$, $\lambda$, and $p$ are the free parameters of the equilibrium flow rate curve $Q_\text{eq}(\rho)$ in the family . Moreover, $w_\text{min}$ and $w_\text{max}$ denote the boundaries of the empty road velocity for the GARZ model, as described in §\[sec:data-fitting\_GARZ\].[]{data-label="tab:model_parameters"}
To quantify the predictive accuracy of the different models, we turn to the error metric . Figure \[fig:NGSIM\_errors\] shows the model errors for the NGSIM data sets: 4:00pm–4:15pm (top), 5:00pm–5:15pm (middle), and 5:15pm–5:30pm (bottom). In each panel, the time-evolution of the spatially averaged error is shown (top-left box), as well as the spatio-temporal average error (top-right box). The models are: Interpolation (thick solid yellow), LWR (solid red), ARZ (dashed blue), and GARZ (thin solid black). In each bottom-left box, the time-evolution of the average traffic density is shown. The numerical values of the model errors are given in rows 2–4 of Table \[tab:model\_errors\].
[|ll||lr|lr|lr|lr|]{} Data set
------------------------------------------------------------------------
&& Interp. & &LWR & & ARZ & & GARZ &\
NGSIM
------------------------------------------------------------------------
& 4:00–4:15 &0.151 &(+10%) & 0.181 &(+31%) & 0.153 &(+11%) & 0.138 &\
NGSIM & 5:00–5:15 &0.160 &(+25%) & 0.161 &(+26%) & 0.174 &(+35%) & 0.129 &\
NGSIM & 5:15–5:30 &0.168 &(+30%) & 0.162 &(+25%) & 0.228 &(+76%) & 0.129 &\
RTMC
------------------------------------------------------------------------
& congested &0.203 &(+14%) & 0.228 &(+24%) & 0.208 &(+13%) & 0.184 &\
RTMC & non-cong. &0.108 &(+63%) & 0.081 &(+26%) & 0.067 & (+4%) & 0.064 &\
The time-evolution of the errors confirm several of the above observations, such as the highly oscillatory nature of the Interpolation predictor, and the good accuracy of the GARZ model. Another particularly visible effect is the bad performance of the ARZ model for large densities: for 4:00–4:15, the ARZ model yields smaller errors than LWR; in contrast, for 5:15–5:30, the ARZ leads to larger errors than LWR. Moreover, the peaks in the ARZ errors coincide with local maxima in the traffic density. These observations confirm that: a) the ARZ model has the potential to improve upon the LWR model, and b) the non-uniform stagnation density of the ARZ model significantly affects its predictive accuracy. In contrast, the GARZ model inherits the advantages of the ARZ model for low densities, and furthermore remedies its shortcoming for high densities.
Regarding the performance of the Interpolation predictor; its accuracy is, except for the large oscillations, surprisingly good. Specifically, its average errors are similar to those of the LWR and ARZ models. The following explanations can be provided to address this observation:
1. The considered road segment is very short. Hence, the coherence between the boundaries and the inside of the domain is high, and the actual delays due to finite-speed information propagation are small. One can expect that on longer road segments, the finite-speed transport of information becomes more relevant, and therefore interpolation significantly loses in accuracy.
2. As described in §\[subsec:list\_of\_models\], the Interpolation predictor picks up twice (four times) as much data from the boundaries as the second (first) order models. One could therefore argue that the traffic models are as accurate as the interpolation, while utilizing less input data.
3. The Interpolation predictor is on par with an “incomplete” model (LWR does not evolve velocities) and a “defective” model (ARZ is flawed for large densities). In turn, it does not achieve the accuracy of the GARZ model.
![Model comparison on March 26, 2003 in the RTMC data set. In each panel, the time-evolutions (4:05pm–5:00pm) of the model predictions (colored dashed curve) and measured data (solid gray curve) at the middle sensor are shown (top box: $\rho$, bottom box: $u$). The four panels correspond to: Interpolation (top left, green), LWR (top right, blue), ARZ (bottom left, red), GARZ (bottom right, black).[]{data-label="fig:RTMC_evolution"}](fig_prediction_rtmc_interp){width="\textwidth"}
![Model comparison on March 26, 2003 in the RTMC data set. In each panel, the time-evolutions (4:05pm–5:00pm) of the model predictions (colored dashed curve) and measured data (solid gray curve) at the middle sensor are shown (top box: $\rho$, bottom box: $u$). The four panels correspond to: Interpolation (top left, green), LWR (top right, blue), ARZ (bottom left, red), GARZ (bottom right, black).[]{data-label="fig:RTMC_evolution"}](fig_prediction_rtmc_lwr){width="\textwidth"}
![Model comparison on March 26, 2003 in the RTMC data set. In each panel, the time-evolutions (4:05pm–5:00pm) of the model predictions (colored dashed curve) and measured data (solid gray curve) at the middle sensor are shown (top box: $\rho$, bottom box: $u$). The four panels correspond to: Interpolation (top left, green), LWR (top right, blue), ARZ (bottom left, red), GARZ (bottom right, black).[]{data-label="fig:RTMC_evolution"}](fig_prediction_rtmc_arz){width="\textwidth"}
![Model comparison on March 26, 2003 in the RTMC data set. In each panel, the time-evolutions (4:05pm–5:00pm) of the model predictions (colored dashed curve) and measured data (solid gray curve) at the middle sensor are shown (top box: $\rho$, bottom box: $u$). The four panels correspond to: Interpolation (top left, green), LWR (top right, blue), ARZ (bottom left, red), GARZ (bottom right, black).[]{data-label="fig:RTMC_evolution"}](fig_prediction_rtmc_garz){width="\textwidth"}
![Comparison of models for the RTMC sensor data on 45 week days with congested traffic (top) and 29 days with non-congested traffic (bottom). The left boxes show the time-averaged error for each day , and the right boxes show the fully averaged errors of congested and non-congested days. The errors for Interpolation, LWR, ARZ, and GARZ are depicted in log-scale.[]{data-label="fig:RTMC_errors"}](fig_results_rtmc_congestion){width=".97\textwidth"}
![Comparison of models for the RTMC sensor data on 45 week days with congested traffic (top) and 29 days with non-congested traffic (bottom). The left boxes show the time-averaged error for each day , and the right boxes show the fully averaged errors of congested and non-congested days. The errors for Interpolation, LWR, ARZ, and GARZ are depicted in log-scale.[]{data-label="fig:RTMC_errors"}](fig_results_rtmc_free){width=".97\textwidth"}
Model Validation with the RTMC Sensor Data
------------------------------------------
In line with the test described in [@FanSeibold2013], we consider a 1.214km long segment of highway on the I-35W, Minneapolis. Two traffic sensors are at the ends of the study segment, and one sensor is inside of the segment (roughly in the middle). The parameters of the data-fitted models, obtained from one-year data at the middle sensor, are given in the RTMC row of Table \[tab:model\_parameters\]. On each weekday (Monday–Friday) between 01/01/2003 and 04/14/2003, we consider the onset of afternoon rush hour between 4pm and 5pm, and we divide the 74 days into 45 days with congested traffic (the space-time averaged traffic density exceeds 20veh/km/lane) and 29 days with non-congested traffic (the remaining ones). As described in §\[subsec:data\_treatment\], the traffic models are run through an initialization phase 4:00pm–4:05pm, in which boundary data create a realistic state inside the segment. The actual validation is then conducted in 4:05pm–5:00pm.
Analogously as for the NGSIM data, we first consider the temporal evolution of the traffic states that the models predict and study the qualitative behavior of the models. We look at a day (03/26/2003) on which congestion builds up rapidly (between 4:35pm and 4:40pm) at the sensor position, so that any delays in the model behavior become visible. The results, shown in Fig. \[fig:RTMC\_evolution\], confirm the observations made for the NGSIM data: a) LWR and ARZ predict similar densities, but ARZ does a better job at also capturing velocities correctly; b) both LWR and ARZ propagate information too slowly, resulting in the onset of congestion to be predicted 5 minutes too late; c) the GARZ model is not perfect (it still predicts the onset of congestion 2 minutes too late), but it captures the general trends in both variables quite nicely. This last point is particularly visible in the velocity profile in the congested state in 4:42pm–5:00pm.
One aspect that is different from the NGSIM test is that the Interpolation predictor performs visibly worse than the traffic models during the low density and high velocity period 4:05pm–4:35pm. This demonstrates the assertion made in §\[subsec:validation\_ngsim\], that simple interpolation performs much worse on longer road segments.
The average model errors obtained with the RTMC data are shown in Fig. \[fig:RTMC\_errors\]. The top panel collects the 45 congested days, and the bottom panel contains the 29 non-congested days. In the left boxes the time-averaged errors in 4:05pm–5:00pm are shown for each day. The right boxes show the resulting total errors , resulting from averaging over all study days. The numerical values, as well as the excess errors of the models relative to GARZ, are shown in rows 5–6 of Table \[tab:model\_errors\].
The results of the low density days demonstrate quite clearly that a) traffic models yield noticeably better predictions than simple interpolation; and b) second-order models reproduce the real traffic behavior better than the first-order LWR model. The fact that the GARZ model does not differ much from the ARZ model results from the fact that for low densities, the two families of flow rate curves are very similar (see Fig. \[fig:models\_flow\_rate\_curves\]). In turn, the results of the high density days confirm that a) as expected, the quality of the ARZ model worsens relative to the other models; while b) the GARZ model does not suffer from the same amount of accuracy deterioration than ARZ. A somewhat unexpected observation is that for the high-density days, the Interpolation predictor does not perform worse than LWR and ARZ. The reason for this lies is the fact that LWR and ARZ propagate information too slowly, and thus capture features with a delay (see Fig. \[fig:RTMC\_evolution\]). In contrast, interpolation propagates information instantaneously—which is obviously unrealistic, but here happens to lead to less severe errors than the spurious delays in LWR and ARZ.
Inhomogeneous ARZ and GARZ Models {#sec:inhomogeneous_models}
=================================
Thus far we have restricted our attention to homogeneous second-order models, in which each vehicle remains for all times attached to its specific velocity curve $u_w(\rho) = V(\rho,w)$. However, it is plausible that in real traffic, drivers vary their empty road velocity $w$ over time, and that the overall traffic dynamics tend towards an equilibrium velocity curve $U_\text{eq}(\rho) = V(\rho,w_\text{eq})$. We therefore extend our model validations to the inhomogeneous ARZ model , denoted ARZ-$\tau$, and the inhomogeneous GARZ model , denoted GARZ-$\tau$. In both cases the equilibrium velocity curve is the LWR velocity curve.
As argued in §\[subsubsec:relaxtion\_GARZ\_LWR\], for a Cauchy problem the solutions of the inhomogeneous second-order models converge (as $t\to\infty$ for $\tau$ fixed; or as $\tau\to 0$ for $t_\text{final}$ fixed) to solutions of the first-order LWR model. In the presence of boundary data, the same statement holds; however, the limits of the second-order model solutions are LWR solutions *with different boundary conditions* than the LWR solutions that we consider here. Consequently, the models ARZ-$\tau$ and GARZ-$\tau$ are not merely perturbations of the LWR model, but instead could reproduce the dynamics of real traffic better than LWR and better than homogeneous second-order models.
![Model errors of the inhomogeneous ARZ and GARZ models, ARZ-$\tau$ (dashed blue) and GARZ-$\tau$ (solid black), as functions of the relaxation time $\tau$. Shown are the time-averaged errors for one day in the RTMC data set (top left), and spatio-temporally averaged errors for the three NSGIM data sets: 4:00pm–4:15pm (top right), 5:00pm–5:15pm (bottom left), and 5:15pm–5:30pm (bottom right). Also shown are the errors with the homogeneous (i.e., $\tau\to\infty$) ARZ model (blue square) and GARZ model (black circle), the error obtained with the LWR model (red square), and the choices of $\tau$ that yield the smallest model error (red star and red plus).[]{data-label="fig:error_tau"}](fig_relaxation_rtmc_congestion){width="\textwidth"}
![Model errors of the inhomogeneous ARZ and GARZ models, ARZ-$\tau$ (dashed blue) and GARZ-$\tau$ (solid black), as functions of the relaxation time $\tau$. Shown are the time-averaged errors for one day in the RTMC data set (top left), and spatio-temporally averaged errors for the three NSGIM data sets: 4:00pm–4:15pm (top right), 5:00pm–5:15pm (bottom left), and 5:15pm–5:30pm (bottom right). Also shown are the errors with the homogeneous (i.e., $\tau\to\infty$) ARZ model (blue square) and GARZ model (black circle), the error obtained with the LWR model (red square), and the choices of $\tau$ that yield the smallest model error (red star and red plus).[]{data-label="fig:error_tau"}](fig_relaxation_ngsim_1){width="\textwidth"}
![Model errors of the inhomogeneous ARZ and GARZ models, ARZ-$\tau$ (dashed blue) and GARZ-$\tau$ (solid black), as functions of the relaxation time $\tau$. Shown are the time-averaged errors for one day in the RTMC data set (top left), and spatio-temporally averaged errors for the three NSGIM data sets: 4:00pm–4:15pm (top right), 5:00pm–5:15pm (bottom left), and 5:15pm–5:30pm (bottom right). Also shown are the errors with the homogeneous (i.e., $\tau\to\infty$) ARZ model (blue square) and GARZ model (black circle), the error obtained with the LWR model (red square), and the choices of $\tau$ that yield the smallest model error (red star and red plus).[]{data-label="fig:error_tau"}](fig_relaxation_ngsim_2){width="\textwidth"}
![Model errors of the inhomogeneous ARZ and GARZ models, ARZ-$\tau$ (dashed blue) and GARZ-$\tau$ (solid black), as functions of the relaxation time $\tau$. Shown are the time-averaged errors for one day in the RTMC data set (top left), and spatio-temporally averaged errors for the three NSGIM data sets: 4:00pm–4:15pm (top right), 5:00pm–5:15pm (bottom left), and 5:15pm–5:30pm (bottom right). Also shown are the errors with the homogeneous (i.e., $\tau\to\infty$) ARZ model (blue square) and GARZ model (black circle), the error obtained with the LWR model (red square), and the choices of $\tau$ that yield the smallest model error (red star and red plus).[]{data-label="fig:error_tau"}](fig_relaxation_ngsim_3){width="\textwidth"}
The inhomogeneous second-order models are based on the same data-fitted functions as their homogeneous counterparts. The only new parameter is the relaxation time $\tau$. Since historic fundamental diagram data is commonly devoid of temporal information relevant for traffic dynamics, there is no canonical way to obtain the value of $\tau$ from historic data. Regarding realistic choices for $\tau$, the only information found in the literature is that it cannot be much smaller than 3 seconds due to physical restrictions of the vehicle engines. Due to this absence of a good modeling principle for the value of $\tau$, we conduct our model validation procedure for multiple instances of the inhomogeneous ARZ and GARZ models, where we let $\tau$ range from milliseconds to days. For each test, whichever choice of $\tau$ yields the smallest model error is then the optimal ARZ-$\tau$ (respectively optimal GARZ-$\tau$) model.
The described $\tau$-study is conducted for one day (January 8, 2003) in the RTMC data set, and for the three NGSIM data sets. For the RTMC data we evaluate the temporally averaged error , and for the NSGIM data the spatio-temporally averaged errors . The results of the study are shown in Fig. \[fig:error\_tau\]. In each test case, the ARZ-$\tau$ and GARZ-$\tau$ models are computed for many values of $\tau$ (blue and black functions, respectively). In addition, the (homogeneous) ARZ and GARZ models are computed (blue square and black circle). As expected, they agree with ARZ-$\tau$ and GARZ-$\tau$, respectively, for $\tau\gg 1$. Moreover, the LWR model is computed (red square). As argued above, due to the presence of boundary data with $u\neq U_\text{eq}(\rho)$, its results are different from ARZ-$\tau$ and GARZ-$\tau$ for $\tau\ll 1$. Finally, in each test case the particular $\tau = \tau_\text{opt}$ is marked (red star and red plus) for which ARZ-$\tau$ and GARZ-$\tau$, respectively, yield the smallest model errors. It is apparent that in all four cases shown in Fig. \[fig:error\_tau\], the inhomogeneous models possess an optimal relaxation time $0<\tau_\text{opt}<\infty$.
[|ll||lr|lr|lr|lr|l]{} Data set
------------------------------------------------------------------------
&& ARZ&& ARZ-$\tau_{\text{opt}}$ & & GARZ & & GARZ-$\tau_\text{opt}$ &\
NGSIM
------------------------------------------------------------------------
& 4:00–4:15 & 0.153 &(+13%) & 0.141 & (+4%) & 0.138 & (+2%) & 0.135 &\
NGSIM & 5:00–5:15 & 0.174 &(+43%) & 0.137 & (+13%) & 0.129 & (+6%) & 0.122 &\
NGSIM & 5:15–5:30 & 0.228 &(+87%) & 0.165 & (+35%) & 0.129 & (+6%) & 0.122 &\
RTMC
------------------------------------------------------------------------
& congested &0.208 &(+18%) & 0.192 &(+8%) & 0.184 &(+4%) & 0.177 &\
RTMC & non-cong. &0.067 &(+6%) & 0.066 &(+5%) & 0.064 & (+2%) & 0.063 &\
As one can see in Fig. \[fig:error\_tau\], the error-minimizing relaxation times are $\tau_\text{opt}\approx 25\text{s}$ for the ARZ model (the result for NGSIM 4:00pm–4:15pm is not as reliable, because the minimum is weakly pronounced), and $\tau_\text{opt}\approx 50\text{s}$ for the GARZ model in the NGSIM data. Moreover, for the RTMC data, the GARZ model yields $\tau_\text{opt}\approx 150\text{s}$. If the values of $\tau_\text{opt}$ give an indication about the time scales on which driving behavior evolves in reality, then their relatively large values give rise to the interesting observation that real driving behavior changes rather slowly; much slower than the time scales on which vehicles are able to accelerate due to engine power. In addition, one can observe that the GARZ model’s optimal relaxation time is noticeably slower for the RTMC data than it is for the NGSIM data. While it is in principle possible that this difference stems from differences in driving behavior between Minnapolis vs. the San Francisco Bay Area, more plausibly the difference stems from the combination of a relatively flat minimum of the black error curve, and from aggregation and lack of precision effects in the sensor data.
In order to quantify how much the inhomogeneous second-order models can improve the model accuracy relative to homogeneous second-order models, we consider the model errors of the ARZ-$\tau$ and GARZ-$\tau$ models (with $\tau = \tau_\text{opt}$ for each test case) in comparison with the errors of the ARZ and GARZ models. The resulting error values, together with the excess errors relative to the GARZ-$\tau_\text{opt}$ model shown in parentheses, are shown in Table \[tab:model\_errors\_tau\]. One can see that the addition of a relaxation term can improve the accuracy of the ARZ model noticeably, in particular for flow at high traffic densities (see NSGIM 5:15–5:30). This is in part due to the fact that the ARZ model’s unrealistic spread in flow rate curves for large densities is ameliorated by a relaxation towards the LWR curve. In contrast, the addition of a relaxation term (of the considered form) to the GARZ model does hurt the model accuracy, but it does not lead to significant improvements either; the GARZ model appears to be quite good already in its homogeneous form, at least for the data sets studied here.
Conclusions {#sec:conclusions}
===========
We have presented a systematic approach to construct a data-fitted generalized Aw-Rascle-Zhang (GARZ) model of traffic flow from historic fundamental diagram data. The modeling advantages and the mathematical properties of the GARZ model have been discussed. Moreover, the predictive accuracy of the GARZ model has been compared with other macroscopic traffic models via a three-detector test on vehicle trajectory and stationary sensor data. The actual model comparison has been carried out in a macroscopic sense, i.e., discretization effects are kept small, and thus the accuracy of the actual PDE is investigated.
The model comparison considers a hierarchy of models: (i) the first-order Lighthill-Whitham-Richards (LWR) model, that is defined via a single curve $Q(\rho)$ in the fundamental diagram; (ii) the second-order Aw-Rascle-Zhang (ARZ) model, that is defined via a family of curves in the fundamental diagram $Q_w(\rho) = Q(\rho)+\rho(w-Q'(0))$, each of which is the LWR curve plus a linear function; (iii) the second-order GARZ model, that is defined via a family of curves that are not simple transformations of each other, i.e., $Q_w(\rho) = \rho V(\rho,w)$, where $V(\rho,w)$ is a two-parameter generalized velocity function; and finally, the model validation is augmented by (iv) an Interpolation predictor that reconstructs the traffic density and velocity in the study domain via a simple linear interpolation of the boundary data. Moreover, the second-order traffic models are considered in their homogeneous form, ARZ and GARZ, as well as inhomogeneous versions thereof, ARZ-$\tau$ and GARZ-$\tau$, for which a relaxation term towards the LWR equilibrium state is added.
The general conclusions that can be drawn from the comparisons are as follows. First, and most importantly, the GARZ model yields the most accurate predictions, and it reproduces the behavior of the real data in the best fashion. Second, the ARZ model is superior to the LWR model for low traffic densities; but its model accuracy suffers noticeably as traffic becomes more congested. This observation can be interpreted as a manifestation of the ARZ model’s multiple stagnation densities—a drawback that is addressed by the GARZ model. Third, in terms of averaged errors, the Interpolation predictor yields model errors that are similar to, or even slightly lower than, those of the LWR and ARZ models. At the same time, predictions based on mere interpolation are more oscillatory and less sharp than predictions obtained via traffic models. In addition, interpolation performs less well on longer road segments. Fourth, it is observed that the addition of a relaxation term can further improve the accuracy of second-order models. However, a noticeable improvement is only observed when the relaxation time $\tau$ is chosen well. In turn, when $\tau$ is selected too small, the inhomogeneous model could be less accurate than the corresponding homogeneous model.
The studies of the inhomogeneous second-order models reveal that the optimal relaxation times lie in the range 14–60 seconds (with some values even larger). This is two orders of magnitude larger than the drivers’ reaction time, and one order of magnitude larger than what the vehicles’ engine capabilities would allow. A possible explanation for such seemingly slow relaxation is that in the ARZ and the GARZ models, it is the drivers’ empty road velocities that relax, i.e., their general driving behavior, and not the actual vehicle velocities.
One modeling shortcoming of the ARZ model that is not addressed by the GARZ model studied in this paper is that no unique functional relationship between flow rate and density for low densities (“free flow regime”) is allowed. To address this point, phase transition models (cf. [@Colombo2003; @Goatin2006]) and variations of the ARZ model that result from introducing an upper bound on the vehicle velocity [@ColomboMarcelliniRascle2003] have been proposed. Moreover, a specific phase transition model has been applied in a practical context [@BlandinWorkGoatinPiccoliBayen2011]. The possibility to allow for a unique flow rate vs. density relationship may be of importance in certain applications. As described in a companion paper [@FanPiccoliSeibold2014], this task can also be achieved within the GARZ framework, by collapsing the fundamental diagram curves into a single function in the free flow regime.
Acknowledgments {#acknowledgments .unnumbered}
===============
M. Herty was supported by Excellenz Cluster EXC128, DAAD 55866082, and BMBF KinOpt 05M2013. B. Seibold would like to acknowledge the support by the National Science Foundation. This work was supported through grant DMS–1318709, and partially supported through grants DMS–1115269 and DMS–1318641.
[^1]: The function $h(\rho)$ is sometimes called “pressure”, and denoted $p(\rho)$, even though it does not play the role of a pressure in the equations.
| 0 |
---
abstract: |
Boundary effects produced by a Chern-Simons (CS) extension to electrodynamics are analyzed exploiting the Green’s function (GF) method. We consider the electromagnetic field coupled to a $\theta$-term in a way that has been proposed to provide the correct low energy effective action for topological insulators (TI). We take the $\theta$-term to be piecewise constant in different regions of space separated by a common interface $\Sigma$, to be called the $\theta$-boundary. Features arising due to the presence of the boundary, such as magnetoelectric effects, are already known in CS extended electrodynamics and solutions for some experimental setups have been found with specific configuration of sources. In this work we illustrate a method to construct the GF that allows to solve the CS modified field equations for a given $%
\theta$-boundary with otherwise arbitrary configuration of sources. The method is illustrated by solving the case of a planar $\theta$-boundary but can also be applied for cylindrical and spherical geometries for which the $\theta$-boundary can be characterized by a surface where a given coordinate remains constant. The static fields of a point-like charge interacting with a planar TI, as described by a planar discontinuity in $\theta$, are calculated and successfully compared with previously reported results. We also compute the force between the charge and the $\theta$-boundary by two different methods, using the energy momentum tensor approach and the interaction energy calculated via the GF. The infinitely straight current-carrying wire is also analyzed.
author:
- 'A. Martín-Ruiz'
- 'M. Cambiaso'
- 'L. F. Urrutia'
title: 'Green’s function approach to Chern-Simons extended electrodynamics: an effective theory describing topological insulators'
---
Introduction
============
The relevance of Chern-Simons (CS) forms [@Chern:1974ft] in several branches of theoretical physics is well accounted for. In quantum field theory in regards to anomalies [@ABJ] they played a key role and in particle physics they proved important as well [@Peccei:1977hh; @'tHooft:1986nc; @Wilczek:1987mv]. In general relativity they also enjoy a prominent position as clearly reviewed in [@Zanelli:2012px]. Further studies involve its uses in topological quantum field theory [@Witten:1988ze], topological string theory [Marino:2004uf]{} and as a quantum gravity candidate [@Bonezzi:2014nua].
In general, CS forms are amenable for capturing topological features of the physical system they describe, which is why in the last decade their importance has also become apparent in the field of condensed matter physics for describing what came to be known as topological phases. von Klitzing’s discovery of the astonishing precision with which the Hall conductance of a sample is quantized [@von; @Klitzing:1980kg], despite the varying irregularities and geometry of the sample, turned out to have a topological origin. The reason for this lies in the band structure of the sample, but ultimately, the Hall conductance can be expressed as an invariant integral over the frequency in momentum space, more precisely as an integral of the Berry curvature over the Brillouin zone [@SQ; @Shen], inasmuch as the genus of a manifold can be expressed in terms of an invariant integral of the local curvature over the surface enclosing it. This quantity plays the role of a topological order parameter uniquely determining the nature of the quantum state, as the order parameter in Landau-Ginzburg effective field theory determines the usual phases of quantum matter.
In this work we will be concerned with a simple case of CS theories, to which we will refer as $\theta$-electrodynamics or simply $\theta$-ED and it amounts to extending Maxwell electrodynamics by a parity violating term of the form $$\Delta \mathcal{L}_\theta = \theta (\alpha / 4 \pi^2) \mathbf{E} \cdot
\mathbf{B} = - {\frac{\theta }{4}} (\alpha / 4 \pi^2) F_{\mu \nu} \tilde
F^{\mu \nu},\label{FF*}$$ where $\tilde{F}^{\mu \nu }=\frac{1}{2}\epsilon ^{\mu \nu \alpha \beta }
F_{\alpha \beta }$ and $\epsilon ^{\mu \nu \alpha \beta }$ is the Levi-Cività symbol. In general $\theta$ can be a dynamical field, however, we will take it as a constant scalar, making Eq. (\[FF\*\]) a pseudo-scalar. Note that this extension is a total derivative, producing no contribution to the field equations when usual boundary conditions are met. If $\theta$ is not globally constant in the manifold where the theory is defined, then the CS-term fails to be a topological invariant and therefore the corresponding modifications to the field equations must be taken into consideration.
Here we will study Maxwell theory extended by Eq. (\[FF\*\]) defined on a manifold in which there are two domains defined by their different constant values of $\theta$ that are separated by a common interface or boundary $\Sigma$. The $\theta$-term can be thought of as an effective parameter characterizing properties of a novel electromagnetic vacuum possibly arising from a more fundamental theory or, as applied to material media, as an effective macroscopic parameter to describe novel quantum degrees of freedom of matter apart from the usual permittivity $\varepsilon$ and permeability $\mu$. The former approach has been taken in the context of classical $\theta$-ED [@ZH] and in the quantum vacuum framework [@Canfora:2011fd]. For related analyses, in several contexts, see [@related]. The latter approach has been used to describe TIs. Concretely, the low-energy limit of the electrodynamics of TIs can be described by extending Maxwell electrodynamics by Eq. (\[FF\*\]), originally formulated in 4+1 dimensions but appropriately adapted to lower dimensions by dimensional reduction [@Qi:2008ew]. Thus, $\theta$-ED as a topological field theory (TFT), serves as model for many theoretical [@theo_topoins] and experimental realizations for studying detailed properties of topological states of quantum matter [@tech-apps; @topo_reviews].
The formulation of $\theta $-ED pursued in this work can be considered as a particularly simple version of the so called Janus field theories [@CFKS; @DEG; @Chen; @Witten2; @Kim1; @Kim2]. Generally speaking, such theories are characterized by having space-time dependent coupling constants, as is $\theta$ in our model. They have been actively explored in the context of the AdS/CFT correspondence. Nevertheless, as we have already mentioned, in the case of $\theta $-ED this idea is applied to a simpler but more realistic system, that constitutes an effective low energy theory which allows to compute the response of topological insulators to arbitrary external sources and currents in a planar geometry, with direct extensions to cylindrical and spherical geometries. Janus field theories were motivated, from the gravitational sector of the AdS/CFT correspondence, by an exact and non-singular solution for the dilatonic field in type II-B supergravity, which was found in a simple deformation of the $AdS_{5}\times S^{5}$ geometry [@Bak]. Even though the solution breaks all the original supersymmetries it proves to be stable under a large class of perturbations [@Bak; @FNSS; @CCDAVPZ]. The dilaton acquires a constant value at the boundary, where $AdS_{5}$ is recovered, but adopts different values at each half of the boundary. On the other hand, the AdS/CFT correspondence requires the existence of a dual gauge theory on the boundary for every non-singular solution of type IIB supergravity with appropriate asymptotics which in this case is a four dimensional $\mathcal{N}=4$ SYM theory living in the boundary [@Bak]. In other words, a running dilaton induces space-time dependent coupling constants in the gauge theories in the dual sector, which defines what is called a Janus field theory. In our case, the four dimensional $\mathcal{N}=4$ SYM theory is replaced by the CS modified ED, where we take the electromagnetic coupling to be globally constant, while the topological coupling to the Pontryagin invariant has different constant values at each side of a plane interface and suffers a jump across such boundary. In relation to $\theta $-ED, it is interesting to recall that the authors of Ref. [@CFKS] proposed a model for the dual theory arising from the Janus solution, where the $\mathcal{N}=4$ SYM coupling $g(z)$ affects only the kinetic term of the non-Abelian gauge field, together with the interaction terms in the original Lagrangian for the standard $\mathcal{N}=4$ SYM theory. The model completely breaks the $16$ original supersymmetries of the $\mathcal{N}=4$ SYM theory. Moreover, $g(z)$ is taken as constant on each side of a planar interface $(z=0)$, with a sharp jump across it. In this way, the gauge field part of the action is the non-abelian generalization of the Maxwell action in an inhomogeneous medium with permitivity $\epsilon $ and permeability $\mu $ related by $\epsilon (z)=1/\mu (z)=1/g^{2}(z)$. The YM fields satisfy boundary conditions at the interface, which are derived by integrating the equations of motion over the standard infinitesimal pill-shaped regions across the boundary, in a way similar to standard electrodynamics. The YM Green’s function is also obtained by using image methods. Nevertheless, let us emphasize that this model does not include a coupling to the YM Pontryagin invariant, so its Abelian limit does not reproduce $\theta $-ED. The inclusion of the topological coupling $\theta (z)$ in addition to the YM coupling $g(z)$ is developed in Refs. [@Witten2; @Kim2], where 1/2 BPS vacuum configurations are studied in particular. As shown in Ref. [@Witten2] half of the original supersymmetries can be maintained provided such couplings are constrained by the relations $1/g^{2}(z)=D\sin 2\psi (z)$ and $\theta (z)=\theta_{0}+8\pi ^{2}D\cos 2\psi (z)$. The case of a sharp interface respecting the above constraints is also considered in Ref. [@Kim2] and it is studied in the abelian Coulomb phase, by setting two different constant values $\psi _{1}$ and $\psi _{2}$ at each side of a planar boundary. However, in the case of $\theta $-ED, supersymmetry does not enter and we are choosing the electromagnetic coupling to be globally constant, *i.e.* $g_{1}=g_{2}=e$, while only the topological coupling $\theta (z)$ becomes discontinuous at the sharp boundary, with constant values $\theta_{1}\neq \theta _{2}$ in each side. As can be seen already, these two systems are not equivalent and we will later discuss this issue in more detail.
The paper is organized as follows. In section \[model\] we review the basics of Chern-Simons electrodynamics defined on a four dimensional spacetime in which the $\theta $-value is piecewise constant in different regions of space separated by a common boundary $\Sigma $. In section \[Green\] we restrict ourselves to the static case and we construct the GF matrix for the planar geometry corresponding to a $\theta $-boundary located at $z=a$. Section \[applications\] is devoted to different applications, *e.g.*, the problems of a point-like charge and a current-carrying wire near a planar $\theta $-boundary. The interaction energy (and forces) between a charge-current distribution and a $\theta $-interface is briefly discussed. Contact between the results obtained with our method and others in the existing literature is made. A concluding summary of our results comprises the last section \[summary\]. Throughout the paper, Lorentz-Heaviside units are assumed ($\hbar =c=1$), the metric signature will be taken as $\left( +,-,-,-\right) $ and the convention $\epsilon ^{0123}=+1$ is adopted.
$\protect\theta$-Electrodynamics in a bounded region {#model}
====================================================
Our model is based on Maxwell electrodynamics coupled to a gauge invariant $\theta$-term as described by the following action $$\mathcal{S}=\int_{\mathcal{M}}d^{4}x\left[ -\frac{1}{16\pi }F_{\mu \nu
}F^{\mu \nu }-\frac{1}{4}{\theta }\frac{\alpha }{4\pi^2 }F_{\mu
\nu }\tilde{F}^{\mu \nu }-j^{\mu }A_{\mu }\right] , \label{action}$$where $\alpha =e^{2}/\hbar c$ is the fine structure constant and $j^{\mu }\;$is a conserved external current. The coupling constant for the $\theta$-term, $\alpha / 4 \pi ^{2}$, is chosen in such a way that the total electric charge $q
_{e} = \frac{1}{4 \pi} \int d\textbf{S} \cdot \textbf{D}$ has to be an integer multiple of the electron charge $e$, whereas the magnetic charge $q _{m} = \frac{1}{4 \pi}
\int d\textbf{S} \cdot \textbf{B}$ should be an integer multiple of $g
= e / 2 \alpha$ by the Dirac quantization condition [@Wilczek:1987mv]. Later we will recall the reasoning which shows that, quantum mechanically, the allowed values of $\theta$ are $0$ or $\pi$ (mod $2 \pi$).
The $(3+1)$-dimensional spacetime is $\mathcal{M}=\mathcal{U}\times \mathbb{R%
}$, where $\mathcal{U}$ is a three-dimensional manifold and $\mathbb{R}$ corresponds to the temporal axis. We make a partition of space in two regions: $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ in such a way that manifolds $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ intersect along a common two-dimensional boundary $\Sigma $, to be called the $\theta $-boundary, so that $\mathcal{U}=\mathcal{U}_{1}\cup \mathcal{U}_{2}$ and $\Sigma =\mathcal{%
U}_{1}\cap \mathcal{U}_{2}$, as shown in Fig. \[regions\]. We also assume that the field $\theta $ is piecewise constant in such way that it takes the constant value $\theta =\theta _{1}$ in region $\mathcal{U}_{1}$ and the constant value $\theta =$ $\theta _{2}$ in region $\mathcal{U}_{2}$. This situation is expressed in the characteristic function $$\theta \left( \textbf{x} \right) =\left\{
\begin{array}{c}
\theta _{1}\;\;\;,\;\;\; \textbf{x} \in \mathcal{U}_{1} \\
\theta _{2}\;\;\;,\;\;\; \textbf{x} \in \mathcal{U}_{2}%
\end{array}%
\right. . \label{theta}$$In this scenario the $\theta $-term in the action fails to be a global topological invariant because it is defined over a region with the boundary $%
\Sigma $. Varying the action gives rise to a set of Maxwell equations with an effective additional current density with support at the boundary $$\partial _{\mu }F^{\mu \nu }=\tilde{\theta}\delta \left( \Sigma \right)
n_{\mu }\tilde{F}^{\mu \nu }+4\pi j^{\nu }, \label{FieldEqs}$$where $n_{\mu }$ is the outward normal to $\Sigma $, and $\tilde{\theta}%
= \alpha \left( \theta _{1}-\theta _{2}\right) /\pi $, which enforces the invariance of the classical dynamics under the shifts of $\theta$ by any constant, $\theta \rightarrow \theta + C$. Current conservation can be verified directly by taking the divergence at both sides of Eq. ([FieldEqs]{}) and using symmetry properties.
![[Region over which the electromagnetic field theory is defined.]{}[]{data-label="regions"}](regions.eps)
The set of Eqs. (\[FieldEqs\]) for $\theta $-ED can be written as $$\begin{aligned}
\nabla \cdot \mathbf{E} &=&\tilde{\theta}\delta \left( \Sigma \right)
\mathbf{B}\cdot \mathbf{n}+4\pi \rho , \label{GaussE} \\
\nabla \times \mathbf{B}-\frac{\partial \mathbf{E}}{\partial t} &=&\tilde{%
\theta}\delta \left( \Sigma \right) \mathbf{E}\times \mathbf{n}+4\pi \mathbf{%
J}, \label{Ampere}\end{aligned}$$while the homogeneous equations are included in the Bianchi identity $\partial _{\mu }\tilde{F}^{\mu \nu }=0$. Here $\mathbf{n}$ is the unit normal to $\Sigma $ shown in the Fig. [regions]{}. In this work we consider a simple geometry corresponding to a surface $\Sigma$ taken as the plane $z=a$.
As we see from Eqs. (\[GaussE\]-\[Ampere\]) the behavior of $\theta $-ED in the bulk regions $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ is the same as in standard electrodynamics. The $\theta $-term modifies Maxwell equations only at the surface $\Sigma $. Here $F^{i0}=E^{i}$, $F^{ij}=-\varepsilon
^{ijk}B^{k}$ and $\tilde{F}^{i0}=B^{i}$, $\tilde{F}^{ij}=\varepsilon
^{ijk}E^{k}$. Equations (\[GaussE\]-\[Ampere\]) also suggest that the electromagnetic response of a system in the presence of a $\theta $-term can be described in terms of Maxwell equations in matter $$\nabla \cdot \textbf{D} = 4 \pi \rho \;\;\;\ , \;\;\;\ \nabla \times \textbf{H} = 4 \pi \textbf{J} , \label{MaxEqMatter}$$ with constitutive relations $$\mathbf{D}=\mathbf{E}+\frac{\alpha }{\pi }\theta \left(z\right) \mathbf{B}%
,\;\;\;\;\;\;\;\;\;\mathbf{H}=\mathbf{B}-\frac{\alpha }{\pi }\theta \left(
z\right) \mathbf{E}, \label{CONST_REL}$$where $\theta \left(z\right) $ is given in Eq. (\[theta\]). If $\theta (z)$ is globally constant in $\mathcal{M}$, there is no contribution to Maxwell equations from the $\theta $-term in the action, even though $\theta$ still is present in the constitutive relations. In fact, the additional contributions of a globally constant $\theta$ to each of the modified Maxwell equations, (\[GaussE\]) and (\[Ampere\]) cancel due to the homogeneous equations.
Now we return to the problem of the allowed values of $
\theta$ to describe topological insulators. $U(1)$ gauge theories with nonzero $\theta$ ($\theta$-ED) exhibit an $SL(2,
\mathbb{Z})$ duality group which strongly constrains the quantum physics [@Witten3; @Karch]. This group is obtained by repeated applications of the $S$ and $T$ generators of electric-magnetic duality. The $S$ generator is associated with the invariance of classical Maxwell equations in matter (\[MaxEqMatter\]) (supplemented with magnetic charge and current densities) under duality rotations. Only the special case of a duality transformation by $\pi / 2$ is consistent with the requirement that the electric charge and the magnetic charge are quantized.
The aforementioned rescaling symmetry $\theta \to \theta +C$ would allow to set $\theta$ to zero at the classical level. Quantum mechanically, however, given that for properly quantized electric and magnetic fluxes $S _{\theta} / \hbar$ is an integer multiple of $\theta$, only $C = 2 \pi n$ for integer $n$ is an allowed symmetry, otherwise non-trivial contributions to the path integral would result. Furthermore, since $\mathbf{E}\cdot \mathbf{B}$ is odd under $t \to -t$, only $\theta = 0$ and $\theta = \pi$ give a time-reversal symmetric theory. Thus, time reversal takes $\theta$ into $- \theta$, so $\theta = 0$ is time reversal invariant per se, whereas $\theta = \pi$ is invariant after the shift $\theta \rightarrow \theta + 2 \pi$. This is typically referred as the $T$ generator of the electric-magnetic duality. The two transformations $S$ and $T$ generate the $SL(2, \mathbb{Z})$ symmetry group acting on the fields [@Karch].
Next we study the effects of a $\theta$-interface in the propagation of the fields. Assuming that the time derivatives of the fields are finite in the vicinity of the surface $\Sigma $, the field equations imply that the normal component of $\mathbf{E}$, and the tangential components of $\mathbf{B}$, acquire discontinuities additional to those produced by superficial free charges and currents, while the normal component of $\mathbf{B}$, and the tangential components of $\mathbf{E}$, are continuous. For vanishing free sources on the surface the boundary conditions read $$\begin{aligned}
\mathbf{E}_{z}\big|^{z=a^+}_{z=a^-} &=&\tilde{\theta}\mathbf{B}_{z}\big|%
_{z=a},\;\;\;\;\;\;\;\;\; \mathbf{B}_{\parallel }\big|^{z=a^+}_{z=a^-}=-\tilde{%
\theta}\mathbf{E}_{\parallel }\big|_{z=a}, \label{Ampere-BC} \\
\mathbf{B}_{z}\big|^{z=a^+}_{z=a^-} &=&0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\; \mathbf{E}%
_{\parallel }\big|^{z=a^+}_{z=a^-}=0. \label{Faraday-BC}\end{aligned}$$ The continuity conditions (\[Faraday-BC\]) imply that the right hand sides of Eqs. (\[Ampere-BC\]) are well defined and they represent self-induced surface charge and current densities, respectively. An immediate consequence of the boundary conditions (\[Ampere-BC\]-\[Faraday-BC\]) is that the presence of a magnetic field crossing the surface $\Sigma $ is sufficient to generate an electric field, even in the absence of free electric charges. Many interesting magnetoelectric effects due to a $\theta$-boundary have been highlighted using different approaches. For example, electric charges close to the interface $\Sigma$ induce magnetic mirror monopoles (and vice versa) [@science; @Kim1; @Kim2]. Also, the propagation of electromagnetic waves across a $\theta$-boundary have been studied finding that a non trivial Faraday rotation of the polarizations appears [@ZH; @Kim1; @Kim2; @Hehl]. It is worth mentioning that with the modified boundary conditions, several properties of conductors still hold for static fields as far as the conductor does not lie in the $\Sigma $-boundary, in particular, conductors are equipotential surfaces and the electric field just outside the conductor is normal to its surface.
The Green’s function method {#Green}
===========================
In this section we use the GF method to solve static boundary-value problems in $\theta $-ED in terms of the electromagnetic potential $A^{\mu}$. Certainly one could solve for the electric and magnetic fields from the modified Maxwell equations together with the boundary conditions (\[Ampere-BC\]-\[Faraday-BC\]), however, just as in ordinary electrodynamics, there might be occasions where information about the sources is unknown and rather we are provided with information of the 4-potential at the given boundaries. In these cases, the GF method provides the general solution, to a given boundary-value problem (Dirichlet or Neumann) for arbitrary sources. Nevertheless, in the sequel we restrict ourselves to contributions of free sources outside the $\theta$-boundary, and without boundary conditions imposed on additional surfaces, except for the standard boundary conditions at infinity.
Since the homogeneous Maxwell equations that express the relationship between potentials and fields are not modified, the electrostatic and magnetostatic fields can be written in terms of the $4$-potential $A ^{\mu} = \left( \phi ,
\mathbf{A} \right)$ according to $\mathbf{E} = - \nabla \phi $ and $\mathbf{B%
} = \nabla \times \mathbf{A}$ as usual. In the Coulomb gauge $\nabla \cdot
\mathbf{A} = 0 $, the $4$-potential satisfies the equation of motion $$\left[ - \eta ^{\mu} _{\; \nu} \nabla ^{2} - \tilde{\theta}\delta \left(
z-a\right) \epsilon _{\;\;\;\;\ \nu }^{3 \mu \alpha }\partial _{\alpha }%
\right] A^{\nu }=4 \pi j^{\mu }, \label{FiedlEqPlaneConfig}$$ together with the boundary conditions $$A ^{\mu} \big| _{z= a^{-}} ^{z = a^{+}} = 0 \;\;\;\;\ , \;\;\;\;\ \left(
\partial _{z} A ^{\mu} \right) \big| _{z= a^{-}} ^{z = a^{+}} = - \tilde{\theta}
\epsilon ^{3 \mu \alpha} _{\;\;\;\;\ \nu} \partial _{\alpha} A ^{\nu } \big| %
_{z=a} . \label{BC4Pot}$$ One can further check that these boundary conditions for the $4$-potential corresponds to the ones obtained in Eqs. (\[Ampere-BC\]-\[Faraday-BC\]).
To obtain a general solution for the potentials $\phi$ and $\mathbf{A}$ in the presence of arbitrary external sources $j^\mu (\textbf{x})$, we introduce the GF $G ^{\nu} _{\; \sigma} \left( \mathbf{x} ,
\mathbf{x} ^{\prime} \right)$ solving Eq. (\[FiedlEqPlaneConfig\]) for a point-like source, $$\left[ - \eta ^{\mu} _{\; \nu} \nabla ^{2} - \tilde{\theta}\delta \left(
z-a\right) \epsilon _{\;\;\;\;\ \nu }^{3 \mu \alpha }\partial _{\alpha }%
\right] G ^{\nu} _{\; \sigma} \left( \mathbf{x} , \mathbf{x} ^{\prime}
\right) = 4 \pi \eta ^{\mu} _{\; \sigma} \delta ^{3} \left( \mathbf{x} -
\mathbf{x} ^{\prime} \right) , \label{EqsGreenMatrix}$$ together with the boundary conditions (\[BC4Pot\]), in such a way that the general solution for the $4$-potential in the Coulomb gauge is $$A ^{\mu} \left( \mathbf{x} \right) = \int d ^{3} \mathbf{x} ^{\prime} \; G
^{\mu} _{\; \nu} \left( \mathbf{x} , \mathbf{x} ^{\prime} \right) j^{\nu}
\left( \mathbf{x} ^{\prime} \right) . \label{GreenMatrix}$$ According to Eqs. (\[EqsGreenMatrix\]) the diagonal entries of the GF matrix are related with the electric and magnetic fields arising from the charge and current density sources, respectively, although they acquire $\theta$-dependence. However, the non-diagonal terms encode the magnetoelectric effect, *i.e.* the charge (current) density contributing to the magnetic (electric) field.
In the following we concentrate in constructing the solution to Eq. (\[EqsGreenMatrix\]). The GF we consider has translational invariance in the directions parallel to $\Sigma $, that is in the transverse $x$ and $y$ directions, while this invariance is broken in the $z$ direction. Exploiting this symmetry we further introduce the Fourier transform in the direction parallel to the plane $\Sigma$, taking the coordinate dependence to be $\left( \mathbf{x}-\mathbf{x}^{\prime }\right)
_{\parallel }$ $=(x-x^{\prime },\;y-y^{\prime })$ and define $$G _{\;\nu }^{\mu }\left( \mathbf{x} , \mathbf{x} ^{\prime }\right) = 4\pi
\int \frac{d^{2}\mathbf{p}}{\left( 2\pi \right) ^{2}}e^{i\mathbf{p}\cdot
\left( \mathbf{x}-\mathbf{x}^{\prime }\right) _{\parallel }} g_{\;\nu }^{\mu
}\left( z,z^{\prime }\right) , \label{RedGreenDef}$$where $\mathbf{p}=(p_{x},p_{y})$ is the momentum parallel to the plane $%
\Sigma $ [@CED]. In Eq. (\[RedGreenDef\]) we have suppressed the dependence of the reduced GF $g_{\;\nu }^{\mu }$ on $\mathbf{p}$.
Due to the antisymmetry of the Levi-Cività symbol, the partial derivative appearing in the second term of the GF Eq. ([EqsGreenMatrix]{}) does not introduce derivatives with respect to $z$, but only in the transverse coordinates. This allows us to write the full reduced GF equation as $$\left[ \partial ^{2}\eta _{\;\nu }^{\mu }+i\tilde{\theta}\delta \left(
z-a\right) \epsilon _{\;\;\;\;\ \nu }^{3 \mu \alpha }p_{\alpha }\right]
g_{\;\ \sigma }^{\nu }\left( z,z^{\prime }\right) =\eta _{\;\ \sigma }^{\mu
}\delta \left( z-z^{\prime }\right) , \label{RedGreenFunc}$$where $\partial ^{2}= \mathbf{p} ^{2}-\partial _{z}^{2}$, $p^{\alpha
}=\left( 0 , \mathbf{p} \right) $ and $\mathbf{p} ^{2} = - p ^{\alpha} p
_{\alpha}$. The solution to Eq. (\[RedGreenFunc\]) is simple, but not straightforward. For solving it we employ a method similar to that used for obtaining the GF for the one-dimensional $\delta$-function potential in quantum mechanics, where the free GF is used for integrating the GF equation with $\delta$-interaction. To this end we consider a reduced free GF having the form $\mathcal{G}_{\;\nu }^{\mu }\left(
z,z^{\prime }\right) =\mathfrak{g}\left( z,z^{\prime }\right) \eta _{\;\nu
}^{\mu }$, associated with the operator $\partial ^{2}$ previously defined, that solves $$\partial ^{2}\mathcal{G}_{\;\nu }^{\mu }\left( z,z^{\prime }\right) =\eta
_{\;\nu }^{\mu }\delta \left( z-z^{\prime }\right), \label{RedGreenFuncA1}$$satisfying the standard boundary conditions at infinity, where $$\mathfrak{g}(z,z^{\prime })=\frac{1}{2p}e^{-p|z-z^{\prime }|} \label{RFSGF}$$ and $p = \vert \textbf{p} \vert$. Note that Eq. (\[RedGreenFuncA1\]) demands the derivative of $\mathfrak{g}$ to be discontinuous at $z=z^{\prime}$, *i.e.,* $\partial _{z} \mathfrak{g} \left( z ,
z ^{\prime} \right) \big| _{z= z^{\prime -}} ^{z=z^{\prime +}} = -1$, and then the continuity of $\mathfrak{g}$ at $z=z^{\prime}$ follows [@CED].
Now we observe that Eq. (\[RedGreenFunc\]) can be directly integrated by using the free GF Eq. (\[RedGreenFuncA1\]) together with the properties of the Dirac delta-function, thus reducing the problem to a set of coupled algebraic equations, $$g_{\;\ \sigma }^{\mu }\left( z,z^{\prime }\right) =\eta _{\;\sigma }^{\mu }%
\mathfrak{g}\left( z,z^{\prime }\right) -i\tilde{\theta}\epsilon _{\;\;\;\;\
\nu }^{3 \mu \alpha }p_{\alpha }\mathfrak{g}\left( z,a\right) g_{\;\ \sigma
}^{\nu }\left( a,z^{\prime }\right) . \label{RedGreenFuncA3}$$ Note that the continuity of $\mathfrak{g}$ at $z=z'$ implies the continuity of $g_{\;\
\sigma }^{\mu }$, but the discontinuity of $\partial _{z} \mathfrak{g}$ at the same point yields $$\partial _{z} g_{\;\ \sigma }^{\mu } \left( z , z^{\prime }\right) \big| %
_{z= a ^{-}} ^{z = a^{+}} = -i \tilde{\theta}\epsilon _{\;\;\;\;\ \nu }^{3 \mu
\alpha }p_{\alpha } \partial _{z} \mathfrak{g}\left( z,a\right) \big| _{z=
a^{-}} ^{z = a^{+}} g_{\;\ \sigma }^{\nu }\left( a,z^{\prime }\right) = i \tilde{%
\theta}\epsilon _{\;\;\;\;\ \nu }^{3 \mu \alpha } p_{\alpha } g_{\;\ \sigma
}^{\nu }\left( a,z^{\prime }\right) , \label{BCRedGreenFunc}$$ from which the boundary conditions for the 4-potential in Eq. (\[BC4Pot\]) are recovered. In this way the solution (\[RedGreenFuncA3\]) guarantees that the boundary conditions at the $\theta$-interface are satisfied.
Now we have to solve for the various components $g_{\;\ \sigma }^{\mu }$. To this end we split Eq. (\[RedGreenFuncA3\]) into $\mu = 0$ and $\mu =
j=1,2,3 $ components; $$\begin{aligned}
g ^{0} _{\; \sigma} \left( z , z ^{\prime} \right) &=& \eta ^{0} _{\;
\sigma} \mathfrak{g} \left( z , z ^{\prime} \right) - i \tilde{\theta}
\epsilon ^{3 0 i} _{\;\;\;\ j} p _{i} \mathfrak{g} \left( z , a \right) g
^{j} _{\; \sigma} \left( a , z ^{\prime} \right) , \label{g0S} \\
g ^{j} _{\; \sigma} \left( z , z ^{\prime} \right) &=& \eta ^{j} _{\;
\sigma} \mathfrak{g} \left( z , z ^{\prime} \right) - i \tilde{\theta}
\epsilon ^{3 j i} _{\;\;\;\ 0} p _{i} \mathfrak{g} \left( z , a \right) g
^{0} _{\; \sigma} \left( a , z ^{\prime} \right) . \label{gjS}\end{aligned}$$ Now we set $z = a$ in Eq. (\[gjS\]) and then substitute into Eq. (\[g0S\]) yielding $$g ^{0} _{\; \sigma} \left( z , z ^{\prime} \right) = \eta ^{0} _{\; \sigma}
\mathfrak{g} \left( z , z ^{\prime} \right) - i \tilde{\theta} \epsilon ^{3
0 i} _{\;\;\;\ j} p _{i} \eta ^{j} _{\; \sigma} \mathfrak{g} \left( z , a
\right) \mathfrak{g} \left( a , z ^{\prime} \right) - \tilde{\theta} ^{2}
p ^{2} \mathfrak{g} \left( z , a \right) \mathfrak{g} \left( a , a
\right) g ^{0} _{\; \sigma} \left( a , z ^{\prime} \right) , \label{g0S-2}$$ where we use the result $\epsilon ^{3 0 i} _{\;\;\;\ j} \epsilon ^{3 j
k} _{\;\;\;\ 0} p _{k} p _{i} = p ^{2}$. Solving for $g ^{0} _{\; \sigma} \left( a , z ^{\prime}
\right)$ by setting $z=a$ in Eq. (\[g0S-2\]) and inserting the result back in that equation, we obtain $$g ^{0} _{\; \sigma} \left( z , z ^{\prime} \right) = \eta ^{0} _{\; \sigma}
\left[ \mathfrak{g} \left( z , z ^{\prime} \right) + \tilde{\theta} p ^{2} \mathfrak{g} \left( a , a \right) A \left( z , z ^{\prime} \right) %
\right] + i \epsilon ^{30i} _{\;\;\;\ \sigma} p _{i} A \left( z , z
^{\prime} \right) , \label{g0S-3}$$ where $$A\left( z,z^{\prime }\right) =-\tilde{\theta}\frac{\mathfrak{g}\left(
z,a\right) \mathfrak{g}\left( a,z^{\prime }\right) }{1+ p ^{2}
\tilde{\theta}^{2}\mathfrak{g}^{2}\left( a,a\right) }. \label{A(Z,Z)}$$ The remaining components can be obtained by substituting $g ^{0} _{\;
\sigma} \left( a , z ^{\prime} \right)$ in Eq. (\[gjS\]). The result is $$g ^{j} _{\; \sigma} \left( z , z ^{\prime} \right) = \eta ^{j} _{\; \sigma}
\mathfrak{g} \left( z , z ^{\prime} \right) + i \epsilon ^{3 j k} _{\;\;\;\
0} p _{k} \left[\eta ^{0} _{\; \sigma} - i \tilde{\theta} \epsilon ^{3 0 i}
_{\;\;\;\ \sigma} p _{i} \mathfrak{g} \left( a , a \right) \right] A \left(
z , z ^{\prime} \right) . \label{gjS-2}$$ Equations (\[g0S-3\]) and (\[gjS-2\]) allow to write the general solution as $$g_{\;\nu }^{\mu }\left( z,z^{\prime }\right) =\eta _{\;\ \nu }^{\mu }%
\mathfrak{g}\left( z,z^{\prime }\right) +A \left( z,z^{\prime }\right)
\left\lbrace \tilde{\theta}\mathfrak{g}\left( a,a\right) \left[ p^{\mu}
p_{\nu } + \left( \eta _{\;\ \nu }^{\mu } + n^{\mu }n_{\nu } \right) p ^{2} \right] +i \epsilon _{\;\ \nu }^{\mu \;\ \alpha 3}p_{\alpha}
\right\rbrace , \label{GenSolGreenPlaneConf}$$where $n _{\mu} =\left( 0,0,0,1\right) $ is the normal to $\Sigma $ The reciprocity between the position of the unit charge and the position at which the GF is evaluated $G_{\mu \nu }(\mathbf{x},\mathbf{x}
^{\prime }) = G_{\nu \mu }(\mathbf{x} ^{\prime},\mathbf{x})$ is one of its most remarkable properties. From Eq. ([RedGreenDef]{}) this condition demands $$g_{\mu \nu }(z,z^{\prime }, \mathbf{p})=g_{\nu \mu }(z^{\prime },z,- \mathbf{%
p}),$$which we verify directly from Eq. (\[GenSolGreenPlaneConf\]). The symmetry $g_{\mu \nu }\left( z,z^{\prime }\right) =g_{\nu \mu
}^{\ast }\left( z,z^{\prime }\right) =g_{\mu \nu }^{\dagger }\left(
z,z^{\prime }\right) $ is also manifest.
The various components of the static GF matrix in coordinate representation are obtained by computing the Fourier transform defined in Eq. (\[RedGreenDef\]), with the reduced GF given by Eq. (\[RFSGF\]). The details are presented in Appendix \[IntegralsStaticCase\]. The final results are $$\begin{aligned}
G_{\;0}^{0}\left( \mathbf{x},\mathbf{x}^{\prime }\right) &=&\frac{1}{|%
\mathbf{x}-\mathbf{x}^{\prime }|}-\frac{\tilde{\theta}^{2}}{4+\tilde{\theta}%
^{2}}\frac{1}{\sqrt{R^{2}+Z^{2}}}, \label{G00} \\G_{\; i}^{0}\left( \mathbf{x},\mathbf{x}^{\prime }\right) &=& - \frac{2 \tilde{\theta}}{4+\tilde{\theta}^{2}} \frac{\epsilon _{0ij3} R ^{j}}{R ^{2}}\left( 1-\frac{Z}{\sqrt{R^{2}+Z^{2}}}\right) , \label{G0i} \\ G _{\; j}^{ i }\left( \mathbf{x},\mathbf{x}^{\prime }\right) &=& \eta ^{i} _{\; j} G _{\; 0}^{ 0}\left( \mathbf{x},\mathbf{x}^{\prime }\right) - \frac{i}{2}\frac{\tilde{\theta}^{2}}{4+ \tilde{\theta}^{2}} \partial ^{i} K _{j} \left( \mathbf{x},\mathbf{x}^{\prime }\right) , \label{Gij}\end{aligned}$$ where $Z=| z - a | + | z^{\prime } - a |$, $R ^{j} = \left( \mathbf{x-x}^{\prime }
\right) _{\parallel } ^{j} = \left( x - x^{\prime } , y - y^{\prime} \right)$, $R=|\left( \mathbf{x-x}^{\prime }\right) _{\parallel }|\;$and$$K ^{j} \left( \mathbf{x},\mathbf{x}^{\prime }\right) =2i\frac{\sqrt{%
R^{2}+Z^{2}}-Z}{R^{2}} R ^{j} .$$ Finally, we observe that Eqs. (\[G00\]-\[Gij\]) contain all the required elements of the GF matrix, according to the choices of $z$ and $z^{\prime
}$ in the function $Z$.
Applications
============
Point-like charge near a planar $\theta$-boundary {#charge_near_plane}
-------------------------------------------------
Let us consider a point-like electric charge $q$ located at a distance $b>0$ from the $z=0$ plane, where we have chosen $a=0$. Also, the region $z<0$ is filled with a topologically non-trivial insulator whereas the region $%
z>0$ is the vacuum ($\theta_2=0$). For simplicity we choose the coordinates such that $x^{\prime }=y^{\prime }=0$. Therefore, the current density is $j^\mu(\mathbf{x}{}^{\prime })=q \eta ^{\mu} _{\; 0} \delta \left( x^{\prime
}\right) \delta \left( y^{\prime }\right) \delta \left( z^{\prime }-b\right)
$. According to Eq. ([GreenMatrix]{}), the solution for this problem is $$A^{\mu }\left( \mathbf{x}\right) =qG_{\;0}^{\mu }\left( \mathbf{x},\mathbf{r}%
\right) , \label{SolPointCharge}$$where $\mathbf{r}=b\hat{\mathbf{e}}_{z}$. We first study the electrostatic potential. From Eq. (\[G00\]), $$\begin{aligned}
z>0\;\;\;\;\ &:&\;\;\;\;\ G_{\;0}^{0}\left( \mathbf{x},\mathbf{r}\right) =%
\frac{1}{|\mathbf{x}-\mathbf{r}|}-\frac{\tilde{\theta}^{2}}{4+\tilde{\theta}%
^{2}}\frac{1}{|\mathbf{x}+\mathbf{r}|}, \label{G00z>} \\
z<0\;\;\;\;\ &:&\;\;\;\;\ G_{\;0}^{0}\left( \mathbf{x},\mathbf{r}\right) =%
\frac{4}{4+\tilde{\theta}^{2}}\frac{1}{|\mathbf{x} - \mathbf{r}|} .
\label{G00z<}\end{aligned}$$For $z>0$ the GF yields the electric potential $%
A^{0}\left( \mathbf{x}\right) =qG_{\;0}^{0}\left( \mathbf{x},\mathbf{r}%
\right)$ which can be interpreted as due to two point-like electric charges, one of strength $q$ at $\mathbf{%
r}$, and the other, the image charge, of strength $-q\tilde{\theta}^{2}/(4+%
\tilde{\theta}^{2})$, at the point $-\mathbf{r}$. For $z<0$ only one point-like electric charge appears, of strength $4q/(4+\tilde{\theta}^{2})$ located at $\mathbf{r}$.
>From Eq. (\[SolPointCharge\]) we see that two components of the magnetic vector potential are nonzero, $A^{1}\left( \mathbf{x}\right)
=qG_{\;0}^{1}\left( \mathbf{x},\mathbf{r}\right) $ and $A^{2}\left( \mathbf{x%
}\right) =qG_{\;0}^{2}\left( \mathbf{x},\mathbf{r}\right) $. The corresponding GF components for each region are given by $$\begin{aligned}
G_{\;0}^{1}\left( \mathbf{x},\mathbf{r}\right) &=&-\frac{2\tilde{\theta}}{4+%
\tilde{\theta}^{2}}\frac{y}{R^{2}}\left\{
\begin{array}{c}
1-\frac{z+b}{|\mathbf{x}+\mathbf{r}|}\;\;\;\ ,\;\;\;\ z>0 \\
1+\frac{z-b}{|\mathbf{x}+\mathbf{r}|}\;\;\;\ ,\;\;\;\ z<0%
\end{array}%
\right.
\label{G01} \\
G_{\;0}^{2}\left( \mathbf{x},\mathbf{r}\right) &=&+\frac{2\tilde{\theta}}{4+%
\tilde{\theta}^{2}}\frac{x}{R^{2}}\left\{
\begin{array}{c}
1-\frac{z+b}{|\mathbf{x}+\mathbf{r}|}\;\;\;\ ,\;\;\;\ z>0 \\
1+\frac{z-b}{|\mathbf{x}+\mathbf{r}|}\;\;\;\ ,\;\;\;\ z<0%
\end{array}%
\right. \label{G02}\end{aligned}$$according to Eqs. (\[G0i\]). It is difficult to interpret the components of the vector potential directly, however the magnetic field $\mathbf{B}=\nabla \times \mathbf{A}$ is illuminating. In fact $$\begin{aligned}
z>0\;\;\;\;\ &:&\;\;\;\;\ \mathbf{B}\left( \mathbf{x}\right) =\frac{2q\tilde{%
\theta}}{4+\tilde{\theta}^{2}}\frac{\mathbf{x}+\mathbf{r}}{|\mathbf{x}+%
\mathbf{r}|^{3}}, \\
z<0\;\;\;\;\ &:&\;\;\;\;\ \mathbf{B}\left( \mathbf{x}\right) =\frac{-2q%
\tilde{\theta}}{4+\tilde{\theta}^{2}}\frac{\mathbf{x}-\mathbf{r}}{|\mathbf{x}%
-\mathbf{r}|^{3}}.\end{aligned}$$Thus we observe that the magnetic field for $z>0$ is that due to a magnetic monopole of strength $2q \tilde{\theta} / ( 4 + \tilde{\theta}^{2}) $ located at $-
\mathbf{r}$. For $z<0$ we have a magnetic monopole of strength -$2q\tilde{%
\theta} / ( 4 + \tilde{\theta}^{2})$ located at $\mathbf{r}$.
The solution shows that, for an electric charge near the planar surface of a topological insulator, both an image magnetic charge and a image electric charge will be induced. The appearance of magnetic monopoles in this solution seems to violate the Maxwell law $\nabla \cdot \mathbf{B}=0$, which remained unaltered in the case of $\theta$-ED. Nevertheless, recalling that $\left( \mathbf{x}\pm \mathbf{r}%
\right) /|\mathbf{x}\pm \mathbf{r}|^{3}\sim \mathbf{\nabla }_{x}(1/|\mathbf{x%
}\pm \mathbf{r}|)$, we have $\nabla \cdot \mathbf{B\sim \nabla }_{x}^{2}(1/|%
\mathbf{x}\pm \mathbf{r}|) \sim \delta (\mathbf{x} \pm \mathbf{r})$ in a region where $\mathbf{x\neq }\pm \mathbf{r}$. Physically, the magnetic field is induced by a surface current density $$\mathbf{J}=\tilde{\theta}\delta \left( z\right) \mathbf{E}\times \mathbf{n}=%
\frac{4q\tilde{\theta}}{4+\tilde{\theta}^{2}}\frac{R}{\left(
R^{2}+b^{2}\right) ^{3/2}} \delta \left( z\right) \hat{\varphi},$$ that is circulating around the origin. However such induced field has the correct magnetic field dependence expected from a magnetic monopole. This current is nothing but the Hall current [@science].
It is worth mentioning that these results were also obtained using different approaches. On the one hand the authors in Ref. [@science] used the image method to show that an electric charge near a topological surface state induces an image magnetic monopole due to the magneto-electric effect and, of course, emphasized the possible experimental verification via a gas of quantum particles carrying fractional statistics, consisting of the bound states of the electric charge and the image monopole charge.
At this stage we clarify the differences between the $\theta$-ED approach we are following and the $1/2$ BPS construction in the sharp interface discussed in Ref. [@Kim2]. As we mentioned in the Introduction the 8 remaining supersymmetries in the latter case are enforced by demanding the couplings to be related in the following way $$\frac{1}{e^2}= D \sin 2\psi(z), \qquad \theta=\theta_0+ 8 \pi^2 D \cos 2\psi,
\label{SUSYCONST}$$ where one chooses the constant values $\psi_1$ and $\psi_2$ for $z>0$ and $z<0$, respectively. The constraint (\[SUSYCONST\]) does not allow to simultaneously set $e_1=e_2$ and $\theta_1\neq \theta_2$, which corresponds to the case of $\theta$-ED, where supersymmetry is irrelevant. In other words, the limit $g=0$ in the electric and magnetic fields of the single dyon at $z=a$ (Eqs. (5.10) of Ref. [@Kim2]), which were calculated using the method of images, do not reproduce the corresponding fields obtained from our Eqs. (\[G00z>\]), (\[G01\]) and (\[G02\]). Also, the transmitted an reflected fields of massless waves propagating across the interface reported in Ref. [@Kim2] do not correspond to those calculated for $\theta$-ED in Refs. [@Hehl; @ZH]. It is worth recalling that these couplings enter through the complexified paramenter $\tau=\theta/2\pi+ 4\pi i/g^2$, which is familiar in the study of the action of the group $SL(2, \mathbb{Z})$ on a topological insulator with nontrivial permitivity, permeability and $\theta$-angle [@Karch].
Force between a charge and a planar $\theta$-boundary {#force}
-----------------------------------------------------
In this section we formulate the interaction energy and the forces arising between external sources and a TI as represented by $\theta$-boundary with a planar symmetry. We use both, the GF matrix and the stress-energy tensor.
The interaction energy between a charge-current distribution and a topological insulator is $$E_{int}=\frac{1}{2}\int d\mathbf{x}\int d\mathbf{x}^{\prime }j^{\mu }\left(
\mathbf{x}\right) \left[ G_{\mu \nu }\left( \mathbf{x},\mathbf{x}^{\prime
}\right) -\eta _{\mu \nu }\mathcal{G}\left( \mathbf{x},\mathbf{x}%
^{\prime }\right) \right] j^{\nu }\left( \mathbf{x}^{\prime }\right) ,$$where $\mathcal{G}\left( \mathbf{x},\mathbf{x}^{\prime }\right) = 1 /
\vert \mathbf{x} - \mathbf{x} ^{\prime} \vert $ is the GF in vacuum. The first contribution represents the total energy of a charge-current distribution in the presence of the $\theta$-boundary, including mutual interactions. We evaluate this energy for the case considered in the previous subsection of a point-like electric charge at position $\mathbf{r}=b \hat{\mathbf{e}} _{z}$. Making use of Eq. (\[G00z>\]), the interaction energy is $$E_{int}=\frac{q^{2}}{2}\left[ G_{00}\left( \mathbf{r},\mathbf{r}\right) -%
\mathcal{G} \left( \mathbf{r},\mathbf{r}\right) \right] =-\frac{q^{2}}{2%
}\frac{\tilde{\theta}^{2}}{4+\tilde{\theta}^{2}}\frac{1}{2b} .$$ Our result implies that the force on the charge exerted by the $\theta$-boundary is $$\mathbf{F} = - \frac{\partial E_{int}}{\partial b} \hat{\mathbf{e}} _{z}=-%
\frac{q^{2}}{\left( 2b\right) ^{2}} \frac{\tilde{\theta}^{2}}{4+\tilde{\theta%
}^{2}} \hat{\mathbf{e}} _{z} , \label{force1}$$ noting that it is always attractive. This can be interpreted as the force between the charge $q$ and the image charge $-q\tilde{\theta}^{2}/(4+\tilde{\theta}^{2})$ according to Coulomb law.
The field point of view provides an alternative derivation of this result. To compute the force on the charge we calculate the normal component of the flow of momentum into the $\theta $-boundary. In terms of the stress-energy tensor this force is $$\mathbf{F}=-\hat{\mathbf{e}}_{z}\int_{\Sigma ^{+}}dST_{zz}\left( \Sigma
^{+}\right) , \label{INTST}$$where the integration is over the surface $\Sigma ^{+}$, just outside the $%
\theta $-interface, at $z=0^{+}$. The identification of the stress tensor in the case of $\theta $-electrodynamics proceeds along the standard lines of electrodynamics in a medium (see for example Ref.[@CED]), where we read the rate at which the electric field does work on the free charges $$\mathbf{J\cdot E}=-\mathbf{\nabla \cdot }\left( \frac{1}{4\pi }\mathbf{%
E\times H}\right) -\frac{1}{4 \pi}\left( \mathbf{E\cdot }\frac{\partial \mathbf{D%
}}{\partial t}+\mathbf{H\cdot }\frac{\partial \mathbf{B}}{\partial t}\right)
\label{ECONS}$$and the rate at which momentum is transferred to the charges$$\left( \rho \mathbf{E}+\mathbf{J\times B}\right) _{k}=-\frac{%
\partial }{\partial t}\left( \frac{1}{4\pi }\mathbf{D\times B}\right) _{k}-%
\frac{1}{4\pi }\left[ D_{i}\partial _{k}E_{i}-\partial _{i}\left(
D_{i}E_{k}\right) \right] -\frac{1}{4\pi }\left[ B_{i}\partial
_{k}H_{i}-\partial _{i}\left( B_{i}H_{k}\right) \right] . \label{MOMCONS}$$Using the constitutive relations in Eq. (\[CONST\_REL\]), we recognize from Eq. (\[ECONS\]) the energy flux $\mathbf{S}$ and the energy density $U$ as $$\mathbf{S=}\frac{1}{4\pi }\mathbf{E\times B,\;\;\;\;\;\;\;\;}U\mathbf{=}%
\frac{1}{8\pi }(\mathbf{E}^{2}+\mathbf{B}^{2}),$$while from Eq. (\[MOMCONS\]) we obtain the momentum density $\mathbf{G}$ and we identify the stress tensor $T_{ij}$ as, $$\mathbf{G}=\frac{1}{4\pi}\mathbf{E\times B,}\qquad T_{ij}=\frac{1}{8\pi }(%
\mathbf{E}^{2}+\mathbf{B}^{2})\delta _{ij}-\frac{1}{4\pi }%
(E_{i}E_{j}+B_{i}B_{j}).\mathbf{\;\;\;}$$Outside the free sources, the conservation equations reads $$\mathbf{\nabla \cdot S+}\frac{\partial U}{\partial t}=0,\;\;\;\;\;\;\;\;\;\;%
\frac{\partial G_{k}}{\partial t}+\partial _{i}T_{ik}=\frac{\alpha }{\pi }%
\left( E_{i}B_{i}\right) \partial _{k}\theta (z).$$In other words, the stress tensor has the same form as in vacuum, but, as expected, it is not conserved on the $\theta $-boundary because of the self-induced charge and current densities arising there.
Thus, the required expression for $T_{zz}\left( \Sigma ^{+}\right) \;$in Eq. (\[INTST\]) is the standard one $$T_{zz}=\frac{1}{8\pi }\left[ E_{\parallel }^{2}-E_{z}^{2}+B_{\parallel
}^{2}-B_{z}^{2}\right] ,$$where $E_{z}$ ($B_{z}$) denotes the electric (magnetic) field component normal to the surface and $E_{\parallel }$ ($B_{\parallel }$) is the component of the electric (magnetic) field parallel to the surface. According to our results in the previous section, the electric and magnetic fields for $z>0$ are $$\begin{aligned}
\mathbf{E}\left( \mathbf{x}\right) &=&q\frac{\mathbf{x}-\mathbf{r}}{|\mathbf{%
x}-\mathbf{r}|^{3}}-q\frac{\tilde{\theta}^{2}}{4+\tilde{\theta}^{2}}\frac{%
\mathbf{x}+\mathbf{r}}{|\mathbf{x}+\mathbf{r}|^{3}}, \label{Efield} \\
\mathbf{B}\left( \mathbf{x}\right) &=&\frac{2q\tilde{\theta}}{4+\tilde{\theta%
}^{2}}\frac{\mathbf{x}+\mathbf{r}}{|\mathbf{x}+\mathbf{r}|^{3}}.
\label{Bfield}\end{aligned}$$ Thus we find $$\mathbf{F}=\frac{1}{4}\frac{q^{2}}{(4+\tilde{\theta}^{2})^{2}}\hat{\mathbf{e}%
}_{z}\int_{0}^{\infty }dR\frac{R}{\left( R^{2}+b^{2}\right) ^{3}}\left[
16R^{2}-(4+2\tilde{\theta}^{2})^{2}b^{2}+4\tilde{\theta}^{2}\left(
R^{2}-b^{2}\right) \right] =-\frac{q^{2}}{\left( 2b\right) ^{2}}\frac{\tilde{%
\theta}^{2}}{4+\tilde{\theta}^{2}}\hat{\mathbf{e}}_{z},$$in agreement with Eq. (\[force1\]).
Infinitely straight current-carrying wire near a planar $\theta$-boundary {#infinite_wire}
-------------------------------------------------------------------------
Let us consider now an infinitely straight wire parallel to the $x$ axis and carrying a current $I$ in the $+x$ direction. The wire is located in vacuum ($\theta _{2} = 0$) at a distance $b$ from an infinite topological medium with $\theta _{1} \neq 0$ in the region $z<0$. For simplicity we choose the coordinates such that $y^{\prime }=0$. Therefore the current density is $ j^\mu \left( \mathbf{x}^{\prime }\right) = I \eta ^{\mu} _{\; 1} \delta
\left( y^{\prime }\right) \delta \left( z^{\prime }-b\right)$.
The solution for this problem is given by $$A^{\mu }\left( \mathbf{x}\right) =I\int_{-\infty }^{+\infty }G_{\;1}^{\mu
}\left( \mathbf{x},\mathbf{r}\right) dx^{\prime } , \label{GenSolCurrent}$$where $\textbf{x} - \mathbf{r}= \left( x - x^{\prime} \right) \hat{\mathbf{e}} _{x} + y
\hat{\mathbf{e}} _{y} + \left( |z|+b\right) \hat{\mathbf{e}} _{z}$. Clearly the nonzero component $A^{0}\left( \mathbf{x}\right) $ arising from the GF implies that an electric field is induced. The required component of the GF, $G_{\;1}^{0}$, defined in Eq. (\[G0i\]) is given by $$G_{\;1}^{0}\left( \mathbf{x},\mathbf{r}\right) =-\frac{2\tilde{\theta}}{4+%
\tilde{\theta}^{2}}\frac{y}{R^{2}}\left[ 1-\frac{|z|+b}{\sqrt{R^{2}+\left(
|z|+b\right) ^{2}}}\right] . \label{01GreenCurrent}$$Substituting Eq. (\[01GreenCurrent\]) in Eq. (\[GenSolCurrent\]) yields the electric potential, which lacks an immediate interpretation. We can directly compute the electric field as $\mathbf{E}%
\left( \mathbf{x}\right) = - \nabla A ^{0} \left( \mathbf{x} \right)$, with the result $$\mathbf{E}\left( \mathbf{x}\right) =\frac{4\tilde{\theta}I}{4+\tilde{\theta}%
^{2}}\left[ \frac{|z|+b}{y^{2}+\left( |z|+b\right) ^{2}}\mathbf{\hat{e}}_{y}-%
\frac{y\;\mbox{sign}\left( z\right) }{y^{2}+\left( |z|+b\right) ^{2}}\mathbf{%
\hat{e}}_{z}\right].$$ We observe that the electric field for $z > 0$ is that due to a magnetic current located at $z = - b$, $\textbf{j} _{m,>} = - 4\tilde{\theta} I / (4+\tilde{\theta}^{2}) \hat{\textbf{e}} _{x}$. For $z < 0$ we have a magnetic current located at $z=b$ of the same strength $\textbf{j} _{m,<} = - \textbf{j} _{m,>}$. Note that $\textbf{j} _{m,>}$ is antiparallel to the current of the wire, while $\textbf{j} _{m,<}$ is parallel.
Similarly we compute the magnetic field. This is $$\mathbf{B}\left( \mathbf{x}\right) =\nabla \times \left[ I\hat{\mathbf{e}}
_{i} \int_{-\infty }^{+\infty } G_{\;1}^{i} \left( \mathbf{x},\mathbf{r}%
\right) dx^{\prime }\right],$$with $i=1,2$, where the corresponding GF are given by Eqs. (\[Gij\]). The result is $$\mathbf{B}\left( \mathbf{x}\right) = 2 I \mbox{sign}\left( z \right) \left[ -%
\frac{|b-z|}{y^{2}+\left( b-z\right) ^{2}}+\frac{\tilde{\theta}^{2}}{4+%
\tilde{\theta} ^{2}}\frac{\left( |z|+b\right) }{y^{2}+\left( |z|+b\right)
^{2}} \right] \mathbf{\hat{e}}_{y}+ 2Iy \left[ \frac{1}{y^{2}+\left(
b-z\right) ^{2}}- \frac{\tilde{\theta}^{2}}{4+\tilde{\theta}^{2}}\frac{1}{%
y^{2}+\left( |z|+b\right) ^{2}}\right] \mathbf{\hat{e}}_{z}.$$ For $z > 0$ the magnetic field corresponds to the one produced by an image electric current located at $z = - b$, flowing in the opposite direction to the current of the wire, $\textbf{j} _{e,>} = - 2\tilde{\theta} I / (4+\tilde{\theta}^{2}) \hat{\textbf{e}} _{x}$. For $z < 0$ we have an electric current located at $z=b$ of the same strength and flowing in the same direction of the current in the wire.
Summary and outlook {#summary}
===================
Classical electrodynamics is a fascinating field theory on which a plethora of technological devices rely. Advances in our theoretical understanding ignite new technological developments and sometimes new discoveries demand extending the limits of theories that lead to them. Chern-Simons forms and topologically ordered materials are a good example of the above. In this work we study a particular kind of Chern-Simons extension to electrodynamics that consists of Maxwell Lagrangian supplemented by a parity-violating Pontryagin invariant coupled to a scalar field $\theta$, restricted to the case where $\theta$ is piecewise constant in different regions of space separated by a common interface $\Sigma$.
It is well known that in this scenario the field equations in the bulk remain the standard Maxwell equations but the discontinuity of $\theta$ alters the behavior of the fields at the interface $\Sigma$, giving rise to effects such as: induced effective charge and currents at $\Sigma$ that are determined by the fields at the interface, electric charges near a planar $%
\theta$-boundary induce magnetic mirror monopoles (and vice versa) and nontrivial additional Faraday- and Kerr-like rotation of the plane of polarization of electromagnetic waves traversing the interface $\Sigma$.
Here we focus on the Green’s function method applied for the static case in $\theta$-electrodynamics. The method is illustrated by the case of a planar $\theta$-interface, where the corresponding Green’s function is calculated. The integral equation which defines the Green’s function becomes an algebraic equation due to the delta interaction arising in the $\theta$-boundary plus the symmetries present in the parallel directions. We show how to compute the electromagnetic fields, on either side of the interface from the Green’s function. Next we compute the force between a point-like charge and a topological insulator. To this end we use the Green’s function to compute the interaction energy between a charge-current distribution and a $\theta$-boundary that mimics the topological insulator, with vacuum energy removed. It can be shown that the above leads to the same interaction force as that computed by momentum flux perpendicular to the interface, for which the energy-momentum tensor and ensuing conservation laws of $\theta$-electrodynamics were analyzed. Finally, we use the Green’s function to obtain the electromagnetic fields for an infinitely straight current-carrying wire parallel to the interface.
For the case of the point-like charge in front of the $\theta$-interface, our results allow us to interpret the fields as those produced by the charge, its image, an induced magnetic monopole, and a circulating current density at the interface, in agreement with previously existing results. Similarly the fields produced by the infinitely straight current-carrying wire and the $\theta$-boundary can be interpreted in terms of electric and magnetic current densities.
Let us emphasize that for or a given $\theta$-boundary, the fields produced by arbitrary external sources can be calculated once the Green’s function is known. Our method can be applied to a broader kind of geometries determined by the $\theta$-boundary. In fact, we can provide the Green’s function for the spherical and the cylindrical cases [@AMU_LARGO]. Given that our results depend on $\tilde{\theta} = \alpha (\theta _{1} - \theta _{2}) / \pi$, it is worth mentioning that they satisfy the quantum-mechanical periodicity condition $\theta \rightarrow \theta + 2 \pi n$, with $\theta=0, \pi$.
The Green’s function method should also be useful for the extension to the dynamic case. In this respect, to our knowledge, little efforts have been done in the context of topological insulators. Furthermore, Green’s functions are also relevant for the computation of other effects, such as the Casimir effect. Therefore, we expect our method and results will be of considerable relevance and that they may constitute the basis for numerous other researches.
LFU acknowledges J. Zanelli for introducing him to the $\theta$-theories. LFU also thanks Alberto Güijosa for useful comments and suggestions. M. Cambiaso has been supported in part by the project FONDECYT (Chile) Initiation into Research Grant No. 11121633 and also wants to thank the kind hospitality at Instituto de Ciencias Nucleares, UNAM. LFU has been supported in part by the project No. IN104815 from Dirección General Asuntos del Personal Académico (Universidad Nacional Autónoma de México) and the project CONACyT (México) \# 237503. LFU and AMR thank the warm hospitality at Universidad Andres Bello.
GF for planar configuration in coordinate representation {#IntegralsStaticCase}
========================================================
Here we derive Eqs. (\[G00\]-\[Gij\]) by computing explicitly the Fourier transform of the reduced GF, whose formula we take from (\[GenSolGreenPlaneConf\]). In the standard case ($\tilde{\theta}=0$), the reduced vacuum GF is [@CED] $$\mathfrak{g}\left( z,z^{\prime }\right) =\frac{1}{2p}e^{-p|z-z^{\prime }|}.
\label{gFree}$$ In coordinate representation, the corresponding GF is obtained by Fourier transforming (\[gFree\]) as defined in Eq. (\[RedGreenDef\]), $$\mathcal{G}\left( \mathbf{x},\mathbf{x}^{\prime }\right) =4\pi \int \frac{%
d^{2}\mathbf{p}}{\left( 2\pi \right) ^{2}}e^{i\mathbf{p}\cdot \left( \mathbf{%
x}-\mathbf{x}^{\prime }\right) _{\parallel }}\frac{1}{2p}e^{-p|z-z^{\prime }|}.
\label{FTgFree}$$This double integral becomes easier to perform if we express the area element in polar coordinates, $d^{2}\mathbf{p}%
=pdpd\varphi $ (instead of the Cartesian ones), and choose the $p_{x}$-axis in the direction of the vector $\mathbf{R}=\left( \mathbf{x}-\mathbf{x}%
^{\prime }\right) _{\parallel }$, as shown in Fig. \[planeint\]. Noting that $\mathbf{p}\cdot \mathbf{R}=pR\cos \varphi $, we can write $$\mathcal{G}\left( \mathbf{x},\mathbf{x}^{\prime }\right) =\int_{0}^{\infty
}dpe^{-p|z-z^{\prime }|}\left\{ \frac{1}{2\pi }\int_{0}^{2\pi }e^{ipR\cos
\varphi }d\varphi \right\} . \label{FTgFree2}$$where $R=|\left( \mathbf{x}-\mathbf{x}^{\prime }\right) _{\parallel }|$. The braces in this equation enclose an integral representation of the Bessel function $J_{0}\left( pR\right) $. The resulting integral, $$\mathcal{G}\left( \mathbf{x},\mathbf{x}^{\prime }\right) =\int_{0}^{\infty
}J_{0}\left( pR\right) e^{-p|z-z^{\prime }|}dp, \label{FTgFree3}$$is well-known, see for example Ref. [@Gradshteyn]. The final result is $$\mathcal{G}\left( \mathbf{x},\mathbf{x}^{\prime }\right) =\frac{1}{\sqrt{%
R^{2}+|z-z^{\prime }|^{2}}}=\frac{1}{|\mathbf{x}-\mathbf{x}^{\prime }|},
\label{FTgFree4}$$which is the vacuum GF in coordinate representation [@CED].
In the following we use a similar procedure to compute the required integrals for establishing Eqs. (\[G00\]-\[G0i\]). We first consider the component$\;G_{\;0}^{0}$. From Eq. ([GenSolGreenPlaneConf]{}) we find $$g_{\;0}^{0}\left( z,z^{\prime }\right) =\mathfrak{g}\left( z,z^{\prime
}\right) +A\left( z,z^{\prime }\right) p ^{2}\tilde{\theta}\mathfrak{g}\left(
a,a\right) , \label{staticg00}$$where the function $A\left( z,z^{\prime }\right) $ is $$A\left( z,z^{\prime }\right) =-\frac{\tilde{\theta}}{4+\tilde{\theta}^{2}}%
p ^{-2}e^{-pZ}, \label{A-static}$$with the notation $Z=|z-a|+|z^{\prime }-a|$. In this way, the component $%
G_{\;0}^{0}$ is given by $$G_{\;0}^{0}\left( \mathbf{x},\mathbf{x}^{\prime }\right) =\mathcal{G}\left(
\mathbf{x},\mathbf{x}^{\prime }\right) -\frac{\tilde{\theta}^{2}}{4+\tilde{%
\theta}^{2}}\int_{0}^{\infty }J_{0}\left( pR\right) e^{-pZ}dp,
\label{Fourier00}$$in coordinate representation. As before we use the integral representation of the Bessel function $J_{0}\left( pR\right) $ to perform the angular integration. The resulting integral is the same as in (\[FTgFree3\]), thus we obtain $$G_{\;0}^{0}\left( \mathbf{x},\mathbf{x}^{\prime }\right) =\frac{1}{|\mathbf{x%
}-\mathbf{x}^{\prime }|}-\frac{\tilde{\theta}^{2}}{4+\tilde{\theta}^{2}}%
\frac{1}{\sqrt{R^{2}+Z^{2}}}. \label{G00Coordinates}$$
Now we evaluate the components $G_{\;1}^{0}$ and $G_{\;2}^{0}$. The corresponding reduced GF are $$g_{\; i}^{0}\left( z,z^{\prime }\right) = - i \epsilon _{0ij3} p ^{j} A\left( z,z^{\prime }\right) , \label{staticg01-g02}$$ with $A\left( z,z^{\prime }\right) $ given by (\[A-static\]). For convenience we define the vector $$\mathbf{I} \left( \mathbf{x},\mathbf{x}^{\prime }\right) =(I ^{1} , I ^{2}) = 4\pi \int \frac{%
d^{2}\mathbf{p}}{\left( 2\pi \right) ^{2}}e^{i\mathbf{p}\cdot \left( \mathbf{%
x}-\mathbf{x}^{\prime }\right) _{\parallel }}\mathbf{p\;}%
p^{-2}e^{-pZ}, \label{vectorI}$$with $\mathbf{p}=\left( p_{x},p_{y}\right) $, in terms of which we have$$G_{\; i}^{0}\left( \mathbf{x},\mathbf{x}^{\prime }\right) =i \frac{\tilde{\theta}}{4+\tilde{\theta}^{2}} \epsilon _{0ij3} I ^{j} \left( \mathbf{x},\mathbf{x}^{\prime }\right) .$$ We calculate the integral (\[vectorI\]) in the same coordinate system as before (see Fig. \[planeint\]), and then we rewrite the result in a vector form. The integral can be written as $$\mathbf{I}_{p}\left( \mathbf{x},\mathbf{x}^{\prime }\right) = 2 \int_{0}^{\infty }dpe^{-pZ}\left\{ \frac{1}{2\pi }\int_{0}^{2\pi }\left[
\begin{array}{c}
\cos \varphi \\
\sin \varphi%
\end{array}%
\right] e^{ipR\cos \varphi }d\varphi \right\} , \label{vectorI2}$$where the subscript $p$ indicates that the vector$\;\mathbf{p}$ is written in the particular coordinate system of Fig. \[planeint\]. Both the required angular and radial integrals are well-known and the result is $$\mathbf{I}_{p}\left( \mathbf{x},\mathbf{x}^{\prime }\right) = 2i \widehat{\mathbf{R}}\int_{0}^{\infty }J_{1}\left( pR\right) e^{-pZ}dp=\frac{2 i}{R}\left( 1-\frac{Z}{\sqrt{R^{2}+Z^{2}}}\right) \widehat{\mathbf{R}}.
\label{vectorI3}$$As a consequence of the chosen coordinate system we find that $I_2=0$, in such a way that the vector $%
\mathbf{I}_{p}$ becomes parallel to $\widehat{\mathbf{R}}$. However this can be generalized in a direct way to an arbitrary coordinate system as $$\mathbf{I}\left( \mathbf{x},\mathbf{x}^{\prime }\right) = 2i \frac{\textbf{R}}{%
R^{2}}\left( 1-\frac{Z}{\sqrt{R^{2}+Z^{2}}}\right) .$$Thus we find $$\begin{aligned}
G_{\; i}^{0}\left( \mathbf{x},\mathbf{x}^{\prime }\right) = - \frac{2 \tilde{\theta}}{4+\tilde{\theta}^{2}} \frac{\epsilon _{0ij3} R ^{j}}{R ^{2}}\left( 1-\frac{Z}{\sqrt{R^{2}+Z^{2}}}\right) .\end{aligned}$$ In order to evaluate the components $G ^{i} _{\; j}$ we first observe that the corresponding reduced GF can be written as $$g ^{i} _{\; j} \left( z , z ^{\prime} \right) = \eta ^{i} _{\; j} g ^{0} _{\; 0} \left( z , z ^{\prime} \right) + \tilde{\theta} \mathfrak{g} (a,a) A \left( z , z ^{\prime} \right) p ^{i} p _{j} , \label{staticgij}$$ where $g ^{0} _{\; 0}$ is given by Eq. (\[staticg00\]). Now we need to compute the Fourier transformation of Eq. (\[staticgij\]) as defined in Eq. (\[RedGreenDef\]). However the first term was studied before and the result is given by Eq. (\[G00Coordinates\]), thus leading to study only the last term. To this end we introduce the vector $$\mathbf{K}\left( \mathbf{x},\mathbf{x}^{\prime }\right) = ( K ^{1} , K ^{2} ) = 4 \pi \int \frac{d^{2}\mathbf{p}}{\left( 2\pi \right) ^{2}}e^{i\mathbf{p}\cdot \left( \mathbf{x}-\mathbf{x}^{\prime }\right) _{\parallel }}\frac{\mathbf{p}}{p} p^{-2}e^{-pZ}, \label{Kvector1}$$ from which the required integral will be calculated by taking the spatial derivative. The integral (\[Kvector1\]) can be computed again in the particular coordinate system of the Fig. \[planeint\]. In the polar coordinates defined in the $\mathbf{p}$-plane the integral reads $$\mathbf{K}_{p}\left( \mathbf{x},\mathbf{x}^{\prime }\right)
=2\int_{0}^{\infty }\frac{dp}{p}e^{-pZ}\left\{ \frac{1}{2\pi }\int_{0}^{2\pi
}\left[
\begin{array}{c}
\cos \varphi \\
\sin \varphi%
\end{array}%
\right] e^{ipR\cos \varphi }d\varphi \right\} . \label{Kaux}$$ Note that the braces in this equation enclose an integral representation of the Bessel function $J_{1}\left( pR\right) $. The resulting integral is well-known and the final result is $$\mathbf{K}_{p}\left( \mathbf{x},\mathbf{x}^{\prime }\right)
=2 i \int_{0}^{\infty }\frac{dp}{p}J_{1}\left( pR\right) e^{-pZ}\hat{\mathbf{R}} = 2i\frac{\sqrt{%
R^{2}+Z^{2}}-Z}{R}\hat{\mathbf{R}}, \label{Kvector4}$$ where $\hat{\mathbf{R}}$ is the unit vector shown in Fig. \[planeint\]. The generalization to an arbitrary coordinate system is then $$\mathbf{K}\left( \mathbf{x},\mathbf{x}^{\prime }\right) =2i\frac{\sqrt{%
R^{2}+Z^{2}}-Z}{R^{2}} \textbf{R} . \label{Kvector5}$$ Note that the required integral involve the term $p ^{i} p _{j}$ which can be generated from (\[Kvector1\]) as follows $$\begin{aligned}
i \partial _{j} K ^{i } \left( \mathbf{x},\mathbf{x}^{\prime }\right) = 4 \pi \int \frac{d^{2} \mathbf{p}}{\left( 2\pi \right) ^{2}}e^{i\mathbf{p}\cdot \left( \mathbf{x}-\mathbf{x}^{\prime }\right) _{\parallel }}\frac{p ^{i} p _{j}}{p} p^{-2}e^{-pZ} .\end{aligned}$$By using the final form of $\mathbf{K}\left( \mathbf{x},\mathbf{x}^{\prime
}\right) $, given by Eq. (\[Kvector5\]), one can further check the consistency condition $ \partial _{1} K ^{2}\left( \mathbf{x},\mathbf{x} ^{\prime }\right) = \partial _{2} K ^{1}\left( \mathbf{x},\mathbf{x} ^{\prime }\right)$ required by the cross terms involving $p ^{1} p _{2} = p ^{2} p _{1} = - p _{x} p _{y}$. From the previous results, the $G ^{i} _{\; j}$ components of the GF matrix in coordinate representation can be written as $$\begin{aligned}
G _{\; j}^{ i }\left( \mathbf{x},\mathbf{x}^{\prime }\right) &=& \eta ^{i} _{\; j} G _{\; 0}^{ 0}\left( \mathbf{x},\mathbf{x}^{\prime }\right) - \frac{i}{2}\frac{\tilde{\theta}^{2}}{4+ \tilde{\theta}^{2}} \partial _{j} K ^{i} \left( \mathbf{x},\mathbf{x}^{\prime }\right) .\end{aligned}$$ These results establish Eqs. (\[Gij\]).
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---
abstract: 'We performed fluctuation analysis by means of the local scaling dimension for the strength function of the isoscalar (IS) giant quadrupole resonance (GQR) in $^{208}$Pb where the strength function is obtained by the shell model calculation including 1p1h and 2p2h configurations. It is found that at almost all energy scales, fluctuation of the strength function obeys the Gaussian orthogonal ensemble (GOE) random matrix theory limit. This is contrasted with the results for the GQR in $^{40}$Ca, where at the intermediate energy scale about 1.7 MeV a deviation from the GOE limit was detected. It is found that the physical origin for this different behavior of the local scaling dimension is ascribed to the difference in the properties of the damping process.'
author:
- Hirokazu Aiba
- Masayuki Matsuo
- Shigeru Nishizaki
- Toru Suzuki
title: 'Fluctuation properties of strength function associated with the giant quadrupole resonance in $^{208}$Pb'
---
Introduction {#sec:intro}
============
Giant resonances, excited by various probes, show, at an initial stage of the excitation process, a regular motion with a definite vibrational frequency [@speth; @harakeh]. These regular motions are then damped due to the coupling with a huge number of background states, and finally the so called compound states are realized.
We now have understood the both ends of these processes: The frequency of the giant resonance, for instance, can be well evaluated by the random phase approximation (RPA). Compound states, on the other hand, are also well described by the random matrix theory with the Gaussian orthogonal ensemble (GOE) [@dyson; @mehta], which characterizes a classical chaotic motion.
It is still not well understood, however, how the dynamics changes from regular to chaotic [@mottelson]. In order to answer this question, it is very useful to study the fluctuation properties of the strength functions: The structure at the large energy scale of the strength function corresponds to the behavior of the initial stage, while the fluctuation properties at small energy scale correspond to the long time behavior.
We proposed and have used a novel fluctuation analysis based on the quantity we call the local scaling dimension to study the fluctuation properties of the strength functions [@aiba]. This method is devised to quantitatively characterize the fluctuation at each energy scale, and is suitable for the investigation of the fine structure of the strength function.
The strength distribution of giant resonances and its fluctuation have also been studied experimentally. Recently, the fine structure of the strength distribution of the giant quadrupole resonance (GQR) in $^{208}$Pb [@shevchenko; @shevchenko2; @lacroix] or the Gamow-Teller resonance (GTR) in $^{90}$Zr [@kalmykov] were measured and theoretical analysis has also been done.
In the previous paper [@aiba2], we investigated the GQR in $^{40}$Ca, where the strength function was calculated by means of the second Tamm-Dancoff approximation (TDA), namely, the 1p1h and 2p2h model space is included. The results of the local scaling dimension analysis were as follows: At small energy scale, the behavior of the local scaling dimension is almost the same as that of the GOE, which exhibits the complexity of 2p2h background states. On the other hand, a clear deviation from the GOE was found at the intermediate energy scale and it was found that this energy corresponds to the spreading width of 1p1h states. Hence, we can say that the spreading width of 1p1h states is detected as deviation from the GOE limit in $^{40}$Ca.
For $^{40}$Ca the Landau damping is important for the damping process of the giant resonance. Namely, the strength is first fragmented over a wide range of 1p1h states, and this fragmentation characterizes a global profile of the total strength function.
However, as the mass of nuclei increases, the relative importance of the Landau damping may change. Accordingly, 2p2h states may also contribute to the global profile of the strength function. Therefore, it is very important to investigate how the difference between the damping process of light nuclei and that of heavy nuclei does affect the properties of the fluctuation of the strength function.
In this paper, we study the isoscalar (IS) GQR of $^{208}$Pb, where the strength function is calculated with the second TDA in the same manner as in $^{40}$Ca, and study the fluctuation of the strength function by means of the local scaling dimension. Comparing results with those of $^{40}$Ca we would like to clarify which properties of the damping process are reflected in the fluctuation of the strength function and make clear the physical origin of the difference.
This paper is organized as follows: In Sec. \[sec:lsd\], we briefly explain the local scaling dimension. The strength function for IS GQR in $^{208}$Pb is calculated in Sec. \[sec:numerical\], where the adopted Hamiltonian and the model space are shown. In Sec. \[sec:measures\], we discuss the nearest-neighbor level spacing distribution, $\Delta_3$ statistics as well as a histogram of the strength distribution. In Sec. \[sec:results\], we apply the local scaling dimension to the IS GQR strength function in $^{208}$Pb. Detail of damping process is studied in Sec. \[sec:damping\], where the physical origin for the difference of the fluctuation property of the strength function between $^{40}$Ca and $^{208}$Pb is also discussed. Finally, Sec. \[sec:conclusion\] is devoted to conclusion.
Local Scaling Dimension {#sec:lsd}
=======================
We briefly explain the local scaling dimension. See Refs. [@aiba; @aiba2] for details.
The strength function is expressed as [@Bohr-Mottelson2] $$S(E)=\sum_i S_i\delta(E-E_i+E_0).
\label{defstr}$$ Here $E_i$ and $S_i$ denote the energy and the strength of exciting the $i$th energy level, respectively. Strengths are normalized as $\sum_i S_i=1$.
To study the fluctuation at each energy scale, we consider binned distribution of the strength by dividing whole energy interval under consideration into $N$ bins with length $\epsilon$. Strength contained in $n$th bin is denoted by $p_n$, $$p_n\equiv\sum_{i\in n{\rm th~ bin}}S_i.
\label{defp}$$ To characterize the distribution of the binned strengths, we introduce the moments of $p_n$, which are called in literature the partition function $\chi_m(\epsilon)$ defined by $$\chi_m(\epsilon)\equiv \sum_{n=1}^N p_n^m \\
=N\langle p_n^m\rangle.
\label{partition}$$ Finally, by extending the idea of the generalized fractal dimensions [@hentschel; @halsey] to non-scaling cases in a straightforward way, we can define the local scaling dimension as, $$D_m(\epsilon)\equiv \frac{1}{m-1}
\frac{\partial\log\chi_m(\epsilon)}
{\partial\log\epsilon}.
\label{scaledim}$$ Since the local scaling dimension has a definite physical meaning similar to that of the generalized fractal dimension, the value of $D_m(\epsilon)$ can quantitatively characterize the fluctuation of the strength function at each energy scale $\epsilon$.
In the actual calculation of the local scaling dimension, we define it by means of the finite difference under the change of a factor 2, $$D_m(\sqrt{2}\epsilon) = \frac{1}{m-1}\frac{
\log\chi_m(2\epsilon)-
\log\chi_m(\epsilon)}
{\log 2},
\label{approscaledim}$$ rather than the derivative in Eq. (\[scaledim\]).
Numerical Calculation of Strength Function {#sec:numerical}
==========================================
We calculated the strength function of the IS GQR in $^{208}$Pb within the second TDA including the 1p1h and 2p2h excitations. Single-particle wave-functions and energies were obtained for a Woods-Saxon potential including the Coulomb interaction. The effective mass parameter $m^*/m$, which scales the Woods-Saxon single-particle energies $\varepsilon_{\rm WS}$ as $\varepsilon_{\rm HF}=\varepsilon_{\rm WS}/(m^*/m)$ to simulate the bare (Hartree-Fock) single-particle energies $\varepsilon_{\rm HF}$, is set to be 1 in this calculation.
As the residual interaction, the Landau-Migdal-type interaction [@Schwe] including the density-dependence was adopted. The model space was constructed in terms of single-particle states within the four major shells, two below and two above the Fermi surface, and included all 1p1h states and 2p2h states whose unperturbed energies are less than 15MeV. Resultant number of 1p1h states and 2p2h states are 39 and 8032, respectively. We diagonalized the Hamiltonian within this model space and obtained the strength function for the isoscalar quadrupole operator.
![ Calculated strength function of the IS GQR in$^{208}$Pb. Dotted curve shows the smooth strength function by means of the Strutinsky method with the smoothing width 0.2 MeV. []{data-label="fig_strfun"}](fig1.eps){width="6cm"}
Figure \[fig\_strfun\] shows the calculated strength function. The average of the excitation energy weighted by the strength is about 10.5 MeV, and the standard deviation around the average is about 2.6 MeV, where all levels are considered. The peak position lies at the same value as the average. These values are consistent with the (p,p’) experimental data [@shevchenko]. Moreover, the agreement of the global shape with the experimental data is also good. The dotted curve in Fig. \[fig\_strfun\] represents the smooth strength function by means of the Strutinsky method [@ring-schuck] with the smoothing width 0.2 MeV. The value of the FWHM of this smooth strength function is 0.63 MeV. In order to quantitatively characterize the spreading of the strength function around the largest peak, the FWHM is more appropriate than the standard deviation [@bertsch2]. Thus, we use the FWHM as a measure of the total width $\Gamma$ of the strength function, which gives $\Gamma=0.63$ MeV.
Hereafter, when we estimate the value of the FWHM, the same procedure as above is adopted, namely, we calculate the FWHM for the smooth strength function by means of the Strutinsky method with the smoothing width 0.2 MeV.
Fluctuation at small scale {#sec:measures}
==========================
![ The nearest-neighbor level spacing distribution for (a) $^{40}$Ca and (b) $^{208}$Pb. For $^{208}$Pb 3321 levels between 9.9 MeV and 13.1 MeV, while for $^{40}$Ca 804 levels between 20 MeV and 30 MeV are considered. Level spacings were unfolded by the Strutinsky method with a smoothing width 0.5 MeV for $^{208}$Pb and 5.0 MeV for $^{40}$Ca, respectively. The solid curve represents the Wigner distribution. []{data-label="fig_nns"}](fig2.eps){width="9.2cm"}
Before going to the detailed discussion of the local scaling dimension, we briefly show the results for other fluctuation measures: the nearest-neighbor level spacing distribution (NND), the strength distribution, and $\Delta_3$ statistics. Here, the NND and the strength distribution are measures characterizing the fluctuation at small energy scale limit. We present the results of $^{40}$Ca as well as those of $^{208}$Pb for the sake of comparison.
Figure \[fig\_nns\] shows the NND. For both nuclei the NND follows the Wigner distribution well. We present the strength distribution in Fig. \[fig\_strdis\] where a histogram of the square-root of normalized strengths is plotted. We also find that for both $^{208}$Pb and $^{40}$Ca the distribution follows the Porter-Thomas one rather well. These two figures indicate that for both nuclei the fluctuation of the strength as well as that of the energy level spacing is governed by the GOE at least at small energy scale limit as expected.
Figure \[fig\_delta3\] shows the $\Delta_3$ statistics. We again find that at small energy range the $\Delta_3$ follows the GOE line for both $^{208}$Pb and $^{40}$Ca, although at intermediate energy scales, $L_{\rm max}\simeq 20$ or 15 for $^{208}$Pb or $^{40}$Ca, respectively, the $\Delta_3$ starts to deviate from the GOE line to upward.
![ The histogram of the square-root of normalized strengths $\bar{S}_i^{1/2}$ associated with IS GQR in (a) $^{40}$Ca and (b) $^{208}$Pb. The solid curve represents the Porter-Thomas distribution which becomes a Gaussian when plotted as a function of $\bar{S}_i^{1/2}$. See the caption of Fig. \[fig\_nns\] for the number of considered levels and also see Sec. \[sec:cal\_lsd\] for the normalization of the strengths. []{data-label="fig_strdis"}](fig3.eps){width="9.2cm"}
![ The $\Delta_3$ statistics for (a) $^{40}$Ca and (b) $^{208}$Pb. The horizontal axis $L$ shows the value of the energy interval for the unfolded spectrum. The solid curve represents the $\Delta_3$ for the GOE level fluctuation. See Fig. \[fig\_nns\] for other parameters. []{data-label="fig_delta3"}](fig4.eps){width="9.2cm"}
Results of local scaling dimension {#sec:results}
==================================
Calculation of the local scaling dimension {#sec:cal_lsd}
------------------------------------------
Since we are not interested in the global shape of the strength function, we actually adopt the normalized strength function $\bar{S}(E)$ for the fluctuation analysis as in the case of $^{40}$Ca [@aiba2]. The normalized strength function $\bar{S}(E)$ is given by $${\bar S}(E)=\sum_i{\bar S}_i\delta(E-{\bar E}_i+{\bar E}_0),
\label{eq_norstrfun}$$ where the normalized strength $\bar{S}_i $ of the $i$th level is defined by $$\bar{S}_i \equiv{\cal N}\frac{S_i\tilde{\rho}(E_i)}{
\tilde{S}(E_i)}.
\label{eq_norstr}$$ Here, $\tilde{\rho}(E)$ and $\tilde{S}(E)$ denote the level density and the strength function, respectively, smoothed by the Strutinsky method [@ring-schuck]. ${\cal N}$ is a normalization factor to guarantee $\sum_i\bar{S}_i=1$.
We determine the width parameter $\omega$ of the Strutinsky smoothing function as follows: We note that the smoothed strength function $\tilde{S}(E)$ should represent the global profile of the original strength function $S(E)$ at large energy scale, but at the same time, we would like to choose $\omega$ as large as possible since we do not want to wash out the fluctuations at smaller energy scales. Figure \[fig\_fwhm\] shows the FWHM of the smoothed strength function $\tilde{S}(E)/\tilde{\rho}(E)$ as the function of the smoothing width $\omega$. The linear increase of the FWHM at large values of $\omega \agt 0.6$ MeV indicates that the value of $\omega$ is too large, while with smaller values $\omega \alt 0.5$ MeV the FWHM stays at an approximately constant value, reflecting the total width. We therefore adopt 0.5 MeV as the value of the smoothing width $\omega$ in order to satisfy the above requirements.
We use the equidistant energy level ${\bar E}_i$ in Eq. (\[eq\_norstrfun\]), namely, ${\bar E}_i=id$, where $d$ denotes the average level spacing. Finally, we adopted the energy range from 9.9 MeV to 13.1 MeV, where 3321 levels are included.
![ FWHM of the smoothed strength function $\tilde{S}(E)/\tilde{\rho}(E)$ of IS GQR in $^{208}$Pb as a function of smoothing width $\omega$ used in the Strutinsky method. The dotted line is fitted to data and gives $\sim 0.98\omega+0.2$. []{data-label="fig_fwhm"}](fig5.eps){width="6cm"}
![ Normalized strength function Eq. (\[eq\_norstrfun\]) of IS GQR in$^{208}$Pb. Smoothing width $\omega=0.5$ MeV was used. []{data-label="fig_norstrfun"}](fig6.eps){width="6cm"}
The normalized strength function is plotted in Fig. \[fig\_norstrfun\]. The local scaling dimension is derived from this normalized strength function.
Behavior of the local scaling dimension
---------------------------------------
![ Partition function (a) and local scaling dimension (b) for the IS GQR in $^{208}$Pb, and those in $^{40}$Ca are also shown at (c) and (d). Curves in each figure correspond to $m=2$ - 5 from upper to lower. Dotted curves in (b) and (d) represent $D_2(\epsilon)$ for the GOE. []{data-label="fig_pat_ldim"}](fig7.eps){width="9.2cm"}
Figure \[fig\_pat\_ldim\] (a) and (b) represent the partition function and the local scaling dimension, respectively, of IS GQR in $^{208}$Pb. The horizontal axes in both figures represent the bin width $\epsilon$ of energy in unit of $d$, where $d$ represents the average level spacing over the energy range 9.9 - 13.1 MeV ($d=0.96$ keV). The partition function clearly deviates from the linear relation in the log-log plot. This means that for the GQR strength function the self-similar property does not hold. We can also see a more detailed structure in the figure of the local scaling dimension. At the smallest energy scale $\epsilon\simeq d$, the value of the local scaling dimension is small, $D_2\simeq 0.35$, which means that the fluctuation is very large at small energy scales. As the energy scale or the bin width increases, the values of $D_m(\epsilon)$ monotonically increase. Finally, at about $\epsilon\simeq 100d$ the values of $D_m(\epsilon)$ converges to unity, which indicates that at large energy scales, the strength function appears smooth. The most important feature in Fig. \[fig\_pat\_ldim\] (b) is that the local scaling dimension for $^{208}$Pb almost follows the GOE line at almost all the energy scales.
This should be contrasted with the case of $^{40}$Ca [@aiba2]: The partition function and the local scaling dimension for $^{40}$Ca are shown in Fig. \[fig\_pat\_ldim\] (c) and (d), respectively, for a comparison. When the energy scale is small, the local scaling dimension almost follows the GOE line. As the energy scale increases, however, we can find a dip and a deviation from the GOE line at about 1.7 MeV (Note that $d=12$ keV for $^{40}$Ca). We verified that an occurrence of the dip is not due to a statistical error. Moreover, further studies indicate that the energy where the minimum is located is approximately related to the value of the spreading width of 1p1h states.
Note that if we look only at the small energy scale limit or large energy scale limit, we can not find the difference between $^{208}$Pb and $^{40}$Ca. Studies of fluctuation at intermediate energy scales lead to the finding of the difference. In the following we shall investigate the mechanism which brings about the difference in fluctuations at intermediate energy scales.
Studies of damping process {#sec:damping}
==========================
Let us now investigate origins of the difference between the cases of $^{40}$Ca and $^{208}$Pb. In our previous study of the GQR in $^{40}$Ca, we have shown that the behavior of the local scaling dimension, shown in Fig. \[fig\_pat\_ldim\] (d), can be interpreted in terms of the doorway damping mechanism. We here employ the same picture in order to clarify the damping mechanism of the GQR in $^{208}$Pb.
The doorway damping mechanism consists of a two-step process which is illustrated in Fig. \[fig\_pic\_Ca\]. The giant resonance is spread over the 1p1h states due to the Landau damping, the width of which is denoted by $\Gamma_{\rm L}$. The average spacing of 1p1h states is denoted by $D_{\rm 1p1h}$. The 1p1h states are considered here as the “doorway" states of the damping process. The 1p1h states then couple to more complicated background states (2p2h states) through the residual two-body interaction. The coupling causes the spreading width of 1p1h states, which we denote $\gamma_{12}$. We define the GQR TD state as the Tamm-Dancoff (TD) state with the largest quadrupole strength among all TD states, where the TD states mean the states obtained in the TDA, i.e., by the diagonalization within the model space limited to the 1p1h configurations. The GQR TD state also couples to 2p2h states, and hence it should have the spreading width due to the coupling. This is similar to $\gamma_{12}$, but we introduce a separate symbol $\Gamma_2$ since the GQR TD state is a special state consisting of a coherent superposition of many unperturbed 1p1h excitations. $d_{\rm 2p2h}$ is the average spacing of background 2p2h states. The residual interaction also acts among the 2p2h states, and the mixing among the 2p2h states causes a spreading width of the 2p2h states, which we denote $\gamma_{22}$.
In the following we shall evaluate all these quantities in order to clarify the damping mechanism of the GQR in $^{208}$Pb (Sec. \[sec:mechanism\] and Sec. \[sec:spreading\_width\] ). We also study whether there are specific states among 2p2h states which strongly couple with the GQR mode (Sec. \[sec:surfacce\_vib\]) and then discuss the difference of the nature associated with the fluctuation of strength function between $^{40}$Ca and $^{208}$Pb (Sec \[sec:comparison\]).
![ Schematic drawing of the doorway damping mechanism of the giant resonance, and related quantities. []{data-label="fig_pic_Ca"}](fig8.eps){width="7cm"}
Mechanism producing the total width {#sec:mechanism}
-----------------------------------
### Landau damping
For $^{40}$Ca, the Landau damping is important, so that the strengths are already fragmented in the 1p1h levels. Therefore we first would like to investigate in $^{208}$Pb, how the strength is distributed in the TDA where only the 1p1h states are included.
![ TDA strength function for the IS quadrupole operator in $^{208}$Pb. See Fig. \[fig\_strfun\] for the dotted curve. []{data-label="fig_tdastr"}](fig9.eps){width="6cm"}
Figure \[fig\_tdastr\] shows the TDA strength function, which is obtained by means of the TDA, namely by neglecting 2p2h states, of the IS quadrupole operator. Different from the case of $^{40}$Ca, strengths in the GQR region is considerably concentrated on the single peak located at about 10.7 MeV. Because of this, the TDA strength function is very different from the full strength function in Fig. \[fig\_strfun\]. At the same time, we also see only a small effect of the Landau damping. In fact, the strength concentration on the single peak at $E=10.7$ MeV is 59% of the strengths in the energy interval 9 - 13 MeV. The Landau damping width $\Gamma_{\rm L}$ may be evaluated in terms of a smoothed profile of the strength function plotted with the dotted curve in Fig. \[fig\_tdastr\]. Its FWHM reads 0.21 MeV. On the other hand, if we closely look at Fig. \[fig\_tdastr\], we find that there is the second largest peak just below the largest one and that these two levels dominate the whole structure. The level spacing between these two levels can be considered as a typical spreading of strength and may be a more direct quantitative measure of the Landau damping width $\Gamma_{\rm L}$: The level spacing 0.18 MeV gives $\Gamma_{\rm L}=0.18$ MeV.
### damping due to 2p2h states
The Landau damping width $\Gamma_{\rm L}=0.18$ MeV is not enough to explain the total width $\Gamma=0.63$ MeV of Sec. \[sec:numerical\]. Then, we would like to study a role of 2p2h states in the damping process, namely, the fragmentation of the GQR TD state located at $E=10.7$ MeV in Fig. \[fig\_tdastr\] over 2p2h states. We shall investigate the damping width $\Gamma_2$ caused by the coupling to 2p2h states. To estimate this width, we perform a calculation where we include only the GQR TD state and 2p2h states, where the coupling between the GQR TD state and 2p2h states as well as the interaction among 2p2h states are taken into account.
![ Strength function by neglecting all TD states except the GQR TD state. 3342 2p2h states lying in 9 MeV - 13 MeV are considered. See Fig. \[fig\_strfun\] for the dotted curve. []{data-label="fig_GQRTDstr"}](fig10.eps){width="6cm"}
Figure \[fig\_GQRTDstr\] shows the resulting strength function. The estimated FWHM is 0.41 MeV, i.e., $\Gamma_2=0.41$ MeV.
If the Landau damping and the 2p2h damping are independent of each other, and neighboring TD states around the GQR TD states also have the same spreading width as $\Gamma_2$, the following approximate relation holds: $$\Gamma\simeq\Gamma_{\rm L}+
\Gamma_2.
\label{Gamma}$$ The values, $\Gamma_{\rm L}=0.18$ MeV and $\Gamma_2=0.41$ MeV, estimated above indeed satisfy this relation. Consequently, the total width $\Gamma=0.63$ MeV is approximately explained as a sum of the Landau damping width $\Gamma_{\rm L} $ and the 2p2h damping width $\Gamma_2$.
The importance of the 2p2h damping is contrasted with the case of $^{40}$Ca, where the total width can be explained essentially by the Landau damping width, i.e., $\Gamma\simeq \Gamma_\text{L}$.
Spreading width of 1p1h states and 2p2h states {#sec:spreading_width}
----------------------------------------------
For the case of $^{40}$Ca, the strength is fragmented over many 1p1h states by the Landau damping, and strength in each 1p1h state is further spread due to the coupling with 2p2h states. Let us evaluate the spreading width $\gamma_{12}$ of the 1p1h states due to this coupling. We shall also evaluate the spreading width $\gamma_{22}$ of 2p2h states, which is caused by the residual coupling among 2p2h states.
![ Averaged strength function of (a) TD states and (b) 2p2h states. Average was performed over levels lying in 9 MeV - 13 MeV. The number of levels is 12 and 3342 for TD states and 2p2h states, respectively. []{data-label="fig_avedoorstr"}](fig11.eps){width="9.2cm"}
We evaluate $\gamma_{12}$ by using the strength functions of TD states as in Ref. [@aiba2]. Namely, we calculate the strength function of each TD state. Averaging the strength functions over whole TD states, we obtain Fig. \[fig\_avedoorstr\] (a). The FWHM of this averaged strength function gives an evaluation of the spreading width $\gamma_{12}$. We read $\gamma_{12 }=0.38$ MeV. (Note that we define $\gamma_{12 }$ as the spreading width of TD states instead of that of unperturbed 1p1h states.) The value of spreading width of 2p2h states $\gamma_{22}$ is also evaluated in the same manner. From Fig. \[fig\_avedoorstr\] (b) we also obtain $\gamma_{22}=0.75$ MeV as the estimate of the spreading width of 2p2h states. These results will be used in Sec. \[sec:comparison\]
For the sake of comparison, let us estimate the spreading width by assuming the Fermi golden rule. The root mean square of matrix elements between 1p1h states and 2p2h states is calculated as $(\overline{ \langle {\rm 1p1h} |V_{\rm 12}|{\rm 2p2h}\rangle^2})^{1/2}=9.3\times10^{-3}$ MeV. Similarly, we calculate $(\overline{ \langle {\rm 2p2h} |V_{\rm 22}|{\rm 2p'2h'}\rangle^2})^{1/2}=1.0\times10^{-2}$ MeV. Since the level spacing of 2p2h states is $d_{\rm 2p2h}=1.2$ keV, the spreading widths $\gamma_{12}$ and $\gamma_{22}$ are approximately estimated in the Fermi golden rule as $\gamma^{\rm FG}_{12}=2\pi \overline{ \langle {\rm 1p1h} |V_{\rm 12}|{\rm 2p2h}\rangle^2}/d_{\rm 2p2h}=0.46$ MeV and $\gamma^{\rm FG}_{22}=2\pi \overline{ \langle {\rm 2p2h} |V_{\rm 22}|{\rm 2p'2h'}\rangle^2}/d_{\rm 2p2h}=0.53$ MeV, respectively, which are in approximate agreement with the direct evaluation within 30%.
Search for strongly coupled states in 2p2h states {#sec:surfacce_vib}
-------------------------------------------------
In the picture of Fig. \[fig\_pic\_Ca\] 2p2h states are assumed to play a role as the chaotic background and provide the GOE fluctuation to the strength function. However, if the GQR TD state couples with not all 2p2h states equally but specific states in 2p2h states strongly, there is a possibility for this hierarchical structure in 2p2h states to give rise to a deviation from the GOE fluctuation. We, here, would like to investigate whether whole 2p2h states are rather equally coupled with the GQR TD state or whether there are specific states in 2p2h states which strongly couple with that state.
As a candidate of such specific states, we can consider the low-energy surface vibration plus 1p1h states: In Refs. [@bertsch2; @bertsch; @broglia; @bortignon; @lacroix2], the importance of the coupling to the surface vibration in the wide range of damping phenomena including the damping of a single particle motion as well as that of giant resonances was discussed. As for the giant resonance, which is composed of a coherent superposition of 1p1h states, this means that the damping occurs via the coupling with the specific 2p2h states, namely, the surface vibration plus 1p1h (s.v.+1p1h) states.
Since our model does not assume the particle-vibration coupling a priori, it is not trivial whether our model also has a mechanism that enhances the coupling with the low-energy surface vibration. Therefore, we would like to study whether the s.v.+1p1h states are particularly strongly coupled with the GQR TD state within our model. To do so, we calculate the FWHM of the following approximate strength function: $$S(E)=-\frac{1}{\pi}{\rm Im}\left(
E-E_c-\sum_\alpha \frac{V_{c\alpha}^2}{E-\omega_\alpha+i\gamma_{22}/2}\right)^{-1},
\label{doorwaystr}$$ where, $E_c$ and $\omega_\alpha$ denote the energy of the GQR TD state and the energy of the $\alpha$th s.v.+1p1h state, respectively. $V_{c\alpha}$ represents the coupling matrix element between the GQR TD state and the s.v.+1p1h state $\alpha$.
Only $J^\pi=2^+$, $3^-$ modes are included as surface vibrations: We took only the lowest TD state as $J^\pi=2^+$ surface vibrational mode. On the other hand, we must pay attention to the collectivity of the octupole mode. Figure \[fig\_octstr\] shows the TDA strength function for the IS octupole operator. Compared with the experimental data [@spear], the energy of the lowest state is too high, and strengths are fragmented over several states. Thus, we took into account the lowest nine states for the octupole mode. Note that s.v.+1p1h states thus defined are not orthogonal. In this sense Eq. (\[doorwaystr\]) is an approximation which neglects the non-orthogonality.
![ TDA strength function for the IS octupole operator in $^{208}$Pb. []{data-label="fig_octstr"}](fig12.eps){width="6cm"}
The strength function based on Eq. (\[doorwaystr\]) is presented in Fig. \[fig\_doorwaystr\]. The width $\Gamma_2^{\rm (s.v.)}$ estimated by the FWHM is 0.074 MeV. This value is significantly smaller than the width $\Gamma_2=0.41$ MeV of the GQR TD state caused by the coupling to the whole 2p2h states.
![ Strength function of the GQR TD state evaluated by considering only surface vibration plus 1p1h (s.v.+1p1h) states based on Eq. (\[doorwaystr\]). $\gamma_{22}=0.75$ MeV is used. []{data-label="fig_doorwaystr"}](fig13.eps){width="6cm"}
From the estimate by the Fermi golden rule, we can give more detailed comparison between the width for the case of s.v.+1p1h states and that for the whole 2p2h states. It is noted in Table \[table1\] that the spreading width $\Gamma_2^{\rm (s.v.)}=0.074$ MeV and $\Gamma_2=0.41$ MeV are well accounted for by the estimate. In the Fermi golden rule the spreading width is governed by two factors; 1) the average value of squared coupling matrix elements $\overline{ V_{c\alpha}^2}$ between the GQR TD state and the states that couple to it, and 2) the level density of the coupling states. From Table \[table1\], we see that the large difference between the two widths simply reflects the difference between the number of s.v.+1p1h states 909 and 2p2h states 3142 whereas the coupling strength of s.v.+1p1h states $\overline{ V_{c\alpha}^2}=0.65\times10^{-4}$ MeV$^2$ is comparable to the coupling strength $\overline{V_{c\alpha}^2}=0.72\times10^{-4}$ MeV$^2$ for the whole 2p2h states.
Table \[table1\] and Fig. \[fig\_doorwaystr\] suggest that our model does not contain the enhancement of the coupling with the surface vibrations in the damping of the GQR. Therefore we consider in the following the 2p2h states as background states which do not have specific structures.
\# $\overline{V_{c\alpha}^2}$ (MeV$^2$) $\Gamma_2^{\rm FG}$ (MeV) $\Gamma_2$ (MeV)
----------- ------ -------------------------------------- --------------------------- ------------------
s.v.+1p1h 909 $0.65\times 10^{-4}$ 0.092 0.074
2p2h 3342 $0.72\times 10^{-4}$ 0.38 0.41
: Averaged value of squared coupling matrix elements $\overline{V_{c\alpha}^2}$ between the GQR TD state and surface vibration plus 1p1h states or the whole 2p2h states(third column), the associated spreading width $\Gamma_2^{\rm FG}$ of the GQR TD state evaluated by the Fermi golden rule (fourth column), and the spreading width $\Gamma_2$ estimated by the FWHM of the strength function based on Eq. (\[doorwaystr\]) (fifth column). Second column shows the number of states considered. The second row shows the results obtained by including only the s.v.+1p1h states while the third row shows those for the case of the whole 2p2h states. []{data-label="table1"}
Physical origin of the difference between $^{40}$Ca and $^{208}$Pb {#sec:comparison}
------------------------------------------------------------------
In the above subsections, we have evaluated the physical quantities such as the various spreading widths, with which we have discussed the damping process of $^{40}$Ca and $^{208}$Pb, especially the mechanism of producing the total width of the strength function. Here, using these quantities we would like to discuss the physical origin of the difference between the fluctuation of the strength fluctuation of $^{40}$Ca and that of $^{208}$Pb. Table \[table2\] summarizes the values of the above physical quantities related to the initial stage of the damping process for both $^{40}$Ca and $^{208}$Pb.
We have shown in our previous study [@aiba] that the damping process through the doorway states causes large fluctuations which have characteristic energy scales, and that the fluctuations emerge in the local scaling dimension. For instance, the energy scale of the spreading width $\gamma_{12}$ of the doorway states is the quantity which shows up prior to the other quantities. It is noted, however, the size of the fluctuations depends on the mutual relations among the quantities mentioned above, and indeed we have examined in [@aiba] the relations which are needed to detect the effect of the spreading width $\gamma_{12}$.
$\Gamma$ $\Gamma_{\rm L}$ $\Gamma_2$ $\gamma_{12}$ $D_{\rm 1p1h}$ $\gamma_{22}$ $d_{\rm 2p2h}$
------------ ---------- ------------------ ------------ --------------- ---------------- --------------- ----------------
$^{40}$Ca 4000 4000 1500 1500 500 5200 11
$^{208}$Pb 630 180 410 380 230 750 1.2
: Values of physical quantities related to the damping of the GQR for $^{40}$Ca and $^{208}$Pb. Unit of the energy is keV for all cases. []{data-label="table2"}
It is trivial that the local scaling dimension can detect the spreading width when the spreading of 1p1h states does not cause the overlap of these states, namely when $\gamma_{12} < D_{\rm 1p1h}$. In addition to this case, the local scaling dimension still keeps the information of the spreading width even if the 1p1h states start to overlap with each other, i.e. $\gamma_{12} \simeq D_{\rm 1p1h}$. Studying more quantitatively with the use of the doorway damping model of Ref. [@aiba], we found the condition to detect the effect of the spreading width as
1. $\gamma_{12} \le 4D_{\rm 1p1h}$.
Furthermore, we need the second condition:
1. $\gamma_{12} < \Gamma_{\rm L}$.
This simply means that the spreading width $\gamma_{12}$ of the doorway states (1p1h states) need to be smaller than the total width $\Gamma$. Since $\Gamma\simeq\Gamma_\text{L}+\Gamma_2$ and $\gamma_{12}\simeq\Gamma_2$, the requirement $\gamma_{12} < \Gamma$ can be written as (B). In addition to (A) and (B), we need the third condition:
1. $D_{\rm 1p1h} < \Gamma_{\rm L}$.
This is because we need more than one doorway states within the the energy interval $\Gamma_{\rm L}$ in order to have fluctuating behavior in the strength function.
Let us first look at the case of $^{40}$Ca. From Table \[table2\], the relation $\gamma_{12} =3.0 D_{\rm 1p1h}$ is derived, and this relation fulfills the condition (A). On the other hand, relations $\gamma_{12} =0.38 \Gamma_{\rm L}$ and $D_{\rm 1p1h} =0.13 \Gamma_{\rm L}$ are also derived from Table \[table2\], and these relations satisfy both conditions (B) and (C). As a result, in the case of $^{40}$Ca, we can see a deviation from the GOE fluctuation in the local scaling dimension and indeed the energy scale where the deviation is seen is related to the value of $\gamma_{12}$.
For $^{208}$Pb, on the other hand, we find in Table \[table2\] that $\gamma_{12}=1.7D_{\rm 1p1h}$, while $\Gamma_L$ is smaller than $\gamma_{12}$ and $D_{\rm 1p1h}$, i.e., $\gamma_{12} =2.1 \Gamma_{\rm L}$ and $D_{\rm 1p1h} =1.3 \Gamma_{\rm L}$. The first relation satisfies the condition (A). The latter two relations, however, break the condition (B) and (C). Accordingly, for the case of $^{208}$Pb, the deviation from the GOE due to the effect of $\gamma_{12}$ can not be seen. The situation in $^{208}$Pb is illustrated in Fig.\[fig\_pic\_Pb\]. Essential physical origin of this difference is that for $^{208}$Pb the Landau damping width is small compared with that of $^{40}$Ca. The smallness or largeness of the value of the Landau damping width affects the fluctuation property of the strength function.
![ Schematic picture of the initial stage of the damping process for GQR in $^{208}$Pb. []{data-label="fig_pic_Pb"}](fig14.eps){width="8cm"}
Conclusion {#sec:conclusion}
==========
We studied the fluctuation properties of the strength function of IS GQR for $^{208}$Pb by means of the local scaling dimension, and compared the results with those of $^{40}$Ca. The strength function was obtained by the second TDA including 2p2h states as well as 1p1h states. For $^{40}$Ca, we find a fluctuation different from GOE around the energy scale which is approximately related to the spreading width of the 1p1h states. On the other hand, for $^{208}$Pb we can not find the fluctuation different from the GOE at almost all the energy scales.
The different behavior of the fluctuation detected by the local scaling dimension analysis is due to the difference of the ratio of the Landau damping width $\Gamma_{\rm L}$ to the spreading width of the 1p1h states $\gamma_{12}$.
Recently, the analysis of the strength function of the IS GQR in $^{208}$Pb obtained by (p,p’) inelastic scattering experiment was performed by means of the wavelet transform [@shevchenko]. The authors suggest from the positions of the local maxima in the wavelet power that there exist three energy scales in the fluctuation of the strength function: I. 120 keV, II. 440, 850 keV, III. 1500 keV. Existence of higher two energy scales is not inconsistent with our results, since our analysis says nothing about the fluctuation at about energy scale II, which may correspond to the total width $\Gamma$ in our model, or higher energy scales. However, the existence of the smallest energy scale $\sim 120$ keV may conflict with our results: If there is such an energy scale in our strength function, our analysis must detect it as a deviation from the GOE fluctuation. Therefore, it is very important to study the origin of this discrepancy. In particular, it is interesting to clarify the relation between two method, namely, the local scaling dimension and the wavelet power. Studies in this direction are now in progress.
The authors acknowledge helpful discussion with K. Matsuyanagi. We are also indebted to A. Richter, and P. von Neumann-Cosel for many fruitful discussion. The numerical calculations were performed at the Yukawa Institute Computer Facility as well as at the RCNP Computer Facility.
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abstract: 'We investigate the phase transitions of a nonlinear, parallel version of the Ising model, characterized by an antiferromagnetic linear coupling and ferromagnetic nonlinear one. This model arises in problems of opinion formation. The mean-field approximation shows chaotic oscillations, by changing the couplings or the connectivity. The spatial model shows bifurcations in the average magnetization, similar to what seen in the mean-field approximation, induced by the change of the topology, after rewiring short-range to long-range connection, as predicted by the small-world effect. These coherent periodic and chaotic oscillations of the magnetization reflect a certain degree of synchronization of the spins, induced by long-range couplings. Similar bifurcations may be induced in the randomly connected model by changing the couplings or the connectivity and also the dilution (degree of asynchronism) of the updating. We also examined the effects of inhomogeneity, mixing ferromagnetic and antiferromagnetic coupling, which induces an unexpected bifurcation diagram with a “bubbling” behavior, as also happens for dilution.'
author:
- Franco Bagnoli
- Raúl Rechtman
title: Stochastic Bifurcations in the Nonlinear Parallel Ising Model
---
Introduction
============
There are quite a large number of studies about opinion formation in uniform societies [@Hegselmann; @Deffuant; @review; @Stauffer; @GalamReview; @Galam1; @guazzini; @BagnoliGuazziniLio; @GrottoGuazziniBagnoli; @GuazziniCiniBagnoliRamasco; @ViloneCarlettibagnoliGuazzini]. Many such models adopt an approach similar to that of the Ising model. In such cases one has two opinions, say A and B or -1 and 1, and one is interested in the establishment of a majority (magnetic phase transitions) or in the effects of borders, or in the influence of some leader (social impact theory) [@latane]. This opinion space can be seen as the first ingredient of these models.
The second ingredient is how to model the response to an external influence. It is common to classify the attitude of people (agents) as either conformist or contrarian (also known as nonconformist). A conformist tends to agree with his neighbors and a contrarian to disagree.
It is also easy to map this attitude onto Ising terms: conformist agents correspond to ferromagnetic coupling and contrarians to antiferromagnetic ones [@review]. The effects induced by the presence of contrarian agents in a society have been studied in models related to the voter model [@masuda2013; @crokidakis2014; @Independence; @schneider2004; @delalama2005; @corcos02; @galam04; @Biswas; @Galam-Gemrev; @Galam-chaotic; @sudoyi2013; @bagnoli2013; @bagnoli2015].
In general, agents that are under a strong social pressure tend to agree with the great majority even when they are certain that the majority’s opinion is wrong, as shown by Asch [@Asch]. Under a strong social pressure a contrarian may agree with a large majority, an phenomenon that may be modelled using non-linear interactions.
The strategy of following an overwhelming majority may be ecological, since it is probable that this coherent behavior is due to some unknown piece of information, and in any case the competitive loss is minimal since it equally affects all other agents.
A binary opinion model where an agent tends to align with the largest neighboring cluster, similar to an Ising model with plaquette interactions, was studied in Ref. [@biswas2009]. In this model, a single dissenting agent immersed in a cluster of different opinions cannot overcome the social pressure, and therefore the model exhibits absorbing homogeneous phases. The possibility of dissenting, distributed as a quenched disorder, was introduced in Ref [@biswas2011a].
The third ingredient is the connectivity, i.e., how the neighborhood of a given agent is composed. Traditionally, magnetic systems have been studied either on regular lattices, trees or with random connections, whose behavior is similar to that of the mean-field approach. In recent years, much attention has been devoted to other network topologies, like the small-world [@WattsStrogatz] to scale-free [@Barabasi] etc.
The Ising model has been studied in different topologies [@Klemm; @WuZhou; @HolmeNewman; @barre], in particular, the topological details may affect the critical dynamics [@goswami2011] and the zero-temperature quench [@biswas2011].
In contrast with the Ising model, in the study of opinion formation there is no compulsory obligation to have symmetric interactions, each agent is influenced by those in his neighborhood, which are not necessarily influenced by the first agent.
Each individual may be a conformist or a contrarian and this character does not change in time. In these terms, the simple ferromagnetic Ising model represents a uniform society of conformists with local symmetric interactions.
The fourth ingredient is the update scheduling, that may be completely asynchronous, like in standard Monte Carlo simulations, or completely parallel, like in Cellular Automata, or something in between [@Derrida; @NewmannDerrida; @Cirillo]. It is not clear which scheme is the most representative of reality. Real human interactions are indeed continuous, but also clocked by days, elections, etc.
An effect that is favoured by parallelism is synchronization in the presence of complex dynamics. As happens in physics, a macroscopic irregular behavior (macroscopic chaos) implies a coherent, although irregular motion of many elements (the microscopic constituents).
One of most intriguing effects is the hipster’s one, in which a society of contrarians tends to behave in a uniform way [@hipster]. Clearly, “conformist hipsters” always change their behavior, when they realize to be still in the mainstream, but since they do so all together, they remain synchronized: the parallelism is a crucial element of such a behavior.
Finally, the fifth ingredient is homogeneity. There are many possibilities of introducing mixtures of agents or spins with different coupling. We investigate what happens when one mixes ferromagnetic and antiferromagnetic interactions, and we shall show that this mixture promotes a “bubbling” behavior in the bifurcation, meaning that the bifurcation appears first for intermediate values of the parameters, similar to what happens with asynchronism.
In previous studies we presented “reasonable contrarian” agents whose response to the average opinion of their neighbors is nonlinear and discusssed the collective behavior of societies composed of reasonable contrarians only and by mixtures of these agents and nonlinear conformists [@bagnoli2013; @bagnoli2015]. The rationale was that in some cases, and in particular in the presence of frustrated situations like in minority games [@minority; @minority1; @minority2; @minority3], it is not convenient to always follow the majority, since in this case one is always on the “losing side” of the market. This is one of the main reasons for the emergence of a contrarian attitude. On the other hand, if all or almost all agents in a market take the same decision, it is often wise to follow such a trend. We can denote such a situation with the word “social norm”.
A society composed by a strong majority of reasonable contrarians exhibits interesting behaviors when changing the topology of the connections. On a one-dimensional regular lattice, there is no long-range order, the evolution is disordered and the average opinion is always halfway between the extreme values [@bagnoli2005]. However, adding long-range connections or rewiring existing ones, we observe the Watts-Strogatz “small-world” effect, with a transition towards a mean-field behavior. But since in this case the mean-field equation is, for a suitable choice of parameters, chaotic, we observe the emergence of coherent oscillations, with a bifurcation cascade eventually leading to a chaotic-like behavior of the average opinion.
The small-world transition is essentially a synchronization effect. Similar effects with a bifurcation diagram resembling that of the logistic map have been observed in a different model of “adapt if novel - drop if ubiquitous” behavior, upon changing the connectivity [@Dodds; @Harris].
The main goal of the present study is that of reformulating the opinion formation models mentioned above [@bagnoli2013; @bagnoli2015], in terms of a parallel, nonlinear Ising model both on a regular lattice, where the spin at any site is influenced by its nearest neighbors, and on small-world networks.
In the first case the mean-field behavior of the magnetization is described by a nonlinear equation for which chaos can be evaluated by the Lyapunov exponent [@ott02], which is a measure of the stability of trajectories.
The Lyapunov exponent is the time average of the growth rate of an initial infinitesimal perturbation of a trajectory. Clearly, this quantity cannot be simply defined for stochastic systems, since in this case one would essentially measure the effects of the noise. However, in many cases and in particular the present one, we would like to compare the dynamical properties of a stochastic microscopic model and its mean-field approximation. We show here that the Boltzmann’s entropy of an aggregate variable like the magnetization is a quantity that can be defined for both deterministic and stochastic systems. In the first case, Boltzmann’s entropy can be used as a measure of chaos [@Boltzmann].
The scheme of the paper is the following. We discuss the “nonlinear” parallel Ising model in Section \[sec:parallelising\]. We can therefore introduce the mean-field approximation of the model in Section \[sec:meanfield\], showing the bifurcation phase diagrams as a function of the parameters. The definition of the entropy and the results of microscopic simulations $\eta$ are reported in Section \[sec:simulations\]. Finally, conclusions are drawn in the last Section. In this Section we discuss also the differences between the present and the original model of Refs. [@bagnoli2013; @bagnoli2015].
![\[plaquette\] (Color online.) A $K=3$ spin neighborhood, with the interaction terms corresponding to the external field $\tilde H$, the two-spin $\tilde J$, three-spin $\tilde Z$, and four-spin $\tilde W$ interaction constants.](plaquette){width="\columnwidth"}
Parallel nonlinear Ising model {#sec:parallelising}
==============================
We consider a system with $N$ sites, each one in a state $s_i\in
\{-1,1\}$, $i=1,\dots,N$. The state of the system is $@s=(s_1,\dots,s_{N})$. The topology of the system is defined by the adjacency matrix $a$ with $a_{ij}=1$ if site $j$ belongs to site $i$’s neighborhood and is zero otherwise. The connectivity $k_i$, the local field $\tilde{h}_i$, and the rescaled local field $h_i$ at site $i$ are $$k_i=\sum_j a_{ij},\qquad
\tilde{h}_i=\sum_j a_{ij}s_j, \qquad
h_i=\dfrac{\tilde{h}_i}{k_i},$$ with $h_i\in [-1,1]$. In this paper we shall use a uniform connectivity $k_i=K\,\,\forall\, i$. The magnetization $m$ is defined as $$m=\dfrac{1}{N}\sum_is_i,$$ with $m\in [-1,1]$.
In the following we consider multi-spin (plaquette) interactions. We moreover consider only completely asymmetric interactions [@Suzuki; @Sherrington], arranged to give a preferred direction that corresponds to time in the standard cellular automata language [@bagnoli2013; @SS-bif].
Considering up to 4-spin interactions, the Hamiltonian is $$\begin{aligned}
\label{Hamiltonian}
\mathcal{H} (@s) =& -\sum_i s_i'\Bigl(\tilde H+ \tilde J\sum_j a_{ij} s_j +%
\tilde Z \sum_{jk} a_{ij} a_{ik}s_j s_k + \nonumber\\
&\tilde W \sum_{jkl} a_{ij} a_{ik} a_{il} s_j s_k s_l\Bigr),\end{aligned}$$ where $\tilde H$ is the external field, and $\tilde J$, $\tilde Z$, $\tilde W$ the two-spin, three-spin and four-spin interaction constants respectively as shown in Fig. \[plaquette\].
It is possible to recast the interaction constants in terms of the local field $h_i$. The terms containing $s'_i$ at “time” $t+1$ are, $$\begin{split}
&\text{2-spin: }\; s_i'\sum_j a_{ij} s_j = s_i' \tilde h_i,\\
&\text{3-spin: }\; s_i'\sum_{jk} a_{ij} a_{jk}s_j s_k=s_i'Q^{(2)}_i,\\
&\text{4-spin: }\; s_i'\sum_{jkl} a_{ij} a_{ik} a_{il} s_j s_k s_l = s_i'Q^{(3)}_i.\\
\end{split}$$ These expressions define $ Q^{(2)}$ and $ Q^{(3)}$. Since $$\begin{split}
&\tilde{h_i}^2 = K + 2Q_i^{(2)},\\
&\tilde{h_i}^3 = (3K-2)\tilde h_i + 6 Q_i^{(3)},\\
\end{split}$$ the Hamiltonian can be written as $$\mathcal{H} (@s) = -\sum_i s'_i
(H + J h_i + Zh_i^2 + W h_i^3),$$ where the correspondences among coupling constants are $$\begin{split}
H &= \tilde H - \frac{1}{2} K\tilde Z,\\
J &= K\left(\tilde J - \frac{3K-2}{6}\tilde W\right),\\
Z &= \frac{1}{2} K^2\tilde Z, \\
W &= \frac{1}{6}K^3\tilde W.\\
\end{split}$$ In the following, we shall consider pair ($J$) and four-spin ($W$) terms, *i.e.*, $H=Z=0$, in agreement with previous investigations [@bagnoli2013].
The coupling term $J$ modulates the “linear” effects of neighbors, so $J>0$ gives a conformist (ferromagnetic) behavior and $J<0$ a contrarian (antiferromagnetic) one. The term $W$ modulates the nonlinear effects of the crowd. In this way one can model the Asch effect by inserting $J<0$ (contrarian attitude) and $W>0$ (social norms).
The time evolution of the spins is given by the parallel application of the transition probabilities $\tau(s_i'|h_i)$ that gives the probability that the spin at site $i$ and time $t+1$ takes value $s_i'$ given the local field $h_i$ at time $t$, see Fig. \[plaquette\]. The local transition probability is defined by a heat bath probability $$\begin{aligned}
\label{eq:ntau}
\tau(s'_i| h_i) &= \dfrac{1}{1+\exp(-2 s'_i (J h_i + W h_i^3))}\nonumber\\
&=\frac{1}{2}\left[1+s'_i\tanh (J h_i + W h_i^3)\right].\end{aligned}$$
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(a) (b)
![\[fig:tau\] (Color online). (a) The transition probability $\tau=\tau(1|h)$, Eq. as a function of the local field $h$ for three values of the coupling constant $J$. (b) Graph of the magnetization $m'$ at time $t+1$ as a function of the magnetization $m$ at time $t$, Eq. , with some iterates for $J=-7.5$. ](PI00-o1-k20-J-10p00-W15p0 "fig:"){width="0.45\columnwidth"} ![\[fig:tau\] (Color online). (a) The transition probability $\tau=\tau(1|h)$, Eq. as a function of the local field $h$ for three values of the coupling constant $J$. (b) Graph of the magnetization $m'$ at time $t+1$ as a function of the magnetization $m$ at time $t$, Eq. , with some iterates for $J=-7.5$. ](m-m-I00-o1-k20-J-7p50 "fig:"){width="0.45\columnwidth"}
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The parallel version of the linear ($W=0$) Ising model does not show many differences with respect to the standard serial one [@Derrida]. The observables that depend only on single-site properties take the same values in parallel or sequential dynamics [@NewmannDerrida], although differences arise for two-site correlations [@SS-meta]. In general the resulting dynamics is no more reversible with respect to the Gibbs measure induced by any Hamiltonian [@Cirillo].
In the following, unless otherwise specified, we always use $W=15$ and $K=20$.
Mean-field approximation {#sec:meanfield}
========================
The mean-field approximation for the magnetization $m$ of the fully parallel case with fixed connectivity $K$, follows from the Markov equation assuming no spatial correlations. Then $$\begin{aligned}
\label{eq:mf}
m'=&f(m) = \frac{1}{2^{K}}\sum_{k=0}^K\binom{K}{k}(1+m)^k(1-m)^{K-k}\nonumber\\
&\times\tanh\left[J\left(\frac{2k}{K}-1\right)+W\left(\frac{2k}{K}-1\right)^3\right],\end{aligned}$$ with $m=m(t)$ and $m'=m(t+1)$. We show in Fig. \[fig:tau\] (b) the graph of $m'$ together with some iterates of the map.
The mean-field magnetization exhibits chaos that can be characterized by the Lyapunov exponent $\lambda$ [@ott02]. However, on spatially extended networks $m$ changes stochastically and cannot be characterized in the same way.
In order to compare microscopic and mean-field models within the same framework we use the Boltzmann’s entropy $\eta$ [@Boltzmann; @bagnoli2013] of the magnetization $m$. The interval $[-1,1]$ is partitioned in $L$ disjoint intervals $I_i$ of equal size and the probability $q_i$ of $I_i$ is the fraction of visits to $I_i$ after $T$ time steps with $T\gg 1$.
Once these probabilities are known, $\eta$ is defined by $$\label{eq:s}
{\eta}=- \dfrac{1}{\log L}\sum_{i=1}^Lq_i\log q_i,$$ so that $0\leq \eta\leq 1$, the lower bound corresponding to a fixed point, the upper one to the uniform distribution $q_i=1/L$.
For a periodic orbit of period $2^p$ and $L=2^b$, $\eta=p/b$. For low-dimensional dynamical systems, like the mean-field equation, the mid-value threshold $\eta=0.5$ effectively separates the contracting dynamics (cycles) from the chaotic ones. For spatially-extended systems, there is always a stochastic noise that increases the value of the entropy in the “fixed-point” part of the parameter space. This base-level value is related to the size of the sample, and slowly vanishes for large samples.
In order to use finite-size samples, we set the onset of the phase in the stochastic systems corresponding to the chaotic phase in the deterministic ones to the mid-value of the range of $\eta$. Taking the limits $T\to\infty$, $L\to\infty$ leads to the Kolmogorov-Sinai entropy [@kolmogorov58; @kolmogorov59; @sinai59; @ott02].
Before presenting the different scenarios, let us illustrate the type of bifurcations that are present. In Figs. \[fig:mf-bif-c0p60\] (a) and (b) we show parts of the bifurcation diagram of the map of Eq. as a function of $J$ staring with different values of the initial magnetization $m_0$.
Referring to the values in the Figure, at $J=J_0$ there is a pitchfork bifurcation, i.e., a separation of basins, that reunite at $J=J_1$, which is another pitchfork bifurcation, in the reverse direction. Intermixed, there are period-doubling bifurcations. There are other pitchfork bifurcations for different intervals of $J$.
In Fig. \[fig:mf-bif-c0p60\] (c) we show the return map of the mean-field map for $J=-5.2$. We can see that there are four basins of attraction. For small values of the initial magnetization $m_0$, the orbit is attracted to $m=-1$ and for large $m_0$ to $m=1$. The regions where this occurs are marked by the vertical dotted lines in the Figure. For other values of $m_0$, $m$ ends in one of two period-two orbits. This figure shows, in the lower part, the two basins of attraction that are symmetric in the sense that if $m_0$ belongs to one basin of attraction, $-m_0$ belongs to the other one.
In what follows we present the bifurcation diagrams of the mean-field map Eq. by varying the coupling constants $J$ and $W$. Unless otherwise noticed, we computed the Lyapunov exponent $\lambda$ by averaging over 10,000 time steps after a transient of another $10,000$ steps. The entropy $\eta$ was computed using 256 boxes and $25,000$ time steps.
![\[fig:mf-J\] (Color online.) (a) Bifurcation diagram of the mean-field map of the magnetization $m$, Eq. , as a function of the linear coupling $J$. In (b) the corresponding Lyapunov exponent $\lambda$ and in (c) the entropy $\eta$. The vertical dotted lines are drawn at the estimated values of $J$ for which $\lambda=0$. The horizontal dotted lines in (c) correspond to period 2, $\eta=1/8$, and period 4, $\eta=1/4$, orbits. For every value of $J$, two initial values of the magnetization were used, $m_0=-0.3$ and $m_0=0.3$.](mf-m-lambda-eta-PI00-o2-k20-W15p00){width="0.9\columnwidth"}
![\[fig:mf-W\] (Color online.) (a) Bifurcation diagram of the mean-field map of the magnetization $m$, Eq. , as a function of the coupling constant $W$ with $J=-10$. In (b) The corresponding Lyapunov exponent $\lambda$ and in (c) the entropy $\eta$. The vertical dotted lines are drawn at the estimated values of $W$ for which $\lambda=0$. The horizontal dotted lines in (c) correspond to period 2, $\eta=1/8$, and period 4, $\eta=1/4$, orbits. For every value of $W$, two initial values of the magnetization were used, $m_0=-0.3$ and $m_0=0.3$.](mf-lambda-eta-PI00-o13--14-k20-J-10){width="0.9\columnwidth"}
In Fig. \[fig:mf-J\] (a) we show the bifurcation diagram of the magnetization $m$, Eq. , as a function of $J$ with $W$ and $K$ fixed. The diagram exhibits a period doubling cascade towards chaos with periodic windows and pitchfork bifurcations.
The bifurcation diagram of $m$ as a function of $W$ with $J$ and $K$ fixed is shown in Fig. \[fig:mf-W\] (a). In this case, there is an inverse period doubling cascade to chaos with pitchfork bifurcations.
The next row of figures, Figs. \[fig:mf-J\] (b) and Fig. \[fig:mf-W\] (b), show the corresponding Lyapunov exponent $\lambda$, and the last one, Figs. \[fig:mf-J\] (c) and Fig. \[fig:mf-W\] (c), the entropy $\eta$. The dotted vertical lines are drawn at some of the values of $J$ or $W$ where $\lambda$ passes from a negative to a positive value or vice versa. These values coincide to jumps of $\eta$ from values smaller to $1/2$ to larger ones or vice versa and mark the appearance of chaos or periodic windows in the bifurcation diagrams.
Therefore, $\eta$ can be used as a measure of chaos. To stress this, we show in Figs. \[fig:mfpdWJ\] the mean-field phase diagrams of $\lambda$ (top) and $\eta$ (bottom). These diagrams are similar. The horizontal lines at $W=15$ correspond to Figs. \[fig:mf-J\] (b) and (c) and the vertical ones at $J=-10$ to Figs. \[fig:mf-W\] (b) and (c) respectively. We find a similar behavior of the mean-field map as $K$ varies with fixed $J$ and $W$. The three quantities $J$, $W$ and $K$ are related by scaling relations, as shown in the Appendix.
![\[fig:mfpdWJ\] (Color online) (a) Mean-field phase diagram of the Lyapunov exponent $\lambda$ showing the values of $(-J,W)$ where $\lambda>0$. (b) The mean-field phase diagram of the entropy showing the values of $(-J,W)$ where $\eta>1/2$. ](lambda-J-W-PI00-o15-k20 "fig:"){width="0.9\columnwidth"}\
![\[fig:mfpdWJ\] (Color online) (a) Mean-field phase diagram of the Lyapunov exponent $\lambda$ showing the values of $(-J,W)$ where $\lambda>0$. (b) The mean-field phase diagram of the entropy showing the values of $(-J,W)$ where $\eta>1/2$. ](eta-J-W-PI00-o15-m8-k20 "fig:"){width="0.9\columnwidth"}
The dotted horizontal lines in Fig. \[fig:mf-J\] (c) and Fig. \[fig:mf-W\] (c) correspond to period 2, $\eta=1/8$, and period 4, $\eta=1/4$ orbits. Looking at the bifurcation diagram in Fig. \[fig:mf-J\] (a), for small $-J$, the map has a fixed point and as $-J$ grows there is a first bifurcation to a period 2 orbit and another one to what looks like a period 4 orbit, but $\eta=1/8$ instead of $\eta=1/4$ for period 4 orbits. What appears like a bifurcation to period four orbits is actually a pitchfork bifurcation to two period-two orbits that depend on the initial magnetization $m_0$ as mentioned before. There are other pitchfork bifurcations for other values of $J$ with $W$ fixed and also for values of $W$ with $J$ fixed.
Small-World stochastic bifurcations {#sec:simulations}
===================================
In the Watts-Strogatz small-world model [@WattsStrogatz], starting with a network where every site has $K$ nearest neighbors, at any site $i$, with probability $p$, known as the long-range connection probability, each one of its $K$ neighbors is replaced by a random one. Then the spin at each site is updated according to Eq. . As $p$ grows, coherent oscillations of a majority of spins begin to appear so that the magnetization $m$ shows noisy periodic or irregular oscillations. The noise is the manifestation of the stochasticity of the updating rule. Similar patterns can be seen in Ref. [@bagnoli2005], where the effect of the size of neighborhood is studied.
As shown in the following, by changing several parameters, we can obtain stochastic bifurcation diagrams similar to the mean-field ones. The following microscopic simulations were carried using lattices of $N=10,000$ sites, with a transient of $10,000$ time steps. The entropy $\eta$ was computed with 256 boxes and $25,600$ time steps.
In Fig. \[bifp\] we show the bifurcation map and the entropy $\eta$ as functions of $p$. There is always some disorder, even for small values of $p$ where $m\sim 0$, and as $p$ grows we find bifurcations and more disorder. Indeed, the entropy $\eta$ is a good measure of this behavior, small values of $\eta$ corresponding to noisy “periodic orbits” while larger ones to disorder (“chaos”).
In the mean-field case, we found $\eta=1/2$ to be a good threshold to separate order from chaos. For the stochastic dynamics on small-world networks we choose as the threshold the approximate value of the entropy at the first bifurcation as shown in the figures. For values smaller than this threshold there are noisy periodic orbits. The bifurcation diagram of the figure is reminiscent of the mean-field one, Fig. \[fig:mf-J\].
Notice that pitchfork bifurcations (dependence on the initial magnetization) are present also in the microscopic simulations, as shown in Fig. \[bifp\]
In Fig. \[fig:sw-J\] we show the bifurcation diagrams and entropy of the magnetization $m$ for the small-world networks obtained for different values of $p$. As before, the entropy is a good indicator of disorder.
![\[bifp\] (Color online) Small-world stochastic bifurcation diagram of the magnetization $m$, dots (in magenta), and the entropy $\eta$, continuous curve (in green), as functions of the long-range connection probability $p$ with $J=-10$. The “jump” of $m$ for $p\simeq 0.45$ corresponds to a pitchfork bifurcation (dependence on the initial magnetization).](sw-bif-eta-PI-1-o2-N10000-k20-J-10p00-W15p00-c0p50){width="\columnwidth"}
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$p=0.3$ $p=0.5$
![\[fig:sw-J\] (Color online) Small-world bifurcation diagram of the magnetization $m$, dots (in magenta) and the corresponding entropy $\eta$, continuous line (in green), as functions of of the linear coupling constant $J$ and different values of the long-range connection probability $p$.](sw-bif-PI-1-o5-N10000-W15p00-k20-p0p30-c0p50 "fig:"){width="0.45\columnwidth"} ![\[fig:sw-J\] (Color online) Small-world bifurcation diagram of the magnetization $m$, dots (in magenta) and the corresponding entropy $\eta$, continuous line (in green), as functions of of the linear coupling constant $J$ and different values of the long-range connection probability $p$.](sw-bif-PI-1-o5-N10000-W15p00-k20-p0p50-c0p50 "fig:"){width="0.45\columnwidth"}
$p=0.8$ $p=1.0$
![\[fig:sw-J\] (Color online) Small-world bifurcation diagram of the magnetization $m$, dots (in magenta) and the corresponding entropy $\eta$, continuous line (in green), as functions of of the linear coupling constant $J$ and different values of the long-range connection probability $p$.](sw-bif-PI-1-o5-N10000-W15p00-k20-p0p80-c0p50 "fig:"){width="0.45\columnwidth"} ![\[fig:sw-J\] (Color online) Small-world bifurcation diagram of the magnetization $m$, dots (in magenta) and the corresponding entropy $\eta$, continuous line (in green), as functions of of the linear coupling constant $J$ and different values of the long-range connection probability $p$.](sw-bif-PI-1-o5-N10000-W15p00-k20-p1p00-c0p50 "fig:"){width="0.45\columnwidth"}
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Clearly, by setting the rewiring probability $p$ large enough, one can also recover the mean-field bifurcation diagrams as function of $J$, $K$ and $W$, with a good correspondence of the critical values of parameters.
Partial asynchronism (dilution)
-------------------------------
The dilution $d$ is the fraction of sites chosen at random that are not updated at every time step. We define the diluted rule as $$\label{p}
s_{i}(t+1) = \begin{cases}
1 & \text{with probability $ (1-d) \tau(1|h_i)$,}\\
-1 & \text{with probability $(1-d) \left[1-\tau(1|h_i)\right]$,}\\
s_{i}(t) & \text{otherwise, i.e., with probability $d$,}
\end{cases}$$ so that for $d=0$ one has the standard parallel updating rule. One time step is defined when on the average every site of the lattice is updated once. For a system with $N$ sites, the smallest value of the dilution is $d=1/N$ and then $t_d=1/d$ updates are needed to complete one time step. If $d=1/2$, $t_d=2$, etc.
The mean-field equation corresponding to dilution is $$m(t+1) = (1-d)m(t) + d f(m(t));$$ where $f$ is the the map of Eq. . The mean-field phase diagram is reported in Fig. \[fig:mfpddJ\]. Notice that the border at $d=0$ corresponds to the horizontal line in Fig. \[fig:mfpdWJ\].
![\[fig:mfpddJ\] (Color online) Mean-field bifurcation diagram of the magnetization $m$ (dots in magenta, two initial conditions), and the entropy $\eta$ (continuous curve in green), as functions of the dilution probability $d$ with $J=-10$. ](m-f-dil-k20-J-10p00-W15p00){width="\columnwidth"}
The bifurcation diagrams and the entropy $\eta$ of the magnetization as functions of of the dilution $d$ are shown in Fig. \[bifd\] for different value of the long-range connection probability $p$. It is interesting to note the “bubbling” transition: the oscillations are favored, for intermediate values of the rewiring $p$, by a non-complete parallelism.
As shown in the figure, for values of $p$ larger than $0.1$, the dilution is able to trigger bifurcations also in the spatial model. In contrast with the linear Ising model [@Cirillo], where even a small amount of asynchronism is able to destroy the “effective” antiferromagnetic coupling, here the behavior is smooth with respect to dilution. See Ref. [@SS-meta] for a study about metastable effects in the linear model.
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$p=0.1$ $p=0.2$
![\[bifd\] (Color online). Bifurcation diagrams of the magnetization $m$ on small-world networks, dots (in magenta), and the entropy $\eta$, continuous curve (in green), as functions of the dilution $d$ with $J=-10$, and different values of the long-range connection probability $p$.](PI5-o4-N10000-k20-J-10p00-W15p00-p0p10 "fig:"){width="0.45\columnwidth"} ![\[bifd\] (Color online). Bifurcation diagrams of the magnetization $m$ on small-world networks, dots (in magenta), and the entropy $\eta$, continuous curve (in green), as functions of the dilution $d$ with $J=-10$, and different values of the long-range connection probability $p$.](PI5-o4-N10000-k20-J-10p00-W15p00-p0p20 "fig:"){width="0.45\columnwidth"}
$p=0.5$ $p=1.0$
![\[bifd\] (Color online). Bifurcation diagrams of the magnetization $m$ on small-world networks, dots (in magenta), and the entropy $\eta$, continuous curve (in green), as functions of the dilution $d$ with $J=-10$, and different values of the long-range connection probability $p$.](PI5-o4-N10000-k20-J-10p00-W15p00-p0p50 "fig:"){width="0.45\columnwidth"} ![\[bifd\] (Color online). Bifurcation diagrams of the magnetization $m$ on small-world networks, dots (in magenta), and the entropy $\eta$, continuous curve (in green), as functions of the dilution $d$ with $J=-10$, and different values of the long-range connection probability $p$.](PI5-o4-N10000-k20-J-10p00-W15p00-p1p00 "fig:"){width="0.45\columnwidth"}
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Heterogeneity
-------------
In order to measure the effects of heterogeneity, we let a fraction $\xi$ of spins interact ferromagnetically ($J>0$) with their $K$ neighbors and a fraction $1-\xi$ interact antiferromagnetically ($J<0$). We show in Fig. \[fig:mix\] the bifurcation diagrams of the magnetization $m$ together with the entropy $\eta$ as functions of $\xi$ for different values of $p$. Again, the entropy is a good measure of disorder. One can see a “bubbling” effect very similar to what observed by changing the dilution. In other words, oscillations, which are a product of antiferromagnetism and parallelism, are actually favoured by a small percentage of asynchronism and/or of ferromagnetic nodes, for a partial long-range rewiring of links.
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$p=0.05$ $p=0.2$
![\[fig:mix\] (Color online.) Small-world ferro-anti ferro bifurcation diagram (left axis, dots in magenta) and entropy $\eta$ (right axis, continuous curve in green) of the magnetization $m$ as a function of $\xi$ for different values of the long range probability $p$ and $N=10,000$, $|J|=10$,. for $P=0.05$ there is no threshold for $\eta$, for $p=0.2$ it is $\eta=0.7$, for $p=0.8$ and $p=1.0$ it is $\eta=0.6$.](sw-bif-eta-m-xi-PI5-o8-N10000-k20-J-10p00-W15p00-p0p05 "fig:"){width="0.45\columnwidth"} ![\[fig:mix\] (Color online.) Small-world ferro-anti ferro bifurcation diagram (left axis, dots in magenta) and entropy $\eta$ (right axis, continuous curve in green) of the magnetization $m$ as a function of $\xi$ for different values of the long range probability $p$ and $N=10,000$, $|J|=10$,. for $P=0.05$ there is no threshold for $\eta$, for $p=0.2$ it is $\eta=0.7$, for $p=0.8$ and $p=1.0$ it is $\eta=0.6$.](sw-bif-eta-m-xi-PI5-o8-N10000-k20-J-10p00-W15p00-p0p20 "fig:"){width="0.45\columnwidth"}
$p=0.8$ $p=1.0$
![\[fig:mix\] (Color online.) Small-world ferro-anti ferro bifurcation diagram (left axis, dots in magenta) and entropy $\eta$ (right axis, continuous curve in green) of the magnetization $m$ as a function of $\xi$ for different values of the long range probability $p$ and $N=10,000$, $|J|=10$,. for $P=0.05$ there is no threshold for $\eta$, for $p=0.2$ it is $\eta=0.7$, for $p=0.8$ and $p=1.0$ it is $\eta=0.6$.](sw-bif-eta-m-xi-PI5-o8-N10000-k20-J-10p00-W15p00-p0p80 "fig:"){width="0.45\columnwidth"} ![\[fig:mix\] (Color online.) Small-world ferro-anti ferro bifurcation diagram (left axis, dots in magenta) and entropy $\eta$ (right axis, continuous curve in green) of the magnetization $m$ as a function of $\xi$ for different values of the long range probability $p$ and $N=10,000$, $|J|=10$,. for $P=0.05$ there is no threshold for $\eta$, for $p=0.2$ it is $\eta=0.7$, for $p=0.8$ and $p=1.0$ it is $\eta=0.6$.](sw-bif-eta-m-xi-PI5-o8-N10000-k20-J-10p00-W15p00-p1p00 "fig:"){width="0.45\columnwidth"}
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Conclusions {#sec:conclusions}
===========
We investigated the phase transitions of a nonlinear, parallel version of the Ising model, characterized by a linear coupling $J<0$ and a nonlinear one $W>0$. The mean-field approximation shows chaotic oscillations, by changing the couplings $J$ and $W$ or the connectivity $K$. We showed in the Appendix that there is a scaling relation among these parameters.
The nonlinear Ising model was studied on small-world networks, where $p$ is the probability of long-range rewiring of links. Here, entropy of the magnetization becomes a measure of disorder which is adequate once a threshold between the presence and absence of noisy periodic orbits is established. The noisy periodic and disordered behavior of $m$ imply a certain degree of synchronization of the spins, induced by long-range couplings.
We have shown also that similar bifurcations may be induced in the randomly connected model by changing the parameters $J$, the dilution factor $d$ and the heterogeneity $\xi$, by mixing ferromagnetic and antiferromagnetic interactions.
In particular, we observed that a small percentage of asynchronism or of ferromagnetic nodes favours the first period-doubling bifurcation, which appears first for intermediate values of $d$ and $\xi$.
This model is a generalization of an opinion formation model presented in Refs. [@bagnoli2013] and [@bagnoli2015]. In contrast with those investigations, we developed here the whole model within the framework of the parallel Ising model, with transition probabilities that are continuous and smooth, derived from Hamiltonian couplings. We found the mean-field bifurcation and phase diagrams as functions of $J$ and $W$, and discussed the dynamics on small-world networks as functions of the long-range rewiring probability $p$ and the heterogeneity $\xi$ The results are qualitatively similar to those found before, showing a certain degree of universality regardless of the details of the model.
We obtained also new results, such as the mapping among the parameters (only possible within this continuous approach) and the stochastic bifurcation phase diagram as function of the asynchronism (or dilution) $d$ of the updating rule.
The different diagrams show a striking similarity, implying that it should be possible to map one bifurcation onto the other, as we did withing the mean-field approach among $J$, $W$ and $K$.
The present model aims at incorporating the Asch effect [@Asch] in mean-field and microscopic simulations, [*i.e*]{}, the influence of social pressure and its dominance over the “linear” contrarian predisposition. The resulting mean-field approach exhibits a chaotic behavior, which is our knowledge was rarely (if ever) observed before.
What is remarkable is the appearance of coherent oscillations of the whole population also for the microscopic model, in the presence of long-range connections (small-world). This may have important consequences for social scientists: the conflict between a liberal education (contrarian attitude) and the ever present social pressure may lead to unpredictable oscillation triggered by many quantities, in our widely connected world.
Appendix
========
Continuous approximation and parameter mapping {#sec:scaling}
----------------------------------------------
![\[scaling\] (Color online.) Scaling relation Eq. for three values of the connectivity $K$, with $J(K=20)=-10$ and $W(K=20)=15$m\], and the other values of $J$ and $W$ obtained from Eq. .](scaling){width="0.85\columnwidth"}
The similarities among bifurcation diagrams with different connectivity $K$ and coupling parameters $J$, $W$ can be explained by using a continuous approximation of the mean-field equation.
By using Stirling’s approximation for the binomial coefficients in Eq. , for small values of $m$ [@ott02], we obtain $$\label{approx}
\begin{split}
&\frac{1}{2^{K}}\binom{K}{k} (1+m)^{k}(1-m)^{K-k} \simeq\\
&\qquad \sqrt{\frac{2K}{\pi (1-m^2)}} \exp\left[-\frac{2K}{1-m^2}\left(\frac{k}{K}-\frac{1+m}{2}\right)^2\right]
\end{split}$$ and therefore, substituting $2k/K-1=x$, $$\label{eq:continuous}
\begin{split}
m' =& \sqrt{\frac{K}{2\pi(1-m^2)}}\\
& \times \int \limits_{-\infty}^{\infty} \mathrm{d}x\;\exp\left(-\frac{K(x-m)^2}{2(1-m^2)}\right)
\tanh(Jx+Wx^3),
\end{split}$$ from which the convolution structure is evident.
By developing $\tanh(-Jx+Wx^3)$ at first order (i.e., large $K$ and small values of $m$), we can compute the convolution, obtaining after remapping the first terms in powers of $m$, $$\label{eq:tanh}
m' = \tanh(\tilde{J} m + \tilde{W} m^3)$$ with $$\label{eq:scaling}
\begin{split}
\tilde{J} &= J + \dfrac{3W}{K},\\
\tilde{W} &= W\left(1-\dfrac{3}{K}\right).
\end{split}$$ Notice that in the limit $K\rightarrow\infty$, $\tilde J\rightarrow J$ and $\tilde W\rightarrow W$.
The relation between parameters $J,K$ and $J_1,K_1$ of two mean-field approximations with different connectivities $K$ and $K_1$ is $$\label{eq:correspondence}
\begin{split}
J_1 &= J +\dfrac{3}{K}W\left(1-\dfrac{K-3}{K_1-3}\right),\\
W_1 & =W \dfrac{K_1}{K}\dfrac{K-3}{K_1-3}.
\end{split}$$
Since the approximation of the hyperbolic tangent is valid for small $x$, we expect that this scaling is better for large $K$, for which the convolution length is small. In Fig. \[scaling\] we report the scaling correspondence for some values of $K$.
Since $J<0$ and $K>0$, the effect of this scaling is that of lowering the absolute value of $\tilde{J}$ and $\tilde{K}$ for small $K$ (larger than 3), so, given that for a large value of $K$ and certain values of $J$ and $W$ the mean-field equation is chaotic, it may be reduced to a fixed point graph by lowering the connectivity $K$.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by project PAPIIT-DGAPA-UNAM IN109213. F.B. acknowledges partial financial support from European Commission (FP7-ICT-2013-10) Proposal No. 611299 SciCafe 2.0.
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| 0 |
---
abstract: 'We study statistics of resonances in a one-dimensional disordered chain coupled to an outer world simulated by a perfect lead. We consider a limiting case for weak disorder and derive some results which are new in these studies. The main focus of the present study is to describe statistics of the scattered complex energies. We derive compact analytic statistical results for long chains. A comparison of these results has been found to be in good agreement with numerical simulations.'
address:
- '$^1$Instituto de Ciencias F'' isicas, Universidad Nacional Aut'' onoma de M'' exico, Cuernavaca, M'' exico'
- '$^2$Department of Physics, Technion-Israel, Institute of Technology, Haifa 32000, Israel'
author:
- 'Vinayak$^{1,2}$'
title: 'Statistics of resonances in a one-dimensional chain: a weak disorder limit'
---
Introduction
============
Resonant phenomena have received much attention in atomic and nuclear physics and more recently in chaotic and disordered systems [@disordered; @GG:00; @Casati:99; @KS:06; @KS:08; @JF:09; @chaos]. Complex energies, $\tilde{{E}}_{\alpha}=E_{\alpha}-\frac{i}{2}\,\Gamma_{\alpha}$, which correspond to poles of the scattering matrix on the unphysical sheet, characterize resonances [@LandauL]. Resonances correspond to the long-lived quasi-stationary states which eventually decay to continuum while distribution of resonance widths, $P(\Gamma)$, determines decay of the corresponding survival probability with time.
In recent years, $P(\Gamma)$ has been a subject of investigations [@disordered; @GG:00] for a, simple but much studied, discrete tight-binding one dimensional random chain which is coupled to a perfect lead at one side. A numerical study [@GG:00] shows that in a broad range of $\Gamma$, $P(\Gamma)\sim \Gamma^{-\gamma}$, where the exponent $\gamma$ is very close to $1$. Intuitively the $1/\Gamma$ behaviour can be deduced by assuming a uniform distribution for the localization centers of exponentially localized states [@Casati:99]. However, from analytic point of view one usually considers an infinitely long chain in which case the average density of resonances (DOR) has a well defined limit. For a finite size system, the difference between the DOR and $P(\Gamma)$ is the normalization by the system size [@KS:06; @KS:08]. Recently, Kunz and Shapiro have derived analytic expression of the DOR for a semi-infinite disordered chain [@KS:08]. They have obtained an exact integral representation of the DOR which is valid for arbitrary lead-chain coupling strength. This has been further simplified for small lead-chain coupling strength where a universal scaling formula is found. In this limit they have proved the $1/\Gamma$-behavior of the DOR [@KS:06; @KS:08]. Besides, for the continuous limit of this model an integral representation of DOR has been obtained [@JF:09].
Kunz and Shapiro’s work has established a universal $1/\Gamma$ law for arbitrary strength of disorder in a semi-infinite chain. Numerically one can verify $1/\Gamma$ law of the DOR, similar to what has been done by Terraneo and Guarnery [@GG:00] in finite samples for $P(\Gamma)$. Such verifications require the localization lengths to be much smaller than the size of the sample. In case of weak disorder an analytic result for the localization lengths is particularly useful. It comes from a second order perturbation theory. It states that the localization length is maximum near the middle of the energy band and is proportional to $W^{-2}$ where $W$ is the width of the disorder [@thou; @krammac; @KWegner:1981]. On the other hand, this result also leads to an interesting limiting situation where the localization lengths are much longer than the sample size. This is what we refer to as a weak disorder limit in this paper. This limit has scarcely been studied hitherto although it is relevant in the study of localization through resonances. Besides, there has been a believe for some sort of universality in the weak disorder limit. In this paper we address to this limit and derive analytic results which describe the statistics of resonances. Our work probes a fresh area and studies a weak disorder limit which has never been addressed before.
For open systems, instead of studying the scattering matrix in a complex plane we follow an alternative approach where one solves the Schrödinger equation by describing a particle ejected from the system or equivalently with a boundary condition of outgoing waves (Siegert boundary condition [@Siegert]). In this approach one naturally turns up to a problem of solving a non-Hermitian effective Hamiltonian which admits complex eigenvalues $\tilde{{E}}_{\alpha}$ [@GG:00; @KS:06; @KS:08; @HKF:09]. For details of such non-Hermitian effective Hamiltonians, we refer to a recent study [@JF:10] and references therein.
We derive the statistics which describe scattered complex energies of disordered chain around those regular ones which correspond to an open chain without any disorder (clean chain). For instance, we derive average of square of the absolute values of the shifts in complex energies from the regular ones over all realizations of the set of random site energies. Similarly we obtain results for the statistics of real and imaginary parts of those shifts. These results lead to compact expressions for long chains. To show the generality of our approach we also derive these results for the so-called parametric resonances which have been particularly useful in numerical studies [@GG:00]. Finally, we give numerical verifications of our analytic results.
The paper is organized as follows. Although the system and its effective Hamiltonian have been nicely explained earlier in [@GG:00; @KS:06; @KS:08], for the sake of completeness of this paper we will describe these briefly in section II. In the same section we will also describe the exact and the parametric resonances. In Sec. III we will derive result for resonances in an open-clean chain of finite length, in terms of a polynomial equation. For long chains, we will solve this polynomial equation in the leading order of the inverse of the length. In Sec. IV we will use the perturbation theory to obtain the first and the second order corrections in the complex energies for a weak disorder. In Sec. V we will calculate statistics of the scattered complex energies. In the same section we will simplify our results for long chains and obtain compact expressions. In Sec. VI we will briefly discuss about the numerical methods to calculate complex energies of non-Hermitian effective Hamiltonians and numerically verify our analytical results. This will be followed by the conclusion in Sec. VII.
![A one-dimensional disordered chain with $N$ sites, represented in the figure by black dots, is coupled to a lead. Open circles represent sites of the lead. The outgoing plane wave is shown by the arrow where $0<\Re\{\tilde{k}\}<\pi$ and $\Im \{\tilde{k}\}<0$, so that it propagates left in the lead and its amplitude grows in the lead.[]{data-label="System"}](system.eps){width="50.00000%"}
Model and Its Effective Hamiltonian
===================================
A discrete tight-binding one dimensional chain of length $N$ (shown by positive integers, $n=1,\,2,...,\,N$, used for indexing the sites of the chain in Fig. \[System\]) is connected to an outer world (represented by a perfect lead which sites are shown by a zero and negative integers, $n=0,\,-1,\,-2,...$). Each site of the chain has the site energy $\epsilon_{n}$ where $\epsilon_{n}$ are statistically independent random variables chosen from some symmetric distribution. Each nearest neighbor site of the chain as well as of the lead is coupled by a hopping amplitude $t$. The hopping amplitude for the pair $n=0$ and $n=1$ is $t'$ which takes values from $t'=0$ (closed chain) to $t'=t$ (fully coupled chain). With this hopping, a particle, which is initially located somewhere in the chain, eventually escapes to the outer world.
Now we write down the Schrödinger equation for the entire system, $$\begin{aligned}
\label{Sch1}
-t\psi_{n+1}-t\psi_{n-1}&=&\tilde{\mathcal{E}}\psi_{n},~~~~~~~~\text{for $n<0$},
\\
\label{Sch2}
-t\psi_{-1}-t'\psi_{1}&=&\tilde{\mathcal{E}}\psi_{0}, ~~~~~~~~\text{for $n=0$},
\\
\label{Sch3}
-t'\psi_{0}-t\psi_{2}+\epsilon_{1}\psi_{1}&=&\tilde{\mathcal{E}}\psi_{1},~~~~~~~~\text{for $n=1$},
\\
\label{Sch4}
-t\psi_{n-1}-t\psi_{n+1}+\epsilon_{n}\psi_{n}&=&\tilde{\mathcal{E}}\psi_{n},~~~~~~~~\text{for $2\leq n\leq N$.}\end{aligned}$$ In order to avoid cluttering of notations we always represent quantities corresponding to disordered system by [*script letters*]{} while quantities for the clean system are represented in usual math notations. Tilde is used to discriminate the open system case from the closed one. Equation (\[Sch1\]) is for the lead where $\epsilon_{n}=0$. Equations (\[Sch2\], \[Sch3\]) describe the lead-chain coupling and Eq. (\[Sch4\]) is for the chain. As in, [@KS:08] we solve Eqs. (\[Sch1\]-\[Sch4\]) with a boundary condition of an outgoing plane wave in the lead, i.e., $\psi_{n_{\leq 0}}\propto \exp(-i\tilde{k}n)$ where $0<\Re\{\tilde{k}\}<\pi$ and $\Im \{\tilde{k}\}<0$. The condition on $\Re\{\tilde{k}\}$ ensures that the outgoing wave propagates to left in the lead. The condition on $\Im \{\tilde{k}\}$ is considered so that the amplitude of the resonance wave function grows in the lead. It comes from Eq. (\[Sch1\]) that the complex energy $\tilde{\mathcal{E}}$ is related to the complex wave vector $\tilde{k}$ via the dispersion relation $\tilde{\mathcal{E}}=-2 t \cos(\tilde{k})$. Now we eliminate all $\psi_{n}$ for $n<1$ from Eqs. (\[Sch1\]-\[Sch4\]) and obtain $$\label{reduced}
-t\psi_{n+1}-t\psi_{n-1}+\tilde{\epsilon}_{n}\psi_{n}=\tilde{\mathcal{E}}\psi_{n},$$ where $$\label{energy}
\tilde{\epsilon}_{n}=\epsilon_{n}-t\eta\, \exp(i\tilde{k})\delta_{n1},$$ for $n=1,\,2,...,\,N$. The parameter $\eta=(t'/t)^2$ measures the coupling strength to the outside world.
An effective Hamiltonian defined by the Eq. (\[reduced\]) is non-Hermitian. For instance, if $\mathcal{H}$ is the $N\times N$ tridiagonal Hermitian matrix which represents the Hamiltonian of the closed-disordered chain then one may write the effective Hamiltonian, $\tilde{\mathcal{H}}$, as $$\label{Hamil}
\tilde{\mathcal{H}}=\mathcal{H}-t\,\eta\,\lambda(\tilde{k})\, P.$$ Here $P=|1\rangle\langle1|$ is the projection for site $n=1$ and $\lambda=\exp(i\tilde{k})$. The above non-Hermitian effective Hamiltonian has been first obtained by Terraneo and Guarnery [@GG:00]. The underlying result here is that the same relation (\[Hamil\]) is valid for any Hermitian $\mathcal{H}$ representing a (closed) quantum system [@JF:10] which has $N$-dimensional state space. Resonances are characterized by the complex eigenvalues, $\tilde{\mathcal{E}}_{\alpha}$, of $\tilde{\mathcal{H}}$.
Note here dependency of $\tilde{\mathcal{H}}$ on the complex wave vector $\tilde{k}$ which is related to the complex energies via the dispersion relation mentioned above - this is not a standard eigenvalue problem. To standardize this problem “parametric resonances” are often used as an alternative. In this approach the dependence of $\lambda$ on $\tilde{k}$ is typically neglected, reducing thereby the problem of finding the eigenvalues of the effective Hamiltonian at chosen value of $\tilde{k}$. As expected, parametric resonances yield approximate statistical results which are close to those for the exact resonances in strongly localized regime [@GG:00]. Parametric resonances depend on a chosen parameter, for instance let $\tilde{k}=k_{0}$ and we fix it in the middle of the energy band, $k_{0}=\pi/2$. Writing explicitly $$\lambda(\tilde{k})=
\begin{cases}
\exp(i \tilde{k}), & \text {for exact resonances,}
\\
i, & \text {for parametric resonances.}
\end{cases}$$
From now on we set the energy scale by taking $t=1$, denoting the complex variable $\tilde{\mathcal{E}}/t$ by $\tilde{\mathcal{Z}}$. We denote the Hamiltonian matrix representing the closed-clean chain by $H$. It differs from $\mathcal{H}$ only at the diagonal as, for the clean chain case, all the site energies are zero. Calculation of the eigenvalues of $H$ is a standard exercise where one derives $z_{\alpha}=-2 \cos[\alpha\pi/(N+1)]$ for $\alpha=1,...,N$.
Before going into a detail treatment to the problem, we should first sketch the outline of our approach. We are interested in a weak disorder regime. Since our approach rely on perturbation theory, we need complex energies of open-clean chain, i.e., the $\tilde{z}_{\alpha}$s. So we will begin with calculating the resonances for open-clean chain of finite length. Then we will do the perturbation series expansion up to the second order of strength of the disorder. This will be followed by the derivation of the statistical results. Finally, we will consider the large-$N$ limit of these results.
Open-clean Chain
================
We begin with defining the resolvent $\tilde{\mathcal{G}}(z)=(z-\tilde{\mathcal{H}})^{-1}$. Using Eq. (\[Hamil\]) we may also write $$\label{resolvent}
\tilde{\mathcal{G}}(z)=(z-\mathcal{H}+\eta\,\lambda\,P)^{-1}.$$ For the open-clean chain we define the resolvent $$\begin{aligned}
\tilde{G}(z)&=&(z-H+\eta\,\lambda\,P)^{-1}
\nonumber
\\
&=&
(1+\eta\,\lambda\,GP)^{-1}\,G,\end{aligned}$$ where we have introduced $G(z)=(z-H)^{-1}$ as the resolvent for the “unperturbed" closed-clean chain. Resonances correspond to the singularities of the matrix $\tilde{G}_{mn}(z)$, or to the roots of the secular equation $$\label{charcpol}
F(z)=0
=1+\eta\lambda G_{11}(z),$$ where $G_{11}$ is the $\{1,\,1\}$ element of the matrix $G$ in site representation. ($G_{nm}(z)=\langle n|(z-H)^{-1}|m\rangle$.)
![Comparison of the result (\[resz\]) (pluses) with the numerical solution of the polynomial equation (\[fincharpol\]) (circles, squares and diamonds) for the exact resonances where $\eta=0.5,\,0.81$ and $0.99$. We have considered $N=100$.[]{data-label="ansatz-exact"}](ansatz-exact.eps){width="75.00000%"}
To obtain $G_{11}$ for finite $N$, we use the ordinary difference equation (ODE), $$\label{ODE}
\psi_{n+1}+\psi_{n-1}+z\,\psi_{n}=0,$$ with the boundary conditions $\psi_{0}=\psi_{N+1}=0$. This equation is obtained from Eq. (\[reduced\]) by setting all $\epsilon_{n}=0$. Next we consider $u_{n}(z)$ and $v_{n}(z)$ to be the two linearly independent functions which satisfy the ODE $$\begin{aligned}
\label{un}
u_{n+1}+u_{n-1}+z\,u_{n}&=&0,
\\
\label{vn}
v_{n+1}+v_{n-1}+z\,v_{n}&=&0,\end{aligned}$$ where $u_{0}=v_{N+1}=0$. Since norm of $u_{n},\,v_{n}$ is arbitrary, we fix $u_{1}=v_{N}=1$. Further we claim that the resolvent is given by $$\label{G0uv}
G_{nm}=-\dfrac{u_{n}\,v_{m}\Theta(m-n)+u_{m}\,v_{n}\Theta(n-m)}{W_{n}}.$$ Here $\Theta(n)$ is the unit-step function and $W_{n}=u_{n}v_{n-1}-u_{n-1}v_{n}$ is the Wronskian. Using Eqs. (\[un\], \[vn\]) it is straight forward to see that the Wronskian is independent of $n$. One can also check that $$G_{n+1m}+G_{n-1m}+z\,G_{nm}=\delta_{nm}.$$ We now set $u_{n}=v_{N+1-n}$ to match the initial value problem (\[un\], \[vn\]) to the boundary value problem (\[ODE\]). We find $$\label{Gun}
G_{11}=-\dfrac{u_{N}}{u_{N+1}}.$$ The ODE (\[un\]) is satisfied by the Chebyshev polynomial of the second kind, $U_{m}(-z/2)$, defined as $$U_{m}(x)=\dfrac{\sin[(m+1)\cos^{-1}(x)]}{\sin[\cos^{-1}(x)]},$$ for $U_{0}(x)=1$ and $U_{1}(x)=2x$. Since we have fixed $u_{1}=1$, therefore $u_{n}=U_{n-1}$, thus we can write Eq. (\[Gun\]) as $$\label{GF}
G_{11}=-\dfrac{U_{N-1}(-z/2)}{U_{N}(-z/2)}=-\dfrac{\sin[N\,k]}{\sin[(N+1)k]}.$$ Here the last equality follows from the energy dispersion relation. Using Eq. (\[GF\]) in Eq. (\[charcpol\]), we end up with an algebraic equation $$\label{fincharpol}
F(z)
=0=
1-\eta\lambda\dfrac{U_{N-1}(-z/2)}{U_{N}(-z/2)}.$$
![Repeated on the same pattern of Fig. \[ansatz-exact\] but for parametric resonances.[]{data-label="ansatz-parametric"}](ansatz-parametric.eps){width="75.00000%"}
Zeros of $F(z)$ are the roots of a polynomial of order $N$. For exact resonances Eq. (\[fincharpol\]) can be easily transformed into $$\label{Az}
[\mathsf{a}(z)]^{(2N+1)}=
\dfrac{\mathsf{a}(z)^{-1}-\eta\,\mathsf{a}(z)}{1-\eta},$$ where $$\label{Aexp}
\mathsf{a}(z)=-\exp[ik(z)].$$ In order to solve Eq. (\[Az\]), we propose an ansatz assuming that opening of the system at one end causes $\mathcal{O}(N^{-1})$ complex corrections to the $k_{\alpha}$’s. Let $$\label{ansatz}
\tilde{k}_{\alpha}=k_{\alpha}+\dfrac{\Phi_{\alpha}}{N},$$ where $\Phi_{\alpha}$ is a complex quantity and $k_{\alpha}=\alpha\pi/(N+1)$. Inserting this ansatz into Eqs. (\[Az\], \[Aexp\]) we obtain $$\label{res}
\tilde{k}_{\alpha}= k_{\alpha}-
\dfrac{i}{2N}\,
\text{ln}\left[\Omega(k_{\alpha};\eta)\right]+\mathcal{O}\left(\dfrac{1}{N^2}\right),$$ where $$\label{Omega}
\Omega(k;\eta)=
\dfrac{1-\eta \,e^{2i\,k_{\alpha}}}{1-\eta}.$$ Now, up to $\mathcal{O}(N^{-1})$, $\tilde{z}_{\alpha}$ may be written as $$\label{resz}
\tilde{z}_{\alpha}=-2\cos(k_{\alpha})-
\dfrac{i\sin(k_{\alpha})}{N}\,
\text{ln}(\Omega).$$ The same result can be obtained for the parametric resonances, after repeating the similar steps, but with different $\Omega$: $$\label{Omgpara}
\Omega(k_{\alpha};\eta)=
\dfrac{1-i\eta \,e^{i\,k_{\alpha}}}
{1-i\eta \,e^{-i\,k_{\alpha}}}.$$
![Scatter plot for exact resonances where $N=100$, $\eta=0.81$. Dense points in the graph represent exact resonances in the disordered chain for $2500$ realizations where $W=0.015$. These are scattered around dots which represent exact resonances in the clean chain.[]{data-label="scatter_exact"}](scatter-exact.eps){width="75.00000%"}
One should bear in mind that there is no resonance for $\eta=1$, as the system is fully coupled to the lead. However, for parametric resonances, one artificially gets resonances even when $\eta=1$. Note that the result (\[resz\]) is symmetric about the imaginary axis for both cases. In Fig. \[ansatz-exact\] and Fig. \[ansatz-parametric\] we compare the numerical solutions of the polynomial equation (\[fincharpol\]) with our results (\[resz\], \[Omega\], \[Omgpara\]), for $N=100$, $\eta=0.50,\,0.81$ and $0.99$ and $N=100$, respectively for exact and parametric resonances. Eq. (\[fincharpol\]) has been solved by using the Newton’s method with the initial guess $\tilde{k}_{\alpha}=k_{\alpha}$. These figures show that our result (\[resz\]) is close to the numerical solution. The agreement gets better as $\eta\rightarrow1$ (not shown here separately). However, the ansatz (\[ansatz\]) is not valid near the band edges. Moreover, the agreement fails for parametric resonances near the middle of the band as $\eta\rightarrow 1$; see Fig. \[ansatz-parametric\] for $\eta=0.99$.
The Weak disorder limit
=======================
![Scatter plot for parametric resonances where $N=100$, $\eta=0.81$ and $W=0.015$, for $5000$ realizations. As in Fig.\[scatter\_exact\], here also dots represent the clean chain and points represent the disordered chain.[]{data-label="scatter_parametric"}](scatter-parametric.eps){width="75.00000%"}
In the next stage of the problem we switch on a very weak disorder in the chain. From a second order perturbation theory we know that for a disordered infinitely long chain the localization length, $\xi(E)$, is maximum at the middle of the band. For small $W$ it is given by [@krammac; @thou] $$\begin{aligned}
\label{xi}
\xi(E)=\dfrac{24(4t^2-E^2)}{W^2},\end{aligned}$$ implying thereby, $\xi(0)=\dfrac{96 t^2}{W^2}$. However, the exact result shows a small deviation at the band center due to the breakdown of the second-order perturbation theory [@KWegner:1981]. We consider a limiting situation when $\xi(0)/N >> 1$. For instance, in Fig. \[scatter\_exact\] and in Fig. \[scatter\_parametric\], we show the scatter plot ($\Re\{\tilde{\mathcal{Z}}_{\alpha}\}$ vs $\Im\{\tilde{\mathcal{Z}}_{\alpha}\}$) for exact and parametric resonances respectively. In both cases we have considered $N=100$, $\eta=0.81$ and $W=0.015$ so that $\xi(0)>>N$. As seen in these figures, complex energies of the disordered chain are scattered around the $\tilde{z}_{\alpha}$s.
We now calculate the corrections to $\tilde{z}_{\alpha}$ for such weak disorder case. It is suggestive here to deal with the self-energy. Let $\mathcal{S}_{1}(\epsilon_{2},...,\epsilon_{N};z)$ be the self-energy for the first site, defined via $$\begin{aligned}
\label{GS}
\mathcal{G}_{11}(z)=\dfrac{1}{z-\epsilon_{1}-\mathcal{S}_{1}(\{\epsilon\};z)}.\end{aligned}$$ Here $\{\epsilon\}$ denotes the set $\epsilon_{2},...,\epsilon_{N}$ and $\mathcal{G}_{11}$ is the $\{1,\,1\}$ element of the resolvent $\mathcal{G}(z)=(z-\mathcal{H})^{-1}$, defined for the Hermitian matrix $\mathcal{H}$. For the later convenience we write $$\begin{aligned}
\label{Hborn}
\mathcal{H}=H+\mathcal{W}, \end{aligned}$$ where $\mathcal{W}=\sum_{\ell=1}^{N}\epsilon_{\ell}P_{\ell}$ and $P_{\ell}=|\,\ell\,\rangle\langle\,\ell\,|$ is the projection for the $\ell$’th site.
In the rest of the paper we will work out results only for the exact resonances. For the parametric resonance theses results can be carried out following similar steps, so we skip all the intermediate steps merely by stating the result at the end.
As before in Eq. (\[charcpol\]), for disordered chain, resonances correspond to the roots of the secular equation $$\label{secular}
\mathcal{F}(z)=0=z-\epsilon_{1}-\mathcal{S}_{1}(\{\epsilon\};z)+\lambda\eta.$$ Preserving $\tilde{z}_{\alpha}$ as the roots of Eq. (\[charcpol\]), we define $\tilde{\mathcal{Z}}_{\alpha}$ as the roots of Eq. (\[secular\]). Now we expand the roots $\tilde{\mathcal{Z}}_{\alpha}=\tilde{z}_{\alpha}+(\delta_{1} \tilde{\mathcal{Z}}_{\alpha})+(\delta_{2} \tilde{\mathcal{Z}}_{\alpha})$, assuming that $(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})$ are linear while $(\delta_{2} \tilde{\mathcal{Z}}_{\alpha})$ are quadratic in the $\epsilon_{j}$ , for $j=1,...,N$. Then for $\mathcal{S}_{1}(\{\epsilon\};\tilde{\mathcal{Z}}_{\alpha})$, up to $\mathcal{O}(\{\epsilon\}^{2})$, we get $$\begin{aligned}
\label{Selfenergy}
\mathcal{S}_{1}(\{\epsilon\};\tilde{\mathcal{Z}}_{\alpha})
&=&
S_{1}(\{0\};\tilde{z}_{\alpha})+
\sum_{n=2}^{N}\epsilon_{n}
\left(\dfrac{\partial \mathcal{S}_{1}(\{\epsilon\};z)}{\partial\epsilon_{n}}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
\nonumber
\\
&+&
(\delta_{1} \tilde{\mathcal{Z}}_{\alpha}+\delta_{2} \tilde{\mathcal{Z}}_{\alpha})\left(\dfrac{\partial \mathcal{S}_{1}(\{\epsilon\};z)}{\partial z}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
\nonumber
\\
&+&
\dfrac{1}{2}\sum_{n,m=2}^{N}\epsilon_{n}\epsilon_{m}
\left(\dfrac{\partial^{2} \mathcal{S}_{1}(\{\epsilon\};z)}{\partial\epsilon_{n}\partial\epsilon_{m}}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
\nonumber
\\
&+&
\dfrac{1}{2}(\delta_{1} \tilde{\mathcal{Z}}_{\alpha})^{2}\left(\dfrac{\partial^{2} \mathcal{S}_{1}(\{\epsilon\};z)}{\partial^{2} z}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}.\end{aligned}$$ We will use this expansion in Eq. (\[secular\]). Before that we evaluate $$\begin{aligned}
\label{SGdz}
1-\left(\dfrac{\partial \mathcal{S}_{1}(\{\epsilon\};z)}{\partial z}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
=
\dfrac{\partial}{\partial z} \dfrac{1}{G_{11}(z)}\bigg|_{z=\tilde{z}_{\alpha}},\end{aligned}$$ and, $$\begin{aligned}
\label{dGdz}
\dfrac{\partial \mathcal{S}_{1}(\{\epsilon\};z)}{\partial \epsilon_{n}}\bigg |_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
&=&
\dfrac{1}{\left(G_{11}\right)^2}\,
\dfrac{\partial \mathcal{G}_{11}}{\partial \epsilon_{n}}\bigg|_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}},
\nonumber
\\\end{aligned}$$ for $n\geq2$. These equalities come from Eq. (\[GS\]). Finally, we calculate derivatives of $\mathcal{G}_{11}$, at $\{\epsilon\}=0$ and $z=\tilde{z}_{\alpha}$ with respect to $\{\epsilon\}$ by using Eq. (\[Hborn\]) for the [*Born-series*]{} expansion of $\mathcal{G}(z)$. We find $$\begin{aligned}
\label{dGdeps}
\dfrac{\partial \mathcal{G}_{11}}{\partial \epsilon_{n}}\bigg|_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
&=&
G_{1n}G_{n1}\bigg|_{z=\tilde{z}_{\alpha}}.\end{aligned}$$ Grouping all these, for the first order corrections, we obtain $$\begin{aligned}
\label{FPT1}
&&(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})
=
\dfrac{\epsilon_{1}+\sum_{n=2}^{N}\epsilon_{n}
\dfrac{G_{1n}G_{n1}}
{\left[G_{11}\right]^{2}}\Bigg|_{z=\tilde{z}_{\alpha}}}
{\dfrac{\partial}{\partial z} \dfrac{1}{G_{11}(z)}\bigg|_{z=\tilde{z}_{\alpha}}
+
\dfrac{i\eta\exp[i\tilde{k}_{\alpha}]}
{2\sin(\tilde{k}_{\alpha})}
}. \end{aligned}$$ Similarly for the second order corrections we get $$\begin{aligned}
\label{dz2}
(\delta_{2} \tilde{\mathcal{Z}}_{\alpha})
&=&
\Bigg[\sum_{n,m=2}^{N}\epsilon_{n} \epsilon_{m}
\left\lbrace
\dfrac{G_{1n}G_{nm}G_{m1}}{[G_{11}]^{2}}
-
\dfrac{[G_{1n}G_{1m}]^{2}}
{[G_{11}]^{3}}
\right\rbrace
\nonumber
\\
&-&
\dfrac{(\delta_{1} \tilde{\mathcal{Z}}_{\alpha})^{2}}{2}
\left\lbrace\left(\dfrac{\partial^{2}}{\partial^{2} z}\dfrac{1}{G_{11}}\right)+\eta\left(\dfrac{d^{2}\exp(ik(z)}{d^{2}z}\right)\right\rbrace
\Bigg]_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
\nonumber\\
&\times &
\Bigg[{\dfrac{\partial}{\partial z} \dfrac{1}{G_{11}(z)}\bigg|_{z=\tilde{z}_{\alpha}}
+
\dfrac{i\eta\exp[i\tilde{k}_{\alpha})}
{2\sin(\tilde{k}_{\alpha})}
}\Bigg]^{-1}.
\nonumber\\\end{aligned}$$
Note that $(\delta_{1} \tilde{\mathcal{Z}}_{\alpha})$ and $(\delta_{2} \tilde{\mathcal{Z}}_{\alpha}) $ have been obtained in terms of the resolvent of the closed-clean chain which we already know in terms of Chebyshev polynomials; see Eq. (\[G0uv\]) and the relation between $u_{n}$ and $v_{n}$ with Chebyshev polynomials.
Statistics of the Scattered Complex Energies
============================================
We are interested in the statistics of the scattered complex energies. For instance, using the first order result (\[FPT1\]) of the perturbation theory, we calculate average of square of absolute shift in complex energies defined as, $\langle|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2\rangle\equiv\langle|(\tilde{\mathcal{Z}}_{\alpha}-\tilde{z}_{\alpha})|^2\rangle$. The angular brackets are used here to represent the averaging over many realizations of set of all random site energies $\{\epsilon_{n}\}$. This quantity gives a statistical account for the scattered complex energies. We also calculate $\langle\,( \Re\{\Delta\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$ and $\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$, viz, average of square of the real and the imaginary part of the shift $(\tilde{\mathcal{Z}}_{\alpha}-\tilde{z}_{\alpha})$, respectively. To obtain the latter quantities we need first to calculate $\langle\,(\Delta \tilde{\mathcal{Z}}_{\alpha})\,^{2}\rangle$ and $\langle\,[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}\,\rangle$, since $$\begin{aligned}
(\Re\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}=\dfrac{(\Delta \tilde{\mathcal{Z}}_{\alpha})\,^{2}+[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}+2(|\Delta \tilde{\mathcal{Z}}_{\alpha})\,|^{2})}{4},
\nonumber
\\
\\
(\Im\{\Delta\tilde{\mathcal{Z}}_{\alpha}\})^{2}=-\dfrac{(\Delta \tilde{\mathcal{Z}}_{\alpha})\,^{2}+[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}-2(|\Delta \tilde{\mathcal{Z}}_{\alpha})\,|^{2})}{4}.
\nonumber
\\\end{aligned}$$ Here we have used $\{^{*}\}$ to represent the complex conjugate (c.c.).
For all these three statistics we simplify $(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})$, given in Eq. (\[FPT1\]), in terms of Chebyshev polynomials as $$\begin{aligned}
\label{FPT2}
(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})
&=&
\Bigg[\dfrac{\tilde{z}_{\alpha}^2-4}{2}
\sum_{n=1}^{N}\epsilon_{n}
\left(U_{N-n}(\tilde{z}_{\alpha}/2)\right)^{2}
\Bigg]
\nonumber\\
&\times&
\Bigg[U_{N-1}(\tilde{z}_{\alpha}/2)T_{N+1}(\tilde{z}_{\alpha}/2)-N
\nonumber\\
&-&
i\eta\exp[i\tilde{k}_{\alpha}]\sin(\tilde{k}_{\alpha})[U_{N-1}(\tilde{z}_{\alpha}/2)]^2
\Bigg]^{-1},\end{aligned}$$ where $T_{m}(z)=\cos[m\cos^{-1}(z)]$ is the Chebyshev polynomial of the first kind. Further simplifications occur when these polynomials are expressed in their trigonometric forms. For instance, let’s calculate $|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2$, with $ \tilde{z}_{\alpha}/2=\cos( \tilde{\theta}_{\alpha})$ where $\tilde{\theta}_{\alpha}=\pi- \tilde{k}_{\alpha}$. We obtain $$\begin{aligned}
\label{abdz2}
\dfrac{|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2}{4}
=
\dfrac{\sum_{n,m=1}^{N}\epsilon_{n'}\epsilon_{m'}\sin^{2}(n' \tilde{\theta}_{\alpha})\sin^{2}(m'\tilde{\theta}^{*}_{\alpha})}
{|D(\tilde{z_{\alpha}})|^{2}}.\end{aligned}$$ Here $n'$ and $m'$ are respectively $N+1-n$ and $N+1-m$, and $D(\tilde{z_{\alpha}})$ is simply the quantity in the second bracket of Eq. (\[FPT2\]). Averaging releases one of the summation as the $\epsilon_{j}'$s are statistically independent-identically-distributed (i.i.d.) random variables. We simply have $$\begin{aligned}
\label{abdz3}
\langle|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2\rangle
=
\sigma^{2}\dfrac{\sum_{n=1}^{N}4\sin^{2}(n\tilde{\theta}_{\alpha})\sin^{2}(n\tilde{\theta}^{*}_{\alpha})}
{|D|^{2}},\end{aligned}$$ where $\sigma^{2}$ is variance of the $\epsilon_{j}'$s.
Summation in the above equality can be performed by using trigonometric identities. For instance, we first write $$\begin{aligned}
\label{abdz4}
4\sin^{2}(n\theta)\sin^{2}(n\theta^{*})
&=&
1-\cos(2n\theta)-\cos(2n\theta^{*})
\nonumber\\
&+&
\dfrac{\cos(4\,n\,\Re\{\theta\})+\cos(4\,i\,n\,\Im\{\theta\})}{2},\end{aligned}$$ and we use the summation formula $$\begin{aligned}
\label{abdz5}
\sum_{n=1}^{N}\cos(n\theta)
&=&
\dfrac{1}{2}
\left[
\dfrac{\sin[(N+1/2)\theta]}{\sin(\theta/2)}-1
\right].\end{aligned}$$ It turns out after some trigonometry that one can write the summation in a closed form. We find $$\begin{aligned}
\sum_{n=1}^{N}4\sin^{2}(n\theta)\sin^{2}(n\theta^{*})
&=&N+\dfrac{1}{2}-\dfrac{U_{2N}+U_{2N}^{*}}{2}+
\dfrac{T^{*}_{2N+2}T_{2N}-T_{2N+2}T^{*}_{2N}}{2[T^{*}_{2}-T_{2}]}.
\nonumber
\\\end{aligned}$$ Here the argument of the polynomials is $\tilde{z}_{\alpha}/2$ and for their complex conjugate it is $\tilde{z}^{*}_{\alpha}/2$. Finally, we write down finite-$N$ result for average of the absolute square of the shift, $$\begin{aligned}
\label{abdz6}
\langle|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2\rangle
=
\sigma^{2}\dfrac{N+\dfrac{1}{2}-\dfrac{U_{2N}+U_{2N}^{*}}{2}
+
\dfrac{T^{*}_{2N+2}T_{2N}-T_{2N+2}T^{*}_{2N}}{2[T^{*}_{2}-T_{2}]}}
{|D|^{2}}.\end{aligned}$$
We now turn our attention to large-$N$ behavior of the result (\[abdz6\]). For this purpose we use the ansatz (\[ansatz\]) and result the (\[res\]) for $\tilde{k}_{\alpha}$. Large-N behavior for the Chebyshev polynomials, with argument $\tilde{z}_{\alpha}$, may be calculated as $$\begin{aligned}
T_{2N}(\tilde{z}_{\alpha}/2)&=&\dfrac{\exp[2iN\tilde{\theta}_{\alpha}]+\exp[-2iN\tilde{\theta}_{\alpha}]}{2}
\nonumber
\\
&\approx&
\dfrac{\Omega({k}_{\alpha};\eta)\exp(-2i{k}_{\alpha})+\left[\Omega({k}_{\alpha};\eta)\right]^{-1}\exp(2i{k}_{\alpha})}{2},$$ $$\begin{aligned}
T_{2N+2}(\tilde{z}_{\alpha}/2)
=
\dfrac{\Omega(k_{\alpha};\eta)+\left[\Omega(k_{\alpha};\eta)\right]^{-1}}{2}+\mathcal{O}(N^{-1}),\end{aligned}$$ $$\begin{aligned}
U_{2N}(\tilde{z}_{\alpha}/2)&=&\dfrac{\exp[i(2N+1)\tilde{\theta}_{\alpha}]-\exp[-i(2N+1)\tilde{\theta}_{\alpha}]}
{\exp(i\tilde{\theta}_{\alpha})-\exp(-i\tilde{\theta}_{\alpha})}
\nonumber
\\
&\approx&
\dfrac{\Omega({k}_{\alpha};\eta)\exp(-i{k}_{\alpha})-\left[\Omega({k}_{\alpha};\eta)\right]^{-1}\exp(i{k}_{\alpha})}{\exp(-i{k}_{\alpha})-\exp(i{k}_{\alpha})}.\end{aligned}$$ Finally, $$\begin{aligned}
T_{2}(\tilde{z}^{*}_{\alpha}/2)- T_{2}(\tilde{z}_{\alpha}/2)
&=& -\dfrac{2i\,\Im\{\Phi_{\alpha}\}}{N}\,
{z}_{\alpha}+\mathcal{O}(N^{-2})
\nonumber
\\
&\approx&
\dfrac{4i}{N}\, \cos(k_{\alpha})\,\Im\{\Phi_{\alpha}\},\end{aligned}$$ where we have used the ansatz (\[ansatz\]) in the second order polynomial $T_{2}(z)=2z^{2}-1$ and $\Im\{\Phi_{\alpha}\}=-\sin({k}_{\alpha})\,\text{ln}(|\Omega|)$, as obtained from Eqs. (\[ansatz\], \[res\]).
We can now plug in these results in Eq. (\[abdz6\]). These asymptotic results gives the numerator as ($N+a1+a2/(a3/N)$) where $a1,\,a2/a3$ are $\mathcal{O}(N^{0})$. Similarly we obtain denominator as ($\,N^{2}+b1\,N+b2$) where $b1$ and $b2$ are $\mathcal{O}(N^{0})$; see Appendix A for details. Thus in the leading order, we obtain $$\begin{aligned}
\label{abdz7}
\langle|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2\rangle
&=&
\dfrac{\sigma^{2}}{N}\,
\left(
1+\dfrac{1}{8}\,
\dfrac{
\left(
|\Omega|^{2}-|\Omega|^{-2}
\right)}
{\text{ln}(|\Omega|)}
\right).\end{aligned}$$
![Asymptotic results for $\langle\,|\delta_{1}\tilde{\mathcal{Z}}_{\alpha}|^{2}\,\rangle/\sigma^{2}$, $\langle\,(\Re\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ and $\langle\,(\Im\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$, shown by filled circles, squares and diamonds, vs energy index $\alpha$. We have compared here the finite-$N$ results, shown by open circles, for the exact resonances where $N=100$ and $\eta=0.81$. In the set we show these results for 14 energy indices near the middle of the energy band but on a different scale.[]{data-label="ex_finitevsLarge"}](ex-finitevsLarge.eps){width="75.00000%"}
What follows next is the calculation of large-$N$ results for $\langle\,( \Re\{\Delta\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$ and $\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$. Since we need first to calculate $\langle\,(\Delta \tilde{\mathcal{Z}}_{\alpha})\,^{2}\rangle$ and $\langle\,[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}\,\rangle$, from Eq. (\[FPT2\]) we obtain after averaging $$\begin{aligned}
\label{dzsq}
\langle\,(\Delta \tilde{\mathcal{Z}}_{\alpha})^{2}\,\rangle
=\sigma^{2}\dfrac{\sum_{n=1}^{N}4\sin^{4}(n\tilde{\theta}_{\alpha})}
{D^{2}},\end{aligned}$$ and $$\begin{aligned}
\label{cdzsq}
\langle\,[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}\,\rangle
=\sigma^{2}\dfrac{\sum_{n=1}^{N}4\sin^{4}(n\tilde{\theta}^{*}_{\alpha})}
{(D^{*})^{2}}.\end{aligned}$$
![Shown on the same pattern of Fig. \[ex\_finitevsLarge\] but for the parametric resonances where $N=500$ and $\eta=0.81$. In this figure $\langle\,|\delta_{1}\tilde{\mathcal{Z}}_{\alpha}|^{2}\,\rangle/\sigma^{2}$, $\langle\,(\Re\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ and $\langle\,(\Im\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ are shown respectively by pluses, crosses and stars. The inset is shown for the indices near the middle of the band.[]{data-label="pa_finitevsLarge"}](pa-finitevsLarge.eps){width="75.00000%"}
For summation we use the formula [@GR] $$\begin{aligned}
\label{sumsin4}
\sum_{n=1}^{N} \sin^{4}(n\,\theta)
&=&
\dfrac{1}{8}
\Bigg[
3N-\dfrac{\sin(N\theta)}{\sin(\theta)}
\big(4\cos[(N+1)\theta]
\nonumber
\\
&-&
\dfrac{\cos[2(N+1)\theta]\,\cos(N\theta)}{\cos(\theta)}
\big)\,
\Bigg],
\nonumber
\\
\sum_{n=1}^{N} \sin^{4}(n\,\tilde{\theta}_{\alpha})&=&
\dfrac{1}{8}
\left[
3N-4U_{N-1}T_{N+1}+\dfrac{T_{2N+2}U_{2N-1}}{\tilde{z}_{\alpha}}
\right],\end{aligned}$$ where in the second equality the polynomials have argument $\tilde{z}_{\alpha}/2$ with $2\cos(\tilde{\theta}_{\alpha})=\tilde{z}_{\alpha}$. Similarly for the summation in Eq. (\[cdzsq\]) one gets the polynomials with argument $\tilde{z}^{*}_{\alpha}/2$. One can now use the equality (\[sumsin4\]) in Eqs. (\[cdzsq\], \[cdzsq\]) in order to derive finite-$N$ result for $\langle\,(\Re\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$ and $\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$. For large-$N$ we make use of the ansatz (\[ansatz\]) and calculate the leading order contribution as $$\begin{aligned}
\label{LargeNdx}
\langle\,(\Re\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle
&=&
\dfrac{\sigma^{2}}{2N}
\Bigg\{\dfrac{5}{2}+\dfrac{1}{8}\,
\dfrac{
\left(
|\Omega|^{2}-|\Omega|^{-2}
\right)}
{\text{ln}(|\Omega|)}
+
g(k_{\alpha})\Bigg\},
\\
\nonumber
\\
\label{LargeNdy}
\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle
&=&
-\dfrac{\sigma^{2}}{2N}
\Bigg\{\dfrac{1}{2}-\dfrac{1}{8}\,
\dfrac{
\left(
|\Omega|^{2}-|\Omega|^{-2}
\right)}
{\text{ln}(|\Omega|)}
+
g(k_{\alpha})\Bigg\},\end{aligned}$$ where $$\begin{aligned}
g(k_{\alpha})&=&
\dfrac{1}{8}
\left(
\dfrac{e^{-2ik_{\alpha}}\Omega^{2}-e^{2ik_{\alpha}}\Omega^{-2}}
{4i \sin(k_{\alpha})\left(2N\cos(k_{\alpha})+i\sin(k_{\alpha})\text{ln}(\Omega)\right)}
\right)
+
(\text{c.c.}).\end{aligned}$$
Equations (\[abdz7\], \[LargeNdx\], \[LargeNdy\]) are our main analytical results and they are also valid for parametric resonances with the $\Omega$ given in Eq. (\[Omgpara\]). In Fig. \[ex\_finitevsLarge\] we verify the asymptotic results (\[abdz7\], \[LargeNdx\], \[LargeNdy\]) against their finite-$N$ counterparts, for exact resonances with $N=100$ and $\eta=0.81$. Fig. \[pa\_finitevsLarge\] is repeated on the same pattern but for parametric resonances where $N=500$ and $\eta=0.81$. They confirm that the asymptotic results give a good account for the finite-$N$ results. However, there are some exception near the edges (not visible on the scale of the plot) where the ansatz (\[ansatz\]) is not valid.
It turns out that in order to calculate the DOR we need the second order corrections $(\delta_{2}\tilde{\mathcal{Z}}_{\alpha})$, derived in Eq. (\[dz2\]). We have followed the method used earlier [@FZ:99] for Hatano-Nelson Model [@HN:97]. However, we have not been able to obtain a closed expression of the DOR. This is discussed in Appendix B where we leave the calculations with a formal expression for the DOR.
Numerical Methods and Verification of The Eqs. (\[abdz7\], \[LargeNdx\], \[LargeNdy\])
======================================================================================
![Comparison of asymptotic results with numerics, for exact resonances where $N=100$, $W=0.015$ and $\eta=0.81$. In this figure, filled circles, squares and diamonds are the numerical results respectively for $\langle\,|\delta_{1}\tilde{\mathcal{Z}}_{\alpha}|^{2}\,\rangle/\sigma^{2}$, $\langle\,(\Re\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ and $\langle\,(\Im\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ while open circles are the rescaled theories (\[abdz7\], \[LargeNdx\]) and (\[LargeNdy\]). In the inset we show a comparison for 14 indices near the middle of the energy band on a different scale of the plot.[]{data-label="ex_numericsvsLarge"}](ex-numericsvsLarge.eps){width="75.00000%"}
Numerical simulations for parametric resonance are always cost efficient. The reason being that there one deals with standard eigenvalue problem for which many fast subroutine packages are available, for instance LAPACK. On the other hand to verify the results (\[abdz7\], \[LargeNdx\], \[LargeNdy\]) for exact resonances, where one needs to obtain numerical solutions of a characteristic polynomial equation of order $N$ in a complex plane, there is no as good algorithm. In this paper we show results for the exact resonances by calculating roots of the characteristic polynomial where we have used a cost efficient numerical subroutine [*ezero*]{}. The subroutine is available on the CPC program library. There is one major advantage of using this subroutine over other methods, for instance the Newton’s method. This subroutine does not require initial guesses for the roots but only the contour which encloses all the roots of the polynomial. Besides, it also avoids calculating the derivatives which may result into numerical overflow.
In alternative to [*ezero*]{} we have used a different approach for calculating the roots. We survey the complex $\tilde{k}$-plane for the zeros of the $\text{Det}[M(\tilde{k})M(\tilde{k})^{\dagger}]$ where $M_{rs}=-2\cos(\tilde{k})\delta_{rs}-\tilde{\mathcal{H}}_{rs}$ for $r,s=1,...,N$ [@Neuberger]. (In our system $-\pi<\Re\{\tilde{k}\}<\pi$ and $\Im\{\tilde{k}\}<0$.) These zeros give the eigenvalues of $\tilde{\mathcal{H}}$. However, in the latter approach it is advisable to disintegrate the complex plane into small cells at first and then at every iteration into smaller one - only for $N$ cells which contain minima of the lowest eigenvalue and throwing the rest out. In this way one makes the algorithm faster and obtain the zeros in a reasonable precision. For a tridiagonal matrix this algorithm consumes a time which roughly grows with $N^3$. However, while comparing the two methods on a simple machine we find that the method used in [*ezero*]{} is much faster than the method described here. We refer to [@ezero] for further details of this subroutine.
In Fig. \[ex\_numericsvsLarge\], we compare asymptotic results with simulation done for the total number of realizations $L=2500$, for exact resonances. In Fig. \[pa\_numericsvsLarge\] we compare numerical results obtained for parametric resonances, where $N=500$, $\eta=0.81$ and $L=5000$, with our theory for large-$N$. Though we have considered only the flat disorder yet our results are valid for the Gaussian or other symmetric distribution functions. These figures show that our asymptotic results are in fair agreement with the numerical results for almost all $\alpha$. For instance, near the middle of the energy band it describes reasonably well a dip and a peak, respectively in the $\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$ and $\langle\,(\Re\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$. These two opposite effects, however, cancel out in $\langle\,(|\{\Delta \tilde{\mathcal{Z}}_{\alpha}\}|)^{2}\,\rangle$.
![ Shown on the same pattern of Fig. \[ex\_numericsvsLarge\] but for the parametric resonances where $N=500$, $W=0.015$ and $\eta=0.81$. These numerical results are obtained from the diagonalization of $N$-dimensional matrices for $5000$ realizations. In this figure $\langle\,|\delta_{1}\tilde{\mathcal{Z}}_{\alpha}|^{2}\,\rangle/\sigma^{2}$, $\langle\,(\Re\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ and $\langle\,(\Im\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ are shown respectively by pluses, crosses and stars while open circles are the rescaled theories (\[abdz7\], \[LargeNdx\]) and (\[LargeNdy\]).[]{data-label="pa_numericsvsLarge"}](pa-numericsvsLarge.eps){width="75.00000%"}
Conclusion
==========
In conclusion, we have studied resonances in a one dimensional discrete tight-binding open chain in a weak disorder limit. In this study we have calculated complex energies in an open-clean chain of finite length. The result we obtain is a polynomial equation which we have been able to solve for long chains using an ansatz for the solution. To the best of our knowledge, this result has never been derived before. We have used a perturbation theory up to the second order where we have derived the first and the second order corrections to the complex energies in terms of Chebyshev polynomials. The first order corrections have been useful to obtain closed form of the statistical results for the scattered complex energies. These results have been further simplified for long chains where we obtain compact results. The asymptotic results have been verified against numerics. Evidently, in the weak disorder limit the perturbation theory predicts nice statistical results. Our results are new in these studies and they could be useful in the further studies of such systems.
It would be interesting to study statistics of resonances in the weak disorder limit for higher dimensional models as well as for the cases when the site energies are not independent random variables but they are correlated with each other [@Izrailev:99]. Besides, there has been growing interest for the case when $M$ sites are connected to the outer world where $1\le M\le N$ [@Borgonovi:2012]. We believe that our methods could be useful for the study of such models. Finally, we mention the case where $\xi(0)\sim\mathcal{O}(N)$. It requires a separate investigation as our perturbative analysis fails in this limit.
The author is thankful to Boris Shapiro for suggesting the problem to him. The author would also like to give credit to Joshua Feinberg for the derivation of some of the equations in Secs. III and IV and also for the help in Appendix B. Discussions with both of them are gratefully acknowledged. The author also acknowledges Marko Žnidarič and Thomas H. Seligman for reading the manuscript.
Support from ISF-1067 and generous hospitality of Technion Institute are also acknowledged. Additional support by the project 79613 by CONACyT, Mexico, is acknowledged.
Large-$N$ behavior of the denominator in (\[FPT2\])
===================================================
The denominator in Eq. (\[FPT2\]) can be simplified as follows: $$\begin{aligned}
\label{app1}
D(z_{\alpha})
&=&
-N+U_{N-1}(z_{\alpha}/2)T_{N+1}(z_{\alpha}/2)-i\eta\exp[I(z_{\alpha})]
\nonumber
\\
&\times&
\sin[k(z_{\alpha})][U_{N-1}(z_{\alpha}/2)]^{2}
\nonumber
\\
&\approx&
-N+\Big[-\exp(-ik_{\alpha})[\Omega(1-\eta)+1]
\nonumber
\\
&+&
\exp(ik_{\alpha})[\Omega^{-1}(1+\eta\exp(2ik_{\alpha}))+1-2\eta]
\Big]
\nonumber
\\
&\times&
\Big[2[\exp(-ik_{\alpha})-\exp(ik_{\alpha})]\Big]^{-1}.\end{aligned}$$ Using now $$\Omega(1-\eta)+1=2-\eta \exp(2ik_{\alpha}),$$ and $$\Omega^{-1}(1+\eta\exp(2ik_{\alpha}))+1-2\eta
=
\dfrac{2-3\eta+\eta^{2}\exp(2ik_{\alpha})}
{1-\eta\exp(2ik_{\alpha})},
\nonumber
\\$$ in (\[app1\]) we get $$\begin{aligned}
D
&\simeq&
-N-\dfrac{1}{1-\eta\exp(2ik_{\alpha})}.\end{aligned}$$ For $|D|^{2}$ this yields $$\begin{aligned}
|D|^{2}
&=&
N^{2}+N\,\dfrac{1-\eta\cos(2k_{\alpha})}{1+\eta^{2}-2\eta\cos(2k_{\alpha})}
\nonumber
\\
&+&
\dfrac{1}{1+\eta^{2}-2\eta\cos(2k_{\alpha})}.\end{aligned}$$
Similarly for the parametric resonances, the denominator for large $N$ is given by $$\begin{aligned}
D
&=&
-N+U_{N-1}(z_{\alpha}/2)T_{N+1}(z_{\alpha}/2)
\nonumber
\\
&\simeq&
-(N+1/2)+
\dfrac{\exp(-ik_{\alpha})\Omega-\exp(ik_{\alpha})\Omega^{-1}}{\exp(ik_{\alpha})-\exp(-ik_{\alpha})}
\nonumber
\\
&\simeq&
-(N+1/2)
+\dfrac{(1+\eta^{2})\,\Omega}{2[\eta\exp(ik_{\alpha})+i]^{2}}.\end{aligned}$$ Thus for $|D|^{2}$ we get $$\begin{aligned}
|D|^{2}&=&N^{2} +N\,\dfrac{(1-\eta^{4})}{(1+\eta^{2})^{2}-4\eta^{2}\sin^{2}(k_{\alpha})}
\nonumber
\\
&+&
\dfrac{(1+\eta^{2})^{2}}{4[1+\eta^{2})^{2}-4\eta^{2}\sin^{2}(k_{\alpha})]}
\,
\dfrac{1+\eta^{2}+2\eta\sin(k_{\alpha})}
{1+\eta^{2}-2\eta\sin(k_{\alpha})}.
\nonumber
\\\end{aligned}$$ Clearly in both cases $|D|^{2}$ has a form $N^{2}+b1\, \{N^{1}\}+b2\,\{N^{0}\}$.
Calculation of DOR
==================
We define the average DOR as $$\begin{aligned}
\label{DOS1}
\langle\,\rho(x,y)\,\rangle=\left\langle\,\sum_{\alpha=1}^{N}\delta(x-\mathcal{\tilde{X}}_{\alpha})\delta(y-\mathcal{\tilde{Y}}_{\alpha})\,\right\rangle,\end{aligned}$$ where $\mathcal{\tilde{X}}_{\alpha}\equiv\Re\{\tilde{\mathcal{Z}}_{\alpha}\}$ and $\mathcal{\tilde{Y}}_{\alpha}\equiv\Im\{\tilde{\mathcal{Z}}_{\alpha}\}$. Next we define $\tilde{x}_{\alpha}=\Re\{\tilde{z}_{\alpha}\}$, $\tilde{y}_{\alpha}=\Im\{\tilde{z}_{\alpha}\}$, $(\delta_{1}\tilde{x}_{\alpha})=\Re\{(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})\}$, $(\delta_{1}\tilde{y}_{\alpha})=\Im\{(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})\}$, $(\delta_{2}\tilde{x}_{\alpha})=\Re\{(\delta_{2}\tilde{\mathcal{Z}}_{\alpha})\}$ and $(\delta_{2}\tilde{y}_{\alpha})=\Im\{(\delta_{2}\tilde{\mathcal{Z}}_{\alpha})\}$ and then use the expansion $\mathcal{\tilde{X}}_{\alpha}=\tilde{x}_{\alpha}+(\delta_{1}\tilde{x}_{\alpha})+(\delta_{2}\tilde{x}_{\alpha})$ and $\mathcal{\tilde{Y}}_{\alpha}=\tilde{y}_{\alpha}+(\delta_{1}\tilde{y}_{\alpha})+(\delta_{2}\tilde{y}_{\alpha})$ in (\[DOS1\]). We find $$\begin{aligned}
\label{DOS2}
&&\langle\,\rho(x,y)\,\rangle
=
\sum_{\alpha=1}^{N}\delta(x-\tilde{x}_{\alpha})\delta(y-\tilde{y}_{\alpha})
\nonumber\\
&+&
\sum_{\alpha=1}^{N}
\Big[
\langle
(\delta_{1}\tilde{x}_{\alpha})(\delta_{1}\tilde{y}_{\alpha})
\rangle
\delta'(x-\tilde{x}_{\alpha})\delta'(y-\tilde{y}_{\alpha})
-
\langle
(\delta_{2}\tilde{x}_{\alpha})
\rangle
\delta'(x-\tilde{x}_{\alpha})\delta(y-\tilde{y}_{\alpha})
\nonumber
\\
&-&
\langle
(\delta_{2}\tilde{y}_{\alpha})
\rangle
\delta(x-\tilde{x}_{\alpha})\delta'(y-\tilde{y}_{\alpha})
+
\dfrac{1}{2}
\langle(\delta_{1}\tilde{x}_{\alpha})^{2}
\rangle
\delta''(x-\tilde{x}_{\alpha})\delta(y-\tilde{y}_{\alpha})
\nonumber\\
&+&
\dfrac{1}{2}
\langle(\delta_{1}\tilde{y}_{\alpha})^{2}
\rangle
\delta(x-\tilde{x}_{\alpha})\delta''(y-\tilde{y}_{\alpha})
\Big]. \end{aligned}$$ Here $\delta'(x)=d\,\delta(x)/dx$ and similarly $\delta''(x)$ is the second derivative of the Dirac-Delta function with respect to the argument. We have already shown that $\langle(\delta_{1}\tilde{x}_{\alpha})^{2}\rangle$ and $\langle(\delta_{1}\tilde{y}_{\alpha})^{2}\rangle$ are of $\mathcal{O}(\sigma^{2}N^{-1})$ while $\langle(\delta_{1}\tilde{x}_{\alpha})(\delta_{1}\tilde{y}_{\alpha})\rangle$ is also $\mathcal{O}(\sigma^{2}N^{-1})$ since $$\begin{aligned}
\langle(\delta_{1}\tilde{x}_{\alpha})(\delta_{1}\tilde{y}_{\alpha})\rangle
=
\dfrac{\langle(\delta_{1}\mathcal{\tilde{Z}}_{\alpha})^{2}\rangle-\langle([\delta_{1}\mathcal{\tilde{Z}}_{\alpha}]^{*})^{2}\rangle}
{4i}. \end{aligned}$$
Motivated from [@FZ:99], we calculate the coefficient of $\epsilon_{n}^{2}$ in (\[dz2\]) to obtain $\langle(\delta_{2}\tilde{x}_{\alpha})\rangle$ and $\langle(\delta_{2}\tilde{y}_{\alpha})\rangle$. Let’s write $$\begin{aligned}
\label{cn1}
\delta_{1}\tilde{\mathcal{Z}}_{\alpha}
=
\sum_{n=1}^{N}\epsilon_{n}c_{n;\alpha}, \end{aligned}$$ and $$\begin{aligned}
\label{cn2}
\delta_{2}\tilde{\mathcal{Z}}_{\alpha}
=
\sum_{n,m=1}^{N}\epsilon_{n}\epsilon_{m}d_{nm;\alpha}.\end{aligned}$$ Expressing the polynomials in (\[FPT2\]) in terms of sinusoidal functions, we simply read-off $c_{n;\alpha}$: $$\begin{aligned}
\label{cn3}
c_{n;\alpha}&=&
\dfrac{-2\sin^{2}(n'\theta_{\alpha})}{D_{\alpha}}
\nonumber
\\
&=&
\dfrac{\cos(2n'\theta_{\alpha})-1}{D_{\alpha}}.\end{aligned}$$ Here $n'=N+1-n$ and $D_{\alpha}$ is the denominator inside the brackets of (\[FPT2\]). The denominator is $\mathcal{O}(N)$ thus $c_{n;\alpha}\sim \mathcal{O}(N^{-2})$, hence will be dropped off. Using now our ansatz, for $d_{nn;\alpha}$ we obtain $$\begin{aligned}
\label{dnn2}
d_{nn;\alpha}
&=&
\dfrac{(1-\cos(2r\Psi_{\alpha}))}{4\,N\,\sin(\Psi_{\alpha}/N)\sin(\Psi_{\alpha})}
\Big\{
4\cos(2 r \Psi_{\alpha})
\nonumber
\\
&-&
\cos[(1+2r)\Psi_{\alpha}]-3\cos[(1-2r)\Psi_{\alpha}]
\Big\}, \end{aligned}$$ where $r=n'/N$ and $\Psi_{\alpha}=Nk_{\alpha}$ with $k_{\alpha}$ given in (\[res\]). There is no further simplification of this result to obtain a compact and simple expression for real and imaginary parts of $d_{nn;\alpha}$ (as the authors [@FZ:99] have been able to do for with the result they obtain for the Hatano-Nelson model [@HN:97]). So we leave the density formally as $$\begin{aligned}
&&\langle\rho(x,y)\rangle
=
\rho_{0}(x,y)-\sigma^{2}\sum_{\alpha=1}^{N}
\Big(\delta'(x-x_{\alpha})\delta(y-y_{\alpha})
\nonumber
\\
&\times&
\sum_{n=1}^{N}\Re\{d_{nn;\alpha}\}
+
\delta(x-x_{\alpha})\delta'(y-y_{\alpha})
\sum_{n=1}^{N}\Im\{d_{nn;\alpha}\}
\Big),\end{aligned}$$ where $\rho_{0}(x,y)$ is first term of Eq. (\[DOS2\]).
References {#references .unnumbered}
==========
M. Weiss, J. A. Mendez-Bermudez, and T. Kottos 2006 045103; T. Kottos 2005 10761; F. A. Pinheiro, M. Rusek, A. Orlowski, and B. A. van Tiggelen 2004 026605; M. Titov and Y. V. Fyodorov R2444); C. Texier and A. Comtet 1999 [*Phys. Rev. Lett.*]{} [**82**]{} 4220 (1999).
M. Terraneo and I. Guarneri 2000 303. G. Casati, G. Maspero and D. Shepelyansky 1999 524. H. Kunz and B. Shapiro 2006 10155. H. Kunz and B. Shapiro 2008 054203. J. Feinberg 2009 565.
J. P. Keating, S. Nonnenmacher, M. Novaes and M. Sieber 2008 2591-2624; S. Wimberger, A. Krug and A. Buchleitner 2002 263601; Y. V. Fyodorov, H.-J. Sommers 1997 , 1918. L. D. Landau and E. M. Lifshitz 1997 [*Quantum Mechanics*]{} (Pergamon, Oxford).
D. J. Thouless Ill-condensed Matter ed. R. Balian, R. Maynard and G. Toulouse 1979 (Amsterdam: North-Holland). B. Kramer and A. MacKinnon 1993 [*Rep. Prog. Phys.*]{} [**56**]{} 1469. M. Kappus and F. Wegner, Z. 1981 [*Phys. B*]{} [**45**]{} 15; C. J. Lambert 1984 [*J. Phys. C: Solid State Phys.*]{} [**17**]{} 2401; B. Derida and E. Gardner 1984 [*J. Physique*]{} [**45**]{} 1283; F. M. Izrailev, S. Ruffo and L. Tessieri 1998 [*J. Phys A: Math. Gen.*]{} [**31**]{} 5263.
A. J. F. Siegert 1939 [*Phys. Rev.*]{} [**56**]{}, 750; R. E. Peierls 1959 [* Proc. Roy. Soc. London A*]{} [**253**]{} 16–35. N. Hatano, T. Kawamoto and J. Feinberg 2009 553. J. Feinberg 2010 1116.
I. S. Gradshteyn and I. M. Ryzhik 2007 [*Table of Integrals, Series, and Products*]{} (Elsevier - Academic press, Seventh Edition). C. J. Gillian, A. Schuchinsky and I. Spence 2006 304. The method is suggested by H. Neuberger and then developed and tested by the author on the system studied herein. J. Feinberg and A. Zee 1999 6433. N. Hatano and D. R. Nelson, Phys. Rev. Lett. [**77**]{} 570 (1997).
F. M. Izrailev and A. A. Krokhin 1999 4062. G. L. Celardo, A. Biella, L. Kaplan and F. Borgonovi 2012 .
| 0 |
---
author:
- Timothy Porter
title: '$\mathcal{S}$-categories, $\mathcal{S}$-groupoids, Segal categories and quasicategories'
---
The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in Hagen, it is safe to assume that the notes assume a reasonable knowledge of our book, , or any equivalent text if one can be found!
What do the notes set out to do?
“Aims and Objectives!” or should it be “Learning Outcomes”? {#aims-and-objectives-or-should-it-be-learning-outcomes .unnumbered}
===========================================================
- To revisit some oldish material on abstract homotopy and simplicially enriched categories, that seems to be being used in today’s resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions;
- To introduce Segal categories and various other tools used by the Nice-Toulouse group of abstract homotopy theorists and link them into some of the older ideas;
- To introduce Joyal’s quasicategories, (previously called weak Kan complexes but I agree with André that his nomenclature is better so will adopt it) and show how that theory links in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cordier and myself;
- To ask lots of questions of myself and of the reader.
The notes include some material from the ‘Cubo’ article, [@cubo], which was itself based on notes for a course at the *Corso estivo Categorie e Topologia* in 1991, but the overlap has been kept as small as is feasible as the purpose and the audience of the two sets of notes are different and the abstract homotopy theory has ‘moved on’, in part, to try the new methods out on those same ‘old’ problems and to attack new ones as well.
As usual when you try to specify ‘learning outcomes’ you end up asking who has done the learning, the audience? Perhaps. The lecturer, most certainly!
**Acknowledgements**
I would like to thank Heiner Kamps and his colleagues at the Fern Univeristät for the invitation to give the talks of which these notes are a summary and to the Fern Univeristät for the support that made the visit possible, to José Manuel García-Calcines, Josué Remedios and their colleagues and for the Departamento de Mathematica Fundamental in the Universidad de La Laguna, Tenerife, simlarly and also to Carlos Simpson, Bertrand Toen, André Joyal, Clemens Berger, André Hirschowitz and others at the Nice meeting in May 2003, since that is where bits of ideas that I had gleaned over a longish period of time fitted together so that I think I begin to understand the way that a lot of things interlock in this area better than I did before!
These notes have also benefitted from comments by Jim Stasheff and some of his colleagues on an earlier version.
$\mathcal{S}$-categories
========================
Categories with simplicial ‘hom-sets’
-------------------------------------
We assume we have a category $\mathcal{A}$ whose objects will be denoted by lower case letter, $x$,$y$,$z$, …, at least in the generic case, and for each pair of such objects, $(x,y)$, a simplicial set $\mathcal{A}(x,y)$ is given; for each triple $x, y, z$ of objects of $\mathcal{A}$, we have a simplicial map, called *composition* $$\mathcal{A}(x,y)\times \mathcal{A}(y,z)\longrightarrow \mathcal{A}(x,z);$$ and for each object $x$ a map $$\Delta[0] \to \mathcal{A}(x,x)$$ that ‘names’ or ‘picks out’ the ‘identity arrow’ in the set of 0-simplices of $\mathcal{A}(x,x)$. This data is to satisfy the obvious axioms, associativity and identity, suitably adapted to this situation. Such a set up will be called a *simplicially enriched category* or more simply *an $\mathcal{S}$-category*. Enriched category theory is a well established branch of category theory. It has many useful tools and not all of them have yet been exploited for the particular case of $\mathcal{S}$-categories and its applications in homotopy theory.
Some authors use the term simplicial category for what we have termed a simplicially enriched category. There is a close link with the notion of simplicial category that is consistent with usage in simplicial theory *per se*, since any simplicially enriched category can be thought of as a simplicial object in the ‘category of categories’, but a simplicially enriched category is not just a simplicial object in the ‘category of categories’ and not all such simplicial objects correspond to such enriched categories. That being said that usage need not cause problems provided the reader is aware of the usage in the paper to which reference is being made.
**Examples**
\(i) $\mathcal{S}$, the category of simplicial sets:\
here $$\mathcal{S}(K,L)_n:= S(\Delta[n] \times K,L);$$ Composition : for $f \in \mathcal{S}(K,L)_n$, $ g \in \mathcal{S}(L,M)_n$, so $f : \Delta[n] \times K \to L$, $g : \Delta[n] \times L \to M$, $$g\circ f := (\Delta[n] \times K \stackrel{diag \times K}{\longrightarrow} \Delta[n] \times\Delta[n] \times K\stackrel{\Delta[n] \times f}{\longrightarrow }\Delta[n] \times L \stackrel{g}{\to} M);$$ Identity : $id_K : \Delta[0] \times K \stackrel{\cong}{\to} K$,
\(ii) $\mathcal{T}op$, ‘the’ category of spaces (of course, there are numerous variants but you can almost pick whichever one you like as long as the constructions work): $$\mathcal{T}op(X,Y)_n := Top(\Delta^n \times X, Y)$$ Composition and identities are defined analogously to in (i).
\(iii) For each $X$, $Y \in Cat$, the category of small categories, then we similarly get $\mathcal{C}at(X,Y)$, $$\mathcal{C}at(X,Y)_n = {Cat}([n] \times X, Y).$$ We leave the other structure up to the reader.
\(iv) $\mathcal{C}rs$, the category of crossed complexes: see for background and other references, and Tonks, [@andythesis] for a more detailed treatment of the simplicially enriched category structure; $$\mathcal{C}rs(A,B) := Crs(\pi(n)\otimes C, D)$$ Composition has to be defined using an approximation to the identity, again see [@andythesis].
\(v) $\mathcal{C}h^+_K$, the category of positive chain complexes of modules over a commutative ring $K$. (Details are left to the reader, or follow from the Dold-Kan theorem and example (vi) below.)
\(vi) $\mathcal{S}(Mod_K)$, the category of simplicial $K$-modules. The structure uses tensor product with the free simplicial $K$-module on $\Delta[n]$ to define the ‘hom’ and the composition, so is very much like (i).
In general any category of simplicial objects in a ‘nice enough’ category has a simplicial enrichment, although the general argument that gives the construction does not always make the structure as transparent as it might be.
There is an evident notion of $\mathcal{S}$-enriched functor, so we get a category of ‘small’ $\mathcal{S}$-categories, denoted $\mathcal{S}\!-\!Cat$. Of course, none of the above examples are ‘small’. (With regard to ‘smallness’, although sometimes a smallness condition is essential, one can often ignore questions of smallness and, for instance, consider simplicial ‘sets’ where actually the collections of simplices are not truly ‘sets’ (depending on your choice of methods for handling such foundational questions).)
From simplicial resolutions to $\mathcal{S}$-cats.
--------------------------------------------------
The forgetful functor $U : Cat \to DGrph_0$ has a left adjoint, $F$. Here $DGrph_0$ denotes the category of directed graphs with ‘identity loops’, so $U$ forgets just the composition within each small category but remembers that certain loops are special ‘identity loops’. The free category functor here takes, between any two objects, all strings of composable *non-identity* arrows that start at the first object and end at the second. One can think of $F$ identifying the old identity arrow at an object $x$ with the empty string at $x$.
This adjoint pair gives a comonad on $Cat$ in the usual way, and hence a functorial simplicial resolution, which we will denote $S(\mathbb{A})\to \mathbb{A}$. In more detail, we write $T = FU$ for the functor part of the comonad, the unit of the adjunction $\eta : Id_{DGrph_0} \to UF$ gives the comultiplication $F\eta U: T \to T^2$ and the counit of the adjuction gives $\varepsilon : FU \to Id_{Cat}$, that is, $\varepsilon : T \to Id$. Now for $\mathbb{A}$ a small category, set $S(\mathbb{A})_n = T^{n+1}(\mathbb{A})$ with face maps $d_i : T^{n+1}(\mathbb{A}) \to T^n(\mathbb{A})$ given by $d_i = T^ {n-i}\varepsilon T^i$, and similarly for the degeneracies which use the comultiplication in an analogous formula. (Note that there are two conventions possible here. The other will use $d_i = T^i\varepsilon T^{n-i}$. The only effect of such a change is to reverse the direction of certain ‘arrows’ in diagrams later on. The two simplicial structures are ‘dual’ to each other.)
This $S(\mathbb{A})$ is a simplicial object in $Cat$, $S(\mathbb{A}) : \mathbf{\Delta}^{op} \to Cat$, so does not immediately gives us a simplicially enriched category, however its simplicial set of objects is constant because $U$ and $F$ took note of the identity loops.
In more detail, let $ob : Cat \to Sets$ be the functor that picks out the set of objects of a small category, then $ob(S(\mathbb{A})) : \mathbf{\Delta}^{op} \to Sets$ is a constant functor with value the set $ob(\mathbb{A})$ of objects of $\mathbb{A}$. More exactly it is a discrete simplicial set, since all its face and degeneracy maps are bijections. Using those bijections to identify the possible different sets of objects, yields a constant simplicial set where all the face and degeneracy maps are identity maps, i.e. we do have a *constant* simplicial set.
\
Let $\mathcal{B} : \mathbf{\Delta}^{op} \to Cat$ be a simplicial object in $Cat$ such that $ob(\mathcal{B})$ is a constant simplicial set with value $B_0$, say. For each pair $(x,y)\in B_0$, let $$\mathcal{B}(x,y)_{n} = \{\sigma \in \mathcal{B}_{n} | ~ {\rm dom}(\sigma) = x, {\rm codom}(\sigma) = y\},$$ where, of course, ${\rm dom}$ refers to the domain function in $\mathcal{B}_n$ since otherwise ${\rm dom}(\sigma)$ would have no meaning, similarly for $\rm codom$.
\(i) The collection $\{\mathcal{B}(x,y)_n |~ n \in \mathbb{N}\}$ has the structure of a simplicial set $\mathcal{B}(x,y)$ with face and degeneracies induced from those of $\mathcal{B}$.
\(ii) The composition in each level of $\mathcal{B}$ induces $$\mathcal{B}(x,y) \times \mathcal{B}(y,z)\to \mathcal{B}(x,z).$$ Similarly the identity map in $\mathcal{B}(x,x)$ is defined as $id_x$, the identity at $x$ in the category $\mathcal{B}_0$.
\(iii) The resulting structure is an $\mathcal{S}$-enriched category, that will also be denoted $\mathcal{B}$.$\blacksquare$
The proof is easy. In particular, this shows that $S(\mathbb{A})$ is a simplicially enriched category. The description of the simplices in each dimension of $S(\mathbb{A})$ that start at $a$ and end at $b$ is intuitively quite simple. The arrows in the category, $T(\mathbb{A})$ correspond to strings of symbols representing non-identity arrows in $\mathbb{A}$ itself, those strings being ‘composable’ in as much as the domain of the $i^{th}$ arrow must be the codomain of the $(i-1)^{th}$ one and so on. Because of this we have:\
$S(\mathbb{A})_0$ consists exactly of such composable chains of maps in $\mathbb{A}$, none of which is the identity;\
$S(\mathbb{A})_1$ consists of such composable chains of maps in $\mathbb{A}$, none of which is the identity, together with a choice of bracketting;\
$S(\mathbb{A})_2$ consists of such composable chains of maps in $\mathbb{A}$, none of which is the identity, together with a choice of two levels of bracketting;\
and so on. Face and degeneracy maps remove or insert brackets, but care must be taken when removing innermost brackets as the compositions that can then take place can result in chains with identities which then need removing, see [@cordier82], that is why the comandic description is so much simpler, as it manages all that itself.
To understand $S(\mathbb{A})$ in general it pays to examine the simplest few cases. The key cases are when $\mathbb{A} = [n]$, the ordinal $\{0< \ldots <n\}$ considered as a category in the usual way. The cases $n=0$ and $n=1$ give no surprises. $S[0]$ has one object 0 and $S[0](0,0$ is isomorphic to $\Delta[0]$, as the only simplices are degenerate copies of the identity. $S[1]$ likewise has a trivial simplicial structure, being just the category $[1]$ considered as an $\mathcal{S}$-category. Things do get more interesting at $n = 2$. The key here is the identification of $S[2](0,2)$. There are two non-degenerate strings or paths that lead from 0 to 2, so $S[2](0,2)$ will have two vertices. The bracketted string $((01)(12))$ on removing inner brackets gives $(02)$ and outer brackets, $(01)(12)$ so represents a 1-simplex $$\xymatrix@+15pt{(01)(12)\ar[r]^{\quad (01)(12))}&(02)}$$ Other simplicial homs are all $\Delta[0]$ or empty. It thus is possible to visualise $S[2]$ as a copy of $[2]$ with a 2-cell going towards the bottom: $$\xymatrix{ &1\ar[dr]&\\
0\ar[rr]\ar[ur]_{\hspace{.5cm}\Downarrow}&&2}$$ The next case $n = 3$ is even more interesting. $S[3](i,j)$ will be (i) empty if $j<i$, (ii) isomorphic to $\Delta[0] $ if $i = j$ or $i = j-1$, (iii) isomorphic to $\Delta[1]$ by the same reasoning as we just saw for $j = i + 2$ and that leaves $S[3](0,3)$. This is a square, $\Delta[1]^2$, as follows: $$\xymatrix@+25pt{(02)(23)\ar[r]^{((02)(23))}&(03)\\
(01)(12)(23)\ar[u]^{((01)(12))((23))}_{\hspace{.5cm} a}\ar[ur]^{diag}\ar[r]_{((01))((12)(23))}&(01)(13)\ar[u]_{((01)(13))}^{b\hspace{6mm}}
}$$ where the diagonal $diag = ((01)(12)(23))$, $a = (((01)(12))((23)))$ and $b = (((01))((12)(23)))$. (It is instructive to check that this is correct, firstly because I may have slipped up (!) as well as seeing the mechanism in action. Removing the innermost brackets is $d_0$, and so on.)
The case of $S[4]$ is worth doing. I will not draw the diagrams here although aspects of it have implications later, but suggest this as an exercise. As might be expected $S[4](0,4)$ is a cube.
**Remark**
The history of this construction is interesting. A variant of it, but with topologically enriched categories as the end result, is in the work of Boardman and Vogt, [@boardmanvogt] and also in Vogt’s paper, [@vogt73]. Segal’s student Leitch used a similar construction to describe a homotopy commutative cube (actually a *homotopy coherent cube*), cf. [@leitch], and this was used by Segal in his famous paper, [@segal], under the name of the ‘explosion’ of $\mathbb{A}$. All this was still in the topological framework and the link with the comonad resolution was still not in evidence. Although it seems likely that Kan knew of this link between homotopy coherence and the comonadic resolutions by at least 1980, (cf. ), the construction does not seem to appear in his work with Dwyer as being linked with coherence until much later. Cordier made the link explicit in [@cordier82] and showed how Leitch and Segal’s work fitted in to the pattern. His motivation was for the description of homotopy coherent diagrams of topological spaces. Other variants were also apparent in the early work of May on operads, and linked in with Stasheff’s work on higher associativity and commutativity ‘up to homotopy’.
Cordier and Porter, [@cordierporter86], used an analysis of a locally Kan simplicially enriched category involving this construction to prove a generalisation of Vogt’s theorem on categories of homotopy coherent diagrams of a given type. (We will return to this aspect a bit later in these notes, but an elementary introduction to this theory can be found in .) Finally Bill Dwyer, Dan Kan and Justin Smith, [@DKS], introduced a similar construction for an $\mathbb{A}$ which is an $\mathcal{S}$-category to start with, and motivated it by saying that $\mathcal{S}$-functors with domain this $\mathcal{S}$-category corresponded to *$\infty$-homotopy commutative $\mathbb{A}$-diagrams*, yet they do not seem to be aware of the history of the construction, and do not really justify the claim that it does what they say. Their viewpoint is however important as, basically, within the setting of Quillen model category structures, this provides a cofibrant replacement construction. Of course, any other cofibrant replacement could be substituted for it and so would still allow for a description of homotopy coherent diagrams in that context. This important viewpoint can also be traced to Grothendieck’s *Pursuing Stacks*, [@stacks].
The DKS extension of the construction, [@DKS], although simple to do, is often useful and so will be outlined next. If $\mathbb{A}$ is already a $\mathcal{S}$-category, think of it as a simplicial category, then applying the $\mathcal{S}$-construction to each $\mathbb{A}_n$ will give a bisimplicial category, i.e. a functor $S(\mathbb{A}) : \mathbf{\Delta}^{op} \times \mathbf{\Delta}^{op} \to Cat$. Of this we take the diagonal so the collection of $n$-simplices is $S(\mathbb{A})_{n,n}$, and by noticing that the result has a constant simplicial set of objects, then apply the lemma.
The Dwyer-Kan ‘simplicial groupoid’ functor.
--------------------------------------------
Let $K$ be a simplicial set. Near the start of simplicial homotopy theory, Kan showed how, if $K$ was reduced (that is, if $K_0$ was a singleton), then the free group functor applied to $K$ in a subtle way, gave a simplicial group whose homotopy groups were those of $K$, with a shift of dimension. With Dwyer in , he gave the necessary variant of that construction to enable it to apply to the non-reduced case. This gives a ‘simplicial groupoid’ $G(K)$ as follows:
The object set of all the groupoids $G(K)_n$ will be in bijective correspondence with the set of vertices $K_0$ of $K$. Explicitly this object set will be written $\{\overline{x} ~|~ x \in K_0\}$.
The groupoid $G(K)_n$ is generated by edges $$\overline{y} : \overline{d_1d_2 \ldots d_{n+1} y} \to \overline{d_0d_2\ldots d_{n+1}y} \quad \mbox{ for } y \in K_{n+1}$$with relations $\overline{s_0x} = id_{\overline{d_1d_2 \ldots d_n x}}$. Note since these just ‘kill’ some of the generating edges, the resulting groupoid $G(K)_n$ is still a free groupoid.
Define $\sigma_i\overline{x} = \overline{s_{i+1}x} \quad \mbox{ for } i\geq 0$, and, for $i > 0$, $\delta_i\overline{x} = \overline{d_{i +1}x}$, but for $i = 0$, $\delta_0\overline{x} = (\overline{d_1x})(\overline{d_0x})^{-1}$.
These definitions yield a simplicial groupoid as is easily checked and, as is clear, its simplicial set of objects is constant, so it also can be considered as a simplicially enriched groupoid, $G(K)$.
(NB. Beware there are several ‘typos’ in the original paper relating to these formulae for the construction and in some of the related material.)
As before it is instructive to compute some examples and we will look at $G(\Delta[2])$ and $G(\Delta[3]$. ) simplicially enriched groupoid are free groupoids in each simplicial dimension, their structure can be clearly seen from the generating graphs. For instance, $G(\Delta[2])_0$ is the free groupoid on the graph $$\xymatrix{&\overline{1}\ar[dr]^{\overline{12}}&\\
\overline{0}\ar[rr]_{\overline{02}}\ar[ur]^{\overline{01}}&&\overline{2}}$$ whilst $G(\Delta[2])_1$ is the free groupoid on the graph $$\xymatrix{&\overline{1}\ar[dr]^{\overline{122}}&\\
\overline{0}\ar[rr]_{\overline{022}}\ar[ur]<.5ex>^{\overline{011}}\ar[ur]<-.5ex>_{\overline{012}}&&\overline{2}}$$ Here it is worth noting that $\delta_0(\overline{012}) = (\overline{02}).(\overline{12})^{ -1}$. Higher dimensions do not have any non-degenrate generators.
Again with $G(\Delta[3])$, in dimension 0 we have the free groupoid on the directed graph give by the 1-skelton of $\Delta[3]$. In dimension 2, the generating directed graph is $$\xymatrix@+20pt{&\overline{1}\ar[dr]^{\overline{133}}\ar[dd]<.5ex>\ar[dd]<-.5ex>&\\
\overline{0}\ar[ur]<.7ex>\ar[ur]\ar[ur]<-.7ex>\ar'[r][rr]\ar[dr]<.5ex>^{\overline{023}}\ar[dr]<-.5ex>_{\overline{022}}&&\overline{3}\\
&\overline{2} \ar[ur]_{\overline{233}} & }$$ Here only a few of the arrow labels have been given. Others are easy to provide (but moderately horrible to typeset in a sensible way!). Those from $\overline{0}$ to $\overline{1}$ are $\overline{012}$, $\overline{011}$ and $\overline{013}$; those from $\overline{1}$ to $\overline{2}$ are $\overline{122}$ and $\overline{123}$, and finally from $\overline{0}$ to $\overline{3}$, we have $\overline{033}$.
The next dimension is only a little more complicated. It has extra degenerate arrows such as $\overline{0112}$ and $\overline{0122}$ from $\overline{0}$ to $\overline{1}$ but also between these two vertices has $\overline{0123}$, coming from the non-degenerate three simplex of $\Delta[3]$. The full diagram is easy to draw (and again a bit tricky to typeset in a neat way), and is therefore left ‘as an exercise’.
**Remarks**
\(i) The functor $G$ has a right adjoint $\overline{W}$ and the unit $K\to\overline{W}G(K)$ is a weak equivalence of simplicial sets. This is part of the result that shows that simplicially enriched groupoids model all homotopy types, for which see the original paper, or the book by Goerss and Jardine, [@GoerssJardine].
\(ii) It is tantalising that the definition of $S(\mathbb{A})$ for a category $\mathbb{A}$ and of $G(K)$ for a simplicial set $K$ are very similar yet very different. Why does the twist occur in the $d_0$ of $G(K)$? Why do the source and target maps at each level in $G(K)$ end up at the zeroth and first vertex rather than the seemingly more natural zeroth and $n^{th}$ that occur in $S(\mathbb{A})$? In fact is there a variant of the $G(K)$ construction that is nearer to the $S(\mathbb{A})$ construction without being merely artificially so?
Structure
=========
The ‘homotopy’ category
-----------------------
If $\mathcal{C}$ is an $\mathcal{S}$-category, we can form a category $\pi_0\mathcal{C}$ with the same objects and having $$(\pi_0\mathcal{C})(X,Y) =
\pi_0({\mathcal{C}}(X,Y)).$$ For instance, if $\mathcal{C} =
\mathcal{CW}$, the category of CW-complexes, then $\pi_0\mathcal{CW} = {Ho\mathcal(CW)}$, the corresponding homotopy category. Similarly we could obtain a groupoid enriched category using the fundamental groupoid (cf. Gabriel and Zisman, [@gabrielzisman]).
One can ‘do’ some elementary homotopy theory in any $\mathcal{S}$-category, $\mathcal{C}$, by saying that two maps $f_0, f_1 : X \to Y$ in $\mathcal{C}$ are homotopic if there is an $H \in {\mathcal{C}}(X,Y)_1$ with $d_0H =
f_1$, $d_1H = f_0$.
This theory will not be very rich unless at least some low dimensional Kan conditions are satisfied. The $\mathcal{S}$-category, $\mathcal{C}$, is called *locally Kan* if each $\mathcal{C}(X,Y)$ is a Kan complex, *locally weakly Kan* if …, etc. (If you have not met ‘weak Kan complexes’ before, you will soon meet them in earnest!)
The theory is ‘geometrically’ nicer to work with if $\mathcal{C}$ is *tensored* or *cotensored*.
Tensoring and Cotensoring
-------------------------
**Tensored**
If for all $K \in \mathcal{S}$, $X, Y, \in \mathcal{C}$, there is an object $K\bar{\otimes} X$ in $\mathcal{C}$ such that $$\mathcal{C}(K\bar{\otimes}X,Y) \cong
\mathcal{S}(K,\mathcal{C}(X,Y)$$naturally in $K$, $X$ and $Y$, then $\mathcal{C}$ is said to be *tensored* over $\mathcal{S}$.
**Cotensored**
Dually, if we require objects $\bar{\mathcal{C}}(K,Y)$ such that $$\mathcal{C}(X,\bar{\mathcal{C}}(K,Y)) \cong
\mathcal{S}(K,\mathcal{C}(X,Y)$$then we say $\mathcal{C}$ is *cotensored* over $\mathcal{S}$.
To gain some intuitive feeling for these, think of $K\bar{\otimes} X$ as being $K$ product with $X$, and $\bar{\mathcal{C}}(K,Y)$ as the object of functions from $K$ to $Y$. These words do not, as such, make sense in general, but do tell one the sort of tasks these constructions will be set to do. They will not be much used explicitly here however.
(cf. Kamps and Porter, )\
If $\mathcal{C}$ is a locally Kan $\mathcal{S}$-category tensored over $\mathcal{S}$ then taking $I\times X = \Delta[1]\bar{\otimes}X$, we get a good cylinder functor such that for the cofibrations relative to $I$ and weak equivalences taken to be homotopy equivalences, the category $\mathcal{C}$ has a cofibration category structure.$\blacksquare$
A cofibration category structure is just one of many variants of the abstract homotopy theory structure introduced to be able to push through homotopy type arguments in particular settings. There are variants of this result, due to Kamps, see references in , where $\mathcal{C}$ is both tensored and cotensored over $\mathcal{S}$ and the conclusion is that $\mathcal{C}$ has a Quillen model category structure. The examples of locally Kan $\mathcal{S}$-categories include $\mathcal{T}op$, $\mathcal{K}an$, $\mathcal{G}rpd$ and $\mathcal{C}rs$, but not $\mathcal{C}at$ or $\mathcal{S}$ itself.
Nerves and Homotopy Coherent Nerves.
====================================
Kan and weak Kan conditions
---------------------------
Before we get going on this section, it will be a good idea to bring to the fore the definitions of *Kan complex* and *weak Kan complex* (or *quasi-category*).
As usual we set $\Delta[n] = {\mathbf \Delta}( - , [n]) \in \mathcal{S}$, then for each $i$, $0 \leq i \leq n$, we can form a subsimplicial set, $\Lambda^i[n]$, of $\Delta[n] $ by discarding the top dimensional $n$-simplex (given by the identity map on $[n]$) and its $i^{th}$ face. We must also discard all the degeneracies of these simplices. This informal definition does not give a ‘picture’ of what we have, so we will list the various cases for $n = 2$. $$\Lambda^0[2] = \xymatrix{
&1\ar@{ ..}[dr]^{\leftarrow 0^{th} \mbox{~\scriptsize{face missing}}}&\\
0\ar[ur]\ar[rr]&&2}$$ $$\Lambda^1[2] = \xymatrix{
&1\ar[dr]^{\hspace{2.3cm}}&\\
0\ar[ur]\ar@{ ..}[rr]&&2}$$ $$\Lambda^2[2] = \xymatrix{
&1\ar[dr]^{\hspace{2.3cm}}&\\
0\ar@{ ..}[ur]\ar[rr]&&2}$$ A map $p: E \rightarrow B$ is a *Kan fibration* if given any $n$, $i$ as above and any $(n,i)$-horn in $E$, i.e. any map $f_1 : \Lambda^i[n]
\rightarrow E$, and $n$-simplex, $f_0 : \Delta[n] \rightarrow B$, such that $$\xymatrix{\Lambda^i[n]\ar[r]^{f_1}\ar[d]_{inc}& E\ar[d]^p\\
\Delta[n]\ar[r]_{f_0}&B}$$ commutes, then there is an $f : \Delta[n] \rightarrow E$ such that $pf = f_0$ and $f.inc = f_1$, i.e. $f$ lifts $f_0$ and extends $f_1$.
A simplicial set, $K$, is a *Kan complex* if the unique map $K
\rightarrow \Delta[0]$ is a Kan fibration. This is equivalent to saying that every horn in $K$ has a filler, i.e. any $f_1 : \Lambda^i[n]\rightarrow Y$ extends to an $f : \Delta[n] \rightarrow Y$. This condition looks to be purely of a geometric nature but in fact has an important algebraic flavour; for instance if $f_1 : \Lambda^1[2]\rightarrow Y$ is a horn, it consists of a diagram $$\xymatrix{ & \ar[dr]^b&\\\ar[ur]^a&&}$$ of ‘composable’ arrows in $K$. If $f$ is a filler, it looks like $$\xymatrix{ & \ar[dr]^b&\\ \ar[rr]_c\ar[ur]^a_{\quad f}&&}$$ and one can think of the third face $c$ as a composite of $a$ and $b$. This ‘composite’ $c$ is not usually uniquely defined by $a$ and $b$, but is ‘up to homotopy’. If we write $c = ab$ as a shorthand then if $g_1 :\Lambda^0[2]
\rightarrow K$ is a horn, we think of $g_1$ as being $$\xymatrix{ & &\\ \ar[ur]^d\ar[rr]_e&&}$$and to find a filler is to find a diagram $$\xymatrix{ & \ar@{--}[dr]^x&\\ \ar[ur]^d\ar[rr]_e&&}$$ and thus to ‘solve’ the equation $dx = e$ for $x$ in terms of $d$ and $e$. It thus requires in general some approximate inverse for $d$, in fact, taking $e$ to be a degenerate 1-simplex puts one in exactly such a position.
In many useful cases, we do not always have inverses and so want to discard any requirement that would imply they always exist. This leads to the weaker form of the Kan condition in which in each dimension no requirement is made for the existence of fillers on horns that miss out the zeroth or last faces. More exactly:
**Definition**
A simplicial set ${\bf K}$ is *a weak Kan complex* or *quasi-category* if for any $n$ and $0< k < n$, any $(n,k)$-horn in $K$ has a filler.
**Remarks**
\(i) Joyal, [@joyal], uses the term *inner horn* for any $(n,k)$-horn in $K$ with $0< k < n$. The two remaining cases are then conveniently called *outer horns*.
\(ii) For any space $X$, its singular complex, $Sing(X)$ is given by $Sing(X)_n = Top(\Delta^n,X)$ with the well known face and degeneracy maps. This simplicial set is always a Kan complex as is the related $\mathcal{T}op(X,Y)$ as mentioned above.
Nerves
------
The categorical analogue of the singular complex is, of course, the nerve: if $\mathbb{C}$ is a category, its *nerve*, $Ner(\mathbb{C})$, is the simplicial set with $Ner(\mathbb{C})_n = Cat([n],\mathbb{C})$, where $[n]$ is the category associated to the finite ordinal $[n] = \{ 0 < 1< \ldots < n\}$. The face and degeneracy maps are the obvious ones using the composition and identities in $\mathbb{C}$.
The following is well known and easy to prove.
\
(i) $Ner(\mathbb{C})$ is always weakly Kan.\
(ii) $Ner(\mathbb{C})$ is Kan if and only if $\mathbb{C}$ is a groupoid.$\blacksquare$
Of course more is true. Not only does any inner horn in $Ner(\mathbb{C})$ have a filler, it has exactly one filler. To express this with maximum force the following idea, often attributed to Graeme Segal or to Grothendieck, is very useful.
Let $p>0$, and consider the increasing maps $e_i : [1] \to [p]$ given by $e_i(0) = i$ and $e_i(1) = i+1$. For any simplicial set $A$ considered as a functor $A : \mathbf{\Delta}^{op} \to Sets$, we can evaluate $A$ on these $e_i$ and, noting that $e_i(1) = e_{i+1}(0)$, we get a family of functions $A_p \to A_1$, which yield a cone diagram, for instance, for $p =3$: $$\xymatrix{A_p \ar[drrr]^{A(e_1)}\ar[ddrr]^{A(e_2)}\ar[dddr]_{A(e_3)}&&&\\
&&&A_1\ar[d]^{d_0}\\
&&A_1\ar[r]^{d_1}\ar[d]^{d_0}&A_0\\
&A_1\ar[r]_{d_1}&A_0&&}$$ and in general, thus yield a map $$\delta[p]: A_p \to A_1\times_{A_0}A_1 \times_{A_0}\ldots \times_{A_0}A_1.$$ The maps, $\delta[p]$, have been called the *Segal maps* and will recur throughout the rest of these notes.
\
If $A = Ner(\mathbb{C})$ for some small category $\mathbb{C}$, then for $A$, the Segal maps are bijections.
**Proof**
A simplex $\sigma \in Ner(\mathbb{C})_p$ corresponds uniquely to a composable $p$-chain of arrows in $\mathbb{C}$, and hence exactly to its image under the relevant Segal map. $\blacksquare$
Better than this is true:
\[GrotSegal\] \
If $A$ is a simplicial set such that the Segal maps are bijections then there is a category structure on the directed graph $$\xymatrix{A_1 \ar[r]<1ex> \ar[r]&A_0\ar[l]<1ex>}.$$making it a category whose nerve is isomorphic to the given $A$.
**Proof**
To get composition you use $$A_1\times_{A_0}A_1 \stackrel{\cong}{\to} A_2\stackrel{d_1}{\to}A_1.$$ Associativity is given by $A_3$. The other laws are easy, and illuminating, to check. $\blacksquare$
The condition ‘Segal maps are a bijection’ is closely related to notions of ‘thinness’ as used by Brown and Higgins in the study of crossed complexes and their relationship to $\omega$-groupoids, see, for instance, , and also to Duskin’s ‘hypergroupoid’ condition, [@duskin].
Another result that is sometimes useful, is a refinement of ‘groupoids give Kan complexes’:. The proof is ‘the same’:
\
Let $A = Ner(\mathbb{C})$, the nerve of a category $\mathbb{C}$.
\(i) Any $(n,0)$-horn $$f : \Lambda^0[n]\rightarrow A$$ for which $f(01)$ is an isomorphism has a filler. Similarly any $(n,n)$-horn $g : \Lambda^n[n]\rightarrow A$ for which $g(n-1~n)$ is an isomorphism, has a filler.
\(ii) Suppose $f$ is a morphism in $\mathbb{C}$ with the property that for any $n$, any $(n,0)$-horn $\phi : \Lambda^0[n]\rightarrow A$ having $f$ in the $(0,1)$ position, has a filler, then $f$ is an isomorphism. (Similarly with $(n,0)$ replaced by $(n,n)$ with the obvious changes.) $\blacksquare$
**Remark**
Joyal in [@joyal] suggested that the name ‘weak Kan complex’, as introduced by Boardman and Vogt, [@boardmanvogt], could be changed to that of ‘quasi-category’ to stress the analogy with categories *per se* as ‘*Most concepts and results of category theory can be extended to quasi-categories*’, [@joyal].
It would have been nice to have explored Joyal’s work on quasi-categories more fully, e.g. [@joyal], but time did not allow it. The following few sections just skate the surface of the theory.
Quasi-categories
----------------
Categories yield quasi-categories via the nerve construction. Quasi-categories yield categories by a ‘fundamental category’ construction that is left adjoint to nerve. This can be constructed using the free category generated by the 1-skeleton of $A$, and then factoring out by a congruence generated by the basic relations : $gf \equiv h$, one for each commuting 1-sphere $(g,h,f)$ in $A$. By a *1-sphere* is meant a map $a : \partial \Delta[2] \to A$, thus giving three faces, $(a_0,a_1,a_2)$ linked in the obvious way. The 1-sphere is said to be *commuting* if there is a 2-simplex, $b\in A_2$, such that $a_i = d_ib$ for $i = 0,1,2$.
This ‘fundamental category’ functor also has a very neat description due to Boardman and Vogt. (The treatment here is adapted from [@joyal].)
We assume given a quasi-category $A$. Write $gf \sim h$ if $(g,h,f)$ is a commuting 1-sphere. Let $x,y \in A_0$ and let $A_1(x,y) = \{f\in A_1 ~|~ x = d_1f, y = d_0f\}$. If $f,g \in A_1(x,y)$, then, suggestively writing $s_0x = 1_x$,
\
The four relations $f1_x \sim g$, $g1_x\sim f$, $1_yf\sim g$ and $1_yg\sim f$ are equivalent.$\blacksquare$
The proof is easy.
We will say $f\simeq g$ if any of these is satisfied and call $\simeq$, the *homotopy relation*. It is an equivalence relation on $A_1(x,y)$. Set $ho\,A_1(x,y) = A_1(x,y)/\simeq$.
If $f \in A_1(x,y)$, $g \in A_1(y,z)$ and $h\in A_1(x,z)$, then the relation $gf \sim h$ induces a map: $$ho\,A_1(x,y)\times ho\,A_1(y,z) \to ho\,A_1(x,z).$$
\
The maps $$ho\,A_1(x,y)\times ho\,A_1(y,z) \to ho\,A_1(x,z)$$give a composition law for a category, $ho\,A$, the homotopy category of $A$.$\blacksquare$
Of course, $ho\,A$ is the fundamental category of $A$ up to natural isomorphism. From previous comments we have:
\
A quasi-category $A$ is a Kan complex if and only if $ho\,A$ is a groupoid.$\blacksquare$
Homotopy coherent nerves
------------------------
Before introducing this topic, recall some of the intuition behind homotopy coherent (h.c.) diagrams. (Again there is an overview of this theory in [@cubo] and a thorough introduction in .)
**Examples of h.c. diagrams in $Top$.**
1\) A diagram indexed by the small category, $[2]$, $$\xymatrix{ &X(1)\ar[dr]^{X(12)}&\\
X(0)\ar[rr]_{X(02)}\ar[ur]^{X(01)}_{\hspace*{.5cm} X(012)}&&X(2)}$$ is h.c. if there is specified a homotopy $$X(012) : I\times X(0) \to X(2),$$ $$X(012) : X(02) \simeq X(12)X(01).$$
2\) For a diagram indexed by $[3]$: Draw a 3-simplex, marking the vertices $X(0)$, …, $X(3)$, the edges $X(ij)$, etc., the faces $X(ijk)$, etc. The homotopies $X(ijk)$ fit together to make the sides of a square $$\xymatrix{X(13)X(01)\ar[rr]^{X(123)X(01)} && X(23)X(12)X(01)\\
X(03)\ar[u]^{X(013)}\ar[rr]_{X(023)}&&X(23)X(02)\ar[u]_{X(23)X(012)}}$$ and the diagram is made h.c. by specifying a second level homotopy $$X(0123) : I^2\times X(0)\to X(3)$$filling this square.
These can be continued for larger $[n]$. Of course, this is not how the theory is formally specified, but it provides some understanding of the basic idea.
The theory was initially developed by Vogt, [@vogt73], following methods introduced with Boardman, [@boardmanvogt] (see also the references in that source for other earlier papers on the area). Cordier [@cordier82] provides a simple $\mathcal{S}$-category theory way of working with h.c. diagrams and hence released an ‘arsenal’ of categorical tools for working with h.c. diagrams. Some of that is worked out in the papers,
**Some Results**
\(i) If $X : {\mathbb{A}}\to {\mathcal{T}op}$ is a commutative diagram and we replace some of the $X(a)$ by homotopy equivalent $Y(a)$ with specified homotopy equivalence data: $$f(a) : X(a) \to Y(a), \quad g(a) : Y(a) \to X(a)$$ $$H(a) : g(a)f(a) \simeq Id, \quad K(a) : f(a)g(a) \simeq Id,$$ then we can combine these data into the construction of a h.c. diagram $Y$ based on the objects $Y(a)$ and homotopy coherent maps $$f : X\to Y, \quad g :
Y \to X, \mbox { etc.,}$$making $X$ and $Y$ homotopy equivalent as h.c. diagrams.
(This is ‘really’ a result about quasi-categories, see [@joyal].)
\(ii) [Vogt]{}, [@vogt73].
If ${\mathbb{A}}$ is a small category, there is a category ${Coh(\mathbb{A},\mathcal{T}op)}$ of h.c. diagrams and homotopy classes of h.c. maps between them. Moreover there is an equivalence of categories $${Coh(\mathbb{A},\mathcal{T}op)} \stackrel{\simeq}{\to} {Ho(\mathcal{T}op^\mathbb{A})}$$ This was extended replacing $\mathcal{T}op$ by a general locally Kan simplicially enriched complete category, $\mathcal{B}$, in [@cordierporter86].
\(iii) [Cordier (1980)]{}, [@cordier82].
Given ${\mathbb{A}}$, a small category, then the $\mathcal{S}$-category ${S(\mathbb{A})}$ is such that a h.c. diagram of type ${\mathbb{A}}$ in ${\mathcal{T}op}$ is given precisely by an $\mathcal{S}$-functor $$F : {S(\mathbb{A})} \to {\mathcal{T}op}$$ This suggested the extension of h.c. diagrams to other contexts such as a general locally Kan $\mathcal{S}$-category, $\mathcal{B}$ and suggests the definition of homotopy coherent diagram in a $\mathcal{S}$-category and thus a h.c. nerve of an $\mathcal{S}$-category.
**Definition** ([Cordier (1980)]{}, [@cordier82], based on earlier ideas of Vogt, and Boardman-Vogt.)
Given a simplicially enriched category $\mathcal{B}$, the *homotopy coherent nerve* of $\mathcal{B}$, denoted $Ner_{h.c.}(\mathcal{B})$, is the simplicial ‘set’ with $$Ner_{h.c.}(\mathcal{B})_n = \mathcal{S}\!-\!Cat(S[n],\mathcal{B}).$$
To understand simple h.c. diagrams and thus $Ner_{h.c.}(\mathcal{B})$, we will unpack the definition of homotopy coherence.
The first thing to note is that for any $n$ and $0\leq i< j\leq n$, $S[n](i,j) \cong \Delta[1]^{j-i-1}$, the $(j-i-1)$-cube given by the product of $j-i-1 $ copies of $\Delta[1]$. Thus we can reduce the higher homotopy data to being just that, maps from higher dimensional cubes.
Next some notation:
Given simplicial maps $$f_1: K_1 \to \mathcal{B}(x,y),$$ $$f_2: K_2 \to \mathcal{B}(y,z),$$ we will denote the composite $$K_1 \times K_2 \to \mathcal{B}(x,y)\times \mathcal{B}(y,z) \stackrel{c}{\to} \mathcal{B}(x,z)$$ just by $f_2.f_1$ or $f_2f_1$. (We already have seen this in the h.c. diagram above for $\mathbb{A} = [3]$. $X(123)X(01)$ is actually $X(123)(I \times X(01) )$, whilst $X(23)X(012)$ is exactly what it states.)
Suppose now that we have the h.c. diagram $F : S(\mathbb{A}) \to \mathcal{B}$. This is an $\mathcal{S}$-functor and so:\
to each object $a$ of $\mathbb{A}$, it assigns an object $F(a)$ of $\mathcal{B}$;\
for each string of composable maps in $\mathbb{A}$, $$\sigma = (f_0, \ldots, f_n)$$ starting at $a$ and ending at $b$, a simplicial map $$F(\sigma) : S(\mathbb{A})(0,n+1) \to \mathcal{B}(F(a), F(b)),$$ that is, a higher homotopy $$F(\sigma) : \Delta[1]^n \to \mathcal{B}(F(a), F(b)),$$ such that
\(i) if $f_0 = id$, $F(\sigma) = F(\partial_0\sigma)(proj \times \Delta[1]^{n-1})$
\(ii) if $f_i = id$, $0< i < n$ $$F(\sigma) = F(\partial_i\sigma(.(I^i \times m \times I^{n-i}),$$ where $m : I^2 \to I$ is the multiplicative structure on $I = \Delta[1]$ by the ‘max’ function on $\{0,1\}$;
\(iii) if $f_n = id$, $F(\sigma) = F(\partial_n \sigma)$;
(iv)$_{i}$ $F(\sigma)|(I^{i-1}\times \{0\} \times I^{n-i}) = F(\partial_i\sigma), 1 \leq i \leq n-1$;
(v)$_{i}$ $F(\sigma)|( I^{i-1}\times \{1\} \times I^{n-i}) = F(\sigma^\prime_i) . F(\sigma_i)$, where $\sigma_i = (f_0, \ldots, f_{i-1}$ and $\sigma^\prime = (f_i, \ldots, f_n)$. We have used $\partial_i$ for the face operators in the nerve of $\mathbb{A}$.
The specification of such a homotopy coherent diagram can be split into two parts:\
(a) specification of certain homotopy coherent *simplices*, i.e. elements in $Ner_{h.c.}(\mathcal{B})$;\
and\
(b) specification, via a simplicial mapping from $Ner(\mathbb{A})$ to $Ner_{h.c.}(\mathcal{B})$, of how these individual parts (from (a)) of the diagram are glued together.
The second part of this is easy as it amounts to a simplicial map $Ner(\mathbb{A}) \to Ner_{h.c.}(\mathcal{B})$, and so we are left with the first part. The following theorem was proved by Cordier and myself, but the idea was essentially in Boardman and Vogt’s lecture notes, like so much else!
([@cordierporter86])\
If $\mathcal{B}$ is a locally Kan $\mathcal{S}$-category then $Ner_{h.c.}(\mathcal{B})$ is a quasi-category.$\blacksquare$
It seems to be the case that if $\mathcal{B}$ is only locally weakly Kan, then $Ner_{h.c.}(\mathcal{B})$ need not be a quasi-category.
The proof of the theorem is in the paper, [@cordierporter86] and is not too complex. The essential feature is that the very definition (unpacked version) of homotopy coherent diagram makes it clear that parts of the data have to be composed together, (recall the composition of simplicial maps $$f_1: K_1 \to \mathcal{B}(x,y),$$ $$f_2: K_2 \to \mathcal{B}(y,z),$$ above and how important that was in the unpacked definition).
We thus have that a homotopy coherent diagram ‘is’ a simplicial map $F : Ner(\mathbb{A}) \to Ner_{h.c.}(\mathcal{B})$ and that $Ner_{h.c.}(\mathcal{B})$ is a quasi-category. Of course, the usual proof that, if $X$ and $Y$ are simplicial sets, and $Y$ is Kan, then $\mathcal{S}(X,Y)$ is Kan as well, extends to having $Y$ a quasi-category and the result being a quasi-category. Earlier we referred to $Coh(\mathbb{A},\mathcal{B})$ in connection with Vogt’s theorem. The neat way of introducing this is as $ho\, \mathcal{S}(Ner(\mathbb{A}), Ner_{h.c.}(\mathcal{B}))$, the fundamental category of the function quasi-category. In fact, this is essentially the way Vogt first described it.
Before we leave homotopy coherence, there is a point that is worth noting for the links with algebraic and categorical models for homotopy types. The $\mathcal{S}$-categories, $S[n]$, contain a lot of the information needed for the construction of such models. A good example of this is the interchange law and its links with Gray categories and Gray groupoids.
Consider $S[4]$. The important information is in the simplicial set $S[4](0,4)$. This is a 3-cube, so is still reasonably easy to visualise. Here it is. The notation is not intended to be completely consistent with earlier uses but is meant to be more self explanatory. $$\xymatrix{& (01)(13)(34)\ar[rr]&&(01)(12)(23)(34)\\
(01)(14)\ar[rr]\ar[ur]&&~~(01)(12)(24)\quad\ar[ur]&\\
& (03)(34)\ar'[r][rr]\ar'[u][uu]&&(02)(23)(34\ar[uu])\\
(04)\ar[rr]\ar[ur]\ar[uu]&&~~(02)(24)\quad\ar[ur]\ar[uu]&}$$ This looks mysterious! A 4-simplex has 5 vertices, and hence 5 tetrahedral faces. Each of the 5 tetrahedral faces will contribute a square to the above diagram, yet a cube has 6 square faces! (Things get ‘worse’ in $S[5](0,5)$, which is a 4-cube, so has 8 cubes as its faces, but $\Delta[5]$ has only 6 faces.) Back to the extra face, this is $$\xymatrix{(01)(12)(24)\ar[rr]^{(01)(12)(234)}&&(01)(12)(23)(34)\\
&&\\
(02)(24)\ar[rr]_{(02)(234)}\ar[uu]^{(012)(24)}&\ar@{}[uu]|{(012)(234)}&(02)(23)(34)\ar[uu]_{(012)(23)(34)}.}$$ The arrow $(012) : (02) \rightarrow (01)(12)$ will, in a homotopy coherent diagram, make its appearence as the homotopy,$$X(012) : I\times X(0) \to X(2),$$ $$X(012) : X(02) \simeq X(12)X(01),$$ thus this square implies that the homotopies $X(012)$ and $X(234)$ interact minimally. Drawing them as 2-cells in the usual way, the square we have above is the interchange square and the interchange law will hold in this system provided this square is, in some sense, commutative. In models for homotopy $n$-types for $n \geq 3$, these interchange squares give part of the pairing structure between different levels of the model. They are there in, say, the Conduché model (2-crossed modules) as the Peiffer lifting, (cf. Conduché, [@conduche]) and in the Loday model, (crossed squares, cf. [@loday]), as the $h$-map. In a general dimension, $n$, there will be pairings like this for any splitting of $\{0,1, \ldots ,n\}$ of the form $\{0.1. \ldots, k\}$ and $\{k, \ldots, n\}$.
Dwyer-Kan Hammock Localisation: more simplicially enriched categories.
======================================================================
In his original contribution [@quillen] to abstract homotopy theory, Quillen introduced the notion of a *model category*. Such a context is a category, $C$, together with three classes of maps: weak equivalences, $W = C_{w.e.}$; fibrations, $fib = C_{fib}$; and cofibrations, $cofib = C_{cofib}$, satisfying certain axioms so as to give a general framework for ‘doing homotopy theory’. One of the constructions he used was a categorical localisation already well known from Gabriel’s thesis and the work of the French school of algebraic geometers, (Grothendieck, Verdier, etc.) and concurrently with the publication of [@quillen], studied in some depth by Gabriel and Zisman, [@gabrielzisman]. The main point was that the analogues of homotopy equivalences, in important instances of homotopical or homological algebra, were only ‘weak equivalences’ so whilst with a homotopy equivalence between two spaces, you are given two maps, one in each direction, plus of course some homotopies, when you have, for instance, a quasi-isomorphism between two chain complexes, you only had one map in one direction: $f: C\to D$ together with the knowledge that $f_*: H_*(C) \to H_*(D)$ was an isomorphism. The partial solution was to go to the ‘homotopy category’ by formally inverting the weak equivalences, thus getting formal maps going in the opposite direction! (This may look like cheating, but really is no worse than introducing fractions into the integers, so as to be able to solve certain equations, and of course the detailed construction is closely related!) We thus end up with a category $C[W^{-1}]$.
This construction is very useful, but this homotopy category does *not* capture the higher order homotopy information implicit in $C$. For instance, the problem of the ‘best’ way to handle homotopy limits and colimits, and more generally derived Kan extensions, in a model category setting is still central to much of the work on abstract homotopy theory, (cf. *Les Dérivateurs*, by George Maltsiniotis, [@malt1] see also [@malt2], Denis-Charles Cisinski’s thesis, and subsequent work, (cf. [@cisinski1; @cisinski2] and related papers), the resumé of Thomason’s note books published by Chuck Weibel, [@weibel] and Carlos Simpson’s, [@carlos9708010], for example). In a series of articles published in 1980, Dwyer and Kan proposed a neat solution to this problem, simplicial localisations. We will limit ourselves to one of the two versions here, the hammock localisation.
Hammocks
--------
Given a category $C$, and a subcategory $W$, having the same class of objects, construct a $\mathcal{S}$-category $L^H(C,W)$ or $L^HC$ for short, the *hammock localisation of $C$ with respect to $W$*, as follows:
The objects of $L^HC$ are the same as those of $C$
Given two objects $X$ and $Y$, the $k$-simplices of $L^HC(X,Y)$ will be the “reduced hammocks of width $k$ and any length” between $X$ and $Y$. Such a thing is a commutative diagram of form $$\xymatrix{&C_{0,1}\ar[d]\ar@{-}[r]&C_{0,2}\ar[d]\ar@{-}[r]&\ldots\ar@{-}[r]&C_{0,n-1}\ar[d]\ar@{-}[ddr]&\\
&C_{1,1}\ar[d]\ar@{-}[r]&C_{1,2}\ar[d]\ar@{-}[r]&\ldots\ar@{-}[r]&C_{1,n-1}\ar[d]\ar@{-}[dr]&\\
X\ar@{-}[dr]\ar@{-}[ddr]\ar@{-}[uur]\ar@{-}[ur]& \vdots\ar[d] & \vdots\ar[d]&&\vdots\ar[d]& Y\\
&C_{k-1,1}\ar[d]\ar@{-}[r]&C_{k-1,2}\ar[d]\ar@{-}[r]&\ldots\ar@{-}[r]&C_{k-1,n-1}\ar[d]\ar@{-}[ur]&\\
&C_{k,1}\ar@{-}[r]&C_{k,2}\ar@{-}[r]&\ldots\ar@{-}[r]&C_{k,n-1}\ar@{-}[uur]&}$$ in which\
(i) the length $n$ of the hammock can be any integer $\geq 0$,\
(ii) all the vertical maps are in $W$,\
(iii) in each column of horizontal maps, all maps go in the same direction; if they go left, then they have to be in $W$;\
plus two reduction conditions,\
(iv) the maps in adjacent columns go in different directions,\
and\
(v) no column contains only identity maps.
(If in manipulating hammocks, these last two conditions become violated. then it is simple to reduce the hammock by, for example, composing adjacent columns if they point in the same direction or by removing a column of identities. Repeated use of the reductions may be needed. One reduction may create a need for another one. It is often useful to work with unreduced hammocks and then to reduce.)
The face and degeneracy maps are defined in the obvious way, (remember the vertices of such a simplex are the ‘zigzags’ from $X$ to $Y$), however they may result in a non-reduced hammock.
Composition is by concatenation followed by reduction: $$L^HC(X,Y) \times L^HC(Y,Z)\to L^HC(X,Z),$$ expanding the intervening $Y$ node into a vertical line with identities and then reducing if need be.
Each $L^HC(X,Y)$ is the direct limit of nerves of small categories in an obvious way, i.e. increasing the length $n$ of the hammocks, and so is itself a quasi-category.
Hammocks in the presence of a calculus of left fractions.
---------------------------------------------------------
If the pair $(C,W)$ satisfies any of the usual ‘calculus of fractions’ type conditions, then the homotopy type of those nerves already stabilises early on in the process (i.e. for small $n$). The argument given in is indirect, so let us briefly see why one of these claims is true. Suppose that $(C,W)$ satisfies a calculus of left fractions, then\
(i) whenever there is a diagram $X^\prime \stackrel{u}{\leftarrow} X \stackrel{f}{\to} Y$ in $C$ with $u \in W$, then there is a diagram $X^\prime \stackrel{f^\prime}{\to}Y^\prime \stackrel{v}{\leftarrow} Y$ so that $v\in W$ and $vf = f^\prime u$,\
and similarly\
(ii) if $f,g: X\to Y \in C$ and $u : X\to X^\prime \in W$ is such that $fu = gu$, then there is a $v\in W$ such that $vf = vg$.\
(By this means any word in arrows of $C$ and $W^{-1}$ can be rewritten to get all the occurrences of arrows from $W^{-1}$ to the left of those ‘ordinary’ arrows from $C$. Each of the two substrings can then be composed to reduce the word to one of the form $w^{-1}c$, i.e. a left fraction.) To understand how this reacts with hammocks, consider a simple case where the chosen vertex of the hammock $L^HC(X,Y)$ is simply $$\xymatrix{X&C\ar[l]_w\ar[r]^c& Y &Y\ar[l]_{id}}$$provide with $w\in W$. We construct a new diagram, using the left fractions rule (i), giving a 1-simplex with the given vertex at one end: $$\xymatrix{X\ar@{=}[d]&C\ar[l]_w\ar[r]^c\ar[d]^w&Y\ar[d]^{w^\prime} &Y\ar[l]_{id}\ar@{=}[d]\\
X &X\ar[l]_{id}\ar[r]_{c^\prime} & C^\prime&Y\ar[l]_{w^\prime} },$$ so was homotopic to a ‘left biased’ hammock $(w^\prime)^{-1}c^\prime$.
Of course, if the length of the hammock had been greater then the chain of ‘moves’ to link it to the ‘left biased ’ form would be longer. Again of course, although combinatorially feasible a detailed proof that the left baissed hammocks with vertices of the form $$X\to C \leftarrow Y$$ provide a deformation retract of $L^HC(X,Y)$ is technically quite messy. Even with a better knowledge of what the $L^HC(X,Y)$ looks like, there is still the problem of composition. Two left biased hammocks compose by concatenation to give a more general form of hammock that then gets reduced by the left fractions rules, but these rules do *not* give a normal form for the composite. Much as in the composite of arrows in a quasi-category, the composite here is only defined up to homotopy.
Suppose we let $L^1(X,Y)$ be the simplicial set of such left biased hammocks, then it is a deformation retract of $L^HC(X,Y)$. After composition we reduce to get a diagram $$\xymatrix{L^1(X,Y)\times L^1(Y,Z) \ar[r]\ar@{^{(}->}[d]_{\simeq}\ar[dr]^{concat} &L^1(X,Z)\ar@{^{(}->}[d]^\simeq \\
L^HC(X,Y)\times L^HC(Y,Z) \ar[r] &L^HC(X,Z)\ar[u]<1ex>^{reduce}}$$ This looks as if it should work well, but if we look at the associativity axiom, it is represented by a commutative diagram, and we have replaced each of the nodes of that diagram by a homotopy equivalent object, so we risk getting a homotopy coherent diagram, not a commutative one. This is happening inside $L^HC$, so this does not matter so much. Although attempting to cut down the size of the ‘hom-sets’ does allow us more control over some aspects of the situation, it also has its downside.
The solution is to study the homotopy theory of $\mathcal{S}$-categories as such. This will lead us back towards the Segal maps as well as continuing to interact with homotopy coherence.
For a short time, for the purpose of exposition, we will restrict ourselves to small $\mathcal{S}$-categories with a fixed set of objects, $O$, say, and $\mathcal{S}$-functors will be the identity on objects. We will denote the category of such things by $S\!-\!Cat/O$. (The material here is adapted from [@DKS].) This category has a closed simplicial model category structure in which the simplicial structure is more or less obvious, in which a map $D\to D^\prime$ is a weak equivalence (resp. a fibration), whenever, for every pair of objects, $x,y \in O$, the restricted map $$D(x,y)\to D^\prime(x,y)$$ is a weak equivalence (resp. fibration). (Note, that several of the constructions we have been looking at gave us weak equivalences in this sense, for instance, $S(\mathbb{A})\to \mathbb{A}$ is one such and that the fibrant objects are the locally Kan $\mathcal{S}$-categories over $O$).
Now as we know any of the categories $S\!-\!Cat/O$ form subcategories of the category of simplicial categories, $Cat^{\mathbf{\Delta}^{op}}$. This category also has a closed simplicial model category structure and the nerve and categorical realisation functors induce an equivalence of homotopy categories (even of the simplicial localisations if you want) between $Cat^{\mathbf{\Delta}^{op}}$ and the category of bisimplicial sets $S^{\mathbf{\Delta}^{op}}$. Within $Cat^{\mathbf{\Delta}^{op}}$ we are used to considering $S\!-\!Cat$ as a full subcategory. Related to the problem of reducing the size of the $L^HC(X,Y)$s is the question of determining the result of restricting the induced nerve functor to $S\!-\!Cat$. The solution is rather surprising:
Consider the full subcategory of $S^{\mathbf{\Delta}^{op}}$ determined by those objects $X$ such that (i) $X[0]$ is a discrete simplicial set (cf. the condition on the object simplicial set in an $\mathcal{S}$-category);\
and\
(ii) for every integer $p \geq 2$, the Segal map $$\delta[p]: X[p] \to X[1]\times_{X[0]}X[1]\times_{X[0]}\ldots \times_{X[0]}X[1]$$ is a weak equivalence of simplical sets.
For reasons that will become clearer later, we will call these objects *Segal categories* or sometimes *Segal 1-categories*. Of course, there is a notion of Segal 0-categories, but these are just nerves of ordinary categories. We will denote the category of these Segal 1-categories by $Segal\!-\!Cat$. The result of Dwyer, Kan and Smith, [@DKS], is that the nerve from $Cat^{\mathbf{\Delta}^{op}}$ to $S^{\mathbf{\Delta}^{op}}$, restricts to given an equivalence of homotopy categories between $S\!-\!Cat$ and $Segal\!-\!Cat$. In particular this says that any Segal category is weakly equivalent to a bisimplicial set that is a nerve of a simplicially enriched category. Segal categories are weakened simplicial versions of the algebraic structures given by the categorical axioms, so this is in many ways a coherence theorem for Segal categories rather like the coherence theorems for bicategories, etc.
$\mathbf{\Gamma}$-spaces, $\mathbf{\Gamma}$-categories and Segal categories.
============================================================================
The reason that Segal categories arise as they do is best sought in the paper [@segal] by Segal, although it is not there but rather in [@DKS] that they were introduced, but not named as such. (In fact their first naming seems to be in Simpson’s [@carlos9710011].) In [@segal], one of the main aims was to get ‘up-to-homotopy’ models for algebraic structures so as to be able to iterate classifying space constructions, to form spectra for studying corresponding cohomology theories and to help ‘delooping’ spaces where appropriate. Various approaches had been tried, notably that of Boardman and Vogt, [@boardmanvogt]. In each case the idea was to mirror the homotopy coherent algebraic structures that occurred in loop spaces, etc.
As an example of the problem, Segal mentions the following: Suppose $\mathcal{C}$ is a category and that coproducts exist in $\mathcal{C}$. How is this reflected in the nerve of $\mathcal{C}$? It very nearly acquires a composition law, since from $X_1$ and $X_2$, one gets $X_{12} = X_1\sqcup X_2$, and two 1-simplices $$X_1\rightarrow X_{12} \leftarrow X_2,$$ but $X_{12}$ is only determined up to isomorphism. Let $\mathcal{C}_2$ be the category of such diagrams, i.e. in which the middle is the coproduct of the ends. There is a functor $$\delta_2: \mathcal{C}_2 \to \mathcal{C}\times \mathcal{C}$$and this is an equivalence of categories, but there is also a ‘composition law’ $$m : \mathcal{C}_2 \to \mathcal{C}$$ given by picking out the coproduct. This looks fine but in fact this tentative multiplication again hits the problem of associativity. The theory of monoidal categories was not as developed then in 1974 as it is now, and Segal’s neat solution was to side-step the issue. He formed a category $\mathcal{C}_3$ consisting of all diagrams of form $$\xymatrix{X_1\ar[rr]\ar[drr]\ar[ddr]&&X_{12}\ar[d]&&X_2\ar[ll]\ar[dll]\ar[ddl]\\
&& X_{123}&&\\
&X_{13}\ar[ur]&&X_{23}\ar[ul]&\\
&&X_3\ar[ul]\ar[uu]\ar[ur]&&}$$ the notation indicating that each split line corresponds to the middle term being the coproduct of the two ends. All the usual natural isomorphisms between multiple coproducts are encoded in the one category. There is an equivalence of categories $$\delta_3: \mathcal{C}_3 \to \mathcal{C}\times \mathcal{C}\times \mathcal{C}$$ sending the above diagram to $(X_1,X_2,X_3)$ and a ‘ternary operation’ $\mathcal{C}_3 \to \mathcal{C}$ sending the diagram to $X_{123}$ compatibly, up to specifiable homotopies, with the structure outlined earlier. The advantage is that all of this can be encoded by the nerve and thus by the classifying space structure as a $\mathbf{\Gamma}$-space. The $\mathbf{\Gamma}$-space machinery is now quite well known as it has, for instance, considerable importance in symmetric operad theory, but what are the definitions in that theory and how do they relate to our main theme.
$\mathbf{\Gamma}$-spaces, $\mathbf{\Gamma}$-categories.
-------------------------------------------------------
**Definition**
\(i) The category $\mathbf{\Gamma}$ is the category whose objects are all finite sets and whose morphisms from $S$ to $T$ are the maps $\theta :S \to \mathcal{P}(T)$ such that when $a\neq b\in S$ then $\theta(a)\cap \theta(b) = \emptyset$. The composite of $\theta: S\to \mathcal{P}(T)$ and $\phi :T \to \mathcal{P}(U)$ is $\psi : S \to \mathcal{P}(U)$ where $\psi(a) = \bigcup_{b\in\theta(A)}\phi(b)$.
\(ii) A *$\mathbf{\Gamma}$-space* is a functor $A: \mathbf{\Gamma}^{op}\to Top$ such that\
(a) $A(0)$ is contractible;\
and\
(b) for any $n$, the map $p_n : A(n) \to\underbrace{ A(1)\times \cdots \times A(1)}_n$ induced by the maps $i_k : 1\to n$ in $\mathbf{\Gamma}$ where $i_k(1) = \{k\}\subset n$, is a homotopy equivalence.
There is an obvious functor $\mathbf{\Delta} \to \mathbf{\Gamma}$ which takes $[m]$ to the set $m$ and $f : [m]\to [n]$ to $\theta (i) = \{j \in n ~|~f(i-1)< j \leq f(i)\}$. Composing a $\mathbf{\Gamma}$-space $A$ with (the opposite of) this functor gives an underlying simplicial space for the $\mathbf{\Gamma}$-space. (As we have tended to concentrate on simplicial theory rather than on the topological side, we could equally well have replaced ‘space’ as meaning topological space by ‘space’ as meaning simplicial set, in which case a $\mathbf{\Gamma}$-space would be a bisimplicial set with side conditions.)
Our task is not to review the contents of Segal’s 1974 paper, so the next point to note is the definition of a $\mathbf{\Gamma}$-category. This follows the same model:
A $\mathbf{\Gamma}$-category is a functor $\mathcal{C}: \mathbf{\Gamma}^{op}\to Cat$ such that (a) $\mathcal{C}(0)$ is equivalent to a one arrow category, and (b) as before except weak homotopy equivalence is replaced by equivalence of categories.
It is not surprising that if $\mathcal{C}$ is a $\mathbf{\Gamma}$-category, applying nerve and then geometric realisation (i.e. taking its classifying space) gives a $\mathbf{\Gamma}$–space.
The most fundamental $\mathbf{\Gamma}$-category is when $Sets_{fin}$ is the category of finite sets under disjoint union, and we take an object $n$ for each natural number $n$. The resulting $\mathbf{\Gamma}$-category is closely related to the disjoint union of the symmetric groups and thus to free symmetric monoidal closed categories, but exploring in that direction would take us too far afield.
**Remarks**
\(i) It is not obvious to start with why the structure of a $\mathbf{\Gamma}$-space is built on $\mathbf{\Gamma}$ and not just on $\mathbf{\Delta}$. The point is, and this is important for the alternative ideas that we will be looking at later, if $A$ is a $\mathbf{\Gamma}$-space, we can form a classifying $\mathbf{\Gamma}$-space $BA$. Any $\mathbf{\Gamma}$-space yields a simplicial space as above and hence a space $|A|$ by using the geometrical realisation (technically one uses the form of realisation that does not use the degeneracies, cf. [@segal] again). The classifying space $BA$ is given by assigning to a finite set $S$, the realisation of the $\mathbf{\Gamma}$-space $T\longmapsto A(S\times T)$. This would not be possible if one considered just the underlying simplicial space, but suggests that the classifying space might be thought of as a bisimplicial space.
\(ii) It is worth recalling the structure of Lawvere’s algebraic theories, [@lawvere], for comparison. One way to view these is to use some elementary ideas from topos theory. The category of finite sets is given the structure of a basic algebraic site $\mathcal{T}_0$ by taking epimorphic families as covering sieves. The minimal covering sieves induce colimit cones $1\sqcup 1 \cdots \sqcup 1 \to n$, so that sheaves on the algebraic site are graded sets $\{X(n) ~|~ n\geq 0\}$, endowed with bijections $X(n) \cong X(1)^n$ for $n\geq 0$. An algebraic theory is then a ‘coproduct preserving’ extension $\mathcal{T}_1$ of $\mathcal{T}_0$ and a $\mathcal{T}_1$-model or $\mathcal{T}_0$-algebra is a presheaf on $\mathcal{T}_1$, which restricts to a sheaf on $\mathcal{T}_0$. (Some of the links between the modern theory of weak $\omega$-categories, and operads as a generalisation of Lawvere’s algebraic theories are considered in Berger, [@berger], and in related articles mentioned there. This is very relevant to the final theme of these notes, namely the link with algebraic models for homotopy types and the corresponding higher order categories.) It is clear that Segal’s $\mathbf{\Gamma}$-spaces are a lax or ‘up-to-homotopy’ version of algebraic theories. We should also note out that there are links with operads of various types, but a discussion of these would take us too far afield.
Segal categories
----------------
The theory of $\mathcal{S}$-categories is too strict for convenience. As we have seen the notion of Segal 1-category should be a weakened version of that of $\mathcal{S}$-category and we have discussed them informally earlier. A more formal treatment is given in several places. The following is adapted from Toen’s [@toen].
**Definitions**
- A *Segal 1-precategory* is specified by a functor $$A : \mathbf{\Delta}^{op} \to \mathcal{S}$$ (i.e. a bisimplicial set) such that $A_0$ is a constant simplicial set called *the set of objects of* $A$.
- A *morphism* between two Segal 1-precategories is a natural transformation between the functors from $ \mathbf{\Delta}^{op}$ to $\mathcal{S}$.
- A Segal 1-precategory $A$ is a *Segal 1-category* if for each $[p]$, the Segal map $$\delta[p]: A_p \to A_1\times_{A_0}A_1 \times_{A_0}\ldots \times_{A_0}A_1,$$ is a homotopy equivalence of simplicial sets.
- For any Segal 1-category $A$, we define its *homotopy category* $Ho(A)$ as the category having $A_0$ as its set of objects and $\pi_0(A_1)$ as its set of morphisms.
- A morphism of Segal 1-categories $f : A \to B$ is an *equivalence* if it satisfies the following conditions:
1. For each $[p]\in \mathbf{\Delta}$, the morphism $f_p : A_p \to B_p$ is an equivalence of simplicial sets;
2. The induced functor $Ho(f) : Ho(A) \to Ho(B)$ is an equivalence of categories.
The category of Segal 1-precategories will be denoted $1-PrCat$. It contains the category of (small) $\mathcal{S}$-categories as we have seen. It also has a Quillen closed model category structure in which the cofibrations are the monomorphisms, and the fibrant objects are exactly the Segal 1-categories, moreover a morphism between Segal 1-categories is a weak equivalence in $1-PrCat$ if and only if it is an equivalence of Segal 1-categories in the above sense. (For more on this structure, see the paper by Simpson, [@carlos9704006].)
**Remark.**
Although quite a useful intermediate concept. I feel that ‘1-precategory’ seems a slight misnomer, it is as if we called a directed graph a ‘precategory. There is no algebraic structure involved in the notion. I have stuck with the terminaology for want of a better term.
We have seen that $\mathcal{S}$-categories and Segal categories model parts of homotopy theory well. At the same time $G(K)$ is a $\mathcal{S}$-groupoid and we noted that these model all homotopy types. In any of these models, it is always relatively easy to extract information in low dimensions, but the level of ‘algebraicity’ in such models is limited and many people have searched for $n$-categorical models of homotopy theory or of homotopy types that incorporate both geometric and algebraic aspects of the theory. Based on Segal-categories and his delooping machine of [@segal], one gets the Tamsamani models, [@tam], for weak $n$-categories and weak $n$-groupoids for any $n$.
Tamsamani weak $n$-categories
=============================
The problem of finding good $n$-categorical models for the geometric data encoded in a homotopy type came to the fore with Grothendieck’s notes, [@stacks] and in particular his letter to Quillen therein. In this he suggested that there should be models of homotopy types that (i) were less redundant in their ‘data storage’ than, say, simplicial sets or simplicial groups, (ii) had the advantages of the mixed algebraic combinatoric power of categories (cf. the way in which the fundamental groupoid of a simplicial complex encodes the graph theoretic structure of the 1-skeleton and the algebraic nature of path concatentation), (iii) had a highly developed homotopy theory that was consistent with enough homotopy coherence and category theory to enable homotopy analogues of sheaf theoretic constructions to be made (stack theory), and (iv) for which could be developed a higher order version of the unified Galois-Poincaré theory that encompasses both classical Galois theory and the Galois correspondence that classifies covering spaces in terms of $\pi_1(X)$–sets, (cf. Borceux and Janeldize, [@BJ]).
There are now well known problems in using the hierarchy of strict $n$-categories for such a task, as the strictness kills certain structure that is needed if one is to model spaces which have, for instance, non-trivial Whitehead products. (The crossed complex models of , for example, are able to have a relatively simple structure because they do not try to model Whitehead products. They can thus be thought of either as incomplete models for all homotopy types or complete models for a restricted class of homotopy types, namely those with trivial Whitehead products.) From a purely categorical position, the problem arises in dimension 3, where 3-groupoids cannot capture all homotopy 2-types. The problem is the interchange law. We saw how it corresponded to two homotopies ‘sliding’ over each other, but that homotopy commutative square is not a trivial one and *does* encode structure. One solution is to used Gray groupoids, (see the discussion and references in , for instance). Another related one is to use tri-groupoids, the next level up from bicategories and bigroupoids. Tamsamani’s approach uses Segal category ideas to encode this necessary lack of ‘strictness’ and obtains weak $n$-categories generalising bicategories for all $n$..
Bisimplicial models for a bicategory.
--------------------------------------
As a starting point for our look at Tamsamani’s models, consider the 2-dimensional analogue of the Grothendieck-Segal condition given in Proposition \[GrotSegal\]. That showed that if $A$ was a simplicial set such that for all $p\ge 2$, the Segal maps $$\delta[p]: A_p \to A_1\times_{A_0}A_1 \times_{A_0}\ldots \times_{A_0}A_1,$$ were bijections, then $A$ was the nerve of a category. (We will just say that $A$ is a *1-nerve*.)
Going up one dimension, a useful process is ‘categorification’. This horrible word is in fact a good term for an important process. A monoid is a set with some extra structure. Categorify it and it is a small category with a single object. A monoidal category is a category with a multiplication. Categorify it and it is a bicategory with a single object. In the categorification process another idea also occurs. Equality is a great idea for elements, but a poorly behaved one for monoids where isomorphism is more natural. Categorify and isomorphism is not so natural, equivalence between categories is what is needed. As one categorifies the key to success is to use equivalences except possibly in the top dimension available.
Categorifying the Grothendieck-Segal condition, let $A$ be a bisimplicial set. We can think of this as $$A: (\mathbf{\Delta}^{op})^2\to Sets$$ or $$A : \mathbf{\Delta}^{op} \to \mathcal{S}.$$ Both ways are useful, but a word is needed on notation. If $A : \mathbf{\Delta}^{op} \to \mathcal{S},$ then for $[p] \in \mathbf{\Delta}$, $A_p : \mathbf{\Delta}^{op}\to Sets$ and its value on $[q]$ is $A_{p,q}$ in the other notation. Confusion can arise about which ‘variable’ is changing and which constant, so the notation $A_{p/}$ has been introduced to indicate that anything after the / is varying. In dimension 2, this is not that essential but in higher dimensions it is a great help.
Assume that (i) $A_{0/}$ is a constant simplicial set, written simply $A_0$ and called the *set of objects* of $A$, (ii) for each $p \in \mathbf{\Delta}$, $A_{p/} \in \mathcal{S}$ is a 1-nerve, and (iii), again for each $p \in \mathbf{\Delta}$, the morphism $$\delta[p]: A_{p/} \to A_{1/}\times_{A_0}A_{1/} \times_{A_0}\ldots \times_{A_0}A_{1/},$$ is an equivalence of categories.
In other words $A$ is a ‘Segal 2-category’ or perhaps ‘Tamsamani-Segal weak 2-category’. What does such a beastie look like?
For any pair $(p,q) \in \mathbb{N}^2$, we can think of any $a\in A_{p,q}$ as being a $(p,q)$-prism, $$a: \Delta[p]\times\Delta[q]\to A.$$ The condition that $A_{0/}$ is constant, implies that $A_{0,q}$ is the same for all $q$. For instance, a $(1,1)$-prism would naturally look like a square, but since it is constant in the second direction we get as a better picture:
 fig. 1
The iterated face maps from $A_{p/}$ to $A_0$ yield a map $$A_{p/} \to A_0\times \ldots A_0$$ and if $(x_0, \ldots, x_p) \in A_0\times \ldots A_0$, then we may think of the inverse image of $(x_0, \ldots, x_p)$ by this map as the 1-nerve (and thus category) of those prisms having the $(x_0, \ldots, x_p)$ as their vertices in the first direction. We will denote this by $A_{p/}(x_0, \ldots, x_p)$. In our simple case we have $p = 1$, and $A_{1/}(x_0,x_1)$ consisting of things like:
 fig. 2
The simplicial set $A_{1/}$ is a 1-nerve, so *is* a category: for each $q\geq 2$, the Segal map $$\delta[q]: A_{1,q} \to A_{1,1}\times_{A_0}A_{1,1} \times_{A_0}\ldots \times_{A_0}A_{1,1},$$ is a bijection. This is inherited by the individual $A_{1/}(x_0,x_1)$s. For instance, things in $A_{1,2}$ look like:
 fig. 3
and as the diagram is intended to indicate, composition works perfectly.
The Segal maps $$\delta[p]: A_{p/} \to A_{1/}\times_{A_0}A_{1/} \times_{A_0}\ldots \times_{A_0}A_{1/},$$ are all equivalences of categories. To start to understand this we look at $p = 2$. The prism for $q=1$ looks like:
 fig. 4
\[cushion\] We have three ‘lozenge’ shaped vertical faces, (each with a 2-cell/arrow going from top to bottom) and the ‘horizontal’ triangular 2-cells (no arrow shown in the diagram).
If $A_{2/}$ was *isomorphic* to $A_{1/}\times_{A_0}A_{1/}$ then the front two faces, the 2-cells $a$ and $b$, would uniquely determine the third face, and thus a composite, but all we have is that $\delta[2]$ is an *equivalence* of categories. We let $\gamma[2] : A_{1/}\times_{A_0}A_{1/}\to A_{2/}$ be an inverse equivalence. There are natural isomorphisms $$\alpha_2: \delta[2]\gamma[2] \stackrel{\cong}{\to} Id, \mbox{\quad and \quad }\gamma[2]\delta[2] \stackrel{\cong}{\to} Id.$$ The obvious definition for the composite would be $d_1^1\gamma[2](a,b)$, but it is better to preprocess things a bit as this does not give quite the right answer! Suppose that within $A_{1/}(x_0,x_1)$, $a: f \Longrightarrow f^\prime$ and that within $A_{1/}(x_1,x_2)$, $b: g \Longrightarrow g^\prime$. (We will use ‘2-cell’ arrows since that is what we hope they will be later.) There are isomorphisms within $A_{1/}\times_{A_0}A_{1/}$, $$\alpha_2(f,g) : \delta[2]\gamma[2](f,g)\Longrightarrow (f,g),$$and $$\alpha_2(f^\prime,g^\prime) : \delta[2]\gamma[2](f^\prime,g^\prime)\Longrightarrow (f^\prime,g^\prime),$$ and hence a composite $$\delta[2]\gamma[2](f,g)\stackrel{\alpha_2(f,g)}{\Longrightarrow} (f,g)\stackrel{(a\times b)}{\Longrightarrow} (f^\prime,g^\prime)\stackrel{\alpha_2(f^\prime,g^\prime)^{-1}}{\Longrightarrow}\delta[2]\gamma[2](f^\prime,g^\prime).$$ Writing $\sigma = \gamma[2](f,g)$ and similarly $\sigma^\prime = \gamma[2](f^\prime,g^\prime)$, we have two objects of the category $A_{2/}$ and as $\delta[2]$ is an equivalence, it is fully faithful and essentially surjective. However we have our composite in $A_{1/}\times_{A_0}A_{1/}(\delta[2]\sigma,\delta[2]\sigma^\prime)$, so there is a unique $\varepsilon \in A_{2/}(\sigma,\sigma^\prime)$ mapped down to it by $\delta[2]$. We set $f\#_0g = d_1^1(\sigma)$, and $f^\prime\#_0g^\prime = d_1^1(\sigma^\prime)$, giving a sensible definition of *horizontal* composition of 1-arrows, and then $a\#_0 b = d^1_1(\varepsilon)$ as composite 2-cell.
We will not give the complete proof of the fact that any Tamsamani-Segal 2-category defines a bicategory and *vice versa*. The above just gives some of the flavour of the method. In this approach it is very easy to make a slip so let us just note that the arguments being used are more or less identical to those used to prove the result on replacement up to specified homotopy equivalence of spaces in a commutative diagram so as to get a homotopy coherent diagram. It should thus be possible to use ideas from that area of homotopy coherence, and experience of similar problems in other areas to shorten the proof given by Tamsamani (the details are omitted from [@tam]), but are included in the thesis on which that article is based.
Tamsamani-Segal weak $n$-categories
-----------------------------------
We will not give a detailed treatment of the extension of these ideas to $n$ dimensions. It would take up too much space here. Rather we will follow the summary of that extension given by Simpson in [@carlos9708010]. (One word of warning, in that source the only types of $n$-categories that occur are the weak $n$-categories being developed in that setting, so the author omits the adjective ‘weak’ consistently. He states this early on but it means that it is dangerous to ‘dip’ into this paper although well worth reading from the start for the ideas it discusses and develops. We will call these objects ‘T-S weak $n$-categories’ to distinguish them from the other models available.) We will launch in at ‘the deep end’!
**Definition**
A *T-S weak $n$-category* is a functor $$A : \mathbf{\Delta}^{op} \to weak(n-1)Cat,$$ where $weak(n-1)Cat$ is the category of T-S weak $(n-1)$- categories, such that
- $A_0$ is a set (i.e. a discrete T-S weak $(n-1)$-category);
- for each $p \geq 2$, the Segal map $$\delta[p]: A_{p/} \to A_{1/}\times_{A_0}A_{1/} \times_{A_0}\ldots \times_{A_0}A_{1/},$$ is an $(n-1)$-equivalence of weak $(n-1)$-categories.
**Remarks**
Clearly some remarks are called for:\
(i) It is clear that any T-S weak $n$-category is a special $n$-simplicial set and could be equally well specified by $$A : (\mathbf{\Delta}^{op})^n\to Sets.$$ The notation $A_{p/}$ is the obvious extension of that which was used in case $n = 2$. Thus the $(n-1)$-simplicial set $A_{p/} $ satisfies $$A_{p/}([q_1], [q_2], \ldots, [q_{n-1}]) = A_{p,q_1,q_2,\ldots, q_{n-1}}.$$ A further extension of the notation is also useful. Let $M = ([m_1], \ldots ,[m_{n-1}])$ be an object of $\mathbf{\Delta}^{n-1}$ and $[m] \in \mathbf{\Delta} $, then $(M,m)$ or more correctly $(M,[m])$, denotes an obvious object of $\mathbf{\Delta}^{n}$. We will write $A_M$ for the obvious simplicial set, in which the first $(n-1)$-variables are ‘clamped’ at $M$. Finally if $M \in\mathbf{\Delta}^{n-h}$ and $M^\prime \in\mathbf{\Delta}^{h}$ for some $h$ then a notation $A_{(M,M^\prime)}$ will be used in the obvious way. This is particularly useful when $M^\prime = 0_h:= (0, \ldots, 0) \in \mathbf{\Delta}^{h}$ because of the next remark.\
(ii) The requirement that $A_0 := A_{0/}$ be a set is to be interpreted as meaning that it is a constant $(n-1)$-simplicial set. It is not hard to show that as this will be the case for all intermediate definitions of T-S weak $k$-category, $0<k<n$, this condition implies that in the $n$-simplicial set $A$, if $p_i = 0$ for some $i$, then $A_{p_1,\ldots, p_n} = A_{p_1, \ldots, p_{i-1},0_{n-i}}$, i.e. it is independent of the values of $p_{i+1}, \ldots, p_n$. We will use the term *$n$-precategory* for a $n$-simplicial set, $A : (\mathbf{\Delta}^{op})^n\to Sets,$ which satisfies the first condition: *$A_0$ is a set* and hence has the property of ‘constancy’ mentioned above.\
(iii) The category of T-S weak $n$-categories is defined in the obvious way, but, of course, *its* definition will depend on that of weak $(n-1)$-categories and the category they form and so on. As we have a notion of weak 2-category in this setting, that does not cause a problem. What is more of a bother is that the definition also depends recursively on a notion of equivalence of weak $(n-1)$-categories.
**Truncation**
A $n$-precategory will be said to be *1-truncatable* if for all $M \in (\mathbf{\Delta})^{n-1}$, $A_M$ is a category (i.e. is the nerve of a category).
If $A$ is 1-truncatable, then we can form a $(n-1)$-precategory $TA$, called the *1-truncation* of $A$, which associates to $M$ the set of isomorphism classes of objects of the category, $A_M$. If $A$ is 1-truncatable, there is a natural morphism of $(n-1)$-simplicial sets, $$\tau(A)(M): A_{(M,0)}\to T(A)_{(M)},$$ which sends an object of the category $A_M$ to its isomorphism class.
We will say that $A$ is *$k$-truncatable* for $2 \leq k \leq n$ if and only if $A$ is $(k-1)$-truncatable and $T^{k-1}A$ is 1-truncatable. Now suppose that $A$ is $k$-truncatable, and that $1 \leq h\leq k <n$. Then there is a morphism of $(n-h)$-simplicial sets, $$\tau^h(A)_M : A_{(M,0_h)}\to T^h(A)_{(M)},$$ for $M\in \mathbf{\Delta}^{n-h}$, and abbreviating $(M,0_i)$ to $M_i$, given as the composite $$\begin{aligned}
A_{(M_h)}\stackrel{\tau(A)_{(M_{h-1})}}{\longrightarrow}T(A)_{(M_{h-1})}\stackrel{\tau(TA)_{(M_{h-2})}}{\longrightarrow}&T^2(A)_{(M_{h-2})}&\longrightarrow \ldots \\
\ldots \longrightarrow &T^{h-1}(A)_{(M_1)}&\stackrel{\tau(T^{h-1}A)_{(M})}{\longrightarrow}T(A)_{(M_{h-1})}.\end{aligned}$$
**$n$-equivalence**
As we have already noted, an equivalence of categories is a functor, $f : A \to B$ that is (i) fully faithful and (ii) essentially surjective. If we want to define the notion of $n$-equivalence, one approach is to generalise these two properties. For $n=2$, $A$ and $B$ are now bicategories, ‘fully faithful’ now means that for each pair of objects, $x,y \in Ob(A)$. the induced functor $$A(x,y) \to B(fx,fy)$$ is an equivalence of categories and similarly ‘essentially surjective’ generalises to saying that every object in $B$ is equivalent to an object of the form $fx$ for some object $x$ of $A$. With our link between T-S weak 2-categories and bicategories and also the truncation, we can see that a bicategory can be truncated twice. If one applies $T$ once then we replace each category $A(x,y)$ by its set of isomorphism classes, thus $T(A)$ is a category and $T^2(A)$ is the set of isomorpism classes in $T(A)$. The claim is that $f : A \to B$ is essentially surjective if and only if $T^2(f)$ is surjective.
Suppose $T^2(f)$ is surjective, and $b$ is an object of $B$, then there is an object $a$ in $A$ and an isomorphism in $T(B)$ between $b$ and $f(a)$. That interprets as saying there is a pair of 1-cells $\beta: b \to f(a)$, and $\alpha: f(a)\to b$, whose composites are isomorphic to the respective identities, i.e. there are diagrams $$\xymatrix{&f(a)\ar[dr]^\alpha\ar@{}[d]|\Updownarrow&\\
b\ar[rr]_{id}\ar[ur]^\beta && b}\hspace{2cm}\xymatrix{&b\ar[dr]^\beta\ar@{}[d]|\Updownarrow&\\
f(a)\ar[rr]_{id}\ar[ur]^\alpha && f(a)},$$ which seems like a pretty good version of equivalent objects. The converse is similar.
Tamsamani, [@tam], generalises this idea to essential surjectivity in higher dimensions, and, as Simpson notes in [@carlos9708010], this can be viewed as saying that $f: A \to B$ between two T-S weak $n$-categories is a essentially surjective if $T^n(f)$ is surjective. In fact, he points out that if $f$ is an $n$-equivalence, then $T^n(f)$ is a bijection.
Thus we have as a definition that a morphism $f: A\to B$ of T-S weak $n$-categories is an $n$-equivalence if and only if (i) each pair of objects, $x,y \in Ob(A)$. the induced morphism $$A_{1/}(x,y) \to B_{1/}(fx,fy)$$ is an $(n-1)$-equivalence of weak $(n-1)$-categories, and (ii) $T^n(f)$ is surjective.
This essentially finishes the definition of T-S weak $n$-category as we now have a working definition for all the terms involved.
It is amusing and quite useful that $T$ gives a functor from $weak~n~Cat$ to $weak~(n-1)~Cat$.
The Poincaré weak $n$-groupoid of Tamsamani
-------------------------------------------
Tamsamani defines a fundamental Poincaré $n$-groupoid for an arbitrary space $X$ and any $n$. There are some errors in the published version, since an attempt at an iterative definition fails since no topology has been specified on the set of simplices involved, however this slip is rectified later on and the full approach would seem to give a fairly clear method for defining the gadget. It has to be remembered that the result will be a weak $n$-groupoid, so in dimension $n=2$ we get a bigroupoid and not a 2-groupoid or a double groupoid with connection. This means that the object is rather large. Here we will attempt to describe the bigroupoid by taking apart the construction in that case. We initially give the construction in general as it is easier to do it that way.
Let $X$ be a space and $M = (m_1, \ldots, m_n)$, an object of $\mathbf{\Delta}^n$. Let $\Delta ^M : = \Delta^{m_n} \times \dots \times \Delta^{m_1}$, (note the reversal of order). If $0\leq k\leq n-1$, set $M_{n-k} = (m_1, \ldots, m_{n-k-1})$ and $M^\prime_{n-k} = (m_{n-k+1}, \ldots , m_n)$ and for each $0\leq i\leq m_{n-k}$, let $v_i$ denote the $i^{th}$ vertex of $\Delta^{m_{n-k}}$.
Let $$\begin{aligned}
X_M = \{f: \Delta^M \to X\hspace{-.3cm} &|&\hspace{-.3cm} \mbox{ for all }k, ~ 0\leq k\leq n-1, \mbox{ for all }i,~ 0\leq i\leq m_{n-k}\\ &&\hspace{1cm}\mbox{ and for all }\mathbf{x}\in \Delta^{M_{n-k}}, \mathbf{x}^\prime \in \Delta^{M^\prime_{n-k}}, \\
&&\hspace{3cm}f(\mathbf{x},v_i,\mathbf{x}^\prime) = f_i(\mathbf{x^\prime}) \mbox{ where }f_i \in X_{M_{n-k}}\}\end{aligned}$$ Note that if $m_{n-k} = 0$, then there is one vertex only of the corresponding $\Delta^0$ so then $X_M = X_{M_{n-k}}$. This encodes into a subcomplex of the $n$-simplicial singular complex, the constancy rule that we need for a weak $n$-category.
**Homotopy in $X_M$**
Let $f,g$ be two elements of $X_M$. We will say that $f$ and $g$ are homotopic and write $f\simeq g$ or $\overline{f} = \overline{g}$ if there is a $\gamma \in X_{M,1}$ such that $\delta_0(\gamma) = f$ and $\delta(\gamma) = g$. Homotopy is an equivalence relation and we will denote by $\overline{X}_M$ the set of homotopy classes.
With the obvious identifications, we now define, $\Pi_n(X)_M := \overline{X}_M$ with the induced face and degeneracy maps.
[@tam]\
The $n$-simplicial set $\Pi_n(X)$ is a T-S weak $n$-groupoid. $\blacksquare$
**So what does it look like in dimension 2?**
The definition of $X_M$ has quite a few subcases even with $n = 2$, so we take them one at a time. We have ‘for all $k$, $0\leq k\leq 1$’:
$k = 0$: for all $i$, $0\leq i\leq m_2$, $v_i$, the $i^{th}$ vertex of $\Delta^{m_2}$, and for all $x_1 \in \Delta^{m_1}$, one has $f(v_i,x_1) = f_i(x_1)$. However this places no restriction on $f$ since there are no variables in front of the $v_i$. It merely states that $f_i = f(v_i, \_)$.
$k = 1$: for all $i$, $0\leq i\leq m_1$, $v_i$, the $i^{th}$ vertex of $\Delta^{m_1}$, and for all $x_2 \in \Delta^{m_2}$, one has $f(x_2,v_i) = f_i$, but here $f_i$ has *no* variables, it is a constant value. The picture for $M = (1,1)$ is of a square with constant values on the vertical sides as in our discussion of the bicategory model earlier in these notes.
 fig. 5
The picture for an element in $X_{(2,1)}$ is similar, a vertical prism, $\Delta^2\times \Delta^1$, $\Delta^2$ as base and with ‘constant’ vertical edges, i.e. like the shape we saw earlier in the discussion of bicategories, i.e. fig 4 on page . The case $M=(1,2)$ is a horizontal prism with constant ends.
To study the homotopy relation, we need to look at $X_{M,1}$.
In particular we look at $X_{1,1,1}$: there are three cases $k = 0,1,2.$ As above the case $k = 0$ imposes no condition on the singular multi-prism $f$, since the rule merely states $f(v_i,\_,\_) = f_i(\_,\_)$ in a sense just defining $f_i$. The condition for $k=1$, however implies that for all vertices $v_i$ of $\Delta^{m_2}$, $$f(x_3,v_i,x_1) = g_i(x_1),$$ i.e. is independent of the variable from $\Delta^{m_3}$. Finally for $k = 2$, the restriction is that $f(x_3,x_2,v_i)$ is a constant function. This means that a homotopy can be represented as a singular cube in $X$ in which the left and right vertical faces are constant, the top and bottom are independent of the third direction and the other two faces have no other restriction, i.e. the homotopy is precisely a homotopy relative to the boundary of the squares in dimension $(1,1)$. This makes it look as if the Tamsamani bigroupoid is essentially the same as that of Hardie, Kamps and Kieboom, (HKK), [@HKK:bigroupoid]. It would be interesting to see if Tamsamani’s weak 3-groupoid could be adapted, as can this bigroupoid case, to allow for a notion of thinness followed by a quotient to, say, a Gray groupoid, analogously to the way in which the HKK bigroupoid leads in [@HKK:2-groupoid], to a 2-groupoid, or a double groupoid with connections.
Conclusion?
===========
We have looked at the way in which $\mathcal{S}$-categories, their homotopy coherent form, Segal 1-categories, and, perhaps, their iterated form, the T-S weak $n$-categories, enter into the two key areas of abstract homotopy theory. Some of the sources used have been fairly recent, so there is still a lot to do. Here is a list of some questions, some better than others:
1\. What is the precise link between the Dwyer-Kan $\mathcal{S}$-groupoid and simplicial coherence? What do homotopy coherent simplices in $G(K)$ tell one about the models? Do they lead to good descriptions of higher interchange elements analogously to the way in which maps from $S[4]$ to a $\mathcal{S}$-category produce that interchange square? (The work of Ali Mutlu and myself on higher order Peiffer pairings in simplicial groups may be of relevance here, cf, [@atmp1; @atmp2].)
2\. The Tamsamani method starts with a space $X$ and produces a multi-simplicial singular complex. That method could be applied to other types of object, for example, a simplicial group, a category, and so on. What do the corresponding Poincaré weak $n$-groupoids tell one? (Remember that categories model *all* homotopy types. They just do it in rather a difficult way from the point of view of calculations!)
3\. Is it true that the Dwyer-Kan hammock localisation of $\mathcal{B}^\mathbb{A}$ with respect to level homotopy equivalences is ‘closely related to’ $\mathcal{S}(S(\mathbb{A}), Ner_{h.c.}(\mathcal{B}))$ for $\mathcal{B}$ locally Kan? If so ‘how close’? A lot of light on this problem has been shed by Vogt in [@vogt92] in the topological setting. The question would seem to be of particular importance given the upsurge of interest in $A_\infty$-categories resulting from new approaches to quantum deformation.
4\. Can one construct a homotopy coherent nerve for a general Segal category (probably yes) and what are its properties? In general what is the precise relationship between quasi categories (as a weakening of categories) and Segal categories (also a weakening of categories)? (This question is vague, of course, and would lead to many interpretations.)
5\. Can one unpack a T-S weak 3-category in a sensible way? Is it sensible to try?.
6\. How powerful is the DKS coherence theorem for Segal categories? Can a clear ‘stand-alone’ proof be given that does not depend on a lot of extra machinery? How constructive can it be made?
7\. If you take a T-S weak $n$-category as an $n$-simplicial set and extract (i) its diagonal and (ii) its Artin Mazur codiagonal, what does the structure that results look like? Is it related, perhaps just in the groupoid case, to hypercrossed complexes or hypergroupoids, [@duskin].
8\. To complete the ‘Grothendieck programme’ of pursuing stacks, (i.e. to construct (and study) stacks of models for homotopy $n$-types and to prove for the locally constant $n$-stacks, some form of Galois-Poincaré correspondence theorem between equivalence classes of $n$-stacks and the corresponding $n+1$-type model of the base space or topos), one needs a good theory of homotopy coherence with those models. The current attempts by Simpson, Toen and others (see papers in the references) go a long way towards such a goal, and in work by others in theoretical physics, similar approaches have been tried in very special cases (abelian models, via chain complexes etc.). These latter approaches use a lot of the machinery sketched in these notes. It is probably fair to say that all these approaches suffer from the gulf between the technical nature of the machinery and the simple intuitions behind them.
This last question is thus to ask is it possible to give a clear intuitive approach, to say, 2-stacks using the Segal-category machinery that does not get bogged down in a technical morass of Quillen model category theory, an enormous amount of weak infinity category theory or similar machinery. This is not to say that approaches using those ideas are not providing a necessary step on the road to understanding the Grothendieck problem, but to ask for a simple approach that will aid the geometric intuition.
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T.Porter,\
Mathematics Department,\
School of Informatics,\
University of Wales Bangor,\
Bangor,\
Gwynedd, LL57 1UT,\
United Kingdom.
| 0 |
---
author:
- 'G.C. Van de Steene'
- 'G.H. Jacoby [^1]'
date: 'received / accepted'
title: Radio observations of new galactic bulge planetary nebulae
---
Introduction
============
Planetary Nebulae (PNe) are bright emission line objects, observable throughout the Galaxy. They are excellent probes of abundance gradients, the chemical enrichment history of the interstellar medium, the effects of metallicity on stellar evolution, and kinematics.
Most small PNe ($\sim$90$\%$) within 10 of the galactic center are physically close to it (Pottasch & Acker 1989). Since they can be assumed to be at the same known distance of $\sim$7.8 kpc, their distance-dependent parameters, such as luminosity and size, can be determined. These parameters are needed to define the underlying population. The chemical composition and the central star parameters are computed via self-consistent photo-ionization modelling of the nebula. Because luminosity correlates with the central star mass, which correlates with the progenitor mass, which, in turn, correlates with stellar age, the relationship between age and composition can be deduced (Dopita et al. [@Dopita97], Walsh et al. [@Walsh00]). The chemical enrichment history of the Bulge could be tracked using PNe.
We surveyed a 4 x 4 degree field centered on the galactic center in \[S[iii]{}\]$\lambda$9532 and a continuum band at KPNO with the 60-cm Schmidt telescope and a 2048 x 2048 pixels thick STIS CCD in July 1994 and June 1995. The field of view was 65x65 and the pixel size 2. This survey has uncovered 95 new PN candidates in addition to the 34 previously known in this region (Acker et al. 1992, Kohoutek 1994). 45 PNe were confirmed via optical spectroscopy with the 1.52-m ESO telescope and the Boller & Chivens spectrograph, while 19 fainter ones were confirmed at the CTIO 4-m with the RC spectrograph (Van de Steene & Jacoby 2001, in preparation). Accurate radio flux densities and angular diameters are crucial to obtain a good photo-ionization model of the PNe (van Hoof & Van de Steene [@vHoof99]). The very high extinction causes the H$\beta$ line to be faint or even undetected in the optical spectra. Hence the radio flux density is needed to determine the extinction and the total ionizing flux.
In this article we present the radio continuum observations of 64 PNe confirmed spectroscopically with the ESO 1.52-m and CTIO 4-m telescopes. We describe the observations in Sect. \[observations\] and the data reduction in Sect. \[reduction\]. The results are presented in Sect. \[results\]. The improved method for determining the distances, based on a relationship between radii and radio surface brightness (Van de Steene & Zijlstra [@VdSteene95]) is used in Sect. \[distances\] to determine the distances of the PNe and discuss their distribution in the galactic bulge. In Sect. \[extinction\] we determine the extinction values of these new bulge PNe.
Observations
============
Radio continuum observations for 64 new PNe which had been confirmed spectroscopically at the ESO 1.52-m and CTIO 4-m telescopes were obtained with the Australia Telescope Compact Array. The array was in configuration $\#$ 6A. The shortest baseline was 337 m and the longest 5939 m. The bandwidth was 128 MHz centered at 4800 MHz and 8640 MHz, corresponding to 6 and 3 cm respectively. The synthesized beam for the 6 km array at 6 cm is $\sim$ 2 and $\sim$1 at 3 cm. According to the ACA manual, the largest well imaged structure in a 60 min observation is 30 at 6 cm and 15 at 3 cm. Consequently we expected all our sources to be well imaged at 6 cm and virtually all at 3 cm. Forty PNe were observed at 3 and 6 cm simultaneously for 12 h on each of 12, 13, 16 & 17 February 1997 and 36 PNe were observed during 10 h on each of 24, 25, and 26 May 1999. In 1997 each subsample contained 10 PNe and each PN was observed for 5 min every hour. In 1999 each subsample contained 6 PNe and each PN was observed for 8 min every hour. The total quality integration time per PN was at least 50 min in 1997 and 80 min in 1999. To avoid artifacts at the center of the field we offsetted the positions 30 in declination. At the beginning and the end of each 12 h shift the primary flux density calibrator 1934-638 was observed for 5 to 10 min. The phase calibrator VLA 1730-130 and VLA 1748-253 were observed about every 20 min for 2 min.
Data Reduction {#reduction}
==============
The data were reduced using the package [Miriad]{} following standard reduction steps as described in the reference guide by Bob Sault and Neil Killeen (Sault et al. 1995) . The data were loaded, bad points flagged, and then split into single source files. The bandpass function and the antenna gains were determined as a function of time to calibrate the data. From the visibilities images were made using the multi-frequency synthesis technique and robust weighting with a robustness parameter of 0.5, which gives nearly the same sensitivity as natural weighting but with a significantly better beam. First a low resolution image of the primary beam was made to identify confusing sources. Next a high resolution image was made including all sources. These maps were [CLEAN]{}ed. Because we had offsetted the PNe from the field center, we applied the primary beam correction which amounted to about 1% at 6 cm and 2% at 3 cm. Self-calibration was not applied because the flux values of the PNe are too low. Contour plots of the detected PNe are presented in the Appendix.
Analysis and Results {#results}
====================
Positions
---------
In Table \[tabpos\] we list the position of the peak flux density per beam of the PN at 6 cm together with the optical positions as determined from the H$\alpha$ or \[S III\]$\lambda$9532 images (Jacoby & Van de Steene, 2001, in preparation). The peak of the radio emission is adopted as the PN position. If the PN is extended in the radio, the radio position may be off center. The radio position will be better determined for higher peak flux values and smaller PNe. It is also for this reason that we chose to use the value at 6 cm and not at 3 cm, where the resolution is twice as high and the signal to noise per beam lower for extended sources.
We note that the radio positions have a tendency to be offset towards the west of the optical position. In declination there is no clear tendency noticeable in the offsets.
The optical and radio positions agree very well. PNe for which the radio position differs more than 2 in RA or DEC from the optical position are extended and the peak in the radio is usually not centered.
Of the 64 PNe observed, 7 were not detected: JaSt 7, 21, 45, 80, 88, 92, and 96. Most likely they are very extended and have a surface brightness that is too low to be detected in the radio. They are also very faint in the H$\alpha$ images and their H$\alpha$ flux values are very uncertain (Jacoby & Van de Steene 2001, in preparation). All but JaSt 96 were also faint and extended in the \[S III\]$\lambda$9532 images.
------ ------------- -------------- -------------- --------------- ------- --------- --- --------------
JaSt $\Delta$RA $\Delta$DEC run comment
RA(h m s) DEC ( ) RA(h m s) DEC ( )
1 17 34 43.64 -29 47 05.03 17 34 43.60 -29 47 06.03 -0.60 1.00 2
2 17 35 00.96 -29 22 15.72 17 35 00.96 -29 22 15.85 0.00 0.13 1
3 17 35 22.90 -29 22 17.58 17 35 22.79 -29 22 17.08 -1.65 -0.50 1
4 17 35 37.47 -29 13 17.67 17 35 37.39 -29 13 17.81 -1.20 0.14 1
5 17 35 52.51 -28 58 27.95 17 35 52.44 -28 58 27.45 -1.05 -0.50 1
7 17 38 26.69 -28 47 06.48 2 not detected
8 17 38 27.74 -28 52 01.31 17 38 27.58 -28 52 01.36 -2.40 0.05 2
9 17 38 45.64 -29 08 59.27 17 38 45.54 -29 08 55.65 -1.50 -3.62 2
11 17 39 00.55 -30 11 35.23 17 39 00.48 -30 11 32.23 -1.05 -3.00 2
16 17 39 22.70 -29 41 46.08 17 39 22.64 -29 41 45.35 -0.90 -0.70 1
17 17 39 31.32 -27 27 46.78 17 39 31.21 -27 27 47.28 -1.65 -0.50 1
19 17 39 39.38 -27 47 22.58 17 39 39.31 -27 47 22.21 -1.05 -0.37 1
21 17 39 52.92 -27 44 20.54 2 not detected
23 17 40 23.17 -27 49 12.04 17 40 23.08 -27 49 12.29 -1.35 0.25 1
24 17 40 28.23 -30 13 51.30 17 40 28.19 -30 13 51.00 -0.60 -0.30 1
26 17 40 33.52 -29 46 14.98 17 40 33.25 -29 46 12.48 -4.05 -2.50 2
27 17 40 42.34 -28 12 31.90 17 40 42.16 -28 12 30.81 -2.7 -1.1 1
31 17 41 27.93 -28 52 51.61 17 41 27.89 -28 52 50.61 -0.60 -1.00 1
34 17 41 54.80 -27 03 20.33 17 41 54.69 -27 03 18.33 -1.65 -2.00 2
36 17 42 25.20 -27 55 36.36 17 42 25.13 -27 55 36.36 -1.05 0.00 1
37 17 42 28.60 -30 09 34.93 17 42 28.52 -30 09 32.93 -2.00 -2.00 1
38 17 42 32.41 -27 33 15.18 17 42 32.29 -27 33 16.26 -1.80 1.08 1
41 17 42 49.96 -27 21 19.68 17 42 49.85 -27 21 19.31 -1.65 -0.37 1
42 17 43 17.06 -26 44 17.67 17 43 16.98 -26 44 18.25 -1.20 0.58 1
44 17 43 23.48 -27 34 06.03 17 43 23.52 -27 34 06.55 0.60 0.52 1
45 17 43 23.44 -27 11 16.91 2 not detected
46 17 43 30.43 -26 47 32.33 17 43 30.36 -26 47 31.83 -1.05 -0.50 1
49 17 44 04.34 -28 15 57.86 17 44 04.23 -28 15 56.99 -1.65 -0.87 1
52 17 44 37.30 -26 47 25.23 17 44 37.23 -26 47 25.23 -1.05 0.00 1
54 17 45 11.06 -27 32 36.80 17 45 11.02 -27 32 36.80 -0.60 0.00 2
55 17 45 37.36 -27 01 18.44 17 45 37.15 -27 01 18.16 -3.15 -0.28 1
56 17 45 47.05 -27 30 42.03 17 45 46.96 -27 30 42.01 -1.35 -0.02 2
58 17 46 52.20 -30 37 42.83 17 46 52.32 -30 37 42.33 1.80 -0.50 2
60 17 47 53.91 -29 36 49.67 17 47 53.91 -29 36 49.67 0.00 0.00 2
63 17 48 46.27 -27 25 37.22 17 48 46.27 -27 25 36.72 0.00 -0.50 1
64 17 48 56.04 -31 06 41.95 17 48 56.04 -31 06 42.45 0.00 0.50 1
65 17 49 20.02 -30 36 05.57 17 49 20.02 -30 36 05.08 0.00 0.23 1
66 17 49 22.15 -29 59 27.02 17 49 22.11 -29 59 27.02 -0.60 0.00 2
67 17 49 28.10 -29 20 47.56 17 49 28.02 -29 20 47.56 -1.20 0.00 2
68 17 49 50.87 -30 03 10.47 17 49 50.83 -30 03 10.97 -0.60 0.50 2
69 17 50 10.04 -29 19 05.14 17 50 10.11: -29 19 08.82: 1.05 3.68 1
70 17 50 21.07 -28 39 02.46 17 50 21.03 -28 39 01.46 -0.60 1.0 2
71 17 50 23.32 -28 33 10.95 17 50 23.21 -28 33 10.45 -1.65 -0.50 1
73 17 50 47.74 -29 53 16.01 17 50 47.82 -29 53 14.01 1.20 -1.99 2
74 17 50 46.85 -28 44 35.42 17 50 46.77 -28 44 34.92 -1.20 -0.50 1
75 17 50 48.08 -29 24 43.60 17 50 48.00 -29 24 43.18 -1.20 -0.42 1
76 17 50 56.47 -28 31 24.63 17 50 56.33 -28 31 24.13 -2.10 -0.50 2
77 17 51 11.65 -28 56 27.20 17 51 11.58 -28 56 27.20 -1.20 0.00 1
78 17 51 24.68 -28 35 40.34 17 51 24.79 -28 35 39.44 1.65 -0.90 1
79 17 51 53.63 -29 30 53.41 17 51 53.55 -29 30 53.41 -1.20 0.00 1
80 17 51 55.54 -27 48 02.46 1 not detected
81 17 52 04.35 -27 36 39.28 17 52 04.28 -27 36 38.28 -1.05 1.00 1
83 17 52 45.17 -29 51 05.21 17 52 45.17 -29 51 03.30 0.00 -1.91 1
85 17 52 49.05 -29 41 54.92 17 52 48.97 -29 41 55.92 -1.20 1.0 2
------ ------------- -------------- -------------- --------------- ------- --------- --- --------------
\[tabpos\]
------ ------------- -------------- -------------- --------------- ------- --------- --- --------------
JaSt $\Delta$RA $\Delta$DEC run comment
RA(h m s) DEC ( ) RA(h m s) DEC ( )
86 17 52 52.20 -29 30 00.07 17 52 52.16 -29 30 01.08 -0.60 1.08 1
88 17 53 00.89 -29 05 44.08 2 not detected
89 17 53 06.67 -28 18 07.91 17 53 06.79 -28 18 10.47 1.80 2.56 1
90 17 53 17.77 -28 04 33.20 17 53 17.73 -28 04 32.67 -0.60 -0.53 2
92 17 53 19.81 -28 27 14.67 2 not detected
93 17 53 24.14 -29 49 48.45 17 53 24.05: -29 49 51.15: -1.35 2.70 1
95 17 53 35.38 -28 28 51.02 17 53 35.51 -28 28 51.96 1.95 0.94 1
96 17 53 57.16 -29 20 14.97 2 not detected
97 17 54 13.36 -28 05 16.82 17 54 13.28 -28 05 16.82 -1.20 0.00 1
98 17 55 46.39 -27 53 38.91 17 55 46.28 -27 53 38.41 -1.65 -0.50 1
------ ------------- -------------- -------------- --------------- ------- --------- --- --------------
------ ------ -------- ------- ------- ------ -------- ------- ------- --------------------------------- ------------ --------- ------------ ---------- --
Name 6cm 3cm size beam FWHM method $\theta$
JaSt Peak Flux error noise Peak Flux error noise $\Delta\delta$ x $\Delta\alpha$ maj x min diam
mJy/ mJy mJy mJy/ mJy/ mJy mJy mJy/ x x
beam beam beam beam
1 0.85 1.6 0.2 0.10 0.47 1.4 0.6 0.12 4.5 x 2.6 4.3 x 1.9 3.4 G 5.5
2 1.8 4.3 0.5 0.22 0.82 4.2 1.4 0.12 5.0 x 4.0 4.7 x 1.8 4.5 C 6.3
3 4.0 12.4 1.2 0.17 1.6 10.0 1.0 0.13 5.8 x 4.4 4.8 x 2.0 5.1 C 9.2
4 1.2 4.1 0.4 0.20 0.65 3.5 1.2 0.15 7.7 x 7.1 4.7 x 1.8 7.4 C 11.2
5 4.0 11.3 1.1 0.15 1.8 10.5 1.0 0.15 6.2 x 5.1 4.9 x 2.0 5.6 C 8.4
8 1.2 3.8 0.4 0.08 0.5 2.6 0.6 0.04 3.6 x 5.8 4.4 x 1.9 4.6 C 7.2
9 0.96 4.0 0.4 0.08 0.38 2.5 0.8 0.08 5.6 x 6.2 4.7 x 1.8 5.9 C 8.1
11 0.76 3.3 0.3 0.07 0.41 $>$2.0 0.12 6.9 x 5.9 4.6 x 1.8 6.4 C 10.9
16 13.6 27.0 2.8 0.27 7.2 24.7 2.5 0.30 3.4 x 2.4 4.8 x 2.0 2.9 G 4.8
17 2.5 9.4 1.0 0.15 1.2 8.2 0.9 0.13 5.4 x 5.3 5.7 x 1.8 5.3 C 7.9
19 2.1 6.8 0.8 0.29 1.0 5.5 0.6 0.14 3.6 x 3.7 5.3 x 1.6 3.6 C 6.8
23 3.3 3.7 0.4 0.25 2.6 3.9 0.4 0.21 PS 3.1 x 1.1 $<$1.8 G$_{3cm}$
24 6.3 16.0 1.7 0.27 2.7 14.1 1.4 0.23 4.8 x 2.7 4.9 x 1.7 3.6 G 5.8
26 2.4 14.1 1.4 0.09 1.0 9.0 0.9 0.08 9.4 x 9.3 5.1 x 1.7 8.5 C 11.1
27 0.8 2.5 0.3 0.14 0.5 0.15 9.9 x 5.3 5.3 x 1.7 8.2 C 8.7
31 3.3 11.5 1.2 0.22 2.4 $>$3.0 0.61 5.0 x 5.1 5.1 x 1.7 5.0 C 9.0
34 0.58 1.7 0.2 0.07 0.34 $>$1.5 0.09 3.2 x 6.0 5.1 x 1.7 4.4 C 7.1
36 18.6 31.1 3.1 0.33 11.7 31.1 3.1 0.35 2.6 x 1.9 3.1 x 1.1 2.2 G$_{3cm}$ 3.6
37 5.2 16.0 1.7 0.70 2.5 $>$5.5 0.35 5.7 x 3.8 4.6 x 1.5 4.6 C 7.4
38 1.2 3.0 0.6 0.28 0.9 2.0 0.7 0.15 6.8 x 2.2 5.3 x 1.7 3.9 C 7.0
41 8.0 16.7 1.7 0.20 4.6 17.6 1.8 0.18 3.4 x 3.2 5.6 x 1.8 3.3 C 6.3
42 5.6 12.9 1.3 0.21 3.3 15.5 1.6 0.26 4.0 x 2.5 5.5 x 1.6 3.1 G 5.0
44 1.3 3.8 1.0 0.27 0.9 $>$1.8 0.27 12.2: x 6.0: 5.4 x 1.6 8.6: C 8.6:
46 14.6 20.8 2.1 0.40 12.3 20.5 2.1 0.30 2.0 x 1.1 3.3 x 1.1 1.5 G 2.5
49 3.1 8.2 0.9 0.27 2.1 10.8 1.1 0.21 4.6 x 3.6 5.2 x 1.6 4.0 C 6.6
52 20.8 24.0 2.4 0.34 16.9 22.8 2.3 0.40 2.5 x 0.4 3.3 x 1.1 1.0 G$_{3cm}$ 1.7
54 21.9 82.3 8.2 0.08 7.5 75.2 7.5 0.09 4.7 x 4.3 5.1 x 1.8 4.5 G 7.2
55 2.0 9.2 0.9 0.19 0.9 10.0 2.0 0.17 5.5 x 6.4 5.4 x 1.6 6.0 C 10.7
56 5.8 14.1 1.4 0.09 2.4 12.2 1.2 0.07 3.8 x 2.7 2.8 x .95 3.2 G$_{3cm}$ 5.0
58 3.5 29.5 2.9 0.06 1.3 21.3 2.1 0.07 8.9 x 8.0 4.6 x 1.7 8.4 C 15.5
60 6.1 6.5 0.7 0.06 7.0 7.9 0.8 0.07 .85 x .28 2.6 x 1.0 0.5 G$_{3cm}$ 0.9
63 1.1 3.9 0.4 0.11 0.7 3.6 1.2 0.13 6.1 x 6.0 5.5 x 1.8 6.0 C 8.4
64 23.8 28.3 2.9 0.42 15.8 25.6 2.6 0.58 2.0 x 0.7 5.0 x 1.9 1.4 G 2.4
65 3.3 3.4 0.4 0.23 6.3 6.6 0.7 0.30 PS 2.7 x 1.0 $<$ 1.6 G$_{3cm}$
66 45.9 65.3 6.5 0.15 26.0 60.9 6.1 0.10 2.0 x 1.5 2.6 x 1.00 1.7 G$_{3cm}$ 2.8
67 24.6 27.8 2.8 0.08 17.7 26.0 2.6 0.10 1.3 x .9 2.5 x 1.0 1.1 G $_{3cm}$ 1.8
68 5.4 5.9 0.6 0.06 7.7 8.8 0.9 0.10 .8 x .3 2.5 x 1.0 1.5 G$_{3cm}$ 0.9
69 0.9 $>$2.0 0.26 0.5 $>$2.0 0.15 4.9 x 1.7
70 4.0 13.0 1.3 0.11 1.9 11.5 1.2 0.06 4.3 x 5.3 4.6 x 1.7 4.8 C 8.5
71 20.2 34.0 3.5 0.50 10.0 36.0 3.6 0.55 4.6 x 1.6 5.4 x 1.9 2.7 G 4.5
73 11.7 13.7 1.4 0.09 9.4 13.0 1.3 0.11 1.3 x .7 2.6 x 1.0 0.95 G$_{3cm}$ 1.6
74 12.7 22.7 2.3 0.28 6.1 22.2 2.2 0.25 4.5 x 2.2 3.3 x 1.0 3.1 G$_{3cm}$ 4.9
75 16.8 22.5 2.3 0.53 11.0 19.0 2.0 0.55 3.0 x 0.9 5.0 x 1.6 1.6 G 2.7
76 5.4 7.4 0.7 0.07 2.5 6.7 0.7 0.08 3.9 x 1.8 2.4 x 1.0 2.6 G$_{3cm}$ 4.7
77 39.4 49.0 5.0 0.85 34.0 51.2 5.1 0.98 PS 3.1 x 1.1 $<$1.5 G$_{3cm}$
78 1.5 12.3 1.2 0.18 0.9 9.0 2.3 0.15 12.0 x 9.0 5.1 x 1.6 10.4 C 14.6
79 4.1 4.1 0.4 0.14 6.7 7.0 0.7 0.23 PS 2.7 x 1.2 $<$1.5 G$_{3cm}$
------ ------ -------- ------- ------- ------ -------- ------- ------- --------------------------------- ------------ --------- ------------ ---------- --
\[tabflux\]
------ ------ ------ ------- ------- ------ -------- ------- ------- --------------------------------- ----------- --------- ----------- ---------- --
Name 6cm 3cm size beam FWHM method $\theta$
JaSt Peak Flux error noise Peak Flux error noise $\Delta\delta$ x $\Delta\alpha$ maj x min diam
mJy/ mJy mJy mJy/ mJy/ mJy mJy mJy/ x x
beam beam beam beam
81 17.8 20.8 2.0 0.25 14.1 20.8 2.1 0.29 1.8 x 0.7 5.1 x 2.0 1.3 G 2.2
83 1.7 5.0: 2.5 0.40 0.6 $>$0.6 0.16 5.3: x 5.7: 4.5 x 1.8 $>$5.5: C $>$6.3
85 0.6 4.4 1.2 0.09 13.1: x 13.1: 4.2 x 1.9 13.1: C 17.5:
86 1.7 8.5 0.9 0.17 0.9 $>$2.5 0.17 7.5 x 7.7 4.7 x 1.8 7.6 C 11.4
89 1.8 8.5 0.9 0.27 0.8 $>$2.0 0.19 9.5 x 5.5 4.8 x 1.7 7.2 C 8.6
90 0.51 1.0 0.2 0.07 0.13 4.5 x 2.5 4.4 x 1.9 3.4 G 5.5
93 0.8 6.3 0.9 0.15 0.6 $>$2.1 0.13 20.2 x 8.0 4.8 x 2.0 12.7: C 17.8:
95 1.1 5.5 1.0 0.25 0.7 4.5: 1.5 0.18 14.6 x 7.5 4.8 x 1.8 10.5 C 12.6
97 2.7 11.5 1.2 0.16 1.2 11.8 1.2 0.18 8.0 x 5.9 5.2 x 1.9 6.8 C 11.4
98 21.0 21.8 2.2 0.38 20.3 24.0 2.4 0.46 PS 2.9 x 1.1 $<$1.5 G$_{3cm}$
------ ------ ------ ------- ------- ------ -------- ------- ------- --------------------------------- ----------- --------- ----------- ---------- --
Radio flux densities
--------------------
Table \[tabflux\] gives the radio flux densities at 6 and 3 cm.
If a Gaussian model provided a satisfactory fit to the surface brightness the total Gaussian flux density was adopted. This was mostly the case for small, unresolved PNe. If the PN was extended the intensities within the 2 or 3 $\sigma$ level contour were summed. This value was compared with the statistics over a larger region across the nebula to obtain an error estimate (Fomalont, 1989).
For JaSt 69 only a small 3 $\sigma$ blob at the right position indicated the presence of a PN. No flux or size values could be determined. In some cases several blobs indicated the presence of a PN at 3 cm and hence their flux is a lower limit. Some objects detected at 6 cm, but having low peak flux density per beam, could not be seen at 3 cm.
PNe are normally optically thin at 6 cm in which case its 3 cm flux density is about 95% of the 6 cm flux density. In our case this means that the flux densities are similar within the error-bars. However when a PN is already well resolved at 6 cm, more flux may have been missed at 3 cm where the beam is half the size, especially if the nebula is extended and of low surface brightness. It is clear that due to these factors the flux density at 3 cm is generally less well-determined than at 6 cm, except for the bright and compact PNe. When a PN is optically thick at 6 cm, its 3 cm flux density is expected to be three times the flux density at 6 cm. There are some PNe like JaSt 65 and JaSt 79 for which the 3 cm flux density is clearly higher than the 6 cm flux density and which are point sources. In this case the PNe may not yet be completely optically thin at 6 cm and should be quite young.
In Fig. \[fluxhist\] we plotted the histogram of the radio flux of the known and new galactic bulge PNe. We selected a sample of known galactic bulge PNe for which radio flux densities and angular diameters are available, as in Van de Steene & Zijlstra ([@VdSteene95]). The new bulge PNe are within 2 degrees of the galactic center. None has a radio flux larger than 100 mJy and their angular sizes are smaller than 20. Hence they fulfill the same selection criteria as these previously known bulge PNe.
There is a larger number of PNe with low flux densities among the new bulge PNe than among the known ones. 67 % of the new PNe have a radio flux less than 15 mJy, while this is only 45 % for the known ones. Of the 7 known ones within 2 degrees of the galactic center only 2 have a radio flux below 15 mJy. The median flux for the new PNe is 11.3 mJy, while the median for the known bulge PNe is 17.0 mJy. Our rms noise level in the maps is similar to the 1 $\sigma$ noise of 0.1 mJy in the 6 cm maps of Zijlstra et al. ([@Zijlstra89]). Apparently these faint and small PNe have just been missed in optical surveys done to date.
\[fluxhist\]
Angular Size
------------
Table \[tabflux\] also gives the angular size of the detected PNe. We chose to determine the angular size at 6 cm because at this wavelength the resolution was lower and thus gives the best signal to noise ratio for the extended nebulae. The diameter at 3 cm is given if it is better determined than at 6 cm, such as for very small and bright PNe.
Diameters may differ considerably depending upon how they are calculated. The diameter was derived by one of several ways depending on the structure of the brightness distribution. If a two-dimensional Gaussian fit provided a satisfactory model to the observed structureless surface distribution, its deconvolved FWHM major and minor axis are given. The equivalent diameter is the square root of their product. To obtain the full diameter, this value must be multiplied with a conversion factor which is a function of the beam FWHM and depends upon the intrinsic surface distribution of the source. We assumed a spherical constant emissivity shell of 0.5 and used formula 5 and Table 1 from van Hoof ([@vHoof00]) to estimate the true radii. For small objects, if the Gaussian deconvolution was well determined and similar at 3 and 6 cm, the FWHM at 3 cm is given. If the deconvolution produced a point source at 6 cm, the source size at 3 cm is given. If the source was still a point source at 3 cm, the beam-size is an upper limit. If the source was extended, a Gaussian model was usually not a good representation of the radio source. The diameter of the PN was measured on the contour at 50 % of the peak and deconvolved with the beam size. To determine the full diameter we determined the ratio of the flux density within the 3 $\sigma$ contour with the flux density within the 50 % contour. Hence, we assumed that the flux decreased linearly with radius outside the 50 % contour. We checked that this procedure seemed to give very good agreement with the size measured based on the 3 $\sigma$ contour. We didn’t use the contour level at 10% of the peak (Zijlstra et al. [@Zijlstra89], Aaquist & Kwok [@Aaquist90]), because this was often below noise level.
It was noticed in the review paper by Pottasch ([@Pottasch92]) that there is a selection against discovering both large and small PNe in the galactic bulge. 36 % of the new PNe have a diameter smaller than 5 , while this is 71 % for the known ones. The median is 66 for the new PNe and only 32 for the known bulge PNe with radio data. We seem to identify relatively more larger PNe than in previous surveys. In regions with large extinction the \[SIII\]$\lambda$9532 line appears efficient in picking out the larger, low surface brightness PNe, and not only the small and dusty ones. Obviously these are the PNe which may have been missed in optical surveys.
\[diamhist\]
Distances
=========
\[disthist\]
\[distsch\]
Based on the flux densities and the angular equivalent diameters we calculated the statistical distances to the PNe as established in Van de Steene & Zijlstra ([@VdSteene95]). The relation is not strictly valid for optically thick and very small nebulae which may have their distance overestimated. The distances will tend to be overestimated for PNe with lower surface brightness than average, while PNe with a higher surface brightness than average will tend to have their distances underestimated. However, statistically the distance distribution will be representative. The distances are presented in Table \[quantities\]. A histogram of the distance distribution is presented in Fig. \[disthist\]. The median distance is 7.2 kpc, about 8% closer than the galactic center; presumably, the more distant PNe suffer greater extinction and fell below our detection limit in our survey.
In Fig. \[distsch\] we plotted the scale height versus the distance. It seems that we observe PNe at the edge of the galactic bulge. As we look further inward, the extinction probably becomes too large. The PNe further away are at larger scale height where, apparently, the extinction is less. The known PNe are generally further away from the galactic center, surrounding our new PNe. The median absolute scale height is 136 pc.
------ ------- -------- ------- ------------ -------- ---------
JaSt Dist z R M$_{ion}$ E(B-V) Comment
kpc pc pc M$_{\sol}$
1 14.6: 401.6: 0.20: 0.25: 2.5:
2 10.0 305.7 0.15 0.19 2.0
3 6.2 183.0 0.14 0.17 2.3
4 8.5 258.9 0.23 0.29 1.9
5 6.6 210.4 0.13 0.17 2.3
8 10.0 241.9 0.17 0.22 2.1
9 9.5 194.9 0.19 0.23 2.9
11 9.2 91.6 0.24 0.31 3.7
16 5.9 77.6 0.07 0.08 3.5
17 7.1 242.5 0.14 0.17 1.8
19 8.4 264.7 0.14 0.17 1.8
23 2.4 PS
24 6.6 31.4 0.09 0.11 X
26 5.6 49.7 0.15 0.19 X
27 10.8 246.2 0.23 0.29 2.3
31 6.4 91.2 0.14 0.17 3.7
34 13.2 392.9 0.23 0.29 2.1
36 6.1 120.2 0.05 0.06 2.7
37 6.1 -6.3 0.11 0.13 X
38 10.9: 251.7: 0.19: 0.23: 2.0:
41 6.3 150.4 0.10 0.12 2.1
42 7.4 207.0 0.09 0.11 2.1
44 9.6: 190.2: 0.20: 0.26: 1.9:
46 7.9 207.1 0.05 0.06 1.9
49 7.9 89.4 0.13 0.16 X
52 8.5 190.6 0.03 0.04 2.1
54 3.5 50.2 0.06 0.07 X
55 6.5 115.9 0.17 0.21 2.5
56 7.2 89.9 0.09 0.11 X
58 3.9 -77.8 0.15 0.18 4.1
60 16.2 -213.6 0.04 0.04 X
63 9.4 31.5 0.19 0.25 2.6
64 7.2 -216.3 0.04 0.05 2.6
65 1.3 PS
66 5.5 -111.1 0.03 0.04 2.8
67 7.9 -127.0 0.03 0.04 4.6: Z
68 16.7 -388.0 0.04 0.04 2.8
70 6.2 -82.0 0.13 0.16 X
71 5.5 -67.4 0.06 0.07 3.6
73 10.5 -261.7 0.04 0.05 1.9
74 6.2 -93.9 0.07 0.09 3.6
75 7.5 -157.5 0.05 0.06 2.7
76 16.8 -223.1 0.03 0.03 5.0: Z
77 2.4 PS
78 5.4 -87.5 0.19 0.24 3.4: Z
79 1.5 PS
81 8.2 -75.8 0.04 0.05 X
85 7.2: -224.1 0.30 0.39: 1.2
------ ------- -------- ------- ------------ -------- ---------
: In this table we present the statistical distance in column 2, the scale height, radius, and ionized mass calculated using this distance in columns 3, 4, and 5 respectively, and E(B-V) $=$ c$_{H_{\beta}}$ / 1.46 in column 6. If the object was not detected in H$\alpha,$ E(B-V) is marked with X. If no H$\alpha$ image was obtained E(B-V) is marked with Z. In this case the E(B-V) value was derived from its spectrum. Uncertain values, due to uncertain radio flux or H$\alpha$ flux values, are marked with a colon. PS stands for Point Source.
\[quantities\]
------ ------- --------- ------- ------------ -------- ---------
JaSt Dist z R M$_{ion}$ E(B-V) Comment
kpc pc pc M$_{\sol}$
86 6.6 -193.0 0.18 0.23 1.9
89 7.2 -138.3 0.15 0.19 2.1
90 17.2: -303.3: 0.23: 0.29: 2.1
93 6.5: -217.2: 0.27: 0.35: 1.9: Z
95 7.4: -157.7: 0.23: 0.28: 1.3
97 5.9 -125.5 0.16 0.21 2.6
98 2.7 Z PS
------ ------- --------- ------- ------------ -------- ---------
: Table \[quantities\] continued
Extinction
==========
\[ext\_lb\]
\[ext\_schdist\]
We calculated the extinction E(B-V) comparing the radio flux densities with the total H$\alpha$ flux values, which were obtained from images obtained with the NOAO 8k x 8k Mosaic Imager on the 0.9-m telescope at KPNO (Jacoby & Van de Steene 2001, in preparation). The flux values were corrected for the contribution of \[N II\]$\lambda$6548 & $\lambda$6584 based on spectra obtained at ESO and CTIO (Van de Steene & Jacoby, 2001, in preparation). The H$\beta$ flux and radio flux density have the same dependency on electron density and the expected ratio is only a weak function of the electron temperature and helium abundance (Pottasch, [@Pottasch84]). The radio flux density and H$\beta$ flux can be used to determine the extinction c$_{H\beta}$ $=$ 1.46 E(B-V). Assuming the standard ratio of H$\alpha$ / H$\beta$ $=$ 2.85, we used the H$\alpha$ flux to predict the extinction: c$_{H\alpha}$ = E(B-V). The values of E(B-V) are presented in Table \[quantities\]. All but one PN have values between 1 and 4. For the PNe detected in the radio but not in H$\alpha$ the extinction is likely to be higher.
As mentioned before: some PNe were detected in H$\alpha$ but not in the radio, probably due to too low surface brightness. Eight PNe were detected in the radio, but not in the H$\alpha$ images, probably due to too high extinction. These PNe also show no H$\alpha$ emission in their spectra. They are visible at longer wavelengths such as \[S III\]$\lambda$9532.
From Fig. \[ext\_lb\] and \[ext\_schdist\] it appears indeed that the extinction increases towards the galactic center. No extinction value could be determined for objects closest to the galactic center. The distances of JaSt 54 and JaSt 58 appear to be underestimated, unless their internal extinction is very large.
Conclusions
===========
We obtained the radio flux densities and diameters for 64 new galactic bulge PNe with the Australia Telescope Compact Array.
1. We have a larger ratio of larger PNe with angular diameters around 10 and radio flux densities below 15 mJy, than in previous surveys. Our survey seems to have picked out the larger low surface brightness PNe which were missed in optical surveys to date.
2. We calculated their distances according to the method presented in Van de Steene & Zijlstra ([@VdSteene95]). The new galactic bulge PNe are mainly located around the galactic center, and closer to it than the previously known PNe.
3. We calculated the E(B-V) extinction value based on the radio flux densities and the total H$\alpha$ flux values from imaging. Only 1 PN was detected with a E(B-V) value larger than 4. Generally speaking, when\
E(B-V) $>$ 4.0 the galactic bulge PNe become undetectable in H$\alpha$.
Aaquist, A., Kwok, S., 1990, A&AS 84, 229
Acker, A., Marcout, J., Ochsenbein, F., Stenholm, Tylenda, R., 1992, Strasbourg-ESO catalogue of galactic planetary nebulae, ESO
Dopita, M.A., Vassiliadis, E., Wood, P.R., Meatheringham, S.J., Harrington, J.P., Bohlin, R.C., Ford, H.C., Stecher, T.P., Maran, S.P., 1997, ApJ 474, 188
Fomalont, E., Synthesis imaging in radio astronomy, 1989, PASP 6, ed. A. Perley, F.R. Schwab, and A. Bridle, p. 222
Kohoutek, L., 1994, Astron. Nachr. 315, 235
Pottasch, S.R., 1984, “Planetary Nebulae”, D. Reidel (Dordrecht) , p.93, p.97
Pottasch, S.R., Acker a., 1989, A&A 221,123
Pottasch,S.R., 1992, A&ARv 4, 256
Sault R.J., Teuben P.J., Wright M.C.H., 1995, “A retrospective of Miriad”, Astronomical Data Analysis Softwared and Systems IV, ed. R. Shaw, H.E. Payne, J.J.E. Hayes, ASP Conf. Ser., 77, 433
Van de Steene, G.C., Zijlstra, A.A., 1995, A&A, 293, 541
van Hoof, P.A.M., Van de Steene, G.C., 1999, MNRAS, 308, 623
van Hoof, P.A.M., 2000, MNRAS, 314, 99
Walsh, J.R., Jacoby, G.H., Peletier, R.F., Walton, N.A., 2000, SPIE, Discoveries and Research Prospects from 8 to 10-Meter Class Telescopes, Vol 4005, p. 131
Zijlstra, A.A., Pottasch, S.R., Bignell, C., 1989, A&AS, 79, 329
Contour Plots: available upon request from [email protected]
=========================================================
[^1]: Based on data acquired at the Australia Telescope Compact Array. The Australia Telescope is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.
| 0 |
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abstract: 'Layers of transition metal dichalcogenides (TMDs) combine the enhanced effects of correlations associated with the two-dimensional limit with electrostatic control over their phase transitions by means of an electric field. Several semiconducting TMDs, such as MoS$_2$, develop superconductivity (SC) at their surface when doped with an electrostatic field, but the mechanism is still debated. It is often assumed that Cooper pairs reside only in the two electron pockets at the K/K’ points of the Brillouin Zone. However, experimental and theoretical results suggest that a multi-valley Fermi surface (FS) is associated with the SC state, involving 6 electron pockets at the Q/Q’ points. Here, we perform low-temperature transport measurements in ion-gated MoS$_2$ flakes. We show that a fully multi-valley FS is associated with the SC onset. The Q/Q’ valleys fill for doping$\gtrsim2\cdot10^{13}$cm$^{-2}$, and the SC transition does not appear until the Fermi level crosses both spin-orbit split sub-bands Q$_1$ and Q$_2$. The SC state is associated with the FS connectivity and promoted by a Lifshitz transition due to the simultaneous population of multiple electron pockets. This FS topology will serve as a guideline in the quest for new superconductors.'
author:
- Erik Piatti
- Domenico De Fazio
- Dario Daghero
- Srinivasa Reddy Tamalampudi
- Duhee Yoon
- 'Andrea C. Ferrari'
- 'Renato S. Gonnelli'
title: 'Multi-Valley Superconductivity In Ion-Gated MoS$_2$ Layers'
---
[^1]
[^2]
Transition metal dichalcogenides (TMDs) are layered materials with a range of electronic properties. Depending on chemical composition, crystalline structure, number of layers (N), doping, and strain, different TMDs can be semiconducting, metallic and superconducting[@FerrN2015]. Amongst semiconducting TMDs, MoS$_2$, MoSe$_2$, WS$_2$ and WSe$_2$ have sizeable bandgaps in the range$\sim$1-2eV[@WangNN2012]. When exfoliated from bulk to single layer (1L), they undergo an indirect-to-direct gap transition[@MakPRL2010; @SpleNL2010; @WangNN2012], offering a platform for electronic and optoelectronic applications[@FerrN2015; @WangNN2012; @MakNP2016], such as transistors[@PodzAPL2004; @RadiNN2011; @FangNL2012], photodetectors[@GourSEMSC1997; @LeeNL2012; @YinACS2012; @KoppNN2014], modulators[@SunNP2016] and electroluminescent devices[@CarlPRB2002; @SundNL2013].
For all TMDs with 2H crystal structure, the hexagonal Brillouin Zone (BZ) features high-symmetry points $\Gamma$, M, K and K’[@SpleNL2010; @BrummePRB2015], Fig.\[figure:bands\]a. The minima of the conduction band fall at K, K’, as well as at Q, Q’, approximately half-way along the $\Gamma$-K(K’) directions[@SpleNL2010; @BrummePRB2015], Fig.\[figure:bands\]a. In absence of an out-of-plane electric field, the relative position of Q and Q’ depends on N and strain[@SpleNL2010; @LambrechtPRB2012; @BrummePRB2015]. The global minimum of the conduction band sits at K/K’ in 1L-MoS$_2$ and at Q/Q’ in few layer (FL)-MoS$_2$ with N$\geq$4[@SpleNL2010]. When an electric field is applied perpendicular to the MoS$_2$ plane, inversion symmetry is broken and the global minimum of the conduction band is shifted to K/K’ in any FL-MoS$_2$[@BrummePRB2015], Figs.\[figure:bands\]b-d. The valleys at K/K’ and at Q/Q’ are characterized by a different electron-phonon coupling (EPC)[@GePRB2013] and, when inversion symmetry is broken, by a different spin-orbit coupling (SOC)[@KadantsevSSC2012]. In particular, both EPC and SOC are larger in the Q/Q’ valleys[@GePRB2013; @KadantsevSSC2012].
{width="80.00000%"}
The field-effect transistor (FET) architecture is ideally suited to control the electronic properties of 1L flakes, as it simultaneously provides an electrostatic control of the transverse electric field and the carrier density. In the electric-double-layer (EDL) technique[@FujimotoReview2013], the standard solid gate dielectric is replaced by an ionic medium, such as an ionic liquid or electrolyte. In this configuration, the EDL that forms at the ionic liquid/electrode interfaces supports electric fields in excess of$\sim$10MV/cm[@UenoReview2014], corresponding to surface carrier densities $n_{2d}\gtrsim10^{14}$cm$^{-2}$[@UenoReview2014]. Ionic-liquid gating has been used to tune the Fermi level, $E_F$, in TMDs and explore transport at different carrier concentrations[@SaitoReview2016; @YeScience2012; @BragNL2012; @YuNN2015; @XiPRL2016]. The vibrational properties of TMDs can also be controlled by means of the EDL technique, as suggested by gate-induced softening of Raman-active modes in 1L-MoS$_2$[@ChakrabortyPRB2012], while the opposite is observed in gated 1L[@DasNatureNanotech2008] and two-layer (2L)[@DasPRB2009] graphene. Ref. reported a gate-induced superconducting state at the surface of liquid-gated MoS$_2$ flakes with N$\gtrsim$25[@YeScience2012], while Ref. detected this down to N=1.
Most of these results have been interpreted in terms of the population of the conduction band minima at K/K’[@YeScience2012; @LuScience2015; @SaitoNatPhys2016; @ChenPRL2017], which are global minima in both 1L-MoS$_2$[@MakPRL2010; @GePRB2013] and electrostatically-doped FL-MoS$_2$[@YeScience2012; @LuScience2015; @SaitoNatPhys2016; @BrummePRB2015], Fig.\[figure:bands\]b. Theoretical investigations however suggested that the population of the high-energy minima at Q/Q’ may have an important role in determining the properties of gated MoS$_2$ flakes, by providing contributions both to EPC[@GePRB2013; @BrummePRB2015] and SOC[@KadantsevSSC2012; @WuNatCommun2016]. Ref. predicted that when the Q/Q’ valleys of 1L-MoS$_2$ are populated (Fig.\[figure:bands\]c,d), EPC strongly increases (from$\sim0.1$ to$\sim18$), leading to a superconducting transition temperature $T_c\sim20$K for a doping level $x=0.18$ electrons(e$^-$)/unit cell (corresponding to $E_F=0.18\pm0.02$eV at K/K’ and $0.08\pm0.02$eV at Q/Q’)[@BrummePRB2015]). However, Ref. measured $T_c\sim2$K for $x\sim0.09\div0.17$ e$^-$/unit cell in e$^-$-doped 1L-MoS$_2$. This mismatch may be associated with the contribution of e$^-$–e$^-$ interactions, whose role in the determination of $T_c$ is still under debate[@RoldanPRB2013; @DasPRB2015]. Overall, the agreement between the model of Ref. and the trend of $T_c$ with e$^-$ doping in Ref. suggests that the mechanism of Ref. for EPC enhancement when the Q/Q’ valleys are crossed may also hold for FL-MoS$_2$.
Inversion symmetry can be broken in MoS$_2$ either by going to the 1L limit[@GePRB2013], or by applying a transverse electric field[@KormanyosPRB2013; @YuanPRL2014]. This leads to a finite SOC[@KormanyosPRB2013; @YuanPRL2014], which lifts the spin degeneracy in the conduction band and gives rise to two spin-orbit-split sub-bands in each valley[@KormanyosPRB2013; @BrummePRB2015], as shown in Fig.\[figure:bands\]b-d for FL-MoS$_2$. When the system is field-effect doped, the inversion symmetry breaking increases with increasing transverse electric field [@LuScience2015; @ChenPRL2017], due to the fact that induced e$^-$ tend to become more localized within the first layer [@LuScience2015; @ChenPRL2017; @RoldanPRB2013]. Hence, the SOC and the spin-orbit splitting between the bands increase as well, as was calculated in Ref..
When combined to the gate-induced SC state[@YuanPRL2014], this can give rise to interesting physics, such as spin-valley locking of the Cooper pairs[@SaitoNatPhys2016] and 2d Ising superconductivity (SC)[@LuScience2015] with a non-BCS-like energy gap[@CostanzoNatNano2018], suggested to host topologically non-trivial SC states[@RoldanPRB2013; @HsuNatComms2017; @NakamuraPRB2017]. Refs. predicted SOC and spin-orbit splitting between sub-bands to be significantly stronger for the Q/Q’ valleys than for K/K’, thus supporting spin-valley locking at Q/Q’ as well[@WuNatCommun2016]. A dominant contribution of the Q/Q’ valleys in the development of the SC state would be consistent with the high ($\gtrsim50$T) in-plane upper critical field, $H_{c2}^{||}$, observed in ion-gated MoS$_2$ [@LuScience2015; @SaitoNatPhys2016] and WS$_2$ [@LuPNAS2018]. The $H_{c2}^{||}$ enhancement is caused by locking of the spin of the Cooper pairs in the out-of-plane direction in a 2d superconductor in the presence of finite SOC, and is therefore promoted by increasing the SOC. However, $H_{c2}^{||}$ for MoS$_2$ and WS$_2$ is higher than in metallic TMD Ising superconductors (such as NbSe$_2$ and TaS$_2$), where $H_{c2}^{||}\lesssim30$T [@delaBarreraNatCommun2018], despite the SOC in the K/K’ valleys being much smaller ($\sim3$meV for MoS$_2$[@KormanyosPRB2013]). Spin-valley locking in the Q/Q’ valleys may thus explain this apparent inconsistency in the physics of ion-gated semiconducting TMDs under magnetic field.
From the experimental point of view, the possible multi-valley character of transport in gated TMDs is currently debated. Refs. measured the Landau-level degeneracy at moderate $n_{2d}\sim10^{12}-10^{13}$cm$^{-2}$, finding it compatible with a carrier population in the Q/Q’ valleys. However, Ref. argued that this would be suppressed for larger $n_{2d}\gtrsim10^{13}$cm$^{-2}$, typical of ion-gated devices and mandatory for the emergence of SC) due to stronger confinement within the first layer[@ChenPRL2017]. In contrast, angle-resolved photoemission spectroscopy in surface-Rb-doped TMDs[@KangNanoLett2017] highlighted the presence of a non-negligible spectral weight at the Q/Q’ valleys only for $n_{2d}\gtrsim8\cdot10^{13}$cm$^{-2}$ in the case of MoS$_2$. Thus, which valleys and sub-bands are involved in the gate-induced SC state still demands a satisfactory answer.
Here we report multi-valley transport and SC at the surface of liquid-gated FL-MoS$_2$. We use a dual-gate geometry to tune doping across a wide range of $n_{2d}\sim5\cdot10^{12}-1\cdot10^{14}$cm$^{-2}$, induce SC, and detect characteristic “kinks” in the transconductance. These are non-monotonic features that emerge in the $n_{2d}$-dependence of the low-temperature ($T$) conductivity when $E_F$ crosses the high-energy sub-bands[@BrummePRB2016], irrespectively of their specific effective masses, Fig.\[figure:bands\]e. We show that the population of the Q/Q’ valleys is fundamental for the emergence of SC. The crossing of the first sub-band Q$_1$ (Fig.\[figure:bands\]c) occurs at small $n_{2d}\lesssim2\cdot10^{13}$cm$^{-2}$, implying that multi-valley transport already occurs in the metallic phase over a wide range of $n_{2d}\sim2-6\cdot10^{13}$cm$^{-2}$. We also show that the crossing of the second sub-band Q$_2$ occurs after a finite $T_c$ is observed, while a full population of both spin-orbit-split sub-bands (Fig.\[figure:bands\]d) in the Q/Q’ valleys is required to reach the maximum $T_c$. These results highlight how SC can be enhanced in MoS$_2$ by optimizing the connectivity of its Fermi Surface (FS), i.e. by adding extra FSs in different BZ regions to provide coupling to further phonon branches[@PickettBook]. Since the evolution of the band structure of MoS$_2$ with field-effect doping is analogous to that of other semiconducting TMDs[@BrummePRB2015; @BrummePRB2016; @DasPRB2015; @WuNatCommun2016; @KangNanoLett2017], a similar mechanism is likely associated with the emergence of SC in TMDs in general. Thus, optimization of the FS connectivity can be a viable strategy in the search of new superconductors.
![a) Hall bar FL-MoS$_2$ flake with voltage probes ($V_i$), source (S), drain (D) and liquid-gate (LG) electrodes. A ionic liquid droplet covers the flake and part of the LG electrode. The sample is biased with a source-drain voltage ($V_{SD}$) and dual gate control is enabled by a voltage applied on the liquid gate ($V_{LG}$) and on the solid back gate ($V_{BG}$). b) Optical image of Hall bar with six voltage probes. The LG electrode is on the upper-right corner. c) AFM scan of the MoS$_2$ Hall bar after ionic liquid removal.[]{data-label="figure:device"}](Figure_device_6.eps){width="0.8\columnwidth"}
We study flakes with N=4-10, as Refs. predicted that flakes with N$\geq$4 are representative of the bulk electronic structure, and Ref. experimentally observed that both $T_c$ and the critical magnetic field $H_{c2}$ in 4L flakes are similar to those of 6L and bulk flakes. Our devices are thus comparable with those in literature[@YeScience2012; @LuScience2015; @SaitoNatPhys2016; @CostanzoNatNano2016; @FuQuantMater2017]. We do not consider 1L flakes as they exhibit a lower $T_c$ and their mobility is suppressed due to disorder[@CostanzoNatNano2016; @FuQuantMater2017].
FL-MoS$_2$ flakes are prepared by micro-mechanical cleavage[@NovoPNAS2005] of 2H-MoS$_2$ crystals from SPI Supplies. The 2H phase is selected to match that in previous reports of gate-induced SC[@YeScience2012; @CostanzoNatNano2016]. Low resistivity ($<0.005\Omega\cdot$cm) Si coated with a thermal oxide SiO$_2$ is chosen as a substrate. We tested both 90 or 285nm SiO$_2$ obtaining identical SC results. Thus, 90nm SiO$_2$ is used to minimize the back gate voltage $V_{BG}$ , while 285nm is used to minimize leakage currents through the back gate $I_{BG}$. Both SiO$_2$ thicknesses provide optical contrast at visible wavelengths[@CasiraghiNanoLett2007]. A combination of optical contrast, Raman spectroscopy and atomic force microscopy (AFM) is used to select the flakes and determine N.
Electrodes are then defined by patterning the contacts area by e-beam lithography, followed by Ti:$10$nm/Au:$50$nm evaporation and lift-off. Ti is used as an adhesion layer[@Kutz2002], while the thicker Au layer provides the electrical contact. Flakes with irregular shapes are further patterned in the shape of Hall bars by using polymethyl methacrylate (PMMA) as a mask and removing the unprotected MoS$_2$ with reactive ion etching (RIE) in a $150$mTorr atmosphere of CF$_4$:O$_2$=5:1, as shown in Figs.\[figure:device\]a,b. A droplet of 1-Butyl-1-methylpiperidinium bis(trifluoromethylsulfonyl)imide (BMPPD-TFSI) is used to cover the FL-MoS$_2$ surface and part of the side electrode for liquid gate operation (LG), as sketched in Fig.\[figure:device\]a.
AFM analysis is performed with a Bruker Dimension Icon in tapping mode. The scan in Fig.\[figure:device\]c is done after the low-T experiments and removal of the ionic liquid, and confirms that the FL-MoS$_2$ sample does not show topographic damage after the measurement cycle.
![Representative Raman spectra at 514nm of a 4L-MoS$_2$ flake before (blue) and after (red) device fabrication, deposition of the ionic liquid droplet and low-T transport measurements.[]{data-label="figure:Raman"}](Figure_Raman_1.eps){width="\columnwidth"}
We use Raman spectroscopy to characterize the devices both before and after fabrication and BMPPD-TFSI deposition. Raman measurements are performed with a Horiba LabRAM Evolution at 514nm, with a 1800grooves/mm grating and a spectral resolution$\sim0.45$cm$^{-1}$. The power is kept below 300$\mu$W to avoid any damage. A representative Raman spectrum of 4L-MoS$_2$ is shown in Fig.\[figure:Raman\] (blue curve). The peak at$\sim$455cm$^{-1}$ is due to a second-order longitudinal acoustic mode at the M point[@StacJPCS1985]. The E$_{2g}^1$ peak at$\sim$385cm$^{-1}$ and the A$_{1g}$ at$\sim$409cm$^{-1}$ correspond to in-plane and out-of plane vibrations of Mo and S atoms[@VerbPRL1970; @WietPRB1971]. Their difference, Pos(E$_{2g}^1$)-Pos(A$_{1g}$), is often used to monitor N[@LeeACS2010]. However, for N$\geq$4, the variation in Pos(E$_{2g}^1$)-Pos(A$_{1g}$) between N and N$+$1 approaches the instrument resolution[@LeeACS2010] and this method is no longer reliable. Thus, we use the low frequency modes ($<100$cm$^{-1}$) to monitor N[@ZhanPRB2013; @TanNM2012]. The shear (C) and layer breathing modes (LBM) are due to the relative motions of the atomic planes, either perpendicular or parallel to their normal[@ZhanPRB2013]. Pos(C) and Pos(LBM) change with N as[@ZhanPRB2013; @TanNM2012]: $$\label{Eq1}
\mathrm{Pos(C)_N}=\frac{1}{\sqrt{2}\pi c}\sqrt{\frac{\alpha_{\parallel}}{\mu_m}}\sqrt{1+\cos\left(\frac{\pi}{N}\right)}$$ $$\label{Eq2}
\mathrm{Pos(LBM)_N}=\frac{1}{\sqrt{2}\pi c}\sqrt{\frac{\alpha_{\perp}}{\mu_m}}\sqrt{1-\cos\left(\frac{\pi}{N}\right)}$$ where $\alpha_{\parallel}\sim$2.82$\cdot$10$^{19}$N/m$^3$ and $\alpha_{\perp}\sim$8.90$\cdot$10$^{19}$N/m$^3$ are spring constants for C and LBM modes, respectively, $c$ is the speed of light in vacuum, $\mu_m\sim$3$\cdot$10$^{-6}$Kg/m$^2$ is the 1L mass per unit area[@ZhanPRB2013; @TanNM2012]. Fig.\[figure:Raman\] shows a C mode at$\sim$30cm$^{-1}$ and an LBM at$\sim$22cm$^{-1}$. These correspond to N$=$4 using Eqs.\[Eq1\],\[Eq2\]. Fig.\[figure:Raman\] also plots the Raman measurements after device fabrication, deposition of the ionic liquid, low-T measurements, $V_{LG}$ removal and warm-up to room T (red curve). We still find Pos(C)$\sim$30cm$^{-1}$ and Pos(LBM)$\sim$22cm$^{-1}$, the same as those of the pristine flake, suggesting no damage nor residual doping.
Four-probe resistance and Hall measurements are then performed in the vacuum chambers of either a Cryomech pulse-tube cryocooler, $T_{min}$=2.7K, or a Lakeshore cryogenic probe-station, $T_{min}$=8K, equipped with a 2T superconducting magnet. A small ($\sim1\mu$A) constant current is applied between S and D (Fig.\[figure:device\]a) by using a two-channel Agilent B2912A source-measure unit (SMU). The longitudinal and transverse voltage drops are measured with an Agilent 34420 low-noise nanovoltmeter. Thermoelectrical and other offset voltages are eliminated by measuring each resistance value and inverting the source current in each measurement[@DagheroPRL2012]. Gate biases are applied between the corresponding G and D with the same two-channel SMU (liquid gate) or a Keithley 2410 SMU (back gate). Samples are allowed to degas in vacuum ($<10^{-5}$mbar) at room $T$ for at least$\sim1$h before measurements, in order to remove residual water traces in the electrolyte.
{width="80.00000%"}
We first characterize the T dependence of the sheet resistance, $R_s$, under the effect of the liquid top gate. We apply the liquid gate voltage, $V_{LG}$, at 240K, where the electrolyte is still liquid, and under high-vacuum ($<10^{-5}$mbar) to minimize unwanted electrochemical interactions and extend the stability window of the ionic liquid[@UenoReview2014]. After $V_{LG}$ is applied, we allow the ion dynamics to settle for$\sim$10min before cooling to a base T=2.7K.
Fig.\[figure:transport\]a plots the T dependence of $R_s$ measured in a four-probe configuration, for different $V_{LG}$ and induced carrier density $n_{2d}$. Our devices behave similarly to Ref., undergoing first an insulator-to-metal transition near $R_s\sim h/e^2$ at low $n_{2d}<1\cdot 10^{13}$cm$^{-2}$, followed by a metal-to-superconductor transition at high $n_{2d}>6\cdot 10^{13}$cm$^{-2}$. The saturating behavior in the $R_s$ vs T curves in Fig.\[figure:transport\]a for $\mathrm{T}\lesssim50$K, close to the insulator-to-metal transition, is typically observed in systems at low $n_{2d}$ characterized by a fluctuating electrostatic potential, such as that due to charged impurities[@ZabrodskiiJETP1984]. This applies to ion-gated crystalline systems at low $V_{LG}$, since the doping is provided by a low density of ions in close proximity to the active channel. These ions induce a perturbation of the local electrostatic potential, locally inducing charge carriers, but are otherwise far apart. The resulting potential landscape is thus inhomogeneous. This low-doping ($\lesssim1\times10^{13}$cm$^{-2}$) density inhomogeneity is a known issue in ion-gated crystalline systems, but becomes less and less relevant at higher ionic densities[@RenNanoLett2015]. We employ Hall effect measurements to determine $n_{2d}$ as a function of $V_{LG}$ (see Fig.\[figure:transport\]b), and, consequently, the liquid gate capacitance $C_{LG}$. $C_{LG}$ for the BMPPD-TFSI/MoS$_2$ interface ($\sim3.4\pm0.6\mu$F/cm$^{2}$) is of the same order of magnitude as for DEME-TFSI/MoS$_2$ in Ref. ($\sim8.6\pm4.1\mu$F/cm$^2$), where DEME-TFSI is the N,N-Diethyl-N-methyl-N-(2-methoxyethyl)ammonium bis(trifluoromethanesulfonyl)imide ionic liquid[@ShiSciRep2015].
Fig.\[figure:transport\]a shows that, while for T$\gtrsim$100K $R_s$ is a monotonically decreasing function of $n_{2d}$, the same does not hold for T$\lesssim$100K, where the various curves cross. In particular, the residual $R_s$ in the normal state $R_s^0$ (measured just above $T_c$ when the flake is superconducting) varies non-monotonically as a function of $n_{2d}$. This implies the existence of multiple local maxima in the $R_s^0 (n_{2d})$ curve. Consistently with the theoretical predictions of Ref., we find two local maxima. The first and more pronounced occurs when the flake is superconducting, i.e. for $n_{2d}>6\cdot 10^{13}$cm$^{-2}$. This feature was also reported in Refs., but not discussed. The second, less pronounced kink, is observed for $1\cdot 10^{13} \lesssim n_{2d} \lesssim 2\cdot 10^{13}$cm$^{-2}$, not previously shown. Both kinks can be seen only for T$\lesssim$70K and they are smeared for T$\gtrsim$150K.
The kink that emerges in the same range of $n_{2d}$ as the superconducting dome extends across a wide range of $V_{LG}$ ($3 \lesssim V_{LG} \lesssim 6$V) for $n_{2d}\gtrsim6\cdot10^{13}$cm$^{-2}$, and can be accessed only by LG biasing, due to the small capacitance of the solid BG. This prevents a continuous characterization of its behavior, as $n_{2d}$ induced by LG cannot be altered for $T\lesssim 220$K, as the ions are locked when the electrolyte is frozen. The kink that appears early in the metallic state, on the other hand, extends across a small range of $n_{2d}$ ($1\lesssim n_{2d} \lesssim 2\cdot10^{13}$cm$^{-2}$), and is ideally suited to be explored continuously by exploiting the dual-gate configuration.
We thus bias our samples in the low-density range of the metallic state ($n_{2d}\sim7\cdot 10^{12}$cm$^{-2}$) by applying V$_{LG}=0.9$V, and cool the system to 2.7K. We then apply $V_{BG}$ and fine-tune $n_{2d}$ across the kink. We constantly monitor $I_{BG}$ to avoid dielectric breakdown. Fig.\[figure:transport\]c plots $\sigma_{2d}$ of a representative device subject to multiple $V_{BG}$ sweeps, as $n_{2d}$ is tuned across the kink. This reproduces well the behavior observed for low $V_{LG}$ ($1\lesssim n_{2d} \lesssim 2\cdot10^{13}$cm$^{-2}$). The hysteresis between increasing and decreasing $V_{BG}$ is minimal. This kink is suppressed by increasing T, similar to LG gating.
$V_{BG}$ provides us an independent tool to estimate $n_{2d}$: If $V_{LG}$ is small enough ($V_{LG}\lesssim 1$V) so that conduction in the channel can be switched off by sufficiently large negative $V_{BG}$ ($V_{BG}\lesssim -25$V), we can write $n_{2d}=C_{ox}/e \cdot (V_{BG}-V_{th})$. Here, $C_{ox}=\epsilon_{ox}/d_{ox}$ is the back gate oxide specific capacitance, $e=1.602\cdot10^{-19}$C is the elementary charge and $V_{th}$ is the threshold voltage required to observe a finite conductivity in the device. We neglect the quantum capacitance $C_q$ of MoS$_2$, since $C_q\gtrsim$100$\mu$F/cm$^2\gg C_{ox}$[@BrummePRB2015]. By using the dielectric constant of SiO$_2\,\epsilon_{ox}$=3.9[@ElKarehBook1995] and an oxide thickness $t_{ox}=90$nm (or $t_{ox}=285$nm, depending on the experiment) we obtain the $n_{2d}$ scale in the top axis of Fig.\[figure:transport\]c, in good agreement with the corresponding values in Fig.\[figure:transport\]a, estimated from the Hall effect measurements in Fig.\[figure:transport\]b.
The bandstructure of field-effect doped NL-MoS$_2$ depends on N[@BrummePRB2015] and strain[@BrummePRB2016]. A fully relaxed N-layer flake, with N$\geq$3, has been predicted to behave as follows[@BrummePRB2015; @BrummePRB2016]: For small doping ($x\lesssim 0.05$e$^-$/unit cell, Figs.\[figure:bands\]b and \[figure:kinks\]a) only the two spin-orbit split sub-bands at K/K’ are populated. At intermediate doping ($0.05\lesssim x \lesssim 0.1$ e$^-$/unit cell, Figs.\[figure:bands\]c and \[figure:kinks\]b), $E_F$ crosses the first spin-orbit split sub-band at Q/Q’ (labeled Q$_1$). For large doping ($x\gtrsim 0.1$ e$^-$/unit cell, Figs.\[figure:bands\]d and \[figure:kinks\]c) $E_F$ crosses the second sub-band (Q$_2$) and both valleys become highly populated[@BrummePRB2015]. Even larger doping ($x\gtrsim 0.35$ e$^-$/unit cell) eventually shifts the K/K’ valleys above $E_F$[@BrummePRB2015].
{width="80.00000%"}
When $E_F$ crosses these high-energy sub-bands at Q/Q’, sharp kinks are expected to appear in the transconductance of gated FL-MoS$_2$[@BrummePRB2016] (see Fig.\[figure:bands\]e). These are reminiscent of a similar behavior in liquid-gated FL graphene, where their appearance was linked to the opening of interband scattering channels upon the crossing of high-energy sub-bands[@YePNAS2011; @Gonnelli2dMater2017; @PiattiAppSS2017]. Even in the absence of energy-dependent scattering, Ref. showed that $\sigma_{2d}$ can be expressed as: $$\sigma_{2d}=e^{2}\tau\langle v_{\parallel}^{2}\rangle N\left(E_{F}\right)\propto e^{2}\langle v_{\parallel}^{2}\rangle\label{eq:sigma_velocity}$$ where $\tau\propto N(E_F)^{-1}$ is the average scattering time, and $N\left(E_{F}\right)$ is the density of states (DOS) at $E_F$. This implies that $\sigma_{2d}$ is proportional to the average of the squared in-plane velocity $\text{\ensuremath{\langle}}\ensuremath{v_{\parallel}^{2}}\text{\ensuremath{\rangle}}$ over the FS[@BrummePRB2016]. Since $\text{\ensuremath{\langle}}\ensuremath{v_{\parallel}^{2}}\text{\ensuremath{\rangle}}$ linearly increases with $n_{2d}$ and drops sharply as soon as a new band starts to get doped[@BrummePRB2016], the kinks in $\sigma_{2d}$ (or, equivalently, $R_s$) at $T\lesssim15$K can be used to determine the onset of doping of the sub-bands in the Q/Q’ valleys. At T=0, the kink is a sharp drop in $\sigma_{2d}$, emerging for the doping value at which $E_F$ crosses the bottom of the next sub-band. This correspondence is lost due to thermal broadening for T$>$0, leading to a smoother variation in $\sigma_{2d}$. If T is sufficiently large the broadening smears out any signature of the kinks, Fig.\[figure:transport\]. Ref. calculated that, at finite T, the conductivity kinks define a *doping range* where the sub-band crossing occurs (between $R_s$ minimum and maximum, i.e. the *lower* and *upper* bounds of each kink sets the resolution of this approach). Each sub-band crossing starts after the $R_s$ minimum at lower doping, then develops in correspondence of the inflection point, and is complete once the $R_s$ maximum is reached.
We show evidence for this behavior in Fig.\[figure:kinks\], where we plot $T_c$ (panel d) and $R_s^0$ (panel e) as a function of $n_{2d}$. The electric field is applied both in liquid-top-gate (filled dots and dashed line) and dual-gate (solid red line) configurations. For comparable values of $n_{2d}$, the liquid-gate geometry features larger $R_s^0$ than back-gated. This difference is due to increased disorder introduced when $n_{2d}$ is modulated via ionic gating[@Gonnelli2dMater2017; @PiattiAppSS2017; @PiattiAPL2017; @GallagherNatCommun2015; @OvchinnikovNatCommun2016]. Two kinks appear in the $n_{2d}$ dependence of $R_s$: a low-doping one for $1.5\cdot 10^{13} \lesssim n_{2d} \lesssim 2\cdot 10^{13}$cm$^{-2}$, and a high-doping one for $7\cdot 10^{13} \lesssim n_{2d} \lesssim 9\cdot 10^{13}$cm$^{-2}$. The plot of the SC dome of gated MoS$_2$ on the same $n_{2d}$ scale shows that the low-doping kink appears well before the SC onset, while the second appears immediately after, before the maximum $T_c$ is reached.
These results can be interpreted as follows. When $n_{2d} \lesssim 1\cdot10^{13}$cm$^{-2}$, only the spin-orbit split sub-bands at K/K’ are populated, and the FS is composed only by two pockets, Fig.\[figure:kinks\]a. For $n_{2d}$ between$\sim1.5$ and $2\cdot 10^{13}$cm$^{-2}$, $E_F$ crosses the bottom of the Q$_1$ sub-band and two extra pockets appear in the FS at Q/Q’[@GePRB2013; @BrummePRB2015], Fig.\[figure:kinks\]b. The emergence of these pockets induces a Lifshitz transition, i.e. an abrupt change in the topology of the FS[@LifshitzJETP1960]. Once Q$_1$ is populated and $E_F$ is large enough ($n_{2d}\sim6\cdot 10^{13}$cm$^{-2}$), the system becomes superconducting[@LuScience2015; @YeScience2012]. For slightly larger $E_F$ ($7\cdot 10^{13} \lesssim n_{2d} \lesssim 9\cdot 10^{13}$cm$^{-2}$), $E_F$ crosses the bottom of Q$_2$ resulting in a second Lifshitz transition, and other two pockets emerge in the FS at Q/Q’[@BrummePRB2015], Fig.\[figure:kinks\]c.
We note that the experimentally observed kinks are at different $n_{2d}$ with respect to the theoretical ones for 3L-MoS$_2$[@BrummePRB2016]. Ref. predicted that for a $1.28\%$ in-plane tensile strain, Q$_1$ and Q$_2$ should be crossed for $n_{2d}\sim5\cdot 10^{13}$ and $\sim1\cdot 10^{14}$. Since the positions of the sub-band crossings are strongly dependent on strain[@BrummePRB2016], we estimate the strain in our devices by monitoring the frequency of the E$_{2g}^1$ mode via Raman spectroscopy.
![a) Raman spectra of the 4L-MoS$_2$ device in Fig.\[figure:device\]c from 4 to 292K. b) Shift in the position of the E$_{2g}^1$ mode as a function of T for as-prepared bulk flake (black circles), a 4L-MoS$_2$ flake (blue circles), and a 4L-MoS$_2$ device with Au contacts (red circles).[]{data-label="fig:strain"}](Figure_strain_2.eps){width="\columnwidth"}
Strain can arise due to a mismatch in the thermal expansion coefficients (TECs) of MoS$_2$[@GanPRB2016], SiO$_2$ substrate[@NIST_standard1991] and Au electrodes[@NixPR1941]. Upon cooling, MoS$_2$, SiO$_2$ and Au would normally undergo a contraction. However the flake is also subject to a tensile strain due to TEC mismatch[@YoonNL2011]. The strain, $\epsilon_{MoS_2}$, due to the MoS$_2$-SiO$_2$ TEC mismatch is: $$\epsilon_{MoS_2}=\int_{T}^{292K} [\alpha_{MoS_2}(T)-\alpha_{SiO_2}(T)] dT$$ whereas the strain, $\epsilon_{Au}$, due to the Au contacts is: $$\epsilon_{Au}=\int_{T}^{292K} [\alpha_{Au}(T)-\alpha_{SiO_2}(T)] dT$$ $\epsilon_{MoS_2}$ and $\epsilon_{Au}$ are$\sim$0.1% and$\sim$0.3% at $\sim$4K, respectively[@YoonNL2011].
Any FL-MoS$_2$ on SiO$_2$ will be subject to $\epsilon_{MoS_2}$ at low T. When the flake is contacted, an additional contribution is present due to $\epsilon_{Au}$. This can be more reliably estimated performing T-dependent Raman scattering and comparing the spectra for contacted and un-contacted flakes[@YoonNL2011; @MohiPRB2009]. Figs.\[fig:strain\]a,b show how a T decrease results in the E$_{2g}^1$ mode shifting to higher frequencies for both as-prepared and contacted 4L-MoS$_2$, due to anharmonicity[@KlemPR1966]. However, in the as-prepared 4L-MoS$_2$, the up-shift is$\sim$1cm$^{-1}$ larger with respect to the contacted one. This difference points to a further tensile strain. Refs. suggested that uniaxial tensile strain on 1L-MoS$_2$ induces a E$_{2g}^1$ softening and a splitting in two components: E$_{2g}^{1+}$ and E$_{2g}^{1-}$[@LeeNC2017; @ConlNL2013]. The shift rates for E$_{2g}^{1+}$ and E$_{2g}^{1-}$ are from -0.9 to -1.0cm$^{-1}$/% and from -4.0 to -4.5cm$^{-1}$/%, respectively[@LeeNC2017; @ConlNL2013]. We do not observe splitting, pointing towards a biaxial strain. As for Ref., we calculate a shift rate of E$_{2g}^1$ for biaxial strain from -7.2 to -8.2cm$^{-1}$/%. The amount of tensile strain on the 4L-MoS$_2$ device can thus be estimated. The E$_{2g}^1$ up-shift difference between contacted and as-prepared 4L-MoS$_2$, $\Delta$Pos(E$_{2g}^1$), at 4K is$\sim$-1.0cm$^{-1}$, corresponding to an additional $\sim$0.13% biaxial tensile strain. Thus, assuming a 0.1% strain for the as-prepared 4L-MoS$_2$ due to TEC mismatch with SiO$_2$, we estimate the total strain in the contacted 4L-MoS$_2$ to be$\sim$0.23% at$\sim$4K.
![a) Surface carrier densities required to cross the Q$_1$ sub-band in FL-MoS$_2$ as a function of tensile strain. Theoretical values for 1L (black dots and line) and 3L (red triangle and line) from Ref.; values for 4L (green diamonds and line) are by linear extrapolation. Blue diamond is the present experiment. b) EPC enhancement due to the crossing of the Q$_2$ sub-band, $\Delta \lambda$, as a function of $n_{2d}$, assuming $\omega_{ln}=230\pm30$K and $\mu^*=0.13$[@GePRB2013]. Filled blue circles are our experiments. Black and magenta open circles from Refs.. The blue dashed line is a guide to the eye.[]{data-label="figure:coupling"}](Figure_EPC_4.eps){width="\columnwidth"}
Fig.\[figure:coupling\]a shows that, for $0.23$% tensile strain, the experimentally observed positions of the kinks agree well with a linear extrapolation of the data of Ref. to 4L-MoS$_2$ (representative of our experiments) and for in-plane strain between $0\%$ (bulk) and $1.28\%$ (fully relaxed). These findings indicate that, while the mechanism proposed in Ref. qualitatively describes the general behavior of gated FL-MoS$_2$, quantitative differences arise due to the spin-orbit split of the Q$_1$ and Q$_2$ sub-bands. The main reason for the EPC (and, hence, $T_c$) increase is the same, i.e. the increase in the number of phonon branches involved in the coupling when the high-energy valleys are populated[@GePRB2013]. However, the finite spin-orbit-split between the sub-bands significantly alters the FS connectivity upon increasing doping[@BrummePRB2015]. If we consider the relevant phonon wave vectors ($q$=$\Gamma$,K,M,$\Gamma$K$/2$) for 1L- and FL-MoS$_2$[@MolinaPRB2011; @AtacaJPCC2011], and only the K/K’ valleys populated, then only phonons near $\Gamma$ and K can contribute to EPC[@GePRB2013]. The former strongly couple e$^-$ within the same valley[@GePRB2013], but cannot contribute significantly due to the limited size of the Fermi sheets[@GePRB2013]. The latter couple e$^-$ across different valleys[@GePRB2013], and provide a larger contribution[@GePRB2013], insufficient to induce SC. MoS$_2$ flakes are metallic but not superconducting before the crossing of Q$_1$. When this crossing happens, the total EPC increases due to the contribution of longitudinal phonon modes near K[@GePRB2013] (coupling states near two different Q or Q’), near $\Gamma$K$/2$[@GePRB2013] (coupling states near Q to states near Q’), and near M[@GePRB2013] (coupling states near Q or Q’ to states near K or K’). However, this first EPC increase associated with Q$_1$ is not sufficient to induce SC, as the SC transition is not observed until immediately before the crossing of the spin-orbit-split sub-band Q$_2$ and the second doping-induced Lifshitz transition. Additionally, the SC dome shows a maximum in the increase of $T_c$ with doping ($dT_c/dn_{2d}$) across the Q$_2$ crossing, i.e. when a new FS emerges. Consistently, the subsequent reduction of $T_c$ for $n_{2d}\geq 13\cdot 10^{13}$cm$^{-2}$ can be associated with the FS shrinkage and disappearance at K/K’[@GePRB2013; @BrummePRB2015], and might also be promoted by the formation of an incipient Charge Density Wave[@RosnerPRB2014; @Piattiarxiv2018] (characterized by periodic modulations of the carrier density coupled to a distortion of the lattice structure[@GrunerBook2009]).
Since the evolution of the bandstructure with doping is similar in several semiconducting TMDs[@BrummePRB2015; @DasPRB2015; @BrummePRB2016; @WuNatCommun2016; @KangNanoLett2017], this mechanism is likely not restricted to gated MoS$_2$. The $T_c$ increase in correspondence to a Lifshitz transition is reminiscent of a similar behavior observed in CaFe$_2$As$_2$ under pressure[@GonnelliSciRep2016], suggesting this may be a general feature across different classes of materials.
We note that the maximum $T_c\sim11$K is reached at $n_{2d}\simeq 12\cdot 10^{13}$cm$^{-2}$, as reported in Ref.. This is a doping level larger than any doping level which can be associated with the kink. Thus, the Q$_2$ sub-band must be highly populated when the maximum $T_c$ is observed. We address this quantitatively with the Allen-Dynes formula[@AllenDynes], which describes the dependence of $T_c$ by a numerical approximation of the Eliashberg theory accurate for materials with a total $\lambda\lesssim1.5$[@AllenDynes]: $$\label{EqAllenDynes}
T_c(n_{2d})=\frac{\omega_{ln}}{1.2}\mathrm{exp}\left\lbrace\frac{-1.04\left[1+\lambda(n_{2d})\right]}{\lambda(n_{2d})-\mu^*\left[1+0.62\lambda(n_{2d})\right]}\right\rbrace$$ where $\lambda(n_{2d})$ is the total EPC as a function of doping, $\omega_{ln}$ is the representative phonon frequency and $\mu^*$ is the Coulomb pseudo-potential. It is important to evaluate the increase in EPC between the non-superconducting region ($n_{2d}\lesssim6\times10^{13}$cm$^{-2}$) and the superconducting one, i.e. the enhancement in $\lambda$ due to the crossing of the sub-band at Q$_2$. $\Delta\lambda=\lambda(T_c)-\lambda(T_c=0)$ indicates the EPC increase due to the appearance of e$^-$ pockets at Q$_2$. By setting $\omega_{ln}=230\pm30$K and $\mu^*=0.13$ (as for Ref.), and using Eq.\[EqAllenDynes\], we find that the limit of $\lambda(T_c)$ for $T_c\to0$ is $\sim0.25$. The corresponding $\Delta\lambda$ vs. $n_{2d}$ dependence is shown in Fig.\[figure:coupling\]b. The crossing at Q$_2$ results in a maximum $\Delta\lambda=0.63\pm0.1$, with a maximum EPC enhancement of $350\pm40\%$ with respect to the non-superconducting region. This indicates that the largest contribution to the total EPC, hence to the maximum $T_c\sim11$K, is associated with the population of the Q$_2$ sub-band. This is consistent with the reports of a reduced $T_c\sim2$K in 1L-MoS$_2$[@CostanzoNatNano2016; @FuQuantMater2017], shown to be superconducting for smaller $n_{2d}\sim5.5\cdot10^{13}$cm$^{-2}$[@FuQuantMater2017], hence likely to populate Q$_1$ only. $n_{2d}\sim5\cdot10^{13}$cm$^{-2}$ is also the doping expected for the crossing of Q$_1$ in 1L-MoS$_2$ in presence of a low-T strain similar to that in our 4L-MoS$_2$ devices (see Fig.\[figure:coupling\]a).
In summary, we exploited the large carrier density modulation provided by ionic gating to explore sub-band population and multivalley transport in MoS$_2$ layers. We detected two kinks in the conductivity, associated with the doping-induced crossing of the two sub-bands at Q/Q’. By comparing the emergence of these kinks with the doping dependence of $T_c$, we showed how superconductivity emerges in gated MoS$_2$ when the Q/Q’ valleys are populated, while previous works only considered the filling of K/K’. We highlighted the critical role of the population of the second spin-orbit-split sub-band, Q$_2$, (and the consequent increase of the FS available for EPC) in the appearance of superconductivity and in the large enhancement of $T_c$ and of EPC in the first half of the superconducting dome. Our findings explain the doping dependence of the SC state at the surface of gated FL-MoS$_2$, and provide a key insight for other semiconducting transition metal dichalcogenides.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank M. Calandra for useful discussions. We acknowledge funding from EU Graphene Flagship, ERC Grant Hetero2D, EPSRC Grant Nos. EP/509K01711X/1, EP/K017144/1, EP/N010345/1, EP/M507799/ 5101, and EP/L016087/1 and the Joint Project for the Internationalization of Research 2015 by Politecnico di Torino and Compagnia di San Paolo.
The authors declare no competing financial interests.
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[^1]: These authors contributed equally to this work.
[^2]: These authors contributed equally to this work.
| 0 |
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abstract: 'It is well known that reverberation mapping of active galactic nuclei (AGN) reveals a relationship between AGN luminosity and the size of the broad-line region, and that use of this relationship, combined with the Doppler width of the broad emission line, enables an estimate of the mass of the black hole at the center of the active nucleus based on a single spectrum. This has been discussed in numerous papers over the last two decades. An unresolved key issue is the choice of parameter used to characterize the line width; generally, most researchers use FWHM in favor of line dispersion (the square root of the second moment of the line profile) because the former is easier to measure, less sensitive to blending with other features, and usually can be measured with greater precision. However, use of FWHM introduces a bias, stretching the mass scale such that high masses are overestimated and low masses are underestimated. Here we investigate estimation of black hole masses in AGNs based on individual or “single epoch” observations, with a particular emphasis in comparing mass estimates based on line dispersion and FWHM. We confirm the recent findings that, in addition to luminosity and line width, a third parameter is required to obtain accurate masses and that parameter seems to be Eddington ratio. We present simplified empirical formulae for estimating black hole masses from the $\lambda4861$ and $\lambda1549$ emission lines.'
author:
- Elena Dalla Bontà
- 'Bradley M. Peterson'
- 'Misty C. Bentz'
- 'W. N. Brandt'
- 'S. Ciroi'
- Gisella De Rosa
- Gloria Fonseca Alvarez
- 'Catherine J. Grier'
- 'P. B. Hall'
- 'Juan V. Hernández Santisteban'
- 'Luis C. Ho'
- 'Y. Homayouni'
- Keith Horne
- 'C. S. Kochanek'
- 'Jennifer I-Hsiu Li'
- 'L. Morelli'
- 'A. Pizzella'
- 'R. W. Pogge'
- 'D. P. Schneider'
- Yue Shen
- 'J. R. Trump'
- Marianne Vestergaard
title: 'THE SLOAN DIGITAL SKY SURVEY REVERBERATION MAPPING PROJECT: ESTIMATING MASSES OF BLACK HOLES IN QUASARS WITH SINGLE-EPOCH SPECTROSCOPY'
---
Introduction {#section:intro}
============
Reverberation-Based Black Hole Masses
-------------------------------------
The presence of emission lines with Doppler widths of thousands of kilometers per second is one of the defining characteristics of active galactic nuclei [@Burbidge67; @Weedman76]. It was long suspected that the large line widths were due to motions in a deep gravitational potential and this implied very large central masses [e.g., @Woltjer59], as did the Eddington limit [@Tarter73]. Under a few assumptions, the central mass is $M
\propto V^2 R$, where $V$ is the Doppler width of the line and $R$ is the size of the broad-line region (BLR). It is the latter quantity that is difficult to determine. An early attempt to estimate $R$ by [@Dibai80] was based on the assumption of constant emissivity per unit volume, but led to an incorrect dependence on luminosity as in this case, luminosity is proportional to volume, so $R \propto L^{1/3}$. [@Wandel85] inferred the BLR size from the luminosity. Other attempts were based on photoionization physics [see @Ferland85; @Osterbrock85]. [@Davidson72] found that the relative strength of emission lines in ionized gas could be characterized by an ionization parameter $$U= \frac{Q({\rm H})}{4\pi R^2 c n_{\rm H}},
\label{eq:Udef}$$ where $Q({\rm H})$ is the rate at which H-ionizing photons are emitted by the central source and $n_{\rm H}$ is the particle density of the gas. The ionization parameter $U$ is proportional to the ratio of ionization rate to recombination rate in the BLR clouds. The similarity of emission-line flux ratios in AGN spectra over orders of magnitude in luminosity suggested that $U$ is constant, and the presence of C[iii]{}\]$\lambda1909$ set an upper limit on the density $n_{\rm H} \la 10^{9.5}$ [@Davidson79]. Since $L \propto Q({\rm H})$, this naturally led to the prediction that the BLR radius would scale with luminosity as $R \propto L^{1/2}$. Unfortunately, best-estimate values for $Q({\rm
H})$ and $n_{\rm H}$ led to a significant overestimate of the BLR radius [@Peterson85] as a consequence of the simple but erroneous assumption that all the broad lines arise cospatially (i.e., models employed a single representative BLR cloud).
With the advent of reverberation mapping [hereafter RM; @Blandford82; @Peterson93], direct measurements of $R$ enabled improved black hole mass determinations. Attempts to estimate black hole masses based on early RM results and the $R \propto L^{1/2}$ prediction included those of [@Padovani88], [@Koratkar91], and [@Laor98]. The first multiwavelength RM campaigns demonstrated ionization stratification of the BLR [@Clavel91; @Krolik91; @Peterson91] and this eventually led to identification of the virial relationship, $R \propto V^{-2}$ [@PetersonWandel99; @PetersonWandel00; @Onken02; @Kollatschny03; @Bentz10], that gave reverberation-based mass measurements higher levels of credibility. Of course, the virial relationship demonstrates only that the central force has a $R^{-2}$ dependence, which is also characteristic of radiation pressure; whether or not radiation pressure from the continuum source is important has not been clearly established [@Marconi08; @Marconi09; @Netzer10]. If radiation pressure in the BLR turns out to be important, then the black hole masses, as we discuss them here, are underestimated.
Masses of AGN black holes are computed as $$\label{eq:masseqn}
M_{\rm BH} = f \left(\frac{V^2 R}{G}\right),$$ where $V$ is the line width, $R$ is the size of the BLR from the reverberation lag, and $G$ is the gravitational constant. The quantity in parentheses is often referred to as the virial product $\mu$; it incorporates the two observables in RM, line width and time delay $\tau = R/c$, and is in units of mass. The scaling factor $f$ is a dimensionless quantity of order unity that depends on the geometry, kinematics, and inclination of the AGN. Throughout most of this work, we ignore $f$ (i.e., set it to unity) and work strictly with the virial product.
While reverberation mapping has emerged as the most effective technique for measuring the black hole masses in AGNs [@Peterson14], it is resource intensive, requiring many observations over an extended period of time at fairly high cadence. Fortunately, observational confirmation of the $R$–$L$ relationship [@Kaspi00; @Kaspi05; @Bentz06a; @Bentz09a; @Bentz13] enables “single-epoch” (SE) mass estimates because, in principle, a single spectrum could yield $V$ and also $R$, through measurement of $L$ [e.g., @Wandel99; @McLure02; @Vestergaard02; @Corbett03; @Vestergaard04; @Kollmeier06; @Vestergaard06; @Fine08; @Shen08a; @Shen08b; @Vestergaard08]. Of the three strong emission lines generally used to estimate central black hole masses, the $R$–$L$ relationship is only well-established for $\lambda4861$ [@Bentz13 and references therein, but see the discussion in §\[section:hbpredictor\]]. Empirically establishing the $R$–$L$ relationship for $\lambda2798$ [@Homayouni20] and $\lambda1549$ [@Peterson05; @Kaspi07; @Trevese14; @Lira18; @Grier19; @Hoormann19] has been difficult.
Masses based on the $\lambda1549$ emission line, in particular, have been somewhat controversial. Some studies claim that there is good agreement between masses based on and those measured from other lines [@Vestergaard06; @Greene10b; @Assef11]. On the other hand, there are several claims that there is inadequate agreement with masses based on other emission lines [@Baskin05; @Netzer07; @Sulentic07; @Shen08b; @Shen12; @Trak12]. [@Denney09a] and [@Denney13], however, note that there are a number of biases that can adversely affect single-epoch mass estimates, with low $S/N$ “survey quality” data being an important problem with some of the studies for which poor agreement between and other lines is found. It has also been argued, however, that some fitting methodologies are more affected by this than others [@Shen19]. There have been more recent papers that attempt to correct mass determinations to better agree with those based on other lines [e.g., @Runnoe13; @Coatman17; @Mejia18; @Marziani19].
Characterizing Line Widths
--------------------------
It is our suspicion that the apparent difficulties with -based masses trace back not only to the $S/N$ issue, but also to how the line widths are characterized. It has been customary in AGN studies to characterize line widths by one of two parameters, either FWHM or the line dispersion , which is defined by $${\mbox{$\sigma_{\rm line}$}}= \left[ \frac{ \int (\lambda - \lambda_0)^2 P(\lambda)\,d\lambda}{\int P(\lambda)\,d\lambda} \right]^{1/2},
\label{eq:Defsigl}$$ where $P(\lambda)$ is the emission-line profile as a function of wavelength and $\lambda_0$ is the line centroid, $$\lambda_0 = \frac{\int \lambda P(\lambda)\,d\lambda}{\int P(\lambda)\,d\lambda}.
\label{eq:Deflambda0}$$ While both FWHM and have been used in the virial equation to estimate AGN black hole masses, they are not interchangeable. It is well known that AGN line profiles depend on the line width [@Joly85]: broader lines have lower kurtosis, i.e., they are “boxier” rather than “peakier.” Indeed, for AGNs, the ratio ${\rm FWHM}/{\mbox{$\sigma_{\rm line}$}}$ has been found to be a simple but useful characterization of the line profile [@Collin06; @Kollatschny13].
Each line-width measure has practical strengths and weaknesses [@Peterson04; @Wang20]. The line dispersion is more physically intuitive, but it is sensitive to the line wings, which are often badly blended with other features. All three of the strong lines usually used to estimate masses — $\lambda4861$, $\lambda2798$, and $\lambda1549$ — are blended with other features: the $\lambda4570$ and $\lambda\lambda$5190, 5320 complexes [@Phillips78] and $\lambda4686$ in the case of , the UV complexes in the case of , and $\,\lambda1640$ in the case of . The FWHM can usually be measured more precisely than (although @Peterson04 note that the opposite is true for the rms spectra, described below, which are sometimes quite noisy), but it is not clear that FWHM yields more [*accurate*]{} mass measurements. In practice, FWHM is used more often than because it is relatively simple to measure and can be measured more precisely while often requires deblending or modeling the emission features, which does not necessarily yield unambiguous results.
There are, however, a number of reasons to prefer to FWHM as the line-width measure for estimating AGN black hole masses. [@Fromerth00] point out that better characterizes an arbitrary or irregular line profile. [@Peterson04] note that produces a tighter virial relationship than FWHM, and [@Denney13] find better agreement between -based and -based mass estimates by using rather than FWHM (these latter two are essentially the same argument). In the case of NGC 5548, for which there are multiple reverberation-based mass measures, a possible correlation with luminosity is stronger for FWHM-based masses than for -based masses, suggesting that the former are biased as the same mass should be recovered regardless of the luminosity state of the AGN [@Collin06; @ShenKelly12]. A possibly more compelling argument for using instead of FWHM is bias in the mass scale that is introduced by using FWHM as the line width. [@Steinhardt10] used single-epoch masses for more than 60,000 SDSS quasars [@Shen08b] with masses computed using FWHM. They found that, in any redshift bin, if one plots the distribution of mass versus luminosity, the higher mass objects lie increasingly below the Eddington limit; they refer to this as the “sub-Eddington boundary.” There is no physical basis for this. [@Rafiee11] point out, however, that if the quasar masses are computed using instead of FWHM, the sub-Eddington boundary disappears: the distribution of quasar black hole masses approaches the Eddington limit at all masses. Referring to Figure 1 of [@Rafiee11], the distribution of quasars in the mass vs. luminosity diagram is an enlongated cloud of points whose axis is roughly parallel to the Eddington ratio when is used to characterize the line width. However, when FWHM is used, the axis of the distribution rotates as the higher masses are underestimated and the lower masses are overestimated. However, the apparent rotation of the mass distribution is in the same sense that is expected from the Malmquist bias and a bottom heavy quasar mass function [@Shen13]. Unfortunately, these arguments are not statistically compelling. Examination of the –$\sigma{*}$ relation using FWHM-based and -based masses is equally unrevealing [@Wang19].
In reverberation mapping, a further distinction among line-width measures must be drawn since either FWHM or can be measured in the mean spectrum, $$\overline{F}(\lambda) =\frac{1}{N} \sum_{1}^{N} F_i(\lambda),
\label{eq:meanspec}$$ where $F_i(\lambda)$ is the flux in the $i^{th}$ spectrum of the time series at wavelength $\lambda$ and $N$ is the number of spectra, or they can be measured in the rms residual spectrum (hereafter simply “rms spectrum”), which is defined as $$\sigma_{\rm rms}(\lambda) = \left\{ \frac{1}{N-1}
\sum_{1}^{N}\left[ F_i(\lambda) -
\overline{F}(\lambda)\right]^2 \right\}^{1/2}.
\label{eq:rmsspec}$$ In this paper, we will specifically refer to the measurements of in the mean spectrum as and in the rms spectrum as . Similarly, refers to FWHM of a line in the mean spectrum or a single-epoch spectrum and is the FWHM in the rms spectrum. It is common to use as the line-width measure for determining black hole masses from reverberation data — it is intuitatively a good choice as it isolates the gas in the BLR that is actually responding to the continuum variations. As noted previously, the strong and strongly variable broad emission lines can be hard to isolate as they are blended with other features. In the rms spectra, however, the contaminating features are much less of a problem because they are generally constant or vary either slowly or weakly and thus nearly disappear in the rms spectra.
Since the goal is to measure a black hole mass from a single (or a few) spectra, we must use a proxy for . Here we will attempt to determine if either or in a single or mean spectrum can serve as a suitable proxy for ; we know [*a priori*]{} that there are good, but non-linear, correlations between and both and . It therefore seems likely that either or could be used as a proxy for .
Investigation of the relationship among the line-width measures motivated a broader effort to produce easy-to-use prescriptions for computing [*accurate*]{} black hole masses using and emission lines and nearby continuum fluxes measurements for each line. We note that we do not discuss RM results in this contribution as the present situation has been addressed rather thoroughly by [@Bahk19] and new SDSS-RM results will be published soon [@Homayouni20]. In §[2]{}, the data used in this investigation are described. In §[3]{}, the relationship between the reverberation lag and different measures of the AGN luminosity are considered, and we identify the physical parameters to lead to accurate black-hole mass determinations. In §[4]{}, we will similarly discuss masses based on . In §[5]{}, we present simple empirical formulae for estimating black hole masses from or . The results of this investigation and our future plans to improve this method are outlined in §[6]{}. Our results are briefly summarized in §[7]{}. Throughout this work, we assume $H_0 = 72$Mpc$^{-1}$, $\Omega_{\rm matter} =0.3$ and $\Omega_\Lambda = 0.7$.
Observational Database and Methodology
======================================
Data {#section:Data}
----
We use two high-quality databases for this investigation:
1. Spectra and measurements for previously reverberation-mapped AGNs, for (Table A1) and for (Table A2). These are mostly taken from the literature (see also @Bentz15 for a compilation[^1]). Sources without estimates of host-galaxy contamination to the optical luminosity $L(5100\,{\rm \AA})$ have been excluded. This database provides the fundamental $R$–$L$ calibration for the single-epoch mass scale. In this contribution, we will refer to these collectively as the “reverberation-mapping database (RMDB)”.
2. Spectral measurements from the Sloan Digital Sky Survey Reverberation Mapping Program [@Shen15 hereafter “SDSS-RM” or more compactly simply as “SDSS”]. We use both (Table A3) and (Table A4) data from the 2014–2018 SDSS-RM campaign [@Grier17b; @Shen19; @Grier19]. Each spectrum is comprised of the average of the individual spectra obtained for each of the 849 quasars in the SDSS-RM field.
In addition, because RM measurements remain rather scarce, we augmented the sample with measurements from [@Vestergaard06] (hereafter VP06), who combined single-epoch luminosity and line-width measurements from archival UV spectra with -based mass measurements of the objects in Table A1. The UV parameters are given in Table A5; we note, however, that we have excluded 3C273 and 3C390.3 because they both have uncertainities in their virial product larger than 0.5dex; the former was a particular problem because there were far more measurements of UV parameters for this source than for any other and the combination of a large number of measurements and a poorly constrained virial product conspired to disguise real correlations.
All SDSS-RM spectra have been reduced and processed as described by [@Shen15] and [@Shen16b], including post-processing with [PrepSpec]{} (Horne, in preparation). We note that only lags ($\tau$), line dispersion in the rms spectrum (${\mbox{$\sigma_{\rm R}$}}$), and virial products (${\mbox{$\mu_{\rm RM}$}}= {\mbox{$\sigma_{\rm R}$}}^2 c \tau/G$) are taken from [@Grier17b] and [@Grier19]; all luminosities and other line-width measures are from [@Shen19] (Tables A3 and A4 are included here for the sake of clarity).
For each SDSS AGN, there are two determinations of both and ; one is the best-fit (BF) to the mean spectrum, and the other is the mean of multiple Monte Carlo (MC) realizations. For each MC realization, $N$ independent random selections of the $N$ spectra are combined and the line width is measured for both and . After a large number of realizations, the mean $\langle V
\rangle$ and rms $\Delta V$, for $V = {\mbox{${\rm FWHM}_{\rm M}$}}$ and $V = {\mbox{$\sigma_{\rm M}$}}$ are computed, and the rms values are adopted as the uncertainties in each line-width measure.
For the purpose of mass estimation, we need to establish relationships based on the most reliable data. Many of the SDSS average spectra are still quite noisy, so we imposed quality cuts. Even though we are for the most part restricting our attention to the SDSS-RM quasars for which there are measured lags for (44 quasars) or (48 quasars), we impose these cuts on the entire sample for the sake of later discussion. The first quality condition is that $$\label{eq:minwidth}
V \geq 1000\,{\mbox{\rm km~s$^{-1}$}}$$ for both $V = {\mbox{${\rm FWHM}_{\rm M}$}}$ and $V ={\mbox{$\sigma_{\rm M}$}}$, since AGNs with lines narrower than 1000 are probably Type 2 AGNs; there are some Type 1 AGNs with line widths narrower than this, including several in Table A1, but these are low-luminosity AGNs [e.g., @Greene07], not SDSS quasars. The second quality condition is that the best fit value $V({\rm BF})$ must lie in the range $$\label{eq:consistency}
\langle V \rangle - \Delta V \leq V({\rm BF}) \leq \langle V \rangle + \Delta V$$ for both FWHM and . A third quality condition is a “signal-to-noise” ($S/N$) requirement that the line width must be significantly larger than its uncertainty. Some experimentation showed that $$\label{eq:s2n}
\frac{V}{\Delta V} \geq 10$$ is a good criterion for both $V = {\mbox{${\rm FWHM}_{\rm M}$}}$ and $V = {\mbox{$\sigma_{\rm M}$}}$ to remove the worst outliers from the line-width comparisons discussed in §[\[section:hblinewidths\]]{} and §[\[section:civfundamental\]]{}.
Finally, we removed quasars that were flagged by [@Shen19] as having broad absorption lines (BALs), mini-BALs, or suspected BALs in .
The effect of each quality cut on the size of the database available for each emission line is shown in Table 1. Of the 44 SDSS-RM quasars with measured lags, 12 failed to meet at least one of the quality criteria, usually the $S/N$ requirement, thus reducing the SDSS-RN sample to 32 quasars. Three quasars with reverberation measurements (RMID 362, 408, and 722) were rejected for significant BALs, thus reducing the SDSS-RM reverberation sample to 45 quasars. As we will show in §[\[section:massformulae\]]{}, another effect of imposing quality cuts is, not surprisingly, that it removes some of the lower luminosity sources from the sample.
[lcc]{} Original sample & 221 & 540\
(a) Minimum Line Width (eq. \[eq:minwidth\]) & 199 & 520\
(b) Consistency (eq. \[eq:consistency\]) & 194 & 368\
(c) $S/N$ (eq. \[eq:s2n\]) & 121 & 462\
(a) + (b) & 174 & 352\
(a) + (c) & 108 & 450\
(b) + (c) & 107 & 309\
(a) + (b) + (c) & 96 & 299\
All + BAL removal & 96 & 248 \[table:qcuts\]
Fitting Procedure
-----------------
Throughout this work, we use the fitting algorithm described by [@Cappellari13] that combines the Least Trimmed Squares technique of [@Rousseeuw06] and a least-squares fitting algorithm which allows errors in all variables and includes intrinsic scatter, as implemented by [@DallaBonta18]. Briefly, the fits we perform here are of the general form $$\label{eq:powerlaw}
y = a + b\left(x - x_0 \right),$$ where $x_0$ is the median value of the observed parameter $x$. The fit is done iteratively with $5 \sigma$ rejection (unless stated otherwise) and the best fit minimizes the quantity $$\label{eq:chi2line}
\chi^2=\sum_{i=1}^N \frac{[a+b (x_i-x_0) - y_i]^2}
{(b \Delta x_i)^2 + (\Delta y_i)^2 + \varepsilon_y^2},$$ where $\Delta x_i$ and $\Delta y_i$ are the errors on the variables $x_i$ and $y_i$, and $\varepsilon_y$ is the sigma of the Gaussian describing the distribution of intrinsic scatter in the $y$ coordinate; $\varepsilon_y$ is iteratively adjusted so that the $\chi^2$ per degree of freedom $\nu=N-2$ has the value of unity expected for a good fit. The observed scatter is $$\label{eq:Deltadef}
\Delta = \left\{ \frac{1}{N-2} \sum_{i=1}^N
\left[y_i - a - b\left(x_i - x_0\right) \right]^2 \right\}^{1/2}.$$ The value of $\varepsilon_y$ is added in quadrature when $y$ is used as a proxy for $x$.
The bivariate fits are intended to establish the physical relationships among the various parameters and also to fit residuals. The actual mass estimation equations that we use will be based on multivariate fits of the general form $$\label{eq:twoparameters}
z = a + b\left( x - x_0 \right) +c\left(y - y_0\right),$$ where the parameters are as described above, plus an additional observed parameter $y$ that has median value $y_0$. Similarly to linear fits, the plane fitting minimizes the quantity $$\label{eq:chi2plane}
\chi^2=\sum_{i=1}^N \frac{[a + b (x_i-x_0) + c (y_i-y_0) - z_i]^2}
{(b \Delta x_i)^2 + (c \Delta y_i)^2 + (\Delta z_i)^2 + \varepsilon_z^2},$$ with $\Delta x_i$, $\Delta y_i$ and $\Delta z_i$ as the errors on the variables $(x_i,y_i,z_i)$, and $\varepsilon_z$ as the sigma of the Gaussian describing the distribution of intrinsic scatter in the $z$ coordinate; $\varepsilon_z$ is iteratively adjusted so that the $\chi^2$ per degrees of freedom $\nu=N-3$ has the value of unity expected for a good fit. The observed scatter is $$\label{eq:twoparmDeltadef}
\Delta = \left\{ \frac{1}{N-3} \sum_{i=1}^N
\left[y_i - a - b\left(x_i - x_0\right)
-c\left(y_i - y_0\right)\right]^2 \right\}^{1/2}.$$
Masses Based on
================
The $R$–$L$ Relationships
-------------------------
In this section, we examine the calibration of the fundamental $R$–$L$ relationship using various luminosity measures. The analysis in this section is based only on the RMDB sample in Table A1 because all these sources have been corrected for host-galaxy starlight. To obtain accurate masses from , contaminating starlight from the host galaxy must be accounted for in the luminosity measurement, or the mass will be overestimated. For reverberation-mapped sources, this has been done by modeling unsaturated images of the AGNs obtained with the [*Hubble Space Telescope*]{} [@Bentz06a; @Bentz09a; @Bentz13]. The AGN contribution was removed from each image by modeling the images as an extended host galaxy plus a central point source representing the AGN. The starlight contribution to the reverberation-mapping spectra is determined by using simulated aperture photometry of the AGN-free image. In the left panel of Figure \[Figure:HbRL\], we show the lag as a function of the AGN continuum with the host contribution removed in each case. This essentially reproduces the result of [@Bentz13] as small differences are due solely to improvements in the quality and quantity of the RM database \[cf. Table A1\]; we give the best-fit values to equation (\[eq:powerlaw\]) in the first row of Table 2.
![Left: The rest-frame lag in days is shown as a function of the AGN luminosity $L_{\rm AGN}(5100\,{\rm \AA})$ in . The host-galaxy starlight contribution has been removed by using unsaturated [[*HST*]{}]{} images [see @Bentz13]. Right: The lag in days is shown as a function of the broad luminosity $L({\mbox{\rm H$\beta$}}_{\rm broad})$ in . The narrow component of has been removed in each case where it was sufficiently strong (i.e., easily identifiable) to isolate. In both panels, the solid line shows the best-fit to the data using equation (\[eq:powerlaw\]) with coefficients given in Table 2. The short dashed lines show the $\pm1\,\sigma$ uncertainty (equivalent to enclosing 68% of the values for a Gaussian distribution) and the long dashed lines show the $2.6\sigma$ uncertainties (equivalent to enclosing 99% of the values for a Gaussian distribution). []{data-label="Figure:HbRL"}](Figure1.eps)
[llccccc]{} $\log L_{\scriptsize\rm AGN}(5100\,{\rm \AA})$ & $\log \tau({\mbox{\rm H$\beta$}})$ & $1.228 \pm 0.025$ & $0.482 \pm 0.029$ & $43.444$ & $0.213 \pm 0.021$ & $0.241$\
$\log L({\mbox{\rm H$\beta$}}_{\scriptsize\rm broad})$ & $\log \tau({\mbox{\rm H$\beta$}})$ & $1.200\pm 0.025$ & $0.492 \pm 0.030$ & $41.746$ & $0.218 \pm 0.022$ & $0.244$\
$\log L(1350\,{\rm \AA})$ & $\log \tau({\mbox{\rm C\,{\sc iv}}})$ & $1.915 \pm 0.047$ & $0.517 \pm 0.036$ & $45.351$ & $0.336 \pm 0.041$ & $0.361$\
$\log L_{\scriptsize\rm AGN}(5100\,{\rm \AA})$ & $\log L({\mbox{\rm H$\beta$}}_{\rm broad})$ & $41.797 \pm 0.017$ & $0.960 \pm 0.020$ & $43.444$ & $0.158 \pm 0.014$ &$0.171$\
$\log L({\mbox{\rm H$\beta$}}_{\rm broad})$ &$\log L_{\scriptsize\rm AGN}(5100\,{\rm \AA})$ & $43.396 \pm 0.018$ & $1.003 \pm 0.022$ & $41.746$ & $0.161 \pm 0.015$ & $0.174$ \[table:RL\]
Accounting for the host-galaxy contribution in the same way for large number of AGNs, such as those in SDSS-RM (not to mention the entire SDSS catalog), is simply not feasible. It is well-known, however, that there is a tight correlation between the AGN continuum luminosity and the luminosity of [e.g., @Yee80; @Ilic17], and it has indeed been argued that the emission-line luminosity can be used as a proxy for the AGN continuum luminosity for reverberation studies [@Vestergaard06; @Greene10a]. However, in some of the reverberation-mapped sources, narrow-line contributes significantly to the total flux; NGC 4151 is an extreme example [e.g., @Antonucci83; @Bentz06a; @Fausnaugh17]. Whenever the narrow-line component can be isolated, it has been subtracted from the total flux. In Figure \[Figure:LAGNLHb\], we show the tight relationship between $L_{\rm AGN}(5100\,{\rm \AA})$ and $L({\mbox{\rm H$\beta$}}_{\rm broad})$; the best-fit coefficients for this relationship are given in Table 2.
In the right panel of Figure \[Figure:HbRL\], we show the lag as a function of the luminosity of the broad component of , with the narrow component removed whenever possible. We give the best-fit values to the equation (\[eq:powerlaw\]) in the second row of Table 2, which shows that the slope of this relationship is nearly identical to the slope of the $R$–$L$ relationship using the AGN continuum. The luminosity of the broad component is thus an excellent proxy for the AGN luminosity and requires only removal of the narrow component (at least when it is significant) which is much easier than estimating the starlight contribution to the continuum luminosity at 5100Å. Moreover, by using the line flux instead of the continuum flux, we can include core-dominated radio sources where the continuum may be enhanced by the jet component [@Greene05]. This is therefore the $R$–$L$ relationship we prefer for the purpose of estimating single-epoch masses and we will focus on this relationship through the remainder of this contribution.
![The relationship between the broad emission line luminosity and the starlight-corrected AGN luminosity for the sources in Table A1. The black solid line is the regression of $L({\mbox{\rm H$\beta$}}_{\rm broad})$ on $L_{\rm AGN}(5100\,{\rm \AA})$; the red dotted line is the regression of $L_{\rm AGN}(5100\,{\rm \AA})$ on $L({\mbox{\rm H$\beta$}}_{\rm broad})$, which we use in equation (\[eq:LHbLAGN\]). The coefficients for both fits are given in Table 2.[]{data-label="Figure:LAGNLHb"}](Figure2.eps)
Line-Width Relationships {#section:hblinewidths}
------------------------
We now consider the use of and as proxies for [cf. @Collin06; @Wang19]. The left panel of Figure \[Figure:windowhbwidths\] shows the relationship between ${\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})$, the line dispersion in the rms spectrum, and ${\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}),$ the line dispersion in the mean spectrum. The relationship is nearly linear (slope $ = 1.085\pm0.045$) and the intrinsic scatter is small ($0.079$dex). The fit coefficients are given in the first row of Table 3.
![The relationship between line dispersion in the rms ${\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})$ and mean ${\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}})$ spectra is shown on the left. The relationship between line dispersion in the rms spectrum () and FWHM in the mean spectrum () is shown on the right. Blue filled circles are for the RMDB sample (Table A1) and open green triangles are for the SDSS sample (Table A3). The solid lines are best fits to equation (\[eq:powerlaw\]) with coefficients in Table 3. The short dashed and long dashed lines indicate the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively, and the red dotted lines indicate where the two line-width measures are equal. Crosses are points that were rejected at the 2.6$\sigma$ (99%) level and are colored-coded like the circles. The relationship on the left is nearly linear (slope $= 1.085 \pm 0.045$) and the scatter $\varepsilon_y$ is low (0.079dex). It is clear in the right panel that () and () are well-correlated, but the relationship is significantly non-linear (slope $= 0.535 \pm 0.042$), the scatter $\varepsilon_y$ is slightly larger (0.106dex), and there are several significant outliers.[]{data-label="Figure:windowhbwidths"}](Figure3.eps)
[llccccc]{} $\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}})$ & $\log {\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})$ & $3.260 \pm 0.008$ & $1.085 \pm 0.045$ & 3.297 & $0.079 \pm 0.006$ & 0.087\
$\log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}})$ & $\log {\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}}) $ & $3.205 \pm 0.011$ & $0.535 \pm 0.042$ & 3.559 & $0.106 \pm 0.001$ & 0.114\
$\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}})$ & $\log {\mbox{$\sigma_{\rm R}$}}({\mbox{\rm C\,{\sc iv}}})$ & $3.436 \pm 0.009$ &$0.822 \pm 0.059$ & 3.394 & $0.064 \pm 0.008$ & 0.067\
$\log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}})$ & $\log {\mbox{$\sigma_{\rm R}$}}({\mbox{\rm C\,{\sc iv}}})$ & $3.447 \pm 0.016$ & $0.445 \pm 0.101$ & 3.580 & $0.121 \pm 0.014$ & 0.121 \[table:LW\]
We also show in the right panel of Figure \[Figure:windowhbwidths\] the relationship between () and the FWHM of in the mean spectrum, (). The fit coefficients are given in the second row of Table 3. The relationship is far from linear (slope $= 0.535 \pm 0.042$), and the scatter $\varepsilon_y$ is larger than it is for the ()–() relationship, even after removal of the notable outliers. While it is clear that () is an excellent proxy for (), the value of () is less clear. Nevertheless we will fit both versions in order to understand the relative merits of each.
Single-Epoch Predictors of the Virial Product {#section:hbpredictor}
---------------------------------------------
In the previous subsections, we have re-established the correlations between $\tau({\mbox{\rm H$\beta$}})$ and $L({\mbox{\rm H$\beta$}}_{\rm broad})$ and between () and both () and (). As a first approximation for a formula to estimate single-epoch masses, we fit the following equations: $$\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}}) = a + b\left[\log L({\mbox{\rm H$\beta$}}_{\rm broad}) - x_0\right]
+c\left[\log{\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}) - y_0\right],
\label{eq:Fit_139}$$ and $$\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}}) = a + b\left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - x_0\right]
+c\left[ \log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}}) - y_0\right].
\label{eq:Fit_141}$$
The results of these fits based on the combined RMDB data (Table A1) and SDSS data (Table A3) are given in the first two rows of Table 4, and illustrated in the upper panels of Figure \[Figure:Fit\_139\_141\]. Using these coefficients, we have initial fits $$\begin{aligned}
\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm H$\beta$}}) & = &
6.975 + 0.566\left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.857\right] \nonumber \\
& & + 1.757\left[ \log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}) - 3.293\right],
\label{eq:SEsigm}\end{aligned}$$ and $$\begin{aligned}
\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm H$\beta$}}) & = &
6.981 + 0.587\left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.857\right] \nonumber \\
& & + 1.039\left[ \log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}}) - 3.599\right],
\label{eq:SEfwm}\end{aligned}$$ for () and (), respectively. The luminosity coefficient $b$ and the line-width coefficient $c$ are roughly as expected from the virial relationship and the $R$–$L$ relationship, and we note that the line-width coefficient for ($ c = 1.039$) is much smaller than that of ($c = 1.757$), as expected from Figure \[Figure:windowhbwidths\]. It is clear that both equations (\[eq:SEsigm\]) and (\[eq:SEfwm\]) overestimate masses at the low end and underestimate them at the high end, thus biasing the prediction. This suggests that another parameter is required for the single-epoch virial product prediction.
![The two upper panels show single-epoch (SE) virial product predictions based on equations (\[eq:Fit\_139\]) and (\[eq:Fit\_141\]) on the left and right, respectively, with coefficients from Table 4 compared with the actual RM measurements for the same sources. Blue filled circles represent RMDB data (Table A1) and green open triangles represent SDSS data (Table A3). The solid line shows the best fit to the data, and the red dotted line shows where the two values are equal. The short and long dashed lines show the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively. It is clear that this is an inadequate virial product predictor as it systematically underestimates higher masses and overestimates lower masses. The two lower panels show the same relationship after the empirical corrections as embodied in equations (\[eq:predictHbsigm\]) and (\[eq:predictHbFWHM\]) for and , respectively. The best fit lines cover the equality lines. []{data-label="Figure:Fit_139_141"}](Figure4.eps)
[lllccccccc]{} $\log L({\mbox{\rm H$\beta$}}_{\rm broad})$ & $\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}})$ & $\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}})$& $6.975 \pm 0.029$ & $0.566 \pm 0.035$ & $1.757 \pm 0.160$ & $41.857$ & $3.293$ & $0.273 \pm 0.025$ & $0.314$\
$\log L({\mbox{\rm H$\beta$}}_{\rm broad})$ & $\log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}})$ & $\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}})$ & $6.981 \pm 0.033$ & $0.587 \pm 0.040$ & $1.039\pm 0.128$ & $41.857$ & $3.559$ & $0.323 \pm 0.028$ & $0.352$\
$\log L(1350\,{\rm \AA})$ & $\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}})$ & $\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm C\,{\sc iv}}})$ & $7.664 \pm 0.039$ & $0.599 \pm 0.033$ & $1.014 \pm 0.265$ & $44.706$ & $3.502$ & $0.364 \pm 0.033$ & $0.397$ \[table:mfits\]
We investigated the possibility of another parameter by plotting the residuals $\Delta \log \mu =
\log {\mbox{$\mu_{\rm RM}$}}- \log {\mbox{$\mu_{\rm SE}$}}$ against other parameters, specifically luminosity, mass (virial product), Eddington ratio, emission-line lag, line width and line-width ratio ${\rm FWHM}/{\mbox{$\sigma_{\rm line}$}}$ for both mean and rms spectra. The most significant correlation between the virial product residuals and other parameters was for Eddington ratio, which has been a result of other recent investigations [@Du16; @Grier17b; @Du18; @Du19; @Alvarez19; @Martinez19]. To determine the Eddington ratio, we start with the Eddington luminosity $$\label{eq:Eddingtonlimit}
L_{\rm Edd} = \frac{4 \pi G c m_e M}{\sigma_e}
= 1.257 \times 10^{38} \left(\frac{M}{{\mbox{$M_\odot$}}}\right),$$ where $m_e$ is the electron mass and $\sigma_e$ is the Thomson cross-section. The black hole mass is $\log M = \log f + \log \mu$ and, as explained in the Appendix, we assume $\log f = 0.683 \pm 0.150$ [@Batiste17] so the Eddington luminosity is $$\label{eq:Ledd}
\log L_{\rm Edd} = \log f + 38.099 + \log {\mbox{$\mu_{\rm RM}$}}= 38.782 + \log {\mbox{$\mu_{\rm RM}$}}.$$ The bolometric luminosity can be obtained from the observed 5100Å AGN luminosity plus a bolometric correction $$\label{eq:bolometriclum}
\log L_{\rm bol} = \log L_{\rm AGN}(5100\,{\rm \AA})+ \log k_{\rm bol} .$$ We ignore inclination effects and, following [@Netzer19], we use $$\label{eq:bolometriccorrection}
\log k_{\rm bol} = 10 - 0.2\log L_{\rm AGN}(5100\,{\rm \AA}).$$ Since we are using $L({\mbox{\rm H$\beta$}}_{\rm broad})$ as a proxy for $L_{\rm AGN}(5100\,{\rm \AA})$, we substitute $L({\mbox{\rm H$\beta$}}_{\rm broad})$ for $L_{\rm AGN}(5100\,{\rm \AA})$ by fitting the luminosities in Table A1, yielding (see Table 2) $$\log L_{\rm AGN}(5100\,{\rm \AA}) = 43.396
+ 1.003\left[\log L({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.746\right],
\label{eq:LHbLAGN}$$ so we can write the bolometric luminosity as $$\log L_{\rm bol} = 44.717 + 0.802\left[\log({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.746 \right].
\label{eq:boldef}$$ The Eddington ratio is given by[^2] $$\log {\mbox{$\dot{m}$}}= \log L_{\rm bol} - \log L_{\rm Edd}.
\label{eq:Eddratio}$$ Using equations (\[eq:boldef\]) and (\[eq:Ledd\]), the Eddington ratio can then be written as $$\log \dot{m} = 5.935 + 0.802 \left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.746\right]
- \log {\mbox{$\mu_{\rm RM}$}}.
\label{eq:Eddalt}$$
[lllccccc]{} All & $\log \dot{m}$ & $\Delta \log \mu ({\mbox{$\sigma_{\rm M}$}})$ & $-0.010 \pm 0.022$ & $-0.422 \pm 0.045$ & $-0.951$ & $0.187 \pm 0.021$ & $0.246$\
All & $\log \dot{m}$ & $\Delta \log \mu ({\mbox{${\rm FWHM}_{\rm M}$}})$ & $-0.007 \pm 0.023$ & $-0.543 \pm 0.046$ & $-0.951$ & $0.191 \pm 0.021$ & $0.247$\
All & $\log \dot{m}$ & $\Delta \log \mu$ & $-0.049 \pm 0.026$ & $-0.557 \pm 0.048$ & $-1.155$ & $0.213 \pm 0.027$ & $0.282$\
All & $\log {\mbox{$\mu_{\rm RM}$}}$ & $\Delta \log \mu$ & $-0.012 \pm 0.026$ & $0.297 \pm 0.024$ & $7.481$ & $0.000 \pm 0.000$ & $0.139$\
All & $\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}},{\mbox{$\sigma_{\rm M}$}})$ & $\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm H$\beta$}},{\mbox{$\sigma_{\rm M}$}})$ & $7.025 \pm 0.025$ & $0.805 \pm 0.038$ & $7.041$ & $0.249 \pm 0.021$ & $0.279$\
All & $\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}},{\mbox{${\rm FWHM}_{\rm M}$}})$ & $\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm H$\beta$}},{\mbox{${\rm FWHM}_{\rm M}$}})$ & $7.012 \pm 0.028$ & $0.749 \pm 0.042$ & $7.007$ & $0.278 \pm 0.023$ & $0.290$\
All & $\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm C\,{\sc iv}}}) $ & $\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm C\,{\sc iv}}}) $ & $7.483 \pm 0.033$ & $0.787 \pm 0.041$ & $7.481$ & $0.321 \pm 0.028$ & $0.347$ \[table:residfits\]
To correct the single-epoch masses for Eddington ratio, we fit the equation $$\Delta \log \mu = \log {\mbox{$\mu_{\rm RM}$}}- \log {\mbox{$\mu_{\rm SE}$}}=
a + b(\log \dot{m} - x_0),
\label{eq:residuals}$$ and use this as a correction to our initial fits, equations (\[eq:SEsigm\]) and (\[eq:SEfwm\]). The best-fit parameters for and -based predictors of are given in Table 5 and shown in Figure \[Figure:windowhbresiduals\]. Combining the correction equation (\[eq:residuals\]) with the best-fit coefficients in Table 5 and equations (\[eq:SEsigm\]) and (\[eq:SEfwm\]) yield the corrected single-epoch masses $$\begin{aligned}
\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm H$\beta$}}) & = &
6.965 + 0.566\left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.857\right] \nonumber \\
& & + 1.757\left[ \log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}) - 3.293\right]
- 0.422\left[ \log {\mbox{$\dot{m}$}}+ 0.951 \right],
\label{eq:SEsigmcorr}\end{aligned}$$ and $$\begin{aligned}
\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm H$\beta$}}) & = &
6.974 + 0.587\left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.857\right] \nonumber \\
& & + 1.039\left[ \log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}}) - 3.599\right]
- 0.543 \left[\log {\mbox{$\dot{m}$}}+0.951 \right],
\label{eq:SEfwmcorr}\end{aligned}$$ for and , respectively.
Once the dependence on Eddington ratio is removed, the residuals do not appear to correlate with other properties. The intrinsic scatter about the final residuals is $0.197$dex for -based masses and $0.204$dex for -based masses.
![Upper left: fit to mass residuals (equation \[eq:residuals\]) vs. Eddington ratio $\dot{m}$ (equation \[eq:Eddalt\]) for single-epoch masses based on (). Upper right: fit to mass residuals vs. Eddington ratio $\dot{m}$ for single-epoch masses based on (). Lower panels: residuals after subtraction of the best fit in the panel above. The $\varepsilon_y$ scatter in the residuals is 0.197dex for the -based virial products and 0.204dex for the -based virial products. In all panels, the solid blue circles represent RMDB data (Table A1) and the open green triangles represent SDSS data (Table A3). The solid line shows the best fit to the data. The short dashed and long dashed lines are the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively. The coefficients of the fits are given in Table 5. Error bars are measurement uncertainties only, without systematic errors.[]{data-label="Figure:windowhbresiduals"}](Figure5.eps)
Masses Based on
================
Fundamental Relationships {#section:civfundamental}
-------------------------
As noted in §\[section:intro\], the veracity of -based mass estimates is unclear and remains controversial. The ideal situation would be to have a large number of AGNs with both and reverberation measurements to effect a direct comparison. There are, unfortunately, very few AGNs that have both; indeed Table A2 of the Appendix lists all results for which there are corresponding measurements in Table A1. For the few sources with both and reverberation measurements, we plot the virial products $\mu_{\rm RM}({\mbox{\rm C\,{\sc iv}}})$ and $\mu_{\rm RM}({\mbox{\rm H$\beta$}})$ in Figure \[Figure:RMcompare\]; these are in each case a weighted mean value of $$\label{eq:mudef}
\mu_{\rm RM} = \left( \frac{c \tau {\mbox{$\sigma_{\rm R}$}}^2}{G}\right)$$ for each of the observations of and for the AGNs that appear in both Tables A1 and A2. The close agreement of these values reassures us that the -based RM masses can be trusted, at least over the range of luminosities sampled.
![Virial products based on and for the few cases in the RMDB sample for which both are available. The solid line is the locus where the two virial products are equal. The values are weighted means of ${\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}})$ and ${\mbox{$\mu_{\rm RM}$}}({\mbox{\rm C\,{\sc iv}}})$ for individual AGNs that appear in both Tables A1 and A2. The Spearman rank coefficient for these data is $\rho = 0.805$. []{data-label="Figure:RMcompare"}](Figure6.eps)
We now need to consider whether or not luminosities and mean line widths are suitable proxies for emission-line lag and rms line widths in the case of . In Figure \[Figure:civradlum\], we show the relationship between the UV continuum luminosity $L(1350\,{\rm \AA})$ and the emission line lag $\tau({\mbox{\rm C\,{\sc iv}}})$ based on the data in Table A2, plus the SDSS-RM data in Table A4. The coefficients of the fit are given in Table 2. We note again that we have removed from the [@Grier19] sample in Table A4 three quasars with BALs, thus reducing the sample size from 48 to 45. The slope of the $R$–$L$ relation ($0.517$) is consistent with that of ($0.492$), though the $\varepsilon_y$ scatter is substantially greater ($0.336$dex for compared to $0.213$dex for ). Definition of the relationship does not depend on the two separate measurements of very short lag measurements for the dwarf Seyfert NGC 4395 [@Peterson05]. Thus it seems clear that we can use $L(1350\,{\rm \AA})$ as a reasonable proxy for $\tau({\mbox{\rm C\,{\sc iv}}})$.
![Relationship between the rest-frame emission-line lag $\tau({\mbox{\rm C\,{\sc iv}}})$ and the continuum luminosity at 1350Å. Blue filled circles represent RMDB data (Table A2) and green open triangles represent SDSS data (Table A4). The solid line is the best fit to the data using equation (\[eq:powerlaw\]) with coefficients given in Table 2. The short dashed and long dashed lines are the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively. The Spearman rank coefficient for these data is $\rho = 0.503$. If the two lowest luminosity points (both measurements of the dwarf Seyfert NGC4395) are omitted, the Spearman rank coefficient is decreased to $\rho = 0.481$. []{data-label="Figure:civradlum"}](Figure7.eps)
We show the relationship between the line dispersion measured in the rms spectrum () and the line dispersion in the mean spectrum () in Figure \[Figure:windowcivwidths\]. The best-fit coefficients are given in Table 3. The correlation is good. However, the correlation between () and (), also shown in Figure \[Figure:windowcivwidths\], is rather poor [see also @Wang20] and demonstrates that () is a dubious proxy for (). Measurement of () is clearly not a reliable predictor of (), so we will not consider () further.
![Left: Relationship between line dispersion in the mean and rms spectra of reverberation-mapped AGNs. The Spearman rank coefficient is $\rho = 0.873$. Right: Relationship between () and () for reverberation-mapped AGNs. The Spearman rank coefficient for these data is $\rho = 0.524.$ In both panels, blue filled circles represent RMDB sources in Table A2 and green open triangles represent SDSS-RM sources in Table A4. The red dotted line shows the locus where the two line-width measures are equal. The solid line is the best fit to equation (\[eq:powerlaw\]) and the coefficients are given in Table 3. The short dashed and long dashed lines show the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively.[]{data-label="Figure:windowcivwidths"}](Figure8.eps)
Single-Epoch Masses {#section:civ}
-------------------
Following the same procedures as with , we use the RMDB data (Table A2) and the SDSS-RM data (Table A4) to fit the equation $$\log {\mbox{$\mu_{\rm RM}$}}= a + b\left[ \log L(1350\,{\rm \AA}) - x_0 \right]
+ c \left[ \log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}}) - y_0 \right].
\label{eq:Fit_244}$$ The resulting fit is shown in Figure \[Figure:civvps\] and the best-fit coefficients are given in Table 4.
![Left: Comparison of single-epoch virial products () and reverberation measurements () for the data in Table A2 (blue filled circles), the SDSS-RM reverberation data from Table A4 (green open triangles), and data from Table A5 (red open circles). The solid line is the best fit to the data and has slope $0.787 \pm 0.041$. As was the case with , masses are overestimated at the low end and underestimated at the high end, excepting the three very low mass measurements. Right: Comparison of single-epoch virial products after empirical correction as given in equation (\[eq:predictciv\]). In both panels, the solid line is the best fit to equation (\[eq:Fit\_244\]). The short dashed and long dashed lines define the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively. The diagonal red dotted line is the locus where and are equal. []{data-label="Figure:civvps"}](Figure9.eps)
With the coefficients from this fit and equation (\[eq:Fit\_244\]), we can generate predicted virial masses ${\mbox{$\mu_{\rm SE}$}}({\mbox{\rm C\,{\sc iv}}})$. We compare the measured reverberation mass with the single-epoch prediction based on this fit in the left panel of Figure \[Figure:civvps\]. As was the case for (Figure \[Figure:Fit\_139\_141\]), the distribution of points is slightly skewed relative to the diagonal, and, guided by our result for , we plot the residuals in $\log {\mbox{$\mu_{\rm RM}$}}- \log {\mbox{$\mu_{\rm SE}$}}$ versus Eddington ratio in the upper left panel of Figure \[Figure:window244\_247\]. The Eddington ratio for the UV data is $$\log {\mbox{$\dot{m}$}}= -33.737 + 0.9 \log L(1350\,{\rm \AA}) - \log \mu_{\rm RM},
\label{eq:Edddefciv}$$ where again we have used a bolometric correction from $L(1350\,{\rm \AA})$ from [@Netzer19], $$\log k_{\rm bol} = 5.045 - 0.1 \log L(1350\,{\rm \AA}).
\label{eq:bolometricciv}$$ We fitted equation (\[eq:residuals\]) for and the coefficients of the fit are given in Table 5.
![Mass residuals $\Delta \log \mu =\log {\mbox{$\mu_{\rm RM}$}}- \log {\mbox{$\mu_{\rm SE}$}}$ versus Eddington rate (left column) and virial product (right column). The upper left panel shows the residuals between () and () versus Eddington ratio (equation \[eq:Eddratio\]). The upper right panel shows the residuals versus virial product . The middle panels show the residuals versus (left) and (right) after subtracting the fit in the upper left panel. The middle right panel shows a best fit to the residuals versus mass; coefficients are given in Table 5. Note that the intrinsic scatter in this relationship is $\epsilon_y = 0.000 \pm 0.000$ because the error bars are so large. The bottom panels show the mass residuals versus and after subtracting the fit in the middle right panel. The scatter in the bottom panels is 0.138dex. In all panels, the blue filled circles represent RMDB data (Table A2), the green open triangles are SDSS data (Table A4), and the red open circles are VP06 data (Table A5). Best fits are shown as solid lines and the short dashed and long dashed lines indicate the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes. []{data-label="Figure:window244_247"}](Figure10.eps)
The offset between the residuals in the upper left panel of Figure \[Figure:window244\_247\] between the RMDB and VP06 data on one hand and the SDSS data on the other might seem to be problematic and we were initially concerned that this might be a data integrity issue. However, upon examining the distribution of mass and luminosity for these three samples as seen in Figure \[Figure:masshist\], we see clearly that the mass distribution of the SDSS sources is skewed toward much higher values than for the RMDB and VP06 sources, which are relatively local and less luminous than the SDSS quasars. We will thus proceed by examining mass residuals versus both Eddington ratio and .
![Distribution in virial product for the RMDB (Table A2, blue solid line), SDSS (Table A3, green dotted line), and VP06 (Table A4, red solid line) samples. The VP06 sample is a subset of the RMDB sample, which is dominated by the relatively low-mass Seyfert galaxies that were the first sources studied by reverberation mapping. The SDSS quasars are comparatively more massive and more luminous. []{data-label="Figure:masshist"}](Figure11.eps)
Figure \[Figure:window244\_247\] illustrates the process by which we eliminate the mass residuals in successive iterations. We compute the mass residuals $\Delta \log \mu = \log {\mbox{$\mu_{\rm RM}$}}- \log {\mbox{$\mu_{\rm SE}$}}$ from equation (\[eq:Fit\_244\]); these are shown versus (left column) and ${\mbox{$\mu_{\rm RM}$}}$ (right column). We fit these residuals versus (top left) and subtract the best fit to equation (\[eq:residuals\]), whose coefficients are given in Table 5. We subtract this fit from the mass residuals to get the corrected residuals in the middle panels. Examination of these residuals as a function of other parameters revealed that they are still correlated with (middle right), suggesting that the importance of the Eddington ratio depends on the black hole mass. We therefore fit the residuals a second time, this time as $$\Delta \log \mu = a + b(\log {\mbox{$\mu_{\rm RM}$}}\ - x_0).
\label{eq:massresiduals}$$ The best fit to this equation is shown in the middle right panel and the coefficients are given in Table 5. Subtraction of the best fit yields the residuals shown in the bottom two panels. We would under most circumstances consider this procedure with some trepidation from a statistical point of view, since ${\mbox{$\mu_{\rm RM}$}}$ appears explicitly in one correction and is implicitly in the Eddington ratio. A generalized solution would have multiple degeneracies as both mass and luminosity appear in multiple terms. However, the residual corrections are physically motivated; several previous investigations have also concluded that Eddington ratio is correlated with the deviation from the [@Bentz13] $R$–$L$ relationship, and the middle panels of Figure \[Figure:window244\_247\] suggests that the impact of Eddington ratio varies slightly with mass. Nevertheless, one would prefer to work with parameters that are correlated with or indicators of and , as we will discuss in §[\[section:discussion\]]{}.
It is worth noting in passing that after correcting for Eddington ratio (Figure \[Figure:windowhbresiduals\]), the residuals in the -based mass estimates show no correlation with either mass or luminosity.
Computing Single-Epoch Masses {#section:massformulae}
=============================
To briefly reiterate our approach so far, we started with the assumption that ${\mbox{$\mu_{\rm SE}$}}= f(R,L)$ only. This proved to be inadequate, so we examined the residuals in the $\log {\mbox{$\mu_{\rm SE}$}}$–$\log {\mbox{$\mu_{\rm RM}$}}$ relationship and found that these correlated best with Eddington ratio : fundamentally, at increasing , the [@Bentz13] $R$–$L$ relationship overpredicts the size of the BLR $R$ [@Du19]. In the case of , we found additional residuals that correlated with , although we cannot definitively demonstrate that some part of this is not attributable to inhomogeneities in the data base (a point that will be pursued in the future). While we believe this analysis identifies the physical parameters that affect the mass estimates, there are multiple degeneracies, with both mass and luminosity appearing in more than one term.
Instead of trying to fit coefficients to all the physical parameters that have been identified, we can do a purely empirical correction to equations (\[eq:Fit\_139\]), (\[eq:Fit\_141\]), and (\[eq:Fit\_244\]) since the residuals in the $\log {\mbox{$\mu_{\rm RM}$}}$–$\log {\mbox{$\mu_{\rm SE}$}}$ relationships (upper panels in Figure \[Figure:Fit\_139\_141\] and left panel of Figure \[Figure:civvps\]) are rather small. We can combine the basic $R$–$L$ fits (equations \[eq:Fit\_139\], \[eq:Fit\_141\], and \[eq:Fit\_244\]) with the residual fits (equations \[eq:residuals\] and \[eq:massresiduals\]) to obtain prescriptions that work over the mass range sampled. Renormalizing for convenience, we can estimate single-epoch masses based on () from $$\begin{aligned}
\log M_{\rm SE} & = & \log f + 7.530
+ 0.703 \left[ \log L({\mbox{\rm H$\beta$}})-42 \right]
+ 2.183 \left[\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}) - 3.5 \right],
\label{eq:predictHbsigm}\end{aligned}$$ with associated uncertainty $$\label{eq:predictHbsigmerror}
\Delta \log M_{\rm SE} = \left\{ (\Delta \log f)^2 +
\left[0.703\ \Delta \log L({\mbox{\rm H$\beta$}}) \right]^2 +
\left[2.183\ \Delta \log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}})\right]^2 \right\}^{1/2}
\; .$$ Here $f$ is the scaling factor which is discussed briefly in the Appendix, and $\Delta \log P$ is the uncertainty in the parameter $\log P$. The intrinsic scatter in this relationship is $0.309$dex, and this must be added in quadrature to the random error. For the case of (), a single-epoch mass estimate is obtained from $$\begin{aligned}
\log M_{\rm SE} & = & \log f + 7.015 +
0.784 \left[ \log L({\mbox{\rm H$\beta$}})-42 \right]
+ 1.387 \left[\log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}}) - 3.5 \right],
\label{eq:predictHbFWHM}\end{aligned}$$ with associated uncertainty $$\label{eq:predictHbFWHMmerror}
\Delta \log M_{\rm SE} = \left\{ (\Delta \log f)^2 +
\left[0.784\ \Delta \log L({\mbox{\rm H$\beta$}}) \right]^2 +
\left[1.387\ \Delta \log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}})\right]^2 \right\}^{1/2} \;
.$$ In this case, the intrinsic scatter is $0.371$dex.
A comparison of the reverberation-based virial products () and the single-epoch masses () based on equations (\[eq:predictHbsigm\]) and (\[eq:predictHbFWHM\]) is shown in the lower two panels of Figure \[Figure:Fit\_139\_141\].
Similarly, single-epoch masses based on can be computed from $$\begin{aligned}
\log M_{\rm SE} & = & \log f +7.934
+ 0.761 \left[ \log L(1350\,{\rm \AA})-45 \right]
+ 1.289 \left[\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}}) - 3.5 \right],
\label{eq:predictciv}\end{aligned}$$ with associated uncertainty $$\label{eq:predictciverror}
\Delta \log M_{\rm SE} = \left\{ (\Delta \log f)^2+
\left[0.761\ \Delta \log L(1350\,{\rm \AA} \right]^2 +
\left[1.284\ \Delta \log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}})\right]^2 \right\}^{1/2}\;
.$$ The intrinsic scatter in this relationship is 0.408dex. Single-epoch predictions and reverberation-based masses for the AGNs in Tables A2, A4, and A5 are compared in the right panel of Figure \[Figure:civvps\].
In Figure \[Figure:zdistsdssselect\], we show the distribution in bolometric luminosity and black hole mass for the entire sample of SDSS-RM quasars for which or single-epoch masses can be estimated.
![Distribution of masses (upper panel) and bolometric luminosities (lower panel) for the entire SDSS-RM sample for which or single-epoch masses can be computed using equations (\[eq:predictHbsigm\]) and (\[eq:predictciv\]). Here we assume $f=4.28$ [@Batiste17]. Bolometric corrections were made using equations (\[eq:bolometriccorrection\]) and (\[eq:bolometricciv\]). On the left side, the quality cuts of §\[section:Data\] have been imposed. On the right side, no quality cuts have been made. []{data-label="Figure:zdistsdssselect"}](Figure12.eps)
Discussion {#section:discussion}
==========
Single-Epoch Masses {#single-epoch-masses}
-------------------
Our primary goal has been to find simple, yet unbiased, prescriptions for estimating the masses of the black holes that power AGNs. Our underlying assumption has been that the most accurate measure of the virial product is given by using the emission-line lag $\tau$ and line width in the rms spectrum (e.g., equation \[eq:explicitmu\] in the Appendix) as that quantity produces, upon adjusting by the scaling factor $f$, an $M_{\rm BH}$–$\sigma_*$ relationship for AGNs that is in good agreement with that for quiescent galaxies. Given that both $\tau$ and average over structure in a complex system [cf., @Barth15], it is somewhat surprising that this method of estimation works as well as it does.
Here we have shown that the broad component of the emission line is a good proxy for the starlight-corrected AGN luminosity (Figure \[Figure:HbRL\]). This is useful since it eliminates the difficult task of accurately modeling the host-galaxy starlight contribution to the continuum luminosity. Moreover, the line luminosity and reflect the BLR state at the same time; a measurement of the continuum luminosity, by contrast, better represents the state of the BLR at a time $\tau$ in the future on account of the light travel-time delay within the system [@Pogge92; @Gilbert03; @Barth15]; this is, however, generally a very small effect. For the sake of completeness, we also note that there is a small, but detectable, lag between continuum variations at shorter wavelengths and those at longer wavelengths [@McHardy14; @Shappee14; @Edelson15; @Fausnaugh16; @Edelson17; @McHardy18; @Edelson19].
We have also confirmed that, for the case of , both and are reasonable proxies for , though is somewhat better than .
On the other hand, the case of remains problematic, as it differs in a number of ways from the other strong emission lines:
1. The equivalent width of decreases with luminosity, which is known as the Baldwin Effect [@Baldwin77]; is driven by higher-energy photons than, say, the Balmer lines and the Baldwin Effect reflects a softening of the high-ionization continuum. This could be due to higher Eddington ratio [@Baskin04] or because more massive black holes have cooler accretion disks [@Korista98].
2. The emission line is typically blueshifted with respect to the systemic redshift of the quasar, which is attributed to outflow of the BLR gas [@Gaskell82; @Wilkes84; @Wilkes86; @Espey89; @Wills93; @Sulentic07; @Richards11; @Coatman16; @Shen16a; @Bisogni17; @Vietri18].
3. BALs in the short-wavelength wing of , another signature of outflow, are common [@Weymann91; @Hall02; @Hewett03; @Allen11]. We remind the reader that in §[\[section:Data\]]{} we removed $\sim17$% of our SDSS sample because the presence of BALs precludes accurate line-width measurements.
4. The pattern of “breathing” in is the opposite of what is seen in [@Wang20]. Breathing refers to the response of the emission lines, both lag and line width, to changes in the continuum luminosity. In the case of , an increase in luminosity produces an increase in lag and a decrease in line width [@Gilbert03; @Goad04; @Cackett06]. In the case of , however, the line width increases when the continuum luminosity increases, contrary to expectations from the virial theorem (equation \[eq:masseqn\]).
We must certainly be mindful that outflows can affect a mass measurement, though the effect is small if the gas is at escape velocity. Notably, in the cases studied to date there is good agreement between -based and -based virial products (Figure \[Figure:civradlum\]), though, again, these are local Seyfert galaxies that are not representative of the general quasar population.
The breathing issue is addressed in detail by [@Wang20], building on evidence for a non-reverberating narrow core or blue excess in the emission line presented by [@Denney12]. In this two-component model, the variable part of the line is much broader than the non-variable core. As the continuum brightens, the variable broad component increases in prominence, resulting in a larger value of . As the broad component reverberates in response to continuum variations, will track much better than , thus explaining the breathing characteristics and why is a poor line-width measure for estimating black hole masses. Physical interpretation of the non-varying core remains an open question: [@Denney12] suggests that it might be an optically thin disk wind or an inner extension of the narrow-line region.
The Role of Eddington Ratio
---------------------------
It is well known that there are strong correlations and anticorrelations among the UV-optical spectral features of AGNs as revealed by Principal Component Analysis (PCA) [@Boroson92; @Sulentic00; @Boroson02; @Shen14; @Sun15; @Marziani18 and references therein]. The strongest of these correlations, , is most clearly characterized by the anticorrelation between (a) the strength of the $\lambda4570$ and $\lambda\lambda5190$, 5320 complexes on either side of the broad complex and (b) the strength of the $\lambda\lambda 4959$, 5007 doublet. There is consensus in the literature that is driven by Eddington ratio; our own analysis supports this. The studies cited above have noted that an Eddington ratio correction is required for single-epoch masses based on . We find, as did [@Marziani19], that a similar correction is required for -based masses as well.
One extreme of is populated by sources with strong and very weak . The broad emission lines in the spectra of these objects also have relatively small line widths. By combining the $R$–$L$ relation with eq. (\[eq:masseqn\]), the line width dependence is seen to be $$\label{eq:linewidth}
V \propto\left( \frac{M}{L^{1/2}} \right)^{1/2} \propto
\left( \frac{M}{{\mbox{$\dot{m}$}}} \right)^{1/4},$$ where ${\mbox{$\dot{m}$}}\propto L/M$ is the Eddington ratio (eq. \[eq:Eddratio\]). Thus AGNs with the highest Eddington ratios have the smallest broad-line widths; many such sources are classified as “narrow-line Seyfert 1 (NLS1) galaxies” [@OsterbrockPogge85]. The Super-Eddington Accreting Massive Black Holes (SEAMBH) collaboration has focused on high- candidates in their reverberation-mapping program [@Du14; @Du16; @Du18; @Du19]. An important result from these studies, as we have noted earlier, is that the lags are smaller than predicted by the current state-of-the-art $R$–$L$ relationship [@Bentz13]. This implies that in these objects the ratio of hydrogen-ionizing photons to optical photons is lower than in the lower sources; this is also consistent with the relative strength of the low-ionization lines such as in SEAMBH sources, the weakness of high-ionization lines, such as , and their soft X-ray spectra [@Boller96]. [@Du19] choose to make their correction to the BLR radius through adding a term that correlates with the deficiency of ionizing photons. In our approach, we absorb the correction directly into the virial product computation.
As noted in §\[section:civ\], from a statistical point of view, it would be preferrable to replace the Eddington ratio with a parameter strongly correlated with it. The PCA studies referenced above find that the ratio of the equivalent widths (EW) or fluxes of to , ${\cal R} = {\rm EW}({\mbox{\rm Fe\,{\sc ii}}})/{\rm EW}({\mbox{\rm H$\beta$}})$, correlates well with Eddington ratio. In the UV, it is also found that the blueshift correlates with Eddington ratio [@Baskin05; @Coatman16; @Sulentic17]. However, we find that the scatter in these relationships is so large that any gain in the accuracy of black hole mass estimates is offset by a large loss of precision. We therefore elect at this time to focus on the empirical formulae given in §\[section:massformulae\].
Future Improvements
-------------------
While we believe our current single-epoch prescription for estimating quasar black hole masses is more accurate than previous prescriptions, we also recognize that there are additional improvements that can be made to improve both accuracy and precision, some of which we became aware of near the end of the current project. We intend to implement these in the future. Topics that we will investigate in the future include the following:
1. Replace those reverberation lag measurements made with the interpolated cross-correlation function [@Gaskell87; @White94; @Peterson98b; @Peterson04] with lag measurements and uncertainties from [JAVELIN]{} [@Zu11]. Recent tests [@Li19; @Yu20] show that while the [JAVELIN]{} and interpolation cross-correlation lags are generally consistent, the uncertainties predicted by [JAVELIN]{} are more reliable.
2. Utilize the expanded SDSS-RM database, which now extends over six years, not only to make use of additional lag detections, but to capitalize on the gains in $S/N$ that will increase the overall quality of the lag and line-width measurements and result in fewer rejections of poor data.
3. Expand the database in Table A1 with recent results and other previous results that we excluded because they did not have starlight-corrected luminosities.
4. Update the VP06 database used to produce Table A5. There are now additional reverberation-mapped AGNs with archived [*HST*]{} UV spectra. Some of the poorer data in Table A5 can be replaced with higher-quality spectra.
5. Consider use of other line-width measures that may correlate well with , but are less sensitive to blending in the wings. Mean absolute deviation is one such candidate.
6. Improve line-width measurements. There appear to be some systematic differences among the various data sets, probably due to different processes for measuring ; for example, the bottom panels of Figure \[Figure:window244\_247\] show that the SE mass estimates for the VP06 sample are slightly higher than those from SDSS (compare also the last two columns in Table A5). Work on deblending alogrithms would aid more precise measurement of , in particular.
Summary
=======
The main results of this paper are:
1. We confirm that the luminosity of the broad component of the emission line $L({\mbox{\rm H$\beta$}}_{\rm broad})$ is an excellent substitute for the AGN continuum luminosity $L_{\rm AGN}(5100\,{\rm \AA})$ for predicting the emission-line reverberation lag $\tau({\mbox{\rm H$\beta$}})$. It has the advantage of being easier to isolate than $L_{\rm AGN}(5100\,{\rm \AA})$, which requires an accurate estimate of the host-galaxy starlight contribution to the observed luminosity.
2. We confirm that the line dispersion of the broad component () and the full-width at half maximum for the broad component () in mean, or single-epoch, spectra are both reasonable proxies for the line dispersion of in the rms spectrum () for computing single-epoch virial products (). We find that () gives better results than (), but both are usable.
3. In the case of , we find that the line dispersion of the emission line () in the mean, or single-epoch, spectrum is a good proxy for the line dispersion in the rms spectrum () for estimating single-epoch virial products (). We find that (), however, does not track () well enough to be used as a proxy.
4. Although the $R$–$L$ relationship based on the continuum luminosity $L(1350\,{\rm \AA})$ and emission-line reverberation lag $\tau({\mbox{\rm C\,{\sc iv}}})$ is not as well defined as that for , the relationship appears to have a similar slope and it appears to be suitable for estimating virial products ().
5. We confirm for both and that combining the reverberation lag estimated from the luminosity with a suitable measurement of the emission-line width together introduces a bias where the high masses are underestimated and the low masses are overestimated. We confirm that the parameter that accounts for the systematic difference between reverberation virial product measurements and those estimated using only luminosity and line width is Eddington ratio. Increasing Eddington ratio causes the reverberation radius to shrink, suggesting a softening of the hydrogen-ionizing spectrum.
6. While the virial product estimate from combining luminosity and line width causes a systematic bias, the relationship between the reverberation virial product and the single-epoch estimate is still a power-law, but with a slope somewhat less than unity (upper panels of Figure \[Figure:Fit\_139\_141\], left panel of Figure \[Figure:civvps\]). We are therefore able to empirically correct this relationship to an unbiased estimator of by fitting the residuals and essentially rotating the power-law distribution to have a slope of unity (lower panels of Figure \[Figure:Fit\_139\_141\], right panel of Figure \[Figure:civvps\]). We present these empirical estimators for () and () in §[\[section:massformulae\]]{}.
EDB is supported by Padua University through grants DOR1715817/17, DOR1885254/18, and DOR1935272/19 and by MIUR grant PRIN 2017 20173ML3WW\_001. EDB and BMP are grateful for the hospitality of STScI early in this investigation. JVHS and KH acknowledge support from STFC grant ST/R000824/1. YS acknowledges support from an Alfred P. Sloan Research Fellowship and NSF grant AST-1715579. CJG, WNB, JRT, and DPS acknowledge support from NSF grants AST-1517113 and AST-1516784. KH acknowledges support from STFC grant ST/R000824/1. PBH acknowledges support from NSERC grant 2017-05983. YH acknowledges support from NASA through STScI grant HST-GO-15650. SW, LJ, and LCH acknowledge support from the National Science Foundation of China (11721303, 11890693, 11991052) and the National Key R&D Program of China (2016YFA0400702, 2016YFA0400703). MV gratefully acknowledges support from the Independent Research Fund Denmark via grant number DFF 8021-00130. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.
Database of Reverberation-Mapped AGNs {#database-of-reverberation-mapped-agns .unnumbered}
=====================================
Reverberation-mapped AGNs provide the fundamental data that anchor the AGN mass scale. We selected all AGNs from the literature (as of 2019 August) for which unsaturated host-galaxy images acquired with [[*HST*]{}]{}are available, since removal of the host-galaxy starlight contribution to the observed luminosity is critical to this calibration, and measurements of time lags. It is worth noting, however, that since our analysis shows that the broad flux is a useful proxy for the 5100Å continuum luminosity, this criterion is over-restrictive and we will avoid imposing it in future compilations. In many cases, there is more than one reverberation-mapping data set available in the literature. In a few cases, the more recent data were acquired to replace, say, a more poorly sampled data set or one for which the initial result was ambiguous for some reason. In other cases, there are multiple data sets of comparable quality for individual AGNs, and in these cases we include them all. The particularly well-studied AGN NGC 5548 has been observed many times and in some sense has served as a “control” source that provides our best information about the repeatability of mass measurements as the continuum and line widths show long-term (compared to reverberation time scales) variations.
The final reverberation-mapped sample for is given in Table A1. It consists of 98 individual time series for 50 individual low-redshift ($z <
0.3$) AGNs. They span a range of AGN luminosity $41.46 \leq \log
L(5100\,{\rm \AA}) \leq 45.81$, in . Luminosities have been corrected for Galactic absorption using extinction values on the NASA Extragalactic Database, which are based on the [@Schlafly11] recalibration of the [@Schlegel98] dust map. Line-width and time-delay measurements are in the rest-frame of the AGNs. Luminosity distances are based on redshift, except the cases noted by [@Bentz13], for which the redshift-independent distances quoted in that paper are used. For two of these sources, NGC 4051 and NGC 4151, we use preliminary Cepheid-based distances (M.M. Fausnaugh, private communication), and for NGC 6814, we use the Cepheid-based distance from [@Bentz19]. Individual virial products for these sources are easily computed using the time lags (Column 6) and line dispersion measurements (Column 12) and the formula $$\label{eq:explicitmu}
\mu = 0.1952\left( \frac{\tau({\mbox{\rm H$\beta$}})}{{\rm days}}\right)
\left( \frac{{\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})} {{\rm km\,s}^{-1}} \right)^2\,{\mbox{$M_\odot$}}.$$
Further conversion to mass requires multiplication by the virial factor $f$, i.e. $\log M = \log f + \log \mu$, a dimensionless factor that depends on the inclination, structure, and kinematics of the broad--emitting region — indeed, detailed modeling of 9 of these objects [@Pancoast14; @Grier17a] shows that $f$ depends most clearly on inclination [@Grier17a]. Since such models are available for only a very limited number of AGNs, it is more common to use a statistical estimate of a mean value of $f$ based on a secondary mass indicator, specifically the well-known $M_{\rm BH}$–$\sigma_*$ relationship [@Ferrarese00; @Gebhardt00; @Gultekin09], where $\sigma_*$ is the host-galaxy stellar bulge velocity dispersion. The required assumption is that the AGN $M_{\rm BH}$–$\sigma_*$ is identical to that of quiescent galaxies [@Woo13]. In fact, it is found that the $\mu$–$\sigma_*$ has a slope consistent with the $M_{\rm BH}$–$\sigma_*$ slope for quiescent galaxies [@Grier13], and the zero points disagree by only a multiplicative factor, which is taken to be $f$. Here we take $\langle \log f \rangle =
0.683 \pm 0.150$ [@Batiste17] where the error on the mean is $\Delta \log f = 0.030$ — this error must be propagated into the mass measurement error when comparing AGN reverberation-based masses to those based on other methods.
[lccccccccccc]{} \[table:RMDBhb\] Mrk335 &1 & 49156-49338& 0.02579& 109.5& $ 16.8^{+4.8}_{-4.2}$&$ 43.802 \pm 0.010$ &$ 43.703 \pm 0.013$ &$ 42.083\pm 0.010$ &$1792 \pm 3 $&$ 1380 \pm 6$ & $ 917 \pm 52$\
Mrk335 &1 & 49889-50118& 0.02579& 109.5& $ 12.5^{+6.6}_{-5.5}$&$ 43.861 \pm 0.010$ &$ 43.777 \pm 0.013$ &$ 42.124\pm 0.010$ &$1679 \pm 2 $&$ 1371 \pm 8$ & $ 948 \pm 113 $\
Mrk335 &1 & 55431-55569& 0.02579& 109.5& $ 14.3^{+0.7}_{-0.7}$&$ 43.791 \pm 0.007$ &$ 43.683 \pm 0.061$ &$ 41.940\pm 0.009$ &$1273 \pm 3 $&$1663 \pm 6$ & $ 1293 \pm 64 $\
Mrk1501 &2 & 55430-55568& 0.08934& 402.5& $ 12.6^{+3.9}_{-3.9}$&$ 44.314 \pm 0.011$ &$ 43.980 \pm 0.053$ &$ 42.719\pm 0.015$ &$3106 \pm 15 $&$3494 \pm35$ & $ 3321 \pm 107 $\
PG0026+129 &3 & 48545-51084& 0.14200& 653.1& $ 111.0^{+24.1}_{-28.3}$&$ 44.977 \pm 0.010$ &$ 44.911 \pm 0.011$ &$ 42.867\pm 0.016$ &$2544 \pm 56 $&$1738 \pm 100$ &$ 1773 \pm 285 $\
PG0052+251 &3 & 48461-51084& 0.15445& 751.9& $ 89.8^{+24.5}_{-24.1}$&$ 44.964 \pm 0.013$ &$ 44.791 \pm 0.020$ &$ 43.113\pm 0.016$ &$5008 \pm 73 $&$2167 \pm 30$ & $ 1783 \pm 86 $\
Fairall9 &4 & 49475-49743& 0.04702& 202.8& $ 17.4^{+3.2}_{-4.3}$&$ 44.224 \pm 0.007$ &$ 43.920 \pm 0.026$ &$ 42.393\pm 0.007$ &$5999 \pm 60 $&$ 2347\pm 16$ &$ 3787 \pm 197 $\
Mrk590 &1 & 48090-48323& 0.02639& 112.1& $ 20.7^{+3.5}_{-2.7}$&$ 43.842 \pm 0.010$ &$ 43.544 \pm 0.029$ &$ 41.855\pm 0.011$ &$2788 \pm 29 $&$ 1942 \pm 26$ & $ 789 \pm 74 $\
Mrk590 &1 & 48848-49048& 0.02639& 112.1& $ 14.0^{+8.5}_{-8.8}$&$ 43.666 \pm 0.011$ &$ 43.075 \pm 0.073$ &$ 41.522\pm 0.011$ &$3729 \pm 426$&$ 2168 \pm 30$ & $ 1935 \pm 52 $\
Mrk590 &1 & 49183-49338& 0.02639& 112.1& $ 29.2^{+4.9}_{-5.0}$&$ 43.743 \pm 0.010$ &$ 43.320 \pm 0.043$ &$ 41.690\pm 0.010$ &$2743 \pm 79 $&$1967 \pm 19$ & $ 1251 \pm 72 $\
Mrk590 &1 & 49958-50122& 0.02639& 112.1& $ 28.8^{+3.6}_{-4.2}$&$ 43.865 \pm 0.010$ &$ 43.589 \pm 0.026$ &$ 41.857\pm 0.010$ &$2500 \pm 43 $&$ 1880 \pm 19$ & $ 1201 \pm 130 $\
3C120 &1 & 47837-50388& 0.03301& 140.9& $ 38.1^{+21.3}_{-15.3}$&$ 44.078 \pm 0.012$ &$ 44.010 \pm 0.014$ &$ 42.306\pm 0.012$ &$2327 \pm 48 $&$ 1249\pm21$ & $ 1166 \pm 50 $\
3C120 &5 & 54726-54920& 0.03301& 140.9& $ 27.9^{+7.1}_{-5.9}$&$ 44.116 \pm 0.013$ &$ 44.094 \pm 0.013$ &$ 42.453\pm 0.012$ &$2386 \pm 52 $& $\ldots$ & $ 1689 \pm 68 $\
3C120 &2 & 55430-55569& 0.03301& 140.9& $ 25.9^{+2.3}_{-2.3}$&$ 43.993 \pm 0.012$ &$ 43.903 \pm 0.052$ &$ 42.298\pm 0.015$ &$1430 \pm 16 $&$1687 \pm 4$ & $ 1514 \pm 65 $\
Akn120 &1 & 48148-48344& 0.03271& 139.6& $ 47.1^{+8.3}_{-12.4}$&$ 44.254 \pm 0.010$ &$ 43.921 \pm 0.032$ &$ 42.553\pm 0.010$ &$6042 \pm 35 $&$1753 \pm 6$ & $ 1959 \pm 109 $\
Akn120 &1 & 49980-50175& 0.03271& 139.6& $ 37.1^{+4.8}_{-5.4}$&$ 44.131 \pm 0.010$ &$ 43.569 \pm 0.067$ &$ 42.390\pm 0.010$ &$6246 \pm 78 $&$1862 \pm 13$ & $ 1884 \pm 48 $\
MCG+08-11-011&6& 56639-56797& 0.02048& 86.6 & $ 15.72^{+0.50}_{-0.52}$&$ 43.574 \pm 0.009$ &$ 43.282 \pm 0.045$ &$ 41.706\pm 0.006$&$1159\pm 8$&$1681 \pm 2$ &$ 1466 \pm 143 $\
Mrk6 &7 & 49250-49872& 0.01881& 80.6 & $ 21.2^{+4.}_{-3.2}$&$ 43.576 \pm 0.009$ &$ 43.351 \pm 0.033$ &$ 41.591\pm 0.011$&$ \ldots $&$ 2813 \pm 13$ &$ 2836 \pm 48 $\
Mrk6 &7 & 49980-50777& 0.01881& 80.6 & $ 20.7^{+3.0}_{-2.4}$&$ 43.578 \pm 0.009$ &$ 43.354 \pm 0.033$ &$ 41.632\pm 0.010$&$ \ldots $&$2804 \pm 6$ &$ 2626 \pm 37 $\
Mrk6 &7 & 50869-51516& 0.01881& 80.6 & $ 20.5^{+5.6}_{-7.0}$&$ 43.523 \pm 0.011$ &$ 43.258 \pm 0.042$ &$ 41.584\pm 0.013$&$ \ldots $&$2808 \pm 14$ &$ 2626 \pm 37 $\
Mrk6 &7 & 51557-53356& 0.01881& 80.6 & $ 23.9^{+17.0}_{-7.3}$&$ 43.431 \pm 0.007$ &$ 43.070 \pm 0.058$ &$ 41.449\pm 0.018$&$ \dots $&$2870 \pm 13$ &$ 3222 \pm 39 $\
Mrk6 &7 & 53611-54804& 0.01881& 80.6 & $ 20.4^{+4.6}_{-4.1 }$&$ 43.613 \pm 0.005$ &$ 43.413 \pm 0.027$ &$ 41.579\pm 0.012$&$ \ldots $&$2807\pm 8$&$ 2864 \pm 35 $\
Mrk6 &2 & 55340-55569& 0.01881& 80.6 & $ 10.1^{+1.1}_{-1.1}$&$ 43.719 \pm 0.008$ &$43.507 \pm 0.029$ &$ 41.849\pm 0.012$&$2619 \pm 24 $& $4006 \pm 6$ &$ 3714 \pm 68 $\
Mrk79 &1 & 47838-48044& 0.02219& 94.0 & $ 9.0^{+8.3}_{-7.8}$&$ 43.668 \pm 0.011$ &$ 43.569 \pm 0.014$ &$ 41.818\pm 0.011$&$5056 \pm 85 $&$2314 \pm 23$ & $ 2137 \pm 375 $\
Mrk79 &1 & 48193-48393& 0.02219& 94.0 & $ 16.1^{+6.6}_{-6.6}$&$ 43.754 \pm 0.010$ &$ 43.675 \pm 0.012$ &$ 41.851\pm 0.010$&$4760 \pm 31$ & $ 2281 \pm 26$ & $ 1683 \pm 72 $\
Mrk79 &1 & 48905-49135& 0.02219& 94.0 & $ 16.0^{+6.4}_{-5.8}$&$ 43.695 \pm 0.010$ &$ 43.602 \pm 0.013$ &$ 41.820\pm 0.010$&$4766 \pm 71$ & $2312 \pm 21$ &$ 1854 \pm 72 $\
Mrk374 &6 & 56663-56795& 0.04263& 183.3& $ 14.84^{+5.76}_{-3.30}$&$ 43.994 \pm 0.009$ &$ 43.752 \pm 0.036$ &$ 41.764\pm 0.013$&$3250 \pm 19 $ &$1490 \pm 4$ &$ 1329 \pm 373 $\
PG0804+761 &3 & 48319-51085& 0.10000& 447.5& $ 146.9^{+18.8}_{-18.9}$&$ 44.905 \pm 0.011$ &$ 44.849 \pm 0.011$ &$ 43.230\pm 0.012$&$3053 \pm 38 $&$1434 \pm 18$ &$ 1971 \pm 105 $\
NGC2617 &6 & 56639-56797& 0.01421& 59.8 & $ 4.32^{+1.1}_{-1.35}$&$ 43.099 \pm 0.011$ &$ 42.610 \pm 0.096$ &$ 41.173\pm 0.012$&$5303 \pm 48$&$2709 \pm 6$ &$ 2424 \pm 89 $\
Mrk704 &8 & 55932-55980& 0.02923& 124.5& $ 12.65^{+1.49}_{-2.14}$&$ 43.708 \pm 0.005$ &$ 43.517 \pm 0.025$ &$ 41.800\pm 0.007$&$3502 \pm 31$ &$ 2650 \pm 4$ & $ 1860 \pm 120 $\
Mrk110 &1 & 48953-49149& 0.03529& 150.9& $ 24.3^{+5.5}_{-8.3}$&$ 43.711 \pm 0.011$ &$43.618 \pm 0.014$ &$ 42.055\pm 0.011$&$1543 \pm 5$&$962 \pm 15$ &$ 1196 \pm 141 $\
Mrk110 &1 & 49751-49874& 0.03529& 150.9& $ 20.4^{+10.5}_{-6.3}$&$ 43.771 \pm 0.010$ &$43.691 \pm 0.012$ &$ 41.960\pm 0.010$&$1658 \pm 3 $ & $953 \pm 10$ &$ 1115 \pm 103 $\
Mrk110 &1 & 50010-50262& 0.03529& 150.9& $ 33.3^{+14.9}_{-10.0}$&$ 43.594 \pm 0.012$ &$ 43.468 \pm 0.017$ &$ 41.905\pm 0.012$&$1600 \pm 39$&$987 \pm 18$ &$ 755 \pm 29 $\
Mrk110 &9 & 51495-51678& 0.03529& 150.9& $ 23.4^{+3.6}_{-3.2}$&$ 43.340 \pm 0.007$ &$ 43.225 \pm 0.011$ &$ 41.769\pm 0.007$&$ \ldots $ & $\ldots$ &$ \ldots$\
PG0953+414 &3 & 48319-50997& 0.23410& 1137.2& $ 150.1^{+21.6}_{-22.6}$&$ 45.193 \pm .010$ &$ 45.126 \pm 0.011$ &$ 43.390\pm 0.012$&$3071 \pm 27$ &$ 1659 \pm 31$ &$ 1306 \pm 144 $\
NGC3227 &10& 54184-54269& 0.00386& 23.7 & $ 3.75^{+0.76}_{-0.82}$&$ 42.629 \pm 0.035$ &$ 42.243 \pm 0.068$ &$ 40.387\pm 0.035$&$3972 \pm 25$ &$ 1749 \pm 4$ &$ 1376 \pm 44 $\
NGC3227 &8 & 55933-56048& 0.00386& 23.7 & $ 1.29^{+1.56}_{-1.27 }$&$42.757 \pm 0.006$ &$ 42.424 \pm 0.051$ &$ 40.487\pm 0.010$&$1602 \pm 2$ &$1402 \pm 2$ &$ 1368 \pm 38 $\
Mrk142 &11& 54506-54618& 0.04494& 193.5& $ 2.74^{+0.73}_{-0.83}$&$ 43.709 \pm 0.010$ &$ 43.543 \pm 0.015$ &$ 41.639\pm 0.010$&$1462 \pm 2 $ &$ 1116 \pm 22$ &$ 859 \pm 102 $\
Mrk142 &12& 56237-56413& 0.04494& 193.5& $ 6.4^{+0.8}_{-2.2}$&$ 43.610 \pm 0.010$ &$ 43.443 \pm 0.016$ &$ 41.586\pm 0.010$&$1647 \pm 69 $& $\ldots$ & $ \ldots $\
NGC3516 &14,15& 54181-54300& 0.00884& 37.1 & $ 11.68^{+1.02}_{-1.53}$&$43.299 \pm 0.055$ &$ 42.726 \pm 0.133$ &$ 40.995\pm 0.057$&$5236 \pm 12$&$ 1584 \pm 1$ &$ 1591 \pm 10 $\
NGC3516 &8 & 55932-56072& 0.00884& 37.1 & $ 5.74^{+2.26}_{-2.04}$&$ 43.272 \pm 0.007$ &$ 42.529 \pm 0.196$ &$ 41.022\pm 0.008$&$3231 \pm 14 $ & $ 2633 \pm 3 $&$ 2448 \pm 69 $\
SBS1116+583A&11& 54550-54618& 0.02787& 118.5& $ 2.31^{+0.62}_{-0.49 }$&$ 42.995 \pm 0.021$ &$ 42.076 \pm 0.224$ &$ 40.788\pm 0.015$&$3668 \pm 186$ &$ 1552 \pm 36$ &$ 1528 \pm 184 $\
Arp151 &11,13& 54506-54618& 0.02109& 89.2 & $ 3.99^{+0.49}_{-0.68}$&$ 42.979 \pm 0.010$ &$ 42.497 \pm 0.047$ &$ 40.931\pm 0.011$&$3098 \pm 69$&$2006 \pm 24$ &$ 1252 \pm 46 $\
NGC3783 &14,15& 48607-48833& 0.00973& 25.1 & $ 10.2^{+3.3}_{-2.3}$&$ 42.791 \pm 0.025$ &$ 42.559 \pm 0.051$ &$ 41.009\pm 0.021$&$3770 \pm 68$&$1691 \pm 19 $&$ 1753 \pm 141 $\
Mrk1310 &11& 54550-54618& 0.01956& 82.7 & $ 3.66^{+0.59}_{-0.61 }$&$ 42.937 \pm 0.018$ &$ 42.231 \pm 0.120$ &$ 40.646\pm 0.012$&$2409 \pm 24$&$ 1209 \pm 42 $&$ 755 \pm 138 $\
NGC4051 &16& 54180-54311& 0.00234& 15.0 & $ 1.87^{+0.54}_{-0.50}$&$42.290 \pm 0.015$ &$ 41.847 \pm 0.080$ &$ 40.079\pm 0.018$&$799 \pm 2 $ & $ 1045 \pm 4$ &$ 927 \pm 64 $\
NGC4051 &6 & 56645-56864& 0.00234& 15.0 & $ 2.87^{+0.86}_{-1.33}$&$42.265 \pm 0.005$ &$ 41.732 \pm 0.106$ &$ 39.882\pm 0.012$&$765 \pm 3 $ &$470 \pm 2 $&$ 493 \pm 35 $\
NGC4151 &17& 53430-53472& 0.00332& 15.0 & $ 6.59^{+1.12}_{-0.76}$&$42.549 \pm 0.012$ &$ 42.004 \pm 0.113$ &$ 40.499\pm 0.013$&$5840 \pm 863 $&$ 6158 \pm 47 $&$ 2680 \pm 64 $\
NGC4151 &6 & 55931-56072& 0.00332& 15.0 & $ 6.82^{+0.48}_{-0.57}$&$42.685 \pm 0.007$ &$ 42.315 \pm 0.060$ &$ 40.956\pm 0.008$&$992 \pm 4 $& $1833 \pm 2 $&$ 1894 \pm 9 $\
Mrk202 &11& 54550-54617& 0.02102& 88.9 & $ 3.05^{+1.73}_{-1.12}$&$42.946 \pm 0.016$ &$ 42.198 \pm 0.126$ &$ 40.477\pm 0.010$&$1471 \pm 18 $&$ 867 \pm 40 $&$ 659 \pm 65 $\
NGC4253 &11& 54509-54618& 0.01293& 54.4 & $ 6.16^{+1.63}_{-1.22}$&$42.948 \pm 0.012$ &$ 42.509 \pm 0.044$ &$ 40.873\pm 0.010$&$1609 \pm 39 $&$ 1088 \pm 37 $&$ \ldots $\
PG1226+023 &3 & 48361-50997& 0.15834& 737.7& $ 306.80^{+68.5}_{-90.9}$&$45.935\pm 0.011$ &$ 45.907 \pm 0.011$ &$ 44.072\pm 0.014$&$3509 \pm 36$ &$ 1778 \pm 17 $&$ 1777 \pm 150 $\
3C273 &18& 54795-58194& 0.15834& 737.7& $146.3^{+8.3}_{-12.1}$& $45.864 \pm 0.011$ & $45.848 \pm 0.011$ &$44.056 \pm 0.010$&$3256 \pm 36$ & $1701 \pm 15$ &$1090 \pm 121 $\
PG1229+204 &3 & 48319-50997& 0.06301& 274.9& $ 37.8^{+27.6}_{-15.3}$&$44.053 \pm 0.010$ &$ 43.636 \pm 0.040$ &$ 42.275\pm 0.011$&$3828 \pm 54 $& $1608 \pm 24 $&$ 1385 \pm 111 $\
NGC4593 &19& 53391-53580& 0.00900& 37.7 & $ 3.73^{+0.75}_{-0.75}$&$43.242 \pm 0.013$ &$ 43.005 \pm 0.035$ &$ 41.237\pm 0.013$&$5143 \pm 16 $&$1790 \pm 3 $&$ 1561 \pm 55 $\
NGC4748 &11& 54550-54618& 0.01463& 61.6 & $ 5.55^{+1.62}_{-2.22}$&$43.072 \pm 0.012$ &$ 42.557 \pm 0.060$ &$ 41.047\pm 0.010$&$1947 \pm 66$ & $ 1009 \pm 27$ &$ 657 \pm 91 $\
PG1307+085 &3 & 48319-51042& 0.15500& 718.7& $ 105.6^{+36.0}_{-46.6}$&$44.849 \pm 0.012$ &$ 44.790 \pm 0.013$ &$ 43.096\pm 0.020$&$5059 \pm 133 $ & $ 1963 \pm 47 $&$ 1820 \pm 122 $\
MCG-06-30-15&20& 55988-56079& 0.00775& 25.5 & $ 5.33^{+1.86}_{-1.75}$&$42.393 \pm 0.009$ &$ 41.651 \pm 0.197$ &$ 39.793\pm 0.011$&$1958 \pm 75 $ &$ 976 \pm 8$ &$ 665 \pm 87 $\
NGC5273 &21& 56774-56838& 0.00362& 15.3 & $ 2.21^{+1.19}_{-1.60}$&$42.000 \pm 0.009$ &$ 41.465 \pm 0.106$ &$ 39.702\pm 0.010$&$5688 \pm 163 $&$ 1821 \pm 53 $&$ 1544 \pm 98 $\
Mrk279 &22& 50095-50289& 0.03045& 129.7& $ 16.7^{+3.9}_{-3.9}$&$43.882 \pm 0.021$ &$ 43.643 \pm 0.036$ &$ 42.242\pm 0.021$&$5354 \pm 32$&$ 1823 \pm 11 $&$ 1420 \pm 96 $\
PG1411+442 &3& 48319-51038& 0.08960& 398.2& $ 124.3^{+61.0}_{-61.7}$&$44.603 \pm 0.012$ &$ 44.502 \pm 0.014$ &$ 42.792\pm 0.014$&$2801 \pm 43 $&$1774 \pm 29$ &$ 1607 \pm 169 $\
NGC5548&23,24,25&47509-47809& 0.01718& 72.5 & $ 19.7^{+1.5}_{-1.5}$&$ 43.534 \pm 0.021$ &$ 43.328 \pm 0.042$ &$ 41.728\pm 0.018$&$4674 \pm 63$&$ 1934 \pm 5 $&$ 1687 \pm 56 $\
NGC5548 &24,25& 47861-48179& 0.01718& 72.5 & $ 18.6^{+2.1}_{-2.3}$&$ 43.390 \pm 0.029$ &$ 43.066 \pm 0.068$ &$ 41.546\pm 0.029$&$5418 \pm 107$ &$ 2227 \pm 20 $&$ 1882 \pm 83 $\
NGC5548 &24,26& 48225-48534& 0.01718& 72.5 & $ 15.9^{+2.9}_{-2.5}$&$ 43.496 \pm 0.017$ &$ 43.264 \pm 0.042$ &$ 41.645\pm 0.026$&$5236 \pm 87$ &$ 2205 \pm 16 $&$ 2075 \pm 81 $\
NGC5548 &24,26& 48623-48898& 0.01718& 72.5 & $ 11.0^{+1.9}_{-2.0}$&$ 43.360 \pm 0.020$ &$ 42.999 \pm 0.070$ &$ 41.457\pm 0.030$&$5986 \pm 95$&$ 3110 \pm 53 $&$ 2264 \pm 88 $\
NGC5548 &24,27& 48954-49255& 0.01718& 72.5 & $ 13.0^{+1.6}_{-1.4}$&$ 43.497 \pm 0.016$ &$ 43.267 \pm 0.040$ &$ 41.691\pm 0.016$&$5930 \pm 42$&$ 2486 \pm 13 $&$ 1909 \pm 129 $\
NGC5548 &24,28& 49309-49636& 0.01718& 72.5 & $ 13.4^{+3.8}_{-4.3}$&$ 43.509 \pm 0.022$ &$ 43.287 \pm 0.043$ &$ 41.649\pm 0.022$&$7378 \pm 39 $&$ 2877 \pm 17 $&$ 2895 \pm 114 $\
NGC5548 &24,28& 49679-50008& 0.01718& 72.5 & $ 21.7^{+2.6}_{-2.6}$&$ 43.604 \pm 0.012$ &$ 43.436 \pm 0.026$ &$ 41.746\pm 0.013$&$6946 \pm 79$&$ 2432 \pm 13 $&$ 2247 \pm 134 $\
NGC5548 &24,28& 50044-50373& 0.01718& 72.5 & $ 16.4^{+1.2}_{-1.1}$&$ 43.527 \pm 0.020$ &$ 43.317 \pm 0.039$ &$ 41.656\pm 0.018$&$6623 \pm 93 $&$ 2276 \pm 15 $&$ 2026 \pm 68 $\
NGC5548 &24,29& 50434-50729& 0.01718& 72.5 & $ 17.5^{+2.0}_{-1.6}$&$ 43.413 \pm 0.018$ &$ 43.113 \pm 0.054$ &$ 41.622\pm 0.015$&$6298 \pm 65 $&$ 2178 \pm 12$ &$ 1923 \pm 62 $\
NGC5548 &24,29& 50775-51085& 0.01718& 72.5 & $ 26.5^{+4.3}_{-2.2}$&$ 43.620 \pm 0.020$ &$ 43.459 \pm 0.032$ &$ 41.762\pm 0.018$&$6177 \pm 36 $&$ 2035 \pm 11 $&$ 1732 \pm 76 $\
NGC5548 &24,29& 51142-51456& 0.01718& 72.5 & $ 24.8^{+3.2}_{-3.0}$&$ 43.565 \pm 0.017$ &$ 43.376 \pm 0.034$ &$ 41.719\pm 0.016$&$6247 \pm 57 $&$ 2021 \pm 18 $&$ 1980 \pm 30 $\
NGC5548 &24,29& 51517-51791& 0.01718& 72.5 & $ 6.5^{+5.7}_{-3.7}$&$ 43.327 \pm 0.019$ &$ 42.918 \pm 0.081$ &$ 41.521\pm 0.017$&$6240 \pm 77 $&$2010 \pm 30 $& $ 1969 \pm 48 $\
NGC5548 &24,29& 51878-52174& 0.01718& 72.5 & $ 14.3^{+5.9}_{-7.3}$&$ 43.321 \pm 0.027$ &$ 42.903 \pm 0.089$ &$ 41.428\pm 0.026$&$6478 \pm 108 $&$ 3111 \pm 131 $&$ 2173 \pm 89 $\
NGC5548 &24,30& 53432-53472& 0.01718& 72.5 & $ 6.3^{+2.6}_{-2.3}$&$ 43.263 \pm 0.016$ &$ 42.526 \pm 0.211$ &$ 40.967\pm 0.017$&$6396 \pm 167$&$3210 \pm 642$ & $ 2388 \pm 373 $\
NGC5548 &10,24& 54180-54332& 0.01718& 72.5 & $ 12.4^{+2.7}_{-3.9}$&$ 43.287 \pm 0.008$ &$ 42.665 \pm 0.140$ &$ 40.660\pm 0.070$&$12575\pm 47$&$4736 \pm 23 $&$ 1822 \pm 35 $\
NGC5548 &11,24& 54508-54618& 0.01718& 72.5 & $ 4.18^{+0.86}_{-1.30}$&$43.214 \pm 0.010$ &$ 42.621 \pm 0.129$ &$ 41.157\pm 0.017$&$12771 \pm 71$& $ 4266 \pm 65 $&$ 4270 \pm 292 $\
NGC5548 &8,24 & 55931-56072& 0.01718& 72.5 & $ 2.83^{+0.88}_{-0.90}$&$43.433 \pm 0.005$ &$ 43.070 \pm 0.058$ &$ 41.543\pm 0.010$&$10587 \pm 82$&$ 3056 \pm 4 $&$ 2772 \pm 34 $\
NGC5548 &31 & 56663-56875& 0.01718& 72.5 & $ 4.17^{+0.36}_{-0.36}$&$43.612 \pm 0.003$ &$ 43.404 \pm 0.027$ &$ 41.666\pm 0.004$&$9496 \pm 418$&$ 3691 \pm 162 $&$ 4278 \pm 671 $\
NGC5548 &32& 57030-57236& 0.01718& 72.5 & $ 7.18^{+1.38}_{-0.70}$&$43.175 \pm 0.005$ &$ 42.787 \pm 0.063$ &$ 41.630\pm 0.003$&$9912 \pm 362 $&$ 3350 \pm 272 $&$ 3124 \pm 302 $\
PG1426+015 &3 & 48334-51042& 0.08657& 383.9& $ 95.0^{+29.9}_{-37.1}$&$44.690 \pm 0.012$ &$ 44.568 \pm 0.019$ &$ 42.764\pm 0.015$&$7113 \pm 160 $&$ 2906 \pm 80$&$ 3442 \pm 308 $\
Mrk817 &1 & 49000-49212& 0.03146& 134.2& $ 19.0^{+3.9}_{-3.7}$&$ 43.848 \pm 0.010$ &$ 43.726 \pm 0.015$ &$ 42.010\pm 0.010$&$4711 \pm 78$&$ 1984\pm 8 $&$ 1392 \pm 78 $\
Mrk817 &1 & 49404-49528& 0.03146& 134.2& $ 15.3^{+3.7}_{-3.5}$&$ 43.761 \pm 0.087$ &$ 43.608 \pm 0.124$ &$ 41.936\pm 0.089$&$5237 \pm 67 $&$ 2098 \pm 13 $&$ 1971 \pm 96 $\
Mrk817 &1 & 49752-49924& 0.03146& 134.2& $ 33.6^{+6.5}_{-7.6}$&$ 43.762 \pm 0.009$ &$ 43.609 \pm 0.016$ &$ 41.860\pm 0.010$&$4767 \pm 72 $&$ 2195 \pm 16 $&$ 1729 \pm 158 $\
Mrk817 &10& 54185-54301& 0.03146& 134.2& $ 14.04^{+3.41}_{-3.47}$&$43.901 \pm 0.006$ &$ 43.776 \pm 0.010$ &$ 41.710\pm 0.016$&$5906 \pm 34$&$2365 \pm 9 $&$ 2025 \pm 5 $\
Mrk290 &10& 54180-54321& 0.02958& 126.0& $ 8.72^{+1.21}_{-1.02}$&$ 43.451 \pm 0.028$ &$ 43.157 \pm 0.036$ &$ 41.747\pm 0.030$&$4521 \pm 24 $&$ 2071 \pm 24 $&$ 1609 \pm 47 $\
PG1613+658 &3 & 48397-51073& 0.12900& 588.4& $ 40.1^{+15.0}_{-15.2}$&$ 44.948 \pm 0.010$ &$ 44.713 \pm 0.019$ &$ 42.943\pm 0.014$&$9074 \pm 103 $&$ 3084 \pm 33$&$ 2547 \pm 342 $\
PG1617+175 &3 & 48362-51085& 0.11244& 507.4& $ 71.5^{+29.6}_{-33.7}$&$ 44.445 \pm 0.011$ &$ 44.330 \pm 0.014$ &$ 42.682\pm 0.023$&$6641 \pm 190 $&$ 2313 \pm 69$&$ 2626 \pm 211 $\
PG1700+518 &3 & 48378-51084& 0.29200& 1463.3& $ 251.8^{+45.9}_{-38.8}$&$45.600 \pm 0.010$ &$ 45.528 \pm 0.011$ &$ 43.717\pm 0.020$&$2252\pm 85$&$ 3160 \pm 93 $&$ 1700 \pm 123 $\
3C382 &6 & 56679-56864& 0.05787& 251.5& $ 40.49^{+8.02}_{-3.74}$&$44.193 \pm 0.008$ &$ 43.792 \pm 0.069$ &$ 42.264\pm 0.011$&$3619\pm 203 $&$ 3227 \pm 7$&$ 4552 \pm 190 $\
3C390.3 &33& 49718-50012& 0.05610& 243.5& $ 23.60^{+6.2}_{-6.7}$&$43.902 \pm 0.018$ &$ 43.620 \pm 0.039$ &$ 42.222\pm 0.015$&$12694 \pm 13 $&$ 3744 \pm 42$ &$ 3105 \pm 81 $\
3C390.3 &34& 50100-54300& 0.05610& 243.5& $ 97.0^{+17.0}_{-17.0}$&$44.028 \pm 0.016$ &$ 43.913 \pm 0.020$ &$ 42.287\pm 0.021$&$11918\pm 325$& $\ldots $&$ \ldots $\
3C390.3 &35& 53631-53714& 0.05610& 243.5& $ 46.4^{+3.8}_{-3.2}$&$ 44.485 \pm 0.007$ &$ 44.434 \pm 0.008$ &$ 42.695\pm 0.012$&$13211\pm 278 $&$ 5377 \pm 37 $&$ 5455 \pm 278 $\
NGC6814 &11& 54545-54618& 0.00521& 21.6 & $ 6.64^{+0.87}_{-0.90}$&$ 42.500 \pm 0.017$ &$ 42.058 \pm 0.057$ &$ 40.443\pm 0.010$&$3323 \pm 7 $&$1918 \pm 36 $&$ 1610 \pm 108 $\
Mrk509 &1 & 47653-50374& 0.03440& 147.0& $ 79.6^{+6.1}_{-5.4}$&$ 44.240 \pm 0.027$ &$ 44.130 \pm 0.028$ &$ 42.545\pm 0.027$&$3015 \pm 2 $&$ 1555 \pm 7 $&$ 1276 \pm 28 $\
PG2130+099 &36& 54352-54450& 0.06298& 274.7& $ 22.9^{+4.7}_{-4.6}$&$ 44.406 \pm 0.012$ &$ 44.368 \pm 0.012$ &$ 42.667\pm 0.011$&$2853 \pm 39 $&$ 1485 \pm 15 $&$ 1246 \pm 222 $\
PG2130+099 &2 & 55430-55557& 0.06298& 274.7& $ 9.6^{+1.2}_{-1.2}$&$ 44.237 \pm 0.032$ &$ 44.150 \pm 0.033$ &$ 42.584\pm 0.033$&$1781 \pm 5 $&$ 1769\pm 2 $ &$ 1825 \pm 65 $\
NGC7469 &37& 55430-55568& 0.01632& 68.8 & $ 10.8^{+3.4}_{-1.3}$&$ 43.768 \pm 0.009$ &$ 43.444 \pm 0.051$ &$ 41.557\pm 0.013$&$4369 \pm 6 $&$ 1095 \pm 5 $&$ 1274 \pm 126 $ \[table:hbrm\]
[lccccccccc]{} DESJ003-42&1 &56919-57627 & 2.593 & 20723 & $123^{+43}_{-42}$ &$46.510 \pm 0.020$& $4944 \pm 93$ & $3917 \pm 29 $& $6250 \pm 64$\
Fairall9 &2,3 & 49473-49713 &0.04702&202.8&$29.6^{+12.9}_{-14.4}$ &$44.530 \pm 0.030$ &$2968 \pm 37 $& $3068 \pm 27$ & $3201 \pm 285 $\
DESJ228-04&1&56919-57627 & 1.905 & 1686.4 &$95^{+16}_{-23}$ & $46.430 \pm 0.098$ & $5232 \pm 57$ &$3932 \pm 22$ & $6365 \pm 66$\
CT286 &4 &54821-57759& 2.556 & 20,366 &$459^{+71}_{-92}$ & $46.798 \pm 0.009$ & $6256$ & $\ldots$ &$\ldots$\
CT406 &4 &54355-57605& 3.183 & 26,533 &$115^{+64}_{-86}$&$46.910 \pm0.040$ & $6236$ &$\ldots$ &$\ldots$\
NGC3783 &5,3 & 48611-48833 &0.00973&25.1 &$3.8^{+1.0}_{-0.9}$ &$43.081 \pm 0.017$ &$2784 \pm 24$& $2476 \pm 18$ &$2948 \pm 160 $\
NGC4151 &6,7 & 47494-47556 &0.00332&15.0 &$3.44^{+1.42}_{-1.24}$ &$42.412 \pm 0.016$ &$2929 \pm 154$&$4922 \pm 51$ &$5426 \pm 196 $\
NGC4395 &8 & 53106 &0.00106&4.0 &$0.033^{+0.017}_{-0.013}$ &$39.494 \pm 0.007$ &$1214 \pm 2$&$1727 \pm 78$ &$3025 \pm 201 $\
NGC4395 &8 & 53190 &0.00106&4.0 & $0.046^{+0.017}_{-0.013}$ &$40.030 \pm 0.012$ &$1532 \pm 6$&$1662\pm34$ &$2859 \pm 376 $\
NGC5548 &9,3 & 47510-47745 &0.01718&72.5 &$9.8^{+1.9}_{-1.5}$ &$43.635 \pm 0.016$ &$5248 \pm 428$&$4351 \pm 37$& $3842 \pm 210 $\
NGC5548 &10,3 & 49060-49135&0.01718&72.5 &$6.7^{+0.9}_{-1.0}$ &$43.552 \pm 0.007$ &$4201 \pm 101$&$3738\pm17$ & $3328 \pm 104 $\
NGC5548 &11 & 56690-56866&0.01718&72.5 &$5.8^{+0.5}_{-0.5}$ &$43.625 \pm 0.007$ &$5236 \pm 87 $&$2205 \pm16$ &$2075 \pm 81 $\
3C390.3 &12,3 & 49718-50147 &0.05610&243.5&$35.7^{+11.4}_{-14.6}$&$44.013 \pm 0.045$ &$6180 \pm 638$&$4578 \pm 65$ &$4400 \pm 186 $\
J214355 &4 & 54729-57605& 2.620&20,985& $128^{+91}_{-82}$& $46.962 \pm 0.048$ & $6895$ & $\ldots$ &$\ldots$\
J221516 &4 & 54232-57689& 2.706&21821& $165^{+98}_{-13}$ & $47.155 \pm 0.057$& $5888$ & $\ldots$ &$\ldots$\
NGC7469 &13,3 & 50245-50293&0.01632&68.8 &$2.5^{+0.3}_{-0.3}$ &$43.719 \pm 0.016$ &$3112 \pm 54 $&$3650 \pm 27$ &$2619 \pm 118 $ \[table:civrm\]
[lcccccccc]{} 16 & $0.848$& 5240.9 & $32.0^{+11.6 }_{-15.5}$ & $44.7779 \pm 0.0012$ & $43.0718 \pm 0.0600$ & $7042 \pm 43 $ & $4804 \pm 41 $ & $6477 \pm 54$\
17 & $0.456$& 2466.9 & $25.5^{+10.9 }_{-5.8 }$ & $44.3552 \pm 0.0005$ & $42.1756 \pm 0.0064$ & $7847 \pm 203 $ & $4295 \pm 47 $ & $6101 \pm 48$\
101 & $0.458$& 2479.8 & $21.4^{+4.2 }_{-6.4 }$ & $44.3758 \pm 0.0005$ & $42.7316 \pm 0.0449$ & $2207 \pm 7 $ & $1178 \pm 5 $ & $976 \pm 32$\
160 & $0.359$& 1859.7 & $21.9^{+4.2 }_{-2.4 }$ & $43.7613 \pm 0.0009$ & $42.0456 \pm 0.0047$ & $3988 \pm 23 $ & $2914 \pm 36 $ & $1909 \pm 12$\
177 & $0.482$& 2635.8 & $10.1^{+12.5 }_{-2.7 }$ & $44.1735 \pm 0.0009$ & $42.2813 \pm 0.0125$ & $4808 \pm 32 $ & $2224 \pm 32 $ & $2036 \pm 39$\
191 & $0.442$& 2377.0 & $8.5 ^{+2.5 }_{-1.4 }$ & $43.9111 \pm 0.0015$ & $41.7344 \pm 0.0131$ & $2023 \pm 32 $ & $1078 \pm 79 $ & $1030 \pm 18$\
229 & $0.47 $& 2557.5 & $16.2^{+2.9 }_{-4.5 }$ & $43.8259 \pm 0.0017$ & $41.9083 \pm 0.0166$ & $3089 \pm 261 $ & $2178 \pm 156$ & $1781 \pm 38$\
265 & $0.734$& 4388.8 & $8.5 ^{+3.2 }_{-3.9 }$ & $44.3809 \pm 0.0019$ & $42.4400 \pm 0.0273$ & $3655 \pm 323 $ & $2526 \pm 55 $ & $7165 \pm 36$\
267 & $0.587$& 3342.0 & $20.4^{+2.5 }_{-2.0 }$ & $44.3013 \pm 0.0008$ & $42.5166 \pm 0.0237$ & $2395 \pm 23 $ & $1229 \pm 32 $ & $1202 \pm 33$\
272 & $0.263$& 1298.0 & $15.1^{+3.2 }_{-4.6 }$ & $43.9119 \pm 0.0009$ & $42.3449 \pm 0.0017$ & $2595 \pm 10 $ & $1590 \pm 5 $ & $1697 \pm 10$\
300 & $0.646$& 3754.6 & $30.4^{+3.9 }_{-8.3 }$ & $44.6130 \pm 0.0008$ & $42.5889 \pm 0.0379$ & $2376 \pm 33 $ & $1303 \pm 29 $ & $1232 \pm 30$\
305 & $0.527$& 2933.9 & $53.5^{+4.2 }_{-4.0 }$ & $44.2995 \pm 0.0008$ & $42.5025 \pm 0.0365$ & $2208 \pm 28 $ & $1647 \pm 20 $ & $2126 \pm 35$\
316 & $0.676$& 3968.3 & $11.9^{+1.3 }_{-1.0 }$ & $44.9958 \pm 0.0004$ & $43.4279 \pm 0.0020$ & $2988 \pm 10 $ & $1884 \pm 5 $ & $7195 \pm 40$\
320 & $0.265$& 1309.4 & $25.2^{+4.7 }_{-5.7 }$ & $43.6876 \pm 0.0010$ & $41.8663 \pm 0.0096$ & $4061 \pm 26 $ & $3110 \pm 37 $ & $1462 \pm 26$\
371 & $0.472$& 2570.5 & $13 ^{+1.4 }_{-0.8 }$ & $44.0638 \pm 0.0009$ & $42.3726 \pm 0.0086$ & $3506 \pm 26 $ & $1682 \pm 18 $ & $1443 \pm 11$\
373 & $0.884$& 5516.4 & $20.4^{+5.6 }_{-7.0 }$ & $44.9025 \pm 0.0012$ & $42.7743 \pm 0.0191$ & $5987 \pm 268 $ & $1897 \pm 48 $ & $2491 \pm 26$\
377 & $0.337$& 1727.4 & $5.9 ^{+0.4 }_{-0.6 }$ & $43.7819 \pm 0.0011$ & $41.5130 \pm 0.0156$ & $2746 \pm 118 $ & $1576 \pm 23 $ & $1789 \pm 23$\
392 & $0.843$& 5202.8 & $14.2^{+3.7 }_{-3.0 }$ & $44.4249 \pm 0.0032$ & $42.4894 \pm 0.0427$ & $2419 \pm 82 $ & $2446 \pm 110$ & $3658 \pm 56$\
399 & $0.608$& 3487.6 & $35.8^{+1.1 }_{-10.3}$ & $44.3272 \pm 0.0020$ & $42.2823 \pm 0.0281$ & $2689 \pm 88 $ & $1989 \pm 89 $ & $1619 \pm 38$\
428 & $0.976$& 6233.7 & $15.8^{+6.0 }_{-1.9 }$ & $45.4013 \pm 0.0015$ & $43.2816 \pm 0.0048$ & $2795 \pm 29 $ & $1836 \pm 18 $ & $7568 \pm 70$\
551 & $0.68 $& 3997.0 & $6.4 ^{+1.5 }_{-1.4 }$ & $44.1196 \pm 0.0021$ & $42.4389 \pm 0.0842$ & $2101 \pm 45 $ & $1255 \pm 59 $ & $1298 \pm 36$\
589 & $0.751$& 4513.8 & $46 ^{+9.5 }_{-9.5 }$ & $44.4877 \pm 0.0015$ & $42.6421 \pm 0.0107$ & $3738 \pm 62 $ & $2835 \pm 62 $ & $5013 \pm 49$\
622 & $0.572$& 3238.9 & $49.1^{+11.1 }_{-2.0 }$ & $44.3737 \pm 0.0006$ & $42.5966 \pm 0.0062$ & $2389 \pm 36 $ & $1147 \pm 11 $ & $1423 \pm 32$\
645 & $0.474$& 2583.6 & $20.7^{+0.9 }_{-3.0 }$ & $44.1342 \pm 0.0008$ & $42.2965 \pm 0.0047$ & $6428 \pm 163 $ & $2799 \pm 13 $ & $1438 \pm 17$\
720 & $0.467$& 2538.0 & $41.6^{+14.8 }_{-8.3 }$ & $44.3176 \pm 0.0008$ & $42.4324 \pm 0.0029$ & $2829 \pm 15 $ & $1679 \pm 17 $ & $1232 \pm 16$\
772 & $0.249$& 1219.6 & $3.9 ^{+0.9 }_{-0.9 }$ & $43.7867 \pm 0.0005$ & $41.5251 \pm 0.0081$ & $2381 \pm 33 $ & $1983 \pm 40 $ & $1026 \pm 14$\
775 & $0.172$& 805.9 & $16.3^{+13.1 }_{-6.6 }$ & $43.7943 \pm 0.0003$ & $41.7848 \pm 0.0021$ & $2744 \pm 36 $ & $2028 \pm 10 $ & $1818 \pm 8 $\
776 & $0.116$& 524.6 & $10.5^{+1.0 }_{-2.2 }$ & $43.3829 \pm 0.0004$ & $41.4179 \pm 0.0220$ & $3060 \pm 20 $ & $3178 \pm 19 $ & $1409 \pm 11$\
781 & $0.263$& 1298.0 & $75.2^{+3.2 }_{-3.3 }$ & $43.7604 \pm 0.0034$ & $41.8863 \pm 0.0155$ & $2506 \pm 19 $ & $1290 \pm 17 $ & $1089 \pm 22$\
782 & $0.362$& 1877.9 & $20 ^{+1.1 }_{-3.0 }$ & $44.0941 \pm 0.0006$ & $41.9722 \pm 0.0044$ & $3027 \pm 35 $ & $1527 \pm 16 $ & $1353 \pm 23$\
790 & $0.237$& 1153.2 & $5.5 ^{+5.7 }_{-2.1 }$ & $43.8222 \pm 0.0014$ & $41.8443 \pm 0.0272$ & $8365 \pm 44 $ & $5069 \pm 47 $ & $6318 \pm 38$\
840 & $0.244$& 1191.8 & $5 ^{+1.5 }_{-1.4 }$ & $43.6987 \pm 0.0005$ & $41.5724 \pm 0.0074$ & $6116 \pm 267 $ & $3286 \pm 254$ & $4457 \pm 60$ \[table:sdssrmhb\]
[lccccccc]{} 0 & 1.463 &10283 &$131.1 _{-36.6 }^{+42.9}$& $44.847 \pm 0.004$& $3967 \pm 107$& $1968 \pm 160$& $2144 \pm 46 $\
32 & 1.72 &12554 &$22.8 _{-3.6 }^{+3.5 }$& $44.492 \pm 0.021$& $2999 \pm 34 $& $1770 \pm 24 $& $2017 \pm 10 $\
36 & 2.213 &17094 &$188.4 _{-29 }^{+15.6}$& $45.909 \pm 0.001$& $4830 \pm 24 $& $2890 \pm 24 $& $3900 \pm 34 $\
52 & 2.311 &18020 &$56.5 _{-5.9 }^{+3.1 }$& $45.499 \pm 0.002$& $2258 \pm 14 $& $1809 \pm 15 $& $1322 \pm 22 $\
57 & 1.93 &14461 &$208.3 _{-5.6 }^{+10.6}$& $45.393 \pm 0.003$& $2692 \pm 11 $& $1626 \pm 8 $& $1682 \pm 12 $\
58 & 2.299 &17906 &$186.1 _{-7.4 }^{+5.9 }$& $45.353 \pm 0.002$& $3627 \pm 45 $& $2611 \pm 31 $& $3412 \pm 30 $\
130 & 1.96 &14737 &$224.3 _{-37.9 }^{+12.4}$& $45.534 \pm 0.001$& $5619 \pm 30 $& $4078 \pm 55 $& $4324 \pm 36 $\
144 & 2.295 &17868 &$179.4 _{-42.3 }^{+31.2}$& $45.516 \pm 0.001$& $6153 \pm 53 $& $2762 \pm 19 $& $2792 \pm 19 $\
145 & 2.138 &16390 &$180.9 _{-4.7 }^{+4.7 }$& $45.113 \pm 0.004$& $4472 \pm 74 $& $3287 \pm 40 $& $3408 \pm 16 $\
158 & 1.477 &10405 &$36.7 _{-26.1 }^{+18.6}$& $44.999 \pm 0.004$& $3603 \pm 101$& $2099 \pm 60 $& $2136 \pm 31 $\
161 & 2.071 &15764 &$180.1 _{-6.4 }^{+5.6 }$& $45.491 \pm 0.001$& $3163 \pm 28 $& $2323 \pm 25 $& $2524 \pm 20 $\
181 & 1.678 &12177 &$102.6 _{-10.1 }^{+5 }$& $44.545 \pm 0.015$& $2998 \pm 35 $& $2127 \pm 44 $& $2721 \pm 34 $\
201 & 1.797 &13248 &$41.3 _{-19.5 }^{+32 }$& $46.240 \pm 0.001$& $5438 \pm 56 $& $1833 \pm 9 $& $2408 \pm 117$\
231 & 1.646 &11892 &$80.4 _{-7.5 }^{+6.3 }$& $45.736 \pm 0.001$& $5975 \pm 98 $& $3267 \pm 102$& $3803 \pm 18 $\
237 & 2.394 &18810 &$49.9 _{-4.4 }^{+6.6 }$& $45.866 \pm 0.001$& $5455 \pm 39 $& $2734 \pm 18 $& $2779 \pm 23 $\
245 & 1.677 &12168 &$107.1 _{-28.6 }^{+22.9}$& $45.351 \pm 0.004$& $9496 \pm 107$& $4174 \pm 54 $& $3953 \pm 86 $\
249 & 1.721 &12562 &$24.9 _{-3.1 }^{+9.7 }$& $44.984 \pm 0.010$& $1871 \pm 15 $& $1432 \pm 12 $& $1640 \pm 15 $\
256 & 2.247 &17414 &$43 _{-11.9 }^{+16.3}$& $45.089 \pm 0.003$& $2544 \pm 54 $& $1742 \pm 29 $& $1802 \pm 24 $\
269 & 2.4 &18868 &$197.2 _{-12.6 }^{+2.4 }$& $45.193 \pm 0.003$& $3930 \pm 312$& $3280 \pm 50 $& $3547 \pm 30 $\
275 & 1.58 &11307 &$81 _{-24.4 }^{+8.2 }$& $45.611 \pm 0.001$& $3213 \pm 20 $& $2108 \pm 9 $& $2406 \pm 5 $\
295 & 2.351 &18400 &$163.8 _{-5.3 }^{+8.2 }$& $45.605 \pm 0.001$& $4311 \pm 41 $& $2501 \pm 23 $& $2446 \pm 19 $\
298 & 1.633 &11777 &$106.1 _{-31.7 }^{+18.7}$& $45.596 \pm 0.001$& $3160 \pm 30 $& $2066 \pm 26 $& $2549 \pm 35 $\
312 & 1.929 &14452 &$56.9 _{-6.7 }^{+11.4}$& $45.077 \pm 0.004$& $7663 \pm 166$& $4273 \pm 74 $& $4291 \pm 30 $\
332 & 2.58 &20598 &$81.6 _{-11.4 }^{+5.6 }$& $45.551 \pm 0.002$& $3799 \pm 14 $& $3009 \pm 63 $& $4277 \pm 33 $\
346 & 1.592 &11413 &$71.9 _{-11.3 }^{+23.8}$& $44.905 \pm 0.003$& $3389 \pm 168$& $2220 \pm 131$& $3055 \pm 29 $\
386 & 1.862 &13838 &$38.2 _{-19.3 }^{+13.2}$& $45.279 \pm 0.002$& $2972 \pm 40 $& $1782 \pm 38 $& $2187 \pm 41 $\
387 & 2.427 &19126 &$30.3 _{-3.4 }^{+19.6}$& $45.687 \pm 0.001$& $3676 \pm 24 $& $2123 \pm 14 $& $2451 \pm 23 $\
389 & 1.851 &13738 &$224.3 _{-18 }^{+7.1 }$& $45.564 \pm 0.002$& $5222 \pm 111$& $3839 \pm 16 $& $4064 \pm 15 $\
401 & 1.823 &13484 &$47.4 _{-8.9 }^{+15.2}$& $45.564 \pm 0.002$& $3273 \pm 21 $& $2457 \pm 12 $& $3321 \pm 12 $\
411 & 1.734 &12679 &$248.3 _{-39 }^{+21.1}$& $44.887 \pm 0.007$& $4256 \pm 67 $& $2511 \pm 61 $& $2490 \pm 39 $\
418 & 1.419 &9903 &$82.5 _{-16.9 }^{+27.6}$& $45.040 \pm 0.003$& $3143 \pm 44 $& $2662 \pm 94 $& $3110 \pm 23 $\
470 & 1.883 &14030 &$19.9 _{-4 }^{+43.2}$& $44.821 \pm 0.006$& $4022 \pm 52 $& $2441 \pm 34 $& $2317 \pm 60 $\
485 & 2.557 &20376 &$133.4 _{-5.2 }^{+22.6}$& $46.119 \pm 0.001$& $5342 \pm 48 $& $2924 \pm 32 $& $3961 \pm 41 $\
496 & 2.079 &15839 &$197.9 _{-6.6 }^{+9.7 }$& $45.560 \pm 0.001$& $2364 \pm 27 $& $2137 \pm 34 $& $2409 \pm 45 $\
499 & 2.327 &18172 &$168.5 _{-35.9 }^{+20.4}$& $45.058 \pm 0.003$& $3261 \pm 41 $& $2968 \pm 41 $& $3085 \pm 26 $\
506 & 1.753 &12850 &$231.6 _{-11.1 }^{+13.3}$& $45.075 \pm 0.003$& $5046 \pm 52 $& $3507 \pm 27 $& $3510 \pm 24 $\
527 & 1.651 &11937 &$52.3 _{-12.2 }^{+15.1}$& $44.788 \pm 0.003$& $5154 \pm 110$& $3384 \pm 62 $& $3587 \pm 34 $\
549 & 2.277 &17698 &$69.8 _{-7.2 }^{+5.3 }$& $45.369 \pm 0.002$& $3907 \pm 59 $& $1818 \pm 47 $& $2176 \pm 21 $\
554 & 1.707 &12437 &$194 _{-12.2 }^{+20.4}$& $45.573 \pm 0.002$& $3690 \pm 65 $& $2253 \pm 47 $& $2229 \pm 35 $\
562 & 2.773 &22476 &$158.5 _{-34.2 }^{+18.2}$& $46.302 \pm 0.001$& $4379 \pm 113$& $2036 \pm 29 $& $2078 \pm 27 $\
686 & 2.13 &16315 &$64.7 _{-6.3 }^{+12.6}$& $45.444 \pm 0.002$& $3827 \pm 34 $& $2135 \pm 25 $& $2203 \pm 27 $\
689 & 2.007 &15170 &$157.6 _{-42.2 }^{+22.9}$& $45.223 \pm 0.003$& $2258 \pm 23 $& $1292 \pm 8 $& $1407 \pm 5 $\
734 & 2.324 &18144 &$87.2 _{-11 }^{+13.9}$& $45.530 \pm 0.001$& $5701 \pm 121$& $2982 \pm 65 $& $3405 \pm 40 $\
809 & 1.67 &12106 &$108.6 _{-50.7 }^{+27.7}$& $45.204 \pm 0.005$& $4811 \pm 38 $& $5210 \pm 60 $& $4749 \pm 96 $\
827 & 1.966 &14792 &$137.7 _{-19.4 }^{+18.3}$& $44.999 \pm 0.006$& $2542 \pm 35 $& $971 \pm 13 $& $1443 \pm 13 $ \[table:sdssrmciv\]
[lccccc]{} Mrk335 &$ 2291 \pm 27 $ & $ 2116 \pm 160 $&$ 44.173 \pm 0.020$ & $ 6.663 \pm 0.337 $ &$7.079 \pm 0.145$\
Mrk335 &$ 1741 \pm 99 $ & $ 1806 \pm 360 $&$ 44.291 \pm 0.078$ & $ 6.588 \pm 0.375 $ &$7.080 \pm 0.187$\
Mrk335 &$ 2023 \pm 17 $ & $ 2140 \pm 93 $&$ 44.262 \pm 0.013$ & $ 6.720 \pm 0.332 $ &$7.153 \pm 0.140$\
PG0026+129 &$ 1837 \pm 136 $ & $ 3364 \pm 70 $&$ 45.165 \pm 0.025$ & $ 7.591 \pm 0.331 $ &$8.092 \pm 0.140$\
PG0052+251 &$ 3983 \pm 370 $ & $ 5118 \pm 486 $&$ 45.265 \pm 0.037$ & $ 8.009 \pm 0.341 $ &$8.402 \pm 0.150$\
PG0052+251 &$ 5192 \pm 251 $ & $ 5083 \pm 437 $&$ 45.176 \pm 0.041$ & $ 7.956 \pm 0.339 $ &$8.331 \pm 0.149$\
Fairall9 &$ 2593 \pm 65 $ & $ 2981 \pm 197 $&$ 44.470 \pm 0.028$ & $ 7.118 \pm 0.335 $ &$7.496 \pm 0.144$\
Fairall9 &$ 2831 \pm 40 $ & $ 3532 \pm 92 $&$ 44.582 \pm 0.011$ & $ 7.325 \pm 0.331 $ &$7.676 \pm 0.139$\
Fairall9 &$ 2370 \pm 151 $ & $ 2978 \pm 508 $&$ 44.759 \pm 0.126$ & $ 7.270 \pm 0.368 $ &$7.715 \pm 0.193$\
Mrk590 &$ 4839 \pm 59 $ & $ 3574 \pm 141 $&$ 44.119 \pm 0.029$ & $ 7.089 \pm 0.332 $ &$7.330 \pm 0.141$\
3C120 &$ 3302 \pm 75 $ & $ 3199 \pm 169 $&$ 44.943 \pm 0.039$ & $ 7.430 \pm 0.334 $ &$7.895 \pm 0.144$\
3C120 &$ 3278 \pm 105 $ & $ 3409 \pm 286 $&$ 44.617 \pm 0.056$ & $ 7.312 \pm 0.339 $ &$7.682 \pm 0.152$\
Ark120 &$ 3989 \pm 451 $ & $ 3795 \pm 165 $&$ 44.634 \pm 0.021$ & $ 7.414 \pm 0.332 $ &$7.755 \pm 0.141$\
Ark120 &$ 3945 \pm 42 $ & $ 3240 \pm 149 $&$ 44.482 \pm 0.022$ & $ 7.197 \pm 0.333 $ &$7.551 \pm 0.141$\
Mrk79 &$ 3182 \pm 521 $ & $ 3344 \pm 222 $&$ 43.879 \pm 0.039$ & $ 6.904 \pm 0.336 $ &$7.110 \pm 0.146$\
Mrk79 &$ 3049 \pm 128 $ & $ 2971 \pm 248 $&$ 43.495 \pm 0.058$ & $ 6.598 \pm 0.339 $ &$6.752 \pm 0.152$\
Mrk79 &$ 3113 \pm 122 $ & $ 3803 \pm 388 $&$ 43.726 \pm 0.065$ & $ 6.935 \pm 0.343 $ &$7.065 \pm 0.157$\
Mrk110 &$ 2990 \pm 64 $ & $ 2601 \pm 272 $&$ 43.770 \pm 0.050$ & $ 6.628 \pm 0.343 $ &$6.887 \pm 0.155$\
Mrk110 &$ 1638 \pm 59 $ & $ 2576 \pm 231 $&$ 43.876 \pm 0.081$ & $ 6.676 \pm 0.342 $ &$6.962 \pm 0.159$\
PG0953+414 &$ 2873 \pm 57 $ & $ 3512 \pm 361 $&$ 45.588 \pm 0.031$ & $ 7.853 \pm 0.342 $ &$8.438 \pm 0.151$\
NGC3516 &$ 4675 \pm 538 $ & $ 3311 \pm 372 $&$ 42.830 \pm 0.093$ & $ 6.340 \pm 0.348 $ &$6.306 \pm 0.167$\
NGC3516 &$ 4875 \pm 17 $ & $ 3132 \pm 64 $&$ 42.823 \pm 0.017$ & $ 6.288 \pm 0.331 $ &$6.270 \pm 0.139$\
NGC3516 &$ 5147 \pm 103 $ & $ 3245 \pm 84 $&$ 43.192 \pm 0.013$ & $ 6.514 \pm 0.331 $ &$6.570 \pm 0.139$\
NGC3516 &$ 4729 \pm 28 $ & $ 3430 \pm 92 $&$ 43.143 \pm 0.013$ & $ 6.536 \pm 0.331 $ &$6.564 \pm 0.139$\
NGC3516 &$ 4525 \pm 97 $ & $ 3137 \pm 79 $&$ 43.030 \pm 0.012$ & $ 6.399 \pm 0.331 $ &$6.428 \pm 0.139$\
NGC3516 &$ 3940 \pm 18 $ & $ 2834 \pm 95 $&$ 42.485 \pm 0.034$ & $ 6.022 \pm 0.332 $ &$5.957 \pm 0.142$\
NGC3516 &$ 4912 \pm 23 $ & $ 3973 \pm 36 $&$ 42.793 \pm 0.012$ & $ 6.479 \pm 0.330 $ &$6.380 \pm 0.138$\
NGC3783 &$ 2831 \pm 22 $ & $ 3273 \pm 100 $&$ 43.601 \pm 0.014$ & $ 6.738 \pm 0.331 $ &$6.886 \pm 0.139$\
NGC3783 &$ 2308 \pm 17 $ & $ 3179 \pm 185 $&$ 43.744 \pm 0.022$ & $ 6.789 \pm 0.334 $ &$6.979 \pm 0.143$\
NGC4051 &$ 1319 \pm 13 $ & $ 1713 \pm 227 $&$ 41.373 \pm 0.058$ & $ 4.995 \pm 0.351 $ &$4.830 \pm 0.163$\
NGC4151 &$ 6929 \pm 76 $ & $ 5220 \pm 123 $&$ 43.224 \pm 0.010$ & $ 6.944 \pm 0.331 $ &$6.860 \pm 0.139$\
NGC4151 &$ 5418 \pm 150 $ & $ 4604 \pm 249 $&$ 43.340 \pm 0.019$ & $ 6.896 \pm 0.333 $ &$6.878 \pm 0.142$\
NGC4151 &$ 5062 \pm 51 $ & $ 4651 \pm 371 $&$ 43.396 \pm 0.029$ & $ 6.935 \pm 0.338 $ &$6.926 \pm 0.147$\
NGC4151 &$ 5246 \pm 44 $ & $ 4675 \pm 397 $&$ 43.396 \pm 0.031$ & $ 6.939 \pm 0.339 $ &$6.929 \pm 0.148$\
NGC4151 &$ 5752 \pm 144 $ & $ 4585 \pm 321 $&$ 43.418 \pm 0.023$ & $ 6.934 \pm 0.336 $ &$6.935 \pm 0.144$\
NGC4151 &$ 5173 \pm 593 $ & $ 4664 \pm 475 $&$ 43.354 \pm 0.044$ & $ 6.915 \pm 0.342 $ &$6.896 \pm 0.153$\
NGC4151 &$ 3509 \pm 10 $ & $ 4384 \pm 66 $&$ 43.038 \pm 0.006$ & $ 6.694 \pm 0.330 $ &$6.621 \pm 0.138$\
PG1229+204 &$ 3391 \pm 205 $ & $ 3241 \pm 457 $&$ 44.654 \pm 0.028$ & $ 7.288 \pm 0.352 $ &$7.682 \pm 0.160$\
PG1307+085 &$ 3465 \pm 168 $ & $ 3687 \pm 290 $&$ 45.012 \pm 0.039$ & $ 7.590 \pm 0.338 $ &$8.027 \pm 0.148$\
Mrk279 &$ 4126 \pm 487 $ & $ 3118 \pm 414 $&$ 43.795 \pm 0.118$ & $ 6.799 \pm 0.355 $ &$7.007 \pm 0.181$\
Mrk279 &$ 3876 \pm 99 $ & $ 3286 \pm 511 $&$ 43.754 \pm 0.127$ & $ 6.823 \pm 0.363 $ &$7.005 \pm 0.189$\
NGC5548 &$ 4790 \pm 67 $ & $ 4815 \pm 257 $&$ 43.654 \pm 0.022$ & $ 7.102 \pm 0.333 $ &$7.142 \pm 0.142$\
NGC5548 &$ 4096 \pm 14 $ & $ 3973 \pm 34 $&$ 43.568 \pm 0.006$ & $ 6.889 \pm 0.330 $ &$6.969 \pm 0.138$\
NGC5548 &$ 3280 \pm 27 $ & $ 5050 \pm 787 $&$ 43.773 \pm 0.069$ & $ 7.206 \pm 0.359 $ &$7.259 \pm 0.171$\
PG1426+015 &$ 3778 \pm 448 $ & $ 4101 \pm 391 $&$ 45.295 \pm 0.023$ & $ 7.832 \pm 0.340 $ &$8.301 \pm 0.149$\
Mrk817 &$ 4027 \pm 71 $ & $ 4062 \pm 289 $&$ 44.123 \pm 0.022$ & $ 7.203 \pm 0.336 $ &$7.404 \pm 0.145$\
PG1613+658 &$ 5902 \pm 136 $ & $ 3965 \pm 215 $&$ 45.221 \pm 0.023$ & $ 7.764 \pm 0.334 $ &$8.226 \pm 0.142$\
PG1617+175 &$ 4558 \pm 1763 $ & $ 3383 \pm 1036 $&$ 44.784 \pm 0.108$ & $ 7.394 \pm 0.428 $ &$7.805 \pm 0.234$\
Mrk509 &$ 5035 \pm 298 $ & $ 3558 \pm 205 $&$ 44.641 \pm 0.029$ & $ 7.362 \pm 0.334 $ &$7.725 \pm 0.143$\
Mrk509 &$ 4345 \pm 49 $ & $ 3426 \pm 115 $&$ 44.532 \pm 0.015$ & $ 7.272 \pm 0.331 $ &$7.621 \pm 0.140$\
Mrk509 &$ 4973 \pm 233 $ & $ 3647 \pm 172 $&$ 44.803 \pm 0.020$ & $ 7.469 \pm 0.333 $ &$7.862 \pm 0.141$\
Mrk509 &$ 4961 \pm 218 $ & $ 3127 \pm 226 $&$ 44.552 \pm 0.033$ & $ 7.203 \pm 0.336 $ &$7.585 \pm 0.146$\
Mrk509 &$ 3716 \pm 228 $ & $ 3174 \pm 448 $&$ 44.706 \pm 0.071$ & $ 7.297 \pm 0.354 $ &$7.710 \pm 0.168$\
PG2130+099 &$ 2113 \pm 119 $ & $ 2390 \pm 184 $&$ 44.692 \pm 0.025$ & $ 7.044 \pm 0.337 $ &$7.541 \pm 0.146$\
NGC7469 &$ 3094 \pm 53 $ & $ 3379 \pm 182 $&$ 43.774 \pm 0.016$ & $ 6.858 \pm 0.333 $ &$7.036 \pm 0.142$\
NGC7469 &$ 2860 \pm 12 $ & $ 3266 \pm 110 $&$ 43.679 \pm 0.015$ & $ 6.778 \pm 0.331 $ &$6.945 \pm 0.140$ \[table:civVP06\]
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[^1]: The database is regularly updated at http://www.astro.gsu.edu/AGNmass
[^2]: Strictly speaking, the Eddington ratio is defined as ${\mbox{$\dot{m}$}}= \dot{M}/\dot{M}_{\rm Edd}$. Since ${\mbox{$\dot{m}$}}= L_{\rm bol}/\eta c^2$, ${\mbox{$\dot{m}$}}= L_{\rm bol}/L_{\rm Edd}$ as long as the efficiency $\eta$ is constant and not a function of the accretion rate, which we will assume for simplicity.
| 0 |
---
abstract: |
We use the very large Millennium Simulation of the concordance $\Lambda$CDM cosmogony to calibrate the bias and error distribution of Timing Argument estimators of the masses of the Local Group and of the Milky Way. From a large number of isolated spiral-spiral pairs similar to the Milky Way/Andromeda system, we find the interquartile range of the ratio of timing mass to true mass to be a factor of $1.8$, while the 5% and 95% points of the distribution of this ratio are separated by a factor of $5.7$. Here we define true mass as the sum of the “virial” masses $M_{200}$ of the two dominant galaxies. For current best values of the distance and approach velocity of Andromeda this leads to a median likelihood estimate of the true mass of the Local Group of $5.27\times 10^{12}{\rm \,M_{\odot}}$, or $\log M_{LG}/M_\odot = 12.72$, with an interquartile range of $[12.58, 12.83]$ and a 5% to 95% range of $[12.26, 13.01]$. Thus a 95% lower confidence limit on the true mass of the Local Group is $1.81\times 10^{12}{\rm \,M_{\odot}}$. A timing estimate of the Milky Way’s mass based on the large recession velocity observed for the distant satellite Leo I works equally well, although with larger systematic uncertainties. It gives an estimated virial mass for the Milky Way of $2.43 \times
10^{12}{\rm \,M_{\odot}}$ with a 95% lower confidence limit of $0.80 \times
10^{12}{\rm \,M_{\odot}}$.
author:
- |
Yang-Shyang Li$^{1}$[^1] and Simon D. M. White$^{2}$[^2]\
$^{1}$Kapteyn Astronomical Institute, PO Box 800, 9700 AV, Groningen, The Netherlands\
$^{2}$Max–Planck–Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany
date: 'Accepted. Received ; in original form '
title: Masses for the Local Group and the Milky Way
---
\[firstpage\]
Galaxy: formation – galaxies: Local Group – galaxies: kinematics and dynamics – dark matter.
Introduction {#intro_section}
============
During the 1970’s it became generally accepted that most, perhaps all, galaxies are surrounded by extended distributions of dark matter, so-called dark halos [@eks74; @opy74]. These were soon understood to play an essential role in driving the formation and clustering of galaxies [@wr78]. With the introduction of the Cold Dark Matter (CDM) paradigm, these ideas took more concrete form, allowing quantitative predictions to be made both for the population properties [@blumenthal84] and for the large-scale clustering [@davis85] of galaxies.
Measurements of the fluctuation spectrum of the Cosmic Microwave Background [@smoot92; @spergel03] and of the apparent acceleration of the cosmic expansion [@riess98; @perlmutter99] elevated the CDM model, in its variant with a cosmological constant ($\Lambda$CDM), to the status of a standard paradigm. At the same time improving numerical techniques and faster computers have enabled detailed simulation of the formation and evolution of the galaxy population within this paradigm throughout a significant fraction of the observable Universe [@mr_nature]. Nevertheless, direct observational evidence for halos as extended as the paradigm predicts around galaxies like our own has so far come only from statistical analyses of the dynamics of satellite galaxies [e.g. @zaritsky97; @prada03] and of the gravitational lensing of background galaxies [e.g. @seljak02; @mandelbaum06] based on large samples of field spirals.
The earliest observational indication that the effective mass of the Milky Way must be much larger than its stellar mass came from the Timing Argument (hereafter TA) of @kw59. These authors noted that the Local Group is dominated by the two big spirals, and that these are currently approaching each other at about $100{{\rm \,km\,s^{-1}}}$. (The next most luminous galaxy is M33 which is probably about a factor of $10$ less massive than M31 and the Galaxy.) This reversal of the overall cosmic expansion must have been generated by gravitational forces, and since the distance to the nearest external bright galaxy is much greater than that between M31 and the Milky Way, these forces are presumably dominated by material associated with the two spirals themselves.
@kw59 set up a simple model to analyse this situation – two point masses on a radial orbit. These were at pericentre (i.e. at zero separation) at the Big Bang and must have passed through apocentre at least once in order to be approaching today. Clearly this requires an apocentric separation larger than the current separation and an orbital period less than twice the current age of the Universe. Together these requirements put a lower limit on on the total mass of the pair. A more precise estimate of the minimum possible mass is obtained from the parametric form of Kepler’s laws for a zero angular momentum orbit: $$r=a(1-\cos\chi)$$
$$t=\bigg(\frac{a^{3}}{GM}\bigg)^{1/2}(\chi-\sin\chi)$$
$$\frac{dr}{dt}=\sqrt{\frac{GM}{a}}\frac{\sin\chi}{1-\cos\chi}$$
where $r$ is the current separation, $dr/dt$ is the current relative velocity, $a$ is the semi-major axis, $\chi$ is the eccentric anomaly, $t$ is the time since the Big Bang (the age of the universe) and $M$ is total mass [@lb81]. Given observationally determined values for $r$, $dr/dt$ and $t$, these equations have an infinite set of discrete solutions for $\chi$, $a$ and $M$ labelled by the number of apocentric passages since the Big Bang. The solution corresponding to a single apocentric passage gives the smallest (and only plausible) estimate for the mass, which is about $5\times 10^{12}{\rm \,M_{\odot}}$ for current estimates of $r$, $dr/dt$ and $t$. Note that this is still a lower limit on the total mass, even within the simple point-mass binary model, since any non-radial motions in the system would increase its present kinetic energy and so increase the mass required to reverse the initial expansion and bring the pair to their observed separation by the present day [see @el82].
As @kw59 realised, this timing estimate of the total mass of the Local group exceeds by more than an order of magnitude the mass within the visible regions of the galaxies, as estimated from their internal dynamics, in particular, from their rotation curves. Thus 90% of the mass must lie outside the visible galaxies and be associated with little or no detectable light. Modern structure formation theories like $\Lambda$CDM predict this mass to be in extended dark matter halos with $M(r)$ increasing very roughly as $r$ out to the point where the halos of the two galaxies meet. Such structures have no well-defined edge, so any definition of their total mass is necessarily somewhat arbitrary. In addition, their dynamical evolution from the Big Bang until the present is substantially more complex than that of a point-mass binary. Thus the mass value returned by the Timing Argument cannot be interpreted without some calibration against consistent dynamical models with extended dark halos.
A first calibration of this type was carried out by @kc91 using simulations of an Einstein-de Sitter CDM cosmogony. Here we use the very much larger Millennium Simulation [@mr_nature] to obtain a more refined calibration based on a large ensemble of galaxy pairs with observable properties similar to those of the Local Group. We find that the standard timing estimate is, in fact, an almost unbiased estimate of the sum of the conventionally defined virial masses of the two large galaxies.
@zaritsky89 attempted to measure the halo mass of the Milky Way alone by measuring radial velocities for its dwarf satellites and assuming the population to be in dynamical equilibrium in the halo potential. They noted, however, that one of the most distant dwarfs, Leo I, has a very large recession velocity and as a result provides a interesting lower limit on the Milky Way’s mass by a variant of the original Timing Argument. To reach its present position and radial velocity, Leo I must have passed pericentre at least once since the Big Bang and now be receding from the Galaxy for (at least) the second time.
Applying the point-mass radial orbit Equations (1) – (3) to this case gives a lower bound of about $1.6\times 10^{12}{\rm \,M_{\odot}}$. This seems likely to be a significant underestimate, since Leo I could not have passed through the centre of the Milky Way without being tidally destroyed so its orbit cannot be purely radial. Below we calibrate the Timing Argument for this case also, finding it to work well although with significantly more scatter than for the Local Group as a whole. This is because the $\Lambda$CDM paradigm predicts that the dynamics on the scale of Leo I’s orbit ($\sim 200{\rm \,kpc}$) is typically more complex than on the scale of the Local Group as a whole ($\sim 700{\rm \,kpc}$).
Our paper is organised as follows. In Section \[data\_section\], we briefly describe the Millennium Simulation and the selection criteria we use to define various samples of ‘Local Group-like’ pairs and of ‘Milky Way-like’ halos. In Section \[result\_section\], we plot the ratio of true total mass to Timing Argument mass estimate for these samples, and we use its distribution to define an unbiased TA estimator of true mass with its associated confidence ranges. In Section \[app\_LG\_section\] this is then applied to the Local Group in order to obtain an estimate its true mass with realistic uncertainties. Section \[app\_MW\_section\] attempts to carry out a similar calibration for the TA-based estimate of the Galaxy’s halo mass from the orbit of Leo I. We conclude in Section \[conclusions\] with a summary and brief discussion of our results.
The Millennium Simulation {#data_section}
=========================
The *Millennium Simulation* is an extremely large cosmological simulation carried out by the Virgo Consortium [@mr_nature]. It followed the motion of $N=2160^{3}$ dark matter particles of mass $8.6 \times 10^{8}
~h^{-1}{\rm \,M_{\odot}}$ within a cubic box of comoving size $500~h^{-1}{\rm \,Mpc}$. Its comoving spatial resolution (set by the gravitational softening) is 5 $h^{-1}{\rm \,kpc}$. The simulation adopted the concordance $\Lambda$CDM model with parameters $\Omega_{m}=0.25, \Omega_{b}=0.045, h=0.73, \Omega_{\Lambda}=0.75,
n=1$ and $\sigma_{8}=0.9$, where, as usual, we define the Hubble constant by $H_{0}=100h{{\rm \,km\,s^{-1}}}{\rm \,Mpc}^{-1}$. The current age of the universe is then $13.6
\times 10^{9}$ yr. The positions and velocities of all particles were stored at 63 epochs spaced approximately logarithmically in expansion factor at early times and at approximately 300 Myr intervals after $z=2$. For each such snapshot a friends-of-friends group-finder was used to locate all virialised structures, and their self-bound substructures (subhalos) were identified using [SUBFIND]{} [@swtk01]. Halos and subhalos in neighbouring outputs were then linked in order to build formation history trees for all the subhalos present at each time. These data are publicly available at the Millennium release site[^3]. A “Milky Way” halo at $z=0$ typically contains a few thousand particles and several resolved subhalos.
Galaxy formation was simulated within these merging history trees by using semi-analytic models to follow the evolution of the baryonic components associated with each halo/subhalo. Processes included are radiative cooling of diffuse gas, star formation, the growth of supermassive black holes, feedback of energy and heavy elements from supernovae and AGN, stellar population evolution, galaxy merging and effects due to a reionising UV background. The $z=0$ galaxy catalogue we analyse here corresponds to the specific model of @croton06 and details of its assumptions and parameters can be found in that paper. Data for the galaxy population at all redshifts are available at the Millennium web site for the updated model of @deluciab07, as well as for the independent galaxy formation model of @bower06. All these models are tuned to fit a wide variety of data on the nearby galaxy population, and in addition fit many (but not all!) available data at higher redshift [see, for example, @kw06]. The details of the galaxy formation modelling are not, however, important for the dynamical issues which are the focus of our own paper.
At $z=0$ there are $18.2 \times 10^{6}$ halos/subhalos identified in the simulation to its resolution limit of $20$ particles. The galaxy formation model populates these with $8,394,180$ galaxies brighter than an absolute magnitude limit of $M_{B} = -16.7$ above which the catalogue can be considered reasonably complete. These catalogues list a number of properties for the halos, subhalos and galaxies which will be important for us. Galaxies are categorised into three types according to the nature of their association with the dark matter. A Type 0 galaxy sits at the centre of the dominant or main subhalo and can be considered the central galaxy of the halo itself (formally, of the FOF group). A Type 1 galaxy sits at the centre of one of the smaller non-dominant subhalos associated with a FOF group. Finally, a Type 2 galaxy is associated to a specific particle and no longer has an associated subhalo because the object within which it formed was tidally disrupted after accretion onto a larger halo. Such galaxies merge with the central galaxy of their new halo after waiting for a dynamical friction time.
Each galaxy in the catalogue has an associated “rotation velocity” $V_{max}$. This is the maximum of the circular velocity $V_c(r) =
(GM(r)/r)^{1/2}$ of its subhalo for Types 0 and 1; for Type 2 objects $V_{max}$ is frozen to its value at the latest time when the galaxy still occupied a subhalo. Type 0 and 1 galaxies also have an associated mass $M_{halo}$ which is the mass of the self-bound subhalo which surrounds them. Finally, halos of Type 0 galaxies have a conventional “virial mass” $M_{200}$, defined as the total mass within the largest sphere surrounding them with an enclosed mean density exceeding 200 times the critical value. Below we will consider both $M_{halo}$ and $M_{200}$ as possible definitions for the “true” masses of M31 and the Galaxy.
We use the Millennium Simulation to construct samples of mock Milky Way/Andromeda galaxies and of mock Local Groups as follows. We begin by identifying all Type 0 or Type 1 galaxies with characteristic “rotation velocity” either in the narrow range $200 \le V_{max}< 250{{\rm \,km\,s^{-1}}}$ or in the wider range, $150 \le V_{max}< 300{{\rm \,km\,s^{-1}}}$. This produces samples of $166,090$ and $699,177$ galaxies respectively. The exclusion of Type 2 galaxies reduces the samples by about 5-6% in each case, but the excluded galaxies are in any case not plausible analogues for the Local Group giants since they are almost all members of large groups or clusters. We also consider subsamples in which the morphologies predicted by the semianalytic model are forced to approximate those of M31 and the Galaxy. Specifically, we require a bulge-to-total luminosity ratio in the range $1.2 \le M_{B,bulge}-M_{B,total}
< 2.5$ so that the disks are 2 to 9 times brighter than the bulges in the *B*-band. This morphology cut reduces the samples in the two $V_{max}$ ranges to $62,605$ and $271,857$ galaxies respectively.
We then identify Local Group analogues in each of these four samples by identifying isolated pairs with separations in the range of $500-1,000{\rm \,kpc}$ and with negative relative radial velocities. (Note that this is the true relative velocity rather than the relative peculiar velocity, i.e. we have added the Hubble expansion to the relative peculiar velocity and have required the result to be negative.) We identify isolated pairs by keeping only those which have no “massive” companion, defined as a galaxy with $V_{max} \ge
150{{\rm \,km\,s^{-1}}}$, within a sphere of 1 Mpc radius centred on the mid-point of the binary, and no nearby cluster, defined as a halo with $M_{200}>3\times
10^{13}{\rm \,M_{\odot}}$ within 3 Mpc of the mid-point of the binary. These cuts ensure that the dynamics are dominated by mass associated with the two main systems, as appears to be the case for the Local Group. For galaxies selected in the narrower $V_{max}$ range we then find $178$ pairs when the morphology cut is applied and $1,128$ pairs when it is not. For the wider $V_{max}$ range the corresponding numbers are $2,815$ pairs and $16,479$ pairs respectively.
When calibrating the TA estimator it proves advantageous to use simulated pairs with dynamical state quite close to that of the real Local Group. As we will see below, this eliminates some systems where the dominant motion is not in the radial direction and the TA therefore significantly underestimates the mass. We therefore make one final cut which requires the approach velocity of the two galaxies to lie between $0.5$ and $1.5$ times the value measured for the real system ($-130{{\rm \,km\,s^{-1}}}$). This results in our final sets of Local Group lookalikes. For the narrower $V_{max}$ range we end up with $117$ pairs when the morphology cut is applied and $758$ pairs when it is not, while for the wider $V_{max}$ range the corresponding numbers are $1,273$ pairs and $8,449$ pairs respectively.
When we study the application of the Timing Argument to the Milky Way–Leo I system, we consider individual galaxies from both our $V_{max}$ ranges. We require these to be isolated by insisting that there should be no bright/massive companion (with luminosity exceeding 10% of that of the host or $V_{max} >
150{{\rm \,km\,s^{-1}}}$) closer than 700 kpc and no massive group (defined as above) closer than 3 Mpc. This produces samples of $137,926$ and $266,229$ potential hosts in the cases with and without the morphology cut for the wider $V_{max}$ range, and $29,245$ and $57,816$ potential hosts for the narrower range. We then search for Leo I analogues around these hosts by identifying companions in the separation range 200 to 300 kpc with $V_{max}({\rm comp}) \leq 80{{\rm \,km\,s^{-1}}}$, $M_{B} < -16.7$ and $V_{ra} \ge 0.7 V_{max}({\rm host})$ where $V_{ra}$ is the relative radial velocity of the two objects and the last condition reflects the fact that Leo I is useful for estimating the Milky Way’s mass only because its recession velocity is comparable to the Galactic rotation velocity ($V_{ra} \sim 0.8 V_{max}({\rm host})$ for the real Leo I–Milky Way system). Pairs sharing the same MW-like host are excluded in the final list.
With these cuts we find $344$ and $896$ satellite-host pairs in the samples with and without the morphology cut for the looser $V_{max}$ range, and $168$ and $374$ for the tighter range. These relatively small numbers reflect the fact that only about 10% of potential hosts actually have a faint companion in this distance range which is still bright enough to be resolved, and fewer than 5% of these satellites are predicted to have positive recession velocities comparable to that observed.
Results {#result_section}
=======
Calibration of the Timing Argument mass for the Local Group
-----------------------------------------------------------
For each simulated Local Group analogue the separation and relative radial velocity of the two galaxies can be combined with the age of the Universe (taken to be 13.6 Gyr) to obtain a Timing Argument mass estimate $M_{TA}$ (Equations (1) to (3)). The true mass of the pair $M_{tr}$ is harder to define because of the extended and complex mass distributions predicted by the $\Lambda$CDM model. The mass of an individual dark halo is often taken to be $M_{200}$ the mass within a sphere of mean density 200 times the critical value, so a natural choice for $M_{tr}$ is the sum of $M_{200}$ for the two galaxies. The Millennium Simulation database only lists $M_{200}$ for Type 0 galaxies, those at the centre of the main subhalo of each friends-of-friends particle group. Many of our LG analogues lie within a single FOF group. One of the pair is then a Type 1 galaxy, the central object of a subdominant subhalo, and so has no listed value for $M_{200}$. In such cases we have gone back to the particle data for the simulation in order to measure $M_{200}$ directly also for these galaxies.
An alternative convention is to define $M_{tr}$ as the sum of the values of $M_{halo}$, the maximal self-bound mass of each subhalo; this is included in the database for both Type 0 and Type 1 galaxies. In the following we use the notation $M_{tr,200}$ and $M_{tr,halo}$ to distinguish these two definitions. For either we can calculate the ratio of true mass to Timing Argument estimate, $$A_x = M_{tr,x}/M_{TA},$$ where the suffix $x$ is $200$ or $halo$ depending on the definition adopted for $M_{tr}$. If the Timing Argument is a good estimator of true mass, our samples of LG analogues should produce a narrow distribution of $A$ values. This distribution then allows the TA mass estimate for the real Local Group to be converted into a best estimate of its true mass, together with associated confidence intervals.
Our preferred sets of Local Group analogues contain simulated galaxy pairs which mimic the real system in terms of morphology, isolation, pair separation and pair approach velocity. In addition, they require the halos of the simulated galaxies to have $V_{max}$ values within about $\pm 10\% $ and $\pm
35\%$ of those estimated for M31 and the Galaxy for the tight and loose ranges of $V_{max}$, respectively. In order to understand the influence of these constraints we give results below not only for our “best” samples but also for samples where the various constraints are relaxed. Thus, we consider samples in which 1) both morphology and isolation requirements are applied (our preferred case), 2) the isolation requirement is removed, 3) the morphology requirement is removed, and 4) both morphology and isolation requirements are removed. For each case, we compare results for the two allowed ranges of $V_{max}$ and we also examine the effect of loosening the radial velocity constraint to $V_{ra} < 0$.
Fig. \[hist\] gives histograms of the distribution of $A_{200}$ for a sample in the narrow $V_{max}$-range with our preferred isolation, morphology and radial velocity cuts, as well as for three samples with the same $V_{max}$ and $V_{ra}$ cuts but with reduced morphology and isolation requirements. Fig. \[narrow\_ratio\_cumu\_dist\] presents these same distributions in cumulative form and compares them with the corresponding distributions for samples with the loosened circular velocity requirement, $150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$. In both plots black curves refer to class (1) samples for which both isolation and morphology cuts are imposed, while red, green and blue curves refer to samples in classes (2), (3) and (4) respectively. Results for the broader $V_{max}$ selection are indicated by dashed curves in Fig. \[narrow\_ratio\_cumu\_dist\]. We give numerical results for various percentile points of these distributions in Table \[tb\_a200\_vra\], and repeat all these in Table \[tb\_a200\] for samples where the separation velocity requirement has been loosened to $V_{ra}
< 0$.
![Normalised histograms of $A_{200}$, the ratio of true mass to Timing Argument estimate, for samples of Local Group analogues with $200{{\rm \,km\,s^{-1}}}\le V_{max} < 250{{\rm \,km\,s^{-1}}}$ and $-195{{\rm \,km\,s^{-1}}}<V_{ra} < -65{{\rm \,km\,s^{-1}}}$. The black histogram refers to our preferred selection where both isolation and morphology requirements are imposed. For the red histogram the isolation requirement has been removed, for the green histogram the morphology requirement, and for the blue histogram both requirements.[]{data-label="hist"}](narrow_radial_a200_hist.ps){width="50.00000%"}
![Cumulative distributions of $A_{200}$ the ratio of true mass to Timing Argument estimate for Local Group analogues with $-195{{\rm \,km\,s^{-1}}}<
V_{ra} < -65{{\rm \,km\,s^{-1}}}$. The solid curves correspond to the four samples already plotted in Fig. \[hist\] while the dashed curves are for samples with $150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$. The colour coding is the same as in Fig. \[hist\]; black indicates samples with our preferred isolation and morphology constraints.[]{data-label="narrow_ratio_cumu_dist"}](radial_a200_cumuhist.ps){width=".5\textwidth"}
The first and most important point to note from these these figures and tables is that the median value of $A_{200}$ is very robust and only varies between $0.98$ and $1.34$ for our full range of sample selection criteria. With our preferred cuts the median values are $1.15$ and $0.99$ for the narrow and wide $V_{max}$ samples respectively. The best estimate of the true mass of the Local Group (for this definition) is thus very similar to its Timing Argument mass estimate, and depends very little on the calibrating sample of simulated pairs.
The second important point is that the width of the distribution of $A_{200}$ does depend on how the calibrating sample is defined. In particular, it is narrower for samples with the more restrictive $V_{max}$ range, and for given $V_{max}$ range it is smallest for samples with our preferred cuts, those which match the dynamical and morphological properties of the Local Group most closely. For the narrow $V_{max}$ sample the interquartile range is a factor of just $1.6$, and the upper and lower 5% points are separated by a factor of $3$. For the wider velocity range the interquartile range is a factor of $1.8$ and the 5% points are separated by a factor of $5.7$. This shows the Timing Argument to be remarkably precise for systems similar to the Local Group.
The broadening of the distribution as the selection requirements are relaxed is easy to understand. Removing the isolation requirements allows third bodies to play a significant role in the dynamics. This can extend the upper tail of the $A_{200}$ distribution if mass from the third body falls inside $R_{200}$ for one of the pair galaxies or if the gravity of the third galaxy produces a tidal field which opposes the attraction between the pair members. It can extend the lower tail if the mass of the third body lies between the pair members but outside their $R_{200}$ spheres, thus enhancing their mutual attraction without adding to their mass. Removing the morphology constraint moves the whole distribution towards larger values and this effect is most pronounced in the large $A_{200}$ tail. This is because objects with more dominant bulges have more complex merger histories. They typically form in denser regions and their halos tend to be more massive and to have more complex structure.
Loosening the requirements on $V_{max}$ affects the distribution in a complex way. There is a tight relation between $V_{max}$ and $M_{200}$ (also $M_{halo}$) in the $\Lambda$CDM structure formation model [e.g. @nfw97]. Thus if we place tight restrictions on the $V_{max}$ values of our galaxies, we will obtain a sample of Local Group analogues with a narrow range of $M_{tr}$ values. If, in addition, we force the parameters which enter in the Timing Argument (the pair separation and relative radial velocity) to lie in narrow ranges, then the TA mass estimate itself is tightly constrained. The distribution of $A_{200}$ is thus forced to be narrow as a consequence of our selection criteria.
A second effect is that most of the new pairs added by widening the requirement on $V_{max}$ have at least one galaxy with $150{{\rm \,km\,s^{-1}}}\le V_{max} < 200{{\rm \,km\,s^{-1}}}$, thus with relatively low $M_{tr}$. This simply reflects the strong dependence of halo abundance on $V_{max}$. Given that halo mass scales approximately as $V_{max}^3$ it is striking that the addition of so many pairs containing a “low mass” galaxy reduces the median value of $A_{200}$ by just 15%. The low tail of the distribution is more strongly affected, by almost a factor of $2$ at the lower 5% point, but the upper end of the distribution is barely affected at all. This demonstrates that the main body of the distribution is weakly affected by restrictions on $V_{max}$, but that the lower tail (which is needed to place a lower limit on the true mass of the Milky Way) is suppressed if $V_{max}$ is not allowed to take small values.
5% 25% 50% 75% 95% \# of pairs
---------------------------------------------------------------- ------ ------ ------ ------ ------ -------------
$200{{\rm \,km\,s^{-1}}}\le V_{max} < 250{{\rm \,km\,s^{-1}}}$
morphology, isolation 0.67 0.93 1.15 1.47 2.05 117
morphology, no isolation 0.61 0.93 1.14 1.52 2.09 155
no morphology, isolation 0.67 0.97 1.20 1.50 2.32 758
no morphology, no isolation 0.63 0.96 1.22 1.55 2.54 1015
$150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$
morphology, isolation 0.34 0.72 0.99 1.27 1.93 1273
morphology, no isolation 0.33 0.68 0.98 1.29 2.00 1650
no morphology, isolation 0.41 0.81 1.09 1.40 2.21 8449
no morphology, no isolation 0.35 0.77 1.08 1.43 2.41 11838
5% 25% 50% 75% 95% \# of pairs
---------------------------------------------------------------- ------ ------ ------ ------ ------ -------------
$200{{\rm \,km\,s^{-1}}}\le V_{max} < 250{{\rm \,km\,s^{-1}}}$
morphology, isolation 0.54 0.97 1.33 1.66 3.93 178
morphology, no isolation 0.45 0.94 1.26 1.66 3.93 241
no morphology, isolation 0.54 1.01 1.34 1.82 4.62 1128
no morphology, no isolation 0.42 0.96 1.34 1.93 5.11 1596
$150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$
morphology, isolation 0.28 0.85 1.19 1.64 3.30 2815
morphology, no isolation 0.22 0.77 1.16 1.64 3.38 3532
no morphology, isolation 0.31 0.89 1.23 1.76 4.06 16479
no morphology, no isolation 0.18 0.79 1.19 1.78 4.46 23429
In Table \[tb\_a200\] we show the effect of weakening the cut on relative radial velocity to require only that the two main galaxies be approaching. Again this has remarkably little effect on the median $A_{200}$ values. A comparison with Table \[tb\_a200\_vra\] shows them all to be increased by about 10%-15%. The effects on the tails of the distributions are more substantial. The 95% point is typically increased by about a factor of $2$. This is because the sample now includes a substantial number of pairs with small $V_{ra}$ (and thus smaller TA mass estimate) for which tangential motion is important for their current orbit. The 5% point of the distribution is significantly reduced, reflecting the fact that our restrictions on relative approach velocity exclude a non negligible number of systems with approach velocities [*larger*]{} than $195{{\rm \,km\,s^{-1}}}$, and thus with large TA mass estimates. Such systems must have more mass [*outside*]{} the conventional virial radii of the two galaxies than do typical Local Group analogues in our samples.
In conclusion, we believe our most precise and robust estimate of the distribution of $A_{200}$ to be that obtained with our preferred morphology, isolation and radial velocity cuts for $150{{\rm \,km\,s^{-1}}}\le V_{max} <
300{{\rm \,km\,s^{-1}}}$, and we will use this distribution in the next section to estimate the true mass of the Local Group. Although the tails of the distribution are suppressed still further for a narrower range of $V_{max}$, this is at least in part due to the artificial effects mentioned above. In addition the number of Local Group analogues is too small in this case for the tails of the distribution to be reliably determined. From Table \[tb\_a200\_vra\] we see that the best estimate of the true mass of the Local Group (which we take to be that obtained using the median value of $A_{200}$) is almost identical to the direct TA estimate. The most probable range of true mass (given by the quartiles of $A_{200}$) extends to values about 30% above and below this, while the 95% confidence lower limit on the true mass (given by the 5% point of the $A_{200}$ distribution) is a factor of $2.9$ smaller.
Application to the Local Group {#app_LG_section}
------------------------------
The three observational parameters needed to make a Timing Argument mass estimate for the Local Group are the separation between the two main galaxies, their radial velocity of approach and the age of the Universe. The latter is now determined to high precision through measurements of microwave background fluctuations. @spergel07 give $13.73\pm 0.16$ Gyr. The distance to M31 is also known to high precision. We adopt the value $784 \pm 21{\rm \,kpc}$ given by @sg98 based on red clump stars, noting that it agrees almost exactly with the slightly less precise value obtained by @holland98 from fits to the colour-magnitude diagrams of M31 globular clusters. Although the heliocentric recession velocity of M31 is known even more precisely ($-301\pm
1{{\rm \,km\,s^{-1}}}$ according to @cvdb99) the approach velocity of the two giant galaxies is less certain because of the relatively poorly known rotation velocity of the Milky Way at the Solar radius. @vdmg07 go through a careful analysis of the uncertainties and conclude that $V_{ra} = 130\pm
8{{\rm \,km\,s^{-1}}}$. Inserting these modern values into Equations (1) to (3) we obtain our Timing Argument estimate of the mass of the Local Group: $$M_{LG, TA} \approx 5.32 \pm 0.48 \times 10^{12} {\rm \,M_{\odot}}\, ,$$ where the uncertainty is dominated by that in the relative radial velocity. This uncertainty is still small in comparison to the scatter in the ratio of true mass to TA estimate, so we will neglect it in the following. The apocentric distance of the implied relative orbit of the two galaxies is $1103\pm 30{\rm \,kpc}$.
We now combine this Timing Argument estimate with the distribution of $A_{200}$ obtained in the last section for our most precise and reliable sample of Local Group analogues (the sample with our preferred morphology, isolation and radial velocity cuts and with the wider allowed range of $V_{max}$) to obtain our best estimate of the true mass of the Local Group, defined here as the sum of the $M_{200}$ values of the two main galaxies: $$M_{LG,true} = 5.27 \times 10^{12} {\rm \,M_{\odot}}\, ,$$ or $\log M_{LG,true}/M_{\odot} = 12.72$. The most plausible range for this quantity is then $[12.58, 12.83]$ with a 95% confidence lower limit of $12.26$, i.e $ M_{LG,true} > 1.81 \times 10^{12} {\rm \,M_{\odot}}$ at 95% confidence.
Application to the Milky Way {#app_MW_section}
----------------------------
We now calibrate the @zaritsky89 Timing Argument which estimates the mass of the Milky Way from the position and velocity of Leo I. This again assumes a radial Keplerian orbit, but Leo I is taken to have passed through pericentre and to be currently moving towards apogalacticon for the second time. Equations (1) to (3) then give a unique mass estimate $M_{MW,TA}$ for the system for any assumed distance and radial velocity. This is taken as the Milky Way’s mass since the mass of Leo I is negligible in comparison.
Proceeding as for the Local Group, we select Milky Way – Leo I analogues from the Millennium Simulation in order to study the relation of this TA estimate to the true mass of the Milky Way, which we again take to be $M_{200}$. Thus we define the ratio $$B_{200} = M_{200}/M_{MW,TA}$$ and investigate its distribution in various samples of analogue host-satellite systems. In particular, we consider samples of isolated host galaxies (as defined in Section \[data\_section\]) using both our looser and tighter $V_{max}$ ranges, both with and without cuts on central galaxy morphology, and requiring the distance, radial velocity and maximum circular velocity of the satellite to satisfy $200{\rm \,kpc}< r < 300{\rm \,kpc}$, $V_{ra}\ge 0.7V_{max}({\rm host})$ and $V_{max}({\rm comp}) \le 80{{\rm \,km\,s^{-1}}}$.
Results for these four samples are given in Table \[tb\_b200\] and the corresponding cumulative distributions of $B_{200}$ are plotted in Fig. \[B\_cumul\]. Scatter plots of $M_{200}$ against $M_{MW,TA}$ for the four samples are shown in Fig. \[mw\_mass\_scatter\]. The behaviour is quite similar to that of the Local Group TA mass estimator $A_{200}$. The median value of $B_{200}$ is robust and varies very little as the definition of the analogue sample is changed. Again it is 10 – 15% smaller for samples with the looser $V_{max}$ selection. Unlike the Local Group case, the median value of $B_{200}$ is about $1.6$ and so is significantly larger than unity. This shows that $M_{MW,TA}$ is biased low as an estimator of true Milky Way mass, reflecting the fact that tangential motions are often significant for satellites at the distance of Leo I. Assuming a purely radial orbit then results in an underestimate of the mass.
5% 25% 50% 75% 95% \# of pairs
---------------------------------------------------------------------------- ------ ------ ------ ------ ------ -------------
$200{{\rm \,km\,s^{-1}}}\le V_{max}({\rm host}) < 250{{\rm \,km\,s^{-1}}}$
morphology 0.71 1.27 1.71 2.01 2.62 168
no morphology 0.71 1.25 1.67 2.01 2.55 374
$150{{\rm \,km\,s^{-1}}}\le V_{max}({\rm host}) < 300{{\rm \,km\,s^{-1}}}$
morphology 0.39 1.04 1.50 1.89 2.47 344
no morphology 0.51 1.14 1.55 1.98 2.66 896
The width of the distribution of $B_{200}$ is significantly greater for samples with the looser $V_{max}$ selection, primarily through an extension of the tail towards low values. This resembles the behaviour we saw above for $A_{200}$ but it must have a different cause, since our selection criteria for Milky Way analogues put no upper bound on $V_{ra}$, instead placing a lower limit on $V_{ra}/V_{max}$. As a result they do not force an upper limit on $M_{MW,TA}$ of the kind imposed on $M_{LG,TA}$ by our upper limit on $V_{ra}$ for Local Group analogues. Fig. \[mw\_mass\_scatter\] shows that the tail of low $B_{200}$ values for the wider $V_{max}$ range is caused a relatively small number of systems for which $M_{MW,TA}$ is anomalously large. These are objects with anomalously large values of $V_{ra}$ and seem to occur preferentially at small $M_{200}$, corresponding to values of $V_{max}$ below $200{{\rm \,km\,s^{-1}}}$.
The bulk of the points in Fig. \[mw\_mass\_scatter\] scatter fairly symmetrically about the median relation $M_{200}= 1.6 M_{MW,TA}$ which we show as a dashed straight line. Their mean slope is somewhat steeper than strict proportionality because our distance constraint on “Leo I’s” is expressed in units of kpc rather than of $R_{200}$ or $V_{max}/H_0$. Distant outliers occur only the low $M_{200}$ side of this relation, suggesting that they may be a consequence of resolution problems in the Millennium Simulation. For $V_{max}\sim 150{{\rm \,km\,s^{-1}}}$, typical halos are represented by fewer than $1,000$ particles and it seems likely that difficulties in describing the dynamics of their satellite substructures may begin to surface. In addition, the sample sizes are relatively small, particularly when we impose a morphology cut, so that the estimates of the tails of the distributions may be noisy. This may explain in part the apparent excess of outliers in the morphology-selected sample with the wider $V_{max}$ range.
The observational data needed to obtain the TA estimate of the Milky Way’s mass are the age of the Universe and the Galactocentric distance and radial velocity of Leo I. As above, we take the age of the Universe to be $13.73\pm
0.16$ Gyr from @spergel07. For the heliocentric distance to Leo I we adopt $254\pm 19{\rm \,kpc}$ from @bellazzini04. The heliocentric radial velocity of Leo I is very precisely determined, $283\pm 0.5{{\rm \,km\,s^{-1}}}$ according to @mom07. Based on an assumed Galactic rotation speed at the Sun of $220\pm 15{{\rm \,km\,s^{-1}}}$, we derive a corresponding Galactocentric radial velocity of $175\pm 8{{\rm \,km\,s^{-1}}}$. When substituted into Equations (1) to (3), these parameters produce a TA estimate for the Milky Way’s mass of $$M_{MW,TA} = 1.57 \pm 0.20 \times 10^{12} {\rm \,M_{\odot}}\, .$$ As was the case for the Local Group, the fundamental observational quantities are so well defined that the uncertainty of this estimate is much smaller than the expected scatter in $B_{200}$. We will therefore neglect it in the following. The implied apocentric distance of Leo I is $619\pm 26{\rm \,kpc}$. Since this is about half the apocentric distance of the M31 – Milky Way relative orbit in the TA model of Section \[app\_LG\_section\], perturbations of the orbit of Leo I due to the larger scale dynamics of the Local Group seem quite likely.
![Cumulative distributions of $B_{200}$ the ratio of true Milky Way mass (taken to be $M_{200}$) to TA estimate for four samples of isolated Milky Way – Leo I analogues from the Millennium Simulation. The red curve refers to Milky Way analogues with $200{{\rm \,km\,s^{-1}}}\le V_{max}({\rm host}) < 250{{\rm \,km\,s^{-1}}}$ and with morphology matching the Milky Way. For the black curve the circular velocity requirement is loosened to $150{{\rm \,km\,s^{-1}}}\le V_{max}({\rm host}) < 300{{\rm \,km\,s^{-1}}}$, for the blue curve the morphology requirement is removed, and for the green curve both requirements are relaxed. In all cases we require $V_{ra}\ge
0.7 V_{max}({\rm host})$.[]{data-label="B_cumul"}](b200_cumuhist.ps){width=".5\textwidth"}
{width=".5\textwidth"}{width=".5\textwidth"}
For the reasons discussed above, we consider our most precise and robust estimate for the distribution of $B_{200}$ to be that obtained for host galaxies with $150{{\rm \,km\,s^{-1}}}\le V_{max}({\rm host}) < 300{{\rm \,km\,s^{-1}}}$ and with no morphology cut. The median of this distribution then gives our best estimate of the true halo mass of the Milky Way: $$M_{200,MW} = 2.43\times 10^{12} {\rm \,M_{\odot}}\, ,$$ or $\log M_{200}/M_{\odot} = 12.39$. The quartiles of the distribution imply $[12.25,12.49]$ for the most probable range of this quantity, while the 5% point implies a lower limit of $11.90$ at 95% confidence. Thus the implied mass of the Milky Way is roughly half that of the Local Group as a whole, as might be expected on the basis of the similarity of the two giant galaxies. It is quite similar to other recent estimates based on applying equilibrium dynamics to the system of distant Milky Way satellites and halo stars [e.g. @we99; @scb03]. A significantly smaller estimate came from the analysis of the high-velocity tail of the local stellar population by @smith07, but we note that such analyses, in reality, only place a lower limit on the mass of the halo, since the distribution of solar neighbourhood stars may well be truncated at energies significantly below the escape energy.
An alternative mass measure?
----------------------------
The halo masses we have quoted so far have been based on the “virial masses” $M_{200}$ of simulated halos. This choice is, of course, somewhat arbitrary, and it may not correspond particularly well to the radii within which individual isolated halos are approximately in static equilibrium. As an alternative convention, we here consider defining the mass of an individual halo to be that of the corresponding self-bound subhalo identified by the [SUBFIND]{} algorithm of @swtk01. This algorithm typically includes material outside the radius $R_{200}$ within which $M_{200}$ is measured, but it excludes any material which is identified as part of a smaller subhalo orbiting within the larger system. In this paper we denote this subhalo mass as $M_{halo}$.
In the left panel of Fig. \[mass\_scatter\] we plot $M_{halo}$ against $M_{200}$ for all halos in our preferred sample of Local Group analogues, that with our preferred morphology, isolation and radial velocity cuts and with $150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$. Black and red points in this plot refer to Type 0 and Type 1 subhalos respectively. The right panel of Fig. \[mass\_scatter\] is a similar plot for the “Milky Way” halos in our preferred sample of Milky Way – Leo I analogues, again the sample which is matched in morphology and which has the wider $V_{max}$ range. In both panels it is clear that the correspondence between the two mass definitions is quite tight, and that $M_{halo}$ tends to be somewhat larger than $M_{200}$. In addition the left panel shows that Type 1 halos have smaller $M_{halo}$ for given $M_{200}$ than do Type 0 halos, as would naively be expected. The average value of $\log M_{halo}/M_{200}$ for the halos in the left panel is $0.100$ for the Type 0’s and $-0.004$ for the Type 1’s, while it is $0.079$ for the “Milky Way” halos in the right panel.
{width="50.00000%"}{width="50.00000%"}
This close correspondence between the two mass definitions carries over to the distribution of our ratios of “true” mass to timing mass. In Table \[tb\_a200\_vra\_halo\] we give percentage points of the $A_{halo}$ distribution for the 8 samples of Local Group analogues already considered above. They can be compared directly with the numbers given in Table \[tb\_a200\_vra\] for these samples. To a good approximation the distribution of $A_{halo}$ agrees with that of $A_{200}$ except that all values are shifted upwards by about 16-20%.
The same is also true for estimates of the Milky Way’s mass obtained using the TA applied to Leo I. This can be seen from Table \[tb\_b200\_halo\] which repeats Table \[tb\_b200\] except that we now give percentage points for $B_{halo}$ rather than $B_{200}$. Clearly it is of rather little importance which definition of halo mass we adopt: the results obtained with our two definitions are very similar.
5% 25% 50% 75% 95% \# of pairs
---------------------------------------------------------------- ------ ------ ------ ------ ------ -------------
$200{{\rm \,km\,s^{-1}}}\le V_{max} < 250{{\rm \,km\,s^{-1}}}$
morphology, isolation 0.78 1.11 1.36 1.72 2.38 117
morphology, no isolation 0.69 1.14 1.36 1.77 2.39 155
no morphology, isolation 0.76 1.17 1.44 1.75 2.74 758
no morphology, no isolation 0.71 1.15 1.44 1.82 2.97 1015
$150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$
morphology, isolation 0.41 0.88 1.20 1.53 2.26 1273
morphology, no isolation 0.38 0.84 1.19 1.54 2.38 1650
no morphology, isolation 0.49 0.99 1.31 1.67 2.60 8449
no morphology, no isolation 0.37 0.92 1.30 1.71 2.85 11838
5% 25% 50% 75% 95% \# of pairs
---------------------------------------------------------------- ------ ------ ------ ------ ------ -------------
$200{{\rm \,km\,s^{-1}}}\le V_{max} < 250{{\rm \,km\,s^{-1}}}$
morphology 0.90 1.53 1.98 2.40 3.14 168
no morphology 0.90 1.50 1.97 2.39 3.21 374
$150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$
morphology 0.42 1.25 1.81 2.31 3.09 344
no morphology 0.61 1.37 1.88 2.36 3.23 896
Discussion and conclusions {#conclusions}
==========================
The statistical argument underlying the analysis of this paper is more subtle than it may at first appear, so it is worth restating it somewhat more formally in order to understand what is being assumed in deriving the mass estimates for the Local Group and for the Milky Way given above.
We believe that the mass distributions around galaxies are much more extended than the visible stellar distributions, and that these have been assembled from near-uniform “initial” conditions in a manner at least qualitatively resembling that in a $\Lambda$CDM universe. Thus the assembly histories of the Local Group and of the Milky Way’s halo differ in major ways from those assumed by the original Timing Arguments of @kw59 and @zaritsky89. In addition, the meaning of the derived mass values needs clarification. We wish to use the Millennium Simulation to calibrate the TA estimates against conventional measures of halo mass, and to test the general applicability of the Timing Argument. However, we want to do this in a way which avoids any significant dependence on the details of the $\Lambda$CDM model, for example, on the exact density profiles, abundances and substructure properties which it predicts for halos.
Our method uses the simulation to estimate the distribution of the ratio of “true” mass to TA mass estimate for samples of objects whose properties “resemble” those of the observed Local Group and Milky Way – Leo I systems. Our restrictions on separation and radial velocity implement this similarity requirement in a straightforward way, but our constraints on $V_{max}$ have a more complex effect. Although the true $V_{max}$ values for M31 and the Milky Way are very likely within our looser range ($150{{\rm \,km\,s^{-1}}}\le V_{max}
< 300{{\rm \,km\,s^{-1}}}$) the simulation exhibits a tight correlation between $V_{max}$ and $M_{200}$ . Imposing fixed limits on $V_{max}$ is thus effectively equivalent to choosing a fixed range of $M_{200}$. As a result, we are in practice estimating the distribution of $A_{200}$ or $B_{200}$ for systems of given [*true*]{} mass, subject to the assumed constraints on separation and radial velocity. However, when we apply our results to estimate true masses for the Local Group and the Milky Way, we implicitly assume that our distributions of $A_{200}$ and $B_{200}$ are appropriate for samples of given TA mass estimate, again subject to our constraints on separation and radial velocity. It is thus important to understand when these two distributions can be considered the same.
The relation can be clarified as follows. From the simulation we compile the distribution of $M_{tr}/M_{TA}$, or equivalently of $\Delta \equiv \ln M_{tr}
- \ln M_{TA}$, for systems with $\ln M_{tr}$ in a given range. We then implicitly assume that this distribution does not depend on $M_{tr}$, at least over this range, so that the result can be taken as an estimate of the probability density of $\Delta$ at given $M_{tr}$. Bayes Theorem then gives us the probability density function (pdf) for $\Delta$ at fixed $M_{TA}$: $$\begin{aligned}
f[\Delta | \ln M_{TA}] = \frac{f[\Delta, \ln M_{TA}]}{f[\ln M_{TA}]}\nonumber\\
= \frac{f[\Delta, \ln M_{tr}]}{f[\ln M_{TA}]}
\nonumber\\
= \frac{f[\Delta | \ln M_{tr}]\, f[\ln
M_{tr}]}{ f[\ln M_{TA}]}\nonumber\\
= f[\Delta | \ln M_{tr}]
\label{Bayes}\end{aligned}$$ The first line here simply writes the conditional pdf of $\Delta$ at given $M_{TA}$ in terms of the joint pdf of the two quantities and the pdf of $M_{TA}$. The second line then rewrites the joint pdf in terms of the equivalent variables $\Delta$ and $M_{tr}$, using the fact that the Jacobian of the transformation is unity. The third line re-expresses the joint pdf as the product of the pdf of $\Delta$ at given $M_{tr}$ times the pdf of $M_{tr}$. The final line then follows from the normalisation condition, [*provided*]{} that $f[\ln M_{tr}]$ is constant and $f[\Delta | \ln M_{tr}]$ is independent of $M_{tr}$. Thus, when estimating $M_{tr}$ from $M_{TA}$, we assume a uniform prior on $\ln M_{tr}$ and that the distribution of $\Delta$ does not depend on true mass. Both these assumptions appear natural and appropriate.
The analysis underlying the Timing Argument (Equations (1) – (3)) assumes that the relative orbit of the two objects is bound and has conserved energy since the Big Bang. Recently, @sales07 have shown that in $\Lambda$CDM models this assumption is significantly violated for a substantial number of satellites within halos comparable to that of the Milky Way. In particular, they demonstrate the presence of a tail of unbound objects which are being ejected from halos as a result of $3$-body “slingshot” effects during their first pericentric passage. These objects are typically receding rapidly from their “Milky Way”, as assumed in the @zaritsky89 argument, but they violate its assumption that the present orbital energy can be used to infer the period of the initial orbit (i.e. the time from the Big Bang to first pericentric passage). Clearly such objects should also be present in the Millennium Simulation, although lack of resolution might make them under-represented in comparison to the simulations analysed by @sales07. Thus our analysis takes the possibility of such ejected satellites into account, at least in principle. Objects of this type will show up as systems with anomalously large TA estimates for their halo mass, and Fig. \[mw\_mass\_scatter\] shows a number of outliers which could well be explained in this way. Issues of this kind do not effect TA-based estimates of the mass of the Local Group since the two big galaxies are currently approaching for the first time.
The only kinematic information about the relative orbit of M31 and the Milky Way used in our analysis is their current approach velocity. @vdmg07 show that geometric arguments can already constrain the transverse component also, and future astrometry missions such as SIM might allow $V_{tr}$ to be measured directly. Thus it is interesting to ask if our TA mass estimate could be significantly refined by measuring the full 3-D relative motion of the two galaxies, rather than just its radial component. We address this in Fig. \[a200\_vtr\_scatter\] which plots $A_{200}$, the ratio of true mass to TA estimate, against $V_{tr}$ for a sample of Local Group analogues with our preferred morphology, isolation and radial velocity cuts, and with $150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$. The median $V_{tr}$ for this sample is $86 {{\rm \,km\,s^{-1}}}$, comparable to the @vdmg07 estimate for the real system. There is no apparent correlation of $A_{200}$ with $V_{tr}$, and indeed, the medians of $A_{200}$ for the high and low $V_{tr}$ halves of the sample are both close to 1 and do not differ significantly. Pairs with high $V_{tr}$ do show larger [*scatter*]{} in $A_{200}$ than pairs on near-radial orbits. For given separation, radial velocity and age, the Kepler model implies a mass which increases monotonically with $V_{tr}$. The absence of a detectable trend in Fig. \[a200\_vtr\_scatter\] shows that uncertainties in $V_{tr}$ do not dominate the scatter in our TA mass estimate, and that a measurement of $V_{tr}$ will not substantially increase the precision with which the true mass can be measured.
![Scatter plot of $A_{200}$ versus transverse velocity for Local Group analogues in our sample with preferred morphology, isolation and radial velocity cuts and with $150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$. The vertical dashed line indicates the median $V_{tr}$. The distributions on either side of this line are each further split in half at the median values of $V_{tr}$ (the solid horizontal lines.) There is essentially no correlation in this plot, indicating that a measurement of the transverse velocity will not significantly improve the TA mass estimate.[]{data-label="a200_vtr_scatter"}](a200_vtr.ps){width="50.00000%"}
In conclusion, our analysis shows the Timing Argument to produce robust estimates of true mass both for the Local Group and for the Milky Way, as long as “true mass” is understood to mean the sum of the conventional masses of the major halos. For the Local Group as a whole, the estimate and confidence limits given in Section \[app\_LG\_section\] and in the Abstract appear reliable given the excellent statistics provided by the Millennium Simulation, the lack of any substantial dependence on our isolation and morphology cuts, and the relatively simple dynamical situation. Although the results based on Leo I’s orbit also appear statistically sound, the more complex dynamical situation offers greater scope for uncertainty, particularly when trying to place a lower limit on the mass of the Milky Way’s halo. On the other hand, our best estimate of this mass is just under half of our estimate of the sum of the halo masses of M31 and the Galaxy. Thus the picture presented by the data appears quite consistent, and gives no reason to be suspicious of the Milky Way results.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Andreas Faltenbacher for help in calculating the $M_{200}$ values of Type 1 halos; the referee, John Dubinski, for useful suggestions regarding the transverse velocity. YSL also thanks Martin C. Smith, Gabriella De Lucia and Amina Helmi for useful discussions, and the Max Planck Institute for Astrophysics for support during her visits there. SW thanks the Kapteyn Astronomical Institute for support as Blaauw Lecturer during extended visits to Groningen.
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\[lastpage\]
[^1]: Email: [email protected]
[^2]: Email: [email protected]
[^3]: http://www.mpa-garching.mpg.de/millennium
| 0 |
---
abstract: 'In a previous paper \[J. Chem. Phys. [**121**]{} 4501 (2004)\] a unique bipolar decomposition, $\Psi = \Psi_1 + \Psi_2$ was presented for stationary bound states $\Psi$ of the one-dimensional equation, such that the components $\Psi_1$ and $\Psi_2$ approach their semiclassical WKB analogs in the large action limit. Moreover, by applying the Madelung-Bohm ansatz to the components rather than to $\Psi$ itself, the resultant bipolar Bohmian mechanical formulation satisfies the correspondence principle. As a result, the bipolar quantum trajectories are classical-like and well-behaved, even when $\Psi$ has many nodes, or is wildly oscillatory. In this paper, the previous decomposition scheme is modified in order to achieve the same desirable properties for stationary scattering states. Discontinuous potential systems are considered (hard wall, step, square barrier/well), for which the bipolar quantum potential is found to be [*zero*]{} everywhere, except at the discontinuities. This approach leads to an exact numerical method for computing stationary scattering states of any desired boundary conditions, and reflection and transmission probabilities. The continuous potential case will be considered in a future publication.'
author:
- Corey Trahan and Bill Poirier
title: |
Reconciling Semiclassical and Bohmian Mechanics:\
II. Scattering states for discontinuous potentials
---
INTRODUCTION {#intro}
============
Much attention has been directed by theoretical/computational chemists towards developing reliable and accurate means for solving dynamical quantum mechanics problems—i.e., for obtaining solutions to the time-dependent equation—for molecular systems. Insofar as “exact” quantum methods are concerned, two traditional approaches have been used: (1) representation of the system Hamiltonian in a finite, direct-product basis set; (2) discretization of the wavefunction onto a rectilinear grid of lattice points over the relevant region of configuration space. Both approaches, however, suffer from the drawback that the computational effort scales exponentially with system dimensionality.[@bowman86; @bacic89] Recently, a number of promising new methods have emerged with the potential to alleviate the exponential scaling problem once and for all. These include various basis set optimization methods,[@poirier99qcII; @poirier00gssI; @yu02b; @wangx03b] and build-and-prune methods,[@dawes04] such as those based on wavelet techniques.[@poirier03weylI; @poirier04weylII; @poirier04weylIII]
On the other hand, a completely different approach to the exponential scaling problem is to use basis sets or grid points, that themselves evolve over time. The idea is that at any given point in time, one need sample a much smaller Hilbert subspace, or configuration space region, than would be required at all times—thus substantially reducing the size of the calculation. For basis set calculations, much progress along these lines has been achieved by the multi-configurational time-dependent Hartree (MCTDH) method, developed by Meyer, Manthe and co-workers.[@meyer90; @manthe92] More recently, time-evolving grid, or “quantum trajectory” methods[@lopreore99; @mayor99; @wyatt99; @wyatt01b; @wyatt01c; @wyatt] (QTMs) have also been developed, and for certain types of systems, successfully applied at quite high dimensionalities.[@wyatt01c; @wyatt]
QTMs are based on the hydrodynamical picture of quantum mechanics, developed over half a century ago by Bohm[@bohm52a; @bohm52b] and Takabayasi,[@takabayasi54] who built on the earlier work of Madelung[@madelung26] and van Vleck.[@vanvleck28] QTMs are inherently appealing for a number of reasons. First, they offer an intuitive, classical-like understanding of the underlying dynamics, which is difficult-to-impossible to extract from more traditional fixed grid/basis methods. In effect, quantum trajectories are like ordinary classical trajectories, except that they evolve under a modified potential $V+Q$, where $Q$ is the wavefunction-dependent “quantum potential” correction. Second, QTMs hold the promise of delivering exact quantum mechanical results without exponential scaling in computational effort. Third, they provide a pedagogical understanding of entirely quantum mechanical effects such as tunneling[@lopreore99; @wyatt] and interference.[@poirier04bohmI; @zhao03] They have already been used to solve a variety of different types of problems, including barrier transmission,[@lopreore99] non-adiabatic dynamics,[@wyatt01] and mode relaxation.[@bittner02b] Several intriguing phase space generalizations have also emerged,[@takabayasi54; @shalashilin00; @burghardt01a; @burghardt01b] of particular relevance for dissipative systems.[@trahan03b; @donoso02; @bittner02a; @hughes04]
Despite this success, QTMs suffer from a significant numerical drawback, which, to date, precludes a completely robust application of these methods. Namely: QTMs are numerically unstable in the vicinity of amplitude nodes. This “node problem” manifests in several different ways:[@wyatt01b; @wyatt] (1) infinite forces, giving rise to kinky, erratic trajectories; (2) compression/inflation of trajectories near wavefunction local extrema/nodes, leading to; (3) insufficient sampling for accurate derivative evaluations. Nodes are usefully divided into two categories,[@poirier04bohmI] depending on whether $Q$ is formally well-behaved (“type one” nodes) or singular (“type two” nodes). For stationary state solutions to the equation, for instance, all nodes are type one nodes. In principle, type one nodes are “gentler” than type two nodes; however, from a numerical standpoint, even type one nodes will give rise to the problems listed above, because the slightest numerical error in the evaluation of $Q$ is sufficient to cause instability.
In the best case, the node problem simply results in substantially more trajectories and time steps than the corresponding classical calculation; in the worst case, the QTM calculation may fail altogether, beyond a certain point in time. Several numerical methods, both “exact” and approximate, are currently being developed to deal with this important problem. The latter category includes the artificial viscosity[@kendrick03; @pauler04] and linearized quantum force methods,[@garashchuk04] both of which have proven to be very stable. While such approximate methods may not capture the hydrodynamic fields with complete accuracy in nodal regions, they do allow for continued evolution and long-time solutions, often unattainable via use of a traditional QTM. The “exact” methods include the adaptive hybrid methods,[@hughes03] and the complex amplitude method.[@garashchuk04b] In the adaptive hybrid methods, for which hydrodynamic trajectories are evolved everywhere except for in nodal regions, where the time-dependent Schrodinger equation is solved instead to avoid node problems. Although they have been applied successfully for some problems, these methods are difficult to implement numerically, since not only must the hydrodynamic fields be somehow monitored for forming singularities, but there must also be an accurate means for interfacing and coupling the two completely different equations of motion. The complex amplitude method is cleaner to implement, but is only exact for linear and quadratic Hamiltonians.
In a recent paper,[@poirier04bohmI] hereinafter referred to as “paper I,” one of the authors (Poirier) introduced a new strategy for dealing with the node problem, based on a bipolar decomposition of the wavefunction. The idea is to partition the wavefunction into two (or in principle, more) component functions, i.e. $\Psi = \Psi_1
+ \Psi_2$. One then applies QTM propagation separately to $\Psi_1$ and $\Psi_2$, which can be linearly superposed to generate $\Psi$ itself at any desired later time. In essence, this works because the equation itself is linear, but the equivalent Bohmian mechanical, or quantum Hamilton’s equations of motion (QHEM) are not.[@poirier04bohmI] In principle, therefore, one may improve the numerical performance of QTM calculations simply by judiciously dividing up the initial wavepacket into pieces.
Although bipolar decompositions have been around for quite some time,[@floyd94; @brown02] their use as a tool for circumventing the node problem for QTM calculations is quite recent. Two promising new exact methods that seek to accomplish this are the so-called “counter-propagating wave” method (CPWM),[@poirier04bohmI] and the “covering function” method (CFM).[@babyuk04] In the CPWM, the bipolar decomposition is chosen to correspond to the semiclassical WKB approximation,[@poirier04bohmI] for which all of the hydrodynamic field functions are smooth and classical-like, and the component wavefunctions are node-free. Interference is achieved naturally, via the superposition of left- and right-traveling (i.e. positive- and negative-momentum) waves. For one-dimensional (1D) stationary bound states, it can be shown that the resultant bipolar quantum potential $q(x)$ becomes arbitrarily small in the large action limit, even though the number of nodes becomes arbitrarily large. (Note: in accord with the convention established in [Ref. ]{}, upper/lower case will be used to denote the unipolar/bipolar field quantities). In the CFM, the idea is to superpose some well-behaved large-amplitude wave, with the actual ill-behaved (nodal or wildly oscillatory) wave, so as to “dilute” the undesirable numerical ramifications of the latter.
This paper is the second in a series designed to explore the CPWM approach, introduced in paper I. As discussed there in greater detail, there are many motivations for this approach, but the primary one is to reconcile the semiclassical and Bohmian theories, in a manner that preserves the best features of both, and also satisfies the correspondence principle. For our purposes, this means that the Lagrangian manifolds (LMs) for the two theories should become identical in the large action limit (Sec. \[scattering\]). As described above, a key benefit of the CPWM decomposition is an elegant treatment of interference, the chief source of nodes and “quasi-nodes”[@wyatt] (i.e. rapid oscillations) in quantum mechanical systems. An interesting perspective on the role of interference in semiclassical and Bohmian contexts is to be found in a recent article by Zhao and Makri.[@zhao03]
Whereas paper I focused on stationary bound states for 1D systems, the present paper (paper II) and the next in the series (paper III)[@poirier05bohmIII] concern themselves with stationary scattering states. The CPWM decomposition of paper I is uniquely specified for any arbitrary 1D state—bound or scattering—and in the bound case, always satisfies the correspondence principle. However, the non-$L^2$ nature of the scattering states is such that the paper I decomposition generally does [*not*]{} satisfy the correspondence principle in this case. Simply put, the quantum trajectories and LMs exhibit oscillatory behavior in at least one asymptotic region (thereby manifesting reflection), whereas the semiclassical LMs do not. This is not a limitation of the CPWM, but is rather due to the fundamental failure of the basic WKB approximation to predict any reflection whatsoever for above-barrier energies, as has been previously well established.[@berry72; @froman; @heading] In semiclassical theory, a modification must therefore be made to the basic WKB approximation, in order to obtain meaningful scattering quantities. As discussed in Sec. \[scattering\] and in paper III, our approach will be to apply a similar modification to the exact quantum decomposition (actually, a [*reverse*]{} modification) such that the correspondence principle remains satisfied, and the two theories thus reconciled, even for scattering systems.
It will be shown the modified CPWM gives rise to bipolar Bohmian LMs that are [*identical*]{} to the semiclassical LMs, regardless of whether or not the action is large. Put another way, this means that the bipolar quantum potentials $q$ effectively [*vanish*]{}, so that the resultant quantum trajectory evolution is [*completely classical*]{}. Moreover, the resultant component wavefunctions, $\Psi_1(x)$ and $\Psi_2(x)$, correspond asymptotically to the familiar “incident,” “transmitted,” and “reflected” waves of traditional scattering theory. Thus, the modified CPWM implementation of the bipolar Bohmian approach provides a natural generalization of these conceptually fundamental entities [*throughout all of configuration space*]{}, not just in the asymptotic regions, as is the case in conventional quantum scattering theory.
The above conclusions will be demonstrated for both discontinuous and continuous potential systems, in papers II and III, respectively. Discontinuous potentials—e.g. the hard wall, the step potential, and the square barrier/well—serve as a useful benchmark for the modified CPWM approach, because the scattering component waves (e.g. “incident wave,” etc.) in this case [*are*]{} well-defined throughout all of configuration space, according to a conventional scattering treatment. Although this is no longer true for continuous potentials, the foundation laid here in paper II can be extended to the continuous (and also time-dependent) case as well, as described in paper III. Additional motivation for the development of a scattering version of the CPWM, vis-a-vis the relevance for chemical physics applications, is provided in paper III. Additional motivation for the consideration of discontinuous potentials is provided in Sec. \[scattering\] of the present paper.
THEORY
======
Background
----------
### Bohmian mechanics {#Bohmian}
According to the Bohmian formulation,[@wyatt; @holland] the QHEM are derived via substitution of the 1D (unipolar) wavefunction ansatz, (x,t) = R(x,t) e\^[i S(x,t)/]{} into the time-dependent equation. For the 1D Hamiltonian, H = -[\^2 2 m]{} [\^2 x\^2]{} + V(x), \[hameqn\] this results in the coupled pair of nonlinear partial differential equations, where $m$ is the mass, $V(x)$ is the system potential, and primes denote spatial partial differentiation.
The first of the two equations above is the quantum Hamilton-Jacobi equation (QHJE), whose last term is equal to $-Q(x,t)$, i.e. comprises the quantum potential correction. The second equation is a continuity equation. When combined with the quantum trajectory evolution equations, i.e. the continuity equation ensures that the probability \[i.e. density, $R(x,t)^2$, times volume element\] carried by individual quantum trajectories is conserved over the course of their time evolution.
### CPWM decomposition for stationary states {#bipolar}
In paper I, we derived a unique bipolar decomposition, (x) = \_+(x) + \_-(x), \[bipolardecomp\] for stationary eigenstates $\Psi(x)$ of 1D Hamiltonians of the form, such that:
1. [$\Ppm(x)$ are themselves (non-$L^2$) solutions to the equation, with the same eigenvalue, $E$, as $\Psi(x)$ itself.]{}
2. [The invariant flux values, $\pm F$, of the two solutions, $\Ppm(x)$, equal those of the two semiclassical (WKB) solutions.]{}
3. [The median of the enclosed action, $x_0$, equals that of the semiclassical solutions.]{}
There are other important properties of the $\Ppm(x)$,[@poirier04bohmI] as discussed in Sec. \[intro\], and in [Ref. ]{}. Nevertheless, the above three conditions are sufficient to uniquely specify the decomposition. In the special case of bound (i.e. $L^2$) stationary states, the real-valuedness of $\Psi(x)$ implies that the $\Ppm(x)$ are complex conjugates of each other.
Scattering systems {#scattering}
------------------
It is natural to ask to what extent the above analysis may be generalized for scattering potentials. Certainly, $\Psi(x)$ itself is no longer $L^2$, nor even real-valued, and there are generally two linearly independent solutions of interest for each $E$, instead of just one. Condition (1) above poses no difficulty for $\Ppm(x)$, as these component functions are non-$L^2$ and complex-valued, even in the bound eigenstate case. In principle, condition (2) is not difficult either; although the flux value depends on the normalization of $\Psi$ itself, which is not $L^2$, certain well-established normalization conventions for scattering states exist, that can be applied equally well to semiclassical and exact quantum solutions. There is no action median [*per se*]{} for scattering states, as the action enclosed within the $\Psi_+(x)$ and $\Psi_-(x)$ phase space Lagrangian manifolds[@poirier04bohmI; @keller60; @maslov; @littlejohn92] (LMs) is infinite; however, the scattering analog of condition (3) is related to the asymptotic boundary conditions, and it is here that one encounters difficulty. Moreover, an additional concern is raised by the doubly-degenerate nature of the continuum eigenstates, namely: should each scattering $\Psi(x)$ have its [*own*]{} $\Ppm(x)$ decomposition, or should there be a single $\Ppm(x)$ pair, from which all degenerate $\Psi(x)$’s may be constructed via arbitrary linear superposition?
To resolve these issues, we will adopt the same general strategy used in paper I, i.e. we will resort to semiclassical theory as our guide, wherever possible. We will also exploit certain special features of the scattering problem not found in generic bound state systems, such as the asymptotic potential condition $V'(x) \ra 0$ as $x \ra \pm \infty$ (where primes denote spatial differentiation), and its usual implications for scattering theory and applications.[@taylor]
The basic WKB solutions are given by \^(x) = r\_(x) e\^[i s\_(x)/]{}, \[scsoln\] where r\_(x) = s’\_(x) = \[scrs\] The corresponding positive and negative momentum functions, specifying the semiclassical LMs, are given by $p^\sc_\pm(x) = \pm
s'_\sc(x)$. Equations (\[scsoln\]) and (\[scrs\]) apply to both bound and scattering cases; note that for both, $\Ppm^\sc(x)$ are complex conjugates of each other. The asymptotic potential condition ensures that these approach exact quantum plane waves asymptotically, with the usual scattering interpretations, i.e. $\Psi_+(x)$ in the $x\ra-\infty$ asymptotic region is the incoming wave from the left (usually taken to be the incident wave), $\Psi_+(x)$ as $x\ra\infty$ is the outgoing wave from the left (the usual transmitted wave), etc.
Insofar as determining the corresponding exact quantum solutions $\Ppm(x)$, the procedure described in paper I is still appropriate for bound and semi-bound (i.e. on one side only) states, in that the results satisfy the correspondence principle globally, as desired (for semi-bound examples, consult the Appendix). For true scattering states, however, this procedure fails, in the sense that if $\Psi_+(x)$ is chosen to match the normalization and flux of $\Psi^\sc_+(x)$ in the $x\ra\infty$ asymptote, then it will necessarily approach a nontrivial linear superposition of $\Psi^\sc_+(x)$ and $\Psi^\sc_-(x)$ in the $x \ra -\infty$ asymptote, and vice-versa. There is therefore an ambiguity as to how the corresponding quantum $\Ppm(x)$’s should be defined, i.e. which asymptotic region should be used to effect the correspondence. More significantly though, [*either*]{} choice will result in component functions $\Ppm(x)$ with substantial interference in one of the two asymptotic regions. This is due to partial reflection of the exact quantum scattering states, which is not predicted by the basic WKB approximation. Thus, in the large action limit, the exact quantum solutions manifest large-magnitude quantum potentials, $q_\pm(x)$, and rapidly oscillating field functions $q_\pm(x)$, $r_\pm(x)$, and $p_\pm(x)$—exactly the undesirable behavior that the CPWM was introduced to avoid—whereas the corresponding basic WKB functions are smooth, and asymptotically uniform.
The lack of any partial reflection is a well-understood shortcoming of the WKB approximation[@berry72; @froman; @heading; @poirier03capI]—i.e., the basic $\Ppm^\sc(x)$ components, though elegantly constructed from smooth classical functions $r_\sc(x)$ and $s_\sc(x)$, do not in and of themselves correspond to any actual quantum scattering solutions $\Psi(x)$. In light of the bipolar decomposition ideas introduced in paper I, however, our perspective is the reverse one: for any [*actual*]{} quantum $\Psi(x)$, can one determine an decomposition such that the resultant $\Ppm(x)$ resemble their well-behaved semiclassical counterparts, and is such a decomposition unique? Among other properties,[@poirier04bohmI] the $\Ppm(x)$ LM’s should become identical to the semiclassical LM’s in the large action limit, so as to satisfy the correspondence principle. Based on the considerations of the previous paragraph it is clear that the paper I decomposition does not achieve this goal, when applied to stationary scattering states.
We defer a full accounting of these issues—in the context of completely arbitrary continuous potentials $V(x)$—to paper III, wherein it will be demonstrated how to compute exact quantum reflection and transmission probabilities (and stationary scattering states) using only classical trajectories, and without the need for explicit numerical differentiation of the wavefunction. In the present paper, we lay the foundation for paper III, by focusing attention onto two key aspects whose development comprises an essential prerequisite.
First, as the paper III approach treats $V(x)$ as a sequence of steps,[@poirier05bohmIII] the present paper II will focus exclusively on the step potential and related discontinuous potential systems, for which $V(x) = \text{const}$ in between successive steps. Discontinuous potentials are important for chemical physics, because they model steep repulsive wells, and are used in statistical theories of liquids. Moreover, they hold a special significance for QTM methods, for which they serve as a “worst-case scenario” benchmark. Indeed, conventional QTM techniques [*always*]{} fail when applied to discontinuous potentials. To date, The only such calculations that have been performed[@holland] have computed the quantum potential from a completely separate time-dependent fixed-grid calculation (the “analytical approach”)[@wyatt] rather than directly from the quantum trajectories themselves. Even if one [*could*]{} propagate trajectories for discontinous systems using a traditional QTM, the trajectories that would be generated would be very kinky and erratic,[@holland] and a great many time trajectories and time steps would thus be required.
Second, since the new $\Ppm(x)$ do [*not*]{} satisfy condition (1), unlike the paper I CPWM decomposition, the time evolution of these two component functions is clearly not that of the time-dependent equation. Moreover, since the $|\Ppm|^2$ are constant over time \[because $|\Psi|^2$ itself is stationary, and is presumed unique\], [*the two $\Ppm(x,t)$ time evolutions must be coupled together*]{}. It is essential that the nature of this coupling be completely understood, in order that the present approach may be generalized to non-stationary state situations—e.g. to wavepacket scattering, as will be discussed in future publications. The ramifications for QTMs are equally important. Accordingly, the present paper focuses on the QTM propagation of the wavefunction and its bipolar components—with a keen eye towards generality and physical interpretation—even though the states involved are stationary. This approach leads to a pedagogically useful reinterpretation of “incident,” “transmitted,” and “reflected” waves—very reminiscent of ray optics in electromagnetic theory—which is applicable much more generally than traditional usage might suggest.
Basic applications {#timeevol}
------------------
The necessary theory will be developed over the course of a consideration of various model application systems of increasing complexity.
### free particle system {#free}
Let us first consider the simplest case imaginable, the free particle system, $V(x)=0$. In this case, the exact solutions $\Ppm(x) = \Ppm^\sc(x)$ clearly satisfy the conditions of Sec. \[bipolar\], and the bipolar quantum potentials $q_\pm(x)$ are zero everywhere. Thus, the bipolar decomposition developed for bound states in paper I can be used directly with this continuum system, requiring only the slight modification that arbitrary linear combinations of $\Psi_+^\sc(x)$ and $\Psi_-^\sc(x)$ are to be allowed, in order to construct arbitrary scattering solutions $\Psi(x)$. For convenience, the linear combination coefficients will from here on out be directly incorporated into the amplitude functions, $r_\pm(x)$, and phase functions, $s_\pm(x)$, so that is still correct.
If from all solutions $\Psi(x)$ one considers only that which satisfies the usual scattering boundary conditions (i.e. incident wave incoming from the left) then the negative momentum wave $\Psi_-$ vanishes, and $\Psi(x)=\Psi_+(x)$. There is zero reflection, and $100\%$ transmission. Put another way, the incident flux, $\lim_{x\ra -\infty} j_+(x)$, is equal to the transmitted flux, $\lim_{x\ra +\infty} j_+(x)$, where \[both flux values are equal to $F$, as in \].
In the quantum trajectory description, flux manifests as probability-transporting trajectories, which move along the LMs. For the boundary conditions described above, there are only positive momentum trajectories, moving uniformly from left to right with momentum $p_+(x) = \sqrt{2mE}$. If a $\Psi_-(x)$ contribution were present, its trajectories would move uniformly in the opposite direction \[$p_-(x)=-\sqrt{2mE}$.\] Since the two components $\Ppm(x)$ are in this case uncoupled, the positive and negative momentum trajectories would have no interaction with each other.
### hard wall system {#hardwall}
We next consider the hard wall system: V(x) = In the $x\le0$ region, the two $\Ppm(x)$ components are exactly the same as in the free particle case, except that the $\Psi(0)=0$ boundary condition imposes the additional constraints, s\_-(0) = s\_+(0) + (2 ) ; r\_-(0) = r\_+(0). \[hwconst\] This also results in only [*one*]{} linearly independent solution instead of two, i.e. $\Psi(x) \propto \sin(k x)$, with $k =
\sqrt{2 m E}/\hbar$. Regarding the LMs and trajectories, in the $x<0$ region, these are identical to those of Sec. \[free\], e.g. the $\Psi_+(x)$ LM trajectories move uniformly to the right, towards the hard wall at $x=0$.
It is natural to ask what happens when the $\Psi_+(x)$ LM trajectories actually reach $x=0$. There are two reasonable interpretations. The first is that the trajectories keep moving uniformly into the $x>0$ region of configuration space. This approach treats the hard wall system as if it were the free particle system, but with the $x>0$ region effectively ignored.[@poirier00qcI] This underscores the fact that unlike $\Psi(x)$ itself, the individual $\Ppm(x)$ components [*per se*]{} are unconstrained at the origin—though the constraint implies a unique correspondence between the two. This interpretation also makes it clear that for the hard wall system, the paper I decomposition is essentially identical to the present decomposition, as is worked out in detail in the Appendix.
In the second interpretation, the effect of the hard wall at $x=0$ is to cause instantaneous elastic reflection of a $\Psi_+(x)$ LM trajectory momentum, from $p = p_+ = +\sqrt{2 m E}$ to $p = p_- =
-\sqrt{2 m E}$. Afterwards, the reflected trajectory propagates uniformly backward, along the $\Psi_-(x)$ LM. In this interpretation, the trajectories never leave the allowed configuration space, $x\le0$. However, wavepacket reflection is essentially achieved via trajectory [*hopping*]{} from one LM to the other—not unlike that previously considered, e.g., in the context of non-adiabatic transitions.[@tully71] The trajectory hopping interpretation is adopted in the present paper, and in paper III, but the first interpretation will also be reconsidered in later publications. Note that for discontinuous potentials—and indeed more generally[@poirier05bohmIII]—one can regard trajectory hopping as the [*source*]{} of $\Ppm(x)$ interaction coupling.
For the hard wall case, trajectory hopping only manifests at $x=0$, the sink of all $\Psi_+(x)$ LM trajectories, and the source of all $\Psi_-(x)$ LM trajectories. If these trajectories are to be regarded as one and the same via hopping, then a unique field transformation for $r$, $s$, and all spatial derivatives, must be specified. Fortunately, the unique correspondence between $\Psi_+(x)$ and $\Psi_-(x)$ described above, enables one to do just that. In particular, specifies the correct transformations for $r$ and $s$, as transported by the quantum trajectories. All spatial derivatives of arbitrary orders can then be obtained via spatial differentiation of —although in the hard wall case, only the $s'$ condition, $p_-(0) = - p_+(0)$ is relevant, because all higher order derivatives are identically zero.
Since the magnitudes of the $p$ and $r$ fields associated with a given quantum trajectory are unchanged as a result of the trajectory hop, implies that the incident and reflected flux values are the same (apart from sign), and so the scattering system exhibits 100% reflection and zero transmission (along each LM, the flux is invariant[@poirier04bohmI]). These basic facts of the hard wall system are of course well understood. The point, though, is that we have now obtained the information in a time-[*dependent*]{} quantum trajectory manner, rather than through the usual route of applying boundary conditions to time-[*independent*]{} piecewise component functions. In other words, now refers to individual [*quantum trajectories*]{}, rather than to wavefunctions.
This shift of emphasis is very important, and leads to quite a number of conceptual and computational advantages. For instance, the standard description of the hard wall stationary states would decompose these into plane wave components interpreted as “incident” and “reflected” waves. This language suggests a process, or change over time—i.e. a state that is initially incident, at some later time is somehow transformed into a reflected state. Nothing in the standard description, however, would seem to render transparent the usage of such terminology, i.e. $\Psi(x)$ is stationary, and so the reflected and transmitted components are in fact [*both*]{} present for all times. Of course, a localized superposition of stationary states, i.e. a wavepacket, may well exhibit such an explicit transformation over the course of the time evolution, as such a state is decidedly non-stationary. Indeed, wavepackets are relied upon by the more rigorous formulations of scattering theory, in order to justify the use of terms such as “reflected wave,” even in a stationary context.[@taylor] Such formulations, though certainly legitimate, seem always to require a clever use of limits, the subtle distinction between unitary and isometric transformations, and other esoteric mathematical tricks.
On the other hand, the time-dependent bipolar quantum trajectory hopping picture presented above provides a physicality to such language that is immediately apparent. Over the course of the time evolution, although the wavefunction as a whole is stationary, each individual [*trajectory*]{} is first incident from the left, then collides with the hard wall, and is subsequently reflected back towards the left (i.e. towards $x\ra-\infty$). The bipolar quantum trajectories are all classical, as the bipolar quantum potentials, $q_\pm(x)$, are zero everywhere except at the wall itself. Interference arises naturally from the superposition of the two LMs—i.e., from the trajectories that have already progressed to the point of reflecting, vs. those that have not reflected yet. In contrast, since $\Psi(x)$ itself exhibits very substantial interference, and an infinite number of nodes, the traditional unipolar QTM treatment would be very ill-behaved, i.e. $R(x)$ would oscillate wildly in the large $k$ limit, and $Q(x)$ would be numerically unstable near the nodes. Apart from these important pragmatic drawbacks, the incident/reflected interpretation of the quantum trajectories would also be lost.
The bipolar quantum trajectory description of the hard wall system is very reminiscent of ray optics, as used to describe the reflection of electromagnetic waves off of a perfectly reflecting surface.[@jackson] Indeed, much can be gained from applying a ray optics analogy to quantum scattering applications, especially where discontinuous potentials are concerned. One can construct a simple gedankenexperiment as follows. Let $x_L<0$ denote some effective left edge of the system, well to the left of the interaction region. At some initial time $t=0$, all trajectories on the positive LM lying to the right of $x_L$ are [*ignored*]{}, as is the negative LM altogether. One then evolves the retained trajectories over time, and monitors the contribution that just these trajectories make to the total wavefunction. In some respects, it is as if the point $x_L$ were serving as the initial wavefront for some incoming wave, that at $t=0$ had not yet reached the hard wall/reflecting surface. Of course, if the actual wave were in fact truncated in this fashion, then the discontinuity in the field functions at the wavefront would result in a very non-trivial propagation over time, owing to the high-frequency components implicitly present. For actual waves, the precise nature of the wavefront is known to have a tremendous impact on the resultant dynamics.[@jackson; @brillouin14] We avoid such complicating details by always interpreting the “actual wave” to be the full stationary wave itself, i.e. the truncation is conceptual only.
In the ray optics analogy, the above situation is like a source of light located at $x_L$, which is suddenly “turned on” at $t=0$. It takes time for the wavefront to propagate to the reflecting surface, and additional time for the reflected wavefront to make its way back to $x=x_L$. Prior to the latter point in time, the evolution of the truncated electromagnetic wave is decidedly [*non*]{}-stationary; afterwards however, a stationary wave is achieved, at least within the region of interest, $x_L\le x \le 0$, as the wavefront has by this stage propagated beyond this region. The same qualitative comments apply to the bipolar quantum case, although of course the evolution equations are different.
A similar prescription may be used to achieve rudimentary “wavepacket dynamics,” even in the context of purely stationary states. Instead of retaining [*all*]{} initial trajectories that lie to the left of $x_L$, one retains only those that lie within some finite interval. The resulting time evolution is analogous to a light source that is turned on at $t=0$, and then turned off at some later time (prior to when the wavefront arrives at the reflecting surface). The initial “wavepacket” has uniform density, and moves with uniform speed towards the hard wall. Interference fringes then form after the foremost trajectories have been reflected onto the negative LM. Eventually, all trajectories within the interval are reflected, at which point interference ceases (the nodes are “healed”[@wyatt]), uniform density is restored, and the reflected wave travels with uniform speed in the reverse direction, back towards the starting point $x_L$. Qualitatively, this behavior is clearly similar to that undergone by actual wavepackets reflecting off of barrier potentials.
More complicated applications {#complicated}
-----------------------------
The ideas described above can be easily extended to more complicated discontinuous potential systems, such as up- and down-step potentials, and any combination of multiple steps, e.g. square barriers and square wells. In paper III, they will even be extended to arbitrary continuous potentials.[@poirier05bohmIII] In every case, the ray optics analogy from electromagnetic theory may also be extended accordingly. This approach provides a useful perspective on global reflection and transmission in scattering systems, and in particular, demonstrates how such quantities may be obtained from a single, universal expression for local reflection and transmission.
### step potential system—above barrier energies {#stepabove}
We next consider the step potential system: V(x) = \[steppot\] Classically, this system exhibits 100% transmission if the trajectory energy is above the barrier (i.e. $E>V_0$), and 100% reflection if the trajectory energy is below the barrier ($E<V_0$). Quantum mechanically, all above barrier trajectories are found to exhibit partial reflection and partial transmission, although there is a general increase in transmission probability with increasing energy. The below barrier quantum trajectories exhibit 100% reflection, as in the classical case; however, they also manifest tunneling into the classically forbidden $x>0$ region. Thus even quantum mechanically, the the above and below barrier cases must be handled somewhat differently.
To begin with, we consider the above-barrier case. Note that the LM’s are unbounded in either direction, i.e. the classically allowed region extends to both asymptotes, $x\ra \pm \infty$. Incoming trajectories can therefore originate from either asymptote, thus giving rise to two linearly independent solutions, $\Psi(x)$. This is in stark contrast to the hard wall system, for which incoming trajectories could only originate from $x\ra -\infty$, thus resulting in only one linearly independent solution for $\Psi(x)$.
In the standard time-independent picture, one starts with the four piecewise solutions, (x) = e\^[i p\_A x/]{} (x) = e\^[i p\_B x/]{}, \[steppieces\] where region $A$ corresponds to $x\le 0$,region $B$ to $x \ge 0$. The momenta values are classical, i.e. p\_A = p\_B = . \[steppees\] Matching $\Psi(x)$ and $\Psi'(x)$ boundary conditions at $x=0$, and specifying asymptotic boundary conditions for $\Psi(x)$, then enables a unique determination of the four complex coefficients $A_\pm$ and $B_\pm$ in (x) = . \[stepwhole\]
In general, the solution coefficients depend on the particular stationary solution of interest. For the usual scattering convention of an incident wave incoming from the left (Fig. \[stepabovefig\]) the solutions are
where $R$ and $T$ are respectively, reflection and transmission amplitudes. When flux is properly accounted for, the resultant reflection and transmission probababilities (which add up to unity) are given by P\_ = |R|\^2 ; P\_ = |T|\^2. \[steprefltrans\] Note that above are correct for both an “up-step” and a “down-step”—i.e. for $V_0$ positive or negative. We can also apply these equations to the “opposite” boundary conditions, i.e. to an incident wave incoming from the right, by simply transposing $A$ and $B$ subscripts, and $+$ and $-$ subscripts ($p_A$ and $p_B$ are still positive). This is important, because any stationary solution $\Psi(x)$ can be obtained as some linear superposition of left-incident and right-incident solutions.
![Component waves for a left-incident stationary eigenstate of the up-step barrier problem with $E>V_0$, as described in Sec. \[upstep\]. Solid and dashed lines represent real and imaginary contributions, respectively. The dot-dashed line denotes the location of the step.[]{data-label="stepabovefig"}](figures/Fig1.eps)
Regarding the time-dependent interpretation, it is evident that upon reaching the step discontinuity, left-incident trajectories must be partially reflected and partially transmitted. The trajectory is suddenly split into two, one that continues to propagate along the positive LM for the transmitted $B$ region (i.e. the $B+$ LM) and the other being instantaneously reflected down to the $A-$ LM. Moreover, since probability carried by individual quantum trajectories is conserved,[@wyatt; @holland] this splitting must be done in a manner that preserves both probability and flux. In other words, the local splitting of the trajectory at $x=0$ must correspond to , which is now regarded as a [*local*]{} condition, giving rise to local reflection and transmission amplitudes, $R$ and $T$. For the present step potential case, these local quantities are directly related to the global $P_{\text{refl}}$ and $P_{\text{trans}}$ values via . For multiple step potentials (Sec. \[steps\]), the global expressions above \[\] no longer apply; however, a local, time-dependent trajectory version of turn out to be correct.
Such an expression, immediately applicable to all single and multiple step systems, can be written as follows: In the above equations, “inc” refers to any trajectory, locally incident on some particular step from some particular direction, which spawns both a locally reflected trajectory, “refl,” and a locally transmitted trajectory, “trans”. The quantity $p_{\text{i/r}}$ is the (positive) momentum associated with the locally incident/reflected trajectory; similarly, $p_{\text{trans}}$ (also positive) is associated with the locally transmitted trajectory. For above-barrier incident trajectories, note that the local reflection and transmission amplitudes are both real, thus ensuring the reality of $r$ and $s$ for the spawned trajectories.
Returning to the step potential system, the ray optics picture can once again shed some interesting light. The optical analog of the step is an interface between two media with different indices of refraction. Light incident on such an interface will partially reflect back towards the original source, and partially refract forwards into the new medium. The refraction is completely analogous to the discontinuous change in momentum, $(p_B-p_A)$, that suddenly occurs as one crosses the step (Fig. \[trajfig\]). In any event, the ray optics gedankenexperiment described in Sec. \[hardwall\] can also be applied to the step potential system, in order to obtain a particular stationary solution $\Psi(x)$ with any desired boundary conditions.
For instance, suppose one is interesting in constructing the left-incident wave solution, i.e. that of . At $t=0$, only the $\PAp$ wave is considered, and only those trajectories for which $x\le x_L$, as before. As the incident trajectories reach the step, two new waves are dynamically created from the spawned trajectories: a transmitted wave traveling to the right, and a reflected wave traveling to the left. A plot of the overall density $|\Psi(x,t)|^2$ so obtained will change over time, as the transmitted and reflected wavefronts propagate into their respective regions (Fig. \[sbbelowfig\]). Eventually, however, these wavefronts will propagate beyond the region of interest, i.e. $x_L
\le x \le x_R$, where $x_R>0$ is the right edge of the region of interest. When this occurs, the solution for $\Psi(x)$ obtained within the region of interest will be exactly equal to the stationary solution with the desired boundary condition.
As in the hard wall case, one can also perform step potential “wavepacket dynamics” by restricting consideration to just those initial $\PAp$ trajectories lying within some coordinate interval. The wavepacket will propagate towards the step with uniform density and speed. As the first few trajectories hit the step, a uniform transmitted wave will be formed in the $B$ region. In the $A$ region, the sudden appearance of a $\PAm$ wave will introduce interference wiggles in the overall density plot (although no nodes [*per se*]{}, owing to partial reflection only). Eventually, after all trajectories have progressed beyond the step, well-separated transmitted and reflected wavepackets emerge, propagating in their respective spaces and directions. There is no longer any interference in the $A$ region, as the incident wave is now gone, having been completely divided into the two final contributions. $P_{\text{refl}}$ and $P_{\text{trans}}$ values may be determined via monitors placed at $x_L$ and $x_R$, either by integrating probability over time as the respective wavepackets travel through, or by recording the (constant) amplitude values $R$ and $T$, and applying .
### step potential system—below barrier energies {#stepbelow}
The case for which the incident trajectory energies are below $V_0$ requires special discussion. In this case, the classical LMs and trajectories are confined to the $A$ region only (i.e. to $x\le 0$), as the entire $B$ region is classically forbidden. In the language of Sec. \[scattering\], these below barrier states are therefore semi-bound, implying that there is only one linearly independent stationary solution, $\Psi(x)$ which without loss of generality, must be real-valued. This in turn implies that the $\Ppm(x)$ are complex conjugates of each other, as in the bound state case discussed in paper I. Indeed, one option is to simply apply the paper I decomposition to such problems. This approach is discussed in detail in the Appendix, wherein it is shown to provide a natural extension of classical trajectories into the tunneling region.
![Wavefunction plot for a left-incident stationary eigenstate of the up-step barrier problem with $E<V_0$, as described in Sec. \[upstep\]. Solid and dashed lines represent real and imaginary contributions, respectively, for the analytical solution. Squares and circles denote corresponding numerical results. The shaded box represents the tunneling region.[]{data-label="stepbelowfig"}](figures/Fig2.eps)
On the other hand, the trajectory hopping-based decomposition scheme offers a different, but also very natural means to accomplish the same task—which has the added advantage that all bipolar quantum potentials [*vanish*]{}, except at $x=0$. The idea is simply to treat all expressions in Sec. \[stepabove\] as being literally correct for the below barrier case as well, with the understanding that the requisite quantities need no longer be real. In particular, $p_{\text{trans}} = i \hbar \kappa$ \[\] becomes pure positive imaginary, implying that the transmitted trajectories “turn a corner” in the complex plane, and start heading off in the positive imaginary direction, with speed $\hbar \kappa/m$ (Fig. \[complexfig\]). Along this path, the transmitted wave is an ordinary plane wave; however, when analytically continued to the real axis in the $x>0$ region (via a $90^\circ$ clockwise rotation in the complex plane), the familiar exponentially damped form results (Fig. \[stepbelowfig\]).
For the reflected wave, states that the reflected “phase” remains unchanged. However, $r_{\text{refl}}$ is now [*complex*]{}, leading to an effective phase shift of $2\delta$, where $\delta$ is defined in the Appendix \[\]. For localized wavepackets, a physical significance can be attributed to this phase shift, in both quantum mechanics and electromagnetic theory; it is the source of the Goos-Hänchen effect,[@jackson; @hirschfelder74] a time delay observed in conjunction with total internal reflection. Consequently, in the time-dependent wavepacket context, it may be more appropriate to associate the phase shift with [*time*]{}, rather than with $s$ or $r$—specifically, with the delay time needed to accrue sufficient action so as to compensate for the shift. For stationary states, however, such a time delay would be inconsequential, because all trajectories are identical apart from overall phase. Consequently, we do not consider such time delays explicitly in this paper, though we will return to this issue in future publications.
### multiple step systems {#steps}
The most interesting case is that for which there are multiple discontinuities, occurring at arbitrary locations $x_k$ (with $k=1,2,\ldots,l$), and dividing up configuration space into $l+1$ regions, labeled $A$, $B$, $C$, etc. In each region, the potential energy has a different constant value, i.e. $V(x) = V_A$ in region $A$, etc. From an optics point of view, this system is analogous to a stack of different materials, each with its own thickness, and index of refraction. Our primary focus in this paper will be square barrier/well systems for which $l=2$, and $V_A = V_C$. However, all of the present analysis extends to the more general case described above.
In the standard time-independent picture, the solution is obtained via a straightforward generalization of Eqs. (\[steppieces\]), (\[steppees\]), and (\[stepwhole\]). However, even when comparable left-incident boundary conditions are specified as in Sec. \[stepabove\]—i.e. $A_+ = 1$, and (for $l=2$) $C_- =
0$—the remaining coefficient values are fundamentally different from those of the single-step case. To begin with, only the $l$’th step exhibits the characteristics of a (locally) left-incident single-step solution; all other steps involve four non-zero coefficients, corresponding locally to some superposition of left- and right-incident waves. Even more importantly, however, the expressions for the coefficient values as a function of system parameters [*in no way*]{} resembles ; in particular, these now depend explicitly on the $x_k$ values, as well as on $V_A$, $V_B$, etc. The same is also true for the global $P_{\text{refl}}$ and $P_{\text{trans}}$ expressions, as compared with .
It is this dependence on the other steps that gives rise to the global nature of the time-independent solutions; i.e. the coefficient values at one step depend in principle on the properties of all of the other steps, no matter how far away these might be located. Consequently, a reflection probability as obtained from the $A_-$ value associated with the first, $k=1$ step, cannot be determined without extending the analysis out to the final step at $x= x_l$, in the standard time-independent picture. On the other hand, a primary goal of the time-[*dependent*]{} approach is to construct a completely [*local*]{} theory, for which local reflection and transmission amplitudes associated with any given trajectory, as it encounters a given step $k$, depend [*only*]{} on the properties of the $k$’th step (i.e. on $x_k$, and on the $p$ or $V$ values to the immediate left and right of $x_k$). In fact, from the point of view of the given trajectory, it must be immaterial whether the potential contains other steps or not—implying that the correct local relations for the spawned trajectories, if they exist at all, must be exactly those already specified in .
How is it possible that for stationary wavefunctions, whose time evolution is presumably trivial, an inherently global problem can be converted to a local one, simply by switching from a time-independent to a time-dependent perspective? This is because of the bipolar decomposition, which provides each step with not one, but two sets of incident trajectories, one from the left, and one from the right. When there are multiple steps, not only does this result in a non-trivial superposition for the resultant locally reflected and transmitted waves, but the trajectories themselves are subject to multiple spawnings, which effectively enable them to traverse back and forth over the same regions of configuration space an arbitrary number of times (Fig. \[trajfig\]). This crucial feature ultimately gives rise to the rich global scattering behavior observed even in two-step systems. However, it is wholly missed by any time-independent treatment, even a bipolar one, which can only summarize the net superposition of all left-traveling and right-traveling waves.
![Bipolar trajectory plot for the square barrier problem with $E>V_0$, as described in Sec. \[squarebarrier\]. One trajectory in five is indicated in the figure. The black/gray solid lines indicate positive/negative LM trajectories, respectively. The open circles represent recombination points. The dashed lines denote the two barrier edges, $x_1=0$ and $x_2=1$.[]{data-label="trajfig"}](figures/Fig3.eps)
We now discuss how the local time-dependent theory described above gives rise to the correct stationary solutions, which is readily understood by invoking the ray optics description introduced earlier. For simplicity and definiteness, we consider only the square potential case, which is optically analogous to say, a single pane of glass surrounded by vacuum. If a single step gives rise to a single reflection, then two steps, like a pair of mirrors, results in an [*infinite*]{} number of reflections. The same is true of a pane of glass, within which a single beam of light will be reflected back and forth at the edges an arbitrary number of times. Of course, these reflections are not perfect; a portion of the incident flux always escapes as transmission into the surrounding vacuum. Consequently, each successive internal reflection is exponentially damped, in accord with .
If the globally incident wave is incoming from the left, then at $x_1$, there are two contributions to $\Psi_{B+}$. One contribution is the portion of the left-incident $\Psi_{A+}$ wave that is [*locally transmitted*]{} through the first step. Apart from a phase factor, the resultant $B_+$ value would be given by if this were the only contribution. However, there is also a contribution that arises from the [*locally reflected*]{} part of the right-incident wave, $\Psi_{B-}$. This contribution is zero for a single step system, but of course non-zero in the multiple step case. Although the second contributing wave is right-incident, we can still use to compute the contribution to $\Psi_{B+}$, as discussed in Sec. \[stepabove\]. For the second, $k=l=2$ step at $x_2$, there are only left-incident waves; consequently, $\Psi_{C+}$ and $\Psi_{B-}$ are obtained from a single source each, i.e. $\Psi_{B+}$ (Fig. \[trajfig\]).
![Seven snapshots of the superposition wavefunction, $\Psi(x,t)$, for the $E>V_0$ square barrier problem, as computed using the numerical algorithm of Sec. \[numerics\]. The shaded box represents the barrier region. All units are atomic. The evolving discontinuities found at intermediate times in these curves denote wavefronts for the newly created reflected/transmitted components. Over time, the magnitudes of these discontinuities (i.e. corrections) become arbitrarily small, signifying that numerical convergence to the left-incident stationary solution has been achieved.[]{data-label="sbabovefig"}](figures/Fig4.eps)
The above description refers to the stationary state result, obtained by our gedankenexperiment in the large time limit only. In practice this result would be achieved in stages. As in the previous examples, we imagine that at time $t=0$, one retains only those trajectories for which $x\le x_L < x_1$. This one-sided trajectory restriction is somewhat analogous to continuous wave cavity ring-down spectroscopy.[@wheeler98] When the wavefront first hits the first interface at $x=x_1$, there is partial reflection and transmission, exactly identical to what would happen for a single step system. The reflected wavefront propagates beyond the left edge of the region of interest at $x=x_L$, and for some time, the reflected amplitude passing through this left edge is constant. The initially transmitted wavefront eventually reaches the second step at $x=x_2$ (i.e. the far side of the pane of glass), leading to a second transmission into the $C$ region, and a second reflection back through the $B$ region. Eventually, the second transmitted wavefront reaches the right edge of interest at $x = x_R$, after which the transmitted amplitude remains constant for some time.
Neither the globally transmitted nor reflected amplitudes for the times indicated above, as determined via monitors at $x=x_L$ and $x
= x_R$ respectively, are correct. However, we have not yet described the steady state solution. To do so requires an accounting of the second reflected wavefront, which eventually reaches the $x=x_1$ step again, this time incident from the right. The resultant locally transmitted wave becomes an instantaneous second contribution to $\Psi_{A-}$, and the locally reflected wave plays the same role for $\Psi_{B+}$. These new contributions give rise to discontinuities in these waves, that subsequently propagate to the left and right, respectively (Fig.\[sbabovefig\]). The new $\Psi_{A-}$ wave discontinuity eventually reaches $x=x_L$, where it is recorded by the monitor, giving rise to a sudden change in the reflection probability value.
The $\Psi_{B+}$ discontinuity propagates to the second step, where it spawns new discontinuities in $\Psi_{C+}$ and $\Psi_{B-}$. The former constitutes the border between first- and second-order transmitted waves, registered at sufficiently later time by the monitor at $x=x_R$. The latter, second-order $\Psi_{B-}$ wave heads back towards the first step, to give rise to third-order waves, with commensurate discontinuities, etc. In principle, this process continues indefinitely, resulting over time in global transmitted and reflected waves of arbitrarily high order. However, and the relation $P_{\text{refl}} + P_{\text{trans}} = 1$ ensure that the result converges to a stationary solution exponentially quickly. Moreover, since $C_-$ is necessarily zero throughout this process, it is clear that the stationary state that is converged to is indeed the one corresponding to the desired boundary condition of a globally incident wave that is incoming from the left.
Note that in an actual optical system as described above, the spatial dimensionality is three rather than one, and the incident wave would usually be taken at some angle to the normal. If in addition, the beam has a finite width, then one would observe separate reflected beams for each order, of exponentially decreasing brightness. The one-dimensional quantum case, however, is analogous to a normal incident beam, for which all orders of reflection are superposed. In addition to providing a pedagogical understanding of the dynamics that is very much analogous to the optical example provided, the picture above also suggests a practical numerical method that may be used to obtain stationary scattering states of any desired boundary condition (via superposition of globally left- and right-incident wave solutions, obtained independently).
Note that the “wavepacket dynamics” version of the ray optics analogy may also be applied. In this case, the resultant initial square wavepacket is somewhat reminiscent of pulsed wave cavity ring-down spectroscopy.[@wheeler98] Once the wavepacket has penetrated the middle region $B$ (i.e. the pane of glass), it reflects back and forth between the two edges, with each reflection giving rise to a left- or right-propagating outgoing square wavepacket in region $A$ or $C$, and a temporary interference pattern in region $B$. The amplitude of the central wavepacket dissipates exponentially in time. All of this complicated behavior is indeed qualitatively observed in actual wavepacket dynamics for such systems, but in the present context, is reconstructed entirely from a single stationary state.
Numerical Details {#numerics}
=================
In this section, we discuss several remaining issues pertaining to the numerical methods used to generate and propagate the various bipolar component waves, for the examples discussed in Secs. \[complicated\] and \[results\]. For all of these examples, the numerical algorithm used corresponds to the gedankenexperiment with one-sided truncation, i.e. to continuous wave cavity ring-down. In essence, this consists of just two basic operations: (1) each piece-wise bipolar component of the wavefunction \[i.e. $\PApm(x)$, $\PBpm(x)$, etc.\] is independently propagated in time over its appropriate region of space, using the standard QHEMs and QTMs; (2) whenever a trajectory reaches a turning point, it is immediately deleted, and replaced with two new trajectories, spawned in the appropriate locally transmitted and reflected component LMs.
The first operation above, i.e. QTM propagation of the wavefunction components, is very straightforward. Note that for simplicity, we have throughout this paper used time-independent expressions for $\Psi(x)$ and its components, but in reality these evolve over time—even for stationary states, via $\dot s = \partial s(x,t) /
\partial t = - E$. We have therefore been rather lax in distinguishing Hamilton’s principle function from Hamilton’s characteristic function, although from a trajectory standpoint, it is always the former that is implied. Since each component is stationary in its own right, the time evolution of the hydrodynamic fields is governed by the quantum stationary Hamilton-Jacobi equation (QSHJE), rather than the QHJE. Moreover, the fact that the piecewise $r$ is constant implies that the component quantum potentials are [*zero*]{}, resulting in classical HJE’s and trajectories. These conclusions are trivially correct for the present paper, for which all components are plane waves; however, the arguments also extend to arbitrary continuous potential systems.[@poirier05bohmIII]
From a numerical perspective, the use of classical trajectories offers many advantages over a conventional QTM propagation. To begin with, the trajectories themselves are always smooth if $V(x)$ is smooth, resulting in far fewer trajectories and larger time steps than would otherwise be the case. Even more importantly, however, since the quantum potential is not required, there is no need to compute on-the-fly numerical spatial derivatives of the local hydrodynamic fields. Consequently, for a given component, the trajectories are completely independent and need not communicate—again resulting in fewer of them. Indeed, it is possible to perform an essentially exact computation using only a [*single trajectory per wavefunction component*]{}. This feature is particularly important for the very frontmost trajectory of the initial ensemble, which for brief periods at later times, will (via spawning) come to be the [*only*]{} trajectory to occupy a given component LM. The subsequent evolution of these lone trajectories does not require the presence of nearby trajectories.
The spawning of new trajectories, i.e. operation (2) above, also bears further discussion. In principle, this is always achieved via application of . For the above barrier case, $R$ and $T$ are both real, ensuring the reality of $r$ and $s$ for the spawned trajectories—although in the case of a down step, $R<0$, resulting in a negative $r_{\text{refl}}$. This is in accord with the conventions discussed in paper I. However, in this paper, we find it numerically convenient to adopt the more usual $r>0$ convention. Thus, if yields a negative $r_{\text{refl}}$, it is replaced with $-r_{\text{refl}}$, and $\pi$ is added to $s_{\text{refl}}$. A similar, but more complicated modification is also applied to the below-barrier trajectories, for which yield complex amplitudes. In this case, the phase shift is $2 \delta$, as discussed in Sec. \[stepbelow\] and the Appendix.
For a single step system, the algorithm is now essentially complete. At the initial time $t=0$, a variable number of particles (or synonymously, grid points) are distributed uniformly along the $\Psi_{A+}$ manifold, to the left of $x= x_L$. The extent of these points must be large enough that at the end of the propagation, there are still $\Psi_{A+}$ grid points that have not yet reached $x_L$. The grid spacing is mostly arbitrary, but must be small enough that at sufficiently later times, there is always at least one trajectory per component LM. The propagation is considered complete when the reflected and transmitted wavefronts travel beyond $x_L$ and $x_R$, respectively.
For multiple step systems, the situation is similar, but somewhat more complex. The primary new feature is the [*recombination*]{} of wavefunction components arising from two sources, i.e. from two locally incident waves coming from opposite directions. The present algorithm would seem to yield [*two*]{} subcomponent wavefunctions for every component, each with its own set of trajectories. If left “unchecked,” this would lead to undesirable further multifurcations for higher orders/later times. A simple solution would be to propagate each subcomponent long enough that there is at least one trajectory for each, then extrapolate the corresponding subcomponent wavefunctions to a common position, where a new trajectory is constructed for the superposed component wavefunction, which is then propagated in lieu of the subcomponent trajectories. This requires dynamical fitting (see below), or at the very least, extrapolation. Although these numerical operations would be very stable in the present context, to rule these out altogether as sources of error in Sec. \[results\], we have adopted a much simpler approach—i.e. the grid spacing is chosen such that trajectories from the two component waves incident on a given step always arrive at the same time (Fig. \[trajfig\]). The corresponding subcomponent wavefunction values are then simply added together when forming the spawned trajectory. Adopting once again the $r>0$ convention for the superposed component wave, $\Psi_\pm$, the corresponding field values are then obtained via $r=\sqrt{\Psi_\pm^{*}\Psi_\pm}$ and $s=\arctan\sof{\im(\Psi_\pm)/\re(\Psi_\pm)}$.
As discussed in Sec. \[steps\], multiple step systems allow for infinite reflections that perpetually modify $|\Psi(x,t)|^2$, in principle for all time. In practice however, there is exponential convergence within the region of interest, $x_L \le x \le x_R$, so that one would not run the calculation indefinitely, but only until the desired accuracy is reached. Accurate “error bars” on the computed global $P_{\text{refl}}$ and $P_{\text{trans}}$ values are conveniently provided by the magnitudes of the most recent discontinuous jumps as recorded by the monitors at $x_L$ and $x_R$. Note that the number of digits of accuracy scales only [*linearly*]{} with propagation time. However, the rate of convergence depends on the energy value. Near the barrier height, in particular, convergence may take quite a long time, as the exponent is close to zero. For all other energies, only a few “cycles” should be required, depending on the level of accuracy desired.
If in addition to reflection and transmission probabilities, the actual stationary solution over the region of interest is also desired, then it is necessary to reconstruct $\Psi(x)$. This is obtained from the final grid, after the propagation is finished, using a multiple step generalization of . The first step is to reconstruct the component wavefunctions $\PApm(x)$, $\PBpm(x)$, etc., via interpolation or fitting of the hydrodynamic field values from the corresponding dynamical grid points onto a much finer common grid (used e.g. for plotting purposes). The second step is to linearly superpose the $\pm$ components onto the plotting grid, and to assemble the pieces together over the coordinate range of interest. For the discontinuous systems considered here, the number of dynamical grid points per component can be as small as one—i.e. much smaller, even, than the number of wavelengths! To our knowledge, such performance has never been achieved previously by a QTM; however, it does require that the plotting grid be much finer than the dynamical grid, e.g. at least several points per wavelength, in order to adequately represent the interference fringes of the the superposed solution, $\Psi(x)$.
RESULTS
=======
In this section, we apply the numerical algorithm previously described to three different applications: the up-step potential, the square barrier, and the square well.
Up-step Potential {#upstep}
-----------------
The first system considered is the up-step potential, i.e. with $V_0 >0$. Since there are no multiple reflections, this is in principle a trivial application for the current algorithm; it therefore serves as a useful numerical test. Both above barrier (Sec. \[stepabove\]) and below barrier (Sec. \[stepbelow\]) energies are considered. We choose molecular-like values for the constants, i.e. $V_0 = 0.009$ hartree, and $m=2000$ a.u. The left and right edges of the region of interest are taken to be $x_L = -1.0$ a.u. and $x_R = 1.0$ a.u., respectively. At the initial time, $t=0$, 51 trajectory grid points are distributed uniformly over the interval $-4 \le x \le -1$ (grid spacing of $0.06$ a.u.). This number is far greater than what would be needed for dynamical purposes, but is chosen so as to avoid construction of a separate plotting grid (Sec. \[numerics\]). The hydrodynamic field functions for the initial $\Psi_{A+}(x)$ wavepacket over the above interval are taken to be $r(x)=1$ a.u.$^{-1}$ and $s(x) = \sqrt{2mE} x$.
For the above barrier calculation, the energy $E = 2 V_0 = 0.018$ hartree was used. The trajectory propagation and termination were performed exactly as described in Sec. \[numerics\]. The real and imaginary parts of all three resultant wavefunction components \[i.e. $\PApm(x)$ and $\PBp(x)]$ at the final time, $t=550$ a.u. are presented in Fig. \[stepabovefig\]. All three components exhibit the desired plane wave behavior, e.g. no interference is evident within a given component. The resultant $\Psi(x)$ does exhibit interference in the $A$ region, however, arising from the superposition of $\PAp$ and $\PAm$.
For the below barrier calculation, the system was given an energy equal to one half of the barrier height, i.e. $E = V_0/2 = 0.0045$. As per the discussion in Secs. \[stepbelow\] and \[numerics\], tunneling into the forbidden region $B$ is achieved, not through a quantum potential, but via analytic continuation. At sufficiently large time ($t=1100$ a.u.) the final wavefunction is reconstructed from the components, i.e. $\Psi_A(x) = \PAp(x) + \PAm(x)$, and $\Psi_B(x) = \PBp(x)$. The real and imaginary parts of the reconstructed wavefunction are presented in Fig. \[stepbelowfig\]. In the figure, squares and circles denote the numerical results obtained via the present algorithm, whereas the solid and dashed lines represent the well-known analytic solutions. The agreement is essentially exact. Note that the real and imaginary parts are in phase throughout the coordinate range—i.e., $S(x)$ is a constant, so apart from a phase factor, $\Psi(x)$ is real. Note also that the tunneling region exhibits the desired exponential decay.
Square Barrier {#squarebarrier}
--------------
The second system considered is the square barrier. This is a two-step potential ($l=2$), with $V_A = V_C = 0$, and $V_B = V_0>0$. The two steps comprise the left and right edges of the barrier, at $x_1 = 0$ and $x_2=w$, respectively. The constants are chosen as follows: $V_0 = 0.018$ hartree; $m=2000$ a.u.; $w=1$; $x_L=-1$ a.u., $x_R=2$ a.u. Initially, 75 trajectory grid points are distributed uniformly over the interval $-5 \le x \le -1$ (grid spacing of $0.05$ a.u.), which again, is far more than are dynamically required. The same initial hydrodynamic field functions are used as in Sec. \[upstep\].
Both above barrier ($E>V_0$) and below barrier ($E<V_0$) energies are considered. For the above barrier case, $E = 2 V_0 = 0.036$ hartree. Once again, the trajectory propagation and termination were performed exactly as described in Sec. \[numerics\]. In order to converge $P_{\text{trans}}$ to $10^{-4}$, a propagation time of $3000$ a.u. was required. This corresponds to 3 complete cycles, i.e. a 3rd-order calculation. Figure \[trajfig\] is a plot of the quantum trajectories for this calculation, in which every fifth trajectory for each of the five component wavefunctions is indicated. Trajectory spawning at the two steps is very clearly evident, as is recombination of pairs of incident waves (indicated by circles). On the whole, this figure demonstrates all of the anticipated analogues with ray optics, i.e. parallel trajectories, reflection and refraction.
In Fig. \[sbabovefig\], the time evolution of the superposition state $\Psi(x,t)$ is represented, via snapshots of the real and imaginary parts at seven different times. At $t=0$ a.u., the incident wavefront is located at $x=-1$ a.u. By $t=250$ a.u., the wavefront has spawned $\PAm(x)$ and $\PBp(x)$ trajectories; the former gives rise to the kink (really a discontinuity) somewhat to the left of the first step. By $t=550$ a.u. and $t=650$ a.u., the $\PAm(x)$ wavefront has moved outside the region of interest, though the $\PBp(x)$ wavefront has not quite reached the second step. After it does so, two new wavefronts are propagated along the $\PCp(x)$ and $\PBm(x)$ LMs (e.g. $t=800$ a.u.), the former of which propagates beyond the region of interest by $t=900$ a.u. Subsequent discontinuity magnitudes become exponentially smaller, so that at sufficiently large time (i.e. $t=2000$ a.u.), the resultant $\Psi(x)$ has converged to the correct stationary solution.
For the below barrier case, $E = V_0/2 = 0.009$ hartree, $x_L=-0.5$ a.u., and the other parameters are as above except $w=0.5$ a.u. Within the barrier, there are in principle an arbitrary number of reflections back and forth as before. However, substantial amplitude loss occurs due to tunneling, in addition to partial reflection, as a result of which fewer cycles are required in order to achieve the same $10^{-4}$ level of convergence ($t=1400$ a.u., or 2 cycles). Fig. \[complexfig\] indicates how the tunneling dynamics is achieved. After the wavefront hits the first step, the transmitted $\PBp$ wave is propagated along the imaginary axis ($iy$), until the point $y=w$ is reached. When this occurs, it is necessary to analytically continue $\PBp$ down to the real axis, in order to compute amplitudes for the new $\PCp(x)$ and $\PBm(x)$ trajectories that are spawned at the second step. The latter component propagates in the (negative) imaginary direction, along $w - iy'$, until $y'=w$, at which point analytic continuation is once more applied (this time for the first step), and the pattern repeated.
![Schematic of the algorithm used to propagate trajectories into the classically forbidden region of the $E<V_0$ square barrier problem, as discussed in Secs. \[numerics\] and \[squarebarrier\].[]{data-label="complexfig"}](figures/Fig5.eps)
Seven snapshots of the superposition density, $|\Psi(x,t)|^2$, are displayed in Fig. \[sbbelowfig\]. The initial density—equal to just the $|\PAp(x)|^2$ density—is uniform. After the wavefront encounters the first step, a reflected $\PAm(x)$ emerges, giving rise to clearly evident interference in the region $A$. The $\Psi_B(x)=\PBp(x)$ wave is at this stage perfectly exponentially damped. Upon encountering the second step, a second contribution, $\PBm(x)$ emerges; however, this first-order correction is already extremely small, owing to the large amount of tunneling that has occurred. The global transmitted wave, $\PCp(x)$, though small, is clearly seen to have uniform density.
In addition to the two detailed trajectory calculations described above, we computed $P_{\text{refl}}$ and $P_{\text{trans}}$ for a large range of $w$ and $E$ values, so as to fully explore (without loss of generality) the entire range of the square barrier problem. The numerical results are presented, and compared with known analytical values,[@gasiorowitz] in Fig. \[sbtransreflfig\]. Two aspects of this study bear comment. First, for all $w$ and $E$ values considered, the computed $P_{\text{refl}}$ and $P_{\text{trans}}$ values agree with the exact values to within an error comparable to that predicted by the level of numerical convergence. In particular, the oscillatory energy dependence is perfectly reproduced. Second, the closer the barrier peak is approached from either above or below in energy, the longer the time required to achieve a given level of convergence, as predicted in Sec. \[numerics\]. In particular, for the calculations closest to the barrier peak, 5-7 cycles were required in order to approximately maintain a $10^{-4}$ convergence of the transmission probability. Although a greater number of particles are required in this case, this poses no great limitation in practice, since one would presumably never require a calculation precisely at the peak energy.
![Seven snapshots of the superposition density, $|\Psi(x,t)|^2$, for the $E<V_0$ square barrier problem, as computed using the numerical algorithm of Sec. \[numerics\]. The shaded box represents the barrier region. All units are atomic. Note that interference manifests in the incident (left) region only after some incident trajectories have struck the left barrier edge, causing reflected trajectories to be created (i.e. just prior to $t=195$, initially). The most advanced reflected trajectory defines the reflected wavefront, manifesting as the left-moving discontinuity, e.g. at $t=195$ and $t=287$.[]{data-label="sbbelowfig"}](figures/Fig6.eps)
![Transmission and reflection probabilities as a function of energy, for square barrier potentials of three different widths, $w$, as discussed in Sec. \[squarebarrier\]. Solid lines denote analytical results; open/closed circles denote numerical results, as obtained via algorithm of Sec. \[numerics\]. The vertical dot-dashed lines represent the barrier height, i.e. $E=V_0$.[]{data-label="sbtransreflfig"}](figures/Fig7.eps)
Square Well {#squarewell}
-----------
As the final system, we consider the square-well potential, i.e. the square barrier but with $V_0<0$. In the scattering state context, there is no tunneling for this system, but in other respects it resembles the square barrier. From an optics point of view, the square well corresponds to a central medium with larger index of refraction than its surroundings, whereas the square barrier corresponds to a smaller index of refraction, giving rise to the possibility of total internal reflection (i.e. tunneling). The parameters are as in Sec. \[squarebarrier\], except that $V_0=0.009$ hartree, and three different $w$ values are considered: $w=2$ a.u., $w=4$ a.u., and $w=16$ a.u.
As the time evolution and trajectory pictures are similar to those of the previous sections, we focus only on the $P_{\text{refl}}$ and $P_{\text{trans}}$ calculations, for which once again, a large range of energies was considered ($0.0005<E<0.2$ hartree). The number of initial trajectories ranged from 50 to 200, for the highest to the lowest energies, respectively. The computed transmission/reflection probabilities were again converged to $10^{-4}$. The energy-resolved reflection and transmission probabilities are presented in Fig. \[swtransreflfig\].
![Transmission and reflection probabilities as a function of energy, for square well potentials of three different widths, $w$, as discussed in Sec. \[squarewell\]. Solid lines denote analytical results; open/closed circles denote numerical results, as obtained via algorithm of Sec. \[numerics\].[]{data-label="swtransreflfig"}](figures/Fig8.eps)
As in the square barrier case, excellent agreement is achieved with the exact analytical results, i.e. on the order of the level of convergence. This is true despite the fact that the square well energy curves are decidedly more oscillatory than the square barrier curves—particularly for wide barriers, for which the $E$ dependence is very sensitive indeed. One particularly important feature exhibited by the exact curves is the so-called Ramsauer-Townsend effect,[@gasiorowitz] i.e. the phenomenon of 100% transmission and zero reflection, even at very low energies. This occurs when $\sin(2k_{B}\,w)=0$, and may be regarded as a purely quantum mechanical resonance phenomenon. Yet it is reproduced perfectly here in the bipolar decomposition, using [*classical trajectories*]{}. Indeed, the bipolar picture provides an interesting physical explanation, i.e. the right-incident and left-incident waves of the first step give rise to spawned contributions to $\PAm(x)$ that exactly cancel each other out via destructive interference.
SUMMARY AND CONCLUSIONS {#conclusion}
=======================
As described in paper I, the equation is linear, yet the equivalent QHEM—obtained via substitution of the Madelung-Bohm ansatz into the equation—are not. This aspect of Bohmian mechanics suggests that it can be beneficial, both from a pedagogical and a computational perspective, to apply a suitable bifurcation (or “multifurcation”) to the wavefunction prior to applying the QHEM. Indeed, following the paper I CPWM bipolar decomposition for 1D stationary bound states,[@poirier04bohmI] the quantum trajectories become more well-behaved and classical-like, in precisely the limit in which there are more nodes, and the usual unipolar calculation breaks down. Moreover, the resultant component LMs admit a natural physical interpretation in terms of the corresponding semiclassical LMs.
In the generalization to the stationary scattering states considered here, a somewhat different bipolar decomposition is found to be required. The new decomposition is still unique, at least for discontinuous potentials. However, the resultant components $\Ppm(x)$ are no longer solutions to the equation in their own right, as a result of which their time evolution is coupled. The new scheme—though fundamentally different from the old one—nevertheless bears a correspondence to a modified version of semiclassical theory appropriate for scattering systems. Curiously, this semiclassical modification is [*not*]{} simply a higher order treatment in $\hbar$;[@berry72] if it were, the corresponding exact quantum modification considered here would not exist.
In any event, the new decomposition also gives rise to its own physical interpretation, specific to the scattering context. In particular, for right-incident boundary conditions, the left and right asymptotes of $\Psi_+(x)$ respectively represent incident and transmitted waves, whereas the left asymptote of $\Psi_-(x)$ represents the reflected wave \[the right asymptote of $\Psi_-(x)$ approaches zero\]. That these conceptually useful asymptotic bipolar assignments—found even in the most elementary treatments of scattering—may be extended [*throughout configuration space*]{} \[even for continuous potentials (paper III)\] represents an important leap forward, especially for QTMs.
Another pedagogically and numerically useful development from this approach is the inherently time-dependent ray optics interpretation that naturally arises, particularly in the discontinuous potential context. The ray optics approach is anticipated to be a relevant guiding force in subsequent generalizations of the present methodology, i.e. to continuous, multidimensional potential systems, and—as per the discussion in Secs. \[hardwall\] and \[stepabove\]—for non-stationary wavepacket dynamics. This approach also provides a much simpler trajectory-based explanation of scattering terminology as applied in a stationary context—e.g. why the “reflected wave” is so-called, despite being present from the earliest times—than those traditionally used.[@taylor]
The discontinuous potential applications considered in this paper—hard wall (Sec. \[hardwall\]), step potential (Secs. \[stepabove\], \[stepbelow\], and \[upstep\]), and square barrier/well (Secs. \[steps\], \[squarebarrier\], and \[squarewell\])—are significant for several reasons. To begin with, these are the first discontinuous applications of a genuine QTM calculation that have ever been performed, to the authors’ knowledge. The singular derivatives associated with discontinuous potential functions would wreak havoc with standard numerical differentiation routines. Second, the use of a time-[*dependent*]{} method for stationary, or time-[*independent*]{}, applications, is also significant. Ordinarily, the time dependence of stationary states is regarded as trivial. In the present context, this is true in a sense for the hard wall and step potential systems, because the correct answer is “built in” the method itself. For multiple step systems, however, the dynamical truncated wave approach, i.e. the gedankenexperiment introduced in Sec. \[hardwall\] and further developed in later sections, yields decidedly nontrivial results.
In particular, the algorithm uses only [*single step*]{} scattering coefficients to obtain global scattering quantities for [*multiple step*]{} systems. In effect, the time dependent nature of this approach allows computation of [*global*]{} properties using a completely [*local*]{} method. Not only were exact quantum results obtained for a full range of system parameters, but the numerical resources necessary to achieve this—i.e. the number of trajectories and time steps—were decidedly minimal. Indeed, the algorithm lives up to the promise made in paper I, of performing an accurate quantum calculation with [*fewer trajectories than nodes*]{}—a prospect virtually unheard of in a unipolar context.
In future publications, we will naturally attempt to generalize the methodology described here and in paper III, for the type of multidimensional time-dependent wavepacket dynamics relevant to chemical physics applications. In this context, the scattering version of the CPWM decomposition developed here is an absolutely essential first step, as reactive scattering is the underpinning of all chemical reactions. Additional discussion and motivation will be provided in paper III.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
This work was supported by awards from The Welch Foundation (D-1523) and Research Corporation. The authors would like to acknowledge Robert E. Wyatt and Eric R. Bittner for many stimulating discussions. David J. Tannor and John C. Tully are also acknowledged.
Appendix: Bipolar decomposition of semi-bound states {#appendix-bipolar-decomposition-of-semi-bound-states .unnumbered}
====================================================
As discussed in Secs. \[scattering\], \[hardwall\], and \[stepbelow\], semi-bound stationary states in 1D are bounded on one side only, as a result of which they are real-valued and singly-degenerate, like bound states. Consequently, they are amenable to the CPWM bipolar decomposition scheme introduced in paper I. In this appendix, we apply this decomposition to two semi-bound systems: the hard wall system, and the below-barrier up-step system.
Hard wall system {#hard-wall-system}
----------------
From [Ref. ]{}, the most general bipolar decomposition of a hard wall stationary state—corresponding to Sec. \[bipolar\] condition (1) only—is found to satisfy - = , where $s(x) = s_+(x) = -s_-(x)$, and $r_+(x) = r_-(x)$ is obtained from $s(x)$ via (without “sc” subscripts). The arbitrary parameters $F$ and $B$ are the invariant flux and median action parameters associated with conditions (2) and (3), respectively,[@poirier04bohmI] although the definition of $F$ has been changed slightly to account for the scattering normalization convention, $r_+(x) = 1$. Note that [*only*]{} the semiclassical values for these parameters yields a solution that satisfies the correspondence principle in the large action (i.e. $k$) limit. In particular, the choice $B=0$ and $F= \hbar k/m$ yields the desired semiclassical result, $s(x) = \hbar k x$; all other choices exhibit undesirable oscillatory behavior in $r_\pm(x)$, $s_\pm(x)$, and $q_\pm(x)$.
Up-step system
--------------
For the hard wall system considered above—which is just the special case of the up-step potential in the limit $V_0 \ra
\infty$—exact agreement is achieved between semiclassical and quantum LM’s in the $x<0$ region. This is the only region of interest for the hard wall system; however, for finite $V_0$ values—i.e. for general below-barrier up-step stationary states—there is of course also tunneling into the forbidden region, which must be accounted for. The paper I bipolar decomposition therefore results in LM’s that span the [*entire*]{} coordinate range $-\infty < x < \infty$. These LMs are given by the following analytical expression: p(x) = , \[ptun\] where $\delta$ is given by = [k]{} = , \[deltaeqn\] and = . \[kappaeqn\] The $p(x)$ LM function of is continuous everywhere, including at the potential discontinuity at $x=0$. In the $A$ region, it agrees exactly with the semiclassical solution; in the $B$ region, it decays exponentially to zero.
The paper I approach thus yields a very natural way to extend trajectories into the tunneling region. Note that the quantum potential in this region is not zero; indeed, it exhibits a discontinuity at $x=0$ that exactly balances that of $V(x)$ itself, so that the bipolar modified potential is continuous across the step. Unlike the above-barrier case, the paper I solution does [*not*]{} manifest oscillatory behavior in the large action limit, and so this approach would at first glance appear to be ideal. There are two reasons, however, why it is not pursued here. The first reason is that $r(x)$ diverges asymptotically as $x\ra\infty$, which according to preliminary numerical investigations, appears to lead to numerical instabilities for completely QTM-based propagation schemes. Second, if the barrier were to fall off again at larger $x$ values, so that the tunneling region were finite, then the asymptotic behavior would be once again undesirably oscillatory. This would be the case, for example, for the below-barrier energies of the square barrier system of Sec. \[squarebarrier\].
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abstract: 'For the first time, we construct a catalog of compact groups selected from a complete, magnitude-limited redshift survey. We select groups with $N \geq 3$ members based on projected separation and association in redshift space alone. We evaluate the characteristics of the Redshift Survey Compact Groups (RSCG’s). Their physical properties (membership frequency, velocity dispersion, density) are similar to those of the Hickson \[ApJ, 255, 382 (1982)\] Compact Groups. Hickson’s isolation criterion is a strong function of the physical and angular group radii and is a poor predictor of the group environment. In fact, most RSCG’s are embedded in dense environments. The luminosity function for RSCG’s is mildly inconsistent with the survey luminosity function — the characteristic luminosity is brighter and the faint end shallower for the RSCG galaxies. We construct a model of the selection function of compact groups. Using this selection function, we estimate the abundance of RSCG’s; for groups with $N \geq 4$ members the abundance is $3.8 \times 10^{-5}\ {h}^3\ {\rm Mpc}^{-3}$. For all RSCG’s ($N \geq 3$) the abundance is $1.4 \times 10^{-4}\ {h}^3\ {\rm Mpc}^{-3}$.'
author:
- 'Elizabeth Barton and Margaret J. Geller'
- Massimo Ramella
- 'Ronald O. Marzke'
- 'L. Nicolaci da Costa'
title: Compact Group Selection From Redshift Surveys
---
Introduction
============
Compact groups are the densest known systems of galaxies in the universe. Rose (1977) and Hickson (1982) made the first large-scale, systematic searches for dense systems on the sky. Subsequent studies of the properties and environments of Hickson’s 100 Compact Groups indicate that they are probably not dynamically simple, isolated systems.
Here, for the first time, we select compact systems from a complete, magnitude-limited redshift survey. We select groups based on physical extent, rather than angular size. This approach eliminates some of the systematic biases intrinsic to identification of systems on the sky. Because we begin with a complete, magnitude-limited survey we can model the residual selection effects from first principles. We use the redshifts of galaxies surrounding our compact groups to explore the embedding of the Redshift Survey Compact Groups (RSCG’s, hereafter) in their environments.
Hickson’s work sparked debates about the physics of compact groups. The existence of compact groups is a challenge for dynamical models because their measured crossing times are generally much smaller than the Hubble time. Numerical simulations indicate that compact group galaxies merge to form bright elliptical galaxies on timescales comparable to the group crossing times (Barnes 1989; Carnevali, Cavaliere & Santangelo 1981; Cavaliere et al. 1983; Governato, Bhatia & Chincarini 1991). To resolve this problem, Mamon (1986) suggested that a large fraction of compact groups are merely unbound chance superpositions of galaxies within loose groups (Mamon 1986, 1987). Hernquist, Katz & Weinberg (1995) suggested that compact groups are unbound superpositions of galaxies viewed along filaments. In contrast, simulations by Hickson & Rood (1988) and Diaferio, Geller & Ramella (1995) and observational work by Zepf (1993) and Pildis, Bregman & Schombert (1995) suggest that many of Hickson’s compact groups (HCG’s, hereafter) are, in fact, bound systems.
The short crossing times of HCG’s are not a problem if the groups form continually in dense environments like loose groups (Barnes 1989, Diaferio, Geller & Ramella 1994). Studies of the environments around some of the HCG’s reveal that many of them are indeed embedded in larger, looser systems (Ramella et al. 1994; de Carvalho, Ribeiro & Zepf 1994).
Compact groups have implications for cosmology, the development of large-scale structure and the evolution of the galaxy population. Compact groups are more likely to form during the present epoch in a dense universe (Diaferio 1994, Governato, Tozzi & Cavaliere 1995). Diaferio, Geller & Ramella (1994) and Ramella et al. (1994) realized that compact groups may be a clue to the evolutionary state of loose groups. If compact groups are dense environments, galaxy interactions and mergers within them are likely (Barnes 1989).
Here, we apply an objective group-finding algorithm for identifying HCG-like systems in the CfA2+SSRS2 Redshift Survey. In §2 we describe previous sky-selected compact group surveys. In §3 we describe the redshift surveys. §4 contains a description of our group-finding algorithm and the biases it introduces. In §5 we present the catalog and evaluate the physical properties of RSCG’s. In §6 we apply Hickson’s (1982) selection criteria to RSCG’s. §7 contains an analysis of the environments of RSCG’s. In §8 we evaluate the luminosity function of galaxies in RSCG’s. We then calculate the selection function and compute the resulting abundance of RSCG’s.
Compact Group Selection on the Sky
==================================
Examination of photographic plates led to the identification of compact groups as unusually dense, apparently isolated knots of galaxies on the sky (Shakhbazyan 1973; Vorontsov-Velyaminov 1959, 1977; Burbidge & Burbidge 1961; Arp 1966). Rose (1977) made the first large statistical study of these systems. Hickson (1982) later defined quantitative criteria for population, isolation on the sky, and compactness. He identified 100 HCG’s on the POSS R plates which satisfy 3 criteria:
- $N \geq 4$ where $N$ is the number of members within 3 magnitudes of the brightest galaxy, $m_{1}$.
- $\theta_N \geq 3 \theta_G$ where $\theta_{G}$ is the angular radius of the smallest circle which contains the geometric centers of all the suggested group members. The radius $\theta_N$ is the angular radius of the largest concentric circle which contains no further galaxies with $m < m_1 + 3$.
- $\mu_G < 26$ where $\mu_G$ is the total magnitude of the galaxies in the R band averaged over the circle of radius $\theta_G$, in magnitudes arcsecond$^{-2}$.
Hickson examined the entire POSS including regions at low Galactic latitude. Later, Hickson et al. (1992) measured radial velocities of galaxies in HCG’s, and defined a set of 92 HCG’s with $N \geq 3$ where $N$ now refers to the number of members within 1000 km s$^{-1}$ of the median group velocity. The median redshift of these HCG’s is 0.030. Prandoni, Iovino & MacGillivray (1994) applied Hickson’s original criteria to automated plate scans and identified 59 candidate compact systems.
These catalogs of systems on the sky have a number of unavoidable systematic problems. First, in spite of their large projected surface density, the group candidates may include interlopers with redshifts substantially different from the systemic mean. Among the 100 systems originally identified by Hickson, 92% have $N \geq 3$; only 69% actually have $N \geq 4$ (Hickson et al. 1992). Selection on the sky unavoidably introduces correlations of group properties with distance. For example, nearby systems with a large angular scale are underrepresented in these catalogs.
The CfA2 + SSRS2 Redshift Survey
================================
We select compact groups from redshift surveys which include 14,383 galaxies. CfAnorth covers the declination range $8.5^\circ \leq \delta \leq 44.5^\circ$ and right ascension range $8^{h} \leq \alpha \leq 17^{h}$ (B1950) and includes 6500 galaxies (Geller & Huchra 1989; Huchra et al. 1990; Huchra et al. 1995). CfAsouth covers the region $-2.5^\circ \leq \delta \leq 48^\circ$ and $20^{h} \leq \alpha \leq 4^{h}$ and includes 4283 galaxies (Giovanelli & Haynes 1985; Giovanelli et al. 1986; Haynes et al. 1988; Giovanelli & Haynes 1989; Wegner, Haynes & Giovanelli 1993; Giovanelli & Haynes 1993; Vogeley 1993). SSRS2 includes 3600 galaxies and is complete over 1.13 steradians of the of the southern galactic cap in the declination range $-40^\circ \leq \delta \leq -2.5 ^\circ$ and $b \leq -40^\circ$ (da Costa et al. 1994). Both SSRS2 and CfA2 are magnitude-limited to $m_{B_{0}} \leq 15.5$. SSRS2 is derived from plate scans. CfA2 is based on the Zwicky catalog. We use only the $cz$ range $300\ {\rm km\ s}^{-1}$ to $15,000\ {\rm km\ s}^{-1}$; the full sample we examine includes 14,011 galaxies.
The coordinate uncertainties in the redshift catalog are about one arcminute. We test the possible effects of the uncertainties on the very dense RSCG’s. They do not affect our compact group catalog.
Compact Group Selection in Redshift Space
=========================================
We develop an algorithm for identifying compact systems from a complete redshift survey. Our criteria mimic the ones established by Hickson. Application of our procedure to a [*distance limited*]{} redshift catalog would yield a set of systems with properties independent of distance. To obtain the largest possible sample of systems, we analyze a [*magnitude limited*]{} redshift survey. This limitation introduces some biases as a function of distance, but they are less severe than those introduced by selection on the sky. Because we start with a complete survey we can account for these biases.
The objective algorithm we use to select compact group candidates from the redshift surveys is a modification of the friends-of-friends algorithm developed by Huchra & Geller (1982). We use a new group-finding code from Ramella, Pisani & Geller (1996). We identify groups of galaxies as linked sets of “neighboring” galaxies. To determine whether two galaxies belong to a group, we consider both their projected separation, $\Delta D$, and their line-of-sight velocity difference, $\Delta V$. The projected separation of the pair is $\Delta D=2\left(\frac{v}{H_{0}}\right)\sin\left(\frac{\Delta \theta}{2}
\right)$, where $\Delta \theta$ is the angular separation on the sky and $v=cz$ is the average redshift. Throughout the paper we use $H_{0} = 100\ {\rm km \ s}^{-1}{\rm Mpc}^{-1}$.
We restrict the size of our groups by specifying limiting parameters, $D_{0}$ and $V_{0}$. If $\Delta D \leq D_{0}$ and $\Delta V \leq V_{0}$, the galaxies are neighbors. We search each galaxy in a pair of neighbors for additional neighbors. Linked sets of neighbors are groups. Groups with three or more members constitute our objective sample of compact groups. We select values of $V_{0}$ and $D_{0}$ which produce a catalog of systems with properties similar to those of Hickson Compact Groups. We keep $D_{0}$ fixed to ensure that we identify only systems where inter-galaxy separations are comparable with their physical size.
The line-of-sight velocity difference between galaxies in a gravitationally bound system is a measure of their relative peculiar velocity. The median radial velocity dispersion in HCG’s is only $\sim$ 200 km s$^{-1}$ (Hickson et al. 1992). We choose the value $V_{0}$ = 1000 km s$^{-1}$, which is large enough to include most physically associated galaxies in compact groups. This value of $V_{0}$ is, however, not so large that the resulting “groups” accidentally span voids in the galaxy distribution (Geller & Huchra 1989). This value is consonant with Hickson’s procedure of rejecting galaxies with velocities different from the median group velocity by $\geq$ 1000 km s$^{-1}$ (Hickson et al. 1992).
The parameter $D_{0}$ directly limits the physical extent of our compact groups. We use $D_{0}$ to match our sample to the HCG’s. As a measure of the spatial extent of HCG’s, we use the distribution of all projected separations, $\Delta D$, among Hickson group members. We compare the distribution to the distribution of all projected separations in compact group catalogs extracted from the CfAnorth catalog. We apply the friends-of-friends algorithm to construct catalogs of compact group candidates at several values of $D_{0}$. Figure \[fig:pairsep\] shows that the value $D_{0}$ = 50 kpc yields the best match between CfAnorth (solid line) and the HCG’s (dashed line). A K-S test indicates that the null hypothesis that the density values are drawn from the same distribution is acceptable at the 19% confidence level. On the basis of this match, the catalog with $V_{0}$ = 1000 km s$^{-1}$ and $D_{0}$ = 50 kpc is our objectively selected catalog of compact groups, the RSCG’s. Table \[tab:sepdist\] lists the median, first and third quartiles of the separation distributions for the HCG catalog and the CfAnorth catalog of compact groups. Table \[tab:sepdist\] also includes physical parameters for the RSCG’s we extract from CfAsouth, SSRS2 and CfA2+SSRS2 using $D_{0}=50$ kpc and $V_{0}=1000$ km s$^{-1}$. Note that the third quartile value for the SSRS2 sample (and for the CfA + SSRS2 sample) is inflated by the presence of a single group of 13 galaxies.
The physical properties of RSCG’s are similar to the HCG’s, even though we do not implement all of Hickson’s selection criteria directly. First, we do not take galaxy magnitudes into account in the group selection; some of our systems have fewer than three galaxies in the interval \[$m_{1}$,$m_{1}+3$\]. Furthermore, many of our groups have $m_{1} + 3 > m_{\rm lim}$, where $m_{\rm lim,Zw}=15.5$ is the limiting magnitude of the survey. Thus we do not necessarily include all of the galaxies which might be in the interval \[$m_{1}$,$m_{1}+3$\]. We later examine POSS images of the groups and find few fainter galaxies inside the group radii.
Second, we do not implement Hickson’s isolation criteria in our initial group selection. Third, we do not reject groups on the basis of surface brightness. However, all but four very nearby groups in our catalog automatically satisfy the surface brightness criterion because we require galaxy projected separations comparable with the size of a galaxy. Finally, we note that unlike the HCG’s, our sample is a complete listing of compact groups of three or more galaxies. Hickson includes a triplet only when one member of his initial group has a discordant redshift.
In general, our limits on the physical extent of compact groups are more restrictive than Hickson’s. 23 HCG’s are within the redshift survey boundaries and contain at least three galaxies brighter than the magnitude limit. We detect parts or all of 15 of these HCG’s. We fail to detect the other HCG’s because the physical or velocity separation of member galaxies exceeds our limits, or because of magnitude errors.
Both our selection procedure and Hickson’s introduce biases as a function of redshift or distance. The most important bias is the dependence of density on distance. At large distances, any magnitude-limited survey samples only the bright end of the luminosity function. Both our sample of galaxies and Hickson’s are magnitude-limited; Hickson’s limit is fainter. Because our selection criteria do not vary with redshift to compensate for the magnitude limit, the systems we detect at large distances are generally denser than the average nearby system. Hickson’s sample suffers from a similar bias. His distant groups are more likely to have larger total populations because of the magnitude limit, but there is no fixed upper limit to the physical inter-galaxy separations. We model these effects in Section 8.2.
Nearby groups with large angular radii are absent from the HCG catalog because of the large probability of interlopers within the annulus $\theta_{G} \leq \theta \leq 3 \theta_{G}$. These nearby groups are also more difficult to spot by eye. Figure \[fig:n\_of\_z\] shows the redshift distribution of HCG’s (dotted line) and RSCG’s (solid line). We identify groups where Hickson’s approach is least effective; the two compact group surveys complement one another.
The Compact Group Catalog
=========================
The search algorithm, applied to the CfA2+SSRS2 survey, yields a catalog of 89 groups of three or more galaxies with properties similar to the HCG’s. There are 50 groups in CfAnorth, 23 groups in CfAsouth and 16 groups in the SSRS. 15 of these groups are HCG’s or subsets of HCG’s. Table \[tab:prop\] lists the locations and basic properties of these groups. The median redshift of our sample of compact groups is $z=0.014$, only half of the median HCG redshift, $z=0.030$.
Figures \[fig:rscg7\] and \[fig:rscg47\] show Digitized Sky Survey images of RSCG 7 and RSCG 47, respectively. Neither RSCG is an HCG. In each image, the inner circle is the smallest circle containing the centers of all member galaxies. It has angular radius $\theta_G = 2.94$ arcmin for RSCG 7 and $\theta_G=
2.14$ acrmin for RSCG 47. The outer circle has radius $3\theta_G$. RSCG 7, a group with $N = 3$, has galaxies in its isolation annulus. The two brightest galaxies in this annulus are in the survey, but their velocities are about 1600 km s$^{-1}$ from the group median. The other galaxies are too faint to be in the survey. RSCG 47, a group with $N = 4$ galaxies, is isolated according to Hickson’s criterion.
Because of the large scale structure in the redshift surveys, the distribution of groups with redshift is nonuniform and varies among the three portions of the survey. The median group redshifts for individual portions of the survey are: $z=0.007$ for CfAnorth, $z=0.017 $ for CfAsouth and $z=0.012$ for SSRS2.
Next, we explore the properties of the RSCG’s and compare them with the HCG’s. Although the selection algorithms for HCG’s and RSCG’s are not identical, the systems in the two catalogs have similar physical properties.
Membership Frequency
--------------------
Figure \[fig:freq\]a shows the frequency of occurrence of groups of all populations $N$ in our catalog (solid line) and in the HCG catalog (dashed line). The overall distributions do not match. Hickson only finds triplets when his groups selected on the sky contain interlopers; in other words, triplets are underrepresented relative to our sample. However, the match is excellent for groups with $N \geq 4$ (Figure \[fig:freq\]b), when the distribution is renormalized. Both distributions also agree with models of compact groups formed as subsystems within loose groups (Diaferio, Geller & Ramella 1994). In this section, we consider the distributions of various compact group properties separately for groups of $N= 3$ and groups of $N \geq 4$. One physical motivation is the much greater likelihood that tight groups of $N \geq 4$ are bound. Diaferio, Geller & Ramella (1994) find that within loose groups only 50% of triplets are bound systems, whereas 80% of systems with four or more galaxies are bound systems.
Velocity Dispersion
-------------------
In Figures \[fig:veldisp\]a and b, we plot the distribution of the velocity dispersions of the HCG’s (dashed line) and of the 89 RSCG’s (solid line). Figure \[fig:veldisp\]a is the distribution for groups of 3 and Figure \[fig:veldisp\]b includes only groups of $N \geq 4$. Table \[tab:veldisp\] lists the median, first and third quartiles of these distributions. The distributions are similar. K-S tests indicate that the probability these samples are drawn from the same parent distribution is $86.5\% $ for groups with $N=3$ and $26.5\% $ for groups with $N \geq 4$.
Density
-------
We also compare the group densities. We choose the density statistic $n=\frac{3N}{4 \pi R^{3}}$, with $N$ the number of detected galaxies and $R$ the radius of the smallest circle which encloses all the galaxy centers in the RSCG. Figures \[fig:dens\]a and \[fig:dens\]b show the distribution of $\log(n)$ for both the HCG’s (solid line) and all RSCG’s (dotted line). Figure \[fig:dens\]a includes only groups with $N= 3$ and Figure \[fig:dens\]b includes only groups with $N \geq 4$. Table \[tab:dens\] lists the median, first and third quartiles of these distributions. As expected, the RSCG density distributions agree well with the HCG density distributions at high densities.
The density distributions of our compact groups fall off sharply below $\log(n) \approx 4$, where $n$ is in Mpc$^{-3}$, especially for groups with $N = 3$. This cutoff is an artifact of our search algorithm. Any galaxy in a compact group has a projected separation of at most $50\ {\rm kpc} = 0.05\ {\rm Mpc}$ from at least one other galaxy in its group. For a group of 3, the minimum density allowed is then 3 galaxies in a line, separated by 0.05 Mpc; the group radius is $R=0.05$ Mpc. The minimum group density is then $n= 5.7 \times 10^{3}$ Mpc$^{-3}$; $\log(n_{\rm min}) = 3.76$. A similar calculation for a group of 4 galaxies yields $\log(n_{\rm min}) = 3.35$. In general, the search algorithm introduces a complex selection as a function of density, but this selection is only relevant over a small range of densities. The HCG’s do not display this sharp cutoff because selection is based on angular rather than physical extent. These criteria lead to a range of HCG densities corresponding to a range of distances.
Hickson’s Criteria
==================
Unlike Hickson, we do not include isolation and luminosity criteria in our selection algorithm. Here we investigate the physical implications of the application of Hickson’s more restrictive criteria.
Population and Surface Brightness Criteria
------------------------------------------
Hickson limited group membership to galaxies with $m \leq m_{1}+3$ because galaxies with similar magnitudes are more likely to be at the same redshift than galaxies with a broader magnitude distribution. This problem is irrelevant for groups selected in redshift space. Furthermore, faint galaxies are probably less massive and hence less important dynamically. This effect is relevant to very nearby groups in our sample. Seven of our groups (all with median velocities $\leq 1600\ {\rm km \ s}^{-1}$) contain fewer than three members in the interval \[$m_{1}$,$m_{1} +3$\]. Most of these groups are located in the outskirts of the Virgo cluster \[see, for example, Mamon (1989), who finds RSCG 65 in Virgo\]. In general, these nearby systems contain one or two large galaxies along with small satellites; they are therefore different in character from the other RSCG’s and from the HCG’s. To exclude these systems, in later Sections we ignore nearby groups ($cz \leq 2300$ km s$^{-1}$) when exploring the environments of RSCG’s and when calculating the compact group selection function.
Hickson required $\overline{\mu}_{G} < 26.0$, where $\overline{\mu}_{G}$ is the surface brightness in E, for the POSS plates, of the group in magnitudes arcsec$^{-2}$. We translate this criterion as $\overline{\mu}_{G,{\rm Zw}} < 27.7$ using arguments from Prandoni, Iovino & MacGillivray (1994) that a surface brightness of 26.0 magnitudes arcsec$^{-1}$ in $E$ is equivalent to a surface brightness of 27.7 magnitudes arcsec$^{-1}$ in $b_{j}$ which is similar to the Zwicky magnitude scale. Four of our compact groups do not meet this requirement. All of these groups are in the CfAnorth survey. They are nearby ($cz \leq$ 1200 km s$^{-1}$), and have large angular radii ($\theta_{G} > 14 '$). All of these groups fail the isolation criterion as well.
Isolation Criterion
-------------------
Hickson’s isolation criterion has complex physical implications. In a survey of compact groups selected on the sky, interloping galaxies can appear in the isolation annulus ($\theta_{G} < \theta < 3\theta_{G}$) and thus can cause bonafide systems to fail the isolation criterion.
Because we have a complete catalog in redshift space, we can apply a cleaner isolation criterion in three dimensions. We compute the isolation ratio, $\Upsilon \equiv \theta_{N}/\theta_{G}$, for all groups in the sample. $\theta_{G}$ is again the radius of the smallest circle which contains the centers of all the group members and $\theta_{N}$ is the angular radius of the largest concentric circle which contains no further galaxies within 1200 km s$^{-1}$ of the median group velocities. Our velocity separation cutoff is somewhat arbitrary; it is more relaxed than the group selection cutoff to allow for a larger velocity dispersion in the group environment. Our isolation criterion is not completely equivalent to Hickson’s criterion because we only search for surrounding galaxies brighter than the magnitude limit, $m_{\rm Zw}=15.5$. Thus, for groups with $m_{1} > 12.5$, we do not search for surrounding galaxies throughout $[m_{1},m_{1}+3]$. Twenty-five RSCG’s fail the “isolation criterion” which requires $\Upsilon > 3$. Six of these groups have $N=3$, 19 have $N \geq 4$.
We explore the differences between “isolated” groups and groups with $\Upsilon < 3$. The isolation parameter $\Upsilon$ is a [*strong*]{} function of group radius. Figure \[fig:isovrad\] shows $\log(\Upsilon)$ as a function of projected group radius in kpc. The horizontal line is $\Upsilon = 3$. Groups with large angular radii are not isolated simply because their isolation annuli are larger. Because of this dependence, $\Upsilon$ alone does not specify the physical environment of a particular RSCG. In fact, there is evidence that most HCG’s are also embedded in larger systems (de Carvalho et al. 1994; Ramella et al. 1994).
The correlation of $\Upsilon$ with radius results in a correlation of $\Upsilon$ with group density. Figures \[fig:isodens\]a and \[fig:isodens\]b show the density distributions of the two subsamples (isolated and not isolated) for groups with $N=3$ and with $N \geq 4$, respectively. We apply a K-S test to the density distributions of the two subsamples in each case. The null hypothesis that the density values are drawn from the same distribution is acceptable only at the 3.6% significance level for groups with $N=3$ and the 0.46% level for groups with $N \geq 4$. Thus isolation does imply higher density, as expected. The correlation is not physical; it is a result of the correlation of $\Upsilon$ with group radius.
The velocity distributions are consistent for the two samples. The velocity dispersion distributions of the subsamples agree at the 99% confidence level for $N=3$ and at the 49% confidence level for $N \geq 4$. Figures \[fig:isovel\]a and \[fig:isovel\]b show the velocity distributions for groups with $N=3$ and groups with $N \geq 4$, respectively. We note, however, that the subsamples of RSCG’s in these comparisons are small. There are 11 groups with $N \geq 4$ and $\Upsilon > 3$, 19 groups with $N \geq 4$ and $\Upsilon < 3$, 53 groups with $N=3$ and $\Upsilon > 3$, and only 6 groups with $N=3$ and $\Upsilon < 3$.
We visually examine regions around the 48 RSCG’s with $\Upsilon > 3$ and $cz > 2300\ {\rm km\ s}^{-1}$ using Digitized Sky Survey images. About 1/3 of these groups have faint galaxies (with $m \lesssim m_{1}+3$) in their isolation annuli without measured redshifts. In other words, at least 30 of the RSCG’s are isolated according to Hickson’s criteria. However, this apparent isolation is not a good predictor of the environment of compact systems and contains no information about the physics of the systems.
Environments of Compact Groups
==============================
Previous studies of the environments of compact groups of galaxies yield mixed messages; some investigators find that compact groups are in dense regions and others conclude that they are not (Rood & Williams 1989; Rubin, Hunter & Ford 1991; de Carvalho, Ribeiro & Zepf 1994; Ramella et al. 1994). These discrepant conclusions are at least in part a result of incomplete data sets. We investigate the environments of the RSCG’s in redshift space. We find that most of them are embedded in loose groups or clusters.
Using the method of Ramella et al. (1994), we count the number of galaxies (including the galaxies in the RSCG), $N_{n}$, in the redshift survey within 1.5 $h^{-1}$ Mpc (projected) of the group center and within 1500 km s$^{-1}$ of the median group velocity, $v_{\rm med}$. These scales are well-matched to the size of loose groups (Ramella, Geller & Huchra 1989).
For the remaining analyses, we exclude the 31 RSCG’s with redshifts less than 2300 km s$^{-1}$. Many of these systems differ significantly from more distant RSCG’s, making them difficult to model. They are often composed of a bright galaxy and its very faint satellites (see §6.1), a combination that we cannot detect at higher redshift. In addition, these groups are hard to model because they are within the local supercluster.
Table \[tab:env\] lists $\frac{N_{n}}{N_{\rm int,env}}$ for RSCG’s with average redshifts greater than 2300 km s$^{-1}$. $N_{\rm int,env}$ is the expected number of interlopers within 1.5 $h^{-1}$ Mpc of the group center in the velocity interval $[v_{\rm med}-1500,v_{\rm med}+1500]$; $$N_{\rm int,env} \approx \int_{V_{\rm env}} \overline{\rho} dV,$$ where $\overline{\rho}$ is the average density of the relevant portion of the survey and $V_{\rm env}$ is the volume of the environment of the group. Similarly, $$N_{n} \approx \int_{V_{\rm env}} \rho_{\rm env} dV,$$ where $\rho_{\rm env}$ is the density of the environment of the group. Therefore, $$\frac{N_{n}}{N_{\rm int,env}} \approx \frac{ \langle\rho_{\rm env}\rangle}
{\langle \overline{\rho} \rangle},$$ where the brackets denote averaging over the volume of the environment of the RSCG. To calculate this expected number of interlopers we integrate over the [*true*]{} luminosity function (Table \[tab:lumrd\]). We use the true luminosity function instead of the observed one for ease of calculation; the error introduced is small. Therefore, $$N_{\rm int,env}=\int_{v_{\rm med}-1500}^{v_{\rm med}+1500}
\int_{-\infty}^{M_{\rm lim}(v)} \pi \left( \frac{1.5\ {\rm Mpc} \times v}{v_{\rm med}}\right)^{2}
\Phi_{\rm true}(M) dM \frac{dv}{H_{0}}, \label{eqn:nint}$$ where $M_{\rm lim}(v)$ is the limiting absolute magnitude at redshift $v$, $\Phi_{\rm true}$ is the true luminosity function of the relevant survey and $v_{\rm med}$ is the median RSCG velocity. When calculating $N_{\rm int, env}$ we integrate over 1.5 $h^{-1}$ projected Mpc, although some groups are centered on the edge of the survey where we do not observe all galaxies in the neighborhood. We indicate these groups in the table.
We also list $N_{\rm int,cg}$ in Table \[tab:env\]. We calculate this number by replacing 1.5 Mpc with $R_{\rm cg}$, the group radius in Mpc, in Equation \[eqn:nint\]. We then compute the overdensity of the RSCG, $\frac{N_{\rm cg}}{N_{\rm int,cg}} \approx \frac{\langle \rho_{\rm cg} \rangle}
{\langle \overline{\rho} \rangle }$. $N_{\rm cg}$ is the number of galaxies in the RSCG and $\rho_{\rm cg}$ is the density of the compact group. The RSCG’s have $460 < \frac{\langle \rho_{\rm cg} \rangle}
{\overline{\langle \rho}\rangle } < 150,000$.
For $cz \geq 2300$ km s$^{-1}$, 43 RSCG’s (74%) have $\langle \rho_{\rm env} \rangle \ \gg \ \langle \overline{\rho} \rangle$. This result gives the same impression as the conclusion of Ramella et al. that most but not all HCG’s are embedded in looser systems.
All 10 RSCG’s with $cz \geq 2300\ {\rm km\ s}^{-1}$ which fail the “isolation” criterion (§6) satisfy $N_{n} > N_{\rm int} + 3\sqrt{N_{\rm int}}$, as expected. However, 30 [*isolated*]{} RSCG’s with $cz \geq 2300\ {\rm km\ s}^{-1}$ also satisfy $N_{n} > N_{\rm int} + 3\sqrt{N_{\rm int}}$ (for $m_{\rm Zw} \leq$ 15.5) — another indication that “isolation” is not physically meaningful.
Figures \[fig:no1\] – \[fig:ss2\] show examples of the embedding of RSCG’s within the large-scale structure of the nearby universe. They show galaxies with redshifts in the interval (300 km s$^{-1}$, 15,000 km s$^{-1}$) in the redshift surveys ($\cdot$) and RSCG centers ($\times$) (including RSCG’s with $cz < $ 2300 km s$^{-1}$). Compact groups generally follow the large-scale distribution of galaxies, often appearing in the densest regions of the survey. However, Figure \[fig:ss2\] shows the seemingly rare example of a compact group in an apparent void.
Most RSCG’s are embedded in larger systems regardless of how “isolated” they appear. The isolation criterion is unphysical: groups with smaller radii are more likely to appear “isolated” mainly because their isolation annuli are smaller. This variation with radius leads to a dependence of density, $n$, on isolation only because density is computed using the group radius; thus the correlation of $\Upsilon$ with $n$ is unphysical. From here on we include all RSCG’s whether they satisfy the isolation criterion or not.
Abundance of Compact Groups
===========================
The abundance of compact groups has important physical implications for cosmology. The compact group abundance can indicate whether compact groups are bound systems or chance alignments of galaxies within loose groups or along filaments. If CG’s are truly compact systems, the abundance of these systems is dependent on the density parameter, $\Omega_0$. If $\Omega_0 \sim 1$ these systems should be forming in the present epoch. Furthermore, the abundance of compact systems should be related to the merger rate at the current epoch.
We calculate the volume number density of RSCG’s. We compute abundances using all RSCG’s and using only RSCG’s with $N \geq 4$. We consider only systems with median redshifts $>$ 2300 km s$^{-1}$.
The Luminosity Distribution of Galaxies in Compact Groups
---------------------------------------------------------
To model the selection function of compact groups, we have to evaluate the luminosity function (LF) for compact group members. We compute the Schechter (1976) function parameters (Marzke, Huchra & Geller 1994, Efstathiou, Ellis & Peterson 1988, Loveday et al. 1992), $\alpha_{\rm cg}$ and $M_{\star, {\rm cg}}$, for the CfAnorth, CfAsouth and SSRS2 RSCG samples. Table \[tab:lum\] lists $\alpha_{\rm cg}$ and $M_{\star,{\rm cg}}$ for the CfAnorth, CfAsouth and SSRS2 catalogs of compact group galaxies. The samples include 84 galaxies, 77 galaxies and 34 galaxies, respectively. In all cases, $M_{\star,{\rm cg}} < M_{\star,{\rm survey}}$ and $\alpha_{\rm survey} < \alpha_{\rm cg}$, where $M_{\star,{\rm survey}}$ and $\alpha_{\rm survey}$ are the Schechter function parameters for each region of the complete redshift survey (Table \[tab:lumrd\]). In other words, the characteristic luminosity is brighter and the faint end shallower for the compact groups. Figure \[fig:lum1\] illustrates the significance of these differences; it shows 1$\sigma$ contours for the compact group galaxies (dotted lines) and 2$\sigma$ contours for all CfAnorth, CfAsouth, and SSRS2 survey galaxies (solid lines). These differences are similar for all portions of the redshift survey. The 1$\sigma$ error ellipse for the compact group galaxies does not overlap the 2$\sigma$ survey error ellipse for any of the surveys.
Mendes de Oliveira & Hickson (1991), Prandoni, Iovino & MacGillivray (1994), and Ribeiro, de Carvalho & Zepf (1994) have also compared compact group LF’s with the “field”. Mendes De Oliveira & Hickson (1991) used simulations of compact groups to derive an HCG galaxy LF. They fit a Schechter function with parameters $\alpha = -0.2^{+0.8}_{-0.9}$ and $L_{\star}=1.1 \pm 0.2 \times 10^{10} h^{-2} L_{\sun}$ ($M_{\star,B} =-19.6 \pm 0.2$). We show the value of Mendes de Oliveira & Hickson, with error bars, in Figure \[fig:lum1\]. The error bars of Mendes de Oliveira & Hickson overlap the 1$\sigma$ LF parameter error ellipses of all three RSCG subsamples. Mendes De Oliveira & Hickson claim a significant deficiency of low-luminosity galaxies, as compared to field, loose group and cluster galaxies. Prandoni, Iovino & MacGillivray (1994) and Ribeiro, de Carvalho & Zepf do not confirm this deficiency, although their samples may be contaminated by superimposed background galaxies. Ribeiro, de Carvalho & Zepf compare faint galaxy counts inside and outside 22 HCG’s. They find that the LF inside HCG’s is consistent with the field LF.
The Selection Function of Compact Groups
----------------------------------------
For the following analysis, we use the survey LF’s when estimating the compact group selection function because the survey LF’s are much better determined and better understood. We thus underestimate the selection function of compact groups somewhat and we overestimate the compact group density slightly, assuming that the differences in luminosity functions are real.
We require that each compact group contain $j$ or more galaxies coincident in redshift space, where $j=3$ is our criterion for selection; we also restrict our sample for comparison to Hickson’s (1982) by adopting $j=4$. We thus estimate the selection function for RSCG’s analytically, by modeling the probability of detecting the $j$th brightest galaxy. We assume that the galaxies in RSCG’s are drawn at random from a magnitude distribution of fixed form: $\overline{\Phi}(M)$. We calculate $P_{j}(M)dM$, the probability that the $j$th brightest member of a group of galaxies lies in the interval \[$M$, $M+dM$\]; this probability is proportional to $P_{(j-1)<M}(M) \overline{\Phi}(M) dM$, where $P_{(j-1)<M}(M)$ is the probability that exactly $j-1$ members of the group are brighter than $M$. If $\lambda_{M}$ is the average number of galaxies in a group brighter than $M$, then, $$\lambda_{M}=\kappa \int_{-\infty}^{M}
\overline{\Phi}(M')dM', \label{eqn:kappa}$$ where, in general, $\kappa = \kappa(M)$ is a normalization parameter. Then from the Poisson distribution we derive $$P_{(j-1)<M}=\frac{e^{-\lambda_{M}}\lambda_{M}^{(j-1)}}{(j-1)!}.
\label{eqn:2gtm}$$ For a survey with a limiting apparent magnitude of 15.5, the limiting absolute magnitude at any redshift $v=cz$ is $M_{\rm lim}(v)=-9.5-5\log_{10}(\frac{v}{H_{0}})$, where the dimensions of $\frac{v}{H_{0}}$ are Mpc. Therefore, the probability of detecting $j$ or more members of a group of galaxies is $$P_{\rm detection}(v)=\frac{1}{\aleph} \int_{-\infty}^{M_{\rm lim}(v)} P_{j}(M)dM=
\frac{1}{\aleph}
\int_{-\infty}^{M_{\rm lim}(v)} \frac{e^{-\lambda_{M}}\lambda_{M}^{(j-1)}}{(j-1)!}
\overline{\Phi}(M) dM.
\label{eqn:pdet}$$ The factor $\aleph$ is the normalization of $P_{\rm detection}(v)$ which ensures that $P_{\rm detection}(0)=1$.
### Estimating the $\kappa$ parameter
In order to evaluate the probability of detecting a compact group \[Equation \[eqn:pdet\]\] we assume that $\kappa(M)$ in Equation \[eqn:kappa\] is a constant. In other words, we assume that the number of galaxies brighter than $M$ in a compact group, $\lambda_M$, is proportional to the integral of the luminosity distribution of galaxies and that the normalization ($\kappa$) is the same for every compact group. These assumptions are not strictly correct. The inaccuracy introduced by approximating $\lambda_M$ by a smooth function, rather than a step function, is large because the numbers of galaxies involved is small.
Below, we compute the density of RSCG’s twice, using systems of $N \geq 3$ \[$j=3$ in Equation \[eqn:pdet\]\] and then using systems with $N \geq 4$ \[$j=4$ in Equation \[eqn:pdet\]\]. If Equation \[eqn:kappa\] were strictly true for a constant $\kappa$, we would obtain the same density for the two samples. However, our RSCG’s with $N=3$ include both groups with a small true $\kappa$ value (these groups contain three and only three bright members) and more distant groups with a larger true $\kappa$ value (these groups contain more than three bright members, but only three above the magnitude limit). Thus the two density estimates are estimates of two fundamentally different quantities. Our compact groups are not uniformly dense, and they do not occupy the same volumes; thus their $\kappa$ values actually vary.
The properties of distant RSCG’s differ from those of the nearby systems. If the population of galaxies in an RSCG brighter than $M_{\rm lim}(v)$ is in fact proportional to $\int_{-\infty}^{M_{\rm lim}(v)} \overline{\Phi}(M) dM $ as we have assumed, then some distant systems are far more populated, and probably much denser, than the nearby RSCG’s. Thus assuming a single value of $\kappa$ for all RSCG-type systems is only an approximation. We do so to construct a manageable model of the selection function for these systems.
Some assumption about the number of bright galaxies in compact groups is necessary in order to construct a selection function. Monte Carlo simulations of compact groups aimed at computing a selection function make implicit assumptions (e.g., Mendes de Oliveira & Hickson 1991). Our approach enables us to make the assumptions explicit and their effects can be easily explored.
In a magnitude-limited redshift survey, we detect only galaxies brighter than the limiting magnitude $M_{\rm lim}(v)$. Because we have no information about fainter galaxies we must estimate $\kappa$ for each compact group; $$\kappa_{\rm estim} = \left[ \frac{V_{\rm cg} \rho_{{\rm cg}, < M}}
{\int_{-\infty}^{M_{\rm lim}(v)}
\overline{\Phi}(M') dM'} \right]
= \left[ \frac{N}{\int_{-\infty}^{M_{\rm lim}(v)} \overline{\Phi}(M') dM'} \right].
\label{eqn:kestimate}$$ Here, $\rho_{{\rm cg}, < M}$ is the compact group density of galaxies brighter than $M=M_{\rm lim}(v)$, $v$ is the median galaxy velocity in the RSCG, $V_{\rm cg}$ is the volume of the RSCG and $N$, as usual, is the number of detected galaxies in the RSCG. There is a lower bound and an effective upper bound to the actual number of observed galaxies in a compact group. We select only systems with $N \geq 3$. We expect from theoretical arguments (Diaferio, Geller & Ramella 1994) and observation (Hickson 1982) only systems with $N \lesssim 6$. For $N \gtrsim 6$ the probability is so small that we expect none even in a volume much larger than the region we survey. Therefore, $N_{\rm min} =3$ and $N_{\rm max} \approx 6$. As $N$ falls only in the range $N_{\rm min} \leq N \lesssim N_{\rm max}$, $\kappa_{\rm estim}$ is limited to the range $\kappa_{\rm min}(v) \leq \kappa_{\rm estim} \lesssim \kappa_{\rm max}(v)$ where $$\kappa_{\rm min}(v) = \left[ \frac{3}{\int_{-\infty}^{M_{\rm lim}(v)}
\overline{\Phi}(M') dM'} \right]$$ and $$\kappa_{\rm max}(v) = \left[ \frac{6}{\int_{-\infty}^{M_{\rm lim}(v)}
\overline{\Phi}(M') dM'} \right].$$ Figure \[fig:kaprange\] shows $\kappa_{\rm min}(v)$ and $\kappa_{\rm max}(v)$ computed assuming the CfAnorth survey LF.
Figure \[fig:k\_vs\_z\] shows $\kappa_{\rm estim}$ as a function of redshift for RSCG’s with $cz < 10,000 {\rm~km~s}^{-1}$ in all of the surveys; we compute $\kappa_{\rm estim}$ using Equation \[eqn:kestimate\]. Figure \[fig:k\_vs\_z\]a shows only the 15 groups with $N \geq 4$; Figure \[fig:k\_vs\_z\]b shows all of the RSCG’s. The median, first and third quartiles of the distribution of $\kappa$ in Figure \[fig:k\_vs\_z\]a are 7.7, 4.2, and 19, respectively; for Figure \[fig:k\_vs\_z\]b they are 5.5, 3.6, and 11, respectively. The points are confined to a region similar to the region between the curves in Figure \[fig:kaprange\].
Figure \[fig:kaprange\] illustrates an important difficulty in [*choosing*]{} an appropriate $\kappa$. Assume that hypothetical “true” compact groups have a distribution of $\kappa$ values that is, for simplicity, a gaussian with a non-negligible width. The width must be large enough to allow all the measured values of $\kappa_{\rm estim}$. Then the “true” distribution of $\kappa$ for RSCG-type systems might approximate the distribution of points in Figure \[fig:kaprange\]. This figure depicts a random sampling of a Gaussian distribution of “$\kappa$” values with a peak at the median measured value of $\kappa$ for all RSCG’s and a width equal to the interquartile range. The sampling is weighted according to the volume sampled, with a minimum velocity of 2300 km s$^{-1}$ and a maximum velocity of 10,000 km s$^{-1}$. Because we use a magnitude-limited sample we measure $\kappa$ only between $\kappa_{\rm min}(v)$ and $\kappa_{\rm max}(v)$. Therefore, it is difficult to reconstruct the “true” $\kappa$ distribution. We are guided only by the available estimators, the median, first and third quartiles of the observed $\kappa_{\rm estim}$ distribution.
### Calculation of the selection function
Assuming a constant value of $\kappa$ simplifies the selection function. In this case, we substitute the differential form of Equation \[eqn:kappa\] into Equation \[eqn:pdet\] with $j=3$ and integrate over $\lambda_M$ to find the probability of detecting three or more galaxies as a function of redshift, $$P_{\rm detection}(v)=1-\frac{1}{2}e^{-\lambda_{M_{\rm lim}(v)}}
\left[ \lambda^2_{M_{\rm lim}(v)} + 2\lambda_{M_{\rm lim}(v)} +2 \right].
\label{eqn:3det}$$ Similarly, we calculate the probability of detecting four or more galaxies by using $j=4$ in Equation \[eqn:pdet\], $$P_{\rm detection}(v)=1-\frac{1}{6}e^{-\lambda_{M_{\rm lim}(v)}}
\left[ \lambda^3_{M_{\rm lim}(v)} + 3\lambda^2_{M_{\rm lim}(v)}
+ 6\lambda_{M_{\rm lim}(v)} + 6 \right].
\label{eqn:4det}$$ When computing $\lambda_{M_{\rm lim}}(v)$ we use the observed LF, which is the true LF convolved with a gaussian of width of about the size of the magnitude errors (Efstathiou, Ellis & Peterson 1988).
Abundance of Compact Groups
---------------------------
The selection function is the first step toward calculating the volume number density of compact groups. In the case of a uniform spatial distribution of compact groups, the product of $P_{\rm detection}$, the survey volume at a particular redshift, and the average volume number density of compact groups ($\varrho_{\rm cga}$) is the differential expected number of compact groups, $$\frac{dN_{\rm cg}}{dv}=\varrho_{\rm cga} \frac{\Omega}{H_{0}}
\left( \frac{v}{H_{0}} \right)^{2} P_{\rm detection}(v). \label{eqn:dndz}$$ We average the estimated density over the solid angle of each survey, $\Omega$. For a survey bounded by redshifts $v_{i}$ and $v_{f}$, the density is approximated by $$\varrho_{\rm cga,s}=\frac{N_{s}}
{\Omega \int_{v_{i}}^{v_{f}} \left( \frac{v}{H_{0}} \right)^{2} P_{\rm detection}(v) d \left( \frac{v}{H_{0}} \right) }.
\label{eqn:bin}$$ $N_{s}$ is the number of compact groups with an average redshift in the interval \[$v_{i},v_{f}$\].
Using Equation \[eqn:bin\] altered to select groups of $N \geq 4$ ($j=4$), we calculate the space density of RSCG’s with $N \geq 4$. There are 15 of these groups in our catalog. Figure \[fig:rho\_hick\]a shows $\varrho_{\rm cga}$ for these groups as a function of $\kappa$. The figure includes data from all three surveys. Vertical lines indicate the median, first and third quartiles of the relevant $\kappa$ distribution. Table \[tab:rho\_hick\] lists the density of compact groups at each of these values of $\kappa$. The combined-survey density estimate at the median value of $\kappa$ is $ 3.8 \times 10^{-5}$ Mpc$^{-3}$.
In selection from a redshift survey we have greater confidence in the triplets than Hickson had for selection on the sky. We repeat the calculation of the space density of compact groups using all RSCG’s with average velocities greater than 2300 km s$^{-1}$. We use a selection function computed with $j=3$ in Equation \[eqn:pdet\] to include the abundant observed triplets. The density of these compact systems exceeds both the original HCG abundance estimate of Mendes de Oliveira & Hickson (1991) and our value for the density of RSCG’s derived from groups with $N \geq 4$. Figure \[fig:rho\_hick\]b shows $\varrho_{\rm cga}$ for these groups (including all surveys) as a function of $\kappa$. Once again, we use vertical lines to indicate the median, first and third quartiles of the relevant $\kappa$ distribution. Table \[tab:rho\_all\] lists the density of compact groups at these $\kappa$ values for each survey and for all surveys combined. The combined-survey density estimate at the median value of $\kappa$ is $ 1.41 \times 10^{-4}$ Mpc$^{-3}$, almost four times the estimate for $N \geq 4$.
Figure \[fig:combssmod\] shows our model for the number density of all RSCG’s as a function of redshift, $\frac{dN_{\rm cg}}{dv}$ from Equation \[eqn:dndz\], for all the surveys combined. The figure shows the models for the median (solid line), first quartile (dotted line), and third quartile (dashed line) $\kappa$ values. The models assume a uniform space density of compact groups; we adopt the $\varrho_{\rm cga}$ values in Table \[tab:rho\_all\]. We plot a histogram of the RSCG’s for comparison. The distribution corresponds roughly to a $\kappa$ value in the proper range. However, the Figure shows variation in the volume number density of compact groups with redshift which results from the large-scale structure in the nearby universe.
We estimate the volume number density of RSCG’s with $N \geq 4$ and all RSCG’s again, at each redshift bin. For this purpose, we choose a bin size of 1000 km s$^{-1}$ for groups with $N \geq 4$ and 500 km s$^{-1}$ for the set of all RSCG’s. Figure \[fig:nosobin\] shows $\varrho_{\rm cga}(v)$ calculated for each bin using Equation \[eqn:bin\]. We display the results for the median value of $\kappa$. There is a large variation in RSCG density with redshift partly because of the small number of groups. However, Figure \[fig:nosobin\]b shows that the density of RSCG’s exceeds the value of Mendes de Oliveira & Hickson (1991), the horizontal line, in most (nearby) redshift bins.
Hickson, Kindl & Auman (1989) arrive at a compact group selection function as a function of apparent magnitude empirically. The results are not directly comparable to our $P_{\rm detection}$ because we calculate a selection function as a function of redshift; they calculate a [*distribution*]{} as a function of redshift. Our density estimate derived from RSCG’s with $N \geq 4$ roughly agrees with the estimated density of HCG’s. Our estimate of $\varrho_{\rm cga}$ for all compact systems, $1.4 \times 10^{-4}\ {h}^{3}\ {\rm Mpc}^{-3}$, greatly exceeds previous estimates of the abundance of compact systems in the universe. In particular, it places stronger demands on theories in which compact groups are chance alignments in loose groups or “filaments” in the universe.
Conclusion
==========
We apply the friends-of-friends group-finding algorithm to the CfA2+SSRS2 Redshift Survey to identify systems similar to Hickson’s compact groups. The result is the first objectively defined and well-controlled sample of compact groups selected in redshift space. We evaluate the physical characteristics of the RSCG’s and find the following:
- The physical properties of the RSCG’s (membership frequency, velocity dispersion, density) are similar to those of the HCG’s.
- The isolation of a compact group is a strong function of group radius and is a poor predictor of the group environment; it probably has little relevance to the dynamical history of these systems.
- Most RSCG’s are embedded in dense environments. Their distribution generally follows the large-scale structure evident in the redshift survey.
- The luminosity distribution of galaxies in RSCG’s is mildly inconsistent with the survey LF. The characteristic luminosity is brighter and the faint end shallower for the RSCG galaxies.
- We model the selection function of compact groups to estimate the abundance of RSCG’s. When we include only groups with $N \geq 4$ the abundance is $ 3.8 \times 10^{-5}\ {h}^3\ {\rm Mpc}^{-3}$; when we include all RSCG’s the abundance is $1.4 \times 10^{-4}\ {h}^3\ {\rm Mpc}^{-3}$.
We plan to measure redshifts of fainter galaxies in and around some RSCG’s to explore their embedding in the surrounding environment. In addition, we will compare the RSCG catalog to a similar objectively-defined catalog of loose groups in the same region (Ramella et al. 1996). We are also searching the [*ROSAT*]{} all-sky survey for x-ray emission from RSCG’s. Eventually, we intend to apply the compact group search algorithm to a deeper redshift survey.
M.J.G., M.R. and L.daC. thank their collaborators for use of the CfA2 and SSRS2 data in advance of publication. We thank Mike Kurtz and Emilio Falco for providing much assistance, and George Rybicki, James Moran, Antonaldo Diaferio, and Reinaldo de Carvalho for useful discussions and advice. We thank the referee, Stephen Zepf, for suggestions which clarified several important points. E. B. is a National Science Foundation Graduate Fellow.
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\[tab:sepdist\]
\[tab:prop\]
\[tab:veldisp\]
\[tab:dens\]
\[tab:env\]
\[tab:lumrd\]
\[tab:lum\]
\[tab:rho\_hick\]
\[tab:rho\_all\]
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abstract: |
We consider the operator
$$L = - (d/dx)^2 + x^2 y + w(x) y \quad \text{in } L^2(\mathbb{R}),$$ where $$w(x) = s \left[ \delta(x - b) - \delta(x + b) \right], \quad b \neq 0 \, \, \text{real}, \quad s \in \mathbb{C}.$$ This operator has a discrete spectrum: eventually the eigenvalues are simple and $$\lambda_n = (2n + 1) + s^2\, \frac{\kappa(n)}{n} + \rho(n) \label{eq:abstractlam}$$ where $$\kappa(n) = \frac{1}{{2\pi}} \left[(-1)^{n + 1} \sin \left( 2 b \sqrt{2n} \right) - \frac{1}{2} \sin \left( 4 b \sqrt{2n} \right) \right]$$ and $$\vert \rho(n) \vert \leq C \frac{\log n}{n^{3/2}}. \label{eq:abstracterr}$$ If $s = i \gamma$, $\gamma$ real, the number $T(\gamma)$ of non-real eigenvalues is finite, and $$T(\gamma) \leq \left( C (1 + \vert \gamma \vert) \log (e + \vert \gamma \vert) \right)^2.$$ The analogue of – is given in the case of any two-point interaction perturbation $$w(x) = c_+ {\delta \left( x - b \right) } + c_- {\delta \left( x + b \right) }, \quad c_+, c_- \in {\mathbb{C}}.$$
address: '231 West 18th Avenue, The Ohio State University, Columbus, OH 43201'
author:
- 'Boris S. Mityagin'
bibliography:
- 'harmonicrefs.btype.bib'
nocite: '[@*]'
title: The spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions
---
Introduction
============
The operator $$L^0 = - \, \frac{d^2}{dx^2} + x^2, \quad x \in {\mathbb{R}}^1,$$ is the one-dimensional harmonic oscillator; this is an unbounded self-adjoint operator acting in $L^2({\mathbb{R}})$. As one can see in any introductory book on quantum mechanics, $L^0$ has a discrete spectrum $\Lambda^0 = {\left\lbrace z_n \right\rbrace}_{n = 0}^{\infty}$, $$z_n = 2n + 1, \quad n = 0, 1, \dotsc$$ and a compact resolvent $$\label{eq:firstresolve}
R^0(z) = (z - L^0)^{-1} , \quad z \not\in \Lambda^0.$$ A normalized orthogonal system of eigenfunctions can be chosen as the Hermite functions $$\label{eq:hefcnintro}
h_n(x) = \left( \pi^{1/2} 2^n n! \right)^{-1/2} e^{-x^2/2} H_n(x), \quad n = 0, 1, \dotsc$$ where $$\label{eq:hepolyintro}
H_n(x) = e^{x^2/2} \left(e^{-x^2/2}\right)^{(n)}$$ are Hermite polynomials.
Spectral analysis of perturbed operators $$\label{eq:elldecompintro}
L = L^0 + W$$ with special $W$, in particular, the point interaction peturbations $$\label{eq:wformintro}
Wf = wf, \quad w(x) = \sum_{j = 1}^J c_j {\delta \left( x - b_j \right) }, \quad J \text{ finite}$$ was studied in many mathematical and physical papers.
S. Fassari, F. Rinaldi and G. Inglese series of papers [@FassIng94], [@FassIng97], [@FassRin] investigate the spectrum of $L \in$ when the perturbation $$\label{eq:wspecificintro}
W = - \tau \left( {\delta \left( x - b \right) } + {\delta \left( x + b \right) } \right), \quad \tau, \beta > 0,$$ i.e., $L^0$ is perturbed by a pair of attractive point interactions of equal strength whose centers are situatied at the same distance from the origin. In this case the operator $L = L^0 + W$ is self-adjoint; the techniques used are based on Green’s function analysis.
D. Haag, H. Cartarius, and G. Wunner [@HCW], motivated by analysis of Bose-Einstein condensates with $\mathcal{PT}$-symmetric loss and gain, focused on the case of non-Hermitian perturbations $$W = i \gamma \left[ {\delta \left( x - b \right) } - {\delta \left( x + b \right) } \right].$$ Their numerical estimates showed that for small $\gamma$ the spectrum of $L = L^0 + W$ is on the real line ${\mathbb{R}}$, and they gave some predictions on the state of decay of the disk radii where the eigenvalues of the operator $L$ are located. Now we provide a rigorous mathematical analysis of the asymptotics of eigenvalues $\lambda_n = \lambda_n(L^0 + W)$.
We follow the techniques used in [@DM06B], [@DM13Diff], [@AdMipub], [@MiSi], [@Elton03], [@Elton04] and based on careful estimates related to the resolvent representation
$$\begin{gathered}
\label{eq:rudefintro}
R = R^0 + \sum_{j = 1}^{\infty} U_j, \\
U_0 = R^0 , \quad U_k = R^0 W U_{k - 1} = U_{k - 1} W R^0, \quad k \geq 0.\end{gathered}$$
Moreover, we essentially use the property of perturbations $W \in $ to have such a matrix $$w_{jk} = \left\langle W h_j, j_k \right\rangle, \quad j, k = 0, 1, \dotsc$$ that for some $\alpha > 0$ there exists $M > 0$ such that $$\label{eq:wcondintro}
\vert w_{jk} \vert \leq \frac{M}{(1 + j)^{\alpha} (1 + k)^{\alpha}}, \quad j, k = 0, 1, \dotsc ,$$ Detailed results on the spectrum and convergence of spectral decompositions of $L = L^0 + W$ for a general $W$ under the condition were given by B. Mityagin and P. Siegl [@MiSi]. In the case of the finite point interaction perturbations ${\displaystyle}\alpha = \frac{1}{4}$.
Preliminaries, Technical Introduction, Review the Results {#sec:prelims}
=========================================================
Our main concern is the harmonic oscillator operator and its special perturbation $W$. We will focus on this case, although many constructions are very general and could be performed in analysis of other differential operators — see [@DM06B], [@AM12], [@MiSi], [@Elton03].
Let $L^0$ be an operator in $\ell^2({\mathbb{Z}}_+)$, $$L^0 e_k = z_k e_k, \quad z_k = (2k + 1), \quad k = 0, 1, \dotsc,
\label{eq:ell0}$$ and $W = (w_{jk})_0^{\infty}$ a matrix such that for some $\alpha > 0$ and $C_0 > 0$, $$\vert w_{jk} \vert \leq \frac{C_0}{(1 + j)^{\alpha} (1 + k)^{\alpha}} \label{eq:wcond}$$ Then the quadratic-form method [see @ReedSimon Section VII.6] leads to the definition of the closed operator $$L = L^0 + W \label{eq:elldef}$$ with a dense domain — see details in [@MiSi]. Let us recall some facts, introduce notations and explain a few elementary but important inequalities.
To adjust our constructions to the set of eigenvalues of the unperturbed operator , let us define strips $$\begin{aligned}
H_n & = {\left\lbrace z \in {\mathbb{C}}: \vert {\operatorname{Re}}z - z_n \vert \leq 1 \right\rbrace}, \quad n \geq 1\\
H_0 & = {\left\lbrace z \in {\mathbb{C}}: {\operatorname{Re}}z - z_0 \leq 1 \right\rbrace}
\end{aligned} \label{eq:hns}$$ and the squares $$\mathcal{D}_n = {\left\lbrace z \in H_n: \vert {\operatorname{Re}}z - z_n \vert \leq \frac{1}{2}, \vert {\operatorname{Im}}z \vert \leq \frac{1}{2} \right\rbrace}, \quad n \geq 0 \label{eq:dn}$$ around eigenvalues $ {\left\lbrace z_n \right\rbrace}_{n = 1}^{\infty} = {\left\lbrace 2n + 1 \right\rbrace}_{n = 0}^{\infty}$ in $H_n$.
The resolvent $$R(z) = (z - L^0 - W)^{-1} \label{eq:res}$$ of the operator is well-defined in the right half-plane $${\left\lbrace z: {\operatorname{Re}}z \geq 2 N_{*} \right\rbrace} \setminus \bigcup_{k = N_*}^{\infty} \mathcal{D}_k \label{eq:resdom}$$ outside of the disks $\mathcal{D}_k$, $k \geq N_*$, if $N_*$ is large enough.
It follows from the Neumann-Riesz decomposition $$\begin{aligned}
\begin{aligned}
R & = R^0 + R^0 W R^0 + R^0 W R^0 W R^0 + \dotsb\\
& = R^0 + \sum_{j = 1}^{\infty} U_j,
\end{aligned} \label{eq:rdecomp} \\
\intertext{where}
\begin{aligned}
U_0 &= R^0 = (z - L^0)^{-1},\\
U_j & = R^0 W U_{j - 1} = U_{j - 1} W R^0, \quad j \geq 1.
\end{aligned} \label{eq:udef}\end{aligned}$$ Of course, the convergence of the series should be explained at least in . This is done in [@AM12], [@MiSi]; now I’ll remind only the estimates of $N_*$ because it will be important later (see Theorem \[thm:eigendistr\], and Corollary XXX) in accounting for points of the spectrum $\sigma(L)$ outside of the real line.
Define a diagonal operator $K$, $$K e_j = \frac{1}{\sqrt{z - z_j}} e_j,\, \, j = 0, 1, \dotsc, \, \, {\operatorname{Im}}z \neq 0 \label{eq:kdef}$$ with understanding that $$\sqrt{\xi} = r^{1/2} e^{i \varphi / 2} \text{ if } \xi = r e^{i \varphi}, \quad - \pi < \varphi \leq \pi. \label{eq:sqrootdef}$$ Then $K^2 = R^0$, $z \in {\mathbb{C}}\setminus {\mathbb{R}}$; maybe, we lose analyticity but rough estimates – when just the absolute values of matrix elements work well – are good enough.
Indeed, , could be rewritten as $$\begin{gathered}
R^0 = K^2, \quad U_j = K (K W K)^j K, \label{eq:ujrevise}\\
R = R^0 + \sum_{j = 1}^{\infty} K (K W K)^j K,\end{gathered}$$ where $$(K W K)_{k m} = \frac{1}{\sqrt{z - z_k}} W_{km} \frac{1}{\sqrt{z - z_m}},\, \, k, m = 0, 1, 2, \dotsc, \, \, z \in {\mathbb{C}}\setminus {\mathbb{R}}. \label{eq:kwkform}$$
\[lem:hs\] Under the assumptions , and , with ${\displaystyle}0 < \alpha < \frac{1}{2}$, if $z \in H_n \setminus \mathcal{D}_n$, then $KWK$ is a Hilbert-Schmidt operator, and $$\ell {\equiv}\Vert K W K \Vert_{\text{HS}} \leq \frac{C_0 M(\alpha) \log (en)}{n^{2\alpha}} \label{eq:HSnorm}, \quad M(\alpha) {\equiv}6 + \frac{4/3}{1- 2 \alpha} + \frac{1}{3\alpha}$$
If $z \in {\partial (\mathcal{D}_n )}$, i.e, $$\begin{aligned}
\begin{aligned}
z = (2n + 1) + \xi + i \eta , & & &\quad \vert \xi \vert = \frac{1}{2}, \, \vert \eta \vert \leq \frac{1}{2},\\
&& \text{ or } & \quad \vert \xi \vert \leq \frac{1}{2}, \, \vert \eta \vert = \frac{1}{2}; \quad \xi, \eta \in {\mathbb{R}},
\end{aligned} \label{eq:bdloc}\end{aligned}$$ then $$\frac{1}{2} \leq \vert z - z_j \vert \leq 3, \, j = n, n \pm 1, \label{eq:zbd}$$ and if $\vert n - j \vert \geq 2$, $$\frac{3}{2} \vert n - j \vert \leq 2 \vert n - j \vert - 1 \leq \vert z - z_j \vert \leq 2 \vert n - j \vert + 1 \leq \frac{5}{2} \vert n - j \vert. \label{eq:njest}$$ Therefore, by , , $$\ell^2 = \sum_{j, k = 1}^{\infty} \frac{\vert w_{jk} \vert^2}{\vert z - z_j \vert \vert z - z_k \vert} \leq C_0^2 \mu^2 \label{eq:nuexpr}$$ with $$\mu = \sum_{j = 0}^{\infty} \frac{1}{(1 + j)^{2 \alpha} \vert z - z_j \vert}. \label{eq:mudef}$$ The sum of three terms for $j = n, n \pm 1$ in by does not exceed $$3 \cdot \frac{1}{n^{2\alpha}} \cdot 2 = \frac{6}{n^{2\alpha}}, \label{eq:smallvarianceterms}$$ and by , the remaining part of $\mu$, namely, $ {\displaystyle}\sum_{j = 0}^{n - 2} + \sum_{j = n + 2}^{\infty} $, by the integral test does not exceed $$\begin{aligned}
&\frac{2}{3} \left[ \frac{1}{n} + \frac{1}{n^{2\alpha}} + \int_0^{n - 1} \frac{dx}{(1 + x)^{2\alpha} (n - x)} \right]\\
+ & \frac{2}{3} \left[ \frac{1}{2} \cdot \left( \frac{1}{n + 3} \right)^{2\alpha} + \int_{n + 2}^{\infty} \frac{dy}{(1 + y)^{2\alpha} (y - n)} \right].
\end{aligned} \label{eq:bounda}$$ The first integral (after the change of variables $x = n \xi$) is $$\begin{aligned}
&\frac{1}{n^{2 \alpha}} \int_0^{1 - (1/n)} \frac{n \, d\xi}{n(1 - \xi) \left( \frac{1}{n} + \xi \right)^{2\alpha}} \leq \\
\leq & \frac{1}{n^{2 \alpha}} \left[ 2 \int_0^{1/2} \frac{d\xi}{\xi^{2 \alpha}} + 2^{2\alpha} \int_{1/2}^{1 - (1/n)} \frac{d\xi}{1 - \xi} \right] =\\
= & \left( \frac{2}{n} \right)^{2\alpha} \left[ \frac{1}{1-2\alpha} + \log \frac{n}{2} \right].
\end{aligned} \label{eq:bounda2.1}$$ The second integral in is equal to $$\begin{aligned}
&\frac{1}{n^{2\alpha}} \int_{1 + (2/n)}^{\infty} \frac{d\eta}{(\eta - 1)\left( \frac{1}{n} + \eta \right)^{2\alpha}} \leq \\
\leq & \frac{1}{n^{2 \alpha}} \left[ \int_{1 + (2/n)}^{2} \frac{d\eta}{\eta - 1} + \int_{2}^{\infty} \frac{d\eta}{(\eta - 1)^{2\alpha + 1}} \right] =\\
= & \frac{1}{n^{2\alpha}} \left[ \log \frac{n}{2} + \frac{1}{2\alpha} \right].
\end{aligned} \label{eq:bounda2.2}$$ If we collect the inequalities , we get (with $2^{2\alpha} \leq 2$) $$\begin{aligned}
&\begin{aligned}
\mu &\leq \frac{2}{3} \frac{1}{n^{2\alpha}} \left[ 9 + \frac{3}{2} + \frac{2}{1 - 2\alpha} + 3 \log \frac{e n}{2} + \frac{1}{2\alpha} \right] \leq \\
&\leq \frac{1}{n^{2\alpha}} [M(\alpha) + 2 \log n].
\end{aligned} \label{eq:muest}\\
&\text{ where } M(\alpha) = 6 + \frac{4/3}{1- 2 \alpha} + \frac{1}{3\alpha}. \label{eq:malphdef}\end{aligned}$$ With and we come to .
Of course, the constant factors in inequalities – are not sharp but we get some idea on their magnitude. If $\alpha = {\displaystyle}\frac{1}{4}$ we have $$\begin{gathered}
M\left( \frac{1}{4} \right) = 6 + \frac{4}{3} \cdot 2 + \frac{2}{3} < 10, \text{ and} \label{eq:alph1qtr1}\\
\mu \leq \frac{2}{\sqrt{n}} (5 + \log n) \label{eq:alph1qtr2}\end{gathered}$$ This case is important in analysis of the harmonic oscillator and its perturbations . The estimates will be used later as well.
\[rem:sbds\] Let $\displaystyle s \equiv \sum_{\substack{ j = 0 \\ j \neq n }}^{\infty} \frac{1}{(1 + j)^{\beta}} \cdot \frac{1}{\vert n - j \vert}$. Then
$$\begin{aligned}
s & \leq \frac{M(\beta)}{n^{\beta}} \log en, && \text{ if } 0 < \beta \leq 1, \label{eq:sbetasmall}\\
s & \leq \frac{M}{n}, && \text{ if } \beta > 1. \label{eq:sbetalarge}\end{aligned}$$
\[eq:sconds\]
The case $\beta = 2 \alpha < 1$ is done in the proof of Lemma \[lem:hs\]. Other cases could be explained in the same way; we omit details.
By the operator $K$ is bounded if $z \in H_n \setminus \mathcal{D}_n$ and by , its norm $$\Vert K \Vert \leq \sqrt{2}. \label{eq:knorm}$$ Therefore, for $U_j \in $ if $j \geq 2$ $$\Vert U_j \Vert_1 \leq 2 \Vert K W K \Vert_2^j \leq 2 \nu^j \leq 2 \left[ M(\alpha) \frac{\log e n}{n^{2\alpha}} \right]^j. \label{eq:ujest}$$ But we can claim that $U_1$ is a trace-class operator as well, and $$\Vert U_1 \Vert_1 = \Vert K(KWK) K \Vert_1 \leq \Vert K \Vert_4 \Vert KWK \Vert_2 \Vert K \Vert_4 \label{eq:u1bds}$$ because $K \in {\mathfrak{S}_{4}}$ \[or any ${\mathfrak{S}_{p}}$, $p > 2$, as a matter of fact\]: just notice that by , $$\begin{aligned}
\Vert K \Vert_4^4& = \sum_{j = 0}^{\infty} \frac{1}{\vert z - z_j \vert^2} \leq \\
&\leq 3/4 + 2 \sum_{k = 2}^{\infty} \left( \frac{2}{3} \right)^2 \cdot \frac{1}{k^2} < 20 < \left( \frac{11}{5} \right)^4,
\end{aligned} \label{eq:k4boundset}$$ so $$\Vert K \Vert_4 \leq \frac{11}{5}; \quad \Vert K \Vert_4^2 \leq 5. \label{eq:k4bound}$$ Therefore we can claim the following.
\[prop:normbds\] Under the assumptions , , $0 < \alpha < \frac{1}{2}$, suppose that $N_{*} = N_*(\alpha)$ is chosen in such a way that $$M(\alpha) \frac{\log en}{n^{2\alpha}} \leq \frac{1}{2} \quad \text{ for all } \quad n \geq N_*. \label{eq:nstardef}$$ Then for $n > N_*(\alpha)$ if $z \in {\partial (\mathcal{D}_n )}$ all the operators $U_j \in $ are of the trace class, their norms satisfy inequalities $$\begin{aligned}
\Vert U_j \Vert_1 &\leq 2 \left[ M(\alpha) \frac{\log en}{n^{2\alpha}}\right]^j, \quad j \geq 2, \label{eq:ujnorm}\\
\Vert U_1 \Vert_1 & = \Vert R^0 W R^0 \Vert_1 \leq \frac{5 M(\alpha) \log en}{n^{2\alpha}} \label{eq:u1norm}\end{aligned}$$ and the Neumann - Riesz series for the difference of two resolvents $$\begin{aligned}
R - R^0 &= \sum_{j = 1}^{\infty} U_j \label{eq:rdiff}\\
\intertext{converges by the ${\mathfrak{S}_{1}}$-norm and}
\Vert R - R^0 \Vert_1 &\leq 7 M(\alpha) \frac{\log en}{n^{2\alpha}}\,\label{eq:rdiffnorm}\\
\intertext{and}
{\left\Vert \sum_{j = m}^{\infty} U_j \right\Vert}_1 & \leq 4 \left( M(\alpha) \frac{\log en}{n^{2\alpha}} \right)^m,\quad m \geq 2. \label{eq:ujsumnorm}\end{aligned}$$
Inequality is identical with (proven) line . come if we combine , , and . Therefore, for $m \geq 2$, by and , $$\begin{aligned}
{\left\Vert \sum_{j = m}^{\infty} U_j \right\Vert}_1 & \leq 2 \sum_{j = m}^{\infty} \left( M(\alpha) \frac{\log en}{n^{2\alpha}} \right)^j \leq \\
& \leq 4 \left( M(\alpha) \frac{\log en}{n^{2\alpha}} \right)^m.
\end{aligned} \label{eq:umanynorm}$$ Then by $$\begin{aligned}
\Vert R - R^0 \Vert_1 & \leq \Vert U_1 \Vert_1 + {\left\Vert \sum_{j = 2}^{\infty} U_j \right\Vert}_1 \leq \\
& \leq M(\alpha) \frac{\log en}{n^{2\alpha}} \cdot \left(5 + 4 M(\alpha) \frac{\log en}{n^{2\alpha}} \right)\\
& \leq 7M(\alpha) \frac{\log en}{n^{2\alpha}}.
\end{aligned} \label{eq:rdiffnormest}$$
Deviations of eigenvalues of the harmonic oscillator operator and its perturbations {#sec:evalperturb}
===================================================================================
Although the constructions and methods of this section are general and applicable to many operators with discrete spectrum and their perturbations, we’ll focus later in this section on the case of Harmonic Oscillator operator and its functional representation $$L^0 y = - y^{\prime \prime} + x^2 y \label{eq:ell0reit}$$ in $L^2({\mathbb{R}})$.
The Riesz-Neumann Series , — as soon as its convergence in ${\mathfrak{S}_{1}}$ is properly justified — can be used to evaluate eigenvalues of the operator $L = L^0 + W$.
Under proper conditions, if $n \geq N_*$, the operator $L$ has the only eigenvalue $\lambda_n$ in $H_n$; moreover, $\lambda_n$ is simple and $\lambda_n \in \mathcal{D}_n$. Therefore, both of the projections $$\begin{aligned}
P_n^0 & = \frac{1}{2\pi i} \int\limits_{{\partial (\mathcal{D}_n )}} R^0(z) \, dz = {\left\langle \cdot, h_n \right\rangle} h_n \label{eq:pnbase}\\
\intertext{and}
P_n & = \frac{1}{2\pi i} \int\limits_{{\partial (\mathcal{D}_n )}} R(z) \, dz = {\left\langle \cdot, \psi_n \right\rangle} \phi_n \label{eq:pn}\end{aligned}$$ are of rank $1$. \[In , $\phi_n$ is an eigenfunction of $L$ and $\psi_n$ is an eigenfunction of $L^* = L^0 + W^*$, with an eigenvalue $\mu_n = \overline{\lambda}_n$ in $\mathcal{D}_n$. We will not use this specific information so nothing more is explained now.\]
Therefore, $$\begin{aligned}
\operatorname{Trace}P_n^0 = \operatorname{Trace}P_n &= 1, \label{eq:trabase}\\
\operatorname{Trace}\frac{1}{2\pi i} \int\limits_{{\partial (\mathcal{D}_n )}} (R(z) - R^0(z)) \, dz &= 0. \label{eq:trafull}\end{aligned}$$ and $$\begin{aligned}
z_n & = \operatorname{Trace}\frac{1}{2\pi i} \int\limits_{{\partial (\mathcal{D}_n )}} z R^0(z) \, dz = 2n + 1, \label{eq:lambase}\\
\lambda_n & = \operatorname{Trace}\frac{1}{2\pi i} \int\limits_{{\partial (\mathcal{D}_n )}} z R(z) \, dz, \label{eq:lamfull}\end{aligned}$$ So imply $$\begin{aligned}
\lambda_n - z_n &= \operatorname{Trace}\frac{1}{2\pi i} \int\limits_{{\partial (\mathcal{D}_n )}} (z - z_n) [R(z) - R^0(z)] \, dz\\
& = \sum_{j = 1}^{\infty} T_j(n)
\end{aligned} \label{eq:lamdiff}$$ where we put \[with $z_n = \lambda_n^0 = 2n + 1$\] $$T_j(n) = T_j(n; W) = \frac{1}{2\pi i} \operatorname{Trace}\int\limits_{{\partial (\mathcal{D}_n )}} (z - z_n) U_j(z) \, dz \label{eq:tjdef}$$ Proposition \[prop:normbds\] is used in , . *Trace* is a linear bounded functional of norm $1$, on the space ${\mathfrak{S}_{1}}$ of trace-class operators. It implies the following.
\[cor:tjnorm\] Under the assumptions of Proposition \[prop:normbds\], with $n \geq N_*$, we have $$\begin{aligned}
\vert T_j(n) \vert & \leq \left[ M(\alpha) \frac{\log en}{n^{2\alpha}}\right]^j, \quad j \geq 2 \label{eq:tjnorm}
\intertext{and}
\vert T_1(n) \vert & \leq \frac{9}{4} M(\alpha) \frac{\log en}{n^{2\alpha}} \label{eq:tjnorm.b}\end{aligned}$$
With $\vert \operatorname{Trace}A \vert \leq \Vert A \Vert_1$ and $$\begin{aligned}
\vert z - z_n \vert \leq \frac{1}{\sqrt{2}},& \quad z \in {\partial (\mathcal{D}_n )},\\
\text{length}(\mathcal{D}_n) = 4
\end{aligned} \label{eq:data2}$$ rough estimates of integrals with $j \geq 2$ and $j = 1$ based on and lead to and .
\[cor:lamest\] Under the assumptions of Proposition \[prop:normbds\], with $n \geq N_*$, $$\lambda_n = (2n + 1) + \sum_{j = 1}^q T_j(n) + r_q(n), \quad q \geq 1, \label{eq:lamest}$$ where $$\vert r_q(n) \vert \leq 2 \left( M(\alpha) \frac{\log en}{n^{2\alpha}} \right)^{q + 1} \label{eq:rqest}$$
The presentation of $\lambda_n$ and the inequality follow from and if we put $m = q + 1$ in and notice that $2 \sqrt{2} < \pi$ when we multiply the constant factors in inequalities.
Analysis of the function N sub \* alpha.
----------------------------------------
This function is determined by the inequality . Later we consider potentials with the coupling coefficient $s$ \[see , \] so it is useful to know the behavior of $X = X_{\beta}(t)$, the solution of the equation $$t \frac{\log eX}{X^\beta} = \frac{1}{2}, \quad \beta = 2\alpha, \text{ for large }t. \label{eq:tfromndef}$$ Let us rewrite it as $$\begin{aligned}
\tau \log Y = Y, \, \, & \text{where } Y = (eX)^{\beta} \label{eq:tfromndef2.1}\\
& \text{and } \tau = \frac{2}{\beta} e^{\beta}t \label{eq:tfromndef2.2}\end{aligned}$$ The equation has solutions $1 < y < Y$, such that $$\begin{aligned}
y(\tau) & = 1 + \frac{1}{\tau} + O {\left( \frac{1}{\tau^2} \right)}, &&\tau \to \infty \label{eq:y1def}\\
\intertext{and}
Y(\tau) & \to \infty, && \tau \to \infty. \label{eq:y2def}\end{aligned}$$
The solution $Y$ is $$\begin{aligned}
Y(\tau) &= \tau \log \tau \cdot (1 + r(\tau)) \label{eq:y2behave}\\
\intertext{where}
r(\tau) &= \frac{\log \log \tau}{\log \tau} \left( 1 + o(1) \right) \label{eq:y2r}\\
\intertext{so for any $\delta$ we can find $\tau^*$ such that}
Y(\tau) &\leq \tau \log \tau + \tau(1 + \delta) \log \log \tau, \quad \tau \geq \tau^*. \label{eq:y2bdhigh}\\
\intertext{or $\tau_* < \tau^*$ such that}
Y(\tau) &\leq (1 + \delta) \tau \log \tau, \quad \tau \geq \tau_*. \label{eq:y2bdsub}\end{aligned}$$
If we look for $r \geq 0$, in , which solves we have: $$\begin{aligned}
\tau \log \tau [1 + r] & = \tau[\log \tau + \log \log \tau + \log(1 + r)] \label{eq:y2proofbegin}\\
\intertext{or}
r = \varphi(r), &\quad \varphi(X) = \xi + \eta \log (1 + X), \quad r > 0 \label{eq:y2proofbegin.2}\\
\intertext{where}
\xi &= \frac{\log \log \tau}{\log \tau}, \quad \eta = \frac{1}{\log \tau} \label{eq:y2proofbegin.3}\end{aligned}$$ For any ${\displaystyle}0 < \delta\leq \frac{1}{2}$ we can choose $\tau^*$ such that $$0 < \xi \leq \frac{\delta}{2}, \quad 0 < \eta < \frac{\delta}{2} \quad \text{ if }\tau \geq \tau^*.$$ Then the function $\varphi$, $\varphi: [0, \delta] \to [0, \delta]$ is a contraction mapping, and has the unique solution $$r = r(\tau), \quad 0 < r(\tau) \leq \delta. \quad \label{eq:rconfirm}$$ Therefore, $$r = \frac{\log \log \tau}{\log \tau} + \frac{\rho}{\log \tau},\quad 0 < \rho \leq \delta. \label{eq:rform}$$ This implies with $$\frac{\rho}{\log \log \tau} = o(1).$$
\[cor:xtbehave\] The solution $X(t)$ of , $0 < \beta \leq 1$, goes to $\infty$ when $t \to \infty$ and $$X(t) \leq 2^{1/\beta} {\left( t \log \frac{At}{\beta} \right)}^{1/\beta}, \quad A \text{ an absolute constant, }\label{eq:xtest}$$ if $t$ is large enough.
If we put into or elementary simplifications give the inequality .
Inequalities and guarantee that we can use the representation and eventually “asymptotics” if $$C_0 M(\alpha) \frac{\log e n}{n^{2\alpha}} \leq \frac{1}{2}, \quad C_0 \in \text{\eqref{eq:wcond}} \label{eq:somebd}$$ and $M(\alpha)$ by is chosen as $$M(\alpha) = \left[ 6 + \frac{4/3}{1- 2 \alpha} + \frac{1}{3\alpha} \right]. \label{eq:malphhelp}$$ Then , with $\beta = 2\alpha < 1$, $t = 2 C_0 M(\alpha)$, implies that $N_*$ can be chosen as $$N_* = N_*(C_0; \alpha) = \left[ 2 C_0 M(\alpha) \log \left(\frac{A}{2\alpha} 2 C_0 M(\alpha) \right) \right]^{1/(2\alpha)} \label{eq:nstarchoice}$$ Now if $\alpha$ is fixed we are interested in the dependence of $N_*$ on $C_0 \in $.
Recall that if $W$ is a multiplier-operator $$Wf = w(x) f(x), \text{ with } w \in L^p({\mathbb{R}}^1), \quad 1 \leq p < \infty, \label{eq:wmultdef}$$ then as we observed and used in [@MiSi] $$w_{jk} = {\left\langle W h_j, h_k \right\rangle} = \int_{-\infty}^{\infty} w(x) h_j(x) h_k(x) \, dx \label{eq:wmatr}$$ so by Hölder inequality $$\vert w_{jk} \vert \leq {\left\Vert w \right\Vert}_p \cdot {\left\Vert h_j \right\Vert}_{2q} {\left\Vert h_k \right\Vert}_{2q}, \quad \frac{1}{p} + \frac{1}{q} = 1. \label{eq:wmatrbd}$$ with $$q > 1, \quad 2q > 2. \label{eq:qbd}$$ But $${\left\vert h_k(x) \right\vert} \leq C k^{-1/12} \label{eq:hsizebd}$$ so $$\begin{gathered}
\begin{aligned}
\int \vert h_k (x) \vert^{2q} \, dx &= \int \vert h_k (x) \vert^2 \vert h_k (x) \vert^{2(q-1)} \, dx\\
& \leq C^{2(q-1)} k^{-(q-1)/6} \int \vert h_k(x) \vert^2 \, dx
\end{aligned} \label{eq:hintbd}\\
\intertext{and}
{\left\Vert h_k \right\Vert}_{2q} \leq C^{1/p} k^{-1/(12p)}, \quad p \geq 1. \label{eq:hnormbd}\end{gathered}$$ This means that the matrix $W$ satisfies the condition with $\alpha = \frac{1}{12p}$ This observation was crucial in [@MiSi]; it gives a broad class of operators covered by so our claims of this sections are applicable to the operators .
But there are much better estimates of $L^p$ norms of the Hermite functions.
As $n \to \infty$,
$$\begin{aligned}
{\left\Vert h_n \right\Vert}_r &\sim n^{-\frac{1}{2} \left( \frac{1}{2} - \frac{1}{r} \right)}, \quad 1 \leq r < 4 \label{eq:hefcnsmallr}\\
{\left\Vert h_n \right\Vert}_r &\sim n^{- \frac{1}{8} } \log n, \quad r = 4 \label{eq:hefcnmidr}\\
{\left\Vert h_n \right\Vert}_r &\sim n^{-\frac{1}{6} \left( \frac{1}{r} + \frac{1}{2} \right)}, \quad r > 4 \label{eq:hefcnlarger}\end{aligned}$$
\[eq:hefcnvarr\]
See [@Thanga Lemma 1.5.2] for the sketch of the proof and further explanations of these claims.
Therefore, could be improved. If $p > 2$ then by $q < 2$, $2q < 4$ so $$\begin{aligned}
{\left\Vert h_k \right\Vert}_{2q} &\sim k^{-\frac{1}{2} \left( \frac{1}{2} -\frac{1}{2q} \right)} = k^{-\frac{1}{4p}} , && p > 2.\label{eq:hefcnactlargep}\\
\intertext{For $p = 2$ we have $2q = 4$ and}
{\left\Vert h_k \right\Vert}_{4} &\sim k^{-\frac{1}{8} } \log k, && p = 2. \label{eq:hefcnactmidp}\\
\intertext{Finally, if $1 \leq p < 2$ then $2q > 4$ so }
{\left\Vert h_k \right\Vert}_{2q} &\sim k^{-\frac{1}{6} \left( \frac{1}{2q} + \frac{1}{2} \right)} = k^{-\frac{1}{12} \left(2 - \frac{1}{p} \right)}, && 1 \leq p <2 & \qquad \label{eq:hefcnactsmallp}\end{aligned}$$ All these estimates are used in Proposition \[prop:imagw\] , — see Section \[subsect:genpointpot\] below.
Of course, shows that $\delta$-potentials $$w(x) = \sum_{k = 1}^m c_k \delta (x - b_k), \quad m \text{ finite,} \label{eq:wdelt}$$ are good for us as well; in this case, $$\begin{aligned}
{\left\langle W h_j, h_i \right\rangle} &= \sum_{k = 1}^m c_k h_j(b_k) h_i(b_l) \label{eq:wpairs}
\intertext{and}
{\left\vert W_{ji} \right\vert} &\leq CM j^{-1/12} i^{-1/12}, \quad i, j \geq 1, \quad M = \sum_{k = 1}^m \vert c_k \vert. \label{eq:wpairbds}\end{aligned}$$ But with more information on asymptotics of Hermite polynomials and Hermite functions we can be accurate in analysis of point-interaction potentials and spectra of operators $L^0 + W$, $L^0 \in $ , or — equivalently — . This is the main goal of this paper. Now we go to detailed analysis of these operators.
Two-point interaction potentials {#sec:2pt}
================================
{#subsect:genpointpot}
Now we apply general constructions of Sections \[sec:prelims\], \[sec:evalperturb\] to the case of the two-point interaction potentials $$w(x) = c^+ {\delta \left( x - b \right) } + c^- {\delta \left( x + b \right) }, \quad b > 0 \label{eq:pointmass1}$$ and particular cases of an odd potential $$\begin{aligned}
s v^o(x), &\quad s \in {\mathbb{C}}\quad \text{ where }&& v^o(x) = {\delta \left( x - b \right) } - {\delta \left( x + b \right) } \label{eq:pointodd}\\
\intertext{and an even potential}
t v^e(x), &\quad t \in {\mathbb{C}}&& v^e(x) = {\delta \left( x - b \right) } + {\delta \left( x + b \right) } \label{eq:pointeven}\end{aligned}$$ — see [@HCW] and [@FassRin].
Of course, for any *odd* potential or , not just for $v^0 \in$ , the matrix elements $w_{jk}$ have the property . Indeed, with parity $$\begin{aligned}
w_{jk} &= {\left\langle w(x) h_j(x),\, h_k(x) \right\rangle} = \label{eq:woddprop1}\\
&= {\left\langle w(-x) h_j(-x),\, h_k(-x) \right\rangle} = -(-1)^{j + k} w_{jk}\label{eq:woddprop2}\end{aligned}$$ so $$w_{jk} = 0 \text{ if } j + k \text{ even.} \label{eq:orthooddeven} $$ If, however, $w$ in or is *even* then we conclude $$w_{jk} = 0 \text{ if } j + k \text{ odd.} \label{eq:orthoevenodd}$$ These observations lead to information on complex eigenvalues of $L = L^0 + W$.
\[prop:imagw\] Let the potential $$\begin{aligned}
w(x) \in L^p, \, \, 1 \leq p < \infty, &\quad \nu = {\left\Vert w \right\Vert}_p, \text{ or} \label{eq:wtype1}\\
w(x) = \sum_{k = 1}^{\infty} c_k {\delta \left( x - b_k \right) }, &\quad \nu = \sum {\left\vert c_k \right\vert} < \infty, \label{eq:wtype2}\end{aligned}$$ be real and odd. Then the operator $$L = L^0 + i W = - \frac{d^2}{dx^2} + x^2 + i w \label{eq:newell}$$ has at most finitely many non-real eigenvalues, if any, and their number does not exceed $N^*$, where
$$\begin{aligned}
N^* &= D\left (\nu \log (1 + \nu) \right)^{2p}, && p > 2 \label{eq:newnstar.plarge}\\
N^* &= D^* \left (\nu \log^2 (1 + \nu) \right)^{4}, && p = 2 \label{eq:newnstar.pmid}\\
N^* &= D \left (\nu \log (1 + \nu) \right)^{\frac{3}{\left(1 - \frac{1}{2p} \right)}}, && 1 \leq p < 2 \label{eq:newnstar.psmall}\\
N^* &= D_*\left( \nu \log (1 + \nu) \right)^{6}, && \text{ in the case \eqref{eq:wtype2}} \label{eq:newnstar.2}\end{aligned}$$
\[eq:newnstar\]
and $D^*$, $D_*$ are absolute constants although $D = D(p)$ does not depend on the norm $\nu$.
By and estimates in Corollary and in we can use to evaluate $\lambda_n$, at least if $p \neq 2$ when we use . \[Details of this case $p = 2$ are explained in Section \[sec:comment\]
\[lem:tjodd\] If $j$ is odd and $W$ is odd, then for $n \geq N^*$ ($N^*$ as defined in ), $$T_j(n; W) = 0. \label{eq:toddzero}$$
By $$\begin{aligned}
T_j(n) = T_j(n; W) = \frac{1}{2\pi i} \operatorname{Trace}\int\limits_{{\partial (\mathcal{D}_n )}} \, \, (z - z_n) U_j(z) \, dz \label{eq:tjdefrep}\\
\intertext{where}
U_j = R^0 W R^0 W \dotsb W R^0 \label{eq:ujdefrep}\end{aligned}$$ with $j$ “letters” $W$ and $j + 1$ “letters” $R^0$ in this “word” $U$. All these operators are of trace class \[see Corollary \[cor:tjnorm\]\] so $\operatorname{Trace}U_j$ is a sum of the diagonal elements $(U_j)_{mm}$ which in turn are sums of matrix elements $\displaystyle \sum_g u(g)$, where $g = {\left\lbrace g_1, g_2, \dotsc, g_{j-1} \right\rbrace} \in {\mathbb{Z}}_+^{j-1}$ and
$$u(g) = \frac{1}{z - z_m} \cdot W(m, g_1) \cdot \frac{1}{z - z_{g_1}} \cdot W(g_1, g_2) \cdot \frac{1}{z - z_{g_2}} \dotsb W(g_{j-1}, m) \cdot \frac{1}{z - z_m} \label{eq:ugdef}$$
If we put $g_0 = g_j = m$ we have $$\sum_{t = 0}^{j-1} (g_{t + 1} - g_t) = g_j - g_0 = 0 \label{eq:gsum}$$ The sum of all these differences in is even (zero), so if $j$ is odd, then at least one of these differences, say $$g_{\tau + 1} - g_{\tau}, \quad 0 \leq \tau < j,$$ is even, and the factor $W(g_{\tau}, g_{\tau + 1})$ in by is zero. Therefore, $u(g) = 0$, and this holds for all elements $u(g) \in $ so $\sum u(g) = 0$.
\[lem:tjeven\] If $j$ is even and $W$ is real, then for $n \geq N^*$ ($N^*$ as defined in ), $$T_j(n) {\equiv}T_j(n; iW) \text{ is real.} \label{eq:tevenreal}$$
Now the number $j = 2q$ of $W$-factors in the product is even for any $g \in {\mathbb{Z}}_+^{(j-1)}$ so $$p(g) = \prod_{t = 0}^{j-1} iW(g_t, g_{t + 1}) = (-1)^q \cdot \text{(real number)}$$ is real by or . and $$\begin{gathered}
u(g) = p(g) \cdot J(g), \text{ where} \label{eq:udecomp} \\ J(g) = \frac{1}{2\pi i} \int\limits_{{\partial (\mathcal{D}_n )}} F_g(z) \, dz \label{eq:jdef}, \text{ with} \\
F_g(z) = (z - z_n) \cdot \frac{1}{(z - z_m)^2} \prod_{t = 1}^{j-1} \frac{1}{z - z_{g_t}} \label{eq:fgzdef}\end{gathered}$$ For any $g$ this integral is real number \[see the next lemma\]. Therefore, $T_j(n)$ — as a sum of (absolutely) convergent series with real terms — is a real number.
This completes the proof of Proposition 4.1.
We will need more specific information about integrals $J(g)\in$ . The following is true.
\[lem:jgbehave\] If $m = n$, and $n \geq N^*$ ($N^*$ as defined in ), $$\begin{aligned}
J(g) = 0 \quad \text{ if at least one } \quad g (\tilde{t}\, ) = n, \label{eq:jequalzero} \\
\intertext{and}
J(g) = {\left( \prod_{t = 1}^{j-1}2 (n - g_t) \right)}^{-1} \quad \text{otherwise} \label{eq:jequalnonzero}.\end{aligned}$$ If $m \neq n$, $$J(g) = 0 \,\text{ if }\,\# \tau(g) \neq 2, \, \, \text{where $\tau(g) = {\left\lbrace t: g_t = n \right\rbrace}$,} \label{eq:jnonequalzero}$$ and $$J(g) = \frac{1}{4(n-m)^2}{\left( \prod_{t \not\in \tau(g)}^{j-1}2 (n - g_{t}) \right)}^{-1} \text{ if } \# \tau(g) = 2. \label{eq:jnonequalnonzero}$$
The integrand of could have a pole inside of $\mathcal{D}_n$ only at $z_n = 2n + 1$. In the cases and the pole’s order $\geq 2$ or $F_g(z)$ is analytic on $\overline{\mathcal{D}_n}$, so $J(g) = 0$. In the cases , the pole’s order is one and $J(g)$ is the residue of $F_g(z)$ at $z_n$.
An odd potential v-super-o {#subsect:oddpot}
---------------------------
As it is noticed in , $$\begin{gathered}
\begin{aligned}
v^0_{jk} &= {\left\langle ({\delta \left( x-b \right) } - {\delta \left( x + b \right) }) h_j, h_k \right\rangle} \\
& = [1 - (-1)^{j + k}] a_j a_k
\end{aligned} \label{eq:v0set}\\
= \begin{cases} 0, & \text{ if } j + k \text{ even} \\
2 a_j a_k, & \text{ if } j + k \text{ odd} \end{cases} \label{eq:v0case} \\
\intertext{where}
a_k = h_k(b), \quad k = 0, 1, \dotsc \label{eq:akdef}\end{gathered}$$ With $b > 0$ fixed, from now on we will use $(a_k)$ as in . By Lemmas \[lem:tjodd\] and \[lem:jgbehave\] for $n \geq N^*$ $$\begin{aligned}
T_j(n; v^0) {\equiv}T_j(n)& = 0 \, \, \, \, \text{ if } j \text{ odd;} \label{eq:orthooddevenext} \\
\intertext{in particular,}
T_1(n) &= 0 , \quad T_3(n) = 0. \label{eq:orthospec} \end{aligned}$$ To evaluate $T_2(n)$ we’ll sum up (we did it in in general setting) Cauchy integrals of functions $$(z - z_n) \cdot \frac{1}{z - z_m} \cdot v_{mk}^0 \cdot \frac{1}{z - z_k} \cdot v_{km}^{0} \cdot \frac{1}{z - z_m} \label{eq:vgdef}$$ If $m \neq n$ it is analytic for any $k$ so Cauchy integral is zero. If $m = n$ $$v_{mk}^0 = 0 \quad \text{ if } n - k \text{ is even.} \label{eq:orthodiff}$$ Therefore, by Lemma \[lem:jgbehave\], $j = 2$, with $z_n - z_k = 2(n-k)$, $$\begin{aligned}
T_2(n; v^0) {\equiv}T_2(n) & = \sum_{\substack{ k = 0 \\ k - n \text{ odd} }}^{\infty} \frac{v_{nk}^0 v_{kn}^0}{z_n - z_k} = \sum_{\substack{ k = 0 \\ k - n \text{ odd} }}^{\infty} \frac{2 a_n a_k \cdot 2 a_k a_n }{2(n - k)} = 2 a_n^2 {\widetilde{\sigma}}(n) \label{eq:t2resolve}\\
\intertext{where}
{\widetilde{\sigma}}(n) &= \sum_{\substack{ k = 0 \\ n - k \text{ odd} }}^{\infty} \frac{a_k^2 }{n - k} \label{eq:sigdef}\end{aligned}$$ Technical analysis of the sequence ${\widetilde{\sigma}}(n)$ is done in the next section. Of course, it is based on asymptotics of Hermite polynomials (or functions), – . It will bring us the proof of the main result of this paper:
\[thm:eigendistr\] The operator $$L = - \frac{d^2}{dx^2} + x^2 + s [ {\delta \left( x-b \right) } - {\delta \left( x + b \right) }], b > 0, s \in {\mathbb{C}}$$ has a discrete spectrum $\sigma(L)$.
There exists an absolute constant $D$ such that with $$N^* = \left( D \vert s \vert \log e \vert s \vert \right)^2 \label{eq:nstarnewdef}$$ all eigenvalues $\lambda_n = \lambda_n(L)$ in the half-plane ${\left\lbrace z \in {\mathbb{C}}: {\operatorname{Re}}z > N^* \right\rbrace}$ are simple, and $$\begin{aligned}
\lambda_n &= (2n + 1) + s^2 \, \frac{{\kappa}(n)}{n} + \widetilde{\rho}(n), \quad \vert \widetilde{\rho}(n) \vert \leq C\frac{\log n}{n^{3/2}}\label{eq:lamasymp}\\
\intertext{where}
{\kappa}_n &= \frac{1}{{2\pi}} \left[ (-1)^{n + 1} \sin (2 b \sqrt{2n}) - \frac{1}{2} \sin(4b \sqrt{2n}) \right] \label{eq:lamasymp.b}\end{aligned}$$
The proof of the theorem is based on the following lemma.
\[lem:siginfo\] With ${\widetilde{\sigma}}(n) \in$ $$\begin{aligned}
{\widetilde{\sigma}}(n) & = (-1)^{n + 1}\frac{1}{2} \frac{\sin (2 b \sqrt{2n})}{\sqrt{2n}} + \rho(n), \label{eq:sigdata}\\
\vert \rho(n) \vert &\leq C \frac{\log n}{n} \label{eq:rhobd}\end{aligned}$$
The technical analysis of this sequence and its variations is the core of this manuscript. Its proof is given in the sections which follow. The final steps to prove Theorem \[thm:eigendistr\] are done in Section \[subsect:evenintro\], – .
An even potential v-super-e {#subsect:evenpot}
---------------------------
Recall ; now $$\begin{aligned}
v^e_{jk} &= [1 + (-1)^{j + k}] a_j a_k \label{eq:veset}\\
&= \begin{cases} 0, & \text{ if } j + k \text{ odd} \\
2 a_j a_k, & \text{ if } j + k \text{ even} \end{cases} \label{eq:vecase} \\\end{aligned}$$ Therefore, by Lemma \[lem:jgbehave\], $j = 1$, $$T_1(n; v^e) {\equiv}T_1(n) = v_{nn}^e = 2 a_n^2 \label{eq:t1even}$$ and \[compare \] $$\begin{aligned}
T_2(n; v^e) {\equiv}T_2(n) &= 2 a_n^2 {\sigma^{\prime}}(n),\label{eq:t2even} \\
{\sigma^{\prime}}(n) & = \sideset{}{'}\sum_{\substack{k = 0\\ n - k \text{ even}}}^{\infty} \frac{a_k^2}{n - k} \label{eq:sigeven},\end{aligned}$$ where $\sum\nolimits^{'}$ means that $k \neq n$.
But for the even potential there is no trivial claim $T_3(n) = 0$. We could make formal references to Lemma \[lem:jgbehave\] but let us again look into those terms which form the sum-trace $T_3(n)$. We integrate functions $$F = (z - z_n) \cdot \frac{1}{z - z_m} \cdot 2a_m a_k \cdot \frac{1}{z - z_k} \cdot 2 a_k a_{\ell} \cdot \frac{1}{z - z_{\ell}} \cdot 2 a_{\ell} a_m \cdot \frac{1}{z - z_m} \label{eq:fgdef}$$ excluding (by ) triples $(m, k, \ell)$ if at least one of the differences $m - k$, $k - \ell$, $\ell - m$ is odd.
If $m = n$ then we can take only $k, \ell \neq n$, otherwise the order of the pole at $z_n$ would be $\geq 2$ and Cauchy integral be zero. Then the partial sum of over triples $${\left\lbrace (m, k, \ell) \vert m = n, k \neq n, \ell \neq n, k - n, \ell - n \text{ even} \right\rbrace}$$ would be $$\begin{aligned}
2 a_n^2 \sum_{\substack{k, \ell \\ n - k, \\ n - \ell \text{ even}}}^{\prime} \frac{a_k^2 a_{\ell}^2}{(n - k)(n - \ell)} = \label{eq:ansumcase1quest}\\
= 2 a_n^2 \left( {\sigma^{\prime}}(n) \right)^2. \label{eq:ansumcase1ans}\end{aligned}$$
If $m \neq n$ Cauchy integral of $F \in$ is not zero only if $k = \ell = n$, i.e., if we have two (and only two) zeros in the denominator to balance the factor $(z - z_n)$. This set of triples $${\left\lbrace (m, k, \ell) \vert m \neq n, k= \ell = n, m - n \text{ even} \right\rbrace} \label{eq:mklcase2}$$ leads to the subsum in $T_3(n)$ coming from $$2 a_n^4 \sideset{}{'}\sum_{\substack{m = 0 \\ m - n \text{ even}}}^{\infty} \frac{a_m^2}{(n - m)^2} = 2 a_n^4 \tau^{\prime}(n) \label{eq:ansumcase2}$$ If we combine – we conclude that $$T_3(n; v^e) = 2 a_n^2 \left[ \sigma^{\prime}(n)^2 + a_n^2 \tau^{\prime}(n) \right] \label{eq:t3even}$$ We’ll analyze the sequences ${\sigma^{\prime}}$, $\tau^{\prime}$ later as well.
Inequalities and technical analysis of the term T2(n) in the case of an odd two-point delta-potential {#sec:manyineqs}
=====================================================================================================
First of all, let us recall the asymptotics of Hermite polynomials \[see (8.22.8) in Szegő, [@Szego]\].
With $M = 2m + 1$
$$\begin{aligned}
\frac{\Gamma \left( \frac{m}{2} + 1 \right)}{\Gamma (m + 1)} e^{-x^2 / 2} H_m(x) &=& &\cos \left( M^{1/2}x - m \frac{\pi}{2} \right) + \\
&&+& \frac{x^3}{6} M^{-1/2} \sin \left( M^{1/2}x - m \frac{\pi}{2} \right) + O \left( \frac{1}{m} \right).
\end{aligned} \label{eq:hepoly1}$$
The normalized Hermite functions $h_m(x)$, $\int h_m^2(x) \, dx = 1,$ are
$$h_m(x) = (m! 2^m \sqrt{\pi} )^{-1/2} H_m(x) e^{-x^2/2} \label{eq:hefcndef}$$
so $(m \geq 1)$ $$\begin{aligned}
h_m(x) = \frac{2^{1/4}}{\pi^{1/2}} \frac{1}{m^{1/4}}& \left[ \cos \left( x \sqrt{2m + 1} - m \frac{\pi}{2} \right) \right.\\
&+ \left. \frac{x^3}{6} \frac{1}{\sqrt{2m + 1}} \sin \left( x \sqrt{2m + 1} - m \frac{\pi}{2} \right) + O \left(\frac{1}{m} \right) \right]
\end{aligned} \label{eq:hefcnform}$$
Of course, this information is crucial because $$a_j = h_j(b) \label{eq:haeq}$$ and by $${\widetilde{\sigma}}(n) = \sum_{\substack{k = 0 \\ n - k \text{ odd}}} \frac{a_k^2}{n - k} \label{eq:sigoddrevise}$$
{#subsect:chopandpeel}
We will chop and peel this sum by getting “error” terms $\rho$’s (with indices) of order ${\displaystyle}O \left( \frac{\log n}{n}\right).$ After finitely many steps we’ll sum up all these error-terms into $\rho(n)$ in .
First adjustment \[to make us flexible to have the denominator $m^{1/4}$ without $m$ being zero\] is to take away in the term with $k = 0$ if $n$ is odd, i.e. $$\rho_1 = \frac{a_0^2}{n} = \frac{1}{\pi} e^{-b^2} \cdot \frac{1}{n} = O \left( \frac{1}{n}\right). \label{eq:rho1expr}$$ Maybe, more drastic is to change $(a_k)$ to the first term $a_k^{\prime}$ in so
$$\begin{aligned}
a_k^{\prime} {}^2 &= \frac{2^{1/2}}{\pi} \frac{1}{\sqrt{k}} \cos^2 \left( b \sqrt{2k + 1} - k \frac{\pi}{2} \right) \label{eq:aksquareform}\\
& = \frac{1}{\pi} \cdot \frac{1}{\sqrt{2k}} \left( 1 + (-1)^k \cos 2b \sqrt{2k + 1}\right) \label{eq:aksquareform2}\end{aligned}$$
It will help us but let us explain first that the following is true.
\[lem:sumest\]Let ${\alpha}_k$, ${\alpha}_k^{\prime}$ be two sequences such that
$$\begin{aligned}
\vert {\alpha}_k \vert, \vert {\alpha}_k^{\prime} \vert \leq \frac{C}{k^{1/4}} \label{eq:alphcond}\\
\delta_k = {\alpha}_k^{\prime} - {\alpha}_k, \quad \vert \delta_k \vert \leq \frac{C}{k^{3/4}} \label{eq:alphdiffcond} \end{aligned}$$
\[eq:alphakcond\]
Then for their transforms $$A(n) = \sum_{\substack{k = 1\ \\ n - k \text{ odd}}}^{\infty} \frac{{\alpha}_k^2}{n - k}, \quad A^{\prime}(n) = \sum_{\substack{k = 1 \\ n - k \text{ odd}}}^{\infty} \frac{{\alpha}_k^{\prime} {}^2}{n - k} \label{eq:ansdef}$$ we have $$\vert A_n \vert, {\left\vert A_n^{\prime} \right\vert} \leq C \frac{\log n}{\sqrt{n}} \label{eq:ansbds}$$ and $$\vert A_n - A_n^{\prime} \vert \leq C \frac{\log n}{n} \label{eq:andiffsbd}$$
By
$$\begin{split}
\vert A(n) \vert &\leq \sum_{k \neq n} \frac{\vert {\alpha}_k \vert^2}{\vert n - k \vert} \leq \\
& \leq C^2 \sum \frac{1}{k^{1/2}} \frac{1}{{\left\vert n - k \right\vert}} \leq \frac{C^{\prime}}{\sqrt{n}} \log (en)
\end{split} \label{eq:anineqs}$$
The latter comes from with ${\displaystyle}\beta = \frac{1}{2}$; of course, the same is true for $A^{\prime}(n)$.
Next, by , $$\begin{split}
{\left\vert {\alpha}_k^2 - {\alpha}_k^{\prime} {}^2 \right\vert} & = {\left\vert {\alpha}_k + {\alpha}_k^{\prime} \right\vert} {\left\vert {\alpha}_k - {\alpha}_k^{\prime} \right\vert}\leq \\
& \leq \frac{2C}{k^{1/4}} \cdot \frac{C}{k^{3/4}} = \frac{\widetilde{C}}{k}
\end{split} \label{eq:squarediffs}$$ Again, , $\beta = 1$, implies $${\left\vert A(n) - A^{\prime}(n) \right\vert} \leq \frac{C}{n} \log(en). \label{eq:andiffact}$$
Lemma \[lem:sumest\] shows that $$\rho_2(n) = \left\vert \sum_{\substack{k = 0 \\ n - k \text{ odd}}}^{\infty} \frac{a_k^2 - a_k^{\prime} {}^2}{n - k} \right\vert = O \left( \frac{\log n}{n}\right) \label{eq:rho2bd}$$ so $\rho_2$ is under the restriction .
In this formula and later on we follow notation and .
Therefore, we can proceed with a focus on the sequence and the series
$${\widetilde{\sigma}^{\prime}}(n) = \frac{1}{\pi \sqrt{2}} \sigma(n), \quad \sigma(n) = \sum_{\substack{k = 1 \\ n - k \text{ odd}}}^{\infty} \frac{1}{\sqrt{k}} \left[ 1 + (-1)^{n + 1} \cos 2b \sqrt{2k + 1}\right] \frac{1}{n - k} \label{eq:sigrenew}$$
It is important to point out that the coefficient $(-1)^k$ in dependent on $k$ can be written in as $(-1)^{n + 1}$ without dependence on the index $k$ of summation because in this sum only $k \equiv n + 1 \pmod{2}$ are involved.
Put $$\begin{aligned}
\sigma(n) & = \xi(n) + (-1)^{n + 1} \eta(n) \label{eq:sigrenewdecomp}\\
\intertext{where}
\xi(n) & = \sum_{\substack{k = 1 \\ n - k \text{ odd}}}^{\infty} \frac{1}{\sqrt{k}} \cdot \frac{1}{n - k} \label{eq:zetarenew}\\
\eta(n) & = \sum_{\substack{k = 1 \\ n - k \text{ odd}}}^{\infty} \frac{1}{\sqrt{k}} \cdot \frac{\cos 2 b \sqrt{2k + 1}}{n - k}. \label{eq:etarenew}\end{aligned}$$
{#section-7}
We’ve already seen (Lemma \[lem:sumest\]) that ${\displaystyle}{\left\vert \xi(n) \right\vert} = O {\left( \frac{\log n}{\sqrt{n}} \right)}$ but such an estimate is not good for $\rho$ in . We could hope to get rid of $\log n$ because of the sign-change in the term $\frac{1}{n - k}$. However, $\xi$ decays much faster than ${\displaystyle}O {\left( \frac{1}{\sqrt{n}} \right)}$. The following is true.
\[lem:zetabds\] Let $\xi \in$ . Then $${\left\vert \xi(n) \right\vert} \leq \frac{C}{n} \label{eq:zetabd}$$
The formulas are a little bit different for $n$ even and odd. Let us write all the details for $n = 2p + 1$ odd fixed; then only even $k = 2m$, $m = 1, 2, \dotsc $, are involved so $$\begin{aligned}
\xi(n) &= \xi(2p + 1) = \frac{1}{2 \sqrt{2}} \sum_{m = 1}^{\infty} \frac{1}{\sqrt{m}} \cdot \frac{1}{(p - m) + \frac{1}{2}} \label{eq:zetaexpr0}\\
&= 2^{-3/2}[S_1 - S_2] \label{eq:zetaexpr}\end{aligned}$$ where $$\begin{aligned}
S_1 & = \sum_{j = 0}^{p-1} \frac{1}{j + 1/2} \left( \frac{1}{\sqrt{p - j}} - \frac{1}{\sqrt{p + j + 1}} \right) , \\
S_2 & = \sum_{j = p}^{\infty} \frac{1}{j + 1/2} \cdot \frac{1}{\sqrt{p + j + 1}} \, .
\end{aligned} \label{eq:sumforms}$$
Put $$P = p + \frac{1}{2},\quad t_j = j + \frac{1}{2}, \quad \text{ and} \label{eq:somesymbols}$$ $$\varphi(x) = \frac{1}{x} \cdot \frac{1}{\sqrt{P + x}}, \quad x \geq P \label{eq:varphinewdef}$$ Then, with the integral test, we compare the sum $$\begin{gathered}
S_2 = \sum_{j = p}^{\infty} \frac{1}{t_j} \cdot \frac{1}{\sqrt{P + t_j}} = \sum_{j = p}^{\infty} \varphi(t_j) \label{eq:s2def}\\
\intertext{and the integral}
I_2 = \int_{P}^{\infty} \varphi(x) \, dx. \label{eq:i2def}\end{gathered}$$ Monotonicity of $\varphi(x)$ down implies that $$S_2 \geq I_2 \geq S_2 - \varphi(t_p) = S_2 - \frac{1}{\sqrt{2} P^{3/2}} \label{eq:twomono}$$ so $$\begin{aligned}
S_2 & = I_2 + \rho_3(n), \label{eq:s2decomp} \\
0 &\leq \rho_3(n) \leq \frac{1}{\sqrt{2} P^{3/2}}; \label{eq:rho3bd}\end{aligned}$$ $\rho_3$ is well under the restriction .
is evaluated explicitly: $$\begin{gathered}
\begin{split}
I_2 & = \int_P^{\infty} \frac{dx}{x} \cdot \frac{1}{\sqrt{P + x}} = \frac{1}{P^{1/2}} \int_1^{\infty} \frac{dt}{t} \frac{1}{\sqrt{1 + t}} = \\
& = \frac{1}{P^{1/2}} \int_0^1 \, \frac{dv}{v} \frac{\sqrt{v}}{\sqrt{ 1+ v}} = B \cdot \frac{1}{P^{1/2}}
\end{split} \label{eq:i2eval}\\
\begin{split}
\text{where } B & = \int_0^1 \frac{dv}{\sqrt{\left(v + \frac{1}{2} \right)^2 - \left( \frac{1}{2} \right)^2}} = \int_1^3 \frac{du}{\sqrt{u^2 - 1}} = \\
& = \int_0^{{\kappa}} \frac{d(\cosh s)}{\sinh s} = {\kappa}, \quad \cosh {\kappa}= 3, \quad {\kappa}> 0
\end{split} \label{eq:bdef}\\
\begin{split}
\text{i.e. } T + \frac{1}{T} = 6, \quad T = e^{{\kappa}} > 1,\\
T^2 - 6T + 1 = 0 \quad \text{so}
\end{split} \label{eq:Tdef}\\
\begin{split}
T = 3 + \sqrt{8}, \quad \text{or} \quad {\kappa}& = \log (3 + \sqrt{8}) = \\
& = 2 \log(1 + \sqrt{2})
\end{split} \label{eq:constfinal}\end{gathered}$$ $$\begin{split}
I_2 & = \int_P^{\infty} \frac{dx}{x} \cdot \frac{1}{\sqrt{P + x}} = \frac{1}{P^{1/2}} \int_1^{\infty} \frac{dt}{t} \frac{1}{\sqrt{1 + t}} = \\
& = \frac{1}{P^{1/2}} \int_0^1 \, \frac{dv}{v} \frac{\sqrt{v}}{\sqrt{ 1+ v}} = B \cdot \frac{1}{P^{1/2}} \\
\text{where } B & = \int_0^1 \frac{dv}{\sqrt{\left(v + \frac{1}{2} \right)^2 - \left( \frac{1}{2} \right)^2}} = \int_1^3 \frac{du}{\sqrt{u^2 - 1}} = \\
& = \int_0^{{\kappa}} \frac{d(\cosh s)}{\sinh s} = {\kappa}, \quad \cosh {\kappa}= 3, \quad {\kappa}> 0\\
\text{i.e. } &T + \frac{1}{T} = 6, \quad T = e^{{\kappa}} > 1,\\
&T^2 - 6T + 1 = 0 \quad \text{so}\\
T &= 3 + \sqrt{8}, \quad \text{or}\\
\quad {\kappa}&= \log (3 + \sqrt{8}) = 2 \log(1 + \sqrt{2})
\end{split} \label{eq:constBfinal}$$
\[We changed the variables of integration as $$\begin{split}
x = Pt, \quad t = \frac{1}{v}, \quad \frac{u}{2} = v + \frac{1}{2}, \quad u = \cosh s .]
\end{split} \label{eq:varchanges}$$ Finally, $$I_2 = BP^{-1/2}, \quad B = 2 \log (1 + \sqrt{2}). \label{eq:i2final}$$
{#section-8}
To analyze $S_1$ let us introduce functions $$\begin{gathered}
w = \sqrt{P^2 - x^2} \label{eq:wdef}\\
\begin{split}
\psi(x) = \frac{1}{x} \left[ \frac{1}{\sqrt{P - x}} - \frac{1}{\sqrt{P + x}} \right], \\
\frac{1}{2} \leq x \leq P - 1 = p - \frac{1}{2}.
\end{split} \label{eq:psidef}\end{gathered}$$ Then $$\begin{split}
\psi(x) &= \frac{1}{x} \cdot \frac{\sqrt{P + x} - \sqrt{P - x}}{w} = \frac{2}{w \left[ \sqrt{P + x} + \sqrt{P - x} \right]} = \\
&= \frac{\sqrt{2}}{w [P + w]^{1/2}}, \quad \text{ monotone increasing,}
\end{split} \label{eq:psimani}$$ and the sum $$S_1 = \sum_{j = 0}^{p - 1} \psi(t_j), \quad t_j = j + \frac{1}{2} \label{eq:s1def}$$ could be compared with the integral $$I_1 = \int_{1/2}^{p - 1/2} \psi(x) \, dx. \label{eq:i1def}$$ Indeed $$\begin{gathered}
S_1 \geq I_1 \geq S_1 - \psi(t_{p-1}) \label{eq:s1mono}\\
\psi(t_{p - 1}) = \frac{1}{p - \frac{1}{2}} \left[ 1 - \frac{1}{\sqrt{2p}} \right] \leq \frac{1}{p} \label{eq:psiest1}\end{gathered}$$ so \[compare \] $$\begin{gathered}
S_1 = I_1 + \rho_4(n), \label{eq:s1decomp}\\
0 \leq \rho_4(n) \leq \psi(t_{p-1}) \leq \frac{1}{P}, \label{eq:rho4bd}\end{gathered}$$ well under the restriction .
Next, we evaluate $I_1 \in $ explicitly, $w = \sqrt{P^2 - x^2}$, ${\displaystyle}p - \frac{1}{2} = P - 1$, $$\begin{aligned}
\begin{split}
I_1 &= \int_{1/2}^{p - 1/2} \frac{\sqrt{2} \, dx}{w [ P + w]^{1/2}} =\\
&= \int_{\frac{1}{2} P}^{1 - \frac{1}{P}} \frac{\sqrt{2} P \, dt}{P \sqrt{1 - t^2} \cdot P^{1/2} \left( 1 + \sqrt{1 - t^2} \right)^{1/2}} = \end{split} \label{eq:i1eqs}\\
\begin{split} &= P^{-1/2} \left[ \int_0^1 \frac{ \sqrt{2} \, dt}{\widetilde{w} (1 + \widetilde{w})^{1/2}} - \rho_5^{\prime} - \rho_6^{\prime} \right] \quad \text{where} \quad \widetilde{w}(t) = \sqrt{1-t^2} , \end{split} \label{eq:i1decomp}\end{aligned}$$ $$\begin{aligned}
\begin{split}\rho_5^{\prime} &= \int_0^{\frac{1}{2} P} \frac{dt \cdot \sqrt{2}}{\widetilde{w}\left(1 + \widetilde{w}\right)^{1/2}} , \end{split} \label{eq:rho5pdef}\\
\begin{split}\rho_6^{\prime}&= \int_{1 - \frac{1}{P}}^{1} \frac{dt}{\sqrt{ 1- t}} \cdot \frac{1}{\sqrt{1 + t}} \frac{\sqrt{2}}{\left( 1 + \widetilde{w} \right)^{1/2}} \leq \\
& \leq \sqrt{2} \int_0^{1/P} \frac{d\tau}{\sqrt{\tau}} = 3P^{-1/2}.
\end{split} \label{eq:rho6pdef}\end{aligned}$$ If $P \geq 2$, then ${\displaystyle}\widetilde{w} \geq \frac{\sqrt{3}}{2}$ and the integrand of $\rho_5^{\prime}$ does not exceed $2$ if $t \in \left[ 0, \frac{1}{4} \right]$ so $$\rho_5^{\prime} \leq \frac{1}{P} \label{eq:rho5pbd}$$ These inequalities, together with , -, , show that $$I_1 = AP^{-1/2} - \rho_5 - \rho_6 \label{eq:i1truedecomp}$$ where $$\begin{aligned}
0 &\leq \rho_5 = P^{-1/2} \rho_5^{\prime} \leq P^{-3/2}, \label{eq:rho5db}\\
0 &\leq \rho_6 = P^{-1/2} \rho_6^{\prime} \leq 3P^{-1}, \label{eq:rho6db}\end{aligned}$$ and $$A = \int_0^1 \frac{\sqrt{2} \, dt}{\widetilde{w}(t) \sqrt{1 + \widetilde{w}(t)}}, \quad \widetilde{w} = \sqrt{1 - t^2}. \label{eq:a2def}$$ Let us evaluate $A$; with $t = \sin \varphi$ we have: $$\begin{aligned}
A & = \int_0^{\pi/2} \frac{\cos \varphi \, d\varphi}{\cos \varphi} \cdot \frac{1}{\left(\frac{1}{2}(1 + \cos \varphi) \right)^{1/2}}= \\
& = \int_0^{\pi/2} \frac{d\varphi}{\cos\frac{\varphi}{2}} \cdot \frac{\cos \frac{\varphi}{2}}{\cos \frac{\varphi}{2} } = \int_0^{\pi/2} \frac{2 \, d \left( \sin \frac{\varphi}{2} \right)}{1 - \left( \sin \frac{\varphi}{2}\right)^2} \\
& = 2 \int_0^{\sqrt{2}/2} \frac{d\tau}{1 - \tau^2} = \int_0^{1/\sqrt{2}} \left[ \frac{1}{1 - \tau} + \frac{1}{1 + \tau} \right] \, d\tau = \\
& = \left. \log \frac{1 + t}{1 - t} \right\vert_0^{1/\sqrt{2}} = \log (1 + \sqrt{2})^2 = 2 \log(1 + \sqrt{2}), \text{ i.e.,}\end{aligned}$$ $$\label{eq:constArep}
A = 2 \log (1 + \sqrt{2}),$$ so by , $$A = B, \quad \text{ and } I_1 - I_2 = 0. \label{eq:iscancel}$$ By , , , , , , and ,
$$\begin{split}
\xi(n) & = 2^{-3/2} [S_1 - S_2] = \\
& = 2^{-3/2} \left[(\rho_4 + AP^{-1/2} - \rho_5 - \rho_6) - (BP^{-1/2} + \rho_3) \right] \\
& = 2^{-3/2} \left[ \rho_4 - \rho_5 - \rho_6 - \rho_3 \right].
\end{split} \label{eq:zetafinal}$$
If $P \geq 4$, with , , , correspondingly, we have: ${\displaystyle}2^{-3/2} \leq \frac{2}{5}$ and
$$\begin{split}
\vert \xi(n) \vert &\leq \frac{2}{5} \left[ P^{-1} + P^{-3/2} + 3P^{-1} + P^{-3/2} \right]\\
& \leq \frac{2}{5} P^{-1} \left[ 1 + \frac{1}{2} + 3 + \frac{1}{2} \right] = 2P^{-1}.
\end{split} \label{eq:zetaactbd1}$$
With $n = 2p + 1 = 2P$ $$\vert \xi(n) \vert \leq \frac{4}{n} \quad \text{for } n \geq 8. \label{eq:zetaactbd2}$$ This proves , at least for odd $n$.
If $n$ is even, say $n = 2p$, then \[see \], with $k = 2m - 1$,
$$\begin{aligned}
\xi(2p) & = \sum_{m = 1}^{\infty} \frac{1}{\sqrt{2m - 1}} \cdot \frac{1}{2(p - m) + 1} \label{eq:zeta2peval} \\
& = \frac{1}{2 \sqrt{2}} \sum_{m = 1}^{\infty} \frac{1}{\sqrt{m - \frac{1}{2}}} \cdot \frac{1}{(p - m) + \frac{1}{2}}. \label{eq:zeta2peval2}\end{aligned}$$
For any $d$, $m > 3\vert d \vert$, for some $\theta$, $\vert \theta \vert < 1$, $$\begin{split}
\delta_m &\equiv \left\vert \frac{1}{\sqrt{m - d}} - \frac{1}{\sqrt{m}} \right\vert= \frac{\vert d \vert}{2 (m - d \theta)^{3/2}}\leq\\
& \leq \frac{\vert d \vert }{2 (m - \vert d \vert )^{3/2}} \leq \frac{3^{3/2}}{2^{5/2}} \frac{\vert d \vert }{m^{3/2}} < \frac{ \vert d \vert}{m^{3/2}} \, .
\end{split} \label{eq:deltameval}$$ and $$\vert \delta_m \vert \leq \vert d \vert m^{-3/2} \quad \text{ if } m > 3 \vert d \vert \label{eq:deltambd}$$ By Remark \[rem:sbds\], ${\displaystyle}\beta = \frac{3}{2}$, ${\displaystyle}d = \frac{1}{2}$, $$\begin{split}
\sum_m \vert \delta_m \vert \cdot \frac{1}{\vert p - m + \frac{1}{2}\vert} \leq \frac{C}{p} = {O \left( \frac{1}{n} \right)}
\end{split} \label{eq:deltamsum}$$ So $$\begin{split}
\vert \rho_7(n) \vert = \vert \xi(2p) - \xi(2p + 1) \vert \leq \frac{C}{p}
\end{split} \label{eq:zetadiffbd}$$ and by and $$\vert \xi(2p) \vert \leq \frac{C}{n} \label{eq:zetaactbd3}$$ This completes the proof of Lemma \[lem:zetabds\].
Asymptotics of the term eta(n), Part 1 {#sec:etabdpt1}
======================================
Recall : $$\eta(n) = \sum_{\substack{k = 1 \\ n - k \text{ odd}}}^{\infty} \frac{1}{\sqrt{k}} \frac{\cos 2 b \sqrt{2k + 1}}{n - k}. \label{eq:etarepeat}$$
{#section-9}
The second term in or , leads us to the sequence $\eta(n)$ which — after Lemma \[lem:zetabds\]— could become a “leading” term in the asymptotics of ${\sigma^{\prime}}(n)$ or $\sigma(n) \in$ and eventually of $T_2(n)$ in , .
One more adjustment along the lines of Remark \[rem:sbds\] or inequalities , . For any $\omega > 0$ and $d$ real, $k > 2 \vert d \vert$,
$$\begin{split}
\delta_k^{\prime} & \equiv \frac{\cos \omega \sqrt{k + d}}{\sqrt{k}} - \frac{\cos \omega \sqrt{k}}{\sqrt{k}} =\\
& = \frac{-\omega}{\sqrt{k}} \cdot \frac{\sin \omega \sqrt{k + \theta_k d}}{2 \sqrt{k + \theta_k d}}, \quad 0 \leq \theta_k \leq 1,
\end{split} \label{eq:deltakpeval}$$
so $$\vert \delta_k^{\prime} \vert \leq \frac{\omega}{k}, \label{eq:deltakpbd}$$ and if in the definition of $\eta(k)$ we’ll write $\sqrt{2k}$ instead of $\sqrt{2k + 1}$, i.e., analyze $$\eta^{\prime}(n) = \sum_{\substack{k = 1 \\ n - k \text{ odd}}}^{\infty} \frac{1}{\sqrt{k}} \cdot \frac{\cos 2 b \sqrt{2k}}{n - k} \label{eq:etapdef}$$ the error $$\rho_8 = \eta - \eta^{\prime}; \quad {\left\vert \rho_8 \right\vert} \leq \sum_{\substack{k = 1 \\ n - k \text{ odd}}}^{\infty} \vert \delta_k^{\prime} \vert \cdot \frac{1}{{\left\vert n - k \right\vert}} \label{eq:rho8def}$$ by Remark \[rem:sbds\] and its variations has an estimate \[with \] $${\left\vert \rho_8(n) \right\vert} \leq 4 \omega \,\frac{\log n}{n},$$ well under restrictions , so we can analyze $\eta^{\prime} \in $ instead of ${\left\lbrace \eta(n) \right\rbrace}$ itself.
{#subsect:enoddintro}
As in Section \[sec:manyineqs\], let us do details when $n$ is odd, i.e., $n = 2p + 1$, so $k = 2m$ and
$$\begin{split}
\eta^{\prime}(n) &= \sum_{m = 1}^{\infty} \frac{1}{\sqrt{2m}} \cdot \frac{\cos \left(2 b \cdot 2 \sqrt{m}\right)}{2(p-m) + 1} = \\
& = \frac{1}{2\sqrt{2}} \sum_{m = 1}^{\infty} \frac{\cos 4 b \sqrt{m}}{\sqrt{m}} \cdot \frac{1}{(p - m) + \frac{1}{2}}.
\end{split} \label{eq:etapeval}$$
As in $$\eta^{\prime}(n) = 2^{-3/2} [S_1 - S_2] \label{eq:etapdecomp}$$ where $$\begin{aligned}
S_1 & = \sum_{j = 0}^{p-1} \frac{1}{j + \frac{1}{2}} \left[ \frac{\cos 4b \sqrt{p-j}}{\sqrt{p - j}} - \frac{\cos 4b \sqrt{p + j + 1}}{\sqrt{p + j + 1}}\right], \label{eq:es1def}\\
S_2 & = \sum_{j = p}^{\infty} \frac{1}{j + \frac{1}{2}} \frac{\cos 4b \sqrt{p + j + 1}}{\sqrt{p + j + 1}}, \label{eq:es2def}\end{aligned}$$
\[prop:es2bds\]With notations $$\begin{gathered}
\vert S_2 \vert \leq C\frac{1}{p} \label{eq:es2bd}\\
\intertext{and}
S_1 = \pi \, \frac{\sin(4 b \sqrt{p})}{\sqrt{p}} + \rho_9(n) = 2\pi \frac{\sin 2b \sqrt{2n}}{\sqrt{2n}} + {O \left( \frac{\log n}{n} \right)}, \label{eq:es1decomp}\\
\vert \rho_9(n) \vert \leq C \frac{\log p}{p} \label{eq:rho9bd}\end{gathered}$$
Let us start to evaluate $S_2$; this is an easier part. Like in put $$\begin{gathered}
{\displaystyle}\varphi(x) = \frac{1}{x} \frac{\cos r \sqrt{P + x}}{\sqrt{P + x}} = \cos (r \sqrt{P + x})\cdot \psi(x), \quad r = 4b, \label{eq:evarphidef} \\
\psi(x) = \frac{1}{x \sqrt{P + x}}, \quad P = p + \frac{1}{2} = \frac{1}{2} n. \label{eq:epsidef}\end{gathered}$$ The function $\psi$ is monotone decreasing, so $\psi^{\prime}(x)$ is negative and $$\vert \psi^{\prime}(x) \vert = - \psi^{\prime}(x). \label{eq:epsisign}$$
If $$\begin{aligned}
t_j &= j + \frac{1}{2}, \quad j \geq p, \quad \text{ and}\\
I(j) &= [t_j, t_j + 1], \quad \text{then}
\end{aligned} \label{eq:someconsts}$$ $$S_2 = \sum_{j = p}^{\infty} \varphi(t_j) \label{eq:es2sum}$$ and with some $\theta_j$, $t_j \leq \theta_j \leq t_{j + 1}$ $$\begin{split}
\Delta_j & \equiv \varphi(t_j) - \int\limits_{I(j)} \varphi(x) \, dx = \varphi (t_j) - \varphi (\theta_j)\\
& = -\int_{t_j}^{\theta_j} \varphi^{\prime}(x) \, dx.
\end{split} \label{eq:edeljdef}$$ Therefore, $${\left\vert \Delta_j \right\vert} \leq \int\limits_{I(j)} {\left\vert \varphi^{\prime}(x) \right\vert} \, dx , \label{eq:edeljbd}$$ but by $$\begin{aligned}
\varphi^{\prime}(x)& = - \frac{r \sin r \sqrt{P + x}}{2 \sqrt{ P + x}} \psi(x) + \cos r \sqrt{P + x} \cdot \psi^{\prime}(x) \label{eq:evarphider}\\
\intertext{so}
{\left\vert \varphi^{\prime}(x) \right\vert} &\leq \frac{r}{2} \, \frac{1}{x(P + x)} - \psi^{\prime}(x). \label{eq:evarphiderbd}\end{aligned}$$ Then by $$\begin{split}
\sum_{j = p}^{\infty}{\left\vert \Delta_j \right\vert} & \leq \sum_{j = p}^{\infty} \int\limits_{I(j)} {\left\vert \varphi^{\prime}(x) \right\vert} \, dx = \\
& = \int_P^{\infty} {\left\vert \varphi^{\prime}(x) \right\vert} \, dx \leq \frac{r}{2} \int_P^{\infty} \frac{dx}{x(P + x)} + \psi(P)\\
& = \frac{r}{2} \frac{\log 2}{P} + \frac{1}{P \sqrt{2P}} \leq \frac{2b + 1}{P}.
\end{split} \label{eq:edeljsumbds}$$
If $$J_2 = \int_P^{\infty} \varphi(x) \, dx \label{eq:j2def}$$ we conclude from , , that $$S_2 = J_2 + \rho_{10}, \quad \vert \rho_{10} \vert \leq \frac{2b + 1}{P} \label{eq:es2decomp}$$ and instead of the sum $S_2$ we’ll analyze the integral $J_2 \in$ with the integrand . A simple absolute value estimates would give the inequality $$\begin{aligned}
\vert J_2 \vert &\leq \int_P^{\infty} \frac{dx}{x(P + x)^{1/2}} = \mu P^{-1/2} \label{eq:j2bd}\\
\intertext{where}
\mu & = \int_1^{\infty} \frac{d\xi}{\xi(1 + \xi)^{1/2}} \label{eq:emudef}\end{aligned}$$ but this is not good enough.
Recall that by , $$J_2 = \int_{P}^{\infty} \frac{\cos r \sqrt{P + x}}{x \sqrt{P + x}} \, dx; \label{eq:ej2renew}$$ put $$\begin{aligned}
P + x &= P(1 + t)^2 \quad \text{so} \\
x &= Pt(2 + t), \quad dx = 2P(1 + t) \, dt
\end{aligned} \label{eq:evarchange}$$ and $$\begin{split}
J_2 & = \int_{\sqrt{2} - 1}^{\infty} \frac{\cos r P^{1/2} (1 + t)}{Pt(2 + t)P^{1/2}} \cdot \frac{2P (1 + t) \, dt}{(1 + t)} = \\
& = 2P^{-1/2} \int_{\sqrt{2}}^{\infty} \frac{\cos \left(r P^{1/2} \tau \right)\, d\tau}{\tau^2 - 1}
\end{split} \label{eq:ej2eval}$$ Now the integrand is absolutely integrable; moreover, we can integrate by parts with $$\cos rP^{1/2}\tau = \frac{1}{r P^{1/2}} d {\left( \sin r P^{1/2} \tau \right)}, \quad \text{etc.}, \label{eq:ibp1}$$ to get $$\begin{aligned}
\rho_{11} = J_2 &= \frac{2 \sin r \sqrt{2p + 1}}{rP} + O {\left( \frac{1}{P^{3/2}} \right)} \label{eq:rho11def}\\
\intertext{and}
{\left\vert \rho_{11} \right\vert} & \leq \frac{3}{r} \, \frac{1}{P} \, , \label{eq:rho11bd}\end{aligned}$$ certainly, under restrictions of .
Together with $$\begin{aligned}
S_2 & = \rho_{11} + \rho_{10}, \quad \text{ and} \label{eq:es2decompfinal} \\
{\left\vert S_2 \right\vert} & \leq \left[\frac{1}{b} + 2b + 1 \right] \frac{1}{P} \label{eq:es2absbd}\end{aligned}$$ Part is proven.
{#section-10}
Next, we analyze $S_1 \in $ . To avoid zero of $(P - x)$ at $x = P$, we consider $$S_1^{\prime} = \sum_{j = 0}^{p-2} \frac{1}{j + \frac{1}{2}} \left[ \frac{\cos 4b \sqrt{p-j}}{\sqrt{p - j}} - \frac{\cos 4b \sqrt{p + j + 1}}{\sqrt{p + j + 1}}\right], \label{eq:es1pdef}$$ $$\begin{aligned}
S_1 &= S_1^{\prime} + \rho_{12},\\
\rho_{12} & = {\left\lbrace \text{ the }(p-1) \text{-th term in }S_1 \right\rbrace}
\end{aligned} \label{eq:es1decomp2}$$ and $${\left\vert \rho_{12} \right\vert} \leq \frac{1}{p - \frac{1}{2}} \cdot \left[ \frac{\cos r}{1} + \frac{1}{\sqrt{2p}} \right] \leq \frac{2}{P}, \quad \text{ if } p \geq 5. \label{eq:rho12bd}$$
We follow pp. –, Section \[sec:manyineqs\], but now the integrand $$\begin{split}
g(x) & = \frac{1}{x} \left[ \frac{\cos r \sqrt{ P - x}}{\sqrt{P - x}} - \frac{\cos r \sqrt{P + x}}{\sqrt{ P + x}}\right] \, ,\\
P & = p + \frac{1}{2}, \quad \frac{1}{2} \leq x \leq P - 1 = p - \frac{1}{2},
\end{split} \label{eq:cpxintegrand}$$ in the integral $$J_1 = \int_{1/2}^{p - \frac{1}{2}} g(x) \, dx \quad \left[\omega = \frac{1}{2}, q = p -1 \text{ if to use Lemma~\ref{lem:suminterr}}\right] \label{eq:ej1def}$$ is not a monotone function. Now, as on pages – , this section, we will use the following.
\[lem:suminterr\] Let $f$ be a $C^1$ function on the interval $[\omega, \omega + q]$, and $$\begin{aligned}
S &= \sum_{j = 0}^{q-1} f(\omega + j), \label{eq:gensdef}\\
J & = \int_\omega^{\omega + q} f(x) \, dx. \label{eq:genjdef}\end{aligned}$$ Then $$\vert S - J \vert \leq \int_\omega^{\omega + q} \vert f^{\prime}(x) \vert \, dx. \label{eq:gensumintdiffbd}$$
This is a well-known claim in elementary Numerical Analysis.
As on page , define $$I(j) = [\omega + j, \omega + j + 1], \quad 0 \leq j \leq q \label{eq:ijdefrenew}$$ Then with some $\theta_j$, $0 \leq \theta_j \leq 1$ $$\begin{split}
\Delta_j & \equiv f(\omega + j) - \int\limits_{I(j)} f (x) \, dx = \\
&= f(\omega + j) - f(\omega + j + \theta_j) = -\int_{\omega + j}^{\omega + j + \theta_j} f^{\prime}(x) \, dx
\end{split} \label{eq:gendeltaj}$$ so $${\left\vert \Delta_j \right\vert} \leq \int_{\omega + j}^{\omega + j + 1} {\left\vert f^{\prime}(x) \right\vert} \, dx, \label{eq:gendeltajbd}$$ and $$\sum_{j = 0}^{q - 1} {\left\vert \Delta_j \right\vert} \leq \int_\omega^{\omega + q} {\left\vert f^{\prime}(x) \right\vert} \, dx. \label{eq:gendeltajsum}$$ But by the notation $$J - S = \sum_{j = 0}^{q-1} \Delta_j \label{eq:gensumintcompare}$$ so implies .
This lemma justifies our estimates which follow although we will not necessarily give formal references to Lemma \[lem:suminterr\] and its formulas.
{#section-11}
Let us split $g$ of as $$g = g_1 + g_2, \quad \text{where} \label{eq:egsplit}$$ $$\begin{split}
g_1(x) & = \frac{1}{x} \left[ \frac{1}{\sqrt{P - x}} - \frac{1}{\sqrt{P + x}}\right] \cos r \sqrt{P - x} \\
& = \psi_1(x) \cdot \cos r \sqrt{P - x}
\end{split} \label{eq:eg1def}$$ and $$\begin{split}
g_2(x) &= \frac{1}{x\sqrt{P + x}} \left[ \cos r \sqrt{P - x} - \cos r \sqrt{P + x} \right] \\
& = \psi_2(x) \cdot \left[ \cos r \sqrt{P - x} - \cos r \sqrt{P + x} \right]
\end{split} \label{eq:eg2def}$$ Notice that as on pages –, $$\psi_1(x) = \frac{\sqrt{2}}{w(P + w)^{1/2}}, \quad w(x) = \sqrt{P^2 - x^2} \label{eq:psi1def}$$ is a monotone increasing $C^1$-function, and as on p. , Section \[sec:etabdpt1\], , $$\psi_2(x) = \frac{1}{x \sqrt{P + x}} \quad \text{is monotone decreasing,} \label{eq:psi2def}$$ so $$\psi_1^{\prime} > 0 \quad \text{and} \quad \psi_2^{\prime} < 0. \label{eq:psiders}$$
These properties will help us to deal with integration of ${\left\vert \psi_1^{\prime}(x) \right\vert}$ or ${\left\vert \psi_2^{\prime}(x) \right\vert}$ without explicit evaluation of derivatives, just Newton-Leibnitz formula.
Notice that $$w^2(P - 1) = \left( p + \frac{1}{2} \right)^2 - \left( p - \frac{1}{2} \right)^2 = 2p \label{eq:wpteval1}$$ so $$0 \leq \psi_1 (P - 1) = \frac{1}{\sqrt{p} \left( P + \sqrt{2p} \right)^{1/2}} \leq \frac{1}{p} \label{eq:psipteval1}$$ and $$w^2 \left( \frac{1}{2} \right) = P^2 - \frac{1}{4} = \left( p + \frac{1}{2} \right)^2 - \frac{1}{4} = p(p + 1). \label{eq:wpteval2}$$ so $$0 \leq \psi_1 \left( \frac{1}{2} \right) = \frac{\sqrt{2}}{\sqrt{p(p + 1)} \left( P + \sqrt{p(p + 1)}\right)^{1/2}} \leq \frac{1}{p^{3/2}} \label{eq:psipteval2}$$ By , $$g_1^{\prime}(x) = \frac{r \sin r \sqrt{P - x}}{2 \sqrt{P - x}} \psi_1(x) + \psi_1^{\prime}(x) \cos r \sqrt{P - x} \label{eq:eg1der}$$ and by and $$\begin{aligned}
\begin{aligned}
\int_{1/2}^{P - 1} {\left\vert g_1^{\prime}(x) \right\vert} \, dx & \leq \frac{r}{2} \int_{1/2}^{P - 1} \frac{\psi_1(x)}{\sqrt{P - x}} + \psi_1(P - 1) \\
& \leq \frac{r}{2} \int_{1/2}^{P - 1} \frac{dx}{(P - x) \sqrt{P + x} \sqrt{P + w}} + \frac{1}{p} \leq
\end{aligned} \label{eq:g1pinta} \\
\begin{aligned}
2b P^{-1} \int_1^{p} \frac{d\xi}{\xi} + \frac{1}{p} \leq \frac{1}{P} [2b \log p + 1], \quad P \geq 3
\end{aligned} \label{eq:g1pintb}\end{aligned}$$ This error term is $O {\left( \frac{\log n}{n} \right)}$ so it is acceptable for .
For $g_2 \in $ we have: $$\begin{aligned}
\begin{aligned}
g_2^{\prime}(x) &= &- \psi_2(x) \cdot \frac{r}{2} \left[ \frac{\sin r \sqrt{P - x}}{\sqrt{P - x}}\right] - \psi_2(x) \left[\frac{\sin r \sqrt{P + x}}{\sqrt{P + x}} \right]\\
&&+ \psi_2^{\prime}(x) \left[ \cos r \sqrt{P - x} - \cos r \sqrt{P + x} \right]
\end{aligned} \label{eq:eg2der}\\
\begin{aligned}
& \equiv v_1(x) + v_2(x) + v_3(x)
\end{aligned} \label{eq:eg2derdecomp}\end{aligned}$$
Then by , $r = 4b$, $$\begin{aligned}
\begin{split}
\int_{1/2}^{P - 1} \vert v_1(x) \vert \, dx & \leq 2b \int_{1/2}^{P - 1} \frac{dx}{x \sqrt{P + x} \sqrt{P - x}} \leq \\
& \leq 2b P^{-1/2} \int_{\frac{1}{2P}}^{1} \frac{P \, dt}{PtP^{1/2} \sqrt{ 1 - t}} \leq \\
& \leq 2b P^{-1} \left( \frac{3}{2} \int_{\frac{1}{2P}}^{1/2} \frac{dt}{t} + 2 \int_{1/2}^1 \frac{dt}{\sqrt{1 - t}}\right) \leq
\end{split} \label{eq:v1inta}\\
\begin{split}
& \leq \frac{3b}{P} \left[ \log P + 1 \right]
\end{split} \label{eq:v1intb} \end{aligned}$$ Next, $$\begin{aligned}
\begin{aligned}
\int_{1/2}^{P - 1} \vert v_2(x) \vert \, dx & \leq \frac{r}{2} \int_{1/2}^{P - 1} \frac{dx}{x(P + x)} = \\
& = 2b P^{-1} \int_{1/2}^{P - 1} \left[ \frac{1}{x}- \frac{1}{P + x} \right] \, dx \leq
\end{aligned} \label{eq:v2inta}\\
\begin{aligned}
& \leq \frac{2b}{P} \log 2P
\end{aligned} \label{eq:v2intb}\end{aligned}$$ Finally,
$$\begin{aligned}
&{\left\vert \cos r \sqrt{P - x} - \cos r \sqrt{P + x} \right\vert} \leq \label{eq:sometermsbda}\\
\leq &\, \, r (\sqrt{P + x} - \sqrt{P - x}) = \frac{8bx}{\sqrt{P + x} + \sqrt{P - x}} \leq \frac{8bx}{\sqrt{P + x}}. \label{eq:sometermsbdb}\end{aligned}$$
and $$\begin{split}
- \psi_2^{\prime}(x) &= \frac{1}{x^2} \, \frac{1}{\sqrt{P + x}} + \frac{1}{2x(P + x)^{3/2}}, \\
\intertext{so}
\int_{1/2}^{P - 1} {\left\vert v_3(x) \right\vert} & \leq 4 \sqrt{2} b \int_{1/2}^{P - 1} \left[ \frac{1}{x \sqrt{P + x}} \cdot \frac{1}{(P + w)^{1/2}} + \frac{1}{(P + x)^{3/2}} \cdot \frac{1}{(P + w)^{1/2}} \right] \, dx
\end{split} \label{eq:v3inta}$$ and $$\begin{aligned}
\int_{1/2}^{P - 1} \vert v_3(x) \vert \, dx & \leq 8b \int_{1/2}^{P - 1} \left[ \frac{1}{x(P + x)} + \frac{1}{2(P + x)^2}\right] \, dx \label{eq:v3intb} \\
& \leq \frac{8b}{P} \left( \log 2P + \frac{1}{2} \right) \label{eq:v3intc}\end{aligned}$$
With our notations , if we collect inequalities we have: $$\int_{1/2}^{P - 1} {\left\vert g_2^{\prime}(x) \right\vert} \, dx \leq \frac{13b}{P} \log(4P) . \label{eq:eg2derbd}$$ Together with and this implies $$\int_{1/2}^{P - 1} {\left\vert g^{\prime}(x) \right\vert} \, dx \leq \frac{15b}{P} \log (4P) \label{eq:egintbd}$$ By Lemma \[lem:suminterr\], $$\begin{aligned}
S_1^{\prime} & = J_1 + \rho_{14}, \label{eq:es1pdecomp} \\
{\left\vert \rho_{14} \right\vert} & \leq \frac{15b}{P} \log(4P), \label{eq:rho14bd}\end{aligned}$$ well under restrictions .
All of the above make $J_1$ a front-runner in the race to be “a leading term” of asymptotics of $\eta(n)$ and $\sigma(n)$.
{#subsect:f1bds}
To analyze $J_1 \in$ we split the integrand $g \in$ on the interval ${\displaystyle}\left[ \frac{1}{2}, p - \frac{1}{2} \right]$ just as it is written there, i.e., $$g(x) = f_1(x) - f_2(x) \label{eq:egnewdecomp}$$ where $$\begin{aligned}
f_1(x) & = \frac{1}{x} \frac{\cos r \sqrt{P - x}}{\sqrt{P - x}}\, , \label{eq:ef1def} \\
f_2(x) & = \frac{1}{x} \frac{\cos r \sqrt{P + x}}{\sqrt{P + x}} \, . \label{eq:ef2def} \end{aligned}$$
There are no zeros in the denominators; all functions $g$, $f_1$, $f_2$ are $C^{\infty}$ on $\left[ \frac{1}{2}, p - \frac{1}{2} \right]$ and for a while we can manipulate $f_1$, $f_2$ and their integrals separately. When the arguments of $\mathit{cos}$ would be linear functions it is easier to compare two components and balance two integrals $$Q_i = \int_{1/2}^{p - \frac{1}{2}} f_i(x) \, dx, \quad i = 1, 2. \label{eq:qidef}$$ To evaluate $Q_1$ put $$\begin{aligned}
P - x &= P(1-\tau)^2 \quad \text{ so} \label{eq:q1changevara} \\
x & = P\tau(2 - \tau), \quad dx = 2P(1 - \tau) \, d \tau. \label{eq:q1changevarb}\end{aligned}$$ The lower bound ${\displaystyle}x = \frac{1}{2}$ corresponds to $$\begin{gathered}
\tau_* = 1 - \left( 1 - \frac{1}{2P}\right)^{1/2} = \frac{1}{4P} + \epsilon, \label{eq:taulowdef}\\
0 \leq \epsilon \leq \frac{1}{(2P)^2} \label{eq:rhotaubd}\end{gathered}$$ and the upper bound $x = P - 1$ leads to $$\tau^* = 1 - \frac{1}{\sqrt{P}}. \label{eq:tauhighdef}$$ Therefore, $$\begin{aligned}
Q_1 & = \int_{\tau_*}^{\tau^*} \frac{\cos {\left( r P^{1/2} (1 - \tau) \right)} 2P(1 - \tau) \, d\tau}{P\tau(2 - \tau)P^{1/2}(1 - \tau)} \label{eq:q1evala} \\
& = 2P^{-1/2} \int_{\tau_*}^{\tau^*} \frac{\cos {\left( r P^{1/2} (1 - \tau) \right)}}{\tau} \, \frac{d\tau}{2 - \tau} \label{eq:q1evalb} \end{aligned}$$
For unification, let us change $\tau_*$ to $\frac{1}{4P}$ and $\tau^*$ to $1$, so $$\begin{split}
\int_{\tau_*}^{\tau^*} = \int_{\frac{1}{4P}}^{1} - \int_{\tau^*}^{1} - \int_{\frac{1}{4P}}^{\frac{1}{4P} + \epsilon} \equiv \int_{\frac{1}{4P}}^{1} + \rho_{15}^{\prime} + \rho_{16}^{\prime}, \quad \tau_* \geq \frac{1}{4P}
\end{split} \label{eq:q1bdschange}$$ On ${\displaystyle}\left[ \frac{1}{4P}, \frac{1}{4P} + \epsilon \right]$ the absolute value of integrand in does not exceed $4P$ but — see — the length of this interval $\leq \frac{1}{4P^2}$; therefore, $$\begin{aligned}
{\left\vert \rho_{15}^{\prime} \right\vert} &\leq \frac{1}{P} \quad \text{and} \label{eq:rho15pbd}\\
\rho_{15} &= 2P^{-1/2} \rho_{15}^{\prime}, \quad {\left\vert \rho_{15} \right\vert} \leq 2P^{-3/2}. \label{eq:rho15bd}\end{aligned}$$ On $[\tau^*, 1]$ the function - integrand has absolute values $\leq 2$ if $P \geq 4$ (see ) so $$\begin{aligned}
{\left\vert \rho_{16}^{\prime} \right\vert} & \leq 2(1 - \tau^*) = \frac{2}{\sqrt{P}} \label{eq:rho16pbd} \\
\intertext{and}
\rho_{16} &= 2P^{-1/2} \rho_{16}^{\prime}, \quad {\left\vert \rho_{16} \right\vert}. \leq 4P^{-1}. \label{eq:rho16bd}\end{aligned}$$ Therefore, by , $$Q_1 = P^{-1/2} \int_{\frac{1}{4P}}^{1} \frac{\cos {\left( r P^{1/2} (1 - \tau) \right)} }{\tau} \frac{2d\tau}{2 - \tau} + \rho_{15} + \rho_{16}. \label{eq:q1decompfinal}$$ One more adjustment: change factor ${\displaystyle}\frac{2}{2 - \tau}$ to $1$: $$\frac{2}{2 - \tau} = 1+ \frac{\tau}{(2-\tau)}, \label{eq:denomchange}$$ so the integral in is equal to $$\int_{\frac{1}{4P}}^{1} \frac{\cos {\left( r P^{1/2} (1 - \tau) \right)} }{\tau} \, d\tau + \rho_{17}^{\prime} \label{eq:q1denomrevise}$$ where $$\begin{split}
\rho_{17}^{\prime} &= \int_{\frac{1}{4P}}^{1} \frac{\cos {\left( r P^{1/2} (1 - \tau) \right)} }{2 - \tau} \, d\tau = \\
& = \left[ \int_0^1 - \int_0^{\frac{1}{4P}} \right]
\end{split} \label{eq:rho17pdef}$$ The absolute value of the integrand in $\rho_{17}^{\prime}$ is bounded by $1$ so the second integral $${\left\vert \int_0^{\frac{1}{4P}}\frac{\cos {\left( r P^{1/2} (1 - \tau) \right)} }{2 - \tau} \, d\tau \right\vert} \leq \frac{1}{4P} \label{eq:rho17pint2bd}$$ and $$\begin{aligned}
\rho_{17}^{\prime} &= \int_0^1 \frac{\cos \left( r P^{1/2} y \right)}{1 + y} \, dy + \rho_{18}^{\prime}, \label{eq:rho17pdecomp} \\
{\left\vert \rho_{18}^{\prime} \right\vert} & \leq \frac{1}{4} P^{-1} \label{eq:rho18pbd}\end{aligned}$$ Riemann lemma tells us that for any $C^1$-function ${\kappa}$ on the interval $[A, B]$ $${\left\vert \int_A^B e^{iTx} {\kappa}(x) \, dx \right\vert} \leq \frac{1}{T} \left[ {\left\vert {\kappa}(A) \right\vert} + {\left\vert {\kappa}(B) \right\vert} + \int_A^B {\left\vert {\kappa}^{\prime}(x) \right\vert} \, dx\right] \label{eq:riemleb}$$ Therefore, the integral in $\rho_{17}^{\prime} \in $ does not exceed $$\frac{1}{rP^{1/2}} \left[ 1 + \frac{1}{2} + \frac{1}{2} \right] = \frac{1}{2b} \cdot P^{-1/2}, \label{eq:rho17pint1bd}$$ and the adjustment , i.e., $$\begin{aligned}
\rho_{17} &= P^{-1/2} \rho_{17}^{\prime}, \quad \text{satisfies} \label{eq:rho17adjust} \\
{\left\vert \rho_{17} \right\vert} &\leq \frac{1}{2b} \cdot P^{-1} + {\left\vert \rho_{18}^{\prime} \right\vert} \label{eq:rho17bd} \end{aligned}$$
If we collect , , , and , we get $$\begin{split}
\delta \equiv \sum_{i = 15}^{18} {\left\vert \rho_i \right\vert} &\leq P^{-1} \left[ 2P^{-1/2} + 4 + \frac{1}{2b} + \frac{1}{4} P^{-1/2} \right] \\
& \leq {\left( 7 + \frac{1}{2b} \right)} P^{-1}
\end{split} \label{eq:q1deltabd}$$ and with this error, instead of $Q_1 \in$ we can consider $$Q_1^{\prime} = P^{-1/2} \int_{\frac{1}{4P}}^1 \frac{\cos r P^{1/2} (1 - \tau)}{\tau} \, d\tau \, . \label{eq:q1pdef}$$ Indeed, by $${\left\vert Q_1 - Q_1^{\prime} \right\vert} \leq {\left( 7 + \frac{1}{2b} \right)} P^{-1}. \label{eq:q1q1pdiff}$$
{#section-12}
Now we evaluate $$Q_2 = \int_{1/2}^{P - 1} f_2(x) \, dx, \quad f_2 \in \text{ \eqref{eq:ef2def}}.$$ Many details are the same as in Section \[subsect:f1bds\] or (pp. –) Section \[subsect:enoddintro\]. We’ll make explanations short but all the formulas are written explicitly. As in – put $$\begin{aligned}
P + x &= P(1 + t)^2 \quad \text{so} \\
x &= Pt(2 + t), \quad dx = 2P(1 + t) \, dt
\end{aligned} \label{eq:evarchangerenew}$$ and $$Q_2 = \int_{t_*}^{t^*} \frac{\cos \left( r P^{1/2} (1 + t) \right) \, 2P(1 +t) \, dt}{Pt(2 + t) P^{1/2}(1 + t)} \label{eq:q2def}$$ where $$\begin{split}
t_* =\left( 1 + \frac{1}{2P}\right)^{1/2} - 1 = \frac{1}{4P} - \epsilon^{\prime}, \\
0 \leq \epsilon^{\prime} \leq \frac{1}{4P^2};
\end{split} \label{eq:tlowdef}$$ $$\begin{split}
t^* =\left( 2 - \frac{1}{P}\right)^{1/2} - 1 = \sqrt{2} - 1 - \epsilon^{\prime\prime},\\
0 \leq \epsilon^{\prime \prime} \leq \frac{1}{2P}.
\end{split} \label{eq:thighdef}$$
With $$Q_2 = P^{-1/2} \int_{t_*}^{t^*} \frac{\cos \left( r P^{1/2} (1 + t) \right) }{t(2 + t)} \, 2\, dt \label{eq:q2eval}$$ If we change $t_*$ to $\frac{1}{4P}$ we are losing the interval ${\displaystyle}\left[ t_*, \frac{1}{4P} \right]$ but its length is ${\displaystyle}\leq \frac{1}{4P^2}$ \[see \] and the integrand’s absolute value does not exceed $4P$ so the total “error” $\rho_{19}$ is $${\left\vert \rho_{19} \right\vert} \leq 4P \cdot \frac{1}{4P^2} \cdot 2P^{-1/2} = 2P^{-3/2} \label{eq:rho19bd}$$ The change of $t^*$ to $\sqrt{2} - 1$ gives the “error” $\rho_{20}$, and $${\left\vert \rho_{20} \right\vert} \leq \frac{1}{2P} P^{-1/2}= \frac{1}{2}P^{-3/2} \label{eq:rho20bd}$$
Now we change the factor ${\displaystyle}\frac{2}{2 + t}$ to $1$ as in , ; $$\frac{2}{2 + t} = 1 - \frac{t}{2 + t}, \label{eq:denomchange2}$$ and the difference-integral $$\rho_{21} = P^{-1/2} \int_{\frac{1}{4P}}^{\sqrt{2} - 1} \frac{\cos {\left( r P^{1/2} (1 - t) \right)} }{(2 + t)} \, dt \label{eq:rho21def}$$ so by we have $$\begin{aligned}
{\left\vert \rho_{21} \right\vert} \leq \frac{2}{r P^{1/2}} P^{-1/2} = \frac{1}{2b} \cdot \frac{1}{P} \label{eq:rho21bd}\end{aligned}$$ As in $$\begin{split}
\delta^{\prime} \equiv \sum_{i = 19}^{21} {\left\vert \rho_i \right\vert} &\leq \frac{1}{P} \left[ 2P^{-1/2} + \frac{1}{2} P^{-1/2} + \frac{1}{2b} \right] \\
& \leq \frac{1}{P}\left[ 2 + \frac{1}{2b} \right], \quad P \geq 4
\end{split} \label{eq:q2deltabd}$$ and $${\left\vert Q_2 - Q_2^{\prime} \right\vert} \leq \left[ 2 + \frac{1}{2b} \right] \cdot \frac{1}{P} \label{eq:q2q2pdiff}$$ where $$Q_2^{\prime} = P^{-1/2} \int_{\frac{1}{4P}}^{\sqrt{2} - 1}\frac{\cos r P^{1/2} (1 - t) }{1 + t} \, dt \label{eq:q2pdef}$$
{#section-13}
The two previous subsections \[see , , \] explained that $$J_1 = Q_1^{\prime} - Q_2^{\prime} + O {\left( \frac{\log P}{P} \right)} \quad , \label{eq:ej1finaldecomp}$$ even with more accurate estimates of the error term ${\displaystyle}O {\left( \frac{\log P}{P} \right)}$. Before the concluding claims about the $J_1$’s (i.e., $\sigma(n)$) asymptotics let us bring an integration for $Q_1^{\prime}$ and $Q_2^{\prime}$ to the same interval, say, ${\displaystyle}\left[ \frac{1}{4P}, \frac{1}{4} \right]$ with understanding that $\frac{1}{4} < 1$ and $\frac{1}{4} < \sqrt{2} - 1$, so a piece of $Q_1^{\prime}$
$$\begin{aligned}
\rho_{22}^{\prime} &= \int_{1/4}^1 \frac{\cos r P^{1/2}(1 - \tau)}{\tau} \, d\tau \label{eq:rhoq1bc}\\
\intertext{and a piece}
\rho_{23}^{\prime} & = \int_{1/4}^{\sqrt{2} - 1} \frac{\cos r P^{1/2}(1 + \tau)}{\tau} \, d\tau \label{eq:rhoq2bc}\end{aligned}$$
could be estimated by Riemann Lemma . We have $$\begin{split}
\rho_{22} = P^{-1/2} \rho_{22}^{\prime}, \quad &{\left\vert \rho_{22} \right\vert} \leq \frac{4 + 4}{rP} = \frac{1}{b} \cdot \frac{2}{P}
\end{split} \label{eq:rhoq1bcbd}$$ and $$\begin{split}
\rho_{23} = P^{-1/2} \rho_{23}^{\prime}, \quad &{\left\vert \rho_{23} \right\vert} \leq \frac{1}{b} \cdot \frac{2}{P}
\end{split} \label{eq:rhoq2bcbd}$$ Therefore $$J_1 = \Delta + \rho_{22} - \rho_{23} \label{eq:ej1anotherdecomp}$$ where $$\begin{aligned}
\Delta &= P^{-1/2} \delta, \label{eq:anotherDelta}\\
\delta & {\equiv}\int_{\frac{1}{4P}}^{1/4} \frac{1}{t} \left[\cos r P^{1/2}( 1 - t) - \cos r P^{1/2}( 1 + t) \right] \, dt \label{eq:anotherdelta}\end{aligned}$$ An elementary identity $$\cos u - \cos v = 2 \sin {\left( \frac{u + v}{2} \right)} \sin {\left( \frac{v - u}{2} \right)}$$ implies that $$\delta = 2 {\left( \sin r P^{1/2} \right)} \cdot \int_{\frac{1}{4P}}^{1/4} \frac{\sin r P^{1/2} t}{t} \, dt \label{eq:anotherdeltarewrite}$$ But $$\begin{aligned}
\rho_{24}^{\prime} &= {\left\vert \int_0^{\frac{1}{4P}} \frac{\sin r P^{1/2} t}{t} \, dt \right\vert} \leq \frac{1}{4P} \cdot r P^{1/2} \quad \text{and} \label{eq:rho24pbd} \\
\rho_{24} &= 2P^{-1/2} \rho_{24}^{\prime}, \quad {\left\vert \rho_{24} \right\vert} \leq 2 b P^{-1} \label{eq:rho24bd}\end{aligned}$$ Next, $$\begin{aligned}
\int_0^{1/4} \frac{\sin r P^{1/2} t}{t} \, dt & = \int_0^{\frac{1}{4} r P^{1/2}} \frac{\sin x}{x} \, dx\\
&= \frac{\pi}{2} - \rho_{25}^{\prime}, \label{eq:sinint}\end{aligned}$$ with $$\rho_{25}^{\prime} = \int_{\frac{1}{4} r P^{1/2}}^{\infty} \frac{\sin x}{x} \, dx \label{eq:rho25pdef}$$ where by $${\left\vert \rho_{25}^{\prime} \right\vert} \leq \frac{2}{bP^{1/2}}$$ and $$\rho_{25} = 2P^{-1/2} \rho_{25}^{\prime}, \quad {\left\vert \rho_{25} \right\vert} \leq \frac{4}{b} P^{-1}. \label{eq:rho25bd}$$ By , , , we conclude
$$\begin{split}
{\left\vert J_1 - \pi \, \frac{\sin r P^{1/2}}{P^{1/2}} \right\vert} & \leq \sum_{i = 22}^{25} {\left\vert \rho_i \right\vert} \leq\\
& \leq \left( b + \frac{10}{b} \right) \cdot P^{-1}
\end{split} \label{eq:ej1finalest}$$
and with and , , , the statement is proven.
Asymptotics of the term eta(n) – Part 2 {#sec:etabdpt2}
=======================================
{#subsect:oddcomplete}
In Section \[sec:etabdpt1\] from the start — see — we did everything for odd $n$. But if $n = 2p$ is even we have to take in the sum only odd $k = 2m - 1$, $m = 1, 2, \dotsc$ so $$\begin{split}
\eta(n) = \eta(2p) & = \sum_{m = 1}^{\infty} \frac{1}{\sqrt{2m + 1}} \cdot \frac{\cos 2b \sqrt{4m - 1}}{2(p - m) + 1}\\
& = \frac{1}{2\sqrt{2}} \sum_{m = 1}^{\infty} \frac{\cos 4b \sqrt{m - \frac{1}{4}}}{\sqrt{m - \frac{1}{2}}} \cdot \frac{1}{(p - m) + \frac{1}{2}} .
\end{split} \label{eq:etaeven}$$ At the same time \[see \] $$\eta(2p + 1) = \frac{1}{2\sqrt{2}} \sum_{m = 1}^{\infty} \frac{\cos 4b \sqrt{m + \frac{1}{4}}}{\sqrt{m}} \cdot \frac{1}{(p - m) + \frac{1}{2}} . \label{eq:etaodd}$$
We’ve already done such estimates at least twice but — to avoid any confusion — let us notice (with $r = 4b$): the following. $$\label{eq:exprprime}
\left( \frac{\cos r \sqrt{x + \frac{1}{4}}}{\sqrt{x}} \right)^{\prime} = - \, \frac{r \sin r \sqrt{x + \frac{1}{4}}}{2 \sqrt{x + \frac{1}{4}} \cdot \sqrt{x}} - \frac{\cos r \sqrt{x + \frac{1}{4}}}{2x^{3/2}}$$ so if $m \geq 2$ $$\label{eq:eomegamdef}
\begin{aligned}
\omega_m &= \left\vert \frac{\cos r \sqrt{m - \frac{1}{4}}}{\sqrt{m- \frac{1}{2}}} - \frac{\cos r \sqrt{m + \frac{1}{4}}}{\sqrt{m}} \right\vert \leq \\
& \leq \frac{1}{2} \cdot \frac{1}{2} \left[ \frac{r}{m - \frac{1}{2}} + \frac{1}{\left( m - \frac{1}{2} \right)^{3/2}}\right] \leq \\
& \leq \frac{1}{4} \left[ \frac{\frac{4}{3} r}{m} + \left( \frac{4}{3} \right)^2 \frac{1}{m^{3/2}}\right] \leq \\
&\leq \frac{1}{m} (2b + 1).
\end{aligned}$$ These inequalities by Remark \[rem:sbds\], $\beta = 1$, imply \[compare \] that $${\left\vert \eta(2p) - \eta(2p + 1) \right\vert} \leq \frac{1}{2 \sqrt{2}} \sum_{m = 1}^{\infty} \omega_m \cdot \frac{1}{\vert p - m \vert + \frac{1}{2}}\leq C (2b + 1) \frac{\log(ep)}{p}. \label{eq:etaoddevendiff}$$
{#section-14}
Now we are ready to complete the proof of the following.
\[prop:sigmatwiddlebounds\] $${\widetilde{\sigma}}(n) = \frac{(-1)^{n + 1}}{2} \frac{\sin 2 b \sqrt{2n}}{\sqrt{2n}} + O {\left( \frac{\log n}{n} \right)} \label{eq:finalsigmatwiddle}$$
With $u {\sim}v$ meaning that ${\displaystyle}\vert u - v \vert = {O \left( \frac{\log n}{n} \right)}$, we had by , –, , and $$\label{eq:mainchain}
\begin{aligned}
\widetilde{\sigma}(n) &{\sim}\frac{1}{\pi \sqrt{2}} \sigma(n) {\sim}\frac{(-1)^{n + 1}}{\pi \sqrt{2}} \eta(n) {\sim}\\
& {\sim}\frac{(-1)^{n + 1}}{\pi \sqrt{2}} \eta^{\prime}(n) {\sim}\frac{(-1)^{n + 1}}{\pi \sqrt{2}} \cdot 2^{-3/2} S_1 {\sim}\\
& {\sim}\frac{(-1)^{n + 1}}{4\pi} J_1 {\sim}\frac{(-1)^{n + 1}}{4\pi} \cdot 2 \pi \cdot \frac{\sin 2b \sqrt{2n}}{\sqrt{2n}} \\
& {\sim}\frac{(-1)^{n + 1}}{2} \cdot \frac{\sin 2b \sqrt{2n}}{\sqrt{2n}}
\end{aligned}$$
{#subsect:evenintro}
In Section \[sec:2pt\], , we’ve seen that in the case of the odd potential $v^o(x) = {\delta \left( x - b \right) } - {\delta \left( x + b \right) }$ $$\begin{aligned}
T_1(n) = T_3(n) & = 0, \quad \text{and}\\
T_2(n) &= 2 a_n^2 {\widetilde{\sigma}}(n) \label{eq:t2orepeat} \end{aligned}$$ —see and Propositions \[prop:es2bds\] and \[prop:sigmatwiddlebounds\]. Now we know ${\widetilde{\sigma}}(n)$ — see Lemma \[lem:siginfo\] or Prop. \[prop:sigmatwiddlebounds\]. Of course, $(a_n)$ come from (8.22.8) in [@Szego] — see , — so $$\begin{split}
a_n^2 &= \frac{1}{\pi} \cdot \frac{1}{\sqrt{2n}} \left[ 1 + (-1)^n \cos 2b \sqrt{2n} \right] + {O \left( \frac{1}{n} \right)}.
\end{split} \label{eq:an2repeat}$$ Recall that by , , $${\widetilde{\sigma}}(n) = \frac{1}{2} (-1)^{n + 1} \frac{\sin 2 b \sqrt{2n}}{\sqrt{2n}} + {O \left( \frac{\log n}{n} \right)} \label{eq:anothersigmaodd}$$
Therefore, by $$\begin{aligned}
T_2(n) &= \frac{{\kappa}(n)}{n} + O {\left( \frac{\log n}{n^{3/2}} \right)} \label{eq:t2repeatresolve}\\
\intertext{where}
{\kappa}(n) &= \frac{1}{{2\pi}} \left[(-1)^{n + 1} \sin 2 b \sqrt{2n} - \frac{1}{2} \sin 4b \sqrt{2n} \right]. \label{eq:helperterm3}\end{aligned}$$ This proves , i.e., the second part of Theorem \[thm:eigendistr\].
{#section-15}
So far we were dealing with an odd potential $v^o \in$ . But the technical hurdles we’ve overcome help to answer the questions on the asymptotics of eigenvalues in the case of an even perturbation $v^e$, — or $tv^e$, $t \in {\mathbb{C}}$, — see .
In this case (see Section \[subsect:evenpot\]) $$\begin{aligned}
T_1(n) &= 2 a_n^2 \quad \text{ by \eqref{eq:sigeven}} \label{eq:t1erepeat}\\
T_2(n) &= 2 a_n^2 {\sigma^{\prime}}(n), \label{eq:t2erepeat}\\
{\sigma^{\prime}}(n) & = \sum_{\substack{k = 0 \\ n - k \text{ even} \\ k \neq n}}^{\infty} \frac{a_k^2}{n-k} \label{eq:sigmaprepeat}\end{aligned}$$ and $$\begin{aligned}
T_3(n) & = 2 a_n^2 [ {\sigma^{\prime}}(n) + a_n^2 \tau^{\prime}(n)] \label{eq:t3erepeat}\\
\intertext{where}
\tau^{\prime}(n) &= \sideset{}{'}\sum_{\substack{m = 0 \\ m - n \text{ even}}}^{\infty} \frac{a_m^2}{(n - m)^2} \label{eq:tauprepeat}\end{aligned}$$ For a while we’ll consider only two traces $T_1$, $T_2$ to get asymptotics , $$\lambda_n = (2n + 1) + T_1(n) + T_2(n) + O \left( {\left( \frac{\log n}{n} \right)}^3\right) \label{eq:lamemidest}$$ in explicit form.
{#section-16}
Of course, $T_1(n) \in$ comes from (8.22.8) in [@Szego] — see , or $$\begin{split}
a_n = h_n(b) = \frac{2^{1/4}}{\pi^{1/2}} \, \frac{1}{n^{1/4}} & \left[ \cos {\left( b \sqrt{2n + 1} - n \frac{\pi}{2} \right)} \right. \\
&\left. + \frac{b^3}{6} \, \frac{1}{\sqrt{2n + 1}} \sin {\left( b \sqrt{2n + 1} - n \frac{\pi}{2} \right)} + {O \left( \frac{1}{n} \right)} \right].
\end{split} \label{eq:anformmid}$$ But we want to collect in the coefficients for ${\displaystyle}\frac{1}{\sqrt{n}}$ and ${\displaystyle}\frac{1}{n}$ \[ sending “smaller” terms into $\rho = {\displaystyle}O {\left( {\left( \frac{\log n }{n} \right)}^3 \right)}$ \] so now we need carefully to watch factors $$\sqrt{2n} \text{ and } \sqrt{2n + 1} = \sqrt{2n} + \frac{1}{2 \sqrt{2n}} + {O \left( \frac{1}{n^{3/2}} \right)} \label{eq:cares}$$ or functions’ values at the points $\sqrt{2n}, \sqrt{2n + 1}$. It was not necessary in analysis of $v^o$ in Section \[sec:manyineqs\]–\[sec:etabdpt1\] because the accuracy of ${\displaystyle}\rho = O {\left( \frac{\log n}{n} \right)}$ in Lemma \[lem:siginfo\] did not depend on the term ${\displaystyle}\frac{1}{2\sqrt{2n}}$ in and we could be indifferent to any divergence between $\sqrt{2n}$ and $\sqrt{2n + 1}$. However, now we need to write $$\begin{split}
\cos 2b \sqrt{2n + 1} &= \cos {\left( 2b \sqrt{2n} + \frac{b}{\sqrt{2n}} \right)} + {O \left( \frac{1}{n^{3/2}} \right)} \\
& = \cos 2b \sqrt{2n} - \frac{b}{\sqrt{2n}} \sin 2b \sqrt{2n} + O\left( \frac{1}{n} \right)
\end{split} \label{eq:coschange}$$ so in the sequence $$\begin{split}
a_n^2 = & \frac{1}{\pi \sqrt{2n}} \left[ 1 (-1)^n \cos 2b \sqrt{2n} \right.\\
& \left. - \frac{(-1)^n}{\sqrt{2n}} b \left( 1- \frac{b^2}{3} \right) \sin 2b \sqrt{2n} + {O \left( \frac{1}{n} \right)} \right]\\
= &\frac{1}{\pi} \, \frac{1}{\sqrt{2n}} \left[ 1 + (-1)^{n} \cos 2b \sqrt{2n} \right] + \\
& + \frac{(-1)^{n + 1}}{2\pi} \cdot \frac{1}{n}\, b \left(1 - \frac{b^2}{3} \right) \sin {\left( 2b \sqrt{2n} \right)} + {O \left( \frac{1}{n^{3/2}} \right)},
\end{split} \label{eq:a2eeval}$$ an additive term with the coefficient ${\displaystyle}\frac{1}{n}$ is important.
{#subsect:tauomega0}
The term $T_2 \in$ has a factor $a_n^2$ so when we evaluate ${\sigma^{\prime}}(n)$ we can ignore — as we did in Sections \[sec:manyineqs\], \[sec:etabdpt1\] —- the second term of order ${\displaystyle}\frac{1}{n}$ in . In particular, if $n = 2p$ is even we evaluate $${\sigma^{\prime}}(n) = \sum_{\substack{m = 0 \\ m \neq p}}^{\infty} \frac{a_{2m}^2}{2(p-m)}. \label{eq:sigmapt2e}$$ We can follow all the constructions and analogues of inequalities of Sections \[sec:manyineqs\], \[sec:etabdpt1\] but as in Section \[subsect:oddcomplete\] we can avoid a new 30 page writing but observe that this presumably new sum adjusted by Lemma \[lem:sumest\] looks as $$\frac{1}{2\pi} \sum_{\substack{m = 1 \\ m \neq p}}^{\infty} \frac{1}{\sqrt{m}} \, \left[ 1 + \cos 4b \sqrt{m} \right] \cdot \frac{1}{p - m} \label{eq:notsonewsum}$$ (compare or ), i.e., we need to give good estimates for $$\begin{aligned}
\tau_0(p) = \sum_{\substack{m = 1 \\ m \neq p}}^{\infty} \frac{1}{\sqrt{m}} \, \frac{1}{p - m} \label{eq:tau0def}
\intertext{and}
\tau_1(p) = \sum_{\substack{m = 1 \\ m \neq p}}^{\infty} \frac{\cos 4b \sqrt{m}}{\sqrt{m}} \, \frac{1}{p - m}. \label{eq:tau1def}\end{aligned}$$ In Sections \[sec:manyineqs\] and \[sec:etabdpt1\] we’ve analyzed the sequences $$\begin{aligned}
\omega_0(p) &= \sum_{m = 1}^{\infty} \frac{1}{\sqrt{m}} \, \frac{1}{p - m + \frac{1}{2}} \quad \text{(see \eqref{eq:sumforms})}
\intertext{and}
\omega_1(p) &= \sum_{m = 1}^{\infty} \frac{\cos 4b \sqrt{m}}{\sqrt{m}} \, \frac{1}{p - m + \frac{1}{2}} \quad \text{(see \eqref{eq:etapeval})}\end{aligned}$$ The following is true.
\[lem:tauomegadiff\]
$$\begin{aligned}
\vert \tau_0(p) - \omega_0(p) \vert & \leq C \frac{\log p}{p^{3/2}} \label{eq:tauomega0diff}\\
\vert \tau_1(p) - \omega_1(p) \vert & \leq C \frac{\log p}{p} \label{eq:tauomega1diff}\end{aligned}$$
\[eq:tauomegadiffs\]
As in , we rewrite $\tau_0(p)$ as $$\begin{split}
&\sum_{j = 1}^{p - 1} \frac{1}{j} \left[ \frac{1}{\sqrt{p-j}} - \frac{1}{\sqrt{p + j}} \right] - \sum_{j = p}^{\infty} \frac{1}{j} \, \frac{1}{\sqrt{p + j}} \\
\equiv & s_1 - s_2
\end{split} \label{eq:tau0eval}$$ and now compare $s_2, s_1$ with $S_2$, $S_1$ in . To deal with $s_2$, $S_2$ let us notice that $$\begin{split}
S_2 - s_2 & = \sum_{j = p}^{\infty} \left[\frac{1}{j + \frac{1}{2}} \cdot \frac{1}{\sqrt{p + j + 1}} - \frac{1}{j} \, \frac{1}{\sqrt{p + j}} \right] \\
& = \sum_{j = p}^{\infty} \left( \left[ \frac{1}{j + \frac{1}{2}} - \frac{1}{j} \right] \frac{1}{\sqrt{p + j + 1}} + \frac{1}{j} \left[\frac{1}{\sqrt{p + j + 1}} - \frac{1}{\sqrt{p + j}} \right] \right)
\end{split} \label{eq:s2diffseval}$$ so $${\left\vert S_2 - s_2 \right\vert} \leq \sum_{j = p}^{\infty} \left( \frac{1}{j^{5/2}} + \frac{1}{j^{5/2}} \right) \leq \frac{2}{p^{3/2}} \label{eq:s2diffsbd}$$ With $$S_1 = \sum_{j = 0}^{p - 1} \frac{1}{j + \frac{1}{2}} \left( \frac{1}{\sqrt{p - j}} - \frac{1}{\sqrt{p + j + 1}} \right) \label{eq:s1renew}$$ and $s_1$ in we have $$\begin{aligned}
S_1 - s_1 & = \sum_{j = 1}^{p - 1} && \left[ \frac{1}{j + \frac{1}{2}} \left( \frac{1}{\sqrt{p - j}} - \frac{1}{\sqrt{p + j + 1}} \right) \right.\\
&&& \left. + \frac{1}{j} \left( - \frac{1}{\sqrt{p + j + 1}} + \frac{1}{\sqrt{p + j}} \right) \right] = \\
& = \sum_{j = 1}^{p - 1} && \left[ \frac{-1}{j} \cdot \frac{1}{\sqrt{p - j} \sqrt{p + j + 1}} \cdot \frac{1}{\sqrt{p - j} + \sqrt{p + j + 1}} \right.\\
&&& \left. + \frac{1}{j} \frac{1}{\sqrt{p + j} \sqrt{p + j + 1} ( \sqrt{p + j} + \sqrt{p + j + 1})} \right]
\end{aligned} \label{eq:s1diffeval}$$ and $$\begin{split}
{\left\vert S_1 - s_1 \right\vert} &\leq \sum_{j = 1}^{p - 1} \left[ \frac{1}{P^{1/2}} \cdot \frac{1}{j(p - j)} + \frac{1}{P^{1/2}} \cdot \frac{1}{j} \cdot \frac{1}{(p + j)} \right]\\
& \leq \frac{2 \log P}{p^{3/2}}.
\end{split} \label{eq:s1diffsbd}$$ Inequalities and prove , i.e., Part (a) is done.
Part (b) {#subsect:tauomega1}
--------
As in Section \[sec:etabdpt1\] we write, $r = 4b$, $$\begin{aligned}
\begin{aligned}
\tau_1(p) & = &&\sum_{m = 1}^{p - 1} \frac{1}{j} \left[ \frac{ \cos r \sqrt{p - j}}{\sqrt{p - j}} - \frac{\cos r \sqrt{ p + j}}{\sqrt{p + j}} \right] \\
&&&- \sum_{j - p}^{\infty} \frac{1}{j} \frac{\cos r \sqrt{p + j}}{\sqrt{p + j}}
\end{aligned} \label{eq:tau1eval1}\\
\begin{aligned}
& = &&t_1 - t_2
\end{aligned} \label{eq:tau1eval2}\end{aligned}$$ and compare $t_1$, $t_2$ with $S_1, S_2 \in$ , separately.
Again, an “easy” case is to explain $${\left\vert S_2 - t_2 \right\vert} \leq C \frac{\log p}{p} \label{eq:es2diffbd}$$ Put $$\mu(x) = \frac{\cos r \sqrt{p + x}}{\sqrt{p + x}}; \label{eq:mu6def}$$ then $$- \mu^{\prime}(x) = \frac{r \sin r \sqrt{p + x}}{p + x} + \frac{\cos r \sqrt{p + x}}{(p + x)^{3/2}} \label{eq:mu6p}$$ and $$\begin{split}
S_2 - t_2 = \sum_{j = p}^{\infty} & \left[ \left( \frac{1}{j + \frac{1}{2}} - \frac{1}{j} \right) \frac{\cos r \sqrt{p + j + 1}}{\sqrt{p + j + 1}} \right.\\
& + \left. \frac{1}{j} \left( \frac{\cos r \sqrt{p + j + 1}}{\sqrt{p + j + 1}} - \frac{\cos r \sqrt{p + j}}{\sqrt{p + j}}\right) \right].
\end{split} \label{eq:e2sdiffact}$$ The last difference is equal to $\mu^{\prime}(j + \theta_j)$, $0 \leq \theta_j \leq 1$, so $$\begin{aligned}
\begin{aligned}
{\left\vert S_2 - t_2 \right\vert} & = \sum_{j = p}^{\infty} \left[ \frac{1}{\sqrt{p}} \cdot \frac{1}{j(2j + 1)} + \frac{1}{j} \, \frac{r}{(p + j)} + \frac{1}{j(p + j)^{3/2}} \right] \leq \\
& \leq \left( \frac{1}{2} \frac{1}{p^{3/2}} + \frac{r \log P}{P} + \frac{1}{P^{3/2}} \right)\\
\end{aligned} \label{eq:es2diffeval1}\\
\begin{aligned}
& \leq (r + 1) \frac{\log p}{p}
\end{aligned} \label{eq:es2diffeval2}\end{aligned}$$ is proven. Finally, with $$\begin{aligned}
S_1 & = \sum_{j = 1}^{p - 1} \frac{1}{j + \frac{1}{2}} \left[ \frac{\cos r \sqrt{p - j}}{\sqrt{p - j}} - \frac{\cos r \sqrt{p + j + 1}}{\sqrt{p + j + 1}}\right] \label{eq:es1repeat}
\intertext{and}
t_1 & = \sum_{j = 1}^{p - 1} \frac{1}{j} \left[ \frac{\cos r \sqrt{p - j}}{\sqrt{p - j}} - \frac{\cos r \sqrt{p + j}}{\sqrt{p + j}}\right] \label{eq:t1repeat}\end{aligned}$$ $$\begin{split}
S_1 - t_1 = \sum_{j = 1}^{p - 1} &\left\lbrace {\left( \frac{1}{j + \frac{1}{2}} - \frac{1}{j} \right)} \left[ \frac{\cos r \sqrt{p - j}}{\sqrt{p - j}} - \frac{\cos r \sqrt{p + j + 1}}{\sqrt{p + j + 1}}\right] \right.\\
& \left. + \frac{1}{j} \left( \frac{\cos r \sqrt{p + j}}{\sqrt{p + j}} - \frac{\cos r \sqrt{p + j + 1}}{\sqrt{p + j + 1}} \right) \right\rbrace =
\end{split} \label{eq:s1t1diffeval1}$$ $$\begin{split}
= \sum_{j = 1}^{p - 1} {\left\lbrace \frac{1}{j(2j + 1)} \mu^{\prime}(\gamma_j)(2j + 1) + \frac{1}{j} \mu(j + \delta_j) \right\rbrace} \quad \text{where}
\end{split} \label{eq:s1t1diffeval2}$$ $$-j \leq \gamma_j \leq j + 1, \quad 0 \leq \delta_j \leq 1 \label{eq:inbetweenconsts}$$ so $$\begin{split}
{\left\vert S_1 - t_1 \right\vert} \leq \sum_{j = 1}^{p - 1} & \left[ \frac{1}{j} \left( \frac{r}{2(p - j)} + \frac{1}{2(p-j)^{3/2}} \right) \right.\\
& \left. \frac{1}{j} \left( \frac{r}{2(p + j)} + \frac{1}{2(p + j)^{3/2}} \right) \right] \leq \\
\leq \sum_{j = 1}^{p - 1} & \frac{1}{j} \, \frac{r + 1}{2} \left[ \frac{1}{p - j} + \frac{1}{p + j} \right] \leq 2 (r + 1) \, \frac{\log P}{P}
\end{split} \label{eq:s1t1diffact}$$ This inequality, together with , prove that $$\begin{split}
{\left\vert \tau_1(p) - \omega_1(p) \right\vert} & \leq {\left\vert S_1 - t_1 \right\vert} + {\left\vert S_2 - t_2 \right\vert}\\
& \leq 3 (r + 1) \frac{\log P}{P}
\end{split} \label{eq:tau1diffeval}$$ and .
Lemma \[lem:tauomegadiff\] is proven.
This lemma shows that up to the error-term ${\displaystyle}O \left( \frac{\log P}{P} \right)$ the evaluation of ${\sigma^{\prime}}(n) \in$ gives the same result for the potential $v^e$ as ${\widetilde{\sigma}}(n) \in$ for the potential $v^o$, at least, if $n$ is even. \[The adjustment as in Section \[subsect:oddcomplete\] shows that we get the same result for $n$ odd. We omit details.\]
{#section-17}
\[cor:et2n\] If $v^e \in $ then $$\begin{split}
T_2(n) & = 2 a_n^2 {\sigma^{\prime}}(n) = \\
& = \frac{{\kappa}(n)}{n} + O {\left( \frac{\log n}{n^{3/2}} \right)}
\end{split} \label{eq:t2evenbd}$$ where $$\begin{aligned}
{\kappa}(n) = \frac{1}{{2\pi}} \left[ (-1)^{n + 1} \sin 2b \sqrt{2n} - \frac{1}{2} \sin 4b \sqrt{2n} \right] \label{eq:mysterysymb}\end{aligned}$$
We showed in previous subsections , that in the case of an even potential $v^e$ $T_2(n)$ behave in the same way as in the case $v^o$ so $$T_2(n) = 2 a_n^2 {\sigma^{\prime}}(n)$$ where by $$\begin{aligned}
{\sigma^{\prime}}(n) &= \frac{1}{2} (-1)^{n + 1} \frac{\sin 2b \sqrt{2n}}{\sqrt{2n}} + O {\left( \frac{\log n}{n} \right)}. \label{eq:anothersigmaeven}\\
\intertext{Recall that}
2 a_n^2 &= \frac{2}{\pi} \frac{1}{\sqrt{2n}} \left[ 1 + (-1)^n \cos (2 b \sqrt{2n}) \right] + O {\left( \frac{1}{n} \right)}. \label{eq:anotheranform}\end{aligned}$$ or more precisely, see . These identities imply .
{#section-18}
Although for an odd perturbation $v^o$ $$T_3(n) = 0, \quad n \geq N_*,$$ this is not true for an even $v^e$; i.e., $T_3(n; v^e)$ does hardly vanish.
By
$$T_3(n) = 2 a_n^2 {\sigma^{\prime}}(n)^2 + 2 a_n^4 \tau^{\prime}(n), \label{eq:et3even}$$
where $$\tau^{\prime}(n) = \sideset{}{^{\prime}}\sum_{\substack{k = 0 \\ k - n \text{ even}}}^{\infty} \frac{a_k^2}{(n - k)^2}. \label{eq:tau3pdef}$$ Notice that $$\begin{split}
\sum_{\substack{k = 1 \\ k - n \text{ even}}}^{\infty} \frac{1}{\sqrt{k}} \, \frac{1}{(n - k)^2} &= \sum_{i = 1}^{n - 1} \frac{1}{i^2} \left[ \frac{1}{\sqrt{n-i}} + \frac{1}{\sqrt{n+i}} \right] + O {\left( \frac{1}{n^{3/2}} \right)}\\
& = O {\left( \frac{1}{\sqrt{n}} \right)}
\end{split} \label{eq:tau3peval1}$$ It implies (with an analogue of Lemma \[lem:tauomegadiff\] that $$T_3(n) = {O \left( \frac{1}{n^{3/2}} \right)} . \label{eq:t3leading}$$ Without specifics of formulas and , we had a general claim \[see Corollary \[cor:tjnorm\], , ${\displaystyle}\alpha = \frac{1}{4}$, $j = 3$, and Corollary \[cor:lamest\], $q = 2$ and $3$.\]: $${\left\vert T_3(n) \right\vert} \leq C \left( \frac{\log n}{\sqrt{n}} \right)^3 \label{eq:t3earlyest}$$ so brings a slight gain of a factor $\left( \log n \right)^3$.
But $T_4$ and the entire tail $$\sum_{j = 4}^{\infty} {\left\vert T_j(n) \right\vert} \leq \left( C \frac{\log n}{\sqrt{n}} \right)^4 = {O \left( \frac{1}{n^{3/2}} \right)}$$ These observations, in particular , show that in both cases $v^o$ and $v^e$ we have by of Corollary \[cor:lamest\]: $$\lambda_n = (2n + 1) + T_1(n) + T_2(n) + {O \left( \frac{\log n}{n^{3/2}} \right)}$$ Explicit form of $T_2(n)$ for odd $v^o$ is incorporated in of Theorem \[thm:eigendistr\]. Now we are ready to give its analog for even $v^e$.
\[thm:evenasymp\] In the case of the potential perturbation $v^e(x) = {\delta \left( x - b \right) } + {\delta \left( x + b \right) }$ asymptotically $$\lambda_n = (2n + 1) + \frac{1}{\sqrt{2n}} \chi(n) + \frac{(-1)^{n+1}}{n} \left( \zeta(n) + \omega(n) \right) + {O \left( \frac{\log n}{n^{3/2}} \right)} , \label{eq:lameasymp}$$ where $$\begin{aligned}
\chi(n) & = \frac{4}{\pi} \left[ 1 + (-1)^n \cos 2b \sqrt{2n} \right], \label{eq:chidef}\\
\zeta(n) & = \frac{b}{\pi} \left( 1 - \frac{b^3}{3} \right)\sin 2b \sqrt{2n} \label{eq:zetatruedef}\\
\omega(n) & = \frac{1}{8} + (-1)^n \, \frac{1}{2} \sin 4 b \sqrt{2n} \label{eq:omegatruedef}\end{aligned}$$
{#section-19}
Of course, if $$W(x) = tv^e(x) \label{eq:wscaleddef}$$ then $$T_j(n; W) = t^j T_j(n; v^e) \, , \label{eq:tjscaled}$$ and $$\begin{split}
\lambda_n(W) = (2n + 1) + t T_1(n; v^e) + t^2 T_2(n; v^e) + {O \left( \frac{\log n}{n^{3/2}} \right)}
\end{split} \label{eq:lamformscaled}$$ and the term “of order ${\displaystyle}\frac{1}{n}$” in $T_1$ comes with the coefficient $t$ although the term\
“of order ${\displaystyle}\frac{1}{n}$” in $T_2$ comes with the coefficient $t^2$. This observation implies the following.
If the potential is then $$\lambda_n = (2n + 1) + \frac{t}{\sqrt{2n}} \chi(n) + \frac{1}{n} \left[t \zeta(n) + t^2\omega(n) \right] + {O \left( \frac{\log n}{n^{3/2}} \right)} , \label{eq:lamescaledasymp}$$ with $\chi$, $\zeta$, $\omega$ defined in .
Finally, we are ready to get asymptotic of eigenvalues in the case of any two-point ${\left\lbrace \pm b \right\rbrace}$ $\delta$-potential.
\[thm:gen2ptpot\] Let $$w(x) = tv^e(x) + sv^o(x), \quad t, s \in {\mathbb{C}}. \label{eq:gen2ptpot}$$ Then $$L = - \frac{d^2}{dx^2} + x^2 + w(x) \label{eq:elllast}$$ has a discrete spectrum, all its eigenvalues ${\left\lbrace \lambda_n \right\rbrace}$ in the half-plane $$H(N_*) = {\left\lbrace z: {\operatorname{Re}}z \geq 2 N_* \right\rbrace}$$ are simple, and $$\begin{split}
\lambda_n = (2n + 1) + \frac{t}{\sqrt{2n}} \chi(n) + \frac{1}{n} \left[ t \zeta(n) + (t^2 + s^2) \omega(n) \right] + {O \left( \frac{\log n}{n^{3/2}} \right)}
\end{split} \label{eq:lamlastdecomp}$$
$T_1$ is a linear function on $w$’s, so $$\begin{split}
T_1(n; W) &= t T_1(n; v^e) + s T_1(n; v^e) \\
& = t T_1(n; v^e).
\end{split} \label{eq:t1.2pt}$$ This explains two terms with $t$-coefficient in . Next, $$\begin{split}
T_2(n; W) &= \operatorname{Trace}\frac{1}{2\pi i} \int\limits_{{\partial (\mathcal{D}_n )}} (z - z_n) R^0 W R^0 W R^0 \, dz\\
&= t^2 T_2(n; v^e) + s^2 T_2(n; v^0) + tsT^{\prime \prime} \quad \text{where}
\end{split} \label{eq:t2.2pt}$$ $$\begin{split}
T^{\prime \prime} = \operatorname{Trace}\frac{1}{2\pi i} \int\limits_{{\partial (\mathcal{D}_n )}} (z - z_n) \left[ R^0 v^e R^0 v^o R^0 + R^0 v^o R^0 v^e R^0 \right] \, dz = 0.
\end{split} \label{eq:t2.2pt.b}$$ Indeed $$\begin{split}
v^e_{mk} & = 0 \quad \text{if } m - k \text{ odd}\\
\text{and} \quad v_o^{kj} & = 0 \quad \text{if } k - j \text{ even}
\end{split} \label{eq:vdecomp}$$ So in $$v^e_{mk} v^o_{km} = 0 \quad \text{and} \quad v^o_{mj}v^e_{jm} = 0$$ for any ${\left\lbrace m, k \right\rbrace}$ or ${\left\lbrace m, j \right\rbrace}$ because at least one factor is zero.
Comments; miscellaneous {#sec:comment}
=======================
The technical estimates of Sections \[sec:manyineqs\] to \[sec:etabdpt2\] \[mostly Section \[sec:etabdpt1\]\] lead to asymptotics of the eigenvalues in the case of perturbations , at least, if the number of point interactions is finite, i.e., $$w(x) = \sum_{j = 1}^J c_j {\delta \left( x - b_j \right) }, \quad J < \infty. \label{eq:pointmassrenew}$$ Of course, if $K = 2$ and $b_2 + b_1 = 0$ the interactions are well balanced and many terms in the asymptotic formulas vanish. But for any set of points $(b_k)$ and coupling coefficients $(c_k)$ we could give general formulas if we know the values of ${\left\lbrace T_j(n; w) \right\rbrace}$. Let us do this exercise for $j = 1, \, 2$.
Case b = 0, c = 2; i.e., v(x) = 2 delta(x)
------------------------------------------
We put $c = 2$ to make possible a direct comparison with the case of perturbations $v^e$ in . It seems we could use Thm. \[thm:evenasymp\] and its formulas to claim Prop. \[prop:singleasymp\] (see below). But in estimates of Sections \[sec:manyineqs\]–\[sec:etabdpt2\] we often used an assumption $b \neq 0$; see for example, inequalities , ; , ; , which are important in evaluation of the error-term or its analogues. However, if $b = 0$ we do not face such terms and evaluation is straightforward.
Indeed, by if $w = v_*$ $$\begin{aligned}
a_n &= h_n(0) = \frac{2^{1/4}}{\pi^{1/2}} \cdot \frac{1}{n^{1/4}} \cdot \cos {\left( n \, \frac{\pi}{2} \right)} + {O \left( \frac{1}{n^{5/4}} \right)} \label{eq:an0est} \\
\intertext{and}
a_k^2 & = \begin{cases} \displaystyle \frac{2}{\pi} \cdot \frac{1}{\sqrt{2k}} + {O \left( \frac{1}{k^{3/2}} \right)}, & k > 0 \text{ even} \\
0, & k \text{ odd.}\end{cases} \label{eq:an02estmain}\end{aligned}$$ Therefore, if $n \geq N_*$, $$\label{eq:t10decomp}
T_1(n; v_*) = 2 a_n^2 = \begin{cases} \displaystyle \frac{4}{\pi} \cdot \frac{1}{\sqrt{2n}} + \rho_{30}(n), & n \text{ even} \\
0, & n \text{ odd}. \end{cases}$$ where $$\label{eq:rho30bd}
{\left\vert \rho_{30}(n) \right\vert} \leq \frac{C}{n^{3/2}}.$$ Next term by is $$\label{eq:t20decomp}
T_2(n) = 2 a_n^2 \cdot \widetilde{\sigma}(n),$$ where $$\label{eq:varsigmarednew}
\widetilde{\sigma}(n) = \sideset{}{'}\sum_{k = 0}^{\infty} \frac{a_k^2}{n - k}.$$ The subsum $\widetilde{\sigma}_{\text{odd}}$ by and Remark \[rem:sbds\], $\displaystyle \beta = \frac{3}{2} > 1$, $$\label{eq:sigrenewoddbd}
\widetilde{\sigma}_{\text{odd}} = {O \left( \frac{1}{n} \right)},$$ and with the same remarks $$\label{eq:sigrenewevendecomp}
\widetilde{\sigma}_{\text{even}} = \frac{2}{\pi} \sum_{\substack{k = 1 \\ k \text{ even} \\ k \neq n}}^{\infty} \frac{1}{\sqrt{2k}} \cdot \frac{1}{n - k} + {O \left( \frac{1}{n} \right)}.$$ Now we refer to Lemma \[lem:zetabds\], , and Lemma \[lem:tauomegadiff\], , to claim $$\label{eq:sigrenewevenbd}
\widetilde{\sigma}_{\text{even}} = {O \left( \frac{1}{n} \right)}.$$ This leads us to the following.
\[lem:t10complete\] If $w(x) = 2 {\delta \left( x \right) }$
1. $$\label{eq:t10decomprepeat}
T_1(n; v_*) = \begin{cases} \displaystyle \frac{4}{\pi} \cdot \frac{1}{\sqrt{2n}} + \rho_{30}(n) , & n \text{ even}, \\
0, & n \text{ odd}. \end{cases}$$
where $$\label{eq:rho30bdrepeat}
{\left\vert \rho_{30}(n) \right\vert} \leq \frac{C}{n^{3/2}},$$ and
2. $$\label{eq:t20bd}
T_2(n) = {O \left( \frac{1}{n^{3/2}} \right)}$$
Part (i) has been explained by formulas – . With $a_n^2$ given by $$a_n^2 = {O \left( \frac{1}{\sqrt{n}} \right)},$$ and $$\widetilde{\sigma} = {O \left( \frac{1}{n} \right)}$$ by and . Therefore, their product is $\displaystyle {O \left( \frac{1}{n^{3/2}} \right)}$, i.e., holds.
Finally, the term $T_3(n)$ — see , or — is by – and the formulas of the previous subsection is ${\displaystyle}\leq C \left[ \frac{1}{\sqrt{n}} \cdot {\left( \frac{1}{n} \right)}^2 + \frac{1}{n} \cdot \frac{1}{\sqrt{n}} \right]$ so $$\label{eq:t30bd}
{\left\vert T_3(n) \right\vert} \leq \frac{C}{n^{3/2}}.$$ We have proven the following.
\[prop:singleasymp\] If $w = tv_* = 2t{\delta \left( x \right) }$ the eigenvalues of the operator for $n \geq N_*$ are $$\begin{aligned}
\lambda_n & = (2n + 1) + \frac{4t}{\pi} \cdot \frac{1}{\sqrt{2n}} + {O \left( \frac{1}{n^{3/2}} \right)}, & n \text{ even}\\
& = (2n + 1) + {O \left( \frac{1}{n^{3/2}} \right)}, & n \text{ odd}.\end{aligned}$$
{#section-20}
Let us conclude our example with a simple case of one-point interaction $$\label{eq:w1pt}
w(x) = t {\delta \left( x - b \right) }, \quad b \neq 0.$$ Again recall that $$\label{eq:hermfcnformrenew}
\begin{split}
h_k(x) = \frac{2^{1/4}}{\pi^{1/2}} \cdot \frac{1}{k^{1/4}} &\left[ \cos \left( x \sqrt{2k + 1} - k \frac{\pi}{2} \right) + \right.\\
&+ \left. \frac{x^3}{6} \frac{1}{\sqrt{2k + 1}} \sin \left( x \sqrt{2k + 1} - k \frac{\pi}{2} \right) + {O \left( \frac{1}{k} \right)} \right]
\end{split}$$ and if $x = b$ $$\label{eq:ak2fcnformrenew}
\begin{aligned}
h_k(x) &= \frac{2^{1/2}}{\pi} \cdot \frac{1}{\sqrt{k}} \cdot \frac{1}{2} \cdot &&\left[1 + (-1)^k \cos \left(2 b \sqrt{2k + 1} \right) + \right.\\
& &&+ \left. \frac{b^3}{6} \frac{(-1)^k}{\sqrt{2k + 1}} \sin \left(2 b \sqrt{2k + 1} \right) + {O \left( \frac{1}{k} \right)} \right] \\
&= \frac{1}{\pi} \cdot \frac{1}{\sqrt{2k}} \cdot &&\left[1 + (-1)^k \cos \left(2 b \sqrt{2k} \right) - \right.\\
& &&- \left. \frac{b^3}{6} \frac{(-1)^k}{\sqrt{2k}} \cdot b\left( 1 - \frac{b^2}{3} \right) \sin \left(2 b \sqrt{2k } \right) + {O \left( \frac{1}{k} \right)} \right]
\end{aligned}$$ Then $$\begin{aligned}
T_1(n) & = W_{nn} = t a_n^2 \label{eq:w1t1} \\
\intertext{and}
T_2(n) & = \frac{t^2}{2} a_n^2 \sum_{k = } \frac{a_k^2}{n - k} \label{eq:w1t2} \end{aligned}$$ Four terms in brackets on the right side of give impacts into $$\label{eq:sigmabardef}
\overline{\sigma}(n) = \sideset{}{^{\prime}} \sum_{k = 0}^{\infty} \frac{a_k^2}{n - k}$$ which are evaluated as $$\label{eq:w1basicsum}
\sideset{}{^{\prime}} \sum_{k = 1}^{\infty} \frac{1}{\sqrt{k}} \cdot \frac{1}{n - k} = {O \left( \frac{\log n}{n} \right)} \quad \text{by Lemma~\ref{lem:zetabds}, \eqref{eq:zetabd},}$$ $$\label{eq:w1basicsum2}
\begin{split}
\sideset{}{^{\prime}}\sum_{k = 1}^{\infty} \frac{(-1)^k}{\sqrt{k}} \cdot \frac{\cos 2b \sqrt{2k}}{n - k} & = (-1)^n \left[ \, \sideset{}{^{\prime}}\sum_{\substack{k = 1 \\ k - n \text{ even}}}^{\infty} - \sum_{\substack{k = 1 \\ k - n \text{ odd}}}^{\infty} \, \right]\\
& = (-1)^n \left[c_* \frac{\sin 2b \sqrt{2n}}{\sqrt{2n}} - c_* \frac{\sin 2 b \sqrt{2n}}{\sqrt{2n}} + {O \left( \frac{\log n}{n} \right)}\right] \\
& = {O \left( \frac{\log n}{n} \right)}
\end{split}$$ by Prop \[prop:es2bds\], or . \[It is important that coefficient $c_*$ is the same for subsums over even and odd $k$.\]
The third and fourth terms are evaluated just by absolute value so $$\label{eq:thirdterm}
\sideset{}{^{\prime}} \sum_{k = 1}^{\infty} \frac{1}{\sqrt{k}} \cdot \frac{1}{\sqrt{k}} \cdot \frac{1}{\vert n - k \vert} = {O \left( \frac{\log n}{n} \right)} \text{ by Remark~\ref{rem:sbds}, }\beta = 1,$$ and $$\label{eq:fourthterm}
\sideset{}{^{\prime}} \sum_{k = 1}^{\infty} \frac{1}{\sqrt{k}} \cdot \frac{1}{k} \cdot \frac{1}{\vert n - k \vert} = {O \left( \frac{1}{n} \right)} \text{ by Remark~\ref{rem:sbds}, }\beta = \frac{3}{2}.$$ The estimates – explain that $$\label{eq:sigmabarbd}
\overline{\sigma}(n) = {O \left( \frac{\log n}{n} \right)}$$ so by and $$\label{eq:w1t2bd}
T_2(n) = {O \left( \frac{\log n}{n^{3/2}} \right)}$$ We have proven the following
\[prop:w1lamform\] If $w = t {\delta \left( x - b \right) }$ then the eigenvalues of the operator for $n \geq N_*$ are $$\label{eq:w1lamform}
\begin{split}
\lambda_n = (2n + 1) &+ \frac{1}{\pi \sqrt{2n}} \left[ 1 + (-1)^n \cos (2b \sqrt{2n}) \right] \\
& - \frac{1}{\pi} \frac{(-1)^n}{2n} b \left( 1 - \frac{b^2}{3} \right) \sin (2b \sqrt{2n}) + {O \left( \frac{\log n}{n^{3/2}} \right)}.
\end{split}$$
{#subsect:extralog}
One more technical adjustment should be to cover the case $p = 2$ in Proposition \[prop:imagw\]. If the condition would be changed to $$\label{eq:wcondadjust}
\vert w_{jk} \vert \leq C_0 \cdot \frac{\log(e + j)}{(1 + j)^{\alpha}} \cdot \frac{\log(e + k)}{(1 + k)^{\alpha}}$$ we could follow the lines of Section \[sec:prelims\] but instead of , we have to evaluate $$\label{eq:muadjust}
\mu = \sum_{j = 0}^{\infty} \frac{\log(e + j)}{(1 + j)^{2\alpha}} \cdot \frac{1}{{\left\vert z - z_j \right\vert}}, \quad z \in {\partial (\mathcal{D}_n )}$$ We omit technicalities which mimick steps – ; they certainl lead us to the estimates (compare , ) $$\label{eq:muadjustest}
\mu \leq \frac{G(\alpha)}{n^{2\alpha}} \left[ \log (en) \right]^2, \quad 2\alpha < 1.$$ If ${\displaystyle}\alpha = \frac{1}{8}$ by $$\label{eq:wcondbetter}
\vert w_{jk} \vert \leq C_0 \frac{\log (ej)}{j^{1/8}} \cdot \frac{\log(ek)}{k^{1/8}}$$ and $$\mu \leq \frac{G^*}{n^{1/4}} \left[ \log (en) \right]^2 .$$ Therefore, in Proposition \[prop:imagw\], $p = 2$, $N^*$ should be chosen to guarantee \[compare \] $$\begin{gathered}
C_0 G^* \nu \frac{(\log en)^2}{n^{1/4}} \leq \frac{1}{2}, \quad \text{or}\\
\frac{c}{2} \nu^{1/2} \frac{\log en}{n^{1/8}} \leq \frac{1}{2}, \quad c^2 = (2 C_0 G^*).\end{gathered}$$ By , $\beta = \frac{1}{8}$, and Corollary \[cor:xtbehave\] $N^*$ could be chosen as $$\begin{aligned}
X_{1/8} \left( \frac{c}{2} \nu^{1/2} \right) &\leq 2^8 \left( \frac{c}{2} \nu^{1/2} \log \left( 8A \nu^{1/2} \frac{c}{2} \right) \right)^8\\
& \leq C_2 \left( \nu( \log e\nu)^2 \right)^4, \, \exists C_2.\end{aligned}$$ This explains the claim in Proposition \[prop:imagw\].
Acknowledgements {#acknowledgements .unnumbered}
================
The author is indebted to Charles Baker and Petr Siegl for numerous discussions. Without their support this work would hardly be written, at least in a reasonable period of time. I am also thankful to Daniel Elton, Paul Nevai, and Günter Wunner for valuable comments and information related to topics of this manuscript.
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abstract: 'The mixed morphology class of supernova remnants has centrally peaked X-ray emission along with a shell-like morphology in radio emission. White & Long proposed that these remnants are evolving in a cloudy medium wherein the clouds are evaporated via thermal conduction once being overrun by the expanding shock. Their analytical model made detailed predictions regarding temperature, density and emission profiles as well as shock evolution. We present numerical hydrodynamical models in 2D and 3D including thermal conduction, testing the White & Long model and presenting results for the evolution and emission from remnants evolving in a cloudy medium. We find that, while certain general results of the White & Long model hold, such as the way the remnants expand and the flattening of the X-ray surface brightness distribution, in detail there are substantial differences. In particular we find that the X-ray luminosity is dominated by emission from shocked cloud gas early on, leading to a bright peak which then declines and flattens as evaporation becomes more important. In addition, the effects of thermal conduction on the intercloud gas, which is not included in the White & Long model, are important and lead to further flattening of the X-ray brightness profile as well as lower X-ray emission temperatures.'
author:
- 'Jonathan D. Slavin'
- 'Randall K. Smith'
- Adam Foster
- 'Henry D. Winter'
- 'John C. Raymond'
- 'Patrick O. Slane'
- Hiroya Yamaguchi
bibliography:
- 'snr\_evol.bib'
title: Numerical Simulations of Supernova Remnant Evolution in a Cloudy Interstellar Medium
---
Introduction
============
Supernovae are the primary source of hot gas in galaxies. The shocks created in a supernova explosion can expand to distances of more than 100 pc and create large, hot, low density volumes of gas that persist for millions of years. Supernova remnant (SNR) evolution has been explored theoretically for several decades with increasing levels of complexity. However, the appearance of SNRs and the way the shocks couple with the interstellar medium (ISM) depend crucially on the nature of the medium into which the remnant expands, in particular its density and inhomogeneity.
For many years SNRs were divided into two categories, shell-like and plerionic. The first category has emission that peaks at the edge of the remnant while plerions are centrally brightened. It was believed that in the latter case a pulsar wind powers the emission, while for shell-like remnants, emission from the outer blastwave shock dominates. However, cases that did not fit either morphology were found and referred to as either thermal composite, because of their centrally concentrated and yet thermal X-ray emission along with edge brightened radio emission, or mixed morphology supernova remnants (MMSNR).
@Rho+Petre_1998 first argued that the mixed morphology remnants formed a truly distinct class of SNRs. In more recent years more remnants in this class have been found and evidence has been presented that these remnants are, as a rule, undergoing interactions with dense clouds. In particular @Yusef-Zadeh_etal_2003 showed a strong correlation of OH maser emission with mixed morphology, indicating that shocks in molecular clouds are present around MMSNRs. In addition, shock-cloud interaction indicators such as broad velocity wings [@Frail+Mitchell_1998] or an enhanced $^{12}\mathrm{CO}(J=2-1)/^{12}\mathrm{CO}(J=1-0)$ ratio [@Seta_etal_1998] are observed in many MMSNRs [see review by @Slane_etal_2015].
At present, there are $\sim 40$ known MMSNRs, but the nature of the central X-ray emission is poorly understood. X-ray spectra show evidence for the presence of ejecta in some such MMSNRs, but the inferred mass of the X-ray emitting material appears to be much larger than any reasonable ejecta contributions. The observed temperature profiles show little variation with radius, in contrast to the steep profile associated with the Sedov phase. Possibly related is the observation that the plasma in many MMSNRs is observed to be overionized [e.g., @Kawasaki_etal_2002; @Yamaguchi_etal_2009], indicating a rapid cooling phase [@Uchida_etal_2012; @Moriya_2012].
Nearly half of the known MMSNRs are observed to produce gamma-ray emission, presumably indicating the interaction between protons accelerated by the SNR (or re-accelerated cosmic rays) and molecular cloud material or dense post-shock regions of radiative shocks [@Uchiyama_etal_2010; @Lee_etal_2015].
Even well before many of these observations that characterized MMSNRs in detail, @White+Long_1991 [hereafter WL] put forward an analytical model that attempted to explain remnants with centrally peaked X-ray emission. WL included terms in the fluid equations describing evolution in spherical symmetry of a SNR that allowed for a continual injection of mass and momentum as the result of cloud evaporation. In this way they aimed to describe the evolution and emission distribution generated when a SN explodes in a cloudy medium and where thermal conduction evaporates the clouds that have been enveloped by the blastwave. We note that there was no term in the equations accounting for the transfer of energy within the hot gas by thermal conduction. We will discuss the ramifications of this further below.
Methods {#sec:meth}
=======
WL created their models for SNR evolution in a cloudy medium starting from the usual set of equations for hydrodynamics under the assumption of spherical symmetry but with added terms to account for the effects of cloud evaporation on the mass, momentum and energy in the remnant. Though they included a term for mass injection into the intercloud medium as a result of thermal conduction, as mentioned above, they did not explicitly include thermal conduction. For our numerical hydrodynamical calculations presented here we employ the code FLASH [@Fryxell_etal_2000; @Fryxell_etal_2010 <http://flash.uchicago.edu/site/flashcode/>] without any extra terms added to account for mass loss from clouds since they are unnecessary in multiple dimensions as long as the physics is correctly modeled. An essential aspect of the modeling is the inclusion of thermal conduction, which is accurately and efficiently included via the `Diffuse` module included in version 4.3 of FLASH. We discuss the details of our use of this module in the appendix. The initial conditions for our cloudy medium runs include an initial distribution of clouds in space with varying sizes. The cloud size distribution and density were not important for WL as long as the fundamental assumptions of their calculations were not violated, namely that the clouds are much denser than the intercloud medium, the clouds are uniformly distributed in space and the filling factor of the clouds is small. We fulfill these criteria by randomly placing clouds that are a factor of 100 times denser than the intercloud medium with an overall small volume filling factor (dictated by the $C$ parameter, see below). We set off the SN explosion in this medium by putting the appropriate amount of thermal energy in a small number of parcels at the center of the remnant. We do not include any ejecta mass, which would be important at early times during the ejecta dominated phase. The characteristic time for the transition from the ejecta dominated to Sedov phase is [@Truelove+McKee_1999] $t_{ch} = E^{-1/2} M_\mathrm{ej} \rho^{-1/3}$, which for our values of explosion energy and density, and assuming 5 $M_\odot$ of ejecta, is about 2700 yr. It will take several times this value for the evolution to truly settle into a Sedov solution evolution in the remnant as a whole, though the approach will be faster for the region behind the forward shock. For this reason we do not expect the early time evolution (e.g. X-ray luminosity) to be very realistic (though of course this also applies to the WL model). In our runs the thermal energy leads to rapid expansion and partial conversion of the thermal energy to kinetic energy as expected for Sedov-Taylor expansion [e.g., @Chevalier_1974]. Most of the results we discuss in this paper were carried out in 2D cylindrical symmetry, though we have done runs in 3D as well as a check on the effects of the 2D assumption.
For all of our runs, we use an explosion energy of $10^{51}$ ergs. As with the Sedov-Taylor solution, we expect the explosion energy to set the overall scale of the remnant but not to affect such things as the division of energy between thermal and kinetic forms. We use an ambient intercloud density of 0.25 cm$^{-3}$ and a temperature of $10^4$ K. These values are typical for the warm interstellar medium, though the temperature is perhaps slightly high for realistic heating and cooling balance. The exact value for the temperature should not have a significant affect on our results. The clouds are assumed to have a density of 25 cm$^{-3}$, which then leads to a temperature of 100 K to maintain pressure balance in the ambient medium. (Note that we are not tracking the ionization which would change these values if the ionization level were lower in the clouds than in the intercloud medium.) These values are consistent with typical cold neutral () cloud values. We do not expect the cloud temperature to be important to the remnant evolution. The cloud density could have some effect, since a larger density, for the same ratio of cloud mass to intercloud mass, would imply a lower cloud filling factor, though we think that our assumed density is realistic. For the cloud sizes we use a power law in radius. We took the distribution used in @McKee+Ostriker_1977, which was partly based on observations by @Hobbs_1974 and has lower and upper radius cutoffs of 0.48 and 10 pc and a power law exponent of -3 in radius. We create realizations of the cloud distribution by drawing randomly from this power law distribution and also choosing cloud location randomly within the simulation volume. We discard clouds that overlap with previously generated clouds such that each cloud is distinct from all the others. We also discard clouds that extend outside of the simulation volume. For our 2D cylindrically symmetric runs the clouds are, in effect, tori with circular cross sections. In our 3D runs, which are done in cartesian coordinates, the clouds are true spheres.
The actual cloud size distribution in the Galaxy is poorly constrained and so our adopted parameters of the cloud size distribution may not accurately reflect the true cloud size distribution, [though others have found similar results, e.g. @Gosachinskij+Morozova_1999; @Chieze+Lazareff_1980]. There is evidence that interstellar clouds have a morphology that is closer to filamentary or “clumpy sheets” [@Heiles+Troland_2003], though more regular clouds are observed as well. We do not expect the cloud geometry to have a very substantial effect on SNR evolution or observables since, once shocked, the complex flows inside the remnant create large distortions of the clouds in any case. However, if a remnant encounters a cloud that is comparable in size to the remnant, both the appearance and evolution of the remnant would be strongly affected. In such cases the remnant would have a strongly asymmetric appearance. Exploration of such cases is beyond the scope of this paper.
WL found that their solutions could be parametrized by just two parameters: $C$, the ratio of the mass in clouds to that in the intercloud medium, and $\tau$ the ratio of the cloud evaporation timescale to the remnant age. While $C$ is clearly a fixed value that characterizes the conditions in the ambient medium, WL concede that $\tau$ may not be. Since the temperature of the interior of a remnant evolves with time and the evaporation timescale would generally be expected to depend on the temperature and/or the pressure. In fact it can be shown, using the results of @Dalton+Balbus_1993, that in the limit of highly saturated evaporation, the evaporation timescale is $\tau_\mathrm{evap} \propto R_\mathrm{cl}^{7/6} P^{-5/6}$, with no explicit dependence on temperature. Since the pressure in the Sedov-Taylor solution drops as $t^{-6/5}$, if the medium surrounding the enveloped clouds is close to that for Sedov-Taylor expansion in a uniform medium, we expect that the evaporation timescale should be proportional to the remnant age and so $\tau$ should be constant. Deviations from these assumptions, for example because the evaporation is not highly saturated or because the cloud-cloud or cloud-shock interactions cause substantial deviations from a Sedov-Taylor evolution that lead to different evolution of the evaporation rate, would then be expected to cause variation of $\tau$ with remnant age. We discuss this further below.
Results
=======
In Figure \[fig:snrevol\] we show the density for four snapshots of the evolution for one simulation. For this case we use a value for the WL parameter $C = 10$, ratio of mass in clouds to that in the intercloud medium. Given the definition of $C$, the volume filling factor of clouds is $$f = \frac{C}{\chi + C}$$ where $\chi$ is the ratio of the cloud density to the intercloud density. With our assumption that $\chi = 100$, for the $C = 10$ case, $f = 0.091$. For our 2D cylindrically symmetric runs, the clouds are really toroids. However, for these cases the area filling factor, that is the fraction of the $r-z$ plane filled with clouds, is quite close to the volume filling factor of the toroidal clouds. If the clouds are evenly distributed in cylindrical radial distance this will always be the case. We have explored a range of $C$ values ranging from 3 to 30. We take 10 as our standard case for much of the discussion in this paper. Figure \[fig:slice3D\] shows the density for a slice through one of our 3D runs. That run also had $C = 10$. Note the similar area filling factor for the clouds. The fact that it is a slice through a 3D volume results in the cloud radii appearing systematically smaller than the true cloud radii.
{width="49.00000%"} {width="49.00000%"}
In Figure \[fig:prof\] we show temperature and density profiles for the same 3D run as in Figure \[fig:slice3D\]. The rays shown illustrate the variations along a line of sight from the origin outward along the direction given by the angles listed in the legend. We also show the radially averaged profiles for temperature and density, but for those we exclude cloud material, here defined as parcels with temperature below $10^5$ K. The averaging is done by calculating a volume weighted sum of all parcels within each radial bin that have hot gas in them and dividing by the volume of parcels (whether or not they have hot gas) in the same radial bin. The WL profile uses the actual value of $C$ for the simulation at the given time ($2.5\times10^4$ yr) and given the current shape of the shock front ($C = 9.0$), which is calculated as the mass in clouds within the shock front divided by the mass in the intercloud medium inside the same volume, both for the initial, undisturbed medium. The value of $\tau$ used is that which leads to the same mass of hot gas as for the simulation. We see that, while for much of the outer parts of the remnant, the radially averaged profile is similar to the WL profile, in the inner region the density and temperature stay flat for the simulation while the WL profile has a steeply rising temperature and falling density similar to a Sedov-Taylor type profile. In addition, the density variations, as well as averaged values, are important for the X-ray emission, as discussed below, since emissivity goes as density squared.
![3D rendering of the shock and clouds for one of our 3D runs. The cloud and shock surfaces are shown as temperature contours. The age of the remnant is $1.5\times 10^4$ yr. The box is 30 pc on a side. Note the strong distortions of the shock front. \[fig:3Dshock\]](3D_image-4){width="49.00000%"}
Shock Evolution
---------------
The similarity solution of WL results in expansion of the shock front with the same power law in time as for the Sedov-Taylor solution, i.e. $R_s \propto
t^{2/5}$. For our calculations, evaluating the shock radius is not entirely straightforward since in regions where a cloud is being encountered the shock is slowed and the front in general is complex. In Figure \[fig:3Dshock\] we show 3D contours of the shock and clouds which illustrate the complexity of the shock front. We found that the simplest approach and the one that is closest to what one would find from observations is to find the outermost pressure contour that is substantially above the ambient pressure, $P/k_B =
10^4$ cm$^{-3}$ K for our case with an ambient pressure of 5200 cm$^{-3}$ K, and use that to calculate the volume inside the remnant. The shock radius is then just $R_{s,\mathrm{eff}} = (3 V_s/4\pi)^{1/3}$, where $V_s$ is the volume enclosed by the shock. (Note: the `find_contours` and `grid_points_in_poly` methods in the scikit-image measure python module provide effective methods for finding parcels inside the shock.) This leads to a relatively smooth shock expansion except for some cases in which the shock wraps around a large cloud leading to a jump in the shock radius. An alternative and easier approach is to use the volume of hot gas to define the shock radius, though in that case the volume in the clouds contained in the SNR is not included. In practice for our calculations that volume is small compared with the hot gas volume and the derived shock radius is very similar.
![Evolution of the mean shock radius for different $C = 10$ runs. This illustrates the variations that are possible with different cloud realizations. We note that the shock evolutions for the 3D cases are consistent with the 2D cylindrically symmetric cases as is the 2D case using the full volume (positive and negative $z$). The inset shows a detail to illustrate more clearly the differences during a medium age period. \[fig:rs\_evol\]](comp_C=10_shocks3){width="49.00000%"}
![Comparison of the evolution of the mean shock radius for runs with different values for $C$, the ratio of ISM mass in clouds to that in the intercloud medium. The density of the clouds and the intercloud medium is the same for all of the runs, $25$ cm$^{-3}$ and $0.25$ cm$^{-3}$, respectively. Only the filling factor of the clouds is varied. As expected larger $C$ values lead to slower shock expansion. The dashed curve labeled “$t^{2/5}$ fit” is a fit to the shock radius evolution assuming that $R_s \propto t^{2/5}$ for the $C
= 10$ case, where the fitting was done only for the range in $t$ as plotted. This shows that the evolution approaches the $t^{2/5}$ law found by WL. For this case an effective density, i.e. the density that the corresponding Sedov-Taylor expansion would require, is 0.455 cm$^{-3}$.\[fig:rs\_C\]](comp_rshock_vs_C3){width="49.00000%"}
For our calculations we do find that the shock evolution depends to some degree on the particular cloud realization used. This is illustrated in Figure \[fig:rs\_evol\] which shows the shock evolution for several different cloud realizations. For most of our runs we have used only half of the 2D volume, $z \geq 0$. The figure shows that the shock evolution for different runs using the 2D half volume differ though the variations are at about the 10% level. We have tested the effects of using the 2D half volume by doing a full volume run, shown as the red dash-dotted line in Figure \[fig:rs\_evol\]. We find no significant difference in that run as compared with other runs with the same $C$ value. We have also compared with a 3D runs (using one octant of the space) done in cartesian coordinates. Again, as demonstrated in Figure \[fig:rs\_evol\] we find no significant difference from the 2D half volume runs. Because of the computational and visualization demands of the 3D runs, they were done with only 5 levels of grid refinement rather than the 6 levels used for the 2D runs. We have found, by doing 2D runs at 5 and 6 levels of refinement, that while there are minor differences in the density distributions, with more filamentary structure visible in the higher resolution runs, the overall evolution of the remnants are nearly identical. Thermal conduction aids in this respect since small scale details tend to be smoothed out when it is included.
The expansion law for the runs shown in Figure \[fig:rs\_evol\] is generally not far from the WL $t^{2/5}$ value, though again there are significant variations around that. One way to test that is to compute $v_s t/r_s$, which for a $t^{2/5}$ expansion law should give 2/5. Doing this involves taking numerical derivatives of the shock radius expansion curve which inevitably results in a noisy curve. In general we find that the expansion law is close to but slightly below this value on average, though with excursions above and below it.
In Figure \[fig:rs\_C\] we show the shock radius evolution for runs with different $C$ values. Since the mean density in the medium increases with $C$ it is expected that the shock expansion rate should decrease as $C$ increases. In the results of WL, the expansion rate is characterized via the parameter $K$ (their equation 8) where $$r_s = \left[\frac{25(\gamma + 1) K E}{16 \pi \rho_1}\right]^{1/5} t^{2/5},
\label{eq:rs}$$ $r_s$ is the shock radius, $E$ is the explosion energy, $\gamma = 5/3$, and $\rho_1$ is the ambient intercloud density (0.25 cm$^{-3}$ for our runs). WL lists values for $K/K(\mathrm{Sedov})$ as a function of $\tau$, the ratio of cloud evaporation time to remnant age. $K(\mathrm{Sedov}) = 1.528$ is the $K$ value for the case of no clouds. We can invert equation \[eq:rs\] to determine the effective value of K for our runs. Doing this we find that K varies considerably over a typical run though it flattens at late times. For most of our runs, K defined this way starts high, $> 2$, but ends near 0.9. The WL model predicts, given a value of $\tau$ of $\sim 5 - 10$ (see discussion below) and $C = 10$, $K \sim 0.816 - 1.149$. This is in fairly good agreement with our results. K can also be alternatively be defined using WL’s equation (6) as $$K = \frac{4 \pi \rho_1 r_s^3 v_s^2}{(\gamma + 1)E}.$$ With this definition, we find more stable values for K, which still fall roughly in the predicted range.
Cloud Evaporation Rate
----------------------
Within the WL model the value of $\tau$, the ratio of evaporation timescale to remnant age, is treated as a free parameter. In our numerical work, this is not the case, since given the initial conditions the cloud evaporation timescale will follow from the physics, including the temperature of the surrounding hot gas and the complex heat flow patterns that develop as the clouds are both evaporated and disrupted by shear flows. In addition, clouds close to the explosion center can be heated enough by being shocked that they could be considered part of the hot gas, though still overdense. However, as we discuss below, such clouds are overpressured compared to the surrounding medium and so they expand and cool at later times in the remnant evolution.
Clouds that are farther from the center do not get heated to high temperatures by the expanding shock but are subject to shear flows and thermal conduction. However, even for these clouds, characterizing the evaporation rate is not entirely straightforward since gas exists at a range of temperatures and densities within the remnant at any given time. As a result defining which gas has been evaporated is not entirely clear cut.
We have tried a variety of approaches for determining the effective evaporation timescale for our simulated remnants. One approach is to use a density criterion to decide what is cloud material and what is evaporated. In our case we defined the mass in clouds as the mass of gas with density above a 2.5 cm$^{-3}$, which is the geometric mean of the density of intercloud medium, 0.25 [cm$^{-3}$]{}, and the cloud density, 25 [cm$^{-3}$]{}. The mass of gas evaporated from clouds at any given point in the remnant’s evolution is then the mass that was in clouds initially within the shock volume minus the mass in clouds at the current time. The average evaporation rate is then the evaporated mass divided by the remnant age. Connecting this evaporation rate to that predicted in the WL model is not entirely straightforward however, since they did not tally that quantity. Instead we can connect the intercloud mass, that is all the mass inside the shock that is not in a cloud, with the total integrated mass in the WL model since the density, $\rho$, or scaled density, $g = \rho/\rho_s$ stands for all the non-cloud gas in the model. Thus we compare the intercloud gas in a simulation with $$M_x = \int_0^{r_s} 4 \pi r^2 \rho\,dr = 4\pi r_s^3 \rho_s \int_0^1 g(x)
x^2 \,dx,$$ (from WL) where $g \equiv \rho/\rho_s$ is derived by integrating the set of equations in WL for given values of $\tau$ and $C$. To find the value of $\tau$ corresponding to a particular time for a simulated remnant we use the actual value for $C$, calculated as the initial ratio of mass in clouds to that in the intercloud medium within the shock volume. We then do a search, using a root finding method, wherein we calculate the values of $M_x$ given values of $\tau$ until we match the value for the simulated remnant. This is the method used to produce Figure \[fig:tevap\].
![Evaporation timescale factor, $\tau$, of WL vs. remnant age. In the WL model $\tau$ is a constant, but here we see that it varies considerably over the age of our simulated remnants. The 3D runs show smoother behavior because more clouds are sampled at any given age. Here $\tau$ is calculated using the density criterion for deciding whether gas is intercloud or cloud gas with $n < 2.5$ cm$^{-3}$ as the dividing line. All the runs plotted here have $C$ set to 10. \[fig:tevap\]](tevap-2){width="49.00000%"}
![Evaporation timescale factor, $\tau$, of WL vs. remnant age. In this case we use the temperature criterion to differentiate between cloud and intercloud gas with $T > 10^5$ K as the dividing line. This method ensures that the X-ray emitting gas mass (though not the luminosity) for the corresponding WL model matches that for the simulated remnant. The $\tau$ values calculated this way are substantially below those in Figure \[fig:tevap\], primarily because of hot, but still dense, gas generated by shocking clouds close to the explosion center. Such gas is counted as cloud gas (and thus not evaporated) by the method of the previous figure, but should radiate X-rays. \[fig:tau\_eff\]](tau_xray){width="49.00000%"}
Another approach is to calculate the mass of X-ray emitting gas, essentially all gas in the remnant that is hotter than 10$^5$ K, and match that with the X-ray emitting mass, $M_x$, for a given WL model. That approach leads to values of $\tau$ that are considerably lower than for the density based criterion, mostly because of the production of hot, dense gas via shock heating. The results for $\tau$ using the temperature based criterion are shown in Figure \[fig:tau\_eff\].
The status of the shocked cloud gas is somewhat ambiguous in the model, however since it is overpressured when shocked and some of it re-expands and cools adiabatically at later times below X-ray emitting temperatures. For clouds shocked by slower shocks, only a surface layer is raised to X-ray emitting temperatures. Thus, at later times, the removal of material from clouds is primarily by thermal conduction and cloud evaporation as discussed by WL does apply.
We expect that the evaporation timescale for the remnant as a whole should depend in a similar way to that for individual clouds [though see @Balbus_1985 regarding the effect on evaporation rate of a collection of clouds]. For “classical” evaporation, i.e. cloud evaporation when the heat flux is far from saturation, the evaporation timescale goes as $t_\mathrm{evap} \propto R_\mathrm{cl}^2 n_\mathrm{cl} T_h^{-5/2}$ where $R_\mathrm{cl}$ is the cloud radius, $n_\mathrm{cl}$ is the cloud density and $T_h$ is the temperature of the hot gas far from the cloud. In the case of highly saturated heat flux, which is the more typical condition in SNRs during the non-radiative phase, we get $t_\mathrm{evap} \propto R_\mathrm{cl}^{7/6}
n_\mathrm{cl} P^{-5/6}$, where $P$ is the thermal pressure in the hot gas. Since, ignoring the effects of shocks, the density in the clouds is expected to be roughly constant and, as mentioned above, the pressure should decrease as $t^{-6/5}$, the evaporation timescale should increase proportional to $t$. From this we would expect the WL parameter $\tau \equiv t_\mathrm{evap}/t$ to be roughly constant and to only depend on the density and size of the clouds.
For our assumptions regarding mean mass per particle we find $$t_\mathrm{class} = 3.02\times10^{14}\frac{R_\mathrm{cl}^2
n_\mathrm{cl}}{T_h^{5/2}}\;\mathrm{Myr}$$ and $$t_\mathrm{sat} = 1.48\times10^3\frac{R_\mathrm{cl}^{7/6}
n_\mathrm{cl}}{P_h^{5/6}}\;\mathrm{Myr}$$ where $R_\mathrm{cl}$ is the cloud radius in pc, $n_\mathrm{cl}$ is the cloud density in cm$^{-3}$, $T_h$ is the temperature of the surrounding hot gas and $P_h$ is the thermal pressure (presumed to be roughly equal in the cloud and hot gas) in cm$^{-3}$ K. We find that with these values, it is not typical to get values for $\tau$ less than one. These expressions also make it clear that the primary determinant of the $\tau$ value is the cloud size distribution. We have confirmed this by doing runs with smaller clouds, using 0.2 and 5.0 pc as the lower and upper size limits, but with the same volume filling factor (and thus $C$ value). As expected we find that the clouds evaporate faster, leading to smaller values of $\tau$.
While our discussion above indicates that $\tau$ should be roughly constant, we find that it evolves considerably over the course of SNR evolution. This is not too surprising since, especially for the 2D evolution, a small number of clouds is encountered and the location and size of the clouds affects the evaporation rate. This is demonstrated in Figures \[fig:tevap\] and \[fig:tau\_eff\] where we plot $\tau$, calculated two different ways, as a function of remnant age for a variety of our runs. The differences between the 2D runs can be attributed to the differences in the particular cloud realization for the runs with some with larger clouds placed closer to the origin and the spacing of clouds differing. As can be seen, the 3D runs show less variation between themselves and over time. This can be understood as deriving from the larger number of clouds encountered and thus more complete sampling of the cloud size distribution. A full volume 3D run would likely show an even smoother variation though the finite cloud volume would still be expected to create variations over the course of the cloud evolution. There is some indication here that there is faster evaporation (smaller $\tau$) in the 3D simulations than in the 2D simulations, which could be understood as being caused by the larger surface area to volume ratio effectively for the spherical clouds in 3D than for the toroids under cylindrical symmetry. Nevertheless, the shock expansion does not show any marked difference between 2D and 3D cases nor does the calculated X-ray emission as we discuss in the next section.
{width="98.00000%"}
{width="98.00000%"}
X-ray Emission Distribution
---------------------------
One of the principal motivations for the WL model was to explain the appearance of SNRs with centrally peaked thermal X-ray emission. WL showed that they could achieve a range of different X-ray emission distributions depending on the choice of their parameters $C$ and $\tau$. From their Figure 4 it can be seen that centrally peaked X-ray emission requires values of $C
\gtrsim 20$, depending on the value of $\tau$. For $\tau = 2$ (WL Figure 4a) none of the emission profiles is truly centrally peaked. With $\tau = 10$ (WL Fig. 4b) centrally peaked emission is achieved, though only for fairly high $C$. For our evaluation of the emission in our simulated remnants we have chosen to use a more realistic emissivity than the constant $\Lambda_x =
10^{-22}$ ergs cm$^{3}$ s$^{-1}$ used by WL (note that in the units for this coefficient, the exponent for cm in WL was mistakenly shown as -3 rather than 3). Here we use a photon emissivity for optically thin emission in the $0.5 -
7$ keV band (see Figure \[fig:APEC\_emis\]) as calculated using APEC [@Foster_etal_2012; @Smith_etal_2001] with the assumption of collisional ionization equilibrium (CIE). We discuss the assumption of CIE below.
![Emission coefficient for optically thin radiative cooling by gas in collisional ionization equilibrium. This coefficient is for photons emitted in the $0.5 - 7$ keV band and includes both line and continuum emission. \[fig:APEC\_emis\]](APEC_emis_coeff){width="49.00000%"}
This choice of emission coefficient covers a typical X-ray observation energy range, e.g. with Chandra. From Figure \[fig:APEC\_emis\] it is clear that this emissivity weights the emission profile strongly toward gas in the temperature range of $\sim 10^{6.5} - 10^7$. A detector with sensitivity at lower energies could see the emission from lower temperature gas generated at the boundaries of the clouds which is expected to be strong. The lower right panels of Figures \[fig:bprof\] and \[fig:bprof2\] show a comparison of the WL profile that corresponds with the $\tau$ and $C$ values at that time for each run. Note that here we use the effective value for $C$ rather than the mean value. By this we mean that we use $C =
m_\mathrm{cl}/m_\mathrm{ic}$ where $m_\mathrm{cl}$ is the cloud mass, $m_\mathrm{ic}$ is the intercloud mass and both are for the initial medium contained within the current volume of the shock. This will tend to make the effective $C$ smaller than the mean value over the whole medium because the shock is slowed where it encounters clouds, though in practice we have found this effect to be fairly minor. The same APEC emissivity was used to generate the WL profile as for those from the simulations. The relative flatness of the WL profile is caused by the lower density and higher temperature near the center as compared with the simulations (see Figure \[fig:prof\]).
![Evolution of the $0.5 -7$ keV photon luminosity in our simulations. At early times the luminosity is low because the very high gas temperature and because few clouds have been encountered. The peak in emission comes when the shock encounters clouds and is still fast enough that the shock transmitted into the clouds can heat gas to X-ray emitting temperatures. Later cloud evaporation raises the hot gas density while expansion lowers the pressure, leading to roughly constant luminosity. \[fig:lumin\_evol\]](lumin_evol_fig-2){width="49.00000%"}
![Evolution of the X-ray weighted temperature in simulated remnants. At the time of peak luminosity, much of the emission is coming from relatively low temperature shocked cloud gas, which leads to low temperature emission. At later times more of the emission comes from the hot intercloud gas which is hotter. \[fig:Tlum\_evol\]](Tlum_evol_fig){width="49.00000%"}
The evolution of the $0.5 - 7$ keV photon luminosity is shown in Figure \[fig:lumin\_evol\] for several simulations with $C = 10$. The emissivity-weighted temperature is shown in Figure \[fig:Tlum\_evol\]. The luminosity starts low both because the remnant volume is low (i.e. the shock radius is small) and because the temperature is very high, above the temperature range of highest emission efficiency in the band (see Figure \[fig:APEC\_emis\]). (Note that because we do not include ejecta mass or extra circumstellar material, the very early emission is not expected to be accurately modeled.) The luminosity then reaches a peak at an age of $\sim
2000$ yr. This peak is associated with a time when the SNR encounters its first cloud. The shock is fast enough at this point (in our runs) to heat the entire cloud to X-ray emitting temperatures. Later the clouds close to the explosion site re-expand and cool (adiabatically since we have not included radiative cooling). This is because the shocked clouds are overpressured relative to the intercloud medium and also the pressure in the remnant is decreasing as it expands. An example of this can be seen in Figure \[fig:slice3D\] where a hollow looking cloud can be seen which has re-expanded and cooled. These clouds cool enough that they fall below X-ray emitting temperatures, i.e. $T \lesssim 10^5$ K. For clouds farther from the origin, only their inward facing edges get heated sufficiently to produce X-rays (see the arcs in Figure \[fig:bprof\]). Cloud material at those later times is continually being evaporated off the clouds, which increases the density of the hot gas, but at the same time the expansion of the remnant cools the gas. The net result, as can be seen from the figure, is an almost flat or slightly decreasing X-ray photon luminosity. This contrasts with the behavior predicted by WL (their eq. 21) which is for the luminosity to increase in proportion to the remnant volume. The evolution of the emissivity weighted temperature of the emission has a nearly inverted profile compared with the luminosity evolution. That can be explained by the fact that the shocked cloud gas is relatively cool, though dense, so when that gas is hot enough to emit X-rays, the emissivity-weighted temperature is low. As a remnant transitions to the phase where most of the emission is coming from the lower density and hotter gas, the emission temperature increases and then flattens out.
These differences in luminosity evolution are some of the most substantial differences between our calculated results and the WL model. The domination of the emission by the shocked cloud gas shows that the MM remnant’s brightness distribution, especially at early times, does not depend as much on evaporated mass as on the shock interactions of the blast wave with the clouds. In this sense, the value of $\tau$ derived based on density, as shown in Figure \[fig:tevap\] is not such an important parameter, though it is consistent with our expectations for the evaporation timescale, including the fact that it is roughly constant during remnant evolution. Instead the effective $\tau$, shown in Figure \[fig:tau\_eff\], where the temperature of the gas is the criterion for differentiating cloud gas from intercloud gas is a better measure for understanding the X-ray luminosity evolution, though given that some of that gas later cools via expansion, this $\tau$ is not a good measure of gas that has been thermally evaporated.
Discussion
==========
There are a number of additional processes that could affect SNR-cloud interaction in the ISM. Chief among these is radiative cooling. WL did not include cooling and we do not include it in our results presented here, though we intend to explore it in future work. In preliminary calculations we have found that the shape and size of shocked clouds is strongly affected, but the overall evolution of the remnant, for ages early enough that the remnant has not yet gone radiative as a whole, is not substantially changed. @Korolev_etal_2015 have recently studied the longer term evolution of SNR evolution in a cloudy medium including radiative cooling but not thermal conduction with 2D numerical simulations.
If radiative cooling is important, then non-equilibrium ionization should be taken into account. In SNRs in general and for shocks into clouds in particular, we expect the post shock gas to be underionized and far from CIE. This is also true for gas that is evaporated from the entrained clouds [@Ballet_etal_1986; @Slavin_1989]. It is also expected that the hot gas that was created early in the remnant’s lifetime will be overionized after the remnant has expanded and cooled. This effect will be enhanced by the presence of thermal conduction which draws thermal energy away from the hot interior toward the cooler outer parts of the remnant. In addition there could be turbulent mixing in the medium that would combine hot and cold gas producing yet another type of non-equilibrium ionization state [@Slavin_etal_1993]. All of these effects can have potentially large impacts on the emitted X-ray spectrum if the ionization is sufficiently far from CIE. These effects will need to be taken into account in order to make quantitative predictions for the emission spectrum for SNRs evolving in a cloudy ISM. We are currently in the process of improving the treatment of non-equilibrium ionization by FLASH, which will then allow efficient and accurate calculations of NEI effects.
Another possibly important influence on remnant evolution is the magnetic field. The field can affect the overall expansion of the remnant as well as the evaporation of clouds in the remnant. The anisotropic thermal conduction within the remnant when the clouds are threaded by the field is likely to have complex effects which deserve to be studied in detail. Our aim in the current study has been to examine SNR evolution under conditions similar to those intended in WL and so we do not include the magnetic field at this stage. Finally, we do not include heating via photoionization. This could be important in the post-shock regions of radiative shocks but is likely a minor effect for the young to middle-aged remnants evolving in the low density, warm ISM such as we examine in this work.
Conclusions
===========
The WL model was put forward in an attempt to explain the mixed morphology class of SNRs as being caused by the evaporation of dense clouds that are overrun by the blast wave. We have presented numerical hydrodynamical models of SNRs evolving in a cloudy medium including thermal conduction in order to test whether the results of WL hold up when more of the physics is included. We find that some of their results are broadly confirmed including: remnants expand with roughly the same $t^{2/5}$ power law dependence on age as a standard Sedov-Taylor expansion, 2) the cloud evaporation timescale increases roughly in proportion to the remnant age (i.e. $\tau$ is roughly constant), and 3) the presence of clouds causes the remnant X-ray brightness to be centrally peaked. However, the lack of the inclusion of the dynamical effects of cloud-shock interaction and thermal conduction within the hot gas that are present in multidimensional calculations lead to substantial differences from the results of WL. Probably the largest difference is with the predicted X-ray luminosity evolution which, rather than rising in proportion to the shocked volume, has an early peak and then flattens to a nearly constant level. This can be traced to the shock heating of the clouds close to the explosion site, which provides the peak, and later to the expansion cooling of the shocked clouds along with a transition to evaporative mass loss as the dominant mass loss mechanism for the clouds. The differences in temperature and density distribution in the remnants caused by thermal conduction within the hot gas and the shocking of the clouds leads to a flat X-ray brightness distribution in the remnants. For certain combinations of parameters, the brightness profile for WL models can match that from our models, though the parameters do not correspond to the physics, i.e. evaporation timescale and cloud-to-intercloud mass ratio, actually present in the simulation. In general we find that the X-ray emission weighted temperature is lower for our models than for WL models because of the inclusion of lower temperature hot gas associated with the clouds, either shocked or evaporated gas.
Our results point to important differences from the WL model for SNRs evolving in a cloudy ISM which could have implications for the interpretation of SNR observations, particularly in the X-rays. However, to make more robust predictions, particularly for the X-ray spectrum, will require the inclusion of more physical processes in future simulations. In particular radiative cooling and non-equilibrium ionization could substantially affect the observed spectrum of SNRs such as those we have modeled. We have begun to look at the effects of these processes and those calculations will allow us to make detailed predictions for the spectra of mixed morphology SNRs in future work.
Software used in this work (FLASH) was in part developed by the DOE NNSA-ASC OASCR Flash Center at the University of Chicago. This work was supported by NASA Astrophysics Theory Program grant NNX12AC70G.
FLASH parameters for the use of thermal conduction
==================================================
The `Diffuse` module of FLASH uses a number of parameters that govern how thermal conduction is calculated. Both standard Spitzer conductivity, $\kappa \propto T^{5/2}$, and saturated conductivity [@McKee+Cowie_1977] $\kappa \propto \rho c^3$ are supported with a smooth transition from “classical” to saturated governed by the saturation parameter $\sigma$. The parameter values that we have used in the `flash.par` files for runs with thermal conduction included are listed in table \[tab:condpar\]. The particular unit that we used is included via the switch `-unit=physics/Diffuse/DiffuseMain/Unsplit` to the `setup` command. In addition, to use the power law conductivity, we added `REQUIRES physics/materialProperties/Conductivity/ ConductivityMain/PowerLaw` to the `Config` file in our simulations directory. With these settings, electron thermal conduction is used with a power law conductivity, $\kappa = a T^{5/2}$ modified by the saturation limit from @Cowie+McKee_1977 and using the harmonic mean weighting (chosen through the `diff_eleFlMode` parameter) as in @Balbus+McKee_1982, $q = (1/q_\mathrm{class} + 1/q_\mathrm{sat})^{-1}$, where $q_\mathrm{class}$ is the “classical” Spitzer heat conduction and $q_\mathrm{sat}$ is the “saturated” conduction, which corresponds to the heat flux being transported at the maximum rate possible by the electrons, $q_\mathrm{sat} = 5\phi_s \rho c^3$. @Balbus+McKee_1982 argue that $\phi_s \approx 0.3$ (though with substantial uncertainty) and we use that value in this work. Here $a$ is the Spitzer conductivity coefficient [@Spitzer_1962], $1.84\times 10^{-5}/ln\Lambda$, where $\Lambda$ is the Coulomb logarithm. For conditions that we are exploring, $a \approx 6\times
10^{-7}$. In FLASH the implementation of saturation uses $q_\mathrm{sat}
= \alpha_\mathrm{ele} \rho_\mathrm{ele} c_\mathrm{ele}^3$ effectively, so to use $\phi_s = 0.3$ we set $\alpha_\mathrm{ele} = 0.04491$. This is derived under the assumption of a fully ionized plasma with a He abundance of 10% by number. These assumptions also lead to setting `eos_singleSpeciesA` to 0.6123. In reality the cooler parts of the plasma will most likely be partially ionized or nearly neutral, though in those regions conductivity is very low in any case. The parameter `diff_thetaImplct` sets the scheme for the diffusion solver, where 0.5 is for the Crank-Nicholson method, 0.0 for fully explicit and 1.0 for fully implicit. As indicated in the table, we use the Crank-Nicholson scheme. Since that scheme is unconditionally stable we set the `dt_diff_factor` to $10^{10}$ so as to effectively prevent the very restrictive diffusion timestep constraint from limiting the timestep.
We have tested the thermal conductivity by calculating the steady evaporation of a spherical cloud in cylindrical symmetry (in 2D) under moderately saturated conditions. The resulting mass loss rate and temperature profile closely matched that predicted by the results of @Dalton+Balbus_1993 who found analytical solutions for steadily evaporating clouds as a function of the degree of saturation. These results give us confidence that thermal conduction is functioning correctly in the code.
[ll]{} useConductivity & `.true.`\
useDiffuse & `.true.`\
useDiffuseTherm & `.true.`\
dt\_diff\_factor & 1.E10\
cond\_densityExponent & 0.0\
cond\_temperatureExponent & 2.5\
cond\_K0 & 6.E-7\
diff\_useEleCond & `.true.`\
diff\_eleFlMode & fl\_harmonic\
diff\_eleFlCoef & 0.04491\
diff\_thetaImplct & 0.5\
diff\_eleXlBoundaryType & zero-gradient\
diff\_eleXrBoundaryType & zero-gradient\
diff\_eleYlBoundaryType & zero-gradient\
diff\_eleYrBoundaryType & zero-gradient\
diff\_eleZlBoundaryType & zero-gradient\
diff\_eleZrBoundaryType & zero-gradient\
| 0 |
---
author:
- 'Davide Gaiotto and Theo Johnson-Freyd'
title: Mock modularity and a secondary elliptic genus
---
Introduction
============
In [@WittenTMF] the following puzzle was posed; the goal of this paper is to propose a solution. Let us say that a (1+1)-dimensional quantum field theory with minimal, aka ${\mathcal N}=(0,1)$, supersymmetry is [*null*]{} if supersymmetry is spontaneously broken and [*nullhomotopic*]{} if it can be connected to a null theory by a sequence of deformations, including deformations that may zig-zag up and down along RG flow lines.
\[mainpuzzle\] Show that the supersymmetric quantum field theory of three free antichiral fermions $\bar\psi_1,\bar\psi_2,\bar\psi_3$ and supersymmetry generated by $G = \sqrt{-1}\, {:}\bar\psi_1\bar\psi_2\bar\psi_3{:}$ is not nullhomotopic.
For the remainder of this paper we will write simply “SQFT” for “(1+1)-dimensional quantum field theory with ${\mathcal N}=(0,1)$ supersymmetry”, and “SCFT” for an SQFT which is furthermore superconformal. The SQFT in Puzzle \[mainpuzzle\] is an SCFT, and is the (conjectured) limit under RG flow of the ${\mathcal N}=(0,1)$ sigma model with target the round $S^3$ and minimal nonzero B-field. The puzzle is difficult because, as is shown in [@WittenTMF], the direct sum of 24 copies of the SQFT in Puzzle \[mainpuzzle\] *is* nullhomotopic (as is the ${\mathcal N}=(0,1)$ sigma model with target $S^3$ and B-field of strength $24$). So the puzzle requires constructing a torsion-valued deformation-invariant of SQFTs that is more sensitive than the elliptic genus.
The motivation for the puzzle comes from the theory of Topological Modular Forms (TMF) described for example in [@MR1989190; @MR3223024]. Based on suggestions from [@MR992209], it is conjectured in [@MR2079378; @MR2742432] that every SQFT defines a class in TMF, invariant under deformations of the SQFT. Indeed, [@MR2079378; @MR2742432] conjecture that this TMF class exactly captures the deformation class of the corresponding SQFT. The TMF-valued invariant of an SQFT refines the usual modular-form valued elliptic index. It is known that the TMF-valued invariant of $S^3$ (with minimal nonzero B-field) has exact order $24$, hence the puzzle.
In this paper we will propose a solution to this puzzle. Let ${\mathcal B}$ be an SCFT which, like the one in Puzzle \[mainpuzzle\], has gravitational anomaly $c_R - c_L = 3/2$.[^1] If ${\mathcal B}$ is not initially conformal, flow it to the IR before proceeding. Build:
1. a “generalized mock modular form $f_1$” with source equal to the torus one-point function of the supersymmetry generator of ${\mathcal B}$.[^2] The $q$-expansion of $f_1$ is not determined by ${\mathcal F}$, but the class of $[f_1] \in {{\mathds}C}(\!(q)\!) / {\mathrm{MF}}_2$ is.
2. a nonnegative-integral $q$-series $f_2$ equal to half the graded dimension of the space of [bosonic]{} Ramond-sector ground states in ${\mathcal B}$.[^3] The class $[f_2] \in {{\mathds}C}(\!(q)\!) / 2{{\mathds}Z}(\!(q)\!)$ is a sort of “mod-2 index” of ${\mathcal F}$.
Neither class $[f_1] \in {{\mathds}C}(\!(q)\!) / {\mathrm{MF}}_2$ nor $[f_2] \in {{\mathds}C}(\!(q)\!) / 2{{\mathds}Z}(\!(q)\!)$ is a deformation invariant of ${\mathcal B}$. But we will argue that the class $[f_1] - [f_2] \in {{\mathds}C}(\!(q)\!) / [ {\mathrm{MF}}_2 + 2{{\mathds}Z}(\!(q)\!)]$ is invariant under SQFT deformations. Furthermore, we will compute that for the SQFT in Puzzle \[mainpuzzle\], this invariant is nonzero and in fact has exact order $24$ in ${{\mathds}C}(\!(q)\!) / [ {\mathrm{MF}}_2 + 2{{\mathds}Z}(\!(q)\!)]$. This is our solution to Puzzle \[mainpuzzle\].
Homotopy-theoretic considerations imply the existence (and nontriviality) of an invariant like ours [@DBE2015], which when restricted to sigma models is described both analytically and geometrically in [@MR3278648] (where it is also shown that the invariant of $S^3$ has exact order $24$). But topological arguments do not explain how to compute this invariant of SQFTs except when the SQFT can be deformed to a sigma model. Our description of the invariant connects it explicitly to holomorphic anomalies of noncompact SQFTs, and thereby to mock modularity, which is of great current interest due to the “moonshine” of [@MR2802725; @MR3271175].
Whenever the CFT ${\mathcal B}$ is rational, the source of the holomorphic anomaly equation is a modular-invariant bilinear combination of vector-valued holomorphic and anti-holomorphic modular forms. The corresponding generalized mock modular form is then a “mixed mock modular form”: a bilinear combination of the same vector-valued holomorphic modular form and a true vector-valued mock modular form. In such a situation, our arguments can be seen as a justification for the very existence of mock modular forms with interesting integrality properties.
The paper is structured as follows.
Section \[sec.general\] presents the general story. We first make some brief remarks about gravitational anomalies, and a $\eta(\tau)^n$ normalization factor that we include in the elliptic genus[^4] and in one-point functions in order to correct the multipliers that would otherwise be present. We then discuss the properties of certain SQFTs which violate the compactness constraint in a controllable manner. It turns out to be still possible to define the elliptic genus of such SQFTs [@Eguchi:2006tu; @MR2821103; @Ashok:2013pya; @Gupta:2017bcp]. Such elliptic genus satisfies a “holomorphic anomaly” equation with a source which we characterize in a precise manner in terms of the torus one-point function of the supersymmetry generator in a compact “boundary SQFT”. We use this construction to argue that the torus one-point function of the supersymmetry generator in a nullhomotopic SQFT is the source of a holomorphic anomaly equation for a generalized mock modular form. We then argue that the coefficients of this generalized mock modular form are (even) integral, up to a correction arising as a type of “mod 2 index” of the boundary SQFT. It follows that, if solutions of the holomorphic anomaly equation fail to have integral (plus correction) mock modular parts, the SQFT cannot be nullhomotopic. This is the justification for our invariants. We then recast our invariant in homotopy-theoretic terms, where it becomes the “secondary invariant” of [@MR3278648].
To illustrate our proposed invariant, we focus on two families of examples. First, in Section \[sec.cigar\], we study the sigma models with target $S^1$ and the “cigar.” We start by reviewing the $S^1$ sigma models, with an emphasis on the role that the target-space spin structure plays on the behaviour of the model. This provides a chance to illustrate the mod-2 index, and allows us to compute our invariant for the $T^3$ sigma model with its Lie group framing. We then analyze the “cigar,” which is a noncompact manifold with “boundary” $S^1$, and demonstrate explicitly that the corresponding sigma model enjoys our predicted holomorphic anomaly and integrality.
The second set of examples, which we study in Section \[sec.S3\], are the ones from [@WittenTMF]: the ${\mathcal N}=(0,1)$ sigma model with target the round $S^3$ and with B-field of strength $k$. We first warm up with the $k=1$ case of Puzzle \[mainpuzzle\], and then study the general case. In all cases, we find that our invariant is precisely $k \pmod {24}$, showing that the $S^3_k$ sigma model is nullhomotopic if and only if $k = 0 \pmod {24}$. We then mention a few related constructions and puzzles: we build an antiholomorphic SCFT of central charge $c_R = 27/2$ which we expect to represent the 3-torsion element in $\pi_{27}\mathrm{TMF}$; and we speculate that $S^3_k$ is “flavoured-nullhomoptic” for all $k$, with the nullhomotopy given by a certain “trumpet” geometry with ${\mathcal N}=(0,4)$ supersymmetry.
Let us end this introduction by emphasizing that we expect our invariant captures only some of the torsion in the space of ${\mathcal N}=(0,1)$ SQFTs.[^5] It is known that the TMF classes represented by the group manifolds (with their Lie group framings) $$\mathrm{Sp}(2),\qquad G_2,\qquad G_2\times U(1)$$ are nonzero: their exact orders are, respectively, $$3, \qquad 2, \qquad 2.$$ The same logic as in [@WittenTMF] suggests that the ${\mathcal N}=(0,1)$ sigma models for $\mathrm{Sp}(2)$ and $G_2$ flow in the IR to SCFTs consisting purely of ($10$ and $14$, respectively) antichiral free fermions, with supersymmetries that encode the structure constants of the Lie algebras $\mathfrak{sp}(2)$ and $\mathfrak{g}_2$.[^6] The sigma model with target $G_2\times U(1)$ does not flow to a purely-antichiral theory, but rather to the product of the $\mathfrak{g}_2$-theory and the “standard” circle theory studied in §\[subsubsec.S1-nonbounding\]. The elliptic and mod-2 indexes of all three SQFTs vanish. Moreover, the invariant described in this paper vanishes for all three SQFTs — the first two for degree reasons, but the third nontrivially. Due to the expected relationship between TMF and SQFTs, we expect that these SQFTs are not nullhomotopic. We leave the reader with the following puzzles:
\[sp2puzzle\]
1. Show that the SQFT $\overline{\operatorname{Fer}}(\mathfrak{sp}(2))$ of $10$ antichiral free fermions and supersymmetry encoding the structure constants of $\mathfrak{sp}(2)$ is not nullhomotopic.
2. Show that the SQFT $\overline{\operatorname{Fer}}(\mathfrak{g}_2)$ of $14$ antichiral free fermions and supersymmetry encoding the structure constants of $\mathfrak{g}_2$ is not nullhomotopic.
3. Show that the product of $\overline{\operatorname{Fer}}(\mathfrak{g}_2)$ with a (standard) $S^1$ sigma model is not nullhomotopic.
A torsion invariant of SQFTs {#sec.general}
============================
Gravitational anomalies and spectator fermions {#sec.anomaly}
----------------------------------------------
For a quantum field theory to be [*gravitationally nonanomalous*]{}, its partition function must be valued in numbers (as opposed to a section of some line bundle on the moduli space of spacetimes); it must have a well-defined Hilbert space (as opposed to a section of some gerbe); and so on for higher-codimensional data [@FreedTeleman2012]. In the $(1+1)$-dimensional SQFT case, the [*elliptic genus*]{} $Z_{RR}$ is by definition the partition function on flat tori with nonbounding, aka Ramond–Ramond aka periodic–periodic, spin structures. The moduli space of Ramond–Ramond flat tori is three-real-dimensional — in addition to the complex and anticomplex parameters $(\tau,\bar\tau)$, there is also a “size” parameter — but we will generally compute in the IR aka large-torus limit. In this limit, the partition function of a nonanomalous $(1+1)$-dimensional SQFT will definitely be modular for the full $\mathrm{SL}(2,{{\mathds}Z})$ with weight $(0,0)$ and no multiplier.[^7] Indeed, modularity is transparent from the path-integral description.[^8]
The SQFT in Puzzle \[mainpuzzle\], and more generally any ${\mathcal N}=(0,1)$ sigma model, suffers a gravitational anomaly due to the unpaired antichiral fermions. For an SCFT, the gravitational anomaly is the difference between the left- and right-moving central charges; for an SQFT, these separate central charges are not well-defined, but the total gravitational anomaly is, and is preserved under RG flow. We will normalize the gravitational anomaly so that for an SCFT with left and right moving central charges $c_L$ and $c_R$, the anomaly is $n := 2(c_R - c_L)$. The factor of $2$ is natural because then $n$ ranges over ${{\mathds}Z}$. The gravitational anomaly $n \in {{\mathds}Z}$ plays the role of homotopical degree in §\[sec.BN\], and so we will occasionally refer to it as the [*degree*]{} of the SQFT.
The gravitational anomaly manifests in various ways. First of all, it produces a nontrivial multiplier for the behaviour of the elliptic genus under the $\tau \mapsto \tau+1$, namely $Z_{RR}(\tau+1,\bar\tau+1) = e^{-2\pi i n/24} Z_{RR}(\tau,\bar\tau)$; the path integral description still guarantees that $Z_{RR}$ transforms under $\tau \mapsto -1/\tau$ with weight $(0,0)$ and some multiplier. Second, the gravitational anomaly leads to an ambiguity in the parity of “the” Ramond sector of the theory. This leads to a sign ambiguity when trying to define “the” elliptic index.
Our convention, standard in algebraic topology, will be to trade nontrivial multipliers for nontrivial weights of modular forms, by including a normalization factor of $\eta(\tau)^n$ whenever necessary, where $\eta(\tau) = q^{1/24} \prod_{j=1}^\infty (1-q^j)$ is Dedekind’s eta function. The combination $\eta(\tau)^n Z_{RR}(\tau,\bar\tau)$ is sometimes called the [*Witten genus*]{} of a gravitationally-anomalous SQFT, and we will use that term. It is automatically modular without multiplier, of weight $(\frac n 2, 0)$.
When $n>0$, the Witten genus can be interpreted as follows. Consider the nonanomalous SQFT $\operatorname{Fer}(n) \otimes {\mathcal F}$, where $\operatorname{Fer}(n) = \operatorname{Fer}(1)^{\otimes n}$ means the holomorphic CFT of $n$ chiral fermions $\psi_1,\dots,\psi_n$, acted upon trivially by the right-moving supersymmetry. Deformations of ${\mathcal F}$ correspond to deformations of $\operatorname{Fer}(n) \otimes {\mathcal F}$ which preserve the $\operatorname{Fer}(n)$-subsector. The $n$ free chiral fermions in $\operatorname{Fer}(n)$ are called [*spectators*]{}. Because of the zero modes of the chiral fermions, the plain elliptic genus of $\operatorname{Fer}(n) \otimes {\mathcal F}$ vanishes. But because we have a distinguished $\operatorname{Fer}(n) \subset \operatorname{Fer}(n) \otimes {\mathcal F}$, we find a distinguished observable, namely ${:}\psi_1\cdots\psi_n{:}$. The Witten genus of the gravitationally-anomalous SQFT ${\mathcal F}$ is precisely the one-point function of $(-1)^{\frac{n}{4}} {:}\psi_1\cdots\psi_n{:}$ in $\operatorname{Fer}(n) \otimes {\mathcal F}$. [^9] By “one-point function,” we will always mean the torus one-point function on nonbounding, aka Ramond–Ramond, tori.
The spectator fermions furthermore allow the sign ambiguity to be handled by demanding that $\operatorname{Fer}(n) \otimes {\mathcal F}$ have *trivialized* anomaly, including a choice of parity for its Ramond-sector ground state.[^10] More precisely, the sign ambiguity can be swapped for an ambiguity in the choice of generators of $\operatorname{Fer}(n)$. We will largely ignore the sign ambiguity in this paper, since we will not try to add different SQFTs together (and so we will never risk thinking we have found a cancellation when there was not one).
Another phenomenon in gravitationally-anomalous SQFTs becomes transparent when working with spectator fermions. Consider the Ramond-sector Hilbert space ${\mathcal H}_R$ for the spectated theory $\operatorname{Fer}(n) \otimes {\mathcal F}$. The decoupled $\operatorname{Fer}(n)$ subalgebra of the full observable algebra provides operators on this Hilbert space: specifically, ${\mathcal H}_R$ is naturally a module for the fermion zero modes,[^11] which form a copy of the $n$th Clifford algebra $\operatorname{Cliff}(n)$. Moreover, the SQFT $\operatorname{Fer}(n) \otimes {\mathcal F}$ compactified on $S^1$ automatically possesses a time-reversal structure — showing this is an interesting exercise, solved in [@GPPV §3.2.2] — and hence ${\mathcal H}_R$ possesses a real form, acted on by the real Clifford algebra $\operatorname{Cliff}(n,{{\mathds}R})$. Modules for $\operatorname{Cliff}(n,{{\mathds}R})$ represent degree-$n$ classes in oriented K-theory.
Non-compact SQFTs {#sec.noncompact}
-----------------
The discussion in §\[sec.anomaly\] implicitly assumed that the SQFT in question was “compact” — for instance, that its Hamiltonian had sufficiently discrete spectrum — in order for its elliptic genus and K-theory class to be well-defined. There are at least two distinct ways one can enlarge the space of SQFTs which admit some kind of elliptic genus.
### Flavoured-compact theories {#subsec.flavoured-compact}
The first way is to consider SQFTs equipped with a continuous global symmetry, say $U(1)$ for simplicity, and define a “flavoured” elliptic genus as a partition function on a torus equipped with a flat $U(1)$ connection. The flat connection can be parameterized by a point $\xi$ in the elliptic curve $E_\tau$ of parameter $\tau$. As long as the current sits in a standard $(0,1)$ multiplet, there is a superpartner of the anti-holomorphic part of the current which should guarantee the independence of the partition function on the anti-holomorphic part $\bar \xi$ of the connection, leaving a holomorphic dependence on $\xi$. As a consequence, the flavoured Witten genus is a Jacobi form, of weight $\frac{n}{2}$ and index $\ell$ determined by the ’t Hooft anomaly coefficient for the $U(1)$ global symmetry.
Such a flavoured SQFT can be considered “compact” as long as the Hamiltonian on the circle has sufficiently discrete spectrum for a generic choice of flat $U(1)$ connection on the circle. Then the Witten genus will be well-defined as a meromorphic Jacobi form. If the SQFT is compact even in the un-flavoured sense then the Witten genus will be a holomorphic Jacobi form and admit an expansion in terms of theta functions of index $\ell$, with coefficients which form a vector-valued modular form.
For a sigma model, the calculation of the flavoured Witten genus will involve the equivariant analogues of the characteristic classes involved in the calculation of the standard Witten genus.
The canonical example is a $(0,1)$ sigma model with target ${{\mathds}R}^2$ and a $U(1)$ isometry acting as rotations of ${{\mathds}R}^2$. The corresponding flavoured Witten genus is a meromorphic Jacobi form of weight $1$ and index $-1$:
$$\label{eqn.R2}
Z_{RR}({{\mathds}R}^2)[\xi;\tau] = \frac{\eta(\tau)^3}{\theta(\xi;\tau)} =\frac{1}{x^{\frac12}-x^{-\frac12}}\prod_{n=1}^\infty \frac{(1-q^n)^2}{(1-x q^n)(1-x^{-1} q^n)}$$
with $q = \exp 2 \pi i \tau$ and $x = \exp 2 \pi i \xi$. The analogue of TMF for the flavoured SQFTs does not appear to be well-studied [@GPPV]. It would be very interesting to do so.
### Theories with cylindrical ends
The second way we can enlarge the space of SQFTs which admit some kind of elliptic genus is by considering the SQFT analogue of sigma models on manifolds with an asymptotic boundary region which approaches ${{\mathds}R}^+ \times B$ for some compact manifold $B$. We can formalize this notion by requiring the SQFT ${\mathcal F}$ to be equipped with a local operator $\Phi$ with the following property:
- Consider the direct product of ${\mathcal F}$ and a free Fermi multiplet, i.e. a free chiral fermion $\lambda$ annihilated by the supercharge.[^12]
- Deform the product theory by a “fermionic superpotential” $\lambda (\Phi-p)$, i.e. add the terms $\lambda (\bar G_0 \Phi) - (\Phi-p)^2$ to the Lagrangian, where $p \in {{\mathds}R}$ is a parameter.
- The result is a family of SQFTs ${\mathcal B}_p$ parameterized by a point $p$ in ${{\mathds}R}$. We require ${\mathcal B}_p$ to stabilize to some compact SQFT ${\mathcal B}$ for large positive $p$ and ${\mathcal B}_p$ to spontaneously break supersymmetry for large negative $p$.
Note that in particular the family ${\mathcal B}_p$ built from ${\mathcal F}$ is a nullhomotopy of the compact SQFT ${\mathcal B}$. Conversely, any nullhomotopy of ${\mathcal B}$ can be converted into a noncompact SQFT ${\mathcal F}$ by reversing the steps above, i.e. promoting the parameter of the deformation family to a dynamical $(0,1)$ chiral multiplet.
What is the elliptic genus of ${\mathcal F}$? The question is subtle because, being noncompact, ${\mathcal F}$ has continuous spectrum in its Hamiltonian. Because of this, the usual supersymmetric cancelation argument verifying that the elliptic genus is holomorphic breaks down. Let us work in the IR limit. In this limit, the Witten genus $\eta(\tau)^n Z_{RR}({\mathcal F})(\tau,\bar\tau)$ will automatically transform as a weight $(\frac n2,0)$ modular form under the action of $\mathrm{SL}(2,{{\mathds}Z})$ acting simultaneously on both $(\tau,\bar\tau)$, since modularity is manifest from the path-integral description of the index.
Let us try to prove that $Z_{RR}({\mathcal F})$ is holomorphic. We will fail, and by failing we will instead compute the [*holomorphic anomaly*]{} of ${\mathcal F}$. We will try to apply the usual proof of holomorphicity of the index. For any QFT,[^13] $$\frac{\partial}{\partial\bar{\tau}}Z_{RR}({\mathcal F}) = - 2\pi i (\text{torus one-point function of } \bar{T} \mbox{ in ${\mathcal F}$}).$$ Use the supersymmetry: $\bar{T} = \frac12 [\bar{G}_0,\bar{G}]$, where the commutator $[,]$ means the supercommutator, $\bar{G}_0$ is the generator of ${\mathcal N}=(0,1)$ supersymmetry and $\bar{G}$ the anti-holomorphic component of the supercurrent. In a compact theory, the one-point function of any anti-commutator $[\bar{G}_0, O]$ would vanish. In a non-compact theory, with non-compact direction parameterized by the expectation value of the operator $\Phi$, we can imagine integrating by parts in the space of fields to obtain a term proportional to[^14] the torus one-point function of $O$ evaluated in the boundary theory ${\mathcal B}$, resulting in the following [*holomorphic anomaly equation*]{}. Including the spectator fermions to fix the normalizations, we propose: [^15] $$\begin{gathered}
\label{eqn.holomorphicanomaly}
\sqrt{-8\tau_2} \frac{\partial}{\partial\bar{\tau}} \bigl[\text{Witten genus of }{\mathcal F}\bigr] \\
=(\text{torus one-point function of } (-1)^{\frac{n}{4}}{:}\psi_1\cdots\psi_{n-1}\bar{G}{:} \mbox{ in $\operatorname{Fer}(n-1) \times {\mathcal B}$})\end{gathered}$$ Here $\tau_2 = \frac1{2i}(\tau - \bar\tau)$ is the imaginary part of $\tau$. The sign of the square root $\sqrt{-8}$ is essentially arbitrary, and can be absorbed in the ambiguity in the sign of $\bar{G}$ or in the sign of the Ramond sector of ${\mathcal B}$. As a reality check, note that, in the IR limit, both sides are real-analytic modular of weight $(\frac{n-1}2, \frac32)$ with trivial multipliers.
Being a bit loose with the phase of the torus one-point function, we can write equation (\[eqn.holomorphicanomaly\]) as $$\label{eqn.holomorphicanomaly2}
\sqrt{-8\tau_2} \, \eta(\tau) \frac{\partial}{\partial\bar{\tau}} Z_{RR}({\mathcal F}) = (\text{torus one-point function of } \bar{G} \mbox{ in ${\mathcal B}$}).$$
A simple generalization of this construction is to require ${\mathcal B}_p$ to stabilize to some compact SQFT ${\mathcal B}_+$ for large positive $p$ and to another compact SQFT ${\mathcal B}_-$ for large positive $p$ for large negative $p$. Then we expect, up to a phase factor: $$\begin{gathered}
\label{eqn.bordismZRR}
\sqrt{-8\tau_2} \frac{\partial}{\partial\bar{\tau}}\bigl[\text{Witten genus of }{\mathcal F}\bigr] = \bigl[ (\text{torus one-point function of } {:}\psi_1\cdots\psi_n{:}\bar{G} \mbox{ in ${\mathcal B}_+$}) \\ -(\text{torus one-point function of } {:}\psi_1\cdots\psi_n{:}\bar{G} \mbox{ in ${\mathcal B}_-$}) \bigr]\end{gathered}$$ This means that, although the torus one-point function of $\bar{G}$ is not a deformation-invariant of an SQFT, it changes in a controlled fashion. This control is the basis of our torsion invariant.
Integrality of the $q$-expansion {#sec.int}
--------------------------------
In addition to holomorphicity, the other fundamental fact about the Witten genus $\eta^n Z_{RR}({\mathcal F})$ of a (compact) SQFT ${\mathcal F}$ is the integrality of its $q$-expansion. We briefly review the argument. Let ${\mathcal F}[S^1]$ denote the $S^1$-equivariant supersymmetric quantum mechanics model produced by compactifying ${\mathcal F}$ on a circle (with Ramond aka nonbounding spin structure). Then $\eta^n Z_{RR}({\mathcal F})$ can be interpreted as the supersymmetric index of ${\mathcal F}[S^1]$, with $q$ parameterizing the $S^1$-action, and indices are well-known to be integral, since they merely count with signs the number of supersymmetric ground states. Note that the compactification breaks manifest modularity. Indeed, suppose we didn’t already know that $Z_{RR}({\mathcal F})$ was holomorphic (for ${\mathcal F}$ compact). Then this compactification implements the canonical way to extract a holomorphic function from a real-analytic modular form: analytically continue away from $\bar \tau = \tau^*$ and take the limit $\bar \tau \to -i \infty$.[^16]
Depending on the value of the gravitational anomaly $n$, one can make stronger statements than mere integrality. The $q$-dependence is immaterial — the statements hold in general for SQM models of degree $n$ equipped with a time-reversal symmetry.[^17] There are various ways to define the notion of “degree-$n$ SQM model”, just like the choices in §\[sec.anomaly\] for how to handle the gravitational anomaly. One general way is to work with SQM models that are relative, in the sense of [@FreedTeleman2012], to certain short-range-entangled $(1+1)$-dimensional phases. When $n\geq0$, another explicit method is to employ spectator fermions. Then a [*degree-$n$ SQM model*]{} is an SQM model (i.e. a super Hilbert space ${\mathcal H}$ with an odd operator $G$ generating the supersymmetry; it is “compact” when $G$ is Fredholm) equipped with an action by the $n$th Clifford algebra $\operatorname{Cliff}(n)$ (which should (super)commute with $G$). The presence of a time-reversal symmetry equips ${\mathcal H}$ with a real structure, acted on by the real Clifford algebra $\operatorname{Cliff}(n,{{\mathds}R})$. The supersymmetric ground states are then a finite-dimensional $\operatorname{Cliff}(n,{{\mathds}R})$-module $V$.[^18]
The usual [supersymmetric index]{} of the SQM model ignores the time-reversal symmetry: it depends just on $V\otimes {{\mathds}C}$ as a $\operatorname{Cliff}(n,{{\mathds}C})$-module. When $n$ is even, $\operatorname{Cliff}(n,{{\mathds}C})$ has two irreducible modules, differing by parity. Choose one of them arbitrarily to be “the” irrep $I$; then $V \otimes {{\mathds}C}\cong I^{a|b} = I \otimes_{{\mathds}C}{{\mathds}C}^{a|b}$, where ${{\mathds}C}^{a|b}$ means the (complex) supervector space with graded dimension $(a,b)$. The ordinary [*index*]{} of $V$ is simply $a-b$.[^19] But in the presence of a time-reversal symmetry, we don’t just have the $\operatorname{Cliff}(n,{{\mathds}C})$-module $V \otimes {{\mathds}C}$ — we have the $\operatorname{Cliff}(n,{{\mathds}R})$-module $V$. It turns out that when $n = 2 \pmod 4$, there is only one irreducible $\operatorname{Cliff}(n,{{\mathds}R})$-module $J$, with complexification $J \otimes {{\mathds}C}\cong I^{1|1}$. Thus the index vanishes when $n = 2 \pmod 4$. And when $n = 4 \pmod 8$, there are two irreducible $\operatorname{Cliff}(n,{{\mathds}R})$-modules, $J^{1|0}$ and $J^{0|1}$, but $J^{1|0} \otimes {{\mathds}C}= I^{2|0}$ splits as two copies of the irreducible $\operatorname{Cliff}(n,{{\mathds}C})$-module, and so the index is automatically even. In summary, the index of a degree-$n$ SQM model with time reversal symmetry lives in $m{{\mathds}Z}$ with: $$\label{eqn.mofn}
m = \begin{cases}
1, & n = 0 \pmod 8, \\
2, & n = 4 \pmod 8, \\
0, & \text{else}.
\end{cases}$$ In the SQFT case that we care about, the Witten genus $\eta^n Z_{RR}({\mathcal F})$ has $q$-expansion in $m{{\mathds}Z}(\!(q)\!)$.[^20]
### The mod-2 index
That the indexes of time-reversal SQM models vanish in degrees $2$ and $6$ mod $8$ and are even in degree $4$ mod $8$ is compensated by a more refined “mod-2 index,” which is nontrivial in degrees $n=1$ and $2 \pmod 8$. We will review its construction because it already measures some torsion in the space of SQFTs and because a variation of the mod-2 index appears when trying to understand indexes of noncompact SQM models. We start with the case $n=1$. The even subalgebra of $\operatorname{Cliff}(1,{{\mathds}R})$ is isomorphic to ${{\mathds}R}$, and $\operatorname{Cliff}(1,{{\mathds}R})$ has a unique irreducible module, namely itself. We will call it ${{\mathds}R}^{1|1}$. The supersymmetric ground states $V$ of a degree-$1$ SQM model is then isomorphic to ${{\mathds}R}^{a|a}$ for some nonnegative integer $a$. By definition, the [*mod-2 index*]{} of the SQM model is $a \pmod 2$.
Although the integer $a$ is not a deformation invariant of the degree-$1$ SQM model, the mod-2 index $a\pmod 2$ is. To explain why, we can repeat the logic from §\[sec.noncompact\] to reinterpret a deformation of a degree-$n$ SQM model as a mildly-noncompact degree-$(n+1)$ SQM model: promote the deformation parameter to a dynamical supersymmetric multiplet; the fermion in this multiplet contributes $+1$ to the degree of the model. The upshot is that any deformation of a degree-$n$ SQM model will add or subtract to the ground states $V$ some $\operatorname{Cliff}(n+1,{{\mathds}R})$-module (thought of as a $\operatorname{Cliff}(n,{{\mathds}R})$-module).
But the even subalgebra of $\operatorname{Cliff}(2,{{\mathds}R})$ is isomorphic to ${{\mathds}C}$ thought of as a real algebra.[^21] Because ${{\mathds}C}$ is a field, $\operatorname{Cliff}(2,{{\mathds}R})$ is irreducible as a module over itself. We will call this irreducible module ${{\mathds}C}^{1|1}$. It has even graded dimension when restricted to $\operatorname{Cliff}(1,{{\mathds}R})$, since $\dim_{{\mathds}R}{{\mathds}C}$ is even, and so adding or subtracting it to $V = {{\mathds}R}^{a|a}$ will not change the value of $a \pmod 2$. This is why the mod-2 index of a degree-$1$ SQM model is a deformation invariant.
The same logic also defines a deformation-invariant mod-2 index of a degree-$2$ SQM model. The ground states $V$ are isomorphic, as a $\operatorname{Cliff}(2,{{\mathds}R})$-module, to ${{\mathds}C}^{a|a}$ for some integer $a$, and the [*mod-2 index*]{} is $a \pmod 2$. A deformation will involve adding or subtracting from $V$ the underlying $\operatorname{Cliff}(2,{{\mathds}R})$-module of some $\operatorname{Cliff}(3,{{\mathds}R})$-module. The even subalgebra of $\operatorname{Cliff}(3,{{\mathds}R})$ is isomorphic to the quaternion algebra ${{\mathds}H}$. Because ${{\mathds}H}$ is a skew field, $\operatorname{Cliff}(3,{{\mathds}R})$ is irreducible as a module over itself. Call this irreducible module ${{\mathds}H}^{1|1}$. Since $\dim_{{\mathds}C}{{\mathds}H}$ is even, if you add or subtract some multiple of ${{\mathds}H}^{1|1}$ to ${{\mathds}C}^{a|a}$, you do not change the value of $a \pmod 2$.
The story repeats when $n = 1$ or $2 \pmod 8$, because the category of $\operatorname{Cliff}(n,{{\mathds}R})$-supermodules depends only on the value of $n \pmod 8$. What about when $n = 3$? The ground states $V$ for an SQM model are then isomorphic to ${{\mathds}H}^{a|a}$ for some nonnegative integer $a$, and so we may still contemplate a [*mod-2 index*]{} defined to be the value of $a \pmod 2$. But now this mod-2 index is not a deformation invariant because $\operatorname{Cliff}(4,{{\mathds}R})$ is not irreducible over itself. In fact, both irreps of $\operatorname{Cliff}(4,{{\mathds}R})$ restrict over $\operatorname{Cliff}(3,{{\mathds}R})$ to copies of ${{\mathds}H}^{1|1}$, and so only the dataless “$a \pmod 1$” is a deformation invariant. The same is true when $n = 5$, $6$, or $7 \pmod 8$.
### Noncompact SQM models
We turn now to the index of “mildly noncompact” SQM models. The definition of “mild noncompactness” mirrors §\[sec.noncompact\]: the SQM model ${\mathcal X}$ should come with a local operator $\Phi$ parameterizing the noncompact direction; writing ${\mathcal Y}_p$ for the theory produced from ${\mathcal X}$ by adding a fermion $\lambda$ and a fermionic superpotential $\lambda(\Phi-p)$, we demand that ${\mathcal Y}_p$ stabilizes in the limits $p \to \pm \infty$ to compact SQM models ${\mathcal Y}_\pm$. For definiteness, we will first describe the case when ${\mathcal X}$ has degree $n=4$, in which case each ${\mathcal Y}_p$ is an SQM model of degree $3$.
We lose no generality by assuming that as $p$ varies, the Hilbert space ${\mathcal H}$ of ${\mathcal Y}_p$ is independent of $p$, with the only variation being in the choice of supersymmetry.[^22] Since ${\mathcal Y}_p$ has degree $3$, ${\mathcal H}$ is naturally a module for $\operatorname{Cliff}(3,{{\mathds}R})$. Choose[^23] an isomorphism $\operatorname{Cliff}(3,{{\mathds}R}) \cong {{\mathds}H}\otimes \operatorname{Cliff}(-1,{{\mathds}R})$, and let $\gamma$ denote the generator of $\operatorname{Cliff}(-1,{{\mathds}R})$. The supersymmetry generator in ${\mathcal Y}_p$ is $G(p) = g(p) \gamma$, where $g(p)$ is a quaternionic matrix; the time-reversal structure for SQM models of degree $3$ requires that $g(p)$ be “quaternionically self-adjoint,” meaning that its eigenvalues live in ${{\mathds}R}\subset {{\mathds}H}$. Thus, after a $p$-dependent change of basis, the only thing varying with $p$ is the spectrum of $g(p)$, which by compactness is a discrete subset (with finite multiplicities) of ${{\mathds}R}$.
The “index” of a noncompact SQM model ${\mathcal X}$ can be defined as the Ramond partition function $Z_R({\mathcal X}) = \mathrm{Tr}_{\mathcal H}(-1)^F \cdots$, but the noncompactness means that this partition function will depend nontrivially on the length of the worldline torus. The limit $\bar\tau \to -i\infty$ corresponds to the IR limit of $Z_R({\mathcal X})$, which merely counts supersymmetric ground states. We will use the term “index” to mean this IR limit.
If the limits $g(\pm\infty) = \lim_{p \to \pm \infty} g(p)$ have no kernel, then the index (in the IR sense) of ${\mathcal X}$ is relatively easy to compute: it counts with signs the number of eigenvalues of $g$ that cross $0$, times a factor of 2 coming from the quaternionic nature of degree-$3$ and degree-$4$ SQM models. Indeed, to say that the limits $g(\pm\infty)$ have no kernel is to say that supersymmetry is spontaneously broken in these limits, and ${\mathcal X}$ wasn’t really “noncompact” at all, because it flows to a compact theory, and the factor of $2$ is the one coming from (\[eqn.mofn\]) when $n=4$.
If, on the other hand, supersymmetry is not spontaneously broken in the boundary theories ${\mathcal Y}_\pm$, then the index of ${\mathcal X}$ receives a fractional contribution from each eigenvalue that lands on $0$ in the limits $p \to \pm \infty$. After multiplying by the factor of $2$ coming from (\[eqn.mofn\]), we find that the index of ${\mathcal X}$ is odd depending on the number of supersymmetric ground states in the boundary theories ${\mathcal Y}_\pm$. And this number is exactly the non-deformation-invariant mod-2 index of ${\mathcal Y}_\pm$!
The same argument holds whenever ${\mathcal X}$ has degree $n = 4 \pmod 8$. When $n = 0 \pmod 8$, a similar argument holds without a factor of $2$. To give a unified formula, we complexify and strip off the spectator fermions. Then, after complexifying, the supersymmetric ground states in ${\mathcal Y}_\pm$ form a vector space isomorphic to ${{\mathds}C}^{a|a}$, with $a \in m{{\mathds}Z}$, and the mod-2 index is $\frac a m \pmod 2$. We will call this number $a$ the “bosonic index” of ${\mathcal Y}_\pm$. The end result is: $$\label{eqn.mod2index.n}
\operatorname{Index}({\mathcal X}) \in \frac 1 2 (\text{bosonic index of }{\mathcal Y}_+) - \frac 1 2 (\text{bosonic index of }{\mathcal Y}_-) + m{{\mathds}Z},$$ where the degree $n$ of ${\mathcal X}$ is divisible by $4$ and $m$ depends on $n$ via (\[eqn.mofn\]).
In the non-IR, the more general statement identifies the failure of $Z_R({\mathcal X})$ to be integral with a version of the $\eta$ invariant of [@MR0331443].
The invariant {#sec.invariant}
-------------
Summarizing the previous two sections, let ${\mathcal B}$ be a compact SCFT[^24] with gravitational anomaly $2(c_R - c_L) = n-1$, where $n$ is divisible by $4$. Suppose that ${\mathcal B}$ is nullhomotopic. Then we can use the nullhomotopy to build an SQFT ${\mathcal F}$ with gravitational anomaly $n$ with one noncompact direction and boundary ${\mathcal B}$. Let $\hat{f}(\tau,\bar\tau)$ denote the Witten genus of ${\mathcal F}$. Then:
1. $\hat{f}$ is real-analytic modular of weight $(\frac n 2,0)$. It solves a [*holomorphic anomaly equation*]{} $$\label{eqn.f1}
\sqrt{-8\tau_2} \frac\partial{\partial \bar\tau} \hat f = g(\tau,\bar\tau)$$ where $$\label{eqn.g1}
g(\tau,\bar\tau) = (\text{torus one-point function of } (-1)^{\frac{n}{4}}{:}\psi_1\cdots\psi_{n-1}\bar{G}{:} \mbox{ in $\operatorname{Fer}(n-1) \otimes {\mathcal B}$}).$$
2. The holomorphic part $f(\tau) = \lim_{\bar\tau \to -i\infty} \hat{f}(\tau,\bar\tau)$ has $q$-expansion $$\label{eqn.f2}
f \in f_2(q) + m{{\mathds}Z}(\!(q)\!)$$ where $m$ depends on $n$ via (\[eqn.mofn\]) and $$\label{eqn.g2}
f_2(q) = \frac12 \bigl(\text{bosonic index of } {\mathcal B}[S^1]\bigr).$$
In particular, if ${\mathcal B}$ is nullhomotopic, then there is a function $\hat f$ solving equations (\[eqn.f1\]) and (\[eqn.f2\]).
Conversely, suppose we suspect that ${\mathcal B}$ is not nullhomotopic. Equations (\[eqn.g1\]) and (\[eqn.g2\]) depend on ${\mathcal B}$ alone. We can encode the data of $g$ by finding some (real-analytic modular of weight $(\frac n 2,0)$) solution $\hat{f}_1$ to (\[eqn.f1\]). There is an ambiguity in the choice of solution, given precisely by the holomorphic modular forms of weight $\frac n 2$. Let $f_1$ denote the holomorphic part of $\hat{f}_1$. Then $g$, and hence ${\mathcal B}$, determines the class of $f_1$ in $$[f_1] \in \frac{{{\mathds}C}(\!(q)\!)}{{\mathrm{MF}}_{n/2}}.$$ The theory ${\mathcal B}$ determines the $q$-series $f_2 \in \frac m 2 {{\mathds}Z}(\!(q)\!)$ exactly, but we will use only its class in $$[f_2] \in \frac{{{\mathds}C}(\!(q)\!)}{m{{\mathds}Z}(\!(q)\!)}.$$
As we have seen already, neither of these classes is separately a deformation invariant of ${\mathcal B}$. But we claim that the class $$[f_1] - [f_2] \in \frac{{{\mathds}C}(\!(q)\!)}{ {\mathrm{MF}}_{n/2} + m{{\mathds}Z}(\!(q)\!)} = A_n$$ is a deformation invariant.[^25] Indeed, suppose that ${\mathcal B}$ can be deformed to some other SQFT ${\mathcal B}'$. Then we can build from the deformation a mildly noncompact SQFT ${\mathcal F}$ with boundaries ${\mathcal B}_- = {\mathcal B}$ and ${\mathcal B}_+ = {\mathcal B}'$. Let $f_1$ and $f_2$ denote the $q$-series for ${\mathcal B}$ as defined above, and $f_1'$ and $f_2'$ the corresponding $q$-series for ${\mathcal B}'$. Then $f_1'$ and $f_1 + Z_{RR}({\mathcal F})$ solve the same holomorphic anomaly equation, and so differ by an element of ${\mathrm{MF}}_{n/2}$, whereas $f_2'$ and $f_2 + Z_{RR}({\mathcal F})$ differ by an element in $m{{\mathds}Z}(\!(q)\!)$. This verifies that $$f_1 - f_2 = f_1' - f_2' \mod {\mathrm{MF}}_{n/2} + m{{\mathds}Z}(\!(q)\!).$$
Thus, to solve a puzzle like Puzzle \[mainpuzzle\] or the other related puzzles from [@WittenTMF], it suffices to calculate this invariant $[f_1] - [f_2]$ for the possibly-nullhomotopic SQFT ${\mathcal B}$. In Sections \[sec.cigar\] and \[sec.S3\] we will study examples with ${\mathcal B}$ of degree $3$ (so in the above notation $n=4$ and $m=2$) where $$f_1(q) - f_2(q) = \alpha q^0 \mod {\mathrm{MF}}_{2} + 2{{\mathds}Z}(\!(q)\!),$$ for some rational number $\alpha \in {{\mathds}Q}$. Such examples ${\mathcal B}$ are definitely not nullhomotopic if $\alpha \not\in 2{{\mathds}Z}$. Indeed, it suffices to show that $\alpha q^0 \not\in {\mathrm{MF}}_{2} + 2{{\mathds}Z}(\!(q)\!)$ for $\alpha \not\in 2{{\mathds}Z}$, or equivalently that no weight-2 modular form has $q$-expansion in $\alpha + 2{{\mathds}Z}(\!(q)\!)$. But a weight-2 modular form is determined by its polar part, which would have to be even, but then the constant term would also have to be even.
Relation to secondary invariants of Bunke–Naumann {#sec.BN}
-------------------------------------------------
The $A_n$-valued invariant of SQFTs that we have constructed is intentionally modelled on the “secondary invariant of the Witten genus” constructed in [@MR3278648]. That paper builds, for any $(n-1)$-dimensional manifold $M$ equipped with String structure[^26] with $n$ divisible by $4$, an invariant valued in the group $A_n$ above (called $T_{n/2}$ in [@MR3278648]); they give topological, geometric, and analytic descriptions of the invariant. Their invariant is a cobordism invariant, meaning that string-cobordant manifolds receive the same value. It is in a precise sense a “derived invariant” of the topologically-defined Witten genus.
This secondary invariant has a simple purely homotopy-theoretic description (our exposition follows [@DBE2015]). Let $\mathrm{HMF}_\bullet$ denote ordinary (de Rham) cohomology with coefficients in the graded ring ${\mathrm{MF}}$ of complex-valued modular forms, graded so that ${\mathrm{MF}}_{n/2}$ has cohomological degree $-n$.[^27] Write $\mathrm{H}{{\mathds}C}(\!(q)\!)_\bullet$ for $4$-periodicized ordinary cohomology with ${{\mathds}C}(\!(q)\!)$-coefficients. Let $\mathrm{KO}_\bullet$ denote orthogonal K-theory, and write $\mathrm{KO}(\!(q)\!)_\bullet$ for $S^1$-equivariant KO. Finally, let $\mathrm{MString}_\bullet$ denote the Thom spectrum of String cobordism. [^28] Then the Witten genus fits into a commutative square of spectra: $$\begin{tikzpicture}[anchor=base]
\path (0,0) node (MString) {$\mathrm{MString}_\bullet$}
+(3,0) node (KO) {$\mathrm{KO}(\!(q)\!)_\bullet$}
+(0,-2) node (HMF) {$\mathrm{HMF}_\bullet$}
+(3,-2) node (HCq) {$\mathrm{H}{{\mathds}C}(\!(q)\!)_\bullet$};
\draw[->] (MString) -- (KO);
\draw[->] (MString) -- (HMF);
\draw[->] (KO) -- (HCq);
\draw[->] (HMF) -- (HCq);
\end{tikzpicture}$$ This square is not a homotopy pullback square. Instead, construct the homotopy fiber product $\mathrm{HMF}_\bullet \times^h_{\mathrm{H}{{\mathds}C}(\!(q)\!)_\bullet} \mathrm{KO}(\!(q)\!)_\bullet$, christened “$\mathrm{KMF}_\bullet$” in [@DBE2015]. Then we automatically find a map: $$\label{KMF-diagram}
\begin{tikzpicture}[anchor=base, baseline=(KMF.base)]
\path (0,0) node (MString) {$\mathrm{MString}_\bullet$}
++(1.5,-1) node (KMF) {$\mathrm{KMF}_\bullet$}
+(.5,-.5) node {$\ulcorner$}
+(3,0) node (KO) {$\mathrm{KO}(\!(q)\!)_\bullet$}
+(0,-2) node (HMF) {$\mathrm{HMF}_\bullet$}
+(3,-2) node (HCq) {$\mathrm{H}{{\mathds}C}(\!(q)\!)_\bullet$};
\draw[->] (MString) -- (KO);
\draw[->] (MString) -- (HMF);
\draw[->] (KMF) -- (KO);
\draw[->] (KMF) -- (HMF);
\draw[->] (KO) -- (HCq);
\draw[->] (HMF) -- (HCq);
\draw[->,dashed] (MString) -- (KMF);
\end{tikzpicture}$$ A short calculation verifies that, for $n$ divisible by $4$, $$\pi_{n-1}\mathrm{KMF}_\bullet \cong A_n,$$ and so some (possible trivial) $A_n$-valued cobordism invariant of $(n-1)$-dimensional String manifolds is automatic. The challenge, solved in [@MR3278648], is to describe the invariant in a useful way.
As emphasized in [@DBE2015], following [@MR885560; @MR970288], the maps $\mathrm{MString}_\bullet \to \mathrm{HMF}_\bullet$ and $\mathrm{MString}_\bullet \to \mathrm{KO}(\!(q)\!)_\bullet$ have natural quantum field theoretic descriptions. Indeed, given a (compact Riemannian) $n$-dimensional String manifold $M$, there is an ${\mathcal N}=(0,1)$ supersymmetric sigma model with target $M$, with gravitational anomaly $n$.[^29] String cobordisms produce deformations of SQFTs [@WittenTMF §3.4]. Such a sigma model has a partition function in ${\mathrm{MF}}$; this defines the map $\mathrm{MString}_\bullet \to \mathrm{HMF}_\bullet$. On the other hand, an SQFT compactified on $S^1$ determines a time-reversal $S^1$-equivariant minimally supersymmetric quantum mechanics model, and so a point in $\mathrm{KO}(\!(q)\!)_\bullet$. [^30]
Let $\mathrm{SQFT}_n$ denote the moduli space of SQFTs with gravitational anomaly $-n$. This is not (as of this writing) a mathematically well-defined moduli space, not least because there is not a sufficient mathematical definition of $(1+1)$-dimensional quantum field theory (and even if $\mathrm{SQFT}_n$ could be defined as a set, correctly topologizing it would be hard). That said, if $\mathrm{SQFT}_n$ were a well-defined moduli space, then the spaces $\mathrm{SQFT}_\bullet$ would fit together into an $E_\infty$ ring spectrum. The adjoint to the suspension map $\Sigma \mathrm{SQFT}_n \to \mathrm{SQFT}_{n+1}$ making $\mathrm{SQFT}_\bullet$ into a spectrum is the map $\mathrm{SQFT}_n \to \Omega\mathrm{SQFT}_{n+1}$ which turns an SQFT ${\mathcal F}$ into the family of SQFTs $t \mapsto {\operatorname{Fer}(1)(t)} \otimes {\mathcal F}$, parameterized by $t \in {{\mathds}R}\cup \{\infty\} \cong S^1$, where $\operatorname{Fer}(1)(t)$ consists of a single chiral fermion $\psi$ with supersymmetry $\psi \mapsto t$.[^31] This map $\mathrm{SQFT}_n \to \Omega\mathrm{SQFT}_{n+1}$ is homotopy invertible: the inverse compiles a family of SQFTs $p \mapsto {\mathcal B}_p$ into an a-priori-noncompact SQFT ${\mathcal F}$ just as in §\[sec.noncompact\], but ${\mathcal F}$ is in fact compact since the elements of $\Omega\mathrm{SQFT}_{n+1}$ are the families such that both limits ${\mathcal B}_{\pm}$ are null.
In summary, conditional on giving a mathematical definition of the spaces $\mathrm{SQFT}_n$, the sigma-model and compactification maps allow us to expand (\[KMF-diagram\]) into: $$\label{SQFT-diagram}
\begin{tikzpicture}[anchor=base,baseline=(KMF.base)]
\path (0,0) node (MString) {$\mathrm{MString}_\bullet$}
++(1.5,-1) node (SQFT) {$\mathrm{SQFT}_\bullet$}
++(1.5,-1) node (KMF) {$\mathrm{KMF}_\bullet$}
+(.5,-.5) node {$\ulcorner$}
+(3,0) node (KO) {$\mathrm{KO}(\!(q)\!)_\bullet$}
+(0,-2) node (HMF) {$\mathrm{HMF}_\bullet$}
+(3,-2) node (HCq) {$\mathrm{H}{{\mathds}C}(\!(q)\!)_\bullet$};
\draw[->] (SQFT) -- (KO);
\draw[->] (SQFT) -- (HMF);
\draw[->] (KMF) -- (KO);
\draw[->] (KMF) -- (HMF);
\draw[->] (KO) -- (HCq);
\draw[->] (HMF) -- (HCq);
\draw[->,dashed] (SQFT) -- (KMF);
\draw[->] (MString) -- (SQFT);
\end{tikzpicture}.$$ This produces in particular a $\pi_{n-1}\mathrm{KMF}_\bullet$-valued invariant of SQFTs with gravitational anomaly $n-1$.
In topology, one can construct a diagram identical to (\[SQFT-diagram\]) except with the mathematically ill-defined spectrum $\mathrm{SQFT}_\bullet$ replaced by the mathematically (although non-geometrically) well-defined spectrum $\mathrm{TMF}_\bullet$ of “topological modular forms.” (For details on TMF, see for example [@MR3223024].) Bunke and Naumann prove in [@MR3278648] that their geometrically-defined invariant of String manifolds does factor through $\mathrm{TMF}_\bullet$. This is no accident: building on [@MR992209], Stolz and Teichner proposed in [@MR2079378; @MR2742432] that indeed $\mathrm{SQFT}_\bullet$ and $\mathrm{TMF}_\bullet$ are homotopy equivalent, in which case the former provides a much-needed geometric model of the latter.
Example: $S^1$ and the cigar {#sec.cigar}
============================
SQM with target $S^1$ {#subsec.SQM}
---------------------
The ${\mathcal N}=(0,1)$ sigma model requires a String structure on the target manifold [@MR796163]: a [*String structure*]{} consists of a spin structure together with a trivialization of the characteristic class on spin manifolds called $\frac{p_1}2$; the choice of trivialization is interpreted in the quantum field theory as a B-field [@MR1748791]. To illustrate the dependence on the spin structure, we warm up by discussing ${\mathcal N}=1$ supersymmetric quantum mechanics with target $S^1$.[^32]
The classical degrees of freedom for ${\mathcal N}=1$ mechanics with target $X$ consist of a boson $x$ valued in $M$ (solving a second-order equation of motion) and a single real fermion $\xi$ valued in the tangent bundle $x^* \mathrm{T}_X$ (solving a first-order equation of motion). The classical phase space is the symplectic supermanifold $\mathrm{T}^* X \times_X \Pi \mathrm {T} X$, where $\Pi \mathrm {T}$ means the parity-reversed tangent bundle. The algebra of quantum observables is the canonical quantization of this space. The cotangent bundle $\mathrm{T}^* X$ quantizes to the algebra of differential operators on $X$, and the odd tangent bundle $\Pi \mathrm {T} X$ quantizes to (the global sections of) a bundle of Clifford algebras $\operatorname{Cliff}(\mathrm{T}_X)$. This algebra does not depend on the spin structure. Rather, the spin structure appears when trying to decide the Hilbert space. The (bosonic) algebra of differential operators has a canonical Hilbert space: the $L^2$ functions on $X$. But to choose a Hilbert space for the Clifford algebra factor, we must use the spin structure to choose a bundle of $\operatorname{Cliff}(\mathrm{T}_X)$-modules.
In the $S^1$ case, the tangent bundle $\mathrm{T}_{S^1}$ is trivial, and so the quantum algebra of observables is $\operatorname{DiffOp}(S^1) \otimes \operatorname{Cliff}(1,{{\mathds}R})$. Write ${{\mathds}R}^{1|1}$ for the irreducible $\operatorname{Cliff}(1,{{\mathds}R})$-module. The nonbounding, aka periodic, spin structure corresponds to the Hilbert space ${\mathcal H}_{\text{nonbounding}} = L^2(S^1) \otimes {{\mathds}R}^{1|1}$. The supersymmetry is diagonalized in the momentum representation $${\mathcal H}_{\text{nonbounding}} \cong L^2({{\mathds}Z}) \otimes {{\mathds}R}^{1|1} \cong \bigoplus_{p\in{{\mathds}Z}} {{\mathds}R}^{1|1}.$$ Write $\gamma \in \operatorname{Cliff}(1)$ for the odd generator and $\hat{p} = \frac{\mathrm d}{{\mathrm d}x}$ for the momentum operator. In the momentum representation, the supersymmetry acts on the $p$th direct summand by $p\gamma$. The supersymmetric ground states are therefore the $p=0$ summand ${{\mathds}R}^{1|1}$. Thus the mod-2 index is $1 \in {{\mathds}Z}/2{{\mathds}Z}$. This being nonzero confirms that the nonbounding spin structure is in fact nonbounding.
For the bounding, aka antiperiodic, spin structure, the sections of the Hilbert space have half-integral Fourier modes, and so in the momentum representation $${\mathcal H}_{\text{bounding}} \cong \bigoplus_{p \in {{\mathds}Z}+\frac12} {{\mathds}R}^{1|1},$$ again with supersymmetry $p\gamma$. Since $p$ now ranges over ${{\mathds}Z}+ \frac12$, there are no supersymmetric ground states — supersymmetry is spontaneously broken — and so the mod-2 index vanishes, as it must for a bounding spin structure.
The $S^1$ sigma models {#subsec.S1}
----------------------
We turn now to the ${\mathcal N}=(0,1)$ sigma model with target $S^1$, with its two possible spin structures. Each spin structure on $S^1$ extends to a unique String structure.
### Nonbounding spin structure, aka the standard circle SCFT {#subsubsec.S1-nonbounding}
Recall that the bosonic sigma model unpacks to a quantum mechanics model in the loop space of $S^1$, which is $$\operatorname{Loops}(S^1) \cong S^1 \times {{\mathds}Z}\times {{\mathds}R}^\infty.$$ The $S^1$ factor records the basepoint of the loop, the ${{\mathds}Z}$ factor records the winding number, and the ${{\mathds}R}^\infty$ factor records the vibrations of the loop. In this quantum mechanics interpretation, the sigma model action unpacks to a quantum mechanics action with a quadratic potential energy. This is diagonalized by using the momentum representation for the $S^1$ factor and the position representation for the ${{\mathds}Z}$ factor; the ${{\mathds}R}^\infty$ factor becomes a stack of harmonic oscillators, and is not particularly interesting for the present discussion. The result is that $${\mathcal H}_{\text{bosonic}} \cong L^2({{\mathds}Z}^2) \otimes (\text{infinitely many harmonic oscillators}).$$
If we choose the nonbounding spin structure, the ${\mathcal N}=(0,1)$ sigma model factors as product of an antichiral free fermion and a standard bosonic sigma model with circle target, just as in the SQM case of §\[subsec.SQM\]: $${\mathcal H}^{(0,1)}_{\text{nonbounding}} \cong {\mathcal H}_{\text{bosonic}} \otimes {{\mathds}R}^{1|1} \otimes (\cdots) \cong \bigoplus_{n,w} {{\mathds}R}^{1|1} \otimes (\dots).$$
The states of the bosonic sigma model carry left- and right-moving momenta $$a_0 = \frac{n}{R} + \frac{w R}{2}, \qquad \qquad \bar a_0 = \frac{w R}{2} -\frac{n}{R},$$ for the free-boson zero-modes, where $R$ is the radius of the target circle and the momentum $n$ and winding $w$ quantum numbers are integral. The momenta fill in an even self-dual lattice of signature $(1,1)$.
The zero-mode contributions to the energy and momenta of the states give $L_0 = \frac12 \left(\frac{n}{R} + \frac{w R}{2} \right)^2$ and $\bar L_0 = \frac12 \left(\frac{n}{R} - \frac{w R}{2} \right)^2$. Also, $\bar G_0$ is proportional to $\bar a_0$. For generic values of the radius $R$ there is a single copy of ${{\mathds}R}^{1|1}$ with $n=w=0$. For rational values of $R^2$ one may have more general ground states with $2n = R^2 w$ and $L_0 = \frac{2 n^2}{R^2}$. That gives a single copy of ${{\mathds}R}^{1|1}$ when $n=0$; two copies of ${{\mathds}R}^{1|1}$ when $n^2 >0$. Thus the mod-2 index is radius-independent and equals $1 \in {{\mathds}Z}_2(\!(q)\!)$. This proves that the $S^1$ sigma model with nonbounding spin structure is not nullhomotopic in the moduli space of SQFTs.
For later reference, we can add a spectator fermion $\psi$ and compute the torus one-point function of $\sqrt{-1} {:}\psi \bar{G}{:}$. In order to get a non-zero answer, we can turn on non-trivial fugacities $y_1$, $y_2$ for the momentum and winding symmetries of the theory.
In order to fix our notations, we write the anti-chiral supercurrent as $$\bar G = \sqrt{-1} \bar \psi \bar \partial \phi$$ and anti-chiral stress-tensor $$\bar T = -\frac12 (\bar \partial \phi)^2 - \frac12 \bar \psi \bar \partial \bar \psi.$$ The basic free field OPE for the free boson currents $$\bar \partial \phi(\bar z) \bar \partial \phi(\bar w) \sim - \frac{1}{(\bar z - \bar w)^2}$$ and the anti-chiral free fermion superpartner $$\bar \psi(\bar z) \bar \psi(\bar w) \sim \frac{1}{\bar z - \bar w}$$ give the expected superconformal OPE: $$\bar G(\bar z) \bar G(\bar w)= \frac{1}{(\bar z - \bar w)^3} + \frac{2 T(\bar w)}{\bar z - \bar w}.$$
We thus have $$\sqrt{-1} \psi \bar G = - \psi \bar \psi \bar \partial \phi$$ It is easy to see that the $\psi \bar \psi$ one-point function is $$\label{eqn-ferbarfer}
\mathrm{Tr} \psi_0 \bar \psi_0 (-1)^F q^{L_0-\frac{1}{48}}\bar q^{L_0-\frac{1}{48}} =\sqrt{-1} |\eta(\tau)|^2$$ where the factor $|\eta(\tau)|^2$ comes from the trace over the Fock space of non-zero-modes, while the factor of $ \sqrt{-1}$ is the trace of the product of the zero-modes, i.e. the trace of $\gamma^1 \gamma^2$ in the irreducible module for $\operatorname{Cliff}(2,{{\mathds}C})$.
On the other hand, the $\sqrt{-1} \bar \partial \phi$ one-point function is $$\mathrm{Tr}\, (-\bar a_0) (-1)^F q^{L_0-\frac{1}{24}}\bar q^{L_0-\frac{1}{24}} = \frac{1}{|\eta(\tau)|^2}\sum_{n,w \in {{\mathds}Z}} \left(\frac{n}{R}- \frac{w R}{2} \right) y_1^n y_2^w q^{\frac{1}{2} (\frac{w R}{2} + \frac{n}{R})^2} \bar q^{\frac{1}{2} (\frac{w R}{2} - \frac{n}{R})^2}.$$ We conclude that the one-point function of $\sqrt{-1} {:}\psi \bar{G}{:}$ equals $$\label{eqn-nonboundingS1}
\sum_{n,w \in {{\mathds}Z}} \left(\frac{w R}{2} - \frac{n}{R} \right) y_1^n y_2^w q^{\frac{1}{2} (\frac{w R}{2} + \frac{n}{R})^2} \bar q^{\frac{1}{2} (\frac{w R}{2} - \frac{n}{R})^2}.$$
### Bounding spin structure, aka the exotic circle SCFT {#subsubsec.S1-bounding}
For the bounding spin structure, the fermion groundstate is taken to be anti-periodic along the target circle. In the sector with even winding number, the momentum lattice is shifted by $\frac12$ just as in the SQM case of §\[subsec.SQM\]. In sectors with odd winding number, the fermion ground-state must be anti-periodic as one goes around the space circle. This can be implemented by an extra shift by $\frac12$ of the momentum lattice. That means that the momentum lives in a shifted lattice ${{\mathds}Z}+\frac{w-1}{2}$. For generic values of the radius there is now no ground state at all, while for rational values of $R^2$ one may have more general ground states with $2n = R^2 w$ and $L_0 = \frac{2 n^2}{R^2}$ which give two copies of ${{\mathds}R}^{1|1}$ for each possible value of $n^2$. Thus the mod-2 index is radius-independent and equals $0 \in {{\mathds}Z}_2(\!(q)\!)$.
We can actually give a simple description of the full SCFT associated to the $S^1$ sigma model with bounding spin structure. We claim that it is the product of an anti-chiral free fermion and of an “exotic” free boson spin-CFT, based on an [*odd*]{} self-dual lattice of signature $(1,1)$ (see also [@Karch:2019lnn] for a discussion of free boson spin-CFTs): $$a_0 = \frac{m}{2R} + \frac{w R}{2}, \qquad \qquad \bar a_0 = \frac{w R}{2} -\frac{m}{2R}$$ The integers $(m,w)$ are either both even or both odd in the NS sector of the theory, while they have opposite parity in the R sector. The fermion number can be defined as $(-1)^F = (-1)^w$ or $(-1)^F = (-1)^m$ depending on the choice of overall fermionic parity of the Ramond sector.
An alternative description of the theory is that of a twisted orbifold of a standard free boson CFT of radius $2R$ by a ${{\mathds}Z}_2$ translation symmetry, where the twist involves the non-trivial fermionic 2-cocycle for ${{\mathds}Z}_2$ [@GuWen2014]: in the language of [@GJFIII], it is the fermionic theory whose bosonic neighbours are the standard free boson CFTs of radii $2R$ and $R$.
We conclude that the one-point function of $\sqrt{-1} {:}\psi \bar{G}{:}$ equals $$\label{eqn-boundingS1}
\sum_{m,w \in {{\mathds}Z}|m-w-1 \in 2{{\mathds}Z}} \left(\frac{w R}{2} - \frac{m}{2R} \right) (-1)^w y_1^{\frac{m}{2}} y_2^w q^{\frac{1}{2} (\frac{w R}{2} + \frac{m}{2R})^2} \bar q^{\frac{1}{2} (\frac{w R}{2} - \frac{m}{2R})^2}.$$
$T^3$ with Lie group framing
----------------------------
Take $S^1$ with nonbounding String structure and multiply it by itself three times; the result is the 3-torus $T^3$ with String structure coming from the Lie group framing. As a spin manifold, this $T^3$ is bounding (it bounds a certain “half K3 surface”), but it is nonbounding as a String manifold, and represents a class of exact order $2$ in TMF.
The (unflavoured) one-point function of $\bar G$ vanishes identically in the $T^3$ sigma model. So the class called $[f_1] \in {{\mathds}C}(\!(q)\!) / {\mathrm{MF}}_2$ in §\[sec.invariant\] vanishes. But the class $[f_2]$ does not vanish. Rather, it is the cube of the mod-2 index for the $S^1$ model, which we computed in §\[subsec.S1\] to be $1 \in {{\mathds}Z}_2(\!(q)\!)$. Thus, for the sigma model with target $T^3$ with Lie group framing, $$[f_2] \equiv 1 \in \frac{{{\mathds}C}(\!(q)\!) }{ 2{{\mathds}Z}(\!(q)\!)}.$$ It follows that the deformation-invariant class $[f] = [f_1] + [f_2]$ is $$[f] \equiv 1 \in A_4 = \frac{{{\mathds}C}(\!(q)\!) }{{\mathrm{MF}}_2 + 2{{\mathds}Z}(\!(q)\!)},$$ which has exact order $2$.
In fact, $T^3$ with this String structure is string-cobordant to $S^3$ with B-field of strength $k=12$. We will calculate directly the invariant of $(S^3, k=12)$ in Section \[sec.S3\], and see that it is equal to the invariant of $T^3$.
The ${\mathcal N}=(1,1)$ cigar sigma model
------------------------------------------
The simplest test of our holomorphic anomaly equation (\[eqn.holomorphicanomaly\]) is to consider the elliptic genus of an ${\mathcal N}=(0,1)$ sigma model whose target is the [*cigar*]{}. This is a Kähler manifold which coincides with ${{\mathds}C}$ as a complex manifold but is endowed with a Kähler metric which asymptotes to the flat metric on ${{\mathds}R}\times S^1$ with some radius $R$. Hence the “boundary theory” is an ${\mathcal N}=(0,1)$ circle sigma model.
The one-point function of $\bar{G}$ in the $(0,1)$ circle sigma model unfortunately vanishes: $\bar{G} = \bar \psi \bar \partial \phi$ is the product of the anti-chiral fermion and the anti-chiral $U(1)$ current for translations of the circle, whose torus one-point function vanishes for symmetry reasons. There is a simple way to produce a nontrivial holomorphic anomaly equation: we can look at the flavoured elliptic genus, using the rotational symmetry of the cigar.
The elliptic genus of the ${\mathcal N}=(1,1)$ version of this theory is very well studied [@Eguchi:2006tu; @MR2821103; @Ashok:2011cy; @murthy13], and is already a useful example. It can be computed by localization techniques [@Ashok:2013pya], as the theory is actually endowed with $(2,2)$ SUSY and can be obtained from RG flow of a $U(1)$ gauge theory [@Hori:2001ax]: the cigar geometry is a Kahler quotient of ${{\mathds}C}\times {{\mathds}C}^*$ by $U(1)$, acting as rotations of the first factor and translations of the second factor. The elliptic genus is: $$\label{eqn.cigar1}
Z_{RR}( \mathrm{cig}^{(1,1)})[\xi;\tau] = g^{2} \int_{{{\mathds}R}^2} {\mathrm d}u_1 {\mathrm d}u_2 \frac{\theta(u_1 + \tau u_2 -\xi;\tau)}{\theta(u_1 + \tau u_2;\tau)} e^{- \frac{\pi g^2}{\tau_2} (u_1 + \tau u_2 + \frac{\xi}{g^2})(u_1 + \bar \tau u_2 + \frac{\xi}{g^2})}$$ The $U(1)$ symmetry with fugacity $\xi$ is the chiral R-symmetry. It acts both on the chiral fermions and on the cigar sigma model, with a specific charge. We will often use the notation $u = u_1 + \tau u_2$.
The cancellation of gauge and mixed anomalies is manifested in the invariance of the integrand under the modular transformation $\tau \to - \frac{1}{\tau}$, $u \to \frac{u}{\tau}$, $z \to \frac{z}{\tau}$, up to a factor of $\exp 2 i \pi (\frac12 + \frac{1}{g^2}) \frac{z^2}{\tau}$ encoding the flavour ’t Hooft anomaly coefficient $1 + \frac{2}{g^2}$.
### Integrality of the $q$-expansion {#subsec.nonboundingS1integrality}
As a check of our normalization, we should verify the integrality of the $q$ expansion of the holomorphic part of the elliptic genus. The $\bar \tau \to - i \infty$ limit of (\[eqn.cigar1\]) gives $$\label{eqn.cig11.d}
Z_{RR}^{hol}( \mathrm{cig}^{(1,1)})[\xi;\tau] = g^{2} \int_{{{\mathds}R}\times {{\mathds}R}} {\mathrm d}u_1 {\mathrm d}u_2 \frac{\theta(u_1 + \tau u_2-\xi;\tau)}{\theta(u_1 + \tau u_2;\tau)} e^{2 i \pi g^2 (u_1 + \tau u_2 + \frac{\xi}{g^2})u_2 }$$ We can bring $u_1$ to the range $[0,1]$: $$Z_{RR}^{hol}( \mathrm{cig}^{(1,1)})[\xi;\tau] = \sum_{t \in {{\mathds}Z}} g^{2} \int_{[0,1] \times {{\mathds}R}} {\mathrm d}u_1 {\mathrm d}u_2 \frac{\theta(u_1 + \tau u_2-\xi;\tau)}{\theta(u_1 + \tau u_2;\tau)}e^{2 i \pi g^2 (t+ u_1 + \tau u_2 + \frac{\xi}{g^2})u_2 },$$ which happens to be the Poisson resummation of a simpler sum: $$\label{eqn.cig11.e}
Z_{RR}^{hol}( \mathrm{cig}^{(1,1)})[\xi;\tau] =\sum_{s \in {{\mathds}Z}} \int_0^1{\mathrm d}u_1 \frac{\theta(u_1 + \frac{\tau}{g^2} s-\xi;\tau)}{\theta(u_1 + \frac{\tau}{g^2} s;\tau)}e^{2 i \pi s (u_1 + \frac{\tau}{g^2} s + \frac{\xi}{g^2}) }$$ The integral in (\[eqn.cig11.e\]) computes the Fourier coefficient of $\frac{\theta(z-\xi;\tau)}{\theta(z;\tau)}$ along a circle positioned at $\frac{\tau}{g^2} s$. The Fourier coefficients of such a meromorphic Jacobi form are integral series in $x$ and $q$ which depend sensitively on the integral part of $\frac{s}{g^2}$. If we denote them as $f_s^{(\frac{s}{g^2})}(\xi;\tau)$, we get $$Z_{RR}^{hol}( \mathrm{cig}^{(1,1)})[\xi;\tau] =\sum_{s \in {{\mathds}Z}} f_s^{(\frac{s}{g^2})}(\xi;\tau) e^{2 i \pi s \frac{\xi}{g^2} }.$$
We should remark that a lot of the formulae simplify when $g^2$ is rational. In particular, for integer $g^2 = k$ one obtains so-called Appell–Lerche sums.
### The holomorphic anomaly equation {#sec.cig11anomaly}
The holomorphic anomaly was computed in the literature (see e.g. [@murthy13]). We will redo the calculation here to make sure we keep track carefully of the overall normalization. We can take the $\bar \tau$ derivative directly in the integral formula (\[eqn.cigar1\]).
It is easy to see that the exponential in the integrand in (\[eqn.cigar1\]) is annihilated by $\frac{\partial}{\partial\bar{\tau}} - \frac{i}{2 \pi g^2}\frac{\partial^2}{\partial\bar{u}^2}$, where $\frac{\partial}{\partial\bar{u}} = \frac{\tau \partial_{u_1} - \partial_{u_2}}{\tau - \bar \tau}$. As a result we have $$\label{eqn.cig11.a}
\frac{\partial}{\partial\bar{\tau}} Z_{RR}( \mathrm{cig}^{(1,1)})[\xi;\tau] = \frac{i}{2\pi} \int_{{{\mathds}R}^2} {\mathrm d}u_1 {\mathrm d}u_2 \frac{\theta(u-\xi;\tau)}{\theta(u;\tau)} \frac{\partial^2}{\partial\bar{u}^2} \left[e^{- \frac{\pi g^2}{\tau_2} (u + \frac{\xi}{g^2})(\bar u + \frac{\xi}{g^2})} \right]$$ where $u = u_1 + \tau u_2$ and thus $\frac{\partial u}{\partial\bar{u}} =0$.
Then we can integrate by parts to get $$\frac{\partial}{\partial\bar{\tau}} Z_{RR}( \mathrm{cig}^{(1,1)})[\xi;\tau] = \frac{1}{2\pi i} \int_{{{\mathds}R}^2} {\mathrm d}u_1 {\mathrm d}u_2 \frac{\partial}{\partial\bar{u}} \left[\frac{\theta(u-\xi;\tau)}{\theta(u;\tau)}\right] \frac{\partial}{\partial\bar{u}} \left[e^{- \frac{\pi g^2}{\tau_2} (u + \frac{\xi}{g^2})(\bar u + \frac{\xi}{g^2})}\right] .$$ As $\frac{\partial}{\partial\bar{u}} \frac{1}{u} = \frac{\pi}{\tau_2} \delta(u_1)\delta(u_2)$, the derivative picks the poles of $\frac{\theta(u-\xi;\tau)}{\theta(u;\tau)}$ at $u = n \tau + m$ and converts the integral into a sum $$\frac{\partial}{\partial\bar{\tau}} Z_{RR}( \mathrm{cig}^{(1,1)})[\xi;\tau] = \frac{g^2}{4 \tau^2_2} \frac{\theta(\xi;\tau)}{\eta(\tau)^3} \sum_{n,m \in {{\mathds}Z}} x^{-n} (n \tau + m + \frac{\xi}{g^2}) e^{- \frac{\pi g^2}{\tau_2} (n \tau + m + \frac{\xi}{g^2})(n\bar \tau +m + \frac{\xi}{g^2})} ,$$ where we used $$\theta(\xi + n \tau + m;\tau) = (-1)^{n+m} q^{-\frac{n^2}{2}} x^{-n}\theta(\xi;\tau).$$ Poisson resummation in $m$ finally gives $$\label{eqn.cigar11.b}
\sqrt{-8 \tau_2} \frac{\partial}{\partial\bar{\tau}} Z_{RR}( \mathrm{cig}^{(1,1)})[\xi;\tau] = \frac{\theta(\xi;\tau)}{\eta(\tau)^3} \sum_{n,s \in {{\mathds}Z}} x^{\frac{s}{g^2} +n} \frac{1}{\sqrt{2}} (g n - \frac{s}{g}) q^{\frac{1}{4}(n g +s/g)^2} \bar q^{\frac{1}{4}(n g -s/g)^2}.$$
We recognize on the right-hand side of (\[eqn.cig11.a\]) the sum over momenta and winding of an $S^1$ sigma model. If we define $R^2 = 2 g^2$, rename the summation variables and reshuffle some factors of $\eta$, we can write $$\label{eqn.cigar11.c}
\sqrt{- 8 \tau_2} \frac{\partial}{\partial\bar{\tau}} \eta(\tau)^2 Z_{RR}( \mathrm{cig}^{(1,1)})[\xi;\tau] = \frac{\theta(\xi;\tau)}{\eta(\tau)} \sum_{n,w \in {{\mathds}Z}} (\frac{w R}{2} - \frac{n}{R}) x^{\frac{2 n}{R^2} +w} q^{\frac{1}{2} (\frac{w R}{2} + \frac{n}{R})^2} \bar q^{\frac{1}{2} (\frac{w R}{2} - \frac{n}{R})^2}$$
The right-hand side of (\[eqn.cigar11.c\]) is precisely the product of the flavoured torus partition function of $\operatorname{Fer}(2)$ and the flavoured torus one-point function (\[eqn-nonboundingS1\]) of $\psi \bar G$ in the sigma model with nonbounding $S^1$ target and spectator fermion $\psi$. The left-hand side is the holomorphic anomaly of the $(1,1)$ cigar theory with two extra spectator fermions. It is not a contradiction to find the nonbounding $S^1$ here: the nonbounding $S^1$ can become a boundary when combined non-trivially with two extra chiral fermions.
The ${\mathcal N}=(0,1)$ cigar sigma model
------------------------------------------
The elliptic genus for the ${\mathcal N}=(0,1)$ sigma model (which actually has ${\mathcal N}=(0,2)$ supersymmetry) can be computed in an analogous manner, though one needs to adjust a bit the formulae available in the literature. The main subtlety is the cancellation of gauge anomalies in the ${\mathcal N}=(0,1)$ gauge theory, since, unlike in the ${\mathcal N}=(1,1)$ case, this cancellation depends on a String structure on the target manifold. For the ${\mathcal N}=(0,1)$ cigar sigma model, our localization formula for the elliptic genus is: $$Z_{RR}(\mathrm{cig})[\xi;\tau] = g^{2} \int_{{{\mathds}R}^2} {\mathrm d}u_1 {\mathrm d}u_2 \frac{\eta(\tau)^3}{\theta(u+(1+g^{-2}) \xi;\tau)} e^{- \frac{\pi g^2}{\tau_2} (u + \frac{\xi}{g^2})(\bar u + \frac{\xi}{g^2})- \frac{\pi}{2\tau_2}(u + \frac{\xi}{g^2})^2}$$ The factor $e^{- \frac{\pi}{2\tau_2}(u + \frac{\xi}{g^2})^2 }$ is a correction which seems to be missing from the literature. We take $\xi$ to be real.
The overall normalization can be fixed by the observation that the limit $g^2 \to \infty$ should bring the theory back to the ${\mathcal N}=(0,1)$ sigma model with target ${{\mathds}R}^2$. Indeed, in this limit the exponent goes to a $g^{-2} \delta(u_1) \delta(u_2)$ and the elliptic genus goes to $\frac{\eta(\tau)^3}{\theta(\xi;\tau)}$, as it should.
### Integrality of the $q$-expansion {#integrality-of-the-q-expansion}
The holomorphic part of the elliptic genus is $$Z^{hol}_{RR}(\mathrm{cig})[\xi;\tau] = g^{2} \int_{{{\mathds}R}\times {{\mathds}R}} {\mathrm d}u_1 {\mathrm d}u_2 \frac{\eta(\tau)^3}{\theta(u+(1+g^{-2}) \xi;\tau)} e^{2 i \pi g^2 (u_1 + \tau u_2 + \frac{\xi}{g^2})u_2}$$ which can be manipulated as in §\[subsec.nonboundingS1integrality\]: $$\begin{gathered}
Z^{hol}_{RR}(\mathrm{cig})[\xi;\tau] = g^{2} \sum_{m \in {{\mathds}Z}} \int_{[0,1]\times {{\mathds}R}} {\mathrm d}u_1 {\mathrm d}u_2 \frac{\eta(\tau)^3}{\theta(u+(1+g^{-2}) \xi;\tau)}(-1)^m e^{2 i \pi g^2 (u_1 + m +\tau u_2 + \frac{\xi}{g^2})u_2}\end{gathered}$$ and then $$\begin{gathered}
Z^{hol}_{RR}(\mathrm{cig})[\xi;\tau] = \sum_{n \in {{\mathds}Z}} \int_{[0,1]} {\mathrm d}u_1 \frac{\eta(\tau)^3}{\theta(u_1+\tau (n+\frac12)\frac{1}{g^2} +(1+g^{-2}) \xi;\tau)} \\ \times (-1)^m e^{2 i \pi (u_1 +\tau (n+\frac12)\frac{1}{g^2} + \frac{\xi}{g^2})(n+\frac12)}\end{gathered}$$ which is doing Fourier transforms along a circle located at $\tau (n+\frac12)\frac{1}{g^2} +(1+g^{-2}) \xi$ of $ \frac{\eta(\tau)^3}{\theta(z;\tau)}$. Again, the expansion is manifestly integral.
### The holomorphic anomaly equation {#the-holomorphic-anomaly-equation}
The calculation of the holomorphic anomaly can proceed as in §\[sec.cig11anomaly\]. The exponential in the integrand is annihilated by the usual heat operator $\frac{\partial}{\partial\bar{\tau}} - \frac{i(1+2 g^2)}{4 \pi g^4}\frac{\partial^2}{\partial\bar{u}^2}$, with $\frac{\partial}{\partial\bar{u}} = \frac{\tau \partial_{u_1} - \partial_{u_2}}{\tau - \bar \tau}$. Then we can integrate by parts to get $$\begin{gathered}
\frac{\partial}{\partial\bar{\tau}} Z_{RR}( \mathrm{cig})[\xi;\tau] = \frac{1}{2\pi i} (1+\frac{1}{2 g^2}) \int_{{{\mathds}R}^2} {\mathrm d}u_1 {\mathrm d}u_2\frac{\partial}{\partial\bar{u}} \left[\frac{\eta(\tau)^3}{\theta(u+(1+g^{-2}) \xi;\tau)}\right] \\ \times \frac{\partial}{\partial\bar{u}} \left[ e^{- \frac{\pi g^2}{\tau_2} (u + \frac{\xi}{g^2})(\bar u + \frac{\xi}{g^2})- \frac{\pi}{2\tau_2}(u + \frac{\xi}{g^2})^2}\right] \end{gathered}$$ This picks the poles of $\frac{\eta(\tau)^3}{\theta(u+(1+g^{-2}) \xi;\tau)}$ at $u = n \tau + m-(1+g^{-2}) \xi$ and converts the integral into a sum $$\begin{gathered}
\frac{\partial}{\partial\bar{\tau}}Z_{RR}( \mathrm{cig})[\xi;\tau] =- (g^2+\frac{1}{2}) \frac{1}{4\tau_2^2} \sum_{n,m}(-1)^{n+m} q^{\frac{n^2}{2}} x^{- (1+g^{-2})n} (n \tau + m- \xi )
\\ \times \left[ e^{- \frac{\pi g^2}{\tau_2} (n \tau + m- \xi )(n \bar \tau + m-\xi )- \frac{\pi}{2\tau_2}(n \tau + m- \xi )^2}\right] \end{gathered}$$ where we used $$\theta(\xi + n \tau + m;\tau) = (-1)^{n+m} q^{-\frac{n^2}{2}} x^{-n}\theta(\xi;\tau)$$ Poisson resummation in $m$ finally gives a nice expression. Defining $\gamma^2 = g^2 + \frac12$ we have $$\begin{gathered}
\sqrt{-4\tau_2} \frac{\partial}{\partial\bar{\tau}} \eta(\tau)^2 Z_{RR}( \mathrm{cig})[\xi;\tau]
\\ = - \sum_{n,s \in {{\mathds}Z}} (-1)^n x^{\frac12 -s-n -\frac{1}{g^2} n} \frac{1}{2} (\gamma n - \frac{s+\frac{n}{2}-\frac12}{\gamma}) q^{\frac{1}{4}(\gamma n +\frac{s+\frac{n}{2}-\frac12}{\gamma})^2} \bar q^{\frac{1}{4}(\gamma n - \frac{s+\frac{n}{2}-\frac12}{\gamma})^2} ,\end{gathered}$$ or, equivalently, with $R=\gamma$, $$\begin{gathered}
\label{eqn.cig12.b}
\sqrt{-8 \tau_2} \frac{\partial}{\partial\bar{\tau}} \eta(\tau)^2 Z_{RR}( \mathrm{cig})[\xi;\tau]
\\ = \sum_{\substack{w,m \in {{\mathds}Z}, \\w+m = \mathrm{odd}}} \left(\frac{m}{2R}-\frac{w R}{2} \right) (-1)^w x^{-\frac{w+m}{2} -\frac{1}{g^2} w}q^{\frac{1}{2}(\frac{m}{2R}+\frac{w R}{2} )^2} \bar q^{\frac{1}{2}(\frac{m}{2R}-\frac{w R}{2} )^2} .\end{gathered}$$ The right-hand side of (\[eqn.cig12.b\]) is nothing but the one-point function of $\psi \bar{G}$ of the circle theory with bounding spin structure (and a spectator fermion). The form of the momentum and winding lattice and the extra $(-1)^w$ factor are explained in §\[subsec.S1\].
Example: $S^3$ with WZW coupling $k$ {#sec.S3}
====================================
The ${\mathcal N}=(0,1)$ sigma model with target $S^3$ and WZW coupling $k$ is expected to flow in the IR to an $\mathrm{SU}(2)$ WZW model, which is the same as a bosonic WZW model at level $\kappa = k-1$ together with three anti-chiral free fermions [@WittenTMF]. In the special case $k=1$, the bosonic WZW model at level $\kappa=0$ is the trivial theory, and the model flows to the free-fermion SCFT from Puzzle \[mainpuzzle\]. In all cases, the SQM produced by compactifying of the model on $S^1$ has spontaneous supersymmetry breaking [@WittenTMF §4], and so to compute our invariant we must only compute the $q$-series called $f_1$ in §\[sec.invariant\]. To warm up, we address the $k=1$ case in §\[subsec.S31\]. The general case is in §\[subsec.S3k\].
A warm-up: $k=1$ {#subsec.S31}
----------------
The anti-chiral stress tensor in the $\overline{\operatorname{Fer}}(3)$ theory is $$\bar T = - \frac12 \bar \psi_a \bar \partial \bar \psi_a$$ The supercurrent can be taken to be $$\label{eqn.barG1}
\bar G =\sqrt{-1} \bar \psi_1 \bar \psi_2 \bar \psi_3$$ so that $$\bar G(\bar z) \bar G(\bar w)= \frac{1}{(\bar z - \bar w)^3} + \frac{2 T(\bar w)}{\bar z - \bar w}.$$
Then we have that the one-point function of $-\psi_1 \psi_2 \psi_3 \bar G$ equals $|\eta(\tau)|^6$, though the sign is somewhat conventional.
The function $$\label{F1-formula}
F_1(\tau) = -\frac{1}{24} + \sum_{n=1}^{\infty} n\, \frac{q^n}{1-q^n}+\sum_{n=1}^{\infty} (-1)^{n-1} n \,\frac{q^{\frac{n(n+1)}{2}} }{1-q^n}$$ has a modular, non-holomorphic completion $\hat F_1$ of weight 2 which satisfies $$\label{F1-holomorphicanomaly}
\sqrt{- 8 \tau_2} \frac{\partial \hat F_1}{\partial \bar \tau} = \frac12 |\eta|^6.$$ and thus $2 \hat F_1$ solves our holomorphic anomaly equation.
This shows immediately that the invariant for the $k=1$ case is $-\frac1{12} \pmod 2$, as expected! In particular, only 24 copies of $\overline{\operatorname{Fer}}(3)$ can be null-homotopic.
In order to verify these assertions, first proven by the first named author in collaboration with D. Zagier, one may employ an integral formula for $\hat F_1$: $$\label{F1-integralformula}
\hat F_1 = \frac{1}{8 \pi^2} \int_0^1 \int_0^1 \wp(u_1 + \tau u_2,\tau) H_1(u_1, u_2; \tau, \bar \tau)\, {\mathrm d}u_1 {\mathrm d}u_2 ,$$ where $\wp(u_1 + \tau u_2,\tau)$ is the Weierstrass function and $$\label{F1-formulaforH1}
H_1(u_1, u_2; \tau, \bar \tau) \equiv \sum_{n,m \in {{\mathds}Z}} e^{2 \pi i (n u_2 - m u_1)} (-1)^{n + m + nm} e^{- \frac{\pi}{2 \tau_2}|m \tau + n|^2},$$ which can be Poisson resummed to $$H_1(u_1, u_2; \tau, \bar \tau) \equiv \sqrt{2\tau_2}|\theta(u_1 + \tau u_2,\tau)|^2 e^{- 2 \pi \tau_2 u_2^2} .$$
In the $\bar \tau \to - \infty$ limit, we recover $$F_1 = \frac{1}{8 \pi^2} \int_0^1 \int_0^1 \wp(u_1 + \tau u_2,\tau) \sum_{n,m \in {{\mathds}Z}} e^{2 \pi i (n u_2 - m u_1)} (-1)^{n + m} e^{i \pi m^2 \tau }{\mathrm d}u_1 {\mathrm d}u_2 ,$$ which is the Poisson resummation of $$F_1 = \frac{1}{8 \pi^2} \int_0^1 \wp(u_1 + \frac{\tau}{2} ,\tau) \sum_{m \in {{\mathds}Z}} e^{-2 \pi i m u_1} (-1)^{m} e^{i \pi m^2 \tau }{\mathrm d}u_1 .$$ We can use the Fourier expansion $$\frac{1}{(2 \pi i)^2} \wp(\xi,\tau) = -2 G_2(\tau) + \sum_{n\in {{\mathds}Z}| n\neq 0} \frac{n}{1-q^n}x^n$$ along that circle to get the expected $$\label{F1-formula-redux}
F_1 = G_2 - \sum_{m > 0 } \frac{m}{1-q^m} (-1)^{m} e^{i \pi m(m+1) \tau }$$ where $$G_2 = -\frac{1}{24} + \sum_{n=1}^{\infty} n\, \frac{q^n}{1-q^n}$$ is the second Eisenstein series. Formula (\[F1-formula\]) is a restatement of (\[F1-formula-redux\]).
At finite $\bar \tau$ we can still use the Fourier expansion of $\wp(\xi,\tau)$ to get $$\begin{aligned}
\hat F_1 & = G_2(\tau)-\frac{1}{2} \int_0^1 \sum_{n,m \in {{\mathds}Z}| m\neq 0} \frac{m}{1-q^m}e^{2 \pi i (m \tau +n) u_2 } (-1)^{n + m + nm} e^{- \frac{\pi}{2 \tau_2}|m \tau + n|^2} {\mathrm d}u_2 ,
\\
& = G_2(\tau)-\frac{i}{4 \pi} \sum_{n,m \in {{\mathds}Z}| m \neq 0} (-1)^{n + m + nm} \frac{m}{m \tau + n} e^{- \frac{\pi}{2 \tau_2}|m \tau + n|^2}.\end{aligned}$$ This satisfies a holomorphic anomaly equation $$\partial_{\bar \tau} \hat F_1 = \frac{1}{16 \tau_2^2} \sum_{n,m \in {{\mathds}Z}} (-1)^{n + m + nm} m(m \tau + n) e^{- \frac{\pi}{2 \tau_2}|m \tau + n|^2},$$ which is Poisson resummed to the desired (\[F1-holomorphicanomaly\]).
We can obtain the same result by starting from the heat equation $$\partial_{\bar \tau} H_1(u_1, u_2; \tau, \bar \tau)= \frac{i}{4 \pi} \partial_{\bar u}^2 H_1(u_1, u_2; \tau, \bar \tau)$$ with $\frac{\partial}{\partial\bar{u}} = \frac{\tau \partial_{u_1} - \partial_{u_2}}{\tau - \bar \tau}$ and $H_1$ from (\[F1-formulaforH1\]). Integrating by parts in (\[F1-integralformula\]) then gives $$\partial_{\bar \tau} \hat F_1 = - \frac{i}{32 \pi^3} \int_0^1 \int_0^1\left[ \frac{\partial}{\partial\bar{u}} \wp(u_1 + \tau u_2,\tau)\right] \left[\frac{\partial}{\partial\bar{u}} H_1(u_1, u_2; \tau, \bar \tau)\right] {\mathrm d}u_1 {\mathrm d}u_2 .$$ But $\frac{\partial}{\partial\bar{u}} (u_1 + \tau u_2)=0$ and $$\frac{\partial}{\partial\bar{u}} \frac{1}{(u_1 + \tau u_2)^2} = - \frac{\pi}{\tau_2} \frac{\partial}{\partial{u}}\left[\delta(u_1)\delta(u_2)\right]$$ with $\frac{\partial}{\partial{u}} = \frac{\bar \tau \partial_{u_1} - \partial_{u_2}}{\bar \tau - \tau}$. As the Weierstrass function has a double pole at the origin, with coefficient $1$ and no residue, we have $$\partial_{\bar \tau} \hat F_1 = - \frac{i}{32 \pi^2 \tau_2} \left[ \frac{\partial}{\partial{u}} \frac{\partial}{\partial\bar{u}} H_1(u_1, u_2; \tau, \bar \tau) \right] \big|_{u_1 = u_2 = 0}$$ which gives (\[F1-holomorphicanomaly\]) directly.
General $k$ {#subsec.S3k}
-----------
Generalizing (\[eqn.barG1\]), the supercurrent for the ${\mathcal N}=(0,1)$ WZW model with bosonic level $\kappa=k-1$ is: $$\bar G = \sqrt{-1} \sqrt{\frac{2}{\kappa+2}} \bar \psi_1 \bar \psi_2 \bar \psi_3 + \cdots$$ The ellipsis is proportional to $\sum_a \psi_a J^a_b$, with $J^a_b$ being the currents of the bosonic WZW model, and so it cannot soak the three fermion zeromodes in the torus one-point function. It follows that the one-point function of $-\psi_1 \psi_2 \psi_3 \bar G$ equals $$\label{eqn.gkappa}
g_\kappa(\tau, \bar \tau) = \sqrt{\frac{2}{\kappa+2}} |\eta(\tau)|^6 Z^{\mathrm{WZW}}_\kappa(\tau, \bar \tau)$$ where $$\label{eqn.bosonicWZW}
Z_\kappa^{\mathrm{WZW}}(\tau, \bar \tau) = \sum_{2j+1=1}^{\kappa+1} |\chi_j^{(\kappa)}(\tau)|^2$$ is the torus partition function of the bosonic WZW model, which can be expanded in terms of characters $\chi_j^{(\kappa)}(\tau)$ of the WZW current algebra, as the WZW CFT is rational.
### Characters and source
The characters of $\mathrm{SU}(2)_\kappa$ appearing in (\[eqn.bosonicWZW\]) can be written as $$\label{eqn.SU2char}
\chi_j^{(\kappa)}(\tau) = \frac{ \sum_{m \in Z+ \frac{j+\frac12}{\kappa+2}} q^{(\kappa+2)m^2 } \left[ 2 m(\kappa+2) \right]}{ \prod_{n>0} (1-q^n)^3}.$$ In terms of the traditional definition of weight $3/2$ theta functions $$\Theta_{k,\ell}(\tau) = \sum_{m\in Z + \frac{\ell}{2k}} m q^{k m^2},$$ equation (\[eqn.SU2char\]) is equivalent to $$\chi_j^{(\kappa)}(\tau) = \frac{ 2(\kappa+2)\Theta_{\kappa+2,2j+1}(\tau)}{\eta(\tau)^3},$$ and so (\[eqn.gkappa\]) becomes: $$g_\kappa(\tau, \bar \tau) = 4 (\kappa+2) \sqrt{2(\kappa+2)} \sum_{2j+1=1}^{\kappa+1} |\Theta_{\kappa+2,2j+1}(\tau)|^2.$$
In §\[flavoured-nullhomotopic\] we will use the flavoured WZW characters $$\chi_j^{(\kappa)}(\xi;\tau) = \frac{\vartheta_{\kappa+2,2j+1}(\xi;\tau)- \vartheta_{\kappa+2,2j+1}(-\xi;\tau)}{\theta(\xi;\tau)}.$$ where $$\vartheta_{k,\ell}(\xi;\tau) = \sum_{m\in Z + \frac{\ell}{2k}} x^{k m} q^{k m^2}.$$ It is useful to extend the definition of $\chi_j^{(\kappa)}(\xi;\tau)$ to the full range $0 \leq 2j+1 <2(\kappa+2)$, with $\chi_{-\frac12}^{(\kappa)}(\xi;\tau) =0$ and $\chi_{\frac{\kappa+1}{2}}^{(\kappa)}(\xi;\tau) =0$. Using $$\vartheta_{k,\ell}(-\xi;\tau) =\vartheta_{k,2 k-\ell}(\xi;\tau)$$ one has $$\chi_{\kappa+1-j}^{(\kappa)}(\xi;\tau) = \chi_j^{(\kappa)}(\xi;\tau).$$
### Solution of the holomorphic anomaly equation
There are generalizations $\hat F_k$ of $\hat F_1$ which were described in detail in [@Harvey:2014cva] and such that $2 F_{\kappa+1}$ precisely solves the correctly normalized holomorphic anomaly equation with source $g_\kappa(\tau, \bar \tau)$.
A simple way to arrive at a definition of the $\hat F_k$ is to look for an integral formula analogous to (\[F1-integralformula\]), i.e. of the form: $$\label{Fk-integralformula}
\hat F_k = \frac{1}{8 \pi^2} \int_0^1 \int_0^1 \wp(u_1 + \tau u_2,\tau) H_k(u_1, u_2; \tau, \bar \tau) {\mathrm d}u_1 {\mathrm d}u_2 .$$ Observe that the function $$\label{formulaforHk}
H_k(u_1, u_2; \tau, \bar \tau) \equiv \sqrt{(\kappa+2)\tau_2} e^{- (\kappa +2) \pi \tau_2 u_2^2} \sum_{2j+1=1}^{\kappa+1} |\chi_j^{(\kappa)}(u_1 + \tau u_2;\tau)|^2 |\theta(u_1 + \tau u_2,\tau)|^2$$ satisfies a heat equation $$\partial_{\bar \tau} H_k(u_1, u_2; \tau, \bar \tau)= \frac{i}{2 \pi (\kappa+2)} \partial_{\bar u}^2 H_k(u_1, u_2; \tau, \bar \tau)$$ and furthermore $\partial_u \chi_j^{(\kappa)}(u;\tau)\theta(u,\tau)|_{u=0} = 2 \pi i \chi_j^{(\kappa)}(\tau)$.
That means the same manipulations as in §\[subsec.S31\] give us $$\partial_{\bar \tau} \hat F_k = - \frac{i}{16 \pi^2 (\kappa+2) \tau_2} \left[ \frac{\partial}{\partial{u}} \frac{\partial}{\partial\bar{u}} H_k(u_1, u_2; \tau, \bar \tau) \right] \big|_{u_1 = u_2 = 0}$$ which gives directly the desired $$\sqrt{-2\tau_2}\partial_{\bar \tau} \hat F_k = \frac14 \sqrt{\frac{2}{\kappa+2}}\sum_{2j+1=1}^{\kappa+1} |\chi_j^{(\kappa)}(\tau)|^2.$$
### Holomorphic part
In order to extract the holomorphic part, it is useful to Poisson resum $H_k$ from (\[formulaforHk\]). We can split $H_k$ in two parts: $$\begin{aligned}
H^{(1)}_k(u_1, u_2; \tau, \bar \tau) & \equiv \sqrt{(\kappa+2)\tau_2} e^{- (\kappa +2) \pi \tau_2 u_2^2} \sum_{2j+1=0}^{2\kappa+3} |\vartheta_{\kappa+2,2j+1}(\xi;\tau)|^2,
\\
H^{(2)}_k(u_1, u_2; \tau, \bar \tau) & \equiv - \sqrt{(\kappa+2)\tau_2} e^{- (\kappa +2) \pi \tau_2 u_2^2} \sum_{2j+1=0}^{2\kappa+3}\vartheta_{\kappa+2,2j+1}(\xi;\tau)\overline{\vartheta_{\kappa+2,2 \kappa + 4-2j-1}(\xi;\tau)}.\end{aligned}$$ These Poisson resum to $$\begin{aligned}
H^{(1)}_k(u_1, u_2; \tau, \bar \tau) &= \sum_{n,m \in {{\mathds}Z}} (\kappa+2) e^{2 \pi i (\kappa+2)(n u_2 - m u_1)} e^{- \frac{(\kappa+2)\pi}{\tau_2}|m \tau + n|^2},
\\
H^{(2)}_k(u_1, u_2; \tau, \bar \tau) &= - \sum_{n,m \in {{\mathds}Z}} e^{2 \pi i(n u_2 - m u_1)} e^{- \frac{\pi}{(\kappa+2)\tau_2}|m \tau + n|^2},\end{aligned}$$ which recombine to $$\label{Hk-resummed}
H_k(u_1, u_2; \tau, \bar \tau) = \sum_{n,m \in {{\mathds}Z}} \epsilon_{n,m}^{\kappa+2} e^{2 \pi i(n u_2 - m u_1)} e^{- \frac{\pi}{(\kappa+2)\tau_2}|m \tau + n|^2},$$ where $\epsilon_{n,m}^{\kappa+2} =\kappa+1$ if both $n$ and $m$ are divisible by $\kappa+2$, and $ \epsilon_{n,m}^{\kappa+2} =-1$ otherwise.
In the $\bar \tau \to - i \infty$ limit (\[Hk-resummed\]) simplifies to $$H_k(u_1, u_2; \tau, - i \infty) = \sum_{n,m \in {{\mathds}Z}} \epsilon_{n,m}^{\kappa+2} e^{2 \pi i(n u_2 - m u_1)} e^{\frac{2 i \pi}{\kappa+2}m (m \tau + n)}$$ and thus $$F_k = \frac{1}{8 \pi^2} \int_0^1 \int_0^1 \wp(u_1 + \tau u_2,\tau) \sum_{n,m \in {{\mathds}Z}} \epsilon_{n,m}^{\kappa+2} e^{2 \pi i(n u_2 - m u_1)} e^{\frac{2 i \pi}{\kappa+2}m (m \tau + n)} {\mathrm d}u_1 {\mathrm d}u_2 ,$$ We can again split into two parts: $$\begin{aligned}
F^{(1)}_k & = \frac{\kappa+2}{8 \pi^2} \int_0^1 \int_0^1 \wp(u_1 + \tau u_2,\tau) \sum_{n,m \in {{\mathds}Z}} e^{2 \pi i(\kappa+2)(n u_2 - m u_1)} e^{2 i \pi(\kappa+2) m^2\tau} {\mathrm d}u_1 {\mathrm d}u_2 , \label{Fk(1)}
\\
F^{(2)}_k & = - \frac{1}{8 \pi^2} \int_0^1 \int_0^1 \wp(u_1 + \tau u_2,\tau) \sum_{n,m \in {{\mathds}Z}} e^{2 \pi i(n u_2 - m u_1)} e^{\frac{2 i \pi}{\kappa+2}m^2 \tau} e^{\frac{2 i \pi}{\kappa+2}m n} {\mathrm d}u_1 {\mathrm d}u_2 . \label{Fk(2)}\end{aligned}$$
The sum over $n$ in (\[Fk(1)\]) gives a sum of delta functions at $u_2 = \frac{\ell}{\kappa+2}$, so that $$F^{(1)}_k = \sum_{\ell=0}^{\kappa+1} \sum_{m \in {{\mathds}Z}} q^{(\kappa+2) m^2+ m \ell}\frac{1}{8 \pi^2} \int_0^1 e^{-2 \pi i(\kappa+2) m u_1} \wp(u_1,\tau) {\mathrm d}u_1, \label{Fk(1)-int}$$ while the sum over $n$ in (\[Fk(2)\]) gives a delta function at $u_2 = -\frac{m}{\kappa+2}$ modulo $1$: $$\begin{aligned}
F^{(2)}_k & = - \sum_{m \in {{\mathds}Z}} q^{\frac{m^2}{\kappa+2} +m [-\frac{m}{\kappa+2}]} \frac{1}{8 \pi^2} \int_0^1 e^{-2 \pi i m u_1} \wp(u_1,\tau) {\mathrm d}u_1 , \notag
\\
& = - \sum_{\ell=0}^{\kappa+1} \sum_{m \in {{\mathds}Z}} q^{(\kappa+2) m^2 + m \ell} \frac{1}{8 \pi^2} \int_0^1 e^{-2 \pi i ((\kappa+2) m+\ell) u_1} \wp(u_1,\tau) {\mathrm d}u_1 . \label{Fk(2)-int}\end{aligned}$$ In particular, the $m=0$ contributions add up to $k G_2$, while the other ones give positive powers of $q$ with integral coefficients. We conclude that the invariant for the general $k$ case is $-\frac{k}{12} \pmod 2$, as expected!
An antiholomorphic SCFT of degree $27$
--------------------------------------
Following [@MR3223024 Chapter 13], write $\alpha \in \pi_3\mathrm{TMF}$ and $\beta \in \pi_{10}\mathrm{TMF}$ for the TMF classes represented, respectively, by $\mathrm{SU}(2)$ and $\mathrm{Sp}(2)$ with their Lie group framings. These are the classes that, conjecturally, correspond to the antiholomorphic free-fermion SCFTs described in Puzzles \[mainpuzzle\] and \[sp2puzzle\]; the motivation for those puzzles came from the fact that $\alpha$ and $\beta$ have exact orders $24$ and $3$, respectively. There is no TMF class with Witten genus $\Delta$, but $24\Delta$ is the Witten genus of an element in $\pi_{12}\mathrm{TMF}$; that element is, not surprisingly, called simply $\{24\Delta\}$. As observed in [@GJFIII], this class can also be represented by a purely antiholomorphic SCFT. Specifically, [@MR2352133] constructs a holomorphic ${\mathcal N}=1$ SCFT called $V^{s\natural}$ (the “$s$” stands for “super” and the “$\natural$” stands for “moonshine”), and $\{24\Delta\}$ is represented by a right-moving copy $\bar{V}^{s\natural}$.
It is known that $$\alpha \times \{24\Delta\} = 0 \in \pi_{27}\mathrm{TMF}.$$ This raises already an interesting puzzle:
Find a nullhomotopy for the degree-27 SQFT $\overline{\operatorname{Fer}}(3) \otimes \bar{V}^{s\natural}$.
Although $\alpha \times \{24\Delta\} = 0$, there is an interesting order-$3$ class in $\pi_{27}\mathrm{TMF}$, which could be called $\{8\alpha\Delta\}$.[^33] We will describe an antiholomorphic SCFT which we expect to represent this class. The supersymmetry-preserving automorphism group of $\bar{V}^{s\natural}$ is Conway’s largest simply group $\mathrm{Co}_1$, and the Ramond-sector ground states for $\bar{V}^{s\natural}$ form the $24$-dimensional “Leech lattice” representation of the double cover $\mathrm{Co}_0 = 2.\mathrm{Co}_1$ [@MR2352133]. The conjugacy classes of order $3$ in $\mathrm{Co}_1$ and $\mathrm{Co}_0$ are in bijection; there are four of them, distinguished by their trace on the $24$-dimensional representation. The class called “$3\mathrm{A}$” in [@ATLAS] acts with trace $0$. Now consider the order-$3$ automorphism of $\overline{\operatorname{Fer}}(3) \otimes \bar{V}^{s\natural}$ which acts on $\bar{V}^{s\natural}$ by class $3\mathrm{A}$, and which acts on $\overline{\operatorname{Fer}}(3)$ by cyclicly permuting the three fermions. It is not hard (c.f. [@JFT; @GJFIII]) to calculate that the ’t Hooft anomalies on the $\overline{\operatorname{Fer}}(3)$ and $\bar{V}^{s\natural}$ factors cancel, and the corresponding orbifold $(\overline{\operatorname{Fer}}(3) \otimes \bar{V}^{s\natural}) \sslash {{\mathds}Z}_3$ is our proposed representative of $\{8\alpha\Delta\}$.
Indeed, the one-point function of $\bar{G}$ in this orbifold is $8|\eta|^6\Delta$, and so the calculations in §\[subsec.S31\] give the solution $16 \hat{F}_1\Delta$ to the holomorphic anomaly equation. The holomorphic part of this solution is $$16 F_1\Delta = -\frac23 q + q^2 2{{\mathds}Z}[\![q]\!],$$ showing that the SQFT $(\overline{\operatorname{Fer}}(3) \otimes \bar{V}^{s\natural}) \sslash {{\mathds}Z}_3$ has order (at least) $3$.
Is $S^3_k$ flavoured-nullhomotopic? {#flavoured-nullhomotopic}
-----------------------------------
The flavoured elliptic genus for the $(0,1)$ $\mathrm{SU}(2)_\kappa$ WZW model, with flavour fugacity for the chiral $\mathrm{SU}(2)$ rotations, takes the form (after some reorganization) $$\sqrt{2(\kappa+2)} \sum_{2j+1=0}^{2\kappa+3}\frac{\vartheta_{\kappa+2,2j+1}(2\xi;\tau)}{\theta(2\xi;\tau)}\Theta_{\kappa+2,2j+1}(\bar \tau).$$ It turns out that there are interesting solutions of the holomorphic anomaly equation with this source, which have [*integral*]{} coefficients, but are meromorphic in $\xi$ with a double pole at $\xi=0$.
These are built from the modular completions of the Appell–Lerche sums $${\cal A}_{1,\kappa+2}(\xi;\tau) = \sum_{m\in Z} q^{(\kappa+2)m^2} x^{2 m (\kappa+2)}\frac{1+ x q^m}{1-x q^m}.$$ The precise statement is that ${\cal A}_{1,\kappa+2}(\xi;\tau)$ can be completed to a Jacobi object $\hat {\cal A}_{1,\kappa+2}(\xi;\tau)$ by combining it with a multiple of $$\sum_{2j+1=0}^{2\kappa+3}\vartheta_{\kappa+2,2j+1}(\xi;\tau)\Theta^*_{\kappa+2,2j+1}(\bar \tau)$$ where $$\Theta^*_{k,\ell}(\tau) = \frac12 \sum_{m\in Z + \frac{\ell}{2k}} \mathrm{sign}(m) \mathrm{erfc}(2 |m| \sqrt{\pi k \tau_2}) q^{-k m^2},$$ which satisfies the holomorphic anomaly equation with source $$\sqrt{2(\kappa+2)} \sum_{2j+1=0}^{2\kappa+3}\vartheta_{\kappa+2,2j+1}(2\xi;\tau)\Theta_{\kappa+2,2j+1}(\bar \tau).$$
It follows that $$\label{eqn.Jk}
J_k \equiv \frac{\eta(\tau)^3}{\theta(2\xi;\tau)} {\cal A}_{1,\kappa+2}(\xi;\tau)$$ is the holomorphic part of a solution to the holomorphic anomaly equation sourced by the flavoured $S^3_k$ torus one-point function. It has a double pole as $\xi \to 0$. Its $q$-expansion has integral coefficients.
In particular, (\[eqn.Jk\]) is a natural candidate for the flavoured elliptic genus for some SQFT which has $S^3_k$ as one boundary component and some other boundary component with a geometry analogous to ${{\mathds}R}^2 \times {{\mathds}R}^2$, with both planes rotated by the flavour symmetry, so as to be flavoured-compact in the sense of §\[subsec.flavoured-compact\].
Identify the corresponding $(0,4)$ four-dimensional “trumpet” geometry.
In principle, given a group $G$, one may define a version of “$G$-flavoured topological modular forms” by working with the derived moduli stack of elliptic curves equipped with sufficiently-nondegenerate $G$-bundle. Such a theory should have classes represented by flavoured-compact manifolds, and this “trumpet” should be a “flavoured nullcobordism” of $S^3_k$.
Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. We thank S. Murthy and E. Witten for discussions.
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[^1]: Our invariant applies to SQFTs of gravitational anomaly $c_R - c_L \in -\frac12 + 2{{\mathds}Z}$.
[^2]: Our convention will be to use powers of $\eta$ to correct for multipliers in (mock) modular forms. The “generalized mock modularity” condition we demand for $f_1$ is that $f_1(\tau) = \lim_{\bar\tau \to -i\infty} \hat{f}_1(\tau,\bar\tau)$ where $\hat{f}_1(\tau,\bar\tau)$ is modular invariant with appropriate weights and $$\sqrt{-8\tau_2} \partial_{\bar\tau} \hat{f}_1(\tau,\bar\tau) = \eta(\tau)^3 \times (\text{one-point function of } \bar{G}).$$ As usual, $\tau_2 = \frac1{2i}(\tau - \bar\tau)$. One must make various essentially-arbitrary sign choices, one of which is the choice of square root $\sqrt{-8}$.
[^3]: Because of the gravitational anomaly $c_R - c_L = \frac32 = -\frac12 + 2 \times (\text{odd})$, the vector space of bosonic ground states is automatically pseudoreal aka quaternionic, and so of even complex dimension. If $c_R - c_L$ were instead $-\frac12 + 2 \times (\text{even})$, then the bosonic ground states would be a real vector space, $f_2$ would take half-integral values, and we would consider its class mod ${{\mathds}Z}(\!(q)\!)$ rather than mod $2{{\mathds}Z}(\!(q)\!)$.
[^4]: We will use the term “Witten genus” for the combination $\eta(\tau)^n\times(\text{elliptic genus})$.
[^5]: The analogous invariant of TMF classes captures all of the $3$-torsion in degree $n=-1 \pmod 4$, but only some of the $2$-torsion [@MR3278648].
[^6]: The connection between free fermion SCFTs and Lie algebras is described in [@MR791865]. The SCFT in Puzzle \[mainpuzzle\] corresponds to the Lie algebra $\mathfrak{su}(2)$.
[^7]: Real-analytic modular forms have two weights. A weight $(w,w')$ modular form transforms under $\bigl( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\bigr) \in \mathrm{SL}(2,{{\mathds}Z})$ with a factor of $(c\tau+d)^w(c\bar\tau+d)^{w'}$.
[^8]: A priori, the partition function can blow up at the cusp in a manner controlled by the IR central charges of the theory. If the SQFT is “compact” then its index will be weakly holomorphic by a standard argument.
[^9]: We include the $(-1)^{\frac{n}{4}}$ phase because the torus one-point function of ${:}\psi_1\psi_2{:}$ in $\operatorname{Fer}(2)$ equals $\sqrt{-1}\, \eta(\tau)^2$; compare equation (\[eqn-ferbarfer\]).
[^10]: When $n <0$, one cannot use spectator fermions to handle the sign ambiguity. One may instead call upon an equivalent discussion in terms of relative quantum field theories in the sense of [@FreedTeleman2012].
[^11]: The Fourier expansion of a fermion in the Ramond sector is integrally-graded.
[^12]: In the sense of §\[sec.noncompact\], the fermion $\lambda$ is the “spectator” corresponding to the noncompact direction.
[^13]: In the IR limit, the stress-energy tensor has only two terms, the chiral and antichiral parts $T(z),\bar{T}(\bar z)$. In the nonconformal case, the stress-energy tensor picks up a third component — its “trace” — measuring the dependence of $Z_{RR}$ on the size of the worldsheet torus.
[^14]: The precise proportionality factor is hard to derived from path integral considerations. Rather, the factor of $\sqrt{-8}$ in (\[eqn.holomorphicanomaly\]) comes from careful computation of examples in Section \[sec.cigar\].
[^15]: It would be interesting to work out the relation between this holomorphic anomaly equation and the holomorphic anomaly equation which occurs for [*modified*]{} elliptic genera of the world-volume theories of E-strings [@Minahan:1998vr], MSW $M5$-strings [@Manschot:2007ha] or Vafa-Witten partition functions [@Manschot:2017xcr].
[^16]: The elliptic curve $E_\tau = {{\mathds}C}/ ({{\mathds}Z}\oplus \tau{{\mathds}Z})$ living in the $\bar\tau \to -i\infty$ limit is called the [*Tate curve*]{}. It does not correspond to a Euclidean 2-torus because we explicitly broke the relation between $\tau$ and $\bar\tau$. When $\tau$ is pure-imaginary, the Tate curve may be pictured as a nonrelativistic torus in which the “space” and “time” axes point in the $z$ and $\bar z$ directions, and the “space” direction is infinitely small compared to the “time” direction.
[^17]: The time-reversal structure on ${\mathcal F}[S^1]$ arises from $180^\circ$-rotation of ${\mathcal F}$ through a slightly subtle argument [@GPPV §3.2.2].
[^18]: The standard convention is that $\operatorname{Cliff}(n,{{\mathds}R})$ is the real superalgebra with $n$ odd generators $\gamma_1,\dots,\gamma_n$, anticommuting with each other and each squaring to $-1$. We will later use the reasonably-standard notation $\operatorname{Cliff}(-n,{{\mathds}R})$ to mean the algebra with the same odd generators but with $\gamma_i^2 = +1$. But in fact these conventions are purely arbitrary: one could decide, with no change in the final results, that $\gamma_i^2 = +1$ in $\operatorname{Cliff}(n,{{\mathds}R})$ and that $\gamma_i^2 = -1$ in $\operatorname{Cliff}(-n,{{\mathds}R})$. The reason for this arbitrariness is that the category of real superalgebras admits an automorphism exchanging $\operatorname{Cliff}(n,{{\mathds}R})$ with $\operatorname{Cliff}(-n,{{\mathds}R})$. As with all sign conventions, what is important is to be consistent.
[^19]: Note that we again encounter an ambiguity in the sign of the index.
[^20]: We already knew that the Witten genus vanished when $n$ was not divisible by $4$, since there are no modular forms of weight not divisible by $2$.
[^21]: As a real superalgebra, $\operatorname{Cliff}(2,{{\mathds}R})$ is isomorphic to a “semidirect tensor product” ${{\mathds}C}\rtimes \operatorname{Cliff}(1,{{\mathds}R})$, where the odd generator of $\operatorname{Cliff}(1,{{\mathds}R})$ acts by complex conjugation on ${{\mathds}C}$.
[^22]: Indeed, one may always achieve this by adding massive modes.
[^23]: There are two choices for this isomorphism, yet another manifestation of the sign ambiguity in the definition of the Witten genus.
[^24]: If one cares about an SQFT ${\mathcal B}$ that is not initially conformal, then it should be RG-flowed to an SCFT before proceeding. Without this step, one would also need anomaly equations encoding the dependence of $Z_{RR}({\mathcal F})$ on the size of the worldsheet torus. As with the holomorphic anomaly, there is no “bulk” contribution to this dependence, but there can be a contribution from the “boundary” ${\mathcal B}$. Indeed, if ${\mathcal B}$ were not conformal, then the one-point function $g$ would include a size dependence. These size dependences are fully controlled by the Zamolodchikov c-theorem. One can confidently guess the IR limit of UV SQFTs with enough symmetry, and all examples in this paper are already SCFTs. For instance, the examples in Section \[sec.S3\] are the expected IR limits of the round $S^3$ with various B-fields.
[^25]: The class $[f_2]$ is 2-torsion by construction, and so the sign in $[f_1] - [f_2]$ is irrelevant.
[^26]: A String structure consists of a spin structure together with a trivialization of the characteristic class on spin manifolds called $\frac{p_1}2$. Target-space String structures are required in order to cancel the anomalies when building ${\mathcal N}=(0,1)$ sigma models [@MR796163].
[^27]: The minus sign comes from the using cohomological rather than homological degree.
[^28]: We will not distinguish cohomology theories from their corresponding spectra — the cohomology theory associated to $\mathrm{MString}_\bullet$ is usually called $\Omega^{\mathrm{String}}_\bullet$.
[^29]: The String structure is used to cancel an anomaly in the construction of the sigma model [@MR796163].
[^30]: The KO-valued deformation-invariant of SQM models is used regularly in physics — for instance, we used it in §\[sec.int\]. The main theorem in [@MR2648897] says that it is a complete invariant: two SQM models are in the same deformation class if and only if their KO-valued invariants agree, and this remains true even in families, so that the $E_\infty$ ring spectrum $\mathrm{SQM}_\bullet$ is homotopy-equivalent to $\mathrm{KO}_\bullet$.
[^31]: As a QFT, $\operatorname{Fer}(1)(t) = \operatorname{Fer}(1)$ is conformal, but the supersymmetry breaks conformal invariance.
[^32]: ${\mathcal N}=1$ SQM models require the target to be spin but not String.
[^33]: Compare [@MR3223024 Chapter 13, §1], but note that what is reported there are the localizations on $\pi_*\mathrm{TMF}$ at the primes $2$ and $3$; the interesting $3$-local class is called simply $\{\alpha\Delta\}$, since $8$ is invertible $3$-locally.
| 0 |
---
abstract: |
We present adaptive optics assisted integral field spectroscopy of nine H$\alpha$-selected galaxies at $z$=0.84–2.23 drawn from the HiZELS narrow-band survey. Our observations map the kinematics of these star-forming galaxies on $\sim$kpc-scales. We demonstrate that within the ISM of these galaxies, the velocity dispersion of the star-forming gas ($\sigma$) follows a scaling relation $\sigma\propto\Sigma_{\rm SFR}^{1/n}$+$\,constant$ (where $\Sigma_{\rm SFR}$ is the star formation surface density and the constant includes the stellar surface density). Assuming the disks are marginally stable (Toomre $Q$=1), this follows from the Kennicutt-Schmidt relation ($\Sigma_{\rm SFR}$=$A\Sigma_{\rm
gas}^n$), and we derive best fit parameters of $n$=1.34$\pm$0.15 and $A$=3.4$_{-1.6}^{+2.5}\times$10$^{-4}$M$_{\odot}$yr$^{-1}$kpc$^{-2}$, consistent with the local relation, and implying cold molecular gas masses of M$_{\rm gas}$=10$^{9-10}$M$_{\odot}$ and molecular gas fractions M$_{\rm gas}$/(M$_{\rm
gas}$+M$_{\star}$)=0.3$\pm$0.1, with a range of 10–75%. We also identify eleven $\sim$kpc-scale star-forming regions (clumps) within our sample and show that their sizes are comparable to the wavelength of the fastest growing mode. The luminosities and velocity dispersions of these clumps follow the same scaling relations as local H[ii]{} regions, although their star formation densities are a factor $\sim$15$\pm$5$\times$ higher than typically found locally. We discuss how the clump properties are related to the disk, and show that their high masses and luminosities are a consequence of the high disk surface density.
author:
- 'A.M. Swinbank, Ian Smail, D. Sobral, T. Theuns, P.N. Best, & J.E. Geach,'
title: 'The Properties of the Star-Forming Interstellar Medium at =0.8–2.2 from HiZELS: Star-Formation and Clump Scaling Laws in Gas Rich, Turbulent Disks'
---
Introduction
============
The majority of the stars in the most massive galaxies (M$_{\star}\gsim$10$^{11}$M$_{\odot}$) formed around 8–10billion years ago, an epoch when star formation was at its peak [@Hopkins06; @Sobral12b]. Galaxies at this epoch appear to be gas-rich ($f_{\rm gas}$=20–80%; @Tacconi10 [@Daddi10; @Geach11]) and turbulent [@Lehnert09], with high velocity dispersions given their rotational velocities ($\sigma$=30–100kms$^{-1}$, $v_{\rm max}$/$\sigma\sim
$0.2–1; e.g. @ForsterSchreiber09 [@Genzel08; @Wisnioski11; @Bothwell12]). Within the dense and highly pressurised inter-stellar medium (ISM) of these high-redshift galaxies, it has been suggestes that star formation may be triggered by fragmentation of dynamically unstable gas (in contrast to star-formation occurring in giant molecular clouds in the Milky-Way which continually condense from a stable disk and then dissipate). This process may lead to the to the formation of massive ($\sim $10$^{8-9}$M$_{\odot}$) star-forming regions [e.g. @ElmegreenD07; @Bournaud09] and give rise to the the clumpy morphologies that are often seen in high-redshift starbursts [@Elmegreen09].
In order to explain the ubiquity of “clumpy” disks seen in images of high-redshift galaxies, numerical simulations have also suggested that most massive, star-forming galaxies at $z$=1–3 continually accrete gas from the inter-galactic medium along cold and clumpy streams from the cosmic web [@Keres05; @Dekel09; @Bournaud09; @VandeVoort11]. This mode of accretion is at its most efficient at $z\sim $1–2, and offers a natural route for maintaining the high gas surface densities, star formation rates and clumpy morphologies of galaxies at these epochs. In such models, the gas disks fragment into a few bound clumps which are a factor 10–100$\times$ more massive than star-forming complexes in local galaxies. The gravitational release of energy as the most massive clumps form, torques between in-spiraling clumps and energy injection from star formation are all likely to contribute to maintaining the high turbulence velocity dispersion of the inter-stellar medium (ISM) [e.g. @Bournaud09; @Lehnert09; @Genzel08; @Genzel11].
In order to refine or refute these models, the observational challenge is now to quantitatively measure the internal properties of high-redshift galaxies, such as their cold molecular gas mass and surface density, disk scaling relations, chemical make up, and distribution and intensity of star formation. Indeed, constraining the evolution of the star formation and gas scaling relations with redshift, stellar mass and/or gas fraction are required in order to understand star formation throughout the Universe. In particular, such observations are vital to determine if the prescriptions for star formation which have been developed at $z$=0 can be applied to the rapidly evolving ISM of gas-rich, high-redshift galaxies [@Krumholz10; @Hopkins12b].
To gain a census of the dominant route by which galaxies assemble the bulk of their stellar mass within a well selected sample of high-redshift galaxies, we have conducted a wide field (several degree-scale) near-infrared narrow-band survey (the High-Z Emission Line Survey; HiZELS) which targets H$\alpha$ emitting galaxies in four precise ($\Delta z$=0.03) redshift slices: $z$=0.40, 0.84, 1.47 and 2.23 [@Geach08; @Sobral09; @Sobral10; @Sobral11; @Sobral12a; @Sobral12b]. This survey provides a large, star formation limited sample of identically selected H$\alpha$ emitters with properties “typical” of galaxies which will likely evolve into $\sim $L$_{\star}$ galaxies by $z$=0, but seen at a time when they are assembling the bulk of their stellar mass, and thus at a critical stage in their evolutionary history. Moreover, since HiZELS was carried out in the best-studied extra-galactic survey fields, there is a wealth of multi-wavelength data, including 16–36 medium and broad-band photometry (from rest-frame UV–mid-infrared wavelengths allowing robust stellar masses to be derived), *Herschel* 250–500$\mu$m imaging (allowing bolometric luminosities and star formation rates to be derived) as well as high-resolution morphologies for a subset from the *Hubble Space Telescope* CANDELS and COSMOS ACS surveys.
In this paper, we present adaptive optics assisted integral field spectroscopy of nine star-forming galaxies selected from HiZELS. The galaxies studied here have H$\alpha$-derived star formation rates of 1–27M$_\odot$yr$^{-1}$ and will likely evolve into $\sim
$L$^{\star}$ galaxies by $z$=0. They are therefore representative of the high-redshift star-forming population. We use the data to explore the scaling relations between the star formation distribution intensity and gas dynamics within the ISM, as well as the properties of the largest star-forming regions. We adopt a cosmology with $\Omega_{\Lambda}$=0.73, $\Omega_{m}$=0.27, and H$_{0}$=72kms$^{-1}$Mpc$^{-1}$ in which 0.12$''$ corresponds to a physical scale of 0.8kpc at $z$=1.47, the median redshift of our survey. All quoted magnitudes are on the AB system. For all of the star formation rates and stellar mass estimates, we use a @Chabrier03 initial mass function (IMF).
Observations
============
Details of the target selection, observations and data-reduction are given in @Swinbank12a. Briefly, we selected nine galaxies from HiZELS with H$\alpha$ fluxes 0.7–1.6$\times$10$^{-16}$ergs$^{-1}$cm$^{-2}$ (star formation ratesof SFR$_{\rm
H\alpha}$=1–27M$_{\odot}$yr$^{-1}$) which lie within 30$''$ of bright (R$<$15) stars. We performed natural guide star adaptive optics (AO) observations with the SINFONI IFU between 2009 September and 2011 April in $\sim $0.6$''$ seeing and photometric conditions with exposure times between 3.6 to 13.4ks. At the three redshift slices of our targets, $z$=0.84\[2\], $z$=1.47\[6\] and $z$=2.23\[1\], the H$\alpha$ emission line is redshifted to $\sim
$1.21, 1.61 and 2.12$\mu$m (i.e. into the $J$, $H$ and $K$-bands respectively). The median strehl achieved for our observations is 20% and the median encircled energy within 0.1$''$ (the approximate spatial resolution of our observations) is 25%.
The data were reduced using the SINFONI [esorex]{} data reduction pipeline which extracts, flat-fields, wavelength calibrates and forms the data-cube for each exposure. The final (stacked) data-cube for each galaxy was generated by aligning the individual data-cubes and then combining them using an average with a 3-$\sigma$ clip to reject cosmic rays. For flux calibration, standard stars were observed each night either immediately before or after the science exposures and were reduced in an identical manner to the science observations.
As Fig. \[fig:2dmaps\] shows, all nine galaxies in our SINFONI-HiZELS survey (SHiZELS) display strong H$\alpha$ emission, with luminosities of L$_{\rm H\alpha}\sim $10$^{41.4-42.4}$ergs$^{-1}$. Fitting the H$\alpha$ and \[N[ii]{}\]$\lambda\lambda$6548,6583 emission lines pixel-by-pixel using a $\chi^{2}$ minimisation procedure we construct intensity, velocity and velocity dispersion maps of our sample and show these in Fig. \[fig:2dmaps\] (see also @Swinbank12a for details).
Analysis & Discussion
=====================
Galaxy Dynamics and Star Formation {#sec:dynSF}
----------------------------------
As @Swinbank12a demonstrate, the ratio of dynamical-to-dispersion support for this sample is $v$sin($i$)/$\sigma$=0.3–3, with a median of 1.1$\pm$0.3, which is consistent with similar measurements for both AO and non-AO studies of star-forming galaxies at this epoch [e.g. @ForsterSchreiber09]. The velocity fields and low kinemetry values of the SHiZELS galaxies (total velocity asymmetry, K$_{\rm tot}$=0.2–0.5) also suggest that at least six galaxies (SHiZELS 1, 7, 8, 9, 10, & 11) have dynamics consistent with large, rotating disks, although all display small-scale deviations from the best-fit dynamical model, with $<$data$-$model$>$=30$\pm$10kms$^{-1}$, with a range from $<$data$-$model$>$=15–70kms$^{-1}$ [@Swinbank12a].
We also use the multi-wavelength imaging to calculate the rest-frame SEDs of the galaxies in our sample and so derive the stellar mass, reddenning and estimates of the star-formation history [@Sobral11]. From the broad-band SEDs (Fig. 1 of @Swinbank12a), the average E(B$-$V) for our sample is E(B$-$V)=0.28$\pm$0.10 which corresponds to A$_v$=1.11$\pm$0.27mag and indicates A$_{\rm
H\alpha}$=0.91$\pm$0.21mag. The resulting dust-corrected H$\alpha$ star formation rate for the sample is SFR$_{\rm
H\alpha}$=16$\pm$5M$_{\odot}$yr$^{-1}$, which is consistent with that inferred from the far-infrared SEDs using stacked [*Herschel*]{} SPIRE observations(SFR$_{\rm
FIR}$=18$\pm$8M$_{\odot}$yr$^{-1}$; @Swinbank12a)
Next, to investigate the star formation occurring within the ISM of each galaxy, we measure the star formation surface density and velocity dispersion of each pixel in the maps. Since we do not have spatially resolved reddening maps, for each galaxy we simply correct the star formation rate in each pixel using the best-fit E(B$-$V) for that system. We also remove the rotational contribution to the line width at each pixel by calculating the local $\Delta$V/$\Delta$R across the point spread function (PSF) for each pixel [@Davies11]. In Fig. \[fig:SF\_galgal\] we plot the resulting line of sight velocity dispersion ($\sigma$) as a function of star formation surface density ($\Sigma_{\rm SFR}$) for each galaxy in our sample. We see that there appears to be a correlation between $\Sigma_{\rm SFR}$ and $\sigma$, and as @Krumholz10 show, this power-law correlation may be a natural consequence of the gas and star formation surface density scaling laws. For example, first consider the Toomre stability criterion, $Q$, [@Toomre64]. $$Q\,=\,\frac{\sigma\kappa}{\pi G\Sigma_{\rm disk}}
\label{eqn:toomre}$$ where $\sigma$ denotes the line of sight velocity dispersion, $\Sigma_{\rm disk}$ is the average surface density of the disk, $\kappa$=$a$$v_{\rm max}$/$R$ where $v_{\rm max}$ is the rotational velocity of the disk, $R$ is the disk radius and $a$=$\sqrt2$ for a flat rotation curve. Galaxy whose disks have $Q< $1 are unstable to local gravitational collapse and will fragment into clumps, whereas those with $Q\gsim $1 have sufficient rotational support for the gas to withstand and collapse. As @Hopkins12b [e.g. see also @Cacciato12] point out, gas-rich galaxies are usually driven to $Q\sim $1 since regions with $Q< $1 begin forming stars, leading to super-linear feedback which eventually arrests further collapse due to energy/momentum injection (recovering $Q\sim$ 1). For galaxies with $Q\gg $1, there is no collapse, no dense regions form and hence no star formation (and so such galaxies would not be selected as star-forming systems).
Following @Rafikov01, and focusing on the largest unstable fluctuations, the appropriate combination of gas and stellar surface density ($\Sigma_{\rm gas}$ and $\Sigma_{\star}$ respectively) is $$\Sigma_{\rm disk}\,=\,\Sigma_{\rm gas}\,+\,\left(\frac{2}{1+f_{\sigma}^2}\right)\Sigma_{\star}
\label{eqn:KSlaw}$$ where $f_{\sigma}$=$\sigma_{\star}$/$\sigma_{g}$ is the ratio of the velocity dispersion of the stellar component to that of the gas (see also the discussion in @Romeo11)..
Next, @KS98 show that the gas and star formation surface densities follow a scaling relation $$\left(\frac{\Sigma_{\rm SFR}}{\rm M_{\odot}\,yr^{-1}\,kpc^{-2}}\right)\,=\,A\left(\frac{\Sigma_{\rm gas}}{\rm M_{\odot}\,pc^{-2}}\right)^{\it n}
\label{eqn:KS98}$$ For local, star-forming galaxies, the exponent, $n\sim1.5$ and the absolute star formation efficiency, $A$=1.5$\pm$0.4$\times$10$^{-4}$ [@Kennicutt98] implying an efficiency for star formation per unit mass of $\sim $0.04 which holds across at least four orders of magnitude in gas surface density.
Combining these relations, the velocity dispersion, $\sigma$, should therefore scale as $$\frac{\sigma}{\rm km\,s^{-1}}\,=\,\frac{\pi\,\times\,10^6\,G\,R}{\sqrt2\,v_{\rm max}}
\left(\left(\frac{\Sigma_{\rm SFR}}{A}\right)^{1/n}+\left(\frac{2}{1+f_{\sigma^2}}\right)\frac{\Sigma_{\star}}{10^6}\,\right)
\label{eqn:sS}$$ where $\Sigma_{\rm SFR}$ and $\Sigma_{\star}$ are measured in M$_{\odot}$yr$^{-1}$ and M$_{\odot}$kpc$^{-2}$ respectively, $R$ is in kpc, $v_{\rm max }$ in kms$^{-1}$, and $G$=4.302$\times$10$^{-6}$kpcM$_{\odot}^{-1}$(kms$^{-1}$)$^2$. With a power law index of $n$=1.4, and a marginally stable disk ($Q$=1), for each galaxy we therefore expect a power law relation $\sigma\propto\Sigma_{\rm SFR}^{0.7}$+$constant$ [@Krumholz12].
---------------------- ---------- ------------- ------------------- ------------------------- --------------- ---------------------------- ------------------ ------------- --------------------------------------- -----------------------------------------
ID RA Dec $z_{\rm H\alpha}$ SFR$_{\rm H\alpha}^{a}$ $r_{1/2}^{b}$ $\sigma_{\rm H\alpha}^{c}$ $v_{\rm asym}^d$ E(B$-$V) $\log$($\frac{M_{\star}}{M_{\odot}}$) $\log$($\frac{M_{\rm gas}}{M_{\odot}}$)
(J2000) (J2000) (M$_{\odot}$/yr) (kpc) (kms$^{-1}$) (kms$^{-1}$)
SHiZELS-1 021826.3 $-$044701.6 0.8425 2 1.8$\pm$0.3 98$\pm$15 112$\pm$11 0.4$\pm$0.1 10.03$\pm$0.15 9.4$\pm$0.4
SHiZELS-4 100155.3 $+$021402.6 0.8317 1 1.4$\pm$0.5 77$\pm$20 ... 0.0$\pm$0.2 9.74$\pm$0.12 8.9$\pm$0.4
SHiZELS-7 021700.4 $-$050150.8 1.4550 8 3.7$\pm$0.2 75$\pm$11 145$\pm$10 0.2$\pm$0.2 9.81$\pm$0.28 9.8$\pm$0.4
SHiZELS-8 021821.0 $-$051907.8 1.4608 7 3.1$\pm$0.3 69$\pm$10 160$\pm$12 0.2$\pm$0.2 10.32$\pm$0.28 9.8$\pm$0.4
SHiZELS-9 021713.0 $-$045440.7 1.4625 6 4.1$\pm$0.2 62$\pm$11 190$\pm$20 0.2$\pm$0.2 10.08$\pm$0.28 9.8$\pm$0.4
SHiZELS-10 021739.0 $-$044443.1 1.4471 10 2.3$\pm$0.2 64$\pm$8 30$\pm$12 0.3$\pm$0.2 9.42$\pm$0.33 9.9$\pm$0.4
SHiZELS-11 021821.2 $-$050248.9 1.4858 8 1.3$\pm$0.4 190$\pm$18 224$\pm$15 0.5$\pm$0.2 11.01$\pm$0.24 10.1$\pm$0.4
SHiZELS-12 021901.4 $-$045814.6 1.4676 5 0.9$\pm$0.5 115$\pm$10 ... 0.3$\pm$0.2 10.59$\pm$0.30 9.6$\pm$0.4
SHiZELS-14 100051.6 +02:3334.5 2.2418 27 4.6$\pm$0.4 131$\pm$17 ... 0.4$\pm$0.1 10.90$\pm$0.20 10.1$\pm$0.4
Median ... ... 1.46 7$\pm$2 2.4$\pm$0.7 75$\pm$19 147$\pm$31 0.3$\pm$0.1 10.25$\pm$0.50 9.8$\pm$0.2
\[table:gal\_props\]
---------------------- ---------- ------------- ------------------- ------------------------- --------------- ---------------------------- ------------------ ------------- --------------------------------------- -----------------------------------------
Notes: $^{a}$H$\alpha$ star formation rate using the calibration from @Kennicutt98 with a Chabrier IMF; SFR$_{\rm
H\alpha}$=4.6$\times$10$^{-42}$L$_{\rm H\alpha}$. $^b$H$\alpha$ half light radius, deconvolved for the PSF. $^c$Average velocity dispersion for each galaxy, corrected for beam-smearing due to the PSF. $^d$$v_{\rm asym}$ denotes the best-fit asymptotic rotation speed of the galaxy, and is corrected for inclination (see @Swinbank12a for details on the kinematic modeling of these galaxies).
In order to test whether this model provides an adequate description of our data, we fit the $\Sigma_{\rm SFR}$–$\sigma$ distribution for each galaxy in our sample. To estimate the stellar surface density, $\Sigma_{\star}$, we we follow @Sobral11 and perform a full SED $\chi^2$ fit of the rest-frame UV–mid-infrared photometry using the @Bruzual03 and Bruzual (2007) population synthesis models. We use photometry from up to 36 (COSMOS) and 16 (UDS) wide, medium and narrow bands (spanning [*GALEX*]{} far-UV and near-UV bands to [ *Spitzer*]{}/IRAC) and calculate the rest-frame spectral energy distribution, reddening, star-formation history and stellar mass [@Sobral10]. The stellar masses of these galaxies range from 10$^{9.7-11.0}$M$_{\odot}$ (Table 1; see also @Swinbank12a).
Since the stellar masses are calculated from 2$''$ aperture photometry (and then corrected to total magnitudes using aperture corrections, @Sobral10), to estimate the stellar surface density in the same area as our IFU observations, we assume that stellar light follows an exponential profile with Sersic index, n$_{\rm serc}$=1–2 and calculate the fraction of the total stellar mass within the disk radius, $R$ (which we define as two times the H$\alpha$ half light radius, $r_{\rm h}$). Allowing a range of power-law index from $n$=1.0–1.8 and a ratio of stellar- to gas- velocity dispersion of $f_{\sigma}$=1–2 [@Korchagin03], we calculate the best-fit absolute star formation efficiency, $A$ and in Fig. \[fig:SF\_galgal\] we overlay the best-fit solutions. Over the range $n$=1.0–1.8, the best fit absolute star formation efficiency for the sample is $A$=(4.1$\pm$2.4)$\times$10$^{-4}$M$_{\odot}$yr$^{-1}$kpc$^{-2}$ (where the error-bar incorporates the galaxy-to-galaxy variation, a range of $f_{\sigma}$=1–2, and the errors on the stellar masses of each galaxy). We note that at low star formation rates and stellar masses, there is a non-zero velocity dispersion due to the sound speed ($c_s$) of the gas ($c_s\lsim $10kms$^{-1}$ for the Milky Way at the solar circle) which we have neglected since this is below both the resolution limit of our observations and the minimum velocity dispersion caused the stellar disks in these systems.
We can improve these constraints further assuming that star formation in each galaxy behaves in a similar way. We reiterate that this model assumes the star formation is occurring in a marginally Toomre stable disk, where the star formation follows the Kennicutt-Schmidt Law. Over a range $A$=10$^{-5}$–10$^{-2}$(M$_{\odot}$yr$^{-1}$kpc$^{-2}$) and $n$=0.8–2.5 we construct a likelihood distribution for all nine galaxies and then convolve these to provide a composite likelihood distribution, and show this in Fig. \[fig:KS\]. Although the values of $n$ and $A$ are clearly degenerate, the best-fit solutions have $n$=1.34$\pm$0.15 and $A$=3.4$_{-1.6}^{+2.5}$$\times$10$^{-4}$M$_{\odot}$yr$^{-1}$kpc$^{-2}$ Our derived values for the absolute star-formation efficiency, $A$, and power-law index, $n$ are within the 1-$\sigma$ of the values derived for local galaxies [e.g. @KS98; @Leroy08].
Using the $^{12}$CO to trace the cold molecular gas, @Genzel10 showed that gas and star-formation surface densities of high-redshift ($z\sim $1.5) star-forming galaxies and ULIRGs are also well described by the Kennicutt-Schmidt relation with coefficients $n$=1.17$\pm$0.10 and A=(3.3$\pm$1.5)$\times$10$^{-4}$M$_{\odot}$yr$^{-1}$kpc$^{-2}$, which is comparable to the coefficients we derive from our sample.
In Fig. \[fig:KS\] we plot the star formation and gas-surface surface density for both local and high-redshift star-forming galaxies and ULIRGs from @Genzel10 and overlay the range of acceptable solutions implied by our data. We reiterate that we have adopted $Q$=1 for this analysis and note that if we adopt $Q<$ 1 then the absolute star formation efficiency will be increased proportionally (as shown in Fig. \[fig:KS\]). Nevertheless, this shows that the values of $n$ and $A$ we derive are consistent with the local and high-redshift star-forming galaxies and ULIRGs, but free from uncertainties associated with converting $^{12}$CO luminosities to molecular gas mass, CO excitation or spatial extent of the gas reservoir.
Using the values of $n$ and $A$ we have derived, we infer cold molecular gas masses for the galaxies in our sample of M$_{\rm
gas}$=10$^{9-10}$M$_{\odot}$ with a median M$_{\rm
gas}$=7$\pm$2$\times$10$^{9}$M$_{\odot}$. This suggests a cold molecular gas fraction of M$_{\rm gas}$/(M$_{\rm
gas}$+M$_{\star}$)=0.3$\pm$0.1 but with a range of 10–75%, similar to those derived for other high-redshift starbursts in other surveys [@Tacconi10; @Daddi10; @Swinbank11].
Finally, with estimates of the disk surface density, we can use Eq. \[eqn:toomre\] to construct maps of the spatially resolved Toomre parameter, $Q(x,y)$. Since we set $Q$=1 to derive the coefficients $n$ and $A$, by construction the average $Q$ across the population is unity, but the relative range of $Q(x,y)$ within the ISM of each galaxy is unaffected by this assumption. In Fig. \[fig:2dmaps\] we show the maps of $Q(x,y)$ for each galaxy in our sample (with contours marking $Q(x,y)$=0.5, 1.0 and 2.0). This shows that there is a range of Toomre $Q$ across the ISM, and to highlight the variation with radius, in Fig. \[fig:Q\_r\] we show the Toomre parameter within each pixel of each galaxy as a function of radius (normalised to the half light radius, $r_{\rm h}$). This shows that in the central regions, on average the Toomre $Q$ increases by a factor $\sim $4$\times$ compared to $Q$ at the half light radius, whilst a radii greater than $r_{\rm h}$, $Q$ decreases by approximately the same factor.
### Identification of Star-Forming Regions {#sec:SFregions}
As Fig. \[fig:2dmaps\] shows, the galaxies in our sample exhibit a range of H$\alpha$ morphologies, from compact (e.g. SHiZELS11 & 12) to very extended/clumpy (e.g. SHiZELS7, 8, 9 & 14). To identify star-forming regions on $\sim $kpc scales and measure their basic properties we isolate the star-forming clumps above the background ($\sigma_{\rm bg}$) by first converting the H$\alpha$ flux map into photon counts (accounting for telescope efficiency) and then search for 3$\sigma_{\rm bg}$ over-densities above the radially averaged background light distribution. In this calculation, we demand that any region is at least as large as the PSF. We identify eleven such regions and highlight these in Fig. \[fig:2dmaps\].
It is still possible that selecting star-forming regions in this way may give misleading results due to random associations and signal-to-noise effects. We therefore use the H$\alpha$ surface brightness distribution from the galaxies and randomly generate $10^5$ mock images to test how many times a “clump” is identified. We find that only 2$\pm$1 spurious clumps (in our sample of eleven galaxies) could be random associations.
Next, we extract the velocity dispersion and luminosity of each clump from the using an isophote defining the star-forming region and report their values in Table 2 (the clump velocity dispersions have been corrected for the local velocity gradient from the galaxy dynamics and sizes are deconvolved for the PSF). Using the velocity dispersion and star formation density of each clump, and fixing the power-law index in the Kennicutt-Schmidt relation to $n$=1.34, we compute their absolute star formation efficiencies, deriving a median $A_{\rm
clump}$=5.4$\pm$1.5$\times$10$^{-4}$ (Fig. \[fig:KS\]). This corresponds to an offset (at fixed $n$) from the galaxy-average of $A_{\rm clump}$/$A$=1.3$\pm$0.4. Equivalently, if we fix the absolute star formation efficiency to that of the galaxy-average, then the Toomre parameter in these regions is $Q$=0.8$\pm$0.4.
------------------ ------------------------ ------------------------ ----------------------- ---------------
Galaxy SFR $\sigma_{\rm H\alpha}$ \[N[ii]{}\]/H$\alpha$ $r_{\rm h}$
(M$_{\odot}$yr$^{-1}$) (kms$^{-1}$) (kpc)
SHiZELS-7 0.5$\pm$0.1 40$\pm$10 0.07$\pm$0.03 0.8$\pm$0.2
SHiZELS-7 1.3$\pm$0.1 61$\pm$12 0.34$\pm$0.03 1.0$\pm$0.2
SHiZELS-8 2.0$\pm$0.1 79$\pm$10 0.36$\pm$0.03 0.7$\pm$0.2
SHiZELS-8 1.6$\pm$0.2 95$\pm$14 0.26$\pm$0.04 0.8$\pm$0.2
SHiZELS-8 1.9$\pm$0.1 140$\pm$20 0.21$\pm$0.04 0.9$\pm$0.2
SHiZELS-9 2.1$\pm$0.2 97$\pm$15 0.31$\pm$0.04 0.7$\pm$0.2
SHiZELS-9 2.3$\pm$0.1 80$\pm$10 0.26$\pm$0.03 1.3$\pm$0.2
SHiZELS-9 0.9$\pm$0.1 86$\pm$14 0.40$\pm$0.03 $< $0.7
SHiZELS-14 0.5$\pm$0.1 56$\pm$12 0.12$\pm$0.04 0.9$\pm$0.2
SHiZELS-14 1.1$\pm$0.2 121$\pm$20 0.24$\pm$0.03 $< $0.7
SHiZELS-14 0.2$\pm$0.1 100$\pm$25 $-$0.03$\pm$0.05 0.9$\pm$0.3
Median 1.4$\pm$0.4 88$\pm$9 0.24$\pm$0.06 0.85$\pm$0.10
\[table:clumps\]
------------------ ------------------------ ------------------------ ----------------------- ---------------
\
Notes: Half light radius, r$_h$, is deconvolved for PSF and the velocity dispersion, $\sigma$, is corrected for local velocity gradient (see § \[sec:dynSF\]). The star formation rates (SFR) are calculated from the H$\alpha$ line luminosity using SFR$_{\rm
H\alpha}$=4.6$\times$10$^{-42}$L$_{\rm H\alpha}$.
The Scaling Relations of Local and High-Redshift Star-Forming Regions {#sec:scaling}
---------------------------------------------------------------------
The internal kinematics and luminosities of H[ii]{} regions in local galaxies, derived from the line widths of their emission lines, have been the subject of various studies for some time [e.g. @Terlevich81; @Arsenault90; @Rozas98; @Rozas06; @Relano05]. In particular, if the large line widths of star-forming H[ii]{} regions reflect the virialization of the gas then they can be used to determine their masses. However, it is unlikely that this condition holds exactly at any time during the evolution of a H[ii]{} region due to the input of radiative and mechanical energy, principally from their ionizing stars [e.g. @Castor75]. Nonetheless, the least evolved H[ii]{} regions may well be within a factor of a few (2–3) of having their kinematics determined by their virial masses (at an early stage, the stellar ionizing luminosities are maximized whereas the mechanical energy input is minimized; @Leitherer99). In the case of H[ii]{} regions close to virial equilibrium, the use of the line-width to compute gaseous masses offers a relatively direct means to study the properties since it is independent of the small-scale structure (density, filling factor, etc.).
@Terlevich81 showed that the H$\beta$ luminosity of the most luminous H[ii]{} regions varies as L(H$\beta$)$\propto\sigma^{4.0\pm0.8}$. This result suggests that the most luminous H[ii]{} regions are likely to be virialized, so that information about their masses, and the resultant mass-luminosity relation, could be obtained using the virial theorem (they also claimed a relation between a radius parameter and the square of the velocity dispersion $\sigma$ for H[ii]{} regions, as further evidence for virialization). However, more recent studies, in particular by @Rozas06 suggest that in super-giant H[ii]{} regions, L$\propto\sigma^{2.9\pm0.2}$ may be a more appropriate scaling (the lower exponent arises since H[ii]{} regions with the largest luminosities are generally density-bound, which means that a significant fraction of the ionizing radiation escapes and so does not contribute to the luminosity, making shallower slopes physically possible).
To investigate the scaling relations of star-forming regions, in Fig. \[fig:scaling\] we show the relations between luminosity, size and velocity dispersion of the clumps in our sample compared to Giant Molecular Clouds (GMCs) and H[ii]{} regions in the Milky Way and local galaxies [@Terlevich81; @Arsenault90; @Bordalo11; @Fuentes-Masip00; @Rozas06]. In this plot, we also include the measurements of giant star-forming regions from other high-redshift star-forming galaxies at $z\sim $1 from @Wisnioski11b, the $z\sim $1–2 galaxies from SINS [@Genzel11], and the clumps identified in strongly lensed $z\sim
$1.5–3 galaxies from @Jones10 and @Stark08.
Despite the scatter, the radius–$\sigma$ and $\sigma$–Luminosity relations of the high-redshift clumps approximately follow the same scaling relations as those locally, but extending up to $\sim $kpc scales. Indeed, including all of the data-points in the fits, we derive the scaling between size ($r$), luminosity ($L$) and velocity dispersion ($\sigma$) of $$\log\left(\frac{r}{\rm kpc}\right)\,=\,(1.01\,\pm\,0.08)\,\log\left(\frac{\sigma}{\rm km\,s^{-1}}\right)\,+\,(0.8\,\pm\,0.1)
\label{eqn:rs}$$ and $$\log\left(\frac{L}{\rm erg\,s^{-1}}\right)\,=\,(3.81\,\pm\,0.29)\log\left(\frac{\sigma}{\rm km\,s^{-1}}\right)\,+\,(34.7\pm0.4)
\label{eqn:Ls}$$ Equation \[eqn:rs\] suggests $\sigma\propto R$. If the clouds are self-gravitating clouds with $\sigma\propto R$, then the virial density is constant. The relation L$\propto\sigma^{3.81\pm0.29}$ is in reasonable agreement with the early work from @Terlevich81, and steeper than that found for super-giant H[ii]{} regions in local galaxies [@Rozas06], although the large error bars (on both the local and high-redshift data) preclude any firm conclusions. Clearly a larger sample is required to confirm this result and/or test whether the scatter in the data is intrinsic.
If the star-forming regions we have identified are short lived, then these scaling relations effectively reflect initial collapse conditions of the clump as it formed, since a clump can not evolve far from those initial conditions [e.g. @Ceverino10]. In this case, the relation between radius, velocity dispersion and gas mass should follow $r$=$\sigma^2$/($\pi G \Sigma_{\rm disk}$) (see § \[sec:Sigma\_disk\_Sigma\_clump\]). In Fig. \[fig:scaling\] we therefore overlay contours of constant gas mass in the r–$\sigma$ plane, which suggests that the [*initial*]{} gas masses for the clumps is $M_{\rm
gas}^{initial}$=2$\pm$1$\times$10$^9$M$_{\odot}$ a factor $\sim $1000$\times$ more massive then the star-forming complexes in local galaxies (e.g. see also @Elmegreen09 [@Genzel11; @Wisnioski11b]). Assuming our gas mass estimates from § \[sec:dynSF\], then these star-forming regions may contain as much as $\sim$10–20% of the cold molecular gas in the disk.
Turning to the relation between size and luminosity of the star-forming regions, it is evident from Fig. \[fig:scaling\] that the star formation densities of the high-redshift clumps higher than those locally. Indeed, local star-forming regions follows a scaling relation $$\log\left(\frac{L}{\rm erg\,s^{-1}}\right)\,=\,(2.91\,\pm\,0.15)\log\left(\frac{r}{\rm kpc}\right)\,+\,(32.1\,\pm\,0.3)
\label{eqn:Lr}$$ We do not have sufficient number of objects or the dynamic range to measure both the slope and zero-point of the size-luminosity relation in the high-redshift clumps, and so instead we fix the slope of the local relation (which is $L\propto r^{2.91\pm0.15}$) and fit for the zero-point evolution and obtain $$\log\left(\frac{L}{\rm erg\,s^{-1}}\right)\,=\,(2.91\,\pm\,0.15)\log\left(\frac{r}{\rm kpc}\right)\,+\,(33.2\,\pm\,0.4)
\label{eqn:Lr_hiz}$$ This suggests that high-redshift star-forming regions have luminosities at a fixed size that are on average a factor 15$\pm$5$\times$ larger than those locally (see also @Swinbank09 [@Swinbank10Nature; @Jones10; @Wisnioski11b]). We note that high luminosities at fixed size have been found in local starbursts, such as in the Antennae [@Bastian06], whilst offsets of factors $\sim $50$\times$ have been inferred for star-forming regions in high-redshift galaxies [e.g. @Swinbank09; @Jones10; @Wisnioski11b].
The Relation between the Disk and Clump Properties {#sec:Sigma_disk_Sigma_clump}
--------------------------------------------------
It is possible to relate the properties of the clumps to the overall properties of the disk [e.g. @Hopkins12a]. For example, the velocity dispersion of the fastest growing Jeans unstable mode which can not be stabilised by rotation in a gas disk is given by $$\sigma_t(R)^2\,=\,\pi\,G\,\Sigma_{\rm disk}\,R
\label{eqn:sigma_Sigma}$$ [e.g. @Escala08; @Elmegreen09b; @Dekel09; @Genzel11; @Livermore12a]. The critical density for collapse ($\rho_c$), on scale $R$ from a turbulent ISM is given by $$\rho_c\,=\,\frac{3}{4\,\pi\,R^{3}}\,M_{\rm J}\,\simeq\,\frac{9}{8\,\pi\,R^2\,G}\,\sigma_t(R)^2
\label{eqn:MJeans}$$ where $\sigma_t(R)$ is the line of sight velocity turbulent velocity dispersion and $M_{\rm J}$ is the Jeans mass. The critical density for collapse therefore scales as $$\rho_c(R)\,=\,\frac{9}{8R}\,\Sigma_{\rm disk}
\label{eqn:R_SigmaDisk}$$ Assuming that the cloud contracts by a factor $\simeq $2.5 as it collapses, the post-collapse surface density of the cloud is $$\Sigma_{\rm cloud}\simeq\,10\,\rho_c R\simeq \,10\,\Sigma_{\rm disk}
\label{eqn:Sigmacloud_Sigmadisk}$$ [see also @Livermore12a]. Thus, the surface density of the collapsed cloud is independent of radius and proportional to the surface density of the disk, with the normalisation set by the collapse factor and under the assumption $Q$=1. @Hopkins12b show that this model provides an reasonable fit to giant molecular clouds in the Milky-Way, and further, suggests that the surface density (and hence surface brightness) of clouds should increase with the surface density of the disk.
Using our estimates of the stellar and gas masses and spatial extent of the galaxies in our sample, we derive disk surface densities of $\Sigma_{\rm
disk}$=1.1$\pm$0.4$\times$10$^{9}$M$_{\odot}$kpc$^{-2}$, and hence expect the mass surface densities of the star-forming regions that form to have mass surface densities of $\Sigma_{\rm clump}\sim
$10$^{10}$M$_{\odot}$kpc$^{-2}$. It is instructive to compare this to the average mass surface density of the clumps. For example, assuming that their velocity dispersions are virial and adopting $M_{\rm clump}$=$C\sigma^2 r_h$/$G$, using the average velocity dispersion and size of the clumps (Table 2), we derive an average clump mass surface density of $\Sigma_{\rm
clump}$=8$\pm$2$\times$10$^9$M$_{\odot}$kpc$^{-2}$ with $C$=5 (appropriate for a uniform density sphere). Although this calculation should be considered crude as it is unclear whether the velocity dispersions we measure are virial, it is encouraging that the predicted surface mass densities of the clumps are similar to those inferred from their velocity dispersions and sizes.
Finally, @Hopkins12b (see also @Escala08 and @Livermore12a) show that for a marginally stable disk of finite thickness, density structures on scales greater than $h$ will tend to be stabilised by rotation which leads to an exponential cut off of the clump mass function above $$M_0\,\simeq\,\frac{4\pi}{3}\,\rho_c(h)\,h^3\,=\,\frac{3\,\pi\,G^2}{2}\frac{\Sigma_{\rm disk}^3}{\kappa^4}
\label{eqn:M0_1}$$ or $$\frac{M_0}{M_{\odot}}\,=\,8.6\,\times\,10^{3}\left(\frac{\Sigma_{\rm disk}}{\rm 10\,M_{\odot}\,pc^{-2}}\right)^3\,\left(\frac{\kappa}{\rm 100\,km\,s^{-1}\,kpc}\right)^{-4}
\label{eqn:M0_2}$$ This suggests that the most massive clumps that can form in a disk (“the cut off mass”) depends strongly on the disk surface density – increasing the disk surface density increases mass of the clumps that are able to form [e.g. @Escala11]. However, there is also a competing (stabilising) factor from the epicyclic frequency such that a fixed radius, higher circular velocities reduce the mass of the largest clumps able to form.
Applying equation \[eqn:M0\_2\] to the Milky-Way, with a cold molecular gas fraction of 10%, $f_{\sigma}$=2 [@Korchagin03], the average surface density is $\Sigma_{\rm
disk}$=35M$_{\odot}$pc$^{-2}$ and for $\kappa$=220kms$^{-1}$/8kpc [@Feast97] the cut off mass should be $M_0\sim$10$^7$M$_{\odot}$, in good agreement with the characteristic mass of the largest galactic GMCs [e.g. @Stark06]
How does the cut-off mass for our high-redshift sample compare to local galaxies? For $f_{\sigma}$=2, and using the scaling relations derived in § \[sec:dynSF\] to estimate the gas mass (Table 1), ($A$=3.4$\times$10$^{-4}$M$_{\odot}$yr$^{-1}$kpc$^{-2}$ and $n$=1.34) we derive a range of cut off masses of M$_{0}$=0.3–30$\times$10$^{9}$M$_{\odot}$ (with a median and error of the sample of $M_{0}$=9$\pm$5$\times$10$^{9}$M$_{\odot}$). This is similar to the mass inferred for the brightest star-forming regions seen in high-resolution images of other high-redshift galaxies [@Elmegreen89; @ElmegreenD07; @Elmegreen09; @Bournaud09; @ForsterSchreiber11; @Genzel11; @Wisnioski11b], and a factor $\sim $1000$\times$ higher than the largest characteristic mass of a star-forming region in the Milky-Way.
In Fig. \[fig:mcut\] we plot our estimates of the the cut off mass versus the clump star-formation densities for the galaxies in our sample [see also @Livermore12a]. We use the H$\alpha$ derived star-formation rate for each clump, corrected for galaxy reddening (note that we do not have reddening estimates for individual clumps and so we assume a factor 2$\times$ uncertainty in their star formation surface density). We also include estimates of the cut off mass and star formation surface density from the SINS survey of $z\sim $2 galaxies from @Genzel11 (with dynamics measured from @ForsterSchreiber09 and @Cresci09), as well as measurements from the lensing samples of @Livermore12a ($z\sim
$1) and @Jones10 ($z\sim $2). Although the error-bars on individual measurements are large (particularly due to the uncertainties in deriving the gas surface density from the Kennicutt-Schmidt law), as can be seen from Fig. \[fig:mcut\], galaxies with high cut-off masses tend to have higher clump luminosity surface densities.
It is also useful to adopt simple models for the evolution of galaxy disks and gas fraction to investigate how the cut off mass and clump properties may be expected to evolve with redshift. For example, @Dutton11 present an analytic model for the evolution of disk scaling relations (size, rotational velocity and stellar mass with redshift; see @Dutton11 Table 3). Combining with a simple model for the evolution of the gas fraction $f_{\rm gas}\propto
$(1+$z$)$^{b}$ with $b$=1.5–2.5 [@Geach11] and using Eq. \[eqn:Sigmacloud\_Sigmadisk\] and Eq. \[eqn:M0\_2\] we show the expected evolution of the cut-off mass and clump luminosity surface density with redshift. This shows that as the gas fraction increases (and adopting evolving models for the size, disk and circular velocity of galaxies), then the cut off mass should increase by a factor 10–100,$\times$ over the redshift range $z$=0–2.5 whilst the star formation density of the clumps should increase by a approximately an order of magnitude over the same redshift range. Although this is a simple model, this framework allows us to understand why the properties of the star-forming clumps within the ISM of our sample of high-redshift galaxies are different to those typically found in star-forming galaxies locally.
Conclusions
===========
We have presented resolved spectroscopy of nine star-forming galaxies at $z$=0.84–2.23 selected from the UKIRT/HiZELS survey. These galaxies have reddenning corrected star-formation rates of SFR=16$\pm$5M$_{\odot}$yr$^{-1}$ and so are representative of the high-redshift population [@Sobral12b]. The H$\alpha$ dynamics suggest that the ionised gas in at least six galaxies is in the form of large, rotating disks. We use the inferred rotation speeds of these systems, together with the spatial extent of the H$\alpha$ to investigate the star formation within the ISM, and we derive the following main conclusions:
$\bullet$ The star formation and velocity dispersion within the ISM of these high-redshift galaxies follow a power-law relation of the form $\sigma\propto A\Sigma_{\rm gas}^{1/n}$+$constant$ where the coefficients, $A$ and $n$ are set by the Kennicutt-Schmidt Law ($\Sigma_{\rm SFR}$=$A\Sigma_{\rm gas}^n$) and the constant includes the disk stellar surface density of finite thickness. Assuming the gas disks are marginally stable ($Q$=1) we combine the solutions for each galaxy and derive best-fit parameters of power-law exponent, $n$=1.34$\pm$0.15 and absolute star formation efficiency, $A$=3.4$_{-1.6}^{+2.5}$$\times$10$^{-4}$M$_{\odot}$yr$^{-1}$kpc$^{-2}$. These values are consistent with the parameters derived via $^{12}$CO observations for both local and high-redshift star-forming galaxies, but free from any assumptions about $^{12}$CO–H$_2$ conversion factors, $^{12}$CO excitation or the spatial extent of the gas reservoir.
$\bullet$ Applying these coefficients, we infer cold molecular gas masses in the range M$_{\rm gas}$=10$^{9-10}$M$_{\odot}$ with a median M$_{\rm gas}$=7$\pm$2$\times$10$^{9}$M$_{\odot}$ and hence a cold molecular gas fraction of M$_{\rm gas}$/(M$_{\rm
gas}$+M$_{\star}$)=0.3$\pm$0.1 but with a range of 10–75%.
$\bullet$ Using a simple analytic model, we show that the largest structures that can form within the disk (the cut-off mass, $M_0$) are set by the disk surface density with a competing (stabilising) force from the epicyclic frequency such that M$_0\propto\Sigma_{\rm
disk}^3\kappa^{-4}$. For the galaxies in our sample, we derive cut off masses of $M_0\sim $10$^{9}$M$_{\odot}$, a factor $\sim
$1000$\times$ higher than the largest characteristic mass of GMCs in the Milky-Way.
$\bullet$ Within the ISM of these galaxies, we reliably isolate eleven $\sim $kpc-scale star-forming regions and measure their properites. We show that their luminosities and velocity dispersions follow the same scaling relations between size and velocity dispersion as local H[ii]{} regions. Assuming the line widths are virial, the masses derived for these star-forming regions are consistent with those implied by the cut-off mass. However, we find that the luminosity densities of these star-forming regions are a factor $\sim
$15$\times$ higher than those typically found locally, which we attribute to the requirement that the surface density of the (collapsed) cloud must be $\sim$10$\times$ that of the disk.
Overall, the scaling relations we have derived suggest that the star formation processes in high-redshift disks are similar to those in local spiral galaxies, but occurring in systems with a gas rich and turbulent ISM. Given the paucity of gas-rich, clumpy disk-like high-redshift galaxies [@ElmegreenD07; @Elmegreen09; @ForsterSchreiber11b], the next step in these studies is to spatially resolve the cold molecular gas via CO spectroscopy in a well selected sample in order to better constrain the interation between star-formation and gas dynamics. Through comparisons with cosmologically based numerical simulations [e.g. @Crain09; @VandeVoort11; @Ceverino10], as well as high resolution simulations of individual gas-rich disks [e.g. @Agertz09; @Krumholz10b] such observations may begin to differentiate whether the dominant mode of accretion is via three-dimensional cold gas flows accrete from the inter-galactic medium, or from two-dimensions from outskirts of the disk as gas cools from the hot halo.
acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the anonymous referee for their constructive report which significantly improved the content and clarity of this paper. We thank Mario van der Ancker for help and support with the SINFONI planning/observations, and Richard Bower, Avashi Dekel, Reinhard Genzel, Rachael Livermore, Natascha Förster-Schreiber, and Phil Hopkins for a number of very useful discussions. AMS gratefully acknowledges an STFC Advanced Fellowship. DS is supported by a NOVA fellowship. IRS acknowledges support from STFC and a Leverhume Senior Fellowship. JEG is supported by a Banting Fellowship, administered by the Natural Sciences and Engineering Research Council of Canada. This research was also supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. The data presented here are based on observations with the SINFONI spectrograph on the ESO/VLT under program 084.B-0300.
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abstract: 'Metapopulation theory for a long time has assumed dispersal to be symmetric, i.e. patches are connected through migrants dispersing bi-directionally without a preferred direction. However, for natural populations symmetry is often broken, e.g. for species in the marine environment dispersing through the transport of pelagic larvae with ocean currents. The few recent studies of asymmetric dispersal concluded, that asymmetry has a distinct negative impact on the persistence of metapopulations. Detailed analysis however revealed, that these previous studies might have been unable to properly disentangle the effect of symmetry from other potentially confounding properties of dispersal patterns. We resolve this issue by systematically investigating the symmetry of dispersal patterns and its impact on metapopulation persistence. Our main analysis based on a metapopulation model equivalent to previous studies but now applied on regular dispersal patterns aims to isolate the effect of dispersal symmetry on metapopulation persistence. Our results suggest, that asymmetry in itself does not imply negative effects on metapopulation persistence. For this reason we recommend to investigate it in connection with other properties of dispersal instead of in isolation.'
address:
- 'University of Gothenburg, Department of Marine Ecology, Box 461, SE-405 30 Göteborg, Sweden'
- 'University of Gothenburg, Department of Marine Ecology, Tjärnö Marine Biological Laboratory, SE-452 96 Strömstad, Sweden'
author:
- David Kleinhans
- 'Per R. Jonsson'
title: On the impact of dispersal asymmetry on metapopulation persistence
---
Connectivity matrix, dispersal network, symmetry, metapopulation viability.
Introduction
============
Many species are structured in space with dispersal and migration connecting local populations into metapopulations [@Levins69; @Hanski97]. The fundamental dynamics of metapopulations are determined by local extinction, dispersal from the local populations, and colonisation success leading to the establishment of new sub-populations [@Levins69]. Metapopulation dynamics may determine a range of ecological and evolutionary aspects including population size [@Gyllenberg92], persistence [@Roy05], spatial distribution [@Roy08], epidemic spread [@McCallum02; @Davis08], gene flow [@Sultan02], and local adaptation [e.g. @Hanski98; @Joshi01]. Much interest has focused on the effect of the spatial structure of metapopulations and how local populations are connected through dispersal. Connectivity among subpopulations is also increasingly emphasized in management and conservation, e.g. to prevent fragmentation of landscapes [@Crooks] and in the design of protected areas and nature reserves [@vanTeeffelen06].
Early models [e.g. @Levins69; @Hanski] assumed identical dispersal probability among habitat patches. The initial focus of spatially explicit metapopulation theory was to explore processes that generate spatial patterns in homogeneous landscapes [@Hanski02; @Malchow]. Later, spatially explicit models were designed to let dispersal probability be a function of patch size or the distance between local habitat patches [@Hanski94; @Hanski02]. One aspect of dispersal that only has been implicitly included in realistic models but not studied in isolation is when dispersal is asymmetric. Asymmetric dispersal is expected for many metapopulations, e.g. where dispersal is dominated by wind transport of pollen and seeds [@Nathan01], and for marine species with spores and larvae transported by ocean currents [@Wares01]. Consequently, it is important to understand how asymmetric dispersal may affect the dynamics and persistence of metapopulations with potential implications for the design of nature reserves. Some studies have considered asymmetric dispersal [e.g. @Pulliam91; @Kawecki02; @Artzy-Randrup10] but have not analysed effects on metapopulation viability.
In a recent contribution a conceptual model was developed to explore the effects of dispersal asymmetry on metapopulation persistence [@Vuilleumier06]. The viability of metapopulations was investigated for different dispersal patterns randomly connecting pairs of patches through either unidirectional or bidirectional dispersal routes. @Vuilleumier06 concluded, that asymmetric dispersal leads to a distinct increase in the extinction risk of metapopulations. In a similar study @Bode08 investigated correlations between metapopulation viability and statistics of the dispersal network; they also found that asymmetric dispersal links resulted in higher extinction risk. Another very recently published work investigates metapopulation viability for a selection of asymmetric dispersal patterns and supports the findings of previous works [@Vuilleumier10].
The main objective with this study is to isolate the effect of dispersal asymmetry from other properties of the metapopulation network. When changing the degree of symmetry of dispersal networks this generally may simultaneously influence the number of isolated patches and other aspects of the network such as the balance of dispersal in the individual patches [see e.g. Figure 4 in @Vuilleumier06]. Since metapopulations are known to be sensitive in particular to the density of the dispersal network [@Barabasi04Review], the existence of closed cycles of dispersal [@Armsworth02], and the hierarchy of dispersal in directed networks [@Bode08; @Artzy-Randrup10] these secondary implications could confound any effect of asymmetric dispersal. We resolve the problem by restricting our main analysis to *regular* networks.
In this paper we in particular analyse the effect of asymmetric dispersal on metapopulation persistence in more detail, with an initial focus on regular dispersal networks. We employ models of synthetic dispersal patterns and demonstrate that asymmetric dispersal per se may not lead to an increase in metapopulation extinction risk. The significance of our results, their consequence for general dispersal patterns and their relations to previous works are addressed in detail in Section \[sec:discussion\].
Material and Methods
====================
For ease of discussion we focus on the metapopulation model used in previous approaches [@Vuilleumier06; @Bode08; @Vuilleumier10]. This stochastic patch occupancy model connects a number of $N$ patches through a complex dispersal matrix; the model is detailed in Section \[sec:metapopulation-model-vuilleumier\]. Within the scope of this work the viability of metapopulations exposed to dispersal patterns with different degree of symmetry is investigated. A consistent definition of the degree of symmetry and details on the dispersal patterns are provided in Sections \[sec:symmetry-def\] and \[sec:viab-metap-conn\].
\[sec:metapopulation-model-vuilleumier\]Metapopulation model
------------------------------------------------------------
We consider a metapopulation consisting of $N$ patches of equal quality, where, at a given time, each patch is either empty ($0$) or populated ($1$). Interactions of the patches are specified by means of the $N\times N$ connectivity matrix $D$, where the elements $d_{ij}\in\{0,1\}$ determine whether patch $j$ is connected to patch $i$ ($d_{ij}=1$) or not ($d_{ij}=0$). For ease of discussion we require $d_{ii}=0$ for all $i$ implying that patches are not connected with themselves.
Building on previous works we used a stochastic discrete time model for a metapopulation of $N$ patches and tested metapopulation viability with respect to different connectivity matrices [@Vuilleumier06; @Bode08; @Vuilleumier10]. The model, which is attractive in its simplicity, implements dispersal through the connectivity matrix $D$. Initially all $N$ patches are populated. At each time step two events occur in succession: first, populated patches go extinct at per patch probability $e$. Subsequently, empty patches can be colonised with probability $c$ by each incoming dispersal connection from a populated patch. Newly populated patches cannot give rise to colonisation of other patches at the same time step they have been colonised.
In order to estimate the extinction risk of metapopulations the model is iterated $T$ times. If any populated patch is left after the $T^{\mbox{th}}$ iteration, the metapopulation is termed *viable* and *extinct* otherwise. As @Vuilleumier06 we chose the parameters $e=0.5$ and $T=1,000$, and discuss the probability of extinction of metapopulations consisting of $N=100$ patches as a function of the colonisation probability $c$.
\[sec:symmetry-def\]Symmetry of dispersal patterns
--------------------------------------------------
For characterisation of the symmetry properties of dispersal patterns the connectivity matrix $D$ is divided into its symmetric and anti-symmetric contributions, $S$ and $A$, by defining the matrix elements
\[eq:def-a-s\] $$\begin{aligned}
s_{ij} & := & \min\left(d_{ij},d_{ji}\right)\\
a_{ij} & := & d_{ij}-s_{ij}\quad.
\end{aligned}$$
Based on these matrices the degree of symmetry $\gamma$ of dispersal patterns is defined as the ratio of symmetric connections among all connections: $$\gamma:=\frac{\sum_{i,j}s_{ij}}{\sum_{i,j}a_{ij}+s_{ij}}\quad.\label{eq:def-deg-asymm}$$ Note that $1-\gamma$ is related to the asymmetry $Z$ discussed in [@Bode08].
By means of Equation , the symmetry properties of dispersal patterns are put on a firm footing: Dispersal patterns are called *symmetric* if $\gamma=1$ and *anti-symmetric* if $\gamma=0$. Generally connectivity matrices $D$ with intermediate $\gamma$ are neither symmetric nor anti-symmetric. We term them *asymmetric* if $\gamma<1$ corresponding to dispersal directed at least to some degree.
\[sec:viab-metap-conn\]Viability of metapopulations connected through regular dispersal patterns
------------------------------------------------------------------------------------------------
![\[fig:Algorithm-examples\]Examples of connectivity matrices $D$ generated by the algorithm described in Section \[sec:viab-metap-conn\] and in \[sec:algor-gener-balanc\] for a reduced number of patches, $N=8$, and different combinations of $L/N$ and $\gamma$. Only non-zero entries are printed explicitly. For reasons of clarity symmetric connections are denoted by ’S’ and asymmetric connections by ’A’. The colours indicate separated closed cycles of dispersal that can be identified in the matrices. While the connectivity matrices with $L/N=1$ (upper row) are degenerate into $2$ ($\gamma=0.0$), $3$ ($\gamma=0.5$), and $4$ ($\gamma=1.0$) clusters respectively, the clusters of all three matrices generated with $L/N=2$ (lower row) already extend to the entire metapopulation. In spite of the fact that the matrices displayed here are only *examples* of randomly generated matrices, this trend is representative. For instance all simulations performed for $N=100$ and $L/N>2$ resulted in dispersal matrices with a single cluster only. Note that our results are based on much larger metapopulations consisting of $N=100$ patches.](balanced-example.eps){width="\textwidth"}
Previous works demonstrated that changes in the symmetry of dispersal patterns in particular affect the local symmetry of migrant flow, since asymmetry can result in *donor*- and *recipient*-dominated patches not present in symmetric networks [@Vuilleumier06]. In order to isolate the effect of the degree of symmetry from these secondary effects, we focus on a specific set of dispersal patterns: we restrict our main analysis to dispersal patterns with the number of dispersal connections, $L$, being an integer multiple of $N$ randomly distributed on the patches under the constraint, that each patch obtains exactly $L/N$ in- and outgoing connections with defined degree of symmetry. The random patterns considered, hence, are *regular* with the connections evenly distributed to all patches available [@Artzy-Randrup10; @NetworkAnalysis]. An algorithm efficiently generating regular random dispersal patterns for small and intermediate $L/N$ and arbitrary degrees of symmetry ($\gamma$) is detailed in \[sec:algor-gener-balanc\]. Examples of random connectivity matrices generated for $N=8$ and different combinations of $L/N$ and $\gamma$ are exhibited in Fig.\[fig:Algorithm-examples\]. Please regard that for the simulations metapopulations consisting of $N=100$ are used resulting in connectivity matrices of dimension $100\times100$ instead.
The regular dispersal patterns we use here restrict our analysis to metapopulations with all patches connected at a fixed density independent of the choice of $\gamma$. For $L/N>2$ the largest cluster extends to the entire metapopulation independent from the degree of dispersal symmetry resulting in irreducible connectivity matrices [@Caswell01; @Bode06]. For a detailed discussion of the impact of regularity on our results we refer to Section \[sec:discussion:regul\].
The viability of metapopulations exposed to these dispersal patterns was tested in the following manner: a sample of $100$ dispersal patterns connecting the $N=100$ patches was generated for each combination of $L/N=1,\ldots,10$ and $10$ different values of $\gamma$. For any of these patterns the viability of $10$ independent realisations of metapopulations was tested for different values of the colonisation probability $c$ according to the procedure outlined in Section \[sec:metapopulation-model-vuilleumier\], resulting in a statistics for a total of $1,000$ simulations on $100$ randomly generated connectivity matrices for every choice of $L/N$, $\gamma$, and $c$. For our main analysis we record the number of viable metapopulations out of the $1,000$ simulations and prepare the results for graphical analysis. The sensitivity of this test procedure and its interpretation with respect to the statistics of extinction times is discussed in Section \[sec:discussion:interpret\].
\[sec:results\]Results
======================
![\[fig:balanced\]Results on the viability of metapopulations exposed to dispersal patterns with regular dispersal randomly generated by the algorithm described in Section \[sec:viab-metap-conn\] and \[sec:algor-gener-balanc\]. In the upper row the viability is plotted as a function of the effective number of connections per patch, $L/N$, and the colonisation probability $c$. At every combination of $L/N$ and $c$ the viability of $1,000$ different dispersal patterns has been investigated. Green and red squares indicate parameters, where either all $1,000$ patterns either were viable or not. The intermediate region where some of the patterns were viable and others were not is coloured yellow. The three panels present the results for different degrees of symmetry, increasing from $\gamma=0$ (anti-symmetric dispersal patterns) on the left to $\gamma=1$ (symmetric patterns) on the right hand side. In the lower row the simulation results are presented accordingly as a function of the symmetry $\gamma$ and the colonisation probability $c$ for three different number of connections per patch, $L/N=2,
3$ and $5$. Only a vanishing impact of symmetry is observed for $L/N> 3$.](E50-balanced_fixed_gamma.eps "fig:"){width="\textwidth"}\
![\[fig:balanced\]Results on the viability of metapopulations exposed to dispersal patterns with regular dispersal randomly generated by the algorithm described in Section \[sec:viab-metap-conn\] and \[sec:algor-gener-balanc\]. In the upper row the viability is plotted as a function of the effective number of connections per patch, $L/N$, and the colonisation probability $c$. At every combination of $L/N$ and $c$ the viability of $1,000$ different dispersal patterns has been investigated. Green and red squares indicate parameters, where either all $1,000$ patterns either were viable or not. The intermediate region where some of the patterns were viable and others were not is coloured yellow. The three panels present the results for different degrees of symmetry, increasing from $\gamma=0$ (anti-symmetric dispersal patterns) on the left to $\gamma=1$ (symmetric patterns) on the right hand side. In the lower row the simulation results are presented accordingly as a function of the symmetry $\gamma$ and the colonisation probability $c$ for three different number of connections per patch, $L/N=2,
3$ and $5$. Only a vanishing impact of symmetry is observed for $L/N> 3$.](E50-balanced_fixed_l.eps "fig:"){width="\textwidth"}
For each scenario $(L/N,\gamma,c)$ a total of $1,000$ simulations were performed. For straightforward statistical evaluation of the viability of metapopulations exposed to the respective conditions the simulation results were divided into three different groups, which are colour coded in the graphical presentation of the results: if all $1,000$ simulated metapopulations either went extinct or were viable the scenario is coloured red or green, respectively. Otherwise, i.e. if the number of extinct simulations out of $1,000$ is greater than $0$ but smaller than $1,000$, the scenario was coloured yellow.
The results are illustrated in Fig. \[fig:balanced\]. The three panels in the upper row show the viability of the metapopulation as a function of the number of dispersal connections per patch, $L/N$, and the colonisation probability $c$ for different values of $\gamma$: anti-symmetric dispersal ($\gamma = 0.0$), asymmetric dispersal with intermediate degree of symmetry ($\gamma = 0.5$), and symmetric dispersal ($\gamma = 1.0$). The lower panels of Fig.\[fig:balanced\] contain the same results, but now analysed with respect to the effect of the degree of symmetry, $\gamma$, for three different values of $L/N$. In fact, for $L/N > 3$ no statistically significant impact of symmetry is observed.
\[sec:discussion\]Discussion
============================
\[sec:discussion:interpret\]Interpretation and significance of results
----------------------------------------------------------------------
First of all the results depicted in Fig. \[fig:balanced\] suggest that the impact of the degree of symmetry on metapopulation viability decreases with increasing $L/N$. Already at $L/N>3$ no statistical significant impact of the degree of symmetry, i.e. no systematic differences depending on the degree of symmetry, can be detected on the basis of the scenarios and the statistical evaluation chosen.
At a small number of dispersal connections per patch ($L/N=1,2$) metapopulation viability is significantly reduced for more symmetric dispersal (Fig. \[fig:balanced\], lower panels). The reason for this effect straightforwardly can be understood from considerations concerning the structure of the underlying dispersal patterns: Let us first focus on patterns with $L/N=1$. In this case a metapopulation with a symmetric dispersal pattern necessarily consists of a number of patches only pairwisely connected through dispersal (Figure \[fig:Algorithm-examples\]). The largest closed dispersal cycle (synonymous to the giant component of the dispersal network [@Berchenko09]), hence, involves only two patches. For the particular metapopulation model applied a lower bound for the extinction probability of a pair of patches per time step is $e^2$. On the contrary the mean size of the largest closed dispersal cycle estimated from the $100$ dispersal patterns generated for the same conditions but antisymmetric dispersal ($\gamma=0$) was $62.7$. For $L/N=2$ the mean size of the largest cycles was $77.5$ for the symmetric dispersal matrices generated, whereas for the asymmetric case all dispersal matrices already extended to the entire metapopulation (i.e. their mean size was $100$). Hence we are faced with a percolation problem on random graphs [@Callaway00], where the percolation threshold depends on the symmetry properties of the dispersal pattern. Analysis of the eigenvalues of associated state transition matrices reveals, that the mean time to extinction of a set of patches participating in a closed cycle of dispersal increases with the size of the cycle. For this reason differences in viability at small $L/N$ are attributed to hierarchical differences of the generated matrices at only a few number of connections, namely $L/N\le
3$. This density is much smaller then relevant cases discussed e.g.in [@Vuilleumier06] as will be discussed in more detail in Section \[sec:discussion:prev\].
![\[fig:E50-itime\]Extinction statistics for the metapopulations with different values of the colonisation probability $c$ connected through dispersal matrices with $\gamma=0.5$ and $L/N=4$. The individual lines indicate the number of non-extinct simulations (out of $1,000$) as a function of the simulation time. The dashed line corresponds to the upper bound for the expectation value of the number of extinct simulation for cases where all simulations went extinct, $1,000\exp(-6.9\times
10^{-3} t)$, as derived in the manuscript text. From the figure it becomes evident, that the number of non-extinct simulations after an initial relaxation phase indeed decreases exponentially in time (i.e. linear in this logarithmic plot). The upper bound approaches $1/M$ with $t\to T$, which is a general result for sufficiently large $M$ and $T$ as a Taylor expansion of expression shows. For this reason the boundary line indeed exhibits the border between the cases marked red and yellow in Figure \[fig:balanced\].](E50-itime.eps){width=".6\textwidth"}
How meaningful is the statistical evaluation of the results with respect to the effect of the symmetry of dispersal patterns on expected extinction times of metapopulations? In order to approach this question we aim to derive lower and upper bounds for extinction times in the red and green regions of the figures, which then help to evaluate the graphical presentation of the results in more detail. If we disregard the initial time period of relaxation of the metapopulation to a quasistationary state, we can assume that the statistics of extinction times is exponentially distributed. This exponential distribution complies with a constant risk of metapopulation extinction per time step, which we call $r$. The chance, that a metapopulation has not gone extinct after $T$ time steps then is $(1-r)^T$. For every combination of parameters we perform $M$ simulations with $M=1,000$ in our case[^1]. It is then straightforward to calculate the probability $P(M|r)$ that all $M$ simulations are viable, $$\label{eq:pMr}
P(M|r)=(1-r)^{M T}\quad.$$ Accordingly the chance that a simulation goes extinct during the $T$ simulation steps is $1-(1-r)^T$, resulting in the probability $P(0|r)$ of observing $0$ viable simulations of $$\label{eq:p0r}
P(0|r)=\left(1-(1-r)^{T}\right)^M\quad.$$ More interesting, however, would be the expressions $P(r|M)$ and $P(r|0)$, the probability distributions of the metapopulation extinction risk $r$ given the fact that either all or none of the simulations are viable. These expressions straightforwardly can be calculated using Bayes’ theorem. Using uniform prior distributions we obtain $$\begin{aligned}
\label{eq:prM}
P(r|M)&=&\left(\int_0^1dr' (1-r')^{M T}\right)^{-1}(1-r)^{M T}\quad
\mbox{and}\\
\label{eq:pr0}
P(r|0)&=&\left[\int_0^1dr' \left(1-(1-r')^{T}\right)^M\right]^{-1}\left(1-(1-r)^{T}\right)^M\quad.\end{aligned}$$ Using a maximum likelihood approach confidence intervals for $r$ can be calculated. Applying a confidence level of $95\%$ the upper bound for $r$ in cases where all simulations are viable is $5.1\times
10^{-8}$. As a lower bound for $r$ for cases where all simulations went extinct we obtain $0.057$. Since the latter result strongly depends on the prior distribution we instead use the inflection point of the sigmoid function at $$\label{eq:sig-inflection}
1-\left[(T-1)/(MT-1)\right]^{1/T}$$ as a more conservative estimate, which for the case of our simulations is located at approximately $6.9\times 10^{-3}$. The inverse of $r$ corresponds to the mean time to extinction. From our considerations we, hence, expect the mean time to extinction for the scenarios marked by red squares in Figure \[fig:balanced\] to be below $6.9^{-1}\times 10^3\approx 145$ and the respective value for the conditions marked green to be in the order of $2\times 10^7$ or larger. Intermediate values are expected for the conditions marked yellow in the individual plots. Figure \[fig:E50-itime\] demonstrates, that assumptions we needed to make seem to hold and that the estimates indeed reflect the underlying extinction statistics to a great extent.
Obviously the classification of the conditions by the three scenarios to a meaningful extent reflects the extinction risks of the metapopulation in a sense, that Figure \[fig:balanced\] succeeds to highlight the main results. From the bounds for the mean extinction times to extinction derived above for the respective classes we can conclude that metapopulations in the red regions almost surely go extinct within a short time, whereas metapopulations in the green regions are likely to be persistent. The yellow region decreases in range with increasing $L/N$. That is, the transition between threatened and persistent metapopulation sharpens with increasing $L/N$.
![\[fig:E50-replicate\]Extreme example of the variation in the number of viable replicates between the different samples (here: $\gamma=0.3$, $L/N=10$, $c=0.15$). In particular sample $98$ deviates strongly from the general mean. Since we can assume that the main source of variations is the stochastic simulation procedure rather than qualitative differences between the random dispersal patterns relevant for the present study, we do not investigate the within-sample variations further within the scope of this work.](E50-replicate.eps){width=".5\textwidth"}
The $10$ replicate simulations performed for each parameter set and each dispersal pattern in addition allow to investigate and to discuss the variability within the sample of $100$ dispersal patterns. In the regions marked red and green by definition all samples show the same behaviour. Detailed analysis of the yellow regions shows only very few cases of large variability of the number of extinct replicates between the samples. One example of rather high variability is depicted in Figure \[fig:E50-replicate\]. Overall the differences between the random dispersal patterns generated for each scenario do not seem to be relevant for the present study, which is probably due to the decision of using regular dispersal patterns.
\[sec:discussion:regul\]Impact of regularity on the results
-----------------------------------------------------------
So far we focused on regular dispersal patterns. This approach made it possible to investigate the impact of the degree of symmetry of connectivity matrices on metapopulation viability independently from other possibly confounding effects, which is important in order to assess the role of dispersal symmetry for metapopulations. Our results on regular dispersal patterns show a remarkably low effect of symmetry ($\gamma$) on the viability of metapopulations at intermediate and high density of dispersal paths, $L/N$. At low $L/N$ symmetric dispersal even results in a slightly negative effects on the viability. How do these results relate to the more general case where the dispersal network is not regular?
![\[fig:nonreg\]Results on the viability of metapopulations exposed to general dispersal patterns randomly generated by modification of the algorithm described in \[sec:algor-gener-balanc\]. The analysis and graphical presentation of the simulation results is accordant to the procedure described in the caption of Figure \[fig:balanced\]. Please note that $L/N$ now specifies the mean number of connections per patch, while the actual number of dispersal links now can vary between patches.](NR-E50-nonreg_fixed_gamma.eps "fig:"){width="\textwidth"}\
![\[fig:nonreg\]Results on the viability of metapopulations exposed to general dispersal patterns randomly generated by modification of the algorithm described in \[sec:algor-gener-balanc\]. The analysis and graphical presentation of the simulation results is accordant to the procedure described in the caption of Figure \[fig:balanced\]. Please note that $L/N$ now specifies the mean number of connections per patch, while the actual number of dispersal links now can vary between patches.](NR-E50-nonreg_fixed_l.eps "fig:"){width="\textwidth"}
In order to follow up this question we repeated the simulations accordingly, but now without the constraint of having regular dispersal networks. Technically this was implemented by skipping steps 4c and 4d of the pattern generation algorithm detailed in \[sec:algor-gener-balanc\], which then controls for the desired degree of symmetry only. The parameter $L/N$ now should be understood in a statistical sense, such that $L$ dispersal connections randomly were distributed between the $N$ patches resulting in a mean density of $L/N$ connections per patch. The results are depicted in Figure \[fig:nonreg\]. Interestingly, the minor effect of symmetry at low density of dispersal connections now shifts to a slight advantage for metapopulations with a symmetric dispersal pattern. From $L/N\ge 7$ no significant differences with respect to the simulation results based on regular dispersal patterns (Figure \[fig:balanced\]) are observed.
In non-regular dispersal patterns the existence of isolated patches not participating in dispersal has an impact on the effective density of dispersal connections in the metapopulations [see also @Bode08]. Moreover, in the case of asymmetric dispersal there exist patches that either only receive or only provide migrants, i.e.*sinks* or *sources*, and that cannot actively take part in the metapopulation dynamics [@Artzy-Randrup10]. Since both of these effects are most distinct at small densities of the random dispersal networks, we assume that these differences basically drive the minor differences at low $L/N$ between our results on regular and the general case of random dispersal. Arguments for *not* assigning this effects to asymmetry in dispersal but to examine them separately are made in Section \[sec:conclusions\].
\[sec:discussion:prev\]Relation to previous works
-------------------------------------------------
In general our results suggest essentially no direct negative effect of asymmetric dispersal on metapopulation viability at intermediate and high densities of the dispersal network, at least as far as the stochastic patch occupancy model applied in this work is concerned. This is in contrast to the findings in [@Vuilleumier06] where it was concluded that extinction risk significantly increased when dispersal became asymmetric. The analysis in [@Vuilleumier06] is not restricted to cases with regular dispersal only, although the relaxation of regular dispersal is not not sufficient to explain the qualitative differences in the results as shown in the previous section.
The description of the random patterns investigated in [@Vuilleumier06] does not provide all information necessary for an in-depth comparison with our results. In [@Vuilleumier06] the number of dispersal connections was chosen randomly for each of the $2,000$ metapopulations. Additional information provided on two particular patterns suggest that the densities are comparable or higher than the densities we investigated in our study. From our results we therefore do not expect a significant impact of dispersal asymmetry at these density of connections.
The analysis of the results in [@Vuilleumier06] is based on the number of connected patches in contrast to our analysis using the global mean number of connections $L/N$. The statistics of the number of connected patches seems to differ significantly between the asymmetric and the symmetric connectivity matrices investigated, a phenomenon we were not able to reproduce. In particular the example of a symmetric random pattern with more than $85$ connections per patch but only $96$ connected patches raises questions, since the largest cycle of closed dispersal in non-regular connectivity matrices we generated always extended to at least $99$ patches for densities above $7$ connections per patch with a strong trend towards $100$ patches with increasing density. For this reason we assume, that the effects described in [@Vuilleumier06] originate from differences in network topology between the investigated connectivity matrices rather than differences in dispersal asymmetry.
@Bode08 investigated the same metapopulation model as in the present work in a slightly different setup ($N=10$, $e=0.4$, and $L/N=2.6$). Instead of simulating individual realisations, the probability of metapopulations to go extinct within $100$ time steps was calculated numerically for different dispersal patterns. This method restricts the analysis to rather small metapopulations of $10$ patches. Extinction probabilities were calculated for metapopulations connected through different dispersal patterns generated by the small world algorithm [see e.g. @Watts98; @Kininmonth09] initiated with a particular symmetric dispersal pattern (Bode, pers.communication). @Bode08 concluded from qualitative graphical analysis of their simulation results[^2], that asymmetry reduces persistence and exhibits a distinct threat to metapopulations.
The discussion of our results in Section \[sec:discussion:interpret\] relates our graphical analysis to the extinction probability in a certain number of time steps[^3], which allows for a comparison of the results. From additional simulation data we received from Bode it seems, that the negative effect in their approach is larger than what we would expect from our simulation for the general, non-regular case (Section \[sec:discussion:regul\]). Additional simulations performed for metapopulations likewise subjected to non-regular dispersal patterns but reduced to the size of $10$ patches indicated a general increase in the probability of extinction but no significant impact of metapopulation size on the impact of symmetry. We therefore assume, that the differences related to symmetry observed by @Bode08 partly are owed to the fact, that the patterns in their study were generated from a particular symmetric starting configuration of the small world algorithm and that the similarity of patterns to this starting configuration correlates with the symmetry properties.
Recently another work was devoted to the effect of asymmetry on metapopulation viability [@Vuilleumier10]. This work aims to cover different aspects of asymmetry simultaneously, which makes it difficult to ascribe the variety of effects observed to certain properties of dispersal matrices. One configuration, however, seems to be equivalent to the simulations we performed for general dispersal matrices in Section \[sec:discussion:regul\] for anti-symmetric and symmetric dispersal, respectively [@Vuilleumier10 p. 229, Fig. 2, right column]. The results the authors obtain on these patterns are in agreement with our observations, that the degree of symmetry of dispersal matrices has no significant impact on metapopulation viability at intermediate density of dispersal connections [cp. @Vuilleumier10 p. 213, Fig. 6, difference between the plots in the right column].
\[sec:conclusions\]Conclusions
==============================
We investigated the consequences of the symmetry of dispersal patterns on the viability of metapopulations. Our analyses are based on simulations of a stochastic patch occupancy model.
First we define the degree of dispersal symmetry, $\gamma$, which is based on the symmetry of the connectivity matrix (Equation \[eq:def-deg-asymm\]). In order to be able to minimise possibly confounding effects we restrict our main analysis to regular dispersal patterns, where asymmetry does neither affect the homogeneity of dispersal nor the local balances of incoming and outgoing dispersal connections. For these patterns we do not see any negative effect of dispersal asymmetry. For the more general case of non-regular dispersal patterns minor negative effects of asymmetric dispersal on metapopulation viability are confirmed, but only at rather weak densities of dispersal (cp. Section \[sec:discussion:regul\]). At these densities differences in dispersal symmetry generally are accompanied by other hierarchical differences of the dispersal network. This e.g. becomes evident from a neat example of a two patch metapopulation investigated in detail in [@Bode08 p. 208, Appendix A], where dispersal asymmetry by return results in a source-sink problem.
From first instance it is not self-evident whether these accompanying effects are the origin or a consequence of asymmetric dispersal, since their characteristic strongly depends on how the system of study was constructed and chosen. For realistic dispersal patterns the solution proposed in [@Vuilleumier10], namely to investigate dispersal asymmetry independent from the discussion of sources and sinks, however does not seem to work out, since these effects in general are strongly connected to one another. These correlations in the past made the investigation of asymmetric dispersal highly dependent on the system of study, which was the main difficulty in understanding the role of dispersal asymmetry. In order to resolve this problem we suggest to discuss the symmetry of dispersal patterns at large scales e.g. based on a definition similar to Equation and the statistics of sources and sinks, the homogeneity of the dispersal network, and other features characterising the local flow of migrants *jointly* instead of in isolation.
It was the aim of the present work, to clarify the role of asymmetric dispersal and its impact on metapopulation viability. In contrast to previous studies [@Vuilleumier10; @Vuilleumier06; @Bode08] we see only weak effects of asymmetric connectivity on metapopulation extinction, which suggests that natural populations with asymmetric dispersal may not per se suffer from increased extinction risks. Instead effects observed in simulations, real world data, or in the evaluation of management strategies [see e.g. @Haight08] might be reflected more significantly by other features of complex dispersal patterns. A promising path towards a discussion of potentially important features is taken in the investigations of the viability of metapopulations connected through a variety of different dispersal patterns as provided in [@Bode08; @Artzy-Randrup10]. We expect that eventually only a theoretical analysis of the stochastic metapopulation model applied can reveal the features relevant for metapopulation viability.
\[sec:ack\]Acknowledgements
===========================
We kindly acknowledge comments by Bernt Wennberg on an early version of the manuscript and suggestions by Kerstin Johannesson on a more recent version. We kindly appreciate that Michael Bode contributed simulation results and shared details on his 2008 work, [@Bode08]. Furthermore we are deeply indebted to kind and constructive comments of two anonymous reviewers. This work was supported by a Linnaeus-grant from the Swedish Research Councils, VR and Formas (http://www.cemeb.science.gu.se), by FORMAS through contract 209/2008-1115 (PRJ), and by the Swedish Research Council through contract 275 621-2008-5456 (PRJ).
[34]{} natexlab\#1[\#1]{}url \#1[`#1`]{}urlprefix
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\[sec:algor-gener-balanc\]Algorithm for the generation of regular dispersal patterns
====================================================================================
Since we intended to compare cases primarily differing in their symmetry properties, we focused on *regular* dispersal patterns with fixed number of in- and out-going dispersal routes for every patch. For the connectivity matrices $D$ this is equivalent to the constraint that the sums over every column and every row are equal, that is $$\sum_{k}d_{ik}=\sum_{k}d_{kj}=L/N\label{eq:l-rew-col}$$ for any $i$ and $j$. Here $L$ is the total number of activated dispersal routes.
Random matrices at arbitrary degree of symmetry that are complying with Equation are generated by the following algorithm, that is repeated until a matrix $D$ with $L$ non-zero elements is obtained:
1. Set $D=0$, generate a random matrix $B\in\left[0,1\right]^{N\times N}$, where $b_{ij}$ are random numbers drawn independently from an arbitrary distribution. For instance uniformly distributed random variables are suitable here. Ensure that all elements of $B$ are unique.
2. Set diagonal elements $b_{ii}$ to $10$ for all $i$.
3. Calculate the desired number of symmetric connections, $n_s=\gamma L$
4. Repeat until smallest element of $B$ is larger than $1$ or $\sum_{i,j}d_{ij}=L$:
1. Identify row $i$ and column $j$ of the smallest value of $B$
2. Set $d_{ij}=1$ and $b_{ij}=10$ ([\*]{})
3. If $\sum_{k}d_{ik}=L/N$ set $b_{ik}=10$ for every $k$ ([\*]{})
4. If $\sum_{k}d_{kj}=L/N$ set $b_{kj}=10$ for every $k$ ([\*]{})
5. Switch $i$ and $j$
6. if $n_s >0$ (generate symmetric connection):
- repeat the steps marked by ([\*]{})
- reduce $n_s$ by $2$
else: (generate asymmetric connection)
- set $b_{ij}=10$
5. Reject result if $\sum_{i,j}d_{ij}<L$.
Note that the value $10$, of course, is arbitrary. Any number greater than $1$ is suitable to ensure that the corresponding elements of $D$ are not selected by the algorithm. This algorithm randomly orders the elements of $D$ and activates them step by step. It generates random connectivity matrices with given degree of symmetry and it is sufficiently efficient for small and intermediate $L$.
The implementation of the algorithm in FORTRAN90 is straightforward (compilation tested with the GNU Fortran compiler *gfortran v4.3.3*):
PROGRAM REGULAR_CONNECTIVITY
!======================================================================
! PROGRAM REGULAR_CONNECTIVITY
! GENERATION OF REGULAR RANDOM DISPERSAL PATTERNS
! TESTED WITH GFORTRAN 4.3.3
! (C) 2010 BY DAVID KLEINHANS, UNIVERSITY OF GOTHENBURG, SWEDEN
! DISTRIBUTED UNDER THE CREATIVE COMMONS ATTRIBUTION 3.0 LICENSE
!======================================================================
IMPLICIT NONE
INTEGER,PARAMETER::N=100 !METAPOPULATION SIZE
INTEGER,PARAMETER::MAX_REJECTIONS=1000!MAXIMUM NO OF REJECTED MATRICES
DOUBLE PRECISION::GAMMA !DEGREE OF SYMMETRY
INTEGER::LBYN !NO. OF CONNECTIONS PER PATCH
INTEGER::D(N,N) !CONNECTIVITY MATRIX
DOUBLE PRECISION::RAND(N,N) !RANDOM MATRIX USED FOR ORDERING OF LINKS
DOUBLE PRECISION::REMAINING_SYM !NO OF REMAINING SYMMETRIC CONNECTIONS
INTEGER::REJECTIONS !COUNT NUMBER OF REJECTED DISPERSAL PATTERNS
INTEGER::LOC(2) !LOCATION OF THE SMALLEST ELEMENT OF RAND
INTEGER::I,J !AUXILIARY VARIABLES, USED FOR LOOPS ONLY
LOGICAL::GRIDOK !CHECK IF GRID COMPLIES WITH CONSTRAINTS
! === INITIALIZE RANDOM NUMBER GENERATOR ===
CALL RANDOM_SEED
! === REQUEST PARAMETERS ===
WRITE(*,"(A,I4)")"REGULAR DISPERSAL MATRIX FOR METAPOPULATION OF SIZE N=",N
WRITE(*,"(A)")"PLEASE ENTER PARAMETERS:"
WRITE(*,"(A)")"DEGREE OF SYMMETRY, GAMMA (DOUBLE PRECISION, >=0 AND <=1)?"
READ(*,*)GAMMA
WRITE(*,"(F8.5)")GAMMA
WRITE(*,"(A,I4,A)")"NO OF CONNECTIONS PER PATCH, LBYN (INTEGER, >0 AND <",&
&(N-1)/2,")?"
READ(*,*)LBYN
WRITE(*,"(I4)")
! === GENERATE DISPERSAL PATTERN ===
GRIDOK=.FALSE.
REJECTIONS=0
DO WHILE(.NOT.GRIDOK)
! == STARTING CONFIGURATION: ==
! ALL LINKS INACTIVE
D=0
! CALCULATE NUMBER OF SYMMETRIC LINKS TO BE GENERATED
REMAINING_SYM=NINT(GAMMA*LBYN*N)
! GENERATE RANDOM NUMBER MATRIX FOR ORDERING OF LINKS
! (EXCLUDE DIAGONAL ELEMENTS BY ASSIGNING VALUE OF 10)
DO I=1,N
DO J=1,N
IF(I.NE.J)THEN
CALL RANDOM_NUMBER(RAND(I,J))
ELSE
RAND(I,J)=10.D0
ENDIF
ENDDO
ENDDO
! == ADD CONNECTIONS UNTIL NO LINKS ARE AVAILABLE ANY MORE ==
DO WHILE((MINVAL(RAND).LT.1).AND.(COUNT(D.EQ.1).LT.LBYN*N))
!LOCATE THE SMALLEST ELEMENT OF RAND
LOC=MINLOC(RAND)
!SET RANDOM NUMBER OF THE ELEMENT TO 10 AND ACTIVATE CORRESPONDING LINK
RAND(LOC(1),LOC(2))=10.D0
D(LOC(1),LOC(2))=1
!CHECK WHETHER NUMBER OF DESIRED INCOMING OR OUTGOING LINKS ALREADY
!HAS BEEN REACHED FOR THE PATCH OF FOCUS, PREVENT FURTHER LINKS IF SO
IF(COUNT(D(LOC(1),:).EQ.1).GE.LBYN)RAND(LOC(1),:)=10.D0
IF(COUNT(D(:,LOC(2)).EQ.1).GE.LBYN)RAND(:,LOC(2))=10.D0
!IF SYMMETRIC CONNECTIONS ARE REMAINING: MAKE THE CURRENT A SYMMETRIC ONE,
!ELSE ENSURE THAT THE REVERSE DIRECTION IS NOT ACTIVATED
IF(REMAINING_SYM.GT.0)THEN
D(LOC(2),LOC(1))=1
RAND(LOC(2),LOC(1))=10.D0
IF(COUNT(D(LOC(2),:).EQ.1).GE.LBYN)RAND(LOC(2),:)=10.D0
IF(COUNT(D(:,LOC(1)).EQ.1).GE.LBYN)RAND(:,LOC(1))=10.D0
REMAINING_SYM=REMAINING_SYM-2
ELSE
RAND(LOC(2),LOC(1))=10.
ENDIF
ENDDO
!CHECK WHETHER THE DESIRED NO OF LINKS HAS BEEN GENERATED
!REJECT AND RESTART IF NOT, ACCEPT THE PATTERN OTHERWISE
IF (COUNT(D.EQ.1).EQ.LBYN*N)THEN
GRIDOK=.TRUE.
ELSE
REJECTIONS=REJECTIONS+1
IF(REJECTIONS.LT.MAX_REJECTIONS)THEN
WRITE(*,"(A,I4,A)")"PATTERN ",REJECTIONS,&
&" REJECTED, RESTARTING GRID GENERATION ..."
ELSE
WRITE(*,"(A)")"GRID GENERATION NOT SUCCESSFULL."
WRITE(*,"(A)")"PLEASE TRY LOWER LBYN OR INCREASE MAX_REJECTIONS."
STOP
ENDIF
ENDIF
ENDDO
! === WRITE D TO STANDARD OUTPUT ===
WRITE(*,*)"CONNECTIVITY MATRIX D:"
DO I=1,N
WRITE(*,"(999I1.1)")D(I,:)
ENDDO
WRITE(*,"(A)")"DONE!"
END PROGRAM REGULAR_CONNECTIVITY
[^1]: For reasons of clarity we here assume that simulations are independent of one another although in each case $10$ of them share the same dispersal patterns. This assumption, however, is not expected to be too extensive as the investigation of the replicate statistics at the end of Section \[sec:discussion:interpret\] suggests.
[^2]: In our point of view a correlation between the extinction probability and dispersal asymmetry is not obvious from the Figure the authors refer to [@Bode08 p. 205, Fig. 3]. Bode, however, kindly provided additional data on an accordant simulation, which indeed shows a negative impact of dispersal asymmetry on the metapopulation extinction probability after $100$ time steps.
[^3]: For the parameters marked green within $100$ time units extinctions probabilities below $1-\exp(-5.1\times 10^{-8}\times 100)\approx
5\times 10^{-6}$ are expected, for the red regions an accordant calculation yields probabilities above almost $0.5$.
| 0 |
---
abstract: 'Moment approximation methods are gaining increasing attention for their use in the approximation of the stochastic kinetics of chemical reaction systems. In this paper we derive a general moment expansion method for any type of propensities and which allows expansion up to any number of moments. For some chemical reaction systems, more than two moments are necessary to describe the dynamic properties of the system, which the linear noise approximation (LNA) is unable to provide. Moreover, also for systems for which the mean does not have a strong dependence on higher order moments, moment approximation methods give information about higher order moments of the underlying probability distribution. We demonstrate the method using a dimerisation reaction, Michaelis-Menten kinetics and a model of an oscillating p53 system. We show that for the dimerisation reaction and Michaelis-Menten enzyme kinetics system higher order moments have limited influence on the estimation of the mean, while for the p53 system, the solution for the mean can require several moments to converge to the average obtained from many stochastic simulations. We also find that agreement between lower order moments does not guarantee that higher moments will agree. Compared to stochastic simulations our approach is numerically highly efficient at capturing the behaviour of stochastic systems in terms of the average and higher moments, and we provide expressions for the computational cost for different system sizes and orders of approximation. [We show how the moment expansion method can be employed to efficiently [quantify parameter sensitivity]{}.]{} Finally we investigate the effects of using too few moments on parameter estimation, and provide guidance on how to estimate if the distribution can be accurately approximated using only a few moments.'
author:
- Angelique Ale
- Paul Kirk
- 'Michael P.H. Stumpf'
title: A general moment expansion method for stochastic kinetic models
---
[^1]
[^2]
Introduction
============
Cellular behaviour is shaped by molecular process that can be described by systems of chemical reactions between different molecular species. At the macroscopic scale, the dynamics of these processes are frequently described in terms of mean concentrations of species using deterministic mass action kinetics (MAK). The deterministic solution, however, does not always capture the essential kinetics of the chemical system accurately, because it excludes stochastic effects.
The stochastic kinetics of chemical reaction systems are captured by the chemical master equation (CME), a probabilistic description of the system (also known as Kolmogorov’s forward equation) [@Gardiner2009; @Kampen:1992aa]. The CME is a set of differential or difference equations that capture how the probability over the states (e.g. the abundances of the different molecular species) evolves over time. The CME offers an exact description of a system’s dynamics but can only be solved analytically for very simple systems. Exact single realisations of the CME, corresponding e.g. to observing processes inside a single cell, can be obtained numerically using for example Gillespie’s Stochastic Simulation Algorithm (SSA) [@Gillespie1976]. The SSA is a discrete simulation method that proceeds by randomly selecting a reaction that occurs at each subsequent time step, according to the probability of that reaction occurring next. This method is associated with considerable computational cost, that increases dramatically with the size of the model, which can make it infeasible to comprehensively characterize large systems through simulations.
A number of numerical methods have been developed to approximate the solution of the CME for more complex systems, including methods that approximate the CME by describing the probability distribution in terms of its moments [[@Gillespie2009; @Gomez2007; @Lee2009; @Singh2010; @Singh2006; @Ruess2011; @Goutsias2007; @Barzel2012]]{}. When only the mean (the first moment) is taken into account, the moment expansion reduces to the MAK description. In the linear noise approximation (LNA), the CME is approximated by taking into account the mean (the first moment), and the variance and covariance (second central moments) of the distribution, whereby the second central moments are decoupled form the mean[@Komorowski2009; @Komorowski2010; @Pahle2012; @Wallace2012]. This is valid in the limit of large volumes and molecule numbers, or when the system consists of first-order reactions only.
For smaller systems and more complex reactions, a number of different approaches have been developed that aim to [capture]{} the temporal changes in coupled moments. These expansions can be performed in terms of the moments or the central moments about the mean; the conversion from molecule numbers can be kept implicit in the rate parameters or made more explicit from the outset. the expansion is done in terms of central moments up to third order for first and second order rate equations, and up to second order for general rate equations. In a recent paper [@Grima2012] it was shown that expansion up to three moments tends to deliver more accurate results than expansion up to only second order; in particular the variances are improved by going to higher moments. While the MFK, 2MA and 3MA approaches include expressions for the first, second and third central moments, no general formulation exists in these frameworks to generate higher order central moments in an automatic way.
In this paper we derive a general method for expanding the CME in terms of its central moments about the mean, which does not make extraneous assumptions about the form of the reactions, and that can be evaluated up to any number of moments; in our exposition we follow the notation used in @Gillespie2009. [The new method described here non-trivially generalizes the work of Gomez et al.[@Gomez2007]: in particular we are able to generate arbitrary order higher central moments automatically and in a computationally efficient manner; combining computer algebra systems with numerical simulation engines does allow us to tackle problems that stymy e.g. the linear noise approximation or conventional (3rd order) MFK approaches. The interplay between noise and non-linear dynamics can give rise to very complicated behaviour, and only a handful of systems have been considered. The moments of the distribution described by the CME reflect this intricate relationship between stochasticty and non-linearity and we spend some time discussing this for a typical non-linear biomolecular system. Thus while going to arbitrary moments is straightforward in our framework, we focus our discussion on the lower (up to sixth order) moments.]{}
[Our expansion is based on two successive Taylor expansions that allow us to express the CME of the moment generating function in computationally convenient form]{}; and we truncate the system by setting higher order terms in the Taylor expansions to zero. The outlined method [could]{} fit into a general framework for parameter inference based on maximum entropy distributions derived from the calculated moments. [We furthermore demonstrate that parameter sensitivity analyses may be performed naturally and efficiently in this framework: we can consider the rate of change of the moments with respect to the parameters, which also allows us to study the factors underlying cell-to-cell variability.]{} We illustrate the [general]{} method using three molecular reaction systems: a simple dimerisation reaction, which allows for a detailed investigation; Michaelis-Menten enzyme kinetics; and the p53 system, an oscillating tumour suppressor system [@Batchelor:2009hk].
Moment expansion method
=======================
We consider a system with $N$ different molecular species $(X_1,...X_N)$ that are involved in $L$ chemical reactions with reaction rates $k_l$, $$\begin{aligned}
\underbar{s}_1X_1+...+\underbar{s}_NX_N \xrightarrow{k_l} \bar{s}_1X_1+...+\bar{s}_NX_N,\end{aligned}$$ with $\underbar{s}_i$ and $\bar{s}_i$ the number of molecules of type $X_i$ before and after the reaction, respectively. The time evolution of the system’s state is described by the chemical master equation (CME), $$\begin{aligned}
\frac{dP(\textbf{x})}{dt}=\sum_{l}P(\textbf{x}-\textbf{s}_l)a_l(\textbf{x}-\textbf{s}_l)-P(\textbf{x})a_l(\textbf{x}),\end{aligned}$$ in which $a_l(\textbf{x})$ are the propensity functions, with $a_l(\textbf{x})dt$ defined as the probability of reaction $l$ occurring in an infinitesimal time interval $dt$ when the number of molecules in the system is $\textbf{x}$ , $P(\textbf{x})$ the probability that the system contains $\textbf{x}$ molecules and $\textbf{s}_{l}=\bar{\mathbf{s}}_{l}-\underbar{\textbf{s}}_{l}$.
We start the derivation of the moment expansion method by deriving a moment generating function from the CME. In general, the moment generating function of a random variable **x** is defined as [@Gardiner2009] $$\begin{aligned}
m(\theta,\mathbf{x})=\sum_{\textbf{x}}e^{\theta \textbf{x}}P(\textbf{x}).\end{aligned}$$ The moments, $\left<\textbf{x}^\textbf{n}\right>$, with $\textbf{x}^\textbf{n}=x_1^{n_1}...x_d^{n_d}$, of the probability distribution can be found by taking the $n$-th order derivatives of the moment generating function with respect to $\theta$. The first moment is, of course, equal to the mean, $\mu=\left<x\right>$. The variance, skewness (related to asymmetry of the distribution) and kurtosis (related to the chance of outliers) can be derived from the central moments about the mean, $\left<(\textbf{x}-\mu)^\textbf{n}\right>$.
Using the CME we can write the time dependent moment generating function [@Azunre2007] $$\begin{aligned}
\frac{dm}{dt}=\sum_{l}\left[(e^{\theta \textbf{s}_l}-1)\sum_{\textbf{x}}e^{\theta \textbf{x}}P(\textbf{x})a_l(\textbf{x})\right]
\label{mgf}\end{aligned}$$ The time evolution of the mean concentration $\mu_i$ of species $X_i$ can be obtained by taking the first derivative of Eq. \[mgf\] with respect to $\theta_i$ and subsequently setting $\theta$ to zero, $$\begin{aligned}
\frac{d\mu_{i}}{dt}=\left.\frac{d}{d\theta_i}\frac{dm}{dt}\right|_{\theta=0}= \sum_{l}s_l\left<a_l(\mathbf{x})\right>\end{aligned}$$ This expression can be evaluated by expanding $a_l(\textbf{x})$ in a Taylor expansion about the mean, $$\begin{aligned}
\frac{d\mu_i}{dt}=S\left[\sum_{l}\left.\sum_{n_1=0}^\infty...\sum_{n_d=0}^\infty\frac{1}{\mathbf{n!}}\frac{\mathbf{\partial^{n}}a_l(\textbf{x})}{\partial\mathbf{x^{n}}}\right|_{x=\mu}\mathbf{M_{x^{n}}}\right],
\label{eq6}\end{aligned}$$ where $S$ is the stoichiometry matrix, $$\begin{aligned}
\mathbf{\partial^{n}}=\partial^{n_1+..+n_d}\nonumber\\
\mathbf{n!}=n_1!...n_d!\nonumber\\
\mathbf{M_{x^{n}}}=M_{x_1^{n_1},...,x_d^{n_d}}\nonumber\\
\mathbf{\partial x^n}=\partial x_1^{n_1}...\partial x_d^{n_d},\nonumber
\label{mmgf}\end{aligned}$$ and we have substituted the central moments around the mean for the expected values of the expansion terms $$\begin{aligned}
M_{x_1^{n_1},...,x_d^{n_d}}=\left<(x_1-\mu_1)^{n_1}...(x_n-\mu_n)^{n_2}\right>.\nonumber
\label{mmgf}\end{aligned}$$ The first central moment $\left<\mathbf{x}-\mu\right>$ is zero. Higher order central moments can be derived from the moments, for example the covariance between $x_1$ and $x_2$ can be written as $$\begin{aligned}
\sigma_{x_1x_2}^2=\left<(x_1-\mu_{1})(x_2-\mu_{2})\right>=\nonumber\\
\left<x_1x_2\right>+\mu_1\mu_2-\left<x_1\right>\left<\mu_2\right>-\left<x_2\right>\left<\mu_1\right>.
\label{cmgf2}\end{aligned}$$ In general the relation between the central moments, $\mathbf{M_{x^{n}}}$, and the moments, $\mathbf{\mu^n}$, can be formulated as $$\begin{aligned}
\mathbf{M_{x^{n}}}=\sum_{k_1=0}^{n_1}...\sum_{k_d=0}^{n_d}\binom{\mathbf{n}}{\mathbf{k}}(-1)^{\mathbf{(n-k)}}\underbrace{\mathbf{\mu^{(n-k)}}}_{\alpha}\underbrace{\left<\mathbf{x^{k}}\right>}_{\beta},
\label{eq7}\end{aligned}$$ where $$\begin{aligned}
(-1)^{\mathbf{(n-k)}}=(-1)^{(n_1-k_1)}...(-1)^{(n_d-k_d)}\nonumber\\
\mathbf{\mu^{(n-k)}}=\mu_1^{(n_1-k_1)}...\mu_d^{(n_d-k_d)}\nonumber\\
\mathbf{x^k}=x_1^{k_1}...x_d^{k_d}\nonumber\\
\binom{\mathbf{n}}{\mathbf{k}}=\binom{n_1}{k_1}...\binom{n_d}{k_d}.\nonumber
\label{eq8}\end{aligned}$$ We obtain the time evolution equations of the central moments by taking the time derivative of Eq. \[eq7\], $$\begin{aligned}
\frac{d\mathbf{M_{x^{n}}}}{dt}=\sum_{k_1=0}^{n_1}...\sum_{k_d=0}^{n_d}\binom{\mathbf{n}}{\mathbf{k}}(-1)^\mathbf{{(n-k)}}\left[\alpha\frac{d\beta}{dt}+\beta\frac{d\alpha}{dt}\right].
\label{cmgf}\end{aligned}$$ The term $\alpha$ makes the time evolution of the central moments a function of Eq. \[eq6\], $$\begin{aligned}
\frac{d\alpha}{dt}=\sum_{i=1}^N(n_i-k_i)\mu_i^{-1}\alpha\frac{d\mu_i}{dt}\end{aligned}$$ The term $\beta$ gives rise to mixed moments, and the derivative of $\beta$ with respect to time yields the time evolution equations for the mixed moments. Therefore we also need to include the time derivatives of the mixed moments in our system of equations. We obtain the time derivative of the term $\left<\mathbf{x^{k}}\right>$ by taking higher order derivatives of the moment generating function Eq. \[mgf\], resulting in $$\begin{aligned}
\frac{d\beta}{dt}=\sum_{e_1=0}^{k_1}...\sum_{e_d=0}^{k_d}\mathbf{s^{e}}\binom{\mathbf{k}}{\mathbf{e}}\underbrace{\left<\mathbf{x^{(k-e)}}a(x) \right> }_{\left<F\right>},\end{aligned}$$ where $$\begin{aligned}
\mathbf{s^e}=s_{1}^{e_1}...s_{d}^{e_d}\nonumber\\
\mathbf{x^{(k-e)}}=x_1^{k_1-e_1}...x_d^{k_d-e_d}.\nonumber
\label{mmgf}\end{aligned}$$ By expanding the individual terms of the resulting expressions in a second Taylor expansion, the time evolution of the mixed moments becomes a function of the central moments; the time evolution of the central moments remains a function of the central moments alone, $$\begin{aligned}
\left<F\right>=\left.\sum_{n_1=0}^\infty...\sum_{n_d=0}^\infty\frac{1}{\mathbf{n!}}\frac{\mathbf{\partial^{n}}F(x)}{\mathbf{\partial x^{n}}}\right|_{x=\mu}{\mathbf{M_{x^{n}}}}.\end{aligned}$$
When the model under investigation is non-linear, each central moment will depend on a higher order central moment, which may itself also depend on higher order moments; hence the number of equations we would need to include is in principle infinite. To overcome this, we can obtain a closed set of equations by evaluating the time evolution equations for $\nu$ moments and truncating the Taylor series after the $\nu$th order, thereby setting all higher order [central moments equal to zero. By truncating the Taylor expansion (i.e. setting terms of the Taylor expansion corresponding to $\sum n_i>\nu$ to 0), the equations are only dependent on the central moments up to the selected order $\nu$.]{} Alternatively, the set of equations could be closed using moment closure techniques based on common expressions for well known probability distributions [@Milner2011]. In this paper we will use truncation as well as closure based on a Gaussian distribution.
Results
=======
We illustrate the approach in a range of applications that serve to highlight both the basic properties of the moment expansion method as well as the advantages this method offers in situations where other methods [@Ito2010; @Komorowski2012; @Wallace2012] tend to fail.
Dimerisation
------------
We first illustrate the moment approximation method using a simple dimerisation reaction [@Wilkinson2012]. The system describes the reversible formation and disintegration of a dimeric molecule, $$\begin{aligned}
X_1+X_1 \xrightleftharpoons[k_2]{k_1} X_2.\end{aligned}$$ The system can be written in terms of two propensities $$\begin{aligned}
\begin{array}{cc}
a_1: k_1x_1(x_1-1);&\hspace{7mm} a_2: k_2x_2,
\end{array}\end{aligned}$$ and the stoichiometry matrix $$\begin{aligned}
S=\left[\begin{array}{cc}
-2 & 2\\
1 & -1\end{array} \right],\end{aligned}$$ where the columns correspond to reactions and the rows to variables.
The exact kinetics of the system can be straightforwardly simulated using the Stochastic Simulation Algorithm (SSA) [@Gillespie1976]. One realisation of the SSA is equivalent to observing the kinetics of the system inside a single cell (Figure \[dimer1\]a), whereas the average over many realisations mimics the observation of the average kinetics for a large number of cells (Figure \[dimer1\]b, n=100,000). The system reaches a stationary state after about $4$ seconds.
![\[\]Study of dimerization system, initial values $\mathbf{x}=[301,0]$, parameters $\mathbf{k}=[1.66\cdot10^{-3}, 0.2]$. a) Single SSA realisation. b) Average of 100,000 SSA simulations. c-d) Histogram of SSA runs (grey bars) and probability density of normal distribution (blue line) calculated from mean and variance of SSA runs corresponding to points c and d in figure (b). e) Mean for both variables, calculated using SSA, the moment approximation using 1 moment (deterministic) and two central moments (2m). f) Variance of $x_1$ calculated using SSA, two central moments (2m), three central moments with Gaussian closure (3m) and four central moments (4m). g) Third central moment calculated using SSA and moment approximation method, fourth central moment calculated using SSA and moment approximation method. []{data-label="dimer1"}](Figure1){width="47.00000%"}
In Fig. \[dimer1\]c-d we plotted histograms of the SSA at two time points that correspond to different regimes in the trajectory: transient state with decreasing molecule number (Fig. \[dimer1\]c) and stationary state (Fig. \[dimer1\]d). We calculated the normal probability density function from the mean and variance of the SSA distribution at those two time points (blue line), ignoring higher order moments. The normal probability density functions fit the histograms very well, which indicates that the distribution of molecule numbers is approximately normal over the time course, and higher order moments would have limited influence on describing the kinetics of the mean for this system.
Figure \[dimer1\]d shows the mean molecule numbers calculated with the moment approximation method compared to the SSA results. The results for the deterministic (including only the mean) as well as the two moment approximation are approximately equal to the means calculated with the SSA. We compare the higher order central moments calculated with the general moment approximation method described above with the results from the SSA simulations in Figures \[dimer1\]f-h. In the 3–moment approximation we closed the equations using the Gaussian probability distribution (setting the fourth cumulant equal to zero [@Grima2012]), while in the other approximations we truncated higher order moments. The approximated moments are close to the exact moments calculated from the SSA, which is also clear from the errors displayed in Table \[table1\], calculated as $\epsilon=(1/N)\sum_{n} \sqrt{((M_{SSA}-M_{ma})/M_{SSA})^2}$ with $M_{SSA}$ the moment or central moment calculated based on the SSA, $M_{ma}$ the corresponding value calculated with the moment approximation, and $N$ the number of time points taken into account. The larger error for the mean when using the deterministic approach is due to small differences in the first part of the trajectory, the decreasing part, where the contributions of the higher order central moments are largest. [The larger errors calculated for the third central moment are due to the fluctuations that are still present in the third central moment trajectory calculated based on 100,000 SSA runs. Increasing the number of SSA runs would reduce this effect.]{}
$\epsilon [\%]$ deterministic 2m 3m 4m 5m 6m
----------------- --------------- -------- -------- -------- -------- -------- --
$\mu_1$ 0.300 0.0545 0.0546 0.0546 0.0546 0.0546
$M_{x_1^2}$ 0.961 0.803 0.804 0.801 0.803
$M_{x_1^3}$ 18.3 18.1 18.2 18.2
$M_{x_1^4}$ 2.39 1.33 1.83
: \[table1\]Error between mean, second and third central moment calculated with SSA and approximation methods for the dimerization system.
Michaelis-Menten enzyme kinetics
--------------------------------
We next look at Michaelis-Menten enzyme kinetics, where an enzyme, $E$, and substrate, $S$, first bind to form a complex, $SE$; following this, the complex can dissociate into the original components $S$ and $E$, or $S$ can be converted into the product, $P$, $$\begin{aligned}
S+E \xrightarrow{k_1} SE\nonumber\\
SE \xrightarrow{k_2} S+E\\
SE \xrightarrow{k_3} P+E\nonumber\end{aligned}$$ The system is often reduced to a system of two variables ($S$ and $P$) [@Wilkinson2012], with three reaction propensities $$\begin{aligned}
\begin{array}{c}
a_1: k_1S[E(0)-S(0)+S+P];\nonumber\\
a_2: k_2[S(0)-(S+P)];\nonumber\\
a_3: k_3[S(0)-(S+P)]
\end{array}\end{aligned}$$ and the stoichiometry matrix $$\begin{aligned}
S=\left[\begin{array}{ccc}
-1 & 1 & 0\\
0 & 0 & 1\end{array} \right].\end{aligned}$$
![\[\]Study of Michaelis-Menten kinetics, with parameters $\textbf{k}=[1.66\cdot10^{-3},1\cdot10^{-4},0.01]$, and initial conditions, $S(0)=301$, $P(0)=0$, and $E(0)=120$. (a) Single SSA realisation. Trajectories calculated using (b) moment approximation including only the mean (deterministic). (c-d) Variance of S and covariance between S and P calculated using SSA and approximation using 2 moments. (e) Skewness of S calculated using SSA and approximation with 3 moments, (f) Kurtosis calculated using SSA and approximation up to 4 and 6 moments. []{data-label="mmfigure1"}](Figure2){width="48.00000%"}
One trajectory calculated with Gillespie’s Stochastic Simulation Algorithm is shown in Figure \[mmfigure1\]a, and the mean calculated by solving the ODE system using the deterministic representation of the system is displayed in Figure \[mmfigure1\]b. For this system the deterministic representation is very close to the stochastic solution. The variance of the substrate and the covariance between the substrate and the product (Figure \[mmfigure1\]c-d) calculated based on 100,000 SSA runs can be closely approximated using the moment approximation method expanded up to two moments. Figure \[mmfigure1\]e shows the skewness of the distribution over time, calculated as $$\begin{aligned}
\gamma=\frac{\left<(x_1-\mu_1)^3\right>}{\sigma_{11}^3}=\frac{\left<(x_1-\mu_1)^3\right>}{\left<(x_1-\mu_1)^2\right>^{3/2}}.\end{aligned}$$ For a normal distribution the skewness is zero. The skewness is approximated well using the moment approximation method up to three moments. The kurtosis, given by $$\begin{aligned}
\gamma_2=\frac{\left<(x_1-\mu_1)^4\right>}{\sigma_{11}^4}=\frac{\left<(x_1-\mu_1)^4\right>}{\left<(x_1-\mu_1)^2\right>^{4/2}},\end{aligned}$$ indicates the thickness of the tails of the probability distribution, relating to the probability of outliers. For a normal distribution, the kurtosis is equal to 3. When four moments are used to approximate the system, the approximation of the kurtosis that we obtain from the SSA is not as close as when also higher moments, here six moments, are included in the calculation. This illustrates that agreement between lower-order moments does not guarantee that higher-order moments will also agree. This problem is likely exacerbated for more complex models, e.g. those exhibiting non-linear dynamics.
P53 system
----------
Finally, we investigate the oscillatory p53 system[@Geva2006], which consists of three proteins connected through a nonlinear feedback loop: p53, precursor of Mdm2 and Mdm2. The system can be written in terms of six propensities, $$\begin{aligned}
\begin{array}{cc}\vspace{2mm}
a_1: k_1;&\hspace{7mm} a_2: k_2x\\\vspace{2mm}
a_3: k_{3}\frac{xy}{x+k_{7}};&\hspace{7mm} a_4: k_4x\\
a_5: k_5y_0; &\hspace{7mm} a_6:k_6y,
\label{p53system}
\end{array}\end{aligned}$$ and the stoichiometry matrix $$\begin{aligned}
S=\left[ \begin{array}{cccccc}
1 & -1 & -1 & 0& 0& 0\\
0 & 0 & 0 & 1& -1 & 0\\
0 & 0 & 0 & 0 & 1 & -1\end{array} \right],\end{aligned}$$ where $x$ is the concentration of p53, $y_0$ the concentration of precursor of Mdm2, $y$ is the concentration of Mdm2, $k_1$ is the p53 production rate, $k_2$ is the Mdm2-independent p53 degradation rate, $k_{3}$ the saturating p53 degradation rate, $k_{7}$ is the p53 threshold for degradation by Mdm2, $k_{4}$ is the p53-dependent Mdm2 production rate, $k_5$ is the Mdm2 maturation rate and $k_6$ is the Mdm2 degradation rate.
Figure \[p53figure1\]a shows an SSA simulation of the model for parameters $q_1=$\[90,0.002,1.7,1.1,0.93,0.96,0.01\] and initial values $\textbf{x}(0)=[70, 30, 60]$, which exhibits oscillating behaviour. Because the oscillations for different realisations of the SSA (corresponding to different cells) are stochastically out of phase, the average over 100,000 stochastic simulations shows a damped oscillation, reflecting the presence of a negative feedback loop. Figures \[p53figure1\]c-h show the trajectories of the mean calculated with the moment approximation method. In the deterministic approximation, the oscillations are not damped but expanding, which would indicate a positive instead of negative feedback loop. The LNA and the 2 moment approximation show the same effect. The mean calculated with the LNA is always equal to the mean calculated with the deterministic approximation because the mean is not coupled to the variance. When 3 moments are included, the system shows damped oscillations, although not as damped as the SSA trajectories. By including more moments the trajectories converge to the SSA trajectories. When 6 moments are taken into account, the trajectories calculated with the moment approximation show a similar behaviour to the trajectories calculated with the SSA. This is confirmed by the cumulative difference between the SSA trajectories and the trajectories calculated with the moment approximation shown in Figures \[p53figure1\]i-j, which show a clear decrease in cumulative error for all variables when 6 central moments compared to 2 central moments are included in the approximation. Including more moments would improve the estimation further.
We analyze the distribution of the p53 model over the time course by looking at the central moments (Figure \[p53figure2\]). The variance for the SSA is first increasing, then after about 20 hours it levels off. In Figure \[p53figure2\]b we compare the variance calculated based on the SSA with the LNA and moment approximation method. When up to five central moments are included, the variance keeps increasing and does not level off. Only for the case of 6 moments does the variance reach a stable value, and even then the value is about three times higher than that predicted by the SSA simulations. Figure \[p53figure2\]c shows the comparison of the skewness of the SSA distribution to the skewness of a normal distribution ($\gamma=0$). For three time points where the skewness is relatively large (indicated by d, e, f) we display the histogram of the 100,000 SSA realisations together with the probability density of the normal distribution (cyan line) calculated based on the mean and variance of the SSA and the 6 moment approximation. Additionally, we plot the skew-normal distribution calculated using the mean, variance and third central moment; this is defined by the probability density function $$\begin{aligned}
f(x)=\frac{2}{\omega}\phi\left(\frac{x-\xi}{\omega}\right)\psi\left(\alpha\left(\frac{x-\xi}{\omega}\right)\right), \end{aligned}$$ with $\xi$ a location parameter and $\omega$ a scale parameter. The parameter $\delta$ can be calculated from the estimated skewness using the relation $$\begin{aligned}
|\delta|=\sqrt{\frac{\pi}{2}\frac{|\hat{\gamma}|^{2/3}}{|\hat{\gamma}|^{2/3}+((4-\pi)/2)^{2/3}}}.\end{aligned}$$ The parameter $\alpha$ can be calculated from $\delta$ with $\delta=\alpha/(\sqrt{1-\delta^2})$, and $\omega$ can be calculated from the variance using $\sigma^2=\omega^2(1-2\delta^2/\pi)$. These plots confirm what we saw above, that using only the mean and variance does not capture the full distribution in this case, and also including the skewness is not enough. [When comparing the skewness of the p53 distribution with the skewness of the Michaelis-Menten enzyme kinetics system (Figure \[mmfigure1\]e), we find that the maximum value of the skewness for both systems is approximately equal, and in both systems the skewness does not have a large effect on the mean.]{}
![Analysis of distribution for p53 model. (a) Variance calculated based on SSA runs. (b) Variance calculated with SSA, LNA and moment approximation method. (c) Skewness calculated based on SSA runs (blue line) and skewness for normal distribution (cyan dashed line). (d-f) Histograms calculated based on SSA for points d, e and f in figure (c), and probability densitiy function of normal distribution calculated using mean and variance based on SSA (cyan line).[]{data-label="p53figure2"}](Figure4){width="48.00000%"}
![\[\][]{data-label="contour1"}](Figure5){width="50.00000%"}
Parameter Sensitivity Estimation
--------------------------------
Assessing parameter sensitivity is a key concern when fitting any parametric model [@Saltelli2004; @Erguler2011]. Such analyses allow us to quantify how rapidly the outputs of our model change as we vary its parameters, which can provide insights into the robustness of the model and the relative influence that each parameter has upon the model’s behaviour. However, sensitivity analyses of stochastic models can be difficult and/or computationally costly [@Gunawan2005; @Plyasunov2007], and often involve simulating many times in order to obtain Monte Carlo estimates of sensitivity coefficients. The development of efficient methods for stochastic sensitivity analyses has been the focus of much recent research [@Plyasunov2007; @Komorowski2012b; @Sheppard2012].
In the context of our proposed approach, a natural and straightforward way to assess parameter sensitivity is to consider the rate at which the moments vary with the parameters. This motivates the calculation of simple sensitivity coefficients [@Varma1999; @Saltelli2004] of the form $s_{ij}(t) = \frac{\partial m_i(t, \boldsymbol{\theta})}{\partial \theta_j}$, where $m_i$ is the (estimated) $i$-th moment and $\theta_j$ is the $j$-th parameter. Within our moment approximation framework, the $s_{ij}$’s may either be estimated by perturbing the model’s parameters and computing a finite difference approximation, or obtained automatically by employing the CVODES solver of @Serban2003 when solving the system of ODEs (Equations and ). In Figure \[sens-fig01\], we reconsider the dimerisation model of Section III A. We focus upon the sensitivity of the mean and 2nd and 3rd central moments of the two molecular species to the parameter $k_1$ (similar results are obtained for the sensitivity to $k_2$). Figures \[sens-fig01\]a, d and g show how the moments estimated from 100,000 SSA runs vary when we increase the original value of $k_1$ by 10 percent. Figures \[sens-fig01\]b, e and h show the same for the moments estimated using our proposed approach with 6 central moments (6m). Figures \[sens-fig01\]c, f and i show sensitivity coefficients estimated from both the SSA and 6m outputs using a finite difference approximation (in the 6m case, the sensitivity coefficients may instead be obtained automatically using the CVODES solver, which yields identical results). There is generally good agreement between the coefficients estimated using the two different approaches. However, as we consider higher moments, our ability to assess sensitivity using the SSA output rapidly diminishes, since the variability caused by the change in the parameter value is overwhelmed by the variability in the estimator due to finite sample size. This may be rectified by increasing the number of SSA simulations, but at considerable computational cost. In contrast, the sensitivity coefficients associated with higher moments may still be straightforwardly calculated using the moment expansion approach (although, of course, care must be taken to ensure that appropriately many moments have been taken into account by the approximation — see Section III E).
Simple Heuristics for Moment Expansions
---------------------------------------
Our results for the p53 system clearly demonstrate that failure to take a sufficient number of moments into account can lead to incorrect conclusions and biased parameter estimates. Ideally we would like to know from the outset whether a deterministic approach or a two moment approximation is sufficient to capture the general statistical behaviour of the stochastic system. But without recourse to a large number of SSA runs it is impossible to predict the statistical properties of the solutions to non-linear stochastic systems. And in such cases it is generally not feasible to perform large numbers of SSA simulations and we need a different approach. We should look at the assumption made at the beginning of our derivation, where we assumed that we can approximate the propensity functions with a Taylor expansion. For a single variable a Taylor expansion of a function $f(x)$ about the point $c$ has the general form $$\begin{aligned}
f(x)=f(c)+\frac{f'x}{\partial x}\left(x-c\right)+\frac{f''x}{\partial^2 x}\left(x-c\right)^2+\ldots\end{aligned}$$ By taking into account only the first term, we assume that the function value in point $x$ is the same as for point $c$; by taking into account also the second term we assume that $f(x)$ can be approximated by a straight line between points $x$ and $c$, etc.. Truncating the expansion at a low order will only result in a good approximation in case $x$ is close to point $c$, where we have approximated the true function. In our case $c$ is the mean, $\mu$, implying that an approximation using a few moments will be accurate only in case all observations are close to the mean. In case it is possible to perform a single realisation of the SSA, we can assess this quality by comparing the mean calculated with the deterministic approach with the trajectory calculated with the SSA (Figure \[p53figure5\]).
Figure \[p53figure5\]a displays the deterministic mean of $x_1$ and one SSA simulation for the dimerisation system. In this case the trajectory is close to the mean over the complete time course. A single trajectory for p53, displayed in Figure \[p53figure5\]c, compared to the deterministic result shows that in this case the distance of the trajectory from the mean is much larger. In case a single SSA realisation of the model is not possible, but experimental data are available, the distance form the mean can still be investigated in the same way. Figures \[p53figure5\]b,d show the mean calculated using the deterministic approach together with ‘measured’ values at three time points (obtained with the SSA), repeated three times, resulting in nine data points. Also when looking at the distance form these nine points from the deterministic mean, it is clear that for the dimerisation system $(x-\mu)$ is small, whereas for the p53 system it is [relatively]{} large, indicating that a larger number of moments is necessary to capture the full distribution.
Such simple heuristics have the advantage of being computationally affordable. While inadequate at guaranteeing good performance of an expansion using any finite number of moments, we can use them to capture any gross inadequacy of a given approximation relatively reliably. Such small-scale analyses should precede or accompany moment expansions. More generally, we can consider this problem from the point of view of statistical model checking; see e.g. [@Gelman:2003]. But the question as to how many stochastic simulations need to be averaged over to get a good idea of the mean (or any higher moment) is challenging to answer for all but the most trivial systems [@Toni:2008aa].
![\[\]Study of the deviation from the mean $(x-\mu)$ (a) Deterministic mean and single SSA trajectory for dimerisation system. (b) Deterministic mean and 9 points taken from different SSA trajectories for dimerisation system. (c) Deterministic mean and single SSA trajectory for p53 system. (d) Deterministic mean and 9 points taken from different SSA trajectories for p53 system. []{data-label="p53figure5"}](Figure7){width="48.00000%"}
Computational Complexity
------------------------
The computational complexity of the moment approximation method depends on the number of variables in the system, and the number of terms that need to be evaluated for each central moment. Because of the symmetry of central moments, e.g. $\left<(x_1-\mu_1)(x_2-\mu_2)\right>=\left<(x_2-\mu_2)(x_1-\mu_1)\right>$, there are many terms we do not need to include in the ODE representation of the system. The total number of central moment terms that could be nonzero when approximating a system with $d$ variables and up to $N$ moments is given by $$\begin{aligned}
N_{cm}=\frac{(N+d)!}{N!d!}-d-1\end{aligned}$$ We subtract $d$ terms because the first order central moments are always zero, and one term corresponding to the zeroth order central moment. For the deterministic case, the total number of ODE equations necessary to describe the system is equal to the number of variables, each term describes the mean of one of the variables. For the LNA the total number of equations is equal to the number of equations needed for the two moment approximation. We displayed the number of central moment terms to be evaluated for systems up to 9 variables and up to 6 central moments in Table \[table2\]. When two central moments are included for 2 variables, we need to evaluate two ODE equations corresponding to the mean, and three equations for the second central moments, i.e. the variance of both variables, and their covariance. From Table \[table2\] we can see that the number of equations that need to be included rises quickly with the number of variables in the system.
However, in order to obtain the same information on higher order moments by simulation with the SSA, a large number of simulation runs would have to be performed; in our experience even for relatively simple systems this is of the order of $>10^4-10^6$. We have already seen from the examples described above that the number of simulations necessary to obtain a smooth estimate of the higher order central moments increases with each order. For example, the third central moment in the dimerisation system calculated from 100,000 simulations still displays fluctuations that can only be expected to smooth out when more simulations were performed.
[ l | c | c | c | c | c | c | c | c c ]{}\
$ $& 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\
$2m$ & 3 & 6 & 10& 15 & 21&28 & 36& 45\
$3m$ & 7 & 16 & 30& 50 &77 & 112& 156& 210\
$4m$ & 12 & 31& 65 & 120&203&322& 486& 705\
$5m$ & 18 & 52& 121& 246 &455 &784& 1278& 1992\
$6m$ & 25 & 80 & 205 & 456 & 917 & 1708& 2994& 4995\
Conclusion
==========
In this paper we have described a general moment expansion method for approximating the time evolution of stochastic kinetic systems. We have shown that taking into account more moments improves the estimate of the mean and the higher order moments. In case the deterministic approach delivers an accurate estimation of the mean, expanding the higher order moments still gives additional information on the variance, skewness and kurtosis of the distribution of the variables. Even for very simple systems, e.g. the dimerisation system, we find that higher moments obtained from averaging over 100,000 SSA runs are still fluctuating noticeably.
Instead of performing large numbers of SSA simulations, the time evolution of higher order moments may be obtained much more efficiently from moment approximation methods. Unfortunately it is not possible to predict [*a priori*]{} how many moments are required to fully capture the output distributions of stochastic processes. And while our method can be used to evaluate arbitrary higher moments, the number of equations that need to be solved increases rapidly with the number of molecular species and the number of moments required. This comes with increased computational cost for calculating the expressions as well as for integrating the equations. Therefore it is desirable to identify whether more moments are needed to model the system, and we have provided a simple heuristic to achieve this in many cases. While this can indicate if more than two moments are needed, it will not clearly identify when sufficiently many central moments have been included in the expansion. This is a point for further future investigation. In cases where the distribution is known, but different from normal, knowledge about the distribution can be used to close the set of equations correspondingly. The estimated higher order moments may also by themselves indicate which distribution is a probable candidate for the output of a given dynamical stochastic system; we can, for example, compare obtained estimates of such moments against known values for the higher order moments for particular distributions: if e.g. the estimated skewness and kurtosis are close to zero and three, respectively, a normal distribution might be a good choice and an approximation similar to the simple linear noise approximation may be appropriate.
When the model under investigation is non-linear, lower order moments typically depend on higher order moments, and the system of equations is not closed. Closure can be achieved in several different ways. Here we have closed the system by truncating the Taylor series expansion at a selected order. Several other approaches have been previously evaluated. Chevalier et al [@Chevalier2011], for example, approximated higher order moments using experimental data. Azunre et al[@Azunre2011] showed that for very small molecule numbers, using only two moments can lead to unstable results. Singh et al. [@Singh2011] developed a derivative-matching approach in the context of polynomial rate equations, which proved to work very well in particular for small molecule numbers. [For the examples described in this paper, the proposed method shows differences in behaviour for very low molecule numbers. More specifically, for the dimerisation system, the mean trajectories become unstable below approximately 7 molecules in the system; for the Michaelis-Menten example, the system does not become unstable, but becomes less accurate for very small molecule numbers. The p53 system trajectories for the system size as displayed in Fig. \[p53figure1\] start running out of phase around $t=35$, and when the molecule number decreases, the point where the trajectories run out of phase moves forward, followed by instability in the trajectories. The LNA solution for the p53 system does not become unstable for small molecule numbers but keeps showing the expanding oscillatory behaviour.]{} For other systems that show periodic behaviour it might be more beneficial to approximate the system in the frequency domain. We would recommend use of this method — certainly if no manual closure is attempted, but probably even then — for moderate and large numbers of molecules; a conservative estimate might be to have at least 10-20 molecules. However, for molecular abundances of less than 10 molecules we find that the numerical burden of the SSA compared to the MFK approaches is no longer prohibitive, and, again conservatively, we would consider the use of the SSA.
The present work also provides a natural framework for an inferential framework: our results above suggest that inclusion of higher order moments will lead to increased accuracy of the parameter estimates. Milner et al. [@Milner2011b] derived a likelihood function that included the mean and variance obtained from a moment closure method, assuming a multivariate Gaussian distribution. Kuegler [@Kuegler2012] used both the mean and the variance obtained from a moment closure method for parameter estimation by minimizing an objective function that included both the difference between the observed and estimated mean and the difference between the observed and estimated variance. This showed that more accurate parameter estimates could be obtained when the observed variance is included for parameter estimation. Other analyses have employed approximate Bayesian computation schemes [@Toni:2012bo], [and used moment-based inference employing the mean and variance to infer parameters of a bimodal system[@Zechner2012]]{}. Through recent technological advances in the field of single cell observations [@Klug2011; @Lin2011; @Ozaki:2010p26116], it becomes possible to probe directly the properties of the output distributions of stochastic dynamical systems over time. The additional information about the higher order central moments that can be derived from these datasets can be exploited when higher order empirical moments are also used for parameter inference. However, while likelihoods are trivially constructed when the mean behaviour and variance estimates are available [(via Gaussian assumptions)]{}, conditioning on higher-order moments typically requires some further assumptions; most easily these moments are included in maximum-entropy estimators of the probability distribution. The present approach yields these moments, however, reliably and affordably. Because of their affordability these moments also open up new ways for assessing the sensitivity of stochastic dynamical systems (as outlined in section IIID), including cases where the linear noise approximation tends to break down [@Komorowski2010]. This includes general feedback systems where the notion of sensitivity may be particularly useful but calculation for stochastic systems is fraught with problems and numerically expensive. In conclusion, the general moment expansion method described in this paper provides a flexible and valuable new tool for investigating many stochastic kinetic systems.
Model equations
===============
[In this section we display the model equations used for the dimerisation and Michaelis-Menten system. The complexity of the equations grows with the number of moments included, we display here the equations used for the mean, variance and co-variance when truncating after second order.]{}
Dimerisation
------------
For the dimerisation system, the equations used for the mean, variance and covariance are given by $$\begin{aligned}
&\mu_{x_1}=2k_2x_2-2k_1\sigma^2_{x_1^2}-2k_1x_1(x_1-1)\nonumber\\
&\mu_{x_2}=k_1\sigma^2_{x_1^2}-k_2x_2+k_1x_1(x_1-1)\nonumber\\
&\sigma^2_{x_2^2}=k_1y_{1}^2-k_1x_1+k_2x_2+c1\sigma^2_{x_1^2}-2k_2\sigma^2_{x_2^2}-2k_1\sigma^2_{x_1,x_2}\nonumber\\
& \ \ \ \ \ \ \ \ +4k_1x_1\sigma^2_{x_1,x_2}\nonumber\\
&\sigma^2_{x_1,x_2}=2k_1\sigma^2_{x_1,x_2}+2k_2\sigma^2_{x_2^2}-k_1\sigma^2_{x_1^2}-k_2\sigma^2_{x_1,x_2}-2k_1x_1^2\nonumber\\
& \ \ \ \ \ \ \ \ -2k_1x_1+2k_2x_2+2k_1\sigma^2_{x_1^2}-4k_1x_1\sigma^2_{x_1,x_2}+2k_1x_1\sigma^2_{x_1^2}\nonumber\\
&\sigma^2_{x_1^2}=4k_1\sigma^2_{x_1^2}+4k_2\sigma^2_{x_1,x_2}+4k_1x_1^2-4k_1x_1+4k_2x_2+\nonumber\\
& \ \ \ \ \ \ \ \ 4k_1\sigma^2_{x_1^2}-8k_1x_1\sigma^2_{x_1^2}\nonumber\end{aligned}$$
Michaelis-Menten System
-----------------------
For the Michaelis-Menten system, the equations used for the mean, variance and covariance are given by $$\begin{aligned}
&\mu_{x_1}=-k_2(x_1+x_2-301)-k_1\sigma^2_{x_1,x_2}-\nonumber\\
& \ \ \ \ \ \ \ \ k_1 \sigma^2_{x_1^2}-k_1x_1(x_1+x_2-181)\nonumber\\
&\mu_{x_2}=-c_3(x_1+x_2-301)\nonumber\\
&\sigma^2_{x_2^2}=-2c_3(\sigma^2_{x_2^2}+\sigma^2_{x_1,x_2})-c_3(x_1+x_2-301)\nonumber\\
&\sigma^2_{x_1,x_2}=181k_1\sigma^2_{x_1,x_2}-k_2\sigma^2_{x_2^2}-k_2\sigma^2_{x_1,x_2}-c_3\sigma^2_{x_1,x_2}\nonumber\\
& \ \ \ \ \ \ \ \ -c_3\sigma^2_{x_1^2}-k_1x_1\sigma^2_{x_2^2}-2k_1x_1\sigma^2_{x_1,x_2}-k_1x_2\sigma^2_{x_1,x_2}\nonumber\\
&\sigma^2_{x_1^2}=-(181k_1x_1-301k_2+k_2x_1+k_2x_2\nonumber\\
& \ \ \ \ \ \ \ \ -k_1\sigma^2_{x_1,x_2}-k_1\sigma^2_{x_1^2}-k_1x_1^2-362k_1\sigma^2_{x_1^2}+\nonumber\\
& \ \ \ \ \ \ \ \ 2k_2\sigma^2_{x_1,x_2}+2k_2\sigma^2_{x_1^2}-k_1x_1x_2+2k_1x_1\sigma^2_{x_1,x_2}\nonumber\\
& \ \ \ \ \ \ \ \ +4k_1x_1\sigma^2_{x_1^2}+2k_1x_2\sigma^2_{x_1^2})\nonumber\end{aligned}$$
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| 0 |
ITP–UU–02/43\
SPIN–02/25\
hep-th/0208145
[**Instantons in the Double-Tensor Multiplet**]{}\
Ulrich Theis and Stefan Vandoren\
[ *Spinoza Instituut, Universiteit Utrecht\
Postbus 80.195, 3508 TD Utrecht, The Netherlands\
U.Theis, [email protected]*]{}
------------------------------------------------------------------------
**Abstract**
The double-tensor multiplet naturally appears in type IIB superstring compactifications on Calabi-Yau threefolds, and is dual to the universal hypermultiplet. We revisit the calculation of instanton corrections to the low-energy effective action, in the supergravity approximation. We derive a Bogomol’nyi bound for the double-tensor multiplet and find new instanton solutions saturating the bound. They are characterized by the topological charges and the asymptotic values of the scalar fields in the double-tensor multiplet.
------------------------------------------------------------------------
Introduction
============
Instanton effects in string and M-theory are still relatively poorly understood. This is due to the lack of a conventional instanton calculus as we know it from (supersymmetric) field theory. A well-known open problem is to determine the instanton corrections to the hypermultiplet moduli space of type II superstrings or M-theory compactified on a Calabi-Yau (CY) threefold down to four or five dimensions. Supersymmetry requires the hypermultiplet moduli space ${\cal M}_H$ to be quaternion-Kähler [@BW]. The four- (or five-) dimensional dilaton lives in a multiplet which can be dualized into the universal hypermultiplet. Hence, ${\cal M}_H$ receives quantum corrections, and the instantons correspond to Euclidean $p$-branes wrapping $p+1$ cycles of the CY [@BBS].
The simplest setup for studying this problem, is to consider CY-compactifications of M-theory/type IIA superstrings with Hodge number $h_{2,1}=0$, or, for type IIB, $h_{1,1}=0$ [^1], since this yields a low-energy effective action of $N=2$ supergravity coupled to a single hypermultiplet, such that the moduli space ${\cal M}_H$ has dimension four. From a type IIB perspective, this hypermultiplet arises from dualizing the *double-tensor* multiplet, whose bosonic components descend from the $NS$-$NS$ and $R$-$R$ two-forms and scalars in ten dimensions. This suggests that instanton calculations should be done on the double-tensor multiplet side. In the next section, we shall make another argument, which also applies to type IIA and M-theory, why the double-tensor multiplet is more appropriate for our purposes.
Yet, even in the case of a single hypermultiplet, it is difficult to compute instanton effects directly in string theory, without explicit knowledge of the instanton measure and the details of the wrapped branes along the CY cycles. Therefore, we will study this problem in a pure supergravity context, in which semi-classical instanton calculations can be done in the more conventional and “field-theoretic” way, following a similar strategy as in [@BB; @GS1], or as in [@R] for matter coupled to $N=1$ supergravity. Although being an approximation of the exact result, the hope is that the leading supergravity corrections, combined with the constraints from quaternion-Kähler geometry, and together with some knowledge from string theory on the isometries and singularity structure of ${\cal M}_H$, should fix the answer uniquely. Such a program has worked succesfully in the context of supersymmetric field theories in three dimensions with eight supercharges, where the hypermultiplet moduli space is hyperkähler [@SW; @DKMTV]. See [@OV; @K] for related issues.
In this paper, we carry out the first steps of the supergravity instanton calculation. In section 2, we explain how the Euclidean theory is best understood in terms of the double-tensor multiplet, since then the action is manifestly positive definite, a requirement needed for a semiclassical approximation. In section 3, we derive a Bogomol’nyi bound and show that the instanton action is purely topological and given by a surface term. We then solve the BPS equation explicitly and compute the instanton action for the solutions. A similar approach was followed in [@BB] and [@GS1]. Compared to these papers, we propose a different Euclidean version of the universal hypermultiplet Lagrangian, so our results, where comparable, are somehow different. Moreover, we have found new instanton solutions, which will play an important role in understanding the quantum corrected hypermultiplet moduli space, as explained in the discussion at the end of the paper.
The double-tensor multiplet
===========================
As mentioned in the introduction, we are interested in the case of a single hypermultiplet coupled to $N=2$ supergravity. Classically, the four scalars of the universal hypermultiplet parametrize the homogeneous quaternion-Kähler manifold [@CFG; @FS] $$\label{UHM-QK}
{\mathcal{M}}_H = {\frac{\raisebox{-2pt}{$\mathrm{SU}(1,2)$}}{\mathrm{U}(2)}}\ .$$ In a basis of real fields $\{\phi,\chi,\varphi,\sigma\}$, the bosonic Lagrangian takes the form[^2] $$\label{UHM-action}
{\mathcal{L}}_\mathrm{UH} = - {\mathrm{d}}^D x\, \sqrt{g\,} R + {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 + {\tfrac{1}{2}}\,
{\mathrm{e}}^{-\phi} \big( |{\mathrm{d}}\chi|^2 + |{\mathrm{d}}\varphi|^2 \big) + {\tfrac{1}{2}}\,
{\mathrm{e}}^{-2\phi}\, |{\mathrm{d}}\sigma + \chi {\mathrm{d}}\varphi|^2\ ,$$ with $D=4$ or $5$, depending on whether one is interested in type II or M-theory compactifications. The Lagrangian has a global SU(1,2) isometry group.
For our purposes, it will be convenient to discuss the dual version of ${\mathcal{L}}_\mathrm{UH}$ in terms of a double-tensor multiplet. Consider the first-order Lagrangian $$\label{DTM-action}
{\mathcal{L}}_\mathrm{DT} = - {\mathrm{d}}^Dx\, \sqrt{g\,} R + {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 + {\tfrac{1}{2}}\,
{\mathrm{e}}^{-\phi} |{\mathrm{d}}\chi|^2 + {\tfrac{1}{2}}M_{ab} *\! H^a {\wedge}H^b - \lambda_a\,
{\mathrm{d}}H^a\ ,$$ where the $H^a$ are a doublet of $(D-1)$ forms, the $\lambda_a$ are two scalars, and $$M = {\mathrm{e}}^{\phi} \begin{pmatrix} 1 & - \chi \\[2pt] - \chi & {\mathrm{e}}^{\phi}
+ \chi^2 \end{pmatrix}\ .$$ The two scalars $\phi$ and $\chi$ parametrize the coset SL$(2,\fieldR)/
\mathrm{O}(2)$; in terms of the complex combination $$\label{tau}
\tau \equiv \chi + 2{\mathrm{i}}\, {\mathrm{e}}^{\phi/2}$$ the scalar part of ${\mathcal{L}}_\mathrm{DT}$ can be written as $2|d\tau/\operatorname{Im}\tau|^2$. The tensor terms, however, break the global SL$(2,\fieldR)$ symmetry, leaving only shift symmetries of $\phi$ and $\chi$. The shift in $\chi$ acts as $$\label{shift-chi}
\tau \rightarrow \tau + b\ ,\quad \begin{pmatrix} H^1 \\[2pt] H^2
\end{pmatrix} \rightarrow \begin{pmatrix} 1 & b \\[2pt] 0 & 1
\end{pmatrix} \begin{pmatrix} H^1 \\[2pt] H^2 \end{pmatrix}\ ,
\quad \begin{pmatrix} \lambda^1 \\[2pt] \lambda^2 \end{pmatrix}
\rightarrow \begin{pmatrix} 1 & 0 \\[2pt] -b & 1 \end{pmatrix}
\begin{pmatrix} \lambda^1 \\[2pt] \lambda^2 \end{pmatrix}\ ,$$ whereas the shift in $\phi$ acts as $$\label{shift-phi}
\tau \rightarrow {\mathrm{e}}^\kappa \tau\ ,\quad \begin{pmatrix} H^1 \\[2pt]
H^2 \end{pmatrix} \rightarrow \begin{pmatrix} {\mathrm{e}}^{-\kappa} H^1
\\[2pt] {\mathrm{e}}^{-2\kappa} H^2 \end{pmatrix}\ ,\quad \begin{pmatrix}
\lambda^1 \\[2pt] \lambda^2 \end{pmatrix} \rightarrow \begin{pmatrix}
{\mathrm{e}}^\kappa \lambda^1 \\[2pt] {\mathrm{e}}^{2\kappa} \lambda^2 \end{pmatrix}\ .$$ Note that the latter acts like an SL$(2,\fieldR)$ transformation on $\tau$, but not on the $H^a$. The full type IIB theory compactified to four dimensions (classically) has SL$(2,\fieldR)$ symmetry, due to the presence of additional tensor multiplets which transform nontrivially [@BGHL]. Setting the scalars in these multiplets to nonvanishing constants results in a breakdown of the symmetry and leaves only the above transformations as residual invariances.
The equations of motion for the Lagrange multipliers $\lambda_a$ imply that the $H^a$ are closed. Writing $H^a={\mathrm{d}}B^a$, one obtains the double-tensor multiplet. Integrating out the tensors instead gives the duality relation $$\label{dual-rel}
{\mathrm{d}}\lambda_a = - M_{ab} *\! H^b\ .$$ Substituting this back yields the action for the universal hypermultiplet , upon identifying $$\label{mult1}
\lambda_1 = \varphi\ ,\quad \lambda_2 = \sigma\ .$$ The dual formulation in terms of the double-tensor multiplet is not unique[^3]. We can start with , but write everywhere $\varphi$ instead of $\chi$. Dualizing the tensors and identifying $$\label{mult2}
\lambda_1 = - \chi\ ,\quad \lambda_2 = \sigma + \varphi\, \chi$$ yields the same hypermultiplet action, as one can easily check.
In addition, the dualization procedure yields a boundary term which has to be added to the hypermultiplet action, $$\label{surf-term}
{\mathcal{L}}_\mathrm{bnd} = (-)^D\, {\mathrm{d}}\big[ \lambda_a\, (M^{-1})^{ab} *{\mathrm{d}}\lambda_b\big]\ ,$$ where we used that, when acting on a $p$-form in Minkowski space, $**=
-(-)^{(D-1)p}$. The different choices corresponding to and would now give different boundary terms. However, due to the isometries of the scalar manifold, they are related to each other by a field redefinition of the multipliers, $\tilde{\sigma}=\sigma+\varphi
\chi$, $\tilde{\varphi}=-\chi$, $\tilde{\chi}=\varphi$. Substituting into , we get $${\mathcal{L}}_\mathrm{bnd} = (-)^D\, {\mathrm{d}}\big[ {\mathrm{e}}^{-\phi} \chi *\! {\mathrm{d}}\chi +
{\mathrm{e}}^{-2\phi} \sigma *\! ({\mathrm{d}}\sigma + \chi {\mathrm{d}}\varphi) \big]\ .$$ The total action for the universal hypermultiplet is then $${\mathcal{L}}= {\mathcal{L}}_\mathrm{UH} + {\mathcal{L}}_\mathrm{bnd}\ .$$
The fermions have been suppressed here. For hypermultiplets, the supersymmetry transformation rules and the fermion-terms in the Lagrangian are known in general. For the double-tensor multiplet Lagrangian , the fermion-terms and susy rules can be determined by dualization. However, the most general self-interacting supersymmetric double-tensor multiplet Lagrangian has not been worked out. For a discussion on this in the context of rigid $N=2$ supersymmetry, we refer to [@B].
Euclidean formulation {#euclidean-formulation .unnumbered}
---------------------
To find instanton solutions, we need the Euclidean formulation of the universal hypermultiplet, or, equivalently, the Euclidean double-tensor multiplet Lagrangian. For the latter, apart from the usual complications with the Euclidean Einstein-Hilbert term, the Wick rotation acts in the standard way on the scalars and tensors. While the double-tensor multiplet Lagrangian formally stays the same, $$\label{E-DTM}
{\mathcal{L}}_\mathrm{DT}^E = {\mathrm{d}}^Dx\, \sqrt{g\,} R + {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 + {\tfrac{1}{2}}\,
{\mathrm{e}}^{-\phi} |{\mathrm{d}}\chi|^2 + {\tfrac{1}{2}}M_{ab} *\! H^a {\wedge}H^b\ ,$$ the dual Euclidean universal hypermultiplet Lagrangian has two sign flips in the kinetic terms, due to the fact that we now have $**=(-)^{(D-1)p}$ when acting on a $p$-form in Euclidean space. The dualization procedure yields $$\label{EUHM-action}
{\mathcal{L}}_\mathrm{UH}^E = {\mathrm{d}}^D x\, \sqrt{g\,} R + {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 + {\tfrac{1}{2}}\,
{\mathrm{e}}^{-\phi} \big( |{\mathrm{d}}\chi|^2 - |{\mathrm{d}}\varphi|^2 \big) - {\tfrac{1}{2}}\, {\mathrm{e}}^{-2
\phi}\, |{\mathrm{d}}\sigma + \chi {\mathrm{d}}\varphi|^2\ ,$$ together with the boundary term $$\label{E-BT}
{\mathcal{L}}_\mathrm{bnd}^E = - (-)^D\, {\mathrm{d}}\big[ {\mathrm{e}}^{-\phi} \chi *\! {\mathrm{d}}\chi +
{\mathrm{e}}^{-2\phi} \sigma *\! ({\mathrm{d}}\sigma + \chi {\mathrm{d}}\varphi) \big]\ .$$ By setting $\varphi=\chi=0$, this boundary term is the same as for the $D$-instanton of type IIB in ten dimensions, obtained by dualizing the nine-form field strength into the $R$-$R$ scalar $\sigma$ [@GGP; @GG]. In four dimensions, we generate more terms due to the fact that we dualize two tensors. The sign flips of the kinetic terms of the two dual fields $\lambda_a$ are compatible with the prescription of Wick rotating pseudoscalars $\lambda_a\rightarrow{\mathrm{i}}\lambda_a$ [@vNW]. This is consistent with the duality relation .
A Euclidean version of the universal hypermultiplet action was also proposed in [@GS1]. Both their bulk Lagrangian and boundary term differ from ours. This has important consequences since the instanton action defines the weight in the path integral, and hence correlation functions and eventually the quantum-corrected hypermultiplet moduli space will be different.
Due to the sign changes in , the geometry of the scalar manifold is no longer SU(1,2)/U(2). Instead, it is given by the coset space $$\label{SL3}
{\mathcal{M}}_H^E = {\frac{\raisebox{-2pt}{$\mathrm{SL}(3,\fieldR)$}}{\mathrm{SL}(2,\fieldR) \times
\mathrm{SO}(1,1)}}\ ,$$ which is *not* a quaternion-Kähler manifold. This is not in contradiction with supersymmetry, since only *Minkowskian* supersymmtry requires the target space to be quaternionic [@BW]. A brief discussion on the geometry of the space is given in appendix \[SL3R\].
In four dimensions, the same target space can be obtained by applying the **c**-map [@CFG] to pure $N=2$ Euclidean supergravity [@TvN]. This turns the four bosonic degrees of freedom contained in the metric and graviphoton into the four scalars of the universal hypermultiplet and gives rise to the two sign flips. Moreover, the **c**-map maps Reissner-Nordstrom black hole solutions to D-instantons in the universal hypermultiplet, as was shown in [@BGLMM].
We remark that it is the inverted signs in the Euclidean hypermultiplet action that make instanton solutions in flat space possible. Indeed, the trace of the Einstein equation sets the bulk Lagrangian to zero, hence nontrivial field configurations would require a nonvanishing curvature scalar if the sigma model part of the Lagrangian were positive definite. The negative signs in allow for cancellations that are compatible with $R=0$. Note also that since on the hypermultiplet side the bulk action vanishes for any solution, the instanton action comes entirely from the boundary term discussed above. As already stated, the boundary term is different from the one proposed in [@GS1]. For this reason, we get different results for the instanton action, and eventually for the instanton corrected hypermultiplet moduli space.
What is more important from the point of view of instanton calculations, is that the Euclidean Lagrangian is no longer positive definite. In a path integral formulation, this makes the finite action configurations irrelevant, since the action is not bounded from below. Moreover, perturbative fluctuations around the instanton yield diverging non-Gaussian integrals, and the semiclassical approximation would break down. Similar considerations apply to the $N=2$ tensor multiplet, whose Euclidean action is not positive definite.
On the other hand, the Euclidean double-tensor multiplet Lagrangian is bounded from below, since the matrix $M_{ab}$ is positive definite. This leads to a well-defined semiclassical treatment, in which the instantons dominate the Euclidean path integral. For this reason, it is important to perform all calculations on the double-tensor multiplet side, and after having computed the instanton corrections there, we can dualize to the hypermultiplet formulation.
Instanton solutions
===================
Asymptotics
-----------
Before finding the explicit instanton solutions, it will be useful to discuss the asymptotic behaviour of the fields that can lead to a finite action. Since the Euclidean action consists of three positive definite terms, each term individually should integrate to a finite quantity. For simplicity we consider for the moment flat four-dimensional space. This determines the following behaviour at infinity: $$\label{large-r}
\phi \rightarrow \phi_\infty + \mathcal{O} \Big( {\frac{\raisebox{-2pt}{$1$}}{r^2}} \Big)\
,\quad \chi \rightarrow \chi_\infty + \mathcal{O} \Big( {\frac{\raisebox{-2pt}{$1$}}{r^2}}
\Big)\ ,\quad H_{\mu\nu\rho} \propto {\frac{\raisebox{-2pt}{$1$}}{r^3}}\ .$$ The asymptotic value of $\phi$ is identified with the four- (or five-) dimensional string coupling constant, $$g_s \equiv {\mathrm{e}}^{-\phi_\infty/2}\ .$$ The field strengths determine topological charges, defined by integrating the tensors $H^a$ over spheres at infinity, $$\label{HQ}
\int_{S^{D-1}_\infty}\! H^a = Q^{(a)} \ ,\quad a = 1,2\ .$$ In the dual (hypermultiplet) formulation, topological charges become Noether charges, corresponding to the Peccei-Quinn symmetries which act as constant shifts in the Lagrange multipliers $\lambda_a$. These charges descend from the brane charges in ten or eleven dimensions, and, in the appropriate units, are expected to be quantized.
The Euclidean space we shall concentrate on is actually flat space with a countable number of points, the locations of the instantons, excised[^4], $$\label{space}
{\mathcal{M}}= \fieldR^D - \cup_i\, \{\vec{x}_i\}\ ,$$ such that non-trivial cycles with corresponding charges exist. Stated differently, in the supergravity approximation it will typically not be possible to find regular solutions at the locations of the instantons, as we will explicitly see below. The only singularity which can still lead to a finite action is a logarithmic singularity in $\phi$ at the origin, $$\label{small-r}
\phi \rightarrow c\, \ln r\ ,$$ for some constant $c$. In our examples below, $\chi$ will tend to a constant $\chi_0$, and the tensors have the same $1/r^3$ behaviour such that the charges stay the same when the $H^a$ are integrated around an infinitesimal sphere around the origin.
The Bogomol’nyi bound
---------------------
The Euclidean double-tensor multiplet action is positive semi-definite (apart from the Einstein-Hilbert term). In fact, we can derive a lower bound by writing it as $${\mathcal{L}}_\mathrm{DT}^E = {\mathrm{d}}^Dx\, \sqrt{g\,} R + {\tfrac{1}{2}}*\! \big( N\! *\! H +
O E \big)^t {\wedge}\big( N\! *\! H + O E \big) + (-)^D H^t {\wedge}N^t O E\ .$$ Here we have defined $$H = \begin{pmatrix} H^1 \\[2pt] H^2 \end{pmatrix}\ ,\quad E =
\begin{pmatrix} {\mathrm{d}}\phi \\[2pt] {\mathrm{e}}^{-\phi/2}\, {\mathrm{d}}\chi
\end{pmatrix}\ ,\quad N = {\mathrm{e}}^{\phi/2} \begin{pmatrix} 0 & {\mathrm{e}}^{\phi/2}\,
\\[2pt] 1 & -\chi \end{pmatrix}\ ,$$ such that $N^t N=M$, and $O$ is some orthogonal (scalar) field-dependent matrix, whose appearance is due to the fact that $N$ and the zweibein $E$ are determined only modulo local O(2) transformations.
Clearly, the action is bounded from below by $$\label{bound}
S^E \geq \int_{{\mathcal{M}}} \big( {\mathrm{d}}^Dx\, \sqrt{g\,} R + (-)^D H^t {\wedge}N^t
O E \big)\ ,$$ where the second term is topological, as it is independent of the spacetime metric. The bound is saturated by field configurations satisfying the BPS condition $$\label{BPS}
* H = - N^{-1} O E\ .$$ A similar Bogomol’nyi equation was derived for an $N=1$, $D=4$ tensor multiplet (containing one tensor and one scalar) in [@R]. Notice that, if the matrix $O$ is invariant, this equation transforms covariantly under and .
Equation is a proper BPS condition only if it implies the equations of motion, and this will fix the O(2) degeneracy. It is easily verified that field configurations satisfying have vanishing energy-momentum tensors, hence they can exist only in Ricci-flat spaces. We therefore have to amend our BPS condition by the equation $R_{\mu\nu}(g)=0$.
For the field equations of the tensors, ${\mathrm{d}}(M\!*\!H)=0$, to hold we must have $$\label{closed}
{\mathrm{d}}(N^t O E) = 0\ .$$ This condition also guarantees that the topological term in is closed and hence can locally be written as a total derivative. As a consequence, it does not contribute to the equations of motion such that also the field equations for the scalars are guaranteed to be satisfied. The latter follow from requiring that the solution of correspond to closed forms for $H^a$, $${\mathrm{d}}(N^{-1} O *\! E)=0\ .$$
To determine the O(2) matrices that are compatible with , we parametrize $O$ by $$O = \begin{pmatrix} 1 & 0 \\[2pt] 0 & {\epsilon}\end{pmatrix}
\begin{pmatrix} c & -s \\[2pt] s & c \end{pmatrix}\ ,$$ where the functions $c(\phi,\chi)$ and $s(\phi,\chi)$ are constrained by $c^2+s^2=1$, and ${\epsilon}=\pm 1$ for the two components of O(2) with $\det O={\epsilon}$. Equation then gives rise to the differential equations $$\begin{aligned}
0 & = {\partial}_\phi c - {\mathrm{e}}^{\phi/2} {\partial}_\chi s \notag \\*
0 & = {\partial}_\phi s + {\mathrm{e}}^{\phi/2} {\partial}_\chi c - {\tfrac{1}{2}}(2{\epsilon}- 1) s\ .
\end{aligned}$$ We derive the general solution in appendix \[appO2\]. The result is that there are *three* distinct BPS conditions corresponding to the O(2) matrices $$O_{1,2} = \pm \begin{pmatrix} 1 & 0 \\[2pt] 0 & {\epsilon}\end{pmatrix}\
,\quad O_3 = \pm {\frac{\raisebox{-2pt}{$1$}}{|\tau'|}} \begin{pmatrix} \operatorname{Re}\tau' & -\operatorname{Im}\tau'\, \\[4pt] \operatorname{Im}\tau' & \operatorname{Re}\tau' \end{pmatrix}\ ,$$ invariant under both and . Here $\tau'=\tau-\chi_0$ with $\tau$ as in , $\chi_0$ is a real integration constant, and the plus and minus signs refer to the instanton and anti-instanton, respectively.
For these three O(2) matrices the 1-form $N^t OE$ is exact, $$\label{dY}
N^t O E = \pm\, {\mathrm{d}}Y\ ,$$ where modulo an additive constant $$Y_{1,2} = \begin{pmatrix} {\epsilon}\chi \\[2pt] {\mathrm{e}}^\phi - {\tfrac{1}{2}}{\epsilon}\chi^2
\end{pmatrix}\ ,\quad Y_3 = {\tfrac{1}{2}}\sqrt{4 {\mathrm{e}}^{\phi} + (\chi -
\chi_0)^2\,} \begin{pmatrix} 2 \\ - \chi - \chi_0 \end{pmatrix}\ .$$ It follows that the action for BPS configurations is given by a topological boundary term $$\label{top-BT}
S^E\,|_\mathrm{BPS} = (-)^D \int_{\mathcal{M}}H^t {\wedge}N^t O E = \mp \int_{{\partial}{\mathcal{M}}}
Y^t H\ .$$ The instanton action is therefore determined by the charges $Q^{(a)}$ and the values of the fields $\chi$ and ${\mathrm{e}}^{\phi}$ at the boundaries.
It is easy to find the corresponding BPS equation in the dual hypermultiplet formulation. Using , , and the fact that $M=N^t N$, we find for the Lagrange multipliers $${\mathrm{d}}\lambda = \pm\, {\mathrm{d}}Y\ ,$$ such that, up to a constant, the solutions for the two extra scalars are completely determined in terms of $\phi$ and $\chi$.
Solutions and instanton action
------------------------------
We can solve the BPS condition for the three possible matrices $O$. For $O_1=\pm 1$, the condition reads $$\label{O1_BPS}
* H = \pm \begin{pmatrix} \chi {\overset{\leftrightarrow}{{\mathrm{d}}}} {\mathrm{e}}^{-\phi} \\[2pt] {\mathrm{d}}{\mathrm{e}}^{-\phi} \end{pmatrix}\ .$$ Applying ${\mathrm{d}}*$ to the equation and using the Bianchi identities of $H$, we find that ${\mathrm{e}}^{-\phi}$ must be harmonic, and from the first component it then follows that also $\chi$ satisfies the Laplace equation, $$O_1:\quad {\mathrm{d}}*\! {\mathrm{d}}{\mathrm{e}}^{-\phi} = 0\ ,\quad {\mathrm{d}}*\! {\mathrm{d}}\chi = 0\ .$$ As mentioned above, scalars satisfying these conditions will also solve their field equations.
In the following, we consider for simplicity spherically symmetric configurations (single instantons) in flat space only. The dilaton equation of motion is then solved by $$\label{O1_dil}
{\mathrm{e}}^{-\phi} = {\mathrm{e}}^{-\phi_\infty} + {\frac{\raisebox{-2pt}{$|Q^{(2)}|$}}{\Omega_D\,
r^{D-2}}}\ .$$ Here $\Omega_D=(D-2)\mathrm{Vol}(S^{D-1})$, and we have chosen the location of the instanton ($\vec{x}_1$ in ) as the origin. The integration constant $Q^{(2)}$ appearing in the solution is equal to the topological charge associated with $H^2$, as follows from the second equation of . The ‘selfdual’ instanton (upper sign in ) is taken for negative $Q^{(2)}$, the ‘anti-selfdual’ instanton for positive $Q^{(2)}$.
Since, up to proportionality factors, there is only a unique spherically symmetric harmonic function, $\chi$ must be of the form $\chi=\chi_1
{\mathrm{e}}^{-\phi}+\chi_0$ with $\chi_0$, $\chi_1$ constant. It then follows from that $\chi_0$ is determined by $$\chi_0 = {\frac{\raisebox{-2pt}{$Q^{(1)}$}}{Q^{(2)}}}\ ,$$ and this relation is consistent with the shift symmetries and , since the charges transform non-trivially. The instanton action for $O_1$ is given by $$S_1^E = \mp \int_{{\partial}{\mathcal{M}}} \big[ \chi H^1 + ({\mathrm{e}}^{\phi} - {\tfrac{1}{2}}\chi^2)
H^2 \big]\ ,$$ where the boundary consists of the disjoint union of two spheres, ${\partial}{\mathcal{M}}=S^{D-1}_\infty\,\cup\,S^{D-1}_0$, with radii as indicated. The terms involving $\chi$ will diverge on $S^{D-1}_0$ since $\chi$ is harmonic, so in order to obtain a finite action we have to take $\chi=
\chi_0$ constant. This was already anticipated from the asymptotic behaviour of the fields, discussed in the beginning of this section. The action then reads $$S_1^E = \frac{\big| Q^{(2)} \big|}{g_s^2}\ .$$ This solution was also found in [@BB; @GS1], and should correspond to the fivebrane wrapping the entire Calabi-Yau [@BBS]. The instanton action is positive and hence does not distinguish instantons from anti-instantons. Imaginary theta-angle-like terms will have to be added to make this distinction.
Turning to $O_2$, we have the BPS condition $$\label{H-O2}
* H = \pm\, {\mathrm{d}}\begin{pmatrix} {\mathrm{e}}^{-\phi} \chi \\[2pt] {\mathrm{e}}^{-\phi}
\end{pmatrix}\ .$$ Again, ${\mathrm{e}}^{-\phi}$ is harmonic, and the same now applies to ${\mathrm{e}}^{-\phi}
\chi$. If one imposes rotational symmetry then $$\label{O2-sol}
O_2:\quad {\mathrm{d}}*\! {\mathrm{d}}{\mathrm{e}}^{-\phi} = 0\ ,\quad \chi = \chi_1 {\mathrm{e}}^{\phi} +
\chi_0\ ,$$ and from , it follows again that $Q^{(1)}=\chi_0 Q^{(2)}$. Notice that the field $\chi$ is now completely regular everywhere, and interpolates between the boundaries according to $$\Delta \chi \equiv \chi_\infty - \chi_0 = {\frac{\raisebox{-2pt}{$\chi_1$}}{g_s^2}}\ .$$ The complete solution agrees with the asymptotics derived in and .
For this solution, with the dilaton again given by , the instanton action then becomes $$\label{S2-inst}
S_2^E = \big| Q^{(2)} \big|\, \Big( {\frac{\raisebox{-2pt}{$1$}}{g_s^2}} + {\tfrac{1}{2}}\,
(\Delta \chi)^2 \Big)\ .$$ For the particular case of $\Delta \chi=0$, the solution and instanton action are the same as for the $O_1$ solution. Notice also that both terms are positive and invariant under the shift symmetries and , as guaranteed by the properties of the original action. For $\Delta\chi\neq 0$, our instanton solution is new, and this term in the instanton action does not depend on the string coupling constant $g_s$. The appearance of $\Delta\chi$ in the instanton action is one of the new results in this paper. Its presence was somehow anticipated in [@BBS], and here we have computed it explicitly.
We now turn to $O_3$. The BPS equation for this case reads $$* H = \pm {\frac{\raisebox{-2pt}{$1$}}{|\tau'|}} \begin{pmatrix} -2\, {\mathrm{d}}\phi + {\mathrm{e}}^{-\phi}
(\chi + \chi_0)\, {\mathrm{d}}\chi + \chi (\chi - \chi_0)\, {\mathrm{d}}{\mathrm{e}}^{-\phi}\,
\\[4pt] (\chi - \chi_0)\, {\mathrm{d}}{\mathrm{e}}^{-\phi} + 2 {\mathrm{e}}^{-\phi} {\mathrm{d}}\chi
\end{pmatrix}\ .$$ We have been unable to find the general solution[^5]. Instead, let us consider two Ansätze for which we can explicitly solve the equations. First, we set $\chi=2\chi_1{\mathrm{e}}^{\phi/2}
+\chi_0$. Then the equations simplify to $$* H = \pm 2\, {\mathrm{d}}{\mathrm{e}}^{-\phi/2} \begin{pmatrix} \sqrt{1 + \chi_1^2\,}\,
\\[2pt] 0 \end{pmatrix}\ .$$ It follows that now ${\mathrm{e}}^{-\phi/2}$ is harmonic, with solution $${\mathrm{e}}^{-\phi/2} = {\mathrm{e}}^{-\phi_{\infty}/2} + \frac{\big|Q^{(1)}\big|}
{2 \sqrt{1 + \chi_1^2\,}\, \Omega_D\, r^{D-2}}\ .$$ The scalar $\chi$ is then completely regular and interpolates between the boundaries as $$\Delta \chi = \chi_{\infty} - \chi_0 = \frac{2\chi_1}{g_s}\ .$$ Since $H^2=0$ we have $Q^{(2)}=0$, and for the instanton action we find $$\label{Sinst_O3_1}
S_3^E = \big| Q^{(1)} \big|\, \sqrt{{\frac{\raisebox{-2pt}{$4$}}{g_s^2}} + (\Delta \chi)^2}
= \big| Q^{(1)} \big|\, \big| \tau'_{\infty} \big|\ ,$$ where $\tau'_{\infty}=(\chi_\infty-\chi_0)+2{\mathrm{i}}\,{\mathrm{e}}^{\phi_{\infty}/2}$ is the value of $\tau'$ at infinity. For $\Delta\chi=0$, a similar solution was also found in [@GS1]. Following the discussion in [@BBS], it should correspond, from a IIA point of view, to the D2-brane wrapping a three-cycle in the Calabi-Yau, or to the D1+D3+D5-branes wrapping even cycles in type IIB. Notice again consistency with the symmetries and . Observe also that for $\Delta
\chi=0$, the solution is inversely proportional to $g_s$, and is for small $g_s$ dominating over the fivebrane instanton .
As a second Ansatz, consider $\chi=2\chi_1{\mathrm{e}}^\phi+\chi_0$. This differs from the first Ansatz in the power of ${\mathrm{e}}^\phi$. The BPS condition turns into $$* H = \pm 2\, {\mathrm{d}}\sqrt{{\mathrm{e}}^{-\phi} + \chi_1^2\,} \begin{pmatrix} 1 -
\chi_1 \chi_0 \\[2pt] -\chi_1 \end{pmatrix}\ .$$ Accordingly, the square root must be harmonic, and we find for $\chi_1
\neq 0$ (the case $\chi_1=0$ is included in the previous Ansatz), $${\mathrm{e}}^{-\phi} = (h - \chi_1) (h + \chi_1)\ ,\quad h = \sqrt{
{\mathrm{e}}^{-\phi_\infty} + \chi_1^2\,} + \Big| {\frac{\raisebox{-2pt}{$Q^{(2)}$}}{2\chi_1}} \Big|\,
{\frac{\raisebox{-2pt}{$1$}}{\Omega_D\, r^{D-2}}}\ .$$ The scalar field $\chi$ is regular everywhere and interpolates between zero and infinity as $$\Delta \chi = \frac{2\chi_1}{g_s^2}\ .$$ The BPS equation further fixes the constant $\chi_1$ to be $$\chi_1 = - {\frac{\raisebox{-2pt}{$Q^{(2)}$}}{Q^{(1)} - \chi_0 Q^{(2)}}}\ ,$$ and the instanton action is easily computed from , $$S_3^E = \big| \tau'_\infty \big|\, \Big( \big| \hat{Q}^{(1)} \big|
+ {\tfrac{1}{2}}\big| \Delta \chi\, Q^{(2)} \big| \Big)\ .$$ We have redefined the $Q^{(1)}$ charge according to $$\hat{Q}^{(1)} \equiv Q^{(1)} - \chi_0 Q^{(2)}\ ,$$ such that it is invariant under . For $Q^{(2)}=0$, the instanton action then clearly reduces to .
The obtained results for the instanton action carry over to the hypermultiplet side, because the dualization procedure does not affect the real part of the instanton action.
Discussion
==========
In this paper, we have carried out the first steps of calculating instanton corrections to the hypermultiplet moduli space. An important ingredient was to derive a Bogomol’nyi bound for the double-tensor multiplet Lagrangian, and to solve the corresponding BPS equation. In a supersymmetric formulation, adapted to Euclidean space, we expect our instanton solutions to preserve one half of the supersymmetries. A more general formulation for the double-tensor multiplet Lagrangian, including the fermions and supersymmetry transformation rules, is presently under study. This will be important for finding the fermionic zero modes and eventually for computing instanton corrections to the relevant correlation functions that determine the hypermultiplet quantum-geometry. The exact moduli space must be consistent with the results derived in our paper. In particular, our supergravity instanton solutions should match with the results obtained from wrapping branes in the full ten-dimensional string theory. Stated differently, the universal hypermultiplet metric must contain exponential corrections which, at leading order in the string coupling constant and $\alpha'$, agree with the form of our instanton action. Using some results about quaternionic geometry [@CP; @DWRV], it should be possible to find quaternionic metrics which asymptotically reproduce our results. We intend to report further on these issues in the near future.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Michael Gutperle and Thomas Mohaupt for discussions and reading an earlier draft of this paper. U.T. thanks the Deutsche Forschungsgemeinschaft for financial support.
SL(3,R) / SL(2,R) $\times$ SO(1,1) {#SL3R}
==================================
In this appendix, we discuss some geometrical aspects related to the sigma model corresponding to , with target space . The easiest way to study this space is by using the fact that the Minkowskian version of the universal hypermultiplet moduli space is both Kähler and quaternion-Kähler. Since Kähler geometry is simpler to analyze, we will study the coset from the point of view of Kähler geometry. Because of the sign flips compared to the Minkowskian version, the target space will no longer be Kähler. It is therefore not possible to define complex coordinates together with a Kähler potential that determines the metric. As we show below, it is still possible to define coordinates and a potential from which the metric can be computed. To see this, we first define the fields $$a = \sigma + {\tfrac{1}{2}}\chi \varphi\ ,\quad C_\pm = {\tfrac{1}{2}}(\varphi \pm
\chi)\ ,$$ in terms of which the sigma model part of the Euclidean Lagrangian reads $${\mathcal{L}}_\mathrm{UH}^E = {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 - 2 {\mathrm{e}}^{-\phi} *\! {\mathrm{d}}C_+ {\wedge}{\mathrm{d}}C_- - {\tfrac{1}{2}}{\mathrm{e}}^{-2\phi}\, |{\mathrm{d}}a + C_+ {\overset{\leftrightarrow}{{\mathrm{d}}}} C_-|^2\ .$$ If we further pass to coordinates $u^1_\pm,u^2_\pm\in\fieldR$ via the relations $$S_\pm = {\mathrm{e}}^{\phi} \mp a - C_+ C_- = {\frac{\raisebox{-2pt}{$1 \mp u^1_\pm$}}{1 \pm
u^1_\pm}}\ ,\quad C_\pm = {\frac{\raisebox{-2pt}{$u^2_\pm$}}{1 \pm u^1_\pm}}\ ,$$ then the Lagrangian can be written as $${\mathcal{L}}^E_\mathrm{UH} = 2 g_{ij} *\! {\mathrm{d}}u^i_+ {\wedge}{\mathrm{d}}u^j_-$$ with a metric $$\label{K+-}
g_{ij} = - {\frac{\raisebox{-2pt}{${\partial}^2$}}{{\partial}u^i_+ {\partial}u^j_-}}\, \ln\! \big( 1 + u^1_+
u^1_- + u^2_+ u^2_- \big)\ .$$ The $u^i_\pm$ are inhomogeneous coordinates of the coset space , transforming under $M\in\mathrm{SL}(2,\fieldR)$ as $u_+
\rightarrow M u_+$, $u_-\rightarrow (M^{-1})^tu_-$. We have therefore identified a potential in terms of real coordinates which determines the metric components. Such spaces are called para-Kähler[^6]. The metric for the Minkowskian universal hypermultiplet moduli space is of the same form as in , but with $u^i_\pm$ treated as complex coordinates, where $u^i_-=-
\bar{u}^i_+$ under complex conjugation.
Determination of O(2) matrices {#appO2}
==============================
We need to solve the differential equations $$\begin{aligned}
0 & = {\partial}_\phi c - {\mathrm{e}}^{\phi/2} {\partial}_\chi s \label{O2_1} \\[2pt]
0 & = {\partial}_\phi s + {\mathrm{e}}^{\phi/2} {\partial}_\chi c - {\tfrac{1}{2}}(2{\epsilon}- 1) s\ ,
\label{O2_2}
\end{aligned}$$ where $c$ and $s$ are subject to the constraint $c^2+s^2=1$. We first multiply by $s$ and by $c$, respectively, and use $-s{\partial}s=c{\partial}c$ to write the equations as $$\begin{aligned}
0 & = s\, {\partial}_\phi c + {\mathrm{e}}^{\phi/2} c\, {\partial}_\chi c \label{O2_3} \\[2pt]
0 & = c\, {\partial}_\phi s + {\mathrm{e}}^{\phi/2} c\, {\partial}_\chi c - {\tfrac{1}{2}}(2{\epsilon}- 1)
c s\ . \label{O2_4}
\end{aligned}$$ Multiplying the difference of these equations by $c$ gives $$\label{O2_5}
0 = c \big[ c\, {\overset{\leftrightarrow}{{\partial}_\phi}} s - {\tfrac{1}{2}}(2{\epsilon}- 1) c s \big] =
{\partial}_\phi s - {\tfrac{1}{2}}(2{\epsilon}- 1)\, (1 - s^2) s\ ,$$ which involves only $s$ and can easily be integrated: $${\frac{\raisebox{-2pt}{$s^2$}}{1 - s^2}} = {\frac{\raisebox{-2pt}{$1 - c^2$}}{c^2}} = f^2(\chi)\, {\mathrm{e}}^{(2{\epsilon}- 1)
\phi}\ .$$ The positive integration constant $f^2$ may depend on $\chi$. These expressions we plug into the sum of and , $$\begin{aligned}
0 & = {\mathrm{e}}^{\phi/2} {\partial}_\chi c^2 + {\partial}_{\phi} (cs) - {\tfrac{1}{2}}(2{\epsilon}- 1) cs
\notag \\[2pt]
& = - {\frac{\raisebox{-2pt}{$2 f\, {\mathrm{e}}^{(3 - 4{\epsilon})\phi/2}$}}{(f^2 + {\mathrm{e}}^{(1 - 2{\epsilon})\phi}
)^2}}\ \big[ {\partial}_\chi f \pm {\tfrac{1}{2}}(2{\epsilon}- 1)\, {\mathrm{e}}^{({\epsilon}- 1)\phi}
f^2 \big]\ ,
\end{aligned}$$ where the sign ambiguity originates from taking the square root of $(cs)^2$. The equation is satisfied if the expression in square brackets vanishes. For ${\epsilon}=-1$, this is only possible if $f=0$ since $f$ is independent of $\phi$. For ${\epsilon}=+1$, which corresponds to $O\in
\mathrm{SO}(2)$, we find $$f = 0 \quad\text{or}\quad f = \pm {\frac{\raisebox{-2pt}{$2$}}{\chi - \chi_0}}\ ,$$ with $\chi_0$ an integration constant. $f=0$ implies $c=\pm 1$ and $s=
0$. For nontrivial $f$ we obtain (with the relative sign fixed by the original equations and $\eqref{O2_2}$) $$c = \pm \frac{\chi - \chi_0}{\sqrt{4 {\mathrm{e}}^\phi + (\chi - \chi_0)^2\,}
\,}\ ,\quad s = \pm {\frac{\raisebox{-2pt}{$2\, {\mathrm{e}}^{\phi/2}$}}{\sqrt{4 {\mathrm{e}}^\phi + (\chi -
\chi_0)^2\,}\,}}\ ,$$ or in terms of $\tau'=(\chi-\chi_0)+2{\mathrm{i}}\,{\mathrm{e}}^{\phi/2}$, $$c + {\mathrm{i}}s = \pm {\frac{\raisebox{-2pt}{$\tau'$}}{|\tau'|}}\ .$$
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[^1]: By type IIB with $h_{1,1}=0$ we mean the mirror version of IIA with $h_{1,2}=0$. As explained in [@AG], this model has to be understood in terms of a Landau-Ginzburg description instead of a geometric compactification, since all CY manifolds are Kähler and have $h_{1,1}>0$.
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\chi+\epsilon$, $\sigma\rightarrow\sigma-\epsilon\varphi$.
[^4]: A possible contribution to the action from a Gibbons-Hawking boundary term will then be absent.
[^5]: The most general spherically symmetric solution was later found in [@DdVTV].
[^6]: A more general discussion on para-Kähler manifolds, in the context of Euclidean supergravity coupled to vector multiplets, will be given in [@CHM].
| 0 |
---
abstract: |
We show that, consistently, for some regular cardinals $\theta<\lambda$, there exist a Boolean algebra ${{\mathbb B}}$ such that $|{{\mathbb B}}|=\lambda^+$ and for every subalgebra ${{\mathbb B}}'\subseteq{{\mathbb B}}$ of size $\lambda^+$ we have ${{\rm Depth}}({{\mathbb B}}')=
\theta$.
address:
- |
Department of Mathematics\
University of Nebraska at Omaha\
Omaha, NE 68182-0243, USA\
and Mathematical Institute of Wroclaw University\
50384 Wroclaw, Poland
- |
Institute of Mathematics\
The Hebrew University of Jerusalem\
91904 Jerusalem, Israel\
and Department of Mathematics\
Rutgers University\
New Brunswick, NJ 08854, USA
author:
- 'Andrzej Ros[ł]{}anowski'
- Saharon Shelah
title: 'Historic forcing for ${{\rm Depth}}$'
---
Introduction
============
The present paper is concerned with forcing a Boolean algebra which has some prescribed properties of ${{\rm Depth}}$. Let us recall that, for a Boolean algebra ${{\mathbb B}}$, its depth is defined as follows: $$\begin{array}{lcl}
{{\rm Depth}}({{\mathbb B}})&=&\sup\{|X|: X\subseteq{{\mathbb B}}\mbox{ is well-ordered by the Boolean
ordering}\;\},\\
{{\rm Depth}}^+({{\mathbb B}})&=&\sup\{|X|^+: X\subseteq{{\mathbb B}}\mbox{ is well-ordered by the
Boolean ordering}\;\}.
\end{array}$$ (${{\rm Depth}}^+({{\mathbb B}})$ is used to deal with attainment properties in the definition of ${{\rm Depth}}({{\mathbb B}})$, see e.g. [@RoSh:534 §1].) The depth (of Boolean algebras) is among cardinal functions that have more algebraic origins, and their relations to “topological fellows” is often indirect, though sometimes very surprising. For example, if we define $${{\rm Depth}}_{{\rm H}+}({{\mathbb B}})=\sup\{{{\rm Depth}}({{\mathbb B}}/I): I\mbox{ is an ideal in }{{\mathbb B}}\;\},$$ then for any (infinite) Boolean algebra ${{\mathbb B}}$ we will have that ${{\rm Depth}}_{{\rm
H}+}({{\mathbb B}})$ is the tightness $t({{\mathbb B}})$ of the algebra ${{\mathbb B}}$ (or the tightness of the topological space ${\rm Ult}({{\mathbb B}})$ of ultrafilters on ${{\mathbb B}}$), see [@M2 Theorem 4.21]. A somewhat similar function to ${{\rm Depth}}_{{\rm H}+}$ is obtained by taking $\sup\{{{\rm Depth}}({{\mathbb B}}'): {{\mathbb B}}'$ is a subalgebra of ${{\mathbb B}}\;\}$, but clearly this brings nothing new: it is the old Depth. But if one wants to understand the behaviour of the depth for subalgebras of the considered Boolean algebra, then looking at the following [*subalgebra ${{\rm Depth}}$ relation*]{} may be very appropriate: $$\begin{array}{lr}
{{\rm Depth}}_{\rm Sr}({{\mathbb B}})=\{(\kappa,\mu):&\mbox{there is an infinite subalgebra
${{\mathbb B}}'$ of ${{\mathbb B}}$ such that }\ \\
&|{{\mathbb B}}'|=\mu\mbox{ and }{{\rm Depth}}({{\mathbb B}}')=\kappa\;\}.
\end{array}$$ A number of results related to this relation is presented by Monk in [@M2 Chapter 4]. There he asks if there are a Boolean algebra ${{\mathbb B}}$ and an infinite cardinal $\theta$ such that $(\theta,(2^\theta)^+)\in {{\rm Depth}}_{\rm
Sr}({{\mathbb B}})$, while $(\omega, (2^\theta)^+)\notin{{\rm Depth}}_{\rm Sr}({{\mathbb B}})$ (see Monk [@M2 Problem 14]; we refer the reader to Chapter 4 of Monk’s book [@M2] for the motivation and background of this problem). Here we will partially answer this question, showing that it is consistent that there is such ${{\mathbb B}}$ and $\theta$. The question if that can be done in ZFC remains open.
Our consistency result is obtained by forcing, and the construction of the required forcing notion is interesting [*per se*]{}. We use the method of [*historic forcing*]{} which was first applied in Shelah and Stanley [@ShSt:258]. The reader familiar with [@ShSt:258] will notice several correspondences between the construction here and the method used there. However, we do not relay on that paper and our presentation here is self-contained.
Let us describe how our historic forcing notion is built. So, we fix two (regular) cardinals $\theta,\lambda$ and our aim is to force a Boolean algebra ${\dot{{{\mathbb B}}}^\theta_\lambda}$ such that $|{\dot{{{\mathbb B}}}^\theta_\lambda}|=\lambda^+$ and for every subalgebra ${{\mathbb B}}\subseteq{\dot{{{\mathbb B}}}^\theta_\lambda}$ of size $\lambda^+$ we have ${{\rm Depth}}({{\mathbb B}})=\theta$. The algebra ${\dot{{{\mathbb B}}}^\theta_\lambda}$ will be generated by $\langle x_i:i\in\dot{U}\rangle$ for some set $\dot{U}\subseteq\lambda^+$. A condition $p$ will be an approximation to the algebra ${\dot{{{\mathbb B}}}^\theta_\lambda}$, it will carry the information on what is the subalgebra ${{\mathbb B}}_p=\langle x_i: i\in u^p\rangle_{{\dot{{{\mathbb B}}}^\theta_\lambda}}$ for some $u^p\subseteq\lambda^+$. A natural way to describe algebras in this context is by listing ultrafilters (or: homomorphisms into $\{0,1\}$):
\[0.C\] For a set $w$ and a family $F\subseteq 2^{\textstyle w}$ we define
${{\rm cl}}(F)=\{g\in 2^{\textstyle w}: (\forall u\in [w]^{\textstyle
<\omega})(\exists f\in F)(f{\restriction}u=g{\restriction}u)\}$,
${{\mathbb B}}_{(w,F)}$ is the Boolean algebra generated freely by $\{x_\alpha:\alpha\in w\}$ except that
if $u_0,u_1\in [w]^{\textstyle <\omega}$ and there is no $f\in F$ such that $f{\restriction}u_0\equiv 0$, $f{\restriction}u_1\equiv 1$
then $\bigwedge\limits_{\alpha\in u_1} x_\alpha\wedge
\bigwedge\limits_{\alpha\in u_0} (-x_\alpha)=0$.
This description of algebras is easy to handle, for example:
\[see [[@Sh:479 2.6]]{}\] \[0.D\] Let $F\subseteq 2^{\textstyle w}$. Then:
1. Each $f\in F$ extends (uniquely) to a homomorphism from ${{\mathbb B}}_{(w,F)}$ to $\{0,1\}$ (i.e. it preserves the equalities from the definition of ${{\mathbb B}}_{(w,F)}$). If $F$ is closed, then every homomorphism from ${{\mathbb B}}_{(w,F)}$ to $\{0,1\}$ extends exactly one element of $F$.
2. If $\tau(y_0,\ldots,y_\ell)$ is a Boolean term and $\alpha_0,\ldots,
\alpha_\ell\in w$ are distinct then $$\begin{array}{l}
{{\mathbb B}}_{(w,F)}\models\tau(x_{\alpha_0},\ldots,x_{\alpha_\ell})\neq 0\qquad
\qquad\mbox{ if and only if}\\
(\exists f\in F)(\{0,1\}\models\tau(f(\alpha_0),\ldots,f(\alpha_k))=1).
\end{array}$$
3. If $w\subseteq w^*$, $F^*\subseteq 2^{\textstyle w^*}$ and $$(\forall f\in F)(\exists g\in F^*)(f\subseteq g)\quad\mbox{ and }\quad
(\forall g\in F^*)(g{\restriction}w\in{{\rm cl}}(F))$$ then ${{\mathbb B}}_{(w,F)}$ is a subalgebra of ${{\mathbb B}}_{(w^*,F^*)}$.
So each condition $p$ in our forcing notion ${{{\mathbb P}^\theta_\lambda}}$ will have a set $u^p\in
[\lambda^+]^{<\lambda}$ and a closed set $F^p\subseteq 2^{\textstyle u^p}$ (and the respective algebra will be ${{\mathbb B}}_p={{\mathbb B}}_{(u^p,F^p)}$). But to make the forcing notion work, we will have to put more restrictions on our conditions, and we will be taking only those conditions that have to be taken to make the arguments work. For example, we want that cardinals are not collapsed by our forcing, and demanding that ${{{\mathbb P}^\theta_\lambda}}$ is $\lambda^+$-cc (and somewhat $({<}\lambda)$–closed) is natural in this context. How do we argue that a forcing notion is $\lambda^+$–cc? Typically we start with a sequence of $\lambda^+$ distinct conditions, we carry out some “cleaning procedure” (usually involving the $\Delta$–lemma etc), and we end up with (at least two) conditions that “can be put together”. Putting together two (or more) conditions that are approximations to a Boolean algebra means amalgamating them. There are various ways to amalgamate conditions - we will pick one that will work for several purposes. Then, once we declare that some conditions forming a “clean” $\Delta$–sequence of length $\theta$ are in ${{{\mathbb P}^\theta_\lambda}}$, we will be bound to declare that the amalgamation is in our forcing notion. The amalgamation (and natural limits) will be the only way to build new conditions from the old ones, but the description above still misses an important factor. So far, a condition does not have to know what are the reasons for it to be called to ${{{\mathbb P}^\theta_\lambda}}$. This information is [*the history of the condition*]{} and it will be encoded by two functions $h^p,g^p$. (Actually, these functions will give histories of all elements of $u^p$ describing why and how those points were incorporated to $u^p$. Thus both functions will be defined on $u^p\times{{\rm ht}}(p)$, were ${{\rm ht}}(p)$ is the height of the condition $p$, that is the step in our construction at which the condition $p$ is created.) We will also want that our forcing is suitably closed, and getting “$({<}\lambda)$–strategically closed” would be fine. To make that happen we will have to deal with two relations on on ${{{\mathbb P}^\theta_\lambda}}$: $\leq_{\rm pr}$ and $\leq$. The first (“pure”) is $({<}\lambda)$–closed and it will help in getting the strategic closure of the second (main) one. In some sense, the relation $\leq_{\rm pr}$ represents “the official line in history”, and sometimes we will have to rewrite that official history, see Definition \[defptran\] and Lemma \[3y1\] (on changing history see also Orwell [@Or49]).
The forcing notion ${{{\mathbb P}^\theta_\lambda}}$ has some other interesting features. (For example, conditions are very much like fractals, they contain many self-similar pieces (see Definition \[defcompo\] and Lemma \[3.4x\]).) The method of historic forcing notions could be applicable to more problems, and this is why in our presentation we separated several observations of general character (presented in the first section) from the problem specific arguments (section 2)
[**Notation:**]{}Our notation is standard and compatible with that of classical textbooks on set theory (like Jech [@J]) and Boolean algebras (like Monk [@M1], [@M2]). However in forcing considerations we keep the older tradition that
[*the stronger condition is the greater one.* ]{}
Let us list some of our notation and conventions.
1. Throughout the paper, $\theta,\lambda$ are fixed regular infinite cardinals, $\theta<\lambda$.
2. A name for an object in a forcing extension is denoted with a dot above (like $\dot{X}$) with one exception: the canonical name for a generic filter in a forcing notion ${{\mathbb P}}$ will be called $\Gamma_{{\mathbb P}}$. For a ${{\mathbb P}}$–name $\dot{X}$ and a ${{\mathbb P}}$–generic filter $G$ over ${{\bf V}}$, the interpretation of the name $\dot{X}$ by $G$ is denoted by $\dot{X}^G$.
3. $i,j,\alpha,\beta,\gamma,\delta,\ldots$ will denote ordinals.
4. For a set $X$ and a cardinal $\lambda$, $[X]^{\textstyle<\lambda}$ stands for the family of all subsets of $X$ of size less than $\lambda$. The family of all functions from $Y$ to $X$ is called $X^{\textstyle Y}$. If $X$ is a set of ordinals then its order type is denoted by ${{\rm otp}}(X)$.
5. In Boolean algebras we use $\vee$ (and $\bigvee$), $\wedge$ (and $\bigwedge$) and $-$ for the Boolean operations. If ${{\mathbb B}}$ is a Boolean algebra, $x\in{{\mathbb B}}$ then $x^0=x$, $x^1=-x$.
6. For a subset $Y$ of an algebra ${{\mathbb B}}$, the subalgebra of ${{\mathbb B}}$ generated by $Y$ is denoted by $\langle Y\rangle_{{{\mathbb B}}}$.
[**Acknowledgements:**]{}We would like to thank the referee for valuable comments and suggestions.
The forcing and its basic properties
====================================
Let us start with the definition of the forcing notion ${{{\mathbb P}^\theta_\lambda}}$. By induction on $\alpha<\lambda$ we will define sets of conditions $P^{\theta,
\lambda}_\alpha$, and for each $p\in P^{\theta,\lambda}_\alpha$ we will define $u^p,F^p,{{\rm ht}}(p),h^p$ and $g^p$. Also we will define relations $\leq^\alpha$ and $\leq^\alpha_{\rm pr}$ on $P^{\theta,\lambda}_\alpha$. Our inductive requirements are:
1. for each $p\in P^{\theta,\lambda}_\alpha$:\
$u^p\in [\lambda^+]^{\textstyle<\lambda}$, ${{\rm ht}}(p)\leq\alpha$, $F^p
\subseteq 2^{\textstyle u^p}$ is a non-empty closed set, $g^p$ is a function with domain ${{\rm dom}}(g^p)=u^p\times{{\rm ht}}(p)$ and values of the form $(\ell,
\tau)$, where $\ell<2$ and $\tau$ is a Boolean term, and $h^p:u^p\times
{{\rm ht}}(p)\longrightarrow\theta+2$ is a function,
2. $\leq^\alpha,\leq^\alpha_{\rm pr}$ are transitive and reflexive relations on $P^{\theta,\lambda}_\alpha$, and $\leq^\alpha$ extends $\leq^\alpha_{\rm pr}$,
3. if $p,q\in P^{\theta,\lambda}_\alpha$, $p\leq^\alpha
q$, then $u^p\subseteq u^q$, ${{\rm ht}}(p)\leq{{\rm ht}}(q)$, and $F^p=\{f{\restriction}u^p:
f\in F^q\}$, and if $p\leq^\alpha_{\rm pr} q$, then for every $i\in u^p$ and $\xi<{{\rm ht}}(p)$ we have $h^p(i,\xi)=h^q(i,\xi)$ and $g^p(i,\xi)= g^q(i,\xi)$,
4. if $\beta<\alpha$ then $P^{\theta,\lambda}_\beta
\subseteq P^{\theta,\lambda}_\alpha$, and $\leq^\alpha_{\rm pr}$ extends $\leq^\beta_{\rm pr}$, and $\leq^\alpha$ extends $\leq^\beta$.
For a condition $p\in P^{\theta,\lambda}_\alpha$, we will also declare that ${{\mathbb B}}^p={{\mathbb B}}_{(u^p,F^p)}$ (the Boolean algebra defined in Definition \[0.C\]).
We define $P^{\theta,\lambda}_0=\{\langle\xi\rangle:\xi<\lambda^+\}$ and for $p=\langle\xi\rangle$ we let $F^p=2^{\textstyle\{\xi\}}$, ${{\rm ht}}(p)=0$ and $h^p=\emptyset=g^p$. The relations $\leq^0_{\rm pr}$ and $\leq^0$ both are the equality. \[Clearly these objects are as declared, i.e, clauses (i)$_0$–(iv)$_0$ hold true.\]
If $\gamma<\lambda$ is a limit ordinal, then we put $$\begin{array}{l}
P^*_\gamma=\big\{\langle p_\xi:\xi<\gamma\rangle: (\forall\xi<\zeta<
\gamma)(p_\xi\in P^{\theta,\lambda}_\xi\ \&\ {{\rm ht}}(p_\xi)=\xi \ \&\ p_\xi
\leq^\zeta_{\rm pr} p_\zeta)\big\},\\
P^{\theta,\lambda}_\gamma=\bigcup\limits_{\alpha<\gamma}P^{\theta,
\lambda}_\alpha\cup P^*_\gamma,
\end{array}$$ and for $p=\langle p_\xi:\xi<\gamma\rangle\in P^*_\gamma$ we let $$u^p=\bigcup\limits_{\xi<\gamma} u^{p_\xi},\quad F^p=\{f\in 2^{\textstyle
u^p}:(\forall\xi<\gamma)(f{\restriction}u^{p_\xi}\in F^{p_\xi})\},\quad {{\rm ht}}(p)=
\gamma$$ and $h^p=\bigcup\limits_{\xi<\gamma} h^{p_\xi}$ and $g^p=\bigcup\limits_{\xi<
\gamma} g^{p_\xi}$. We define $\leq^\gamma$ and $\leq^\gamma_{\rm pr}$ by:
$p\leq^\gamma_{\rm pr}q$if and only if
[*either*]{} $p,q\in P^{\theta,\lambda}_\alpha$, $\alpha<\gamma$ and $p\leq^\alpha_{\rm pr}q$,
[*or*]{} $q=\langle q_\xi:\xi<\gamma\rangle\in P^*_\gamma$, $p\in
P^{\theta,\lambda}_\alpha$ and $p\leq^\alpha_{\rm pr}q_\alpha$ for some $\alpha<\gamma$,
[*or*]{} $p=q$;
$p\leq^\gamma q$if and only if
[*either*]{} $p,q\in P^{\theta,\lambda}_\alpha$, $\alpha<\gamma$ and $p\leq^\alpha q$,
[*or*]{} $q=\langle q_\xi:\xi<\gamma\rangle\in P^*_\gamma$, $p\in
P^{\theta,\lambda}_\alpha$ and $p\leq^\alpha q_\alpha$ for some $\alpha<
\gamma$,
[*or*]{} $p=\langle p_\xi:\xi<\gamma\rangle\in P^*_\gamma$, $q=\langle
q_\xi:\xi<\gamma\rangle\in P^*_\gamma$ and $$(\exists\delta<\gamma)(\forall\xi<\gamma)(\delta\leq\xi\ \Rightarrow\
p_\xi\leq^\xi q_\xi).$$
\[It is straightforward to show that clauses (i)$_\gamma$–(iv)$_\gamma$ hold true.\]
Suppose now that $\alpha<\lambda$. Let $P^*_{\alpha+1}$ consist of all tuples $$\langle\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<\theta\rangle\rangle$$ such that for each $\xi_0<\xi_1<\theta$:
1. $\zeta^*<\theta$, $n^*<\omega$, $\tau^*=\tau^*(y_1,\ldots,y_{
n^*})$ is a Boolean term, $u^*\in [\lambda^+]^{\textstyle<\lambda}$,
2. $p_{\xi_0}\in P^{\theta,\lambda}_\alpha$, ${{\rm ht}}(p)=\alpha$, $v_{\xi_0}\in [u^{p_{\xi_0}}]^{\textstyle n^*}$,
3. the family $\{u^{p_\xi}:\xi<\theta\}$ forms a $\Delta$–system with heart $u^*$ and $u^{p_{\xi_0}}\setminus u^*\neq
\emptyset$ and $$\sup(u^*)<\min(u^{p_{\xi_0}}\setminus u^*)\leq\sup (u^{p_{\xi_0}}\setminus
u^*)<\min(u^{p_{\xi_1}}\setminus u^*),$$
4. ${{\rm otp}}(u^{p_{\xi_0}})={{\rm otp}}(u^{p_{\xi_1}})$ and if $H:u^{p_{
\xi_0}}\longrightarrow u^{p_{\xi_1}}$ is the order isomorphism then $H{\restriction}u^*$ is the identity on $u^*$, $F^{p_{\xi_0}}=\{f{\circ}H:f\in F^{p_{\xi_1}}
\}$, $H[v_{\xi_0}]=v_{\xi_1}$ and $$(\forall j\in u^{p_{\xi_0}})(\forall\beta<\alpha)(h^{p_{\xi_0}}(j,\beta)=
h^{p_{\xi_1}}(H(j),\beta)\ \&\ g^{p_{\xi_0}}(j,\beta)=g^{p_{\xi_1}}(H(j),
\beta)).$$
We put $P^{\theta,\lambda}_{\alpha+1}=P^{\theta,\lambda}_\alpha\cup P^*_{
\alpha+1}$ and for $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<
\theta\rangle\rangle\in P^*_{\alpha+1}$ we let $u^p=\bigcup\limits_{\xi<
\theta} u^{p_\xi}$ and $$\begin{array}{ll}
F^p=\{f\in 2^{\textstyle u^p}:& (\forall\xi<\theta)(f{\restriction}u^{p_\xi}\in
F^{p_{\xi}})\mbox{ and for all }\xi<\zeta<\theta\\
\ &f(\sigma_{\rm maj}(\tau_{3\cdot\xi},\tau_{3\cdot\xi+1},\tau_{3\cdot\xi+
2}))\leq f(\sigma_{\rm maj}(\tau_{3\cdot\zeta},\tau_{3\cdot\zeta+1},\tau_{3
\cdot\zeta+2}))\},
\end{array}$$ where $\tau_\xi=\tau^*(x_i:i\in v_\xi)$ for $\xi<\theta$ (so $\tau_\xi$ is an element of the algebra ${{\mathbb B}}^{p_\xi}={{\mathbb B}}_{(u^{p_\xi},F^{p_\xi})}$), and $\sigma_{\rm maj}(y_0,y_1,y_2)=(y_0\wedge y_1)\vee (y_0\wedge y_2)\vee
(y_1\wedge y_2)$. Next we let ${{\rm ht}}(p)=\alpha+1$ and we define functions $h^p,g^p$ on $u^p\times (\alpha+1)$ by $$h^p(j,\beta)=\left\{\begin{array}{lll}
h^{p_{\xi}}(j,\beta)&\mbox{if}&j\in u^{p_\xi},\ \xi<\theta,\ \beta<\alpha,\\
\theta &\mbox{if}&j\in u^*,\ \beta=\alpha,\\
\theta+1 &\mbox{if}&j\in u^{p_{\zeta^*}}\setminus u^*,\
\beta=\alpha,\\
\xi &\mbox{if}&j\in u^{p_\xi}\setminus u^*,\ \xi<\theta,\
\xi\neq\zeta^*,\ \beta=\alpha,
\end{array}\right.$$ $$g^p(j,\beta)=\left\{\begin{array}{lll}
g^{p_{\xi}}(j,\beta)&\mbox{if}&j\in u^{p_\xi},\ \xi<\theta,\ \beta<\alpha,\\
(1,\tau^*) &\mbox{if}&j\in v_\xi,\ \xi<\theta,\ \beta=\alpha,\\
(0,\tau^*) &\mbox{if}&j\in u^{p_\xi}\setminus v_\xi,\ \xi<\theta,\
\beta=\alpha.
\end{array}\right.$$ Next we define the relations $\leq^{\alpha+1}_{\rm pr}$ and $\leq^{\alpha+1}$ by:
$p\leq^{\alpha+1}_{\rm pr}q$if and only if
[*either*]{} $p,q\in P^{\theta,\lambda}_\alpha$ and $p\leq^\alpha_{\rm
pr}q$,
[*or*]{} $q=\langle \zeta^*,\tau^*,n^*,u^*,\langle q_\xi,v_\xi:\xi<\theta
\rangle\rangle\in P^*_{\alpha+1}$, $p\in P^{\theta,\lambda}_\alpha$, and $p\leq^\alpha_{\rm pr} q_{\zeta^*}$,
[*or*]{} $p=q$;
$p\leq^{\alpha+1} q$if and only if
[*either*]{} $p,q\in P^{\theta,\lambda}_\alpha$ and $p\leq^\alpha q$,
[*or*]{} $q=\langle\zeta^*,\tau^*,n^*,u^*,\langle q_\xi,v_\xi:\xi<\theta
\rangle\rangle\in P^*_{\alpha+1}$, $p\in P^{\theta,\lambda}_\alpha$, and $p\leq^\alpha q_\xi$ for some $\xi<\theta$,
[*or*]{} $p=\langle\zeta^{**},\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<
\theta\rangle\rangle$, $q=\langle\zeta^*,\tau^*,n^*,u^*,\langle q_\xi,v_\xi:
\xi<\theta\rangle\rangle$ are from $P^*_{\alpha+1}$ and $$(\forall\xi<\theta)(p_\xi\leq^\alpha q_\xi\ \&\ u^{p_\xi}=u^{q_\xi}).$$
\[Again, it is easy to show that clauses (i)$_{\alpha+1}$–(iv)$_{
\alpha+1}$ are satisfied.\]
After the construction is carried out we let $${{{\mathbb P}^\theta_\lambda}}=\bigcup\limits_{\alpha<\lambda} P^{\theta,\lambda}_\alpha\quad\mbox{
and }\quad{}\leq_{\rm pr}{}={}\bigcup\limits_{\alpha<\lambda}{\leq^\alpha_{
\rm pr}}\quad\mbox{ and }\quad{}\leq{}={}\bigcup\limits_{\alpha<\lambda}
{\leq^\alpha}.$$ One easily checks that $\leq_{\rm pr}$ is a partial order on ${{{\mathbb P}^\theta_\lambda}}$ and that the relation $\leq$ is transitive and reflexive, and that ${\leq_{\rm pr}}
\subseteq {\leq}$.
\[3.1\] Let $p,q\in{{{\mathbb P}^\theta_\lambda}}$.
1. If $p\leq q$ then ${{\rm ht}}(p)\leq{{\rm ht}}(q)$, $u^p\subseteq u^q$ and $F^p=\{
f{\restriction}u^p: f\in F^q\}$ (so ${{\mathbb B}}^p$ is a subalgebra of ${{\mathbb B}}^q$). If $p\leq q$ and ${{\rm ht}}(p)={{\rm ht}}(q)$, then $q\leq p$.
2. For each $j\in u^p$, the set $\{\beta<{{\rm ht}}(p): h^p(j,\beta)<\theta\}$ is finite.
3. If $p\leq_{\rm pr} q$ and $i\in u^p$, then $h^q(i,\beta)\geq\theta$ for all $\beta$ such that ${{\rm ht}}(p)\leq\beta<{{\rm ht}}(q)$.
4. If $i,j\in u^p$ are distinct, then there is $\beta<{{\rm ht}}(p)$ such that $\theta\neq h^p(i,\beta)\neq h^p(j,\beta)\neq\theta$.
5. For each finite set $X\subseteq{{\rm ht}}(p)$ there is $i\in u^p$ such that $$\{\beta<{{\rm ht}}(p):h^p(i,\beta)<\theta\}=X.$$
6. If $p\leq_{\rm pr} q$ then there is a $\leq_{\rm pr}$–increasing sequence $\langle p_\xi:\xi\leq{{\rm ht}}(p)\rangle\subseteq{{{\mathbb P}^\theta_\lambda}}$ such that $p_{{{\rm ht}}(p)}=p$, $p_{{{\rm ht}}(q)}=q$ and ${{\rm ht}}(p_\xi)=\xi$ (for $\xi\leq{{\rm ht}}(p)$). (In particular, if $p\leq_{\rm pr} q$ and ${{\rm ht}}(p)={{\rm ht}}(q)$ then $p=q$.)
7. If ${{\rm ht}}(p)=\gamma$ is a limit ordinal, $p=\langle p_\xi:\xi<\gamma
\rangle$, then for each $i\in u^p$ and $\xi<\gamma$: $$i\in u^{p_\xi}\quad\mbox{ if and only if }\quad (\forall\zeta<\gamma)(\xi
\leq\zeta\ \Rightarrow\ h^p(i,\zeta)\geq\theta).$$
1)Should be clear (an easy induction).
2)Suppose that $p\in{{{\mathbb P}^\theta_\lambda}}$ and $j\in u^p$ are a counterexample with the minimal possible value of ${{\rm ht}}(p)$. Necessarily ${{\rm ht}}(p)$ is a limit ordinal, $p=\langle p_\xi:\xi<{{\rm ht}}(p)\rangle$, ${{\rm ht}}(p_\xi)=\xi$ and $\zeta<
\xi<{{\rm ht}}(p)\ \Rightarrow\ p_\zeta\leq_{\rm pr} p_\xi$. Let $\xi<{{\rm ht}}(p)$ be the first ordinal such that $j\in u^{p_\xi}$. By the choice of $p$, the set $\{\beta\leq\xi: h^p(j,\beta)<\theta\}$ is finite, but clearly $h^p(j,\beta)
\geq\theta$ for all $\beta\in (\xi,{{\rm ht}}(p))$.
3)An easy induction on ${{\rm ht}}(q)$ (with fixed $p$).
4)We show this by induction on ${{\rm ht}}(p)$. Suppose that ${{\rm ht}}(
p)=\alpha+1$, so $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<
\theta\rangle\rangle$, and $i,j\in u^p$ are distinct. If $i,j\in u^{p_\xi}$ for some $\xi<\theta$, then by the inductive hypothesis we find $\beta<
\alpha$ such that $$\theta\neq h^p(i,\beta)=h^{p_\xi}(i,\beta)\neq h^{p_\xi}(j,\beta)=
h^p(j,\beta)\neq\theta.$$ If $i\in u^{p_\xi}\setminus u^*$, $j\in u^{p_\zeta}\setminus u^*$ and $\xi,
\zeta<\theta$ are distinct, then look at the definition of $h^p(i,\alpha)$, $h^p(j,\alpha)$ – these two values cannot be equal (and both are distinct from $\theta$). Finally suppose that ${{\rm ht}}(p)$ is limit, so $p=\langle
p_\xi:\xi<{{\rm ht}}(p)\rangle$. Take $\xi<{{\rm ht}}(p)$ such that $i,j\in u^{p_\xi}$ and apply the inductive hypothesis to $p_\xi$ getting $\beta<\xi$ such that $h^p(i,\beta)\neq h^p(j,\beta)$ (and both are not $\theta$).
5)Again, it goes by induction on ${{\rm ht}}(p)$. First consider a limit stage, and suppose that ${{\rm ht}}(p)=\gamma$ is a limit ordinal, $X\in[\gamma]^{\textstyle{<}\omega}$ and $p=\langle p_\xi:\xi<\gamma
\rangle$. Let $\xi<\gamma$ be such that $X\subseteq\xi$. By the inductive hypothesis we find $i\in u^{p_\xi}$ such that $\{\beta<\xi:h^p(i,\beta)<
\theta\}=X$. Applying clause (3) we may conclude that this $i$ is as required. Now consider a successor case ${{\rm ht}}(p)=\alpha+1$. Let $p=\langle
\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<\theta\rangle\rangle$, and let $\xi<\theta$ be $\zeta^*$ if $\alpha\in X$, and be $\zeta^*+1$ otherwise. Apply the inductive hypothesis to $p_\xi$ and $X\cap \alpha$ to get suitable $i\in u^{p_\xi}$, and note that this $i$ works for $p$ and $X$ too.
6), 7)Straightforward.
\[defiso\] We say that conditions $p,q\in{{{\mathbb P}^\theta_\lambda}}$ are [*isomorphic*]{} if ${{\rm ht}}(p)=
{{\rm ht}}(q)$, ${{\rm otp}}(u^p)={{\rm otp}}(u^q)$, and if $H:u^p\longrightarrow u^q$ is the order isomorphism, then for every $\beta<{{\rm ht}}(p)$ $$(\forall j\in u^p)(h^p(j,\beta)=h^q(H(j),\beta)\ \&\ g^p(j,\beta)=g^p(H(j),
\beta)).$$ \[In this situation we may say that $H$ is the isomorphism from $p$ to $q$.\]
\[3x1\] Suppose that $q_0,q_1\in{{{\mathbb P}^\theta_\lambda}}$ are isomorphic conditions and $H$ is the isomorphism from $q_0$ to $q_1$.
1. If ${{\rm ht}}(q_0)={{\rm ht}}(q_1)=\gamma$ is a limit ordinal, $q_\ell=\langle
q^\ell_\xi: \xi<\gamma\rangle$ (for $\ell<2$), then $H{\restriction}u^{q_\xi^0}$ is an isomorphism from $q^0_\xi$ to $q^1_\xi$.
2. If ${{\rm ht}}(q_0)={{\rm ht}}(q_1)=\alpha+1$, $\alpha<\lambda$, and $q_\ell=
\langle\zeta^*_\ell,\tau^*_\ell,n^*_\ell,u^*_\ell,\langle q^\ell_\xi,
v^\ell_\xi:\xi<\theta\rangle\rangle$ (for $\ell<2$), then $\zeta^*_0=
\zeta^*_1$, $\tau^*_0=\tau^*_1$, $n^*_0=n^*_1$, $H{\restriction}u^{q_\xi^0}$ is an isomorphism from $q^0_\xi$ to $q^1_\xi$ and $H[v^0_\xi]=v^1_\xi$ (for $\xi<
\theta$).
3. $F^{q_0}=\{f{\circ}H:f\in F^{q_1}\}$.
4. Assume $p_0\leq q_0$. Then there is a unique condition $p_1\leq q_1$ such that $H{\restriction}u^{p_0}$ is the isomorphism from $p_0$ to $p_1$.\
1), 2)Straightforward (for (1) use Lemma \[3.1\](7)).\
3), 4)Easy inductions on ${{\rm ht}}(q_0)$ using (1), (2) above.
\[defptran\] By induction on $\alpha<\lambda$, for conditions $p,q\in P^{\theta,
\lambda}_\alpha$ such that $p\leq^\alpha q$, we define [*the $p$–transformation $T_p(q)$ of $q$*]{}.
- If $\alpha=0$ (so necessarily $p=q$) then $T_p(q)=p$.
- Assume that ${{\rm ht}}(q)=\alpha+1$, $q=\langle\zeta^*,\tau^*,n^*,u^*,
\langle q_\xi,v_\xi:\xi<\theta\rangle\rangle$.
If $p\leq q_\xi$ for some $\xi<\theta$, then let $\xi^*$ be such that $p\leq
q_{\xi^*}$. Next for $\xi<\theta$ let $q_\xi'=T_{H_{\xi^*, \xi}(p)}(q_\xi)$, where $H_{\xi^*,\xi}$ is the isomorphism from $q_{\xi^*}$ to $q_\xi$. Define $T_p(q)=\langle\xi^*,\tau^*,n^*,u^*,\langle q_\xi',v_\xi: \xi<\theta\rangle
\rangle$.
Suppose now that $p=\langle\zeta^{**},\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi
<\theta\rangle\rangle$ and $u^{p_\xi}=u^{q_\xi}$, $p_\xi\leq q_\xi$ (for $\xi<\theta$). Let $q_\xi'=T_{p_\xi}(q_\xi)$ and put $T_p(q)=\langle
\zeta^{**},\tau^*,n^*,u^*,\langle q_\xi',v_\xi:\xi<\theta\rangle\rangle$.
- Assume now that ${{\rm ht}}(q)$ is a limit ordinal and $q=\langle q_\xi:\xi<
{{\rm ht}}(q)\rangle$.
If ${{\rm ht}}(p)<{{\rm ht}}(q)$ then $p\leq q_{\varepsilon}$ for some ${\varepsilon}<{{\rm ht}}(q)$, and we may choose $q_\xi'$ (for $\xi<{{\rm ht}}(q)$) such that ${{\rm ht}}(q_\xi')=\xi$, $\xi<
\xi'<{{\rm ht}}(q)\ \Rightarrow\ q_\xi'\leq_{\rm pr} q_{\xi'}'$, and $q_\zeta'
=T_p(q_\zeta)$ for $\zeta\in [{\varepsilon},{{\rm ht}}(q))$. Next we let $T_p(q)=\langle
q_\zeta':\zeta<\theta\rangle$.
If ${{\rm ht}}(p)={{\rm ht}}(q)$, $p=\langle p_\xi:\xi<{{\rm ht}}(p)\rangle$ and $p_\xi\leq
q_\xi$ for $\xi>\delta$ (for some $\delta<{{\rm ht}}(p)$) then we define $T_p(q)=p$.
To show that the definition of $T_p(q)$ is correct one proves inductively (parallely to the definition of the $p$–transformation of $q$) the following facts.
\[3y1\] Assume $p,q\in{{{\mathbb P}^\theta_\lambda}}$, $p\le q$. Then:
1. $T_p(q)\in{{{\mathbb P}^\theta_\lambda}}$, $u^{T_p(q)}=u^q$, ${{\rm ht}}(T_p(q))={{\rm ht}}(q)$,
2. $p\leq_{\rm pr} T_p(q)\leq q\leq T_p(q)$,
3. ${{\rm ht}}(p)={{\rm ht}}(q)\ \Rightarrow\ T_p(q)=p$,
4. if $q'\in{{{\mathbb P}^\theta_\lambda}}$ is isomorphic to $q$ and $H:u^q\longrightarrow u^{q'}$ is the isomorphism from $q$ to $q'$, then $H$ is the isomorphism from $T_p(q)$ to $T_{H(p)}(q')$,
5. if $q\leq_{\rm pr}q'$ then $T_p(q)\leq_{\rm pr} T_p(q')$.
\[3y2\] Every $\leq_{\rm pr}$–increasing chain in ${{{\mathbb P}^\theta_\lambda}}$ of length $<\lambda$ has a $\leq_{\rm pr}$–upper bound, that is the partial order $({{{\mathbb P}^\theta_\lambda}},\leq_{\rm
pr})$ is $(<\lambda)$–closed.
Let us recall that a forcing notion $({{\mathbb Q}},\leq)$ is [*$({<}\lambda)$–strategically closed*]{} if the second player has a winning strategy in the following game $\Game_\lambda({{\mathbb Q}})$.
The game $\Game_\lambda({{\mathbb Q}})$ lasts $\lambda$ moves. The first player starts with choosing a condition $p^*\in{{\mathbb Q}}$. Later, in her $i^{\rm th}$ move, the first player chooses an open dense subset $D_i$ of ${{\mathbb Q}}$. The second player (in his $i^{\rm th}$ move) picks a condition $p_i\in{{\mathbb Q}}$ so that $p_0\geq
p^*$, $p_i\in D_i$ and $p_i\geq p_j$ for all $j<i$. The second player looses the play if for some $i<\lambda$ he has no legal move.
It should be clear that $({<}\lambda)$–strategically closed forcing notions do not add sequences of ordinals of length less than $\lambda$. The reader interested in this kind of properties of forcing notions and iterating them is referred to [@Sh:587], [@Sh:667].
\[3.3\] Assume that $\theta<\lambda$ are regular cardinals, $\lambda^{<\lambda}=
\lambda$. Then $({{{\mathbb P}^\theta_\lambda}},\leq)$ is a $(<\lambda)$–strategically closed $\lambda^+$–cc forcing notion.
It follows from Lemma \[3y1\](2) that if $D\subseteq{{{\mathbb P}^\theta_\lambda}}$ is an open dense set, $p\in{{{\mathbb P}^\theta_\lambda}}$, then there is a condition $q\in D$ such that $p\leq_{\rm
pr} q$. Therefore, to win the game $\Game_\lambda({{{\mathbb P}^\theta_\lambda}})$, the second player can play so that the conditions $p_i$ that he chooses are $\leq_{\rm
pr}$–increasing, and thus there are no problems with finding $\leq_{\rm
pr}$–bounds (remember Proposition \[3y2\]).
Now, to show that ${{{\mathbb P}^\theta_\lambda}}$ is $\lambda^+$–cc, suppose that $\langle p_\delta:
\delta<\lambda^+\rangle$ is a sequence of distinct conditions from ${{{\mathbb P}^\theta_\lambda}}$. We may find a set $A\in [\lambda^+]^{\textstyle \lambda^+}$ such that
- conditions $\{p_\delta:\delta\in A\}$ are pairwise isomorphic,
- the family $\{u^{p_\delta}:\delta\in A\}$ forms a $\Delta$–system with heart $u^*$,
- if $\delta_0<\delta_1$ are from $A$ then $$\sup(u^*)<\min(u^{p_{\delta_0}}\setminus u^*)\leq \sup(u^{p_{\delta_0}}
\setminus u^*)<\min(u^{p_{\delta_0}}\setminus u^*).$$
Take an increasing sequence $\langle\delta_\xi:\xi<\theta\rangle$ of elements of $A$, let $\tau^*={\bf 1}$, $v_\xi=\emptyset$ (for $\xi<\theta$), and look at $p=\langle 0,\tau^*,0,u^*,\langle p_{\delta_\xi},v_\xi:\xi<
\theta\rangle\rangle$. It is a condition in ${{{\mathbb P}^\theta_\lambda}}$ stronger than all $p_{\delta_\xi}$’s.
\[defcompo\] By induction on ${{\rm ht}}(p)$ we define [*$\alpha$–components of $p$*]{} (for $p\in{{{\mathbb P}^\theta_\lambda}}$, $\alpha\leq{{\rm ht}}(p)$).
- First we declare that the only ${{\rm ht}}(p)$–component of $p$ is the $p$ itself.
- If ${{\rm ht}}(p)=\beta+1$, $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,
v_\xi:\xi<\theta\rangle\rangle$ and $\alpha=\beta$, then $\alpha$–components of $p$ are $p_\xi$ (for $\xi<\theta$); if $\alpha<\beta$, then $\alpha$–components of $p$ are those $q$ which are $\alpha$–components of $p_\xi$ for some $\xi<\theta$.
- If ${{\rm ht}}(p)$ is a limit ordinal, $p=\langle p_\xi:\xi<{{\rm ht}}(p)\rangle$ and $\alpha<{{\rm ht}}(p)$, then $\alpha$–components of $p$ are $\alpha$–components of $p_\xi$ for $\xi\in [\alpha,{{\rm ht}}(p))$.
\[3.4x\] Assume $p\in {{{\mathbb P}^\theta_\lambda}}$ and $\alpha<{{\rm ht}}(p)$.
1. If $q$ is an $\alpha$–component of $p$ then $q\leq p$, ${{\rm ht}}(q)=
\alpha$, and for all $j_0,j_1\in u^q$ and every $\beta\in [\alpha,{{\rm ht}}(p))$: $$h^p(j_0,\beta)\neq\theta\ \&\ h^p(j_1,\beta)\neq \theta\quad\Rightarrow
\quad h^p(j_0,\beta)=h^p(j_1,\beta).$$ Moreover, for each $i\in u^p$ there is a unique $\alpha$–component $q$ of $p$ such that $i\in u^q$ and $$(\forall j\in u^q)(\forall\beta\in [\alpha,{{\rm ht}}(p)))(h^p(i,\beta)\geq
\theta\ \Rightarrow\ h^p(j,\beta)\geq\theta).$$
2. If $H$ is an isomorphism from $p$ onto $p'\in{{{\mathbb P}^\theta_\lambda}}$, and $q$ is an $\alpha$–component of $p$, then $H(q)$ is an $\alpha$–component of $p'$. If $q_0,q_1$ are $\alpha$–components of $p$ then $q_0,q_1$ are isomorphic.
3. There is a unique $\alpha$–component $q$ of $p$ such that $q\leq_{\rm
pr} p$.
Easy inductions on ${{\rm ht}}(p)$.
\[closed\] By induction on ${{\rm ht}}(p)$ we define when a set $Z\subseteq \lambda$ is $p$–closed for a condition $p\in{{{\mathbb P}^\theta_\lambda}}$.
- If ${{\rm ht}}(p)=0$ then every $Z\subseteq\lambda$ is $p$–closed;
- if ${{\rm ht}}(p)$ is limit, $p=\langle p_\xi:\xi<{{\rm ht}}(p)\rangle$, then $Z$ is $p$–closed provided it is $p_\xi$–closed for each $\xi<{{\rm ht}}(p)$;
- if ${{\rm ht}}(p)=\alpha+1$, $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle
p_\xi,v_\xi:\xi<\theta\rangle\rangle$ and $\alpha\notin Z$, then $Z$ is $p$–closed whenever it is $p_{\zeta^*}$–closed;
- if ${{\rm ht}}(p)=\alpha+1$, $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle
p_\xi,v_\xi:\xi<\theta\rangle\rangle$ and $\alpha\in Z$, then $Z$ is $p$–closed provided it is $p_{\zeta^*}$–closed and $$\{\beta<\alpha:(\exists j\in v_{\zeta^*}\cup\{\min(u^{p_{\zeta^*}}
\setminus u^*)\})(h^{p_{\zeta^*}}(j,\beta)<\theta)\}\subseteq Z.$$
\[3.5.1\]
1. If $p\in{{{\mathbb P}^\theta_\lambda}}$ and $w\in [{{\rm ht}}(p)]^{\textstyle<\omega}$, then there is a finite $p$–closed set $Z\subseteq{{\rm ht}}(p)$ such that $w\subseteq Z$.
2. If $p,q\in{{{\mathbb P}^\theta_\lambda}}$ are isomorphic and $Z$ is $p$–closed, then $Z$ is $q$–closed. If $Z$ is $p$–closed, $\alpha<{{\rm ht}}(p)$ and $p^*$ is an $\alpha$–component of $p$, then $Z\cap\alpha$ is $p^*$–closed.
Easy inductions on ${{\rm ht}}(p)$ (remember Lemma \[3.1\](2)).
\[UUpsilon\] Suppose that $p\in{{{\mathbb P}^\theta_\lambda}}$ and $Z\subseteq {{\rm ht}}(p)$ is a finite $p$–closed set. Let $Z=\{\alpha_0,\ldots,\alpha_{k-1}\}$ be the increasing enumeration.
1. We define $$U[p,Z]\stackrel{\rm def}{=}\{j\in u^p: (\forall\beta<{{\rm ht}}(p))(h^p(j,\beta)
<\theta\ \Rightarrow\ \beta\in Z)\}.$$
2. We let $$\Upsilon_p(Z)=\langle\zeta_\ell,\tau_\ell,n_\ell,\langle g_\ell,h^\ell_0,
\ldots,h^\ell_{n_\ell-1}\rangle:\ell<k\rangle,$$ where, for $\ell<k$, $\zeta_\ell$ is an ordinal below $\theta$, $\tau_\ell$ is a Boolean term, $n_\ell<\omega$ and $g_\ell,h^\ell_0,\ldots,h^\ell_{
n_\ell-1}:\ell\longrightarrow 2$, and they all are such that for every (equivalently: some) $\alpha_\ell+1$–component $q=\langle\zeta^*,\tau^*,
n^*,u^*,\langle q_\xi,v_\xi:\xi<\theta\rangle\rangle$ of $p$ we have:$\zeta_\ell=\zeta^*$, $\tau_\ell=\tau^*$, $n_\ell=n^*$ and if $v_\xi=\{j_0,
\ldots,j_{n_\ell-1}\}$ (the increasing enumeration) then $$(\forall m<n_\ell)(\forall\ell'<\ell)(h^\ell_m(\ell')=h^q(j_m,
\alpha_{\ell'})),$$ and if $i_0=\min(u^{q_{\zeta^*}}\setminus u^*)$ then $(\forall\ell'<\ell)(
g_\ell(\ell')=h^q(i_0,\alpha_{\ell'}))$. (Note that $\zeta_\ell,\tau_\ell,
n_\ell$, $g_\ell,h^\ell_0,\ldots,h^\ell_{n_\ell-1}$ are well-defined by Lemma \[3.4x\]. Necessarily, for all $m<n_\ell$ and $\beta\in
\alpha_\ell\setminus Z$ we have $h^q(i_0,\beta),h^q(j_m,\beta)\geq\theta$; remember that $Z$ is $p$–closed.)
Note that if $Z\subseteq{{\rm ht}}(p)$ is a finite $p$–closed set, $\alpha=\max(Z)$ and $p^*$ is the $\alpha+1$–component of $p$ satisfying $p^*\leq_{\rm pr} p$ (see \[3.4x\](3)), then $U[p,Z]\subseteq u^{p^*}$.
\[3.6.x\] Suppose that $p\in{{{\mathbb P}^\theta_\lambda}}$ and $Z_0,Z_1\subseteq{{\rm ht}}(p)$ are finite $p$–closed sets such that $\Upsilon_p(Z_0)=\Upsilon_p(Z_1)$. Then ${{\rm otp}}(U[p,Z_0])=
{{\rm otp}}(U[p,Z_1])$, and the order preserving isomorphism $\pi:U[p,Z_0]\longrightarrow U[p,Z_1]$ satisfies
1. $(\forall\ell<k)(h^p(i,\alpha^0_\ell)=h^p(\pi(i),
\alpha^1_\ell))$,
where $\{\alpha^x_0,\ldots,\alpha^x_{k-1}\}$ is the increasing enumeration of $Z_x$ (for $x=0,1$).
We prove this by induction on $|Z_0|=|Z_1|$ (for all $p,Z_0,Z_1$ satisfying the assumptions).
[Step]{} $|Z_0|=|Z_1|=1$; $Z_0=\{\alpha^0_0\}$, $Z_1=\{\alpha^1_0\}$.\
Take the $\alpha^x_0+1$–component $q_x$ of $p$ such that $q_x\leq_{\rm pr}
p$. Then, for $x=0,1$, $q_x=\langle\zeta,\tau,n, u^x,\langle q^x_\xi,
v^x_\xi:\xi<\theta\rangle\rangle$, and for each $i\in v^x_\xi$, $\beta<
\alpha^x_0$ we have $h^{q^x_\xi}(i,\beta)\geq\theta$. Also, if $i^x_0=
\min(u^{q^x_\zeta}\setminus u^x)$ and $\beta<\alpha^x_0$, then $h^{q^x_\zeta}(i^x_0,\beta)\geq\theta$. Consequently, $n=|v^x_\xi|\leq 1$, and if $n=1$ then $\{i^x_0\}=v^x_\zeta$ (remember Lemma \[3.1\](4)). Moreover, $$U[p,Z_x]=U[q_x,Z_x]=\{H^x_{\xi,\zeta}(i^x_0):\xi<\theta\},$$ where $H^x_{\xi,\zeta}$ is the isomorphism from $q^x_\zeta$ to $q^x_\xi$. Now it should be clear that the mapping $\pi:H^0_{\xi,\zeta}(
i^0_0)\mapsto H^1_{\xi,\zeta}(i^1_0):U[p,Z_0]\longrightarrow U[p,Z_1]$ is the order preserving isomorphism (remember clause $(\gamma)$ of the definition of $P^*_{\alpha+1}$), and it has the property described in $(\otimes)$.
[Step]{} $|Z_0|=|Z_1|=k+1$; $Z_0=\{\alpha^0_0,\ldots,\alpha^0_k
\}$, $Z_1=\{\alpha^1_0,\ldots,\alpha^1_k\}$.\
Let $$\Upsilon_p(Z_0)=\Upsilon_p(Z_1)=\langle\zeta_\ell,\tau_\ell,n_\ell,\langle
g_\ell,h^\ell_0,\ldots,h^\ell_{n_\ell-1}\rangle:\ell\leq k\rangle.$$ For $x=0,1$, let $q_x=\langle\zeta,\tau,n, u^x,\langle q^x_\xi,v^x_\xi:\xi<
\theta\rangle\rangle$ be the $\alpha^x_k+1$–component of $p$ such that $q_x
\leq_{\rm pr} p$. The sets $Z_x\cap\alpha^x_k$ (for $x=0,1$) are $q^x_\xi$–closed for every $\xi<\theta$, and clearly $\Upsilon_p(Z_0\cap
\alpha^0_k)= \Upsilon_p(Z_1\cap\alpha^1_k)$. Hence, by the inductive hypothesis, ${{\rm otp}}(U[q^0_\xi,Z_0\setminus\{\alpha^0_k\}])={{\rm otp}}(U[q^1_\xi,Z_1
\setminus\{\alpha^1_k\}])$ (for each $\xi<\theta$), and the order preserving mappings $\pi_\xi:U[q^0_\xi,Z_0\setminus\{ \alpha^0_k\}]\longrightarrow
U[q^1_\xi, Z_1\setminus\{\alpha^1_k\}]$ satisfy the demand in $(\otimes)$. Let $i^x_\xi=\min(u^{q^x_\xi}\setminus u^x)$. Then, as $q^x_\xi$ and $q^x_\zeta$ are isomorphic and the isomorphism is the identity on $u^x$, we have $(\forall\ell<k)(h^p(i^x_\xi,\alpha_\ell^x)=g_k(\ell))$. Hence $\pi_\xi
(i^0_\xi)=i^1_\xi$, and therefore $\pi_\xi[u^0\cap U[q^0_\xi,Z_0\setminus
\{\alpha^0_k\}]]=u^1\cap U[q^1_\xi,Z_1\setminus\{\alpha^1_k\}]$. But since the mappings $\pi_\xi$ are order preserving, the last equality implies that $\pi_\xi\restriction (u^0\cap U[q^0_\xi,Z_0\setminus\{\alpha^0_k\}])=
\pi_\zeta\restriction (u^0\cap U[q^0_\zeta,Z_0\setminus\{\alpha^0_k\}])$, and hence $\pi=\bigcup\limits_{\xi<\theta}\pi_\xi$ is a function, and it is an order isomorphism from $U[q_0,Z_0]=U[p,Z_0]$ onto $U[q_1,Z_1]=U[p,Z_1]$ satisfying $(\otimes)$.
The algebra and why it is OK (in ${{\bf V}}^{{{{\mathbb P}^\theta_\lambda}}}$)
==============================================================================
Let ${\dot{{{\mathbb B}}}^\theta_\lambda}$ and $\dot{U}$ be ${{{\mathbb P}^\theta_\lambda}}$–names such that $${\Vdash}_{{{{\mathbb P}^\theta_\lambda}}}\mbox{`` }{\dot{{{\mathbb B}}}^\theta_\lambda}=\bigcup\{{{\mathbb B}}^p:p\in \Gamma_{{{{\mathbb P}^\theta_\lambda}}}\}\mbox{
''}\quad\mbox{ and }\quad{\Vdash}_{{{{\mathbb P}^\theta_\lambda}}}\mbox{`` }\dot{U}=\bigcup\{u^p:p\in
\Gamma_{{{{\mathbb P}^\theta_\lambda}}}\}\mbox{ ''.}$$ Note that $\dot{U}$ is (a name for) a subset of $\lambda^+$. Let $\dot{F}$ be a ${{{\mathbb P}^\theta_\lambda}}$–name such that $${\Vdash}_{{{{\mathbb P}^\theta_\lambda}}}\mbox{`` }\dot{F}=\{f\in 2^{\textstyle\dot{U}}: (\forall
p\in\Gamma_{{{{\mathbb P}^\theta_\lambda}}})(f\restriction u^p\in\dot{F}^p)\}\mbox{ ''.}$$
\[3.4\] Assume $\theta<\lambda$ are regular, $\lambda^{<\lambda}=\lambda$. Then in ${{\bf V}}^{{{{\mathbb P}^\theta_\lambda}}}$:
1. $\dot{F}$ is a non-empty closed subset of $2^{\textstyle\dot{U}}$, and ${\dot{{{\mathbb B}}}^\theta_\lambda}$ is the Boolean algebra generated ${{\mathbb B}}_{(\dot{U},\dot{F})}$ (see Definition \[0.C\]);
2. $|\dot{U}|=|{\dot{{{\mathbb B}}}^\theta_\lambda}|=\lambda^+$;
3. For every subalgebra ${{\mathbb B}}\subseteq{\dot{{{\mathbb B}}}^\theta_\lambda}$ of size $\lambda^+$ we have ${{\rm Depth}}^+({{\mathbb B}})>\theta$.
2)Note that if $p\in{{{\mathbb P}^\theta_\lambda}}$, $\sup(u^p)<j<\lambda^+$ then there is a condition $q\geq p$ such that $j\in u^q$. Hence ${\Vdash}|\dot{U}|=
\lambda^+$. To show that, in ${{\bf V}}^{{{{\mathbb P}^\theta_\lambda}}}$, the algebra ${\dot{{{\mathbb B}}}^\theta_\lambda}$ is of size $\lambda^+$ it is enough to prove the following claim.
\[3.4.1\] Let $p\in{{{\mathbb P}^\theta_\lambda}}$, $j\in u^p$. Then $x_j\notin\langle x_i:i\in j\cap u^p
\rangle_{{{\mathbb B}}^p}$.
Suppose not, and let $p,j$ be a counterexample with the smallest possible ${{\rm ht}}(p)$. Necessarily, ${{\rm ht}}(p)$ is a successor ordinal, say ${{\rm ht}}(p)=\alpha+1$. So let $p=\langle\zeta^*,\tau^*, n^*,u^*,\langle
p_\xi,v_\xi:\xi<\theta\rangle\rangle$ and suppose that $v\in [u^p\cap
j]^{\textstyle <\omega}$ is such that $x_j\in \langle x_i: i\in
v\rangle_{{{\mathbb B}}^p}$. If $j\in u^*$ then $v\subseteq u^*$ and we immediately get a contradiction (applying the inductive hypothesis to $p_{\zeta^*}$). So let $\xi<\theta$ be such that $j\in u^{p_\xi}\setminus u^*$. We know that $x_j\notin\langle x_i:i\in u^*\cup (v\cap u^{p_\xi})\rangle_{{{\mathbb B}}^{p_\xi}}$ (remember clause $(\gamma)$ of the definition of $P^*_{\alpha+1}$), so we may take functions $f_0,f_1\in F^{p_\xi}$ such that $f_0{\restriction}(u^*\cup (v\cap
u^{p_\xi}))=f_1{\restriction}(u^*\cup (v\cap u^{p_\xi}))$, $f_0(j)=0$, $f_1(j)=1$. Let $g_0,g_1:u^p\longrightarrow 2$ be such that $g_\ell{\restriction}u^{p_\xi}=
f_\ell$, $g_\ell{\restriction}u^{p_\zeta}=f_0{\circ}H_{\zeta,\xi}$ for $\zeta\neq\xi$ (where $H_{\zeta,\xi}$ is the order isomorphism from $u^{p_\zeta}$ to $u^{p_\xi}$). Now one easily checks that $g_0,g_1\in F^p$ (remember the definition of the term $\sigma_{\rm maj}$). By our choices, $g_0(i)=g_1(i)$ for all $i\in v$, and $g_0(j)\neq g_1(j)$, and this is a clear contradiction with the choice of $i$ and $v$.
3)Suppose that $\langle\dot{a}_\xi:\xi<\lambda^+\rangle$ is a ${{{\mathbb P}^\theta_\lambda}}$–name for a $\lambda^+$–sequence of distinct members of ${\dot{{{\mathbb B}}}^\theta_\lambda}$ and let $p\in{{{\mathbb P}^\theta_\lambda}}$. Applying standard cleaning procedures we find a set $A\subseteq\lambda^+$ of the order type $\theta$, an ordinal $\alpha<
\lambda$ and $\tau^*,n^*,u^*$ and $\langle p_\xi,v_\xi:\xi\in A\rangle$ such that $p\leq p_\xi$, ${{\rm ht}}(p_\xi)=\alpha$, $p_\xi{\Vdash}\dot{a}_\xi=
\tau^*(x_i:i\in v_\xi)$ and $$q\stackrel{\rm def}{=}\langle 0,\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi\in
A\rangle\rangle\in P^*_{\alpha+1},$$ where $A$ is identified with $\theta$ by the increasing enumeration (so we will think $A=\theta$). For $\xi<\theta$ let $\tau_\xi=\tau^*(x_i: i\in
v_\xi)\in{{\mathbb B}}^{p_\xi}$. Since $\dot{a}_\xi$ were (forced to be) distinct we know that ${{\mathbb B}}^q\models\tau_\xi\neq\tau_\zeta$ for distinct $\xi,\zeta$. Hence $\tau_\xi\notin\langle x_i:i\in u^*\rangle_{{{\mathbb B}}^{p_\xi}}$ (for each $\xi$) and therefore we may find functions $f^0_\xi,f^1_\xi\in F^{p_\xi}$ such that $f^0_\xi{\restriction}u^*=f^1_\xi{\restriction}u^*$, and $f^0_\xi(\tau_\xi)=0$, $f^1_\xi(\tau_\xi)=1$, and if $\xi<\zeta<\theta$, and $H_{\xi,\zeta}$ is the isomorphism from $p_\xi$ to $p_\zeta$, then $f^\ell_\xi=f^\ell_\zeta{\circ}H_{\xi,\zeta}$. Now fix $\xi<\zeta<\theta$ and let $$g\stackrel{\rm def}{=}\bigcup_{\alpha\leq 3\cdot\xi+2} f^0_\alpha\cup
\bigcup_{3\cdot\xi+2<\alpha<\theta} f^1_\alpha.$$ It should be clear that $g$ is a function from $u^q$ to $2$, and moreover $g\in F^q$. Also easily $$g(\sigma_{\rm maj}(\tau_{3\cdot\xi},\tau_{3\cdot\xi+1},\tau_{3\cdot\xi+
2}))=0\ \mbox{ and }\ g(\sigma_{\rm maj}(\tau_{3\cdot\zeta},\tau_{3\cdot
\zeta+1},\tau_{3\cdot\zeta+2}))\}=1.$$ Hence we may conclude that $${{\mathbb B}}^q\models\sigma_{\rm maj}(\tau_{3\cdot\xi},\tau_{3\cdot\xi+1},\tau_{3
\cdot\xi+2})<\sigma_{\rm maj}(\tau_{3\cdot\zeta},\tau_{3\cdot\zeta+1},\tau_{3
\cdot\zeta+2})$$ for $\xi<\zeta<\theta$ (remember the definition of $F^q$ and Proposition \[0.D\]). Consequently we get $q{\Vdash}{{\rm Depth}}^+(\langle\dot{a}_\xi:\xi<
\lambda^+\rangle_{{\dot{{{\mathbb B}}}^\theta_\lambda}})>\theta$, finishing the proof.
\[3.5\] Assume $\theta<\lambda$ are regular, $\lambda=\lambda^{<\lambda}$. Then ${\Vdash}_{{{{\mathbb P}^\theta_\lambda}}}{{\rm Depth}}({\dot{{{\mathbb B}}}^\theta_\lambda})=\theta$.
By Proposition \[3.4\] we know that ${\Vdash}{{\rm Depth}}^+({\dot{{{\mathbb B}}}^\theta_\lambda})>\theta$, so what we have to show is that there are no increasing sequences of length $\theta^+$ of elements of ${\dot{{{\mathbb B}}}^\theta_\lambda}$. We will show this under an additional assumption that $\theta^+ <\lambda$ (after the proof is carried out, it will be clear how one modifies it to deal with the case $\lambda=\theta^+$). Due to this additional assumption, and since the forcing notion ${{{\mathbb P}^\theta_\lambda}}$ is $(<\lambda)$–strategically closed (by Proposition \[3.3\]), it is enough to show that ${{\rm Depth}}({{\mathbb B}}^p)\leq\theta$ for each $p\in{{{\mathbb P}^\theta_\lambda}}$.
So suppose that $p\in{{{\mathbb P}^\theta_\lambda}}$ is such that ${{\rm Depth}}({{\mathbb B}}^p)\geq\theta^+$. Then we find a Boolean term $\tau$, an integer $n$ and sets $w_\rho\in [u^p]^{
\textstyle n}$ (for $\rho<\theta^+$) such that $$\rho_0<\rho_1<\theta^+\quad\Rightarrow\quad{{\mathbb B}}^p\models\tau(x_i:i\in w_{
\rho_0})<\tau(x_i:i\in w_{\rho_1}).$$ For each $\rho<\theta^+$ use Lemma \[3.5.1\] to choose a finite $p$–closed set $Z_\rho\subseteq{{\rm ht}}(p)$ containing the set $$\{\beta<{{\rm ht}}(p): (\exists j\in w_\rho)(h^p(j,\beta)<\theta)\}.$$ Look at $\Upsilon_p(Z_\rho)$ (see Definition \[UUpsilon\]). There are only $\theta$ possibilities for the values of $\Upsilon_p(Z_\rho)$, so we find $\rho_0<\rho_1<\theta^+$ such that
1. $|Z_{\rho_0}|=|Z_{\rho_1}|$, $\Upsilon_p(Z_{\rho_0})=\Upsilon_p(
Z_{\rho_1})=\langle\zeta_\ell,\tau_\ell,n_\ell,\langle g_\ell,h^\ell_0,
\ldots,h^\ell_{n_\ell-1}\rangle:\ell<k\rangle$,
2. if $\pi^*:Z_{\rho_0}\longrightarrow Z_{\rho_1}$ is the order isomorphism then $\pi^*{\restriction}Z_{\rho_0}\cap Z_{\rho_1}$ is the identity on $Z_{\rho_0}\cap Z_{\rho_1}$,
3. if $\pi:U[p,Z_{\rho_0}]\longrightarrow U[p,Z_{\rho_1}]$ is the order isomorphism, then $\pi[w_{\rho_0}]=w_{\rho_1}$.
Note that, by Lemma \[3.6.x\], ${{\rm otp}}(U[p,Z_{\rho_0}])={{\rm otp}}(U[p,
Z_{\rho_1}])$ and the order isomorphism $\pi$ satisfies $$(\forall j\in U[p,Z_{\rho_0}])(\forall\beta\in Z_{\rho_0})(h^p(j,\beta)=
h^p(\pi(j),\pi^*(\beta))),$$ and hence $\pi$ is the identity on $U[p,Z_{\rho_0}]\cap U[p,Z_{\rho_1}]$ (remember Lemma \[3.1\]).
For a function $f\in F^p$ let $G^{\rho_0}_{\rho_1}(f):u^p\longrightarrow 2$ be defined by $$G^{\rho_0}_{\rho_1}(f)(j)=\left\{\begin{array}{lll}
f(\pi(j))&\mbox{ if }& j\in U[p,Z_{\rho_0}],\\
f(\pi^{-1}(j))&\mbox{ if }& j\in U[p,Z_{\rho_1}]\setminus U[p,\rho_0],\\
0& &\mbox{otherwise}.
\end{array}\right.$$
\[3.5.2\] For each $f\in F^p$, $G^{\rho_0}_{\rho_1}(f)\in F^p$.
By induction on $\alpha\leq{{\rm ht}}(p)$ we show that for each $\alpha$–component $q$ of $p$, the restriction $G^{\rho_0}_{\rho_1}(f){\restriction}u^q$ is in $F^q$.
If $\alpha$ is limit, we may easily use the inductive hypothesis to show that, for any $\alpha$–component $q$ of $p$, $G^{\rho_0}_{\rho_1}(f){\restriction}u^q\in F^q$.
Assume $\alpha=\beta+1$ and let $q=\langle\zeta^*,\tau^*,n^*,u^*,\langle
q_\xi,v_\xi:\xi<\theta\rangle\rangle$ be an $\alpha$–component of $p$. We will consider four cases.
[*Case 1:*]{}$\beta\notin Z_{\rho_0}\cup Z_{\rho_1}$.\
Then $(U[p,Z_{\rho_0}]\cup U[p,Z_{\rho_1}])\cap u^q\subseteq
u^{q_{\zeta^*}}$ and $G^{\rho_0}_{\rho_1}(f){\restriction}(u^{q_\xi}\setminus u^*)
\equiv 0$ for each $\xi\neq\zeta^*$. Since, by the inductive hypothesis, $G^{\rho_0}_{\rho_1}(f){\restriction}u^{q_\xi}\in F^{q_\xi}$ for each $\xi<\theta$, we may use the definition of $P^*_{\beta+1}$ and conclude that $G^{\rho_0}_{\rho_1}(f){\restriction}u^q\in F^q$ (remember the definition of the term $\sigma_{\rm maj}$).
[*Case 2:*]{}$\beta\in Z_{\rho_0}\setminus Z_{\rho_1}$.\
Let $Z_{\rho_0}=\{\alpha_0,\ldots,\alpha_{k-1}\}$ be the increasing enumeration. Then $\beta=\alpha_\ell$ for some $\ell<k$ and $\zeta^*=
\zeta_\ell$, $\tau^*=\tau_\ell$, $n^*=n_\ell$. Moreover, if $v_\xi=
\{j^\xi_0,\ldots,j^\xi_{n_\ell-1}\}$ (the increasing enumeration), $\xi<\theta$, then for $m<n_\ell$: $$(\forall\ell'<\ell)(h^\ell_m(\alpha_{\ell'})=h^q(j^\xi_m,\alpha_{\ell'}))
\quad\mbox{and}\quad(\forall\gamma\in\beta\setminus Z_{\rho_0})(h^q(j^\xi_m,
\gamma)\geq\theta).$$ Note that $U[p,Z_{\rho_1}]\cap u^q\subseteq u^{q_{\zeta^*}}$, so if $U[p,Z_{\rho_0}]\cap u^q=\emptyset$, then we may proceed as in the previous case. Therefore we may assume that $U[p,Z_{\rho_0}]\cap u^q\neq\emptyset$. So, for each $\gamma\in Z_{\rho_0}\setminus\alpha$ we may choose $i_\gamma
\in U[p,Z_{\rho_0}]\cap u^q$ such that $$(\forall i\in U[p,Z_{\rho_0}]\cap u^q)(h^p(i,\gamma)\neq\theta\
\Rightarrow\ h^p(i,\gamma)=h^p(i_\gamma,\gamma))$$ (remember Lemma \[3.4x\](1)). Let $i^*=\max\{i_\gamma:\gamma\in
Z_{\rho_0}\setminus\alpha\}$ (if $\beta=\max(Z_{\rho_0})$, then let $i^*$ be any element of $U[p,Z_{\rho_0}]\cap u^q$). Note that then $$(\forall i\in U[p,Z_{\rho_0}]\cap u^q)(\forall\gamma\in Z_{\rho_0}
\setminus\alpha)(h^p(i,\gamma)\neq\theta\ \Rightarrow\ h^p(i,\gamma)=h^p(
i^*,\gamma))$$ \[Why? Remember Lemma \[3.4x\](1) and the clause $(\gamma)$ of the definition of $P^*_{\beta+1}$.\] By Lemma \[3.4x\], we find a $(\pi^*(
\beta)+1)$–component $q'=\langle\zeta',\tau',n',u',\langle q_{\varepsilon}',
v_{\varepsilon}':{\varepsilon}<\theta\rangle\rangle$ of $p$ such that $\pi(i^*)\in u^{q'}$ and $$(\forall j\in u^{q'})(\forall\gamma\in (\pi^*(\beta),{{\rm ht}}(p)))(h^p(
\pi(i^*),\gamma)\geq\theta\ \Rightarrow\ h^p(j,\gamma)\geq\theta).$$ We claim that then
1. $(\forall j\in U[p,Z_{\rho_0}]\cap u^q)(\pi(j)\in
u^{q'}\cap U[p,Z_{\rho_1}])$.
Why? Fix $j\in U[p,Z_{\rho_0}]\cap u^q$. Let $r,r'$ be components of $p$ such that $r\leq_{\rm pr}p$, $r'\leq_{\rm pr}p$, ${{\rm ht}}(r)=\beta+1$, ${{\rm ht}}(r')=\pi^*(\beta)+1$ (so $r$ and $q$, and $r',q'$, are isomorphic). The sets $Z_{\rho_0}\cap (\beta+1)$ and $Z_{\rho_1}\cap (\pi^*(\beta)+1)$ are $p$–closed, and they have the same values of $\Upsilon$, and therefore $U[p,Z_{\rho_0}\cap(\beta+1)]$ and $U[p,Z_{\rho_1}\cap(\pi^*(\beta)+1)]$ are (order) isomorphic. Also, these two sets are included in $u^r$ and $u^{r'}$, respectively. So looking back at our $j$, we may successively choose $j_0\in
u^r\cap U[p,Z_{\rho_0}\cap (\beta+1)]$, $j_1\in u^{r'}\cap U[p,Z_{\rho_1}
\cap (\pi^*(\beta)+1)]$, and $j^*\in u^q$ such that
- $(\forall\gamma\leq\beta)(h^q(j,\gamma)=h^r(j_0,\gamma))$,
- $(\forall\ell'\leq\ell)(h^r(j_0,\alpha_{\ell'})=h^{r'}(j_1,\pi^*(
\alpha_{\ell'})))$, and
- $(\forall\gamma\leq\pi^*(\beta))(h^{r'}(j,\gamma)=h^{q'}(j^*,
\gamma))$.
Then we have $$(\forall\ell'\leq\ell)(h^q(j,\alpha_{\ell'})=h^{q'}(j^*,\pi^*(
\alpha_{\ell'}))\quad\mbox{and}\quad(\forall\gamma\in\pi^*(\beta)\setminus
Z_{\rho_1})(h^{q'}(j^*,\gamma)\geq\theta).$$ To conclude $(\boxtimes)$ it is enough to show that $\pi(j)=j^*$. If this equality fails, then there is $\gamma<{{\rm ht}}(p)$ such that $\theta\neq h^p(
\pi(j),\gamma)\neq h^p(j^*,\gamma)\neq\theta$. If $\gamma\leq\pi^*(\beta)$, then necessarily $\gamma\in Z_{\rho_1}$, and this is impossible (remember $h^p(j,\alpha_{\ell'})=h^p(\pi(j),\pi^*(\alpha_{\ell'}))$ for $\ell'\leq
\ell$). So $\gamma>\pi^*(\beta)$. If $h^p(\pi(j),\gamma)=\theta+1$, then $h^p(j^*,\gamma)<\theta$ and (by the choice of $q'$) $h^p(\pi(i^*),\gamma)<
\theta$. Then $\gamma\in Z_{\rho_1}$ and $h^p(i^*,(\pi^*)^{-1}(\gamma))<
\theta$, and also $h^p(i^*,(\pi^*)^{-1}(\gamma))=h^p(j,(\pi^*)^{-1}(\gamma))
=\theta+1$ (by the choice of $i^*$), a contradiction. Thus necessarily $h^p(\pi(j),\gamma)<\theta$ (so $\gamma\in Z_{\rho_1}$) and therefore $$\theta>h^p(j,(\pi^*)^{-1}(\gamma))=h^p(i^*,(\pi^*)^{-1}(\gamma))=h^p(
\pi(i^*),\gamma)= h^p(j^*,\gamma)$$ (as the last is not $\theta$), again a contradiction. Thus the statement in $(\boxtimes)$ is proven.
Now we may finish considering the current case. By the definition of the function $\Upsilon$ (and by the choice of $\rho_0,\rho_1$) we have $$\zeta'=\zeta_\ell,\quad\tau'=\tau_\ell,\quad n'=n_\ell,\quad\mbox{and }\
\pi[v_\xi]=v'_\xi\ \mbox{ for }\xi<\theta$$ (and $\pi{\restriction}v_\xi$ is order–preserving). Therefore $$G^{\rho_0}_{\rho_1}(f)(\tau^*(x_i:i\in v_\xi))=f(\tau'(x_i:i\in v'_\xi))
\qquad\mbox{ (for every $\xi<\theta$).}$$ By the inductive hypothesis, $G^{\rho_0}_{\rho_1}(f){\restriction}u^{q_\xi} \in
F^{q_\xi}$ (for $\xi<\theta$), so as $f\in F^p$ (and hence $f{\restriction}u^{q'}\in
F^{q'}$) we may conclude now that $G^{\rho_0}_{\rho_1}(f){\restriction}u^q\in F^q$.
[*Case 3:*]{}$\beta\in Z_{\rho_1}\setminus Z_{\rho_0}$\
Similar.
[*Case 3:*]{}$\beta\in Z_{\rho_0}\cap Z_{\rho_1}$\
If $U[p,Z_{\rho_0}]\cap u^q=\emptyset=U[p,Z_{\rho_1}]\cap u^q$, then $G^{\rho_0}_{\rho_1}(f){\restriction}u^q\equiv 0$ and we are easily done. If one of the intersections is non-empty, then we may follow exactly as in the respective case (2 or 3).
Now we may conclude the proof of the theorem. Since $${{\mathbb B}}^p\models\tau(x_i:i\in w_{\rho_0})<\tau(x_i:i\in w_{\rho_1}),$$ we find $f\in F^p$ such that $f(\tau(x_i:i\in w_{\rho_0}))=0$ and $f(\tau(x_i:i\in w_{\rho_1}))=1$. It should be clear from the definition of the function $G^{\rho_0}_{\rho_1}(f)$ (and the choice of $\rho_0,\rho_1$) that $$G^{\rho_0}_{\rho_1}(f)(\tau(x_i:i\in w_{\rho_0}))=1\quad\mbox{and}\quad
G^{\rho_0}_{\rho_1}(f)(\tau(x_i:i\in w_{\rho_1}))=0.$$ But it follows from Claim \[3.5.2\] that $G^{\rho_0}_{\rho_1}(f)\in F^p$, a contradiction.
It is consistent that for some uncountable cardinal $\theta$ there is a Boolean algebra ${{\mathbb B}}$ of size $(2^\theta)^+$ such that $${{\rm Depth}}({{\mathbb B}})=\theta\quad\mbox{ but }\quad (\omega,(2^\theta)^+)\notin
{{\rm Depth}}_{\rm Sr}({{\mathbb B}}).$$
Assume $\theta<\lambda=\lambda^{<\lambda}$ are regular cardinals. Does there exist a Boolean algebra ${{\mathbb B}}$ such that $|{{\mathbb B}}|=\lambda^+$ and for every subalgebra ${{\mathbb B}}'\subseteq{{\mathbb B}}$ of size $\lambda^+$ we have ${{\rm Depth}}({{\mathbb B}}')=\theta$?
[1]{}
Thomas Jech. . Academic Press, New York, 1978.
Donald Monk. . Lectures in Mathematics. ETH Zurich, Birkhauser Verlag, Basel Boston Berlin, 1990.
Donald Monk. , volume 142 of [ *[Progress in Mathematics]{}*]{}. Birkhäuser Verlag, Basel–Boston–Berlin, 1996.
Geore Orwell. . Harcourt Brace Jovanovich, San Diego, 1977.
Andrzej Roslanowski and Saharon Shelah. . , 155:101–151, 1998, math.LO/9703218[^1].
Saharon Shelah. . , accepted, math.LO/9707225.
Saharon Shelah. . , [submitted]{}, math.LO/9808140.
Saharon Shelah. . , 151:1–19, 1996, math.LO/9601218.
Saharon Shelah and Lee Stanley. . , 36:119–152, 1987.
[^1]: References of the form [math.XX/$\cdots$]{} refer to the [arXiv.org/archive/math]{} archive
| 0 |
---
abstract: 'We consider an infinite chain of coupled harmonic oscillators with a Langevin thermostat attached at the origin and energy, momentum and volume conserving noise that models the collisions between atoms. The noise is rarefied in the limit, [that corresponds to the hypothesis]{} that in the macroscopic unit time only a finite number of collisions takes place (Boltzmann-Grad limit). We prove that, after the hyperbolic space-time rescaling, the Wigner distribution, describing the energy density of phonons in space-frequency domain, converges to a positive energy density function $W(t, y, k)$ that evolves according to a linear kinetic equation, with the interface condition at $y=0$ that corresponds to reflection, transmission and absorption of phonons. The paper extends the results of [@kors], where a thermostatted harmonic chain (with no inter-particle scattering) has been considered.'
author:
- 'Tomasz Komorowski[^1]'
- 'Stefano Olla[^2]'
title: Kinetic limit for a chain of harmonic oscillators with a point Langevin thermostat
---
Introduction {#intro}
============
The mathematical analysis of macroscopic energy transport in anharmonic chain of oscillators [constitutes a very hard mathematical problem]{}, see [@spohn2006]. One approach to it is to replace the non-linearity by a stochastic exchange of momentum between nearest neighbor particles [in such a way that the total kinetic energy and momentum are conserved]{}. This stochastic exchange can be modeled in various ways: e.g. for each couple of nearest neighbor particles the exchange of their momenta can occur independently at an exponential time (which models their elastic collision). Otherwise, for each triple of consecutive particles, exchange of momenta can be performed in a continuous, diffusive fashion, so that its energy and momentum are preserved. In the present article we adopt the latter choice, see Section \[sec2.2.1\] for a detailed description of the dynamics, but, with no significant changes, all our results can be extended to other stochastic noises. A small parameter ${\epsilon}>0$ is introduced to rescale space and time, and the intensity of the noise is adjusted so that in a (macroscopic) finite interval of time, there is only a finite amount of momentum exchanged by the stochastic mechanism. In terms of the random exchanges, it means that, on average, each particle undergoes only a finite number of stochastic collisions in a finite time. Letting ${\epsilon}\to 0$ corresponds therefore to taking the kinetic limit for the system.
The Wigner distribution is a useful tool to localize in space the energy per frequency mode. In the absence of the thermostat, it is proven in [@BOS], that as ${\epsilon}\to 0$, the Wigner distribution converges to the solution of the kinetic transport equation $$\label{eq:bos}
\partial_t W(t,y,k) + \bar\omega'(k) \partial_y W(t,y,k) =
2\gamma_0 \int_{{{\mathbb T}}} R(k,k') \left(W(t,y,k') - W(t,y,k)\right) dk, \qquad (t,y,k)\in [0,+\infty)\times{{\mathbb T}}\times \mathbb R,$$ with an explicitly given scattering kernel $ R(k,k') \ge 0$. It is symmetric and the total scattering kernel behaves as $$\label{Rk}
R(k):=\mathlarger{\int}_{{{\mathbb T}}}R(k,k')dk' \sim |k|^2\quad\mbox{ for }|k| \ll 1.$$ Here ${{\mathbb T}}$ is the unit torus, which is the interval $[-1/2,1/2]$, with identified endpoints. Furthermore $\gamma_0>0$ is the scattering rate for the microscopic chain (see below) and $\bar\omega(k) = \omega(k)/2\pi$, where $\omega(k)$ is the dispersion relation of the chain (see definition ).
In the present paper we are interested in the macroscopic effects of a heat bath at temperature $T$, modeled by a Langevin dynamics, applied to one particle, say the one labelled $0$, with a coupling strength $\gamma_1>0$ (see below for a detailed description). Unlike the conservative stochastic dynamics acting on the bulk, the action of the heat bath is not rescaled with ${\epsilon}$, so in the limit as ${\epsilon}\to 0$ it constitutes a singular perturbation on the dynamics. The effect in the limit is to introduce the following interface conditions at $y=0$ on : $$\label{bc-int}
\begin{split}
W(t,0^+,k) &=p_-(k) W(t,0^+, -k) + p_+(k)W(t, 0^-,k)+{{\fgeeszett}}(k)T, \quad\hbox{ for $0< k\le 1/2$},\\
W(t,0^-, k) &=p_-(k)W(t,0^-,-k) + p_+(k) W(t,0^+, k) +{{\fgeeszett}}(k)T,\quad \hbox{ for $-1/2< k< 0$.}
\end{split}$$ Interpreting $W(t,y,k)$ as the density of the energy of the phonons of mode $k$ at time $t$ and position $y$, then $ p_+(k)$, $ p_-(k)$ and ${{\fgeeszett}}(k)$ are respectively the probabilities for transmission, reflection and absorption of a phonon of mode $k$ when it crosses $y=0$, while ${{\fgeeszett}}(k)T$ is the rate of creation of a phonon of that mode. These probabilities are functions of $k$ and depend only on the dispersion relation $\omega(\cdot)$ and the intensity $\gamma_0$ of the thermostat (cf. ). They are properly normalized, i.e. $ p_+(k) + p_-(k) + {{\fgeeszett}}(k) = 1$, so that $W(t,y,k) = T$ is a stationary solution (thermal equilibrium).
This result was recently proven in the absence of the conservative noise in the bulk (i.e. $\gamma_0= 0$ in ), see [@kors]. Then, the resulting dynamics outside the interface, given by , reduces itself to pure transport as ${\gamma}_0=0$. Obviously, the coefficients appearing in the interface conditions do not depend on the presence of the bulk noise.
The goal of the present paper is to extend the result of [@kors] to the case when the inter-particle noise is present, i.e. ${\gamma}_0>0$, see Theorem \[main-thm\] below for the precise formulation of our main result. We emphasize that in the situation when $\gamma_0=0$, both the equations for the microscopic and macroscopic dynamics, given below by and respectively, can be solved explicitly, in terms of the initial condition, and this fact has been extensively used in the proof in [@kors]. The argument can be extended to the dynamics where only the damping terms of the noise are present, i.e. with no noise input both from the inter-particle scattering and the thermostat, see . The equation for the macroscopic limit of the respective Wigner distribution $W^{\rm un}(t,y,k)$ reads (cf ), see Theorem \[main:thm-un\] below, $$\label{eq:damp}
\partial_t W^{\rm un}(t,y,k) + \bar\omega'(k) \partial_y W^{\rm un}(t,y,k) =
-2 \gamma_0 R(k) W^{\rm un}(t,y,k), \quad y\not=0,$$ with the boundary conditions as in . In the next step we add the stochastic part corresponding to the inter-particle scattering, which corresponds to $T=0$ for the thermostat, see equation formulated for the respective wave function. Next, we use the previously described dynamics to represent the solution of the equation with the help of the Duhamel formula, see . The corresponding representation for the Wigner distribution is given in . Having already established the macroscopic limit for the dynamics with no stochastic noise, we can use the Duhamel representation to identify the kinetic limit of the noisy microscopic dynamics when the thermostat temperature $T=0$, see Theorem \[cor020805-19\]. The extension to the case when the temperature $T>0$ is possible by another application of the Duhamel formula, see Section \[sec10\].
Concerning the organization of the paper, Section \[prelim\] is devoted to preliminaries and the formulation of the main result, see Theorem \[main-thm\]. Among things discussed is the rigorous definition of a solution of a kinetic equation with the interface condition , see Sections \[sec2.6.3\] and \[sec2.6.4\]. Section \[sec3\] deals with the basic properties of the microscopic dynamics obtained by removal of stochastic noises, both between the particles of the chain and the thermostat. This dynamics is an auxiliary tool for the mild formulation of the microscopic dynamics corresponding to the chain with inter-particle scattering and thermostat. We discuss first the case when thermostat is set at $T=0$, see Section \[sec4\]. In this section we obtain also basic estimates for the microscopic Wigner distributions, see Proposition \[prop012105-19\], that follow from the energy balance equation established in . A similar result is also formulated for the auxiliary dynamics with no stochastic noise in Section \[sec5.1\]. In Section \[sec5.3\] we formulate the result concerning the kinetic limit for this dynamics, see Theorem \[main:thm-un\]. Its proof is quite analogous to the argument of [@kors] and is given in Appendix \[appb\]. Section \[sec5.5\] is essentially devoted to the proof of the main result (Theorem \[main-thm\]) for the case $T=0$ and the proof for $T>0$ is presented in Section \[sec10\]. Some properties of the dynamics corresponding to the macroscopic kinetic limit are proven in Appendix \[appa\]. Section \[appC\] of the appendix is devoted to the proof of some properties of the interface coefficients appearing in the limit.
Acknowledgements {#acknowledgements .unnumbered}
================
T.K. acknowledges the support of the National Science Centre: NCN grant DEC-2016/23/B/ST1/00492. S.O. acknowledges the ANR-15-CE40-0020-01 LSD grant of the French National Research Agency.
Preliminaries and statement of the main results {#prelim}
===============================================
Basic notation
--------------
We shall use the following notation: let ${{\mathbb R}}_*:={{\mathbb R}}\setminus\{0\}$, ${{\mathbb R}}_+:=(0,+\infty)$, ${{\mathbb R}}_-:=(-\infty,0)$ and likewise ${{\mathbb T}}_*:={{\mathbb T}}\setminus\{0\}$, ${{\mathbb T}}_+:=(0,1/2)$, ${{\mathbb T}}_-:=(-1/2,0)$. Throughout the paper we use the short hand notation $$\label{011909}
{\frak s}(k):=\sin(\pi k)\quad {\frak c}(k):=\cos(\pi k),\quad k\in{{\mathbb T}}.$$ Let $
e_x(k):=\exp\{2\pi i xk\}$ for $x$ belonging to the set of integers ${{\mathbb Z}}.$ The Fourier series corresponding to a complex valued sequence $(f_x)_{x\in{{\mathbb Z}}}$ belonging to $\ell_2$ - the Hilbert space of square integrable sequences of complex numbers - is given by $$\label{fourier}
\hat f(k)=\sum_{x\in{{\mathbb Z}}}f_xe_x^\star(k), \quad k\in{{\mathbb T}}.$$ Here $z^\star$ is the complex conjugate of $z\in\mathbb C$. By the Parseval identity $\hat f\in L^2({{\mathbb T}})$ - the space of complex valued, square integrable functions - and $\|\hat f\|_{L^2({{\mathbb T}})}=\|f\|_{\ell_2}$. Given ${\epsilon}>0$ we let ${{\mathbb Z}}_{{\epsilon}}:=({\epsilon}/2){{\mathbb Z}}$ and ${{\mathbb T}}_{\epsilon}:=(2/{\epsilon}){{\mathbb T}}$. Let $\ell_{2,{\epsilon}}$ be the space made of all complex valued square integrable sequences $(f_y)_{y\in
{{\mathbb Z}}_{\epsilon}}$ equipped with the norm $$\|f\|_{\ell_{2,{\epsilon}}}:=\left\{\frac{{\epsilon}}{2}\sum_{y\in{{\mathbb Z}}_{\epsilon}}|f_y|^2\right\}^{1/2}.$$ Let $$\label{fouriereps}
\hat f(\eta)=\frac{{\epsilon}}{2}\sum_{y\in{{\mathbb Z}}_{\epsilon}}f_ye_y^\star(\eta), \quad \eta\in{{\mathbb T}}_{\epsilon}.$$ The Parseval identity takes then the form $\|\hat f\|_{L^2({{\mathbb T}}_{\epsilon})}=\|f\|_{\ell_{2,{\epsilon}}}$.
For any non-negative functions $f,g$ acting on a set $A$ the notation $f\preceq g$ means that there exists a constant $C>0$ such that $f(a)\le Cg(a)$ for $a\in A$. We shall write $f\approx g$ if $f\preceq g$ and $g\preceq f$.
Given a function $f:\bar{{\mathbb R}}_+\to\mathbb C$ satisfying $|f(t)|\le
Ce^{Mt}$, for fome $C,M>0$ we denote by $\tilde
f({\lambda})$ its Laplace transform $$\tilde
f({\lambda})=\int_0^{+\infty}e^{-{\lambda}t}f(t)dt,\quad {\rm Re}\,{\lambda}>M.$$
Some function spaces
--------------------
For a given $ G\in {\cal S}({{\mathbb R}}\times{{\mathbb T}})$ - the class of Schwartz functions on ${{\mathbb R}}\times{{\mathbb T}}$ - we let $$\widehat G(\eta,k)=\int_{{{\mathbb R}}}e_y^\star(\eta)G(y,k)dy$$ be its Fourier transform in the first variable. Let $$\label{AC}
{\cal
A}_c:=[G:\,\hat G\in C_c^\infty({{\mathbb R}}\times{{\mathbb T}})].$$ Let ${\cal A}$ be the Banach space obtained by the completion of ${\cal A}_c$ in the norm $$\label{060805-19}
\|G\|_{\cal A}:=\int_{{{\mathbb R}}}\sup_{k\in{{\mathbb T}}}|\widehat G(\eta,k)|d\eta,\quad G\in{\cal
A}_c.$$ Space ${\cal A}'$ - the dual to ${\cal A}$ - consists of all distributions $G\in {\cal
S}'({{\mathbb R}}\times{{\mathbb T}})$ of the form $$\langle G, F\rangle=\int_{{{\mathbb R}}\times {{\mathbb T}}} \widehat G^\star(\eta,k)
\widehat F(\eta,k)d\eta dk,\quad F\in {\cal A}$$ for some measurable function $\widehat G:{{\mathbb R}}\times{{\mathbb T}}\to\mathbb C$, equipped with the norm $$\label{060805-19a}
\|G\|_{{\cal A}'}=\sup_{\eta\in{{\mathbb R}}}\int_{{{\mathbb T}}}|\widehat G(\eta,k)|d k<+\infty.$$
We shall also consider the spaces ${\cal
L}_{2,{\epsilon}}:=\ell_{2,{\epsilon}}\otimes L^2({{\mathbb T}})$. The respective norms of $G:{{\mathbb Z}}_{\epsilon}\times{{\mathbb T}}\to\mathbb C$ and $\widehat
G:{{\mathbb T}}_{\epsilon}\times{{\mathbb T}}\to\mathbb C$ are given by $$\label{011505-19}
\|G\|_{{\cal L}_{2,{\epsilon}}}:=\left\{\frac{{\epsilon}}{2}\sum_{y\in{{\mathbb Z}}_{\epsilon}}\|G_y\|_{L^2({{\mathbb T}})}^2\right\}^{1/2}
=\|\widehat
G\|_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})}.$$
Infinite system of interacting harmonic oscillators
---------------------------------------------------
### Thermostatted Hamiltonian dynamics with momentum and energy conserving noise {#sec2.2.1}
We consider a stochastically perturbed chain of harmonic oscillators thermostatted at a fixed temperature $T\ge 0$ at $x=0$. Its dynamics is described by the system of Itô stochastic differential equations $$\begin{aligned}
&&d{\frak q}_{x}(t)={\frak p}_x(t)dt \nonumber
\\
&& d{\frak p}_x(t)=\left[-(\alpha\star{\frak
q}(t))_x-\frac{{\epsilon}{\gamma}_0}{2}(\theta\star{\frak
p}(t))_x\right]dt+\sqrt{{\epsilon}{\gamma}_0}\sum_{k=-1,0,1}(Y_{x+k}{\frak
p}_x(t))dw_{x+k}(t)\label{eq:bas1} \\
&&
+\left(-{\gamma}_1{\frak p}_0(t)dt+\sqrt{2{\gamma}_1 T}dw(t)\right)\delta_{0,x},\quad x\in{{\mathbb Z}}.\nonumber\end{aligned}$$ Here $$\label{011210}
Y_x:=({\frak p}_x-{\frak p}_{x+1})\partial_{{\frak p}_{x-1}}+({\frak p}_{x+1}-{\frak p}_{x-1})\partial_{{\frak p}_{x}}+({\frak p}_{x-1}-{\frak p}_{x})\partial_{{\frak p}_{x+1}}$$ and $\left(w_x(t)\right)_{t\ge0}$, $x\in{{\mathbb Z}}$ with $\left(w(t)\right)_{t\ge0}$, are i.i.d. one dimensional, real valued, non-anticipative standard Brownian motions, over some filtered probability space $(\Sigma,{\cal F},\left({\cal F}_t\right)_{t\ge0},{{\mathbb P}})$. In addition, $$\theta_x=\Delta\theta^{(0)}_x:=\theta^{(0)}_{x+1}+\theta^{(0)}_{x-1}-2\theta^{(0)}_x$$ with $$\theta^{(0)}_x=\left\{
\begin{array}{rl}
-4,&x=0\\
-1,&x=\pm 1\\
0, &\mbox{ if otherwise.}
\end{array}
\right.$$ A simple calculation shows that $$\label{beta}
\hat \theta(k)=8{\frak s}^2(k)\left(1+2{\frak c}^2( k)\right),\quad k\in{{\mathbb T}}.$$ Parameters ${\epsilon}\gamma_0>0$, ${\gamma}_1$ describe the strength of the inter-particle and thermostat noises, respectively. In what follows we shall assume that ${\epsilon}>0$ is small, that corresponds to the low density hypothesis that results in atoms suffering finitely many ”collisions” in a macroscopic unit of time (Boltzmann-Grad limit). Although the noise considered here is continuous we believe that the results of the present paper extend to other type of noises, such as e.g. Poisson shots.
Since the vector field $Y_x$ is orthogonal both to a sphere ${\frak
p}_{x-1}^2+{\frak p}_x^2+{\frak p}_{x+1}^2\equiv {\rm const}$ and plane ${\frak p}_{x-1}+{\frak p}_x+{\frak p}_{x+1}\equiv {\rm const}$, the inter-particle noise conserves locally the kinetic energy and momentum.
Concerning the Hamiltonian part of the dynamics, we assume (cf [@BOS]) that the coupling constants $({\alpha}_x)_{x\in{{\mathbb Z}}}$ satisfy the following:
- they are real valued and there exists $C>0$ such that $|\alpha_x|\le Ce^{-|x|/C}$ for all $x\in {{\mathbb Z}}$,
- $\hat\alpha(k)$ is also real valued and $\hat\alpha(k)>0$ for $k\not=0$ and in case $\hat \alpha(0)=0$ we have $\hat\alpha''(0)>0$.
The above conditions imply that both functions $x\mapsto\alpha_x$ and $k\mapsto\hat\alpha(k)$ are even. In addition, $\hat\alpha\in
C^{\infty}({{\mathbb T}})$ and in case $\hat\alpha(0)=0$ we have $\hat\alpha(k)=k^2\phi(k^2)$ for some strictly positive $\phi\in
C^{\infty}({{\mathbb T}})$. The dispersion relation ${\omega}:{{\mathbb T}}\to
\bar{{\mathbb R}}_+$, given by $$\label{mar2602}
{\omega}(k):=\sqrt{\hat \alpha (k)}$$ is even. Throughout the paper it is assumed to be unimodal, i.e. increasing on $[0,1/2]$ and then, in consequence, decreasing on $[-1/2,0]$. Its unique minimum and maximum, attained at $k=0$, $k=1/2$, respectively are denoted by ${\omega}_{\rm min}\ge 0$ and ${\omega}_{\rm max}$, correspondingly. We denote the two branches of its inverse by ${\omega}_\pm:[{\omega}_{\rm min},{\omega}_{\rm max}]\to\bar {{\mathbb T}}_\pm$.
### Initial data
We assume that the initial data is random and, given ${\epsilon}>0$, distributed according to probabilistic measure $\mu_{\epsilon}$ and $$\label{mu-eps}
{\cal E}_*:=\sup_{{\epsilon}\in(0,1]}{\epsilon}\sum_{x\in{{\mathbb Z}}}\langle{\frak e}_x\rangle_{\mu_{\epsilon}}<+\infty.$$ Here $\langle\cdot\rangle_{\mu_{\epsilon}}$ is the expectation with respect to $\mu_{\epsilon}$ and the microscopic energy density $${\frak e}_x:=\frac12\left({\frak p}_x^2+\sum_{x'\in{{\mathbb Z}}}{\alpha}_{x-x'}{\frak q}_x{\frak q}_{x'}\right).$$ The assumption ensures that the macroscopic energy density of the chain is finite.
Kinetic scaling of the wave-function
------------------------------------
To observe macroscopic effects of the inter-particle scattering consider time of the order $t/{\epsilon}$. It is also convenient to introduce the wave function that, adjusted to the macroscopic time, is given by $$\label{011307}
\psi^{({\epsilon})}(t):=\tilde{{\omega}} \star {\frak q}\left(\frac{t}{{\epsilon}}\right)+i{\frak p}\left(\frac{t}{{\epsilon}}\right),$$ where $({\frak p}(t),{\frak q}(t))$ satisfies and $\left(\tilde {\omega}_x\right)_{x\in{{\mathbb Z}}}$ are the Fourier coefficients of the dispersion relation, see . We shall consider the Fourier transform of the wave function, given by $$\label{011307a}
\hat\psi^{({\epsilon})}(t,k)={\omega}(k)\hat {\frak q}^{({\epsilon})}\left(t,k\right)+i\hat{\frak p}^{({\epsilon})}\left(t,k\right).$$ Here $\hat {\frak q}^{({\epsilon})}\left(t\right)$, $\hat{\frak
p}^{({\epsilon})}\left(t\right)$ are given by the Fourier series for ${\frak q}^{({\epsilon})}_x(t):={\frak q}_x(t/{\epsilon})$ and ${\frak
p}^{({\epsilon})}_x(t):={\frak p}_x(t/{\epsilon})$, $x\in{{\mathbb Z}}$, respectively. They satisfy $$\begin{aligned}
\label{basic:sde:2a}
&&
d\hat\psi^{({\epsilon})}(t,k)=\left\{-\frac{i}{{\epsilon}} {\omega}(k)\hat\psi^{({\epsilon})}(t,k)
-2i{\gamma}_0R(k)\hat{\frak p}^{({\epsilon})}\left(t,k\right)-\frac{i{\gamma}_1}{{\epsilon}}\int_{{{\mathbb T}}}\hat{\frak p}^{({\epsilon})}\left(t,k'\right)dk'\right\}dt
\nonumber\\
&&
-2\sqrt{{\gamma}_0}\int_{{{\mathbb T}}}r(k,k')\hat{\frak p}^{({\epsilon})}\left(t,k-k'\right)B(dt,dk')+i\sqrt{\frac{2{\gamma}_1T}{{\epsilon}}}dw(t),\\
&&
\hat\psi^{({\epsilon})}(0)= \hat\psi,\nonumber
\end{aligned}$$ where $$\begin{aligned}
&
\hat{\frak p}^{({\epsilon})}\left(t,k\right):=\frac{1}{2i}\left[\hat\psi^{({\epsilon})}(t,k)-\left(\hat\psi^{({\epsilon})}\right)^\star(t,-k)\right],\\
&
r(k,k'):=4 {\frak s}(k) {\frak s}(k-k') {\frak s}(2k-k')
\quad k,k'\in {{\mathbb T}},\\
&
R(k):=\int_{{{\mathbb T}}}r^2(k,k')dk'={\frak s}^2(2k)+2 {\frak s}^2(k)=\frac{\hat\theta(k)}{4}.
\end{aligned}$$ Here $B(t,dk)=\sum_{x\in{{\mathbb Z}}}w_x(t)e_x(k)dk$ is a cylindrical Wiener process on $L^2({{\mathbb T}})$, i.e. $$\label{cnoise}
{\mathbb E}[B(dt,dk)B^\star(ds,dk')]=\delta(k-k')\delta(t-s)dt ds dkdk'.$$
Energy density - Wigner function
--------------------------------
One can easily check that $$\label{e1}
\|\hat\psi^{({\epsilon})}(t)\|^2_{L^2({{\mathbb T}})}=\|\psi^{({\epsilon})}(t)\|^2_{\ell^2}=\sum_{x\in{{\mathbb Z}}}\left({\frak p}^{({\epsilon})}_x(t)\right)^2+\sum_{x,x'\in{{\mathbb Z}}}{\alpha}_{x-x'}{\frak q}^{({\epsilon})}_x(t){\frak q}^{({\epsilon})}_{x'}(t).$$ After straightforward calculations one can verify that $$\label{031709}
d\|\hat\psi^{({\epsilon})}(t)\|^2_{L^2({{\mathbb T}})}
=-\frac{2{\gamma}_1}{{\epsilon}} \left[
\left({\frak p}^{({\epsilon})}_0(t)\right)^2
- T \right]dt
+\left(\frac{2{\gamma}_1T}{{\epsilon}}\right)^{1/2}{\frak p}^{({\epsilon})}_0(t) dw(t),\quad {{\mathbb P}}_{\epsilon}\,\mbox{-a.s.}$$ Here ${{\mathbb P}}_{\epsilon}:={{\mathbb P}}\otimes\mu_{\epsilon}$. By ${\mathbb E}_{\epsilon}$ we denote the expectation with respect to ${{\mathbb P}}_{\epsilon}$. From assumptions and we obtain.
\[prop012409\] Under the kinetic scaling we have $$\label{052709-18}
{\cal E}_*(t):=\sup_{{\epsilon}\in(0,1]}\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\|\hat\psi^{({\epsilon})}(t)\|^2_{L^2({{\mathbb T}})}\le
{\cal E}_*+{\gamma}_1 T t,\quad t\ge0.$$
We can introduce the (averaged) Wigner distribution $
W_{\epsilon}(t)\in {\cal A}'$, corresponding to $\psi^{({\epsilon})}(t)$, by the formula $$\label{wigner-bas}
\langle
W_{\epsilon}(t),G\rangle:=\frac{{\epsilon}}{2}\sum_{x,x'\in{{\mathbb Z}}}{\mathbb E}_{\epsilon}\left[\left(\psi^{({\epsilon})}_{x'}(t)\right)^\star
\psi^{({\epsilon})}_{x}(t)\right]e_{x'-x}(k)G\left({\epsilon}\frac{x+x'}{2},k\right),\quad
G\in {\cal A}.$$ Thanks to it is well defined for any $t\ge0$ and ${\epsilon}\in(0,1]$. Using the Fourier transform in the first variable we can rewrite the Wigner distribution as $$\label{wigner-bas1}
\langle
W_{\epsilon}(t),G\rangle=\int_{{{\mathbb T}}_{\epsilon}\times{{\mathbb T}}}\widehat
W_{\epsilon}^\star(t,\eta,k)\widehat G\left(\eta,k\right)d\eta dk,\quad
G\in {\cal A},$$ where $$\label{W+}
\widehat
W_{\epsilon}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\left[\left(\hat\psi^{({\epsilon})}\right)^\star\left(t,k-\frac{{\epsilon}\eta}{2}\right) \hat\psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right)\right].$$ We shall refer to $\widehat
W_{\epsilon}(t)$ as the Fourier-Wigner function corresponding to the given wave function. For the sake of future reference define also $Y_{\epsilon}(t)$, by its Fourier transform $$\label{Y+}
\widehat
Y_{\epsilon}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\left[\hat\psi^{({\epsilon})}\left(t,-k+\frac{{\epsilon}\eta}{2}\right) \hat\psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right)\right].$$
Kinetic equation
----------------
An important role in our analysis will be played by the function, see Section 2 of [@kors], $$\label{eq:bessel0}
J(t) = \int_{{{\mathbb T}}}\cos\left(\omega(k) t\right) dk.$$ Its Laplace transform $$\label{eq:2}
\tilde J({\lambda}):=\int_0^{\infty}e^{-{\lambda}t}J(t)dt= \int_{{{\mathbb T}}}
\frac{\lambda}{\lambda^2 + \omega^2(k)} dk,\quad {\lambda}\in \mathbb C_+:=[z:\,{\rm Re}\,z>0].$$ One can easily see that ${\rm Re}\,\tilde J({\lambda})>0$ for ${\lambda}\in
\mathbb C_+$, therefore we can define the function $$\label{tg}
\tilde g(\lambda) := ( 1 + \gamma_1 \tilde J(\lambda))^{-1},\quad {\lambda}\in \mathbb C_+.$$ We have $$\label{012410}
|\tilde g(\lambda)|\le 1,\quad {\lambda}\in \mathbb C_+.$$ The function $\tilde g(\cdot)$ is analytic on $ \mathbb C_+$ so, by the Fatou theorem, see e.g. p. 107 of [@koosis], we know that $$\label{nu}
\nu(k) :=\lim_{{\epsilon}\to0+}\tilde g({\epsilon}-i{\omega}(k))$$ exists a.e. in ${{\mathbb T}}$ and in any $L^p({{\mathbb T}})$ for $p\in[1,\infty)$.
Let us introduce $$\label{033110}
\wp(k):=\frac{{\gamma}_1 \nu(k)}{2|\bar{\omega}'(k)|},\quad \fgeeszett(k):=\frac{{\gamma}_1|\nu(k)|^2}{|\bar{\omega}'(k)|},\quad p_+(k):=\left|1-\wp(k)\right|^2 ,\quad
p_-(k):=|\wp(k)|^2 ,$$ where $
\bar{\omega}'(k):={\omega}'(k)/(2\pi)$. We have shown in [@kors] that $$\label{feb1402}
{\rm Re}\,\nu(k)=\left(1+\frac{{\gamma}_1}{2|\bar{\omega}'(k)|}\right)|\nu(k)|^2.$$ The functions $p_\pm(\cdot)$ and $\fgeeszett(\cdot)$ are even. Thanks to we have $$\label{012304}
p_+(k)+p_-(k)+\fgeeszett(k)=1.$$
### Linear kinetic equation with an interface
Let $L$ be the operator given by $$\label{L}
LF(k):= 2\int_{{{\mathbb T}}}R(k,k')
\left[F\left(k'\right) - F\left(k\right)\right]dk',\quad k\in{{\mathbb T}},$$ [for $F\in L^1({{\mathbb T}})$]{} and $$\begin{aligned}
\label{R}
&
R(k,k'):=\frac12\left\{r^2\left(k,k-
k'\right)+r^2\left(k,k+k'\right)\right\}\\
&
=8{\frak s}^2(k){\frak s}^2(k')\left\{{\frak s}^2(k){\frak c}^2(k')+{\frak s}^2(k'){\frak c}^2(k)\right\},\quad k,k'\in{{\mathbb T}}.\nonumber\end{aligned}$$ Note that (cf ) the total scattering kernel equals $$\label{Rk}
R(k):=\int_{{{\mathbb T}}}R(k,k')dk'=\frac{\hat\theta(k)}{4}.$$
\[df012603-19\] Given $T\in{{\mathbb R}}$, let ${\cal C}_T$ be a subclass of $C_b({{\mathbb R}}_*\times{{\mathbb T}}_*)$ that consists of continuous functions $F$ that can be continuously extended to $\bar{{\mathbb R}}_\pm\times{{\mathbb T}}_*$ and satisfy the interface conditions $$\label{feb1408}
F(0^+,k)=p_-(k)F(0^+, -k)+p_+(k)F(0^-,k)+{{\fgeeszett}}(k)T, \quad\hbox{ for $0< k\le 1/2$},\\$$ and $$\label{feb1410}
F(0^-, k)=p_-(k)F(0^-,-k) + p_+(k)F(0^+, k) +{{\fgeeszett}}(k)T,\quad \hbox{ for $-1/2< k< 0$.}$$
Note that $F\in{\cal C}_T$ if and only if $F-T'\in{\cal C}_{T-T'}$ for any $T,T'\in{{\mathbb R}}$, because of (\[012304\]).
Let us fix $T\ge0$. We consider the kinetic interface problem given by equation $$\label{eq:8}
\begin{aligned}
&\partial_tW(t,y,k) + \bar{\omega}'(k) \partial_y W(t,y,k) = {\gamma}_0 L_k W(t,y,k),
\quad (t,y,k)\in{{\mathbb R}}_+\times {{\mathbb R}}_*\times{{\mathbb T}}_*,
\\
&
W(0,y,k)=W_0(y,k),
\end{aligned}$$ with the interface condition $$\label{feb1408aa}
W(t)\in {\cal C}_T,\quad t\ge0.$$ Here $ L_k $ denotes the operator $L$ acting on the $k$ variable. We shall omit writing the subscript if there is no danger of confusion.
### Simplified case. Explicit solution {#sec2.6.2}
We consider first the situation when equation is replaced by $$\label{eq:8p}
\begin{aligned}
&\partial_tW^{\rm un}(t,y,k) + \bar{\omega}'(k) \partial_y W^{\rm un}(t,y,k) = -2{\gamma}_0R(k) W^{\rm un}(t,y,k),
\quad (t,y,k)\in{{\mathbb R}}_+\times {{\mathbb R}}_*\times{{\mathbb T}}_*,
\\
&
W^{\rm un}(0,y,k)=W_0(y,k),
\end{aligned}$$ with the interface conditions – , with $T=0$. It can be solved explicitly, using the method of characteristics, and we obtain $$\begin{aligned}
\label{010304}
&W^{\rm un}(t,y,k)
=
e^{-2{\gamma}_0R(k)t}\left\{\vphantom{\int_0^1}W_0\left(y-\bar{{\omega}}'(k)t,k\right)
1_{[0,\bar{{\omega}}'(k)t]^c}(y)
+p_+(k)W_0\left(y-\bar{{\omega}}'(k)t,k\right)1_{[0,\bar{{\omega}}'(k)t]}(y)
\right. \nonumber
\\
&
+\left. \vphantom{\int_0^1}p_-(k) W_0\left(-y+\bar{\omega}'(k) t,-k\right)1_{[0,\bar {\omega}'(k)t]}(y)\right\}.\end{aligned}$$ Consider a semigroup of bounded operators on $L^\infty({{\mathbb R}}\times {{\mathbb T}}_*)$ defined by $$\label{010304s}
{\frak W}^{\rm un}_t(W_0)\left(y,k\right)
:=
W^{\rm un}(t,y,k),$$ with $W_0\in L^\infty({{\mathbb R}}\times
{{\mathbb T}}_*)$, $t\ge0$ and $(y,k)\in {{\mathbb R}}_*\times {{\mathbb T}}_*$. From formula one can conclude that $\left({\frak
W}^{\rm un}_t\right)_{t\ge0}$ forms a semigroup of contractions on both $L^1({{\mathbb R}}\times{{\mathbb T}})$ and $L^\infty({{\mathbb R}}\times{{\mathbb T}})$. Thus, by interpolation, formula defines a semigorup of contractions on any $L^p({{\mathbb R}}\times{{\mathbb T}})$, $1\le p\le+\infty$.
Note that if $W_0$ is continuous in ${{\mathbb R}}_*\times {{\mathbb T}}_*$, then $ W^{\rm un}(t,y,k)$ satisfies the interface conditions and , with $T=0$ for all $t>0$. Therefore, $( {\frak W}^{\rm un}_t)_{t\ge0}$ is a semigroup on ${\cal C}_0$ with $W^{\rm un}(t,y,k)$ (cf ) satisfying the first equation of , the interface condition - and the initial condition $$\lim_{t\to0+} W^{\rm un}(t,y,k)=W_0(y,k),\quad (y,k)\in{{\mathbb R}}_*\times{{\mathbb T}}_*.$$
### Kinetic equation - classical solution {#sec2.6.3}
\[df013001-19\] We say that a function $W:\bar{{\mathbb R}}_+\times {{\mathbb R}}\times {{\mathbb T}}_*\to {{\mathbb R}}$ is a classical solution to equation with the interface conditions , at $y=0$, if it is bounded and continuous on ${{\mathbb R}}_+\times{{\mathbb R}}_*\times {{\mathbb T}}_*$, and the following conditions hold:
- the restrictions of $W$ to ${{\mathbb R}}_+\times{{\mathbb R}}_\iota\times {{\mathbb T}}_{\iota'}$, $\iota,\iota'\in\{-,+\}$, can be extended to bounded and continuous functions on the respective closures $\bar{{\mathbb R}}_+\times\bar{{\mathbb R}}_{\iota}\times \bar{{\mathbb T}}_{\iota'}$,
- for each $(t,y,k)\in {{\mathbb R}}_+\times{{\mathbb R}}_*\times {{\mathbb T}}_*$ fixed, the function $W(t+s,y+\bar{\omega}'(k) s,k)$ is of the $C^1$ class in the $s$-variable in a neighborhood of $s=0$, and the directional derivative $$\label{Dt}
D_tW(t,y,k)=\left(\partial_t+\bar{\omega}'(k)\partial_y\right)W(t,y,k):=\frac{d}{ds}_{|s=0} W(t+s,y+\bar{\omega}'(k) s,k)$$ is bounded in ${{\mathbb R}}_+\times{{\mathbb R}}_*\times {{\mathbb T}}_*$ and satisfies $$\label{eq:8a}
\begin{array}{ll}
D_tW(t,y,k) = {\gamma}_0 L_k W(t,y,k), &
\quad (t,y,k)\in{{\mathbb R}}_+\times {{\mathbb R}}_*\times{{\mathbb T}}_*,
\end{array}$$
- $W(t)$ satisfies , and $$\label{010102-19}
\lim_{t\to0+}W(t,y,k)=W_0(y,k),\quad (y,k)\in{{\mathbb R}}_*\times{{\mathbb T}}_*.$$
The following result has been shown in [@koran], see Proposition 2.2.
\[prop013001-19\] Suppose that $W_0\in{\cal C}_T$. Then, under the above hypotheses on the scattering kernel $R(k,k')$ and the dispersion relation ${\omega}(k)$, there exists a unique classical solution to equation with the interface conditions and in the sense of Definition $\ref{df013001-19}$.
### $L^2$ solution {#sec2.6.4}
We assume that $T=0$. Define ${\frak W}_t(W_0):=W(t)$. Thanks to Proposition \[prop013001-19\] the family $\left({\frak W}_t\right)_{t\ge0}$ forms a semigroup on the linear space ${\cal C}_0$. Furthermore, we let $$\label{cR}
{\cal R}F(k):= \int_{{{\mathbb T}}}R(k,k')
F\left(k'\right) dk',\quad k\in{{\mathbb T}},\quad F\in L^1({{\mathbb T}}).$$ Let $
{\cal C}_0':= {\cal C}_0\cap L^2({{\mathbb R}}\times {{\mathbb T}}).
$ The following result holds.
\[prop011406-19\] We have ${\frak W}_t\left( {\cal C}_0'\right)\subset {\cal C}_0'$ for all $t\ge0$. The semigroup $\left({\frak W}_t\right)_{t\ge0}$ extends by the $L^2$ closure from ${\cal C}_0'$ to a $C_0$-continuous semigroup of contractions on $L^2({{\mathbb R}}\times{{\mathbb T}})$. Moreover, it is the unique solution in $L^2({{\mathbb R}}\times{{\mathbb T}})$ of the integral equation $$\label{integral}
{\frak W}_t={\frak W}_t^{\rm un}+2{\gamma}_0\int_0^t {\frak W}_{t-s}^{\rm
un}{\cal R} {\frak W}_sds,\quad t\ge0.$$
The proof of this result is contained in Appendix \[appa\]. We shall refer to the semigroup solution described in Proposition \[prop011406-19\] as the $L^2$-solution of quation with the interface conditions and for $T=0$. To extend the definition of such a solution to the case of an arbitrary $T\ge0$ we proceed as follows. Suppose that $W_0\in L^2({{\mathbb R}}\times{{\mathbb T}})$. Let $\chi\in C^\infty_c({{\mathbb R}})$ be an arbitrary real valued, even function that satisfies $$\label{011406-19}
\chi(y)=\left\{\begin{array}{ll}
1,&\mbox{ for }|y|\le 1/2, \\
0,&\mbox{ for }|y|\ge 1,\\
\mbox{belongs to }[0,1],& \mbox{ if otherwise.}
\end{array}\right.$$
\[df011406-19\] We say that $W(t,y,k)$ is the $L^2$-solution of quation with the interface conditions and for a given $T\ge0$ and an initial condition $W_0\in L^2({{\mathbb R}}\times{{\mathbb T}})$, if it is of the form $$\label{021406-19}
W(t,y,k):={\frak W}_t(\widetilde W_0)(y,k)+\int_0^t {\frak
W}_s(F)(y,k)ds+T\chi(y),\quad (t,y,k)\in \bar{{\mathbb R}}_+\times{{\mathbb R}}\times{{\mathbb T}}.$$ Here $$\label{eqF}
F(y,k):=-T\bar{\omega}'(k)\chi'(y),\quad \widetilde W_0(y,k):= W_0(y,k)-T\chi(y).$$
\[rm011406-19\] [*Note that the definition of the solution does not depend on the choice of function $\chi$ satisfying .* ]{}
\[rm021406-19\] [*Suppose that $W_0\in {\cal C}_T$. Then, $W(t,y,k)$ given by is the classical solution of with the interface conditions and , in the sense of Definition $\ref{df013001-19}$.*]{}
Asymptotics of the Wigner functions - the statement of the main result
----------------------------------------------------------------------
Thanks to we conclude that $$\label{030406-19}
\sup_{{\epsilon}\in(0,1]}\| W_{{\epsilon}}\|_{L^\infty([0,\tau];{\cal
A}')}<+\infty,\quad \mbox{for any }\tau>0.$$ Therefore $\left(W_{{\epsilon}}(\cdot)\right)$ is sequentially $\star$-weakly compact in $L^\infty_{\rm loc}([0,+\infty),{\cal A}')$, i.e. from any sequence ${\epsilon}_n\to0$ we can choose a subsequence, that we still denote by the same symbol, for which $\left( W_{{\epsilon}_n}(\cdot)\right)$ $\star$-weakly converges in $\left(L^1([0,t],{\cal
A})\right)'$ for any $t>0$. In our main result we identify the limit as the $L^2$ solution of the kinetic equation with the interface conditions and , in the sense of Definition $\ref{df011406-19}$.
\[main-thm\] Suppose that there exist $C,\kappa>0$ such that $$\label{011812aa}
|\widehat W_{\epsilon}(0,\eta,k)|+|\widehat Y_{\epsilon}(0,\eta,k)|\le
C\varphi(\eta),\quad (\eta,k)\in{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}}, \,{\epsilon}\in(0,1],$$ where $$\label{011812c}
\varphi(\eta):=\frac{1}{(1+\eta^2)^{3/2+\kappa}},$$ and $$ W_{{\epsilon}}(0)\mathop{\stackrel{{\rm \tiny w}^\star}{\longrightarrow}}\limits_{\tiny{{\epsilon}\to0+}} W_0\quad\mbox{ in
}{\cal A}'.$$ Then, $W_0\in L^2({{\mathbb R}}\times{{\mathbb T}})$ and for any $G\in
L^1_{\rm loc}\left([0,+\infty);{\cal A}\right)$ we have $$\label{061406-19}
\lim_{{\epsilon}\to0+}\int_0^{\tau}\langle W_{{\epsilon}}(t), G(t)\rangle dt=\int_0^{\tau}\langle W(t), G(t)\rangle dt,\quad\tau>0.$$ Here $W(t)$ is the $L^2$ solution of the kinetic equation with the interface conditions and satisfying $W(0)=W_0$.
In the case $T=0$ the theorem is a direct consequence of Theorem \[cor020805-19\] proved below. The more general case $T\ge0$ is treated in Section \[sec10\].
Deterministic wave equation corresponding to (\[basic:sde:2a\]) {#sec3}
===============================================================
Here we consider a deterministic part of the dynamics described in (\[basic:sde:2a\]). Its corresponding energy density function will converge to the solution of . In microscopic time the evolution of the wave function is given by $$\begin{aligned}
\label{012503-19}
&&
\frac{d}{dt}\hat\phi(t,k)=-i{\omega}(k)\hat\phi(t,k)
-2i{\epsilon}{\gamma}_0R(k)\hat p\left(t,k\right)-i{\gamma}_1\int_{{{\mathbb T}}}\hat{ p}\left(t,k'\right)dk',
\\
&&
\hat\phi(0,k)= \hat\psi(k),\nonumber
\end{aligned}$$ where $$\hat p(t,k):=\frac{1}{2i}\left[\hat\phi(t,k)-\left(\hat\phi(t,-k)\right)^\star\right].$$ In fact it is convenient to deal with the vector formulation of the equation for $$\label{Ut}
\hat\Phi(t,k)=\left[
\begin{array}{c}
\hat\phi_+(t,k)\\
\\
\hat\phi_-(t,k),
\end{array}\right],\quad \hat\Psi(k)=\left[
\begin{array}{c}
\hat\psi_+(k)\\
\\
\hat\psi_-(k)
\end{array}\right].$$ Here, we use the convention $\hat\phi_+(t,k)=\hat\phi(t,k)$ and $\hat\phi_-(t,k):=\left(\hat\phi(t,-k)\right)^\star$ and similarly for $\hat\psi_\pm(k)$. The equation then takes the form $$\begin{aligned}
\label{basic:sde:2av}
&&
\frac{d}{dt}\hat\Phi(t,k)={\Omega}_{\epsilon}(k) \hat\Phi(t,k) - i\gamma_1 {\frak f}{p}_0(t)
,\nonumber\\
&&
\hat\Phi(0,k)= \hat\Psi(k).
\end{aligned}$$ Here $$\label{Omk}
{\Omega}_{\epsilon}(k):=\left[
\begin{array}{cc}
-{\gamma}_0{\epsilon}R(k)-i{\omega}(k)&{\gamma}_0{\epsilon}R(k)\\
{\gamma}_0{\epsilon}R(k)&-{\gamma}_0{\epsilon}R(k)+i{\omega}(k)
\end{array}\right]={\Omega}_0(k)-{\gamma}_0{\epsilon}R(k){\bf D}.$$ and $${\frak f}:=\left[
\begin{array}{r}
1\\
-1
\end{array}\right],\quad {\bf D}:={\frak f}^T\otimes {\frak f}=\left[
\begin{array}{rr}
1&-1\\
-1&1
\end{array}\right].$$ The momentum at $x=0$ equals $$\label{p0}
{p}_0(t)
:=\frac{1}{2i}\int_{{{\mathbb T}}}\hat\Phi(t,k)\cdot {\frak f}dk=\frac{1}{2i}\int_{{{\mathbb T}}}\left[\hat\phi (t,k)-\left(\hat\phi (t,-k)\right)^\star\right]dk.$$ The eigenvalues of the matrix ${\Omega}_{\epsilon}(k)$ equal $
{\lambda}_\pm(k)=-{\gamma}_0{\epsilon}R(k)\pm i{\omega}_{\epsilon}(k),
$ where $$\label{030904-19}
\beta(k)=\frac{{\gamma}_0 R(k)}{{\omega}(k)},\qquad{\omega}_{\epsilon}(k):={\omega}(k) \sqrt{1-({\epsilon}\beta(k))^2}.$$ Note that $
{\lambda}_+^\star={\lambda}_-.
$
Solution of (\[basic:sde:2av\]) {#solution-of-basicsde2av .unnumbered}
-------------------------------
By the Duhamel formula, from we get $$\label{022503-19}
\hat\Phi(t,k)=e_{{\Omega}_{\epsilon}}(k,t) \hat\Psi(k) - i\gamma_1 \int_0^t e_{{\Omega}_{\epsilon}}(k,t-s){\frak f}{p}_0(s)ds .$$ Here $$\begin{aligned}
\label{011902-19ac}
&e_{{\Omega}_{\epsilon}}(k,t):=\exp\left\{{\Omega}_{\epsilon}(k)t\right\}=\left[
\begin{array}{cc}
e_{{\Omega}_{\epsilon}}^{1,1}(k,t)&e_{{\Omega}_{\epsilon}}^{1,2}(k,t)\\
e_{{\Omega}_{\epsilon}}^{1,2}(k,t)&[e_{{\Omega}_{\epsilon}}^{1,1}]^\star(k,t)
\end{array}\right]\end{aligned}$$ and $$\begin{aligned}
\label{011902-19a}
&
e_{{\Omega}_{\epsilon}}^{1,1}(k,t):=\dfrac14\left\{\left(1+\sqrt{1-({\epsilon}\beta(k))^2}\right)^2e_-(k,t)-({\epsilon}\beta(k))^2e_+(k,t)\right\}, \nonumber\\
&e_{{\Omega}_{\epsilon}}^{1,2}(k,t):=\dfrac{i{\epsilon}\beta(k)}{4} \left(1+\sqrt{1-({\epsilon}\beta(k))^2}\right)(e_-(k,t)-e_+(k,t))
,\\
&
e_\pm(k,t):=e^{{\lambda}_\pm(k)
t}.\nonumber\end{aligned}$$ Note that $e_\pm^\star(k,t)=e_\mp(k,t)$.
Multiplying scalarly both sides of by ${\frak f}$ and integrating over $k$ we conclude that $$\label{042503-19}
p_0(t)+{\gamma}_1J_{\epsilon}\star p_0(t)=p_0^0(t)$$ Here $$\label{null}
p_0^0(t):=\frac{1}{2i}\int_{{{\mathbb T}}}e_{{\Omega}_{\epsilon}}(k,t)\hat\Psi(k) \cdot {\frak f}dk$$ and $$\label{J-eps}
{J}_{\epsilon}(t):=\frac12\int_{{{\mathbb T}}} \exp\left\{{\Omega}_{\epsilon}(k)t\right\}{\frak
f}\cdot {\frak f} dk=\int_{{{\mathbb T}}}j_{\epsilon}(t,k)dk,$$ where $$\label{j}
j_{\epsilon}(t,k):=\frac12\exp\left\{{\Omega}_{\epsilon}(k)t\right\}{\frak f}\cdot
{\frak f}
=e^{-{\epsilon}{\gamma}_0 R(k) t}\left\{\sqrt{1-({\epsilon}\beta(k))^2}\cos\left({\omega}_{\epsilon}(k)t\right)-{\epsilon}\beta(k)\, \sin\left({\omega}_{\epsilon}(k)t\right)\right\}.$$ Taking the Laplace transforms of the both sides of we obtain $$\label{062503-19}
\tilde p_0({\lambda})(1+{\gamma}_1\tilde J_{\epsilon}({\lambda}))=\tilde p_0^0({\lambda}).$$ By a direct calculation one concludes that $$\label{052503-19}
{\rm Re}\, \tilde {J}_{\epsilon}({\lambda})>0,\quad\mbox{for any }{\lambda}\in \mathbb
C_+.$$ Since $J_{\epsilon}(\cdot)$ is real valued, we have $$\tilde {J}_{\epsilon}^\star({\lambda})=\tilde {J}_{\epsilon}({\lambda}^\star)\quad\mbox{for any }{\lambda}\in \mathbb
C_+.$$ From we get $$\label{072503-19}
\tilde p_0({\lambda})=\tilde g_{\epsilon}({\lambda})\tilde p_0^0({\lambda}),$$ with $\tilde g_{\epsilon}({\lambda})$ defined by $$\label{012302-19}
\tilde {g}_{\epsilon}({\lambda}):=\left(1+{\gamma}_1\tilde {J}_{\epsilon}({\lambda})\right)^{-1},\quad {\rm Re}\,{\lambda}>0.$$ Thanks to we obtain $$\label{G}
|\tilde {g}_{\epsilon}({\lambda})|\le 1,\quad {\rm Re}\,{\lambda}>0$$ and, as a result, $$\label{JG}
{\gamma}_1|\tilde {J}_{\epsilon}({\lambda})\tilde {g}_{\epsilon}({\lambda})|\le 2,\quad {\rm Re}\,{\lambda}>0.$$ The following result shows in particular that $\tilde
{g}_{\epsilon}({\epsilon}-i{\omega}(k))$ approximates in some sense $\nu(k)$, as ${\epsilon}\to0+$ (see ).
\[cor010304-19\] Suppose that $K:{{\mathbb T}}\to{{\mathbb R}}_+$ is a uniformly continuous and bounded function satisfying $$\label{lim-is}
\inf_{k\in{{\mathbb T}}}K(k)>0.$$ Then, $$\label{010304-19}
\tilde g_{\epsilon}({\lambda})=\tilde g({\lambda})+{\epsilon}\tilde r_{\epsilon}({\lambda}),\quad
{\lambda}\in\mathbb C_+,$$ where $$\lim_{{\epsilon}\to0+}{\epsilon}^p\int_{{{\mathbb T}}}|\tilde r_{\epsilon}({\epsilon}K(k)-i{\omega}(k))|^pdk=0,$$ for any $p\in[1,+\infty)$.
The proof of Proposition \[cor010304-19\] is shown in Appendix \[appC\].
Define by $g_{\epsilon}(ds)$ the distribution such that $$\label{g-eps}
\tilde {g}_{\epsilon}({\lambda})=\int_0^{+\infty}e^{-{\lambda}t}g_{\epsilon}(ds).$$ From it satisfies $$\label{g-eps1}
g_{\epsilon}(ds)=\delta(ds)-{\gamma}_1 J_{\epsilon}\star g_{\epsilon}(s)ds.$$ The Volterra equation has a unique real-valued solution and ${\gamma}_1 J_{\epsilon}\star g_{\epsilon}(s)$ is a $C^\infty$ smooth function, see e.g. the argument made in Section 3 of [@kors]. The solution of can be then written as follows $$\begin{aligned}
\label{eq:10}
&
\hat\Phi(t,k) =U(t) \hat\Psi(k):=e_{{\Omega}_{\epsilon}}(k,t)\hat\Psi(k) - i\gamma_1 \int_0^t
e_{{\Omega}_{\epsilon}}(k,t-s){\frak f}p_0^0\star g_{\epsilon}(s) ds \\
&
=e_{{\Omega}_{\epsilon}}(k,t) \hat\Psi(k) - \frac{\gamma_1}{2} \int_0^tds\int_0^sg_{\epsilon}(ds_1)\int_{{{\mathbb T}}}
e_{{\Omega}_{\epsilon}}(k,t-s){\bf D}e_{{\Omega}_{\epsilon}}(k,s-s_1) \hat\Psi(\ell) d\ell .\nonumber
\end{aligned}$$
Dynamics of the energy density when $T=0$ {#sec4}
=========================================
Starting with the present section untill Section \[sec10\] we shall assume that the thermostat temperature $T=0$, see . We maintain this assumption untill Section \[sec10\]. Let $$\label{Psit}
\hat\Psi^{({\epsilon})}(t,k)=\left[
\begin{array}{c}
\hat\psi^{({\epsilon})}_+(t,k)\\
\hat\psi^{({\epsilon})}_-(t,k)
\end{array}\right],$$ where $\hat\psi^{({\epsilon})}_+(t,k):=\hat \psi^{({\epsilon})}(t,k)$ and $\hat\psi^{({\epsilon})}_-(t,k):=\left(\hat
\psi^{({\epsilon})}\right)^\star(t,-k)$ (cf ). From we get $$\begin{aligned}
\label{basic:sde:2av1}
&&
d \hat\Psi^{({\epsilon})}(t,k)=\frac{1}{{\epsilon}}{\Omega}_{\epsilon}(k) \hat\Psi^{({\epsilon})}(t,k) dt
+i\sqrt{{\gamma}_0}\int_{{{\mathbb T}}}r(k,k'){\bf
D} \hat\Psi^{({\epsilon})}\left(t,k-k'\right)B(dt,dk')-\frac{i{\gamma}_1}{{\epsilon}}{\frak p}_0(t){\frak
g}dt,\nonumber\\
&&
\Psi^{({\epsilon})}(0,k)= \hat\Psi(k).
\end{aligned}$$
With some abuse of notation we denote by ${\cal A}$ the Banach space of all matrix valued functions obtained by the completion of functions of the form $$\begin{aligned}
\label{bFS}
{\bf F}(y,k)=\left[\begin{array}{ll}
G(y,k)&H(y,k)\\
H^\star(y,k)&G(y,-k)
\end{array}\right],\quad (y,k)\in{{\mathbb R}}\times {{\mathbb T}},\end{aligned}$$ with $C^\infty$ smooth entries satisfying $G$ is real valued and $H$ is even in $k$. The completion is taken in the norm given by the maximum of the ${\cal A}$ norms of the entries, see .
The Wigner distribution, corresponding to the wave function $\psi^{({\epsilon})}(t)$, is a $2\times 2$-matrix tensor ${\bf W}_{\epsilon}(t)$, whose entries are distributions, given by their respective Fourier transforms $$\begin{aligned}
\label{hbw}
&
\widehat {{\bf W}}_{\epsilon}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}\left[\hat\Psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right)\otimes \left(\hat\Psi^{({\epsilon})}\right)^\star\left(t,k-\frac{{\epsilon}\eta}{2}\right)\right]
\\
&
=\left[\begin{array}{ll}
\widehat W_{{\epsilon},+}(t,\eta,k)&\widehat Y_{{\epsilon},+}(t,\eta,k)\\
\widehat Y_{{\epsilon},-}(t,\eta,k)&\widehat W_{{\epsilon},-}(t,\eta,k)
\end{array}\right], \quad (\eta,k)\in{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}},\end{aligned}$$ with $$\begin{aligned}
&
\widehat W_{{\epsilon},+}(t,\eta,k):=\widehat W_{{\epsilon}}(t,\eta,k)=\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\left[\hat\psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right) \left(\hat\psi^{({\epsilon})}\right)^\star\left(t,k-\frac{{\epsilon}\eta}{2}\right)\right],
\\
&
\widehat Y_{{\epsilon},+}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\left[\hat\psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right) \hat\psi^{({\epsilon})}\left(t,-k+\frac{{\epsilon}\eta}{2}\right)\right],
\\
&
\widehat Y_{{\epsilon},-}(t,\eta,k):=\widehat
Y_{{\epsilon},+}^\star(t,-\eta,k),\quad
\widehat W_{{\epsilon},-}(t,\eta,k):=\widehat W_{{\epsilon},+}(t,\eta,-k).\end{aligned}$$ Then ${\bf W}_{\epsilon}(t)$ belongs to $ {\cal A}'$ - the dual to ${\cal A}$ that is made of all distributions ${\bf W}$, whose Fourier transform in the first variable equals $$\begin{aligned}
\label{bW}
\widehat{\bf W}(\eta,k)=\left[\begin{array}{ll}
\widehat W_{+}(\eta,k)&\widehat Y_{+}(\eta,k)\\
\widehat Y_{-}(\eta,k)&\widehat W_{-}(\eta,k)
\end{array}\right],\quad (\eta,k)\in{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}},\end{aligned}$$ whose entries belong to ${\cal A}'$ and satisfy $$\begin{aligned}
&\widehat W_{+}^\star(\eta,k)=\widehat W_{+}(-\eta,k),\quad \widehat
Y_{+}(\eta,k)=\widehat Y_{+}(\eta,-k),\\
&
\widehat W_{-}(\eta,k)=\widehat W_{+}(\eta,-k),\quad \widehat Y_{-}(\eta,k)=\widehat
Y_{+}^\star(-\eta,k).\end{aligned}$$ The duality pairing between ${\cal A}'$ and ${\cal A}$ is determined by the relation $$\begin{aligned}
\label{pairing}
&
\left\langle {\bf F},{\bf
W}\right\rangle:=\int_{{{\mathbb T}}_{\epsilon}\times{{\mathbb T}}}\widehat {\bf
F}(\eta,k)\cdot {\bf
W}(\eta,k)d\eta dk\\
&
=2\int_{{{\mathbb T}}_{\epsilon}\times{{\mathbb T}}}\left\{\widehat F_{+}(\eta,k)\widehat
W_{+}^\star(\eta,k)+{\rm Re}\left(\widehat H_{+}(\eta,k)\widehat
Y_{+}^\star(\eta,k)\right)\right\},\nonumber\end{aligned}$$ wher the scalar product of two matrices is given by $$\label{scalar}
\widehat {\bf
F}\cdot \widehat {\bf
W}=\sum_{\iota=\pm}\left(\widehat {
F}_\iota \widehat {
W}^\star_\iota+\widehat {
H}_\iota \widehat {
Y}^\star_\iota\right).$$ The norm $\|{\bf
W}\|_{{\cal A}'}$ is therefore the sum of the norm of its entries.
Thanks to and we conclude that $$\label{030406-19}
\sup_{{\epsilon}\in(0,1]}\sup_{t\ge0}\|{\bf W}_{{\epsilon}}(t)\|_{{\cal A}'}=:A_*'<+\infty.$$ Therefore $\left({\bf W}_{{\epsilon}}(\cdot)\right)$ is bounded in $L^\infty([0,+\infty),{\cal A}')$. In consequence, from any sequence ${\epsilon}_n\to0$ we can choose a subsequence, that we still denote by the same symbol, such that $\left({\bf W}_{{\epsilon}_n}(\cdot)\right)$ is $\vphantom{1}^\star$-weakly convergent in $\left(L^1([0,+\infty),{\cal A})\right)'$.
In what follows we shall also consider the Hilbert spaces ${\cal
L}_{2,{\epsilon}}$ with the scalar product $\langle\cdot,\cdot\rangle_{{\cal L}_{2,{\epsilon}}}$ given by the formula . The respective Hilbert space norms are $$\|{\bf W}\|_{{\cal L}_{2,{\epsilon}}}:=\left\{2\left(\| W_{+}\|_{{\cal
L}_{2,{\epsilon}}}^2+\| Y_{+}\|_{{\cal
L}_{2,{\epsilon}}}^2\right)\right\}^{1/2}. $$ We introduce the following notation, given a function $f:{{\mathbb T}}\to\mathbb C$, we let $$\label{bar-f}
\bar f(k,\eta):=\frac{1}{2}\left[f\left(k+\frac{\eta}{2}\right)+f\left(k-\frac{ \eta}{2}\right)\right]$$ and the difference quotient for the dispersion relation $$\label{d-om}
\delta_{{\epsilon}}{\omega}(k,\eta):=\frac{1}{{\epsilon}}\left[{\omega}\left(k+\frac{{\epsilon}\eta}{2}\right)-{\omega}\left(k-\frac{{\epsilon}\eta}{2}\right)\right].$$ Equipped with this notation we introduce $$\begin{aligned}
\label{bH}
\widehat{\bf H}_{\epsilon}(\eta,k)=\left[\begin{array}{cc}
-i\delta_{{\epsilon}}{\omega}(k;\eta)&-\dfrac{2i}{{\epsilon}} \bar{\omega}(k,{\epsilon}\eta)\\
&\\
\dfrac{2i}{{\epsilon}} \bar{\omega}(k,{\epsilon}\eta)&i\delta_{{\epsilon}}{\omega}(k;\eta)
\end{array}\right],\quad (\eta,k)\in{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}}\end{aligned}$$ and $$\begin{aligned}
\label{021605-19}
&&
{L}_\eta f(k):=2{\cal R}_{ \eta} f(k) - 2\bar R (k,\eta) f(k),\qquad
{L}_{\eta}^\pm f(k):=2{\cal R}_{\eta} f(k) -2R\left(k\pm\frac
\eta2\right) f(k),\nonumber\\
&&
{\cal R}_\eta f(k):=\int_{{{\mathbb T}}} R(k,k',\eta)f(k')dk',
\\
&&
R(k,k',\ell):=\frac12\sum_{\iota=\pm1}r\left(k-\frac{ \ell}{2},k-\iota
k'\right)r\left(k+\frac{ \ell}{2},k-\iota k'\right),\quad
k,k'\in{{\mathbb T}},\,\ell\in 2{{\mathbb T}}. \nonumber\end{aligned}$$ We denote by ${\frak
L}_{{\epsilon}\eta}$, ${\frak H}_{\epsilon}$, ${\frak T}_{\epsilon}$ the operators, acting on ${\cal L}_{2,{\epsilon}}$, defined by $$\label{011906-19}
\widehat{{\frak
L}_{{\epsilon}\eta}{\bf W}}=\widehat{\frak
L}_{{\epsilon}\eta}\widehat{\bf W},\quad \widehat{{\frak
H}_{{\epsilon}}{\bf W}}=\widehat{\frak
H}_{{\epsilon}}\widehat{\bf W},\quad \widehat{{\frak
T}_{{\epsilon}}{\bf W}}=\widehat{\frak
T}_{{\epsilon}}\widehat{\bf W}.$$ Here $\widehat{\bf W}$ is the Fourier transform of ${\bf W}\in {\cal L}_{2,{\epsilon}}$, given by . Operator $\hat{\frak H}_{\epsilon}$, acting on $L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})$, is given by $$\begin{aligned}
\label{fH}
\widehat{\frak H}_{\epsilon}\widehat {{\bf W}}(\eta,k):=\widehat{\bf H}_{\epsilon}(\eta,k) \circ\widehat {{\bf W}}(\eta,k)=\left[\begin{array}{cc}
-i\delta_{{\epsilon}}{\omega}(k;\eta)\widehat W_{+}(\eta,k)&-\dfrac{2i}{{\epsilon}} \bar{\omega}(k,{\epsilon}\eta)\widehat Y_{+}(\eta,k)\\
&\\
\dfrac{2i}{{\epsilon}} \bar{\omega}(k,{\epsilon}\eta)\widehat Y_{-}(\eta,k)&i\delta_{{\epsilon}}{\omega}(k;\eta)\widehat W_{-}(\eta,k)
\end{array}\right],\end{aligned}$$ with $\circ$ denoting the Hadamard’s product of $2\times 2$ matrices. Moreover, $\widehat{\frak
L}_{{\epsilon}\eta}$ and $\widehat{\frak
T}_{{\epsilon}}$ act on $L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})$ via the formulas $$\begin{aligned}
\label{LHT}
\widehat{\frak
L}_{{\epsilon}\eta}\widehat {{\bf W}}(\eta,k):=\left[\begin{array}{cc}
\widehat W'_{+}(\eta,k)&\widehat Y'_{+}(\eta,k)\\
\widehat Y'_{-}(\eta,k)&\widehat W'_{-}(\eta,k)
\end{array}\right],\qquad \widehat{\frak
T}_{{\epsilon}}\widehat {{\bf W}}(\eta,k):=\left[\begin{array}{cc}
\widehat W''_{+}(\eta,k)&\widehat Y''_{+}(\eta,k)\\
\widehat Y''_{-}(\eta,k)&\widehat W''_{-}(\eta,k)
\end{array}\right],\end{aligned}$$ with $$\begin{aligned}
\label{011605-19z}
&&\widehat W_{\pm}'(\eta,k)={L}_{{\epsilon}\eta}\widehat W_{\pm} (\eta,k)-\frac{1}{2}\sum_{\iota=\pm}{L}^{\pm}_{\iota{\epsilon}\eta}\widehat Y_{-\iota}(\eta,k),
\\
&&
\widehat Y_{\pm}'(\eta,k)={L}_{{\epsilon}\eta}\widehat Y_{\pm}(\eta,k)
+ {\cal R}_{{\epsilon}\eta}(\widehat
Y_{\mp}-\widehat Y_{\pm} )(\eta,k)
-\frac{1}{2}\sum_{\iota=\pm} {L}^\pm_{\iota{\epsilon}\eta}
\widehat W_{-\iota}(\eta,k)
\nonumber \end{aligned}$$ and $$\begin{aligned}
\label{011605-19z}
&&\widehat W_{\pm}''(\eta,k)=
\frac{1}{2{\epsilon}}\int_{{{\mathbb T}}}\left[\widehat Y_{\pm}\left(\eta-\frac{2k'}{{\epsilon}},k+k'\right)
+\widehat Y_{\mp}\left(\eta+\frac{2k'}{{\epsilon}},k+k'\right)
\right. \nonumber\\
&&
\left.-\widehat W_{\pm}\left(\eta-\frac{2k'}{{\epsilon}},k+k'\right)-\widehat W_{\pm}\left(\eta+\frac{2k'}{{\epsilon}},k+k'\right)\right]dk',
\\
&&
\widehat Y_{\pm}''(\eta,k)=-\frac{1}{2{\epsilon}}\int_{{{\mathbb T}}}\left[\widehat Y_{\pm}\left(\eta+\frac{2k'}{{\epsilon}},k+k'\right)
+\widehat Y_{\pm}\left(\eta-\frac{2k'}{{\epsilon}},k+k'\right)
\right.\nonumber \\
&&
\left.-\widehat W_{\mp}\left(\eta+\frac{2k'}{{\epsilon}},k+k'\right)-\widehat W_{\pm}\left(\eta-\frac{2k'}{{\epsilon}},k+k'\right)\right]dk'.
\nonumber \end{aligned}$$ Using we obtain the following system of equations for the evolution of the tensor $\widehat {{\bf W}}_{\epsilon}(t,\eta,k)$: $$\label{011806-19}
\frac{d}{dt}\widehat {{\bf W}}_{\epsilon}(t,\eta,k)=\left({\gamma}_0\widehat{\frak
L}_{{\epsilon}\eta}+\widehat{\frak H}_{\epsilon}+{\gamma}_1\widehat{\frak
T}_{\epsilon}\right)\widehat {{\bf W}}_{\epsilon}(t,\eta,k).$$ The respective semigroups on ${\cal
L}_{2,{\epsilon}}$ and $L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})$ shall be denoted by $$\label{frakW}
{\frak W}_{\epsilon}(t):=\exp\left\{\left({\gamma}_0{\frak
L}_{{\epsilon}\eta}+{\frak H}_{\epsilon}+{\gamma}_1{\frak T}_{\epsilon}\right)t\right\},\quad \widehat{\frak W}_{\epsilon}(t):=\exp\left\{\left({\gamma}_0\hat{\frak
L}_{{\epsilon}\eta}+\hat{\frak H}_{\epsilon}+{\gamma}_1\hat{\frak
T}_{\epsilon}\right)t\right\},\quad t\ge0.$$
Let us introduce the Hilbert space norms $$\label{H0eps}
\|{\bf W}\|_{{\cal
H}_{0,{\epsilon}}}:=\left\{\int_{{{\mathbb T}}_{{\epsilon}}\times{{\mathbb T}}}R\left(k-\frac{{\epsilon}\eta}{2}\right)\left|
\widehat W_{+} (\eta,k)
-\widehat
Y_{+} (\eta,k)\right|^2d\eta dk\right\}^{1/2}$$ and $$\label{H2eps}
\|{\bf W}\|_{{\cal
H}_{1,{\epsilon}}}:=\left\{\int_{{{\mathbb T}}}dk\left|\int_{{{\mathbb T}}_{{\epsilon}}}\left[\widehat
W_{+}\left(t,\eta, k-\frac{{\epsilon}\eta}{2}\right)-\widehat
Y_{+}\left(t, \eta, k-\frac{{\epsilon}\eta}{2}\right)\right]d\eta\right|^2\right\}^{1/2}.$$ By a direct calculation one can verify the following identity $$\begin{aligned}
&\|{ {{\bf W}}}_{\epsilon}(t)\|^2_{{\cal
L}_{2,{\epsilon}}}+8{\gamma}_0\int_0^t\|{ {{\bf W}}}_{\epsilon}(s)\|^2_{{\cal
H}_{0,{\epsilon}}}ds+4{\gamma}_1\int_0^t\|{ {{\bf W}}}_{\epsilon}(s)\|^2_{{\cal
H}_{1,{\epsilon}}}ds
=\|{ {{\bf W}}}_{\epsilon}(0)\|^2_{{\cal
L}_{2,{\epsilon}}}\nonumber
\\
\label{energy-balance100}
&
+4{\gamma}_0\int_0^tds\int_{{{\mathbb T}}_{{\epsilon}}\times{{\mathbb T}}^2}R(k,k',{\epsilon}\eta)
{\rm Re}\,\left\{2\widehat
W_{{\epsilon},+} (s,\eta,k) \left(\widehat
W_{{\epsilon},+}\right)^\star (s,\eta,k') \right.
\\
&
\left.+Y_{{\epsilon},+} (s,\eta,k) \left(\widehat
Y_{{\epsilon},+}\right)^\star (s,\eta,k') +Y_{{\epsilon},-} (s,\eta,k) \left(\widehat
Y_{{\epsilon},+}\right)^\star (s,\eta,k')
\right\}d\eta dk dk'\nonumber
\\
&
-8{\gamma}_0\int_0^tds\int_{{{\mathbb T}}_{{\epsilon}}\times{{\mathbb T}}^2}R(k,k',{\epsilon}\eta)
{\rm Re}\,\left\{\widehat
W_{{\epsilon},+} (s,\eta,k) \left(\widehat
Y_{{\epsilon},+}\right)^\star (s,\eta,k')
\right. \nonumber
\\
&
\left.+\widehat Y_{{\epsilon},+} (s,\eta,k) \left(\widehat
W_{{\epsilon},+}\right)^\star (s,\eta,k') \right\}d\eta dk dk',\quad t\ge0,\,{\epsilon}\in(0,1]. \nonumber\end{aligned}$$ In particular we have $$\label{012105-19}
\frac{d}{dt}\|{ {{\bf W}}}_{\epsilon}(t)\|^2_{{\cal
L}_{2,{\epsilon}}}\le 16 R_*\|{ {{\bf W}}}_{\epsilon}(t)\|^2_{{\cal
L}_{2,{\epsilon}}}$$ and $$\label{012105-19a}
2{\gamma}_0\|{ {{\bf W}}}_{\epsilon}(t)\|^2_{{\cal
H}_{0,{\epsilon}}}+{\gamma}_1\|{ {{\bf W}}}_{\epsilon}(t)\|^2_{{\cal
H}_{1,{\epsilon}}}\le 4 R_*\|{ {{\bf W}}}_{\epsilon}(t)\|^2_{{\cal
L}_{2,{\epsilon}}},$$ where $R_*:=\sup_{k,k'\in {{\mathbb T}},\,\ell\in 2{{\mathbb T}}}|R(k,k',\ell)|$. By the Gronwall inequality we conclude the following.
\[prop012105-19\] We have $$\label{022105-19}
\| {\bf W}_{{\epsilon}}(t)\|_{{\cal
L}_{2,{\epsilon}}}\le \|{\bf W}_{{\epsilon}}(0)\|_{{\cal
L}_{2,{\epsilon}}}e^{8 R_* t}$$ and $$\label{022105-19b}
2{\gamma}_0\|{ {{\bf W}}}_{\epsilon}(t)\|^2_{{\cal
H}_{0,{\epsilon}}}+{\gamma}_1\|{ {{\bf W}}}_{\epsilon}(t)\|^2_{{\cal
H}_{1,{\epsilon}}}\le 4 R_*\|{\bf W}_{{\epsilon}}(0)\|_{{\cal
L}_{2,{\epsilon}}}^2e^{16 R_* t},\quad t\ge0.$$
A direct consequence of Proposition \[prop012105-19\] is the following.
\[cor011706-19\] The semigroup $\left({\frak W}_{\epsilon}(t)\right)_{t\ge0}$ is uniformly continuous on ${\cal L}_{2,{\epsilon}}$. Moreover, its norm satisfies $$\label{022105-19a}
\| {\frak W}_{{\epsilon}}(t)\|_{{\cal L}_{2,{\epsilon}}}\le e^{8 R_* t},\quad t\ge0,\quad {\epsilon}\in(0,1].$$
\[cor010406-19\] Suppose that ${\bf W}(\cdot)$ is a $\vphantom{1}^\star$-weak limit of $\left({\bf W}_{{\epsilon}_n}(\cdot)\right)$ in $\left(L^1([0,+\infty),{\cal A})\right)'$ and assume that $$\label{Wstar}
W_*:=\limsup_{{\epsilon}\to0+}\|{\bf W}_{{\epsilon}}(0)\|_{{\cal
L}_{2,{\epsilon}}}<+\infty.$$ Then, ${\bf
W}(\cdot)\in L^\infty_{\rm loc}([0,+\infty);L^2({{\mathbb R}}\times {{\mathbb T}}))$. In fact we have $$\label{040406-19}
\left\|{\bf W}(\cdot) \right\|_{L^\infty([0,\tau];L^2({{\mathbb R}}\times
{{\mathbb T}}))}\le e^{8R_*\tau}W_*,\quad \tau\ge0.$$
[[*Proof.* ]{}]{} Fix some $\tau>0$. Suppose that ${\bf G}\in{\cal A}_c$. Suppose that $A\subset [0,\tau]$ is a Borel measurable set. We know that $$\lim_{n\to+\infty}\int_A du\int_{{{\mathbb R}}\times{{\mathbb T}}}\widehat{\bf G}(\eta,k)\cdot\widehat{\bf W}_{{\epsilon}_n}(u,\eta,k) d\eta dk=\int_ Adu\int_{{{\mathbb R}}\times{{\mathbb T}}}\widehat{\bf G}(\eta,k)\cdot\widehat{\bf W}(u,\eta,k) \hat
d\eta dk.$$ Suppose that $n_0$ is such that $
{{{\rm supp\,}}} \widehat {\bf G}\subset
[-{\epsilon}_n^{-1},{\epsilon}_n^{-1}]\times{{\mathbb T}}$, $n\ge n_0.
$ Then for these $n$ we can write $$\begin{aligned}
&
\left|\int_A du\int_{{{\mathbb R}}\times{{\mathbb T}}}\widehat{\bf
G}(\eta,k)\cdot\widehat{\bf W}_{{\epsilon}_n}(u,\eta,k)
d\eta dk\right|
=\left|\int_A du\int_{{{\mathbb T}}_{{\epsilon}_n}\times{{\mathbb T}}}\widehat{\bf G}(\eta,k)\cdot\widehat{\bf W}_{{\epsilon}_n}(u,\eta,k) d\eta dk\right|\\
&
\le \|{\bf G}\|_{L^2({{\mathbb R}}\times{{\mathbb T}})}\int_A \|{\bf W}_{{\epsilon}_n}(u) \|_{{\cal L}_{2,{\epsilon}_n}}du.\end{aligned}$$ Therefore, by , $$\left|\int_A du\int_{{{\mathbb R}}\times{{\mathbb T}}}\widehat{\bf G}(\eta,k)\cdot\widehat{\bf W}(u,\eta,k) d\eta dk\right|\le e^{8R_*\tau}m_1(A)\| {\bf G}\|_{L^2({{\mathbb R}}\times{{\mathbb T}})}\limsup_{{\epsilon}\to0+}\|{\bf W}_{{\epsilon}}(0)\|_{{\cal L}_{2,{\epsilon}}}.$$ By the density argument, the above inequality holds for all ${\bf G}\in L^2({{\mathbb R}}\times{{\mathbb T}})$. We can further extend the above estimate by taking a simple function of the form ${\bf F}(t):=\sum_{i=1}^{n}{\bf G}_i1_{A_i}(t)$, where ${\bf G}_i\in {\cal A}_2$ and $A_1,\ldots, A_n$ are disjoint, Borel measurable subsets of $[0,\tau]$. The above argument generalizes easily and we conclude that $$\left|\int_ 0^{\tau}\int_{{{\mathbb R}}\times{{\mathbb T}}}\widehat
{\bf F}(u,\eta,k) \cdot\widehat{\bf W}(u,\eta,k) d\eta dk\right|\le e^{8R_*\tau}\|
{\bf F}\|_{L^1([0,\tau]:L^2({{\mathbb R}}\times{{\mathbb T}}))}\limsup_{{\epsilon}\to0+}\|{\bf W}_{{\epsilon}}(0)\|_{{\cal L}_{2,{\epsilon}}}.$$ Since the functions ${\bf F}(\cdot)$ are dense in $L^1([0,\tau]; L^2({{\mathbb R}}\times{{\mathbb T}}))$ we conclude the proof of the corollary. Estimate is a consequence of the results of Section IV.1 of [@diestel]. [$\Box$ ]{}
Fourier-Wigner functions for the deterministic dynamics
=======================================================
Throughout the present section we shall assume that $T=0$.
Dynamics of the Wigner distributions for the solution of (\[012503-19\]) {#sec5.1}
------------------------------------------------------------------------
We consider the Wigner tensor ${{\bf W}}^{\rm un}_{{\epsilon}}(t)$, corresponding to the wave function $\hat\Phi^{({\epsilon})}(t,k)=\hat\Phi(t/{\epsilon},k)$, where $\hat\Phi(t,k)$ is given by . Its Fourier transform (in the first variable) is given by (see ) $$\widehat{{\bf W}}^{\rm un}_{{\epsilon}}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}\left[\hat\Phi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right)\otimes \left(\hat\Phi^{({\epsilon})} \right)^\star\left(t,k-\frac{{\epsilon}\eta}{2}\right)\right]
=\left[\begin{array}{ll}
\widehat W^{\rm un}_{{\epsilon},+}(t,\eta,k)&\widehat Y^{\rm un}_{{\epsilon},+}(t,\eta,k)\\
\widehat Y^{\rm un}_{{\epsilon},-}(t,\eta,k)&\widehat W^{\rm un}_{{\epsilon},-}(t,\eta,k)
\end{array}\right],$$ with $$\begin{aligned}
&
\widehat W^{\rm un}_{{\epsilon},+}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}\left[\hat\phi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right) \left(\hat\phi^{({\epsilon})}\right)^\star\left(t,k-\frac{{\epsilon}\eta}{2}\right)\right],
\\
&
\widehat Y^{\rm un}_{{\epsilon},+}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}\left[\hat\phi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right) \hat\phi^{({\epsilon})}\left(t,-k+\frac{{\epsilon}\eta}{2}\right)\right],
\\
&
\widehat Y^{\rm un}_{{\epsilon},-}(t,\eta,k):=\left(\widehat Y^{\rm un}_{{\epsilon},+}(t,-\eta,k)\right)^\star,\quad
\widehat W^{\rm un}_{{\epsilon},-}(t,\eta,k):=\widehat W^{\rm un}_{{\epsilon},+}(t,\eta,-k).\end{aligned}$$
Using we conclude that the dynamics of the tensor ${{\bf W}}^{\rm un}_{{\epsilon}}(t)$ is described by the ${\cal L}_{2,{\epsilon}}$ strongly continuous semigroup $$\label{frakWun}
{\frak W}_{\epsilon}^{\rm un}(t):=\exp\left\{\left({\gamma}_0{\frak
L}_{{\epsilon}\eta}'+{\frak H}_{\epsilon}+{\gamma}_1{\frak T}_{\epsilon}\right)t\right\},\quad t\ge0,$$ where ${\frak H}_{\epsilon}$, ${\frak T}_{\epsilon}$ are defined in , and . On the other hand ${\frak
L}_{{\epsilon}\eta}'$ is given by the respective Fourier transform $$\begin{aligned}
\label{LHT1}
\widehat{\frak
L}_{{\epsilon}\eta}'\widehat {{\bf W}}(\eta,k):=\left[\begin{array}{cc}
\widehat W'_{+}(\eta,k)&\widehat Y'_{+}(\eta,k)\\
\widehat Y'_{-}(\eta,k)&\widehat W'_{-}(\eta,k)
\end{array}\right],\end{aligned}$$ with $$\begin{aligned}
\label{exp-wigner-eqt-1a}
&&\widehat W'_{\pm}(\eta,k):=-2\bar R(k,{\epsilon}\eta)\widehat W^{\rm
un}_{\pm} (\eta,k)+\left\{R\left(k\mp\frac{{\epsilon}\eta}{2}\right)\widehat Y^{\rm un}_{+}
(\eta,k)+R\left(k\pm\frac{{\epsilon}\eta}{2}\right)\widehat Y^{\rm
un}_{-}(\eta,k)\right\}, \nonumber\\
&&
\widehat Y'_{\pm} (\eta,k)=-2\bar R(k,{\epsilon}\eta)\widehat Y _{\pm} (\eta,k)
+ R\left(k\mp\frac{{\epsilon}\eta}{2}\right)\widehat W_{+}(\eta,k)+
R\left(k\pm\frac{{\epsilon}\eta}{2}\right)\widehat
W_{-}(\eta,k).\end{aligned}$$ By a direct calculation we can verify the following.
\[prop011505-19\] The following identity holds $$\begin{aligned}
\label{energy-balance10}
&\|{ {{\bf W}}}_{\epsilon}^{\rm un}(t)\|^2_{{\cal
L}_{2,{\epsilon}}}+8{\gamma}_0\int_0^t\|{ {{\bf W}}}_{\epsilon}^{\rm un}(s)\|^2_{{\cal
H}_{0,{\epsilon}}}ds+4{\gamma}_1\int_0^t\|{ {{\bf W}}}_{\epsilon}^{\rm un}(s)\|^2_{{\cal
H}_{1,{\epsilon}}}ds
=\|{ {{\bf W}}}^{\rm un}_{\epsilon}(0)\|^2_{{\cal
L}_{2,{\epsilon}}},\quad t\ge0,\,{\epsilon}\in(0,1].\end{aligned}$$
A direct consequence of the proposition is the following.
\[cor011906-19\] $\left({\frak W}_{\epsilon}^{\rm un}(t)\right)_{t\ge0}$ forms a uniformly continuous semigroup of contractions on ${\cal L}_{2,{\epsilon}}$ for any ${\epsilon}\in(0,1]$.
Let ${ p}_0^{({\epsilon})}\left(t\right):={ p}_0\left(t/{\epsilon}\right)$ (see ) and $$\begin{aligned}
\label{091505-19}
&
d_{\epsilon}(t,k):=i{\mathbb E}_{\epsilon}\left[
\left(\hat\phi^{({\epsilon})}\right)^\star\left(t,k\right){
p}_0^{({\epsilon})}\left(t\right)\right] .\end{aligned}$$ By a direct calculation we obtain $$\label{021906-19}
\left\|{\bf W}^{\rm un}_{\epsilon}(t)\right\|_{{\cal
H}_{1,{\epsilon}}}=2\|d_{\epsilon}(t)\|_{L^2({{\mathbb T}})},\quad t\ge0,\,{\epsilon}\in(0,1].$$ From it follows directly.
\[cor011505-19\] We have $$\label{1015-5-19}
\int_0^{+\infty}\|d_{\epsilon}(t)\|_{L^2({{\mathbb T}})}^2dt\le \frac{\|{{{\bf W}}}_{\epsilon}(0)\|^2_{{\frak
A}_{2,{\epsilon}}}}{16{\gamma}_1},\quad\,{\epsilon}\in(0,1].$$
Duhamel representation of the energy density dynamics
-----------------------------------------------------
Using the Duhamel formula we can reformulate as follows $$\begin{aligned}
\label{eq:sol1}
&
\hat\Psi^{({\epsilon})}(t,k)= U\left(\frac{t}{{\epsilon}}\right)\hat\Psi(k)
+\sqrt{{\gamma}_0}\int_0^{t} U\left(\frac{t-s}{{\epsilon}}\right)\left(i\int_{{{\mathbb T}}} r(\cdot,k'){\bf
D}\hat\Psi^{({\epsilon})}\left(s,\cdot-k'\right)B(ds,dk')\right).\end{aligned}$$ Here $U(t)$ is given by and . Using to express the Fourier-Wigner tensor ${{\bf W}}_{\epsilon}(t)$ we obtain the following equality $$\label{012904-19}
{{\bf W}}_{\epsilon}(t)={{\bf W}}_{\epsilon}^{\rm un}(t)+ {{\bf W}}_{\epsilon}'(t),$$ where $$\label{012904-19-1}
{{\bf W}}_{\epsilon}^{\rm un}(t)={\frak W}_{\epsilon}^{\rm
un}(t)\left({{\bf W}}_{\epsilon}(0)\right)$$ and $$\widehat {{\bf W}}_{\epsilon}'(t,\eta,k)
=\frac{{\epsilon}{\gamma}_0}{2}\sum_{n}\int_0^{t}{\mathbb E}\left\{\left(U\left(\frac{t-s}{{\epsilon}}\right)\big(\hat\Psi_n^{{\epsilon}}(s)\big)\right)^\star\left(k-\frac{{\epsilon}\eta}{2}\right)\otimes U\left(\frac{t-s}{{\epsilon}}\right)\big(\hat\Psi_n^{{\epsilon}}(s)\big) \left(k+\frac{{\epsilon}\eta}{2}\right)\right\}ds.$$ Here $$\hat\Psi_n^{({\epsilon})}\big(s,k\big)=i\int_{{{\mathbb T}}} r(k,k'){\bf
D}\hat\Psi^{({\epsilon})}\left(s,k-k'\right)e_n(k')dk',$$ where $(e_n)$ is an orthonormal base in $L^2({{\mathbb T}})$. Using we conclude that $$\begin{aligned}
&\widehat {{\bf W}}_{\epsilon}'(t,\eta,k) ={\gamma}_0 \int_0^{t}\widehat {\frak W}_{\epsilon}^{\rm
un}(t-s)\big({\bf V}_{\epsilon}(s)\big)(\eta,k)ds.\end{aligned}$$ Here $\widehat {\frak W}_{\epsilon}^{\rm
un}(t)$ is the Fourier transform of the semigroup and ${\bf V}_{\epsilon}(t)$ is given by its Fourier transform $\widehat {\bf V}_{\epsilon}(t,\eta,k)
:=\widehat{\frak R}_{\epsilon}\widehat {\bf W}_{\epsilon}(t,\eta,k)
$, where $\widehat{\frak R}_{\epsilon}:\hat{\cal L}_{2,{\epsilon}}\to \hat{\cal
L}_{2,{\epsilon}}$ is defined by $$\begin{aligned}
\label{030805-19}
&
\widehat{\frak R}_{\epsilon}\widehat {\bf W}:=\widehat V_{\epsilon}(\eta,k){\bf D},\\
&
\widehat V_{{\epsilon}}(\eta,k)=\int_{{{\mathbb T}}}r\left(k-\frac{{\epsilon}\eta}{2},k-k'\right)
r\left(k+\frac{{\epsilon}\eta}{2},k-k'\right)
\left[\widehat W_{+}(\eta,k')+\widehat W_{-}(\eta,k')-\widehat Y_{+}(\eta,k')-\widehat Y_{-}(\eta,k')\right]dk'.\nonumber\end{aligned}$$ Summarizing, we have shown that $$\label{020805-19}
{{\bf W}}_{\epsilon}(t)={\frak W}_{\epsilon}^{\rm un}(t)({{\bf W}}_{\epsilon}(0))+ {\gamma}_0 \int_0^{t} {\frak W}_{\epsilon}^{\rm
un}(t-s)\big({\frak R}_{\epsilon}{{\bf W}}_{\epsilon}(s)\big)(\eta,k)ds,\quad t\ge0,\,{\epsilon}\in(0,1].$$ Operator ${\frak R}_{\epsilon}$ is given by its Fourier transform, see . A direct calculation yields the following.
\[prop030706-19\] Operators ${\frak R}_{\epsilon}$ can be defined by as bounded operators both on ${\cal L}_{2,{\epsilon}}$ and $L^2({{\mathbb R}}\times{{\mathbb T}})$. They are uniformly bounded in ${\epsilon}\in(0,1]$. More precisely, there exists ${\frak r}_*>0$ such that $$\label{050706-19}
\|{\frak R}_{\epsilon}\|_{{\cal L}_{2,{\epsilon}}}\le {\frak r}_*,\quad \|{\frak
R}_{\epsilon}\|_{L^2({{\mathbb R}}\times{{\mathbb T}})}\le {\frak r}_*,\quad {\epsilon}\in(0,1].$$ In addition, we have $$\label{050702-19}
\lim_{{\epsilon}\to0+}{\frak R}_{\epsilon}{\bf W}={\frak R}{\bf W},\quad\mbox{in }L^2({{\mathbb R}}\times{{\mathbb T}})$$ for any ${\bf W}\in L^2({{\mathbb R}}\times{{\mathbb T}})$. Here (cf ) $$\label{fR}
{\frak R}{\bf W}(y,k)={\bf D}{\cal R}\left(W_{+}(y,\cdot)+
W_{-}(y,\cdot)-Y_{+}(y,\cdot)-
Y_{-}(y,\cdot)\right)(k),\quad (y,k)\in{{\mathbb R}}\times{{\mathbb T}}.$$
Theorem \[main-thm\] is a direct corollary from the following.
\[cor020805-19\] Under the assumptions of Theorem \[main-thm\] for any $G\in L^1\left([0,+\infty);{\cal A}\right)$ we have $$\label{012404-19z}
\lim_{{\epsilon}\to0+}\int_0^{+\infty}dt\int_{{{\mathbb R}}\times{{\mathbb T}}} \widehat{Y}^\star_{{\epsilon},\pm}(t,\eta,k) \widehat G(t,\eta,k)d\eta dk=0$$ and $$\label{012404-19}
\lim_{{\epsilon}\to0+}\int_0^{+\infty}dt\int_{{{\mathbb R}}\times{{\mathbb T}}} \widehat{ W}^\star_{{\epsilon},\pm}(t,\eta,k) \widehat G(t,\eta,k)d\eta dk=
\int_0^{+\infty}dt\int_{{{\mathbb R}}\times{{\mathbb T}}} \widehat{ W}^\star(t,\eta,\pm k) \widehat G(t,\eta,k)d\eta dk,$$ where $
W(t,y,k)
$ is the unique solution of the equation with the initial condition $W_0$.
The proof is presented in Section \[sec5.4\].
Laplace transform {#sec5.3}
-----------------
As we have already mentioned, see , $\left({\bf W}_{{\epsilon}}(\cdot)\right)$ is $\vphantom{1}^\star$-weakly sequentially compact in the dual to $L^1([0,+\infty),{\cal A})$, therefore the proof of Theorem \[cor020805-19\] comes down to showing uniqueness of limiting points, as ${\epsilon}\to0+$. For that purpose it is convenient to work with the Laplace transform $$\begin{aligned}
\label{012906-19}
&
{\bf w}_{{\epsilon}}({\lambda})=\left[\begin{array}{ll}
w_{{\epsilon},+}({\lambda})&y_{{\epsilon},+}({\lambda})\\
y_{{\epsilon},-}({\lambda})&w_{{\epsilon},-}({\lambda})
\end{array}\right]:=\int_0^{+\infty}e^{-{\lambda}t}{\bf W}_{{\epsilon}}(t)dt.\end{aligned}$$ Sometimes, when we wish to highlight the dependence on the initial data, we shall also write $${\bf w}_{{\epsilon}}({\lambda}; {\bf W}_{{\epsilon}}(0))=:\widetilde{\frak W}_{\epsilon}({\lambda})( {\bf W}_{{\epsilon}}(0)).$$
Thanks to the Laplace transform is well defined as an element in ${\cal A}'$ for any ${\rm Re}\,{\lambda}>0$. From Proposition \[prop012105-19\] we conclude the following estimate $$\label{032105-19}
\| \widetilde{\frak W}_{\epsilon}({\lambda})\|_{{\cal
L}_{2,{\epsilon}}}\le {[({\rm Re}\,{\lambda}-8R_*){\rm Re}\,{\lambda}]^{-1/2}},\quad {\rm Re}\,{\lambda}>8R_*.$$ The argument used in the proof of Corollary \[cor010406-19\] shows that $$\begin{aligned}
&
{\bf w}({\lambda})
=\left[\begin{array}{cc}
{w}_{+}({\lambda})&{y}_{+}({\lambda})\\
{y}_{-}({\lambda})&{w}_{-}({\lambda})
\end{array}\right],\end{aligned}$$ - any $\vphantom{1}^\star$-weak limiting point in ${\cal A}'$ of ${\bf
w}_{{\epsilon}}({\lambda})$, as ${\epsilon}\to0+$ - belongs to $L^2({{\mathbb R}}\times{{\mathbb T}})$ and satisfies the estimate $$\label{032105-19a}
\| {\bf w}({\lambda})\|_{L^2({{\mathbb R}}\times{{\mathbb T}})}\le {[({\rm Re}\,{\lambda}-8R_*){\rm Re}\,{\lambda}]^{-1/2}}\limsup_{{\epsilon}\to0+}\| {\bf W}_{\epsilon}(0)\|_{L^2({{\mathbb R}}\times{{\mathbb T}})},\quad {\rm Re}\,{\lambda}>8R_*.$$
Equation leads to the following equation for the Laplace transform $$\label{020805-19a}
{\bf w}_{\epsilon}({\lambda}; {\bf W}_{{\epsilon}}(0))={\bf w}_{\epsilon}^{{\rm
un}}({\lambda}; {\bf W}_{{\epsilon}}(0))+{\gamma}_0 {\bf w}_{\epsilon}^{\rm
un}\left({\lambda}; {\bf v}_{\epsilon}\left({\lambda}\right)\right),\quad {\rm Re}\, {\lambda}>0.$$ Here, $$\label{010701-19}
{\bf w}_{\epsilon}^{{\rm
un}}({\lambda};{\bf W}):= \widetilde{\frak W}_{\epsilon}^{\rm un}({\lambda}) {\bf
W}:=\int_0^{+\infty}e^{-{\lambda}t}{\frak W}_{\epsilon}^{\rm un}(t) {\bf
W}dt,\quad {\bf W}\in {\cal L}_{2,{\epsilon}}$$ and $$\label{020701-19}
{\bf v}_{\epsilon}\left({\lambda}\right):={\frak R}_{\epsilon}{\bf w}_{\epsilon}({\lambda}; {\bf W}_{{\epsilon}}(0)).$$ Let $W_0\in L^2({{\mathbb R}}\times{{\mathbb T}})$. We define $$w^{\rm un}_{+}({\lambda}, W_0)=\widetilde {\frak W}^{\rm un}({\lambda})W_0:=\int_0^{+\infty}e^{-{\lambda}t}{{\frak
W}^{\rm un}_tW_0}(\eta,k)dt.$$ It follows directly from that its Fourier transform satisfies the equality $$\begin{aligned}
\label{020911absz}
& \left(\vphantom{\int_0^1}{\lambda}+2{\gamma}_0 R(k)+
i{\omega}'(k)\eta\right)\widehat w^{\rm un}_{+}({\lambda},\eta,k; W_0)=\widehat W_0(\eta,k)
\\
&
-{\gamma}_1\left\{ (1-p_+(k))\int_{{{\mathbb R}}}\frac{ \widehat W_0(\eta',k)d\eta' dk}{{\lambda}+2{\gamma}_0 R(k)+i{\omega}'(k)\eta'}-p_-(k) \int_{{{\mathbb R}}}\frac{
\widehat W_0(\eta',-k)d\eta' }{ {\lambda}+2{\gamma}_0 R(k)-i{\omega}'(k)\eta'}\right\}.\nonumber\end{aligned}$$ It is clear from the above formula that $w^{\rm un}_{+}({\lambda}, W_0) \in {\cal A}'$, provided that $W_0\in {\cal A}'$. Thanks to formulas and it can also be seen that $w^{\rm un}_{+}({\lambda}, W_0) \in L^2({{\mathbb R}}\times{{\mathbb T}})$, if $W_0 \in L^2({{\mathbb R}}\times{{\mathbb T}})$.
The first step towards the limit identification of $\left({\bf
W}_{{\epsilon}}(\cdot)\right)$ consists in showing the following result.
\[main:thm-un\] Under the assumptions on the initial data made in Theorem \[main-thm\], the family $\left({\bf w}^{\rm
un}_{{\epsilon}}({\lambda}; {\bf W}_{{\epsilon}}(0))\right)$ converges $\vphantom{1}^\star$-weakly to ${\bf w}^{\rm un}({\lambda};W_0)$ of the form $$\begin{aligned}
&
{\bf w}^{\rm un}({\lambda},y,k;W_0)
=\left[\begin{array}{cc}
w^{\rm un}_{+}({\lambda},y,k;W_0)&0\\
0&w^{\rm un}_{+}({\lambda},y,-k;W_0)
\end{array}\right],\quad (y,k)\in {{\mathbb R}}\times{{\mathbb T}},\end{aligned}$$ where the Fourier transform of $w^{\rm un}_{+}({\lambda},y,k;W_0)$ is given by .
The proof of this result follows closely the argument contained in [@kors]. We present its outline in Appendix \[appb\].
The identification of the limit of $\left({\bf W}_{\epsilon}(t)\right)$ is possible thanks to the following result.
\[main:thm2\] Assume the hypotheses of Theorem $\ref{main-thm}$. Furthermore, suppose that ${\bf w}({\lambda})$ is the $\vphantom{1}^\star$-weak limit of $\left({\bf
w}_{{\epsilon}_n}({\lambda})\right)$ in ${\cal A}'$ for some sequence ${\epsilon}_n\to0+$. Then, there exists ${\lambda}_0>0$ such that ${\bf w}({\lambda}) \in L^2({{\mathbb R}}\times{{\mathbb T}})$ for ${\rm Re}\,{\lambda}>{\lambda}_0$ and $$\label{010702-19}
{\bf w}({\lambda},y,k)
=\left[\begin{array}{cc}
{w}_{+}({\lambda},y,k)&0\\
0&{w}_{+}({\lambda},y,-k)
\end{array}\right],\quad (y,k)\in L^2({{\mathbb R}}\times{{\mathbb T}}),$$ where ${w}_{+}({\lambda},y,k)$ satisfies the equation $$\label{integral-bis}
w_+({\lambda})=w^{\rm un}_{+}({\lambda};W_0)+2{\gamma}_0\widetilde{\frak W}^{\rm
un}({\lambda}){\cal R} { w}_+({\lambda}),$$ and ${\cal R}:L^2({{\mathbb R}}\times{{\mathbb T}})\to L^2({{\mathbb R}}\times{{\mathbb T}})$ (cf ) is given by $$\label{cRW}
{\cal R}F(y,k):= \int_{{{\mathbb T}}}R(k,k')
F\left(y,k'\right) dk',\quad (y,k)\in{{\mathbb R}}\times {{\mathbb T}},\quad F\in L^2({{\mathbb R}}\times {{\mathbb T}}).$$
We present the proof of Theorem \[main:thm2\] in Section \[sec5.5\] below.
Proof of Theorem \[cor020805-19\] {#sec5.4}
---------------------------------
Since $\left({\frak W}^{\rm
un}_t\right)_{t\ge0}$ is a semigroup of contractions, see Proposition \[prop011406-19\], we have $$\label{032105-19b}
\|\widetilde{\frak W}^{\rm
un}({\lambda})\|_{L^2({{\mathbb R}}\times{{\mathbb T}})}\le \frac{1}{{\rm
Re}\,{\lambda}},\quad {\rm
Re}\,{\lambda}>0.$$ Thanks to the fact that ${\cal R}$ is bounded, it is straightforward to see that equation has a unique $L^2({{\mathbb R}}\times{{\mathbb T}})$ solution for ${\lambda}$ with a sufficiently large real part. Therefore, the Laplace transform ${w}_{+}({\lambda})$ of any limiting point of $\left(W_{\epsilon}(t)\right)$ is uniquely determined by . This in turn implies the conclusion of the theorem.
Proof of Theorem \[main:thm2\] {#sec5.5}
------------------------------
To avoid using double subscript notation we assume that ${\bf w}({\lambda})=\lim_{{\epsilon}\to0+}{\bf
w}_{{\epsilon}}({\lambda})$. We wish to show that the limit is of the form with ${w}_{+}({\lambda})$ satisfying .
### Proof of (\[010702-19\]) {#sec5.5.1}
We prove that $$\label{031405-19z}
\lim_{{\epsilon}\to0+}\int_{{{\mathbb R}}\times{{\mathbb T}}}\widehat
y_{{\epsilon},\iota}^\star({\lambda},\eta,k)\widehat G(\eta,k)d\eta dk=0,\quad \iota=\pm$$ for any $G\in {\cal A}_c$ (see ) and ${\rm Re}\,{\lambda}>0$. Consider only the case of $\iota=+$, the other one is analogous. Assume that $K>0$ is fixed and ${\epsilon}$ is so small that $$\label{GK}
{\rm supp}\,\hat
G\subset[-K,K]\times{{\mathbb T}}\subset
[{\epsilon}^{-1},{\epsilon}^{-1}]\times{{\mathbb T}}.$$ Let $$\label{chik}
\chi_K(\eta,k):=1_{[-K,K]}(\eta)1_{{{\mathbb T}}}(k).$$ By virtue of estimate we conclude that $\left(\widehat y_{{\epsilon},+} ({\lambda})\chi_K\right)$ is bounded, thus weakly compact in $L^2({{\mathbb R}}\times{{\mathbb T}})$ for a fixed $K>0$. It converges weakly in $L^2({{\mathbb R}}\times{{\mathbb T}})$, due to its $\vphantom{1}^\star$- weak convergence in ${\cal A}'$. Denote the limit, belonging to $L^2({{\mathbb R}}\times{{\mathbb T}})$, by $\widehat y_{+} ({\lambda})\chi_K$.
Let $$\begin{aligned}
\label{fd-eps}
&
{\frak d}_{\epsilon}({\lambda},k):=i\int_0^{+\infty}e^{-{\lambda}t}{\mathbb E}_{\epsilon}\left[
\left(\hat\psi^{({\epsilon})}\right)^\star\left(t,k\right){
\frak p}_0^{({\epsilon})}\left(t\right)\right]dt \\
&
=\frac{1}{2}\int_{{{\mathbb T}}_{{\epsilon}}}\left[\widehat
y_{{\epsilon},+}\left({\lambda}, \eta,
k-\frac{{\epsilon}\eta}{2}\right)-\widehat
w_{{\epsilon},+}\left({\lambda},\eta, k-\frac{{\epsilon}\eta}{2}\right)\right]d\eta .\nonumber\end{aligned}$$ Taking the Laplace transforms of both sides of and multiplying by $\chi_K$ we obtain in particular the equation $$\begin{aligned}
\label{020911ab1z}&
-{\epsilon}{\gamma}_0 R\left(k-\frac{{\epsilon}\eta}{2}\right)\widehat w_{{\epsilon},+}({\lambda})\chi_K +\left(\vphantom{\int_0^1}{\epsilon}{\lambda}+2{\gamma}_0{\epsilon}\bar R(k,{\epsilon}\eta)+ 2i\bar{\omega}(k,
{\epsilon}\eta)\right) \widehat y_{{\epsilon},+}({\lambda}) \chi_K-{\gamma}_0{\epsilon}R\left(k+\frac{{\epsilon}\eta}{2}\right) \widehat w_{{\epsilon},-}({\lambda}) \chi_K
\nonumber\\
&
={\epsilon}\widehat Y_{{\epsilon},+}(\eta,k) \chi_K +\frac{{\epsilon}{\gamma}_1}{2}\left\{{\frak
d}_{\epsilon}^\star\left({\lambda},-k+\frac{{\epsilon}\eta}{2}\right)+{\frak
d}_{\epsilon}^\star\left({\lambda},k+\frac{{\epsilon}\eta}{2}\right)\right\}\chi_K\\
&
+{\gamma}_0{\cal R}_{{\epsilon}\eta}\left\{\vphantom{\int_0^1}\widehat y_{{\epsilon},+}({\lambda})
+\widehat
y_{{\epsilon},-}({\lambda})-
\widehat w_{{\epsilon},+}({\lambda}) -
\widehat w_{{\epsilon},-}({\lambda})\right\}\chi_K.\nonumber\end{aligned}$$ Thanks to estimate , see also , we conclude that $$\label{121505-19a}
\|{\frak d}_{\epsilon}({\lambda})\|_{L^2({{\mathbb T}})}\le \left(\frac{R_*}{ {\gamma}_1{\rm Re}\,{\lambda}({\rm Re}\,{\lambda}-8R_*)}\right)^{1/2}\|{ {{\bf W}}}_{\epsilon}(0)\|_{{\cal
L}_{2,{\epsilon}}},\quad
{\epsilon}\in(0,1],\,{\rm Re}\,{\lambda}>8R_*.$$ From the strong convergence in $L^2({{\mathbb R}}\times{{\mathbb T}})$ of the coefficients of and estimate we conclude that $2i{\omega}(k) \widehat y_{+} ({\lambda})\chi_K=0$ for any ${\lambda}$ such that ${\rm
Re}\,{\lambda}>8R_*$. This in turn implies that $\widehat
y_{+}({\lambda}) =0$ for such ${\lambda}$-s, which by analytic continuation, implies .
### Proof of (\[integral-bis\])
To avoid writing double subscript we maintain the convention to denote a subsequence of $\left({\bf w }_{\epsilon}({\lambda})\right)$ by the same symbol as the entire sequence. From the already proved part of the theorem we know that the limiting element is of the form . We let the test matrix valued function from ${\cal A}_c$ be of the form $${\bf G}(y,k):=\left[\begin{array}{cc}
G(y,k)&0\\
0&G(y,-k)
\end{array}\right],\quad (y,k)\in {{\mathbb R}}\times{{\mathbb T}},$$ with $G$ satisfying . Using the inclusion $[-K,K]\subset {{\mathbb T}}_{\epsilon}$ we can treat ${\bf G}$ as an element of ${\cal L}_{2,{\epsilon}}$. Applying both sides of to this test matrix we obtain the following equality $$\label{020702-19}
\left\langle{\bf w}_{{\epsilon}}({\lambda}),
{\bf
G}\right\rangle_{{\cal L}_{2,{\epsilon}}}=I_{\epsilon}+I\! I_{\epsilon},$$ where $$I_{\epsilon}:=\left\langle{\bf w}_{{\epsilon}}^{\rm un}({\lambda}),{\bf
G}\right\rangle_{{\cal
L}_{2,{\epsilon}}},\quad I\! I_{\epsilon}:={\gamma}_0\left\langle{\bf w}_{{\epsilon}}({\lambda}),
{\frak R}_{{\epsilon}}^\star\left(\widetilde {\frak W}_{{\epsilon}}^{\rm
un}({\lambda})\right)^\star{\bf
G}\right\rangle_{{\cal L}_{2,{\epsilon}}}.$$ Here $ {\frak R}_{{\epsilon}}^\star$ and $\left(\widetilde {\frak
W}_{{\epsilon}}^{\rm un}({\lambda})\right)^\star$ are the adjoints of the respective operators (see and ), in ${\cal
L}_{2,{\epsilon}}$. Invoking Theorem \[main:thm-un\] we conclude that $$\label{030702-19}
\lim_{{\epsilon}\to0+}I_{\epsilon}=2\int_{{{\mathbb R}}\times {{\mathbb T}}}\left({ w}^{\rm un}_+({\lambda},y,k;W_0)\right)^\star{
G}(y,k)dy dk,\quad {\rm Re}\,{\lambda}>0.$$ Concerning the term $ I\! I_{\epsilon}$, we are going to show that there exists ${\lambda}_0>0$ such that $$\label{wish}
\lim_{{\epsilon}\to0+}I\!I_{\epsilon}=4{\gamma}_0\int_{{{\mathbb R}}\times{{\mathbb T}}} w_+({\lambda},k,y){\cal R} \bar{ g}_+({\lambda})(y,k) dydk$$ for ${\rm Re}\,{\lambda}>{\lambda}_0$. Here $$\bar{ g}_+({\lambda},y,k):=\left(\widetilde{\frak W}^{\rm un} ({\lambda})\right)^\star(G)(y,k).$$ Denote by $$\widehat{\bf g}_{\epsilon}({\lambda},\eta,k):=\left[\begin{array}{cc}
\widehat{ g}_{{\epsilon},+}({\lambda},\eta,k)& \widehat{ h}_{{\epsilon},+} ({\lambda},\eta,k)\\
\widehat{ h}_{{\epsilon},-}({\lambda},\eta,k)& \widehat{ g}_{{\epsilon},-} ({\lambda},\eta,k)
\end{array}\right],$$ the Fourier transform of the distribution $
{\bf g}_{\epsilon}({\lambda})=\left(\widetilde{\frak W}^{\rm
un}_{\epsilon}({\lambda})\right)^\star {\bf G}
$ and let $$\label{whF}
\widehat{\bf f}_{\epsilon}({\lambda},\eta,k):=
\widehat{\frak R}_{\epsilon}^\star\widehat{\bf g}_{\epsilon}({\lambda})(\eta,k),\quad {\epsilon}\in(0,1].$$ Note that $$\begin{aligned}
\label{030805-19}
&
\widehat{\bf f}_{\epsilon}({\lambda},\eta,k)=\widehat f_{\epsilon}({\lambda},\eta,k){\bf D},\quad\mbox{with}\nonumber\\
&
\widehat f_{{\epsilon}}({\lambda},\eta,k):=\int_{{{\mathbb T}}}r\left(k'-\frac{{\epsilon}\eta}{2},k'-k\right)
r\left(k'+\frac{{\epsilon}\eta}{2},k'-k\right)\\
&
\times\left[\vphantom{\int_0^1} g_{{\epsilon},+}({\lambda},\eta,k')+ g_{{\epsilon},-}({\lambda},\eta,k')- h_{{\epsilon},+}({\lambda},\eta,k')- h_{{\epsilon},-}({\lambda},\eta,k')\right]dk'.\nonumber\end{aligned}$$
The argument used in the proof of Theorem \[main:thm-un\] (it suffices only to replace the terms containing the dispersion relation by their conjugates) shows that in fact $$\label{060702-19a}
\lim_{{\epsilon}\to0+}\left\langle {\bf g}_{\epsilon}({\lambda}),{\bf
F}\right\rangle= \left\langle \bar{\bf g}({\lambda}), {\bf
F}\right\rangle$$ for any ${\bf F}\in {\cal A}_c$ and ${\rm Re}\,{\lambda}>0$. Here $$\bar {\bf g}({\lambda},y,k):=\left[\begin{array}{cc}
\bar { g}_+({\lambda},y,k)&0\\
0&\bar { g}_+({\lambda},y,-k)
\end{array}\right].$$
Suppose now that $\varphi$ is an arbitrary, bounded and compactly supported measurable function. Then, using , we conclude that $$\label{050703-19}
\lim_{{\epsilon}\to0+}\varphi(\eta)\widehat{\bf f}_{\epsilon}({\lambda},\eta,k)=2
\varphi(\eta){\bf D}{\cal R}\widehat{\bar { g}}_+({\lambda},\eta,k),\quad\mbox{weakly in $L^2({{\mathbb R}}\times{{\mathbb T}})$},$$ where $\widehat{\bar { g}}_+({\lambda})$ is the Fourier transform of ${\bar { g}}_+({\lambda})$ in the first variable.
In order to show , we prove the following two results.
\[lm010606-19\] There exists $C>0$ such that $$\label{111505-19a1}
\left\| \nabla\widehat{\bf f}_{\epsilon}({\lambda})\right\|_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})}\le \frac{C}{ {\rm Re}\,{\lambda}},\quad
{\epsilon}\in(0,1],\,{\rm Re}\,{\lambda}>0.$$ The gradient operator in is in the $\eta$ and $k$ variables.
\[lm011006-19\] For any $\rho>0$ there exists $M>0$ such that $$\label{011006-19}
\limsup_{{\epsilon}\to0+}\int_{[(\eta,k)\in{{\mathbb T}}_{\epsilon}\times{{\mathbb T}},\,|\eta|>M]}\left|\widehat{\bf w}_{{\epsilon}}({\lambda},\eta,k)\cdot
\widehat {\bf
f}_{\epsilon}({\lambda},\eta,k)\right| d\eta dk<\rho$$
Having the above results we can write that $
I\!I_{\epsilon}=I\!I_{{\epsilon},1}+I\!I_{{\epsilon},2},
$ where $$\begin{aligned}
&
I\!I_{{\epsilon},1}:=\int_{{{\mathbb T}}_{{\epsilon}}\times{{\mathbb T}}}\varphi^2(\eta)\widehat{\bf w}_{{\epsilon}}({\lambda},\eta,k)
\cdot
\widehat {\bf
f}_{\epsilon}({\lambda},\eta,k) d\eta dk,\\
&
I\!I_{{\epsilon},2}:=\int_{{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}}}[1-\varphi^2(\eta)]\widehat{\bf w}_{{\epsilon}}({\lambda},\eta,k)
\cdot
\widehat {\bf
f}_{\epsilon}({\lambda},\eta,k) d\eta dk.\end{aligned}$$ Here $\varphi:{{\mathbb R}}\to[0,1]$ is a $C^\infty$ smooth function satisfying $\varphi(\eta)\equiv 1$, $|\eta|\le M$ and $\varphi(\eta)\equiv 0$, $|\eta|\ge 2M$.
Choose an arbitrary $\rho>0$. Using Lemma \[lm011006-19\] we conclude that $M$ can be adjusted in such a way that $$\label{031006-19}
\limsup_{{\epsilon}\to0+}|I\!I_{{\epsilon},2}|<\rho.$$ On the other hand, by virtue of Lemma \[lm010606-19\] the set $\left(\varphi \widehat{\bf
f}_{\epsilon}({\lambda}) \right)$, ${\epsilon}\in(0,1]$ is compact in $L^2({{\mathbb R}}\times{{\mathbb T}})$ in the strong topology. Since it converges also in the weak topology (see ), it has to also converge strongly in $L^2({{\mathbb R}}\times{{\mathbb T}})$. Combining this with the fact that $\left(\varphi\widehat{\bf w}_{{\epsilon}}({\lambda})\right)$ is weakly compact in $L^2({{\mathbb R}}\times{{\mathbb T}})$ we conclude that $$\label{031006-19}
\limsup_{{\epsilon}\to0+}I\!I_{{\epsilon},1}=
4\int_{{{\mathbb R}}\times {{\mathbb T}}}\varphi^2(\eta) \widehat w_{+}({\lambda},\eta,k)d\eta dk\left\{\int_{{{\mathbb T}}}R(k,k')\widehat{\bar g} _{+}^\star({\lambda},\eta,k')dk'\right\}.$$ This ends the proof of . The only items yet to be proven are Lemmas \[lm010606-19\] and \[lm011006-19\].
### Proof of Lemma \[lm010606-19\]
Let ${ \bf G}_{\epsilon}(t):=\left({\frak W}_{\epsilon}^{\rm un}\right)^\star(t){\bf
G}$ and $\widehat{ \bf G}_{\epsilon}(t,\eta,k)$ be its Fourier transform in the first variable. From we conclude that it satisfies $$\label{frakWun1}
\frac{d}{dt}\widehat{ \bf G}_{\epsilon}(t)=\left({\gamma}_0 \widehat{\frak
L}_{{\epsilon}\eta}'+\widehat{\frak H}_{\epsilon}^\star+{\gamma}_1 \widehat{\frak T}_{\epsilon}\right) \widehat{ \bf G}_{\epsilon}(t).$$ One can formulate the respective the energy balance equation, see , $$\begin{aligned}
\label{energy-balance11}
&\|\widehat{ \bf G}_{\epsilon}(t)\|^2_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})}+8{\gamma}_0\int_0^t\|{ \bf G}_{\epsilon}(s)\|^2_{{\cal
H}_{0,{\epsilon}}}ds+4{\gamma}_1\int_0^t\|{ \bf G}_{\epsilon}(s)\|^2_{{\cal
H}_{1,{\epsilon}}}ds
=\|\widehat{ \bf G}\|^2_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})},\quad t\ge0,\,\end{aligned}$$ and ${\epsilon}$ sufficiently small that holds. It allows us to obtain estimates $$\label{060703-19}
\|\widehat{\bf g}_{\epsilon}({\lambda})\|_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})}\le\frac{1}{{\rm Re}\,{\lambda}}\|{\bf
G}\|_{L^2({{\mathbb R}}\times{{\mathbb T}})}$$ for $ {\rm Re}\,{\lambda}>0$ and ${\epsilon}$ sufficiently small, as above.
Thanks to and estimate we conclude that $$\label{111505-19a2}
\left\| \partial_k\widehat{\bf f}_{\epsilon}({\lambda})\right\|_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})}\le \frac{R_*' \|{\bf
G}\|_{L^2({{\mathbb R}}\times{{\mathbb T}})}}{ {\rm Re}\,{\lambda}},\quad
{\epsilon}\in(0,1],\,{\rm Re}\,{\lambda}>0,$$ with $R'_*:=\sup_{k,k'\in{{\mathbb T}},\ell\in
2{{\mathbb T}}}|\partial_{k'}R(k,k',\ell)|$ (cf ).
To estimate the $L^2$-norm of $ \partial_\eta\widehat{\bf
f}_{\epsilon}({\lambda},\eta,k)$ we differentiate in $\eta$ both sides of and obtain $$\label{frakWun1p}
\frac{d}{dt}\widehat{ \bf G}_{{\epsilon},\eta}'(t)=\left({\gamma}_0 \widehat{\frak
L}_{{\epsilon}\eta}'+\widehat{\frak H}_{\epsilon}^\star+{\gamma}_1 \widehat{\frak T}_{\epsilon}\right) \widehat{ \bf G}_{{\epsilon},\eta}'(t)+\left({\gamma}_0\partial_\eta \widehat{\frak
L}_{{\epsilon}\eta}'+\partial_\eta \widehat{\frak H}_{\epsilon}^\star\right) \widehat{ \bf G}_{{\epsilon}}(t).$$ Both here and below $$\widehat{ \bf G}_{{\epsilon},\eta}'(t):= \partial_\eta \widehat{ \bf
G}_{{\epsilon}}(t) =
\left[\begin{array}{cc}
\widehat G_{{\epsilon},\eta,+} (t,\eta,k)&\widehat H_{{\epsilon}, \eta,+} (t,\eta,k)\\
\widehat H_{{\epsilon}, \eta,-} (t,\eta,k)&\widehat G_{{\epsilon}, \eta,-} (t,\eta,k)
\end{array}\right]$$ and $\widehat{ \bf G}_{\eta}':= \partial_\eta \widehat{ \bf
G}$. Let ${ \bf G}_{{\epsilon}}^{(1)}(t)$ denote the inverse Fourier transform of $\widehat{ \bf G}_{{\epsilon},\eta}'(t)$. Analogously to we conclude the following identity $$\begin{aligned}
\label{energy-balance21}
&\|\widehat{ \bf G}_{{\epsilon},\eta}'(t)\|^2_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})}+8{\gamma}_0\int_0^t\|{ \bf G}_{{\epsilon}}^{(1)}(s)\|^2_{{\cal
H}_{0,{\epsilon}}}ds+4{\gamma}_1\int_0^t\|{ \bf G}_{{\epsilon}}^{(1)}(s)\|^2_{{\cal
H}_{1,{\epsilon}}}ds
=\|\widehat{ \bf G}_{\eta}\|^2_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})}\nonumber\\
&
+2 {\rm Im}\,\left\{\int_0^tds \int_{{{\mathbb T}}_{{\epsilon}}\times{{\mathbb T}}} \left\{\left[{\omega}'\left(k+\frac{{\epsilon}\eta}{2}\right)+{\omega}'\left(k-\frac{{\epsilon}\eta}{2}\right)\right]\widehat G^{\rm un}_{{\epsilon},+} (s,\eta,k) \left[\widehat G_{{\epsilon},\eta,+}
(s,\eta,k)\right]^\star
\right. \right.\nonumber\\
&
\left. \left.+\left[{\omega}'\left(k+\frac{{\epsilon}\eta}{2}\right)-{\omega}'\left(k-\frac{{\epsilon}\eta}{2}\right)\right]\widehat H^{\rm
un}_{{\epsilon},+} (s,\eta,k) \left[\widehat H_{{\epsilon},\eta,+}
(s,\eta,k)\right]^\star d\eta dk\vphantom{\int_0^1}\right\}\right\}\\
&
-2{\epsilon}{\gamma}_0 \int_0^tds \int_{{{\mathbb T}}_{{\epsilon}}\times{{\mathbb T}}}
[R'(k+{\epsilon}\eta/2)-R'(k-{\epsilon}\eta/2)]\nonumber\\
&
\times {\rm Re}\, \left\{\widehat
G_{{\epsilon},+} (s,\eta,k) \left[\widehat G^{\rm
un}_{{\epsilon},\eta,+} (s,\eta,k)\right]^\star +\widehat H^{\rm
un}_{{\epsilon},+} (s,\eta,k) \left[\widehat H_{{\epsilon},\eta,+}
(s,\eta,k)\right]^\star\vphantom{\int_0^1}\right\} d\eta dk
\nonumber\end{aligned}$$ $$\begin{aligned}
&
-4{\epsilon}{\gamma}_0 \int_0^tds \int_{{{\mathbb T}}_{{\epsilon}}\times{{\mathbb T}}} R'\left(k-\frac{{\epsilon}\eta}{2}\right) {\rm Re}\,\left\{ \widehat H_{{\epsilon},+}(s,
\eta,k) \left[\widehat G_{{\epsilon},\eta,+}
(s,\eta,k)\right]^\star+\widehat G^{\rm
un}_{{\epsilon},+} (s,\eta,k) \left[\widehat H_{{\epsilon},\eta,+}
(s,\eta,k)\right]^\star\right\}d\eta dk .\nonumber\end{aligned}$$
From and it follows directly that for any $\delta>0$ there exists a constant $C>0$ such that $$\begin{aligned}
\label{energy-balance22}
&\|\widehat {\bf G}_{{\epsilon}, \eta}(t)\|^2_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})}+8{\gamma}_0\int_0^t\|{ \bf G}^{(1)}_{{\epsilon}}(s)\|^2_{{\cal
H}_{0,{\epsilon}}}ds+4{\gamma}_1\int_0^t\|{ \bf G}^{(1)}_{{\epsilon}}(s)\|^2_{{\cal
H}_{1,{\epsilon}}}ds\\
&
\le \|\widehat {\bf
G}_{\eta}\|^2_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})}+\delta\int_0^t
\|\widehat {\bf G}_{{\epsilon}, \eta}(s)\|^2_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})} ds+Ct,\quad t\ge0,\,{\epsilon}\in(0,1],\,t\ge0.\nonumber\end{aligned}$$ Therefore, for any ${\lambda}_0>0$ we can find $C>0$ such that $$\begin{aligned}
\label{energy-balance23}
&\|\partial_\eta\widehat{ \bf f}({\lambda})\|^2_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})}\le \frac{C}{{\rm Re}\,{\lambda}-{\lambda}_0},\quad {\rm Re}\,{\lambda}>{\lambda}_0,\,{\epsilon}\in(0,1] \end{aligned}$$ and the conclusion of Lemma \[lm010606-19\] follows.[$\Box$ ]{}
### Proof of Lemma \[lm011006-19\]
We show that for any $\rho>0$ there exists $M>0$ such that $$\begin{aligned}
\label{051006-19}
\limsup_{{\epsilon}\to0+}{\cal I}_{\epsilon}(M)<\rho,\end{aligned}$$ where $$\begin{aligned}
\label{051006-19a}
&
{\cal I}_{\epsilon}(M):=\int_{[\eta \in{{\mathbb T}}_{\epsilon},\,|\eta|>M]}d\eta
\int_{{{\mathbb T}}^2}dk dk'\left|{\frak s}
\left(k-\frac{{\epsilon}\eta}{2}\right)
{\frak s}
\left(k+\frac{{\epsilon}\eta}{2}\right)
{\frak s}
\left(k'-\frac{{\epsilon}\eta}{2}\right)
{\frak s} \left(k'+\frac{{\epsilon}\eta}{2}\right)
\right.\nonumber\\
&
\times \left. {\frak s}\left(k+k'-{\epsilon}\eta\right){\frak
s}\left(k+k'+{\epsilon}\eta\right)\widehat{ w}_{{\epsilon},+}({\lambda},\eta,k)
\widehat{ g}_{{\epsilon},+}^\star({\lambda},\eta,k')\right|.\end{aligned}$$ The proof in the case of the remaining terms appearing in the expression carries out in a similar fashion. We shall also consider the case of an optical dispersion relation, that is somewhat more involved than the accoustic one, as then the dispersion relation has two critical points.
Recalling well known trigonometric identities we can write $${\frak s}\left(k+k'-{\epsilon}\eta\right){\frak
s}\left(k+k'+{\epsilon}\eta\right)=\sum_{\iota_1,\iota_2=0,1}{\frak s}_{\iota_1}
\left(k'-\frac{{\epsilon}\eta}{2}\right) {\frak s}_{1-\iota_1}
\left(k-\frac{{\epsilon}\eta}{2}\right) {\frak s}_{\iota_2}
\left(k'+\frac{{\epsilon}\eta}{2}\right) {\frak s}_{1-\iota_2}
\left(k+\frac{{\epsilon}\eta}{2}\right).$$ Here ${\frak s}_0(k):={\frak c}(k)$ and ${\frak s}_1(k):={\frak s}(k)$. Correspondingly, expression can be rewritten in the form $\sum_{\iota_1,\iota_2\in\{0,1\}}{\cal
I}_{\iota_1,\iota_2}$. The analysis of each term is similar, so we only deal with $\iota_1=\iota_2=1$. The respective expression is of the form $$\begin{aligned}
\label{051006-19b}
&
{\cal I}_{1,1}=\int_{[\eta \in{{\mathbb T}}_{\epsilon},\,|\eta|>M]}d\eta
\int_{{{\mathbb T}}^2}dk dk'\left|\vphantom{\int_0^1}{\frak s}
\left(k-{\epsilon}\eta/2\right) {\frak c}
\left(k-{\epsilon}\eta/2\right){\frak s}
\left(k+{\epsilon}\eta/2\right) {\frak c}
\left(k+{\epsilon}\eta/2\right) \widehat{ w}_{{\epsilon},+}({\lambda},\eta,k)
\right.\nonumber\\
&
\times \left. {\frak s}^2
\left(k'-\frac{{\epsilon}\eta}{2}\right)
{\frak s} ^2 \left(k'+\frac{{\epsilon}\eta}{2}\right)
\widehat{ g}_{{\epsilon},+}^\star({\lambda},\eta,k')\right|.\end{aligned}$$ We partition the domain of integration in into two sets $T_{1}$ and $T_2$. To $T_1$ belong all those $(\eta,k,k')$, for which either $$\label{010707-19}
\left|k\pm \frac{{\epsilon}\eta}{2}\right|\le
\delta, \quad \mbox{or}\quad 1/2-\delta\le \left|k\pm\frac{{\epsilon}\eta}{2}\right|\le
1/2,$$ while to $T_2$ belong all other $(\eta,k,k')$-s. Parameter $\delta\in(0,1/2)$ is to be chosen later on. We can write then ${\cal I}_{1,1}={\cal I}_{1,1}^{1}+{\cal I}_{1,1}^{2}$, where $
{\cal I}_{1,1}^{i}$ correspond to integration over $T_{i}$, $i=1,2$. We can write $$\label{061106-19}
{\cal I}_{1,1}^{1}\preceq \delta \int_{{{\mathbb T}}_{\epsilon}}d\eta
\int_{{{\mathbb T}}^2}dk dk'\left|\widehat{ w}_{{\epsilon},+}({\lambda},\eta,k)
\widehat{ g}_{{\epsilon},+}^\star({\lambda},\eta,k')\right|\preceq \delta,$$ for ${\epsilon}\in(0,1]$, by virtue of and .
Taking the Laplace transforms of both sides of we obtain in particular that $$\begin{aligned}
\label{010704-19}
& \widehat w_{{\epsilon},+}({\lambda},\eta,k) =\left(\vphantom{\int_0^1}{\lambda}+2{\gamma}_0\bar R(k,{\epsilon}\eta)+ i\delta_{\epsilon}{\omega}(k,
\eta)\right)^{-1}D_{{\epsilon}}({\lambda},\eta,k),\end{aligned}$$ where $$\begin{aligned}
&
D_{{\epsilon}}({\lambda},\eta,k)
:=\widehat W_{{\epsilon},+}(\eta,k) -\frac{{\gamma}_1}{2}\left\{\vphantom{\int_0^1}{\frak d}_{\epsilon}\left({\lambda},k-\frac{{\epsilon}\eta}{2}\right)+{\frak
d}_{\epsilon}^\star\left({\lambda},k+\frac{{\epsilon}\eta}{2}\right)\right\}
\nonumber\\
&
+\frac{{\gamma}_0}{2}\left(\vphantom{\int_0^1}{\cal L}^{+}_{{\epsilon}\eta}\widehat y_{{\epsilon},-}({\lambda}, \eta,k)
+{\cal L}^{+}_{-{\epsilon}\eta}\widehat y_{{\epsilon},+}({\lambda},
\eta,k)\right) -
2{\gamma}_0{\cal R}_{{\epsilon}\eta}\widehat w_{{\epsilon},+}
({\lambda},\eta,k).\nonumber\end{aligned}$$ Hence, $$\begin{aligned}
\label{021106-19}
& {\cal I}_{1,1}^2=\int_{T_2}d\eta
dk dk'\left|\frac{{\frak s}
\left(k-{\epsilon}\eta/2\right) {\frak c}
\left(k-{\epsilon}\eta/2\right){\frak s}
\left(k+{\epsilon}\eta/2\right) {\frak c}
\left(k+{\epsilon}\eta/2\right) d_{{\epsilon}}({\lambda},\eta,k)}{{\lambda}+2{\gamma}_0\bar R(k,{\epsilon}\eta)+i\delta_{{\epsilon}}{\omega}(k,\eta)}
\right.\nonumber\\
&
\times \left.
{\frak s}^2
\left(k'-\frac{{\epsilon}\eta}{2}\right)
{\frak s}^2 \left(k'+\frac{{\epsilon}\eta}{2}\right) \widehat{ g}_{{\epsilon},+}^\star({\lambda},\eta,k')\vphantom{\int_0^1}\right|.\end{aligned}$$ Thanks to and there exists ${\lambda}_0$ such that $$\label{011106-19}
d_{*}:=\sup_{{\epsilon}\in
(0,1],\,\eta\in{{\mathbb T}}_{{\epsilon}}}\|D_{{\epsilon}}({\lambda},\eta,\cdot)\|_{L^2({{\mathbb T}})}<+\infty,\quad
\mbox{for }{\rm Re}\,{\lambda}>{\lambda}_0.$$ Since ${\omega}'(k)\not=0$, except for $k=0,1/2$ we can find $c_*(\delta)>0$ such that $$|{\lambda}+2{\gamma}_0\bar R(k,{\epsilon}\eta)+i\delta_{{\epsilon}}{\omega}(k,\eta)|\ge
{\lambda}+c_*(\delta)|\eta|\quad \mbox{ for $(\eta,k)$ such that
\eqref{010707-19} does not hold}.$$ Therefore $$\begin{aligned}
\label{021106-19a}
& {\cal I}_{1,1}^2\le \int_{[\eta \in{{\mathbb T}}_{\epsilon},\,|\eta|>M]}d\eta
\int_{{{\mathbb T}}^2}\frac{|D_{{\epsilon}}({\lambda},\eta,k) \widehat{
g}_{{\epsilon},+}^\star({\lambda},\eta,k')|}{{\lambda}+c_*(\delta)|\eta|}dk dk'\\
&
\le \left\{\int_{[|\eta|>M]}\frac{d\eta }{({\lambda}+c_*(\delta)|\eta|)^2}
\sup_{{\epsilon}\in(0,1],\eta'\in{{\mathbb T}}_{\epsilon}}\int_{{{\mathbb T}}}|D_{{\epsilon}}({\lambda},\eta',k)|^2dk\right\}^{1/2}
\|{
g}_{{\epsilon},+}({\lambda})\|_{{\cal L}_{2,{\epsilon}}}.\nonumber\end{aligned}$$ In light of for any $\rho>0$ we can choose a sufficiently large $M$ so that $\limsup_{{\epsilon}\to0+}{\cal I}_{1,1}^2\le{\rho}/{2}$. Adjusting suitably $\delta>0$, cf , we have also $\limsup_{{\epsilon}\to0+}{\cal I}_{1,1}^1\le{\rho}/{2}$. Combining these two estimates we conclude that there exists ${\lambda}_0$ such that for any $\rho>0$ we can find $M>0$ for which $$\limsup_{{\epsilon}\to0+}{\cal I}_{1,1}<\rho\quad \mbox{for all }{\rm Re}\,{\lambda}>{\lambda}_0$$ and the conclusion of the lemma follows.[$\Box$ ]{}
The case of arbitrary thermostat temperature $T$ {#sec10}
================================================
Setting the thermostat temperature at $T$ leads to the following dynamics of the Wigner functions (cf ) $$\label{011806-19a}
\frac{d}{dt}\widehat {{\bf W}}_{\epsilon}(t,\eta,k)=\left({\gamma}_0\widehat{\frak
L}_{{\epsilon}\eta}+\widehat{\frak H}_{\epsilon}+{\gamma}_1\widehat{\frak
T}_{\epsilon}\right)\widehat {{\bf W}}_{\epsilon}(t,\eta,k) +\frac{{\gamma}_1T}{{\epsilon}}{\bf D}.$$
Suppose that $\chi\in C^\infty_c({{\mathbb R}})$ is an arbitrary real valued, even function satisfying . Then $\widehat \chi\in {\cal S}({{\mathbb R}})$ and let $\widehat \chi_{\epsilon}\in C^\infty({{\mathbb T}}_{{\epsilon}})$ be given by $$\widehat \chi_{\epsilon}(\eta):=\sum_{n\in\mathbb Z}\widehat
\chi\left(\eta+\frac{2n}{{\epsilon}}\right),\quad \eta\in{{\mathbb T}}_{{\epsilon}}.$$ Note that $$\int_{{{\mathbb T}}_{{\epsilon}}}\widehat \chi_{\epsilon}(\eta)d\eta=\int_{{{\mathbb R}}}\widehat \chi(n)d\eta=\chi(0)=1.$$ Define $$\widehat {\bf V}_{\epsilon}(t,\eta,k)=\left[\begin{array}{cc}
\widehat V_{{\epsilon},+}(t,\eta,k)&\widehat U_{{\epsilon},+}(t,\eta,k)\\
\widehat U_{{\epsilon},-}(t,\eta,k)&\widehat V_{{\epsilon},-}(t,\eta,k)
\end{array}\right]:=\widehat {\bf W}_{\epsilon}(t,\eta,k)-T \widehat
\chi_{\epsilon}(\eta){\bf I}_{2},$$ where ${\bf I}_2$ is the $2\times 2$ identity matrix, i.e. $$\widehat V_{{\epsilon},\pm}(t,\eta,k):=\widehat W_{{\epsilon},\pm}(t,\eta,k)-T \widehat
\chi_{\epsilon}(\eta),\quad \widehat U_{{\epsilon},\pm}(t,\eta,k):=\widehat Y_{{\epsilon},\pm}(t,\eta,k),\quad t\ge0,\,(\eta,k)\in{{\mathbb T}}_{{\epsilon}/2}\times{{\mathbb T}}.$$
It satisfies $$\label{011806-19b}
\frac{d}{dt}\widehat {\bf V}_{\epsilon}(t,\eta,k)=\left({\gamma}_0\widehat{\frak
L}_{{\epsilon}\eta}+\widehat{\frak H}_{\epsilon}+{\gamma}_1\widehat{\frak
T}_{\epsilon}\right)\widehat {\bf V}_{\epsilon}(t,\eta,k)+\widehat {\bf F}_{\epsilon}(\eta,k),$$ where $$\widehat {\bf F}_{\epsilon}(\eta,k):=-i\delta_{{\epsilon}}{\omega}(k;\eta)T \widehat
\chi_{\epsilon}(\eta){\bf J}_2$$ and $${\bf J}_2:=
\left[\begin{array}{cc}
1&0\\
0&-1
\end{array}\right].$$ The solution can be then written in the form, cf , $$\label{111206-19}
{\bf V}_{\epsilon}(t)={\frak W}_{\epsilon}(t){\bf V}_{\epsilon}(0)+\int_0^t {\frak
W}_{\epsilon}(s){\bf F}_{\epsilon}ds.$$
Using the already proved part of Theorem \[main-thm\] for $T=0$, we conclude that for any ${\bf
G}\in L^1([0,+\infty),{\cal A})$ we have $$\label{121206-19}
\lim_{{\epsilon}\to0+}\int_0^{+\infty}\langle {\bf V}_{\epsilon}(t),{\bf G}(t)\rangle dt=\int_0^{+\infty}\langle {\bf V}(t),{\bf G}(t)\rangle dt,$$ where $${\bf V}(t,y,k)=
\left[\begin{array}{cc}
V_+(t,y,k)&0\\
0&V_+(t,y,-k)
\end{array}\right]$$ and $$\label{111206-19a}
V_+(t,y,k):={\frak W}(t){ V}_{0,+}(y,k)-\int_0^t {\frak
W}(s)F(y,k)ds.$$ Here $
{ F}$ is given by and ${ V}_{0,+}(y,k):=W_{0,+}(y,k)-T\chi(y)$. This ends the proof of Theorem \[main-thm\] for an arbitrary $T\ge0$. [$\Box$ ]{}
Proof of Proposition \[prop011406-19\] {#appa}
======================================
Using formula we can see that ${\frak W}^{\rm un}_t({\cal
C}_0')\subset {\cal C}_0'$, $t\ge0$. In addition (see Section \[sec2.6.2\]) $\left({\frak W}^{\rm un}_t\right)_{t\ge0}$, given by , is a $C_0$-semigroup of contractions on $L^2({{\mathbb R}}\times{{\mathbb T}})$. According to Section A, of the Appendix of [@koran] the semigroup $\left({\frak W}_t\right)_{t\ge0}$ is defined by the Duhamel series that corresponds to the equation . Since ${\cal R}$ is a bounded operator on $L^2({{\mathbb R}}\times{{\mathbb T}})$ and $\left({\frak W}^{\rm
un}_t\right) _{t\ge0}$ is a semigroups of contractions, the semigroup defined by the series is a $C_0$-semigroup of bounded operators on $L^2({{\mathbb R}}\times{{\mathbb T}})$. From here we conclude also that ${\cal
C}_0'$ has to be invariant under $\left({\frak W}_t\right)_{t\ge0}$. For $W_0\in {\cal C}_0'$ we conclude by a direct calculation that $W(t,y,k):={\frak W}_t(W_0)$ satisfies the following identity $$\begin{aligned}
\label{eq:14a}
& \frac12 \frac{d}{dt} \|W(t)\|_{L^2({{\mathbb R}}\times{{\mathbb T}})}^2 = -\gamma_0\int_{{{\mathbb R}}\times{{\mathbb T}}^2} R(k,k')\left[ W(t,y,k) -W(t,y,k')\right]^2 dydkdk'\\
&
- \frac{1}{2} \int_{{{\mathbb T}}} \;
\bar\omega'(k) \left[W(t,0^-, k)^2-W(t,0^+, k)^2
\right] dk ,\quad t\ge0.\nonumber\end{aligned}$$ Taking into account and we obtain $$\begin{aligned}
\label{010709-19}
&\int_{{{\mathbb T}}} \; \bar\omega'(k) \left\{ [W(t,0^-, k)]^2-[W(t,0^+, k)]^2 \right\} dk
\\
&
=\int_{{{\mathbb T}}_+} \; \bar\omega'(k) \left\{ [W(t,0^-, k)]^2-\left[p_-(k) W(t,0^+, -k)+p_+(k) W(t,0^-,k)\right]^2 \right\} dk\nonumber\\
&
+\int_{{{\mathbb T}}_-} \; \bar\omega'(k) \left\{ \left[p_-(k) W(t,0^-,-k) + p_+(k) W(t,0^+, k)\right]^2-[W(t,0^+, k)]^2 \right\} dk.\nonumber\end{aligned}$$ After straightforward calculations (recall that coefficients $p_\pm(k)
$ are even, while $\bar\omega'(k)$ is odd) we conclude that the right hand side equals $$\begin{aligned}
\int_{{{\mathbb T}}_+} \; \bar\omega'(k) & \left\{ \left(W(t,0^-, k)^2+ W(t,0^+, -k)^2\right)
\left(1-p_+^2(k)-p_-^2(k)\right)\right. \\
&
\left.-
4p_-(k)p_+(k)W(t,0^+, -k)W(t,0^-,k)\right\} dk.\end{aligned}$$ Since $p_+(k)+p_-(k)\le 1$ we have $1-p_+^2(k)-p_-^2(k)\ge 0$. In addition, $$\begin{aligned}
&
{\rm det}
\left[
\begin{array}{ll}
1-p_+^2(k)-p_-^2(k)&-2p_-(k)p_+(k)\\
&\\
-2p_-(k)p_+(k)&1-p_+^2(k)-p_-^2(k)
\end{array}
\right]
\\
&
=\left[1-(p_+(k)+p_-(k))^2\right]\left[1-(p_+(k)-p_-(k))^2\right]\ge0.
\end{aligned}$$ Using we conclude that the quadratic form $$\label{020709-19}
(x,y)\mapsto \left(1-p_+^2(k)-p_-^2(k)\right)(x^2+y^2)-4p_-(k)p_+(k)xy$$ is non-negative definite (since $p_+(k)+p_-(k)\le1$). Hence, in particular $$\frac{d}{dt}
\|W(t)\|_{L^2({{\mathbb R}}\times{{\mathbb T}})}^2\le0,\quad t\ge0,$$ which in turn proves that $\left({\frak W}_t\right)_{t\ge0}$ is a semigroup of contractions.[$\Box$ ]{}
Outline of the proof of Theorem \[main:thm-un\] {#appb}
===============================================
The proof of Theorem \[main:thm-un\], for the most part, follows closely the argument contained in [@kors]. We shall present here its outline, invoking the relevant parts of [@kors] and focus on the necessary modifications. We start with the following.
\[thm021405-19\] Suppose that the initial data satisfies the assumptions of Theorem $\ref{main-thm}$. Then, $$\label{031405-19}
\lim_{{\epsilon}\to0+}\int_{{{\mathbb R}}\times{{\mathbb T}}}\widehat y^{\rm
un}_{{\epsilon},\iota}({\lambda},\eta,k)\widehat G^\star(\eta,k)d\eta dk=0,\quad \iota=\pm$$ for any ${\rm Re}\,{\lambda}>0$ and $G\in {\cal A}$.
The proof of the proposition follows the argument presented in Section \[sec5.5.1\], with the simplification consisting in the fact that the equation of $\widehat y^{\rm
un}_{{\epsilon},+}({\lambda},\eta,k)$ corresponding to does not contain the scattering terms involving the operator ${\cal R}_{{\epsilon}\eta}$.
Next, we show that $$\label{011605-19}
\lim_{{\epsilon}\to0+}\int_{{{\mathbb R}}\times{{\mathbb T}}}\widehat w^{\rm
un}_{{\epsilon},+}({\lambda},\eta,k)\widehat G^\star(\eta,k)d\eta dk=\int_{{{\mathbb R}}\times{{\mathbb T}}}\widehat w^{\rm
un}_{+}({\lambda},\eta,k)\widehat G^\star(\eta,k)d\eta dk,$$ where $w^{\rm
un}_{+}({\lambda})$ is given by and $G\in
{\cal A}_c$.
We let ${\rm supp}\,\hat
G\subset[-K,K]\times{{\mathbb T}}$. Taking the Laplace transforms of both sides of we obtain in particular the equation $$\begin{aligned}
\label{010712-19}
& \left(\vphantom{\int_0^1}{\lambda}+2{\gamma}_0\bar R(k,{\epsilon}\eta)+ i\delta_{\epsilon}{\omega}(k,
\eta)\right)\widehat w^{\rm un}_{{\epsilon},+}({\lambda},\eta,k)
+{\gamma}_0R\left(k-\frac{{\epsilon}\eta}{2}\right) \widehat y^{\rm
un}_{{\epsilon},+} ({\lambda}, \eta,k)+{\gamma}_0R\left(k+\frac{{\epsilon}\eta}{2}\right)
\widehat y^{\rm un}_{{\epsilon},-}({\lambda}, \eta,k)
\\
&
=\widehat W_{{\epsilon},+}(0,\eta,k) -\frac{{\gamma}_1}{2}\left\{\vphantom{\int_0^1}{\frak d}_{\epsilon}^{\rm un}\left({\lambda},k-\frac{{\epsilon}\eta}{2}\right)+\left({\frak
d}_{\epsilon}^{\rm un}\right)^\star\left({\lambda},k+\frac{{\epsilon}\eta}{2}\right)\right\}.\nonumber\end{aligned}$$ Here (cf ) $$\begin{aligned}
\label{fdeps}
&{\frak
d}_{{\epsilon}}^{\rm un}({\lambda},k):=\int_0^{+\infty}e^{-{\lambda}t}d_{\epsilon}(t,k)ds=\frac{1}{2}\int_{{{\mathbb T}}_{{\epsilon}}}\left[\widehat
y_{{\epsilon},+}^{\rm un}\left({\lambda}, \eta,
k-\frac{{\epsilon}\eta}{2}\right)-\widehat
w_{{\epsilon},+}^{\rm un}\left({\lambda},\eta, k-\frac{{\epsilon}\eta}{2}\right)\right]d\eta .\end{aligned}$$ Thanks to we conclude $$\label{121505-19}
\|{\frak d}_{\epsilon}^{\rm un}({\lambda})\|_{L^2({{\mathbb T}})}\le \frac{\|{ {{\bf W}}}_{\epsilon}(0)\|_{{\cal
L}_{2,{\epsilon}}}}{(2^5 {\gamma}_1{\rm Re}\,{\lambda})^{1/2}},\quad
{\epsilon}\in(0,1],\,{\rm Re}\,{\lambda}>0.$$
Suppose that ${\epsilon}_n\to0$ as a sequence that corresponds to a $\vphantom{1}^\star$-weakly convergent in ${\cal A}'$ subsequence $\left(w^{\rm un}_{{\epsilon}_n,+}({\lambda},\cdot)\right)$ and $[-K,K]\subset [-{\epsilon}_n^{-1},{\epsilon}_n^{-1}]$, $n\ge1$. By virtue of Proposition \[prop011505-19\] families $\left(\widehat w^{\rm
un}_{{\epsilon}_n,+}({\lambda})\chi_K\right)$ (see ) and $\left({\frak
d}_{{\epsilon}_n}\chi_K\right)$ are $\vphantom{1}^\star$-weakly compact in $L^2({{\mathbb R}}\times{{\mathbb T}})$. Let $\widehat w^{\rm un}_{+}({\lambda},\cdot)$ and ${\frak
d}({\lambda},\cdot)$ be their respective limits. Multiplying equation by $\chi_K$ and letting ${\epsilon}\to0+$ we conclude that $$\begin{aligned}
\label{020911abs}
& \left(\vphantom{\int_0^1}{\lambda}+2{\gamma}_0 R(k)+
i{\omega}'(k)\eta\right)\widehat w^{\rm un}_{+}=\widehat W_0(\eta,k)
-{\gamma}_1{\rm Re}\,{\frak d}^{\rm un}\left({\lambda},k\right),\end{aligned}$$ where $$\int_{{{\mathbb T}}}{\rm Re}\,{\frak
d}^{\rm un}\left({\lambda},k\right) \widehat G^\star(k)dk=\lim_{{\epsilon}\to0+}\int_{{{\mathbb T}}}{\rm
Re}\,{\frak d}^{\rm un}_{\epsilon}\left({\lambda},k\right) \widehat G^\star(k)dk,\quad \widehat G\in C^\infty({{\mathbb T}}).$$ The conclusion of Theorem \[main:thm-un\] follows from.
\[thm010504-19\] For any ${\lambda}>0$ we have $$\label{013110}
{\rm Re}\,{\frak d}({\lambda},k)=(1-p_+(k))\int_{{{\mathbb R}}}\frac{ \widehat W_0(\eta,k)d\eta dk}{{\lambda}+2{\gamma}_0 R(k)+i{\omega}'(k)\eta}-p_-(k) \int_{{{\mathbb R}}}\frac{
\widehat W_0(\eta,-k)d\eta }{ {\lambda}+2{\gamma}_0 R(k)-i{\omega}'(k)\eta},\quad k\in{{\mathbb T}}.$$
Proof of Theorem \[thm010504-19\]
---------------------------------
To simplify somewhat our presentation we shall assume that $$\label{null}
\langle\hat\psi(k)\hat\psi(\ell) \rangle_{\mu_{\epsilon}}=0,\quad k,\ell\in{{\mathbb T}}.$$ The result remains valid without this hypothesis, although the calculations become more extensive. Assumption results in the condition $$\label{nullY}
\widehat Y_{{\epsilon},\pm}(0,\eta,k)=0,\quad (\eta,k)\in{{\mathbb R}}\times {{\mathbb T}}.$$
Using we may write $$\label{d-12}
{\frak d}_{\epsilon}^{\rm un}\left({\lambda},k\right)= {\frak d}_{\epsilon}^1\left({\lambda},k\right)+{\frak d}_{\epsilon}^2\left({\lambda},k\right).$$ Here, ${\frak d}_{\epsilon}^j\left({\lambda},k\right)$, $j=1,2$ are the respective Laplace transforms of $$\label{012703}
I_{\epsilon}(t,k):= i \int_0^t\left\langle { p}^0_0(t-s) {\frak
e}_1\cdot e_{{\Omega}_{\epsilon}}(k,t)\hat\Psi(k) \right\rangle_{\mu_{\epsilon}}g_{\epsilon}(ds),$$ $$I\!I_{\epsilon}(t,k):=
- \gamma_1 \int_0^{t} g_{\epsilon}\left(ds'\right) \int_0^{t} \Theta\left(t-s,k\right)\langle{ p}_0^0(s){ p}_0^0(t-s')\rangle_{\mu_{\epsilon}}ds,$$ Here $e_{{\Omega}_{\epsilon}}(k,t)$ is given by and $$p_0^0(t)=\frac{1}{2i}\int_{{{\mathbb T}}} e_{{\Omega}_{\epsilon}}(k,t)\hat\Psi(k) \cdot {\frak f}dk,\quad
\Theta(t,k) := \int_0^{t}{\frak
e}_1\cdot e_{{\Omega}_{\epsilon}}(k,t-\tau){\frak f} g_{\epsilon}(d\tau) .$$ We introduce $$\label{feb1416}
{\frak L}^{\epsilon}_{\rm scat}({\lambda}):=-{\gamma}_1\int_{{{\mathbb T}}}\hat G^*(k) {\rm Re}\,{\frak
d}_{\epsilon}\left({\lambda},k\right)dk =\sum_{j=1}^2{\frak L}_{{\rm scat},j}^{\epsilon}({\lambda}).$$ with $${\frak L}_{{\rm scat},j}^{{\epsilon}}({\lambda}):=-{\gamma}_1
\int_{{{\mathbb T}}}\hat G^*(k) {\rm Re}\,{\frak d}_{\epsilon}^j\left({\lambda},k\right)dk,
~~j=1,2.$$
### The limit of ${\frak L}_{{\rm scat},1}^{{\epsilon}}({\lambda})$
We will show the following.
\[lem-feb1504\] For any test function $\widehat G\in C^\infty({{\mathbb T}})$ and ${\lambda}>0$ we have $$\label{020811}
\lim_{{\epsilon}\to0+}{\frak L}_{{\rm scat},1}^{\epsilon}({\lambda})=
-{\gamma}_1 \int_{{{\mathbb T}}}{\rm Re} [\nu(k)]\widehat G^*(k)dk
\int_{{{\mathbb R}}}\frac{\widehat W_0(\eta,k)d\eta}{{\lambda}+2{\gamma}_0R(k)+i{\omega}'(k)\eta}
d\eta'.$$
[[*Proof.* ]{}]{}From and we get $$\begin{aligned}
\label{Ibis}
&I_{\epsilon}(t,k)= \frac{1}{2}\left\{ e_{{\Omega}_{\epsilon}}^{1,1}(k,t)\int_0^{t} g_{\epsilon}\left(ds\right) \int_{{{\mathbb T}}}
e_{{\Omega}_{\epsilon}}(\ell,t-s){\frak e}_1\cdot {\frak f}\langle\hat\psi^\star(k)\hat\psi(\ell) \rangle_{\mu_{\epsilon}}
\right.\nonumber\\
&
\left.+e_{{\Omega}_{\epsilon}}^{1,2}(k,t)\int_0^{t} g_{\epsilon}\left(ds\right) \int_{{{\mathbb T}}}
e_{{\Omega}_{\epsilon}}(\ell,t-s){\frak e}_1\cdot {\frak f}\langle\hat\psi(-k)\hat\psi^\star(-\ell) \rangle_{\mu_{\epsilon}}\right\} d\ell.\end{aligned}$$ Using formula we can write $$\begin{aligned}
\label{010705-19}
&
e_{{\Omega}_{\epsilon}}^{1,1}(k,t)=e^{{\lambda}_+(k)
t}+{\epsilon}\sum_{\iota=\pm} r^{1}_{\iota,{\epsilon}}(k)e^{{\lambda}_\iota
t},\quad
e_{{\Omega}_{\epsilon}}^{1,2}(k,t)={\epsilon}\sum_{\iota=\pm} r^{2}_{\iota,{\epsilon}}(k)e^{{\lambda}_\iota
t},\\
&
e_{{\Omega}_{\epsilon}}(\ell,t-s){\frak e}_1\cdot {\frak f}=e^{{\lambda}_-(\ell)
(t-s)}+{\epsilon}\sum_{\iota=\pm} r^{3}_{\iota,{\epsilon}}(\ell)e^{{\lambda}_\iota(\ell)
(t-s)},\nonumber\end{aligned}$$ where $$\label{020304-19}
\sup_{k\in{{\mathbb T}},{\epsilon}\in(0,1]}|r^{j}_{\iota,{\epsilon}}(k)|=r^{j}_{\iota,*}<+\infty,\quad,
\iota\in\{-,+\},\,j=1,2,3.$$ The following result allows us to replace the entries of $e_{{\Omega}_{\epsilon}}(k,t)$ by the leading terms appearing in . Define $$\begin{aligned}
\label{Ibis1}
&\tilde I^{\iota_1,\iota_2}_{\epsilon}(t,k):= {\epsilon}r_{1,{\epsilon}}(k)
e^{{\lambda}_{\iota_1}(k) t}\int_0^{t} g_{\epsilon}\left(ds\right)
\int_{{{\mathbb T}}}r_{2,{\epsilon}}(\ell) e^{{\lambda}_{\iota_2}(\ell)
(t-s)}\langle\hat\psi^\star(k)\hat\psi(\ell)
\rangle_{\mu_{\epsilon}}d\ell\end{aligned}$$ and $$\tilde I^{\iota_1,\iota_2}_{\epsilon}({\lambda},k):=
\int_0^{+\infty}e^{-{\lambda}t}\tilde
I^{\iota_1,\iota_2}_{\epsilon}\left(\frac{t}{{\epsilon}},k\right)dt .$$
\[lm010304-19\] Suppose that holds, $$\label{020304-19a}
\sup_{k\in{{\mathbb T}},{\epsilon}\in(0,1]}|r_{i,{\epsilon}}(k)|=r_{i,*}<+\infty,\quad i=1,2.$$ Then, for any $\iota_1,\iota_2\in\{-,+\}$ and ${\lambda}>0$ we have $$\label{020304-19a}
\Big\|\tilde I^{\iota_1,\iota_2}_{\epsilon}({\lambda})\Big\|_{L^1({{\mathbb T}})}\preceq
{\epsilon}^{1/2}\log{\epsilon}^{-1},\quad {\epsilon}\in(0,1].$$
The proof of this lemma is presented in Section \[C3\]
Using Lemma \[lm010304-19\] we conclude that $$\begin{aligned}
&&\int_{{{\mathbb T}}}{\frak d}_{\epsilon}^1({\lambda},k)
\widehat G^\star(k)dk =\frac{\epsilon}2\int_{{{\mathbb T}}^2}{\epsilon}\langle\hat \psi^\star (k)\hat \psi(\ell)
\rangle_{\mu_{\epsilon}}\widehat G^\star(k)d\ell dk
\\
&&
\times\int_0^{+\infty}\exp\left\{-{\lambda}_-(\ell)s\right\}g_{\epsilon}(ds)
\int_s^{+\infty}e^{-{\lambda}{\epsilon}t}
\exp\left\{[{\lambda}_+(k)+{\lambda}_-(\ell)]t\right\}dt+O({\epsilon})
\\
&&
=\frac12\int_{{{\mathbb T}}^2}{\epsilon}\langle\hat \psi^\star (k)\hat \psi(\ell)
\rangle_{\mu_{\epsilon}} \widehat G^\star(k)\tilde g_{\epsilon}\Big({\epsilon}({\lambda}+{\gamma}_0R(k))-i{\omega}_{\epsilon}(k) \Big)d\ell
dk
\\
&&
\times\left\{{\epsilon}({\lambda}+{\gamma}_0R(k)+{\gamma}_0R(\ell))+i[{\omega}_{\epsilon}(\ell) -{\omega}_{\epsilon}(k) ]\right\}^{-1}
+O({\epsilon})\end{aligned}$$ Changing variables $
k:=k'-{\epsilon}\eta'/2,$ $\ell:=k'+{\epsilon}\eta'/2$ we can write (cf and ) $$\label{050404-19}
\int_{{{\mathbb T}}}{\frak d}_{\epsilon}^1({\lambda},k)
\widehat G^\star(k)dk
={\cal I}_{\epsilon}+O({\epsilon}),$$ where $${\cal I}_{\epsilon}:=\int_{T_{\epsilon}}\widehat
W_{{\epsilon},+}(0,\eta,k)\tilde g_{\epsilon}\Big({\epsilon}({\lambda}+{\gamma}_0R(k-{\epsilon}\eta/2))-i{\omega}_{\epsilon}(k-{\epsilon}\eta/2) \Big) \Delta_{\epsilon}(k,\eta) d\eta
dk.$$ Here (cf and ) $$\begin{aligned}
&\Delta_{\epsilon}(k,\eta):=
\left\{{\lambda}+2{\gamma}_0\bar R(k,{\epsilon}\eta)+i\delta_{\epsilon}{\omega}_{\epsilon}(k;\eta) \right\}^{-1}\widehat G^\star
(k-{\epsilon}\eta/2)\end{aligned}$$ and $T_{\epsilon}$ is the image of ${{\mathbb T}}^2$ under the inverse map $
k':=(\ell+k)/2,$ $\eta':=(\ell-k)/{\epsilon}$.
We claim that in fact $$\label{020705-19}
\lim_{{\epsilon}\to0+}({\cal I}_{\epsilon}-{\cal I}_{\epsilon}')=0,$$ where the definition of the expression ${\cal I}_{\epsilon}'$ differs from ${\cal I}_{\epsilon}$ only by replacing $\tilde g_{\epsilon}$ by $\tilde g$. Changing variable $k\mapsto k-{\epsilon}\eta/2$ we can write $${\cal I}_{\epsilon}-{\cal I}_{\epsilon}'=\int_{T_{\epsilon}'}\widehat
W_{{\epsilon},+}(0,\eta, k+{\epsilon}\eta/2)(\tilde g_{\epsilon}-\tilde g)\Big({\epsilon}({\lambda}+{\gamma}_0R(k))-i{\omega}_{\epsilon}(k) \Big) \Delta_{\epsilon}\left(k+{\epsilon}\eta/2,\eta\right) d\eta
dk.$$ Here $T_{\epsilon}'$ is the impage of $T_{\epsilon}$ under the change of variable. Equality follows from Proposition \[cor010304-19\] and the fact that the expression under the integral in the right hand side is bounded by an integrable function, see .
Next, thanks to Lemma 7.3 of [@kors], we have $$\begin{aligned}
&\lim_{{\epsilon}\to0}\tilde g\Big({\epsilon}({\lambda}+{\gamma}_0R(k-{\epsilon}\eta/2))-i{\omega}_{\epsilon}(k-{\epsilon}\eta/2) \Big) \Delta_{\epsilon}(k,\eta)
=\frac{\nu(k)\widehat
G^\star(k)}{{\lambda}+2{\gamma}_0R(k)+i{\omega}'(k)\eta}\end{aligned}$$ a.e. in $(\eta,k)$. Using bounds and we can argue, via the dominated convergence theorem, as in the proof of Lemma 5.1 of [@kors], that $$\label{020706b}
\lim_{{\epsilon}\to0+}{\cal I}_{\epsilon}'=\int_{{{\mathbb R}}\times {{\mathbb T}}} \frac{\widehat
W_0(\eta,k) \nu(k)
\widehat
G^*(k) d\eta dk}{{\lambda}+2{\gamma}_0R(k)+i{\omega}'(k)\eta}$$ and conclusion of Lemma \[lem-feb1504\] follows.[$\Box$ ]{}
Concerning the second term in the utmost right hand side of we have the following.
\[lem-feb1502\] For any ${\lambda}>0$ and $G\in C^\infty({{\mathbb T}})$ we have $$\label{021511}
\lim_{{\epsilon}\to 0}{\frak L}_{scat,2}^{\epsilon}({\lambda})=
\frac{\gamma_1}{4}\sum_{\iota=\pm}\int_{{{\mathbb T}}}\frac{\hat G^\star(k)\fgeeszett(k)\widehat
W_0(\eta,\iota k)d\eta dk}{{\lambda}+2{\gamma}_0R(k)+\iota i{\omega}'(k)\eta}
.$$
The proof of the lemma follows very closely the proof of Lemma 6.4 of [@kors]. We present its outline in Section \[sec:6\] below.
The limit of ${\frak L}_{{\rm scat}}^{{\epsilon}}({\lambda})$ {#sec:4.3}
-------------------------------------------------------------
Putting together the results of Lemmas \[lem-feb1504\] and \[lem-feb1502\], we see that $$\lim_{{\epsilon}\to 0}{\frak L}_{scat}^{{\epsilon}}({\lambda})=\int_{{{\mathbb T}}} \Big({\cal W}_{tr}(k)+{\cal W}_{ref}(k)\Big)\hat G^*(k)
dk,
\label{feb1506}$$ with the transmission term $$\begin{aligned}
\label{feb1508}
&&{\cal W}_{tr}(k)={\frac}{\gamma_1}{|\bar\omega'(k)|}
\Big\{- {\rm Re}[\nu(k)]+\frac{{\fgeeszett}(k) }{4}\Big\}
\int_{{{\mathbb R}}}\frac{ \widehat W_0(\eta,k)d\eta }{{\lambda}+2{\gamma}_0
R(k)+i{\omega}'(k)\eta}\\
&&
=
(p_+(k)-1)\int_{{{\mathbb R}}}\frac{ \widehat W_0(\eta,k)d\eta }{{\lambda}+2{\gamma}_0 R(k)+i{\omega}'(k)\eta}.\nonumber\end{aligned}$$ We have used (\[feb1402\]) in the last step. The reflection term ${\cal
W}_{ref}(k)$ equals (cf ) $$\begin{aligned}
\label{feb1510}
{\cal W}_{ref}(k)=\frac{\gamma_1{\fgeeszett}(k)}{4|\bar\omega'(k)| }
\int_{{{\mathbb R}}}\frac{
\widehat W_0(\eta,-k)d\eta }{ {\lambda}+2{\gamma}_0 R(k)-i{\omega}'(k)\eta} =
p_-(k) \int_{{{\mathbb R}}}\frac{
\widehat W_0(\eta,-k)d\eta }{ {\lambda}+2{\gamma}_0 R(k)-i{\omega}'(k)\eta}.\end{aligned}$$ Combining the scattering terms in (\[feb1506\])-(\[feb1510\]) we obtain (\[013110\]). Thus, the proof of Theorem \[thm010504-19\] is reduced to showing Lemma \[lem-feb1502\].
Outline of the proof of Lemma \[lem-feb1502\]: the limit of ${\frak L}_{scat,2}^{{\epsilon}}({\lambda})$ {#sec:6}
--------------------------------------------------------------------------------------------------------
Let $$\begin{aligned}
\label{Ibis1a}
&\widetilde{I\!I}^{\iota_1,\iota_2,\iota_3}_{\epsilon}(t,k):= {\epsilon}r_{1,{\epsilon}}(k)
\int_0^{t} ds \int_0^{t} g_{\epsilon}\left(ds'\right)\int_{0}^{s}g_{\epsilon}\left(ds_1\right)
\\
&
\times \int_{{{\mathbb T}}^2}r_{2,{\epsilon}}(\ell) r_{3,{\epsilon}}(\ell') e^{{\lambda}_{\iota_2}(\ell) (s-s_1)} e^{{\lambda}_{\iota_3}(\ell')
(t-s')} e^{{\lambda}_{\iota_1}(k)
(t-s)}\langle\hat\psi^\star(\ell)\hat\psi(\ell')
\rangle_{\mu_{\epsilon}}d\ell d\ell'\nonumber\end{aligned}$$ and $$\label{020504-19a}
\widetilde{
I\!I}^{\iota_1,\iota_2,\iota_3}_{\epsilon}({\lambda},k):=\int_0^{+\infty}e^{-{\lambda}t}\widetilde{
I\!I}^{\iota_1,\iota_2,\iota_3}_{\epsilon}\left(\frac{t}{{\epsilon}},k\right)dt.$$ We start with the following.
\[lm010504-19\] Suppose that condition holds, $$\label{020504-19a}
\sup_{k\in{{\mathbb T}},{\epsilon}\in(0,1]}|r_{i,{\epsilon}}(k)|=r_{i,*}<+\infty,\quad i=1,2,3.$$ Then, for any $\iota_1,\iota_2,\iota_3\in\{-,+\}$ and ${\rm Re}\,{\lambda}>0$ we have $$\label{020304-19aa}
\lim_{{\epsilon}\to0+} \Big\|\widetilde{
I\!I}^{\iota_1,\iota_2,\iota_3}_{\epsilon}({\lambda})\Big\|_{L^1({{\mathbb T}})} =0.$$
The proof of the lemma is presented in Section \[C3.1\] below.
Using Lemma \[lm010504-19\] we can write $$\label{020904-19}
{\frak L}_{scat,2}^{\epsilon}({\lambda})=\bar{\frak L}_{scat,2}^{\epsilon}({\lambda})+o(1),$$ as ${\epsilon}\to0+$, where $$\label{010904-19}
\bar{\frak L}_{scat,2}^{\epsilon}({\lambda}):=-{\gamma}_1\int_{ {{\mathbb T}}}{\rm Re}\,\bar{\frak
d}_{\epsilon}^2\left({\lambda},k\right)\hat
G^\star(k) dk$$ and $$\begin{aligned}
\bar{\frak
d}_{\epsilon}^2\left({\lambda},k\right):= - \gamma_1{\epsilon}\int_0^{+\infty}e^{-{\epsilon}{\lambda}t}dt\left\langle \bar p_0^0\star
g_{\epsilon}\left(t\right) \int_0^{t}
\bar p^0_0 \star g_{\epsilon}(s)ds\right\rangle_{\mu_{\epsilon}}e^{{\lambda}_+(k)
(t-s)}.\end{aligned}$$ Here $$\begin{aligned}
&
\label{eq:1}
\bar p^0_0(t):=\int_{{{\mathbb T}}} e^{-{\epsilon}{\gamma}_0 R(k)t}{\rm Im}\left(\hat\psi(k) e^{-i\omega(k) t}\right) dk.\end{aligned}$$
Since $ \bar p_0^0\star
g_{\epsilon}$ is real valued we have $$\label{021207-19}
{\rm Re}\,\bar{\frak
d}_{\epsilon}^2\left({\lambda},k\right):= - \gamma_1{\epsilon}\int_0^{+\infty}e^{-{\epsilon}{\lambda}t}dt\left\langle \bar p_0^0\star
g_{\epsilon}\left(t\right) \int_0^{t}
\bar p^0_0 \star g_{\epsilon}(s)ds\right\rangle_{\mu_{\epsilon}}e^{-{\gamma}_0{\epsilon}R(k) (t-s)}\cos({\omega}_{\epsilon}(k)(t-s)).$$ After rather lengthy, cut starightforward calculation, see Section \[d-eps2\] below for details, we get $$\label{010104}
2{\rm Re}\,\bar{\frak d}_{\epsilon}^2({\lambda},k)
=R_{\epsilon}({\lambda},k)+\rho_{\epsilon}({\lambda},k),$$ with $$\begin{aligned}
\label{R-eps}
&&
R_{\epsilon}({\lambda},k):=-\frac{\gamma_1 ({\lambda}+2{\gamma}_0{\epsilon}R(k))}{2^3\cdot\pi{\epsilon}^2}\int_{{{\mathbb R}}}\frac{d\xi}{({\lambda}/2+{\gamma}_0R(k))^2+\xi^2} \int_{{{\mathbb T}}^2}d\ell d\ell' {\epsilon}\langle \hat\psi(\ell)\hat\psi^*(\ell')\rangle_{\mu_{\epsilon}}\\
&&
\times \frac{|\tilde g_{\epsilon}\left({\lambda}{\epsilon}/2-i[{\epsilon}\xi+{\omega}_{\epsilon}(k)]\right)|^2 }{{\lambda}/2-i\{\xi+{\epsilon}^{-1}[{\omega}_{\epsilon}(k)-{\omega}_{\epsilon}(\ell)]\}}
\times \frac{1
}{{\lambda}/2+i\{\xi+{\epsilon}^{-1}[{\omega}_{\epsilon}(k)-{\omega}_{\epsilon}(\ell')]\}}\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{rho-e}
&&
\rho_{\epsilon}({\lambda},k):=-\frac{{\epsilon}\gamma_1 ({\lambda}+2{\gamma}_0{\epsilon}R(k))}{2^4\cdot\pi}\int_{{{\mathbb R}}}\frac{d\xi}{({\lambda}/2+{\gamma}_0R(k))^2+\xi^2}
\int_{{{\mathbb T}}^2}d\ell d\ell' \langle \hat\psi(\ell)
\hat\psi^*(\ell')\rangle_{\mu_{\epsilon}}\nonumber
\\
&&
\times
\Bigg\{ \frac{\tilde g\left({\lambda}{\epsilon}/2-i[{\epsilon}\xi+{\omega}_{\epsilon}(k)]\right) }{{\epsilon}{\lambda}/2-i[{\epsilon}\xi+{\omega}_{\epsilon}(k)-{\omega}_{\epsilon}(\ell)]}\nonumber\\
&&\times
\Big\{\frac{\tilde g\left({\lambda}{\epsilon}/2+i[{\epsilon}\xi+{\omega}_{\epsilon}(k)]\right)
}{{\epsilon}{\lambda}/2+i[{\epsilon}\xi+{\omega}_{\epsilon}(k)+{\omega}_{\epsilon}(\ell')]}+\frac{\tilde
g\left({\lambda}{\epsilon}/2+i[{\epsilon}\xi-{\omega}_{\epsilon}(k)])\right)
}{{\epsilon}{\lambda}/2+i\{{\epsilon}\xi-[{\omega}_{\epsilon}(k)+{\omega}_{\epsilon}(\ell')]\}}\Big\} \nonumber\\
&&
+\frac{\tilde
g\left({\lambda}{\epsilon}/2-i[{\epsilon}\xi-{\omega}_{\epsilon}(k)]\right)}{{\epsilon}{\lambda}/2-i[{\epsilon}\xi+{\omega}_{\epsilon}(k)+{\omega}_{\epsilon}(\ell)]}\\
&&\times
\Big\{\frac{\tilde g\left({\lambda}{\epsilon}/2+i[{\epsilon}\xi+{\omega}_{\epsilon}(k)]\right)}{{\epsilon}{\lambda}/2+i[{\epsilon}\xi+{\omega}_{\epsilon}(k)+{\omega}_{\epsilon}(\ell')]}+\frac{\tilde
g\left({\lambda}{\epsilon}/2+i[{\epsilon}\xi-{\omega}_{\epsilon}(k)])\right)
}{{\epsilon}{\lambda}/2+i\{{\epsilon}\xi-[{\omega}_{\epsilon}(k)+{\omega}_{\epsilon}(\ell')]\}}\Big\}\nonumber\\
&&{-\frac{\tilde g\left({\lambda}{\epsilon}/2-i[{\epsilon}\xi+{\omega}_{\epsilon}(k)]\right)
}{{\epsilon}{\lambda}/2-i[{\epsilon}\xi+{\omega}_{\epsilon}(k)-{\omega}_{\epsilon}(\ell)]}\cdot\frac{\tilde
g\left({\lambda}{\epsilon}/2+i[{\epsilon}\xi-{\omega}_{\epsilon}(k)])\right)
}{{\epsilon}{\lambda}/2+i\{{\epsilon}\xi-[{\omega}_{\epsilon}(k)+{\omega}_{\epsilon}(\ell')]\}}}\nonumber \\
&&
{-\frac{\tilde g\left({\lambda}{\epsilon}/2-i[{\epsilon}\xi-{\omega}_{\epsilon}(k)]\right)}{{\epsilon}{\lambda}/2-i[{\epsilon}\xi+{\omega}_{\epsilon}(k)+{\omega}_{\epsilon}(\ell)]}\cdot\frac{\tilde g\left({\lambda}{\epsilon}/2+i[{\epsilon}\xi\textcolor{red}{-}{\omega}_{\epsilon}(k)]\right)}{{\epsilon}{\lambda}/2+i[{\epsilon}\xi+{\omega}_{\epsilon}(k)-{\omega}_{\epsilon}(\ell')]}}
\Bigg\}.\nonumber\end{aligned}$$ Substituting for ${\rm Re}\,\bar{\frak d}_{\epsilon}^2({\lambda},k)$ from into we obtain $\bar{\frak L}_{scat,2}^{\epsilon}({\lambda})=\bar{\frak
L}_{scat,21}^{\epsilon}({\lambda})+\bar{\frak L}_{scat,22}^{\epsilon}({\lambda})$, where the terms in the right hand side correspond to $R_{\epsilon}({\lambda},k)$, $\rho_{\epsilon}({\lambda},k)$ respectively. As for $\rho_{\epsilon}({\lambda},k)$ we expect its contribution to be small in the limit and $\lim_{{\epsilon}\to0+}\bar{\frak
L}_{scat,22}^{\epsilon}({\lambda})=0$. In fact, we have.
\[lm011311\] For each ${\lambda}>0$ we have $$\label{011311}
\lim_{{\epsilon}\to0+}\int_{{{\mathbb T}}}|\rho_{\epsilon}({\lambda},k)|dk=0.$$
The proof of the lemma follows closely the argument presented in the proof Lemma 6.1 in [@kors], so we will not present it here.
Concerning $\bar{\frak
L}_{scat,21}^{\epsilon}({\lambda})$, we note first that, by the same type of estimate as in , $$\label{031804-19}
\lim_{{\epsilon}\to0+}\Big[\bar{\frak
L}_{scat,21}^{\epsilon}({\lambda})-{\frak
L}_{scat,21}^{\epsilon}({\lambda})\Big]=0,$$ where $$\label{Hpm0}
{\frak
L}_{scat,21}^{\epsilon}({\lambda}):=-\frac{{\gamma}_1}{2}\int_{
{{\mathbb T}}}R_{\epsilon}^0\left({\lambda},k\right) \hat
G^\star(k) dk$$ and $$\begin{aligned}
\label{R-eps0}
&&
R_{\epsilon}^0({\lambda},k):=-\frac{\gamma_1 ({\lambda}+2{\gamma}_0{\epsilon}R(k))}{2^3\cdot\pi{\epsilon}^2}\int_{{{\mathbb R}}}\frac{d\xi}{({\lambda}/2+{\gamma}_0R(k))^2+\xi^2} \int_{{{\mathbb T}}^2}d\ell d\ell' {\epsilon}\langle \hat\psi(\ell)\hat\psi^*(\ell')\rangle_{\mu_{\epsilon}}\\
&&
\times \frac{|\tilde g_{\epsilon}\left({\lambda}{\epsilon}/2-i[{\epsilon}\xi+{\omega}(k)]\right)|^2 }{{\lambda}/2-i\{\xi+{\epsilon}^{-1}[{\omega}(k)-{\omega}(\ell)]\}}
\times \frac{1
}{{\lambda}/2+i\{\xi+{\epsilon}^{-1}[{\omega}(k)-{\omega}(\ell')]\}}.\nonumber\end{aligned}$$
Using and the change of variables $$\label{ell-k}
\ell=:k'+\frac{{\epsilon}\eta'}{2},\quad \ell'=:k'-\frac{{\epsilon}\eta'}{2}$$ we can write $$\begin{aligned}
\label{011811}
&&
{\frak
L}_{scat,21}^{\epsilon}({\lambda}):
=\frac{\gamma_1^2 ({\lambda}+2{\gamma}_0{\epsilon}R(k))}{2^3\pi{\epsilon}}\int_{{{\mathbb R}}}d\xi
\int_{{{\mathbb T}}\times T_{\epsilon}} \frac{\widehat W_{{\epsilon},+}^{\rm un}(0,\eta',k') G^*(k) dk d\eta' dk'
}{{\lambda}/2-i\{\xi+{\epsilon}^{-1}[{\omega}(k)-{\omega}(k'+{\epsilon}\eta'/2)]\}}\nonumber\\
&&
\times \frac{|\tilde g_{\epsilon}\left({\lambda}{\epsilon}/2-i[{\epsilon}\xi+{\omega}(k)]\right)|^2
}{{\lambda}/2+i\{\xi+{\epsilon}^{-1}[{\omega}(k)-{\omega}(k'-{\epsilon}\eta'/2)]\}}\times \frac{1}{({\lambda}/2+{\gamma}_0R(k))^2+\xi^2}
. \end{aligned}$$ Here $T_{\epsilon}$ is the image of ${{\mathbb T}}_{\epsilon}\times{{\mathbb T}}$ under the change of variables. In fact, we may discard the contribution due to large $\eta'$, thanks to assumption . The main contribution to the limit comes therefore from the regions where $\omega(k)\approx\omega(k')$, that is, where either $k\approx k'$ – this generates the transmission term, or $k\approx -k'$ – this is responsible for the reflection term in the limit. The conclusion of Lemma \[lem-feb1502\] follows directly from the following result.
\[lem-feb1508\] We have $$\label{021511c}
\lim_{{\epsilon}\to0+}{\frak
L}_{scat,21}^{\epsilon}({\lambda})=
\frac{\gamma_1}{4}\sum_{\iota=\pm}\int_{{{\mathbb T}}}\frac{\hat G^\star(k)\fgeeszett(k)\widehat
W_0(\eta,\iota k)d\eta dk}{{\lambda}+2{\gamma}_0R(k)+\iota i{\omega}'(k)\eta} .$$
The proof of the lemma follows the argument presented in the proof of Lemma 6.2 in [@kors], so we omit it here.
Proofs of auxiliary results
---------------------------
### Proof of Lemma \[lm010304-19\] {#C3}
We can write $$\begin{aligned}
&
\int_{{{\mathbb T}}}\Big|\tilde I^{\iota_1,\iota_2}_{\epsilon}({\lambda},k)\Big|dk
=
{\epsilon}^2 \int_{{{\mathbb T}}}dk \Big|
\int_{{{\mathbb T}}}d\ell \int_0^{+\infty} \exp\left\{-\{{\epsilon}({\lambda}+{\gamma}_0 R(k))- i\iota_1 {\omega}_{\epsilon}(k)\} s\right\}
g_{\epsilon}\left(ds\right)\\
&\times
\frac{ r_{1,{\epsilon}}(k) r_{2,{\epsilon}}(\ell) }{{\epsilon}({\lambda}+{\gamma}_0R(k)+{\gamma}_0 R(\ell))-i[\iota_1 {\omega}_{\epsilon}(k)+\iota_2 {\omega}_{\epsilon}(\ell)]}
\langle\hat\psi^\star(k)\hat\psi(\ell)
\rangle_{\mu_{\epsilon}}\Big|\end{aligned}$$ $$\begin{aligned}
&
\le
{\epsilon}^2
\int_{{{\mathbb T}}^2}dkd\ell \Big|\tilde g_{\epsilon}\left(\vphantom{\int_0^1}{\epsilon}({\lambda}+{\gamma}_0 R(k))+ i\iota_1 {\omega}_{\epsilon}(k)\right)\Big|
\\
&\times
\frac{| r_{1,{\epsilon}}(k) r_{2,{\epsilon}}(\ell)| }{{\epsilon}({\lambda}+{\gamma}_0R(k)+{\gamma}_0 R(\ell))-i[\iota_1 {\omega}_{\epsilon}(k)+\iota_2 {\omega}_{\epsilon}(\ell)]|}
\Big|\langle\hat\psi^\star(k)\hat\psi(\ell)
\rangle_{\mu_{\epsilon}}\Big|.\end{aligned}$$ The expression in the right hand side can be estimated by $I_{\epsilon}^{\iota_1\iota_2}$, where $$\begin{aligned}
&
I_{\epsilon}^{\iota}:={\epsilon}^2 r_{1,*}r_{2,*}\|\tilde g_{\epsilon}\|_\infty
\int_{{{\mathbb T}}^2}
\frac{\Big|\langle\hat\psi^\star(k)\hat\psi(\ell)
\rangle_{\mu_{\epsilon}}\Big| dkd\ell }{{\epsilon}({\lambda}+{\gamma}_0R(k)+{\gamma}_0 R(\ell))+|{\omega}_{\epsilon}(k)+\iota {\omega}_{\epsilon}(\ell)|}.\end{aligned}$$ We need to show that $$\label{050905-19}
I_{\epsilon}^{\iota}\preceq
{\epsilon}^{1/2}\log{\epsilon}^{-1},\quad {\epsilon}\in(0,1]$$ for $\iota=\pm$. Note that $$\begin{aligned}
\label{011804-19}
&
{\frak r}_{\epsilon}:=\sup_{A\in{{\mathbb R}},k\in{{\mathbb T}}}\left|\frac{1}{|{\epsilon}{\lambda}/2-i\left({\omega}_{\epsilon}(k)-A\right)|}-\frac{1}{|{\epsilon}{\lambda}/2-i\left({\omega}(k)-A\right)|}\right|\\
&
=
\sup_{A\in{{\mathbb R}},k\in{{\mathbb T}}} \frac{{\epsilon}^2dk}{|{\epsilon}{\lambda}/2-i\left({\omega}_{\epsilon}(k)-A\right)||{\epsilon}{\lambda}/2-i\left({\omega}(k)-A\right)|}
\preceq 1, \quad {\epsilon}\in(0,1].\nonumber\end{aligned}$$ Let $$\label{050905-19a}
\tilde I_{\epsilon}^{\iota}:={\epsilon}^2 r_{1,*}r_{2,*}\|\tilde g_{\epsilon}\|_\infty
\int_{{{\mathbb T}}^2}
\frac{\Big|\langle\hat\psi^\star(k)\hat\psi(\ell)
\rangle_{\mu_{\epsilon}}\Big| dkd\ell }{{\epsilon}({\lambda}+{\gamma}_0R(k)+{\gamma}_0 R(\ell))+|{\omega}(k)+\iota {\omega}(\ell)|},\quad {\epsilon}\in(0,1].$$ We have $$\begin{aligned}
&&
|\tilde I^\iota_{{\epsilon}}-I^\iota_{{\epsilon}}|\le {\epsilon}^2 r_{1,*}r_{2,*}{\frak r}_{\epsilon}\|\tilde g_{\epsilon}\|_\infty
\int_{{{\mathbb T}}^2}
\Big|\langle\hat\psi^\star(k)\hat\psi(\ell)
\rangle_{\mu_{\epsilon}}\Big| dkd\ell \preceq {\epsilon},\quad {\epsilon}\in(0,1],\end{aligned}$$ as ${\epsilon}\to0$.
In the case $\iota=+$ we can write $$\begin{aligned}
&\tilde I_{\epsilon}^+
\le 2 r_{1,*}r_{2,*}\|\tilde g_{\epsilon}\|_\infty
\Big\{{\epsilon}\int_{{{\mathbb T}}}\Big|\langle\hat\psi^\star(k) \rangle_{\mu_{\epsilon}}\Big|^2dk\Big\}
\int_{{{\mathbb T}}}\frac{{\epsilon}d\ell }{{\epsilon}{\lambda}+ {\omega}(\ell)}\preceq {\epsilon}\log{\epsilon}^{-1},\quad {\epsilon}\in(),1].\end{aligned}$$ In the case $\iota=-$ we can write $$\begin{aligned}
&\tilde I_{\epsilon}^-
\le 2 r_{1,*}r_{2,*}\Gamma_{\epsilon}\|\tilde g_{\epsilon}\|_\infty
\Big\{{\epsilon}\int_{{{\mathbb T}}}\Big|\langle\hat\psi^\star(k) \rangle_{\mu_{\epsilon}}\Big|^2dk\Big\}
,\end{aligned}$$ with $$\Gamma_{\epsilon}:=\sup_{A\in{{\mathbb R}}}\int_{{{\mathbb T}}}\frac{{\epsilon}dk}{{\epsilon}{\lambda}+|{\omega}(k)-A|}.$$ Note that $
\Gamma_{\epsilon}=\Gamma_{{\epsilon}}^++\Gamma_{{\epsilon}}^-,
$ with $$\Gamma_{{\epsilon}}^\pm:=\sup_{A\in{{\mathbb R}}}\int_{{\omega}_{\rm min}}^{{\omega}_{\rm max}}\frac{{\epsilon}du}{({\epsilon}{\lambda}+|u-A|)|{\omega}'({\omega}_\pm(u))|}.$$ Recall that ${\omega}_-$, ${\omega}_+$ are the decreasing and increasing branches of the inverse function of the dispersion relation ${\omega}(\cdot)$. Our assumptions on the dispersion relation imply that $$\label{020707-19}
|{\omega}'({\omega}_\pm(u))|\approx ({\omega}_{\rm max}-u)^{1/2},
\hbox{ for ${\omega}_{\rm max}-u\ll1$.}$$ The consideration near the minimum of $\omega$ is identical unless $\omega_{\rm min}=0$, in which case $|\omega'(k)|$ stays uniformly positive near the minimum. Therefore, we have $$\Gamma_{{\epsilon}}^\pm\preceq \sup_{A\in [0,1]}\int_0^1\frac{{\epsilon}du}{({\epsilon}+|u-A|)\sqrt{u}}
\preceq {\epsilon}^{1/2}\log{\epsilon}^{-1}$$ and we conclude that holds. [$\Box$ ]{}
### Proof of Lemma \[lm010504-19\] {#C3.1}
We have $$\begin{aligned}
&\Big\|\widetilde{
I\!I}^{\iota_1,\iota_2,\iota_3}_{\epsilon}({\lambda})\Big\|_{L^1({{\mathbb T}})}
={\epsilon}^2 \int_{{{\mathbb T}}}dk\Big|\int_0^{+\infty}\int_0^{+\infty}e^{-{\epsilon}{\lambda}(t+t')/2}\delta(t-t') dt dt'\int_0^{t} ds \int_0^{t'}
g_{\epsilon}\left(ds'\right)\int_{0}^{s}g_{\epsilon}\left(ds_1\right)
\int_{{{\mathbb T}}^2}
d\ell d\ell'\\
&
\times r_{1,{\epsilon}}(k)r_{2,{\epsilon}}(\ell) r_{3,{\epsilon}}(\ell') e^{{\lambda}_{\iota_2}(\ell) (s-s_1)} e^{{\lambda}_{\iota_3}(\ell')
(t'-s')} e^{{\lambda}_{\iota_1}(k)
(t-s)}\langle\hat\psi^\star(\ell)\hat\psi(\ell')
\rangle_{\mu_{\epsilon}}\Big|\end{aligned}$$ $$\begin{aligned}
&
=\frac{{\epsilon}^2}{2\pi}\int_{{{\mathbb T}}}dk\Big|\int_{{{\mathbb R}}}d\xi\int_0^{+\infty}\int_0^{+\infty}e^{-{\epsilon}{\lambda}(t+t')/2}e^{i\xi(t-t')} dt dt'\int_0^{t} ds \int_0^{t'}
g_{\epsilon}\left(ds'\right)\int_{0}^{s}g_{\epsilon}\left(ds_1\right)
\int_{{{\mathbb T}}^2}
d\ell d\ell'\\
&
\times r_{1,{\epsilon}}(k)r_{2,{\epsilon}}(\ell) r_{3,{\epsilon}}(\ell') e^{{\lambda}_{\iota_2}(\ell) (s-s_1)} e^{{\lambda}_{\iota_3}(\ell')
(t'-s')} e^{{\lambda}_{\iota_1}(k)
(t-s)}\langle\hat\psi^\star(\ell)\hat\psi(\ell')
\rangle_{\mu_{\epsilon}}\Big|.\end{aligned}$$ Note that $$\begin{aligned}
&
\int_0^{+\infty}e^{-({\epsilon}{\lambda}/2-i\xi)t} dt\int_0^{t} ds \int_{0}^{s}g_{\epsilon}\left(ds_1\right)
e^{{\lambda}_{\iota_2}(\ell) (s-s_1)} e^{{\lambda}_{\iota_1}(k)
(t-s)}\\
&
=\int_0^{+\infty} ds \int_{0}^{s}g_{\epsilon}\left(ds_1\right)
\frac{e^{{\lambda}_{\iota_2}(\ell) (s-s_1)} \exp\left\{-({\epsilon}{\lambda}/2-i\xi)s\right\}}{{\epsilon}{\lambda}/2-i\xi-{\lambda}_{\iota_1}(k)}\\
&
=\frac{\tilde g_{\epsilon}({\epsilon}{\lambda}/2-i\xi)}{[{\epsilon}{\lambda}/2-i\xi-{\lambda}_{\iota_1}(k)][{\epsilon}{\lambda}/2-i\xi-{\lambda}_{\iota_2}(\ell)]} .\end{aligned}$$ Similarly $$\begin{aligned}
&
\int_0^{+\infty}e^{-({\epsilon}{\lambda}/2+i\xi)
t'}dt' \int_0^{t'}
g_{\epsilon}\left(ds'\right)e^{{\lambda}_{\iota_3}(\ell')
(t'-s')}
=\frac{\tilde g_{\epsilon}({\epsilon}{\lambda}/2+i\xi)}{{\epsilon}{\lambda}/2+i\xi-{\lambda}_{\iota_3}(\ell')}.\end{aligned}$$ Taking the above into account we obtain $$\begin{aligned}
\label{011305-19}
&\Big\|\widetilde{
I\!I}^{\iota_1,\iota_2,\iota_3}_{\epsilon}({\lambda})\Big\|_{L^1({{\mathbb T}})} =\frac{{\epsilon}^2}{2\pi}\int_{{{\mathbb T}}}dk\Big|\int_{{{\mathbb R}}}d\xi
\int_{{{\mathbb T}}^2}
d\ell d\ell' \langle\hat\psi^\star(\ell)\hat\psi(\ell')
\rangle_{\mu_{\epsilon}}|\tilde g_{\epsilon}({\epsilon}{\lambda}/2-i\xi)|^2\\
&
\times\left\{\vphantom{\int_0^1}[{\epsilon}({\lambda}/2+{\gamma}_0R(k))-i(\xi+\iota_1{\omega}_{\epsilon}(k))][{\epsilon}({\lambda}/2+{\gamma}_0R(k))-i(\xi+\iota_2{\omega}_{\epsilon}(\ell))][{\epsilon}( {\lambda}/2+{\gamma}_0R(k))+i(\xi-\iota_3{\omega}_{\epsilon}(\ell'))]\right\}\Big|.\nonumber\end{aligned}$$ Consider only the case $\iota_1=\iota_2=\iota_3=+$, as the other ones can be done in a similar fashion. We omit writing the superscripts in what follows. Change variables $\ell:=k'-{\epsilon}\eta/2$, $\ell'=k'-{\epsilon}\eta/2$ and obtain $$\begin{aligned}
&\Big\|\widetilde{
I\!I}_{\epsilon}({\lambda})\Big\|_{L^1({{\mathbb T}})} =\frac{{\epsilon}^2}{\pi}\int_{{{\mathbb T}}}dk\Big|\int_{{{\mathbb R}}}d\xi
\int_{T_{\epsilon}}
d\eta dk' \widehat W_{\epsilon}(\eta,k')|\tilde g_{\epsilon}({\epsilon}{\lambda}/2-i\xi)|^2 \Big\{[{\epsilon}({\lambda}/2+{\gamma}_0R(k))-i(\xi+{\omega}_{\epsilon}(k))]\\
&
[{\epsilon}({\lambda}/2+{\gamma}_0R(k))-i(\xi+{\omega}_{\epsilon}(k'-{\epsilon}\eta/2))][{\epsilon}( {\lambda}/2+{\gamma}_0R(k))+i(\xi-{\omega}_{\epsilon}(k'+{\epsilon}\eta/2))]\Big\}^{-1}\Big|.\end{aligned}$$ Thanks to and we can estimate $$\begin{aligned}
&\Big\|\widetilde{
I\!I}_{\epsilon}({\lambda})\Big\|_{L^1({{\mathbb T}})} \preceq {\epsilon}^2\int_{{{\mathbb R}}}d\xi\int_{{{\mathbb R}}}\varphi(\eta)d\eta
\int_{{{\mathbb T}}^2}\left|[{\epsilon}({\lambda}/2+{\gamma}_0R(k))-i(\xi+{\omega}_{\epsilon}(k))]\right.
\\
&\left.
\times [{\epsilon}({\lambda}/2+{\gamma}_0R(k))-i(\xi+{\omega}_{\epsilon}(k'-{\epsilon}\eta/2))][{\epsilon}(
{\lambda}/2+{\gamma}_0R(k))+i(\xi-{\omega}_{\epsilon}(k'+{\epsilon}\eta/2))]\right|^{-1}dk
dk',\quad {\epsilon}\in(0,1].\end{aligned}$$ We need to prove that the right hand side vanishes, as ${\epsilon}\to0$. Similarly to what has been done in the proof of Lemma \[lm010304-19\], it suffices only to show that $$\label{J-eps}
\lim_{{\epsilon}\to0}J_{\epsilon}=0,$$ where $$\begin{aligned}
&
J_{\epsilon}:={\epsilon}^2\int_{{{\mathbb R}}}d\xi\int_{{{\mathbb R}}}\varphi(\eta)d\eta
\int_{{{\mathbb T}}^2}\left|[{\epsilon}{\lambda}/2-i(\xi+{\omega}(k))]\right.
\\
&\left.
\times [{\epsilon}{\lambda}/2-i(\xi+{\omega}(k'-{\epsilon}\eta/2))][{\epsilon}{\lambda}/2+i(\xi-{\omega}(k'+{\epsilon}\eta/2))]\right|^{-1}dk dk'\end{aligned}$$ Changing variables $\xi+{\omega}(k):={\epsilon}\xi'$ we conclude that $$\begin{aligned}
&
J_{\epsilon}\approx
\int_{{{\mathbb R}}}d\xi\int_{{{\mathbb R}}}\varphi(\eta)d\eta
\int_{{{\mathbb T}}^2}
\left\{\vphantom{\int_0^1}(1+|\xi|)(
1+|\xi+{\epsilon}^{-1}({\omega}(k'-{\epsilon}\eta/2)-{\omega}(k))|)\right.\\
&
\left. \vphantom{\int_0^1} (1+|\xi+{\epsilon}^{-1}({\omega}(k)-{\omega}(k'+{\epsilon}\eta/2)))\right\}^{-1}dkd
k'
\le J_{\epsilon}^1+J_{\epsilon}^2,\end{aligned}$$ with $$\begin{aligned}
&
J_{\epsilon}^1:=
\frac12\int_{{{\mathbb R}}}d\xi\int_{{{\mathbb R}}}\varphi(\eta)d\eta
\int_{{{\mathbb T}}^2}
\left\{(1+|\xi|)(
1+|\xi+{\epsilon}^{-1}({\omega}(k'-{\epsilon}\eta/2)-{\omega}(k))|)^2\right\}^{-1}dkd
k'\end{aligned}$$ and $$\begin{aligned}
&
J_{\epsilon}^2:=\frac12
\int_{{{\mathbb R}}}d\xi\int_{{{\mathbb R}}}\varphi(\eta)d\eta
\int_{{{\mathbb T}}^2}
\left\{(1+|\xi|)(1+|\xi+{\epsilon}^{-1}({\omega}(k)-{\omega}(k'+{\epsilon}\eta/2)))^2\right\}^{-1}dkd
k'\end{aligned}$$ Using an elementary estimate $$\label{ab}
\int_{{{\mathbb R}}}\frac{1}{1+|x+a|}\times\frac{dx}{1+x^2}\preceq
\frac{1}{1+|a|},\quad a\in{{\mathbb R}}$$ we conclude that $$\begin{aligned}
&
J_{\epsilon}^1\preceq
\frac12\int_{{{\mathbb R}}}\varphi(\eta)d\eta
\int_{{{\mathbb T}}^2}
\left\{
1+|{\epsilon}^{-1}({\omega}(k'-{\epsilon}\eta/2)-{\omega}(k))|\right\}^{-1}dkd
k'\to0,\end{aligned}$$ by virtue of the dominated convergence theorem. Estimates for $J_{\epsilon}^2$ are similar. [$\Box$ ]{}
### Proof of formula (\[010104\]) {#d-eps2}
From we have $$\begin{aligned}
&&
2{\rm Re}\,\bar{\frak d}_{\epsilon}^2({\lambda},k)
=-\gamma_1 {\epsilon}\left\langle\int_0^{+\infty}e^{-{\lambda}{\epsilon}t}e^{-2{\gamma}_0{\epsilon}R(k) t} \frac{d}{dt}\left\{\left[\int_0^{t} e^{{\gamma}_0{\epsilon}R(k) ts}
\cos({\omega}_{\epsilon}(k)s)g_{\epsilon}*{\bar p}_0^0(s)ds\right]^2\right.\right.\\
&&
\left.\left.+\left[\int_0^{t} e^{{\gamma}_0{\epsilon}R(k) ts} \sin({\omega}(k)s)g_{\epsilon}*{\bar p}_0^0(s)ds\right]^2
\right\}dt \right\rangle_{\mu_{\epsilon}}.\end{aligned}$$ Integrating by parts, we obtain $$\begin{aligned}
&&
2{\rm Re}\,{\frak d}_{\epsilon}^2({\lambda},k)
=C_{\epsilon}({\lambda},k)+S_{\epsilon}({\lambda},k).\label{feb1518}\nonumber\end{aligned}$$ The first term in the right side is $$\begin{aligned}
\label{012803}
&
C_{\epsilon}({\lambda},k)
=-\frac{\gamma_1}{4\pi} {\epsilon}({\lambda}+2{\gamma}_0{\epsilon}R(k))\int_{{{\mathbb R}}}d\xi \int_{{{\mathbb T}}^2}{\epsilon}\langle \hat\psi(\ell)\hat\psi^*(\ell')\rangle_{\mu_{\epsilon}} \Xi_{\epsilon}(\ell,k,{\lambda},\xi)\Xi_{\epsilon}^\star(\ell',k,{\lambda},\xi)d\ell d\ell' ,\end{aligned}$$ with $$\Xi_{\epsilon}(\ell,k,{\lambda},\xi):=\int_0^{+\infty}e^{{\gamma}_0{\epsilon}R(k) s}
\cos({\omega}_{\epsilon}(k)s)ds\left\{\int_0^s e^{-i{\omega}_{\epsilon}(\ell)
(s-\tau)}g_{\epsilon}(d\tau)\int_{s}^{+\infty}e^{-[({\lambda}/2+{\gamma}_0 R(k)){\epsilon}-i\xi] t}dt\right\}.$$
Integrating out first the $t$ variable, and then the $s$ varable, we obtain $$\begin{aligned}
&&
\Xi_{\epsilon}(\ell,k,{\lambda},\xi)
=\frac{1}{2[({\lambda}/2+{\gamma}_0R(k)){\epsilon}-i\xi]}\left\{\frac{\tilde g_{\epsilon}\left({\lambda}{\epsilon}/2-i[\xi+{\omega}_{\epsilon}(k)]\right) }{{\lambda}{\epsilon}/2-i(\xi+{\omega}_{\epsilon}(k)-{\omega}_{\epsilon}(\ell))}+\frac{\tilde g_{\epsilon}\left({\lambda}{\epsilon}/2-i[\xi-{\omega}_{\epsilon}(k))]\right)}{{\lambda}{\epsilon}/2-i[\xi+{\omega}_{\epsilon}(k)+{\omega}_{\epsilon}(\ell)]}\right\}.\end{aligned}$$ Hence, after a change of variables $\xi:={\epsilon}\xi'$, we get $$\begin{aligned}
\label{feb1524}
&&
C_{\epsilon}({\lambda},k)
=-\frac{\gamma_1 ({\lambda}+2{\gamma}_0{\epsilon}R(k))}{2^4\cdot\pi{\epsilon}^2}\int_{{{\mathbb R}}}\frac{d\xi}{({\lambda}/2+{\gamma}_0R(k))^2+\xi^2} \int_{{{\mathbb T}}^2}d\ell d\ell' {\epsilon}\langle \hat\psi(\ell)\hat\psi^*(\ell')\rangle_{\mu_{\epsilon}}\\
&&
\times \left\{\frac{\tilde g_{\epsilon}\left({\lambda}{\epsilon}/2-i[{\epsilon}\xi+{\omega}_{\epsilon}(k)]\right) }{{\lambda}/2-i\{\xi+{\epsilon}^{-1}[{\omega}_{\epsilon}(k)-{\omega}_{\epsilon}(\ell)]\}}+\frac{\tilde g_{\epsilon}\left({\lambda}{\epsilon}/2-i[{\epsilon}\xi-{\omega}_{\epsilon}(k)]\right)}{{\lambda}/2-i\{\xi+{\epsilon}^{-1}[{\omega}_{\epsilon}(k)+{\omega}_{\epsilon}(\ell)]\}}\right\}
\nonumber\\
&&
\times \left\{\frac{\tilde g_{\epsilon}\left({\lambda}{\epsilon}/2+i[{\epsilon}\xi+{\omega}_{\epsilon}(k)]\right) }{{\lambda}/2+i\{\xi+{\epsilon}^{-1}[{\omega}_{\epsilon}(k)-{\omega}_{\epsilon}(\ell')]\}}+\frac{\tilde g\left({\lambda}{\epsilon}/2+i[{\epsilon}\xi+{\omega}_{\epsilon}(k)])\right)}{{\lambda}/2+i\{\xi+{\epsilon}^{-1}[{\omega}_{\epsilon}(k)+{\omega}_{\epsilon}(\ell')]\}}\right\}.\nonumber\end{aligned}$$ A similar calculation leads to $$\begin{aligned}
\label{feb1526}
&&
S_{\epsilon}({\lambda},k)
=\frac{\gamma_1 ({\lambda}+2{\gamma}_0{\epsilon}R(k))}{2^4\pi{\epsilon}^2}\int_{{{\mathbb R}}}\frac{d\xi}{({\lambda}/2+{\gamma}_0R(k))^2+\xi^2} \int_{{{\mathbb T}}^2}d\ell d\ell' {\epsilon}\langle \hat\psi(\ell)\hat\psi^*(\ell')\rangle_{\mu_{\epsilon}}\\
&&
\times \left\{\frac{\tilde g_{\epsilon}\left({\lambda}{\epsilon}/2-i[{\epsilon}\xi+{\omega}_{\epsilon}(k)]\right) }{{\lambda}/2-i\{\xi+{\epsilon}^{-1}[{\omega}_{\epsilon}(k)-{\omega}_{\epsilon}(\ell)]\}}-\frac{\tilde g_{\epsilon}\left({\lambda}{\epsilon}/2-i[{\epsilon}\xi-{\omega}_{\epsilon}(k)]\right)}{{\lambda}/2-i\{\xi+{\epsilon}^{-1}[{\omega}_{\epsilon}(k)+{\omega}_{\epsilon}(\ell)]\}}\right\}
\nonumber\\
&&
\times \left\{\frac{\tilde g_{\epsilon}\left({\lambda}{\epsilon}/2+i({\epsilon}\xi-{\omega}_{\epsilon}(k)])\right) }{{\lambda}/2+i\{\xi-{\epsilon}^{-1}[{\omega}_{\epsilon}(k)+{\omega}_{\epsilon}(\ell')]\}}-\frac{\tilde g_{\epsilon}\left({\lambda}{\epsilon}/2+i[{\epsilon}\xi+{\omega}_{\epsilon}(k)]\right)}{{\lambda}/2+i\{\xi+{\epsilon}^{-1}[{\omega}_{\epsilon}(k)-{\omega}_{\epsilon}(\ell')]\}}\right\}.
\nonumber\end{aligned}$$ Putting (\[feb1518\]) – (\[feb1526\]) together, gives (\[010104\]).
Proof of Proposition \[cor010304-19\] {#appC}
=====================================
Let (cf ) $$\begin{aligned}
\label{jl}
&&
\tilde{ j}_{\epsilon}({\lambda},k) :=\int_0^{+\infty}e^{-{\lambda}t}j_{\epsilon}(t,k)dt, \quad {\rm Re}\,{\lambda}>0.\end{aligned}$$ From and it follows that $$\label{jl1}
\tilde{ j}_{\epsilon}({\lambda},k)
=\frac{{\lambda}\sqrt{1-({\epsilon}\beta)^2}}{2(
{\lambda}+{\epsilon}{\gamma}_0R(k) )}\left\{\frac{
1}{{\lambda}+{\epsilon}{\gamma}_0R(k)+i{\omega}_{\epsilon}(k)}+\frac{
1}{{\lambda}+{\epsilon}{\gamma}_0R(k)-i{\omega}_{\epsilon}(k)}\right\}.$$ Also, since ${\Omega}_0(k)-{\Omega}_{\epsilon}(k)={\epsilon}{\gamma}_0R(k){\frak f}\otimes{\frak
g}$ (see ) we have $$\begin{aligned}
\label{020204-19a}
&\tilde{ j}_0({\lambda},k)-\tilde{
j}_{\epsilon}({\lambda},k)=\frac12\left[({\lambda}-{\Omega}_0(k))^{-1}
-({\lambda}-{\Omega}_{\epsilon}(k))^{-1}\right]{\frak f}\cdot{\frak f}\\
&
=\frac12({\lambda}-{\Omega}_0(k))^{-1}({\Omega}_0(k)-{\Omega}_{\epsilon}(k))
({\lambda}-{\Omega}_{\epsilon}(k))^{-1}{\frak f}\cdot{\frak f}
=\frac12 {\epsilon}{\gamma}_0R(k)\tilde{ j}_{\epsilon}({\lambda},k) \tilde{ j}_0({\lambda},k).\nonumber\end{aligned}$$ The following result holds.
\[prop010204-19\] We have $$\label{010204-19}
\tilde J({\lambda})=\tilde J_{\epsilon}({\lambda})+{\epsilon}{\gamma}_0\tilde{\cal R}_{\epsilon}({\lambda}),\quad{\epsilon}\in(0,1],\,
{\lambda}\in\mathbb C_+,$$ where $$\label{010204-19a}
\tilde{\cal R}_{\epsilon}({\lambda})=\int_{{{\mathbb T}}}R(k)\tilde j_{\epsilon}({\lambda},k)\tilde j_0({\lambda},k)dk,\quad
{\lambda}\in\mathbb C_+.$$ In addition, for any $p\in(1,2)$ and a uniformly continuous and bounded function $\xi:{{\mathbb R}}\to{{\mathbb R}}_+$ satisfying $$\label{lim-is}
\xi(\eta)\ge
\xi_0,\quad\,\eta\in{{\mathbb R}}$$ we have $$\label{010204-19b}
\lim_{{\epsilon}\to0}{\epsilon}^p\int_{{{\mathbb R}}}|\tilde{\cal R}_{\epsilon}({\epsilon}\xi(\eta)+i\eta)|^pd\eta=0.$$
[[*Proof.* ]{}]{}Identity follows directly from . Letting ${\lambda}:={\epsilon}\xi(\eta)+i\eta$, using together with and the change of variables $k\mapsto
v={\omega}(k)$ we can write $$\begin{aligned}
\label{030204-19a}
&
\tilde{ J}({\epsilon}\xi(\eta)+i\eta)-\tilde{ J}_{\epsilon}({\epsilon}\xi(\eta)+i\eta)
=\frac12 \sum_{\iota=\pm}\left\{ \int_{{{\mathbb R}}}
\frac{ \chi_{\iota,{\epsilon}}(v)dv }{{\epsilon}\xi(\eta)+i(\eta+v)}+\int_{{{\mathbb R}}}
\frac{ \chi_{\iota,{\epsilon}}(v)dv }{{\epsilon}\xi(\eta)+i(\eta-v)}\right\},\end{aligned}$$ with $$\chi_{\iota,{\epsilon}}(v)=\iota {\epsilon}{\gamma}_0R({\omega}_+(v))\tilde{ j}_{\epsilon}({\epsilon}\xi(\eta)+i\eta, {\omega}_+(v))\frac{1_{[{\omega}_{\rm min}, {\omega}_{\rm max}]}(v)}{{\omega}'({\omega}_+(v))}.$$ Here, (see Section \[sec2.2.1\]) ${\omega}_{\pm}$ are the two branches of inverses of the unimodal dispersion relation ${\omega}$, with ${\omega}_+: [{\omega}_{\rm min}, {\omega}_{\rm max}]\to [0,1/2]$ and ${\omega}_-:=-{\omega}_+$. In addition to , in the optical case, we also have $$\label{020707-19a}
|{\omega}'({\omega}_\pm(u))|\approx (u-{\omega}_{\rm min})^{1/2},
\hbox{ for $u-{\omega}_{\rm min}\ll1$.}$$
Hence, $$\begin{aligned}
\label{040204-19}
&
{\epsilon}{\gamma}_0R(k)|\tilde{ j}_{\epsilon}({\epsilon}\xi(\eta)+i\eta,k)|\\
&\le \frac12\left\{\left|\frac{
{\epsilon}{\gamma}_0R(k)}{{\epsilon}(\xi(\eta)+{\gamma}_0R(k))+i(\eta+{\omega}_{\epsilon}(k))}\right|+\left|\frac{
{\epsilon}{\gamma}_0R(k)}{{\epsilon}(\xi(\eta)+{\gamma}_0R(k))+i(\eta-{\omega}_{\epsilon}(k))}\right|\right\}.\nonumber\end{aligned}$$ The above allows us to conclude that $$\begin{aligned}
&
{\epsilon}{\gamma}_0R(k)|\tilde{ j}_{\epsilon}({\epsilon}\xi(\eta)+i\eta,k)|\le C\end{aligned}$$ for some $C>0$ independent of $\eta,{\epsilon},k$ and, as a result, $$|\chi_{\iota,{\epsilon}}(v)|\le \frac{C1_{[{\omega}_{\rm min}, {\omega}_{\rm max}]}(v)}{{\omega}'({\omega}_+(v))}.$$ Thus $$\label{030404-19}
\lim_{{\epsilon}\to0+}\chi_{\iota,{\epsilon}}(v)=0$$ both a.s. and in the $L^p$ sense for any $p\in[1,2)$, see and . Let $\xi_0$ be as in . Note that for any $\eta\in{{\mathbb R}}$ $$\begin{aligned}
&
\Big|\int_{{{\mathbb R}}}
\frac{ \chi_{\iota,{\epsilon}}(v-\eta)dv }{{\epsilon}\xi(\eta)+iv}\Big|\le
\Big|\int_{[|v|\ge {\epsilon}\xi_0]}
\frac{ \chi_{\iota,{\epsilon}}(v-\eta)dv }{iv}\Big|
\\
&
+ \Big|\int_{[|v|\ge {\epsilon}\xi_0]}
\frac{ \chi_{\iota,{\epsilon}}(v-\eta)dv }{{\epsilon}\xi(\eta)+iv}-\int_{[|v|\ge {\epsilon}\xi_0]}
\frac{ \chi_{\iota,{\epsilon}}(v-\eta)dv }{iv}\Big|+\Big|\int_{[|v|<{\epsilon}\xi_0]}
\frac{ \chi_{\iota,{\epsilon}}(v-\eta)dv }{{\epsilon}\xi(\eta)+iv}\Big|\end{aligned}$$ Denote the expressions in the utmost right hand side by ${\cal J}_{1,{\epsilon}}(\eta)$, ${\cal J}_{2,{\epsilon}}(\eta)$ and ${\cal
J}_{3,{\epsilon}}(\eta)$, respectively.
Using Theorem 3.2, p. 35 of [@stein] we conclude that for any $p\in(1,+\infty)$ there exists a constant $C>0$, independent of ${\epsilon}>0$ such that $$\label{010404-19}
\|{\cal J}_{1,{\epsilon}}\|_{L^p({{\mathbb R}})}\le C \|\chi_{\iota,{\epsilon}}\|_{L^p({{\mathbb R}})}.$$ Considering ${\cal J}_{3,{\epsilon}}$, there exists $C>0$ such that $$\frac{ 1_{[|v|<{\epsilon}\xi_0]} }{|{\epsilon}\xi({\epsilon})+iv|}\le \frac{C 1_{[|v|<{\epsilon}\xi_0]}}{{\epsilon}}$$ thus, again (using Young’s inequality) for any $p\in(1,+\infty)$ there exists $C>0$ such that $$\label{020404-19}
\|{\cal J}_{3,{\epsilon}}\|_{L^p({{\mathbb R}})}\le C \|\chi_{\iota,{\epsilon}}\|_{L^p({{\mathbb R}})} ,\quad{\epsilon}\in(0,1].$$
For $\rho\in(0,1)$ we can write $$\begin{aligned}
&
|{\cal J}_{2,{\epsilon}}(\eta)|\le \int_{[|v|\ge {\epsilon}\xi_0]}
\frac{{\epsilon}\xi(\eta) |\chi_{\iota,{\epsilon}}(v-\eta)|dv
}{[({\epsilon}\xi(\eta))^2+v^2]^{1/2}|v|}\le
\|\xi\|_\infty\int_{[|v|\ge {\epsilon}\xi_0]}
\frac{{\epsilon}^{\rho}|\chi_{\iota,{\epsilon}}(v-\eta)|dv }{|v|^{1+\rho}}.\end{aligned}$$ By an application of Young’s inequality for convolutions we have $$\label{020404-19}
\|{\cal J}_{2,{\epsilon}}\|_{L^p({{\mathbb R}})}\le
2{\epsilon}^{\rho}\|\xi\|_\infty \|\chi_{\iota,{\epsilon}}\|_{L^p({{\mathbb R}})}\int_{{\epsilon}\xi_0}^{+\infty}\frac{dv}{v^{1+\rho}}
\le C \|\chi_{\iota,{\epsilon}}\|_{L^p({{\mathbb R}})},\quad{\epsilon}\in(0,1],$$ with constant $C>0$ independent of ${\epsilon}$.
Taking into account all the above we conclude that $$\lim_{{\epsilon}\to0+}\int_{{{\mathbb R}}}|\tilde{ J}({\epsilon}\xi(\eta)+i\eta)-\tilde{ J}_{\epsilon}({\epsilon}\xi(\eta)+i\eta)|^pd\eta=0$$ for any $p\in(1,2)$. [$\Box$ ]{}
Proposition \[cor010304-19\] follows from the following result.
\[cor010304-19a\] Suppose that $\xi:{{\mathbb R}}\to{{\mathbb R}}_+$ is a uniformly continuous and bounded function satisfying $$\label{lim-is1}
\inf_{\eta\in{{\mathbb R}}}\xi(\eta)>0$$ and $\tilde r_{\epsilon}({\lambda})$ is given by . Then, for any $p\in(1,+\infty)$, we have $$\lim_{{\epsilon}\to0+}{\epsilon}^p\int_{{{\mathbb R}}}|\tilde r_{\epsilon}({\epsilon}\xi(\eta)+i\eta)|^pd\eta=0,$$
Indeed, assume the above result and let $K$ be as in the statement of Proposition \[cor010304-19\]. Define $\xi_\pm(\eta):=K({\omega}_\pm(\eta))$, $\eta\in\bar {{\mathbb T}}_\pm$ and $\xi_{\pm}(\eta)=1$, elsewhere. We conclude that in particular ${\epsilon}|\tilde r_{\epsilon}({\epsilon}\xi_\pm(\eta)+i\eta)$ converges to $0$ in the Lebesgue measure on $[{\omega}_{\rm min},{\omega}_{\rm max}]$. This obviously implies that ${\epsilon}|\tilde r_{\epsilon}({\epsilon}K(k)-i{\omega}(k))$ convergence in the Lebesgue measure on ${{\mathbb T}}$ to $0$. The $L^p$ convergence follows from the fact that the functions are bounded.
Proof of Proposition \[cor010304-19a\] {#proof-of-proposition-cor010304-19a .unnumbered}
--------------------------------------
We have (cf ) $$\tilde g_{\epsilon}({\lambda})-\tilde g({\lambda})
={\epsilon}{\gamma}_0{\gamma}_1\left(1+{\gamma}_1\tilde
{J}_{\epsilon}({\lambda})\right)^{-1}\left(1+{\gamma}_1\tilde
{J}({\lambda})\right)^{-1}\tilde{\cal R}_{\epsilon}({\lambda}).$$ From the above identity we conclude that $$\begin{aligned}
&
|\tilde g_{\epsilon}({\epsilon}\xi(\eta)+i\eta)-\tilde g({\epsilon}\xi(\eta)+i\eta)|\le {\epsilon}{\gamma}_0{\gamma}_1|\tilde{\cal R}_{\epsilon}({\epsilon}\xi(\eta)+i\eta)|.\end{aligned}$$ From we conclude that $$\begin{aligned}
&
\lim_{{\epsilon}\to0+}\int_{{{\mathbb R}}}\Big|\tilde g_{\epsilon}({\epsilon}\xi(\eta)+i\eta)-\tilde g({\epsilon}\xi(\eta)+i\eta)\Big|^pd\eta=0\end{aligned}$$ for any $p\in(1,2)$. On the other hand, thanks to , for $p\ge
2$ we get $$\begin{aligned}
&
\int_{{{\mathbb R}}}|\tilde g_{\epsilon}({\epsilon}\xi(\eta)+i\eta)-\tilde
g({\epsilon}\xi(\eta)+i\eta)|^pd\eta\le 2^{p-3/2}\int_{{{\mathbb R}}}|\tilde g_{\epsilon}({\epsilon}\xi(\eta)+i\eta)-\tilde
g({\epsilon}\xi(\eta)+i\eta)|^{3/2}d\eta\to0\end{aligned}$$ as ${\epsilon}\to0+$. [$\Box$ ]{}
[99]{}
G. Basile, S. Olla, H. Spohn [*Wigner functions and stochastically perturbed lattice dynamics*]{}, Arch.Rat.Mech., Vol. 195, no. 1, 171-203, 2009.
Diestel, J.; Uhl, J. J., Jr. [*Vector measures.* ]{}. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977. xiii+322 pp. T. Komorowski, S. Olla, L. Ryzhik, H. Spohn, [*High frequency limit for a chain of harmonic oscillators with a point Langevin thermostat*]{}, 2018, arxiv.org/1806.02089.
T. Komorowski, S. Olla, L. Ryzhik, [*Fractional diffusion limit for a kinetic equation with an interface*]{}, 2019, arxiv.org/1905.10586. P. Koosis, [*Introduction to $H^p$ spaces.*]{} Cambridge Univ. Press (1980). Spohn, H., *The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics*, J. Stat. Phys. 124, no.2-4, 1041-1104 (2006).
Stein, E., [*Singular Integrals and Differentiability Properties of Functions*]{}, Princeton, 1970.
[^1]: Institute of Mathematics, Polish Academy Of Sciences, ul. Śniadeckich 8, 00-956 Warsaw, Poland, e-mail: [[email protected]]{}
[^2]: CEREMADE, UMR-CNRS, Université de Paris Dauphine, PSL Research University [Place du Maréchal De Lattre De Tassigny, 75016 Paris, France]{}, e-mail:[ [email protected]]{}
| 0 |
---
abstract: 'The current fleet of space-based solar observatories offers us a wealth of opportunities to study solar flares over a range of wavelengths. Significant advances in our understanding of flare physics often come from coordinated observations between multiple instruments. Consequently, considerable efforts have been, and continue to be made to coordinate observations among instruments (*e.g.* through the *Max Millennium Program of Solar Flare Research*). However, there has been no study to date that quantifies how many flares have been observed by combinations of various instruments. Here we describe a technique that retrospectively searches archival databases for flares jointly observed by the *Ramaty High Energy Solar Spectroscopic Imager* (RHESSI), *Solar Dynamics Observatory* (SDO)/*EUV Variability Experiment* (EVE) (*Multiple EUV Grating Spectrograph* (MEGS)-A and MEGS-B), *Hinode*/(EUV Imaging Spectrometer, *Solar Optical Telescope*, and *X-Ray Telescope*), and *Interface Region Imaging Spectrograph* (IRIS). Out of the 6953 flares of GOES magnitude C1 or greater that we consider over the 6.5 years after the launch of SDO, 40 have been observed by six or more instruments simultaneously. Using each instrument’s individual rate of success in observing flares, we show that the numbers of flares co-observed by three or more instruments are higher than the number expected under the assumption that the instruments operated independently of one another. In particular, the number of flares observed by larger numbers of instruments is much higher than expected. Our study illustrates that these missions often acted in cooperation, or at least had aligned goals. We also provide details on an interactive widget () now available in that allows a user to search for flaring events that have been observed by a chosen set of instruments. This provides access to a broader range of events in order to answer specific science questions. The difficulty in scheduling coordinated observations for solar-flare research is discussed with respect to instruments projected to begin operations during Solar Cycle 25, such as the *Daniel K. Inouye Solar Telescope*, *Solar Orbiter*, and *Parker Solar Probe*.'
author:
- 'Ryan O. $^{1,2,3,4}$, Jack $^{3,5}$'
title: 'On the Performance of Multi-Instrument Solar Flare Observations During Solar Cycle 24'
---
Introduction {#s:intro}
============
The study of solar flares is a high-priority research area in the international heliophysics community. Understanding the physics of these energetic events is crucial, not only for the field of space weather, but also in the broader scope of astrophysics where similar processes are believed to occur in stellar flares, black-hole accretion disks, and in the Earth’s magnetotail. Observations of solar flares are made by many different instruments, both in space and on the ground. These instruments provide imaging, photometric, and spectroscopic data over a range of wavelengths, from radio waves through the optical and EUV to X-rays and $\gamma$-rays: often the greatest advances in our understanding of solar flares come through various combinations of these datasets. From @flet11 [Section 7.2]:
> The multifarious observations across the broad spectrum of phenomena each help us to characterize the equilibrium change in the corona and chromosphere that we call a flare, and it should be clear that the multiwavelength approach is crucial in flare studies. It tells us where the flare energy starts and where it ends up, and something about the intermediate steps. It also provides some geometrical and diagnostic information about the flare magnetic environment, at different levels in the atmosphere, and how and when this changes as the flare proceeds. This big picture cannot be reached using one spectral region on its own. The multiwavelength observations have many detailed applications as we try to understand specific mechanisms that are at work in various phases and regions of the flare development.
However, it is difficult to keep track of which flares have been observed by which instruments. While most currently operational missions have their own individual flare lists (*e.g.* Hinode Flare Catalog: @wata12 or *Solar Dynamics Observatory* (SDO)/*EUV Variability Experiment* (EVE): @hock12), it was only recently that the first inter-instrument catalog became available, hosted by New Jersey Institute of Technology (NJIT)[^1] [@sady17]. The *Max Millennium Program for Solar Flare Research* (see @bloo16 for a recent review) and others have aimed to coordinate ground- and space-based instrumentation to observe a flaring active region simultaneously in order to optimize the scientific return. However, this can be difficult due to factors such as coordinating across multiple time zones, planning schedules being uploaded days in advance, ground-based seeing conditions, competing scientific priorities, and so on.
Therefore when a solar flare is known to have been observed by a combination of instruments, the event can receive considerable attention as a consequence. A notable recent example of this is the 29 March 2014 X-class flare, which was observed by four space-based observatories and one ground based telescope[^2]. Consequently there have been 23 refereed publications that discuss this flare, according to a NASA ADS fulltext search on the Solar Object Locator keyword . Similarly, the first X-class flare of Solar Cycle 24 () was simultaneously observed by multiple instruments at high cadence, resulting in 42 refereed publications to date. The exceptional data coverage of each of these events allowed [@klei16] and [@mill14], respectively, to investigate the redistribution of nonthermal electron energy. They were both able to compare radiative losses in the chromosphere across a range of wavelengths with the energy injected by nonthermal particles from hard X-ray observations. In both cases only 15–20% of the nonthermal energy could be accounted for from longer-wavelength measurements. Understanding where this “missing energy’ went to can only be answered by even better data coverage. Clearly there is great scientific merit in multi-instrument observations of the same event.
Likewise for other astronomical research areas where coordinated observations of transient objects (“Targets Of Opportunity”) at different wavelengths are highly desirable. The study of supernovae, for example, is facilitated by the availability of both lightcurves (to understand the evolution) and spectra (to understand the composition and dynamics). The *Open Supernova Catalog* [@guil17] acts as a central repository providing access to data for over 42,000 known supernova events. According to the statistics page of the website[^3] only 6% of these events have *both* photometric and spectroscopic data available (34% only have lightcurves, 7% only have spectra, and 53% have neither).
This article presents an analysis of flare statistics by retrospectively cross-referencing metadata from a suite of instruments that take flare-relevant observations – the *Ramaty High Energy Solar Spectroscopic Imager* (RHESSI: @lin02), the *Multiple EUV Grating Spectrograph* (MEGS; @crot04) -A and -B components of EVE [@wood12], the *EUV Imaging Spectrometer* (EIS: @culh07), the *Solar Optical Telescope* (SOT: @tsun08), the *X-Ray Telescope* (XRT: @golu07), and the *Interface Region Imaging Spectrometer* (IRIS: @depo14) – to search for flaring events observed simultaneously, either intentionally or serendipitously. The purpose of this article is to present an overview of how successful the solar community has been in capturing flare data through coordinated efforts. We also describe a database of these events that give researchers access to multi-wavelength datasets with which to address a given science question. Section \[s:data\_anal\] describes how the various archives from each instrument were exploited. Section \[s:results\] presents the findings. The conclusions and a discussion are presented in Sections \[s:conc\] and \[s:disc\], respectively.
Data Analysis {#s:data_anal}
=============
In order to cross-reference datasets from different instruments to infer which observed a given solar flare simultaneously, it is important to define what exactly constitutes a flare. The most commonly accepted catalog is that of the *Geostationary Operational Environmental Satellite* (GOES) event list provided by NOAA/SWPC. This defines a solar flare as a continuous increase in the one-minute averaged X-ray flux in the long-wavelength channel (1–8Å) of the GOES *X-ray Sensor* (XRS: @hans96) for the first four minutes of the event. The flux in the fourth minute must be at least 1.4 times the initial flux. The start time of the event is then defined as the first of these four minutes. The peak time is when the long-wavelength channel flux reaches a maximum, thus defining its class. The end of an event is defined as the time when the long channel flux reaches a level halfway between the peak and initial values[^4]. However in the vast majority of instances the NOAA catalog does not provide information on the location of a flare on the solar disk. As this is necessary for cross-referencing with the pointing information for reduced field-of-view instruments, the location of each flare was determined from the SSW Latest Events list, which is accessible through the Heliophysics Events Knowledgebase[^5] (HEK). Flare locations are determined by subtracting the SDO/*Atmospheric Imaging Assembly* (AIA: @leme12) 131Å image closest to the GOES start time, from that image closest to the GOES peak time. The flare location is then extracted from the peak intensity of this difference image (S. Freeland; private communication, 2017). Knowing the timing and position of each event then allowed this information to be cross-referenced with the metadata from other instruments to determine whether or not they observed the same location at the same time. *Note that this does not guarantee that a given instrument actually detected flaring emission, but only that the timing and pointing of a given dataset were consistent with the timing and location of the flare.* B-class flares were not included in this study due to discrepancies between flare locations derived from RHESSI and SDO/AIA and were therefore deemed unreliable. The SSW Latest Events list also has several months of data missing[^6]. Nevertheless, out of the 8090 flares of GOES class C1 or greater that appear in the NOAA event list, 6953 (86%) are also in the SSW Latest Events list and include location information.
![Solar Cycles 23 and 24 (average monthly sunspot number) with mission durations overplotted. The two vertical-dotted lines denote the 6.5-year time range considered for this study. Note that SDO/EVE MEGS-A and IRIS only overlapped for $\approx$11 months.[]{data-label="solar_cycle"}](solar_cycle_monthly_ssn-eps-converted-to.pdf){width="\textwidth"}
For the purposes of this study, only flares greater than GOES C1 class that occurred over the 6.5 years of Solar Cycle 24 observed by SDO [@pesn12] were considered. This defines the date range 1 May 2010 to 31 October 2016, as denoted by the vertical-dotted lines in Figure \[solar\_cycle\]. Also shown are the durations of the missions considered in this study. Note that EVE MEGS-A and IRIS were only operational together for around 11 months after IRIS was launched, and before MEGS-A suffered a power anomaly on 26 May 2014[^7].
Figure \[sff\_plot\] shows a sample plot from the widget, which was developed in tandem with this study (see Appendix \[appendixa\]). Plots such as this have been generated for every SSW event since the launch of SDO, and they are being continuously updated. These plots allow the user to readily view the timing and pointing of each instrument during a chosen event. This particular plot shows one flare from this study that was found to have been observed by all seven instruments: an M1.5 flare that occurred on 4 February 2014. The upper-left panel shows the GOES X-ray lightcurves with the start, peak, and end times overlaid (vertical gray-dotted, solid, and dashed lines, respectively). Note that for completeness, the time profiles of GOES EUVS-E (centered on the Lyman-$\alpha$ – [Ly$\alpha$]{} – line of hydrogen at 1216Å; @vier07) are also shown in gray. [@mill16] recently showed that these data are more reliable for flare studies than the EVE MEGS-P data given that the GOES/EUVS-E data exhibit a more impulsive profile – as one would expect for chromospheric emission – whereas current EVE MEGS-P data erroneously show a more gradually varying behavior.
![Sample event from this study that was observed by all instruments; an M1.5 flare that occurred on 4 February 2014. Upper left panel: GOES/XRS lightcurves in 1–8Å (solid black curve) and 0.5–4Å (dotted-black curve), along with the GOES/EUVS-E ([Ly$\alpha$]{}) profile in gray. Vertical dotted, solid, and dashed grey lines denote the start, peak, and end times of the GOES event, respectively. Dotted- and dashed-green ticks mark the start and end times of each *Hinode*/EIS raster, respectively, while red and yellow ticks mark the times of each SOT and XRT image, respectively. Horizontal blue and cyan lines illustrate the times which MEGS-A and MEGS-B were exposed, respectively, while the horizontal purple line shows the time of the corresponding IRIS study. Lower-left panel: RHESSI lightcurves up to the maximum energy detected, with GOES start, peak, and end times overlaid. Right panel: A PROBA2/SWAP 174Å image taken near the peak of the flare. The white circle is 100$''$ wide centered on the location derived from AIA 131Å images, while the black contours mark out the 6–25keV emission observed by RHESSI. The fields of view of EIS, SOT, XRT, and IRIS are overplotted in green, yellow, red, and purple, respectively.[]{data-label="sff_plot"}](20140204_152500_M1_hsi100_megsab_eis_sot_xrt_iris.png){width="\textwidth"}
The *Ramaty High Energy Solar Spectroscopic Imager* {#ss:rhessi}
---------------------------------------------------
RHESSI, launched on 5 February 2002[^8], observes the full disk of the Sun in X-rays and $\gamma$-rays. It orbits the Earth at an inclination angle of 38$^{\circ}$, at an altitude of $\approx$600 km, and as such suffers from eclipse passes and transits through the South Atlantic Anomaly. In order to determine whether or not RHESSI observed a given GOES flare event, the IDL routine was run between the start and end times of each flare. This searches the RHESSI flare catalog[^9] for the largest event detected in the time range of interest. If a RHESSI flare is detected, the fraction of the rise time (GOES start $\rightarrow$ GOES peak) that the RHESSI flare flag was active was also calculated. While RHESSI is a full-disk instrument, its orbit implies that it may have captured anywhere from a few seconds of a given flare up to around an hour (note that some long-duration flares are detectable over several RHESSI orbits).
From the lightcurves presented in the lower-left panel of Figure \[sff\_plot\] it can be seen that RHESSI captured the peak of the M1.5 flare up to an energy of 50–100keV. The contours of the RHESSI quicklook image (6–25keV; black contours overlaid on the EUV image) agree with the flare location computed from the AIA 131Å data (white circle).
The *EUV Variability Experiment* {#ss:sdo_eve}
--------------------------------
The SDO spacecraft is in a geosynchronous orbit allowing it to observe the full disk of the Sun continuously without interruption (except for the occasional lunar and terrestrial eclipses). For simplicity, it was assumed that both AIA and the *Helioseismic and Magnetic Imager* (HMI: @sche12) were observing continuously throughout each event. The EVE instrument, however, is less straightforward. While MEGS-A, which provides spatially integrated Sun-as-a-star spectra over the 60–370Å range every ten seconds, and was exposed to the Sun continuously from launch until it ceased operations on 26 May 2014, the MEGS-B (370–1050Å) and MEGS-P ([Ly$\alpha$]{}) exposure times have been much more erratic due to unforeseen degradation soon after launch. For much of the mission MEGS-B has only been exposed for three hours per day in order to limit degradation, as well as five minutes per hour for the consistency of long term variability studies. During periods of substantial solar activity it would observe continuously for 24–48 hours. Recently the flight software was changed to allow MEGS-B to respond to a flare trigger based on the EVE EUV Solar Photometer (ESP) flux for events $>$M1. Although there is an EVE flare catalog online[^10], this includes events for which MEGS-B may have only been exposed for five minutes. Therefore for the purposes of this study, MEGS-B was considered to have observed a flare if it was exposed to the Sun continuously between the GOES start and GOES peak times as determined from the daily exposure times[^11]. However, this does not necessarily mean that the flare itself will show up in the data, as EVE is often only sensitive to flares $\gtrsim$C5 level. The times at which MEGS-A and MEGS-B were exposed to the Sun around the time of a given flare are illustrated by the horizontal blue and cyan lines, respectively, as shown in the top-left panel of Figure \[sff\_plot\].
*Hinode* {#ss:hinode}
--------
The *Hinode* spacecraft [@kosu07] was launched into a Sun-synchronous orbit on 22 September 2006 and comprises three instruments: EIS, SOT, and XRT. They were designed to study the interplay between the photosphere and the corona by working in unison. However, by January 2008 *Hinode* had lost the use of its X-band transmitter, resulting in a dramatic reduction in the amount of data being transmitted to the ground.
### The Extreme-ultraviolet Imaging Spectrometer {#sss:eis}
EIS is a two-channel, normal-incidence EUV spectrometer. Its two channels cover the wavelength ranges 170–210Å and 250–290Å, selected to cover coronal emission lines with formation temperatures ranging from 8000 K (He [ii]{}) to 16 MK (Fe [xxiv]{}). It has a mirror that is tiltable in the solar X-direction, and is used to build up rastered spectral images of portions of the Sun in up to 25 spectral ranges. Additionally, EIS has both narrow (1$''$ and 2$''$ wide) slits, and wider (40$''$ and 266$''$ wide) imaging slots, with up to 512$''$ in the solar Y-direction.
For this study, a flare successfully observed by EIS must have had at least one raster begin, end, or straddle the GOES start and end times as determined from the routine. If such a raster exists, then all rasters within -30 minutes and +60 minutes of the GOES start and end times, respectively, are returned. The flare location as projected from AIA must have also lain within the EIS field of view. This does not imply that EIS captured any flaring emission; due to the rastering nature of the instrument, the slit may not have been over the flare site at the opportune time. In the example shown in Figure \[sff\_plot\], EIS was running a sequence of $\approx$three-minute rasters (denoted by the vertical green-dotted and dashed ticks) around the peak of the M1.5 flare. The associated regions of the Sun corresponding to each raster are also overlaid on the EUV image as green boxes.
### The Solar Optical Telescope {#sss:sot}
SOT is the first large optical telescope flown in space to observe the Sun. It images sub-full-disk portions of the Sun. Its aperture is 50 cm in diameter, the angular resolution is 0.25$''$ (corresponding to 175 km on the Sun), and the wavelengths covered extend from 4800 to 6500Å. SOT also includes the *Focal Plane Package*, which consists of a vector magnetograph and a spectrograph. The vector magnetograph provides time series of photospheric vector magnetograms, Doppler velocity and photospheric intensity.
In order to determine whether SOT observed a given flare, the routine was run between the GOES start and end times. If the routine returned at least one image, and the flare location fell within the SOT field of view, then all corresponding images between -30 and +60 minutes of the GOES start and end times, respectively, were returned and plotted over the GOES X-ray lightcurves as shown in Figure \[sff\_plot\] (vertical yellow ticks). The associated regions of the Sun corresponding to each SOT image are also overlaid on the EUV image as yellow boxes.
### The X-Ray Telescope {#sss:xrt}
XRT is a high-resolution (1$''$) grazing-incidence Wolter telescope that obtains high-resolution soft X-ray images covering the energy range 0.2 to 2 keV. This reveals magnetic-field configurations and their evolution, allowing the observation of energy buildup, storage, and release process in the corona for any transient event. XRT covers a wide temperature range from 0.5 to 10 million Kelvin allowing it to see coronal features that are not visible with a normal incidence telescope. XRT can observe the full disk of the Sun, but can also return sub-full-disk images, depending on the science goal of the observation.
In order to determine whether XRT observed a given flare, the routine was run between the GOES start and end times. If the routine returned at least one image, and the flare location fell within the XRT field of view, then all corresponding images between -30 and +60 minutes of the GOES start and end times, respectively, were returned and plotted over the GOES X-ray lightcurves as shown in Figure \[sff\_plot\] (vertical red ticks). The associated regions of the Sun corresponding to each XRT image are also overlaid on the EUV image as red boxes.
The *Interface Region Imaging Spectrometer* {#ss:iris}
-------------------------------------------
Launched on 27 June 2013 into a Sun-synchronous polar orbit, IRIS obtains UV spectra and images with high spatial (1/3$''$) and temporal resolution (one-second) focused on the solar chromosphere and transition region. The instrument comprises an ultraviolet telescope combined with an imaging spectrograph. IRIS records observations of material at specific temperatures, ranging from 5000 K and 65,000 K, and up to 10 MK during solar flares. IRIS is a sub-full-disk instrument, imaging portions of the solar disk and limb.
The timing and pointing of IRIS observation studies that were run during the start and end times of a given GOES event were obtained using the routine. This searches the Heliophysics Coverage Registry for the [^12] corresponding to the time of the flare, as shown by the horizontal purple line in the upper-left panel for Figure \[sff\_plot\]. Similar to the previously mentioned instruments with limited fields of view, the pointing information obtained from the was cross-referenced with the flare location to determine if IRIS was pointed at the required location (purple box overlaid on the EUV image in Figure \[sff\_plot\]).
[max width=]{}
------------------- ------------ ---------- --------- ------------ ----------------
Instrument/ Success Rate
Database C-class M-class X-class Total Over 6.5 Years
NOAA/GOES 7360 685 45 8090 100%
SSW Latest Events 6339 581 33 6953 86%
RHESSI 3673 370 23 4066 58%
SDO/EVE MEGS-A 3825 343 19 4187 100%
SDO/EVE MEGS-B 787 97 8 892 12%
*Hinode*/EIS 496 54 6 556 8%
*Hinode*/SOT 1167 177 15 1359 20%
*Hinode*/XRT 3739 357 26 4122 59%
IRIS 523 (3349) 76 (335) 5 (16) 604 (3700) 16%
------------------- ------------ ---------- --------- ------------ ----------------
: Distribution of how many solar flares – and of which class – were observed by individual instruments between 1 May 2010 and 31 October 2016 based on the timing and pointing information available (where applicable). The percentage of SSW Latest Events found is calculated relative to the number of NOAA/GOES events. Percentage of flares captured by each instrument during their respective missions are calculated against the total number of events found via SSW Latest Events.
[MEGS-A was assumed to have observed all flares from launch until it ceased operations on 26 May 2014]{}
[The total number of flares listed in the HEK between the launch of IRIS and 31 October 2016 are given in parentheses]{}
\[tab:instr\_flares\]
Results {#s:results}
=======
[max width=]{}
Degree Number of flares observed % of potentially observable flares
---------------------- --------------------------- ------------------------------------
No instruments 127 1.8%
Exactly 1 instrument 1432 20.6%
Any 2 instruments 2371 34.1%
Any 3 instruments 2035 29.2%
Any 4 instruments 720 10.3%
Any 5 instruments 228 3.3%
Any 6 instruments 37 0.5%
All 7 instruments 3 0.3%
: Number and percentage of total flares observed by different combinations of instruments. Note that there were 6953 flare events that were potentially observable by six or fewer instruments. Only 934 events were potentially observable by all seven of the instruments considered in this study.
[A total of 934 flares were recorded during the 11 months when both MEGS-A and IRIS were operational together.]{}
\[tab:joint\_flares\]
Based on the search criteria defined in Section \[s:data\_anal\], the number of flares, and their percentages of the total number of SSW Latest Events (which itself is a subset – 86% – of the available NOAA/GOES events) that were considered to have been observed by each of the instruments are listed in Table \[tab:instr\_flares\]. The instruments with full-disk capability and high duty cycles (RHESSI, MEGS-A, and *Hinode*/XRT) unsurprisingly were able to capture more than half of the total flares considered. The remaining instruments – which have either limited duty cycles and/or limited fields of view – were only able to capture around 20% or less of all flares during Solar Cycle 24. Similarly, the number of flares and their percentages that were observed by different combinations of instruments are listed in Table \[tab:joint\_flares\]. Around 84% of all flares were observed by between one and three instruments. Most of the remaining 16% were observed by either four or five instruments, while a total of 37 flares were observed by different combinations of six instruments and only three out of 934 were observed by all seven instruments during the 11 months that they were simultaneously operating. Interestingly, 127 flares (1.8%) were not observed at all by *any* of the seven instruments considered.
The findings of how many solar flares were observed by different combinations of instruments are displayed in Figures \[f:upset\] as [^13] plots [@lex14]. This type of plot enables the efficient visualization of common elements of a large number of sets (the more common and familiar Venn diagram approach produces ineffective visualizations for more than $\approx$five sets). The top panel of Figure \[f:upset\] shows the intersections of the various combinations of datasets ordered by decreasing frequency (i.e. the most common combinations are on the left and decrease towards the right), while the bottom panel shows the same information only now ordered by increasing number of instruments (i.e. flares observed by individual instruments alone come first, with flares observed by all seven on the far right). In each plot, the total number of flares observed by each instrument are given by the horizontal black bars in the bottom-left corner. The dots connected by lines at the bottom of each figure denote the combinations of instruments considered, while the histograms above give the number of events corresponding to a given combination. The most common combination of flare datasets was RHESSI+MEGS-A+*Hinode*/XRT (930 flares), due to their large fields of view and high duty cycles as mentioned above.\
![ plots of the intersection of flare datasets from each instrument as ordered by decreasing frequency (top panel) and increasing number of instruments (bottom panel). Zero-element sets are not included in either plot.[]{data-label="f:upset"}](UpSetR_dec_freq_cmx_nozero.pdf "fig:"){width="\textwidth"} ![ plots of the intersection of flare datasets from each instrument as ordered by decreasing frequency (top panel) and increasing number of instruments (bottom panel). Zero-element sets are not included in either plot.[]{data-label="f:upset"}](UpSetR_inc_deg_cmx_nozero.pdf "fig:"){width="\textwidth"}
Evaluation of Measured Versus Expected Success Rates
----------------------------------------------------
It is difficult to give a good estimate of how many flares one would *expect* to see with each instrument, given their different science goals and operational constraints. Table \[tab:estimated\_flares\] summarizes an attempt to estimate this expectation value \[${e}$\] for each instrument considered in this article. The estimates are based on the average field-of-views times the duty cycles of each instrument. The area of consideration is estimated in two ways. The first estimate is simply the area of the full disk of the Sun. The second estimate assumes that there are four active regions on the Sun each with an area of $240''\times240''$, and that the majority of the duty cycle is spent examining the active-region areas. These two estimates give an upper and lower range to the percentage field of view. The percentage field of view is calculated as the percentage of the area of consideration covered by the average field-of-view of the instrument disk. The duty cycle is estimated as the percentage of the time that the instrument could have observed a flare. Crucially, the estimates assume that a *random* location within the area of consideration (either the full disk of the Sun, or an estimated average area of active regions that the instrument could point to, assuming that active regions form the majority of target areas during the duty cycle).
[max width=]{}
---------------- ------------ --------- -------------- -------------- --
Instrument “Expected” Measured
Duty cycle %FOV Success Rate Success Rate
${e}$ ${m}$
RHESSI 50% 100% 50% 58%
SDO/EVE MEGS-A 100% 100% 100% 100%
SDO/EVE MEGS-B 12.5% 100% 12.5% 12%
*Hinode*/EIS 25% 2–25% 0.5–6% 6%
*Hinode*/SOT 50% 1–17% 0.5–8% 13%
*Hinode*/XRT 100% 25–100% 25–100% 57%
IRIS 100% 0.5–3% 0.5–3% 11%
---------------- ------------ --------- -------------- -------------- --
: Estimates of the percentage of flares expected to be observed \[${e}$\] by each instrument based on the product of their duty cycles and field-of-view. The calculation assumes that each instrument points randomly in the area of consideration. The percentage of flares that were actually observed \[${m}$\] is also presented.
[Duty cycle estimated at approximately three hours per day (see text).]{}
[Duty cycle estimated by examining recent EIS planning notes. Field of view estimated at $240''\times 240''$, one quarter the full FOV of EIS.]{}
[Field of view estimated at $200''\times200''$, one quarter the full FOV of SOT.]{}
[Field of view estimated at $1024''\times1024''$, one quarter the full FOV of XRT.]{}
[Field of view estimated at $85''\times85''$, one quarter the full FOV of IRIS.]{}
[Out of the 934 flares listed in the HEK over the 11-month period that all seven instruments were operational. This is around half of all the flares listed in the NOAA/GOES event list (1774).]{}
\[tab:estimated\_flares\]
A very crude estimate of the “expected” success rate \[${e}$\] is therefore the product of the duty cycle and the FOV. This can be readily compared to the measured success rate \[${m}$\] for each instrument individually from the 11-month period during which all seven instruments were operational together. This time period also happened to coincide with the peak of the solar cycle as illustrated in Figure \[solar\_cycle\]. These expected and measured values are presented in the last two columns of Table \[tab:estimated\_flares\]. The success rates over this 11-month period bear a reasonable agreement with the values measured over the entire 6.5 years under study that are presented in the final column of Table \[tab:instr\_flares\], and they can therefore be considered as characteristic of each instrument. They are also consistent with or better than the individual expected value implying that each pointing instrument is performing well. This reflects the fact that solar flares are a high priority science goal for these instruments, and that operators of course do not point their instruments randomly.
The measured success rates of each individual instrument in Table \[tab:estimated\_flares\] can be used to predict the number of flares expected to be seen by different combinations of instruments as follows. The measured success rate of each instrument indicates the probability \[${m}$\] of an instrument observing a flare. Therefore $1-{m}$ indicates the probability of an instrument missing a flare. This suggests the use of the binomial distribution to model the number of flares detected by each instrument individually. The binomial distribution is $$B(k; N, {m}) = {{N}\choose{k}}{m}^{k}(1-{m})^{N-k}
\label{eqn:binomial}$$ where $k$ is the number of successful outcomes, $N$ is the number of trials and $${{N}\choose{k}}=\frac{N!}{(N-k)!k!}
\label{eqn:choose}$$ is the binomial coefficient. Consider the case of two instruments observing the same flare. If the flare is observed by the first instrument with a probability ${m}_{1}$, and by the second instrument with a probability ${m}_{2}$, conditional on the first instrument having observed the flare, then the resulting probability of observing $k$ flares from a possible $N$ is $$B(k; N, {m}_{1}{m}_{2}).
\label{eqn:binomialcombined}$$ This can be used to calculate ${n_\textrm{expected}}$, the expected numbers of flares observed by arbitrary combinations of instruments. We define ${n_\textrm{expected}}$ to be the mean value of the probability mass functions (PMF) defined by Equation \[eqn:binomialcombined\]. This value can be compared to the actual number of flares observed \[${n_\textrm{measured}}$\]. If the actual number of flares observed \[${n_\textrm{measured}}$\] is much larger than the expected number due to chance \[${n_\textrm{expected}}$\] then this can be taken as evidence that instruments were acting in cooperation, or at least had aligned goals. The number of distinct subsets of combinations of $r$ instruments from the seven instruments that we are considering is ${{7}\choose{r}} ($Equation \[eqn:choose\]). For example, there are 21 possible combinations of two out of seven instruments, 35 combinations of three, and so on.
![The probability mass functions for three different combinations of instruments: RHESSI+XRT (left), EIS+SOT (center), and IRIS+EIS+SOT (right).[]{data-label="binomial_pmf"}](plot_binomial_pmf-eps-converted-to.pdf){width="\textwidth"}
Figure \[binomial\_pmf\] shows example PMFs for three combinations of instruments using Equation \[eqn:binomialcombined\]: RHESSI+XRT, EIS+SOT, and IRIS+EIS+SOT. The first two panels show that their respective probabilities display a distribution peaked on the most probable number of flares observed (309 and 7, respectively, out of 934), while the right-hand panel shows that the most probable number of flares observed simultaneously by IRIS+EIS+SOT is zero. Defining ${n_\textrm{expected}}$ to be the mean of the PMF ensures that ${n_\textrm{expected}}>0$. The uncertainty on ${n_\textrm{expected}}$ is estimated by calculating the standard deviation of the PMF.
![The expected mean number of flares observed \[${n_\textrm{expected}}$\] by various combinations of instruments versus the number actually observed \[${n_\textrm{measured}}$\]. The diagonal dotted line marks the 1:1 ratio, while the diagonal dashed line denotes the 10:1 ratio. Some of the data points overlap since some combinations of instruments are subsets of combinations of more instruments. In the case when the mean number minus the standard deviation is less than zero, the error bar is extended to the lower value of the plot range.[]{data-label="meas_pred"}](meas_pred_rates_plot-eps-converted-to.pdf){width="80.00000%"}
A scatter plot showing the expected mean number of flares out of a possible 934 that were observed \[${n_\textrm{expected}}$\] by 2–7 instruments, against the number actually observed \[${n_\textrm{measured}}$\] is shown in Figure \[meas\_pred\]. Points close to the 1:1 (dotted) line indicate that those combinations of instruments are co-observing flares at a rate consistent with Equation \[eqn:binomialcombined\], i.e. the measured co-observation rate is close to that expected by chance.
For combinations of four or more instruments, the number of flares actually observed is far larger than that expected randomly (Equation \[eqn:binomialcombined\]). While the expected mean number of flares observed is small (and sometimes less than unity) for increasing numbers of coordinating instruments the number of flares actually observed is often up to ten times greater than expected if the instruments operated without co-ordination. This shows that when a flare-productive active region is present on the Sun, many instrument planners will choose to track the region – within their operational constraints – thereby greatly increasing the likelihood of jointly observing a given flare in conjunction with other missions. The statistics for the expected and measured number of flares observed by each possible combination of 2–6 instruments is given in Appendix \[appendixb\].
Conclusions {#s:conc}
===========
A statistical analysis of how many solar flares ($\geq$C1) were observed by various combinations of instruments during the 6.5 years after the launch of SDO in Solar Cycle 24 is presented. On average, over the entire 6.5 years, each flare was observed by 2.4 instruments. Out of the 6953 flares considered, only three were observed simultaneously by RHESSI, MEGS-A+B, *Hinode*/EIS+SOT+XRT and IRIS: a C2.3 flare on 1 February 2014, a C4.6 flare on 3 February 2014, and an M1.5 flare on 4 February 2014. The occurrence of these three events coincided with a *Max Millennium Major Flare Watch* campaign that ran from 30 January 2014–8 February 2014 on NOAA active region 11967[^14]. This illustrates the value of tracking the target region as advised by the *Max Millennium Chief Observers* when trying to optimize the scientific return on solar flare datasets. Note that all seven instruments were observing contemporaneously for only 11 months, and in this time 934 events are currently listed as SSW Latest Events; therefore 0.3% of all possible GOES flares were observed with all seven instruments. While this may not seem impressive, the probability of all seven instruments observing a single flare simultaneously by chance is 0.003%. Similarly, for combinations of four, five, or six instruments the number of flares captured is often a factor of ten or more greater than random. This shows that instrument planners are intentionally co-observing the same flare-productive active regions. The analysis in Section \[s:results\] suggests that multi-instrument observations are occurring much more frequently than expected by chance. It should be noted that this conclusion is reached by assuming that each instrument is acting independently of all the others. This is a dubious assumption, as the community is aware of the scientific value of multi-instrument observations of solar flares, and many of the people who operate flare-observing instruments have a professional interest in studying flares. It does not take account of known existing community efforts that are designed to promote flare co-observation such as the *Max Millennium Program*, or joint observing programs such as the *Hinode* Operations Program[^15] that often specifically request that multiple instruments point to the same target. As the number of instruments goes up the deviation towards increased co-observation increases suggesting that the co-observation rate depends not only on which instruments are observing (the $m_{i}$s in Equation \[eqn:binomialcombined\]) but also on the number of instruments. This suggests that instrument operators are more likely to co-observe if a number of other instruments are already co-observing a target.
Unsurprisingly the instruments with the longest duty cycles and largest fields of view (RHESSI, MEGS-A, XRT) performed the best individually. Reduced field-of-view instruments that require operations planning, while performing within their expected success rates, still only captured on the order of 20% or less of all flares. The possible reasons for this have already been touched upon, but may also be due to other specific factors. For example, until recently, EIS (which observed 6–8% of flares) only received 15% of *Hinode’s* total telemetry since January 2008, thereby limiting its daily duty cycle. This has since been revised up to 23–43%[^16] which should improve its statistics. Similarly, MEGS-B was often only exposed to the Sun for three hours per day to minimize detector degradation, but now responds to a flare trigger when the flux level at 1–7Å as measured by the ESP component of EVE exceeds the GOES M1 level [@mill16]. This effectively makes MEGS-B a dedicated flare instrument and it is likely to observe a higher fraction of flares in the future.
Discussion {#s:disc}
==========
This study raises a significant question for the solar-physics community: even though we seem to be co-observing flares at rates larger than those expected by chance, is the number of co-observed flares acceptable or not? This is not an easy question to answer. This study presents some analysis regarding the retrospective behavior of the community regarding co-observation of solar flares, but does not comment on what the solar-physics community wants to do, or should do. It seems obvious that increasing the number of co-observed flares is desirable, but there are three factors for the community to consider in relation to this question.
The first factor is the desire of the community to use limited instrumental resources to study flares compared to other solar features and phenomena. If the community decides that the study of solar flares is relatively more important than the study of other features and phenomena, then their relatively unpredictable occurrence means that a greater fraction of each instrument’s observational resources should be devoted to capturing as much data as possible when a flare-productive region appears (*e.g.* in response to a *Max Millennium Major Flare Watch*). Other more “quiescent” targets (e.g. coronal holes, filaments, plage regions, etc.) are much more commonplace and can be observed at almost any time.
Having decided that flare research is important, the next factor to consider is our ability to predict when and where a flare will occur. Without a reliable method of predicting when and where a solar flare will occur, we are left with trying to optimize instrumental resources in the face of incomplete information as well as each instruments’ operational constraints and competing scientific priorities. The *Max Millennium Program* aims to provide an assessment of the likelihood of a flare in a given region over the following 24 hours. As well as a human assessment of flare likelihood, we suggest that machine-learning techniques be employed as another tool to aid the human flare forecaster. For example, [@bobr15] use support vector-machine-methods to determine flare probabilities. Another possible approach is to aggregate results from all existing flare-prediction tools to provide a single, combined measure assessing flare likelihood. It is also fundamentally important that support for the basic science of understanding how and why a solar flare is (or is not) triggered continues.
With a target selected, the final factor to consider is the number of co-observing instruments required to answer the particular science question. The utility of a set of multi-instrument flare observations depends on the science question being asked. All flares do not have to be observed by all instruments all of the time. For example, to understand the dynamic response of the chromosphere to energy deposited by nonthermal electrons, perhaps only RHESSI, EIS, and IRIS data are necessary [@bros16]. For flare differential emission measure studies, maybe having simultaneous RHESSI and EVE [@casp15] or EIS and XRT observations [@odwy14] are desirable. This is the primary function of the IDL widget described in Appendix \[appendixa\]: to allow users to quickly and easily search for joint datasets of solar-flare observations in order to answer a specific science question. Assessing the scientific impact of flare co-observations is difficult. It may be possible to measure the scientific impact of flare co-observation through citation analysis (for example, @desollaprice510 [@Giles:1998:CAC:276675.276685; @kaur]) of flare articles as a function of number of instruments. This type of study is beyond the scope of this article, but it would provide more information to the community in understanding the scientific impact of flare co-observations.
Understanding how a solar flare operates is a fundamental challenge to our understanding of the Sun and the conditions of the heliosphere. In the upcoming Solar Cycle 25, the *Daniel K. Inouye Solar Telescope*, *Solar Orbiter*, and *Parker Solar Probe* will all be operational. These facilities all have limited duty cycles, and different operational constraints. Optimizing the solar-flare science return from these and other instruments relies on continued inter-instrument co-ordination where possible. We suggest that each instrument’s observational plans be made available online, ideally in a commonly agreed format. Tools should be developed to read and visualize those plans (the Helioviewer Project clients and could be extended to present observation plans) that could take into account solar differential rotation, overplotting it on images from many different instruments. This will enable instrument operators, scientists and other users to understand how and why particular observations are being planned. We suggest that increased planning transparency will inevitably lead to an increased understanding of how an instrument’s operations create the *revealed* science priorities of an instrument, as opposed to its *stated* priorities. From this basis a better understanding of how to co-ordinate co-observations between instruments can be generated. Finally, it should be noted that co-observation of *non-flaring* regions is also of considerable scientific value. Co-observations that do not catch a flare are not without merit; much can be learned about the physics of active regions, the chromosphere–corona connection, polarity inversion lines, sunspots, etc, using observations from multiple instruments.
R.O. Milligan is grateful for financial support from NASA LWS/SDO Data Analysis grant NNX14AE07G, the Science and Technologies Facilities Council for the award of an Ernest Rutherford Fellowship (ST/N004981/1), and to Kim Tolbert for help with developing the widget. J. Ireland acknowledges gratefully the support of the Heliophysics Data Environment Enhancement program and the Solar Data Analysis Center. Both authors are also very grateful to the anonymous referee who provided critical and constructive feedback that greatly improved the quality and scope of this article. Error-bar estimates on the expected number of flares were calculated using [@scipy].
Disclosure of Potential Conflicts of Interest {#disclosure-of-potential-conflicts-of-interest .unnumbered}
=============================================
The authors declare that they have no conflicts of interest.
Solar Flare Finder Widget {#appendixa}
=========================
![Screenshot of the Solar Flare Finder widget in . The sample flare shown is an C2.3 flare that was observed simultaneously by all instruments on 4 February 2014.[]{data-label="sff_idl"}](ssw_sff_all_insts_20140201.png){width="\textwidth"}
The list of solar flares used for the analysis in this article were used as the source data for the , a widget that was developed in tandem with this study that allows users to search for flares observed by a chosen set of instruments. The Solar Flare Finder widget is available now via (@free98; ). A screenshot is shown in Figure \[sff\_idl\]. The widget searches a pre-generated lookup table[^17] (the same lookup table used to generate the plots in Figure \[f:upset\], only with B-class flares included and which is continuously being updated) to return SSW Latest Events simultaneously observed by selected instruments. The widget allows the user to search by GOES class (B, C, M, X), flare location (disk; $>-600''\,--\,<+600''$ or limb; $<-600''\,--\,>+600''$), percentage of the rise phase covered by RHESSI ($>$0% or $>$90%), and by the maximum energy recorded by RHESSI. The widget returns a list of flares conforming to the users specifications (if any), allowing the user to click on a desired event to bring up a plot similar to that shown in Figure \[sff\_plot\] that displays the metadata from all available datasets. These plots, and the associated metadata (in the form of an IDL .sav file), are downloadable via the widget, and are hosted at <http://hesperia.gsfc.nasa.gov/sff/>. Note that no guarantees are made regarding the quality of the data itself. Perhaps the EIS or IRIS slits may not have been precisely aligned with the flare ribbons during the impulsive phase, or the solar background may have been sufficiently high that, say, $<$C5 flares do not show up in EVE data. However, this tool aims to greatly narrow the search for specific events that match a user’s request in order to answer particular science questions.
Expected Versus Measured Number of Flares {#appendixb}
=========================================
The five panels in Figure \[fig:pred\_succ\] compare the expected number of flares \[${n_\textrm{expected}}$: blue triangles\] of a given combination of instruments observing a flare (based on individual measured success rates, ${m}$, from Table \[tab:estimated\_flares\] and Equation \[eqn:binomialcombined\]) with the actual measured values \[${n_\textrm{measured}}$, red diamonds\] during the 11-month period for which all seven of the instruments were operating. The ratio of these values \[${n_\textrm{measured}}/{n_\textrm{expected}}$: solid black circles\] indicates how successful each given combination has performed. A value greater than unity indicates that a given combination has performed better than random. For almost all combinations of instruments (particularly higher $r$ values; $>$4), ${n_\textrm{measured}}>>{n_\textrm{expected}}$. Furthermore, if we consider the probability of all seven instruments targeting the same flare independently, then the product of the individual expected values gives us a probability of 0.003%. Therefore the measured value of three flares (0.3% of 934) is actually considerably better than what one might expect for such a fortuitous combination of observations.
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: 2012, [The Role of Solar Flares in the Variability of the Extreme Ultraviolet Solar Spectral Irradiance]{}. PhD thesis, University of Colorado at Boulder. .
, , , et al.: 2001, *[SciPy: Open source scientific tools for Python]{}*.
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[^1]: [solarflare.njit.edu](solarflare.njit.edu)
[^2]: [www.nasa.gov/content/goddard/nasa-telescopes-coordinate-best-ever-flare-observations/](www.nasa.gov/content/goddard/nasa-telescopes-coordinate-best-ever-flare-observations/)
[^3]: [sne.space/statistics/](sne.space/statistics/)
[^4]: [www.swpc.noaa.gov/products/goes-x-ray-flux](www.swpc.noaa.gov/products/goes-x-ray-flux)
[^5]: [www.lmsal.com/hek/](www.lmsal.com/hek/)
[^6]: October–December 2012; July–November 2013; May 2014; February 2015; March and June 2016.
[^7]: [lasp.colorado.edu/home/eve/2014/05/28/eve-megs-a-power-anomaly/](lasp.colorado.edu/home/eve/2014/05/28/eve-megs-a-power-anomaly/)
[^8]: The first solar flare observation was of a GOES C2 flare on 12 February 2002.
[^9]: [hesperia.gsfc.nasa.gov/hessidata/dbase/hessi\_flare\_list.txt](hesperia.gsfc.nasa.gov/hessidata/dbase/hessi_flare_list.txt)
[^10]: [lasp.colorado.edu/eve/data\_access/evewebdata/interactive/eve\_flare\_catalog.html](lasp.colorado.edu/eve/data_access/evewebdata/interactive/eve_flare_catalog.html)
[^11]: [lasp.colorado.edu/eve/data\_access/evewebdata/interactive/megsb\_daily\_exposure\_hours.html](lasp.colorado.edu/eve/data_access/evewebdata/interactive/megsb_daily_exposure_hours.html)
[^12]: See [iris.lmsal.com/itn26/quickstart.html](iris.lmsal.com/itn26/quickstart.html) for more details.
[^13]: [gehlenborglab.shinyapps.io/upsetr/](gehlenborglab.shinyapps.io/upsetr/)
[^14]: [solar.physics.montana.edu/hypermail/mmmotd/index.html](solar.physics.montana.edu/hypermail/mmmotd/index.html)
[^15]: [www.isas.jaxa.jp/home/solar/guidance/index.html](www.isas.jaxa.jp/home/solar/guidance/index.html)
[^16]: [solarb.mssl.ucl.ac.uk/SolarB/hinode\_revised\_tlm.html](solarb.mssl.ucl.ac.uk/SolarB/hinode_revised_tlm.html)
[^17]: [hesperia.gsfc.nasa.gov/sff/ssw\_sff\_list.txt](hesperia.gsfc.nasa.gov/sff/ssw_sff_list.txt)
| 0 |
---
abstract: 'We study a three-dimensional dynamical system in two slow variables and one fast variable. We analyze the tangency of the unstable manifold of an equilibrium point with “the” repelling slow manifold, in the presence of a stable periodic orbit emerging from a Hopf bifurcation. This tangency heralds complicated and chaotic mixed-mode oscillations. We classify these solutions by studying returns to a two-dimensional cross section. We use the intersections of the slow manifolds as a basis for partitioning the section according to the number and type of turns made by trajectory segments. Transverse homoclinic orbits are among the invariant sets serving as a substrate of the dynamics on this cross-section. We then turn to a one-dimensional approximation of the global returns in the system, identifying saddle-node and period-doubling bifurcations. These are interpreted in the full system as bifurcations of mixed-mode oscillations. Finally, we contrast the dynamics of our one-dimensional approximation to classical results of the quadratic family of maps. We describe the transient trajectory of a critical point of the map over a range of parameter values.'
author:
- Ian Lizarraga
bibliography:
- 'shnfsaa4-preprint3.bib'
title: 'Tangency bifurcation of invariant manifolds in a slow-fast system'
---
> We study a three-dimensional multiple timescale system in five parameters. A startling variety of behaviors can be identified as its five parameters are varied. Organizing this variety are the interactions between classical invariant manifolds (including fixed points, periodic orbits, and their (un)stable manifolds) and locally invariant slow manifolds. Here we focus on the interaction between the two-dimensional unstable manifold of a saddle-focus equilibrium point and a two-dimensional repelling slow manifold, in the presence of a stable periodic orbit of small amplitude.
>
> The images of global return maps, defined on carefully chosen two-dimensional cross-sections, are organized by the interactions of the attracting and repelling slow manifolds with these cross-sections. They are also influenced by the basin of attraction of the periodic orbit. We construct a symbolic map which partitions one such section according to the number and type of turning behaviors of the corresponding trajectories. We locate transverse homoclinic orbits to saddle points. On another cross-section, global returns are well-approximated by one-dimensional, nearly unimodal maps. We show that saddle-node bifurcations of periodic orbits and period-doubling cascades occur. Finally, we describe the dynamics of the critical point of the return map at carefully chosen parameters.
>
> Taking a broader view, our numerical results continue to point to the fruitful connections that exist between multiple-timescale flows and low-dimensional maps.
\[sec:intro\] Introduction
==========================
We study slow-fast dynamical systems of the form
$$\begin{aligned}
{ \varepsilon}\dot{x} &=& f(x,y,{ \varepsilon})\\
\dot{y} &=& g(x,y,{ \varepsilon}),\end{aligned}$$
where $x \in R^m$ is the [*fast*]{} variable, $y \in R^n$ is the [*slow*]{} variable, ${ \varepsilon}$ is the [*singular perturbation parameter*]{} that characterizes the ratio of the timescales, and $f,g$ are sufficiently smooth. The [*critical manifold*]{} $C = \{f = 0\}$ is the manifold of equilibria of the fast subsystem defined by $\dot{x} = f(x,y,0)$. When ${ \varepsilon}> 0$ is sufficiently small, theorems of Fenichel[@fenichel1972] guarantee the existence of locally invariant [*slow manifolds*]{} that perturb from subsets of $C$ where the equilibria are hyperbolic. We may also project the vector field $\dot{y} = g(x,y,0)$ onto the tangent bundle $TC$. Away from folds of $C$, we may desingularize this projected vector field to define the [*slow flow*]{}. The desingularized slow flow is oriented to agree with the full vector field near stable equilibria of $C$. For sufficiently small values of ${ \varepsilon}$, trajectories of the full system can be decomposed into segments lying on the slow manifolds near $C$ together with fast jumps across branches of $C$. Trajectory segments lying near the slow manifolds converge to solutions of the slow flow as ${ \varepsilon}$ tends to 0.
\(a) {width="45.00000%"} (b) {width="45.00000%"}
We now focus on the case of two slow variables and one fast variable ($m=1$, $n=2$). The critical manifold $C$ is two-dimensional and folds of $C$ form curves. Points on fold curves are called [*folded singularities*]{}. when the slow flow is two-dimensional we use the terms “folded node”, “folded focus”, and “folded saddle” to denote folded singularities of node-, focus-, and saddle-type, respectively. In analogy to classical bifurcation theory, folded saddle-nodes are folded singularities having a zero eigenvalue. When they exist, folded saddle-nodes are differentiated by whether they persist as equilibria in the full system of equations. We are interested here in folded-saddle nodes of type II (FSNII), which are true equilibria of the full system. It can be shown that [*singular Hopf bifurcations*]{} occur generically at distances $O({ \varepsilon})$ from the FSNII bifurcation in parameter space.[@guckenheimer2008siam] At this bifurcation, a pair of eigenvalues of the linearization of the flow crosses the imaginary axis, and a small-amplitude periodic orbit is born at the bifurcation point.
Normal forms are used to study the local flow of full systems in neighborhoods of these folded singularities. Previous work by Guckenheimer[@guckenheimer2008chaos] analyzes the local flow maps and return maps of three-dimensional systems containing folded nodes and folded saddle-nodes. There, it is shown that the appearance of these folded singularities can give rise to complex and chaotic behavior. Characterizing the emergence of small-amplitude oscillations near a folded singularity has also been the subject of intense study. In the case of a folded node, Benoît[@benoit1990] and Wechselberger[@wechselberger2005] observed that the maximum number of small oscillations made by a trajectory passing through the folded node region is related to the ratio of eigenvalues of the folded node.
The present paper focuses on a dynamical system, defined in Sec. \[sec:shnf\], which contains folded singularities lying along a cubic critical manifold. The critical manifold serves as a global return mechanism. Parametric subfamilies of this dynamical system have served as important prototypical models of electrochemical oscillations, including the Koper model[@koper1992]. This system serves as a concrete, minimal example of a three-dimensional system having an $S$-shaped critical manifold as a global return mechanism. Trajectories leaving a neighborhood of the folded singularities do so by jumping between branches of the critical manifold, before ultimately being reinjected into the regions containing the folded singularities. This interplay between local and global mechanisms gives rise to [*mixed-mode oscillations*]{} (MMOs), which are periodic solutions of the dynamical system containing large and small amplitudes and a distinct separation between the two. These solutions may be characterized by their signatures, which are symbolic sequences of the form $L_1^{s_1}L_2^{s_2} \cdots$. This notation is used to indicate that a particular solution undergoes $L_1$ large oscillations, followed by $s_1$ small oscillations, followed by $L_2$ large oscillations, and so on. The distinction between ‘large’ and ‘small’ oscillations is dependent on the model. Nontrivial aperiodic solutions are referred to as [*chaotic MMOs*]{}, and may be characterized as limits of families of MMOs as the lengths of the signatures grow very large.
The classification of routes to MMOs with complicated signatures as well as chaotic MMOs continues to garner interest. Global bifurcations have been identified as natural starting points in this direction. Even so, the connection between these bifurcations and interactions of slow manifolds—which organize the global dynamics for small values of ${ \varepsilon}$—remains poorly understood. Period-doubling cascades, torus bifurcations,[@guckenheimer2008siam] and most recently, Shilnikov homoclinic bifurcations,[@guckenheimer2015] have been shown to produce MMOs with complex signatures. In the last case, one-dimensional approximations of return maps were used to analyze a Shilnikov bifurcation in a system which exhibits singular Hopf bifurcation.
In this paper, we use a similar technique to analyze a tangency of invariant manifolds. Our starting point is a study by Guckenheimer and Meerkamp[@guckenheimer2012siam], which comprehensively classifies local and global unfoldings of singular Hopf bifurcation. We describe the changes in the phase space as the unstable manifold of the saddle-focus equilibrium point crosses the repelling slow manifold of the system. Our approach takes for granted the complicated crossings of these two-dimensional manifolds, instead focusing directly on the influence of these crossings on the global dynamics. The main tool in our analysis is the approximation of the two-dimensional return map by a map on an interval, which parametrizes trajectories beginning on the attracting slow manifold. We show that in the presence of a small-amplitude stable periodic orbit, the one-dimensional return map has a rich topology. The domain of the map is disconnected, with components separated by finite-length gaps. Intervals where the return map is undefined correspond to bands of initial conditions in the full system whose forward trajectories asymptotically approach the small-amplitude stable periodic orbit without making a large-amplitude passage. The first and second derivatives of the map grow very large outside of large subintervals where the map is unimodal.
We also interpret classical bifurcations of the one-dimensional map as routes to chaotic behavior in the full system. We show that a period-doubling cascade occurs in this map, which gives rise to chaotic MMOs. This cascade is reminiscent of the classical cascade in the family of quadratic maps, even though on small subsets, our return map is far from unimodal. Saddle-nodes of mixed-mode cycles, defined as fixed points of the return map with unit derivative, are also shown to occur. Finally, we identify a parameter set for which the full dynamics is close to the dynamics of a unimodal map with a critical point having dense forward orbit.
\[sec:shnf\] Three-Dimensional System of Equations
==================================================
We study the following three-dimensional flow: $$\begin{aligned}
{ \varepsilon}\dot{x} &=& y - x^2 - x^3 \nonumber\\
\dot{y} &=& z - x \label{eq:shnf}\\
\dot{z} &=& -\nu - ax -by - cz,\nonumber\end{aligned}$$
where $x$ is the fast variable, $y,z$ are the slow variables, and ${ \varepsilon},\nu,a,b,c$ are the system parameters. This system exhibits a singular Hopf bifurcation.[@braaksma1998; @guckenheimer2008siam; @guckenheimer2012dcds] The critical manifold is the S-shaped cubic surface $C = \{y = x^2 + x^3\}$ having two fold lines $L_0 := S \cap \{x = 0\}$ and $L_{-2/3} := S \cap \{ x = -2/3\}$. When ${ \varepsilon}> 0 $ is sufficiently small, nonsingular portions of $C$ perturb to families of slow manifolds: near the branches $S\cap \{x > 0\}$ (resp. $S \cap \{x < -2/3\}$), we obtain the [*attracting slow manifolds*]{} $S^{a+}_{{ \varepsilon}}$ (resp. $S^{a-}_{{ \varepsilon}}$) and near the branch $S\cap\{-2/3<x<0\}$ we obtain the [*repelling slow manifolds*]{} $S^r_{{ \varepsilon}}$. Nearby trajectories are exponentially attracted toward $S^{a\pm}_{{ \varepsilon}}$ and exponentially repelled from $S^r_{{ \varepsilon}}$. One derivation of these estimates uses the Fenichel normal form.[@jones1994] Within each family, these sheets are $O(-\exp(c/ { \varepsilon}))$ close[@jones1994; @jones1995], so we refer to any member of a particular family as ‘the’ slow manifold. This convention should not cause confusion.
We focus on parameters where forward trajectories beginning on $S^{a+}_{{ \varepsilon}}$ interact with a ‘twist region’ near $L_0$, a saddle-focus equilibrium point $p_{eq}$, or both. A folded singularity $n =(0,0,0) \in L_0$ is the governing center of this twist region. The saddle-focus $p_{eq}$ has a two-dimensional unstable manifold $W^u$ and a one-dimensional stable manifold $W^s$. This notation disguises the dependence of these manifolds on the parameters of the system.
Tangency bifurcation of invariant manifolds
===========================================
Guckenheimer and Meerkamp[@guckenheimer2012siam] drew bifurcation diagrams of the system in a two-dimensional slice of the parameter space defined by ${ \varepsilon}= 0.01$, $b = -1$, and $c = 1$. Codimension-one tangencies of $S^r_{{ \varepsilon}}$ and $W^u$ are represented in Figure 5.1 of their paper by smooth curves (labeled T) in $(\nu,a)$ space. For fixed $a$ and increasing $\nu$, this tangency occurs after $p_{eq}$ undergoes a supercritical Hopf bifurcation. A parametric family of stable limit cycles emerges from this bifurcation. Henceforth we refer to ‘the’ small-amplitude stable periodic orbit $\Gamma$ to refer to the corresponding member of this family at a particular parameter set. The two-dimensional stable manifolds of $\Gamma$ interact with the other invariant manifolds of the system. Guckenheimer and Meerkamp identify a branch of period-doubling bifurcations as $\nu$ continues to increase after the first slow-manifold tangency. We show that the basin of attraction of the periodic orbit has a significant influence on the global returns of the system.
Fixing $a = -0.03$, the tangency occurs within the range $\nu \in \left[ 0.00647, 0.00648\right]$. The location of the tangency may be approximated by studying the asymptotics of orbits beginning high up on $S^{a+}_{{ \varepsilon}}$. Fix a section $\Sigma = S^{a+}_{{ \varepsilon}}\cap \{x = 0.27\}$. Before the tangency occurs, trajectories lying on and sufficiently near $W^u$ must either escape to infinity or asymptotically approach $\Gamma$; these trajectories cannot jump to the attracting branches of the slow manifold, as they must first intersect $S^r_{{ \varepsilon}}$ before doing so. Trajectories beginning in $\Sigma$ first flow very close to $p_{eq}$. As shown in Figure 1, these trajectories then leave the region close to $W^u$. We observe that before the tangency, $W^u$ forms a boundary of the basin of attraction of $\Gamma$. Therefore, all trajectories sufficiently high up on $S^{a+}_{{ \varepsilon}}$ must lie inside the basin of attraction (Figure \[fig:tangbif\]a).
After the tangency has occurred, isolated trajectories lying in $W^u$ will also lie in $S^r_{{ \varepsilon}}$. These trajectories will bound sectors of trajectories which can now make large-amplitude passages. Trajectories within these sectors jump ‘to the left’ toward $S^{a-}_{{ \varepsilon}}$ or ‘to the right’ toward $S^{a+}_{{ \varepsilon}}$. Trajectories initialized in $\Sigma$ that leave neighborhoods of $p_{eq}$ near these sectors contain [*canard*]{} segments, which are solution segments lying along $S^r_{{ \varepsilon}}$. Examples of such trajectories are highlighted in green in Figure 1. We can now establish a dichotomy between those trajectories in $\Sigma$ that immediately flow to $\Gamma$ and never leave a small neighborhood of the periodic orbit, versus those that make a global return. In Figure \[fig:tangbif\]b, only two of the thirty sample trajectories are able to make a global return. Near the boundaries of these subsets, trajectories can come arbitrarily close to $\Gamma$ before escaping and making one large return. Note however that such trajectories might still lie inside the basin of attraction of $\Gamma$, depending on where they return on $\Sigma$. Such trajectories escape via large-amplitude excursions at most finitely many times before tending asymptotically to $\Gamma$. We now focus on the parameter regime where the tangency has already occurred. In Figure 5.1 of the paper of Guckenheimer and Meerkamp, this corresponds to the region to the right of the $T$ (manifold tangency) curve.
![\[fig:retmap\] The return map $R: \Sigma_+ \to \Sigma_+$ of the system with $\Sigma_+ = \{x = 0.3\}$. Points in the two-dimensional section are parametrized by their $z$-coordinates. The dashed black line is the line of fixed points $\{(z,z)\}$. Parameter set: $\nu = 0.00802$, $a = -0.3$, $b = -1$, $c = 1$.](2png.png){width="50.00000%"}
\[sec:maps\] Singular and Regular Returns
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\(a) ![\[fig:retmap2\] (a) Subinterval of the return map $R: \Sigma_+ \to \Sigma)+$ of Eqs. and (b) a refinement of the subinterval. Dashed black line is the line of fixed points $R(z) = z$. (c) Periodic orbit corresponding to fixed point of $R$ at $z \approx 0.05939079$. (d) Time series of the periodic orbit. The orbit is decomposed into red, gray, green, blue, magenta, and black segments (defined as in Sec. \[sec:maps\]). Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](3apng.png "fig:"){width="45.00000%"} (b) ![\[fig:retmap2\] (a) Subinterval of the return map $R: \Sigma_+ \to \Sigma)+$ of Eqs. and (b) a refinement of the subinterval. Dashed black line is the line of fixed points $R(z) = z$. (c) Periodic orbit corresponding to fixed point of $R$ at $z \approx 0.05939079$. (d) Time series of the periodic orbit. The orbit is decomposed into red, gray, green, blue, magenta, and black segments (defined as in Sec. \[sec:maps\]). Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](3bpng.png "fig:"){width="45.00000%"}\
(c) ![\[fig:retmap2\] (a) Subinterval of the return map $R: \Sigma_+ \to \Sigma)+$ of Eqs. and (b) a refinement of the subinterval. Dashed black line is the line of fixed points $R(z) = z$. (c) Periodic orbit corresponding to fixed point of $R$ at $z \approx 0.05939079$. (d) Time series of the periodic orbit. The orbit is decomposed into red, gray, green, blue, magenta, and black segments (defined as in Sec. \[sec:maps\]). Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](3cpng.png "fig:"){width="45.00000%"} (d) ![\[fig:retmap2\] (a) Subinterval of the return map $R: \Sigma_+ \to \Sigma)+$ of Eqs. and (b) a refinement of the subinterval. Dashed black line is the line of fixed points $R(z) = z$. (c) Periodic orbit corresponding to fixed point of $R$ at $z \approx 0.05939079$. (d) Time series of the periodic orbit. The orbit is decomposed into red, gray, green, blue, magenta, and black segments (defined as in Sec. \[sec:maps\]). Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](3dpng.png "fig:"){width="45.00000%"}
Approximating points on $\Sigma$ by their $z$-coordinates, the return map $R: \Sigma \to \Sigma$ is well-approximated by a one-dimensional map on an interval, also denoted $R$. In the presence of the small-amplitude stable periodic orbit $\Gamma$, we now compare our one-dimensional approximation to return maps in the case of folded nodes[@wechselberger2005] and folded saddle-nodes[@guckenheimer2008chaos; @krupa2010]. Where the return map is defined, trajectories beginning in different components of the domain of $R$ make different numbers of small turns before escaping the local region. These subsets are somewhat analogous to the rotation sectors arising from twists due to a folded node.[@wechselberger2005] However, in the present case there is a folded singularity as well as a saddle-focus as well as a small-amplitude periodic orbit. Each of these local objects plays a role in the twisting of trajectories that enter neighborhoods of the fold curve $L_0$.
When the small-amplitude stable periodic orbit exists, the domain of the return map is now disconnected, with components separated by finite-length gaps (Figure \[fig:retmap\]). The gaps where $R$ is undefined correspond to those trajectories beginning on $S^{a+}_{{ \varepsilon}}$ that asymptotically approach $\Gamma$ without making a large-amplitude oscillation. The second difference concerns the extreme nonlinearity near the boundaries of the disconnected intervals where $R$ is defined (Figure \[fig:retmap2\]a). Portions of the image lie below the local minima in these local concave segments, resulting in tiny regions near the boundaries where the derivative changes rapidly. These points arise from canard segments of trajectories resulting in a jump from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$ and hence to $\Sigma$. Fixing the parameters and iteratively refining successively smaller intervals of initial conditions, this pattern of disconnected regions where the derivative changes rapidly seems to repeat up to machine accuracy. One consequence of this structure is the existence of large numbers of unstable periodic orbits, defined by fixed points of $R$ at which $|R'(z)| > 1$. This topological structure also appears to be robust to variations of the parameter $\nu$.
This complicated structure arises from the interaction between the basin of attraction of $\Gamma$, the twist region near the folded singularity and $W^{u,s}$. As an illustration of this complexity, consider an unstable fixed point $z \approx 0.05939079$ of the return map as defined in Figure \[fig:retmap2\](b), interpreted as an unstable periodic orbit in the full system of equations (Figure \[fig:retmap2\](c)-(d)). The orbit is approximately decomposed according to its interactions with the (un)stable manifolds of $p_{eq}$ and the slow manifolds. One possible forward-time decomposition of this orbit proceeds as follows:
- A segment (red) that begins on $S^{a+}_{{ \varepsilon}}$ and flows very close to $p_{eq}$ by remaining near $W^s$,
- a segment (gray) that leaves the region near $p_{eq}$ along $W^u$, then jumping right from $S^r$ to $S^{a+}_{{ \varepsilon}}$,
- a segment (green) that flows from $S^{a+}_{{ \varepsilon}}$ to $S^{r}_{{ \varepsilon}}$, making small-amplitude oscillations while remaining a bounded distance away from $p_{eq}$, then jumping right from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$,
- a segment (blue) that flows back down into the region near $p_{eq}$, making small oscillations around $W^s$, then jumping right from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$,
- a segment (magenta) with similar dynamics to the green segment, making small-amplitude oscillations while remaining a bounded distance away from $p_{eq}$, then jumping right from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$, and
- a segment (black) making a large-amplitude excursion by jumping left to $S^{a-}_{{ \varepsilon}}$, flowing to the fold $L_{-2/3}$, and then jumping to $S^{a+}_{{ \varepsilon}}$.
A linearized flow map can be constructed[@glendinning1984; @silnikov1965] in small neighborhoods of the saddle-focus $p_{eq}$, which can be used to count the number of small-amplitude oscillations contributed by orbit segments approaching the equilibrium point. However, the small-amplitude periodic orbit and the twist region produce additional twists, as observed in the green and magenta segments of the example above.
(a)![\[fig:z0\] (a) Geometry in the section $\Sigma_0 = \{(x,y,z): x \in \left[-0.07,0.11\right], y \in \left[-0.005,0.01\right], z=0\}$. Gray points sample the subset of $\Sigma_0$ whose corresponding forward trajectories tend asymptotically close to the stable periodic orbit without returning to $\Sigma_0$. Green points denote the first forward return of the remaining points in $\Sigma_0$ with the orientation $\dot{z} < 0$. (b) Color plot of maximal height ($y$-coordinate) obtained by trajectories that return to $\Sigma_0$ as defined in (a). Cross-sections of $S^{a+}_{{ \varepsilon}}$ (red) and $S^r_{{ \varepsilon}}$ (black) at $\Sigma_0$ are shown, and the tangency of the vector field with $\Sigma_0$ (i.e. the set $\{ax + by = -\nu\}$) is given by the magenta dashed line. Parameter set: $\nu \approx 0.00870134$, $a = 0.01$, $b = -1$, $c = 1$.](4apng.png "fig:"){width="48.00000%"}\
(b)![\[fig:z0\] (a) Geometry in the section $\Sigma_0 = \{(x,y,z): x \in \left[-0.07,0.11\right], y \in \left[-0.005,0.01\right], z=0\}$. Gray points sample the subset of $\Sigma_0$ whose corresponding forward trajectories tend asymptotically close to the stable periodic orbit without returning to $\Sigma_0$. Green points denote the first forward return of the remaining points in $\Sigma_0$ with the orientation $\dot{z} < 0$. (b) Color plot of maximal height ($y$-coordinate) obtained by trajectories that return to $\Sigma_0$ as defined in (a). Cross-sections of $S^{a+}_{{ \varepsilon}}$ (red) and $S^r_{{ \varepsilon}}$ (black) at $\Sigma_0$ are shown, and the tangency of the vector field with $\Sigma_0$ (i.e. the set $\{ax + by = -\nu\}$) is given by the magenta dashed line. Parameter set: $\nu \approx 0.00870134$, $a = 0.01$, $b = -1$, $c = 1$.](4bpng.png "fig:"){width="48.00000%"}
We will return to one-dimensional approximations of the return map in Sec. \[sec:ret\], but now we focus on two-dimensional maps, and show that we can illuminate key features of their small-amplitude oscillations. Let us fix a cross-section and define the geometric objects whose interactions organize the return dynamics. Define $\Sigma_0$ to be a compact subset of $\{ z = 0\}$ containing the first intersection (with orientation $\dot{z} >0$) of $W^s$ . Let $B_0$ denote the [*immediate basin of attraction*]{} of the stable periodic orbit $\Gamma$, which we define as the set of points in $\Sigma_0$ whose forward trajectories under the flow of Eq. asymptotically approach $\Gamma$ without returning to $\Sigma_0$, and let $\partial B_0$ denote its boundary. The periodic orbit itself does not intersect our choice of cross-section.
Since we wish to study trajectory segments that return to the cross-section, $B_0$ functions as an escape subset. Rigorously, the forward return map $R: \Sigma_0 \to \Sigma_0$ is undefined on the subset $B_0$, and points landing in $B_0$ under forward iterates of $R$ ‘escape’. Obviously the trajectories with initial conditions inside $\cup_{i=0}^{\infty}R^{-i}(B_0)$ are contained within the basin of attraction of $\Gamma$, and furthermore the $j$-th iterate of the return map $R^j$ is defined only on the subset $\Sigma_0 - \cup_{i=0}^j R^{-j}(B_0)$. Finally, we abuse notation slightly and denote by $S^{a+}_{{ \varepsilon}}$ (resp. $S^r_{{ \varepsilon}}$) the intersections of the corresponding slow manifolds with $\Sigma_0$. We also refer to the intersection of $S^{a+}_{{ \varepsilon}}$ (resp. $S^r_{{ \varepsilon}}$) with $\Sigma_0$ as the [*attracting*]{} (resp. [*repelling*]{}) [*spiral*]{} due to its distinctive shape (see Figure \[fig:z0\]). The immediate basin of attraction $B_0$ is depicted by gray points in Figure \[fig:z0\](a). This result implies that the basin of attraction of the periodic orbit contains at least a thickened spiral which $S^{a+}_{{ \varepsilon}}$ intersects transversely in interval segments, accounting for the disconnected images of the one-dimensional return maps.
The slow manifolds also intersect transversely. Segments of the attracting spiral can straddle both $B_0$ and the repelling spiral. In Fig. \[fig:z0\](b), we color initial conditions based on the maximum $y$-coordinate achieved by the corresponding trajectory before its return to $\Sigma_0$. Due to the Exchange Lemma, only thin bands of trajectories are able to remain close enough to $S^r_{{ \varepsilon}}$ to jump at an intermediate height. We choose the maximum value of the $y$-coordinate to approximately parametrize the length of the canards. This parametrization heavily favors trajectories jumping left (from $S^r_{{ \varepsilon}}$ to $S^{a-}_{{ \varepsilon}}$) rather than right (from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$), since trajectories jumping left can only return to $\Sigma_0$ by first following $S^{a-}_{{ \varepsilon}}$ to a maximal height, and then jumping from $L_{-2/3}$ to $S^{a+}_{{ \varepsilon}}$. This asymmetry is useful: in Figure \[fig:z0\], $S^r_{{ \varepsilon}}$ serves as a boundary between the (apparently discontinuous) blue and yellow regions, clearly demarcating those trajectories which turn right rather than left before returning to $\Sigma_0$. Summarizing, $\partial B_0$ and $S^r_{{ \varepsilon}}$ partition this section according to the behavior of orbits containing canards.
\(a) ![\[fig:sao\] (a) Two phase space trajectories beginning and ending on the section $\{z = 0\}$ with stopping condition $\dot{z} < 0$ and (b) the normalized time series of their normalized $y$-coordinates of each trajectory. Initial conditions: blue, $(x,y,z) = (0.000553, 0.000201, 0)$; red, $(x,y,z) = (0.000553, 0.003065, 0)$. Parameter set: $\nu \approx 0.00870134$, $a = 0.01$, $b = -1$, $c = 1$.](5apng.png "fig:"){width="45.00000%"}\
(b) ![\[fig:sao\] (a) Two phase space trajectories beginning and ending on the section $\{z = 0\}$ with stopping condition $\dot{z} < 0$ and (b) the normalized time series of their normalized $y$-coordinates of each trajectory. Initial conditions: blue, $(x,y,z) = (0.000553, 0.000201, 0)$; red, $(x,y,z) = (0.000553, 0.003065, 0)$. Parameter set: $\nu \approx 0.00870134$, $a = 0.01$, $b = -1$, $c = 1$.](5bpng.png "fig:"){width="45.00000%"}
Trajectories beginning in $\Sigma_0$ either follow $W^s$ closely and spiral out along $W^u$ or remain a bounded distance away from both the equilibrium point and $W^s$, instead making small-amplitude oscillations consistent with a folded node. Differences between these two types of small-amplitude oscillations have been observed in earlier work. The transition from the first kind of small-amplitude oscillation to the second is a function of the distance from the initial condition to the intersection of $W^s$ with the cross-section. Two initial conditions are chosen on a vertical line embedded in the section $\{z=0\}$, having the property that the resulting trajectory jumps right from $S^r_{{ \varepsilon}}$ at an intermediate height before returning to the section with orientation $\dot{z} < 0$. These initial conditions are found by selecting points in Figure \[fig:z0\](b) in the blue regions lying on a ray that extends outward from the center of the repelling spiral. The corresponding return trajectories are plotted in Figure \[fig:sao\]. The production of small-amplitude oscillations is dominated by the saddle-focus mechanism: in the example shown, the red orbit exhibits four oscillations before the (relatively) large-amplitude return, whereas the blue orbit exhibits seven oscillations. We can select trajectories with increasing numbers of small-amplitude oscillations by picking points closer to $W^s \cap \{z=0\}$. A complication in this analysis is that jumps at intermediate heights, which are clearly shown to occur in these examples, blur the distinction between ‘large’ and ‘small’ oscillations in a mixed-mode cycle.
\(a) ![\[fig:2dmap\] (a) Partition of a compact subset of the cross-section $\Sigma_0 = \{z = 0\}$. Black dashed line is the tangency of the vector field $\{\dot{z} = 0\}$, separating the subset $\{\dot{z} > 0\}$ (black points) from the other subsets. Yellow (resp. green): points above (resp. below) the line $\{y = 0\}$ with winding number less than three. Red (resp. blue): points whose forward trajectories reach a maximal height greater than (resp. less than) 0.18 and have winding number three or greater. (b) Overlay of red and blue subsets of domain (points) with images of yellow, green, and black subsets (crosses). (c) Overlay of red and blue subsets of domain (points) with the image of the blue subset (crosses). (d) Overlay of attracting spiral (magenta), repelling spiral (dark green), and image of red subset (crosses). Note the change in scale of the final figure. Generated from a $500 \times 500$ grid of initial conditions beginning on $\Sigma_0$. Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](6apng.png "fig:"){width="40.00000%"} (b) ![\[fig:2dmap\] (a) Partition of a compact subset of the cross-section $\Sigma_0 = \{z = 0\}$. Black dashed line is the tangency of the vector field $\{\dot{z} = 0\}$, separating the subset $\{\dot{z} > 0\}$ (black points) from the other subsets. Yellow (resp. green): points above (resp. below) the line $\{y = 0\}$ with winding number less than three. Red (resp. blue): points whose forward trajectories reach a maximal height greater than (resp. less than) 0.18 and have winding number three or greater. (b) Overlay of red and blue subsets of domain (points) with images of yellow, green, and black subsets (crosses). (c) Overlay of red and blue subsets of domain (points) with the image of the blue subset (crosses). (d) Overlay of attracting spiral (magenta), repelling spiral (dark green), and image of red subset (crosses). Note the change in scale of the final figure. Generated from a $500 \times 500$ grid of initial conditions beginning on $\Sigma_0$. Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](6bpng.png "fig:"){width="40.00000%"}\
(c) ![\[fig:2dmap\] (a) Partition of a compact subset of the cross-section $\Sigma_0 = \{z = 0\}$. Black dashed line is the tangency of the vector field $\{\dot{z} = 0\}$, separating the subset $\{\dot{z} > 0\}$ (black points) from the other subsets. Yellow (resp. green): points above (resp. below) the line $\{y = 0\}$ with winding number less than three. Red (resp. blue): points whose forward trajectories reach a maximal height greater than (resp. less than) 0.18 and have winding number three or greater. (b) Overlay of red and blue subsets of domain (points) with images of yellow, green, and black subsets (crosses). (c) Overlay of red and blue subsets of domain (points) with the image of the blue subset (crosses). (d) Overlay of attracting spiral (magenta), repelling spiral (dark green), and image of red subset (crosses). Note the change in scale of the final figure. Generated from a $500 \times 500$ grid of initial conditions beginning on $\Sigma_0$. Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](6cpng.png "fig:"){width="40.00000%"} (d) ![\[fig:2dmap\] (a) Partition of a compact subset of the cross-section $\Sigma_0 = \{z = 0\}$. Black dashed line is the tangency of the vector field $\{\dot{z} = 0\}$, separating the subset $\{\dot{z} > 0\}$ (black points) from the other subsets. Yellow (resp. green): points above (resp. below) the line $\{y = 0\}$ with winding number less than three. Red (resp. blue): points whose forward trajectories reach a maximal height greater than (resp. less than) 0.18 and have winding number three or greater. (b) Overlay of red and blue subsets of domain (points) with images of yellow, green, and black subsets (crosses). (c) Overlay of red and blue subsets of domain (points) with the image of the blue subset (crosses). (d) Overlay of attracting spiral (magenta), repelling spiral (dark green), and image of red subset (crosses). Note the change in scale of the final figure. Generated from a $500 \times 500$ grid of initial conditions beginning on $\Sigma_0$. Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](6dpng.png "fig:"){width="40.00000%"}
\[sec:sao\] Modeling Small-Amplitude Oscillations
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We now study some of the possible concatenations of small-amplitude oscillation segments as seen in Fig. \[fig:sao\]. Tangencies of the vector field with the cross-section are given by curves which partition the section into disconnected subsets. The partition that does not contain the attracting and repelling spirals is mapped with full rank to the remaining partition in one return (Fig. \[fig:2dmap\](b)), allowing us to restrict our analysis to an invariant two-dimensional subset where the vector field is transverse everywhere.
Mixed-rank behavior occurs in this subset, as shown in Figure \[fig:2dmap\]. Note that this set of figures is plotted at a slightly different parameter set from that in Figure \[fig:z0\], the main difference being that the line of vector field tangencies intersects a portion of the attracting spiral. Other features of the dynamics persist, including the fact that trajectories jumping left to $S^{a-}_{{ \varepsilon}}$ reach a greater maximal height ($y$-component) than the trajectories jumping right to $S^{a+}_{{ \varepsilon}}$. As shown in Figure \[fig:2dmap\](a), the region is partitioned according to three criteria: their location with respect to the curve of tangency, and their location with respect to the repelling spiral (corresponding to left or right jumps), and their winding number (defined later in this section).
We will refine this partition in order to create a symbolic model of the dynamics, but we can already state two significant results:
- [*Mixed-rank dynamics*]{}. As shown in Fig. \[fig:2dmap\](c)-(d), the red and blue regions collapse to $S^{a+}_{{ \varepsilon}}$ within one return. This contraction includes those trajectories that return to the cross-section by first jumping right from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$ at an intermediate height. Figure \[fig:2dmap\](b) shows that the yellow subset returns immediately to this low-rank region. The green subset returns either to the low-rank region or to the yellow region. But note that it does not intersect its image, and furthermore, it intersects the yellow region on a portion of the attracting spiral. Therefore, after at most two returns, the dynamics of the points beginning in $\Sigma_0$ (and which did not map to $B_0$) is characterized by the dynamics on the attracting spiral.
- [*Trajectories jumping left or right return differently*]{}. Those trajectories jumping left to $S^{a-}_{{ \varepsilon}}$ return to a tiny segment very close to the center of the $S^{a+}_{{ \varepsilon}}$, as shown in Fig. \[fig:2dmap\](d). In contrast, the trajectories jumping right sample the entire spiral of $S^{a+}_{{ \varepsilon}}$, as shown in Fig. \[fig:2dmap\](c). Thus, multiple intermediate-height jumps to the right are a necessary ingredient in concatenating small- and medium-amplitude oscillations (arising from right jumps) between large-amplitude excursions (arising from left jumps).
We now construct a dynamical partition of the cross-section. First we define the winding of a trajectory. Let $s$ and $u$ denote a stable and unstable eigenvector, respectively, of the linearization of the flow at $p_{eq}$. Then consider a cylindrical coordinate system with basis $(u,s,n)$ centered at $p_{eq}$, where $n = u \times s$. The [*winding*]{} of a given trajectory is the cumulative angular rotation (divided by $2\pi$) of the projection of the trajectory onto the $(u,n)$-plane. The [*winding number*]{} (or simply [*number of turns*]{}) of a trajectory is the integer part of the winding.
The cumulative angular rotation depends on both the initial and stopping condition of the trajectory, which in turn depend on the section used. Close to $p_{eq}$, the winding of a trajectory measures winding around $W^s$. This is desirable since most of the rotation occurs as trajectories enter small neighborhoods of $p_{eq}$ along $W^s$.
If Figure \[fig:turns\](a), we study the winding on a connected subset of the attracting spiral. On this connected subset we may parametrize the spiral by its arclength. The number of turns increases by approximately one whenever $S^{a+}_{{ \varepsilon}}$ intersects $S^{r}_{{ \varepsilon}}$ twice (these intersections occur in pairs since they correspond to bands of trajectories on $S^{a+}_{{ \varepsilon}}$ which leave the region by jumping left to $S^{a-}_{{ \varepsilon}}$). In between these intersections, there are gaps corresponding to regions where $S^{a+}_{{ \varepsilon}}$ intersects $B_0$.
\(a) ![\[fig:turns\] (a) Winding of the attracting spiral as a function of its parametrization by arclength. The starting point $s = 0$ is chosen close to the tangency. Positive values of $s$ track the spiral as it turns inward. (b) Partition of the section $\Sigma_0 = \{z = 0\}$ according to number of turns made by corresponding trajectories as well as whether the trajectories turn left or right from $S^r_{{ \varepsilon}}$. Left-turning trajectories are plotted with circles and right-turning trajectories are plotted with crosses. Color definitions: teal, 3 turns; blue, 4 turns; gray, 5 turns; green, 6 turns; gold, 7 turns; magenta, 8 turns. Also plotted are the slow manifolds $S^{a+}_{{ \varepsilon}}$ (red curve) and $S^r_{{ \varepsilon}}$ (black curve) as well as a saddle-point defined in Figure \[fig:fporbit\] (green square). Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](7apng.png "fig:"){width="44.00000%"} (b) ![\[fig:turns\] (a) Winding of the attracting spiral as a function of its parametrization by arclength. The starting point $s = 0$ is chosen close to the tangency. Positive values of $s$ track the spiral as it turns inward. (b) Partition of the section $\Sigma_0 = \{z = 0\}$ according to number of turns made by corresponding trajectories as well as whether the trajectories turn left or right from $S^r_{{ \varepsilon}}$. Left-turning trajectories are plotted with circles and right-turning trajectories are plotted with crosses. Color definitions: teal, 3 turns; blue, 4 turns; gray, 5 turns; green, 6 turns; gold, 7 turns; magenta, 8 turns. Also plotted are the slow manifolds $S^{a+}_{{ \varepsilon}}$ (red curve) and $S^r_{{ \varepsilon}}$ (black curve) as well as a saddle-point defined in Figure \[fig:fporbit\] (green square). Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](7bpng.png "fig:"){width="45.00000%"}
Sets in the partition are defined according to each trajectory’s winding number and jump direction. This partition uses the attracting and repelling spirals as a guide; small rectangles straddling the attracting spiral are contracted strongly transverse to the spiral and stretched along the attracting spiral, giving the dynamics a hyperbolic structure. In the next section we will compute a transverse homoclinic orbit, where this extreme contraction and expansion is shown explicitly.
We restrict ourselves to a subset $S \subset \Sigma_0$ where returns are sufficiently low-rank (i.e. the union of red and blue regions in Fig. \[fig:2dmap\](a)). Let $L_{n}\subset S$ (resp. $R_n \subset S$) denote those points whose forward trajectories make $n$ turns before jumping left to $S^{a-}_{{ \varepsilon}}$ (resp. right to $S^{a+}_{{ \varepsilon}}$). Then define $L_{tot} = \cup_{n=0}^{\infty} L_n$ and $R_{tot} = \cup_{n=0}^{\infty} R_n$. The collection $\mathcal{P} = \{L_i,R_j\}_{i,j=1}^{\infty}$ partitions $S$.
For a collection of sets $\mathcal{A}$, let $\sigma(\mathcal{A})$ denote the set of all one-sided symbolic sequences $x = x_0 x_1 x_2 \cdots$ with $x_i \in \mathcal{A}$. We can assign to each $x\in S$ a symbolic sequence in $\sigma(\mathcal{P}\cup \{S^c, B_0\})$, also labeled $x$. This sequence is constructed using the return map: $x = \{x_i\}$ is defined by $x_i = \iota(R^i(x))$, where $\iota: \Sigma_0 \to \mathcal{P}\cup\{B_0,S^c\}$ is the natural inclusion map. Note that some symbolic sequences have finite length, as $R$ is undefined over $B_0$. A portion of the partition is depicted in Fig. \[fig:turns\].
The results in figures \[fig:2dmap\] and \[fig:turns\] and the definition of $B_0$ constrain the allowed symbolic sequences. In particular:
- there exists a sufficiently large integer $N$ with $R(L_{tot}) \subset S^{a+}_{{ \varepsilon}} \cap (\cup_{n\geq N} L_n \cup R_n \cup B_0)$ (Figs. \[fig:2dmap\](d) and \[fig:turns\]),
- $R(R_{tot}) \subset S^{a+}_{{ \varepsilon}}\cap \Sigma_0$ (Fig. \[fig:2dmap\](c)),
- $R(S^c) \subset S$ (Fig. \[fig:2dmap\](b)), and
- the set of finite sequences are precisely those containing and ending in $B_0$.
The first result implies that for any integer $n \geq 1$, the block $L_n \alpha_m$ (where $\alpha \in \{L,R\}$) is impossible when $m < N$, since $R(L_n)$ is either $B_0$ or $\alpha_{m\geq N}$. For the parameter set used in Fig. \[fig:2dmap\], our numerics suggest a lower bound of $N = 13$. On the other hand, the second result reminds us that only right-jumping trajectories are able to sample the entire attracting spiral. The first two results then imply that blocks of type $R_i L_j$ or $R_i R_j$ are necessarily present in the symbolic sequences of orbits which concatenate small-amplitude oscillations with medium-amplitude oscillations as shown in Fig. \[fig:sao\], since medium-amplitude oscillations arise precisely from those points on $\Sigma_0$ whose forward trajectories remain bounded away from the saddle-focus (i.e. those points in $\Sigma_0$ sufficiently far from the intersection of $W^s$ with $\Sigma_0$) and jump right.
The first two results also imply that forward-invariant subsets lie inside the intersection of $S^{a+}_{{ \varepsilon}}$ with $\Sigma_0$. In terms of the full system, it follows that the trajectories corresponding to these points each contain segments which lie within a sheet of $S^{a+}_{{ \varepsilon}}$.
The attracting spiral and the nonsingular region $S^c$ have a nontrivial intersection. In view of the second result, it is possible for trajectories that we track to sometimes be mapped outside of the subset $S$ where our partition is defined. The third result implies the symbolic sequence of a point $x\in S$ whose forward returns leave the subset $S$ must contain the block $$\begin{aligned}
x_{n_j-1} S^c x_{n_j},\end{aligned}$$
where the index $n_j$ is defined by the $j$-th instance when the orbit leaves $S$ and $x_{n_j-1} \in \mathcal{P}$. We can also constrain the possible symbols of $x_{n_j}$ as follows. Trajectories which have intersected $\Sigma_0$ in the singular region $S$ can only return to $S^c$ along the curve $S^{a+}_{{ \varepsilon}}$. Furthermore, our numerical results demonstrate that $R(S^c \cap S^{a+}_{{ \varepsilon}})$ nontrivially intersects subsets of $\mathcal{P}\cup\{B_0\}$ only in the subcollection $\mathcal{P}_c = \{L_3,L_4,L_5,R_3,R_4,R_5,B_0\}$. Therefore $x_{n_j} \in \mathcal{P}_c$ whenever $n_j$ is defined.
In view of the last result, for each $i \geq 1$ we define the $i$-th [*escape subset*]{} $E_i$ to be the set of length-$i$ sequences ending in $B_0$. Note that $E_i$ contains the symbol sequences of the points in $R^{-(i-1)}(B_0)$. Section \[sec:ret\] provides a concrete numerical example of a point in $E_{n}$, where $n$ is at least 1284.
Let us summarize the main results of the symbolic dynamics. Points which do not have repeating symbolic sequences (i.e. points which are not equilibria or do not belong to cycles) either terminate in $B_0$, indicating that trajectory tends asymptotically to the small-amplitude stable periodic orbit $\Gamma$, or the sequence is infinitely long. In either case, due to the apparent hyperbolicity of the map we can track neighborhoods of points until they leave the set $S$ where the partition is defined. However, those ‘lost’ points on the orbit return to $S$ in one interate and we can resume tracking them.
\[sec:homorbit2d\] Invariant Sets of the Two-Dimensional Return Map
===================================================================
(a)![\[fig:fporbit\] (a) Mixed-mode oscillation in phase-space corresponding to the saddle point $p \approx (-0.053438, 0.001873)$ of the return map defined on $\Sigma_0 = \{z = 0\}$ and (b) the time series of its $x$-component. Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](8apng.png "fig:"){width="46.00000%"} (b)![\[fig:fporbit\] (a) Mixed-mode oscillation in phase-space corresponding to the saddle point $p \approx (-0.053438, 0.001873)$ of the return map defined on $\Sigma_0 = \{z = 0\}$ and (b) the time series of its $x$-component. Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](8bpng.png "fig:"){width="46.00000%"}
The structure of the invariant sets of the two-dimensional return map is a key dynamical question, and is related to the intersection of the basin of attraction of the small-amplitude stable periodic orbit with $\Sigma_0$. In this section we focus on two types of invariant sets : fixed points and transverse homoclinic orbits.
Certain invariant sets of the map may be used to construct open sets of points all sharing the same initial block in their symbolic sequence. We briefly describe how the simplest kind of invariant set– a fixed point– implies that neighborhoods of points must have identical initial sequences of oscillations. In Figure \[fig:fporbit\] we plot the saddle-type MMO corresponding to a saddle equilibrium point $p$, whose location in the section $\{z = 0\}$ is plotted in Figures \[fig:turns\] and \[fig:homorbit\]. According to Figure \[fig:turns\], $p$ has symbolic sequence $R_5 R_5 R_5 \cdots$, in agreement with the time-series shown in Figure \[fig:fporbit\](b). We also observe that the fixed-point is sufficiently far away from $W^s$ (the stable manifold of the saddle-focus) that the oscillations remain bounded away from $p_{eq}$. Furthermore, the dynamics in small neighborhoods of $p$ are described by the linearization of the map $R$ near $p$. This implies that small neighborhoods of $p$ consist of points with initial symbolic blocks of $R_5$, where the length of this initial block can be as large as desired. We can relax the condition that this be the initial block by instead considering preimages of these neighborhoods.
From this case study we observe that arbitrarily long chains of small-amplitude oscillations can be constructed using immediate neighborhoods of fixed points, periodic points, and other invariant sets lying in $S^{a+}_{{ \varepsilon}} \cap \{z = 0\}$. These in turn correspond to complicated invariant sets in the full three-dimensional system. Consequently, the maximum number of oscillations produced by a periodic orbit having one large-amplitude return can be very large at a given parameter value, depending on the number of maximum possible returns to sections in the region containing these local mechanisms. This situation should be compared to earlier studies of folded-nodes, in which trajectories with a given number of small-amplitude oscillations can be classified[@wechselberger2005]; and the Shilnikov bifurcation in slow-fast systems, in which trajectories have unbounded numbers of small-amplitude oscillations as they approach the homoclinic orbit.[@guckenheimer2015]
In the present system, we observe numerically that the return map $R$ contracts two-dimensional subsets of the cross-section to virtually one-dimensional subsets of $S^{a+}_{{ \varepsilon}}$. Subsequent returns act on $S^{a+}_{{ \varepsilon}}$ by stretching and folding multiple times before further extreme contraction of points transverse to $S^{a+}_{{ \varepsilon}}$.
If horseshoes exist, they will also appear to break the diffeomorphic structure of the return map due to the strong contraction (although topologically the horseshoe is still given by two-dimensional intersections of forward images of sets with their inverse images under the return map). We can draw a useful analogy to the Hénon family $H_{a,b}(x,y) = (1-ax^2+y,bx)$ when a strange attractor exists at parameter values $0 < b \ll 1$. The local structure of the strange attractor, outside of neighborhoods of its folds, is given by $C \times \mathbb{R}$, where $C$ is a Cantor set.
(a)![\[fig:homorbit\] (a) A saddle equilibrium (green point) of the return map defined on $\Sigma_0 = \{z = 0\}$, together with a neighborhood $U$ (blue grid), image $R(U)$ (red), subset of preimage $U \cap R^{-1}(U)$ (yellow), and a branch of its stable manifold $W^s(p)$(black). The intersection of $R(U)$ with $W^s(p)$ is also shown (magenta point). (b) Color plot of $10^4$ initial conditions beginning in $\Sigma_0$ on a $100\times 100$ grid, whose forward trajectories are integrated for the time interval $t \in [0,600]$. Color denotes number of intersections with $\Sigma_0$ with orientation $\dot{z} < 0$. (c) Last recorded intersection (blue circles) of each trajectory defined in (b) with $\Sigma_0$. The attracting and repelling spirals (red and black curves, respectively) are overlaid. Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](9apng.png "fig:"){width="44.00000%"} (b)![\[fig:homorbit\] (a) A saddle equilibrium (green point) of the return map defined on $\Sigma_0 = \{z = 0\}$, together with a neighborhood $U$ (blue grid), image $R(U)$ (red), subset of preimage $U \cap R^{-1}(U)$ (yellow), and a branch of its stable manifold $W^s(p)$(black). The intersection of $R(U)$ with $W^s(p)$ is also shown (magenta point). (b) Color plot of $10^4$ initial conditions beginning in $\Sigma_0$ on a $100\times 100$ grid, whose forward trajectories are integrated for the time interval $t \in [0,600]$. Color denotes number of intersections with $\Sigma_0$ with orientation $\dot{z} < 0$. (c) Last recorded intersection (blue circles) of each trajectory defined in (b) with $\Sigma_0$. The attracting and repelling spirals (red and black curves, respectively) are overlaid. Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](9bpng.png "fig:"){width="44.00000%"}\
(c)![\[fig:homorbit\] (a) A saddle equilibrium (green point) of the return map defined on $\Sigma_0 = \{z = 0\}$, together with a neighborhood $U$ (blue grid), image $R(U)$ (red), subset of preimage $U \cap R^{-1}(U)$ (yellow), and a branch of its stable manifold $W^s(p)$(black). The intersection of $R(U)$ with $W^s(p)$ is also shown (magenta point). (b) Color plot of $10^4$ initial conditions beginning in $\Sigma_0$ on a $100\times 100$ grid, whose forward trajectories are integrated for the time interval $t \in [0,600]$. Color denotes number of intersections with $\Sigma_0$ with orientation $\dot{z} < 0$. (c) Last recorded intersection (blue circles) of each trajectory defined in (b) with $\Sigma_0$. The attracting and repelling spirals (red and black curves, respectively) are overlaid. Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](9cpng.png "fig:"){width="44.00000%"}
Let us now provide a visual demonstration of these issues. Let $U$ be a small neighborhood of the saddle fixed point $p$ that we located in the previous section. In Fig. \[fig:homorbit\] we plot $U$, $R(U)$, $U\cap R^{-1}(U)$, and $W^s(p)$ on the section $\Sigma_0$. The image is a nearly one-dimensional subset of $S^{a+}_{{ \varepsilon}}$ and the preimage is a thin strip which appears to be foliated by curves tangent to $S^r_{{ \varepsilon}}$. The subsets $R(U)$ and $R^{-1}(U)$ contain portions of $W^u(p)$ and $W^s(p)$, respectively. The transversal intersection of $R(U)$ with $W^s(p)$ is also indicated in this figure.
Numerically approximating the diffeomorphism $R^{-1}$ is a challenging problem. Trajectories which begin on the section and approach the attracting slow manifolds $S^{a\pm}_{{ \varepsilon}}$ in reverse time are strongly separated, analogous to the scenario where pairs of trajectories in forward time are strongly separated by $S^r_{{ \varepsilon}}$. This extreme numerical instability means that trajectories starting on the section and integrated backward in time often become unbounded. In order to compute $W^s(p)$, we therefore resort to a continuation algorithm which instead computes orbits in forward time. We take advantage of the singular behavior of the map to reframe this problem as a boundary value problem, with initial conditions beginning in a line on the section and ending ‘at’ $p$. Beginning with a point $y_0$ along $W^s(p)$, we construct a sequence $\{y_0 , y_1, \cdots\}$ along $W^s(p)$ as follows.
(C1) [*Prediction step*]{}. Let $w_i = y_{i-1} + h v_i$, where $h$ is a fixed step-size and $v_i$ is a numerically approximated tangent vector to $W^s(p)$ at $y_{i-1}$.
(C2) [*Correction step*]{}. Construct a line segment $L_i$ of initial conditions perpendicular to $v_i$. Use a bisection method to locate a point $y_i \in L_i$ such that $|R(y_i) - p| < { \varepsilon}$, where ${ \varepsilon}$ is a prespecified tolerance.
The relevant branch of $W^s(p)$ which intersects $R(U)$ lies inside the nearly singular region of the return map, so the segment $L_i$ can be chosen small enough that $R(L_i)$ is, to double-precision accuracy, a segment of $S^{a+}_{{ \varepsilon}}$ which straddles $p$. This justifies our correction step above.
It is usually not sufficient to assert the existence of a transverse homoclinic orbit from the intersection of the image sets. But in the present case, these structures are organized by the slow manifolds $S^{a+}_{{ \varepsilon}}$ and $S^r_{{ \varepsilon}}$. The strong contraction onto $S^{a+}_{{ \varepsilon}}$ in forward time implies that the discrete orbits comprising $W^u(p)$ must also lie along this slow manifold. The unstable manifold $W^u(p)$ lies inside a member of the $O(\exp(-c/{ \varepsilon}))$-close family which comprises $S^{a+}_{{ \varepsilon}}$, so the forward images serve as good proxies for subsets of $W^u(p)$ itself. On the other hand, when $U$ is sufficiently small, its preimage $R^{-1}(U)$ appears to be foliated by a family of curves tangent to $S^r_{{ \varepsilon}}$, such that one of the curves contains $W^s(p)$ itself.
The Smale-Birkhoff homoclinic theorem[@birkhoff1950; @smale1965] then implies that there exists a hyperbolic invariant subset on which the dynamics is conjugate to a subshift of finite type. Note that while we expect fixed points to lie in $S^{a+}_{{ \varepsilon}}$ due to strong contraction, we do not expect fixed points to also lie in $S^r_{{ \varepsilon}}$ in the generic case. We end this result by commenting on its apparent degeneracy of the two-dimensional sets $U,R(U),$ and $R^{-1}(U)$. A classical proof of the Smale-Birkhoff theorem uses the set $V = R^k(U) \cap R^{-m}(U)$ (where $k,m\geq 0$ are chosen such that $V$ is nonempty) as the basis for constructing the Markov partition on which the shift is defined.[@guckenheimer1983] Here, $V$ is well-approximated by a curve segment.
Does a positive-measure set of initial conditions approach this hyperbolic invariant set? In other words, is the set an attractor? While it is difficult to assess the invariance of open sets with finite-time computations, our numerics support the conjecture that most initial conditions lying outside $B_0$ tend to the chaotic invariant set without tending asymptotically to $\Gamma$. In Figs. \[fig:homorbit\](b)-(c), we study the eventual fates of a grid of initial conditions beginning on $\Sigma_0$. Fig. \[fig:homorbit\](b) shows that even after a relatively long integration time of $t = 600$, most initial conditions in $B_0^c$ are able to return repeatedly to $\Sigma_0$. However, it may simply be that the measure of $(R^{-i} (B_0))^c$ decays extremely slowly to $0$ as $i$ tends to infinity. In Fig. \[fig:homorbit\](c), we plot the last recorded intersection with $\Sigma_0$ of those trajectories that do not tend to $\Gamma$ within $t = 600$. Even with a relatively sparse set of $10^4$ points, we observe that these intersections sample much of the attracting spiral. Many of the points are not visible at the scale of the figure because they sample the segment shown in Fig. \[fig:2dmap\](d) (i.e. the penultimate intersections resulted in the trajectory jumping left to $S^{a-}_{{ \varepsilon}}$).
We now turn to a section transverse to $S^{a+}_{{ \varepsilon}}$ and approximate returns to this section by a one-dimensional map. The advantage of this low-dimensional approximation is that we can readily identify classical bifurcations and routes to complex behavior. We may interpret invariant sets of these one-dimensional returns (such as fixed points, periodic orbits, and more complicated sets) as large-amplitude portions of the mixed-mode oscillations in the full system.
\(a) ![\[fig:sn\] Saddle-node bifurcation of periodic orbits in system . (a) $\nu = 0.00801$, (b) $\nu = 0.00802$. Dashed black line denotes line of fixed points $R(z) = z$. Remaining parameters: $a = -0.3, b = -1, c = 1$.](10apng.png "fig:"){width="44.00000%"} (b) ![\[fig:sn\] Saddle-node bifurcation of periodic orbits in system . (a) $\nu = 0.00801$, (b) $\nu = 0.00802$. Dashed black line denotes line of fixed points $R(z) = z$. Remaining parameters: $a = -0.3, b = -1, c = 1$.](10bpng.png "fig:"){width="44.00000%"}
\[sec:bif\] Bifurcations of the One-Dimensional Return Map
==========================================================
Fixed points of a return map defined on the section $\Sigma_+ = \{x = 0.3\}$ are interpreted in the full system as the locations of mixed-mode oscillations, formed from trajectories making one large-amplitude passage after interacting with the local mechanisms near $L_0$. Similarly, periodic orbits of the (discrete) return map can be used to identify mixed-mode oscillations having more than one large-amplitude passage. We demonstrate common bifurcations associated with these invariant objects. First we locate a saddle-node bifurcation of periodic orbits, in which a pair of orbits coalesce and annihilate each other at a parameter value.
Figure \[fig:sn\] demonstrates the existence of a fixed point $z = R(z)$ with unit derivative as $\nu$ is varied within the interval $\left[0.00801, 0.00802\right]$ (remaining parameters are as in Figure \[fig:retmap\]). This parameter set lies on a generically codimension one branch in the parameter space. We also note that saddle-node bifurcations serve as a mechanism to produce stable cycles in the full system, which in turn may undergo torus bifurcations and period-doubling cascades as a parameter is varied.
\(a) ![\[fig:bifdiag\] (a) Bifurcation sequence of the one-dimensional approximation of the return map $R: \Sigma_+ \to \Sigma_+$ as the parameter $\nu$ is varied from $0.008685$ to $0.0087013$. Remaining parameters are as in Figure \[fig:tangbif\]. (b) Magnification of upper branch of first period doubling cascade. ](11apng.png "fig:"){width="43.00000%"} (b) ![\[fig:bifdiag\] (a) Bifurcation sequence of the one-dimensional approximation of the return map $R: \Sigma_+ \to \Sigma_+$ as the parameter $\nu$ is varied from $0.008685$ to $0.0087013$. Remaining parameters are as in Figure \[fig:tangbif\]. (b) Magnification of upper branch of first period doubling cascade. ](11bpng.png "fig:"){width="43.00000%"}
\(a) ![\[fig:crit\] (a) Forward trajectory (red points) of the critical point (green square) under the return map $R: \Sigma_+ \to \Sigma_+$. Red dashed lines indicate the cobweb diagram of the first two iterates of the trajectory to guide the eye. Black dashed line is the line of fixed points. All 1284 forward iterates are plotted. The subsequent iterate lands outside the domain of $R$: the corresponding portion of the full trajectory of tends asymptotically to $\Gamma$ without returning to $\Sigma_+$. (b) Distribution of points in the forward orbit of the critical point. Parameter set as in Figure \[fig:retmap\].](12apng.png "fig:"){width="42.00000%"} (b) ![\[fig:crit\] (a) Forward trajectory (red points) of the critical point (green square) under the return map $R: \Sigma_+ \to \Sigma_+$. Red dashed lines indicate the cobweb diagram of the first two iterates of the trajectory to guide the eye. Black dashed line is the line of fixed points. All 1284 forward iterates are plotted. The subsequent iterate lands outside the domain of $R$: the corresponding portion of the full trajectory of tends asymptotically to $\Gamma$ without returning to $\Sigma_+$. (b) Distribution of points in the forward orbit of the critical point. Parameter set as in Figure \[fig:retmap\].](12bpng.png "fig:"){width="45.00000%"}
The beginning of a period-doubling cascade is identified in the return map $R$ as $\nu$ is varied in the interval $\left[ 0.008685, 0.0087013\right]$ (Figure \[fig:bifdiag\]a). Within this range, period-3, period-5, and period-6 parameter windows are readily identifiable in Figure \[fig:bifdiag\]b. The local unimodality of the return map suggests that our ($\nu$-parametrized) family of return maps share some universal properties with maps of the interval that exhibit period-doubling cascades,[@feigenbaum1978; @coullet1978] despite the nonlinearity at the right boundary of the interval observed in Figure \[fig:retmap2\]. The cascading structure is clearly robust to small boundary perturbations of the quadratic-like maps we consider.
We stress that these bifurcations produce additional [*large-amplitude*]{} oscillations of MMOs. As the parameter is varied between period-doubling events, more small-amplitude twists may be generated. The connection between Figs. \[fig:bifdiag\] and \[fig:turns\] is that between each large-amplitude passage, the number and type of small-amplitude twists is determined by the location of periodic points of the return map $R:\Sigma_0\to\Sigma_0$.
\[sec:ret\] Returns of the critical point
=========================================
We recall a classical result of unimodal dynamics for the quadratic family $f_a(x) = 1 - ax^2$ near the critical parameter $a = 2$, where $f_a: I \to I$ is defined on its invariant interval $I$ (when $a = 2$, $I = \left[ -1,1\right]$). On positive measure sets of parameters near $a = 2$, the map $f_a$ admits absolutely continuous invariant measures with respect to Lebesgue measure.[@jakobson1981] These facts depend on the delicate interplay between stretching behavior away from neighborhoods of the critical point, together with recurrence to the arbitrarily small neighborhoods of the critical point as trajectories are ‘folded back’ by the action of $f$. This motivates our current objective: to locate a parameter set for which (i) there exists a forward-invariant subset $\Sigma_u \subset \Sigma_+ $ where $R: \Sigma_u \to \Sigma_u$ has exactly one critical point $c \in \Sigma_u$, and (ii) $R^2(c)$ is a fixed point of $R$.
We couldn’t locate a parameter set satisfying both (i) and (ii), but we can obtain a parameter set where $R$ has the topology of Figure \[fig:retmap2\] (i.e. is unimodal over a sufficiently large interval) and admits a critical point satisfying (ii). This parameter set is numerically approximated using a two-step bisection algorithm. First, a bisection method is used to approximate the critical point $c$ by refining the region where $R'$ first changes sign up to a fixed error term, which we take to be $10^{-15}$. Another bisection method is used to approximate the parameter value at which $|R^2(c) - R^3(c)|$ is minimized. We were able to minimize this distance to $2.5603\times 10^{-8}$ at the parameter value $\nu = 0.0087013381084$, where the remaining parameters are given in Figure \[fig:retmap\].
Figure \[fig:crit\]a depicts the forward trajectory of the critical point near the line of fixed points at this parameter value. The itinerary of $c$ is finite, eventually landing in a subinterval of $\Sigma_u$ where $R$ is undefined. Even so, its forward orbit is unpredictable and samples the interval $\left[ R(c), R^2(c)\right]$ with a nontrivial ‘transient’ density for 1284 iterates (Figure \[fig:crit\]b). The length of the itinerary is extremely sensitive to tiny ($O(10^{-14})$) perturbations of the parameter $b$, reflecting the sensitive dependence of initial conditions in the selected parameter neighborhood. However, the normalized distributions of the forward iterates behave much more regularly: they are all similar to that shown in Figure \[fig:crit\]b. We conjecture that there exists a path in parameter space such that these distributions rigorously converge to a smooth distribution on the subset where $R$ is defined.
Concluding remarks
==================
We have classified much of the complex dynamics arising from a tangency of a slow manifold with an unstable manifold of an equilibrium point. The key to this analysis has been the identification of global bifurcations in carefully-chosen return maps of the system. In particular, transverse homoclinic orbits and period-doubling cascades are identified as mechanisms leading to chaotic behavior in the present system. Our objective has not been to attempt rigorous proofs of the results. Instead, we show that these bifurcations can be identified with fairly standard numerical integration and bisection procedures. The challenge, which is typical in studies of systems with a strong timescale separation, is to use techniques which bypass the numerical instability that occurs in integrating in forward or reverse time. This is particularly relevant in the continuation procedure that is used to locate transverse homoclinic orbits for the two-dimensional return map, as well as to identify a section high up on $S^{a+}_{{ \varepsilon}}$ which admits an approximation by a one-dimensional map.
We also motivate the study of maps having the topology shown in Figure \[fig:retmap\](a)-(b). These maps are distinguished by two significant features: they admit small disjoint escape subsets, and they are unimodal over most—but not all—of the remainder of the subset over which the map is defined. Sections \[sec:bif\] and \[sec:ret\] can then be regarded retrospectively as an introduction to the dynamics of these maps, especially as they compare to the dynamics of unimodal maps. In particular, we observe that such maps undergo period-doubling cascades (Figure \[fig:bifdiag\]) as a system parameter is varied. The forward trajectory of the critical point is also seen to have a transient density for a range of parameters (for eg., Figure \[fig:crit\]). These results lead us to conjecture whether absolutely continuous invariant measures and universal bifurcations for unimodal maps persist weakly for the family of maps studied in this paper. The geometric theory of rank-one maps pioneered by Wang and Young[@wang2008] is a possible starting point to prove theorems in this direction. This theory has been used successfully to identify chaotic attractors in families of slow-fast vector fields with one fast and two slow variables.[@guckenheimer2006] Their technique is based upon approximating returns by one-dimensional maps.
This work was supported by the National Science Foundation (Grant No. 1006272). The author thanks John Guckenheimer for useful discussions.
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abstract: 'We propose a technique to improve the search efficiency of the bag-of-words method for image retrieval. We introduce a notion of difficulty for the image matching problems and propose methods that reduce the amount of computations required for the feature vector-quantization task in BoW by exploiting the fact that easier queries need less computational resources. Measuring the difficulty of a query and stopping the search accordingly is formulated as a stopping problem. We introduce stopping rules that terminate the image search depending on the difficulty of each query, thereby significantly reducing the computational cost. Our experimental results show the effectiveness of our approach when it is applied to appearance-based localization problem.'
author:
- 'Kiana Hajebi and Hong Zhang[^1]'
title: '**Stopping Rules for Bag-of-Words Image Search and Its Application in Appearance-Based Localization** '
---
INTRODUCTION
============
Bag-of-Words (BoW) was originally proposed for document retrieval. In recent years, the method has been successfully applied to image retrieval tasks in computer vision community [@sivic; @nister]. The method is attractive because of its efficient image representation and retrieval. BoW represents an image as a sparse vector of visual words, and thus images can be searched efficiently using an inverted index file system. Other major application areas of the BoW method are appearance-based mobile robot localization and SLAM[^2] problems.
The fundamental issue involved with the appearance-based approach to both visual SLAM and global localization is the place recognition. Robot should be able to recognize the places it has visited before to localise itself or refine the map of the environment. This task is performed by matching the current view of the robot to the existing map that contains the images of the previously visited locations. In this paper we consider the problem of appearance-based localization in which the map is known *a priori*.
In large-scale environments maps contain too many images to match. The image search in such a large map is still a challenging and open problem. Matching images by comparing the local features of each image directly to the local features of all other images in the map is not practical. Bag-of-words proposes a more efficient approach; first, rather than matching with a pool of million visual features extracted from many thousands of images, the local features are mapped to a smaller number of vocabulary words that are built in an offline phase. This process is called vector-quantization. Once the visual words are identified in the query image, they are used as indices into the image database, to directly retrieve the images that share the same words.
When the vocabulary is large, the vector-quantization process can be a computationally expensive task in real-time localization. Considerable research has been done to speed up the search; some papers [@fabmap] do approximate nearest-neighbor search using structures like vocabulary trees [@nister]; some methods reduce the number of local image features by selecting only a fraction of features that are highly discriminative [@Achar; @BoRF]; another group makes use of more compact feature descriptors like [@bi-BoW; @fabmap]. In this paper, we first show that some image retrieval tasks are [*easier*]{} for BoW method. The hardness criteria, defined later, concerns how distinctive the image query is among all images in the dataset. Given this criteria, we show that the BoW search can be terminated earlier for easier queries. This means, in such queries, mapping only a portion of features can be sufficient to yield a relatively good result. The stopping rule saves considerable amount of computational resources.
The intuition behind this is that when there are many similar images to the query image, in terms of the number of common words they share, the search becomes more difficult as more processing is required to find the closest match candidate. Whereas when the query image has only a few match candidates, i.e., it shares its visual words with only a few images, the search becomes easy as vector-quantizing only a small number of features is sufficient to find the closest match to the query. By exploiting this fact, we can stop the vector-quantization when the search is easy and the nearest neighbor to the query is easy to find. Our method acts as an approximate image search algorithm. Our experimental results show that the accuracy decreases only slightly while the computational cost decreases dramatically.
Our approach can be best compared with the approach of Cummins and Newman [@accel_fabmap] who use concentration inequalities (Bennett’s inequality in their case) for early bail-out in multi-hypothesis testing that excludes unlikely location hypotheses from further evaluation. However, we use a different bail-out strategy for the process of vector-quantization. In the next section, we briefly review the image representation and the inverted-index search algorithm used in BoW framework. This is followed by a review of the localization algorithms that employ BoW for the image search. Our proposed method to improve the efficiency of BoW is described in Section \[sec:method\]. Section \[sec:results\] presents the experimental results and the evaluation criteria, and the result of our comparisons. Finally, we conclude the paper in Section \[sec:conclud\].
BACKGROUND {#sec:bakgnd}
==========
Bag-of-Words for image retrieval {#sec:bow}
--------------------------------
Bag-of-words is a popular model that has been used in image classification, objection recognition, and appearance-based navigation. Because of its simplicity and search efficiency it has been used as a successful method in Web search engines for large-scale image and document retrieval [@sivic; @nister; @Sivic2]. Bag-of-words model represents an image by a sparse vector of visual words. Image features, e.g., SIFTs [@lowe], are sampled and clustered (e.g., using k-means) in order to quantize the space into a discrete set of visual words. The centroids of clusters are then considered as visual words which form the visual vocabulary. Once a new image arrives, its local features are extracted and vector-quantized into the visual words. Each word might be weighed by some score which is either the word frequency in the image (i.e., *tf*) or the “term frequency-inverse document frequency” or *tf-idf* [@sivic]. A histogram of weighted visual words, which is typically sparse, is then built and used to represent the image.
An inverted index file, used in the BoW framework, is an efficient image search tool in which the visual words are mapped to the database images. Each visual word serves as a table index and points to the indices of the database images in which the word occurs. Since not every image contains every word and also each word does not occur in every image, the retrieval through inverted-index file is fast.
Bag-of-Words for Image-based Localization {#sec:relwork}
-----------------------------------------
Bag-of-words model has been extensively used as the basis of the image search in appearance-based localization or SLAM algorithms [@fabmap; @fabmap2; @Achar; @Angeli; @bi-BoW]. Cummins and Newman [@fabmap; @fabmap2] propose a probabilistic framework over the bag-of-words representation of locations, for the appearance-based place recognition. Along with the visual vocabulary they also learn the Chow Liu tree to capture the co-occurrences of visual words. Similarly, Angeli *et al.* [@Angeli] develop a probabilistic approach for place recognition in SLAM. They build two visual vocabularies incrementally and use two BoW representations as an input of a Bayesian filtering framework to estimate the likelihood of loop closures.
Assuming each image has hundreds of SIFT features, mapping the features to the visual words, using a linear search method, is computationally expensive and not practical for real-time localization. Researchers have tackled this problem with different approaches that speeds up the search but at the expense of accuracy. A number of papers have employed compact feature descriptors that speeds up the search. Gálvez-López and Tardós in [@bi-BoW] propose to use FAST [@FAST] and BREIF [@BRIEF] binary features and introduce a BoW model that descritizes a binary space. Similarly, [@fabmap; @fabmap2] use SURF [@Surf] to have a more compact feature descriptor. Another approach is to use approximate nearest neighbor search algorithms, like hierarchical k-means [@nister], KD-trees [@fabmap] or graph-based search methods [@GNNS] and [@SGNNS], to speed up the quantization process.
Achar *et al.* [@Kosecka] and Zhang [@BoRF] propose reducing the number of features in each image, thereby reducing (removing) the vector-quantization process. They keep track of the features that are repeatable over time. However, these approaches are more specific to the navigation problems where the data is sequential and there exists considerable overlaps between consecutive images. Our approach is more similar to this group as we also reduce the amount of feature mapping. However rather than only selecting a small set of features, we use all features but we stop the mapping process when necessary. Our approach is also more general as it does not depend on the sequential property of the data.
PROPOSED METHOD {#sec:method}
===============
Vector-quantization (VQ) is an expensive process when the BoW-based image retrieval is performed in large-scale environments, in which hundreds of features extracted from an image need to be matched against hundreds of thousands of visual words. The question is if we really need to vector-quantize all features? Depending on the difficulty level of the search, the number of features to be converted to visual words may vary. We call a search difficult when there are many similar images to the query and thus finding the nearest neighbor among all those candidates requires more computations[^3]. Whereas in an easy search, the query image is similar to only a few images in the database and it can find its true match after processing only a small percentage of features. Figure \[diff\_search\] and Figure \[easy\_search\] show examples of easy and difficult searches. The histograms show the similarity of each database image to the query based on the *tf-idf* score. The difficult search needs to process at least $89\%$ of features to find the closest match to the query (indicated by the peak of the histogram), however the easy search can stop the search after processing only $12\%$ of features as the peak does not change until the end of the search. Comparing the distance between the peak and average of the histograms, it can be seen that in difficult search this distance is smaller than that in easy search, which is expected. Initially each image starts with no vote. In original BoW, image features are converted to visual words one by one and the histogram (of images’ scores) is built incrementally. Each bin of the histogram corresponds to one of the database images and indicates the score of that image. Each visual word will cast a distance-weighted vote for multiple images, i.e., histogram bins. This process continues until all words cast their votes and then the peak of the histogram (the bin with the highest score) determines the nearest neighbor to the query.[^4]
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![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/846.jpg "fig:"){width=".21\textwidth"} ![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/633.jpg "fig:"){width=".21\textwidth"}
(a) (b)
![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/diff_query11percent_newcol_odd_846_peak633_firsthist.eps "fig:"){width=".22\textwidth"} ![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/diff_query11percent_newcol_odd_846_peak633_lasthist.eps "fig:"){width=".223\textwidth"}
(c) (d)
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![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/996.jpg "fig:"){width=".21\textwidth"} ![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/430.jpg "fig:"){width=".21\textwidth"}
(e) (f)
![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/easy_query88percent_newcol_odd_996_peak430_firsthist.eps "fig:"){width=".22\textwidth"} ![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/easy_query88percent_newcol_odd_996_peak430_lasthist.eps "fig:"){width=".223\textwidth"}
(g) (h)
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[c]{}\
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Our method builds upon the voting scheme of BoW; See Table \[alg:mapping\]. The features are selected randomly to be vector-quantized and are used to score the images (performed by `extract-feature`, `random-permute`, and `vector-quantize` functions in Table \[alg:mapping\]). In an easy search, as the query has many common words with its nearest neighbor and not with other images, after processing only some small percentage of the features the score of the true matching image becomes significantly larger than the mean of the histogram. However, when the query shares its words with many images, i.e., a difficult search, a higher percentage of the words need to cast their vote to find the true match. We stop the feature mapping once the distance between the peak of the histogram to the average of the other bins is greater than some threshold (`stop-time` function). Other stopping rules might be defined, which are described in Section \[sec:rules\].
Our method is different from the naive approach of stopping after quantizing a fixed percentage of the features. Based on our stopping criterion when the search is easy, the VQ stops sooner, otherwise it stops after processing more features. With the naive approach, a search is forced to stop even if more processing is required to find the nearest neighbor.
0 The problem that we study in this paper can be formulated as a stopping problem [@Shiryaev]. The stopping problem is a decision making problem where at each round, the decision maker observes an input and a (possibly noisy) reward and decides whether he wants to see the next input or not. The objective is to maximize the expected reward when he decides to [*stop*]{}. The stopping problem is studied in various settings in statistics, decision theory, and economics.
More specifically, we can formulate our problem as finding the best option from a pool on options based on some measurements. Assume a number of distributions are given, from which we want to find the one with the highest mean. At each round, we observe a new sample from these distributions. As sampling can be expensive, the objective is to stop sampling and choose the distribution with the highest mean as quick as possible . The common approach is to employ certain concentration inequalities (such as Hoeffding’s inequality [@Hoeffding]) to approximate the mean of each distribution after a number of observations. When the approximations are accurate enough and the decision maker has enough confidence, he stops and chooses the distribution with the highest empirical mean [@Hoeffding_races].
In our problem, given a query image, we want to find an image that is closest in terms of an approximate cosine distance. The BoW procedure processes the features one by one, increasing the score of each candidate image by some number. We can view the candidate images as different distributions and the new scores as the new samples. At each round, we have a new estimate for the similarity of each candidate image to the query image. The problem is to stop sampling and return the true match as quick as possible.
The problem that we study in this paper can be formulated as a stopping problem [@Shiryaev]. The stopping problem is a decision making problem where at each round, the decision maker observes an input and a (possibly noisy) reward and decides whether he wants to see the next input or not. The objective is to maximize the expected reward when he decides to [*stop*]{}. The stopping problem is studied in various settings in statistics, decision theory, and economics.
More specifically, we can formulate our problem as finding the best option from a pool of options based on some measurements. Assume a number of distributions are given, from which we want to find the one with the highest mean. At each round, we observe a new sample from each distribution. As sampling can be expensive, the objective is to stop sampling and find the distribution with the highest mean as quick as possible. The common approach is to employ certain concentration inequalities (such as Hoeffding’s Inequality [@Hoeffding]) to construct confidence bands around the empirical mean of each distribution after a number of observations. When the approximations are accurate enough and the decision maker has enough confidence, he stops and chooses the distribution with the highest empirical mean [@Hoeffding_races].
To illustrate the ideas, assume we are given two distributions, $p_1$ and $p_2$, and are asked to find the one with the higher mean. Let the expected values of these distributions be $\mu_1$ and $\mu_2$, respectively. Assume that $\mu_1>\mu_2$. Let $x_s\in[0,1], 1\le s\le t$ be samples from $p_1$ and $y_s\in[0,1], 1\le s\le t$ be samples from $p_2$. Define the empirical means by $\overline{X}_t = \frac{1}{t}\sum_{s=1}^t x_s$ and $\overline{Y}_t = \frac{1}{t}\sum_{s=1}^t y_s$ and the empirical gap by $g_t = \overline{X}_t - \overline{Y}_t$. Without loss of generality, assume that $g_t > 0$. From Hoeffding’s Inequality, we get that
$${{\mathbb P}\left(\overline{X}_t - \mu_1 \le g_t/2\right)} \ge 1 - e^{-g_t^2 t/2}\,,$$ and $${{\mathbb P}\left(\overline{Y}_t - \mu_2 \ge -g_t/2\right)} \ge 1 - e^{-g_t^2 t/2}\;.$$ Thus, with probability at least $1-2e^{-g_t^2 t/2}$, $\overline{X}_t - \mu_1 \le g_t/2$ and $\overline{Y}_t - \mu_2 \ge -g_t/2$, which implies that $$\mu_1 > \mu_2\;.$$ Thus, we have correctly identified the distribution with the highest mean (here $p_1$) with probability at least $1-2e^{-g_t^2 t/2}$. If we demand that this probability be at least $1-\delta$ for some $\delta\in (0,1)$, then we need to have $$\label{eq:num_samples}
g_t\ge \sqrt{\frac{2\log(2/\delta)}{t}}\;.$$ Equation can be used as a stopping condition to make the right decisions with high probability. We might ask how many samples are needed before Condition is satisfied? By applying Hoeffding’s Inequality, it is not difficult to see that if the true gap ($g=\mu_1-\mu_2$) is small, then the number of samples $t$ in Condition needs to be larger. This implies that the identification is more difficult when there is a small gap between the two distributions.
0 $$\label{eq:num_samples}
t\ge \frac{2\log(2/\delta)}{g^2},$$ with probability at least $1-\delta/2$, it holds that $$\label{eq:error1}
\overline{X}_t - \mu_1 \ge -g/2\;.$$ Similarly, it can be shown that if holds, then with probability at least $1-\delta/2$, it holds that $$\label{eq:error2}
\overline{Y}_t - \mu_2 \le g/2\;.$$ Then, by and , we get that if holds, with probability at least $1-\delta$, it holds that $$\overline{X}_t \ge \overline{Y}_t\,,$$ which implies that we have identified the distribution with the highest mean correctly. Equation determines the number of samples that is required to make the right decision with high probability. Notice that this number scales like $1/g^2$, which implies that the identification is more difficult (more samples are required) when there is a small gap between the two distributions.
In our image matching problem, given a query image, we want to find an image that is closest in terms of an approximate cosine distance. The BoW procedure processes the features one by one, increasing the score of each candidate image by some number. We can view the candidate images as different distributions and the new scores as the new samples. At each round, we have a new estimate for the similarity of each candidate image to the query image. The problem is to stop sampling and return the true match as quick as possible.
It might seem natural to use the stopping condition for this problem. However, we found out that these theoretical results are very conservative in practice and often require the processing of a large number of features before stopping, so we do not test Hoeffding’s inequality experimentally. In the next section, we propose a number of rules that show better performance.
EXPERIMENTAL RESULTS {#sec:results}
====================
In this section, we compare the performance of the original BoW method with the BoW method that uses our stopping rules, when applied to the appearance-based localization problem. We will describe the datasets we used for performance evaluation of both methods, followed by discussion of our experimental results. We also describe the different stopping rules that we used in our experiments.
Datasets
--------
We performed our experiments on four datasets. Two datasets have been selected from the Oxford City Center dataset and two from the New College dataset [@fabmap] (see Figure \[fig:datasets\]). Both have been used as the benchmark for localization and SLAM evaluations. The ground truth data is also available for each of them[^5]. Each dataset contains two sets of image sequences. One sequence is taken from the right camera and the other from the left camera mounted on the robot. Each sequence of the City Center dataset contains $1237$ images and each of the New College contains $1073$. The resolution of images is $640\times480$. For each dataset we used the first half of the images as training data on which we performed the *k-means* clustering and generated a vocabulary of $5000$ visual words. We used $128$-dimensional SIFT feature descriptors as the input to the clustering. Each image has $\sim 400$ SIFT descriptors on average.
The other half of the sequences have been used as the test data, i.e., query images. Each query is matched to the earlier images in the sequence. For each query there are multiple matches in ground truth. If the match that we find for each query is among those correct(ground truth) matches, we call the match correct otherwise incorrect.
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![Images from the New College (top row) and City Center (bottom row) sequences used in the reported experiment.[]{data-label="fig:datasets"}](./images/college_0007.jpg "fig:"){width=".15\textwidth"} ![Images from the New College (top row) and City Center (bottom row) sequences used in the reported experiment.[]{data-label="fig:datasets"}](./images/college_0442.jpg "fig:"){width=".15\textwidth"} ![Images from the New College (top row) and City Center (bottom row) sequences used in the reported experiment.[]{data-label="fig:datasets"}](./images/college_1732.jpg "fig:"){width=".15\textwidth"}
![Images from the New College (top row) and City Center (bottom row) sequences used in the reported experiment.[]{data-label="fig:datasets"}](./images/city_0301.jpg "fig:"){width=".15\textwidth"} ![Images from the New College (top row) and City Center (bottom row) sequences used in the reported experiment.[]{data-label="fig:datasets"}](./images/city_1840.jpg "fig:"){width=".15\textwidth"} ![Images from the New College (top row) and City Center (bottom row) sequences used in the reported experiment.[]{data-label="fig:datasets"}](./images/city_0048.jpg "fig:"){width=".15\textwidth"}
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Performance Evaluations
-----------------------
On each dataset we ran the BoW with the original voting scheme (inverted-index) and BoW with our voting scheme that employs stopping rules. Each visual word in the query image has been weighed with the *tf-idf* score and the top-$n$ images with highest scores have been returned as match candidates. We set $n$ to 3, 5 and 10 in our experiments. We select the candidates whose similarity to the query is above some threshold. Our experimental results have been summarized in Table \[exp1:city1\] to Table \[exp1:newcol2\]. The reported recall values are for precisions above 90%. Top 3, top 5 and top 10 in Tables \[exp1:city1\] to \[exp1:newcol2\] indicate the top-3, -5 and -10 nearest neighbors to the query image that we retrieved. We used different stopping thresholds to generate different recalls. The percentage of the features that we processed and the accuracy of the localization have been computed and compared with the original BoW. All the results have been produced by averaging over $10$ Monte Carlo runs. The experiments show that we have improved the computational cost significantly at the expense of slight reduction in accuracy.
4.5pt
-- --------- ------------- -------- -------- -------- --------
BoW
(Inv. Indx)
top 3: 0.7897 0.7861 0.7540 0.7273 0.6934
top 5: 0.8378 0.8342 0.8111 0.7932 0.7647
top 10: 0.9055 0.9037 0.8806 0.8610 0.8503
1 0.8655 0.6879 0.5911 0.4965
- 0.25 0.2 0.18 0.16
-- --------- ------------- -------- -------- -------- --------
: Comparison of our approach to the original BoW on the New College dataset, left-side sequence, 1395 words.[]{data-label="exp1:newcol2"}
[cc]{} &\
4.5pt
-- --------- ------------- -------- -------- -------- --------
BoW
(Inv. Indx)
top 3: 0.8324 0.8253 0.8164 0.7879 0.7219
top 5: 0.8556 0.8538 0.8485 0.8235 0.7772
top 10: 0.8895 0.8877 0.8895 0.8717 0.8307
1 0.8395 0.6606 0.5001 0.3240
- 0.28 0.20 0.15 0.10
-- --------- ------------- -------- -------- -------- --------
: Comparison of our approach to the original BoW on the New College dataset, left-side sequence, 1395 words.[]{data-label="exp1:newcol2"}
[cc]{} &\
4.5pt
-- --------- ------------- -------- -------- -------- --------
BoW
(Inv. Indx)
top 3: 0.9104 0.8920 0.8668 0.8455 0.7743
top 5: 0.9395 0.9322 0.9211 0.8988 0.8416
top 10: 0.9709 0.9642 0.9613 0.9535 0.9138
1 0.7821 0.6670 0.4994 0.3030
- 0.12 0.10 0.08 0.06
-- --------- ------------- -------- -------- -------- --------
: Comparison of our approach to the original BoW on the New College dataset, left-side sequence, 1395 words.[]{data-label="exp1:newcol2"}
[cc]{} &\
4.5pt
-- --------- ------------- -------- -------- -------- --------
BoW
(Inv. Indx)
top 3: 0.9487 0.9397 0.9299 0.8778 0.8150
top 5: 0.9780 0.9707 0.9544 0.9356 0.8818
top 10: 0.9902 0.9894 0.9804 0.9658 0.9487
1 0.8567 0.7374 0.5517 0.3211
- 0.25 0.20 0.15 0.10
-- --------- ------------- -------- -------- -------- --------
: Comparison of our approach to the original BoW on the New College dataset, left-side sequence, 1395 words.[]{data-label="exp1:newcol2"}
Different Stopping Rules {#sec:rules}
------------------------
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![Accuracy (recall) vs. % of features used, City Center dataset[]{data-label="fig:rules2"}](./images/feat_Acc_3rules_top5_im.eps "fig:"){width=".40\textwidth"}
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![Accuracy (recall) vs. % of features used, City Center dataset[]{data-label="fig:rules2"}](./images/feat_Acc_city_odd.eps "fig:"){width=".40\textwidth"}
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We used different stopping rules in our experiments and compared the efficiency of each of them when applied to the localization problem. Three rules have been used that are explained below:
- Rule 1: computes the distance between the peak and the average of the similarity histogram (i.e., the histogram that shows the similarity of database images to the query). Once the distance is above some threshold, it stops the search: Stop if $|\max(hist)-\mbox{mean}(hist)| > T$ and return the image corresponding to the peak.\
- Rule 2: computes the relative distance between the peak and the average of the histogram. Once the distance is above some threshold it stops the search: Stop if $|\max(h)-\mbox{mean}(h)|/\mbox{mean}(h) > T$ and return the image corresponding to the peak.\
- Rule 3: keeps track of the peak of the histogram. If the peak does not change after processing $T$ features, the image corresponding to the peak is returned as the nearest neighbor to query.
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![Our experiment on City Center dataset with our stopping method that shows only $60\%$ of images need to process $90\%$$100\%$ of their features to get the accuracy of BoW.[]{data-label="fig:feat_im"}](./images/feat_im3.eps "fig:"){width=".35\textwidth"}
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The experimental results shown in Tables \[exp1:city1\] to \[exp1:newcol2\] have been performed by Rule 1. Figures \[fig:rules\] and \[fig:rules2\] show the result of applying different rules to the BoW search on the New College (right-side sequence) and City Center (right-side sequence) datasets, respectively. The graphs have been generated by varying the stopping thresholds. As can be seen, stopping rules \#1 and \#3 generate better results. By reducing the number of processed features to half the accuracy only decreases slightly.
Figure \[fig:feat\_im\] shows another experiment that validates our proposed stopping method. Figure shows the relation between the number of images and the percentage of features they need to process to obtain the same result as that of the original BoW. The experiment has been done on City Center dataset (left-side sequence) with the stopping rule \#1 used. We set the threshold to $0.29$ to get exactly the same accuracy of original BoW. As can be seen, only $338$ images need to process $90\%$$100\%$ of their features, another $223$ only need to vector-quantize smaller percentages of features.
CONCLUSION {#sec:conclud}
==========
We observed that the computational requirements of the BoW method can be significantly reduced by allocating less computations to easier search queries. Deciding when to terminate the search is treated as a stopping problem. We proposed several stopping rules and showed their effectiveness on an appearance-based localization problem.
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[^1]: K. Hajebi and H. Zhang are with Faculty of Computing Science, University of Alberta, Canada, [at ualberta.ca]{}
[^2]: SLAM stands for simultaneous localization and mapping
[^3]: The similarity between two images can be simply defined as the number of visual words they share. Other similarity measures like tf\_idf [@sivic] have been used as well.
[^4]: Usually, rather than returning the peak as the true match, a number of highest scored images are selected as match candidates and then the true match is selected after further processing like geometric verification among those candidates. Here for the simplicity, we consider the peak of the histogram as the true match.
[^5]: <http://www.robots.ox.ac.uk/~mobile/IJRR_2008_Dataset/>
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---
abstract: |
We consider multi-variable sigma function of a genus $g$ hyperelliptic curve as a function of two group of variables - jacobian variables and parameters of the curve. In the theta-functional representation of sigma-function, the second group arises as periods of first and second kind differentials of the curve. We develop representation of periods in terms of theta-constants. For the first kind periods, generalizations of Rosenhain type formulae are obtained, whilst for the second kind periods theta-constant expressions are presented which are explicitly related to the fixed co-homology basis.\
We describe a method of constructing differentiation operators for hyperelliptic analogues of $\zeta$- and $\wp$-functions on the parameters of the hyperelliptic curve. To demonstrate this method, we gave the detailed construction of these operators in the cases of genus 1 and 2.
address:
- 'Steklov Mathematical Institute, Moscow'
- 'National University of Kyiv-Mohyla Academy'
- Institute of Magnetism NASU
author:
- 'V.M.Buchstaber'
- 'V.Z. Enolski'
- 'D.V.Leykin'
title: 'Multi-variable sigma-functions: old and new results'
---
..
Introduction
============
Our note belongs to an area in which Emma Previato took active part in the development. Since the time of first publication of the present authors [@bel97] she has inspired them, and given them a lot of suggestions and advices.
The area under consideration is the construction of Abelian functions in terms of multi-variable $\sigma$-functions. Similarly to the Weierstrass elliptic function, the multi-variable sigma keeps the same main property - it remains form-invariant at the action of the symplectic group. Abelian functions appear as logarithmic derivatives like the $\zeta, \wp$-functions of the Weierstrass theory and similarly to the standard theta-functional approach which lead to the Krichever formula for KP solutions, [@kr1977]. But a fundamental difference between sigma and theta-functional theories is the following. They are both constructed by the curve and given as series in Jacobian variables, but in the first case the expansion is purely algebraic with respect to the model of the curve since its coefficients are polynomials in parameters of the defining curve equation, whilst in the second case coefficients are transcendental, being built in terms of Riemann matrix periods which are complete Abelian integrals.
In many publications, in particular see [@oni05; @eemop07; @bel12; @ny12; @ehkklp12; @an13; @bef13; @mp14; @nak16; @nak18] and references therein, it was demonstrated that multi-variable $\wp$-functions represent a language which is very suitable to speak about completely integrable systems of KP type. In particular, very recently, using the fact that sigma is an entire function in the parameters of the curve (in contrast with theta-function), families of degenerate solutions for solitonic equations were obtained [@bl18; @ben18]. In recent papers [@bm17] and [@bm18], an algebraic construction of a wide class of polynomial Hamiltonian integrable systems was given, and those of them whose solutions are given by hyperelliptic $\wp$-functions were indicated.
Revival of interest in multi-variable $\sigma$-functions is in many respects guided by H.Baker’s exposition of the theory of Abelian functions, which go back to K.Weierstrass and F.Klein and is well documented and developed in his remarkable monographs [@bak998; @bak903]. The heart of his exposition is the representation of fundamental bi-differential of the hyperelliptic curve in algebraic form, in contrast with the representation as a double differential of Riemann theta-function developed by Fay in the monograph, [@fay973]. Recent investigations have demonstrated that Baker’s approach can be extended beyond hyperelliptic curves to wider classes of algebraic curves.
The multi-variable $\sigma$-function of algebraic curve $\mathcal{C}$ is known to be represented in terms of $\theta$-function of the curve as a function of two groups of variables - the Jacobian of the curve, $\mathrm{Jac}(\mathcal{C})$ and the Riemann matrix $\tau$. In the vast amount of recent publications, properties of the $\sigma$-function as a function of the first group of variables were discussed, whilst modular part of variables and relevant objects like $\theta$-constant representations of complete Abelian integrals are considered separately. In this paper we deals with $\sigma$ as a function over both group of variables.
Due to the pure modular part of the $\sigma$-variables, we consider problem of expression of complete integrals of first and second kind in terms of theta-constants. A revival of interest in this classically known material accords to many recent publications reconsidering such problems such as the Schottky problem [@fgs17], the Thomae [@eg06; @ef08; @ekz18] and Weber [@nr17] formulae, and the theory of invariants and its applications [@ksv05; @ef16].
The paper is organized as follows. In the Section 2 we consider the hyperelliptic genus $g$ curve and the complete co-homology basis of $2g$ meromorphic differentials, with $g$ of them chosen as holomorphic ones. We discuss expressions for periods of these differentials in terms of $\theta$-constants with half integer characteristics. Theta-constants representations of periods of holomorphic integrals is known from the Rosenhain memoir [@ros851], where the case of genus two was elaborated. We discus this case and generalize the Rosenhain expressions to higher genera hyperelliptic curves. Theta constant representations of second kind periods is known after F.Klein [@klein888], who presented closed formula in terms of derivated even theta-constants for the non-hyperelliptic genus three curve. We re-derive this formula for higher genera hyperelliptic curves.
Section 3 is devoted to a classically known problem, which was resolved in the case of elliptic curves by Frobenius and Stickelberger [@fs882]. The general method of the solution of this problem for a wide class, the so-called $(n, s)$ - curves, has been developed in [@bl08] and represents extensions of the Weierstrass’s method for the derivation of system of differential equations defining the sigma-function. All stages of derivation are given in details and the main result is that the sigma-function is completely defined as the solution of a system of heat conductivity equations in a nonholonomic frame. We also consider there another widely known problem - the description of the dependence of the solutions on initial data. This problem is formulated as a description of the dependence of the integrals of motion, which levels are given as half of the curve parameters, from the remaining half of the parameters. The differential formulae obtained permit us to present effective solutions of this problem for Abelian functions of the hyperelliptic curve. It is noteworthy that because integrals of motion can be expressed in terms of second kind periods in this place, the results of Section 2 are required. The results obtained in Section 3 are exemplified in details by curves of genera one and two. All consideration is based on explicit uniformization of the space of universal bundles of the hyperelliptic Jacobian.
Modular representation of periods of hyperelliptic co-homologies
================================================================
Modular invariance of the Weierstrass elliptic $\sigma$-function, $\sigma=\sigma(u;g_2,g_3)$ follows from its defining in in [@wei885] in terms of recursive series in terms of variables $(u,g_2,g_3)$. Alternatively $\sigma$-function can be represented in terms of Jacobi $\theta$-function and its modular invariance follows from transformation properties of $\theta$-functions. The last representation involves complete elliptic integrals of first and second kind and their representations in terms of $\theta$-constants are classically known. In this section we are studying generalizations of these representations to hyperelliptic curves of higher genera realized in the form $$\begin{aligned}
y^2 = P_{2g+1}(x)= (x-e_1)
\cdots (x-e_{2g+1}) \label{HCurve} \end{aligned}$$ Here $P_{2g+1}(x)$ - monic polynomial of degree $2g+1$, $e_i\in \mathbb{C}$ - branch points and the curve supposed to be non-degenerate, i.e. $e_i\neq e_j$.
Representations of complete elliptic integrals of first and second kind in terms of Jacobi $\theta$-constants are classically known. In particular, if elliptic curve is given in Legendre form[^1] $$y^2=(1-x^2)(1- k^2x^2)$$ where $k$ is Jacobian modulus, then complete elliptic integrals of the first kind $K=K(k)$ is represented as $$\begin{aligned}
K= \int_{0}^1 \frac{\mathrm{d} x}{ \sqrt{ (1-x^2)(1-k^2 x^2) }}= \frac{\pi}{2} \vartheta_3^2(0;\tau), \label{FirstKind}
\end{aligned}$$ and $\vartheta_3=\vartheta_3(0|\tau)$ and $\tau=\imath \frac{K'}{K}$, $K'=K(k'), k^2+{k'}^2=1$.
Further, for elliptic curve realized as Weierstrass cubic $$y^2= 4 x^3-g_2 x - g_3= 4 (x-e_1)(x-e_2)(x-e_3) \label{WCubic}$$ recall standard notations for periods of first and second kind elliptic integrals $$\begin{aligned}
\begin{split}
2\omega &= \oint_{\mathfrak{a}} \frac{\mathrm{d}x}{y}, \quad 2\eta = -\oint_{\mathfrak{a}} \frac{x\mathrm{d}x}{y}\\
2\omega' &= \oint_{\mathfrak{b}} \frac{\mathrm{d}x}{y}, \quad 2\eta' = -\oint_{\mathfrak{b}} \frac{x\mathrm{d}x}{y}
\end{split} \hskip1.5cm \tau = \frac{\omega'}{\omega}\end{aligned}$$ and Legendre relation for them $$\quad\omega\eta'-\eta\omega' = - \frac{\imath \pi}{2} \label{LegendreRel}$$ Then the following Weierstrass relation is valid $$\begin{aligned}
\eta=- \frac{1}{12\omega} \left(\frac{\vartheta_2''(0)}{\vartheta_2(0)}
+ \frac{\vartheta_3''(0)}{\vartheta_3(0)}+ \frac{\vartheta_4''(0)}{\vartheta_4(0)}\right)
\label{SecondKind}\end{aligned}$$ In this section we are discussing generalization of these relations to higher genera hyperelliptic curves realized as in (\[HCurve\]).
In this subsection we reproduce H.Baker [@bak907] notations. Let $\mathcal{C}$ be genus $g$ non-degenerate hyperelliptic curve realised as double cover of Riemann sphere, $$\begin{aligned}
y^2= 4 \prod_{j=1}^{2g+1}(x-e_j)\equiv 4x^{2g+1}+ \sum_{i=0}\lambda_i x^i,\quad e_i\neq e_j,\; \lambda_i
\in \mathbb{C}
\label{hyperelliptic}\end{aligned}$$ Let $({\mathfrak{a}};{\mathfrak{b}})
=( \mathfrak{a}_1,\ldots,\mathfrak{a}_g; \mathfrak{b}_1,\ldots,\mathfrak{b}_g ) $ be canonic homology basis. Introduce co-homology basis ([*Baker co-homology basis*]{}) $$\begin{aligned}
\begin{split} \mathrm{d}{u} (x,y)=( \mathrm{d}u_1 (x,y), \ldots,
\mathrm{d}u_g (x,y))^T, \mathrm{d}{r} (x,y)=( \mathrm{d}r_1 (x,y), \ldots,
\mathrm{d}r_g (x,y))^T\\
\mathrm{d}u_i(x,y)=\frac{x^{i-1}}{y}\mathrm{d}x,\quad \mathrm{d}r_j(x,y)=\sum_{k=j}^{2g+1-j}(k+1-j)
\frac{x^k}{4y}\mathrm{d}x,\quad i,j=1,\ldots,g\end{split}\label{bakerbasis}\end{aligned}$$ satisfying to the generalized Legendre relation, $$\begin{aligned}
\mathfrak{M}^TJ\mathfrak{M}=-\frac{\imath \pi}{2} J,\qquad \mathfrak{M}=\left(\begin{array}{cc} \omega&\omega'\\
\eta& \eta' \end{array}\right), \quad J = \left( \begin{array}{cc} 0_g& 1_g \\ -1_g&0_g
\end{array} \right)\end{aligned}$$ where $g\times g$ period matrices $\omega,\omega', \eta, \eta'$ are defined as $$\begin{aligned}
2\omega=\left( \oint_{\mathfrak{a}_j}\mathrm{d}u_i \right),\quad
2\omega'=\left( \oint_{\mathfrak{b}_j}\mathrm{d}u_i \right), \quad
2\eta=-\left( \oint_{\mathfrak{a}_j}\mathrm{d}r_i \right),\quad
2\eta'=-\left( \oint_{\mathfrak{b}_j}\mathrm{d}r_i \right)\end{aligned}$$ We also denote $\mathrm{d}v=(\mathrm{d}v_1,\ldots,\mathrm{d}v_g )^T = (2\omega)^{-1} \mathrm{d}u$ vector of normalized holomorphic differentials.
Define Riemann matrix $\tau= \omega^{-1}\omega'$ belonging to Siegel half-space $\mathcal{S}_g =\{ \tau^T=\tau,\; \mathrm{Im} \tau>0 \} $. Define Jacobi variety of the curve $\mathrm{Jac}(\mathcal{C})=\mathbb{C}^g/ 1_g\oplus \tau $. Canonic Riemann $\theta$-function is defined on $\mathrm{Jac}(\mathcal{C})\times \mathcal{S}_g$ by Fourier series $$\theta({z};\tau) = \sum_{\mathbb{n} \in \mathbb{Z}^n} \mathrm{e}^{ \imath \pi {n}^T\tau {n}
+ 2\imath \pi {z}^T {n} }\label{thetacan}$$ We will also use $\theta$-functions with half-integer characteristics $[\varepsilon] =
\left[\begin{array}{c} {\varepsilon'}^T\\ {\varepsilon''} \end{array} \right]$, $\varepsilon_i', \varepsilon_j'' = 0$ or $1$ defined as $$\theta[\varepsilon]({z};\tau) = \sum_{\mathbb{n} \in \mathbb{Z}^n} \mathrm{e}^{ \imath \pi
( {n+\varepsilon'}/2)^T
\tau( {n+\varepsilon'}/2)
+ 2\imath \pi ({z+\varepsilon''}/2)^T( n+\varepsilon'/2) }\label{thetachar}$$ Characteristic is even or odd whenever $ {\varepsilon'}^T\varepsilon'' = 0 $ (mod 2) or 1 (mod 2) and $\theta[\varepsilon](z;\tau)$ as function of $z$ inherits parity of the characteristic.
Derivatives of $\theta$-functions by arguments $z_i$ will be denoted as $$\theta_i[\varepsilon](z;\tau) = \frac{\partial}{\partial z_i} \theta[\varepsilon](z;\tau),
\quad \theta_{i,j}[\varepsilon](z;\tau) = \frac{\partial^2}{\partial z_i\partial z_j} \theta[\varepsilon](z;\tau) , \quad\text{etc.}$$
Fundamental bi-differential $\Omega(P,Q)$ is uniquely definite on the product $(P,Q)\in \mathcal{C}\times \mathcal{C}$ by following conditions:
[**i**]{} $\Omega$ is symmetric, $\Omega(P,Q)=\Omega(Q,P)$
[**ii**]{} $\Omega$ is normalized by the condition $$\begin{aligned}
\oint_{\mathfrak{a}_i} \Omega(P,Q)=0, \quad i=1,\ldots,g\end{aligned}$$
[**iii**]{} Let $P=(x,y)$ and $Q=(z,w)$ have local coordinates $\xi_1=\xi(P)$, $\xi_2=\xi(Q)$ in the vicinity of point $R$, $\xi(R)=0$, then $\Omega(P,Q)$ expands to power series as $$\Omega(P,Q)= \frac{\mathrm{d}\xi_1 \mathrm{d}\xi_2}{( \xi_1-\xi_2)^2 } + \; \text{homorphic 2-form}$$
Fundamental bi-differential can be expressed in terms of $\theta$-function [@fay973] $$\Omega(P,Q)= \mathrm{d}_x\mathrm{d}_z\theta\left( \int_{Q}^P\mathrm{d}{v} + {e}\right), \quad P=(x,y), Q=(z,w)$$ where $\mathrm{d}v$ is normalized holomorphic differential and ${e}$ any non-singular point of the $\theta$-divisor $(\theta)$, i.e. $\theta({e})=0$, but not all $\theta$-derivatives, $ \partial_{z_i}\theta({z})\vert_{{z}={e}} $, $i=1,\ldots,g$ vanish.
In the case of hyperelliptic curve $\Omega(P,Q)$ can be alternatively constructed as $$\Omega(P,Q) =\frac12 \frac{\partial}{\partial z} \frac{ y+w }{ y(x-z)} \mathrm{d}x\mathrm{d}z
+ \mathrm{d}{r}(P)^T \mathrm{d}{u} (Q) + 2\mathrm{d}{u}^T (P)\varkappa
\mathrm{d}{u} (Q)\label{omega1}$$ where first two terms are given as rational functions of coordinates $P,Q$ and necessarily symmetric matrix $\varkappa^T=\varkappa$, $\varkappa=\eta(2\omega)^{-1}$ is introduced to satisfy the normalization condition ${\bf ii}$. In shorter form (\[omega1\]) cab be rewritten as $$\Omega(P,Q) =\frac{2 yw +F(x,z)}{ 4 (x-z)^2 yw } \mathrm{d}x\mathrm{d}z + 2\mathrm{d}{u}^T (P)\varkappa
\mathrm{d}{u} (Q)\label{omega1}$$ where $F(x,z)$ is so-called Kleinian 2-polar, given as $$F(x,z)=\sum_{k=0}^g x^kz^k \left( 2\lambda_{2k}+\lambda_{2k+1}(x+z) \right) \label{polar}$$ Recently algebraic representation for $\Omega(P,Q)$ similar to (\[omega1\]) found in [@suz17], [@eyn18] for wide class on algebraic curves, included $(n,s)$-curves [@bel999].
Main relation lying in the base of the theory is Riemann formula represented third kind Abelian integral as $\theta$-quotient written in terms of described above realization of the fundamental differential $\Omega(P,Q)$.
(Riemann) Let $P'=(x',y')$ and $P''=(x'',y'')$ are two arbitrary distinct points on $\mathcal{C}$ and let $\mathcal{D}'=\{ P_1'+\ldots+P_{g}'\}$ and $\mathcal{D}''=\{ P_1''+\ldots+P_{g}''\}$ are two non-special divisors of degree $g$. Then the following relation is valid $$\begin{aligned}
\begin{split}
&\int_{P''}^{P'} \sum_{j=1}^g\int_{P_j'}^{P_j''} \left\{ \frac{2yy_i+F(x,x_i)}{4(x-x_i)^2}\frac{\mathrm{d}x}{y}\frac{\mathrm{d}x_i}{y_i} + 2 \mathrm{d}{u}(x,y)\varkappa \mathrm{d} {u}(x_i,y_i) \right\} \\
&=\mathrm{ln}
\left(\frac{ \theta(\mathcal{A}(P')-\mathcal{A}(\mathcal{D}') +{K}_{\infty} )}
{ \theta(\mathcal{A}(P')-\mathcal{A}(\mathcal{D}'') +{K}_{\infty} ) } \right)
-\mathrm{ln}
\left(\frac{\theta(\mathcal{A}(P'')-\mathcal{A}(\mathcal{D}') +{K}_{\infty} )}
{\theta(\mathcal{A}(P'')-\mathcal{A}(\mathcal{D}'') +{K}_{\infty} ) } \right)
\end{split} \label{Riemann1}\end{aligned}$$ where $\mathcal{A}(P)= \int_{\infty}^{P} \mathrm{d}{v}$ Abel map with base point $\infty$, ${K}_{\infty}$ - vector of Riemann constants with bases point $\infty$ which is a half-period.
Introduce multi-variable fundamental $\sigma$-function, $$\sigma({u}) = C \theta[{K}_{\infty}]( (2\omega)^{-1}{u})
\mathrm{e}^{ {u}^T\varkappa{u} },$$ where $[{K}_{\infty}]$ is characteristic of the vector of Riemann constants, ${u}= \int_{\infty}^{P_1} \mathrm{d}{u} + \ldots + \int_{\infty}^{P_g} \mathrm{d}{u} $ with non-special divisor $P_1+\ldots+P_g$. The constant $C$ is chosen so that expansion $\sigma(\boldsymbol{u})$ near ${u}\sim 0$ starts with a Schur-Weierstrass polynomial [@bel999]. The whole expression is proved to be invariant under the action of symplectic group $\mathrm{Sp}(2g,\mathbb{Z})$. Klein-Weierstrass multi-variable $\wp$-functions are introduced as logarithmic derivatives, $$\begin{aligned}
\wp_{i,j}({u})=-\frac{\partial^2}{\partial u_i\partial u_j}, \quad \wp_{i,j,k}
({u})=-\frac{\partial^3}{\partial u_i\partial u_j\partial u_k},\quad \text{etc.} \quad
i,j,k = 1,\ldots g\end{aligned}$$
For $r\neq s \in \{1,\ldots, g\}$ the following formula is valid $$\sum_{i,j=1}^g\wp_{i,j} \left( \sum_{k=1}^g \int_{\infty}^{(x_k,y_k)} \mathrm{d}{u} \right)x_s^{i-1} x_r^{j-1}
= \frac{F(x_s,x_r)-2y_sy_r}{ 4(x_s-x_r)^2}$$
Jacobi problem of inversion of the Abel map $\mathcal{D} \rightarrow \mathcal{A}( \mathcal{D})$ with $\mathcal{D}= (x_1,y_1)+ \ldots+ (x_g,y_g)$ is resolved as $$\begin{aligned}
\begin{split}
&x^g-\wp_{g,g}({u})x^{g-1} - \ldots -\wp_{g,1}({u})=0\\
&y_k= \wp_{g,g,g}({u})x_k^{g-1}+ \ldots + \wp_{g,g,1}({u})
, \quad k=1,\ldots,g\end{split}\end{aligned}$$
In this section we present generalization of Weierstrass formulae $$\wp(\omega)=e_1,\; \wp(\omega+\omega')=e_2,\; \wp(\omega')=e_3$$ to the case of genus $g$ hyperelliptic curve (\[hyperelliptic\]). To do that introduce partitions $$\begin{aligned}
\begin{split} \{1,\ldots, 2g+1\} = \mathcal{I}_0\cup \mathcal{J}_0, \quad \mathcal{I}_0\cap \mathcal{J}_0=\emptyset\\
\mathcal{I}_0=\{ i_1,\ldots, i_g \}, \quad \mathcal{J}_0=\{ j_1,\ldots, j_{g+1} \} \end{split}
\end{aligned}$$ Then any non-singular even half-period $\Omega_{\mathcal{I}}$ is given as $$\begin{aligned}
\Omega_{\mathcal{I}_0} = \int_{\infty}^{( e_{i_1},0 )}\mathrm{d} {u}
+\ldots+ \int_{\infty}^{( e_{i_g},0 )}\mathrm{d} {u},\quad \mathcal{I}_0= \{i_1,\ldots, i_g\}\subset \{1,\ldots,2g+1\}
\end{aligned}$$
Denote elementary symmetric functions $s_n(\mathcal{I}_0)$, $S_{n}(\mathcal{J}_0)$ of order $n$ built on branch points $\{ e_{i_k} \}$, $i_k\in \mathcal{I}_0$, $\{ e_{j_k} \}$, $j_k\in \mathcal{J}_0$ correspondingly. In particular, $$\begin{aligned}
\begin{split}
s_1(\mathcal{I}_0)&=e_{i_1}+\ldots+ e_{i_g}, \hskip1.65cm S_1(\mathcal{J}_0)=e_{j_1}+\ldots+ e_{j_{g+1}}\\
s_2(\mathcal{I}_0)&=e_{i_1}e_{i_2}+\ldots+e_{i_{g-1}} e_{i_g},
\hskip0.5cm S_2(\mathcal{J}_0)=e_{j_1}e_{j_2}+\ldots+e_{j_{g}} e_{i_{g+1}}\\
&\vdots \hskip5.4cm\vdots\\
s_g(\mathcal{I}_0)&=e_{i_1}\cdots e_{i_g} \hskip 2.62cm
S_{g+1}(\mathcal{J}_0)=e_{j_1}\cdots e_{j_{g+1}}
\end{split}\end{aligned}$$ Because of symmetry, $\wp_{p,q}({\Omega}_{\mathcal{I}_0})=\wp_{q,p}({\Omega}_{\mathcal{I}_0})$ enough to find these quantities for $ p\leq q \in \{1,\ldots,g\} $. The following is valid
(Conjectural Proposition ) Let even non-singular half-period ${\Omega}_{\mathcal{I}_0}$ associate to a partition $\mathcal{I}_0\cup \mathcal{J}_0=\{1,\ldots,2g+1\}$. Then for all $k,j\geq k, k,j = 1\ldots,g$ the following formula is valid
$$\begin{aligned}
\begin{split}
&\wp_{k,j}({\Omega}_{\mathcal{I}_0})\\&=(-1)^{k+j}\sum_{n=1}^k n \left( s_{g-k+n}(\mathcal{I}_0)S_{g-j-n+1}(\mathcal{J}_0)
+s_{g-j-n}(\mathcal{I}_0)S_{g+n-k+1}(\mathcal{J}_0) \right),
\end{split} \label{formula}\end{aligned}$$
Klein formula written for even non-singular half period ${\Omega}_{\mathcal{I}_0}$ leads to linear system of equations with respect to Kleinian two-index symbols $ \wp_{i,j}({\Omega}_{\mathcal{I}_0}) $ $$\begin{aligned}
\sum_{i=1}^g\sum_{j=1}^g \wp_{i,j}({\Omega}_{\mathcal{I}_0}) e_{i_r}^{i-1}e_{i_s}^{j-1} =\frac{F(e_{i_r}, e_{i_s})}
{4(e_{i_r} - e_{i_s})^2}\quad \; i_r, i_s \in \mathcal{I}_0\label{Kleineq}\end{aligned}$$
To solve these equation we note that $$\wp_{k,g}({\Omega}_{\mathcal{I}_0})= (-1)^{k+1}s_k(\mathcal{I}_0), \quad k=1,\ldots,g$$ Also note that $F(e_{i_r}, e_{i_s})$ is divisible by $(e_{i_r}- e_{i_s})^2$ and $$\frac{F(e_{i_r}, e_{i_s})}{4(e_{i_r} - e_{i_s})^2}= e_{i_r}^{g-1} e_{i_s}^{g-1}\mathfrak{S}_1+
e_{i_r}^{g-2} e_{i_s}^{g-2}\mathfrak{S}_2+\ldots + \mathfrak{S}_{2g-1}\label{JIP1}$$ where $\mathfrak{G}_k$ are order $k$ elementary symmetric functions of elements $e_i$ $ i\in \{1, \ldots, 2g+1 \} - \{ i_r,i_s \} $
Let us analyse equations (\[Kleineq\]) for small genera, $g\leq 5$. One can see that plugging to the equation (\[JIP1\]) to (\[Kleineq\]) we get non-homogeneous linear equations solvable by Kramer rule and the solutions can be presented in the form (\[formula\]).
Now suppose that (\[formula\]) is valid for higher $g>5$ were computer power is insufficient to check that by means of computer algebra. But that’s possible to check (\[formula\]) for arbitrary big genera numerically giving to branch points $e_i, i=1,\ldots, i=2g+1$ certain numeric values. Many checking confirmed (\[formula\]) .
Quantities $\wp_{i,j}({\Omega}_{\mathcal{I}_0})$ are expressed in terms of even $\theta$-constants as follows $$\begin{aligned}
\wp_{i,j}({\Omega}_{\mathcal{I}_0})=-2\varkappa_{i,j}
- \frac{1}{\theta[\varepsilon_{\mathcal{I}_0}]({0})}
\partial_{{U}_i,{U_j} }^2 \theta[\varepsilon_{\mathcal{I}_0}]({0}), \quad \forall \mathcal{I}_0,\; i,j=1,\ldots, g.\end{aligned}$$ Here $[\varepsilon_{\mathcal{I}_0}]$ is characteristic of the vector $\left[{\Omega}_{\mathcal{I}_0}+{K}_{\infty} \right]$, where ${K}_{\infty}$ is vector of Riemann constants with base point $\infty$ and $\partial_{{U}}$ is directional derivative along vector $\boldsymbol{U}_i$, that is $i$th column vector of inverse matrix of $\mathfrak{a}$-periods, $\mathcal{A}^{-1} = ( {U}_1,\ldots, {U}_g )$. The same formula is valid for all possible partitions $\mathcal{I}_0\cup \mathcal{J}_0$, there are $N_g$ of that, that is number of non-singular even characteristics, $$N_g= \left( \begin{array}{c} 2g+1\\ g \end{array} \right)$$ Therefore one can write
$$\begin{aligned}
\begin{split}
\varkappa_{i,j} &=\frac{1}{8N_g} \Lambda_{i,j}- \frac{1}{2 N_g}
\sum_{ \text{All even non-singular} \;\; [\varepsilon] } \frac{\partial^2_{{U}_i,{U_j}}
\theta[\varepsilon_{\mathcal{I}_0}]({0})}{\theta[\varepsilon_{\mathcal{I}_0}]({0})}\\
\end{split}\end{aligned}$$
where $$\begin{aligned}
\Lambda_{i,j}=-4\sum_{ \text{All partitions} \; \mathcal{I}_0 } \wp_{i,j}({\Omega}_{\mathcal{I}_0})\end{aligned}$$ Denote by $\Lambda_g$ symmetric matrix $$\Lambda_g = ( \Lambda_{i,j} )_{i,j=1,\ldots,g} \label{Lambda1}$$
Entries $\Lambda_{k,j}$ at $k\leq j$ to the symmetric matrix $\Lambda$ are given by the formula $$\begin{aligned}
\begin{split}
\Lambda_{k,j}&= \lambda_{k+j} \frac{ \left( \begin{array}{c} 2g+1\\g \end{array} \right) }
{ \left( \begin{array}{c} 2g+1\\2g+1-k-j \end{array} \right) }
\sum_{n=1}^k n \left[ \left( \begin{array}{c} g\\ g-k+n \end{array} \right) \left( \begin{array}{c} g+1\\ g-j-n+1 \end{array} \right) \right. \\&\hskip 5.5cm + \left.\left( \begin{array}{c} g \\g-j-n \end{array} \right)
\left( \begin{array}{c} g+1\\ g-k+n+1 \end{array} \right) \right]\end{split}\label{Lambda2}\end{aligned}$$
Execute summation in (\[formula\]) and find that each $\Lambda_{k,j}$ proportional to $\lambda_{k+j}$ with integer coefficient.
Matrix $\Lambda_g$ exhibits interesting properties of sum of anti-diagonal elements implemented at derivations in [@eil16], $$\begin{aligned}
\sum_{i,j, \;i+j=k}\Lambda_{g;i,j}=\lambda_k \frac{N_g}{4g+2}
\left[ \frac12 k(2g+2-k)+\frac14(2g+1)( (-1)^k-1 ) \right]\end{aligned}$$
Lower genera examples of matrix $\Lambda$ were given in [@ehkklp12], [@eil16], but method implemented there unable to get expressions for $\Lambda$ at big genera.
At $g=6$ we get matrix $$\begin{aligned}
\Lambda_6= \left( \begin{array}{cccccc} 792\lambda_2&330\lambda_3&120\lambda_4&36\lambda_5&8\lambda_6&\lambda_7\\
330\lambda_3&1080\lambda_4&492\lambda_5&184\lambda_6&51\lambda_7&8\lambda_8\\
120\lambda_4&492\lambda_5&1200\lambda_6&542\lambda_7&184\lambda_8&36\lambda_9\\
36\lambda_5&184\lambda_6&542\lambda_7&1200\lambda_8&492\lambda_9&120\lambda_{10}\\
8\lambda_6&51\lambda_7&184\lambda_8&492\lambda_9&1080\lambda_{10}&330\lambda_{11}\\
\lambda_7&8\lambda_8&36\lambda_9&120\lambda_{10}&330\lambda_{11}&792\lambda_{12}
\end{array} \right)\end{aligned}$$
Collecting all these together we get the following
$\varkappa$-matrix defining multi-variate $\sigma$-function admits the following modular form representation
$$\begin{aligned}
\varkappa=\frac{1}{8 N_g}\Lambda_g-\frac{1}{2N_g} {(2\omega)^{-1}}^T . \left[
\sum_{N_g\;\text{even}\;[\varepsilon]} \frac{1}{\theta[\varepsilon]}
\left(\begin{array}{ccc}
\theta_{1,1}[\varepsilon]&\cdots&\theta_{1,g}[\varepsilon]\\
\vdots&\cdots&\vdots\\
\theta_{1,g}[\varepsilon]&\cdots&\theta_{g,g}[\varepsilon]\end{array}\right)\right].(2\omega)^{-1}
\label{formula2}\end{aligned}$$
[*where $2\omega$ is matrix of $\mathfrak{a}$-periods of holomorhphic differentials and $ \theta_{i,j}[\varepsilon]= \partial^2_{z_i,z_j}\theta[\varepsilon]({z})_{{z}=0} $.*]{}
Note that modular form representation of period matrices $\eta, \eta'$ follows from the above formula,
$$\eta=2\varkappa\omega, \qquad \eta'=2\varkappa \omega'-\imath\pi{(2\omega)^T}^{-1}\label{etamodular}$$
At $g=2$ for the curve $y^2=4x^5+\lambda_4x^4+\ldots+\lambda_0$ the following representation of $\varkappa$-matrix is valid $$\begin{aligned}
\varkappa= \frac{1}{80}\left( \begin{array}{cc} 4\lambda_2&\lambda_3\\
\lambda_3&4\lambda_4 \end{array} \right)-\frac{1}{20}\sum_{10\;\text{even}\;[\varepsilon]}\frac{1}{\theta[\varepsilon]}
\left(\begin{array}{cc} \partial^2_{{U}_1^2}\theta[\varepsilon]& \partial^2_{{U}_1,{U}_2}
\theta[\varepsilon]\\
\partial^2_{{U}_1,{U}_2}
\theta[\varepsilon]& \partial^2_{{U}_2^2}\theta[\varepsilon]\end{array}\right)\end{aligned}$$ with $\varkappa=\eta(2\omega)^{-1}$, $\mathcal{A}^{-1}=(2\omega)^{-1}=({U}_1,{U}_2 ) $ and directional derivatives $\partial_{{U}_i}$, $i=1,2$.
Representation of $\varkappa$ matrix of genus 2 and 3 hyperelliptic curves in terms of directional derivatives of non-singular odd constant was found in [@eee13]
Co-homologies of Baker and Klein
--------------------------------
Calculations of $\varkappa$-matrix for the hyperelliptic curve (\[hyperelliptic\]) were done in co-homology basis introduced by H.Baker (\[bakerbasis\]). When holomorphic differentials, $ \mathrm{d}{u} (x,y) $ are chosen meromorphic differentials, $ \mathrm{d}{r} (x,y)$ can be find from the symmetry condition $\mathbf{I}$. One can check that symmetry condition also fulfilled if meromorphic differentials will be changed as $$\mathrm{d}{r}(x,y) \rightarrow \mathrm{d}{r}(x,y)+M\mathrm{d}{u}(x,y),$$ where $M$ is arbitrary constant symmetric matrix $M^T=M$. One can then choose $$M=-\frac{1}{8 N_g}\Lambda_g$$ Then $\varkappa$ will change to $$\begin{aligned}
\varkappa=
-\frac1{2}\frac1{N_g}\sum_{N_g\;\text{even}\;\left[\varepsilon_{\mathcal{I}_0}\right]} \frac{1}{\theta[\varepsilon_{\mathcal{I}_0}]({0})} \left(\partial_{{U}_i}\partial_{{U_j}} \theta[\varepsilon_{\mathcal{I}_0}]({0})\right)_{i,j=1,\ldots,g} .
\label{Kleinian}\end{aligned}$$
Following [@eil16] introduce co-homology basis of Klein $$\begin{aligned}
\mathrm{d}{u} (x,y), \quad \mathrm{d}{r} (x,y)
-\frac{1}{8N_g}\Lambda_g\mathrm{d}{u} (x,y)\label{Kleinbasis}\end{aligned}$$ with constant matrix, $\Lambda_g=\Lambda_g({\lambda})$ given by (\[Lambda1\],\[Lambda2\]). Therefore we proved
$\varkappa$-matrix is represented in the modular form (\[Kleinian\]) in the co-homology basis (\[Kleinbasis\]).
Formula (\[Kleinian\]) first appears in F.Klein ([@klein886], [@klein888]), it was recently revisited in a more general context by Korotkin and Shramchenko ([@ksh12]) who extended representation for $\varkappa$ to non-hyperelliptic curves. Correspondence of this representation to the co-homology basis to the best knowledge of the authors was not earlier discussed.
Rewrite formula (\[Kleinian\]) in equivalent form, $$\begin{aligned}
\omega^T\eta=-\frac{1}{4N_g}
\sum_{N_g\;\text{even}\;[\varepsilon]} \frac{1}{\theta[\varepsilon]}
\left(\begin{array}{ccc}
\theta_{1,1}[\varepsilon]&\cdots&\theta_{1,g}[\varepsilon]\\
\vdots&\cdots&\vdots\\
\theta_{1,g}[\varepsilon]&\cdots&\theta_{g,g}[\varepsilon]\end{array}\right)\label{omegaeta}\end{aligned}$$ where $\omega,\eta$ are half-periods of holomorphic and meromorphic differentials in Kleinian basis.
For the Weierstrass cubic $y^2=4x^3-g_2x-g_3$ (\[omegaeta\]) represents Weierstrass relation (\[SecondKind\]).
At $g=2$ (\[omegaeta\]) can be written in the form $$\begin{aligned}
\omega^T\eta = -\frac{\imath \pi}{10} \left(\begin{array}{cc}
\partial_{\tau_{1,1}} & \partial_{\tau_{1,2}}\\
\partial_{\tau_{1,2}} & \partial_{\tau_{2,2}}
\end{array} \right)\; \mathrm{ln}\; \chi_{5}\end{aligned}$$ where $\chi_5$ is relative invariant of weight 5, $$\begin{aligned}
\chi_{5} = \prod_{ 10\;\text{even}\; [\varepsilon] } \theta[\varepsilon]\end{aligned}$$
Worth to mention how equations of KdV flows looks in both bases. For example at $g=2$ and curve $y^2=4x^5+\lambda_4x^4+\ldots+\lambda_0$ in Baker basis we got [@bel997] $$\begin{aligned}
\begin{split}
\wp_{2222}&=6\wp_{2,2}^2+4\wp_{1,2}+\lambda_4\wp_{2,2}+\frac12\lambda_3 \\
\wp_{1222}&=6\wp_{2,2}\wp_{1,2}-2\wp_{1,1}+\lambda_4\wp_{1,2} \end{split}\end{aligned}$$ In Kleinian basis the same equations change only in linear in $\wp_{i,j}$-terms $$\begin{aligned}
\begin{split}
\wp_{2222}&=6\wp_{2,2}^2+4\wp_{1,2}-47\lambda_4\wp_{2,2}+92\lambda_4^2-\frac72\lambda_3\\
\wp_{1222}&=6\wp_{2,2}\wp_{1,2}-2\wp_{1,1}-23\lambda_4\wp_{1,2}-6\lambda_3\wp_{2,2}+23\lambda_3\lambda_4+8\lambda_2 \end{split}\end{aligned}$$
Rosenhain modular form representaion of first kind periods
----------------------------------------------------------
Rosenhain [@ros851] was the first who introduced $\theta$-functions with characteristics at $g=2$. There are 10 even and 6 odd characteristics in that case. Let us denote each from these characteristics as $$\varepsilon_j = \left[\begin{array}{c} {\varepsilon_j'}^T\\ {\varepsilon''_j}^T \end{array} \right],
\quad j=1,\ldots 10$$ where $\varepsilon'_j$ and $\varepsilon''_j$ are column 2-vectors with entries equal to 0 or 1.
Rosenhain fixed the hyperelliptic genus two curve in the form $$y^2=x(x-1)(x-a_1)(x-a_2)(x-a_3)$$ and presented without proof expression $$\begin{aligned}
\mathcal{A}^{-1} =\frac{1}{2\pi^2 Q^2 } \left( \begin{array}{rr} -P\theta_2[\delta_2]&Q\theta_2[\delta_1]\\
P\theta_1[\delta_2]&-Q\theta_1[\delta_1] \end{array} \right) \label{RosenhainFormula}\end{aligned}$$ with $$\begin{aligned}
P&=\theta[\alpha_1]\theta[\alpha_2]\theta[\alpha_3],\quad
Q=\theta[\beta_1]\theta[\beta_2]\theta[\beta_3]\end{aligned}$$ and 6 even characteristics $[\alpha_{1,2,3}], [\beta_{1,2,3}]$ and two odd $[\delta_{1,2}]$ which looks chaotic. One of first proofs can be found in H.Weber [@web859]; these formulae are implemented in Bolza dissertation [@bol885] and [@bol886]. Our derivation of these formulae are based on the [*Second Thomae relation*]{} [@tho870], see [@er08] and [@eil18]. To proceed we give the following definitions.
A triplet of characteristics $[\varepsilon_1]$, $[\varepsilon_2,]$, $[\varepsilon_3]$ is called azygetic if
$$\mathrm{exp}\; \imath \pi \left\{
\displaystyle{\sum_{j=1}^3{\varepsilon_j'}^T \varepsilon''_j + \sum_{i=1}^3{\varepsilon_i'}^T
\sum_{i=1}^3\varepsilon''_i } \right\} =- 1$$
A sequence of $2g + 2$ characteristics $[\varepsilon_1],\ldots,[\varepsilon_{2g+2} ] $ is called a [*special fundamental system*]{} if the first $g$ characteristics are odd, the remaining are even and any triple of characteristics in it is azygetic.
(Conjectural Riemann-Jacobi derivative formula) Let $g$ odd $[\varepsilon_1], \ldots, [\varepsilon_g]$ and $g+2$ even $ [\varepsilon_{g+1}], \ldots, [\varepsilon_{2g+2}]$ characteristics create a special fundamental system. Then the following equality is valid $$\mathrm{Det}\left. \frac{ \partial( \theta[\varepsilon_1]({v}),\ldots,
\theta[\varepsilon_g]({v})) }
{\partial (v_1,\ldots, v_g)}\right|_{{v}=0}= \pm \prod_{k=1\ldots g+2}\theta[\varepsilon_{g+k}]({0})
\label{RiemannJacobi}$$
(\[RiemannJacobi\]) proved up to $g=5$ [@fro885], [@igu980], [@fay979]
Jacobi derivative formula for elliptic curve $$\begin{aligned}
\vartheta'_1(0)=\pi \vartheta_2(0)\vartheta_3(0)\vartheta_4(0)\end{aligned}$$
Rosenhain derivative formula for genus two curve is given without proof in the memoir [@ros851], namely, let $[\delta_1]$ and $[\delta_2]$ are any two odd characteristics from all 6 odd, then $$\begin{aligned}
\theta_1[\delta_1] \theta_2[\delta_2]-\theta_2[\delta_1] \theta_1[\delta_2]
= \pi^2\theta[\gamma_1]\theta[\gamma_2]\theta[\gamma_3]\theta[\gamma_4]
\label{RosenhainDerivative}\end{aligned}$$ where 4 even characteristics $[\gamma_1], \ldots, [\gamma_4]$ are given as $[\gamma_i]=[\delta_1]+[\delta_2]+[\delta_i], \; 3\leq i \leq 6$. There are 15 Rosenhain derivative formulae.
The following geometric interpretation of the special fundamental system can be given in the case of hyperelliptic curve. Consider genus two curve, $$\mathcal{C}:\quad y^2= (x-e_1) \cdots ( x-e_{6})$$ Denote associated homology basis as $( \mathfrak{a}_1, \mathfrak{a}_2;\mathfrak{b}_1,\mathfrak{b}_2 )$. Denote characteristics of Abelian images of branch points with base point $e_6$ as $ \mathfrak{A}_k$, $k=1,\ldots,6$. These are half-periods given by their characteristics, $[\mathfrak{A}_k]$ with $$\begin{aligned}
\mathfrak{A}_k=
\int_{(e_6,0)}^{(e_k,0)} {u}= \frac12 \tau {\varepsilon'}_k
+ \frac12 {\varepsilon''}_k, \quad k=1,\ldots,6\end{aligned}$$
For the homology basis drawn on the Figure we have
0.7mm
(150.00,80.00) (9.,33.)[(1,0)[12.]{}]{} (9.,33.) (21.,33.) (10.,29.)[(0,0)\[cc\][$e_1$]{}]{} (21.,29.)[(0,0)\[cc\][$e_2$]{}]{} (15.,33.)[(20,30.)]{} (8.,17.)[(0,0)\[cc\][$\mathfrak{ a}_1$]{}]{} (15.,48.)[(1,0)[1.0]{}]{} (32.,33.)[(1,0)[9.]{}]{} (32.,33.) (41.,33.) (33.,29.)[(0,0)\[cc\][$e_3$]{}]{} (42.,29.)[(0,0)\[cc\][$e_4$]{}]{} (37.,33.)[(18.,26.)]{} (30.,19.)[(0,0)\[cc\][$\mathfrak{a}_2$]{}]{} (36.,46.)[(1,0)[1.0]{}]{} (100.,33.00) [(1,0)[33.]{}]{} (100.,33.) (133.,33.) (101.,29.)[(0,0)\[cc\][$e_{5}$]{}]{} (132.,29.)[(0,0)\[cc\][$e_{6}$]{}]{} (59.,58.)[(0,0)\[cc\][$\mathfrak{b}_1$]{}]{} (63.,62.)[(1,0)[1.0]{}]{} (15.,33.00)(15.,62.)(65.,62.) (65.00,62.)(119.00,62.00)(119.00,33.00) (15.,33.00)(15.,5.)(65.,5.) (65.00,5.)(119.00,5.00)(119.00,33.00) (70.,44.)[(0,0)\[cc\][$\mathfrak{b}_2$]{}]{} (74.00,48.)[(1,0)[1.0]{}]{} (37.,33.00)(37.,48.)(76.00,48.) (76.00,48.)(111.00,48.00)(111.00,33.00) (37.,33.00)(37.,19.)(76.00,19.) (76.00,19.)(111.00,19.00)(111.00,33.00)
$$\begin{aligned}
[{\mathfrak A}_1]= \frac{1}{2}
\left[\begin{array}{cc}1&0\\
0&0\end{array}\right],\quad
[{\mathfrak A}_2]= \frac{1}{2}
\left[\begin{array}{cc}1&0\\
1&0\end{array}\right],\quad
[{\mathfrak A}_3]= \frac{1}{2}
\left[\begin{array}{cc}0&1\\
1&0\end{array}\right],\end{aligned}$$
$$\begin{aligned}
[{\mathfrak A}_4]= \frac{1}{2}
\left[\begin{array}{cc}0&1\\
1&1\end{array}\right],\quad
[{\mathfrak A}_5]= \frac{1}{2}
\left[\begin{array}{cc}0&0\\
1&1\end{array}\right],\quad
[{\mathfrak A}_6]= \frac{1}{2}
\left[\begin{array}{cc}0&0\\
0&0\end{array}\right]\end{aligned}$$
$${\mathfrak{A}}_k= \int_{\infty}^{e_k}{u}= \frac12 \tau.
{\varepsilon}_k
+ \frac12 {\varepsilon'}_k , \quad [ {\mathfrak{A}}_k ] =
\left[ \begin{array}{c}
{\varepsilon}_k^T\\ {\varepsilon'}_k^T \end{array} \right]$$ One can see that the set of characteristics $[\varepsilon_k] = [ \mathfrak{A}_k]$ of Abelian images of branch point contains two odd $[\varepsilon_2]$ and $[\varepsilon_4]$ and remaining four characteristics are even. One can check that the whole set of these $6=2g+2$ characteristics is azygetic and therefore the set constitutes special fundamental system. Hence one can write Rosenhain derivative formula $$\theta_1[\varepsilon_2]\theta_2[\varepsilon_4]-\theta_2[\varepsilon_2]\theta_1[\varepsilon_4]=\pm \pi^2
\theta[\varepsilon_1]\theta[\varepsilon_3]\theta[\varepsilon_5]\theta[\varepsilon_6]$$ In this way we gave geometric interpretation of Rosenhain derivative formula and associated set of characteristics in the case when one from even characteristic is zero. The same structure is observed for higher genera hyperelliptic curves even for $g>5$.
The characteristics entering to the Rosenhain formula are described as follows. Take any of 15 Rosenhain derivative formula, say, $$\theta_1[p] \theta_2[q]-\theta_2[p] \theta_1[q]
= \pi^2\theta[\gamma_1]\theta[\gamma_2]\theta[\gamma_3]\theta[\gamma_4]$$ Then 10 even characteristics can be grouped as $$\underbrace{ [\gamma_1],\ldots,[\gamma_4]}_4, \; \underbrace{ [\alpha_1],[\alpha_2],[\alpha_3] }_{[\alpha_1]+ [\alpha_2]+ [\alpha_3]=[p]},\; \;
\underbrace{ [\beta_1],[\beta_2],[\beta_3] }_{ [\beta_1]+ [\beta_2]+ [\beta_3]=[q]},$$ Then matrix of $\mathfrak{a}$-periods $$\begin{aligned}
\mathcal{A} &= \frac{2Q}{PR} \left( \begin{array}{rr}
Q\theta_1[q]&Q \theta_2[q]\\
P\theta_1[p]& P\theta_2[p]
\end{array} \right)\end{aligned}$$ with $$\begin{aligned}
P=\prod_{j=1}^3\theta[\alpha_j], \;\;Q=\prod_{j=1}^3\theta[\beta_j],\; \;R=\prod_{j=1}^4\theta[\gamma_j]\label{PQR}\end{aligned}$$
Note, that the 15 curves are given as $$\begin{aligned}
\mathcal{C}_{p,q}: \quad y^2=x(x-1)(x-a_1)(x-a_2)(x-a_3) \label{Bolza} \end{aligned}$$ where branch points are computed by [*Bolza formulae*]{} [@bol886], $$\begin{aligned}
e[\delta_j]= - \frac{\partial_{{U}_1} \theta[\delta_j] }{\partial_{\boldsymbol{U}_2} \theta[\delta_j]} , \quad
j=1,\ldots,6
\label{bolza1}
\end{aligned}$$ where $\partial_{{U}_i}$ directional derivative along vector ${U}_i$, $i=1,2$, $\mathcal{A}^{-1}=({U}_1,{U}_2)$ and $$e[p]=0,\; e[q]=\infty,$$ All 15 curves $\mathcal{C}_{p,q}$ are Möbius equivalent.
-
Generalization of the Rosenhain formula to higher genera hyperelliptic curve was found in [@er08] and developed further in [@eil18].
$$\begin{aligned}
\begin{split} y^2&=\phi(x) \psi(x)\\ \phi(x)&= \prod_{k=1}^g(x-e_{2k}), \;
\psi(x)= \prod_{k=1}^{g+1}(x-e_{2k-1}) \label{curve-g} \end{split} \end{aligned}$$
Denote $ R=\prod_{k=1}^{g+2}\theta[\gamma_k]$ monomial in left had side of Riemann-Jacobi formula (\[RiemannJacobi\])
Let genus $g$ hyperelliptic curve is given in (\[curve-g\]). Then winding vectors $(U_1,\ldots,U_g)=\mathcal{A}^{-1}$ are given by the formula $$\begin{aligned}
{U}_m=\frac{\epsilon}{ 2\pi^g R} \mathrm{Cofactor}\left( \left. \frac{ \partial( \theta[\varepsilon_1]({v}),\ldots,
\theta[\varepsilon_g]({v})) }
{\partial (v_1,\ldots, v_g)}\right|_{{v}=0}
\right)
\left( \begin{array}{c} s_{m-1}^{2} \sqrt[4]{\chi_1}\\ \vdots
\\s_{m-1}^{2g} \sqrt[4]{ \chi_g}\end{array} \right)\end{aligned}$$ Here $s_k^i$ -order $k$ symmetric function of elements $\{e_{2}, \ldots e_{2g} \} / \{ e_{2i} \}$ and $$\chi_{i} = \frac{\psi(e_{2i})}{\phi'(e_{2i})} , \quad i=1,\ldots, g$$ $s_k^i, \chi_i$ are expressible in $\theta$-constants via Thomae formulae [@tho870].
Typical answer of multi-gap integration includes $\theta$-function $ \theta({U}x +{V}t+{W};\tau )$ where winding vectors ${U}, {V}$ expressed in terms of complete holomorphic integrals and constant ${W}$ defined by initial data. Rosenhain formulae and their generalization, express $U,V$ in terms of $\theta$-constants and parameters the equation defining $\mathcal{C}$. In this way the problem of [*effectivization of finite gap solutions*]{} [@dub81] can be solved in that way at least for hyperelliptic curves.
Other application of the Rosenhain formula (\[RosenhainFormula\]) presented in [@be994] where two-gap Lamé and Treibich-Verdier potentials were obtained by the reduction to elliptic functions of general Its-Matveev representation [@im975] of finite-gap potential to the Schrödinger equation in terms of multi-variable $\theta$-functions.
Another application relevant to a computer algebra problem. In the case when, say in Maple, periods of holomorphic differentials are computed them periods of second kind differentials can be obtained by Rosenhain formula (\[RosenhainFormula\]) and its generalization.
Sigma-functions and the problem of differentiatiion of Abelian functions.
=========================================================================
Consider the curve $$\label{F-1}
V_\lambda = \left\{(x,y)\in\mathbb{C}^2\, : y^2 = \mathcal{C}(x;\lambda) = x^{2g+1}+\sum_{k=2}^{2g+1}
\lambda_{2 k} x^{2g - k + 1} \right\}$$ where $g\geqslant 1$ and $\lambda=(\lambda_4,\ldots,\lambda_{4g+2})\in \mathbb{C}^{2g}$ are the parameters. Set $\mathcal{D} = \{ \lambda\in \mathbb{C}^{2g}\,: \mathcal{C}(x;\lambda)\; \text{has multiple roots}\}$ and $\mathcal{B} = \mathbb{C}^{2g}\setminus\mathcal{D}$. For any $\lambda\in \mathcal{B}$ we obtain the affine part of a smooth projective hyperelliptic curve $\overline{V}_\lambda$ of genus $g$ and the Jacobian variety $Jac(\overline{V}_\lambda) = \mathbb{C}^{g}/\Gamma_g$, where $\Gamma_g \subset \mathbb{C}^{g}$ is a lattice of rank $2g$ generated by the periods of the holomorphic differential on cycles of the curve $V_\lambda$.
In the general case, an *Abelian function* is a meromorphic function on a complex Abelian torus $T^g=\mathbb{C}^g\!/\Gamma$, where $\Gamma\subset\mathbb{C}^g$ is a lattice of rank $2g$. In other words, a meromorphic function $f$ on $\mathbb{C}^g$ is Abelian iff $f(u)=f(u+\omega)$ for all $u=(u_1,\ldots,u_g)\in\mathbb{C}^{g}$ and $\omega\in\Gamma$. Abelian functions on $T^g$ form a field $\mathcal{F} = \mathcal{F}_g$ such that:
\(1) let $f\in\mathcal{F}$, then $\partial_{u_i} f\in \mathcal{F}$, $i=1,\dots,g$;
\(2) let $f_1,\dots,f_{g+1}$ be any nonconstant functions from $\mathcal{F}$, then there exists a polynomial $P$ such that $P(f_1,\dots, f_{g+1})(u)=0$ for all $u\in T^g$;
\(3) let $f\in\mathcal{F}$ be a nonconstant function, then any $h\in\mathcal{F}$ can be expressed rationally in terms of $(f,\partial_{u_1}f,\dots,\partial_{u_g}f)$;
\(4) there exists an entire function $\vartheta\colon\mathbb{C}^g\to\mathbb{C}$ such that $\partial_{u_i,u_j}\log\vartheta \in \mathcal{F}$, $i,j=1,\dots,g$.
For example, any elliptic function $f\in \mathcal{F}_1$ is a rational function in the Weierstrass functions $\wp(u; g_2, g_3)$ and $\partial_u\wp(u; g_2, g_3)$, where $g_2$ and $g_3$ are parameters of elliptic curve $$V = \{ (x,y) \in \mathbb{C}^2\;|\; y^2 = 4 x^3 - g_2 x - g_3 \}.$$ It is easy to see that the function $\frac{\partial}{\partial g_2}\wp(u; g_2, g_3)$ will no longer be elliptic. This is due to the fact that the period lattice $\Gamma$ is a function of the parameters $g_2$ and $g_3$. In [@fs882] Frobenius and Stickelberger described all the differential operators $L$ in the variables $u,\;g_2$ and $g_3$, such that $Lf\in \mathcal{F}_1$ for any function $f\in \mathcal{F}_1$ (see below Section 7.3).
In [@bl07; @bl08] the classical problem of differentiation of Abelian functions over parameters for families of $(n,s)$-curves was solved. In the case of hyperelliptic curves this problem was solved more explicitly.
All genus $2$ curves are hyperelliptic. We denote by $\pi\colon \mathcal{U}_g \to \mathcal{B}_g$ the universal bundle of Jacobian varieties $Jac(\overline{V}_\lambda)$ of hyperelliptic curves. Let us consider the mapping $\varphi\colon \mathcal{B}_g\times \mathbb{C}^g \to \mathcal{U}_g$, which defines the projection $\lambda\times \mathbb{C}^g \to \mathbb{C}^g/\Gamma_g(\lambda)$ for any $\lambda\in \mathcal{B}_g$. Let us fix the coordinates $(\lambda;u)$ in $\mathcal{B}_g\times \mathbb{C}^g\subset \mathbb{C}^{2g}\times \mathbb{C}^g$ where $u = (u_1,\ldots,u_{2g-1})$. Thus, using the mapping $\varphi$, we fixed in $\mathcal{U}_g$ the structure of the space of the bundle whose fibers $J_\lambda$ are principally polarized Abelian varieties.
We denote by $F = F_g$ the field of functions on $\mathcal{U}_g$ such that for any $f\in F$ the function $\varphi^*(f)$ is meromorphic, and its restriction to the fiber $J_\lambda$ is an Abelian function for any point $\lambda \in \mathcal{B}_g$.
Below, we will identify the field $F$ with its image in the field of meromorphic functions on $\mathcal{B}\times \mathbb{C}^g$.
The following [**Problem I**]{}:
[*Describe the Lie algebra of differentiations of the field of meromorphic functions on $\mathcal{B}_g\times \mathbb{C}^g$, generated by the operators $L$, such that $Lf\in F$ for any function $f\in F$*]{}
was solved in [@bl07; @bl08].
From the differential geometric point of view, Problem I is closely related to [**Problem II**]{}:
[*Describe the connection of the bundle $\pi\colon \mathcal{U}_g \to \mathcal{B}_g$.*]{}
The solution of Problem II leads to an important class of solutions of well-known equations of mathematical physics. In the case $g=1$, the solution is called the Frobenius-Stikelberger connection (see [@dubr96]) and leads to solutions of Chazy equation.
The space $\mathcal{U}_g$ is a rational variety, more precisely, there is a birational isomorphism $\varphi\colon \mathbb{C}^{3g} \to \mathcal{U}_g$. This fact was discovered by B. A. Dubrovin and S. P. Novikov in [@dn974]. In [@dn974], a fiber of the universal bundle is considered as a level surface of the integrals of motion of $g$th stationary flow of KdV system, that is, it is defined in $\mathbb{C}^{3g}$ by a system of $2g$ algebraic equations. The degree of the system grows with the growth of genus. In [@bel97], [@bel997] the coordinates in $\mathbb{C}^{3g}$ were introduced such that a fiber is defined by $2g$ equations of degree not greater than $3$.
The Dubrovin-Novikov coordinates and the coordinates from [@bel97], [@bel997] are the same for the universal space of genus $1$ curves. But already in the case of genus 2, these coordinates differ (see [@bl08]).
The integrals of motion of KdV systems are exactly the coefficients $\lambda_{2g+4},\ldots,\lambda_{4g+2}$ of hyperelliptic curve $V_\lambda$, in which the coefficients $\lambda_{4},\ldots,\lambda_{2g+2}$ are free parameters (see ). Choosing a point $z\in \mathbb{C}^{3g}$ such that the point $\varphi(z)\in \mathcal{B}_g$ is defined, one can calculate the values of the coefficients $(\lambda_4,\ldots,\lambda_{4g+2}) = \pi(z)\in \mathcal{B}_g$ substituting this point in these integrals. Thus, the solution of the Problem I of differentiation of hyperelliptic functions led to the solution of another well-known [**Problem III**]{}:
[*Describe the dependence of the solutions of $g$-th stationary flow of KdV system on the variation of the coefficients $\lambda_4,\ldots,\lambda_{4g+2}$ of hyperelliptic curve, that is, from variation of values of the integrals of motion and parameters.*]{}
In [@go89] it was obtained results on Problem III, which use the fact that for a hierarchy KdV the action of polynomial vector fields on the spectral plane is given by the shift of the branch points of the hyperelliptic curve along this fields (see [@ss14]). The deformations of the potential corresponding to this action are exactly the action of the nonisospectral symmetries of the hierarchy KdV.
Let us describe a different approach to Problem III, developed in our works. In [@bl02] it was introduced the concept of a polynomial Lie algebra over a ring of polynomials $A$. For brevity, we shall call them Lie A-algebras. In [@bl07; @bl08] it was considered the ring of polynomials $\mathcal{P}$ in the field $F$. This ring is generated by all logarithmic derivatives of order $k\geqslant 2$ from the hyperelliptic sigma function $\sigma(u;\lambda)$. It was constructed the Lie $\mathcal{P}$-algebra $\mathcal{L}=\mathcal{L}_g$ with generators $L_{2k-1},\; k=1,\ldots,g$ and $L_{2l},\; l=0,\ldots,2g-1$. The fields $L_{2k-1}$ define isospectral symmetries, and the fields $L_{2l}$ define nonisospectral symmetries of the hierarchy KdV. The Lie algebra $\mathcal{L}$ is isomorphic to the Lie algebra of differentiation of the ring $\mathcal{P}$ and, consequently, allows to solve the Problem I (see property (3) of Abelian functions). The generators $L_{2k-1},\; k=1,\ldots,g$, coincide with the operators $\partial_{u_{2k-1}}$, and, consequently, commute. Thus, in the Lie $\mathcal{P}$-algebra $\mathcal{L}$ it is defined the Lie $\mathcal{P}$-subalgebra $\mathcal{L}^*$ generated by the operators $L_{2k-1},\; k=1,\ldots,g$. The generators $L_{2l},\; l=0,\ldots,2g-1,$ are such that the Lie $\mathcal{P}$-algebra $\mathcal{L}^*$ is an ideal in the Lie $\mathcal{P}$-algebra $\mathcal{L}$.
The construction of $L_{2k},\; k=0,\ldots,2g-1$, is based on the following fundamental fact (see [@bl04]):
The entire function $\psi(u;\lambda)$, satisfying the system of heat equations in a nonholonomic frame $$\ell_{2i}\psi = H_{2i}\psi,\; i=0,\ldots,2g-1,$$ under certain initial conditions (see [@bl04]) coincides with the hyperelliptic sigma-function $\sigma(u;\lambda)$. Here $\ell_{2i}$ are polinomial linear first-order differential operators in the variables $\lambda=(\lambda_4,\ldots,\lambda_{4g+2})$ and $H_{2i}$ are linear second-order differential operators in the variables $u=(u_1,\ldots,u_{2g-1})$. The methods for constructing these operators are described in [@bl04].
The following fact was used essentially in constructing the operators $\ell_{2i}$:
The Lie $\mathbb{C}[\lambda]$-algebra $\mathcal{L}_\lambda$ with generators $\ell_{2i},\; i=0,\ldots,2g-1$ is isomorphic to an infinite-dimensional Lie algebra $Vect_{\mathcal{B}}$ of vector fields on $\mathbb{C}^{2g}$, that are tangent to the discriminant variety $\Delta$. We recall that the Lie algebra $Vect_{\mathcal{B}}$ is essentially used in singularity theory and its applications (see [@ar90]).
In the Lie $\mathcal{P}$-algebra $\mathcal{L}$ we can choose the generators $L_{2k},\; k=0,\ldots,2g-1$ such that for any polynomial $P(\lambda)\in \mathbb{C}[\lambda]$ and any $k$ the formula $L_{2k}\pi^*P(\lambda) = \pi^*(\ell_{2k}P(\lambda))$ holds, where $\pi^*$ is the ring homomorphism induced by the projection $\pi \colon \mathcal{U}_g \to \mathcal{B}_g$.
Section 3 describes the development of an approach to solving the Problem I. This approach uses:
1. the graded set of multiplicative generators of the polynomial ring $\mathcal{P}=\mathcal{P}_g$;
2. the description of all algebraic relations between these generators;
3. the description of the birational isomorphism $J \colon \mathcal{U}_g \to \mathbb{C}^{3g}$ in terms of graded polynomial rings;
4. the description of the polynomial projection $\pi \colon \mathbb{C}^{3g} \to \mathbb{C}^{2g}$, where $\mathbb{C}^{2g}$ is a space in coordinates $\lambda = (\lambda_4,\ldots,\lambda_{4g+2})$, such that for any $\lambda \in \mathcal{B}$ the space $J^{-1}\pi^{-1}(\lambda)$ is the Jacobian variety $Jac(V_\lambda)$;
5. the construction of linear differential operators of first order $$\widehat{H}_{2k} = \sum_{i=1}^q h_{2(k-i)+1}(\lambda;u)\partial_{u_{2i-1}}, \; q=\min(k,g),$$ such that $L_{2k} = \ell_{2k}-\widehat{H}_{2k},\; k=0,\ldots,2g-1$.
Here:
- $h_{2(k-i)+1}(\lambda;u)$ are meromorphic functions on $\mathcal{B}_g\times \mathbb{C}^{g}$;
- $h_{2(k-i)+1}(\lambda;u)$ are homogeneous functions of degree $2(k-i)+1$ in $\lambda = (\lambda_4,\ldots,\lambda_{4g+2})$, $\deg\lambda_{2k}=2k$, and $u = (u_1,\ldots,u_{2g-1}),\; \deg u_{2k-1}=1-2k$;
- $\partial_{u_{2l-1}}h_{2(k-i)+1}$ are homogeneous polynomials of degree $2(k+l-i)$ in the ring $\mathcal{P}_g$.
This approach was proposed in [@buch16] and found the application in [@bun17]. A detailed construction of the Lie algebra $\mathcal{L}_2$ is given in [@buch16], and the Lie algebra $\mathcal{L}_3$ in [@bun17].
General methods and results (see Section 3.2) will be demonstrated in cases $g=1$ (see Section 3.3) and $g=2$ (see Section 3.4).
For brevity, Abelian functions on the Jacobian varieties of will be called [*hyperelliptic functions*]{} of genus $g$. In the theory and applications of these functions, that are based on the sigma-function $\sigma(u;\lambda)$ (see [@bak998; @bel97; @bel997; @bel12]), the grading plays an important role. Below, the variables $u=(u_1,u_3,\ldots,u_{2g-1})$, parameters $\lambda=(\lambda_4,\ldots,\lambda_{4g+2})$ and functions are indexed in a way that clearly indicates their grading. Note that our new notations for the variables differ from the ones in [@bel97; @bel997; @bel12] as follows $$u_i \longleftrightarrow u_{2(g-i)+1},\, i=1,\ldots,g.$$
Let $$\omega = \left( (2k_1-1)\cdot j_1,\ldots,(2k_s-1)\cdot j_s \right)$$ where $1\leqslant s\leqslant g$, $j_q> 0,\; q=1,\ldots,s$ and $j_1+\ldots+j_s\geqslant 2$. We draw attention to the fact that the symbol “$\cdot$” in the two-component expression $(2k_q-1)\cdot j_q$ is not a multiplication symbol. Set $$\label{f-2}
\wp_\omega(u;\lambda)=
-\partial^{j_1}_{u_{2k_1-1}} \cdots \partial^{j_s}_{u_{2k_s-1}}\,\ln\sigma(u;\lambda).$$ Thus $$\deg \wp_\omega = (2k_1-1)j_1+\cdots+(2k_s-1)j_s.$$
Note that our $\omega$ differ from the ones in [@buch16; @bun17].
Say that a multi-index $\omega$ is given in normal form if $1 \leqslant k_1 < \ldots < k_s$. According to formula , we can always bring the multi-index $\omega$ to a normal form using the identifications: $$\begin{aligned}
\left( (2k_p-1)\cdot j_p,(2k_q-1)\cdot j_q \right) &= \left( (2k_q-1)\cdot j_q,(2k_p-1)\cdot j_p \right), \\
\left( (2k_p-1)\cdot j_p,(2k_q-1)\cdot j_q \right) &= (2k_p-1) \cdot (j_p+j_q),\; \text { if }\, k_p=k_q.\end{aligned}$$
In [@bel12] (see also [@bel97; @bel997]) it was proved that for $1\leqslant i \leqslant k \leqslant g$ all algebraic relations between hyperelliptic functions of genus $g$ follow from the relations, which in our graded notations have the form $$\label{f-3}
\wp_{1\cdot 3,(2i-1)\cdot 1} = 6\left(\wp_{1\cdot 2}\wp_{1\cdot 1,(2i-1)\cdot 1} + \wp_{1\cdot 1,(2i+1)\cdot 1}\right) -
2\left(\wp_{3\cdot 1,(2i-1)\cdot 1} - \lambda_{2i+2} \delta_{i,1}\right).$$ Here and below, $\delta_{i,k}$ is the Kronecker symbol, $\deg \delta_{i,k}=0$. $$\begin{gathered}
\label{f-4}
\wp_{1\cdot 2,(2i-1)\cdot 1}\wp_{1\cdot 2,(2k-1)\cdot 1} = 4\left(\wp_{1\cdot 2}\wp_{1\cdot 1,(2i-1)\cdot 1}\wp_{1\cdot 1,(2k-1)\cdot 1} + \wp_{1\cdot 1,(2k-1)\cdot 1}\wp_{1\cdot 1,(2i+1)\cdot 1}\right. +\\
\left.+\wp_{1\cdot 1,(2i-1)\cdot 1}\wp_{1\cdot 1,(2k+1)\cdot 1} + \wp_{(2k+1)\cdot 1,(2i+1)\cdot 1} \right) - 2\left(\wp_{1\cdot 1,(2i-1)\cdot 1}\wp_{3\cdot 1,(2k-1)\cdot 1} \right. + \\
\left. + \wp_{1\cdot 1,(2k-1)\cdot 1}\wp_{3\cdot 1,(2i-1)\cdot 1} +\wp_{(2k-1)\cdot 1,(2i+3)\cdot 1} +\wp_{(2i-1)\cdot 1,(2k+3)\cdot 1} \right) + \\
+ 2 \left(\lambda_{2i+2}\wp_{1\cdot 1,(2k-1)\cdot 1}\delta_{i,1} + \lambda_{2k+2}\wp_{1\cdot 1,(2i-1)\cdot 1}\delta_{k,1} \right) +
2\lambda_{2(i+j+1)} (2\delta_{i,k} + \delta_{k,i-1} + \delta_{i,k-1}).\end{gathered}$$
\[ex-1\] For all $g\geqslant 1$, we have the formulas:
1. Setting $i=1$ in , we obtain $$\label{ex-1-1}
\wp_{1\cdot 4} = 6\wp_{1\cdot 2}^2 + 4\wp_{1\cdot 1,3\cdot 1} + 2\lambda_4.$$
2. Setting $i=2$ in , we obtain $$\label{ex-1-2}
\wp_{1\cdot 3,3\cdot 1} = 6(\wp_{1\cdot 2} \wp_{1\cdot 1,3\cdot 1} + \wp_{1\cdot 1,5\cdot 1}) - 2\wp_{3\cdot 2}.$$
3. Setting $i=k=1$ in , we obtain $$\label{ex-1-3}
\wp_{1\cdot 3}^2 = 4\left[ \wp_{1\cdot 2}^3 + (\wp_{1\cdot 1,3\cdot 1} + \lambda_4)\wp_{1\cdot 2} + (\wp_{3\cdot 2} -
\wp_{1\cdot 1,5\cdot 1} + \lambda_6) \right].$$
\[T-7.1\] [1.]{} For any $\omega = \left( (2k_1-1)\cdot j_1,\ldots,(2k_s-1)\cdot j_s \right)$ the hyperelliptic function $\wp_\omega(u;\lambda)$ is a polynomial from $3g$ functions $\wp_{1\cdot j,(2k-1)\cdot 1},\;1\leqslant j \leqslant 3,\;
1\leqslant k\leqslant g$.
Note that if $k=1$, we have $\wp_{1\cdot j,1\cdot 1} = \wp_{1\cdot (j+1)}$.
[2.]{} Set $W_{\wp} = \{ \wp_{1\cdot j,(2k-1)\cdot 1},\;1\leqslant j \leqslant 3,\; 1\leqslant k\leqslant g \}$. The projection of the universal bundle $\pi_g\colon \mathcal{U}_g \to \mathcal{B}_g \subset \mathbb{C}^{2g}$ is given by the polynomials $\lambda_{2k}(W_{\wp}),\; k=2, \ldots,2g+1$ of degree at most 3 from the functions $\wp_{1\cdot j,(2k-1)\cdot 1}$.
The proof method of Theorem \[T-7.1\] will be demonstrated on the following examples:
\[ex-2\] [1.]{} Differentiating the relation with respect to $u_1$, we obtain $$\label{ex-2-1}
\wp_{1\cdot 5} = 12\wp_{1\cdot 2}\wp_{1\cdot 3} + 4\wp_{1\cdot 2,3\cdot 1}.$$ [2.]{} According to formula , we obtain $$\label{ex-2-2}
2\lambda_4 = \wp_{1\cdot 4} - 6\wp_{1\cdot 2}^2 - 4\wp_{1\cdot 1,3\cdot 1}.$$ [3.]{} According to formula , we obtain $$\label{ex-2-3}
2\wp_{3\cdot 2} = 6(\wp_{1\cdot 2} \wp_{1\cdot 1,3\cdot 1} + \wp_{1\cdot 1,5\cdot 1}) - \wp_{1\cdot 3,3\cdot 1}.$$ [4.]{} Substituting expressions for $\lambda_4$ (see ) and $\wp_{3\cdot 2}$ (see ) into formula , we obtain an expression for the polynomial $\lambda_6$.
The derivation of formulas - and the method of obtaining the polynomial $\lambda_6$ demonstrate the method of proving Theorem \[T-7.1\]. Below, this method will be set out in detail in cases $g=1$ (see Section 3.3) and $g=2$ (see Section 3.4).
\[cor-7-0\] The operator $L$ of differentiation with respect to $u=(u_1,\ldots, u_{2g-1})$ and $\lambda = (\lambda_1,\ldots, \lambda_{4g+2})$ is a derivation of the ring $\mathcal{P}$ if and only if $L\wp_{1\cdot 1,(2k-1)\cdot 1}\in \mathcal{P}$ for $k = 1,\ldots,g$.
According to part 1 of Theorem \[T-7.1\], it suffices to prove that $L\wp_{1\cdot j,(2k-1)\cdot 1}\in \mathcal{P}$ for $j = 2$ and 3, $k = 1,\ldots,g$. We have $L\wp_{1\cdot j,(2k-1)\cdot 1} = LL_1\wp_{1\cdot (j-1),(2k-1)\cdot 1} = (L_1L + [L,L_1])\wp_{1\cdot (j-1),(2k-1)\cdot 1}$. Using now that $[L,L_1]\in \mathcal{L}^*$ and the assumption of Theorem \[T-7.1\], we complete the proof by induction.
Set $\mathcal{A} = \mathbb{C}[X]$, where $X = \{ x_{i,2j-1},\; 1 \leqslant i \leqslant 3,\; 1 \leqslant j \leqslant g \}$, $\deg x_{i,2j-1} = i + 2j - 1$.
\[cor-7-1\] [1.]{} The birational isomorphism $J \colon \mathcal{U}_g \to \mathbb{C}^{3g}$ is given by the polynomial isomorphism $$J^* \colon \mathcal{A} \longrightarrow \mathcal{P}\; : \; J^*X = W_{\wp}.$$
[2.]{} There is a polynomial map $$p \colon \mathbb{C}^{3g} \longrightarrow \mathbb{C}^{2g},\quad p(X) = \lambda,$$ such that $$p^* \lambda_{2k} = \lambda_{2k}(X),\; k=2,\ldots,2g+1,$$ where $\lambda_{2k}(X)$ are the polynomials from Theorem \[T-7.1\], item 2, obtained by substituting $W_{\wp}~\longmapsto~X$.
The isomorphism $J^*$ defines the Lie $\mathcal{A}$-algebra $\mathcal{L} = \mathcal{L}_g$ with $3g$ generators $L_{2k-1},\; k=1,\ldots,g$ and $L_{2l},\; l=0,\ldots,2g-1$. In terms of the coordinates $x_{i,2j-1}$, we obtain the following description of the $g$-th stationary flow of KdV system.
\[T-7.2\] [1.]{} The commuting operators $L_{2k-1},\; k=1,\ldots,g,$ define on $\mathbb{C}^{3g}$ a polynomial dynamical system $$\label{f-sist}
L_{2k-1}X = G_{2k-1}(X),\; k=1,\ldots,g,$$ where $G_{2k-1}(X) = \{ G_{2k-1,i,2j-1}(X) \}$ and $G_{2k-1,i,2j-1}(X)$ is a polynomial that uniquely defines the expression for the function $\wp_{1\cdot i,(2j-1)\cdot 1,(2k-1)\cdot 1}$ in the form of a polynomial from the functions $\wp_{1\cdot i,(2q-1)\cdot 1}$.
[2.]{} System has $2g$ polynomial integrals $\lambda_{2k} = \lambda_{2k}(X),\; k=2,\ldots,2g+1$.
Consider the curve $$V_\lambda = \{ (x,y) \in \mathbb{C}^2\;:\; y^2 = x^3 + \lambda_4 x + \lambda_6 \}.$$ The discriminant of the family of curves $V_\lambda$ is $$\Delta = \{ \lambda = (\lambda_4,\lambda_6)\in \mathbb{C}^2\;:\; 4\lambda_4^3 + 27\lambda_6^2 = 0 \}.$$ We have the universal bundle $\pi \colon \mathcal{U}_1 \to \mathcal{B}_1 = \mathbb{C}^2\setminus \Delta$ and the mapping $$\varphi \colon \mathcal{B}_1\times\mathbb{C} \to \mathcal{U}_1\;:\; \lambda \times\mathbb{C}
\to \mathbb{C}/\Gamma_1(\lambda).$$
Consider the field $F = F_1$ of functions on $\mathcal{U}_1$ such that the function $\varphi^*(f)$ is meromorphic, and its restriction to the fiber $\mathbb{C}/\Gamma_1(\lambda)$ is an elliptic function for any point $\lambda \in \mathcal{B}_1$. Using the Weierstrass sigma function $\sigma(u;\lambda)$ for $\partial = \frac{\partial}{\partial u}$, we obtain $$\zeta(u) = \partial\ln \sigma(u;\lambda)\quad \text{and}\quad \wp(u;\lambda) = -\partial\zeta(u;\lambda).$$
The ring of polynomials $\mathcal{P} = \mathcal{P}_1$ in $F$ is generated by the elliptic functions $\wp_{1\cdot i},\; i\geqslant 2$. Set $\wp_{1\cdot i} = \wp_{i}$. We have $\wp_{2} = \wp$ and $\wp_{i+1} =
\partial\wp_{i} = \wp_{i}'$. All the algebraic relations between the functions $\wp_{i}$ follow from the relations $$\begin{aligned}
\wp_{4} &= 6\wp_{2}^2 + 2\lambda_4 \quad (\text{see}\;\eqref{f-3}),\label{f-12} \\
\wp_{3}^2 &= 4[\wp_{2}^3 + \lambda_4\wp_{2} + \lambda_6] \quad (\text{see}\;\eqref{f-4}).\label{f-13}\end{aligned}$$ Thus, we obtain a classical result:
[1.]{} There is the isomorphism $\mathcal{P}\simeq \mathbb{C}[\wp,\wp',\wp'']$.
[2.]{} The projection $\pi \colon \mathcal{U}_1 \to \mathbb{C}^2$ is given by the polynomials $$\begin{aligned}
\label{f-14} & \frac{1}{2}\wp'' - 3\wp^2 = \lambda_4,\\
\label{f-15} & \left( \frac{\wp'}{2} \right)^2 + 2\wp^3 - \frac{1}{2}\wp''\wp = \lambda_6.\end{aligned}$$
Consider the linear space $\mathbb{C}^3$ with the graded coordinates $x_2,x_3,x_4,\; \deg x_k=k$. Set $\mathcal{A}_1 = \mathbb{C}[x_2,x_3,x_4]$.
\[cor-7-7\] The birational isomorphism $J\colon \mathcal{U}_1 \to \mathbb{C}^3$ is given by the ring isomorphism $$J^* \colon \mathcal{A}_1 \to \mathcal{P}_1 \;:\; J^*(x_2,x_3,x_4) = (\wp,\wp',\wp'').$$
The ring $\mathcal{P}_1$ is generated by elliptic functions $\wp_{i},\; i \geqslant 2,$ where $\wp_{i+1} = \wp_{i}'$. It follows from formula that $\wp_{5} = 12\wp_{2}\wp_{3}$. Hence, each function $\wp_{i}$ is a polynomial in $\wp_{2},\,\wp_{3}$ and $\wp_{4}$ for all $i \geqslant 5$.
\[cor-7-8\] [1.]{} The operator $L_1 = \partial$ defines on $\mathbb{C}^3$ a polynomial dynamical system $$\label{F-7-sist}
x_2' = x_3, \quad x_3' = x_4, \quad x_4' = 12x_2x_3.$$
[2.]{} The system has 2 polynomial integrals $$\lambda_4 = \frac{1}{2}x_4 - 3x_2^2 \quad \text{and}\quad \lambda_6 = \frac{1}{4}x_3^2 + 2x_2^3 - \frac{1}{2}x_4x_2.$$
Let us consider the standard Weierstrass model of an elliptic curve $$V_g = \{ (x,y) \in \mathbb{C}^2\;:\; y^2 = 4x^3 - g_2x - g_3 \}.$$ The discriminant of this curve has the form $\Delta(g_2,g_3) = g_2^3 - 27g_3^2$. We have $V_\lambda = V_g$ where $g_2 = -4\lambda_4$ and $g_3 = -4\lambda_6$.
The elliptic sigma function $\sigma(u;\lambda)$ satisfies the system of equations $$\label{f-16}
\ell_{2i}\,\sigma = H_{2i}\,\sigma, \; i=0,1,$$ where $$\begin{aligned}
\ell_0 &= 4\lambda_4 \partial_{\lambda_4} + 6\lambda_6 \partial_{\lambda_6};\,\quad H_0 = u\partial - 1;\\
\ell_2 &= 6\lambda_6 \partial_{\lambda_4} - \frac{4}{3}\lambda_4^2 \partial_{\lambda_6};\quad H_2 =
\frac{1}{2} \partial^2 + \frac{1}{6} \lambda_4 u^2.\end{aligned}$$
The operators $\ell_0,\,\ell_2$ and $H_0,\,H_2$, characterizing the sigma function $\sigma(u;g_2,g_3)$ of the curve $V_g$, were constructed in the work of Weierstrass [@weier894]. The operators $L_i \in Der(F_1),\; i=0,1$ and 2, were first found by Frobenius and Stickelberger (see [@fs882]). Below, following work [@bl08], we present the construction of the operators $L_0$ and $L_2$ on the basis of equations .
Let us construct the linear differential operators $\widehat{H}_{2i},\; i=0,1$, of first order, such that $L_{2i} = \ell_{2i} - \widehat{H}_{2i},\; i=0,1,$ are the differentiations of the ring $\mathcal{P}_1 = \mathbb{C}[\wp,\wp',\wp'']$.
.2cm [1.]{}
We have $\ell_0 \sigma = (u\partial - 1)\sigma$. Therefore, $\ell_0\ln \sigma = u\zeta(u)-1$. Applying the operators $\partial$, $\partial^2$ and using the fact that operators $\partial$, $\ell_0$ commute, we obtain: $$\ell_0\zeta = \zeta - u\wp, \quad \ell_0\wp = 2\wp + u\partial\wp.$$ Setting $\widehat{H}_0 = u\partial$, we obtain $L_0 = \ell_0 - u\partial$. Consequently $$L_0\zeta = \zeta, \quad L_0\wp = 2\wp.$$
.2cm [2.]{}
We have $\ell_2 \sigma = \frac{1}{2} \partial^2\sigma - \frac{1}{6} \lambda_4 u^2 \sigma$. Therefore $\ell_2 \ln \sigma = \frac{1}{2} \frac{\partial^2\sigma}{\sigma} - \frac{1}{6} \lambda_4 u^2$. We have $\frac{\partial^2\sigma}{\sigma} = -\wp_2 + \zeta^2$. Thus $$\label{F-18}
\ell_2 \ln \sigma = -\frac{1}{2}\wp_2 + \frac{1}{2}\zeta^2 - \frac{1}{6} \lambda_4 u^2.$$ Applying the operators $\partial$ and $\partial^2$ to , we obtain $$\ell_2 \zeta = -\frac{1}{2}\wp_3 + \zeta\partial\zeta - \frac{1}{3}\lambda_4 u, \qquad
-\ell_2 \wp_2 = -\frac{1}{2}\wp_4 + \wp_2^2 - \zeta\partial \wp_2 - \frac{1}{3}\lambda_4.$$ Setting $\widehat{H}_2 = \zeta\partial$, we obtain $L_2 = \ell_2 - \zeta\partial$. Consequently, $$L_2 \zeta = -\frac{1}{2}\wp_3 - \frac{1}{3}\lambda_4 u, \qquad
L_2 \wp_2 = \; \frac{1}{2}\wp_4 - \wp_2^2 + \frac{1}{3}\lambda_4 = \frac{2}{3}\wp_4 - 2\wp_2^2.$$ Thus, we get the following result:
\[T-7.9\] The Lie $\mathcal{P}_1$-algebra $\mathcal{L}_1$ is generated by operators $L_0,\, L_1$ and $L_2$ such that $$\label{f-19}
[L_0,\, L_k] = kL_k, \; k=1,2, \qquad
[L_1,\, L_2] = \wp_2 L_1,$$ $$\label{f-20}
L_0 \wp_2 = 2\wp_2;\quad L_1 \wp_2 = \wp_3;\quad L_2 \wp_2 = \frac{2}{3}\wp_4 - 2\wp_2^2.$$
Formulas – completely determine the actions of the operators $L_k, \; k=0,1,2,$ on the ring $\mathcal{P}_1$ by the following inductive formula: $$\label{f-21}
L_k \wp_{i+1} = [L_k,\, L_1]\wp_{i} + L_1 L_k \wp_{i}.$$
Substituting $k=2$ and $i=2$ in , we obtain $$L_2 \wp_3 = [L_2,\, L_1]\wp_{2} + L_1 L_2 \wp_{2} = -5\wp_{2}\wp_{3} + \frac{4}{3}\wp_5.$$
For each curve with affine part of the form $$V_{\lambda} = \left\{ (x,y) \in \mathbb{C}^2\,|\, y^2 = x^5 + \lambda_4 x^3 +
\lambda_6 x^2 + \lambda_8 x + \lambda_{10} \right\},$$ one can construct a sigma-function $\sigma(u; \lambda)$ (see [@bel97]). This function is an entire function in $u = (u_1, u_3) \in \mathbb{C}^2$ with parameters $\lambda =
(\lambda_4, \lambda_6, \lambda_8, \lambda_{10})\in \mathbb{C}^4$. It has a series expansion in $u$ over the polynomial ring $\mathbb{Q}[\lambda_4, \lambda_6, \lambda_8, \lambda_{10}]$ in the vicinity of $0$. The initial segment of the expansion has the form $$\begin{gathered}
\label{F-22}
\sigma(u; \lambda) = u_3 - {1 \over 3}\, u_1^3 + {1 \over 6}\,
\lambda_6 u_3^3 - {1 \over 12}\, \lambda_4 u_1^4 u_3 - {1 \over 6}\,
\lambda_6 u_1^3 u_3^2 - \\ - {1 \over 6}\, \lambda_8 u_1^2 u_3^3 -
{1 \over 3}\, \lambda_{10} u_1 u_3^4 + \left({ 1 \over 60}\,
\lambda_4 \lambda_8 + {1 \over 120}\, \lambda_6^2\right) u_3^5 +
(u^7).\end{gathered}$$ Here $(u^k)$ denotes the ideal generated by monomials $u_1^i u_3^j$, $i+j = k$.
The sigma-function is an odd function in $u$, i.e. $\sigma(-u;\lambda)=-\sigma(u;\lambda)$.
Set $$\nabla_{\lambda} = \left(
{\partial \over \partial \lambda_4}, \; {\partial \over \partial
\lambda_6},\; {\partial \over \partial \lambda_8}, \; {\partial
\over \partial \lambda_{10}} \right)\quad \text{and}\quad
\partial_{u_1}={\partial \over \partial u_1}, \; \partial_{u_3}= {\partial \over \partial u_3}.$$
We need the following properties of the two-dimensional sigma-function\
(see [@bel997; @bl05] for details):
[**1.**]{} The following system of equations holds: $$\label{F-23}
\ell_i\sigma = H_i\sigma, \quad i = 0,2,4,6,\qquad$$ where $(\ell_0\;\ell_2\;\ell_4\;\ell_6)^\top=T \, \nabla_\lambda$, $$T =
\begin{pmatrix}
4 \lambda_4 & 6 \lambda_6 & 8 \lambda_8 & 10 \lambda_{10} \\[5pt]
6 \lambda_6 & 8 \lambda_8 - {12 \over 5} \lambda_4^2 & 10 \lambda_{10}
- {8 \over 5} \lambda_4 \lambda_6 & - {4 \over 5} \lambda_4 \lambda_8 \\[5pt]
8 \lambda_8 & 10 \lambda_{10} - {8 \over 5} \lambda_4 \lambda_6 &
4 \lambda_4 \lambda_8 - {12 \over 5} \lambda_6^2 & 6 \lambda_4 \lambda_{10}
- {6 \over 5} \lambda_6 \lambda_8 \\[5pt]
10 \lambda_{10} & - {4 \over 5} \lambda_4 \lambda_8 &
6 \lambda_4 \lambda_{10} - {6 \over 5} \lambda_6 \lambda_8 &
4 \lambda_6 \lambda_{10} - {8 \over 5} \lambda_8^2 \\
\end{pmatrix} \qquad\qquad\quad\quad$$ and $$\begin{aligned}
H_0 &= u_1\partial_{u_1}+3u_3\partial_{u_3}-3, \\
H_2 &= {1 \over 2}\,\partial_{u_1}^2 - {4 \over 5}\lambda_4 u_3 \partial_{u_1}+u_1\partial_{u_3} - {3 \over 10}\lambda_4 u_1^2 + {1 \over 10}(15\lambda_8-4\lambda_4^2)u_3^2, \\
H_4 &= \partial_{u_1}\partial_{u_3} - {6\over 5}\,\lambda_6u_3 \partial_{u_1} + \lambda_4 u_3 \partial_{u_3} - {1 \over 5}\,\lambda_6u_1^2 + \lambda_8u_1u_3 + {1 \over 10}(30\lambda_{10} -
6\lambda_6\lambda_4)u_3^2 - \lambda_4, \\
H_6 &= {1 \over 2}\,\partial_{u_3}^2 - {3 \over 5}\lambda_8 u_3 \partial_{u_1} - {1 \over 10}\,\lambda_8u_1^2 + 2\lambda_{10}u_1u_3 - {3 \over 10}\,\lambda_8\lambda_4 u_3^2 - {1 \over 2}\,\lambda_6.\end{aligned}$$
[**2.**]{} The equation $\ell_0 \, \sigma = H_0 \sigma$ implies that $\sigma$ is a homogeneous function of degree $-3$ in $u_1$, $u_3$, $\lambda_j$.
[**3.**]{} The discriminant of the hyperelliptic curve $V_\lambda$ of genus 2 is equal to $\Delta= {16 \over 5}\, \det T$. It is a homogeneous polynomial in $\lambda$ of degree $40$. Set $\mathcal{B} = \{ \lambda \in \mathbb{C}^4\, :\, \Delta(\lambda) \ne 0 \}$; then the curve $V_\lambda$ is smooth for $\lambda \in \mathcal{B}$.
We have $$\ell_0\,\Delta = 40 \Delta, \quad \ell_2\,\Delta = 0,\quad \ell_4\,\Delta =
12 \lambda_4 \Delta,\quad \ell_6\,\Delta = 4 \lambda_6 \Delta.$$ Thus, the fields $\ell_0,\ell_2,\ell_4$ and $\ell_6$ are tangent to the variety $\{ \lambda\in \mathbb{C}^4\;:\; \Delta(\lambda)=0 \}$.
The present study is based on the following results.
\
The entire function $\sigma(u;\lambda)$ is uniquely determined by the system of equations [(\[F-23\])]{} and initial condition $\sigma(u;0)=u_3-\frac{1}{3}u_1^3$.
We have the universal bundle $\pi \colon \mathcal{U}_2 \to \mathcal{B}_2 = \mathbb{C}^4\setminus\mathcal{D}$ and the mapping $$\varphi \colon \mathcal{B}_2\times\mathbb{C}^2 \to \mathcal{U}_2\;:\; \lambda \times\mathbb{C}^2
\to \mathbb{C}^2/\Gamma_2(\lambda).$$
Consider the field $F = F_2$ of functions on $\mathcal{U}_2$ such that the function $\varphi^*(f)$ is meromorphic, and its restriction to the fiber $\mathbb{C}^2/\Gamma_2(\lambda)$ is an hyperelliptic function for any point $\lambda \in \mathcal{B}_2$.
All the algebraic relations between the hyperelliptic functions of genus 2 follow from the relations, which in our notations have the form: $$\begin{aligned}
\wp_{1\cdot 4} &= 6\wp_{1\cdot 2}^2 + 4\wp_{1\cdot 1,3\cdot 1} + 2\lambda_4,\label{f-26} \\
\wp_{1\cdot 3,3\cdot 1} &= 6\wp_{1\cdot 2} \wp_{1\cdot 1,3\cdot 1} - 2\wp_{3\cdot 2},\label{f-27}\end{aligned}$$ (see for $i=1$ and $i=2$) and $$\begin{aligned}
\wp_{1\cdot 3}^2 &= 4\left[ \wp_{1\cdot 2}^3 + (\wp_{1\cdot 1,3\cdot 1} + \lambda_4)\wp_{1\cdot 2} + \wp_{3\cdot 2}
+ \lambda_6 \right],\label{f-28} \\
\wp_{1\cdot 3}\wp_{1\cdot 2,3\cdot 1} &= 4\wp_{1\cdot 2}^2\wp_{1\cdot 1,3\cdot 1} + 2\wp_{1\cdot 1,3\cdot 1}^2 -
2\wp_{1\cdot 2}^2\wp_{3\cdot 2} + 2\lambda_4\wp_{1\cdot 1,3\cdot 1} + 2\lambda_8, \label{f-29} \\
\wp_{1\cdot 2,3\cdot 1}^2 &= 4(\wp_{1\cdot 2}\wp_{1\cdot 1,3\cdot 1}^2 - \wp_{1\cdot 1,3\cdot 1}\wp_{3\cdot 2} + \lambda_{10}) \label{f-30}\end{aligned}$$ (see for $(i,k) = (1,1),\, (1,2)$ and $(2,2)$).
Consider the linear space $\mathbb{C}^6$ with the graded coordinates $X = (x_2,x_3,x_4),\; Y = (y_4,y_5,y_6)$, $\deg x_k = k,\; \deg y_k = k$. Set $\mathcal{A}_2 = \mathbb{C}[X,Y]$.
\[T-7.13\] [1.]{} The birational isomorphism $J_2 \colon \mathcal{U}_2 \to \mathbb{C}^6$ is given by the isomorphism of polynomial rings $$J_2^* \colon \mathcal{A}_2 \longrightarrow \mathcal{P}_2\;:\; J_2^*X = (\wp_{1\cdot 2},\wp_{1\cdot 3},\wp_{1\cdot 4}),\;
J_2^*Y = (\wp_{1\cdot 1,3\cdot 1},\wp_{1\cdot 2,3\cdot 1},\wp_{1\cdot 3,3\cdot 1}).$$ [2.]{} The projection $\pi_2 \colon \mathbb{C}^6 \to \mathbb{C}^4$ is given by the polynomials $$\begin{aligned}
\lambda_4 &= -3x_2^2 + \frac{1}{2}x_4 - 2y_4, \label{f-31} \\
\lambda_6 &= 2x_2^3 + \frac{1}{4}x_3^2 - \frac{1}{2}x_2x_4 - 2x_2y_4 + \frac{1}{2}y_6, \label{f-32} \\
\lambda_8 &= (4x_2^2 + y_4)y_4 - \frac{1}{2}(x_4y_4 - x_3y_5 + x_2y_6), \label{f-33} \\
\lambda_{10} &= 2x_2y_4^2 + \frac{1}{4}y_5^2 -\frac{1}{2}y_4y_6. \label{f-34}\end{aligned}$$
Using the isomorphism $J_2^*$, we rewrite the relations - in the form $$\begin{aligned}
x_4 &= 6x_2^2 + 4y_4 + 2\lambda_4, \label{f-35} \\
y_6 &= 6x_2y_4 - 2\wp_{3\cdot 2}, \label{f-36} \\
x_3^2 &= 4\left[ x_2^3 + (y_4 + \lambda_4)x_2 + \wp_{3\cdot 2} + \lambda_6 \right], \label{f-37} \\
x_3y_5 &= 2\left[ 2x_2^2y_4 + y_4^2 - x_2^2\wp_{3\cdot 2} + \lambda_4y_4 + \lambda_8 \right], \label{f-38} \\
y_5^2 &= 4\left[ x_2y_4^2 - y_4\wp_{3\cdot 2} + \lambda_{10} \right]. \label{f-39}\end{aligned}$$ Directly from relations - , we obtain the formula for the polynomial mapping $\pi_2$, that is, the proof of assertion 2 of the theorem.
Set $x_{i+1} = \wp_{1\cdot(i+1)},\; y_{i+3} = \wp_{1\cdot i,3\cdot 1},\; i\geqslant 1$. Applying the operator $\partial_{u_1}$ to formula , we obtain $$\label{f-40}
x_5 = 12x_2x_3 + 4y_5 = x_5(X,Y).$$ Substituting the expression for $\lambda_4$ from and the expression for $\wp_{3\cdot 2}$ from into the formula and then applying the operator $\partial_{u_1}$, we obtain $$\label{f-41}
y_7 = 4x_3y_4 + x_2(x_5 + 4y_5 -12x_2x_3) = y_7(X,Y).$$ By induction from formulas and , we obtain the polynomial formulas $$\label{f-42}
x_{i+1} = x_{i+1}(X,Y),\qquad y_{i+3} = y_{i+3}(X,Y).$$ From formula we obtain $$\label{f-43}
\wp_{3\cdot 2} = 3x_2y_4 - \frac{1}{2}y_6 = z_{6}(X,Y), \qquad \wp_{3\cdot (i+2)} = \partial_{u_3}^i z_{6}(X,Y) = z_{3i+6}.$$ The following formulas complete the proof of assertion 1 of the theorem $$\begin{aligned}
\partial_{u_3}x_{i+1} &= \partial_{u_1}\wp_{1\cdot i,3\cdot 1} = \partial_{u_1}y_{i+3}(X,Y), \label{f-44} \\
\partial_{u_3}y_{i+3} &= \partial_{u_3}\wp_{1\cdot i,3\cdot 1} = \wp_{1\cdot i,3\cdot 2} =
\partial_{u_1}^iz_{6}(X,Y). \label{f-45}\end{aligned}$$
In the course of the proof of Theorem \[T-7.13\], we obtained a detailed proof of Theorem \[T-7.1\] in the case $g=2$.
Set $L_1=\partial_{u_1}$ and $L_3=\partial_{u_3}$. We introduce the operators $L_i\in\operatorname{Der}(F_2),\; i=0,2,4,6$, based on the operators $\ell_i-H_i$.
\[T-7.14\] The generators of the $F_2$-module $\operatorname{Der}(F_2)$ are given by the formulas $$L_{2k-1} = \partial_{u_{2k-1}},\; k=1,2, \qquad L_{2k} = \ell_{2k} - \widehat H_{2k},\; k=0,1,2,3,$$ where $$\begin{aligned}
\widehat H_{0} &= u_1\partial_{u_1} + 3u_3\partial_{u_3}, \qquad
\widehat H_{2} = \left( \zeta_1 - \frac{4}{5}\lambda_4u_3 \right)\partial_{u_1} + u_1\partial_{u_3}, \\
\widehat H_{4} &= \left( \zeta_3 - \frac{6}{5}\lambda_6u_3 \right)\partial_{u_1} + (\zeta_1 + \lambda_4u_3)\partial_{u_3}, \qquad
\widehat H_{6} = -\frac{3}{5}\lambda_8u_3\partial_{u_1} + \zeta_3\partial_{u_3}.\end{aligned}$$
We will use the methods of [@bl08] to obtain the explicit form of operators $L_i$ and to describe their action on the ring $\mathcal{P}_2$. Note here that this theorem corrects misprints made in [@bl08; @buch16].
We have $L_1=\partial_{u_1} \in \operatorname{Der}(F_2)$ and $L_3=\partial_{u_3} \in \operatorname{Der}(F_2)$.
Below we use the fact that $[\partial_{u_k},\ell_q]=0$ for $k=1,3$ and $q=0,2,4,6$.
.2cm 1).
Using , we have $\ell_0\sigma=H_0\sigma=(u_1\partial_{u_1}+3u_3\partial_{u_3}-3)\sigma$. Therefore $$\label{F-26}
\ell_0\ln\sigma = u_1\partial_{u_1}\ln\sigma + 3u_3\partial_{u_3}\ln\sigma - 3.$$ Applying the operators $\partial_{u_1}$ and $\partial_{u_3}$ to (\[F-26\]), we obtain $$\begin{aligned}
\ell_0\zeta_1 &= \zeta_1 - u_1\wp_{1\cdot 2} - 3u_3\wp_{1\cdot 1,3\cdot 1}, \label{F-27} \\
\ell_0\zeta_3 &= 3\zeta_3 - u_1\wp_{1\cdot 1,3\cdot 1} - 3u_3\wp_{3\cdot 2}. \label{F-28}\end{aligned}$$ We apply the operator $\partial_{u_1}$ to (\[F-27\]) to obtain $$-\ell_0\wp_{1\cdot 2} = -2\wp_{1\cdot 2} - u_1\wp_{1\cdot 3} - 3u_3\wp_{1\cdot 2,3\cdot 1}.$$ Therefore $$(\ell_0 - u_1\partial_{u_1} - 3u_3\partial_{u_3})\wp_{1\cdot 2} = 2\wp_{1\cdot 2}.$$ Applying the operator $\partial_{u_1}$ to (\[F-28\]), we obtain $$-\ell_0\wp_{1\cdot 1,3\cdot 1} = -\wp_{1\cdot 1,3\cdot 1} - u_1\wp_{1\cdot 2,3\cdot 1} - 3\wp_{1\cdot 1,3\cdot 1} - 3u_3\wp_{1\cdot 1,3\cdot 2}.$$ Therefore, $$(\ell_0 - u_1\partial_{u_1} - 3u_3\partial_{u_3})\wp_{1\cdot 1,3\cdot 1} = 4\wp_{1\cdot 1,3\cdot 1}.$$ Thus, we have proved that $$L_0 = \ell_0-u_1\partial_{u_1}-3u_3\partial_{u_3} \in \operatorname{Der}(F_2).$$
.2cm 2).
Using , we have $$\ell_2\sigma = H_2\sigma = \left(\frac{1}{2}\,\partial_{u_1}^2 - \frac{4}{5}\,\lambda_4u_3\partial_{u_1} +
u_1\partial_{u_3} + w_2\right)\sigma$$ where $$w_2 = w_2(u_1,u_3) = -\frac{3}{10}\,\lambda_4u_1^2 + \frac{1}{10}\,(15\lambda_8 - 4\lambda_4^2)u_3^2.$$ Therefore $$\ell_2\ln\sigma = \frac{1}{2}\,\frac{\partial_{u_1}^2\sigma}{\sigma} - \frac{4}{5}\,\lambda_4u_3\partial_{u_1}\ln\sigma + u_1\partial_{u_3} \ln\sigma + w_2.$$ It holds that $$\frac{\partial_{u_1}^2\sigma}{\sigma} = -\wp_{1\cdot 2,0} + \zeta_1^2.$$ We get $$\label{F-29}
\ell_2\ln\sigma = -\frac{1}{2}\,\wp_{1\cdot 2} + \frac{1}{2}\,\zeta_1^2 - \frac{4}{5}\,\lambda_4u_3\zeta_1 + u_1\zeta_3 + w_2.$$ Applying the operators $\partial_{u_1}$ and $\partial_{u_3}$ to (\[F-29\]), we obtain $$\begin{aligned}
\ell_2\zeta_1 &= - \frac{1}{2}\,\wp_{1\cdot 3} - \zeta_1 \wp_{1\cdot 2} + \frac{4}{5}\,\lambda_4u_3\wp_{1\cdot 2} + \zeta_3 - u_1\wp_{1\cdot 1,3\cdot 1} + \partial_{u_1}w_2, \\
\ell_2\zeta_3 &= - \frac{1}{2}\,\wp_{1\cdot 2,3\cdot 1} - \zeta_1\wp_{1\cdot 1,3\cdot 1} - \frac{4}{5}\,\lambda_4\zeta_1 + \frac{4}{5}\,\lambda_4u_3 \wp_{1\cdot 1,3\cdot 1} -
u_1 \wp_{3\cdot 2} + \partial_{u_3}w_2.\end{aligned}$$ Applying the operator $\partial_{u_1}$ again, we obtain $$-\ell_2\wp_{1\cdot 2} = -\frac{1}{2}\,\wp_{1\cdot 4} + \wp_{1\cdot 2}^2 - \zeta_1\wp_{1\cdot 3} + \frac{4}{5}\,\lambda_4u_3\wp_{1\cdot 3} - 2\wp_{1\cdot 1,3\cdot 1} -
u_1\wp_{1\cdot 2,3\cdot 1} + \partial_{u_1}^2w_2,$$ $$\begin{gathered}
-\ell_2\wp_{1\cdot 1,3\cdot 1} = -\frac{1}{2}\,\wp_{1\cdot 3,3\cdot 1} + \wp_{1\cdot 2}\wp_{1\cdot 1,3\cdot 1} - \zeta_1\wp_{1\cdot 2,3\cdot 1} + \frac{4}{5}\,\lambda_4\wp_{1\cdot 2} +
\frac{4}{5}\,\lambda_4u_3\wp_{1\cdot 2,3\cdot 1} - \\
- \wp_{3\cdot 2} - u_1\wp_{1\cdot 1,3\cdot 2} + \partial_{u_1}\partial_{u_3}w_2.\end{gathered}$$ Thus, we have proved that $$L_2=\left(\ell_2-\zeta_1\partial_{u_1} - u_1\partial_{u_3} + \frac{4}{5}\,\lambda_4 u_3\partial_{u_1}\right) \in \operatorname{Der}(F_2).$$ We have $\partial_{u_1}^2w_2=-\frac{3}{5}\lambda_4$ and $\partial_{u_1}\partial_{u_3}w_2=0$.
.2cm 3).
Using , we have $$\ell_4\sigma = H_4\sigma = \left(\partial_{u_1}\partial_{u_3} - \frac{6}{5}\,\lambda_6u_3\partial_{u_1} +
\lambda_4u_3\partial_{u_3} + w_4\right)\sigma$$ where $$w_4 = -\frac{1}{5}\,\lambda_6u_1^2 + \lambda_8u_1u_3 + \frac{1}{10}\,(30\lambda_{10} - 6\lambda_6\lambda_4)u_3^2 - \lambda_4.$$ Therefore, $$\ell_4\ln\sigma = \frac{\partial_{u_1}\partial_{u_3}\sigma}{\sigma} - \frac{6}{5}\,\lambda_6u_3\partial_{u_1}\ln\sigma + \lambda_4u_3\partial_{u_3} \ln\sigma + w_4.$$ It holds that $$\frac{\partial_{u_1}\partial_{u_3}\sigma}{\sigma} = - \wp_{1\cdot 1,3\cdot 1} + \zeta_1\zeta_3.$$ We obtain $$\label{F-30}
\ell_4\ln\sigma = - \wp_{1\cdot 1,3\cdot 1} + \zeta_1\zeta_3 - \frac{6}{5}\,\lambda_6u_3\zeta_1 + \lambda_4u_3\zeta_3 + w_4.$$ Applying the operators $\partial_{u_1}$ and $\partial_{u_3}$ to (\[F-30\]), we obtain $$\begin{aligned}
\ell_4\zeta_1 &= - \wp_{1\cdot 2,3\cdot 1} - \wp_{1\cdot 2}\zeta_3 - \zeta_1 \wp_{1\cdot 1,3\cdot 1} + \frac{6}{5}\,\lambda_6u_3\wp_{1\cdot 2} - \lambda_4u_3\wp_{1\cdot 1,3\cdot 1} + \partial_{u_1}w_4, \\
\ell_4\zeta_3 &= - \wp_{1\cdot 1,3\cdot 2} - \wp_{1\cdot 1,3\cdot 1}\zeta_3 - \zeta_1\wp_{3\cdot 2} - \frac{6}{5}\,\lambda_6\zeta_1 + \frac{6}{5}\,\lambda_6u_3 \wp_{1\cdot 1,3\cdot 1} + \lambda_4\zeta_3 -
\lambda_4u_3 \wp_{3\cdot 2} + \partial_{u_3}w_4.\end{aligned}$$ Applying the operator $\partial_{u_1}$ again, we obtain $$\begin{gathered}
-\ell_4\wp_{1\cdot 2} = - \wp_{1\cdot 3,3\cdot 1} - \wp_{1\cdot 3}\zeta_3 + \wp_{1\cdot 2}\wp_{1\cdot 1,3\cdot 1} + \wp_{1\cdot 2}\wp_{1\cdot 1,3\cdot 1} - \zeta_1\wp_{1\cdot 2,3\cdot 1} +
\frac{6}{5}\,\lambda_6u_3\wp_{1\cdot 3} - \\ - \lambda_4 u_3\wp_{1\cdot 2,3\cdot 1} + \partial_{u_1}^2w_4,\end{gathered}$$ $$\begin{gathered}
-\ell_4\wp_{1\cdot 1,3\cdot 1} = - \wp_{1\cdot 2,3\cdot 2} - \wp_{1\cdot 2,3\cdot 1}\zeta_3 + \wp_{1\cdot 1,3\cdot 1}^2 + \wp_{1\cdot 2}\wp_{3\cdot 2} -
\zeta_1\wp_{1\cdot 1,3\cdot 2} + \frac{6}{5}\,\lambda_6\wp_{1\cdot 2} + \\
+ \frac{6}{5}\,\lambda_6u_3\wp_{1\cdot 2,3\cdot 1} - \lambda_4\wp_{1\cdot 1,3\cdot 1} - \lambda_4u_3\wp_{1\cdot 1,3\cdot 2} + \partial_{u_1}\partial_{u_3}w_4.\end{gathered}$$ Therefore, we have proved that $$L_4 = \left(\ell_4-\zeta_3\partial_{u_1}-\zeta_1\partial_{u_3} + \frac{6}{5}\,\lambda_6 u_3\partial_{u_1} -
\lambda_4u_3\partial_{u_3}\right) \in \operatorname{Der}(F_2).$$ We have $\partial_{u_1}^2w_4= - \frac{2}{5}\,\lambda_6$ and $\partial_{u_1}\partial_{u_3}w_4=\lambda_8$.
.2cm 4).
Using , we have $$\ell_6\sigma = H_6\sigma = \left(\frac{1}{2}\,\partial_{u_3}^2 - \frac{3}{5}\,\lambda_8u_3\partial_{u_1} + w_6\right)\sigma$$ where $$w_6 = -\frac{1}{10}\,\lambda_8u_1^2 + 2\lambda_{10}u_1u_3 - \frac{3}{10}\,\lambda_8 \lambda_4 u_3^2 - \frac{1}{2}\,\lambda_6.$$ Therefore, $$\ell_6\ln\sigma = \frac{1}{2}\,\frac{\partial_{u_3}^2\sigma}{\sigma} - \frac{3}{5}\,\lambda_8u_3\partial_{u_1}\ln\sigma + w_6.$$ We obtain $$\label{F-31}
\ell_6\ln\sigma = -\frac{1}{2}\,\wp_{3\cdot 2} + \frac{1}{2}\,\zeta_3^2 - \frac{3}{5}\,\lambda_8u_3\zeta_1 + w_6.$$ Applying the operators $\partial_{u_1}$ and $\partial_{u_3}$ to (\[F-31\]), we obtain $$\begin{aligned}
\ell_6\zeta_1 &= - \frac{1}{2}\,\wp_{1\cdot 1,3\cdot 2} - \zeta_3 \wp_{1\cdot 1,3\cdot 1} + \frac{3}{5}\,\lambda_8u_3\wp_{1\cdot 2} + \partial_{u_1}w_6, \\
\ell_6\zeta_3 &= - \frac{1}{2}\,\wp_{3\cdot 3} - \zeta_3\wp_{3\cdot 2} - \frac{3}{5}\,\lambda_8\zeta_1 + \frac{3}{5}\,\lambda_8u_3 \wp_{1\cdot 1,3\cdot 1} + \partial_{u_3}w_6.\end{aligned}$$ Applying the operator $\partial_{u_1}$ again, we obtain $$\begin{aligned}
-\ell_6\wp_{1\cdot 2} &= -\frac{1}{2}\,\wp_{1\cdot 2,3\cdot 2} + \wp_{1\cdot 1,3\cdot 1}^2 - \zeta_3\wp_{1\cdot 2,3\cdot 1} + \frac{3}{5}\,\lambda_8u_3\wp_{1\cdot 3} + \partial_{u_1}^2w_6,\\
-\ell_6\wp_{1\cdot 1,3\cdot 1} &= -\frac{1}{2}\,\wp_{1\cdot 1,3\cdot 3} + \wp_{1\cdot 1,3\cdot 1}\wp_{3\cdot 2} - \zeta_3\wp_{1\cdot 1,3\cdot 2} + \frac{3}{5}\,\lambda_8\wp_{1\cdot 2} +
\frac{3}{5}\,\lambda_8u_3\wp_{1\cdot 2,3\cdot 1} + \partial_{u_1}\partial_{u_3}w_6.\end{aligned}$$ Therefore, we have proved that $$L_6 = \left(\ell_6-\zeta_3\partial_{u_3} + \frac{3}{5}\,\lambda_8 u_3\partial_{u_1}\right) \in \operatorname{Der}(F_2).$$ We have $\partial_{u_1}^2w_6= - \frac{1}{5}\,\lambda_8$ and $\partial_{u_1}\partial_{u_3}w_6 = 2\lambda_{10}$. This completes the proof.
The description of commutation relations in the differential algebra of Abelian functions of genus $2$ was given in [@bl08; @buch16], see also [@bel12]. We obtain this result directly from Theorem \[T-7.14\] and correct some misprints made in [@bl08; @buch16]. To simplify the calculations, we use the following results:
\[lem1\] The following commutation relations hold for $\ell_k$: $$\begin{aligned}
&[\partial_{u_1}, \ell_k] =0, \quad k = 0, 2, 4, 6,
&
&[\partial_{u_3}, \ell_k] =0, \quad k = 0, 2, 4, 6,
\\
&[\ell_0, \ell_k] = k \ell_k, \quad k = 2, 4, 6,
&
&[\ell_2, \ell_4] = {8 \over 5} \lambda_6 \ell_0 - {8 \over 5} \lambda_4 \ell_2 + 2 \ell_6,
\\
&[\ell_2, \ell_6] = {4 \over 5} \lambda_8 \ell_0 - {4 \over 5} \lambda_4 \ell_4,
&
&[\ell_4, \ell_6] = - 2 \lambda_{10} \ell_0 + {6 \over 5} \lambda_8 \ell_2 - {6 \over 5} \lambda_6 \ell_4 + 2 \lambda_4 \ell_6.\end{aligned}$$
This relations follow directly from .
\[lem2\] The operators $L_i$, $i = 0, 1, 2, 3, 4, 6$, act on $- \zeta_1$ and $- \zeta_3$ according to the formulas $$\begin{aligned}
L_0(- \zeta_1) &= - \zeta_1 , & L_0(- \zeta_3) &= - 3 \zeta_3,\\
L_1(- \zeta_1) &= \wp_{1\cdot 2}, & L_1(- \zeta_3) &= \wp_{1\cdot 1,3\cdot 1},\\
L_2(- \zeta_1) &= {1 \over 2} \wp_{1\cdot 3} - \zeta_3 + {3 \over 5} \lambda_4 u_1, &
L_2(- \zeta_3) &= {1 \over 2} \wp_{1\cdot 2,3\cdot 1} + {4 \over 5} \lambda_4 \zeta_1 + \left({4 \over 5} \lambda_4^2 - 3 \lambda_8\right) u_3,\\
L_3(- \zeta_1) &= \wp_{1\cdot 1,3\cdot 1}, & L_3(- \zeta_3) &= \wp_{3\cdot 2},\end{aligned}$$ $$\begin{gathered}
L_4(- \zeta_1) = \wp_{1\cdot 2,3\cdot 1} + {2 \over 5} \lambda_6 u_1 - \lambda_8 u_3, \qquad
L_4(- \zeta_3) = \wp_{1\cdot 1,3\cdot 2} + {6 \over 5} \lambda_6 \zeta_1 - \lambda_4 \zeta_3 - \lambda_8 u_1 +\quad \\
+ 6 \left({1 \over 5} \lambda_4 \lambda_6 - \lambda_{10}\right) u_3,\end{gathered}$$ $$L_6(- \zeta_1) = {1 \over 2} \wp_{1\cdot 1,3\cdot 2} + {1 \over 5} \lambda_8 u_1 - 2 \lambda_{10} u_3, \;\;
L_6(- \zeta_3) = {1 \over 2} \wp_{3\cdot 3} + {3 \over 5} \lambda_8 \zeta_1 - 2 \lambda_{10} u_1 + {3 \over 5} \lambda_4 \lambda_8 u_3.$$
For the operators $L_1, L_3$ this result follows from definitions. For the operators $L_0, L_2, L_4$ and $L_6$ this result follows from the proof of Theorem \[T-7.14\].
The following theorem completes the description of the action of generators of the Lie $\mathcal{P}_2$-algebra $\mathcal{L}_2$ on the ring of polynomials $\mathcal{P}_2$.
\[T-7.17\] The operators $L_i,\; i=0,1,2,3,4,6$ act on $\wp_{1\cdot 2}$ and $\wp_{1\cdot 1,3\cdot 1}$ according to the formulas $$\begin{aligned}
L_0 \wp_{1\cdot 2} &= 2\wp_{1\cdot 2}, \qquad L_0 \wp_{1\cdot 1,3\cdot 1} = 4\wp_{1\cdot 1,3\cdot 1},\\
L_1 \wp_{1\cdot 2} &= \wp_{1\cdot 3}, \qquad \;\; L_1 \wp_{1\cdot 1,3\cdot 1} = \wp_{1\cdot 2,3\cdot 1},\\
L_2 \wp_{1\cdot 2} &= \frac{1}{2}\wp_{1\cdot 4} - \wp_{1\cdot 2}^2 + 2\wp_{1\cdot 1,3\cdot 1} + \frac{3}{5}\lambda_4,\\
L_2 \wp_{1\cdot 1,3\cdot 1} &= \frac{1}{2}\wp_{1\cdot 3,3\cdot 1} - \wp_{1\cdot 2}\wp_{1\cdot 1,3\cdot 1} - \frac{4}{5}\lambda_4\wp_{1\cdot 2} + \wp_{3\cdot 2},\\
L_4 \wp_{1\cdot 2} &= \wp_{1\cdot 3,3\cdot 1} - 2\wp_{1\cdot 2}\wp_{1\cdot 1,3\cdot 1} + \frac{2}{5}\lambda_6,\\
L_4 \wp_{1\cdot 1,3\cdot 1} &= \wp_{1\cdot 2,3\cdot 2} - \wp_{1\cdot 1,3\cdot 1}^2 - \wp_{1\cdot 2}\wp_{3\cdot 2} - \frac{6}{5}\lambda_6\wp_{1\cdot 2} +
\lambda_4\wp_{1\cdot 1,3\cdot 1} - \lambda_8,\\
L_6 \wp_{1\cdot 2} &= \frac{1}{2}\wp_{1\cdot 2,3\cdot 2} - \wp_{1\cdot 1,3\cdot 1}^2 + \frac{1}{5}\lambda_8,\\
L_6 \wp_{1\cdot 1,3\cdot 1} &= \frac{1}{2}\wp_{1\cdot 1,3\cdot 3} - \wp_{1\cdot 1,3\cdot 1}\wp_{3\cdot 2} - \frac{3}{5}\lambda_8\wp_{1\cdot 2} - 2\lambda_{10}.
\end{aligned}$$
Note, that in these formulas the parameters $\lambda_{2k},\; k=4,\ldots,10,$ are considered as polynomials $\lambda_{2k}(W_\wp)$ (see \[f-31\] - \[f-34\]).
See the derivation of the formulas for the operators $L_{2k},\; k=0,2,3,4,$ in the proof of Theorem \[T-7.14\].
The following result is based on the formulas of Theorem \[T-7.14\].
\[t-26\] The commutation relations in the Lie $F_2$-algebra $\operatorname{Der}(F_2)$ of derivations of the field $F_2$ have the form $$\begin{aligned}
[L_0, L_k] &= kL_k, \quad k=1,2,3,4,6; &
[L_1, L_2] &= \wp_{1\cdot 2} L_1 - L_3;\\
[L_1, L_3] &= 0; &
[L_1, L_4] &= \wp_{1\cdot 1,3\cdot 1} L_1 + \wp_{1\cdot 2} L_3; \\
[L_1, L_6] &= \wp_{1\cdot 1,3\cdot 1} L_3; &
[L_3, L_2] &= \left(\wp_{1\cdot 1,3\cdot 1} + {4 \over 5} \lambda_4 \right) L_1; \\
[L_3, L_4] &= \left(\wp_{3\cdot 2} + {6 \over 5} \lambda_6 \right) L_1 + \left(\wp_{1\cdot 1,3\cdot 1} - \lambda_4\right) L_3; &
[L_3, L_6] &= {3 \over 5} \lambda_8 L_1 + \wp_{3\cdot 2} L_3; \\
[L_2, L_4] &=\frac{8}{5}\lambda_6 L_0 -\frac{1}{2}\wp_{1\cdot 2,3\cdot 1}L_1 -\frac{8}{5}\lambda_4 L_2 +\frac{1}{2}\wp_{1\cdot 3}L_3 + 2L_6; \hspace{-20mm} & \\
[L_2, L_6] &= \frac{4}{5}\lambda_8 L_0 -\frac{1}{2}\wp_{1\cdot 1,3\cdot 2}L_1 +\frac{1}{2}\wp_{1\cdot 2,3\cdot 1}L_3 -\frac{4}{5}\lambda_4 L_4; & \\
[L_4, L_6] &=-2\lambda_{10} L_0 - \frac{1}{2}\wp_{3\cdot 3}L_1 +\frac{6}{5}\lambda_8 L_2 +\frac{1}{2}\wp_{1\cdot 1,3\cdot 2}L_3 -\frac{6}{5}\lambda_6 L_4 + 2\lambda_4 L_6.
\hspace{-100mm}&\end{aligned}$$
Due to linearity, the relation $[L_0, L_k] = k L_k, \quad k=1,2,3,4,6,$ can be checked independently for every summand in the expression for $L_k$.
The expressions for $[L_m, L_n]$, where $m$ or $n$ is equal to $1$ or $3$, can be obtained by simple calculations using Theorem \[T-7.14\].
It remains to prove the commutation relations among $L_2, L_4$ and $L_6$. We express $[L_m, L_n]$, where $m < n$ and $m, n = 2, 4, 6$, in the form $$[L_m,L_n]= a_{m,n,0} L_0 + a_{m,n,-1} L_1 + a_{m,n,-2} L_2 + a_{m,n,-3} L_3 + a_{m,n,-4} L_4 + a_{m,n,-6} L_6.$$ We have $\deg a_{i,j,-k} = i+j-k$. Applying both sides of this equation to $\lambda_k$ and using the explicit expressions for $L_k$, we get $$[\ell_m, \ell_n] \lambda_k = (a_{m,n,0} \ell_0 + a_{m,n,-2} \ell_2 + a_{m,n,-4} \ell_4 + a_{m,n,-6} \ell_6) \lambda_k.$$ This formula and Lemma \[lem1\] yield the values of the coefficients $a_{m,n,-k}$, $k = 0, 2, 4, 6$: $$\begin{aligned}
[L_2, L_4] &=\frac{8}{5}\lambda_6 L_0 + a_{2,4,-1} L_1 -\frac{8}{5}\lambda_4 L_2 + a_{2,4,-3} L_3 + 2L_6; & \label{bbb24} \\
[L_2, L_6] &= \frac{4}{5}\lambda_8 L_0 + a_{2,6,-1} L_1 + a_{2,6,-3} L_3 -\frac{4}{5}\lambda_4 L_4; & \label{bbb26} \\
[L_4, L_6] &=-2\lambda_{10} L_0 + a_{4,6,-1} L_1 +\frac{6}{5}\lambda_8 L_2 + a_{4,6,-3} L_3 -\frac{6}{5}\lambda_6 L_4 + 2\lambda_4 L_6. \label{bbb46}\end{aligned}$$
In subsequent calculations we compare the actions of the left- and right-hand sides of the expressions – on the coordinates $u_1$ and $u_3$. To this end we use the expressions , Theorem \[T-7.14\] and Lemma \[lem2\].
We present the calculation of the coefficient $a_{2,4,-1}$. The left-hand side of gives $$\begin{gathered}
[L_2, L_4] u_1 = L_2(-\zeta_3 + \frac{6}{5}\, \lambda_6 u_3) - L_4(-\zeta_1 + \frac{4}{5}\lambda_4u_3) = \\
= L_2(-\zeta_3) + \frac{6}{5} \ell_2(\lambda_6) u_3 - \frac{6}{5} \lambda_6 u_1
- L_4(-\zeta_1) - \frac{4}{5} \ell_4(\lambda_4) u_3 - \frac{4}{5} \lambda_4 (-\zeta_1 - \lambda_4u_3) = \\
= - {1 \over 2} \wp_{1\cdot 2,3\cdot 1} + {8 \over 5} \lambda_4 \zeta_1 - \frac{8}{5} \lambda_6 u_1
+ {2 \over 5} \left(3 \lambda_8 - {16 \over 5} \lambda_4^2\right) u_3.\end{gathered}$$ The right-hand side of gives $$[L_2, L_4] u_1 = a_{2,4,-1} + \frac{8}{5}\lambda_4 \zeta_1 - \frac{8}{5}\lambda_6 u_1
+ {2 \over 5} \left( 3 \lambda_8 -\frac{16}{5}\lambda_4^2\right) u_3.$$ By equating them, we obtain $a_{2,4,-1} = - {1 \over 2} \wp_{1\cdot 2,3\cdot 1}$.
The coefficients $a_{2,4,-3}$, $a_{2,6,-1}$, $a_{2,6,-3}$, $a_{4,6,-1}$ and $a_{4,6,-3}$ are calculated in a similar way.
.2cm
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[^1]: Here and below we punctually follows notations of elliptic functions theory fixed in [@be955]
| 0 |
---
author:
- Alain Dresse
- |
Marc Henneaux[Also at Centro de Estudios Científicos de Santiago, Casilla 16443, Santiago 9, Chile]{}\
Faculté des Sciences, Université Libre de Bruxelles,\
Campus Plaine C.P. 231, B-1050 Bruxelles (Belgium)
title: BRST Structure of Polynomial Poisson Algebras
---
\#1\#2[[\#1\#2]{}]{} =0 ${\global\advance\parenthesis by1\left(}
\def$[by-1)]{} $${\relax} \def$$ \#1 \#1[0=\#1sp0 by 30]{}
Introduction
============
Polynomial algebras with a Lie bracket fulfilling the derivation property $$[f g, h] = f [g,h] + [f,h]g$$ are called polynomial Poisson algebras and play an increasingly important role in various areas of theoretical physics [@Nak:; @Pri:; @Skl:; @Zam:; @FatZam:; @Oh:; @TarTakFad:; @BakMat:; @BhaRam:; @GraZhe:]. In terms of a set of independent generators $G_a$, $a = 1, \ldots, n$, the brackets are given by $$\label{basic_bracket}
[G_a, G_b] = C_{ab}(G)$$ where $C_{ab} = - C_{ba}$ are polynomials in the $G$’s[^1]. If the polynomials $C_{ab}(G)$ vanish when the $G$’s are set equal to zero, i.e. if they have no constant part, the polynomial algebra is said to be first class, in analogy with the terminology for constrained Hamiltonian systems (see e.g. [@HenTei:QuaGauSys]). An important class of first class Poisson algebras are symmetric algebras over a finite dimensional Lie algebra. In that case, the bracket (\[basic\_bracket\]) belongs to the linear span of the $G_a$’s, i.e. the $C_{ab}(G)$ are homogeneous of degree one in the $G$’s, $[G_a, G_b] = C_{ab}{}^c G_c$. We shall call this situation the “Lie algebra case”, and refer to the non Lie algebra case as the “open algebra case” using again terminology from the theory of first class constrained systems [@HenTei:QuaGauSys][^2].
The purpose of this paper is to investigate the BRST structure of first class Poisson algebras. The BRST formalism has turned out recently to be the arena of a fruitful interplay between physics and mathematics (see e.g. [@HenTei:QuaGauSys] and references therein). A crucial ingredient of BRST theory is the recursive pattern of homological perturbation theory [@Sta:] which allows one to construct the BRST generator step by step. In most applications, however, this recursive construction collapses almost immediately, and, to our knowledge, no example has been given so far for which the full BRST machinery is required (apart from the field-theoretical membrane models [@Hen:PhyLet; @FujKub:]). We show in this paper that Poisson algebras—actually, already quadratic Poisson algebras—offer splendid examples illustrating the complexity of the BRST construction. While Lie algebras yield a BRST generator of rank 1 (see e.g. [@HenTei:QuaGauSys]), the BRST charge for quadratic Poisson algebras can be of arbitrarily high rank. We also point out that BRST concepts provide intrinsic characterizations of Poisson algebras.
In the next section, we briefly review the BRST construction. We then discuss how it applies to Poisson algebras, even when the generators $G_a$ are not realized as phase space functions of some dynamical system. We analyze the BRST cohomology and introduce the concepts of covariant and minimal ranks, for which an elementary theorem is proven. Quadratic algebras are then shown to provide models with arbitrarily high rank. These contain “self-reproducing” algebras for which the bracket of $G_a$ with $G_b$ is proportional to the product $G_a G_b$. The first few terms in the BRST generator are also computed for more general algebras by means of a program written in REDUCE. The paper ends with some concluding remarks on the quantum case.
A Brief Survey of the BRST Formalism
====================================
We follow the presentation of [@HenTei:QuaGauSys], to which we refer for details and proofs. Given a set of independent functions $G_a(q,p)$ defined in some phase space $P$ with local coordinates $(q^i, p_i)$ and fulfilling the first class property $[G_a, G_b] \approx 0$, where $\approx$ denotes equality on the surface $G_a(q,p) = 0$, one can introduce an odd generator $\Omega$ (“the BRST generator”) in an extended phase space containing further fermionic conjugate pairs $(\eta^a, {{\cal P}}_a)$ (the “ghost pairs”) which has the following properties : $$\begin{aligned}
[\Omega, \Omega] &=& 0 \label{nilpotency} \\
\Omega &=& G_a \eta^a + \mbox{``more''}.\end{aligned}$$ Here, “more” stands for terms containing at least one ghost momentum ${{\cal P}}_a$. We take the ghosts $\eta^a$ to be real and their momenta imaginary, with graded Poisson bracket $$[{{\cal P}}_a, \eta^b] = - \delta_a{}^b$$
The BRST derivation $s$ in the extended phase space is generated by $\Omega$, $$s \bullet = [ \bullet, \Omega]$$ and is a differential ($s^2 = 0$) because of (\[nilpotency\]). One also introduces a grading, the “ghost number” by setting $${\mbox{gh}}\eta^a = - {\mbox{gh}}{{\cal P}}_a = 1, \quad {\mbox{gh}}q^i = {\mbox{gh}}p_i = 0.$$ The ghost number of the BRST generator is equal to 1.
The BRST generator $\Omega$ is constructed recursively as follows. One sets $$\Omega = {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} + {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}} + \cdots$$ where ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$ contains $k$ ghost momenta. One has ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} = G_a \eta^a$. The nilpotency condition becomes, in terms of ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$, $$\label{delta-om=d}
\delta {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}} + {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}} = 0$$ where ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}}$ involves only the lower order ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (s)} \\ \Omega \end{array}}}$ with $s \leq p$ and is defined by $$\label{d-p}
{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}} = 1 / 2 \left[
\sum^p_{k=0} [{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} , {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p-k)} \\ \Omega \end{array}}}]_{\mbox{orig}} +
\sum^{p-1}_{k=0} [ {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k+1)} \\ \Omega \end{array}}} ,
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p-k)} \\ \Omega \end{array}}}]_{{{\cal P}}, \eta}
\right].$$
Here, the bracket $[\; , \;]_{\mbox{orig}}$ refers to the Poisson bracket in the original phase space, which only acts on the $q^i$ and $p_i$, and not on the ghosts, whereas $[\; , \;]_{{{\cal P}}, \eta}$ refers to the Poisson bracket acting only on the ghost and ghost momenta arguments and not on the original phase space variables. The “Koszul” differential $\delta$ in (\[delta-om=d\]) is defined by $$\label{koszul}
\delta q^i = \delta p_i = 0, \quad \delta \eta^a = 0, \quad \delta
{{\cal P}}_a = - G_a$$ and is extended to arbitrary functions on the extended phase space as a derivation. One easily verifies that $\delta^2 = 0$.
Given ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (s)} \\ \Omega \end{array}}}$ with $s \leq p$, one solves (\[delta-om=d\]) for ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}}$. This can always be done because $\delta
{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}} = 0$, and because $\delta$ is acyclic in positive degree. One then goes on to ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+2)} \\ \Omega \end{array}}}$ etc... until one reaches the complete expression for $\Omega$. The last function ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$ that can be non zero is ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (n-1)} \\ \Omega \end{array}}}$ where $n$ is the number of constraints. Indeed, the product $\eta^{a_1} \cdots \eta^{a_n} \eta^{a_{n+1}}$ of $n+1$ anticommuting ghost variables in ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (n)} \\ \Omega \end{array}}$ is zero. The function ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}$ is determined by (\[delta-om=d\]) up to a $\delta$-exact term. This amounts to making a canonical transformation in the extended phase space.
First Class Polynomial Poisson algebras
=======================================
The standard BRST construction recalled in the previous section assumes that the $G_a$’s are realized as functions on some phase space, and allows the $C^c{}_{ab}$ in $$[G_a, G_b] = C^c{}_{ab} G_c$$ to be functions of $q^i$ and $p_i$. However, when the $C^c{}_{ab}$’s depend on the $q$’s and $p$’s only through the $G_a$’s themselves, as is the case when the $G_a$’s form a first class polynomial Poisson algebra, one can define the BRST generator directly in the algebra $\Bbb{C}\,({{\cal P}}_a) \otimes \Bbb{C}\,(G_a) \otimes \Bbb{C}\,(\eta^a)$ of polynomials in the $G$’s, the $\eta$’s and the ${{\cal P}}$’s without any reference to the explicit realization of the $G$’s as phase space functions[^3]. That is, the BRST generator can be associated with the Poisson algebra itself.
The reason for which this can be done is that both the Koszul differential $\delta$ defined by (\[koszul\]) [*and*]{} the ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}}$ in (\[d-p\]) involve only $G_a$ and not $q^i$ or $p_i$ individually. Thus, ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}$ can be taken to depend only on $G_a$. The BRST generator is defined accordingly in the algebra $\Bbb{C}\,({{\cal P}}_a)
\otimes \Bbb{C}\,(G_a) \otimes \Bbb{C}\,(\eta_a)$.
One can give an explicit solution of (\[delta-om=d\]) in terms of the homotopy $\sigma$ defined on the generators by $$\sigma G_a = - {{\cal P}}_a, \quad \sigma {{\cal P}}_a = \sigma G_a = 0$$ and extended to the algebra $\Bbb{C}\,({{\cal P}}_a) \otimes \Bbb{C}\,(G_a) \otimes
\Bbb{C}\,(\eta_a)$ as a derivation, $$\label{sigma}
\sigma = - {{\cal P}}_a \frac{\partial}{\partial G_a}.$$ One has $$\sigma \delta + \delta \sigma = N$$ where $N$ counts the degree in the $G$’s and the ${{\cal P}}$’s. Hence, if ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D_m \end{array}}$ is the term of degree $m$ in $(G, {{\cal P}})$ of ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}}$, a solution of (\[delta-om=d\]) is given by $$\label{om-p}
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}} = - \sum_m 1/m \left( \sigma {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}}_m
\right)$$ since $\delta {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}}=0$ [@HenTei:QuaGauSys] and $m >
0$ (one has $m \geq p$ and for $p=0$, $m \geq 1$ because $[{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}},
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}]$ contains $G_a$ by the first class property).
It should be stressed that the partial derivations $\partial/\partial
G_a$ in(\[sigma\]) are well defined because the functions on which they act depend only on on $G_a$. For an arbitrary function of $q^i,
p_i$, $\partial F / \partial G_a$ would not be well defined even if the constraints $G_a$ are independent (i.e. irreducible) as here. One must specify what is kept fixed. For example, if there is one constraint $p_1 = 0$ on the four-dimensional phase space $(q^1, p_1),
(q^2, p_2)$, then $\partial p_2 / \partial p_1 = 0$ if one keeps $q^1,
q^2$ and $p_2$ fixed, but $\partial p_2 / \partial p_1 = 1$ if one keeps $q^1, q^2$ and $p_2 - p_1$ fixed. Note that the subsequent developments require only that the $C^a{}_{b c}$ be functions of the $G_a$, but not that these functions be polynomials. We consider here the polynomial case for the sole sake of simplicity.
As mentioned earlier, the solution (\[om-p\]) of the equation (\[delta-om=d\]) is not unique. We call it the “covariant solution” because the homotopy $\sigma$ defined by (\[sigma\]) is invariant under linear redefinitions of the generators.
[**Example:**]{} for a Lie algebra $$[G_a, G_b] = C^c{}_{ab} G_c$$ the covariant BRST generator is given by $$\label{L-A-omega}
\Omega = G_a \eta^a - 1/2 {{\cal P}}_a C^a{}_{bc} \eta^c \eta^b.$$ Its nilpotency expresses the Jacobi identity for the structure constants $C^a{}_{bc}$. One has ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ \Omega \end{array}}} = 0$ for $p \geq
2$.
In general, the BRST generator $\Omega$ for a generic Poisson algebra contains higher order terms whose calculation may be quite cumbersome. However, because the procedure is purely algorithmic, it can be performed by means of an algebraic program like REDUCE.
The cohomology of the Poisson algebra may be defined to be the cohomology of the BRST differential $s$ in the algebra $\Bbb{C}\,({{\cal P}}_a)
\otimes \Bbb{C}\,(G_a) \otimes \Bbb{C}\,(\eta_a)$. Because $s$ contains $\delta$ as its piece of lowest antighost number (with ${\mbox{antigh}}({{\cal P}}_a) = 1, {\mbox{antigh}}(\mbox{anything else}) = 0$), and because $\delta$ provides a resolution of the zero-dimensional point $G_a =
0$, standard arguments show that the cohomology of $s$ is isomorphic to the cohomology of the differential $s'$ in $\Bbb{C}\,(\eta^a)$, $$\label{eq:sPrime}
s' \eta^a = 1/2 C^a{}_{bc} \eta^b \eta^c$$ where $C^a{}_{bc}$ is defined by $$C^a{}_{bc} = \left.\frac{\partial C_{bc}}{\partial G_a}\right|_{G = 0}$$
The $C^a{}_{bc}$ fulfill the Jacobi identity so that $s'^2 = 0$. Hence, they are the structure constants of a Lie algebra, which is called the Lie algebra underlying the given Poisson algebra.
Because of (\[eq:sPrime\]), the BRST cohomology of a Poisson algebra is isomorphic to the cohomology of the underlying Lie algebra. For a different and more thorough treatment of Poisson cohomology, see [@Hue:].
Rank
====
Again in analogy with the terminology used in the theory of constrained systems, we shall call [*“covariant rank”*]{} of a first class polynomial Poisson algebra the degree in ${{\cal P}}_a$ of the covariant BRST generator. This concept is invariant under linear redefinitions of the generators because the covariant BRST generator is itself invariant if one transforms the ghosts and their momenta as $$\begin{aligned}
G_a &\rightarrow& \bar{G}_a = A_a{}^b G_b \\
{{\cal P}}_a &\rightarrow& \bar{{{\cal P}}}_a = A_a{}^b {{\cal P}}_b \\
\eta^a &\rightarrow& \bar{\eta}^a = (A^{-1})_b{}^a \eta^b\end{aligned}$$
We shall call [*“minimal rank”*]{} the degree in ${{\cal P}}_a$ of the solution of $[ \Omega, \Omega] = 0$ of lowest degree in ${{\cal P}}$ (i.e., one chooses at each stage ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}}$ in such a way that $\Omega$ has lowest possible degree in ${{\cal P}}$). It is easy to see that for a Lie algebra, the concepts of covariant and minimal ranks coincide. As we shall see on an explicit example below, they do not in the general case.
Now, for a Lie algebra, the rank is not particularly interesting in the sense that it does not tell much about the structure of the algebra : the rank of a Lie algebra is equal to zero if and only if the algebra is abelian. It is equal to one otherwise. For non linear Poisson algebras, the rank is more useful. All values of the rank compatible with the trivial inequality $$rank \leq n-1$$ may occur. Thus, the rank of the BRST generator provides a non trivial characterization of Poisson algebras. Conversely, non linear Poisson algebras yield an interesting illustration of the full BRST machinery where higher order terms besides ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}}$ are required in $\Omega$ to achieve nilpotency.
Upper bound on the rank
=======================
One can understand the fact that the rank of a Lie algebra is at most equal to one by introducing a degree in $\Bbb{C}\,({{\cal P}}_a) \otimes
\Bbb{C}\,(G_a) \otimes \Bbb{C}\,(\eta_a)$ different from the ghost degree as follows.
Assume that one can assign a “degree” $n_a \geq 1$ to the generators $G_a$ in such a way that the bracket decreases the degree by at least one, $$\label{deg-g=n}
\deg G_a = n_a, \; \deg([G_a, G_b]) \leq n_a + n_b - 1.$$ Then, one can bound the covariant and minimal ranks of the algebra by $\sum_a (n_a - 1) + 1$, $$r \leq \sum_a (n_a - 1) + 1$$
In the case of a Lie algebra, one can take $n_a = 1$ for all the generators since $\deg([G_a, G_b])$ is then equal to one and fulfills (\[deg-g=n\]). The theorem then states that the rank is bounded by one, in agreement with (\[L-A-omega\]).
Assign the following degrees to $\eta^a$ and ${{\cal P}}_a$, $$\deg \eta^a = - n_a + 1, \; \deg {{\cal P}}_a = n_a - 1$$ If $\delta A = B$ and $\deg B = b$, then $\deg A = b - 1$ since $\delta$ increases the degree by one. Now ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} = G_a
\eta^a$ is of degree one. It follows that $[{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}},
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}] = [{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}},{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}]_{\mbox{orig}}$ is of degree $\leq 1$ and hence, by (\[delta-om=d\]) and (\[d-p\]), $\deg {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}
\leq 0$. More generally, one has $\deg {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \leq -k + 1$. Indeed, if this relation is true up to order $k-1$, then it is also true at order $k$ because in $$\delta {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \sim [{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (r)} \\ \Omega \end{array}}},{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (s)} \\ \Omega \end{array}}}]_{\mbox{orig}} + [{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (r')} \\ \Omega \end{array}}},
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (s')} \\ \Omega \end{array}}}]_{{{\cal P}}, \eta}$$ ($r+s = k-1, \; r' + s' = k$), the right hand side is of degree $\leq
-k+2$. Thus $\deg {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \leq -k + 2 - 1 = -k + 1$.
But the element with most negative degree in the algebra is given by the product of all the $\eta$’s, which has degree $-\sum_a(n_a - 1)$. Accordingly, ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$ is zero whenever $-k+1 >= - \sum_a(n_a - 1)$, which implies $r \leq \sum_a(n_a-1)+1$ as stated in the theorem.
[**Remarks:**]{}
1. One can improve greatly the bound by observing that the $\eta$’s do not come alone in ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$. There are also $k$ momenta ${{\cal P}}_a$ which carry positive degree. This remark will, however, not be pursued further here.
2. One can actually assign degrees smaller than one to the generators $G_a$. For instance, in the case of an Abelian Lie algebra, one may take $deg G_a = 1/2, \; \deg \eta^a = 1/2, \deg {{\cal P}}_a = -
1/2$. Because the degree of a ghost number one object is necessarily greater than or equal to $1/2$, the condition $\deg {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \leq -k+1$ (if ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \neq 0$) implies ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} = 0$ for $k > 0$.
Self-reproducing algebras
=========================
While Lie algebras are characterized by the existence of a degreee that is decreased by the bracket, one may easily construct examples of Poisson algebras for which such a degree does not exist. The simplest ones are quadratic algebras for which $[G_a, G_b]$ is proportional to $G_a, G_b$ $$[G_a, G_b] = M_{ab} G_a G_b \quad\quad\mbox{no summation on $a,b$}$$ with $M_{ab} = -M_{ba}$. The Jacobi identity is fulfilled for arbitrary $M$’s. Since $\deg(G_a G_b) = n_a + n_b$, the inequality (\[deg-g=n\]) is violated for any choice of $n_a$. Because $[G_a, G_b]$ is proportional to $G_a G_b$, we shall call these algebras “self-reproducing algebras”.
The most general self-reproducing algebra with three generators is given by $$\begin{aligned}
[G_1, G_2] &=& \alpha\, G_1 G_2 \\{}
[G_2, G_3] &=& \beta \, G_2 G_3 \\{}
[G_3, G_1] &=& \gamma \, G_1 G_3.\end{aligned}$$ This Poisson algebra can be realized on a six-dimensional phase space by setting $$G_1 = \exp(p_2 + \alpha q_3), G_2 = \exp(p_3 + \beta q_1), G_3 =
\exp(p_1 + \gamma q_2).$$ The covariant BRST charge for this model is equal to $$\begin{aligned}
\Omega &=&\eta^1 \, G_1 + \eta^2 \, G_2 + \eta^3 \, G_3 + \\
\nonumber
& & 1/2 \,
(\alpha \,\eta^{2}\,\eta^{1}\,{{\cal P}}_{2}\,G_{1}
-\alpha \,\eta^{2}\,\eta^{1}\,{{\cal P}}_{1}\,G_{2}
-\beta \,\eta^{3}\,\eta^{2}\,{{\cal P}}_{3}\,G_{2} \\ \nonumber
& &\mbox{~~~~~~}
-\beta \,\eta^{3}\,\eta^{2}\,{{\cal P}}_{2}\,G_{3}
+\gamma \,\eta^{3}\,\eta^{1}\,{{\cal P}}_{3}\,G_{1}
+\gamma \,\eta^{3}\,\eta^{1}\,{{\cal P}}_{1}\,G_{3}) + \\ \nonumber
& & 1/12 \, (
( - \alpha \,\beta +2\,\alpha \,\gamma -\beta \,\gamma )
\,\eta^{3}\,\eta^{2}\,\eta^{1}\,{{\cal P}}_{3}\,{{\cal P}}_{2}\,G_{1} + \\
\nonumber & &\mbox{~~~~~~~~~}
( -2\,\alpha \,\beta +\alpha \,\gamma +\beta \,\gamma )
\,\eta^{3}\,\eta^{2}\,\eta^{1}\,{{\cal P}}_{3}\,{{\cal P}}_{1}\,G_{2} +\\
\nonumber & &\mbox{~~~~~~~~~}
( -\alpha \,\beta -\alpha \,\gamma +2\,\beta \,\gamma )
\,\eta^{3}\,\eta^{2}\,\eta^{1}\,{{\cal P}}_{2}\,{{\cal P}}_{1}\,G_{3}
)\end{aligned}$$ and is of rank 2 (the maximum possible rank) unless $\alpha = \beta =
\gamma$, or $\alpha = \beta = 0$, $\gamma \neq 0$, in which case it is of rank 1.
Examples
========
We now give the BRST charge (or the first terms of the BRST charge) for some particular Poisson algebras. The examples have been treated using REDUCE, using the treatment of summation over dummy indices developed in [@Dre:CanExp; @Dre:Imacs]. Details of the implementation of the BRST algorithm can be found in [@BurCapDre:]. All dummy variables are noted as $d_i$ where $i$ is an integer. Unless stated otherwise, there is an implicit summation on all dummy variables. For the examples in which the Jacobi identity is not trivially satisfied, the expressions have been normalized so that no combinations of terms in a polynomial belongs to the polynomial ideal generated by the left hand side of the Jacobi identity. In particular, polynomials in this ideal are represented by identically null expressions.
Self-Reproducing Algebras
-------------------------
As we have just defined, the basic Poisson brackets for the generators $G_d$ of the [*self-reproducing algebra*]{} are given by $$[G_{d_1}, G_{d_2}] = M_{d_1 d_2} G_{d_1} G_{d_2}$$ without summation over the dummy variables $d_1$ and $d_2$. The matrix $M$ is antisymmetric, but otherwise arbitrary.
The seven first orders of the covariant BRST charge are given by $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}=
\[G_{d_{1}}\,\eta^{d_{1}}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}}=
\[\frac{G_{d_{1}}\,M_{d_{1}d_{2}}\,\eta^{d_{1}}\,
\eta^{d_{2}}\,{{\cal P}}_{d_{2}}}{
2}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (2)} \\ \Omega \end{array}}}=
\[\frac{-
\(G_{d_{1}}\,M_{d_{1}d_{2}}\,\eta^{d_{1}}\,
\eta^{d_{2}}\,\eta^{d_{3}}\,{{\cal P}}_{d_{2}}\,
{{\cal P}}_{d_{3}}\,
\(M_{d_{1}d_{3}}
+M_{d_{2}d_{3}}
\)
\)
}{
12}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (3)} \\ \Omega \end{array}}}=
\[\frac{-
\(G_{d_{1}}\,M_{d_{1}d_{2}}\,M_{d_{1}d_{4}}\,
M_{d_{2}d_{3}}\,\eta^{d_{1}}\,\eta^{d_{2}}\,
\eta^{d_{3}}\,\eta^{d_{4}}\,{{\cal P}}_{d_{2}}\,
{{\cal P}}_{d_{3}}\,{{\cal P}}_{d_{4}}
\)
}{
24}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (4)} \\ \Omega \end{array}}}=
\[\(G_{d_{1}}\,\eta^{d_{1}}\,\eta^{d_{2}}\,
\eta^{d_{3}}\,\eta^{d_{4}}\,\eta^{d_{5}}\,
{{\cal P}}_{d_{2}}\,{{\cal P}}_{d_{3}}\,{{\cal P}}_{d_{4}}\,
{{\cal P}}_{d_{5}}\,\nl
\off{3499956}
\(-
\(M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\,M_{d_{1}d_{4}}\,
M_{d_{1}d_{5}}
\)
+4\,M_{d_{1}d_{2}}\,M_{d_{1}d_{4}}\,M_{d_{1}d_{5}}\,
M_{d_{2}d_{3}}\nl
\off{3827636}
+2\,M_{d_{1}d_{2}}\,M_{d_{1}d_{4}}\,M_{d_{2}d_{3}}\,
M_{d_{4}d_{5}}
+M_{d_{1}d_{2}}\,M_{d_{1}d_{5}}\,M_{d_{2}d_{3}}\nl
\off{3827636}
\,M_{d_{2}d_{4}}
-M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{2}d_{5}}
-M_{d_{1}d_{4}}\,M_{d_{1}d_{5}}\,\nl
\off{3827636}
M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}
-2\,M_{d_{1}d_{4}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{4}d_{5}}
+M_{d_{1}d_{5}}\,\nl
\off{3827636}
M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,M_{d_{2}d_{5}}
-M_{d_{1}d_{5}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{4}d_{5}}
\)
\)
/720
\]
\Nl}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (5)} \\ \Omega \end{array}}}=
\[\(G_{d_{2}}\,M_{d_{1}d_{2}}\,M_{d_{3}d_{4}}\,
\eta^{d_{1}}\,\eta^{d_{2}}\,\eta^{d_{3}}\,
\eta^{d_{4}}\,\eta^{d_{5}}\,\eta^{d_{6}}\,
{{\cal P}}_{d_{1}}\,{{\cal P}}_{d_{3}}\,{{\cal P}}_{d_{4}}\,
{{\cal P}}_{d_{5}}\,{{\cal P}}_{d_{6}}\nl
\off{3499956}
\,
\(-
\(M_{d_{2}d_{3}}\,M_{d_{2}d_{5}}\,M_{d_{2}d_{6}}
\)
+2\,M_{d_{2}d_{3}}\,M_{d_{2}d_{5}}\,M_{d_{5}d_{6}}
+M_{d_{2}d_{3}}\nl
\off{4100703}
\,M_{d_{2}d_{6}}\,M_{d_{3}d_{5}}
-M_{d_{2}d_{3}}\,M_{d_{3}d_{5}}\,M_{d_{3}d_{6}}
-M_{d_{2}d_{5}}\nl
\off{4100703}
\,M_{d_{2}d_{6}}\,M_{d_{3}d_{5}}
-2\,M_{d_{2}d_{5}}\,M_{d_{3}d_{5}}\,M_{d_{5}d_{6}}\nl
\off{4100703}
+M_{d_{2}d_{6}}\,M_{d_{3}d_{5}}\,M_{d_{3}d_{6}}
-M_{d_{2}d_{6}}\,M_{d_{3}d_{5}}\,M_{d_{5}d_{6}}
\)
\)
/1440
\]
\Nl}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (6)} \\ \Omega \end{array}}}=
\[\(G_{d_{3}}\,\eta^{d_{1}}\,\eta^{d_{2}}\,
\eta^{d_{3}}\,\eta^{d_{4}}\,\eta^{d_{5}}\,
\eta^{d_{6}}\,\eta^{d_{7}}\,{{\cal P}}_{d_{1}}\,
{{\cal P}}_{d_{2}}\,{{\cal P}}_{d_{4}}\,{{\cal P}}_{d_{5}}\,
{{\cal P}}_{d_{6}}\,{{\cal P}}_{d_{7}}\,\nl
\off{3499956}
\(-
\(M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\,M_{d_{1}d_{6}}\,
M_{d_{2}d_{4}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}
\)
-M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\nl
\off{3827636}
\,M_{d_{2}d_{4}}\,M_{d_{2}d_{6}}\,M_{d_{4}d_{5}}\,
M_{d_{6}d_{7}}
+2\,M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\,M_{d_{2}d_{4}}\,
\nl
\off{3827636}
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}
+M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\,M_{d_{2}d_{6}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
\,M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\,M_{d_{2}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+2\,M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\,M_{d_{2}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
-2\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{2}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
-13\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,
M_{d_{2}d_{4}}\,M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\,
M_{d_{6}d_{7}}\nl
\off{3827636}
+4\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+2\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{6}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
-4\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+4\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
-2\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{4}}\,
M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
+M_{d_{1}d_{3}}\,M_{d_{1}d_{4}}\,M_{d_{1}d_{6}}\,
M_{d_{2}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{3}}\nl
\off{3827636}
\,M_{d_{1}d_{4}}\,M_{d_{2}d_{4}}\,M_{d_{4}d_{5}}\,
M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}
+5\,M_{d_{1}d_{3}}\,M_{d_{1}d_{4}}\,\nl
\off{3827636}
M_{d_{2}d_{7}}\,M_{d_{3}d_{7}}\,M_{d_{4}d_{5}}\,
M_{d_{4}d_{6}}
+M_{d_{1}d_{3}}\,M_{d_{1}d_{6}}\,M_{d_{2}d_{3}}\,
M_{d_{2}d_{4}}\nl
\off{3827636}
\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{3}}\,M_{d_{1}d_{6}}\,M_{d_{2}d_{6}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
\,M_{d_{6}d_{7}}
-5\,M_{d_{1}d_{3}}\,M_{d_{1}d_{6}}\,M_{d_{2}d_{7}}\,
M_{d_{3}d_{7}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
+M_{d_{1}d_{3}}\,M_{d_{1}d_{7}}\,M_{d_{2}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}
-M_{d_{1}d_{3}}\,M_{d_{1}d_{7}}\nl
\off{3827636}
\,M_{d_{2}d_{7}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,
M_{d_{6}d_{7}}
-M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{2}d_{6}}\,\nl
\off{3827636}
M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}
-8\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
+2\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{6}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{3}}\nl
\off{3827636}
\,M_{d_{2}d_{3}}\,M_{d_{2}d_{7}}\,M_{d_{4}d_{5}}\,
M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}
+2\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\nl
\off{3827636}
\,M_{d_{2}d_{7}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,
M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{4}}\,
\nl
\off{3827636}
M_{d_{3}d_{5}}\,M_{d_{3}d_{6}}\,M_{d_{3}d_{7}}
+12\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,
M_{d_{3}d_{4}}\,M_{d_{3}d_{6}}\nl
\off{3827636}
\,M_{d_{3}d_{7}}\,M_{d_{4}d_{5}}
-6\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{4}}\,
M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\nl
\off{3827636}
\,M_{d_{6}d_{7}}
-5\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{4}}\,
M_{d_{3}d_{7}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
+5\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{4}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+5\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{6}}\,
M_{d_{3}d_{7}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
+13\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,
M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,
M_{d_{6}d_{7}}\nl
\off{3827636}
-5\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+5\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
+2\,M_{d_{1}d_{3}}\,M_{d_{2}d_{4}}\,M_{d_{3}d_{4}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
-8\,M_{d_{1}d_{3}}\,M_{d_{2}d_{6}}\,M_{d_{3}d_{4}}\,
M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
+2\,M_{d_{1}d_{3}}\,M_{d_{2}d_{6}}\,M_{d_{3}d_{6}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
-M_{d_{1}d_{3}}\,M_{d_{2}d_{7}}\,M_{d_{3}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+M_{d_{1}d_{3}}\,M_{d_{2}d_{7}}\,M_{d_{3}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{4}}\,\nl
\off{3827636}
M_{d_{2}d_{4}}\,M_{d_{3}d_{4}}\,M_{d_{4}d_{5}}\,
M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}
+M_{d_{1}d_{4}}\,M_{d_{2}d_{6}}\,M_{d_{3}d_{4}}\nl
\off{3827636}
\,M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}
+5\,M_{d_{1}d_{6}}\,M_{d_{2}d_{6}}\,M_{d_{3}d_{4}}\,
M_{d_{3}d_{6}}\nl
\off{3827636}
\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}
-8\,M_{d_{1}d_{6}}\,M_{d_{2}d_{6}}\,M_{d_{3}d_{6}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
\,M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{7}}\,M_{d_{2}d_{7}}\,M_{d_{3}d_{6}}\,
M_{d_{3}d_{7}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
+6\,M_{d_{1}d_{7}}\,M_{d_{2}d_{7}}\,M_{d_{3}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}
-6\,\nl
\off{3827636}
M_{d_{1}d_{7}}\,M_{d_{2}d_{7}}\,M_{d_{3}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}
\)
\)
/60480
\]
\Nl}$$
These expressions are not particularly illuminating but are of interest because they generically do not vanish and hence, define higher order BRST charges. This can be seen by means of the following example, in which only the brackets of the first generator with the other ones are non vanishing, and taken equal to $$\begin{aligned}
\label{eq:maxquad}
[G_1, G_{\alpha}] &=& G_1 G_{\alpha} = - [ G_{\alpha}, G_1]
\quad (\alpha = 2,3,\ldots,n),\\{}
[G_{\alpha}, G_{\beta}] &=& 0.\end{aligned}$$ For this particular self-reproducing algebra, all orders of the covariant BRST charge can be explicitly computed. One finds $$\begin{aligned}
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} &=& \eta^a G_a \\
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} &=& \alpha_k ({\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_1 \end{array}} + (-)^{k+1} {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_2 \end{array}})\end{aligned}$$ where $$\begin{aligned}
{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_1 \end{array}} &=& G_1 \eta^{\alpha_1} \cdots \eta^{\alpha_k} \eta^1
{{\cal P}}_{\alpha_1} \cdots {{\cal P}}_{\alpha_k} \\
{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_2 \end{array}} &=& G_{\alpha_k} \eta^{\alpha_1} \cdots \eta^{\alpha_k} \eta^1
{{\cal P}}_1 {{\cal P}}_{\alpha_1} \cdots {{\cal P}}_{\alpha_{k-1}}\end{aligned}$$ and $$\begin{aligned}
\alpha_1 &=& -1/2, \; \alpha_2 = -1/12, \; \alpha_3 = 0 \\
\alpha_k &=& -\frac{1}{k+1} \sum_{l=1}^{k-3} \alpha_{l+1} \alpha_{k-l-1}
\quad \mbox{for $k > 3$}\label{eq:AlpRec}.\end{aligned}$$
This can be seen as the only non zero brackets involved in the construction of the BRST charge are $$\begin{aligned}
[{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_1 \end{array}}, {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}]_{\mbox{orig}} &=& (-)^{k+1}{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ S \end{array}} \\{}
[{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_1 \end{array}}, {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (l)} \\ T_2 \end{array}}]_{\eta{{\cal P}}} &=&
(-)^{l(k+1)}{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k+l-1)} \\ S \end{array}},\end{aligned}$$ where $${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ S \end{array}} = G_1 G_\alpha \eta^{\alpha_1} \cdots \eta^{\alpha_k} \eta^\alpha
\eta^1
{{\cal P}}_{\alpha_1} \cdots {{\cal P}}{\alpha_k}.$$ We further have $$\label{eq:sigmaS}
\sigma {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ S \end{array}} = (-)^{k+1} {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k+1)} \\ T_2 \end{array}} - {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k+1)} \\ T_1 \end{array}}$$ Given these relations, it is straightforward to verify (39). First, one easily checks that (39) is correct for $k=1$ with $\alpha_1$ equal to 1/2. Let us then assume that (39) is true for $k = 0, 1 ...$ up to $p$. One then obtains $${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}} = \beta_p {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ S \end{array}}$$ with $\beta_p$ given by $$\beta_p = (-)^{p+1} \alpha_p - \sum_{k=0}^{p-1} (-)^{p(k+1)} \alpha_{k+1}
\alpha_{p-k} = - \sum_{k=1}^{p-2} (-)^{p(k+1)} \alpha_{k+1} \alpha_{p-k}$$ from which one gets, using (\[eq:sigmaS\]), that ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}}$ is indeed given by (39) with $\alpha_{p+1}$ equal to $$\alpha_{p+1} = \frac{\beta_p}{p+2}$$ Observe now that $\alpha_k = 0$ for $k$ odd, $k \neq 1$. This can again be shown by recurrence. First note that $\alpha_3$ = 0. Now let $p$ be even, $p > 3$. Suppose $\alpha_k = 0$ for $k$ odd, $1 < k < p$. All terms in the relation defining $\beta_{p}$ are proportional to an $\alpha_m$ with $m$ odd, $1 < m < p$, since $k+1$ and $p-k$ have opposite parities. Therefore, $\beta_p = 0 = \alpha_{p+1}$ and thus $\alpha_k
= 0$ for $k$ odd, $k > 1$. Accordingly only $\alpha_k $ with $k$ even can be different from zero. The expression for $\alpha_k$ reduces then to (43) since $k+1$ must be even in (49).
Although $\alpha_k = 0$ for $k$ odd, $k > 1$, one easily sees that $\alpha_k < 0$ for $k$ even. This is true for $k =
2$ as $\alpha_2 = -1/12$. Let $p$ be even, and suppose $\alpha_m < 0$ for $1 < m < p$, $m$ even. Then, all terms in the sum in the recurrence relation (\[eq:AlpRec\]) are strictly positive, so that $\alpha_p < 0$. Since $\alpha_p \neq 0$, the quadratic algebra (\[eq:maxquad\]) provides examples of systems with arbitrarily high covariant rank.
Note also that the minimal rank is equal to one: indeed, the non covariant BRST charge given by $$\tilde{\Omega}={{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} + {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}} + ({{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ T \end{array}}}_1 - {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ T \end{array}}}_2)/2 =
\eta^a G_a - G_a \eta^a \eta^1 {{\cal P}}_1$$ is nilpotent and $$\delta({{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ T \end{array}}}_1 - {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ T \end{array}}}_2) = 0$$ so $\tilde{\Omega}$ is indeed a valid BRST charge. This shows that the minimal and covariant ranks are in general different.
Finally, it is easy to modify slightly the basic brackets so as to induce non zero covariant ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$ with $k$ odd. One simply replaces (\[eq:maxquad\]) by $$\begin{aligned}
[G_{n-1}, G_n] &=& G_{n-1} G_n = - [G_n, G_{n-1}] \\{}
[G_1, G_n] &=& - G_1 G_n, \\{}
[G_1, G_{\alpha}] &=& G_1 G_{\alpha} \quad (\alpha \neq n).\end{aligned}$$
Purely Quadratic Algebras
-------------------------
A generalization of the above is the pure quadratic algebra. The basic Poisson brackets are then given by $$[G_{d_1}, G_{d_2}] = D_{d_1 d_2}^{d_3 d_4} G_{d_3} G_{d_4}$$ where $D_{d_1 d_2}^{d_3 d_4}$ is antisymmetric in $d_1, d_2$ and symmetric in $d_3, d_4$. The Jacobi identity implies that $$D_{d_4 d_1}^{d_5 d_6} D_{d_2 d_3}^{d_4 d_7} + \mbox{symm}(d_5, d_6,
d_7) + \mbox{cyclic}(d_1,d_2,d_3) = 0$$
The first orders of the covariant BRST charge are given by $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}=
\[G_{d_{1}}\,\eta^{d_{1}}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}}=
\[\frac{D_{d_{1}d_{2}}^{
d_{4}d_{3}}\,G_{d_{4}}\,\eta^{d_{1}}\,
\eta^{d_{2}}\,{{\cal P}}_{d_{3}}}{
2}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (2)} \\ \Omega \end{array}}}=
\[\frac{D_{d_{2}d_{3}}^{
d_{6}d_{5}}\,D_{d_{6}d_{1}}^{
d_{7}d_{4}}\,G_{d_{7}}\,
\eta^{d_{1}}\,\eta^{d_{2}}\,\eta^{d_{3}}\,
{{\cal P}}_{d_{4}}\,{{\cal P}}_{d_{5}}}{
6}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (3)} \\ \Omega \end{array}}}=
\[\frac{-
\(D_{d_{2}d_{3}}^{
d_{8}d_{7}}\,D_{d_{4}d_{1}}^{
d_{9}d_{5}}\,D_{d_{8}d_{9}}^{
d_{10}d_{6}}\,G_{d_{10}}\,\eta^{d_{1}}\,
\eta^{d_{2}}\,\eta^{d_{3}}\,\eta^{d_{4}}\,
{{\cal P}}_{d_{5}}\,{{\cal P}}_{d_{6}}\,{{\cal P}}_{d_{7}}
\)
}{
24}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (4)} \\ \Omega \end{array}}}=
\[\(\eta^{d_{1}}\,\eta^{d_{2}}\,\eta^{d_{3}}\,
\eta^{d_{4}}\,\eta^{d_{5}}\,{{\cal P}}_{d_{6}}\,
{{\cal P}}_{d_{7}}\,{{\cal P}}_{d_{8}}\,{{\cal P}}_{d_{9}}\,
\nl
\off{3499956}
\(3\,D_{d_{1}d_{2}}^{
d_{13}d_{6}}\,D_{d_{4}d_{5}}^{
d_{12}d_{9}}\,
D_{d_{10}d_{3}}^{
d_{11}d_{8}}\,D_{d_{12}d_{13}}^{
d_{10}d_{7}}\,G_{d_{11}}
+4\,D_{d_{3}d_{4}}^{
d_{10}d_{9}}\,D_{d_{5}d_{2}}^{
d_{13}d_{6}}\,
D_{d_{10}d_{11}}^{
d_{12}d_{8}}\nl
\off{3827636}
\,D_{d_{13}d_{1}}^{
d_{11}d_{7}}\,G_{d_{12}}
-4\,D_{d_{4}d_{5}}^{
d_{12}d_{9}}\,D_{d_{10}d_{3}}^{
d_{11}d_{8}}\,
D_{d_{12}d_{1}}^{
d_{13}d_{7}}\,D_{d_{13}d_{2}}^{
d_{10}d_{6}}\,G_{d_{11}}
\)
\)
/360
\]
\Nl}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (5)} \\ \Omega \end{array}}}=
\[\(D_{d_{12}d_{13}}^{
d_{14}d_{10}}\,G_{d_{14}}\,\eta^{d_{1}}\,
\eta^{d_{2}}\,\eta^{d_{3}}\,\eta^{d_{4}}\,
\eta^{d_{5}}\,\eta^{d_{6}}\,{{\cal P}}_{d_{7}}\,
{{\cal P}}_{d_{8}}\,{{\cal P}}_{d_{9}}\,{{\cal P}}_{d_{10}}\,
{{\cal P}}_{d_{11}}\nl
\off{3499956}
\,
\(-
\(2\,D_{d_{1}d_{2}}^{
d_{16}d_{7}}\,D_{d_{5}d_{6}}^{
d_{15}d_{11}}\,
D_{d_{15}d_{3}}^{
d_{12}d_{8}}\,D_{d_{16}d_{4}}^{
d_{13}d_{9}}
\)
+3\,D_{d_{1}d_{3}}^{
d_{16}d_{8}}\,D_{d_{4}d_{5}}^{
d_{12}d_{11}}\,
\nl
\off{4100703}
D_{d_{6}d_{2}}^{
d_{15}d_{7}}\,D_{d_{15}d_{16}}^{
d_{13}d_{9}}
-4\,D_{d_{4}d_{5}}^{
d_{12}d_{11}}\,D_{d_{6}d_{1}}^{
d_{15}d_{7}}\,
D_{d_{15}d_{3}}^{
d_{16}d_{9}}\,D_{d_{16}d_{2}}^{
d_{13}d_{8}}
\)
\)
/720
\]
\Nl}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (6)} \\ \Omega \end{array}}}=
\[\(\eta^{d_{1}}\,\eta^{d_{2}}\,\eta^{d_{3}}\,
\eta^{d_{4}}\,\eta^{d_{5}}\,\eta^{d_{6}}\,
\eta^{d_{7}}\,{{\cal P}}_{d_{8}}\,{{\cal P}}_{d_{9}}\,
{{\cal P}}_{d_{10}}\,{{\cal P}}_{d_{11}}\,{{\cal P}}_{d_{12}}\,
{{\cal P}}_{d_{13}}\,\nl
\off{3499956}
\(6\,D_{d_{1}d_{2}}^{
d_{17}d_{8}}\,D_{d_{6}d_{7}}^{
d_{16}d_{13}}\,
D_{d_{14}d_{5}}^{
d_{15}d_{12}}\,D_{d_{16}d_{17}}^{
d_{18}d_{11}}\,D_{d_{18}d_{4}}^{
d_{19}d_{10}}\,D_{d_{19}d_{3}}^{
d_{14}d_{9}}\,G_{d_{15}}\nl
\off{3827636}
-9\,D_{d_{1}d_{2}}^{
d_{18}d_{8}}\,D_{d_{3}d_{4}}^{
d_{19}d_{9}}\,
D_{d_{6}d_{7}}^{
d_{16}d_{13}}\,D_{d_{14}d_{5}}^{
d_{15}d_{12}}\,D_{d_{16}d_{17}}^{
d_{14}d_{10}}\,D_{d_{18}d_{19}}^{
d_{17}d_{11}}\,G_{d_{15}}\nl
\off{3827636}
+12\,D_{d_{1}d_{2}}^{
d_{18}d_{8}}\,D_{d_{3}d_{4}}^{
d_{19}d_{9}}\,
D_{d_{6}d_{7}}^{
d_{17}d_{13}}\,D_{d_{14}d_{15}}^{
d_{16}d_{12}}\,D_{d_{17}d_{18}}^{
d_{15}d_{11}}\,D_{d_{19}d_{5}}^{
d_{14}d_{10}}\,G_{d_{16}}\nl
\off{3827636}
-10\,D_{d_{1}d_{2}}^{
d_{18}d_{8}}\,D_{d_{6}d_{7}}^{
d_{16}d_{13}}\,
D_{d_{14}d_{5}}^{
d_{15}d_{12}}\,D_{d_{16}d_{3}}^{
d_{17}d_{9}}\,D_{d_{17}d_{19}}^{
d_{14}d_{11}}\,D_{d_{18}d_{4}}^{
d_{19}d_{10}}\,G_{d_{15}}\nl
\off{3827636}
+24\,D_{d_{1}d_{2}}^{
d_{18}d_{8}}\,D_{d_{6}d_{7}}^{
d_{16}d_{13}}\,
D_{d_{14}d_{5}}^{
d_{15}d_{12}}\,D_{d_{16}d_{17}}^{
d_{14}d_{11}}\,D_{d_{18}d_{4}}^{
d_{19}d_{10}}\,D_{d_{19}d_{3}}^{
d_{17}d_{9}}\,G_{d_{15}}\nl
\off{3827636}
-4\,D_{d_{1}d_{2}}^{
d_{18}d_{8}}\,D_{d_{6}d_{7}}^{
d_{17}d_{13}}\,
D_{d_{14}d_{15}}^{
d_{16}d_{12}}\,D_{d_{17}d_{4}}^{
d_{15}d_{9}}\,D_{d_{18}d_{5}}^{
d_{19}d_{11}}\,D_{d_{19}d_{3}}^{
d_{14}d_{10}}\,G_{d_{16}}\nl
\off{3827636}
+4\,D_{d_{1}d_{2}}^{
d_{19}d_{8}}\,D_{d_{6}d_{7}}^{
d_{16}d_{13}}\,
D_{d_{14}d_{5}}^{
d_{15}d_{12}}\,D_{d_{16}d_{17}}^{
d_{18}d_{11}}\,D_{d_{18}d_{3}}^{
d_{14}d_{9}}\,D_{d_{19}d_{4}}^{
d_{17}d_{10}}\nl
\off{3827636}
\,G_{d_{15}}
-6\,D_{d_{1}d_{3}}^{
d_{18}d_{9}}\,D_{d_{5}d_{6}}^{
d_{14}d_{13}}\,
D_{d_{7}d_{2}}^{
d_{17}d_{8}}\,D_{d_{14}d_{15}}^{
d_{16}d_{12}}\,D_{d_{17}d_{18}}^{
d_{19}d_{11}}\nl
\off{3827636}
\,D_{d_{19}d_{4}}^{
d_{15}d_{10}}\,G_{d_{16}}
-4\,D_{d_{1}d_{3}}^{
d_{19}d_{9}}\,D_{d_{5}d_{6}}^{
d_{14}d_{13}}\,
D_{d_{7}d_{2}}^{
d_{17}d_{8}}\,D_{d_{14}d_{15}}^{
d_{16}d_{12}}\,\nl
\off{3827636}
D_{d_{17}d_{18}}^{
d_{15}d_{11}}\,D_{d_{19}d_{4}}^{
d_{18}d_{10}}\,G_{d_{16}}
-16\,D_{d_{5}d_{6}}^{
d_{14}d_{13}}\,D_{d_{7}d_{1}}^{
d_{17}d_{8}}\,
D_{d_{14}d_{15}}^{
d_{16}d_{12}}\nl
\off{3827636}
\,D_{d_{17}d_{4}}^{
d_{18}d_{11}}\,D_{d_{18}d_{3}}^{
d_{19}d_{10}}\,
D_{d_{19}d_{2}}^{
d_{15}d_{9}}\,G_{d_{16}}
+16\,D_{d_{6}d_{7}}^{
d_{16}d_{13}}\,D_{d_{14}d_{5}}^{
d_{15}d_{12}}\nl
\off{3827636}
\,D_{d_{16}d_{4}}^{
d_{17}d_{11}}\,D_{d_{17}d_{3}}^{
d_{19}d_{10}}\,
D_{d_{18}d_{1}}^{
d_{14}d_{8}}\,D_{d_{19}d_{2}}^{
d_{18}d_{9}}\,G_{d_{15}}
\)
\)
/15120
\]
\Nl}$$
Again, these expressions are not particularly illuminating. The point emphasized here is that the calculation of the BRST charge is purely algorithmic and follows a general, well-established pattern.
Since homogeneous quadratic algebras contain the self-reproducing algebras as special case, they are generically of maximal covariant rank. More on this in [@ADresse3].
L-T algebras
------------
We now consider adding a linear term to the quadratic algebra above. The basic Poisson brackets for the generators $G_d$ are given by $$[G_{d_1}, G_{d_2}] = C_{d_1 d_2}^{d_3} \, G_{d_3} +
D_{d_1 d_2}^{d_3 d_4} G_{d_3} G_{d_4}$$ where $ C_{d_1 d_2}^{d_3} $ and $D_{d_1 d_2}^{d_3 d_4}$ are antisymmetric in $d_1, d_2$, and $D_{d_1 d_2}^{d_3 d_4}$ is symmetric in $d_3, d_4$. A particular instance of such an algebra is given by Zamolodchikov algebras [@Zam:]. We will start with a specific example, and consider general quadratically nonlinear Poisson algebras next.
The generators in the example are assumed to split into $L_{a}$ and $T_b$, $a = 1, \ldots n_1$, $b = n_1 + 1, \ldots, n$, with the brackets $$\begin{aligned}
[L_{a_1}, L_{a_2}] &=& \tilde{C}_{a_1 a_2}^{a_3} L_{a_3} \nonumber \\{}
[L_{a_1}, T_{b_1}] &=& \tilde{C}_{a_1 b_1}^{a_2} L_{a_2} +
\tilde{C}_{a_1 b_1}^{b_2} T_{b_2}
\label{eq:zam} \\{}
[T_{b_1}, T_{b_2}] &=& \tilde{C}_{b_1 b_2}^{a_1} L_{a_1} +
\tilde{C}_{b_1 b_2}^{b_3} T_{b_3} +
\tilde{D}_{b_1 b_2}^{a_1 a_2} L_{a_1} L_{a_2}
\nonumber\end{aligned}$$ so that contractions of $\tilde{D}$ are impossible.
Going back to the notations $G_{d_i} = \{L_{a}, T_{b}\}$, $d = 1,
\ldots, n$ the Jacobi identity imply $$C_{d_1 d_2}^{d_4} C_{d_3 d_4}^{d_5} + \mbox{cyclic}(d_1,d_2,d_3) = 0$$ $$\{D_{d_1 d_2}^{d_4 d_5} C_{d_3 d_4}^{d_6} + \mbox{symm}(d_5, d_6) \} + \\
C_{d_1 d_2}^{d_4} D_{d_3 d_4}^{d_5 d_6} + \mbox{cyclic}(d_1,d_2,d_3) = 0$$ and contractions of $D$ vanish.
For instance, the conditions (\[eq:zam\]) are fulfilled if one takes for the $L$’s the generators of a semi-simple Lie algebra and take the $T$’s to commute with the $L$’s and to close on the Casimir element: $$\begin{aligned}
[L_a, T_b] &=& 0 \\{}
[T_{b_1}, T_{b_2}] &=& \delta_{b_1 b_2} k^{a_1 a_2} L_{a_1} L_{a_2}\end{aligned}$$ where $k^{a_1 a_2}$ is the Killing bilinear form. The Jacobi identity is verified because the Casimir element commutes with the $L$’s.
The previous theorem on the rank yields, by taking $n(l) = 1$ and $n(T) = 3/2$, that the rank is bounded by $1/2 m + 1$, where $m$ is the number of T-generators. Actually, the rank is much lower, since the covariant BRST charge is computed to be $$\displaylines{\qdd
\Omega =
\[\frac{1}{2}
\,C_{d_{1}d_{2}}^{
d_{3}}\,\eta^{d_{1}}\,\eta^{d_{2}}\,
{{\cal P}}_{d_{3}}
+
\frac{1}{24}
\,C_{d_{8}d_{9}}^{
d_{6}}\,D_{d_{1}d_{2}}^{
d_{8}d_{7}}\,D_{d_{3}d_{4}}^{
d_{9}d_{5}}\,\eta^{d_{1}}\,\eta^{d_{2}}\nl
\off{2695321}
\,\eta^{d_{3}}\,\eta^{d_{4}}\,{{\cal P}}_{d_{5}}\,
{{\cal P}}_{d_{7}}\,{{\cal P}}_{d_{6}}
+
\frac{1}{2}
\,D_{d_{1}d_{2}}^{
d_{4}d_{3}}\,G_{d_{4}}\,\eta^{d_{1}}\,
\eta^{d_{2}}\,{{\cal P}}_{d_{3}}
+G_{d_{1}}\,\eta^{d_{1}}
\]
\Nl}$$ which is identical to the result in [@SchSevNie:QuaBRSChaQua].
Generalizations
---------------
The previous L-T algebras can be generalized in various directions. One may consider the general quadratic non homogeneous Poisson structure $$\begin{aligned}
& [G_{d_1}, G_{d_2}] = C_{d_1 d_2}^{d_3} \, G_{d_3} +
D_{d_1 d_2}^{d_3 d_4} G_{d_3} G_{d_4} & \\
& C_{d_1 d_2}^{d_4} C_{d_3 d_4}^{d_5} + \mbox{cyclic}(d_1,d_2,d_3)
= 0 & \\
& \{D_{d_1 d_2}^{d_4 d_5} C_{d_3 d_4}^{d_6} + \mbox{symm}(d_5, d_6) \} +
C_{d_1 d_2}^{d_4} D_{d_3 d_4}^{d_5 d_6} + \mbox{} \quad\quad &
\nonumber \\
& \quad \quad \mbox{cyclic}(d_1,d_2,d_3) = 0 &\\
& D_{d_4 d_1}^{d_5 d_6} D_{d_2 d_3}^{d_4 d_7} + \mbox{symm}(d_5, d_6,
d_7) + \mbox{cyclic}(d_1,d_2,d_3) = 0&\end{aligned}$$ with $ C_{d_1 d_2}^{d_3} $ and $D_{d_1 d_2}^{d_3 d_4}$ antisymmetric in $d_1, d_2$, and $D_{d_1 d_2}^{d_3 d_4}$ symmetric in $d_3, d_4$. One may also include higher order terms in the bracket while preserving the existence of a degree decreased by the bracket, as in the so called spin 4 algebra : $$\begin{aligned}
[L_{a_1}, L_{a_2}] &=& C_{a_1 a_2}^{a_3} L_{a_3} \\{}
[L_{a_1}, T_{b_1}] &=& C_{a_1 b_1}^{b_2} T_{b_2} \\{}
[L_{a_1}, W_{c_1}] &=& C_{a_1 c_1}^{c_2} W_{c_2} \\{}
[T_{b_1}, T_{b_2}] &=& C_{b_1 b_2}^{a_1} L_{a_1} + C_{b_1 b_2}^{b_3} T_{b_3} +
D_{b_1 b_2}^{a_1 a_2} L_{a_1} L_{a_2} \\{}
[T_{b_1}, W_{c_1}] &=& C_{b_1 c_1}^{b_2} T_{b_2} +
D_{b_1 c_1}^{a_1 b_2} L_{a_1} T_{b_2} \\{}
[W_{c_1}, W_{c_2}] &=& C_{c_1 c_2}^{a_1} L_{a_1} +
D_{c_1 c_2}^{a_1 a_2} L_{a_1} L_{a_2} +
E_{c_1 c_2}^{a_1 a_2 a_3} L_{a_1} L_{a_2} L_{a_3}.\end{aligned}$$ If one sets $n(L) = 1, n(T) = 3/2, L(W) = 2$, one gets $n([A,B]) \leq
n(A) + n(B)$.
We have checked, using REDUCE, that in both cases the first seven terms in $\Omega$ are generically non zero.
Conclusion
==========
We have shown in this paper that polynomial Poisson algebras provide a rich arena in which the perturbative features of the BRST construction are perfectly illustrated. We believe this to be of interest because models of higher rank are rather rare and are usually thought not to arise in practice. Non polynomial Poisson algebras(e.g. of the type arising in the study of quantum groups) can also be analyzed along the same BRST lines and should provide further models of higher rank.
We have not discussed the quantum realization of Poisson algebras, and whether the nilpotency condition for the BRST generator is maintained quantum-mechanically. This is a difficult question, which is model-dependent. Indeed, while the ghost contribution to $\Omega^2$ can be evaluated independently of the specific form of the $G_a$’s in terms of the canonical variables $(q^i, p_i)$ (once a representation of the ghost anticommutation relations is chosen), the “matter” contribution to $\Omega^2$ depends on the “anomaly” terms in $[G_a, G_b]$, which, in turn, depend on the specific form of the $G_a$’s. It would be interesting to pursue this question further.
Acknowledgments
===============
We are grateful to Jim Stasheff and Claudio Teitelboim for fruitful discussions at the early stages of this research. This work has been supported in part by research funds from FNRS (Belgium) and by a research contract with the Commission of the European Communities.
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[^1]: We shall restrict here the analysis to ordinary polynomial algebras with commuting generators, but one can easily extend the study to the graded case with both commuting and anticommuting generators.
[^2]: It should be stressed that the polynomial algebra generated by the $G$’s, equipped with the bracket (\[basic\_bracket\]) is [*always*]{} an infinite-dimensional Lie algebra, even in the “open algebra” case.
[^3]: In agreement with the notations of [@HenTei:QuaGauSys], we denote the algebra of polynomials in the anticommuting variables ${{\cal P}}_a$ with complex coefficients by $\Bbb{C}\,({{\cal P}}_a)$, and not by the more familiar notation $\Lambda({{\cal P}}_a)$. A typical element of $\Bbb{C}\,({{\cal P}}_1)$ is $a + b {{\cal P}}_1$ with $a, b \in \Bbb{C}\,$ since $({{\cal P}}_1)^2 = 0$.
| 0 |
---
abstract: 'We introduce and analyze a discontinuous Galerkin method for a time-harmonic eddy current problem formulated in terms of the magnetic field. The scheme is obtained by putting together a DG method for the approximation of the vector field variable representing the magnetic field in the conductor and a DG method for the Laplace equation whose solution is a scalar magnetic potential in the insulator. The transmission conditions linking the two problems are taken into account weakly in the global discontinuous Galerkin scheme. We prove that the numerical method is uniformly stable and obtain quasi-optimal error estimates in the DG-energy norm.'
author:
- |
[Ana Alonso Rodríguez]{}[^1], $\,\,$ [Salim Meddahi]{}[^2]\
and\
[Alberto Valli]{}[^3]
title: 'A discontinuous Galerkin method for the time harmonic eddy current problem [^4]'
---
Introduction
============
In this paper, we present a discontinuous Galerkin (DG) approximation of a time-harmonic eddy current problem. The eddy current approximation of Maxwell equations is obtained by disregarding the displacement current term. It is commonly used in applications related with induction heating, transformers, magnetic levitation and non-destructive testing. These problems often involve composite materials and structures, complex transmission conditions and, eventually, boundary layers due to the skin effect. The ability of DG methods to handle efficiently unstructured meshes with hanging nodes combined with $hp$-adaptive strategies make them well-suited for the numerical simulation of physical systems related to eddy currents.
The eddy current problem is generally written in terms of either the electric or the magnetic field, cf. [@AVbook]. These two formulations are equivalent at the continuous level but they lead to different numerical schemes. A discontinuous Galerkin method based on a time-harmonic eddy current problem written in terms of the electric field has been analyzed in the pioneering work of Perugia and Schotzau [@PS03]. For the time domain eddy current problem, Ausserhofer et al. introduced in [@ABP09] a formulation based on a magnetic vector potential and propose a numerical method that combines a DG approximation in the conductor with the usual ${\mathrm{H}}^{1}$-conforming Lagrange finite element approximation in the insulator.
Here, we are interested in imposing the magnetic field as primary unknown. The advantage of this approach rests on the reduction of the number of degrees of freedom resulting from the introduction of a scalar magnetic potential in the nonconducting medium. The global formulation of the problem consists in a ${\mathrm{H}}({\mathbf{curl}})$-elliptic problem for vector fields that are curl-free in the insulator $\Omega_I$. Our DG formulation is obtained by applying for the Laplace equation posed in $\Omega_{I}$ the usual interior penalty finite element method, that can be traced back to [@arnoldIP], see also [@DiPietroErn] and the references cited therein for more details. In the conductor $\Omega_{C}$ we employ, as in [@HPSS05; @PS03], the interior penalty method corresponding to the Nédélec curl-conforming finite element space of the second kind. We point out that the introduction of discrete harmonic fields is necessary when considering domains of general topology. We prove the stability of the resulting combined DG scheme by exploiting the elliptic character of the problem. We also obtain, under adequate regularity assumptions, quasi-optimal asymptotic error estimates. It is worthwhile to notice that the implementation of the DG-method presented here only requires the use of standard shape functions. The curl-conforming finite elements, more precisely, the Nédélec finite elements of the second kind, are only needed for the theoretical convergence results in Section 5.
The outline of this paper is as follows. In Section 2 we derive the model problem used in the finite element approximation. We introduce our DG formulation in Section 3. Finally, Section 4 is devoted to the convergence analysis, and asymptotic error estimates are provided in Section 5.
The model problem {#s2}
=================
Let $\Omega_C\subset \mathbb{R}^3$ be a bounded polyhedral domain with a Lipschitz boundary $\Gamma$. We denote by $\mathbf{n}_\Gamma$ the unit normal vector on $\Gamma$ that points towards $\Omega_e:= \mathbb{R}^3\setminus \overline \Omega_C$. In order to illustrate the impact of the conductor’s topology in our method, we assume that $\Omega_C$ has a toroidal shape. We notice that the eddy current problem is posed in the whole space with asymptotic conditions on the behaviour of the electric and magnetic fields at infinity. Depending on the nature of the eddy current problem being solved and the geometry involved, a discretization method can be obtained for this problem by either applying a pure finite element approach on a truncated domain or by using a combination of boundary (BEM) and finite elements (FEM), see [@AMV; @Hiptmair; @MS03; @AV09]. The FEM-BEM formulation is posed in the conductor but its implementation is more difficult and it leads to more complex algebraic linear systems of equations. The FEM method needs a large computational domain, but it is simpler and it can provide an alternative in many practical situations. It is the option that we will consider in the following. To this end, we introduce a bounded domain $D$ containing in its interior $\overline{\Omega}_{C}$ and whose connected boundary $\Sigma= \partial D$ is located at a large enough distance from the conductor $\Omega_{C}$. The bounded domain $\Omega_I:= D\setminus \overline \Omega_C$ represents then the nonconducting region of the computational domain $D$.
Under our assumptions, the first de Rham cohomology group $\mathcal H^1(\Omega_I)$ of $\Omega_{I}$, namely, the space of curl-free vector fields that are not gradients, has dimension one. If we assume that $\Omega_I$ is a polyhedral domain endowed with a tetrahedral mesh, one can use the technique given in [@BRS02] for the explicit construction of a piecewise-linear vector field $\boldsymbol \rho$ spanning $\mathcal H^1(\Omega_I)$ and satisfying $\boldsymbol \rho \times {\bf n}_\Sigma = {\bf 0}$ on $\Sigma$, where ${\bf n}_\Sigma$ denotes the outward unit normal vector to $\Sigma$. For an alternative construction of $\boldsymbol \rho$ see Alonso Rodríguez et al. [@ABGV13].
The eddy current problem formulated in terms of the magnetic field ${\boldsymbol{h}}$ and the scalar magnetic potential $\psi$ reads as follows: $$\label{model0}
\begin{array}{rcll}
\imath \omega \mu {\boldsymbol{h}}+ {\mathbf{curl}}\, {\boldsymbol{e}}&=& \boldsymbol 0 &\text{in $D$}\\[2ex]
{\boldsymbol{e}}&=& \sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}}) &\text{in $\Omega_C$}\\[2ex]
{\boldsymbol{h}}&=& \nabla \psi + k \boldsymbol \rho &\text{in $\Omega_I$}\\[2ex]
\psi &=& 0 & \text{on } \Sigma \, ,
\end{array}$$ where ${\boldsymbol{j}}$ is the applied current density, $\mu$ is the magnetic permeability and $\sigma$ is the electric conductivity. In what follows, we assume that $\mu$ and $\sigma$ are positive piecewise constant functions in $\Omega_{C}$ and that $\mu_{|\Omega_I}= \mu_0$ is the permeability constant of vacuum. It follows from the first equation that $$\label{lap}
0={\mathrm{div}}({\boldsymbol{h}}_{|\Omega_I})= {\mathrm{div}}(\nabla \psi + k \boldsymbol \rho)\, \hbox{ in } \Omega_I\, .$$ We point out here that the electric field ${\boldsymbol{e}}$ is not uniquely determined in $\Omega_{I}$. Nevertheless, the tangential components of the magnetic field and the tangential components of any admissible representation of the electric field should be continuous across the interface $\Gamma$, i.e., $$\label{magt}
{\boldsymbol{h}}|_{\Omega_{C}} \times {\boldsymbol{n}}_\Gamma = (\nabla \psi + k \boldsymbol \rho) \times {\boldsymbol{n}}_\Gamma \, .$$ and $$\label{mage}
{\boldsymbol{e}}_{|\Omega_C} \times {\boldsymbol{n}}_\Gamma = {\boldsymbol{e}}_{|\Omega_I} \times {\boldsymbol{n}}_\Gamma.$$ The electric field ${\boldsymbol{e}}$ is considered here as an auxiliary variable, it will be removed from the formulation. Hence, we should deduce from a transmission condition relating ${\boldsymbol{h}}$ and $\varphi$ on $\Gamma$. Applying the surface divergence operator ${\mathrm{div}}_\Gamma$ to both side of and recalling that ${\mathrm{div}}_\Gamma({\boldsymbol{e}}\times {\boldsymbol{n}}_\Gamma)= {\mathbf{curl}}\,{\boldsymbol{e}}\cdot {\boldsymbol{n}}_\Gamma$ we deduce that the field ${\mathbf{curl}}\,{\boldsymbol{e}}$ admits continuous normal components across $\Gamma$. As a consequence of the first equation of , $\mu{\boldsymbol{h}}$ should also have continuous normal components across $\Gamma$, i.e., $$\label{transe}
\mu {\boldsymbol{h}}\cdot {\boldsymbol{n}}_\Gamma = \mu_{0}(\nabla \psi + k \boldsymbol \rho) \cdot {\boldsymbol{n}}_\Gamma \,.$$ Finally, we deduce from and the property ${\mathbf{curl}}\, \boldsymbol \rho = {\bf 0}$ that $$\int_\Gamma {\boldsymbol{e}}_{|\Omega_C} \times {\boldsymbol{n}}_\Gamma \cdot \boldsymbol \rho =\int_\Gamma {\boldsymbol{e}}_{|\Omega_I} \times {\boldsymbol{n}}_\Gamma \cdot \boldsymbol \rho = \int_{\Omega_I} {\mathbf{curl}}\, {\boldsymbol{e}}\cdot \boldsymbol \rho,$$ thus $$\label{scal}
\int_\Gamma \sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}}) \cdot (\boldsymbol \rho \times {\boldsymbol{n}}_\Gamma)= \imath \, \omega \int_{\Omega_I} \mu_0 (\nabla \psi + k \boldsymbol \rho) \cdot \boldsymbol \rho \, .$$
From now on, for the sake of simplicity in notations, ${\boldsymbol{h}}$ will stand for ${\boldsymbol{h}}|_{\Omega_{C}}$. Taking into account , , and , we deduce that the eddy current problem can be formulated in terms of the magnetic field and its scalar potential representation in the insulator in the following form: Find ${\boldsymbol{h}}:\, \Omega_{C}\to \mathbb C^{3}$, $\psi:\, \Omega_{I}\to \mathbb C$ and $k\in \mathbb C$ such that, $$\begin{aligned}
\imath \omega \mu {\boldsymbol{h}}+ {\mathbf{curl}}\, [\sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}})] &= \boldsymbol 0 &\text{in $\Omega_C$}\label{ModelProblem1}\\[2ex]
{\boldsymbol{h}}\times {\boldsymbol{n}}_\Gamma & = (\nabla \psi + k \boldsymbol \rho) \times {\boldsymbol{n}}_\Gamma &\text{on $\Gamma$}\label{ModelProblem2}\\[2ex]
\mu\, {\boldsymbol{h}}\cdot {\boldsymbol{n}}_\Gamma & = \mu_0(\nabla \psi + k \boldsymbol \rho) \cdot {\boldsymbol{n}}_\Gamma &\text{on $\Gamma$}\label{ModelProblem3}\\[2ex]
\int_\Gamma \sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}}) \cdot (\boldsymbol \rho \times {\boldsymbol{n}}_\Gamma)&=\imath \, \omega \mu_0 \int_{\Omega_I} (\nabla \psi + k \boldsymbol \rho) \cdot \boldsymbol \rho\label{ModelProblem4}\\[2ex]
{\mathrm{div}}(\nabla \psi + k \boldsymbol \rho) & = 0 &\text{in $\Omega_I$}\label{ModelProblem5}\\[2ex]
\psi &= 0 &\text{on } \Sigma\label{ModelProblem6} \, .
$$
We refer to [@AVbook Section 5] for a proof of the well-posedness of problem -.
The discrete problem {#section3}
====================
Notations
---------
Given a real number $r\geq 0$ and a polyhedron $\mathcal O\subset \mathbb R^d$, $(d=2,3)$, we denote the norms and seminorms of the usual Sobolev space ${\mathrm{H}}^r(\mathcal O)$ by $\|\cdot \|_{r,\mathcal O}$ and $|\cdot|_{r,\mathcal O}$ respectively (cf. [@McLean]). We use the convention ${\mathrm{L}}^2(\mathcal O):= {\mathrm{H}}^0(\mathcal O)$ and ${\bf L}^2(\mathcal O):= [{\mathrm{L}}^2(\mathcal O)]^3$. We recall that, for any $t \in [-1,\: 1 ]$, the spaces ${\mathrm{H}}^{t}(\partial \mathcal O)$ have an intrinsic definition (by localization) on the Lipschitz surface $\partial \mathcal O$ due to their invariance under Lipschitz coordinate transformations. Moreover, for all $0< t\leq 1$, ${\mathrm{H}}^{-t}(\partial\mathcal O)$ is the dual of ${\mathrm{H}}^{t}(\partial\mathcal O)$ with respect to the pivot space ${\mathrm{L}}^2(\partial \mathcal{O})$. Finally we consider $\mathbf{H}({\mathbf{curl}}, \mathcal O):=\{ {\boldsymbol{v}}\in {\mathrm{L}}^2(\mathcal O)^3 \, : \, {\mathbf{curl}}\, {\boldsymbol{v}}\in {\mathrm{L}}^2(\mathcal O)^3\}$ and endow it with its usual Hilbertian norm ${\lVert{\boldsymbol{v}}\rVert}_{\mathbf{H}({\mathbf{curl}}, \mathcal O)}^2:=
{\lVert{\boldsymbol{v}}\rVert}_{0, \mathcal O}^2 + {\lVert{\mathbf{curl}}\, {\boldsymbol{v}}\rVert}_{0, \mathcal O}^2$.
We consider a sequence $\{{\mathcal{T}}_h\}_h$ of conforming and shape-regular triangulations of $\overline \Omega_C \cup \overline \Omega_I$. We assume that each partition ${\mathcal{T}}_h$ consists of tetrahedra $K$ of diameter $h_K$ and unit outward normal to $\partial K$ denoted ${\boldsymbol{n}}_K$. We also assume that for all $K\in {\mathcal{T}}_h$ we have either $K\subset \overline\Omega_C$ or $K\subset \overline \Omega_I$ and denote $${\mathcal{T}}_h^{\Omega_C}:= {\left\{K\in {\mathcal{T}}_h;\quad K\subset \overline\Omega_C\right\}},\qquad {\mathcal{T}}_h^{\Omega_I}:= {\left\{K\in {\mathcal{T}}_h;\quad K\subset \overline\Omega_I\right\}}.$$ We also assume that the meshes $\{{\mathcal{T}}_h^{\Omega_C}\}_h$ are aligned with the discontinuities of the coefficients $\sigma$ and $\mu$. The parameter $h:= \max_{K\in {\mathcal{T}}_h} \{h_K\}$ represents the mesh size.
We denote by ${\mathcal{F}}_h^0(\Omega_C)$ and ${\mathcal{F}}_h^0(\Omega_I)$ the sets of interior faces of the triangulations ${\mathcal{T}}_h^{\Omega_C}$ and ${\mathcal{T}}_h^{\Omega_I}$ respectively. We also introduce the sets of boundary faces $${\mathcal{F}}_h^\Gamma:= {\left\{F = \overline K\cap\overline{K'};\quad K\in {\mathcal{T}}_h^{\Omega_C},\,\, K'\in {\mathcal{T}}_h^{\Omega_I}\right\}}
\quad
\text{and}
\quad
{\mathcal{F}}_h^\Sigma:= {\left\{F = \partial K \cap \Sigma;\quad K\in {\mathcal{T}}_h^{\Omega_I}\right\}}$$ and consider $${\mathcal{F}}_h^{\Omega_C}:={\mathcal{F}}_h^0(\Omega_C) \cup {\mathcal{F}}_h^\Gamma, \quad {\mathcal{F}}_h^{\Omega_I}:={\mathcal{F}}_h^0(\Omega_I) \cup {\mathcal{F}}_h^\Sigma
\quad \text{and} \quad {\mathcal{F}}_h := {\mathcal{F}}_h^{\Omega_C}\cup {\mathcal{F}}_h^{\Omega_I}.$$ We notice that ${\left\{{\mathcal{F}}_h^\Gamma\right\}}_h$ is a shape regular family of triangulations of $\Gamma$ into triangles $T$ of diameter $h_T$. Finally, we consider the set ${\mathcal{E}}_h$ of edges $e = \overline T\cap \overline{T'}$ (where $T$ and $T'$ are two adjacent triangles from ${\mathcal{F}}_h^\Gamma$).
Let $\mathcal{O}_h$ be anyone of the previously introduced partitions of $\overline\Omega_C\cup \overline \Omega_I$, $\overline\Omega_C$, $\overline \Omega_I$ or $\Gamma$ and let $E$ be a generic element of the given partition. We introduce for any $s\geq 0$ the broken Sobolev spaces $${\mathrm{H}}^s(\mathcal{O}_h) := \prod_{E\in \mathcal{O}_h} {\mathrm{H}}^s(E)\quad \text{and} \quad \mathbf{H}^s(\mathcal{O}_h) := \prod_{E\in \mathcal{O}_h} \mathbf{H}^s(E)^3\, .$$
For each $w:= {\left\{w_E\right\}}\in {\mathrm{H}}^s(\mathcal{O}_h)$, the components $w_E$ represents the restriction $w|_E$. When no confusion arises, the restrictions will be written without any subscript.
The space ${\mathrm{H}}^s(\mathcal{O}_h)$ is endowed with the Hilbertian norm $${\lVertw\rVert}_{s,\mathcal{O}_h}^2 := \sum_{E\in \mathcal{O}_h} {\lVertw_E\rVert}^2_{s,E}.$$
We consider identical definitions for the norm and the seminorm on the vectorial version $\mathbf{H}^s(\mathcal{O}_h)$. We use the standard conventions ${\mathrm{L}}^2(\mathcal{O}_h):={\mathrm{H}}^0(\mathcal{O}_h)$ and $\mathbf{L}^2(\mathcal{O}_h):=\mathbf{H}^0(\mathcal{O}_h)$ and introduce the bilinear forms $$(w, z)_{\mathcal{O}_h} = \sum_{E\in \mathcal{O}_h} \int_E w_E z_E, \quad \forall w, z \in {\mathrm{L}}^2(\mathcal{O}_h)$$ and $$(\boldsymbol{w}, \boldsymbol{z})_{\mathcal{O}_h} = \sum_{E\in \mathcal{O}_h} \int_E \boldsymbol{w}_E\cdot \boldsymbol{z}_E, \quad \forall \boldsymbol{w}, \boldsymbol{z}\in \mathbf{L}^2(\mathcal{O}_h).$$
Assume that $({\boldsymbol{v}},\varphi,m)\in \mathbf{H}^{1+s}({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})\times {\mathbb C}$, with $s>1/2$. Moreover, let us recall that $\boldsymbol \rho$ has been constructed as a piecewise-linear vector field, therefore its restriction to any face $F$ has a meaning. We define ${\mathbf{curl}}_h{\boldsymbol{v}}\in \mathbf{H}^s({\mathcal{T}}_h^{\Omega_C})$ by $({\mathbf{curl}}_h {\boldsymbol{v}})|_K = {\mathbf{curl}}\, {\boldsymbol{v}}_K$, for all $K\in {\mathcal{T}}_h^{\Omega_C}$; $\nabla_h \varphi \in \mathbf{H}^s({\mathcal{T}}_h^{\Omega_I})$ by $(\nabla_h \varphi)|_K = \nabla \varphi_K$, for all $K\in {\mathcal{T}}_h^{\Omega_I}$. We define also the averages ${\{{\boldsymbol{v}}\}}_{{\mathcal{F}}}\in \mathbf{L}^2({\mathcal{F}}_h^{\Omega_C})$ and ${\{\nabla_h \varphi+m\boldsymbol \rho\}}_{{\mathcal{F}}}\in \mathbf{L}^2({\mathcal{F}}_h^{\Omega_I})$ by $$\label{average1}
\begin{array}{l}
{\{{\boldsymbol{v}}\}}_{{\mathcal{F}}}|_F := {\{{\boldsymbol{v}}\}}_F \hbox{ with}\\ \\
{\{{\boldsymbol{v}}\}}_F:= \left\{
\begin{array}{ll}
({\boldsymbol{v}}_K + {\boldsymbol{v}}_{K'})/2 & \text{if $F=K\cap K'\in {\mathcal{F}}_h^0(\Omega_C)$}\\[.1cm]
{\boldsymbol{v}}_K & \text{if $F\subset \partial K$ and $F \in {\mathcal{F}}_h^\Gamma$},
\end{array} \right.
\end{array}$$ and $$\label{average2}
\begin{array}{l}
{\{\nabla_h\varphi+m \boldsymbol \rho\}}_{{\mathcal{F}}}|_F := {\{\nabla_h \varphi + m \boldsymbol \rho\}}_F \hbox{ with} \\ \\
{\{\nabla_h \varphi + m \boldsymbol \rho\}}_F :=\left\{ \begin{array}{l}
(\nabla \varphi_K + \nabla \varphi_{K'})/2 + m (\boldsymbol \rho_K+\boldsymbol \rho_{K'})/2 \\ \hspace{3cm} \text{if $F=K\cap K'\in {\mathcal{F}}_h^0(\Omega_I)$}\\[.1cm]
\nabla \varphi_K + m \boldsymbol \rho_K \\
\hspace{3cm} \text{if $F\subset \partial K$ and $F \in {\mathcal{F}}_h^\Sigma$}\, ,
\end{array} \right.
\end{array}$$ and the jumps ${\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_{{\mathcal{F}}}\in \mathbf{L}^2({\mathcal{F}}_h^{\Omega_C})$ and ${\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\in \mathbf{L}^2({\mathcal{F}}_h^{\Omega_I})$ by $$\label{jump1}
\begin{array}{l}
{\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_{{\mathcal{F}}}|_F :={\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_F \hbox{ with}
\\ \\
{\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_F:=\left\{ \hspace{-.2cm} \begin{array}{l}
{\llbracket {\boldsymbol{v}}\times {\boldsymbol{n}}\rrbracket}_F:={\boldsymbol{v}}_K \times {\boldsymbol{n}}_K + {\boldsymbol{v}}_{K'}\times {\boldsymbol{n}}_{K'} \\ \hspace{2cm} \text{if $F=K\cap K'\in {\mathcal{F}}_h^0(\Omega_C)$}\\[.1cm]
{\boldsymbol{v}}_K \times {\boldsymbol{n}}+( \nabla \varphi_{K'} + m \boldsymbol \rho_{K'}) \times {\boldsymbol{n}}_{K'}\\ \hspace{2cm} \text{if }F= K\cap K'\in {\mathcal{F}}_h^\Gamma \text{ with } K\in {\mathcal{T}}_h^{\Omega_C}, \,K'\in {\mathcal{T}}_h^{\Omega_I}\, ,
\end{array} \right.
\end{array}$$ and $$\label{jump2}
\begin{array}{l}
{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}|_F := {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_F \hbox{ with} \\ \\
{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_F := \left\{
\begin{array}{ll}
\varphi_K {\boldsymbol{n}}_K + \varphi_{K'}{\boldsymbol{n}}_{K'} & \text{if $F=K\cap K'\in {\mathcal{F}}_h^0(\Omega_I)$}\\[.1cm]
\varphi_K {\boldsymbol{n}}_\Sigma & \text{if $F\subset \partial K$ and $F \in {\mathcal{F}}_h^\Sigma$}\, .
\end{array} \right.
\end{array}$$ Similarly, we define the edge averages ${\{{\boldsymbol{v}}\}}_{{\mathcal{E}}}\in \mathbf{L}^2({\mathcal{E}}_h)$ by $${\{{\boldsymbol{v}}\}}_{{\mathcal{E}}}|_e :={\{{\boldsymbol{v}}\}}_e \hbox{ with } {\{{\boldsymbol{v}}\}}_e:=({\boldsymbol{v}}_{K_e} + {\boldsymbol{v}}_{K'_e})/2$$ where $K_e, K'_e\in {\mathcal{T}}_h^{\Omega_C}$ are such that $T=\partial K_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$, $T'=\partial K'_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$ and $e = T\cap T'$. We also need to define the edge jumps ${\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\in \mathbf{L}^2({\mathcal{E}}_h)$ by $${\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}|_e := {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_e \hbox{ with }
{\llbracket \varphi{\boldsymbol{t}}\rrbracket}_e :=
\varphi_{K_e} {\boldsymbol{t}}_e + \varphi_{K'_e} {\boldsymbol{t}}'_e\, ,$$ where $K_e, K_e'$ are in this case the elements from ${\mathcal{T}}_h^{\Omega_I}$ such that $T=\partial K_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$, $T'=\partial K'_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$ and $e = T\cap T'$. Here, ${\boldsymbol{t}}_e$, ${\boldsymbol{t}}'_e$ are the tangent unit vectors along the edge $e$ given by ${\boldsymbol{t}}_e = ({\boldsymbol{n}}_\Gamma \times \boldsymbol \nu_{T})|_e$ and ${\boldsymbol{t}}_e = ({\boldsymbol{n}}_\Gamma \times \boldsymbol \nu_{T'})|_e$ where $\boldsymbol \nu_{T}$ and $\boldsymbol \nu_{T'}$ are the outward unit normal vector to $\partial T$ and $\partial T'$ respectively that lies on the tangent plane to $\Gamma$.
The DG formulation
------------------
Hereafter, given an integer $k\geq 0$ and a domain $\mathcal O\subset \mathbb{R}^3$, ${\mathcal{P}}_k(\mathcal O)$ denotes the space of polynomials of degree at most $k$ on $\mathcal O$. For any $m\geq 1$, we introduce the finite element spaces $$\mathbf{X}_h := \prod_{K\in {\mathcal{T}}_h^{\Omega_C}} P_m(K)^3
\quad \text{and} \quad
V_h := \prod_{K\in {\mathcal{T}}_h^{\Omega_I}} \tilde{\mathcal{P}}_{m}(K),$$ where $$\label{tildePm}
\tilde{\mathcal{P}}_{m}(K):=\begin{cases}
{\mathcal{P}}_m(K) & \text{if $\partial K\cap \Gamma \notin {\mathcal{F}}_h^\Gamma$ }\\
{\mathcal{P}}_m(K) + {\mathcal{P}}_{m+1}^T(K) & \text{if $T=\partial K\cap \Gamma \in {\mathcal{F}}_h^\Gamma$}
\end{cases}$$ with ${\mathcal{P}}_{m+1}^T(K)$ representing the subspace of ${\mathcal{P}}_{m+1}(K)$ spanned by the elements of the Lagrange basis corresponding to nodal points located on $T$. It follows that ${\mathcal{P}}_m(K) \subset \tilde{\mathcal{P}}_m(K) \subset
{\mathcal{P}}_{m+1}(K)$ and if $T = \partial K \cap \Gamma \in {\mathcal{F}}_h^\Gamma$ then $\tilde{\mathcal{P}}_m(K)|_T = {\mathcal{P}}_{m+1}(T)$.
Let $h_{\mathcal{F}}\in \prod_{F\in \mathcal{F}_h} {\mathcal{P}}_0(F)$ and $h_{\mathcal{E}}\in \prod_{e\in \mathcal{E}_h} {\mathcal{P}}_0(e)$ be defined by $h_{\mathcal{F}}|_F := h_F$ $, \forall F \in \mathcal{F}_h$ and $h_{\mathcal{E}}|_e := h_e$ $, \forall e \in {\mathcal{E}}_h$ respectively. By virtue of our hypotheses on $\sigma$ and on the triangulation ${\mathcal{T}}_h^{\Omega_C}$, we may consider that $\sigma$ is an element of $\prod_{K\in \mathcal{T}_h^{\Omega_C}} {\mathcal{P}}_0(K)$ and denote $\sigma_K:= \sigma|_K$ for all $K\in \mathcal{T}_h^{\Omega_C}$. We introduce $\mathtt{s}_{\mathcal{F}}\in \prod_{F\in \mathcal{F}_h(\Omega_C)} {\mathcal{P}}_0(F)$ defined by $\mathtt{s}_F := \min(\sigma_K, \sigma_{K'})$, if $F = \partial K \cap \partial K'\in {\mathcal{F}}_h^0(\Omega_C)$ and $\mathtt{s}_F := \sigma_K$, if $F = \partial K \cap \Gamma\in {\mathcal{F}}_h^\Gamma$. We also need to define $\mathtt{s}_{\mathcal{E}}\in \prod_{e\in \mathcal{E}_h} {\mathcal{P}}_0(e)$ given by $\mathtt{s}_e = \min(\sigma_{K_e}, \sigma_{K'_e})$ where $K_e, K_e\in {\mathcal{T}}_h^{\Omega_C}$ are such that $T=\partial K_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$, $T'=\partial K'_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$ and $e = T\cap T'$.
We consider, for $s>1/2$, the Hilbert space $$\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C}) := \left\{{\boldsymbol{v}}\in \mathbf{H}^s({\mathcal{T}}_h^{\Omega_C});\quad {\mathbf{curl}}_h {\boldsymbol{v}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})\right\}$$ and define on $\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C}) \times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I}) \times {\mathbb C}$ the sesquilinear forms $$\begin{aligned}
A_h^{\Omega_C}(({\boldsymbol{u}}, \phi,c), &({\boldsymbol{v}}, \varphi,m)) := \imath \omega \left(\mu {\boldsymbol{u}}, {\boldsymbol{v}}\right)_{{\mathcal{T}}_h^{\Omega_C}} + \left(\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{u}}, {\mathbf{curl}}_h {\boldsymbol{v}}\right)_{{\mathcal{T}}_h^{\Omega_C}} \\ &+ \left({\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{u}}\}}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_C}}
+ \left({\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{u}}, \phi, c) \rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_C}}\\ &+ \mathtt{a}^{\Omega_C} \left( \mathtt{s}^{-1}_{{\mathcal{F}}} h_{{\mathcal{F}}}^{-1} {\llbracket ({\boldsymbol{u}}, \phi, c) \rrbracket}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_C}}\, ,\end{aligned}$$ $$\begin{aligned}
A_h^{\Omega_I}(&({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi, m)):= \imath \omega \mu_0 (\nabla_h \phi+c \boldsymbol \rho, \nabla_h \varphi+m \boldsymbol \rho)_{{\mathcal{T}}_h^{\Omega_I}} + \dfrac{\mathtt{a}^{\Omega_I}}{\omega \mu_0}\left(h_{{\mathcal{F}}}^{-1} {\llbracket \phi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}, {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}}\\
&- \imath \omega \mu_0\left({\{\nabla_h \phi+c \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}}
- \imath \omega \mu_0\left({\{\nabla_h \varphi+m \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \phi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}}
\\
&- \left({\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{u}}\}}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h} - \left({\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}, {\llbracket \phi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h}
+ \alpha \left( \mathtt{s}^{-1}_{{\mathcal{E}}} h_{{\mathcal{E}}}^{-2} {\llbracket \phi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}} \right)_{{\mathcal{E}}_h},\end{aligned}$$ and let $$A_h(({\boldsymbol{u}}, p,c), ({\boldsymbol{v}}, \varphi,m)):= A_h^{\Omega_C}(({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi, m)) + A_h^{\Omega_I}(({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi, m))\, .$$
Let us assume that $\sigma^{-1} {\boldsymbol{j}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})$ with $s>1/2$. Then we can define the linear form $L_h(\cdot)$ on $\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C}) \times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I}) \times {\mathbb C}$ by $$L_h(({\boldsymbol{v}}, \varphi)) := (\sigma^{-1} {\boldsymbol{j}}, {\mathbf{curl}}_h {\boldsymbol{v}})_{{\mathcal{T}}_h^{\Omega_C}} + \left( {\{\sigma^{-1} {\boldsymbol{j}}\}}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}} \right)_{{\mathcal{F}}_h^{\Omega_C}}
- \left({\{\sigma^{-1} {\boldsymbol{j}}\}}_{{\mathcal{E}}}, {\llbracket \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h}.$$
We propose the following DG formulation of problem -: $$\label{DG-FEM}
\begin{array}{l}
\text{Find $({\boldsymbol{h}}_{h}, \psi_h,k_h)\in \mathbf{X}_h\times V_h\times {\mathbb C}$ such that,}\\[2ex]
A_h(({\boldsymbol{h}}_{h}, \psi_h,k_h), ({\boldsymbol{v}}, \varphi, m)) = L_h(({\boldsymbol{v}}, \varphi, m))\quad \forall \, ({\boldsymbol{v}}, \varphi, m)\in \mathbf{X}_h\times V_h\times {\mathbb C}\, .
\end{array}$$
The existence and uniqueness of the solution of this problem is proved in Theorem \[LM\]
We end this section by showing that the DG scheme is consistent.
\[consistency0\] Let $({\boldsymbol{h}}, \psi, k)\in \mathbf{H}({\mathbf{curl}}, {\Omega_C})\times {\mathrm{H}}^1(\Omega_I)\times {\mathbb C}$ be the solution of -. Under the assumption $\sigma^{-1} {\boldsymbol{j}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})$ and the regularity conditions $({\boldsymbol{h}},\psi, k) \in \mathbf{X}^s({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})\times {\mathbb C}$, with $s>1/2$, we have that $$A_h(({\boldsymbol{h}}, \psi, k), ({\boldsymbol{v}}, \varphi, m)) = L_h(({\boldsymbol{v}}, \varphi, m)) \quad
\forall \, ({\boldsymbol{v}},\varphi, m)\in \mathbf{X}_h\times V_h \times {\mathbb C}.$$
Using again the notation ${\boldsymbol{e}}= \sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}})$ and taking into account that ${\llbracket ({\boldsymbol{h}},\psi,k) \rrbracket}_{{\mathcal{F}}} =0$, ${\llbracket \psi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}=0$, and ${\llbracket \psi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}=0$, it is straightforward to show that $$\begin{gathered}
\label{diff}
A_h(({\boldsymbol{h}}, \psi, k), ({\boldsymbol{v}}, \varphi, m)) - L_h(({\boldsymbol{v}}, \varphi, m)) = \imath \omega \int_{\Omega_{C}} \mu {\boldsymbol{h}}\cdot {\boldsymbol{v}}+ \int_{\Omega_{C}} {\boldsymbol{e}}\cdot {\mathbf{curl}}_h {\boldsymbol{v}}\\
+ \imath \omega \mu_0 \int_{\Omega_I}(\nabla \psi + k\boldsymbol \rho)\cdot (\nabla_h \varphi + m \boldsymbol \rho)
+ \left({\{{\boldsymbol{e}}\}}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_{C}}}\\ - \imath \omega \mu_0\left({\{\nabla \psi + k \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}}
- \left({\{{\boldsymbol{e}}\}}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h}.\end{gathered}$$ Integrating by parts in each $K\in {\mathcal{T}}_h^{\Omega_{C}}$ and using yield $$\begin{gathered}
\label{GreenOmegaC}
\int_{\Omega_{C}} {\boldsymbol{e}}\cdot {\mathbf{curl}}_h {\boldsymbol{v}}= \sum_{K\in {\mathcal{T}}_h^{\Omega_{C}}}
\int_K {\mathbf{curl}}\, {\boldsymbol{e}}\cdot {\boldsymbol{v}}-
\sum_{K\in {\mathcal{T}}_h^{\Omega_{C}}} \int_{\partial K} {\boldsymbol{e}}\cdot {\boldsymbol{v}}\times {\boldsymbol{n}}_K \\
= -\imath\omega \int_{\Omega_{C}} \mu{\boldsymbol{h}}\cdot {\boldsymbol{v}}- \sum_{F\in {\mathcal{F}}^0_h(\Omega_{C})}
\int_{F} {\{{\boldsymbol{e}}\}}_F \cdot {\llbracket {\boldsymbol{v}}\times {\boldsymbol{n}}\rrbracket}_F -
\sum_{T\in {\mathcal{F}}_h^\Gamma}
\int_{T} {\boldsymbol{e}}\cdot {\boldsymbol{v}}\times {\boldsymbol{n}}.\end{gathered}$$ Similarly, integrating by parts in each $K\in {\mathcal{T}}_h^{\Omega_I}$ together with and give $$\begin{gathered}
\label{GreenOmegaI}
\imath \omega \mu_{0}\int_{\Omega_I}(\nabla \psi + k\boldsymbol \rho)\cdot (\nabla_h \varphi + m \boldsymbol \rho) = - \imath \omega \mu_{0}\sum_{K\in {\mathcal{T}}_h^{\Omega_I}} \int_K {\mathrm{div}}(\nabla \psi + k \boldsymbol \rho) \varphi \\+\imath \omega \mu_{0}
\sum_{K\in {\mathcal{T}}_h^{\Omega_I}} \int_{\partial K} (\nabla \psi + k\boldsymbol \rho) \cdot {\boldsymbol{n}}_K \varphi
+ m \int_{\Omega_{I}} (\nabla \psi + k\boldsymbol \rho)\cdot \boldsymbol \rho
= \imath \omega \mu_{0} \sum_{F\in {\mathcal{F}}_{h}^0} \int_{F} {\{\nabla \psi + k \boldsymbol \rho\}}_{F}
\cdot {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{F}\\
- \imath \omega \mu_{0} \sum_{T\in {\mathcal{F}}^{\Gamma}_{h}}\int_{F} (\nabla \psi + k \boldsymbol \rho)
\cdot \varphi{\boldsymbol{n}}_{\Gamma} + \imath \omega \mu_{0} \sum_{T\in {\mathcal{F}}^{\Sigma}_{h}}\int_{F} (\nabla \psi + k \boldsymbol \rho) \cdot \varphi{\boldsymbol{n}}_{\Sigma} + m \int_{\Gamma} {\boldsymbol{e}}\cdot (\boldsymbol \rho \times {\boldsymbol{n}}_{\Gamma}).\end{gathered}$$ Substituting back and in we obtain $$\begin{gathered}
\label{diff1}
A_h(({\boldsymbol{h}}, \psi, k), ({\boldsymbol{v}}, \varphi, m)) - L_h(({\boldsymbol{v}}, \varphi, m)) = -
\sum_{T\in {\mathcal{F}}_h^\Gamma} \int_T {\boldsymbol{e}}\cdot {\mathbf{curl}}_T \varphi\\
- \imath \omega \mu_0 \sum_{T\in {\mathcal{F}}_h^\Gamma} \int_T \nabla (\psi+ k \boldsymbol \rho) \cdot \varphi {\boldsymbol{n}}_{\Gamma} - \left({\{{\boldsymbol{e}}\}}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h}.\end{gathered}$$ Finally, using the integration by parts formula $$\sum_{T\in {\mathcal{F}}_h^\Gamma}
\int_{T} {\boldsymbol{e}}\cdot {\mathbf{curl}}_T \varphi =\sum_{T\in {\mathcal{F}}_h^\Gamma}
\int_{T} (\text{curl}_T {\boldsymbol{e}}) \varphi - \sum_{T\in {\mathcal{F}}_h^\Gamma} \int_{\partial T} {\boldsymbol{e}}\cdot \varphi{\boldsymbol{t}}_{\partial T}
=
\int_{\Gamma} (\text{curl}_\Gamma {\boldsymbol{e}}) \varphi -
\left({\{{\boldsymbol{e}}\}}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h},$$ we deduce from that $$\begin{gathered}
\label{diff1+}
A_h(({\boldsymbol{h}}, \psi, k), ({\boldsymbol{v}}, \varphi, m)) - L_h(({\boldsymbol{v}}, \varphi, m)) = -
\int_{\Gamma} (\text{curl}_\Gamma {\boldsymbol{e}}) \varphi\\
- \imath \omega \mu_0 \sum_{T\in {\mathcal{F}}_h^\Gamma} \int_T \nabla (\psi+ k \boldsymbol \rho) \cdot \varphi {\boldsymbol{n}}_{\Gamma}.\end{gathered}$$ and the result follows from the identity $\text{curl}_\Gamma {\boldsymbol{e}}= {\mathbf{curl}}{\boldsymbol{e}}\cdot {\boldsymbol{n}}$, equation and the transmission condition .
Convergence analysis of the DG-FEM formulation {#section4}
==============================================
The aim of this Section is to prove that the DG-FEM formulation is stable in the DG-norm defined on $\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})\times {\mathbb C}$ by $$\begin{aligned}
{\lVert({\boldsymbol{v}}, \varphi,m)\rVert}^2 := & {\lVert(\omega\mu)^{1/2}{\boldsymbol{v}}\rVert}^2_{0,\Omega_C} + {\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{v}}\rVert}^2_{0,\Omega_C} + \omega\mu_0 {\lVert\nabla_h \varphi+ m \boldsymbol \rho\rVert}^2_{0,\Omega_I} \\
+ & {\lVert\mathtt{s}^{-1/2}_{{\mathcal{F}}}h_{{\mathcal{F}}}^{-1/2}{\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_C}} + \omega\mu_0 {\lVerth_{{\mathcal{F}}}^{-1/2}{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}}\\
+ &{\lVert\mathtt{s}^{-1/2}_{{\mathcal{E}}} h_{{\mathcal{E}}}^{-1}{\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}^2_{0,{\mathcal{E}}_h}.\end{aligned}$$ We also need to introduce $$\begin{gathered}
{\lVert({\boldsymbol{v}}, \varphi,m)\rVert}_{\ast}^2 := {\lVert({\boldsymbol{v}}, \varphi,m)\rVert}^2 +
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}{\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_C}}\\
+ {\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}\rVert}^2_{0,{\mathcal{E}}_h}
+{\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla_h \varphi + m \boldsymbol \rho\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}}.\end{gathered}$$
The following discrete trace inequality is standard, (see, e.g. [@DiPietroErn Lemma 1.46]).
For all integer $k\geq 0$ there exists a constant $C^*>0$ independent of $h$ such that, $$\label{discreteTrace3D}
h_Q {\lVert v\rVert}^2_{0,\partial Q} \leq C^* {\lVert v\rVert}^2_{0,Q} \quad \forall \, v\in {\mathcal{P}}_k(Q),\quad
\forall Q\in \{{\mathcal{T}}_h,{\mathcal{F}}_h^\Gamma\}.$$
It is used to prove the following auxiliary result.
\[equivalence\] For all $k\geq 0$, there exist constants $C_{\Omega_C}>0$ and $C_{\Omega_I}>0$ independent of the mesh size and the coefficients such that $$\label{discIneq1}
{\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} \mathbf{w}\}}_{{\mathcal{E}}}\rVert} _{0,{\mathcal{E}}_h} +
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}\mathbf{w}\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} \leq C_{\Omega_C} {\lVert\sigma^{-1/2} \mathbf{w}\rVert}_{0,{\Omega_C}}\, ,$$ for all $\mathbf{w} \in \prod_{K\in {\mathcal{T}}_h^{\Omega_C}}{\mathcal{P}}_k(K)^3$, and $$\label{discIneq2}
{\lVerth_{{\mathcal{F}}}^{1/2} {\{\mathbf{w}\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} \leq C_{\Omega_I} {\lVert\mathbf{w}\rVert}_{0,\Omega_I} \, ,$$ for all $\mathbf{w} \in \prod_{K\in {\mathcal{T}}_h^{\Omega_I}}{\mathcal{P}}_k(K)^3$.
By definition of $\mathtt{s}_{{\mathcal{F}}}$, for any $\mathbf{w} \in \prod_{K\in {\mathcal{T}}_h^{\Omega_C}}{\mathcal{P}}_k(K)^3$, $$\begin{gathered}
\label{transfer0}
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}\mathbf{w}\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_C}} =
\sum_{F\in {\mathcal{F}}_h^{\Omega_C}} h_F {\lVert \mathtt{s}_F^{1/2}{\{\sigma^{-1}\mathbf{w}\}}_F \rVert}^2_{0,F}\\
\leq \sum_{K\in {\mathcal{T}}_h^{\Omega_C}} \sum_{F\in {\mathcal{F}}(K)} h_F {\lVert \mathtt{s}_F^{1/2}\sigma_K^{-1}\mathbf{w}_K \rVert}^2_{0,F}
\leq \sum_{K\in {\mathcal{T}}_h^{\Omega_C}} h_K
{\lVert \sigma_K^{-1/2}\mathbf{w}_K \rVert}^2_{0,\partial K}.\end{gathered}$$
Similarly, $$\begin{gathered}
\label{transfer}
{\lVert \mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} \mathbf{w}\}}_{{\mathcal{E}}} \rVert}^2 _{0,{\mathcal{E}}_h} =
\sum_{e \in {\mathcal{E}}_h} h_e^2 {\lVert\mathtt{s}_{e}^{1/2} {\{\sigma^{-1} \mathbf{w}\}}_e \rVert}^2_{0,e} \\
\leq \sum_{T\in {\mathcal{F}}_h^\Gamma} \sum_{e\in {\mathcal{E}}(T)} h_e^2 {\lVert\mathtt{s}_{e}^{1/2} \sigma_{K_T}^{-1}
\mathbf{w}_{K_T}\rVert}^2_{0,e}
\leq \sum_{T\in {\mathcal{F}}_h^\Gamma }
h_{T}^2 {\lVert\sigma_{K_T}^{-1/2} \mathbf{w}_{K_T}\rVert}^2_{0,\partial T}\, ,\end{gathered}$$ where $K_T\in {\mathcal{T}}_h^{\Omega_C}$ is such that $T=\partial K_T \cap \Gamma$. It follows from that $$\begin{gathered}
{\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} \mathbf{w}\}}_{{\mathcal{E}}}\rVert}^2 _{0,{\mathcal{E}}_h^{\Omega_I}} \leq C^*
\sum_{T \in {\mathcal{F}}_h^\Gamma} h_{T} {\lVert\sigma_{K_T}^{-1/2} \mathbf{w}_{K_T}\rVert}^2_{0, T} \leq C^*
\sum_{K \in{\mathcal{T}}^{\Omega_C}_h} h_{K} {\lVert\sigma_K^{-1/2} \mathbf{w}_K\rVert}^2_{0, \partial K} \end{gathered}$$ and follows by applying again the discrete trace inequality in the last estimate and in . Finally, for any $\mathbf{w} \in \prod_{K\in {\mathcal{T}}_h^{\Omega_I}}{\mathcal{P}}_k(K)^3$, $$\label{transfer1}
{\lVerth_{{\mathcal{F}}}^{1/2} {\{\mathbf{w}\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}} = \sum_{F\in {\mathcal{F}}_h^{\Omega_I}} h_F {\lVert{\{\mathbf{w}\}}_F\rVert}_{0,F}^2
\leq \sum_{K\in {\mathcal{T}}_h^{\Omega_I}} h_K {\lVert \mathbf{w}_K \rVert}^2_{0,\partial K}$$ and follows again from .
\[boundedness\] There exists a constant $M>0$ independent of $h$ such that $$| A_h(({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi, m)) | \leq M {\lVert({\boldsymbol{u}}, \phi, c)\rVert}_* {\lVert({\boldsymbol{v}},\varphi, m)\rVert}$$ for all $({\boldsymbol{u}}, \phi,c)$, $({\boldsymbol{v}},\varphi,m)\in \mathbf{X}^s({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})\times {\mathbb C}$, with $s>1/2$.
By the Cauchy-Schwarz inequality, we have that $$\begin{array}{l}
|A_h^{\Omega_C}(({\boldsymbol{u}}, \phi,c), ({\boldsymbol{v}}, \varphi,m))| \\[.1cm]
\qquad \leq \omega {\lVert\mu^{1/2}{\boldsymbol{u}}\rVert}_{0,\Omega_C} {\lVert\mu^{1/2}{\boldsymbol{v}}\rVert}_{0,\Omega_C} +
{\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{u}}\rVert}_{0,\Omega_C} {\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C}\\[.1cm]
\qquad + {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}{\mathbf{curl}}_h {\boldsymbol{u}}\}}_{{\mathcal{F}}}\rVert}_{0, {\mathcal{F}}_h^{\Omega_C}}
{\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ({\boldsymbol{v}}, \varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0, {\mathcal{F}}_h^{\Omega_C}}\\[.1cm]
\qquad + {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}{\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}\rVert}_{0, {\mathcal{F}}_h^{\Omega_C}}
{\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ({\boldsymbol{u}}, \phi,c) \rrbracket}_{{\mathcal{F}}}\rVert}_{0, {\mathcal{F}}_h^{\Omega_C}}\\[.1cm]
\qquad + \mathtt{a}^{\Omega_C} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{F}}}^{-1/2} {\llbracket ({\boldsymbol{u}}, \phi,c) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{F}}}^{-1/2} {\llbracket ({\boldsymbol{v}}, \varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}}.
\end{array}$$ Applying with $\mathbf{w} = {\mathbf{curl}}_h {\boldsymbol{v}}$ we obtain $$|A_h^{\Omega_C}(({\boldsymbol{u}}, \phi,c), ({\boldsymbol{v}}, \varphi, m))| \leq (1+ C_\Omega + \mathtt{a}^{\Omega_C})\, {\lVert({\boldsymbol{u}}, \phi,c)\rVert}_* {\lVert({\boldsymbol{v}},\varphi,m)\rVert}$$ for all $({\boldsymbol{u}}, \phi,c)$ and $({\boldsymbol{v}},\varphi,m)\in \mathbf{X}^s({\mathcal{T}}_h^\Omega)\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I}) \times {\mathbb C}$. On the other hand, $$\begin{array}{l}
|A_h^{\Omega_I}(({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi,m))| \leq \omega \mu_0 {\lVert\nabla_h \phi+c \boldsymbol \rho\rVert}_{0,\Omega_I} {\lVert\nabla_h \varphi + m \boldsymbol \rho\rVert}_{0,\Omega_I} \\[.1cm]
\quad\qquad+ \omega \mu_0 {\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla_h \varphi+m \boldsymbol \rho\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \phi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} \\[.1cm]
\quad\qquad+ \omega \mu_0 {\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla_h \phi+c \boldsymbol \rho\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}}\\[.1cm]
\quad\qquad+ \alpha {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{E}}}^{-1} {\llbracket \phi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{E}}}^{-1} {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\[.1cm]
\quad\qquad+ {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2}h_{{\mathcal{E}}}{\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{E}}}^{-1} {\llbracket \phi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\[.1cm]
\quad\qquad+ {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2}h_{{\mathcal{E}}}{\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{u}}\}}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{E}}}^{-1} {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\[.1cm]
\quad\qquad+\mathtt{a}^{\Omega_I}{\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \phi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h}
\end{array}$$ and it follows from (applied with $\mathbf{w}= \nabla_h \varphi+ m \boldsymbol \rho$ ) and (applied with $\mathbf{w}= {\mathbf{curl}}_h {\boldsymbol{v}}$) that $$|A_h^{\Omega_I}(({\boldsymbol{u}}, \phi,c), ({\boldsymbol{v}}, \varphi, m))| \leq (1 + C_{\Omega_I} + C_{\Omega}+ \mathtt{a}^{\Omega_I}+ \alpha) \, {\lVert({\boldsymbol{u}}, \phi,c)\rVert}_* {\lVert({\boldsymbol{v}},\varphi,m)\rVert},$$ which gives the result.
\[discElip\] There exists a constant $\alpha_0>0$ independent of the mesh size and the coefficients such that if $\min(\mathtt{a}^{\Omega_C}, \mathtt{a}^{\Omega_I}, \alpha)\geq \alpha_0$ then, $$\label{elip}
\text{Re}\left[ (1 - \imath) A_h(({\boldsymbol{v}}, \varphi,m), (\overline {\boldsymbol{v}}, \overline \varphi,\overline m))\right] \geq \dfrac{1}{2} {\lVert({\boldsymbol{v}}, \varphi,m)\rVert}^2 \qquad \forall ({\boldsymbol{v}}, \varphi,m)\in \mathbf{X}_h\times V_h\times {\mathbb C}.$$
By definition of $A_h(\cdot, \cdot)$, $$\label{elip0}
\begin{array}{l}
\text{Re}\left[(1 - \imath) A_h(({\boldsymbol{v}}, \varphi,m), (\overline {\boldsymbol{v}}, \overline \varphi,\overline m))\right] = \omega {\lVert \mu^{1/2} {\boldsymbol{v}}\rVert}_{0,\Omega_C}^2 +
{\lVert\sigma^{-1/2} {\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C}^2\\[.1cm]
\qquad+ 2 \text{Re} \left( {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}, {\llbracket (\overline {\boldsymbol{v}}, \overline \varphi,\overline m) \rrbracket}_{{\mathcal{F}}} \right)_{{\mathcal{F}}_h^{\Omega_C}}
+ \mathtt{a}^{\Omega_C} {\lVerth_{{\mathcal{F}}}^{-1/2}{\llbracket ({\boldsymbol{v}}, \varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}}^2\\[.1cm]
\qquad +\omega \mu_0 {\lVert\nabla_h \varphi+m \boldsymbol \rho\rVert}_{0,\Omega_C}^2 - 2 \omega \mu_0 \text{Re} \left(
{\{\nabla_h \varphi+ m \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \overline \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}} \\[.1cm]
\qquad+ \mathtt{a}^{\Omega_I}{\lVerth_{{\mathcal{F}}}^{-1/2}{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}}
- 2 \text{Re} \left( {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}, {\llbracket \overline \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}} \right)_{{\mathcal{E}}_h}
+ \alpha {\lVerth_{{\mathcal{E}}}^{-1}{\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}^2_{0, {\mathcal{E}}_h}.
\end{array}$$
It follows from the Cauchy-Schwarz inequality and that, $$\label{elip1}
\begin{array}{l}
2 | \text{Re} \left( {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}, {\llbracket (\overline {\boldsymbol{v}}, \overline \varphi,\overline m) \rrbracket}_{{\mathcal{F}}} \right)_{{\mathcal{F}}_h^{\Omega_C}}|
\\[.1cm]
\qquad \leq 2 {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}}
{\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ( {\boldsymbol{v}}, \varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} \\[.1cm]
\qquad\leq 2C_{\Omega_C} {\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ( {\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} \\[.1cm]
\qquad\leq \frac{1}{4} {\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C}^2 + 4 C_{\Omega_C}^2
{\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ( {\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}}^2.
\end{array}$$ Similarly, by virtue of , $$\label{elip2}
\begin{array}{l}
2 | \text{Re} \left(
{\{\nabla_h \varphi+m \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \overline \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}} | \leq
2{\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla_h \varphi+ m \boldsymbol \rho\}}_{{\mathcal{F}}} \rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \overline \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}} \rVert}_{0,{\mathcal{F}}_h^{\Omega_I}}\\[.1cm]
\qquad \qquad \leq 2 C_{\Omega_I} {\lVert\nabla_h \varphi+m \boldsymbol \rho\rVert}_{0,\Omega} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}} \rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} \\[.1cm]
\qquad\qquad \leq \frac{1}{2} {\lVert\nabla_h \varphi + m \boldsymbol \rho\rVert}_{0,\Omega}^2 + 4 C_{\Omega_I}^2 {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}} \rVert}_{0,{\mathcal{F}}_h^{\Omega_I}}^2.
\end{array}$$ Finally, using we have that $$\label{elip3}
\begin{array}{l}
2 |\text{Re} \left( {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}, {\llbracket \overline \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}} \right)_{{\mathcal{E}}_h}|\leq
2 {\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}}{\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}
{\lVert\mathtt{s}_{{\mathcal{E}}}^{-1/2} h_{{\mathcal{E}}}^{-1}{\llbracket \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} \\[.1cm]
\qquad \leq
2 C_\Gamma {\lVert\sigma^{-1/2} {\mathbf{curl}}_h {\boldsymbol{v}}\rVert}^2_{0,\Omega_C}
{\lVert\mathtt{s}_{{\mathcal{E}}}^{-1/2} h_{{\mathcal{E}}}^{-1}{\llbracket \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\[.1cm]
\qquad\leq \frac{1}{4}
{\lVert\sigma^{-1/2} {\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C}^2 + 4 C_{\Omega_C}^2 {\lVert\mathtt{s}_{{\mathcal{E}}}^{-1/2} h_{{\mathcal{E}}}^{-1}{\llbracket \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}^2.
\end{array}$$ Combining with - and choosing $\alpha_0 = 1/2 + 4 C_\Omega^2 + 4C_{\Omega_I}^2$ we obtain .
We are now in a position to prove the ${\lVert\cdot\rVert}$-stability of the DG scheme .
\[LM\] Assume that $\sigma^{-1} {\boldsymbol{j}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})$ and $\min(\mathtt{a}^\Omega, \mathtt{a}^{\Omega_I}, \alpha)\geq \alpha_0$. Then, there exits a unique $({\boldsymbol{h}}_{h},\psi_h,k_h) \in \mathbf{X}_h\times V_h\times {\mathbb C}$ solution of Problem . Moreover if $({\boldsymbol{h}}, \psi,k)\in [\mathbf{H}({\mathbf{curl}},\Omega)\times {\mathrm{H}}^1(\Omega_I)\times {\mathbb C}] \cap [\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C}) \times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I}) \times {\mathbb C}]$ is the solution to - then $$\label{Cea}
{\lVert({\boldsymbol{h}}- {\boldsymbol{h}}_{h}, \psi - \psi_h, k -k_h)\rVert} \leq (1 + 2\sqrt{2} M) \inf_{({\boldsymbol{v}}, \varphi)\in \mathbf{X}_h\times V_h} {\lVert({\boldsymbol{h}}- {\boldsymbol{v}}, \psi -\varphi, 0)\rVert}_*.$$
The well posedness of Problem follows immediately from Proposition \[discElip\].
Moreover we deduce from Proposition \[discElip\] and the consistency of the scheme that $$\begin{array}{l}
\frac{1}{2} {\lVert({\boldsymbol{h}}_{h}- {\boldsymbol{v}}, \psi_h - \varphi, k_h-m)\rVert}^2 \\[.1cm]
\qquad \leq \text{Re} \left[ (1 - \imath) A_h(({\boldsymbol{h}}_{h}- {\boldsymbol{v}}, \psi_h - \varphi, k_h-m),({\boldsymbol{h}}_{C,h}- {\boldsymbol{v}}, \psi_h - \varphi, k_h-m)) \right] \\[.1cm]
\qquad = \text{Re} \left[ (1 - \imath) A_h(({\boldsymbol{h}}- {\boldsymbol{v}}, \psi - \varphi, k-m),({\boldsymbol{h}}- {\boldsymbol{v}}, \psi - \varphi, k-m)) \right]
\end{array}$$ for all $({\boldsymbol{v}}, \varphi,m)\in \mathbf{X}_h\times V_h\times {\mathbb C}$. Then from Proposition \[boundedness\] we have $${\lVert({\boldsymbol{h}}_{h}- {\boldsymbol{v}}, \psi_h - \varphi, k_h-m)\rVert} \leq 2 \sqrt{2} M {\lVert({\boldsymbol{h}}- {\boldsymbol{v}}, \psi - \varphi,k-m)\rVert}_\ast.$$ The result follows now from the triangle inequality.
Asymptotic error estimates {#sec5}
==========================
We denote by ${\boldsymbol{\Pi}}_{h,m}^{\text{curl}}$ the $m$-order $\mathbf{H}({\mathbf{curl}}, \Omega_C)$-conforming Nédélec interpolation operator of the second kind, see for example [@NED86] or [@Monk Section 8.2]. It is well known that ${\boldsymbol{\Pi}}_{h,m}^{\text{curl}}$ is bounded on $\mathbf{H}({\mathbf{curl}}, \Omega_C)\cap \mathbf{H}^s({\mathbf{curl}}, {\mathcal{T}}_h^{\Omega_C})$ for $s>1/2$, where $$\mathbf{H}^s({\mathbf{curl}}, {\mathcal{T}}_h^{\Omega_C}) := {\left\{{\boldsymbol{v}}\in \mathbf{H}^s({\mathcal{T}}_h^{\Omega_C});\quad {\mathbf{curl}}_h {\boldsymbol{v}}\in \mathbf{H}^{s}({\mathcal{T}}_h^{\Omega_C})\right\}}.$$ Moreover, there exists a constant $C_1>0$ independent of $h$ such that (cf. [@AVbook]) $$\label{errorInterp1}
{\lVert{\boldsymbol{u}}- {\boldsymbol{\Pi}}_{h,m}^{\text{curl}} {\boldsymbol{u}}\rVert}_{0, \Omega_C} + {\lVert{\mathbf{curl}}({\boldsymbol{u}}- {\boldsymbol{\Pi}}_{h,m}^{\text{curl}}{\boldsymbol{u}})\rVert}_{0, \Omega_C} \leq C_1 h^{\min(s, m)} \big( {\lVert{\boldsymbol{u}}\rVert}_{s, {\mathcal{T}}_h^{\Omega_C}} + {\lVert{\mathbf{curl}}_h {\boldsymbol{u}}\rVert}_{s, {\mathcal{T}}_h^{\Omega_C}} \big).$$
We introduce $
\mathbf{L}^2_t(\Gamma) = {\left\{\boldsymbol{\varphi}\in \mathbf{L}^2(\Gamma);\,\, \boldsymbol{\varphi}\cdot {\boldsymbol{n}}= 0\right\}}
$ and consider the $m$-order order Brezzi-Douglas-Marini (BDM) finite element approximation of the space $$\mathbf{H}(\text{div}_\Gamma, \Gamma) := {\left\{ \boldsymbol{\varphi}\in \mathbf{L}_t^2(\Gamma); \quad
\text{div}_\Gamma \boldsymbol{\varphi}\in L^2(\Gamma) \right\}}$$ relatively to the mesh ${\mathcal{F}}_h^\Gamma$ (see, e.g. [@Boffi]). It is given by $$\mathcal{\mathbf{BDM}}({\mathcal{F}}_h^\Gamma) = {\left\{\boldsymbol{\varphi}\in \mathbf{H}(\text{div}_\Gamma, \Gamma); \quad
\boldsymbol{\varphi}|_T \in {\mathcal{P}}_m(T)^2,\quad \forall T\in {\mathcal{F}}_h^\Gamma\right\}}.$$ The corresponding interpolation operator $\Pi_{h,m}^{\text{BDM}}$ is bounded on $
\mathbf{H}(\text{div}_\Gamma, \Gamma)\cap \prod_{T\in {\mathcal{F}}_h^\Gamma} {\mathrm{H}}^\delta(T)^2$ for all $\delta>0$ and we recall that it is uniquely characterized on each $T\in {\mathcal{F}}_h^\Gamma$ by the conditions $$\label{BDMfreedom1}
\int_e \Pi_{h,m}^{\text{BDM}} \boldsymbol{\varphi}\cdot {\boldsymbol{n}}_T q = \int_e \boldsymbol{\varphi}\cdot {\boldsymbol{n}}_T q\quad
\forall q \in {\mathcal{P}}_m(e),\quad \forall e\in {\mathcal{E}}(T),$$ $$\label{BDMfreedom2}
\int_T \Pi_{h,m}^{\text{BDM}} \boldsymbol{\varphi}\cdot \mathbf{q} = \int_T \boldsymbol{\varphi}\cdot \mathbf{q}\quad
\forall \mathbf{q} \in {\mathcal{P}}_{m-2}(T)^2 + \mathbf{S}_{m-1}(T),$$ where $\mathbf{S}_{m-1}(T):= {\left\{\mathbf{q}\in \tilde{\mathcal{P}}_{m-1}(T)^2;\quad \mathbf{q}\cdot \begin{pmatrix}x_1\\x_2
\end{pmatrix} = 0\right\}}$ with $\tilde{\mathcal{P}}_{m-1}(T)$ representing the set of homogeneous polynomials of degree $m-1$ and $\begin{pmatrix}x_1\\x_2\end{pmatrix}$ being the local variable on the plane containing $T$.
The commuting diagram property $$\begin{aligned}
\label{commuting2} ({\boldsymbol{\Pi}}_{h,m}^{\text{curl}} {\boldsymbol{u}}) \times {\boldsymbol{n}}_\Gamma &= \Pi_{h,m}^{\text{BDM}} ({\boldsymbol{u}}\times {\boldsymbol{n}}_\Gamma)\end{aligned}$$ holds true for all ${\boldsymbol{u}}\in \mathbf{H}({\mathbf{curl}}, \Omega_C)\cap \mathbf{H}^s({\mathbf{curl}}, {\mathcal{T}}_h^{\Omega_C})$, $s>1/2$, see [@Hiptmair1 section 9] for more details.
For all $K\in {\mathcal{T}}_h^{\Omega_I}$ we define the local interpolation operator $\tilde\pi_{K,m}:{\mathrm{H}}^{1+s}(K)\to \tilde{\mathcal{P}}_{m}(K)$, $s>1/2$ as follows: recalling the definition of $\tilde{\mathcal{P}}_{m}(K)$ given in
- if $\partial K \cap \Gamma \not \in \mathcal F_h^\Gamma$ then $\tilde{\mathcal{P}}_{m}(K)= {\mathcal{P}}_{m}(K)$ and we take $\tilde\pi_{K,m} = \pi_{K,m}$, where $\pi_{K,m}$ is defined as in [@Monk Section 5.6];
- if $\partial K \cap \Gamma=T \in \mathcal F_h^\Gamma$ then $\tilde{\mathcal{P}}_{m}(K)={\mathcal{P}}_m(K) + {\mathcal{P}}_{m+1}^T(K)$ and $\tilde \pi_{K,m}$ is defined by changing the conditions defining $\pi_{K,m}$ on $T$ and on the edges composing $T$ into $$\label{freedomFb}
\int_T \tilde\pi_{K,m} p q = \int_T p q \qquad \forall q \in {\mathcal{P}}_{m-2}(T)$$ and $$\label{freedomEb}
\int_e \tilde\pi_{K,m} p q = \int_e p q \qquad \forall q \in {\mathcal{P}}_{m-1}(e), \quad \forall e\in {\mathcal{E}}(F)$$ respectively. The remaining degrees of freedom are the same as those defining $\pi_{K,m}$, see [@Monk Section 5.6].
We notice that $
\text{dim}({\mathcal{P}}_m(K) + {\mathcal{P}}_{m+1}^T(K))= \text{dim}({\mathcal{P}}_m(K)) + m+1
$ and the number of degrees of freedom defining $\tilde \pi_{K,m}$ is equal to the number of degrees of freedom of $\pi_{K,m}$ plus $\text{dim}({\mathcal{P}}_{m-2}(T)) - \text{dim}({\mathcal{P}}_{m-3}(T)) = m-1$ additional degrees of freedom on $T$ and one additional degree of freedom on each of the three edges of $T$, which gives a total of $\text{dim}({\mathcal{P}}_m(K)) + m+1$ degrees of freedom. Using this fact, it is straightforward to show that $\tilde \pi_{K,m}$ is uniquely determined on elements $K\in {\mathcal{T}}_h^{\Omega_I}$ with a face $T$ lying on $\Gamma$. Moreover, it is clear that the corresponding global ${\mathrm{H}}^1(\Omega)$-conforming interpolation operator $\tilde \pi_{h,m}$ satisfies the following interpolation error estimate.
If $p\in {\mathrm{H}}^1(\Omega_I)\cap {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})$ with $s>1/2$, there exists a constant $C>0$ independent of $h$ such that $$\label{interp1}
{\lVert\nabla (p - \tilde \pi_{h,m} p)\rVert}_{0,\Omega_I} \leq C h^{\min(m, s)} {\lVertp\rVert}_{1+s, {\mathcal{T}}_h^{\Omega_I}}.$$
See [@Monk Lemma 5.47] and [@Monk Theorem 5.48].
The commuting diagram property stated in the next proposition is the reason for which we use $\tilde \pi_h$ instead of the usual Lagrange interpolation operator.
\[commuting3\] For any $p\in H^1(\Omega)\cap {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})$, with $s>1/2$, it holds $$\nabla \tilde \pi_{h,m} p \times {\boldsymbol{n}}_\Gamma = \Pi_{h,m}^{\text{BDM}} (\nabla p \times {\boldsymbol{n}}_\Gamma).$$
We first notice that $\nabla \tilde \pi_{h,m} p \times {\boldsymbol{n}}_\Gamma \in \mathbf{H}(\text{div}_\Gamma, \Gamma)$ and $\nabla \tilde \pi_{h,m} p \times {\boldsymbol{n}}_\Gamma \in {\mathcal{P}}_m(T)$ for all $T\in {\mathcal{F}}_h^\Gamma$. Hence, $\nabla \tilde \pi_{h,m} p \times {\boldsymbol{n}}_\Gamma \in \mathcal{\mathbf{BDM}}({\mathcal{F}}_h^\Gamma)$. To show that $\nabla \tilde \pi_{h,m} p \times {\boldsymbol{n}}_\Gamma = \Pi_{h,m}^{\text{BDM}} ({\mathbf{curl}}_\Gamma p)$, it is sufficient to compare the degrees of freedom of these two tangential fields on each triangle $T\in {\mathcal{F}}_h^\Gamma$. On the one hand, for all $q\in {\mathcal{P}}_m(e)$, $e\in {\mathcal{E}}(T)$, $$\begin{array}{l}
\displaystyle{\int_e (\nabla \tilde \pi_{h,m} p \times {\boldsymbol{n}}_\Gamma - \Pi_{h,m}^{\text{BDM}} (\nabla p \times {\boldsymbol{n}}_\Gamma))\cdot {\boldsymbol{n}}_F q}\\[.3cm]
\qquad = \displaystyle{\int_e \nabla \left((\tilde\pi_{h,m} p - p)\times n_\Gamma \right)\cdot {\boldsymbol{n}}_F q
= \int_e \frac{ \partial(\tilde\pi_{h,m} p - p)}{\partial{\boldsymbol{t}}_e} q} \\[.3cm]
\qquad=\displaystyle{ - \int_e (\tilde\pi_{h,m} p - p) \frac{ \partial q}{\partial{\boldsymbol{t}}_e} +
(\tilde\pi_{h,m} p - p)(\mathbf{a}_e) q(\mathbf{a}_e) - (\tilde\pi_{h,m} p - p)(\mathbf{b}_e) q(\mathbf{b}_e)=0}\, ,
\end{array}$$ where the last identity follows from the fact that $\tilde\pi_{h,m} p$ and $p$ must coincide at the endpoints $\mathbf{a}_e$ and $\mathbf{b}_e$ of edge $e$ (by definition of the $\tilde \pi_{h,m}$) and from , taking into account that $\frac{ \partial q}{\partial{\boldsymbol{t}}_e}\in {\mathcal{P}}_{m-1}(e)$.
On the other hand, for any $\mathbf{q}\in {\mathcal{P}}_{m-2}(T)^2 + \mathbf{S}_{m-1}(T)$, we have that $$\begin{array}{l}
\displaystyle{\int_T (\nabla \tilde \pi_{h,m} p \times {\boldsymbol{n}}_\Gamma- \Pi_{h,m}^{\text{BDM}} (\nabla p \times {\boldsymbol{n}}_\Gamma))\cdot \mathbf{q} }\\
\qquad \displaystyle{=\int_T \nabla (\tilde \pi_{h,m}p - p) \times {\boldsymbol{n}}_\Gamma \cdot \mathbf{q}=-\int_T \nabla (\tilde \pi_{h,m}p - p) \cdot (\mathbf{q}\times {\boldsymbol{n}}_\Gamma) }\\
\qquad =\displaystyle{ \int_T (\tilde\pi_{h,m} p - p) \,\text{div}_\Gamma (\mathbf{q}\times {\boldsymbol{n}}_\Gamma)
- \sum_{e\in {\mathcal{E}}(T)}\int_{e} (\tilde\pi_{h,m} p - p)\, (\mathbf{q}\times {\boldsymbol{n}}_\Gamma)\cdot \boldsymbol \nu_T }\\
\qquad \displaystyle{=\int_T (\tilde\pi_{h,m} p - p) \,\text{div}_\Gamma (\mathbf{q}\times {\boldsymbol{n}}_\Gamma)
- \sum_{e\in {\mathcal{E}}(T)}\int_{e} (\tilde\pi_{h,m} p - p)\, \mathbf{q} \cdot {\boldsymbol{t}}_e }
= 0
\end{array}$$ by virtue of and , since $\text{div}_\Gamma (\mathbf{q} \times {\boldsymbol{n}}_\Gamma)\in {\mathcal{P}}_{m-2}(F)$ and $\mathbf{q}\cdot {\boldsymbol{t}}_e\in {\mathcal{P}}_{m-1}(e)$.
Finally, we consider the $\mathbf{L}^2({\mathcal{T}}^{\Omega_C}_h)$-orthogonal projection ${\bf P}^k_{{\mathcal{T}}_h^{\Omega_C}}$ onto $\prod_{K\in {\mathcal{T}}^{\Omega_C}_h} {\mathcal{P}}_k(K)^3$ and the $\mathbf{L}^2({\mathcal{T}}^{\Omega_I}_h)$-orthogonal projection ${\bf P}^k_{{\mathcal{T}}_h^{\Omega_I}}$ onto $\prod_{K\in {\mathcal{T}}^{\Omega_I}_h} {\mathcal{P}}_k(K)^3$, $k\geq 0$. We denote indifferently by ${\boldsymbol{\Pi}}^k_K$ the restriction of ${\boldsymbol{\Pi}}^k_{{\mathcal{T}}_h^{\Omega_C}}$ and ${\boldsymbol{\Pi}}^k_{{\mathcal{T}}_h^{\Omega_I}}$ to an element $K$.
\[v\] For all $K\in {\mathcal{T}}_h$ and $\mathbf{w}\in \mathbf{H}^{r}(K)$, $r\geq 1/2$, we have $$\label{proj}
h_F{\lVert\mathbf{w} - {\bf P}^k_K \mathbf{w}\rVert}_{0,\partial F} + h_K^{1/2}{\lVert\mathbf{w} - {\bf P}^k_K \mathbf{w}\rVert}_{0,\partial K} + {\lVert\mathbf{w} - {\bf P}^k_K \mathbf{w}\rVert}_{0,K}
\leq C h_K^{\min\{r,k+1\}} {\lVert\mathbf{w}\rVert}_{r,K},$$ with a constant $C>0$ independent of $h$.
See [@DiPietroErn], Lemma 1.58 and Lemma 1.52.
We are now in a position to prove the main result of this section.
\[mainDGFEM\] Let $({\boldsymbol{h}}, \psi,k)\in \mathbf{H}({\mathbf{curl}}, \Omega_C)\times {\mathrm{H}}^1(\Omega_I)\times {\mathbb C}$ and $({\boldsymbol{h}}_{h}, \psi_h,k_h)\in \mathbf{X}_h\times V_h\times C$ be the solutions to - and respectively. If $\sigma^{-1} {\boldsymbol{j}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})$, $({\boldsymbol{h}}, \psi)\in \mathbf{X}^s({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})$, with $s>1/2$, and $\min(\mathtt{a}^{\Omega_C}, \mathtt{a}^{\Omega_I}, \alpha)\geq \alpha_0$, then $${\lVert({\boldsymbol{h}}- {\boldsymbol{h}}_{h}, \psi - \psi_h, k-k_h)\rVert} \leq C h^{\min(s, m)} \Big(
{\lVert{\boldsymbol{h}}\rVert}_{s, {\mathcal{T}}_h^\Omega} + {\lVert{\mathbf{curl}}\, {\boldsymbol{h}}\rVert}_{1/2 + s, {\mathcal{T}}_h^\Omega} + {\lVert\psi\rVert}_{1+s, {\mathcal{T}}_h^{\Omega_I}}
\Big),$$ where $C>0$ is a constant independent of $h$.
Taking $({\boldsymbol{v}}, \varphi)=({\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}}, \tilde \pi_{h,m} \psi)$ in yields $${\lVert({\boldsymbol{h}}- {\boldsymbol{h}}_{h}, \psi - \psi_h, k-k_h)\rVert} \leq (1 + 2\sqrt{2} M)
{\lVert({\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}}, \psi - \tilde \pi_{h,m} \psi,0)\rVert}_*.$$ All the jumps terms in the right-hand side of the last inequality are zero since the identities $$\label{true?}
({\boldsymbol{\Pi}}^{\text{curl}}_{h,m}\,{\boldsymbol{h}})\times {\boldsymbol{n}}= {\boldsymbol{\Pi}}^{\text{BDM}}_{h,m}({\boldsymbol{h}}\times {\boldsymbol{n}}_\Gamma) = {\boldsymbol{\Pi}}^{\text{BDM}}_{h,m}((\nabla \psi +k \boldsymbol \rho) \times {\boldsymbol{n}}_\Gamma) = \left( \nabla \widetilde{\pi}_{h,m} \psi+ k \boldsymbol \rho \right)\times {\boldsymbol{n}}_{\Gamma}$$ holds true on $\Gamma$ and we also have that $${\llbracket (\psi -\tilde \pi_{h,m} \psi){\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}
=
{\llbracket (\psi -\tilde \pi_{h,m} \psi) {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}} =0\, ,$$ by construction. Note that in the last equality of we have used the fact that $\boldsymbol \rho$ belongs to $H({\mathbf{curl}}; \Omega_I)$ and is a piecewise-linear polynomial. It follows that, $$\begin{array}{l}
{\lVert({\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}}, \psi - \tilde \pi_{h,m} \psi,0)\rVert}_*^2 \\[.2cm]
\qquad=
{\lVert(\omega\mu)^{1/2}({\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}})\rVert}^2_{0,\Omega_C}
+ {\lVert\sigma^{-1/2}{\mathbf{curl}}({\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}})\rVert}^2_{0,\Omega_C} \\[.2cm]
\qquad+ \omega\mu_0 {\lVert\nabla_h (\psi - \tilde \pi_{h,m} \psi)\rVert}^2_{0,\Omega_I} +
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}{\mathbf{curl}}({\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}})\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_C}}\\[.2cm]
\qquad+ {\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} {\mathbf{curl}}({\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}})\}}_{{\mathcal{E}}}\rVert}^2_{0,{\mathcal{E}}_h}
+{\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla (\psi - \tilde \pi_{h,m} \psi)\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}}.
\end{array}$$ We deduce from the triangle inequality that, $$\begin{gathered}
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}{\mathbf{curl}}({\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}})\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} =
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}({\mathbf{curl}}{\boldsymbol{h}}- {\bf P}^{m-1}_{{\mathcal{T}}_h^{\Omega_C}}{\mathbf{curl}}\, {\boldsymbol{h}})\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} \\+
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}({\bf P}^{m-1}_{{\mathcal{T}}_h^{\Omega_C}}{\mathbf{curl}}\, {\boldsymbol{h}}- {\mathbf{curl}}\, {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}})\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} = A_{\Omega_C} + B_{\Omega_C}\, .\end{gathered}$$ Using yields $$\begin{gathered}
B_{\Omega_C} \leq C_{\Omega_C} {\lVert\sigma^{-1/2} ({\bf P}^{m-1}_{{\mathcal{T}}_h^{\Omega_C}}{\mathbf{curl}}\, {\boldsymbol{h}}- {\mathbf{curl}}\, {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}}) \rVert}_{0,\Omega_C}\\ =
C_{\Omega_C} {\lVert\sigma^{-1/2} {\bf P}^{m-1}_{{\mathcal{T}}_h^{\Omega_C}} ({\mathbf{curl}}\, {\boldsymbol{h}}- {\mathbf{curl}}\, {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}}) \rVert}_{0,\Omega_C} \leq
C_{\Omega_C} {\lVert\sigma^{-1/2}{\mathbf{curl}}( {\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m} {\boldsymbol{h}}) \rVert}_{0,\Omega_C}\end{gathered}$$ and by virtue of we obtain $$A_{\Omega_C}^2 \leq \sum_{K\in {\mathcal{T}}_h^{\Omega_C}} h_K
{\lVert \sigma_K^{-1/2}({\mathbf{curl}}\, {\boldsymbol{h}}- {\bf P}^{m-1}_K{\mathbf{curl}}\, {\boldsymbol{h}}) \rVert}^2_{0,\partial K}.$$ Similarly, we consider the splitting $$\begin{gathered}
{\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1}{\mathbf{curl}}({\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}})\}}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} \leq
{\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1}({\mathbf{curl}}\,{\boldsymbol{h}}- {\bf P}^{m-1}_{{\mathcal{T}}_h^{\Omega_C}}{\mathbf{curl}}\, {\boldsymbol{h}})\}}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\ +
{\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1}({\bf P}^{m-1}_{{\mathcal{T}}_h^{\Omega_C}}{\mathbf{curl}}\, {\boldsymbol{h}}- {\mathbf{curl}}\,{\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}})\}}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} = A_\Gamma + B_\Gamma\,\end{gathered}$$ and use to obtain $$B_\Gamma \leq C_\Gamma {\lVert\sigma^{-1/2} ({\bf P}^{m-1}_{{\mathcal{T}}_h^{\Omega_C}}{\mathbf{curl}}\, {\boldsymbol{h}}- {\mathbf{curl}}\,{\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}}) \rVert}_{0,\Omega_C}$$ $$=C_\Gamma {\lVert\sigma^{-1/2} {\bf P}^{m-1}_{{\mathcal{T}}_h^{\Omega_C}} \left( {\mathbf{curl}}( {\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m} {\boldsymbol{h}}) \right) \rVert}_{0,\Omega_C}\leq
C_\Gamma {\lVert\sigma^{-1/2}{\mathbf{curl}}( {\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m} {\boldsymbol{h}}) \rVert}_{0,\Omega_C}.$$ Moreover, it follows from that $$A_\Gamma^2 \leq
\sum_{T\in {\mathcal{F}}_h^\Gamma }
h_{T}^2 {\lVert\sigma_{K_T}^{-1/2} ({\mathbf{curl}}\,{\boldsymbol{h}}- {\bf P}^{m-1}_K{\mathbf{curl}}\, {\boldsymbol{h}})\rVert}^2_{0,\partial T}.$$ Finally, $$\begin{gathered}
{\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla (\psi - \tilde \pi_{h,m} \psi)\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} \leq
{\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla \psi - {\bf P}^m_{{\mathcal{T}}_h^{\Omega_I}} \nabla \psi\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}}\\ +
{\lVerth_{{\mathcal{F}}}^{1/2} {\{{\bf P}^m_{{\mathcal{T}}_h^{\Omega_I}} \nabla \psi - \nabla \tilde \pi_{h,m} \psi)\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} = A_{\Omega_I} + B_{\Omega_I}\end{gathered}$$ and we derive from and the following estimates $$B_{\Omega_I} \leq C_{\Omega_I} {\lVert{\bf P}^m_{{\mathcal{T}}_h^{\Omega_I}} \nabla \psi - \nabla \tilde \pi_{h,m} \psi\rVert}_{0,\Omega_I}\\ \leq
C_{\Omega_I} {\lVert\nabla (\psi - \tilde \pi_{h,m} \psi)\rVert}_{0,\Omega_I},$$ $$A_{\Omega_I}^2 \leq \sum_{K\in {\mathcal{T}}_h^{\Omega_I}} h_K {\lVert\nabla \psi - {\bf P}^m_K \nabla \psi \rVert}^2_{0,\partial K}.$$ Combining the last inequalities we deduce that $$\begin{gathered}
{\lVert({\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}}, \psi - \tilde \pi_{h,m} \psi)\rVert}_*^2 \leq C \Big(
{\lVert{\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}}\rVert}^2_{0,\Omega_C} + {\lVert{\mathbf{curl}}({\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}})\rVert}^2_{0,\Omega_C} \\+ {\lVert\nabla_h (\psi - \tilde \pi_{h,m} \psi)\rVert}^2_{0,\Omega_I} + \sum_{K\in {\mathcal{T}}_h^{\Omega_C}} h_K
{\lVert {\mathbf{curl}}{\boldsymbol{h}}- {\bf P}^{m-1}_K{\mathbf{curl}}{\boldsymbol{h}}\rVert}^2_{0,\partial K} \\+ \sum_{T\in {\mathcal{F}}_h^\Gamma }
h_{T}^2 {\lVert{\mathbf{curl}}{\boldsymbol{h}}- {\bf P}^{m-1}_K{\mathbf{curl}}{\boldsymbol{h}}\rVert}^2_{0,\partial T}+ \sum_{K\in {\mathcal{T}}_h^{\Omega_I}} h_K {\lVert\nabla \psi - {\bf P}^m_K \nabla \psi \rVert}^2_{0,\partial K}
\Big)\end{gathered}$$ with $C>0$ independent of $h$. Applying the interpolation error estimates given by , and we obtain $$\begin{gathered}
{\lVert({\boldsymbol{h}}- {\boldsymbol{\Pi}}^{\text{curl}}_{h,m}{\boldsymbol{h}}, \psi - \tilde \pi_{h,m} \psi)\rVert}_* \leq C \Big( h^{\min(s, m)} (
{\lVert{\boldsymbol{h}}\rVert}_{s, {\mathcal{T}}_h^{\Omega_C}} + {\lVert{\mathbf{curl}}{\boldsymbol{h}}\rVert}_{s, {\mathcal{T}}_h^{\Omega_C}}) +
h^{\min(s, m)} {\lVert\psi\rVert}_{1+s, {\mathcal{T}}_h^{\Omega_I}}\\ + h^{\min(1/2+s, m)}
{\lVert{\mathbf{curl}}{\boldsymbol{h}}\rVert}_{1/2+s, {\mathcal{T}}_h^{\Omega_C}} + h^{\min(s, m+1)} {\lVert\nabla \psi\rVert}_{s, {\mathcal{T}}_h^{\Omega_I}}
\Big)\end{gathered}$$ and the result follows.
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[^1]: Department of Mathematics, University of Trento, Trento, Italy, e-mail: [[email protected]]{}
[^2]: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, Oviedo, España, e-mail: [[email protected]]{}
[^3]: Department of Mathematics, University of Trento, Trento, Italy, e-mail: [[email protected]]{}
[^4]: Support by the University of Trento and by the Spanish Ministry of Economy Project MTM2013-43671-P.
| 0 |
---
author:
- 'Tathagata Banerjee[^1]'
- 'Alex Bernstein[^2]'
- 'Zachary Feinstein[^3]'
bibliography:
- 'bibtex2.bib'
title: Dynamic clearing and contagion in financial networks
---
Introduction {#sec:intro}
============
Financial networks and the contagion of bank failures have been widely studied beginning with the seminal work on financial payment networks by Eisenberg & Noe [@EN01]. The 2007-2009 financial crisis and credit crunch showed the severe impacts that systemic crises can have on the financial sector and the economy as a whole. As the costs of such cascading events is tremendous, the modeling of such events is imperative. Recently there have been significant studies on modeling financial systemic risk and financial contagion. Two major classes of models exist for systemic risk, i.e., those based on network models from [@EN01] and those based on a mean field approach [@fouque2013illustrated; @fouque2015meanfield]. Notably, the network model approach generally is considered in only a static, single time, setting while the mean field approach is considered as a differential system. In this paper we will construct a dynamic extension of the interbank network model of [@EN01] thus closing the gap between these two streams of literature.
Interbank networks were studied first in [@EN01] to model the spread of defaults in the financial system. In the Eisenberg-Noe framework, financial firms must satisfy their liabilities by transferring assets. One firm being unable to meet its liabilities due to a shortfall of assets can cause other firms to default on some of their liabilities as well, causing a cascading failure in the financial system. The existence and uniqueness of the clearing payments of this baseline model was proven in [@EN01]. That paper additionally provides methods for numerically computing the realized interbank payments. This baseline model has been extended in multiple directions, including bankruptcy costs, cross-holdings, and fire sales. We refer to [@AW_15; @Staum] for reviews of the prior literature. In regards to bankruptcy costs in financial networks, we refer to [@E07; @RV13; @EGJ14; @GY14; @AW_15; @CCY16; @veraart2017distress]. Cross-holdings have been studied in [@E07; @EGJ14; @AW_15; @GHM12]. Fire sales for a single (representative) illiquid asset have been studied in [@CFS05; @NYYA07; @GK10; @AFM13; @CLY14; @AW_15; @AFM16] and for multiple illiquid assets in [@feinstein2015illiquid; @feinstein2016leverage; @feinstein2017currency]. These network models have been implemented by central banks and regulators for stress testing of and studying cascading failures in the banking systems under their jurisdiction, see, e.g., [@Anand:Canadian; @HK:Modeling; @BEST:2004; @ELS13; @U11; @GHK2011].
Mean field models have also been considered for studying financial contagion and systemic risk. [@fouque2013illustrated] provides a model of agents who revert to the ensemble mean to provide understanding of “systemic risk events” in which many firms fail. Similar mean field diffusion models without controls were studied in, e.g., [@fouque2013stability; @garnier2013a; @garnier2013b]. In contrast, mean field and stochastic games have been proposed for the study of systemic risk in, e.g., [@fouque2015meanfield; @carmona2016delay]. In such models the firms are allowed to borrow from (or lend to) a central bank, the amount of which is optimized to minimize a quadratic cost function. Thus the choice of borrowing and lending provides an optimal control problem beyond the simpler mean field model of [@fouque2013illustrated]. [@NS17] proposes a separate particle system model with mean field interactions.
The current work will focus on adding the time dynamics, which make the mean field models attractive, to the interbank network approach. In fact, the conclusion of [@EN01] provides a discussion of future extensions, one of which is the inclusion of multiple clearing dates. This has been studied directly in [@CC15; @ferrara16]. Additionally, [@KV16] considers a similar approach to model financial networks with multiple maturities. [@feinstein2017currency] further provides another approach to financial networks with multiple maturities by considering each clearing date as a different asset. All of these works, however, only consider clearing at discrete times. [@sonin2017] presents a continuous-time clearing model that exactly replicates the static Eisenberg-Noe framework. In this work we will present both discrete and continuous-time clearing models. However, our emphasis will be on the derivation and the characterization of the continuous-time model. This in part is motivated by the prospect of unification with the mean-field models as well as traditional financial models which typically employ continuous-time models. Additionally, as we will demonstrate, the continuous-time framework no longer requires monotonicity for existence and uniqueness which is generally assumed for static and discrete-time systems. This is valuable for future works that may model network formation and payments as a non-cooperative game; such games may not satisfy the strong monotonicity assumptions usually considered in static and discrete-time systems, but would likely satisfy the sufficient conditions for the continuous-time framework.
The organization of this paper is as follows. In Section \[sec:setting\] we will provide a review of the static Eisenberg-Noe framework. Of particular interest, in this section, we consider the clearing to be in terms of the equity and losses of the firms, as considered in, e.g., [@veraart2017distress; @barucca2016valuation] rather than payments as originally studied in [@EN01]. In Section \[sec:discrete\] we propose a discrete-time formulation for the Eisenberg-Noe model. In discrete time we provide results on existence and uniqueness, as well as a numerical algorithm based on the fictitious default algorithm of [@EN01]. We then extend our model to a continuous-time setting in Section \[sec:continuous\]. For continuous time we consider existence and uniqueness of the clearing solutions, and a numerical algorithm for finding sample paths of this clearing solution, under cash flows modeled by Itô processes. We additionally provide conditions for the discrete-time setting to converge to the continuous-time solution as the time step limits to 0. Section \[sec:discussion\] provides discussion on the financial implications of time dynamics in interbank networks. In particular, we find that the static Eisenberg-Noe clearing solution can be recovered in the continuous-time setting by choosing the network parameters precisely. This allows for a notion of determining the true order of defaults as opposed to the fictitious default order discussed in the static literature based on [@EN01]. However, if the continuous-time network parameters are determined to not follow the rules for recreating the static Eisenberg-Noe setting, then the dynamic and static clearing solutions will generally not coincide. In fact, the set of defaulting and solvent institutions can be altered by rearranging the timing of obligations. As such, using the static Eisenberg-Noe framework for stress testing may result in an incorrect assessment of the health of the financial system. The proofs of the main results are provided in the Appendix.
Static clearing systems {#sec:setting}
=======================
We begin with some simple notation that will be consistent for the entirety of this paper. Let $x,y \in {\mathbb{R}}^n$ for some positive integer $n$, then $$x \wedge y = \left(\min(x_1,y_1),\min(x_2,y_2),\ldots,\min(x_n,y_n)\right)^{\top},$$ $x^- = -(x \wedge 0)$, and $x^+ = (-x)^-$. Further, to ease notation, we will denote $[x,y] := [x_1,y_1] \times [x_2,y_2] \times \ldots \times [x_n,y_n] \subseteq {\mathbb{R}}^n$ to be the $n$-dimensional compact interval for $y - x \in {\mathbb{R}}^n_+$. Similarly, we will consider $x \leq y$ if and only if $y - x \in {\mathbb{R}}^n_+$.
Throughout this paper we will consider a network of $n$ financial institutions. We will denote the set of all banks in the network by ${\mathcal{N}}:= \{1,2,\ldots,n\}$. Often we will consider an additional node $0$, which encompasses the entirety of the financial system outside of the $n$ banks; this node $0$ will also be referred to as society or the societal node. The full set of institutions, including the societal node, is denoted by ${\mathcal{N}}_0 := {\mathcal{N}}\cup \{0\}$. We refer to [@feinstein2014measures; @GY14] for further discussion of the meaning and concepts behind the societal node.
We will be extending the model from [@EN01] in this paper. In that work, any bank $i \in {\mathcal{N}}$ may have obligations $L_{ij} \geq 0$ to any other firm or society $j \in {\mathcal{N}}_0$. We will assume that no firm has any obligations to itself, i.e., $L_{ii} = 0$ for all firms $i \in {\mathcal{N}}$, and the society node has no liabilities at all, i.e., $L_{0j} = 0$ for all firms $j \in {\mathcal{N}}_0$. Thus the *total liabilities* for bank $i \in {\mathcal{N}}$ is given by $\bar p_i := \sum_{j \in {\mathcal{N}}_0} L_{ij} \geq 0$ and relative liabilities $\pi_{ij} := \frac{L_{ij}}{\bar p_i}$ if $\bar p_i > 0$ and arbitrary otherwise; for simplicity, in the case that $\bar p_i = 0$, we will let $\pi_{ij} = \frac{1}{n}$ for all $j \in {\mathcal{N}}_0 \backslash \{i\}$ and $\pi_{ii} = 0$ to retain the property that $\sum_{j \in {\mathcal{N}}_0} \pi_{ij} = 1$. On the other side of the balance sheet, all firms are assumed to begin with some amount of external assets $x_i \geq 0$ for all firms $i \in {\mathcal{N}}_0$. The resultant *clearing payments*, under a no priority of payments assumption, satisfy the fixed point problem in payments $p \in [0,\bar p]$ $$\label{eq:EN-p}
p = \bar p \wedge \left(x + \Pi^{\top}p\right).$$ That is, each bank pays the minimum of what it owes ($\bar p_i$) and what it has ($x_i + \sum_{j \in {\mathcal{N}}} \pi_{ji} p_j$). The resultant vector of *wealths* for all firms is given by $$\label{eq:equity}
V = x + \Pi^{\top}p - \bar p.$$ Noting that payments can be written as a simple function of the wealths ($p = \bar p - V^-$), we provide the following proposition. We refer also to [@veraart2017distress; @barucca2016valuation; @banerjee2017insurance] for similar notions of utilizing clearing wealth instead of clearing payments.
\[prop:EN-e\] A vector $p \in [0,\bar p]$ is a clearing payments in the Eisenberg-Noe setting if and only if $p = [\bar p - V^-]^+$ for some $V \in {\mathbb{R}}^{n+1}$ satisfying the following fixed point problem $$\label{eq:EN-e}
V = x + \Pi^{\top}[\bar p - V^-]^+ - \bar p.$$ Vice versa, a vector $V \in {\mathbb{R}}^{n+1}$ is a clearing wealths (i.e., satisfying ) if and only if $V$ is defined as in for some clearing payments $p \in [0,\bar p]$ as defined in the fixed point problem .
We will prove the first equivalence only, the second follows similarly.
Let $p \in [0,\bar p]$ be a clearing payment vector. Define the wealth vector $V$ by , then it is clear that $V^- = \bar p - p$ by definition as well, i.e., $p = \bar p - V^- \geq 0$. Thus from we immediately recover that the wealth vector $V$ must satisfy .
Let $p = [\bar p - V^-]^+$ for some wealth vector $V \in {\mathbb{R}}^{n+1}$ satisfying . By construction we find $$\begin{aligned}
p &= [\bar p - V^-]^+ = \bar p - \left(x + \Pi^{\top}[\bar p - V^-]^+ - \bar p\right)^-
= \bar p - \left(x + \Pi^{\top}p - \bar p\right)^- = \bar p \wedge \left(x + \Pi^{\top}p\right).\end{aligned}$$ We note that $\bar p \geq \left(x + \Pi^{\top}[\bar p - V^-]^+ - \bar p\right)^-$ can be shown trivially.
Due to the equivalence of the clearing payments and clearing wealths provided in Proposition \[prop:EN-e\], we are able to consider the Eisenberg-Noe system as a fixed point of equity and losses rather than payments. In [@EN01] results for the existence and uniqueness of the clearing payments (and thus for the clearing wealths as well) are provided. In fact, it can be shown that there exists a unique clearing solution in the Eisenberg-Noe framework so long as $L_{i0} > 0$ for all firms $i \in {\mathcal{N}}$. We will take advantage of this result later in this paper. This is a reasonable assumption (as discussed in, e.g., [@GY14]) as obligations to society include, e.g., deposits to the banks.
Discrete-time clearing systems {#sec:discrete}
==============================
Consider now a discrete set of clearing times ${\mathbb{T}}$, e.g., ${\mathbb{T}}= \{0,1,\dots,T\}$ for some (finite) terminal time $T < \infty$ or ${\mathbb{T}}= {\mathbb{N}}$. Such a setting is presented in [@CC15]. For processes we will use the notation from [@cont2013ito] such that the process $Z: {\mathbb{T}}\to {\mathbb{R}}^n$ has value of $Z(t)$ at time $t \in {\mathbb{T}}$ and history $Z_t := (Z(s))_{s = 0}^t$.
(0,9.5) rectangle (6,10) node\[pos=.5\][**Balance Sheet**]{}; (0,9) rectangle (3,9.5) node\[pos=.5\][**Assets**]{}; (3,9) rectangle (6,9.5) node\[pos=.5\][**Liabilities**]{};
(0,7) rectangle (3,9) node\[pos=.5,style=[align=center]{}\][Cash-Flow @ $t = 0$\
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In this setting, we will consider the external (incoming) cash flow $x: {\mathbb{T}}\to {\mathbb{R}}^{n+1}_+$ and nominal liabilities $L: {\mathbb{T}}\to {\mathbb{R}}^{(n+1) \times (n+1)}_+$ to be functions of the clearing time, i.e., as assets and liabilities with different maturities. The external cash in-flows and nominal liabilities can explicitly depend on the clearing results of the prior times (i.e., $x(t,V_{t-1})$ and $L(t,V_{t-1})$) without affecting the existence and uniqueness results we present, but for simplicity of notation we will focus on the case where the external assets and nominal liabilities are independent of the health and wealth of the firms. Throughout we are considering the discounted cash flows and liabilities so as to simplify notation. In contrast to the static Eisenberg-Noe framework, herein we need to consider the results of the prior times. In particular, if firm $i$ has positive equity at time $t-1$ (i.e., $V_i(t-1) > 0$) then these additional assets are available to firm $i$ at time $t$ in order to satisfy its obligations. Similarly, if firm $i$ has negative wealth at time $t-1$ (i.e., $V_i(t-1) < 0$) then the debts that the firm has not yet paid will roll-forward in time and be due at the next period. For example, consider a network in which obligations come due throughout the day at, e.g., opening, mid-day, and closing, but that all debts must be cleared by the end of the day. In such a way, the current unpaid liabilities may be paid at a future time, but before the terminal time. That is, a firm can be considered in *distress* at a time if it is unable to satisfy its obligations at that time, but only *defaults* if it has negative wealth at the terminal time. Thus in this paper we primarily focus on the intra-day dynamics rather than the inter-day dynamics. See Figure \[fig:discrete-BS\] for a stylized (snapshot of the) balance sheet example for a firm that has positive wealth at time $0$ that rolls forward to time $1$. The full (actualized) balance sheet for this example with only those two time periods is displayed in Figure \[fig:BS\]; we note that the full balance sheet as depicted considers actualized payments rather than the book value of the obligations.
To incorporate the inter-day dynamics in this framework we can “zero out” a firm before the terminal date if it is deemed to default in much the same as in [@banerjee2017insurance]. A broader framework for dealing with various default mechanisms is discussed in Remark \[rem:loans\]. We can further consider the Nash game in which firms decide if they will allow debts to be rolled forward in time. In such a setting, if we include a delay for payment due to, e.g., bankruptcy court so that defaulting firms do not pay any obligations until after the terminal time $T$, then the optimal strategy for all firms (up until the terminal time $T$) would be to always allow other firms to roll all debts forward so as to maximize payments.
\[ass:initial\] Before the time of interest, all firms are solvent and liquid. That is, $V_i(-1) \geq 0$ for all firms $i \in {\mathcal{N}}_0$.
We can now construct the total liabilities and relative liabilities at time $t \in {\mathbb{T}}$ as $$\begin{aligned}
\bar p_i(t,V_{t-1}) &:= \sum_{j \in {\mathcal{N}}_0} L_{ij}(t) + V_i(t-1)^-\\
\pi_{ij}(t,V_{t-1}) &:= \begin{cases} \frac{L_{ij}(t) + \pi_{ij}(t-1,V_{t-2})V_i(t-1)^-}{\bar p_i(t,V_{t-1})} &\text{if } \bar p_i(t,V_{t-1}) > 0 \\ \frac{1}{n} &\text{if } \bar p_i(t,V_{t-1}) = 0, \; j \neq i\\ 0 &\text{if } \bar p_i(t,V_{t-1}) = 0, \; j = i \end{cases} \quad \forall i,j \in {\mathcal{N}}_0.\end{aligned}$$ In this way, coupled with the accumulation of positive equity over time, the clearing wealths must satisfy the following fixed point problem in time $t$ wealths: $$\label{eq:EN-discrete}
V(t) = V(t-1)^+ + x(t) + \Pi(t,V_{t-1})^{\top}\left[\bar p(t,V_{t-1}) - V(t)^-\right]^+ - \bar p(t,V_{t-1}).$$ That is, all firms have a clearing wealth that is the summation of their positive equity at the prior time, the new incoming external cash flow, and the payments made by all other firms minus the total obligations of the firm (including the prior unpaid liabilities). In this way we can construct the wealths of firms forward in time. This can be considered a discrete-time extension of .
We now wish to consider a reformulation of . To accomplish this, we consider a process of cash flows $c$ and functional relative exposures $A$. These we define by $$\begin{aligned}
\nonumber c(t) &:= x(t) + L(t)^{\top}\vec{1} - L(t)\vec{1}\\
\label{eq:discrete-A} a_{ij}(t,V_t) &:= \begin{cases} \pi_{ij}(t,V_{t-1}) &\text{if } \bar p_i(t,V_{t-1}) \geq V_i(t)^-\\ \frac{L_{ij}(t) + \pi_{ij}(t-1,V_{t-2})V_i(t-1)^-}{V_i(t)^-} &\text{if } \bar p_i(t,V_{t-1}) < V_i(t)^-\end{cases} \quad \forall i,j \in {\mathcal{N}}_0.\end{aligned}$$ In the above, $\vec{1} := (1,1,\ldots,1)^{\top}\in {\mathbb{R}}^n$ is the vector of ones. Here we consider $c(t) = x(t) + L(t)^{\top}\vec{1} - L(t)\vec{1} \in {\mathbb{R}}^{n+1}$ to be the vector of book capital levels at time $t$, i.e., the new wealth of each firm assuming all other firms pay in full. We wish to note that the new total liabilities are given by $L(t)\vec{1}$ and the new incoming interbank obligations are given by $L(t)^{\top}\vec{1}$. We can also consider $c_i(t)$ to be the *net cash flow* for firm $i$ at time $t$. Further, we introduce the functional matrix $A: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \to [0,1]^{(n+1) \times (n+1)}$ to be the relative exposure matrix. That is, $a_{ij}(t,V_t)V_i(t)^-$ provides the (negative) impact that firm $i$’s losses have on firm $j$’s wealth at time $t \in {\mathbb{T}}$. This is in contrast to $\Pi$, the relative liabilities, in that it endogenously imposes the limited exposures concept. In this work the two notions will generally coincide, but for mathematical simplicity we introduce this relative exposure matrix. For the equivalence we seek, we define the relative exposures so that $$L(t)^{\top}\vec{1} + A(t-1,V_{t-1})^{\top}V(t-1)^- - A(t,V_t)^{\top}V(t)^- = \Pi(t,V_{t-1})^{\top}[\bar p(t,V_{t-1}) - V(t)^-]^+$$ for any $V(t) \in {\mathbb{R}}^{n+1}$. This formulation is such that if the positive part were removed from the right hand side, the relative exposures $A$ would be defined exactly as the relative liabilities $\Pi$ by construction. In particular, we will define the relative exposures element-wise and pointwise so as to encompass the limited exposures as in . If $\bar p_i(t,V_{t-1}) > 0$ then we can simplify this further as $a_{ij}(t,V_t) = \frac{L_{ij}(t) + a_{ij}(t-1,V_{t-1})V_i(t-1)^-}{\max\{\bar p_i(t,V_{t-1}) , V_i(t)^-\}}$.
Using the notation and terms above we can rewrite with respect to the cash flows $c$ and relative exposures $A$ as $$\begin{aligned}
\nonumber V(t) &= V(t-1)^+ + x(t) + \Pi(t,V_{t-1})^{\top}[\bar p(t,V_{t-1}) - V(t)^-]^+ - \bar p(t,V_{t-1})\\
\nonumber &= V(t-1)^+ + x(t) + L(t)^{\top}\vec{1} + A(t-1,V_{t-1})^{\top}V(t-1)^-\\
\nonumber &\qquad - A(t,V_t)^{\top}V(t)^- - L(t)\vec{1} -V(t-1)^-\\
\nonumber &= V(t-1) + x(t) + L(t)^{\top}\vec{1} + A(t-1,V_{t-1})^{\top}V(t-1)^- - A(t,V_t)^{\top}V(t)^- - L(t)\vec{1}\\
\label{eq:discrete-V} &= V(t-1) + c(t) - A(t,V_t)^{\top}V(t)^- + A(t-1,V_t)^{\top}V(t-1)^-.\end{aligned}$$ For the remainder of this paper we will utilize the cash flow $c$ rather than the external (incoming) cash flow $x$. That is, we will consider financial networks defined by the joint parameters $(c,L)$ as given by the state equations and for wealths and relative exposures.
With this setup we now wish to extend the existence and uniqueness results of [@EN01] to discrete time.
\[thm:discrete\] Let $(c,L): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ define a dynamic financial network such that every bank has cash flow at least at the level dictated by nominal interbank liabilities, i.e., $c_i(t) \geq \sum_{j \in {\mathcal{N}}} L_{ji}(t) - \sum_{j \in {\mathcal{N}}_0} L_{ij}(t)$, and so that every bank owes to the societal node at all times $t \in {\mathbb{T}}$, i.e., $L_{i0}(t) > 0$ for all banks $i \in {\mathcal{N}}$ and times $t \in {\mathbb{T}}$. Under Assumption \[ass:initial\], there exists a unique solution of clearing wealths $V: {\mathbb{T}}\to {\mathbb{R}}^{n+1}$ to .
\[rem:regularnetwork\] The assumption that all firms have obligations to the societal node $0$ at all times $t \in {\mathbb{T}}$ guarantees that the financial system is a “regular network” (see [@EN01 Definition 5]) at all times.
The analysis of the discrete-time framework can be extended to a probabilistic setting over the filtered probability space $(\Omega,{\mathcal{F}},({\mathcal{F}}(t))_{t \in {\mathbb{T}}},{\mathbb{P}})$. That is, we can consider the clearing wealths in the same manner assuming the cash flow $c: {\mathbb{T}}\times \Omega \to {\mathbb{R}}^{n+1}$ and nominal liabilities $L: {\mathbb{T}}\times \Omega \to {\mathbb{R}}^{(n+1) \times (n+1)}_+$ be adapted processes. Let ${\mathcal{L}}_t^0({\mathbb{R}}^m)$ be the space of ${\mathcal{F}}_t$-measurable random vectors in ${\mathbb{R}}^m$. Let ${\mathcal{L}}_t^p({\mathbb{R}}^m) \subseteq {\mathcal{L}}_t^0({\mathbb{R}}^m)$ for $p \in (0,\infty]$ be the space of equivalence classes of ${\mathcal{F}}_t$-measurable functions $X: \Omega \to {\mathbb{R}}^m$ such that $\|X\|_p := \left(\int_\Omega \sqrt{\sum_{k = 1}^m X_k(\omega)^2} d{\mathbb{P}}\right)^{1/p} < \infty$ for $p < \infty$ and $\|X\|_{\infty} := \operatorname*{ess\,sup}_{\omega \in \Omega} \sqrt{\sum_{k = 1}^m X_k(\omega)^2}$ for $p = \infty$. The following corollary considers the boundedness and measurability properties of the discrete-time clearing wealths. Though we will not utilize this discrete-time result in this paper, we consider it important to discuss random events to more closely match reality. Further, this result will implicitly appear in the construction and analysis of the continuous-time Eisenberg-Noe formulation of the next section.
\[cor:discrete\] Consider the setting of Theorem \[thm:discrete\] where the random network parameters $(c,L)$ adapted to the filtered probability space $(\Omega,{\mathcal{F}},({\mathcal{F}}(t))_{t \in {\mathbb{T}}},{\mathbb{P}})$. If $c(s) \in {\mathcal{L}}_s^p({\mathbb{R}}^{n+1})$ and $L(s) \in {\mathcal{L}}_s^p({\mathbb{R}}^{(n+1) \times (n+1)}_+)$ for all times $s \leq t$ for some $p \in [0,\infty]$, then the unique clearing solution at time $t$ has finite $p$-norm, i.e., $V(t) \in {\mathcal{L}}_t^p({\mathbb{R}}^{n+1})$.
With the construction of the existence and uniqueness of the solution we now want to emphasize the *fictitious default algorithm* from [@EN01] to construct this clearing wealths vector over time. This algorithm is presented for the deterministic setting; if a stochastic setting is desired then Algorithm \[alg:discrete\] provides a method for computing a single sample path. We note that at each time $t$ this algorithm takes at most $n$ iterations. Thus with a terminal time $T$, this algorithm will construct the full clearing solution over ${\mathbb{T}}$ in $nT$ iterations.
\[alg:discrete\] Under the assumptions of Theorem \[thm:discrete\] in a deterministic setting the clearing wealths process $V: {\mathbb{T}}\to {\mathbb{R}}^{n+1}$ can be found by the following algorithm. Initialize $t = -1$ and $V(-1) \geq 0$ as a given. Repeat until $t = \max{\mathbb{T}}$:
1. Increment $t = t+1$.
2. \[alg:v\] Initialize $k = 0$, $V^0 = V(t-1) + c(t)$, and $D^0 = \emptyset$. Repeat until convergence:
1. Increment $k = k+1$;
2. Denote the set of insolvent banks by $D^k := \left\{i \in \{1,2,...,n\} \; | \; V_i^{k-1} < 0\right\}$.
3. If $D^k = D^{k-1}$ then terminate and set $V(t) = V^{k-1}$.
4. Define the matrix $\Lambda^k \in \{0,1\}^{n \times n}$ so that $\Lambda_{ij}^k = \begin{cases}1 &\text{if } i = j \in D^k \\ 0 &\text{else}\end{cases}$.
5. \[alg:vk\] Define $V^k = (I - \Pi(t,V_{t-1})^{\top}\Lambda^k)^{-1}\left(V(t-1) + c(t) + A(t-1,V_{t-1})^{\top}V(t-1)^-\right)$.
\[rem:loans\] Note that in the construction of $V^k$ in step of the fictitious default algorithm we utilize the relative liabilities $\Pi(t,V_{t-1})$ in the matrix inverse rather than the relative exposures $A(t,(V_{t-1},V^k))$. This has the added benefit that this definition of $V^k$ is *not* a fixed point problem, which it would be if the relative exposures matrix at time $t$ were considered. This change is possible since, as discussed in the proof of Theorem \[thm:discrete\], any clearing solution must be in the domain so that the relative liabilities and exposures coincide. This additionally provides the invertibility of this matrix using standard input-output results as discussed in [@EN01; @feinstein2017sensitivity].
We wish to finish up our discussion of the discrete-time Eisenberg-Noe framework by considering some extensions involving loans.
The theoretical framework presented in this paper can be easily extended to incorporate the concepts of loans until some (deterministic) insolvency condition is hit. In particular, we will consider loans made from a central bank or lender of last resort who we will assume are part of the societal node $0$. From this perspective we consider three cases that a firm might be in:
- [**solvent and liquid**]{} in which case the firm has positive equity and pays off its obligations in full;
- [**solvent and distressed**]{} in which case the firm has negative equity, but receives an overnight loan (with interest rate set at the risk-free rate for simplicity) to cover all obligations due on that date; and
- [**insolvent**]{} in which the firm will not receive any loans and is sent to a bankruptcy court.
The determination whether a firm is solvent can be done with an appropriate exogenous solvency function. We will assume that once a firm is deemed insolvent it can never recover to solvency again. Two possible systems for considering insolvent firms are:
1. [**Receivership:**]{} In such a system, when a firm is deemed insolvent it is placed in receivership so that obligations are payed out on a first-come first-serve basis.
2. [**Auctions:**]{} In such a system, when a firm is deemed insolvent its future assets are auctioned off in order to pay the future liabilities (in a proportional scheme) at the next time point. This will then affect the cash flows $c$ and nominal liabilities $L$, as such we would need to consider $c(t,V_{t-1})$ and $L(t,V_{t-1})$ to truly consider this case. We refer to [@CC15] for a detailed discussion of the auction model for insolvency. The auction system can be interpreted as an internal mechanism for determining bankruptcy costs in contrast to the exogenous parameter in, e.g., [@RV13].
The existence and uniqueness of the clearing solutions in these scenarios require an additional monotonicity property; we can use the notion a speculative system from [@banerjee2017insurance] to get the desired results. This condition encodes the notion that a firm does not benefit from any firm’s distress.
Continuous-time clearing systems {#sec:continuous}
================================
Consider now a continuous set of clearing times ${\mathbb{T}}$, e.g., ${\mathbb{T}}= [0,T]$ for some (finite) terminal time $T < \infty$ or ${\mathbb{T}}= {\mathbb{R}}_+$. As before, for processes we will use the notation from [@cont2013ito] such that the process $Z: {\mathbb{T}}\to {\mathbb{R}}^n$ has value of $Z(t)$ at time $t \in {\mathbb{T}}$ and history $Z_t := (Z(s))_{s \in [0,t]}$. We will now construct an extension of the continuous-time setting of [@sonin2017] in that we allow for liabilities to change over time and for firms to have stochastic cash flows.
In order to construct a continuous-time model we will begin by considering our network parameters of cash flows and nominal liabilities. Instead of considering $c(t)$ to be the net cash flow at time $t \in {\mathbb{T}}$, we will consider the term $dc(t)$ of marginal change in cash flow at time $t$. Similarly we will consider $dL(t)$ to be the marginal change in nominal liabilities matrix at time $t$; we note that by assumption $dL_{ij}(t) \geq 0$ for all firms $i,j \in {\mathcal{N}}_0$ as, without any payments made, total liabilities should accumulate over time. Our main result in this section (Theorem \[thm:continuous\]) provides existence and uniqueness of the clearing wealths driven by $(dc,dL)$ when $c(t) = \int_0^t dc(s)$ is an Itô process and $L(t) = \int_0^t dL(s)$ is deterministic and continuous (e.g., $dL$ does not include any Dirac delta functions). This setting, and the results on the continuous-time Eisenberg-Noe model, can be extended to the case in which the cash flows and liabilities are additionally functions of the wealths $V$. For simplicity, in this section we will restrict ourselves so that the parameters are independent of the current wealths. In order to construct a continuous-time differential system, we will consider again the discrete-time setting with explicit time steps ${{\Delta t}}$.
\[ass:society\] The cash flows $c$ are defined by the Itô stochastic differential equation $dc(t) = \mu(t,c(t))dt + \sigma(t,c(t))dW(t)$ for $(n+1)$-vector of Brownian motions $W$ over some filtered probability space $(\Omega,{\mathcal{F}},({\mathcal{F}}_t)_{t \in {\mathbb{T}}},{\mathbb{P}})$. Additionally, the drift and diffusion functions $\mu: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \to {\mathbb{R}}^{n+1}$ and $\sigma: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \to {\mathbb{R}}^{(n+1) \times (n+1)}$ are jointly continuous and satisfy the linear growth and Lipschitz continuous conditions, i.e., there exist constants $C,D > 0$ such that for all times $t \in {\mathbb{T}}$ and cash flows $c,d \in {\mathbb{R}}^{n+1}$ $$\begin{aligned}
\|\mu(t,c)\|_1 + \|\sigma(t,c)\|_1^{op} &\leq C(1 + \|c\|_1)\\
\|\mu(t,c) - \mu(t,d)\|_1 + \|\sigma(t,c) - \sigma(t,d)\|_1^{op} &\leq D\|c - d\|_1\end{aligned}$$ where $\|\cdot\|_1$ is the 1-norm and $\|\cdot\|_1^{op}$ is the corresponding operator norm. The nominal liabilities $L: {\mathbb{T}}\to {\mathbb{R}}^{(n+1) \times (n+1)}_+$ are deterministic and twice differentiable; for notation we will define $dL(t) = \dot{L}(t) dt$ and $d^2L(t) = \ddot{L}(t) dt^2$. Further, the relative liabilities to society is bounded from below by a level $\delta > 0$, i.e., $\inf_{t \in {\mathbb{T}}} \frac{dL_{i0}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)} = \delta > 0$ for all banks $i \in {\mathcal{N}}$.
We remark that the assumption on the cash flows can be relaxed so long as the stochastic differential equation has a unique strong solution on ${\mathbb{T}}$ and $\mu,\sigma$ satisfy a local linear growth condition and are locally Lipschitz. This relaxation will be applied in Examples \[ex:differentialEN\] and \[ex:dc-dependence\].
In the prior section on a discrete-time model for clearing wealths, we implicitly assumed a constant time-step between each clearing date of ${{\Delta t}}= 1$ throughout. In order to construct a continuous-time clearing model we will begin by making a discrete-time model with an explicit ${{\Delta t}}>0$ term. In fact, this is immediate from the prior construction with a minor alteration to the cash flow term. Herein we construct the net cash flow at time $t$ to be given by ${\Delta}c(t,{{\Delta t}}) := \int_{t-{{\Delta t}}}^t dc(s)$ and the nominal liabilities at time $t$ are similarly provided by ${\Delta}L(t,{{\Delta t}}) := \int_{t-{{\Delta t}}}^t dL(s)$ where both $dc$ and $dL$ are discussed above (additionally, we set $dc(-t) = 0$ and $dL(-t) = 0$ for any times $t < 0$). The choice of notation for ${\Delta}c$ and ${\Delta}L$ are to make explicit the “change” inherent in the construction.
With these parameters we can construct the ${{\Delta t}}$-discrete-time clearing process $V(t,{{\Delta t}})$ and exposure matrix $A(t,{{\Delta t}},V_t({{\Delta t}}))$ by: $$\begin{aligned}
\label{eq:discrete-Vdt} \begin{split}V(t,{{\Delta t}}) &= V(t-{{\Delta t}},{{\Delta t}}) + {\Delta}c(t,{{\Delta t}}) - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}V(t,{{\Delta t}})^-\\ &\qquad + A(t-{{\Delta t}},{{\Delta t}},V_{t-{{\Delta t}}}({{\Delta t}}))^{\top}V(t-{{\Delta t}},{{\Delta t}})^-\end{split}\\
\label{eq:discrete-Adt} \begin{split}a_{ij}(t,{{\Delta t}},V_t({{\Delta t}})) &= \frac{{\Delta}L_{ij}(t,{{\Delta t}})+a_{ij}(t-{{\Delta t}},{{\Delta t}},V_{t-{{\Delta t}}}({{\Delta t}}))V_i(t-{{\Delta t}},{{\Delta t}})^-}{\max\{\sum_{k \in {\mathcal{N}}_0} {\Delta}L_{ik}(t,{{\Delta t}})+V_i(t-{{\Delta t}},{{\Delta t}})^- , V_i(t,{{\Delta t}})^-\}}1_{\{i \neq 0\}}\\ &\qquad + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}} \qquad \forall i,j \in {\mathcal{N}}_0.\end{split}\end{aligned}$$ Here we assume that $V(t) = V(-1) \geq 0$ for every time $t < 0$ as in Assumption \[ass:initial\]. This construction can be computed either in continuous time $t \in {\mathbb{T}}$ with sliding intervals of size ${{\Delta t}}$ or at the discrete times $t \in \{0,{{\Delta t}},...,T\}$. The existence and uniqueness of this system follow exactly as in Theorem \[thm:discrete\] under Assumption \[ass:society\].
\[cor:discrete-cont\] Let $(dc,dL): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ define a dynamic financial network satisfying Assumption \[ass:society\] such that every bank has cash flow at least at the level dictated by nominal interbank liabilities, i.e., ${\Delta}c_i(t,{{\Delta t}}) \geq \sum_{j \in {\mathcal{N}}} {\Delta}L_{ji}(t,{{\Delta t}}) - \sum_{j \in {\mathcal{N}}_0} {\Delta}L_{ij}(t,{{\Delta t}})$ for all banks $i \in {\mathcal{N}}_0$, times $t \in {\mathbb{T}}$, and step-sizes ${{\Delta t}}> 0$. Under Assumption \[ass:initial\], there exists a unique solution of clearing wealths $V: {\mathbb{T}}\times {\mathbb{R}}_{++} \to {\mathbb{R}}^{n+1}$ to . Further, the clearing wealths are jointly continuous in time and step-size.
Now we want to consider the limiting behavior of this discrete-time system as ${{\Delta t}}$ tends to 0. To do so, first, we will consider the formulation of the relative exposures $a_{ij}$ from bank $i$ to $j$. From Corollary \[cor:discrete-cont\] and Assumption \[ass:society\], we know that for any time $t \in {\mathbb{T}}$ and bank $i \in {\mathcal{N}}$ it must follow that $\sum_{k \in {\mathcal{N}}_0} {\Delta}L_{ik}(t,{{\Delta t}}) + V_i(t-{{\Delta t}},{{\Delta t}})^- \geq V_i(t,{{\Delta t}})^-$ for ${{\Delta t}}> 0$ small enough due to the joint continuity of the wealths in time and step-size. Thus in the limiting case, as ${{\Delta t}}\searrow 0$, we find that we can consider the relative liabilities rather than the relative exposures, i.e., for ${{\Delta t}}$ small enough $$\label{eq:discrete-Adt-limiting}
\begin{split}a_{ij}(t,{{\Delta t}},V_t({{\Delta t}})) &= \frac{{\Delta}L_{ij}(t,{{\Delta t}})+a_{ij}(t-{{\Delta t}},{{\Delta t}},V_{t-{{\Delta t}}}({{\Delta t}}))V_i(t-{{\Delta t}},{{\Delta t}})^-}{\sum_{k \in {\mathcal{N}}_0} {\Delta}L_{ik}(t,{{\Delta t}})+V_i(t-{{\Delta t}},{{\Delta t}})^-}1_{\{i \neq 0\}}\\
&\qquad + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}} \qquad \forall i,j \in {\mathcal{N}}_0.\end{split}$$ Rearranging these terms we are able to deduce that, for any firm $i \in {\mathcal{N}}$, $$\label{eq:discrete-Adt-limiting2}
\begin{split}&[a_{ij}(t,{{\Delta t}},V_t({{\Delta t}})) - a_{ij}(t-{{\Delta t}},{{\Delta t}},V_{t-{{\Delta t}}}({{\Delta t}}))] V_i(t-{{\Delta t}},{{\Delta t}})^- \\ &\qquad = {\Delta}L_{ij}(t,{{\Delta t}}) - a_{ij}(t,{{\Delta t}},V_t({{\Delta t}})) \sum_{k \in {\mathcal{N}}_0} {\Delta}L_{ik}(t,{{\Delta t}}).\end{split}$$
Coupled with the assumption that the societal node always has positive wealth, we are thus able to consider the limiting behavior of as the step-size ${{\Delta t}}$ tends to 0. To do so, consider $$\begin{aligned}
\begin{split}V(t,{{\Delta t}}) &= V(t-{{\Delta t}},{{\Delta t}}) + {\Delta}c(t,{{\Delta t}}) - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}V(t,{{\Delta t}})^-\\ &\qquad + A(t-{{\Delta t}},{{\Delta t}},V_{t-{{\Delta t}}})^{\top}V(t-{{\Delta t}},{{\Delta t}})^-\end{split}\\
\begin{split} &= V(t-{{\Delta t}},{{\Delta t}}) + {\Delta}c(t,{{\Delta t}})\\ &\qquad - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}V(t,{{\Delta t}})^- + A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}V(t-{{\Delta t}},{{\Delta t}})^-\\ &\qquad - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}V(t-{{\Delta t}},{{\Delta t}})^- + A(t-{{\Delta t}},{{\Delta t}},V_{t-{{\Delta t}}})^{\top}V(t-{{\Delta t}},{{\Delta t}})^-\end{split}\\
\begin{split} &= V(t-{{\Delta t}},{{\Delta t}}) + {\Delta}c(t,{{\Delta t}}) - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}[V(t,{{\Delta t}})^- - V(t-{{\Delta t}},{{\Delta t}})^-]\\ &\qquad - {\Delta}L(t,{{\Delta t}})^{\top}\vec{1} + A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}{\Delta}L(t,{{\Delta t}}) \vec{1}.\end{split}\end{aligned}$$ Consider the notation for the matrix of distressed firms from the fictitious default algorithm (Algorithm \[alg:discrete\]), i.e., $\Lambda(V) \in \{0,1\}^{(n+1) \times (n+1)}$ is the diagonal matrix of banks in distress $$\Lambda_{ij}(V) = \begin{cases}1 &\text{if } i = j \neq 0 \text{ and } V_i < 0 \\ 0 &\text{else}\end{cases} \quad \forall i,j \in {\mathcal{N}}_0.$$ We are able to set $\Lambda_{00}(V) = 0$ without loss of generality since, by assumption, the outside node $0$ has no obligations into the system. Thus, as with , by continuity of the clearing wealths and ${{\Delta t}}$ small enough, we can conclude that except at specific event times (to be considered later, see Algorithm \[alg:continuous\]) it follows that $\Lambda(V(t,{{\Delta t}})) = \Lambda(V(t-{{\Delta t}},{{\Delta t}}))$. Thus, with this added notation we can reformulate the clearing wealths equation as $$\begin{aligned}
\begin{split}V(t,{{\Delta t}}) &= V(t-{{\Delta t}},{{\Delta t}}) + A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}\Lambda(V(t,{{\Delta t}})) [V(t,{{\Delta t}}) - V(t-{{\Delta t}},{{\Delta t}})] + {\Delta}c(t,{{\Delta t}}) \\ &\qquad - {\Delta}L(t,{{\Delta t}})^{\top}\vec{1} + A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}{\Delta}L(t,{{\Delta t}}) \vec{1}.\end{split}\end{aligned}$$ For the construction of a differential form we can consider the equivalent formulation $$\begin{aligned}
\label{eq:discrete-Vdt-limiting}
\begin{split} V(t,{{\Delta t}}) &- V(t-{{\Delta t}},{{\Delta t}}) =\\
&[I - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}\Lambda(V(t,{{\Delta t}}))]^{-1} \left(\begin{array}{l}{\Delta}c(t,{{\Delta t}}) - {\Delta}L(t,{{\Delta t}})^{\top}\vec{1}\\ \quad + A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}{\Delta}L(t,{{\Delta t}}) \vec{1}\end{array}\right).\end{split}\end{aligned}$$ Note that $I - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}\Lambda(V(t,{{\Delta t}}))$ is invertible by standard input-output results and as proven in Proposition \[prop:Leontief\].
Utilizing and and taking the limit as ${{\Delta t}}\searrow 0$, we are thus able to construct the joint differential system: $$\begin{aligned}
\label{eq:continuous-V} dV(t) &= [I - A(t)^{\top}\Lambda(V(t))]^{-1} \left(dc(t) - dL(t)^{\top}\vec{1} + A(t)^{\top}dL(t) \vec{1}\right)\\
\label{eq:continuous-A} da_{ij}(t) &= \begin{cases} \frac{d^2L_{ij}(t) - a_{ij}(t)\sum_{k \in {\mathcal{N}}_0} d^2L_{ik}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)} &\text{if } i \in {\mathcal{N}}, \; V_i(t) \geq 0 \\ \frac{dL_{ij}(t) - a_{ij}(t)\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)}{V_i(t)^-} &\text{if } i \in {\mathcal{N}}, \; V_i(t) < 0 \\ 0 &\text{if } i = 0\end{cases} \quad \forall i,j \in {\mathcal{N}}_0\end{aligned}$$ with initial conditions $V(0) \geq 0$ given and $a_{ij}(0) = \frac{dL_{ij}(0)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(0)}1_{\{i \neq 0\}} + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}}$ for all firms $i,j \in {\mathcal{N}}_0$. As in , $I - A(t)^{\top}\Lambda(V(t))$ is invertible by standard input-output results and as proven in Proposition \[prop:Leontief\]. The first case in is constructed by noting that $a_{ij}(t) = \frac{dL_{ij}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)}$ if $V_i(t) \geq 0$ and $i \in {\mathcal{N}}$ and $da_{0j}(t) = 0$ for any firm $j \in {\mathcal{N}}_0$ for all times $t$; the second case in follows from and taking the limit as ${{\Delta t}}\searrow 0$. Note that this differential system is discontinuous, with events at times when firms cross the 0 wealth boundary, i.e., when $\Lambda(V(t)) \neq \Lambda(V(t^-))$. As such, we will consider the differential system on the inter-event intervals, then update the differential system between these intervals. This is made more explicit in the proof of Theorem \[thm:continuous\] and in Algorithm \[alg:continuous\]. As with the discrete-time system , the relative exposures follow the incoming proportional obligations if a firm has a surplus wealth. When a firm is in distress, the relative exposures follow a path that provides the average relative obligations between new liabilities and the prior unpaid liabilities.
As in the discrete-time section we consider the debt to roll forward in this case. In this way we encode the notion of either intra-day dynamics in this model or when bankruptcy court would not settle debts before the terminal time $T$ for the system. To allow for insolvencies, we can consider some (deterministic) mechanism to determine when a bank becomes insolvent and restart the differential system with updated parameters from that time point, e.g., using an instantaneous auction as in [@CC15]; see also Remarks \[rem:loans\] and \[rem:cont-loans\].
We will complete our discussion of the construction of this differential system by providing some properties on the relative liabilities and exposures matrix $A$. Notably, these properties are those that would be expected from the discrete-time setting for the relative exposures. Namely, as a firm recovers from a distressed state its relative liabilities return to be only the fraction of incoming liabilities, that the relative exposures are bounded from below by 0 (and to society by $\delta$ as provided in Assumption \[ass:society\]), and the relative exposure matrix is row stochastic at all times.
\[prop:A\] Let $(dc,dL): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ define a dynamic financial network satisfying Assumption \[ass:society\]. Let $(V,A): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}$ be any solution of the differential system and satisfying Assumption \[ass:initial\]. The relative exposure matrix $A(t)$ satisfies the following properties:
1. \[prop:A1\] For any bank $i \in {\mathcal{N}}$, if $V_i(t) \nearrow 0$ as $t \nearrow \tau$ then $\lim_{t \nearrow \tau} a_{ij}(t) = \frac{dL_{ij}(\tau)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(\tau)}$.
2. \[prop:A2\] For all times $t \in {\mathbb{T}}$ and for any bank $i \in {\mathcal{N}}$, the elements $a_{ij}(t) \geq 0$ for all banks $j \in {\mathcal{N}}$ and $a_{i0}(t) \geq \delta$;
3. \[prop:A3\] For all times $t \in {\mathbb{T}}$ and for any bank $i \in {\mathcal{N}}_0$, the row sums $\sum_{k \in {\mathcal{N}}_0} a_{ik}(t) = 1$;
With this differential construction and , we seek to prove existence and uniqueness of the clearing solutions. For notational simplicity, define the space of relative exposure matrices $${\mathbb{A}}:= \left\{A \in [0,1]^{(n+1) \times (n+1)} \; | \; A\vec{1} = \vec{1}, \; a_{ii} = 0, \; a_{i0} \geq \delta \; \forall i \in {\mathcal{N}}, \; a_{0j} = \frac{1}{n} \; \forall j \in {\mathcal{N}}\right\}.$$ From Proposition \[prop:A\], we have already proven that if $(V,A): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}$ is a solution to the continuous-time Eisenberg-Noe system then $A(t) \in {\mathbb{A}}$ for all times $t \in {\mathbb{T}}$.
\[thm:continuous\] Let ${\mathbb{T}}= [0,T]$ be a finite time period and let $(dc,dL): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ define a dynamic financial network satisfying Assumption \[ass:society\]. There exists a unique strong solution to the clearing wealths and relative exposures $(V,A)$ satisfying and if $V(0) \in {\mathbb{R}}^{n+1}_{++}$.
\[rem:dc\] The restrictions on the cash flows $dc$ made in Assumption \[ass:society\] can be relaxed to depend explicitly on the wealths and relative exposures, i.e., $$dc(t) = \mu(t,c(t),V(t),A(t))dt + \sigma(t,c(t),V(t),A(t))dW(t).$$ This would still guarantee a unique strong solution of the clearing wealths and relative exposures as in Theorem \[thm:continuous\] so long as $\mu,\sigma$ satisfy a local linear growth condition, local Lipschitz condition, and $c(t)$ can be bounded above and below by elements of ${\mathcal{L}}_t^2({\mathbb{R}}^{n+1})$ for all time $t$.
We now present an algorithm for numerically computing an approximation of a single sample path for the continuous-time Eisenberg-Noe clearing system. To do so we consider Euler’s method for differential equations with an event finding algorithm.
\[alg:continuous\] Under the assumptions of Theorem \[thm:continuous\] for a fixed event $\omega \in \Omega$ the clearing wealths process $V: {\mathbb{T}}\to {\mathbb{R}}^{n+1}$ and relative exposures $A: {\mathbb{T}}\to {\mathbb{A}}$ can be found by the following algorithm. Fix a step-size ${{\Delta t}}_0 > 0$. Initialize $t = 0$, $V(0) \geq 0$ given, $a_{ij}(0) = \frac{dL_{ij}(0)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(0)}1_{\{i \neq 0\}} + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}}$, and $\Lambda = \{0\}^{(n+1) \times (n+1)}$. Repeat until $t \geq T$:
1. Initialize $\Lambda^0 \neq \Lambda$ and ${{\Delta t}}= {{\Delta t}}_0$.
2. Sample $Z \sim N(0,I)$.
3. Repeat until $\Lambda^0 = \Lambda$:
1. Set $\Lambda^0 = \Lambda$.
2. Compute $$\begin{aligned}
\bar\mu(t) &= (I - A(t)^{\top}\Lambda)^{-1} \left(\mu(t,c(t)) - \dot{L}(t)^{\top}\vec{1} + A(t)^{\top}\dot{L}(t) \vec{1}\right)\\
\bar\sigma(t) &= (I - A(t)^{\top}\Lambda)^{-1} \sigma(t,c(t)) Z.\end{aligned}$$
3. \[alg:dt\] Loop through each bank $i \in {\mathcal{N}}$:
1. \[alg:dt1\] If $V_i(t) > 0$, $\bar\mu_i(t) < 0$, and $\bar\sigma_i(t)^2 - 4\bar\mu_i(t)V_i(t) \geq 0$ then $${{\Delta t}}= \min\left\{{{\Delta t}}\; , \; \left(\frac{-\bar\sigma_i(t) - \sqrt{\bar\sigma_i(t)^2 - 4\bar\mu_i(t)V_i(t)}}{2\bar\mu_i(t)}\right)^2\right\}.$$
2. \[alg:dt4\] If $V_i(t) < 0$, $\bar\mu_i(t) \neq 0$, and $\bar\sigma_i(t)^2 - 4\bar\mu_i(t)V_i(t) \geq 0$ then $${{\Delta t}}= \min\left\{{{\Delta t}}\; , \; \left(\frac{-\bar\sigma_i(t) + \sqrt{\bar\sigma_i(t)^2 - 4\bar\mu_i(t)V_i(t)}}{2\bar\mu_i(t)}\right)^2\right\}.$$
3. \[alg:dt25\] If $\bar\mu_i(t) = 0$ and $V_i(t)\bar\sigma_i(t) < 0$ then ${{\Delta t}}= \min\left\{{{\Delta t}}\; , \; V_i(t)^2 / \bar\sigma_i(t)^2\right\}$.
4. \[alg:dt7\] If $\bar\mu_i(t)\bar\sigma_i(t) < 0$ then ${{\Delta t}}= \min\left\{{{\Delta t}}\; , \; \bar\sigma_i(t)^2 / \bar\mu_i(t)^2\right\}$.
4. Compute $\Delta V(t) = \bar\mu(t){{\Delta t}}+ \bar\sigma(t) \sqrt{{{\Delta t}}}$.
5. Define the matrix $\Lambda \in \{0,1\}^{(n+1) \times (n+1)}$ such that $$\Lambda_{ij} = \begin{cases}0 &\text{if } i = j \neq 0, \; V_i(t) > 0 \text{ or } [V_i(t) = 0, \; \Delta V_i(t) \geq 0] \\ 1 &\text{if } i = j \neq 0, \; V_i(t) < 0 \text{ or } [V_i(t) = 0, \; \Delta V_i(t) < 0] \\ 0 &\text{else} \end{cases}$$
4. Define the matrix $\bar\Lambda \in \{0,1\}^{(n+1) \times (n+1)}$ so that $\bar\Lambda = \begin{cases}1 &\text{if } i = j \neq 0, \; V_i(t) < 0 \\ 0 &\text{else}\end{cases}$.
5. Set $$\begin{aligned}
c(t + {{\Delta t}}) &= c(t) + \mu(t,c(t)){{\Delta t}}+ \sigma(t,c(t))\sqrt{{{\Delta t}}}Z\\
V(t + {{\Delta t}}) &= V(t) + \Delta V(t)\\
A(t + {{\Delta t}}) &= \bar\Lambda\left[A(t) + \operatorname{diag}(V(t)^-)^{-1}[\dot{L}(t) - A(t) \ast (\dot{L}(t)\mathbbm{1})]{{\Delta t}}\right]\\
&\qquad + (I - \bar\Lambda)\operatorname{diag}(\dot{L}(t)\vec{1})^{-1}\dot{L}(t).\end{aligned}$$ where $\mathbbm{1} = \{1\}^{(n+1) \times (n+1)}$ and $\ast$ denotes the element-wise multiplication operator.
6. Increment $t = t + {{\Delta t}}$.
If $t > T$ then set $$\begin{aligned}
c(T) &= c(t-{{\Delta t}}) + \frac{c(t) - c(t-{{\Delta t}})}{{{\Delta t}}}(T - [t-{{\Delta t}}])\\
V(T) &= V(t-{{\Delta t}}) + \frac{V(t) - V(t-{{\Delta t}})}{{{\Delta t}}}(T - [t-{{\Delta t}}])\\
A(T) &= A(t-{{\Delta t}}) + \frac{A(t) - A(t-{{\Delta t}})}{{{\Delta t}}}(T - [t-{{\Delta t}}]).\end{aligned}$$
In the above event-finding algorithm for the continuous-time Eisenberg-Noe system, the main concern is that we do not increment time too far in any step so as to pass over an event (e.g., a solvent bank becoming a distressed bank). This is accomplished in the loop described in step . In particular, - guarantee that $V_i(t) + \bar\mu_i(t){{\Delta t}}+ \bar\sigma_i(t)\sqrt{{{\Delta t}}}$ is nonnegative if $V_i(t) > 0$ and nonpositive if $V_i(t) < 0$. The additional condition in guarantees that the direction of $\bar\mu_i(t){{\Delta t}}+ \bar\sigma_i(t)\sqrt{{{\Delta t}}}$ is maintained as ${{\Delta t}}$ shrinks, i.e., if ${{\Delta t}}$ is too large then the direction of the change in wealth could be impacted by choosing a smaller (and thus more accurate) step-size. While not strictly necessary, we include step as it improves the accuracy of the algorithm.
\[rem:cont-loans\] As with the discrete-time setting discussed in Remark \[rem:loans\], we can introduce the concept of loans from a central bank to the continuous-time Eisenberg-Noe system. To do so we would need to introduce stopping times associated with each bank becoming insolvent. Notably, the receivership setting would act the same as our described continuous-time Eisenberg-Noe system after insolvencies occur. In contrast, a pure auction model would eliminate all need for continuous-time contagion. At the time of the auction a static system would be considered, e.g., the static Eisenberg-Noe clearing, based on the results of the auction; this would update the cash flow parameters for each firm going forward, but no dynamic contagion would need to be modeled.
We wish to conclude our discussion of the continuous-time Eisenberg-Noe system by providing a result on how the unique solution to the discrete-time solution converges to the continuous-time solution as ${{\Delta t}}\searrow 0$. That is, we wish to consider how the unique clearing wealths and relative exposures solving the discrete-time systems and converge to those in continuous-time Eisenberg-Noe system and as the step-size decreases to 0.
\[lemma:limiting\] Consider the setting of Corollary \[cor:discrete-cont\] and Theorem \[thm:continuous\]. Then the continuous-time clearing solutions at any time $t \in {\mathbb{T}}$ is the limit of the discrete-time solution as the step-size tends to 0, i.e., $(V(t),A(t)) = \lim_{{{\Delta t}}\searrow 0} (V(t,{{\Delta t}}),A(t,{{\Delta t}},V_t({{\Delta t}})))$ where $(V(\cdot,{{\Delta t}}),A(\cdot,{{\Delta t}}))$ satisfy and and $(V,A)$ satisfy and .
Discussion {#sec:discussion}
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In this section we will consider the implications of time on the clearing solutions in the Eisenberg-Noe setting. Specifically, we will focus on the continuous-time formulation, though all conclusions hold in the discrete-time setting as well. Notably, we deduce rules so as to recreate the static Eisenberg-Noe clearing solution via our continuous-time differential system, which (independently) replicates the results from [@sonin2017]. Further, we consider the implications of time dynamics on the health of the financial system by determining bounds on how different the static clearing solution and a dynamic solution might be. This demonstrates the importance of time dynamics on accurately assessing the health and wealth of the financial system.
The static model as a differential system {#sec:discussion-EN}
-----------------------------------------
Herein we will consider the case in which the relative liabilities are constant through time. That is, we consider the setting in which $dL_{ij}(s)/\sum_{k \in {\mathcal{N}}_0} dL_{ik}(s) = dL_{ij}(t)/\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)$ for all times $s,t \in {\mathbb{T}}$ and firms $i,j \in {\mathcal{N}}_0$ so long as $\sum_{k \in {\mathcal{N}}_0} dL_{ik}(s), \sum_{k \in {\mathcal{N}}_0} dL_{ik}(t) > 0$. The key implication of this assumption is that the relative exposures matrix in can be found explicitly to equal the relative liabilities $$a_{ij}(t) = \pi_{ij} := \begin{cases} \frac{dL_{ij}(s_i)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(s_i)} &\text{if } s_i < \sup{\mathbb{T}}\\ \frac{1}{n} &\text{if } s_i = \sup{\mathbb{T}}, \; j \neq i\\ 0 &\text{if } s_i = \sup{\mathbb{T}}, \; j = i\end{cases}$$ for all times $t$ and banks $i,j \in {\mathcal{N}}_0$ where $s_i \in \{t \in {\mathbb{T}}\; | \; \sum_{k \in {\mathcal{N}}_0} dL_{ik}(t) > 0\}$ chosen arbitrarily strictly less than $\sup{\mathbb{T}}$ (and $s_i = \sup{\mathbb{T}}$ if the supremum is taken over the empty set).
Further, expanding and solving the differential system , we deduce that the continuous-time clearing wealths must satisfy the fixed point problem $$\label{eq:continuous-static}
V(t) = V(0) + \int_0^t dc(s) - \Pi^{\top}V(t)^-$$ at all time $t \in {\mathbb{T}}$. Therefore, if $\int_0^t dc(s) \geq \int_0^t dL(s)^{\top}\vec{1} - \int_0^t dL(s)\vec{1}$ at some time $t$, it follows that $V(t)$ are the *static* clearing wealths to the Eisenberg-Noe system with aggregated data with nominal liabilities matrix defined by $\int_0^t dL(s)$ and (incoming) external cash flow given by $\int_0^t dc(s) - \left(\int_0^t dL(s)^{\top}\vec{1} - \int_0^t dL(s)\vec{1}\right)$. Importantly, this means that, if the relative liabilities are kept constant over time, taking aggregated data and considering the static Eisenberg-Noe framework will produce the same *final* clearing wealths as the dynamic Eisenberg-Noe setting presented in this paper. However, though the set of defaulting banks is the same as in the static setting, the order of defaults need not strictly follow the order given in the fictitious default algorithm of [@EN01].
\[defn:order-default\] A bank is called a ***$k$th-order default*** in the static Eisenberg-Noe setting if it is determined to be in default in the $k$th iteration of the fictitious default algorithm (see, e.g., [@EN01 Section 3.1] or the inner loop of Algorithm \[alg:discrete\]).
We note that the first-order defaults are exactly those firms that have negative wealth even if it has no negative exposure to other firms (i.e., all other firms satisfy their obligations in full).
\[prop:order-defaults\] Let $(x,\bar{L}) \in {\mathbb{R}}_+^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ denote the static incoming external cash flow and nominal liabilities. Define a dynamic system over the time period ${\mathbb{T}}= [0,T]$ such that $V(0) \in [0,x]$, $dL(t) = \frac{1}{T}\bar{L}dt$, and $dc(t) = \frac{1}{T}\left(x - V(0) + \bar{L}^{\top}\vec{1} - \bar{L}\vec{1}\right)dt$. The clearing wealths at the terminal time $V(T)$ are equal to those given in the static setting. Additionally, no firm will ever recover from distress in the dynamic setting. Finally, the first $k$th-order default will occur only after the first $(k-1)$th-order default in the static fictitious default algorithm; in particular, the first firm to become distressed will be a first-order default in the static fictitious default algorithm.
The fact that the clearing wealths $V(T)$ are equal to the static Eisenberg-Noe clearing wealths (as defined in Proposition \[prop:EN-e\]) follows from and the logic given in the proof of Lemma \[lemma:limiting\]. Additionally, since $dc(t)$ is constant in time and firms are beginning in a solvent state, over time the unpaid liabilities may accumulate as a negative factor on bank balance sheets, but there is no outlet to allow for a firm to recover from distress. Finally, by definition, a $k$th-order default is only driven into distress through the failure of the $(k-1)$th-order defaults (and *not* solely by the $(k-2)$th-order defaults). Therefore, by way of contradiction, if a $k$th-order default were to occur before any $(k-1)$th-order default then such a firm must default without regard to what happens to the $(k-1)$th-order defaults, i.e., this firm must be a $(k-1)$th-order default. By this same logic, the first firm to become distressed must be a first-order default.
The notion of real defaulting times differing from the order introduced by the fictitious default algorithm of [@EN01] is unsurprising. Consider a financial system with two subgraphs that are only connected through their obligations to the societal node. By construction, the default of a firm in one subgraph will have no impact on the firms in the other subgraph. Thus we can construct a network so that all defaults in one subgraph (including higher order defaults as defined in Definition \[defn:order-default\]) occur before any first-order defaults in the other subgraph.
Notably, Proposition \[prop:order-defaults\] states that, provided the aggregate data (until the terminal time) is kept constant, the clearing wealths at the terminal time will be path-independent in this setting. We will demonstrate this with an illustrative example demonstrating this setting in a small 4 bank (plus societal node) system. In particular, we will consider the cash flows $c$ to be defined as a Brownian bridge so as to provide the appropriate aggregate data at the terminal time.
\[ex:differentialEN\] Consider a financial system with four banks, each with an additional obligation to an external societal node. Consider the time interval ${\mathbb{T}}= [0,1]$ with aggregated data such that the initial wealths are given by $V(0) = (100,1,3,2,5)^{\top}$, cash flows $dc$ are such that $\int_0^1 dc(s) = \bar{L}^{\top}\vec{1} - \bar{L}\vec{1}$, and where the nominal liabilities matrix $dL = \bar{L}dt$ is defined by $$\bar{L} = \left(\begin{array}{ccccc}0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 7 & 1 & 1 \\ 3 & 3 & 0 & 3 & 3 \\ 3 & 1 & 1 & 0 & 1 \\ 3 & 1 & 2 & 1 & 0 \end{array}\right).$$ The *static* Eisenberg-Noe clearing wealths, with nominal liabilities $\bar{L}$ and external assets $V(0)$, are found to be $V(1) \approx (109.38,-6.81,-3.03,-0.32,1.62)^{\top}$. Further, from the static fictitious default algorithm, we can determine that bank 1 is a first-order default, bank 2 is a second-order default, and bank 3 is a third-order default. Consider now three dynamic settings which are differentiated only by the choice of the cash flows $dc$:
1. Consider the deterministic setting introduced in Proposition \[prop:order-defaults\], i.e., $dc(t) = [\bar{L}^{\top}\vec{1} - \bar{L}\vec{1}]dt$ for all times $t \in {\mathbb{T}}$.
2. Consider a Brownian bridge with low volatility, i.e., $dc(t) = \frac{\bar{L}^{\top}\vec{1} - \bar{L}\vec{1} - c(t)}{1-t}dt + dW(t)$ for vector of independent Brownian motions $W$ and with $c(0) = 0$.
3. Consider a Brownian bridge with high volatility, i.e., $dc(t) = \frac{\bar{L}^{\top}\vec{1} - \bar{L}\vec{1} - c(t)}{1-t}dt + 5dW(t)$ for vector of independent Brownian motions $W$ and with $c(0) = 0$.
[0.45]{} ![Example \[ex:differentialEN\]: Comparison of clearing wealths under deterministic and random cash flows that aggregate to the same terminal values.[]{data-label="fig:EN"}](EN-BB0.eps "fig:"){width="\linewidth"}
[0.45]{} ![Example \[ex:differentialEN\]: Comparison of clearing wealths under deterministic and random cash flows that aggregate to the same terminal values.[]{data-label="fig:EN"}](EN-BB1.eps "fig:"){width="\linewidth"}
[0.45]{} ![Example \[ex:differentialEN\]: Comparison of clearing wealths under deterministic and random cash flows that aggregate to the same terminal values.[]{data-label="fig:EN"}](EN-BB5.eps "fig:"){width="\linewidth"}
A single sample path for each dynamic setting is provided. In each plot we reduce the equity of the societal node by 100 so that it begins with an initial wealth of 0, but more importantly so that it can easily be displayed on the same plot as the other 4 institutions. First, we point out that, as indicated by the circles at the terminal time in each plot, the terminal wealths of the continuous-time setting match up with the clearing wealths in the static model. We further note that in the deterministic setting (Figure \[fig:EN-BB0\]) and the low volatility setting (Figure \[fig:EN-BB1\]) the order of defaults is maintained. However, in the high volatility setting (Figure \[fig:EN-BB5\]) the order of defaults given by the fictitious default algorithm no longer holds.
The implications of time dynamics {#sec:discussion-time}
---------------------------------
Now we will consider the case in which the relative liabilities change over time. As in the prior discussion, we will focus on the setting in which the aggregate cash flows and interbank liabilities correspond to a static Eisenberg-Noe model. As the liabilities are now changing over time there is an inherent prioritization in the obligations due to the rolling forward of unpaid debts. Any earlier obligations are more likely to be paid, and accumulate to be paid proportionally with any new obligations. As such, by altering only the rate at which the liabilities are due, the terminal wealths and also the set of defaulting firms can be modified. Proposition \[prop:defaults\] provides analysis on which banks will always be solvent and which will always be in default at the terminal time. In particular, the results of Proposition \[prop:defaults\] show that the *static* Eisenberg-Noe model applied to aggregate data can produce a viewpoint on the health of the financial system that is either incorrectly optimistic or pessimistic; without explicitly knowing the dynamics of the cash flows and liabilities, only rough estimates can be considered. This is in contrast to, e.g., [@GY14] in which data from the European Banking Authority’s 2011 stress test was utilized to assess the health of the European financial system without time dynamics.
\[defn:order-solvent\] In the static Eisenberg-Noe setting a bank is called a ***first-order solvency*** if it has positive wealth even under the maximum negative exposure (i.e., no other firms pay at all).
Note that, by assumption, the societal node $0$ will always be a first-order solvent institution.
\[prop:defaults\] Let $(x,\bar{L}) \in {\mathbb{R}}_+^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ denote the static incoming external cash flow and nominal liabilities. Define a dynamic system over the time period ${\mathbb{T}}= [0,T]$ such that $V(0) \in [0,x]$, $\int_0^T dL(t) = \bar{L}$, and $\int_0^T dc(t) = x - V(0) + \bar{L}^{\top}\vec{1} - \bar{L}\vec{1}$. At time $T$, those banks that are first-order defaults in the static setting will be in default in the dynamic setting. Similarly, those banks that are first-order solvencies in the static setting will be solvent in the dynamic setting at the terminal time.
This result follows from the definition of a first-order default or solvency as such firms allow us to disregard all interbank dynamics.
To conclude this discussion, we will consider two examples with the same aggregate values as given in Example \[ex:differentialEN\]. The first example considers the case in which the nominal liabilities are shifted in time so as to have the maximum possible number of banks be solvent or, vice versa, the maximum number of banks be in default at the terminal time. The second example considers a fixed structure for the nominal liabilities in time (but non-constant relative liabilities), thus demonstrating the path-dependence of the clearing wealths on the cash flows.
\[ex:dL-dependence\] Consider the financial system described in Example \[ex:differentialEN\] over the time interval ${\mathbb{T}}= [0,1]$ with aggregated data such that the initial wealths $V(0) = (100,1,3,2,5)^{\top}$ and where the aggregate nominal liabilities matrix is defined by $\bar{L}$. Further, consider the cash flows $dc(t) = dL(t)^{\top}\vec{1} - dL(t)\vec{1}$ for all times $t \in {\mathbb{T}}$ where $dL$ is either:
1. prioritizing the defaulting firms: $dL(t) = 5\bar{L}\left(E_0 1_{\{t \in (0.8,1]\}} + \sum_{i \in {\mathcal{N}}}E_i 1_{\{t \in (0.2(i-1),0.2i]\}}\right)$, or
2. prioritizing society: $dL(t) = 5\bar{L}\left(E_0 1_{\{t \in [0,0.2)\}} + \sum_{i \in {\mathcal{N}}}E_i 1_{\{t \in (0.2i,0.2(i+1)]\}}\right)$
where the collection of matrices $E_i \in \{0,1\}^{(n+1) \times (n+1)}$ are such that $(E_i)_{ii} = 1$ and all other elements are set to $0$.
[0.45]{} ![Example \[ex:dL-dependence\]: Comparison of clearing wealths under different ordering of the nominal liabilities in time that aggregate to the same terminal values.[]{data-label="fig:dL"}](dL-1Default.eps "fig:"){width="\linewidth"}
[0.45]{} ![Example \[ex:dL-dependence\]: Comparison of clearing wealths under different ordering of the nominal liabilities in time that aggregate to the same terminal values.[]{data-label="fig:dL"}](dL-1Solvent.eps "fig:"){width="\linewidth"}
As in Figure \[fig:EN\], the circles at the terminal time in both plots denote the clearing wealths under the static Eisenberg-Noe setting. It is clear in both examples that the terminal dynamic clearing wealths now are *not* equal to the static wealths. Further, by choosing the liabilities to be introduced in the order provided we provide the settings so that only the first-order defaults, Bank 1, have negative terminal wealth (Figure \[fig:dL-1default\]) or so that only the first-order solvencies, the societal firm, have positive terminal wealth (Figure \[fig:dL-1solvent\]). In Figure \[fig:dL-1default\], we notice that firms 2 and 3 have a terminal wealth of $0$, so although they are not defaulting, they do not have any positive equity either. Further, it is clear that though all financial firms have improved their wealth given this ordering of the nominal liabilities, the societal wealth is decreased (though to a lesser amount than the aggregate improvement for the banks) in comparison to the static results. In contrast, in the second scenario in which obligations to society are first (Figure \[fig:dL-1solvent\]), the societal wealths are greater than those provided in the static setting but all banks have less wealth. Notice further that, even after the obligations to society have “ended” at time 0.2 the societal wealth still increases. This occurs as the banks in distress receive money as their incoming liabilities come due and thus they have cash to immediately transfer to cover the prior unpaid obligations to, e.g., society. Finally, this numerically verifies the results of Proposition \[prop:defaults\] and demonstrates the importance of understanding the order of obligations for an accurate measure of the health of the financial system.
\[ex:dc-dependence\] Consider the financial system described in Example \[ex:differentialEN\] over the time interval ${\mathbb{T}}= [0,1]$ with aggregated data such that the initial wealths $V(0) = (100,1,3,2,5)^{\top}$ and where the aggregate nominal liabilities matrix is defined by $\bar{L}$. Further, consider the nominal liabilities determined by $$dL(t) = \bar{L}\left(\begin{array}{l}E_0 + \frac{1}{0.237}E_1 1_{\{t \in [0.145,0.382]\}} + \frac{1}{0.178}E_2 1_{\{t \in [0.331,0.509]\}}\\ \quad + \frac{1}{0.439}E_3 1_{\{t \in [0.301,0.740]\}} + \frac{1}{0.105}E_4 1_{\{t \in [0.673,0.778]\}}\end{array}\right)$$ where the collection of matrices $E_i \in \{0,1\}^{(n+1) \times (n+1)}$ are such that $(E_i)_{ii} = 1$ and all other elements are set to $0$. Finally, consider the cash flows determined by a Brownian bridge with volatility of 2, i.e., $dc(t) = \frac{\bar{L}^{\top}\vec{1} - \bar{L}\vec{1} - c(t)}{1-t}dt + 2dW(t)$ for vector of independent Brownian motions $W$ and with $c(0) = 0$.
![Example \[ex:dc-dependence\]: Empirical distribution of the terminal societal wealths under random cash flows. The $\times$ marks the societal wealth under the static Eisenberg-Noe framework with aggregated data.[]{data-label="fig:dc"}](dc-10000.eps){width=".5\linewidth"}
Figure \[fig:dc\] depicts the empirical distribution of the terminal societal wealths under 10,000 samples of the Brownian bridge cash flows. The black curve depicts the kernel density for this empirical distribution. The $\times$ illustrates the societal wealth under the static Eisenberg-Noe framework considering the aggregated data (as provided in Example \[ex:differentialEN\]). The key takeaway of this figure is the payments to society range from 8.12 to 10.20 out of an obligated 12, i.e., society can experience anywhere from 16% to 32% shortfall in payments depending on the sample path. This also implies that society can experience anywhere from a 13.4% decrease to an 8.8% increase over the payments found under the static Eisenberg-Noe model. Similar results can be shown for the other firms in the system as well. Notably, firms 2, 3, and 4 all have simulations in which they are solvent at the terminal time and simulations in which they are defaulting on their obligations. Recall none of these three firms are first-order defaults or first-order solvencies. Empirically, firm 2 (a second-order default) is found to default in approximately 98% of the simulations; firm 3 (a third-order default) is found to default in approximately 3.6% of simulations; firm 4 (which does not default in the static setting) is found to default in just 0.03% of the provided simulations (i.e., 3 out of the 10,000 simulations). Therefore, if relative liabilities are not constant over time, the order of the cash flows can have a significant impact on the health of the system.
Conclusion {#sec:conclusion}
==========
In this paper we considered an extension of the financial contagion model of [@EN01] to allow for cash flows and obligations to be dynamic in time. We presented this model in both discrete and continuous time, thus extending the frameworks of [@CC15; @ferrara16; @KV16] which consider only discrete-time clearing. Notably, we determine conditions for existence and uniqueness of the clearing solutions under deterministic and Itô settings. In this way, we have written a dynamical system for the Eisenberg-Noe contagion model that may include an inherent prioritization scheme. Specifically, we determine that if the relative liabilities are constant over time then the dynamic Eisenberg-Noe model presented herein will reproduce the static system at the terminal time in a path-independent manner. Notably, in such a setting, we are able to determine the true defaulting order rather than the fictitious order found in the fictitious default algorithm that is widely used in computing static clearing models. If, however, the relative liabilities are not constant over time, then we determine that the static Eisenberg-Noe model may report an incorrectly optimistic or pessimistic picture of the financial system.
Three clear extensions of this model are apparent to us, and which we foresee creating further divergence between static and dynamic models. The first extension is the inclusion of illiquid assets and fire sales. In the static models, e.g., [@CFS05; @AFM16; @feinstein2015illiquid], there is no first mover advantage to liquidating assets as all firms receive the same price. However, in a dynamic model there may be advantage to liquidating early in order to receive a higher price, but which may precipitate a larger fire sale amongst the other firms. The second extension is the inclusion of contingent payments and credit default swaps. In the static setting this has recently been considered by [@banerjee2017insurance; @SSB16; @SSB16b]. By considering the network dynamics to be dependent on the history of clearing wealths, many of the difficulties reported in the static works are likely to be resolved naturally; we refer to [@banerjee2017insurance] which provides an initial discussion of this extension. The final extension, for which we believe the proposed dynamic model will be especially useful, is in considering strategic or dynamic actions by the market participants, e.g., incorporating bankruptcy costs and strategic decisions on rolling forward of debt. We feel that the continuous-time framework will be particularly suitable for these extensions as it allows us to construct unique clearing solutions without requiring strong monotonicity assumption.
Proof of results in Section \[sec:discrete\] {#sec:discrete-proofs}
============================================
We will prove this result inductively. First consider time $t = 0$. Recall from Assumption \[ass:initial\] that $V(-1) \geq 0$. The clearing wealths at time $0$ follow the fixed point equation $$V(0) = \Phi(0,V(0)) := V(-1) + c(0) - A(0,V_0)^{\top}V(0)^-.$$ Note that, by construction, $A(0,V_0)^{\top}V(0)^- \leq L(0)^{\top}\vec{1}$. Therefore any clearing solution must fall within the compact range $[V(-1) + c(0) -L(0)^{\top}\vec{1},V(-1) + c(0)] \subseteq {\mathbb{R}}^{n+1}$. It is clear from the definition that $\Phi(0,\cdot)$ is a monotonic operator, and thus there exists a greatest and least clearing solution $V^\uparrow(0) \geq V^\downarrow(0)$ by Tarski’s fixed point theorem [@Z86 Theorem 11.E], both of which must fall within this domain. Further, $a_{ij}(0,V_0) = \frac{L_{ij}}{\sum_{k \in {\mathcal{N}}_0} L_{ik}}$ (for $i \in {\mathcal{N}}$ and $j \in {\mathcal{N}}_0$) for any wealth $V(0)$ in this domain since $V(-1) + c(0) - L(0)^{\top}\vec{1} \geq -L(0) \vec{1} = -\bar p(0,V_{-1})$. We will prove uniqueness as it is done in [@EN01] by noting additionally that we can assume that the societal node will always have positive equity (i.e., $V^\downarrow(0) \geq 0$). First, we will show that the positive equities are the same for every firm no matter which clearing solution is chosen, i.e., $V_i^\uparrow(0)^+ = V_i^\downarrow(0)^+$ for every firm $i \in {\mathcal{N}}_0$. By definition $V^\uparrow(0) \geq V^\downarrow(0)$ and using $\sum_{j \in {\mathcal{N}}_0} a_{ij}(0) = 1$ for every firm $i \in {\mathcal{N}}_0$ we recover $$\begin{aligned}
\sum_{i \in {\mathcal{N}}_0} V_i^\uparrow(0)^+ &= \sum_{i \in {\mathcal{N}}_0} \left[V_i^\uparrow(0) + V_i^\uparrow(0)^-\right]\\
&= \sum_{i \in {\mathcal{N}}_0} \left[V_i(-1) + c_i(0) - \sum_{j \in {\mathcal{N}}} a_{ji}(0,V_0^\uparrow) V_j^\uparrow(0)^- + V_i^\uparrow(0)^-\right]\\
&= \sum_{i \in {\mathcal{N}}_0} \left[V_i(-1) + c_i(0)\right] - \sum_{j \in {\mathcal{N}}} V_j^\uparrow(0)^- \sum_{i \in {\mathcal{N}}_0} a_{ji}(0,V_0^\uparrow) + \sum_{i \in {\mathcal{N}}_0} V_i^\uparrow(0)^-\\
&= \sum_{i \in {\mathcal{N}}_0} \left[V_i(-1) + c_i(0)\right] = \sum_{i \in {\mathcal{N}}_0} V_i^\downarrow(0)^+.\end{aligned}$$ Therefore it must be the case that $V_i^\uparrow(0)^+ = V_i^\downarrow(0)^+$ for all firms $i \in {\mathcal{N}}_0$. Since we assume that the societal node will always have positive equity, it must be the case that $V_0^\uparrow(0) = V_0^\downarrow(0)$. Now since we assume that each node $i \in {\mathcal{N}}$ owes to the societal node, if any firm $i \in {\mathcal{N}}$ is such that $0 \geq V_i^\uparrow(0) > V_i^\downarrow(0)$ then it must be that $V_0^\uparrow(0) > V_0^\downarrow(0)$, which is a contraction.
Continuing with the inductive argument, assume that the history of clearing wealths $V_{t-1}$ up to time $t-1$ is fixed and known. The clearing wealths at time $t$ follow the fixed point equation $$V(t) = \Phi(t,V(t)) := V(t-1) + c(t) - A(t,V_t)^{\top}V(t)^- + A(t-1,V_{t-1})^{\top}V(t-1)^-.$$ Note that, by construction, $A(t,V_t)^{\top}V(t)^- \leq L(t)^{\top}\vec{1} + A(t-1,V_{t-1})^{\top}V(t-1)^-$. Therefore any clearing solution must fall within the compact range $[V(t-1) + c(t) - L(t)^{\top}\vec{1} , V(t-1) + c(t) + A(t-1,V_{t-1})^{\top}V(t-1)^-] \subseteq {\mathbb{R}}^{n+1}$. Further, $a_{ij}(t,V_t) = \frac{L_{ij} + a_{ij}(t-1,V_{t-1})V_i(t-1)^-}{\sum_{k \in {\mathcal{N}}_0} L_{ik} + V_i(t-1)^-}$ (for $i \in {\mathcal{N}}$ and $j \in {\mathcal{N}}_0$) for any wealth $V(t)$ in this domain since $V(t-1) + c(t) - L(t)^{\top}\vec{1} \geq -V(t-1)^- - L(t) \vec{1} = -\bar p(t,V_{t-1})$. Thus we can apply the same logic as in the time $0$ case to recover existence and uniqueness of the clearing wealths $V(t)$ at time $t$.
This follows immediately from the proof of Theorem \[thm:discrete\] using induction and noting that the lattice upper and lower bounds for the domain and range spaces of $\Phi(s,\cdot)$ are subsets of ${\mathcal{L}}_s^p({\mathbb{R}}^{n+1})$. Therefore any clearing solution $V(t)$ is bounded above and below by an element of ${\mathcal{L}}_t^p({\mathbb{R}}^{n+1})$ and the result is proven.
Proof of results in Section \[sec:continuous\] {#sec:continuous-proof}
==============================================
Existence and uniqueness of the clearing solutions follows from Theorem \[thm:discrete\]. To prove continuity we will employ an induction argument. To do so, we will consider the reduced domain $V: {\mathbb{T}}\times [\epsilon,\infty) \to {\mathbb{R}}^{n+1}$ for some $\epsilon > 0$. That is, we restrict the step-size ${{\Delta t}}\geq \epsilon$. As we will demonstrate that the continuity argument holds for any $\epsilon > 0$ then the desired result must hold as well. Before continuing, consider an expanded version of the recursive formulation of , i.e., $$\label{eq:discrete-Vdt-expand}
V(t,{{\Delta t}}) = V(-1) + \int_0^t dc(s) - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}V(t,{{\Delta t}})^-
$$ for all times $t \in {\mathbb{T}}$. Fix the minimal step-size $\epsilon > 0$. Note that the relative exposures satisfy $a_{ij}(t,{{\Delta t}},V_t({{\Delta t}})) := \frac{\int_0^t dL_{ij}(s)}{\sum_{k \in {\mathcal{N}}_0} \int_0^t dL_{ik}(s)}$ for any time $t \in [0,\epsilon)$ by the assumption that $V(-1) \geq 0$. Thus we can conclude $V: [0,\epsilon) \times [\epsilon,\infty) \to {\mathbb{R}}^{n+1}$ is continuous by an application of [@feinstein2014measures Proposition A.2]. Now, by way of induction, assume that $V: [0,s) \times [\epsilon,\infty) \to {\mathbb{R}}^{n+1}$ is continuous for some $s > 0$. Again, by [@feinstein2014measures Proposition A.2], we are able to immediately conclude that $V: [0,s+\epsilon) \cap {\mathbb{T}}\times [\epsilon,\infty) \to {\mathbb{R}}^{n+1}$ is continuous. As we are able to always extend the continuity result by $\epsilon > 0$ in time, the result is proven.
1. Consider firm $i \in {\mathcal{N}}$. By assumption we have that $a_{ij}(t)$ for $t \nearrow \tau$ solves the first order differential equation: $$\frac{da_{ij}(t)}{dt}+\frac{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)/dt}{V_i(t)^-}a_{ij}(t)=\frac{dL_{ij}(t)/dt}{V_i(t)^-}.$$ For sake of simplicity, let this differential equation start at time $0$ with $V_i(0) < 0$ and some initial value $a_{ij}(0)$. Then this differential equation can be solved via the integrating factor $\nu(t) := \int_0^t \frac{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(s)}{V_i(s)^-}ds $. Thus for $t \nearrow \tau$ it follow that $$a_{ij}(t)= e^{-\nu(t)}\left[\int_0^t e^{\nu(s)} \frac{dL_{ij}(s)}{V_i(s)^-} + a_{ij}(0)\right].$$ Therefore, utilizing L’Hôspital’s rule, $$\begin{aligned}
\lim_{t \nearrow \tau} a_{ij}(t) &= \lim_{t \nearrow \tau} e^{-\nu(t)}\left[\int_0^t e^{\nu(s)} \frac{dL_{ij}(s)}{V_i(s)^-} + a_{ij}(0)\right]\\
&= \lim_{t \nearrow \tau} \frac{e^{\nu(t)}\frac{dL_{ij}(t)}{V_i(t)^-}}{e^{\nu(t)} \frac{d}{dt} \nu(t)} = \lim_{t \nearrow \tau} \frac{dL_{ij}(t)/V_i(t)^-}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)/V_i(t)^-}
= \frac{dL_{ij}(\tau)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(\tau)}.\end{aligned}$$
2. First, if $V_i(t) \geq 0$ then by construction (and the above result) it follows that $a_{ij}(t) = \frac{dL_{ij}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)} \geq 0$ for any $i,j \in {\mathcal{N}}_0$ and $a_{i0}(t) \geq \delta$ by this construction. Consider now the case for $V_i(t) < 0$ and assume $a_{ij}(t) < 0$. Let $\tau = \sup\{s \leq t \; | \; V_i(s) = 0\}$. Since $a_{ij}(\tau) \in [0,1]$ by construction and the relative exposures are continuous, this implies there exists some time $s \in [\tau,t)$ such that $a_{ij}(s) = 0$. By the definition of the relative exposures, this must follow that $da_{ij}(s) \geq 0$ for any time $a_{ij}(s) \leq 0$ (with $da_{ij}(s) > 0$ if $a_{ij}(s) < 0$), thus $a_{ij}(t) < 0$ can never be reached. Further, assume $a_{i0}(t) < \delta$. By Assumption \[ass:society\], if $a_{i0}(s) \leq \frac{dL_{i0}(s)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(s)}$ then $da_{i0}(s) \geq 0$. In particular, if $a_{i0}(s) \leq \delta$ then $da_{i0}(s) \geq 0$ (with $da_{i0}(s) > 0$ if $a_{i0}(s) < \delta$). Thus, by the same contradiction found in the case for $j \in {\mathcal{N}}$, we are able to bound $a_{i0}(t) \geq \delta$.
3. First, if $i = 0$ then $\sum_{j \in {\mathcal{N}}_0} a_{0j}(t) = 1$ by property that $a_{0j}(t) = \frac{1}{n}1_{\{j \neq 0\}}$ for all times $t$. Now consider $i \in {\mathcal{N}}$, if $V_i(t) \geq 0$ then by construction (and the above result) it follows that $\sum_{j \in {\mathcal{N}}_0} a_{ij}(t) = \sum_{j \in {\mathcal{N}}_0} \frac{dL_{ij}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)} = 1$. Consider now the case for $V_i(t) < 0$ and let $\tau = \sup\{s \leq t \; | \; V_i(s) = 0\}$. Since $\sum_{j \in {\mathcal{N}}_0} a_{ij}(\tau) = 1$ by prior results, we will assume that $\sum_{j \in {\mathcal{N}}_0} a_{ij}(t) = 1$ to deduce $$\begin{aligned}
\sum_{j \in {\mathcal{N}}_0} da_{ij}(t) &= \sum_{j \in {\mathcal{N}}_0} \frac{dL_{ij}(t) - a_{ij}(t) \sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)}{V_i(t)^-}\\
&= \frac{\sum_{j \in {\mathcal{N}}_0} dL_{ij}(t)}{V_i(t)^-} - \frac{\left(\sum_{j \in {\mathcal{N}}_0} a_{ij}(t)\right) \left(\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)\right)}{V_i(t)^-} = 0.\end{aligned}$$ Therefore based on the initial conditions, $a_{ij}(t)$ must evolve so that it maintains the constant row sum of $1$.
Recall that the initial values to the Eisenberg-Noe differential system are $V_i(0) > 0$ and $a_{ij}(0) = \frac{dL_{ij}(0)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(0)}1_{\{i \neq 0\}} + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}}$ for all banks $i,j \in {\mathcal{N}}_0$. For ease of notation, consider $\tau_0 := 0$ and recursively define the stopping times $$\tau_{m+1} := \inf\{t \in (\tau_m,T] \; | \; V_i(\tau_m)V_i(t) < 0 \text{ or } [V_i(\tau_m) = 0, \; dV_i(\tau_m)V_i(t) < 0]\}.$$ That is, $\tau_m \in {\mathbb{T}}$ is the time of the $m$th change in $\Lambda(V)$. Without loss of generality, we will assume that $\tau_m = T$ if the infimum is taken over an empty set. We note that the times $\tau_m$ are all stopping times with respect to the natural filtration.
With these times, note that in particular, on the interval $(\tau_m,\tau_{m+1}]$ we can consider the set of distressed banks to be constant; to simplify, and slightly abuse, notation we can thus consider a constant matrix of distressed firms $\Lambda(\tau_m)$ in the interval $(\tau_m,\tau_{m+1}]$. We will now construct the unique strong solution forward in time over these time intervals, noting that we update $\Lambda$ and $\tau_{m+1}$ once the next event is found.
First, by construction, on $[0,\tau_1]$ there exists a unique solution to the differential system provided by $V(t) = V(0) + c(t)$ and $a_{ij}(t) = \frac{dL_{ij}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)} 1_{\{i \neq 0\}} + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}}$ for all banks $i,j \in {\mathcal{N}}_0$. Assume there exists a strong solution in the time interval $[0,\tau_m]$ for $\tau_m < T$. Now we want to prove the existence and uniqueness for the clearing wealths and relative exposures on the interval $(\tau_m,\tau_{m+1}]$. Expanding $dc(t)$ based on its differential form allows us to consider as $$\begin{aligned}
dV(t) &= [I-A(t)^{\top}\Lambda(\tau_m)]^{-1}(\mu(t,c(t))-[\dot{L}(t)^{\top}- A(t)^{\top}\dot{L}(t)]\vec{1})dt\\
&\quad +[I-A(t)^{\top}\Lambda(\tau_m)]^{-1}\sigma(t,c(t)) dW(t)\\
&= \bar\mu(t,c(t),A(t),V(t))dt+\bar\sigma(t,c(t),A(t),V(t))dW(t).\end{aligned}$$ Let us first consider the linear growth condition for $dV$. Utilizing the $1$-norm and where $\|\cdot\|_1^{op}$ denotes the corresponding operator norm, let $A \in {\mathbb{A}}$ and $V \in {\mathbb{R}}^{n+1}$, then $$\begin{aligned}
\|\bar\mu(t,c,A,V)\|_1 &+ \|\bar\sigma(t,c,A,V)\|_1^{op}\\
&\leq \|(I - A^{\top}\Lambda(\tau_m))^{-1}\|_1^{op} \left(\|\mu(t,c)\|_1 + \|[\dot{L}(t)^{\top}- A^{\top}\dot{L}(t)]\vec{1}\|_1 + \|\sigma(t,c)\|_1^{op}\right)\\
&\leq \sum_{k = 0}^\infty \|[A^{\top}\Lambda(\tau_m)]^k\|_1^{op} \left(\|\mu(t,c)\|_1 + \|\dot{L}(t)^{\top}\vec{1}\|_1 + \|A^{\top}\dot{L}(t)\vec{1}\|_1 + \|\sigma(t,c)\|_1^{op}\right)\\
&\leq \left(1 + \sum_{k = 1}^\infty (1-\delta)^{k-1}\right) \left(\|\mu(t,c)\|_1 + \|\dot{L}(t)^{\top}\vec{1}\|_1 + \|A^{\top}\|_1^{op} \|\dot{L}(t)\vec{1}\|_1 + \|\sigma(t,c)\|_1^{op}\right)\\
&\leq \left(1 + \frac{1}{\delta}\right) \left(\|\mu(t,c)\|_1 + \|[\dot{L}(t)^{\top}\vec{1}\|_1 + \|\dot{L}(t)\vec{1}\|_1 + \|\sigma(t,c)\|_1^{op}\right)\\
&\leq \frac{1 + \delta}{\delta} \sup_{s \in [\tau_m,\tau_{m+1}]} \left(\|\mu(s,c)\|_1 + \|\dot{L}(s)^{\top}\vec{1}\|_1 + \|\dot{L}(s)\vec{1}\|_1 + \|\sigma(s,c)\|_1^{op}\right)\\
&\leq \theta (1 + \|c\|_1)\end{aligned}$$ The second line follows from the triangle inequality and definition of the operator norm. The third line is a result of Proposition \[prop:Leontief\] and further use of the triangle inequality. The fourth line follows from Proposition \[prop:A\] and noting that, by assumption, $\Lambda_{00} = 0$. The upper bound $\theta \geq 0$ can be determined by Assumption \[ass:society\] and since all terms are continuous and being evaluated on a compact interval of time (since $\tau_{m+1} \leq T$ by definition). Further, we wish to prove $\bar\mu: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \times {\mathbb{A}}\times {\mathbb{R}}^{n+1} \to {\mathbb{R}}^{n+1}$ and $\bar\sigma: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \times {\mathbb{A}}\times {\mathbb{R}}^{n+1} \to {\mathbb{R}}^{(n+1) \times (n+1)}$ are jointly locally Lipschitz in $(c,A,V)$. First $(c,A,V) \in {\mathbb{R}}^{n+1} \times {\mathbb{A}}\times {\mathbb{R}}^{n+1} \mapsto \mu(t,c) - [\dot{L}(t)^{\top}- A^{\top}\dot{L}(t)] \vec{1}$ and $(c,A,V) \in {\mathbb{R}}^{n+1} \times {\mathbb{A}}\times {\mathbb{R}}^{n+1} \mapsto \sigma(t,c)$ are Lipschitz continuous by their linear (or constant) forms with Lipschitz constants that can be taken independently of time (via continuity and the compact time domain) as well as the definitions of $\mu$ and $\sigma$. It remains to show that $(c,A,V) \in {\mathbb{R}}^{n+1} \times {\mathbb{A}}\times {\mathbb{R}}^{n+1} \mapsto (I - A^{\top}\Lambda(\tau_m))^{-1}$ is Lipschitz continuous. Let $A,B \in {\mathbb{A}}$, then by the same argument as above on the bounds of the norm of the matrix inverse, $$\begin{aligned}
\|(I - A^{\top}&\Lambda(\tau_m))^{-1} - (I - B^{\top}\Lambda(\tau_m))^{-1}\|_1^{op}\\
&= \|(I - A^{\top}\Lambda(\tau_m))^{-1}[(I - B^{\top}\Lambda(\tau_m)) - (I - A^{\top}\Lambda(\tau_m))](I - B^{\top}\Lambda(\tau_m))^{-1}\|_1^{op}\\
&= \|(I - A^{\top}\Lambda(\tau_m))^{-1}[A - B]^{\top}\Lambda(\tau_m)(I - B^{\top}\Lambda(\tau_m))^{-1}\|_1^{op}\\
&\leq \|(I - A^{\top}\Lambda(\tau_m))^{-1}\|_1^{op} \|(I - B^{\top}\Lambda(\tau_m))^{-1}\|_1^{op} \|\Lambda(\tau_m)\|_1^{op} \|[A - B]^{\top}\|_1^{op}\\
&\leq \left(\frac{1 + \delta}{\delta}\right)^2 \|\Lambda(\tau_m)\|_1^{op} \|A - B\|_{\infty}^{op}\\
&\leq n\left(\frac{1 + \delta}{\delta}\right)^2 \|\Lambda(\tau_m)\|_1^{op} \|A - B\|_1^{op}.\end{aligned}$$ Thus $\bar\mu$ and $\bar\sigma$ are appropriately locally Lipschitz continuous on $[\tau_m,\tau_{m+1}]$.
Now we wish to consider the differential form for the relative exposures matrix . First, if $\Lambda_{ii}(\tau_m) = 0$ (and in particular, $\Lambda_{00}(\tau_m) = 0$ by assumption of the societal node) then $a_{ij}(t) = \frac{dL_{ij}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)}1_{\{i \neq 0\}} + \frac{1}{n}1_{\{i = 0, \; j \neq 0\}}$ is the unique solution for any firm $j \in {\mathcal{N}}_0$ over all times $t \in (\tau_m,\tau_{m+1}]$. In particular, this is independent of the evolution of the wealths $V$, so we need only consider the joint differential equation between the wealths $V$ and the relative exposures $a_{ij}$ where bank $i$ is in distress between times $\tau_m$ and $\tau_{m+1}$, i.e., $\Lambda_{ii}(\tau_m) = 1$. Consider bank $i \in {\mathcal{N}}$ with $\Lambda_{ii}(\tau_m) = 1$. Therefore by construction $V_i(t) < 0$ for all $t \in (\tau_m,\tau_{m+1})$. If $V_i(\tau_{m+1}) = 0$ then from Proposition \[prop:A\], it already follows that the unique solution $a_{ij}(\tau_{m+1}) = \frac{dL_{ij}(\tau_{m+1})}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(\tau_{m+1})}$ must hold, otherwise we can extend $V_i(t) < 0$ for $t \in (\tau_m,\tau_{m+1}]$. The differential form for all relative exposures on the interval $(\tau_m,\tau_{m+1}]$ is provided by $da_{ij}(t) = \frac{dL_{ij}(t) - a_{ij}(t)\sum_{k \in {\mathcal{N}}_0}dL_{ik}(t)}{V_i(t)^-}$. By construction $(a_{ij},V_i) \in [0,1] \times -{\mathbb{R}}_{++} \mapsto \frac{\dot{L}_{ij}(t) - a_{ij} \sum_{k \in {\mathcal{N}}_0} \dot{L}_{ik}(t)}{-V_i}$ is locally Lipschitz and satisfies a local linear growth condition (with constants bounded independent of time as above utilizing continuity of the parameters and the compact time domain).
Combining our results for the joint differential system for the cash flows $c$, clearing wealths $V$ from , and relative exposures $A$ from , we find that this system satisfies a joint local linear growth and local Lipschitz property on the interval $(\tau_m,\tau_{m+1}]$. Therefore, there exists some $\epsilon \in {\mathcal{L}}_{T}^{\infty}({\mathbb{R}}_{++})$ (such that $\tau_m + \epsilon$ is a stopping time) for which a strong solution for $(c,V,A): [\tau_m,\tau_m + \epsilon] \to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{n+1} \times {\mathbb{A}}$ exists and is unique. Using the same logic with local properties, we can continue our unique strong solution sequentially. This can be continued until the stopping time $\tau_{m+1}$ is reached (found along the path of $(c,V,A)$ as a stopping time) or this process reaches some maximal time $T^* < \tau_{m+1}$ for which a unique strong solution exists on the time interval $[\tau_m,T^*)$. First, as $c(t)$ can be calculated separately from the clearing wealths and relative exposures, we can immediately determine that $c(T^*) = \lim_{t \nearrow T^*} c(t)$ exists. Further, we note that any solution $V(t)$ must, almost surely, exist in the (almost surely) compact space $$\left[V(\tau_m) - \left(I + \frac{1+\delta}{\delta} \mathbbm{1}\right)\left(\int_{\tau_m}^t dc(s)^- + \left(L(t) - L(\tau_m)\right)\vec{1}\right) , V(\tau_m) + c(t) - c(\tau_m)\right]
\subseteq {\mathcal{L}}_t^2({\mathbb{R}}^{n+1})$$ where $\mathbbm{1} = \{1\}^{(n+1) \times (n+1)}$. The lower bound is determined to be based on the bounding of the Leontief inverse; the upper bound follows from the continuous-time version of , i.e., $$V(t) = V(0) + c(t) - A(t)^{\top}V(t)^-.$$ Additionally, $a_{ij}(t)$ almost surely exists in the compact neighborhood $[0,1]$ by definition. Therefore $(V(T^*),A(T^*)) = \lim_{t \nearrow T^*} (V(t),A(t))$ exists by continuity of the solutions and compactness of the range space. Thus we can continue the differential equation from time $T^*$ with values $(c(T^*),V(T^*),A(T^*))$ which contradicts the nature that $T^*$ is the maximal time. Notably, if $V_i(T^*) = 0$ for some bank $i$ then it is imperative to check if $\tau_{m+1} = T^*$ to update the set of distressed banks $\Lambda$.
Therefore, by induction, there exists a unique strong solution $(V,A)$ to and on the domain $[0,\tau_m]$ for any index $m \in {\mathbb{N}}$ by use of [@Oksendal Theorem 5.2.1]. In particular this holds up to $\tau^* = \sup_{m \in {\mathbb{N}}} \tau_m$. If $\tau^* \geq T$ then the proof is complete. If $\tau^* < T$, then by the same argument as above we can find $(V(\tau^*),A(\tau^*))$ as we can bound both the wealths and relative exposures into an almost surely compact neighborhood (and a subset of ${\mathcal{L}}_{\tau^*}^2({\mathbb{R}}^{n+1})$). Therefore, as before, we can start the process again at time $\tau^*$, which contradicts the terminal nature of $\tau^*$. This concludes the proof.
Consider the dynamic Eisenberg-Noe systems as fixed point problems for processes. That is, consider the fixed point problem $(V,A) = (\Phi_V({{\Delta t}},V,A),\Phi_A({{\Delta t}},V,A))$ for ${{\Delta t}}\geq 0$ where $\Phi_V: {\mathbb{R}}_+ \times ({\mathbb{R}}^{n+1})^{\mathbb{T}}\times {\mathbb{A}}^{\mathbb{T}}\to ({\mathbb{R}}^{n+1})^{\mathbb{T}}$ and $\Phi_A: {\mathbb{R}}_+ \times ({\mathbb{R}}^{n+1})^{\mathbb{T}}\times {\mathbb{A}}^{\mathbb{T}}\to {\mathbb{A}}^{\mathbb{T}}$ defined by $$\begin{aligned}
\Phi_V({{\Delta t}},A,V) &:= \left(V(-{{\Delta t}}) + \int_0^t dc(s) - A(t)^{\top}V(t)^-\right)_{t \in {\mathbb{T}}}\\
\Phi_{a_{ij}}({{\Delta t}},A,V) &:= \begin{cases} \left(\begin{array}{l}\frac{\int_{t-{{\Delta t}}}^t dL_{ij}(s) + a_{ij}(t-{{\Delta t}})V_i(t-{{\Delta t}})^-}{\max\{\sum_{k \in {\mathcal{N}}_0} \int_{t-{{\Delta t}}}^t dL_{ik}(s) + V_i(t-{{\Delta t}})^- \; , \; V_i(t)^-\}}1_{\{i \neq 0\}}\\ \quad + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}}\end{array}\right)_{t \in {\mathbb{T}}} &\text{if } {{\Delta t}}> 0 \\ \left(\begin{array}{l} \left[\frac{dL_{ij}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)}1_{\{i \neq 0\}} + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}}\right]1_{\{V_i(t) \geq 0\}} \\ + \left[\begin{array}{l}\frac{dL_{ij}(\tau(t))}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(\tau(t))}\\ \quad + \int_{\tau(t)}^t \frac{dL_{ij}(u) - a_{ij}(u)\sum_{k \in {\mathcal{N}}_0} dL_{ik}(u)}{V_i(u)^-}\end{array}\right]1_{\{V_i(t) < 0\}}\end{array}\right)_{t \in {\mathbb{T}}}&\text{if } {{\Delta t}}= 0\end{cases}\end{aligned}$$ where $\tau(t) := \sup\{s < t \; | \; V_i(s) \geq 0\}$ is the last time that bank $i$ was not in distress before time $t$. Note that $\Phi_V$ follows from the logic of by expanding out the recursive formulation or differential systems . By construction $$(\Phi_V(0,A,V),\Phi_A(0,A,V)) = \lim_{{{\Delta t}}\searrow 0} (\Phi_V({{\Delta t}},A_{{{\Delta t}}},V_{{{\Delta t}}}),\Phi_A({{\Delta t}},A_{{{\Delta t}}},V_{{{\Delta t}}}))$$ for paths of convergent relative exposure matrices $A_{{{\Delta t}}} \to A \in {\mathbb{A}}^{\mathbb{T}}$ and wealths processes $V_{{{\Delta t}}} \to V \in ({\mathbb{R}}^{n+1})^{\mathbb{T}}$ (in the product topologies). Thus by the uniqueness of the discrete-time and continuous-time clearing solutions (Corollary \[cor:discrete-cont\] and Theorem \[thm:continuous\]) and an application of [@feinstein2014measures Proposition A.2], the proof is completed.
\[prop:Leontief\] For any relative exposure matrix $A \in {\mathbb{A}}$ and any distress matrix $\Lambda \in \{0,1\}^{(n+1) \times (n+1)}$ such that $\Lambda_{00} = 0$ and $\Lambda_{ij} = 0$ for $i \neq j$, the matrix $I - A^{\top}\Lambda$ is invertible with Leontief form, i.e., $(I-A^{\top}\Lambda)^{-1}=\sum_{k=0}^\infty(A^{\top}\Lambda)^k$.
By inspection, for any $A \in {\mathbb{A}}$, $(I-A^{\top}\Lambda)(I+A^{\top}(I-\Lambda A^{\top})^{-1} \Lambda)=I$, i.e., the form of the inverse is provided by $I + A^{\top}(I - \Lambda A^{\top})^{-1} \Lambda$. We refer to [@feinstein2017sensitivity Theorem 2.6] for a detailed proof that $(I - \Lambda A^{\top})^{-1}$ is nonsingular and is provided by the Leontief inverse. Therefore, by construction $$\begin{aligned}
(I-A^{\top}\Lambda)^{-1}&=(I+A^{\top}(I-\Lambda A^{\top})^{-1} \Lambda)
= I+ A^{\top}\left(\sum_{k=0}^\infty (\Lambda A^{\top})^k\right) \Lambda\\
&= I + \sum_{k=0}^\infty A^{\top}\Lambda (A^{\top})^k \Lambda^k
= I + \sum_{k=0}^\infty (A^{\top}\Lambda)^{k+1}
=\sum_{k=0}^\infty(A^{\top}\Lambda)^k.\end{aligned}$$
[^1]: Washington University in St. Louis, Department of Electrical and Systems Engineering, St. Louis, MO 63130, USA.
[^2]: University of California Santa Barbara, Department of Statistics & Applied Probability, Santa Barbara, CA 93106. Part of this work was undertaken while Alex Bernstein was at Washington University.
[^3]: Washington University in St. Louis, Department of Electrical and Systems Engineering, St. Louis, MO 63130, USA.
| 0 |
---
abstract: 'A relationship of the random walks on one-dimensional periodic lattice and the correlation functions of the $XX$ Heisenberg spin chain is investigated. The operator averages taken over the ferromagnetic state play a role of generating functions of the number of paths made by the so-called “vicious” random walkers (the vicious walkers annihilate each other provided they arrive at the same lattice site). It is shown that the two-point correlation function of spins, calculated over eigen-states of the $XX$ magnet, can be interpreted as the generating function of paths made by a single walker in a medium characterized by a non-constant number of vicious neighbors. The answers are obtained for a number of paths made by the described walker from some fixed lattice site to another sufficiently remote one. Asymptotical estimates for the number of paths are provided in the limit, when the number of steps is increased.'
author:
- |
$${}$$\
[**N. M. Bogoliubov$^\dagger$, C. Malyshev$^\ddagger$**]{}\
[Steklov Mathematical Institute, St.-Petersburg Department, RAS]{}\
[Fontanka 27, St.-Petersburg, 191023, Russia]{}\
\[0.5cm\] $^\dagger$ e-mail: [*[email protected]*]{}\
$^\ddagger$ e-mail: [*[email protected]*]{}
title: |
$${}$$\
[**The correlation functions**]{}\
[**of the ${\bf XX}$ Heisenberg magnet and**]{}\
[**random walks of vicious walkers** ]{}
---
Introduction {#sec1}
============
The random walks is a classical problem both for combinatorics and statistical physics. The problem of enumeration of the paths made by the, so-called, *vicious* walkers on the one-dimensional lattice has been formulated and investigated in details by Fisher [@1]. It is supposed that any two vicious walkers, provided both arrive at the same lattice site, annihilate not only one another but all other walkers as well. The problem mentioned still continues to attract considerable attention both of physicists and mathematicians [@2]–[@9]. Closely related problems arise also in the studies of the self-organized criticality [@10], domain walls [@11], and polymers [@12]. In paper [@13] a random walks of the annihilating particles on a ring was considered. In paper [@14] a random turns walks on a semi-axes with a possible creation of the particles at the origin was studied.
It has been shown in [@15], [@16] that the correlation functions, obtained as an averages over the ferromagnetic state of the $XX$ Heisenberg chain, can be used for enumeration of the paths of random walks of vicious walkers. In the present paper the averages of special type are investigated both for the case of ferromagnetic state and for superposition of the eigen-states of the $XX$ magnet in zero magnetic field. The averages in question play a role of the generating functions of number of paths of the vicious walkers. The calculation of the correlation functions is carried out by means of the functional integration [@17], [@18]. The answers are obtained for the number of paths of a single pedestrian which is travelling from one chosen site to another sufficiently remote lattice site. The asymptotical estimates are obtained for the number of paths in the limit, when the number of steps (and, correspondingly, the number of random turns) is increasing.
The paper is organized as follows. Section \[sec1\] has an introductory character. The Hamiltonian of the model and general calculation of the correlation functions are discussed in Section \[sec2\]. Section \[sec3\] deals with the specific calculations and the corresponding asymptotic estimates. Discussion in Section \[sec4\] concludes the paper.
The model and the correlation functions {#sec2}
=======================================
The $XX$ magnet we are interested in is a particular limit of a more general spin model known as the $XY$ Heisenberg chain, with the Hamiltonian in the transverse magnetic field $h>0$ given by [@19], [@20]: $$\begin{aligned}
&H=H_0+\gamma H_1-hS^z,\\
&H_0\equiv-\sum^{M}_{n,m=1}\Delta^{(+)}_{nm}\sigma^+_{n}\sigma^-_{m},
\\
&H_1\equiv-\frac12\sum_{n,m=1}^M\Delta^{(+)}_{nm}
(\sigma^+_n\sigma^+_{m}+\sigma^-_n\sigma^-_{m}),\qquad
S^z\equiv\frac{1}2\sum_{n=1}^M\sigma^z_n,
\end{aligned}
\label{1}$$ where $S^z$ is $z$-component of the total spin operator, and $\gamma$ is the anisotropy parameter. The local spin operators $\sigma^\pm_n=(\sigma^x_n\pm i\sigma^y_n)/2$ and $\sigma^z_n$ are given by the Pauli matrices, which depend on the lattice argument $n\in\mathcal M\equiv\{1,2,\dots,M\}$, where $M=0\pmod{2}$. The corresponding commutation relations have the form: $$[\sigma^+_k,\sigma^-_l]\,=\,\delta_{k,l}\sigma^z_l\,,\quad
[\sigma^z_k,\sigma^\pm_l]\,=\,\pm2\delta_{k,l}\sigma^{\pm}_l\,.$$ The introduced *hopping matrix* $\Delta^{(s)}$ is defined by the following entries: $$\Delta^{(s)}_{nm}\equiv\frac12\,
(\delta_{|n-m|,1}+s\delta_{|n-m|,M-1}), \label{2}$$ where ${\delta}_{n,l}$ is the Kronecker symbol, and $s$ can take two values: $s=\pm$. It is assumed that the periodic boundary conditions $\sigma^{\alpha}_{n+M}=\sigma^{\alpha}_n$ are imposed for any $n\in\mathcal M$. The Hamiltonian $H$ is reduced to the Hamiltonian of the $XX$ magnet at zero value of the parameter $\gamma$.
The most general definition of the time $t$ and temperature $T\equiv
1/\beta$ dependent correlation functions of the model under consideration looks as follows: $$G_{j;l}^{ab}(t)\equiv\frac{1}Z
\Tr(\sigma^a_{j}(0)\sigma^b_l(t)e^{-\beta H}),\qquad
Z\equiv\Tr(e^{-\beta H}),
\label{3}$$ where $\sigma^b_l(t)\equiv e^{itH}\sigma^b_le^{-itH}$ and $\Tr$ means the averaging with respect to all eigen-states of the Hamiltonian $H$. In addition, the normalization involves the partition function $Z$. Calculation of the correlators has been carried out in [@21] as averaging over all eigen-functions of the Hamiltonian of the $XX$ magnet. In [@21] the main attention has been paid to a relationship between the correlation functions and the Fredholm determinants in the thermodynamic limit. In the present paper we shall consider the $XX$ chain only and denote its Hamiltonian by $H$.
To calculate the averages one can use a representation of the canonical Fermi variables $c_j$, $c^{\dagger}_j$, $j\in{\mathcal M}$ through the spin variables [@19], [@20]. The corresponding Jordan–Wigner transformation has the form: $$\sigma^+_n=\biggl(\,\prod_{j=1}^{n-1}\sigma^z_j\biggr)c_n,\qquad
\sigma^-_n=c_n^{\dagger}\biggl(\,\prod_{j=1}^{n-1}\sigma^z_j\biggr),
\qquad n\in{\mathcal M},
\label{4}$$ where $\sigma^z_j=1-2c_j^{\dagger} c_j$. The periodic boundary conditions for the spin variables lead to the following boundary conditions for the Fermi variables: $$c_{M+1}=(-1)^{\mathcal{N}}c_1,\qquad
c^{\dagger}_{M+1}=c^{\dagger}_1(-1)^{\mathcal{N}},
\label{5}$$ where $\mathcal{N}=\sum_{n=1}^Mc_n^{\dagger} c_n$ is the operator of the total number of particles. The Hamiltonian $H$ takes the following form in the fermion representation [@19], [@20] $$H=H^+P^++H^-P^-,
\label{6}$$ where $P^{+}$ ($P^{-}$) are projectors on the states characterized by an even/odd number of fermions: $$P_++P_-=\mathbb I\,,\quad P_+-P_-=(-1)^{\mathcal N}\,.$$ The operators $H^{\pm}$ are formally identical, their superscripts $s=\pm$ point out an appropriate specification of the boundary conditions : $$c_{M+1}\,=\,-s\,c_1\,,\quad c^{\dagger}_{M+1}\,=\,- s\,c^{\dagger}_1\,.$$ To put it differently, the quadratic in the fermion variables operators $H^\pm$ has the following representation: $$H^{\pm}=c^{\dagger}\widehat H^{\pm} c-\frac{Mh}2,\qquad
\widehat H^{\pm}=-\hat\Delta^{(\mp)}+h{\hat I},
\label{7}$$ where the matrices ${\widehat H}^{\pm}$ are expressed through the hopping matrices and $\hat I$ is the unit matrix: $${\widehat H}^\pm\,=\,\left(
\begin{array}{cccccc} h & -1/2 & & & & \pm1/2 \\
-1/2 & h & -1/2 & & & \\
& -1/2 & h & -1/2 & & \\
& & & \dots & & \\
& & & -1/2 & h & -1/2 \\
\pm 1/2 & & & & -1/2 & h
\end{array}\right)$$ (only non-zero entries are displayed). Besides, the short-hand notations $c^{\dagger}$ and $c$ are used in for the $M$-dimensional row and column with the entries $c_n^{\dagger}$, $c_n$, $n\in\mathcal M$.
In particular, the correlator at $a=b=z$ takes the following form in the representation [@18], [@22], [@23]: $$G_{j;l}^{zz}(t)=1-\frac{2}Z\Tr(c^{\dagger}_{j}c_{j}e^{-\beta H})-
\frac{2}Z\Tr(c^{\dagger}_lc_le^{-\beta H})+
\frac{4}Z\Tr(c^{\dagger}_{j}c_{j}
e^{itH}c^{\dagger}_lc_le^{-(\beta+it)H}).
\label{8}$$ In order to calculate , it is convenient to introduce the generating functional [@18]: $$\mathcal G\equiv\mathcal G(S,T\mid\lambda,\nu)=
\frac{1}Z\Tr(e^Se^{-\lambda H}e^Te^{-\nu H}),
\label{9}$$ where $\lambda$, $\nu$ are the complex parameters, $\lambda+\nu=\beta$. The quadratic operators $S\equiv c^{\dagger}{\widehat S}c$ and $T\equiv c^{\dagger}{\widehat T}c$, used in , are defined by means of the matrices $\widehat S=\diag\{S_1,S_2,\dots,S_M\}$ and $\widehat T=\diag\{T_1,T_2,\dots,T_M\}$. For instance, the last term in right-hand side of is obtained from in the following way: $$\lim_{\substack{S_n,T_n\to0,\\n\in \mathcal M}}\,\lim_{\lambda\to-it}\,
\lim_{\nu\to\beta+it}\frac{\partial}{\partial S_j}\,
\frac{\partial}{\partial T_l}\,\mathcal G(S,T\mid\lambda,\nu).
\label{10}$$ The trace in right-hand side of can be re-written by means of [@18]: $$\Tr(e^Se^{-\lambda H}e^Te^{-\nu H})=
\frac12\,(\mathcal G^+_{\Fa}Z^+_{\Fa}+\mathcal G^-_{\Fa}Z^-_{\Fa}
+\mathcal G^+_{\Ba}Z^+_{\Ba}-\mathcal G^-_{\Ba}Z^-_{\Ba}),
\label{11}$$ where $$\begin{aligned}
&\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}\equiv
\Tr(e^Se^{-\lambda H^{\pm}}e^Te^{-\nu H^{\pm}}),
\\
&\mathcal G^{\pm}_{\Ba}Z^{\pm}_{\Ba}\equiv
\Tr(e^Se^{-\lambda H^{\pm}}e^T(-1)^{\mathcal N}e^{-\nu H^{\pm}}),
\end{aligned}
\label{12}$$ and $$Z^{\pm}_{\Fa}\,=\,\Tr(e^{-\beta H^{\pm}})\,,\quad
Z^{\pm}_{\Ba}\,=\,\Tr((-1)^{\mathcal N}e^{-\beta H^{\pm}})\,.$$ Moreover, for the partition function $Z$ we obtain the representation: $$Z=\frac12\,(Z^+_{\Fa}+Z^-_{\Fa}+Z^+_{\Ba}-Z^-_{\Ba})\,.$$ In the thermodynamic limit the terms with the subscript $\Ba$ are mutually compensated, therefore, in order to obtain $\mathcal G$ it is enough to calculate $\mathcal G^{\pm}_{\Fa}$.
The considered fermion representation is characterized by the existence of the Fock state $|0\rangle$ common for both operators $H^{+}$ and $H^{-}$, and satisfying the relations $c_k|0\rangle=0$, $k\in\mathcal M$. However, the corresponding coherent states over $|0\rangle$, $$\begin{array}{rcl}
&&\mid\!z\big\rangle\,\equiv\,\exp \Big(\sum\limits^M_{k=1} c^\dagger_k
z_k\Big) \mid\!0\big\rangle\,\equiv\,\exp (c^\dagger z)\mid\!0\big\rangle\,,\\ [0.5cm]
&&\big\langle z^*\!\mid\,\equiv\,\big\langle 0\!\mid \exp
\Big(\sum\limits^M_{k=1} z^*_k c_k \Big)\,\equiv\,
\big\langle 0\!\mid\exp (z^* c)\,,
\end{array}$$ are different for $H^+$ and $H^-$. Here the short-hand notations $z^*\equiv(z^*_1,\dots,z^*_M)$ and $z\equiv(z_1,\dots,z_M)$ are used for the sets of independent Grassmann parameters $z_k,z^*_k$, $k\in {\mathcal M}$ (it is appropriate to omit the extra index $\pm$ in $z^*$, $z$). Besides, $\sum_{k=1}^{M} c^{\dagger}_kz_k\equiv c^{\dagger} z$, $\prod_{k=1}^M dz_k\equiv dz$. Let us calculate $\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}$ in using the representation of the trace in the Grassmann integration formalism [@18]: $$\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}=\int dz\,dz^*\,e^{z^*z}
\langle z^*|e^Se^{-\lambda H^{\pm}}e^Te^{-\nu H^{\pm}}|z\rangle.
\label{13}$$
In order to represent the right-hand side of this equality as the functional integral, let us introduce $L$ new copies of the coherent states $|x(I)\rangle$, $\langle x^*(I)|$, where $I\in\{1,2,\dots,L\}$. Each of the $2L$ multi-indices $x^*(I)$, $x(I)$ is expressed by $M$ independent Grassmann parameters. Using the decompositions of unity one can represent the right-hand side of as the $(L+1)$-fold multiple integral. In order to express the quasi-periodicity condition it is convenient to introduce the auxiliary variables: $$-{\widehat E}x(0)=x(L+1)\equiv z,\qquad
-x^*(L+1)=x^*(0){\widehat E}^{-1}\equiv z^*,
\label{14}$$ where ${\widehat E}\equiv e^{\widehat S}e^{-\lambda{\widehat H}^{\pm}}e^{\widehat T}$. Tending $L$ to infinity, we obtain the functional integral over the space of the trajectories $x^*(\tau)$, $x(\tau)$, where $\tau\in {\mathbb R}$: $$\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}=\int e^S\,d\lambda^*\,d\lambda
\prod_{\tau} dx^*(\tau)\,dx(\tau).
\label{15}$$ The action functional $S\equiv\int L(\tau)\,d\tau$ is expressed through the Lagrangian $L(\tau)$: $$L(\tau)\equiv x^*(\tau)\biggl(\frac{d}{d\tau}-
\widehat H^{\pm}\biggr)x(\tau)+
J^*(\tau)x(\tau)+x^*(\tau)J(\tau),$$ where $$J^*(\tau)\equiv\lambda^*(\delta(\tau)\hat I+
\delta(\tau-\nu){\widehat E}^{-1}),\qquad
J(\tau)\equiv(\delta(\tau)\hat I+
\delta(\tau-\nu)\widehat E)\lambda.$$ The integration over the auxiliary Grassmann variables $\lambda^*$, $\lambda$ in guarantees the fulfilment of the constraints . The $\delta$-functions in $J^*(\tau)$, $J(\tau)$ reduce $\tau\in\mathbb R$ to $\tau\in[0,\beta]$. The stationarity conditions $\delta S/\delta x^*=0$, $\delta S/\delta x=0$ result in the following regularized answer [@18]: $$\mathcal G^{\pm}_{\Fa}={\det}\biggl(\hat I+
\frac{e^{(\beta-\nu){\widehat H}^{\pm}}e^{\widehat S}
e^{-\lambda{\widehat H}^{\pm}}e^{\widehat T}-{\hat I}}
{\hat I+e^{\beta{\widehat H}^{\pm}}}\biggr).
\label{16}$$ Furthermore, we substitute into and pass to the momentum representation. The procedure described can also be applied to other correlators $G_{j;l}^{ab}(t)$ , where $a,b\in\{+,-\}$.
Random walks {#sec3}
============
As it has been shown in [@15], [@16], the flips of spins on a one-dimensional lattice may be associated with a random movements of walkers. Indeed, let us consider a state of the $XX$ Heisenberg chain, which corresponds to the ferromagnetic ordering of $M$ spins: $|\!\!\Uparrow\rangle\equiv\bigotimes_{n=1}^M|\!\!\uparrow\rangle_n$ (i.e., all spins are oriented “up”). Consider the average of the following type: $$F_{j;l}(\lambda)\equiv
\langle\Uparrow\!|\sigma_{j}^{+}e^{-\lambda H_0}
\sigma_{l}^{-}|\!\Uparrow\rangle,
\label{17}$$ where the notation $H_0$ implies that the zero magnetic field $h=0$ is taken in the Hamiltonian , (we shall omit the same subscript for the corresponding matrices ${\widehat H}^{\pm}$ ), and $\lambda\in\mathbb C$ is an “evolution” parameter. “Up” (or “down”) direction of spin corresponds to the empty (or filled) site. Differentiating ${F_{j;l}({\lambda})}$ and expanding the commutator $\lbrack H_0,\sigma_j^{+}\rbrack$ we obtain the difference–differential equation: $$\frac{d}{d\lambda}\,F_{j;l}({\lambda})=
\frac12\,(F_{j+1;l}({\lambda})+F_{j-1;l}({\lambda}))
\label{18}$$ (and similar equation can be also obtained for the fixed index $j$). Solution of the given equation is specified by the boundary conditions imposed on the lattice argument, and by the initial condition at $\lambda=0$.
The average $F_{j;l}(\lambda)$ can be considered as the generating function of the trajectories with random turns that start at the $l$-th site and end up at the $j$-th site. Indeed, let us introduce the notation $\mathcal D^K_{\lambda}$ for the operator of differentiation of $K$-th order with respect to $\lambda$ at the point $\lambda=0$. The application of $\mathcal D^K_{\lambda}$ to the average leads to the answer: $$\mathcal D^K_{\lambda}\bigl[F_{j;l}({\lambda})\bigr]=
\langle\Uparrow|\sigma_{j}^{+}(-H_0)^K\sigma_{l}^{-}|\Uparrow\rangle=
\sum_{n_1,\dots,n_{K-1}}\Delta^{(+)}_{jn_{K-1}}\dots
\Delta^{(+)}_{n_2n_1}\Delta^{(+)}_{n_1l}.
\label{19}$$ The right hand side of coincides with the entry at the crossing of the $j$-th row and the $l$-th column of the matrix given by the product of $K$ copies of the hopping matrix . Each matrix in this product corresponds to a transition between the two nearest sites of the lattice. After multiplication by $2^{K}$ (this is due to the accepted normalization of the matrix ), the right-hand side of gives the number of the trajectories that consist of $K$ steps and are connecting the $l$-th and $j$-th sites. Let us denote this number by $|P_K(l\rightarrow j)|$.
Let $|P_K (l_1, \dots, l_N\rightarrow j_1, \dots, j_N)|$ be a number of trajectories consisting of $K$ links made by $N$ vicious walkers in the random turns model. Here, the initial and final positions of the walkers on the sites are given respectively by the sequences $l_1 > l_2 > \dots > l_N$ and $j_1 > j_2 >\dots > j_N$. Let us consider the $N$-point correlation function ($N\leq M$): $$F_{j_1,j_2,\dots,j_N;l_1,l_2,\dots,l_N}({\lambda})=
\langle\Uparrow\!\!|\sigma_{j_1}^{+}\sigma_{j_2}^{+}\dots
\sigma_{j_N}^{+}e^{-\lambda H_0}\sigma_{l_1}^{-}
\sigma_{l_2}^{-}\dots\sigma_{l_N}^{-}|\!\!\Uparrow\rangle.
\label{20}$$ The present correlator is related to enumeration of the admissible trajectories which are traced by $N$ vicious walkers. Indeed, the application of the operator $\mathcal D^K_{\lambda/2}$ to results in the average of the type $$\langle\Uparrow\!\!|\sigma_{j_1}^{+} \sigma_{j_2}^{+} \dots
\sigma_{j_N}^{+}(-2H_0)^K \sigma_{l_1}^{-}\sigma_{l_2}^{-} \dots
\sigma_{l_N}^{-} |\!\!\Uparrow\rangle\,.$$ This average provides the numbers $|P_K (l_1, \dots, l_N\rightarrow j_1, \dots, j_N)|$ that can be established with the help of the commutator $$\lbrack H_0,\sigma_{l_1}^{-}\sigma_{l_2}^{-}\dots\sigma_{l_K}^{-}]=
\sum_{k=1}^K\sigma_{l_1}^{-}\dots\sigma_{l_{k-1}}^{-}
[H_0,\sigma_{l_k}^{-}]\sigma_{l_{k+1}}^{-}\dots\sigma_{l_K}^{-}
\label{21}$$ (in this case, differentiation with respect to $\lambda/2$, instead of $\lambda$, allows to take into account the normalization of the hopping matrix ). The condition of non-intersection of trajectories of the walkers is expressed by the vanishing of the correlation function for any pair of coinciding indices $l_k$ or $j_p$.
Differentiating with respect to $\lambda$ and applying , we obtain the equation: $$\frac{d}{d\lambda}\,F_{j_1,\dots,j_N;l_1,\dots,l_N}(\lambda)=
\frac12\sum_{k=1}^N\bigl(F_{j_1,
\dots,j_N;l_1,l_2,\dots,l_k+1,\dots,l_N}({\lambda})+
F_{j_1,\dots,j_N;l_1,l_2,\dots,l_k-1,\dots,l_N}({\lambda})\bigr).
\label{22}$$ Equation has been considered in [@16] for the case of periodicity with respect to the lattice argument and with the initial condition: $$F_{j_1, \dots, j_N;l_1, \dots, l_N}(0)\,=\,\prod_{m=1}^N\delta_{j_m,l_m}\,.$$ The function $F_{j_1,j_2, \dots, j_N; l_1,l_2, \dots, l_N}(\lambda)$ can be expressed as the determinant of the matrix consisting of the averages of the type of [@16]: $$F_{j_1,\dots,j_N;l_1,\dots,l_N}({\lambda})=
\det\bigl(F_{j_r;l_s}({\lambda})\bigr)_{1\leq r,s\leq N}.
\label{23}$$
Random walks on the axis {#sec3.1}
------------------------
Let us consider an infinite chain ($M\to\infty$). Then, the modified Bessel function $I_{j-l}(\lambda)$ turns out to be a solution of equation , which respects the condition $F_{j;l}({0})={\delta}_{j,l}$ [@15]: $$F_{j;l}({\lambda})=I_{j-l}({\lambda})=\frac1{2\pi
}\int_{-\pi}^{\pi}d\theta\,e^{{\lambda}\cos\theta}e^{i(j-l)\theta}.
\label{24}$$ There exists the following expansion into the power series for $I_{j-l}(\lambda)$: $$I_{j-l}({\lambda})=\sum_{Q\geq|l-j|}
\frac{1}{\bigl(\frac{Q-j+l}2\bigr)!\,\bigl(\frac{Q+j-l}2\bigr)!}
\biggl(\frac{{\lambda}}2\biggr)^Q,
\label{25}$$ where the summation index $Q$ is subjected to the requirement: $Q+|j-l|=0\pmod{2}$. In the limit of large “time” (${\lambda}\rightarrow\infty$) and for moderate values of $m\equiv|l-j|$, using the known asymptotics for the Bessel function, we obtain for the generating function: $$F_{j;l}({\lambda})\simeq\frac{e^{{\lambda}}}{{\sqrt{2\pi{\lambda}}}}
\biggl(1-\frac{4m^2-1}{8{\lambda}}+\dotsb\biggr),$$ where the decay is governed by the critical exponent $\xi=-1/2$.
Let the number $K$ satisfies the relations $K\geq|l-j|$ and ${K+|j-l|=0\nobreak\pmod{2}}$. Then, differentiation of the series leads to the binomial relation ${|P_K(l\rightarrow j)|}=C_K^L$ for the number of all lattice paths of the “length” $K$ between two sites on the infinite axis: $$|P_K(l\rightarrow j)|\equiv
\mathcal D^K_{\lambda/2}[F_{j;l}({\lambda})]=
\frac{(m+2L)!}{L!\,(m+L)!}\,.
\label{26}$$ Here $L$ denotes the one-half of the total number of turns: $L\equiv(K-m)/2$.
Let us consider now the multi-point correlation function $F_{j_1,j_2, \dots, j_N;l_1,l_2, \dots, l_N}({\lambda})$. As it has been shown above, $\mathcal D^K_{\lambda/2} [F_{j_1, \dots, j_N;l_1, \dots, l_N}(\lambda)]$ has the sense of the number of trajectories of $N$ vicious walkers each of which has made $K$ steps. A different combinatorial interpretation of this object, however, can be proposed. Really, let us consider a representation of the multi-point correlator in the form of the determinant . Its entries $F_{j_r;l_s}(\lambda)$ in the case of an infinite chain are given by the Bessel function $I_{j_r-l_s}(\lambda)$ . The operator $\mathcal D^K_{\lambda/2}$ acts on the determinant as the differentiation of the product of $N$ functions: $$(f_1(x)f_2(x)\dots f_N(x))^{(K)}=
\sum_{n_1+n_2+\dots+n_N=K}P(n_1,n_2,\dots,n_N)
f^{(n_1)}_1f^{(n_2)}_2\dots f^{(n_N)}_N.
\label{27}$$ The notation $f^{(n)}\equiv d^nf(x)/dx^n$ is used here, and the coefficients $P(n_1,n_2,\dots,n_N)$ are the numbers of permutations with repeats: $$P(n_1,n_2,\dots,n_N)\equiv
\frac{(n_1+n_2+\dots+n_N)!}{n_1!\,n_2!\,\dots n_N!}.
\label{28}$$ Summation in is over all non-negative values of $n_1,n_2,\dots,n_N$, provided their sum is equal to $K$.
Suppose further, that an $N$-dimensional (hyper-)cubic lattice of infinite extension is given, and each site of this lattice is labelled by a set of $N$ numbers. Let $\mathcal T_K(q_1,q_2, \dots
,q_N)$ be the number of the lattice trajectories that can be traced by some walker from the “initial” point $\textbf{\textit{O}}\equiv(0,0, \dots, 0)$ to a point $(q_1,q_2,\dots,q_N)$ in $K$ steps (by a single step the walker can move to one of the nearest sites). Let all numbers $q_k$ be non-negative, and let the inequality $K\geq q_1+q_2+\dots+q_N$ be fulfilled, which means that the steps that can compensate each other are allowed. Let us denote the number of these steps as $2L$, $$L\equiv\frac{K-q_1-q_2-\dots-q_N}{2}\,.
\label{29}$$ Taking into account , the following formula for the number of paths takes place: $$\mathcal T_K(q_1,q_2,\dots,q_N)=\sum_{L_1+L_2+\dots+L_N=L}
P(q_1+L_1,q_2+L_2,\dots,q_N+L_N,L_1,L_2,\dots,L_N),
\label{30}$$ where summation is taken over all non-negative values of $L_1,L_2,\dots,L_N$, provided that their sum is equal to $L$, and the formula for the number of permutations with repeats is used.
Turning back to the function $F_{j_1,j_2,\dots,j_N;l_1,l_2,\dots,l_N}({\lambda})$ let us define the matrix $(n_{rs})_{1\leq r,s\leq N}$ with the entries $n_{rs}\equiv j_r-l_s$. Then, we arrive to the following
The number of trajectories consisting of $K$ links, which are traced by $N$ vicious walkers on an axis, is expressed through the number of trajectories of the same “length” $K$, which are traced by a single walker travelling over sites of $N$-dimensional lattice of infinite extension: $$\begin{aligned}
|P_K(l_1,\dots,l_N\rightarrow j_1,\dots,j_N)|&\equiv
\mathcal D^K_{\lambda/2}\bigl[F_{j_1,\dots,j_N;l_1,\dots,l_N}(\lambda)\bigr]=
\\
&=
\sum_{S_{a_1,a_2,\dots,a_N}}(-1)^{\mathcal P_S}
\mathcal T_K(n_{a_11},n_{a_22},\dots,n_{a_NN}),
\end{aligned}
\label{31}$$ where summation is taken over all permutations $S_{a_1,a_2,\dots,a_N}\equiv$ $S(\begin{smallmatrix}1,&2,&\dots,&N \\ a_1,&a_2,&\dots,&a_N\end{smallmatrix})$ of the numbers $1,2,\dots,N$, and $\mathcal P_S$ implies a parity of a specific permutation.
In order to verify one should develop the determinant by a row or by a column and then apply the induction using the relations –.
Let us calculate, for instance, at $N=2$: $$\mathcal T_K(q_1,q_2)=C_{q_1+q_2+2L}^{q_1+L}
\sum_{k=0}^LC^{L-k}_{q_1+L}C_{q_2+L}^{k}=C_K^{q_1+L}C_K^L,
\label{32}$$ where $L=(K-q_{1}-q_{2})/2$ denotes one-half of the total number of turns. Then, using we obtain: $$\mathcal D^K_{\lambda/2}\bigl[F_{j_1,j_2;l_1,l_2}(\lambda)\bigr]=
\mathcal T_K(n_{11},n_{22})-\mathcal T_K(n_{21},n_{12})=
\begin{vmatrix}
C_K^L&C_K^{L+n_{21}}
\\[1mm]
C_K^L&C_K^{L+n_{11}}
\end{vmatrix},
\label{33}$$ where $L=(K-n_{11}-n_{22})/2$ and the equality $n_{11}+n_{22}=n_{12}+n_{21}$ is used.
Representation of the entries $F_{j_1,j_2,\dots,j_N;l_1,l_2,\dots,l_N}(\lambda)$ in the integral form allows to obtain the following expression [@15]: $$\begin{aligned}
F_{j_1,\dots,j_N;l_1,\dots,l_N}({\lambda})={}&
\frac{e^{{\lambda}N}}{N!}\prod_{i=1}^N
\biggl(\,\int_{-\pi}^{\pi}\frac{d\theta_i}{2\pi}\biggr)
e^{-{\lambda}\sum_{k=1}^N(1-\cos\theta_k)}\times{}
\\
&\times
{S}_{\boldsymbol{\pi}}(e^{i\theta_1},e^{i\theta_2},\dots,
e^{i\theta_N})\prod_{1\leq j<k\leq N}
|e^{i\theta_j}-e^{i\theta_k}|^2,
\end{aligned}
\label{34}$$ where $S_{\boldsymbol{\pi}}(e^{i\theta_1},e^{i\theta_2},
\dots,e^{i\theta_N})$ is the Schur function [@24], $$S_{\boldsymbol{\pi}}(x_1,x_2,\dots,x_N)\equiv
\frac{\det(x_j^{\pi_k+N-k})_{1\leq j,k\leq N}}
{\det(x_j^{N-k})_{1\leq j,k\leq N}}\,.
\label{35}$$ The Schur function depends on the partition $\boldsymbol{\pi}=(\pi_1,\pi_2,\dots,\pi_N)$ defined by a sequence of non-negative integers, which are ordered according to non-strict decreasing: $\pi_1\geq\pi_2\geq\dots\geq\pi_N\geq0$. In virtue of translational invariance it is always possible to choose the numbers $l_1$ $>$ $l_2>\dots>l_N\geq-N$ for the initial position of the walkers and to define the elements of the partition by the equalities $\pi_k=l_k+k$. In order to calculate the leading asymptotics of the generating function in the limit ${\lambda}\rightarrow\infty$, let us transform the integral into the following integral [@7], [@25]: $$\int d^n\theta\prod_{1\leq j<k\leq N}|\theta_j-\theta_k|^2
e^{-{\lambda}/2\sum_{k=1}^N\theta^2_k}=
\frac{(2\pi)^{N/2}}{{\lambda}^{N^2/2}}\biggl(\,\prod_{p=1}^Np!\biggr).$$ It is a special case of the *Mehta integral*, which arises in the theory of the *Gaussian matrix ensembles*. Finally, we obtain the following asymptotics of the generating function for the trajectories traced by $N$ vicious walkers: $$F_{j_1,\dots,j_N;l_1,\dots,l_N}({\lambda})\simeq
\mathcal A\frac{e^{{\lambda}N}}{{\lambda}^{N^2/2}}\,,\qquad
\mathcal A=\frac{\prod_{p=1}^{N-1}p!}{(2\pi)^{N/2}}
\prod_{1\leq j<k\leq N}\frac{l_j-l_k}{k-j}\,,$$ where the well known formula for ${S}_{\boldsymbol{\lambda}}(1,1,\dots,1)$ is taken into account in $\mathcal A$ [@9], [@25]. Therefore, the power-like behavior of $F_{j_1,\dots,j_N;l_1,\dots,l_N}({\lambda})$ is characterized by the exponent $\xi=-N^2/2$.
Random walks over superposition of the eigen-states {#sec3.2}
---------------------------------------------------
The eigen-functions of the $XX$ Hamiltonian, given by the relations , , are constructed as combinations of the states, obtained by “flipping” of $N$ spins in the state $|\!\!\Uparrow\rangle$ [@21]. Indeed, let us consider all admissible strict partitions $\boldsymbol{\mu}=(\mu_1,\mu_2, \dots, \mu_N)$, where $M\geq\mu_1>\mu_2> \dots >\mu_N\geq1$, and establish a correspondence between each partition and an appropriate sequence of zeros and unities: $\bigl\{e_k\equiv e_k(\boldsymbol{\mu})\bigr\}_{k\in\mathcal M}$, where $e_k=\delta_{k,\mu_n}$, $1\leq n\leq N$. The required eigen-function is defined as: $$|\Psi_N(u_1,\dots,u_N)\rangle=
\sum_{\{e_k(\boldsymbol{\mu})\}_{k\in\mathcal M}}
\Upsilon_N(\{u_k\}\!\mid\boldsymbol{\mu})
(\sigma_{M}^{-})^{e_{M}}(\sigma_{M-1}^{-})^{e_{M-1}}\dots
(\sigma_{1}^{-})^{e_1}|\Uparrow\rangle,
\label{36}$$ where summation is taken over all strict partitions $\boldsymbol{\mu}$ of the given type. The number of such partitions is expressed through the number of permutations with repeats : $P(N,M-N)=C_{M}^N$. The wave functions satisfy the periodic boundary conditions, $$\Upsilon_N(\{u_k\}|\boldsymbol{\mu})\equiv
\det(u_k^{2\mu_l})_{1\leq k,l\leq N}
\label{37}$$ are parametrized by the partitions $\boldsymbol{\mu}$ and by different, up to permutation, sets $\{u_1, \dots, u_N\}$ of solutions of the Bethe equations: $$u_k^{2M}=(-1)^{N-1},\qquad1\leq k\leq N.
\label{38}$$ These solutions have the form: $u_k^2=e^{i2\pi I_k/M}$, where $I_k$ are integers or half-integers (this depends on parity of $N$). Due to the antisymmetry of with the respect to permutations of the parameters $u_k$, it is sufficient to restrict oneself to the strict partitions $M\geq I_1>I_2>\dots>I_N \geq1$ in order to guarantee the single-valuedness of $u_k$. With the help of the corresponding normalized average $$\langle\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_1^{-}\rangle_N\equiv
\frac{\langle\Psi_N|\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_1^{-}
|\Psi_N\rangle}{\langle\Psi_N\mid\Psi_N\rangle}
\label{39}$$ can be represented as a linear combination of $(N+1)$-point generating functions . Therefore, this average is related to the number of random walks of $N+1$ pedestrians. The initial and the final positions of one of them are fixed at $l_1=1$ and $j_1=m+1$, respectively, while for the rest (*virtual*) pedestrians the choice of their initial and the final positions is arbitrary.
Calculation of equation is of interest in the thermodynamic limit, when $M$ and $N$ are growing (their ratio remains finite), which means that the number of virtual pedestrians is increasing. In this limit [@26] $$\widetilde F_{m+1;1}({\lambda})\equiv
\langle\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_1^{-}\rangle_N
\bigr|_{M,N\gg1}\stackrel{\defa}{=}\Tr^{\prime}(\sigma_{m+1}^{+}
e^{-{\lambda}H_0}\sigma_1^{-}),
\label{40}$$ where the notation $\Tr^{\prime}$ points out that the procedure presented in Section \[sec2\] is used for the calculation of the normalized average. The difference-differential relation, analogous to the equation is valid for ${\widetilde F}_{m+1;1}({\lambda})$ : $$\begin{aligned}
&\frac d{d{\lambda}}\,{\widetilde F}_{m+1;1}({\lambda})=
\frac12\,({\widetilde F}_{m;1}({\lambda})+
{\widetilde F}_{m+2;1}({\lambda}))-
\Tr^{{\prime}}(H_0\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_{1}^{-})-{}\notag
\\
&\qquad{}-
\Tr^{\prime}\biggl(\biggl(\frac{1-\sigma_{m+1}^{z}}{2}\biggr)
\sigma_{m}^{+}e^{-{\lambda}H_0}\sigma_{1}^{-}\biggr)-
\Tr^{\prime}\biggl(\biggl(\frac{1-\sigma_{m+1}^{z}}{2}\biggr)
\sigma_{m+2}^{+}e^{-{\lambda}H_0}\sigma_{1}^{-}\biggr).
\label{41}\end{aligned}$$ The form of the present equation makes it possible to suppose that the average ${\widetilde F}_{m+1;1}({\lambda})$ can also be of interest as a generating function of the random walks.
Let us turn to the calculation of ${\widetilde F}_{m+1;1}({\lambda})$ in the fermionic representation . It is convenient to reduce the problem to calculation of the generating function of the form: $$\widetilde{\mathcal G}\equiv\Tr^{\prime}
(e^Sc_{m+1}e^{-\lambda H_{0}}c^{\dagger}_1e^{-\nu H_{0}}),
\label{42}$$ where the operator $S$ is defined just like in (i.e., by means of the matrix $\widehat S = \break \diag\{S_1,S_2,\dots,S_M\}$). Indeed, the functional ${\widetilde F}_{m+1;1}({\lambda})$ corresponds to the choice of $\nu=0$ and $S_k=-i\pi\theta(m-k)$, where $\theta(m-k)$ is the Heavyside function, $\theta(0)=1$. The second term in the right-hand side of corresponds to the differentiation by $\nu$ at the point $\nu=0$, in the third term we put $\nu=0$ and differentiate with respect to $S_{m+1}$. In both cases we put $S_k=-i\pi\theta(m-k)$. Taking into account the fact, that the contribution of the terms labelled by the index $\Ba$ in is negligible at sufficiently large $M$ and $N$, we approximately obtain: $$\begin{aligned}
&\widetilde{\mathcal G}\approx
\biggl[\tr(e^{-{\lambda}{\widehat H}^{0}}{\hat e}_{1,m+1})-
\frac d{d\alpha}\biggr]\det({\hat I}+\widehat{\mathcal M}_1+
\alpha\widehat{\mathcal M}_2)\bigr|_{{\alpha}=0},
\\
&\widehat{\mathcal M}_1+\alpha\widehat{\mathcal M}_2\equiv
e^{-\nu\widehat H^{0}}e^{\widehat S}e^{-\lambda{\widehat H}^{0}}
({\hat I}+\alpha\hat e_{1,m+1}e^{-\lambda{\widehat H}^{0}}),
\end{aligned}
\label{43}$$ where $\hat{ e}_{1,m+1}\equiv
({\delta}_{1,n}{\delta}_{m+1,l})_{1\leq n,l\leq M}$. The matrix $\widehat H^{0}$ is used instead of $\widehat H^{\pm}$ since $s$ can be replaced by zero for the sufficiently large $M$.
The relation is written in the coordinate representation. In order to pass to the momentum representation it is convenient to use certain formulas provided in [@22]. Keeping the matrix notations as in , we obtain the answer for ${\widetilde F}_{m+1;1}({\lambda})$ (in the limit $M\to\infty$, the corresponding operations should be understood in the sense of the operations over the corresponding integral operators [@26]): $${\widetilde F}_{m+1;1}({\lambda})=
\det({\hat I}+\widehat{\mathcal U}_m)
\biggl[\tr(e^{-{\lambda}\hat{\varepsilon}_{0}}\breve e_{1,m+1})-
\tr\biggl(\frac{\widehat{\mathcal V}_{m}}{{\hat I}+
\widehat{\mathcal U}_m}\biggr)\biggr]
\label{44}$$ (the notation $\tr$, for instance, corresponds to the trace of $M\times M$ matrices). The matrices $\widehat{\mathcal U}_m$, $\widehat{\mathcal V}_{m}$, ${\breve e}_{1,m+1}$ are given by the corresponding momentum representations of the matrices $\widehat{\mathcal M}_1$, $\widehat{\mathcal M}_2$, ${\hat
e}_{1,m+1}$ . However, we shall need explicit expressions only for the traces $\tr\widehat{\mathcal U}_m$ and $\tr\widehat{\mathcal V}_{m}$ (see below). In the momentum representation, $\hat{\varepsilon}_{0}$ is a diagonal matrix of the eigen-energies of the $XX$ model at $h=0$ [@21]. Formally expanding ${\widetilde F}_{m+1;1}({\lambda})$ in the powers of $\widehat{\mathcal U}_m$ we shall obtain the answer in two lowest orders: $$\begin{aligned}
&{\widetilde F}_{m+1;1}({\lambda})\approx
F_{m+1;1}({\lambda})+F_{m+1;1}({\lambda})
\tr\widehat{\mathcal U}_m-\tr\widehat{\mathcal V}_{m},
\nonumber
\\
&\tr\widehat{\mathcal U}_m=(M-2m)F_{1;1}({\lambda}),
\label{45}
\\
&\tr\widehat{\mathcal V}_{m}=F_{m+1;1}(2{\lambda})-
2\sum_{l=1}^{m}F_{m+1;l}({\lambda})F_{l;1}({\lambda}),
\nonumber\end{aligned}$$ where the notation $F_{j;l}({\lambda})$ implies the relations . Although $M$ and $m$ are chosen to be large enough, the ratio $m/M$ is assumed to be finite. Equation is fulfilled in each order separately by the terms presented in ${\widetilde F}_{m+1;1}$ .
By an analogy with the ferromagnetic case, let us act on ${\widetilde F}_{m+1,1}({\lambda})$ by the operator $\mathcal D^K_{\lambda/2}$. Then, in the first order we shall obtain the relation . In the second order, the answer is of the following form: $$MC_K^LC_K^L-C_K^L\sum_{l=0}^KC_K^l+2\sum_{l=1}^m
\begin{vmatrix}
C_K^{L+l-1}&C_K^L
\\
C_K^L&C_K^L
\end{vmatrix}.
\label{46}$$ By virtue of , the result of the application of $\mathcal D^K_{\lambda/2}$ to the second order function $F_{j_1,j_2;l_1,l_2}(\lambda)$ is connected, as a particular case of , with the number of the two-dimensional paths $\mathcal T_K$, and is expressed through the corresponding determinant. It means that it will be appropriate to express in the following equivalent form: $$\begin{aligned}
&(M-K)|P_{K}(l\rightarrow l+m)|^2+{}
\notag \\
&\qquad{}+
\mathcal D^K_{\lambda/2}\begin{bmatrix}
2\displaystyle\sum_{l=1}^m
\begin{vmatrix}
F_{m+1;l}({\lambda})&F_{m+1;1}({\lambda})
\\
F_{l;l}({\lambda})&F_{l;1}({\lambda})
\end{vmatrix}
-\displaystyle\sum_{l=0}^K
\begin{vmatrix}
F_{m+L;l}({\lambda})&F_{m+1;1}({\lambda})
\\
F_{l;l}({\lambda})&F_{l;L}({\lambda})
\end{vmatrix}
\end{bmatrix}.
\label{47}\end{aligned}$$ In other words, the result of application of $\mathcal D^K_{\lambda/2}$ to in the second order can be reformulated in terms of the random walks of the two pedestrians (see and ) and the squared number of walks of a single pedestrian. The summation by the index $l$ in $\tr\widehat{\mathcal V}_{m}$ can be interpreted as the summation over positions of a virtual walker in .
Using the equation , one can represent in terms of the number of trajectories on a two-dimensional lattice: $$(M-2(m+1))\mathcal T_K(m,0)+
\sum_{l=0}^m\mathcal T_K(m-l,l)-\sum_{l=1}^L\mathcal T_K(m+l,l)-
\sum_{l=1}^L\mathcal T_K(l,m+l).
\label{48}$$ In this relation various lattice trajectories of $K$ links are enumerated. All these trajectories start at the same point $\textbf{\textit{O}}=(0,0)$ while they terminate on the segments of the dashed broken line which connects the points $(L,L+m)$, $(0,m)$, $(m,0)$, and $(L+m,L)$ (see figure). Formally, the sign of the sum is not definite though its asymptotics is positive, in general. An analogous description is expected in the higher orders as well.
Typical configuration.
Let us estimate the behavior of the number of paths, which is given by the representation , in the limit, when the number of links $K=m+2 L$ increases. We shall assume, that the restriction $1\ll m\ll L$ is valid which means that $m$ increases moderately in the comparison with the increase of the number of turns $L$: for instance, let $L$ increase as $m^2$. Using the known asymptotical expansion of the logarithm of the gamma-function (see Appendix) [@27], one can estimate the binomial coefficient $C_K^L$. Restricting ourselves by the first order of smallness, we obtain: $$C_K^L\approx\frac{2^K}{{\sqrt{\pi L}}}e^{-m^2/(4L)}
\biggl(1-\frac{m}{2L}\biggl(1-\frac{m^2}{4L}\biggr)\biggr)\approx
\frac{2^K}{{\sqrt{\pi L}}}\biggl(1-\frac{m^2}{4L}\biggr)
\sim\frac{2^{2 L}}{{\sqrt{\pi L}\,}}.
\label{49}$$ The second approximate equality in takes place if $L$ is increasing faster than $m^2$. The estimate characterizes an increase of the number of the trajectories for a single pedestrian.
The third term in can be written as $2A(m,L)C_K^LC_K^L$, where $$A(m,L)\equiv-m+\sum_{l=1}^m\frac{(L+m+2-l)_{l-1}}{(L+1)_{l-1}}\,.
\label{50}$$ Standard notation $(\alpha)_{n}$ for the Pochhammer’s symbol is used in [@27]. Applying again an expansion of the logarithm of the gamma-function (A1), we can estimate $A(m,L)$ : $$A(m,L)\simeq mZ_1(m,L)-Z_0(m,L)+\mathcal{O}(m^{-1}),
\label{51}$$ where $$\begin{aligned}
&Z_0(m,L)\equiv e^{m^2/4L}\biggl(1+\frac{m}{L^2}\sum_{l=0}^{m/2}
e^{-l^2/L}\biggl(\frac{m^2}{4}-l^2\biggr)\biggr),
\\
&Z_1(m,L)\equiv-1+e^{m^2/4L}\,\frac{2}{m}\sum_{l=0}^{m/2}e^{-l^2/L}.
\end{aligned}
\label{52}$$ Let the values $m$ and $L$ increase with the ratio $L/m^2$ being finite and of order of unity. It can be shown (by means of numerical check as well), that the coefficient functions $Z_0(m,L)$ and $Z_1(m,L)$ remain finite in this case, and the contribution of $Z_0(m,L)$ is negligible in comparison with $m Z_1(m,L)$ in . One can use Eqs. and in order to estimate in the leading approximation: $$\frac{2^{4L}}{\pi L}\,e^{-m^2/2L}(M+2mZ_1(m,L)-
(\pi L)^{1/2}e^{m^2/4L}).
\label{53}$$ Because of the behavior of the coefficient $Z_1(m,L)$, the corresponding contribution in may turn out to be comparable with $M$. The relation demonstrates that the description of the random walks considered in the representation of the superposition of the eigen-states is more complicated than the one in the ferromagnetic case. This description can be regarded as a simultaneous walks of the initial (i.e., principal) and virtual pedestrians. The ending points of the trajectories belonging to all the three segments of the dashed broken line on the figure (see the representation of two-dimensional random walks ) correspond to comparable contributions into the estimate . In certain cases, characterized by the limiting behavior of the ratio $m^2/L$, the contribution of the segment between the points $(m,0)$ and $(0,m)$ can become dominating.
Conclusion {#sec4}
==========
It is shown that the correlation functions of the $XX$ Heisenberg magnet, calculated over the superposition of the eigen-states, as well as over the ferromagnetic state, are connected with enumeration of the trajectories made by the walkers moving on the lattice. A relationship is established between the number of trajectories made by a several vicious walkers and the number of paths made by a single random turns walker on a lattice of a dimension equal to the number of the vicious walkers. Differentiation of the generating function, calculated over the superposition of the eigen-states, demonstrates a more complicated combinatorial picture than that of the ferromagnetic case. In particular, the set of the paths made by a single pedestrian is replaced by the set of trajectories made simultaneously by the principal and virtual (both vicious) pedestrians. An estimate is obtained for the number of trajectories made both by the principal and the virtual pedestrians.
Acknowledgement {#acknowledgement .unnumbered}
===============
This paper was partially supported by the Russian Foundation for Basic Research, No. 07-01-00358, and by the Russian Academy of Sciences program ,,Mathematical Methods in Non-Linear Dynamics”.
Appendix {#appendix .unnumbered}
========
Asymptotic expansion for the logarithm of the gamma-function at large $|z|$ and $|\arg z |< \pi$ has the form [@27]: $$\begin{array}{rcl}
\log\Gamma
(z\,+\,\alpha)&=&\displaystyle{\Bigl(z\,+\,\alpha\,-\,\frac12\Bigr) \log
z\,-\,z\,+\,\frac12\,\log(2\pi)}\nonumber\\[0.4cm]
&+&\displaystyle{\sum\limits_{p=1}^n
(-1)^{p+1}\,\frac{B_{p+1}(\alpha)}{p(p+1)}\,z^{-p}\,+\,
\mathcal{O}\Bigl(\frac1{z^{n+1}}\Bigr)}\,,
\end{array}\eqno(A1)$$ where $n= 1, 2, 3, \dots$. The Bernoulli polynomials $B_{n}(\alpha)$ ($A1$) are defined as follows: $$B_n(\alpha)\,=\,\sum\limits_{l=0}^n C_n^l\,B_l\,\alpha^{n-l}\,,$$ where $C_n^l$ are the binomial coefficients, and $B_l$ are the Bernoulli numbers. The first Bernoulli polynomials $B_{p}(\alpha)$ look as follows: $$\begin{aligned}
B_0(\alpha)\,=\,1\,,\qquad B_1(\alpha)\,=\,\alpha\,-\,\frac12\,,\qquad
B_2(\alpha)\,=\,\alpha^2\,-\,\alpha\,+\,\frac16\,,\nonumber\\[0.4cm]
B_3(\alpha)\,=\,\alpha^3\,-\,\frac32\,\alpha^2\,+\,\frac12\,\alpha\,,\qquad
B_4(\alpha)\,=\,\alpha^4\,-\,2 \alpha^3\,+\,\alpha^2\,-\,\frac1{30}\,,
\nonumber\end{aligned}$$ where the Bernoulli numbers $B_l$ are used: $$\begin{aligned}
B_0\,=\,1\,,\qquad B_1\,=\,-\,\frac12\,,\qquad
B_2\,=\,\frac16\,,\qquad B_4\,=\,-\,\frac1{30}\,.
\nonumber\end{aligned}$$ 0.5cm
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| 0 |
INTRODUCTION
============
Because of the macroscopic degeneracy of single-particle states in a Landau level, neither disorder nor electron-electron interactions in a two-dimensional electron system (2DES) can be treated perturbatively in the quantum Hall regime. This is the [*raison d’être*]{} for the many interesting and surprising phenomena[@general] which have arisen in quantum Hall physics. Theories of quantum Hall physics usually include either only interactions or only disorder, although both are always present. In particular, it is common to include only disorder in studies of the integer quantum Hall effect (IQHE), which generally focus on the sudden jump in the Hall conductivity between values separated by $e^2/h$, and it is common to include only interactions in studies of the fractional quantum Hall effect (FQHE), which generally focus on the ability of interactions to create charge gaps at partial Landau level fillings. The competition between interactions and disorder has often, but not always,[@Green; @Nederveen] been neglected, in part because of the lack of easily manageable analytical and numerical tools that can deal with both simultaneously. In this paper we address an instance in which this competition is particularly direct and can be successfully addressed with elementary techniques.
At Landau level filling factor $\nu=1$, the ground state of a disorder-free 2DES is a strong ferromagnet,[@Sondhi] [*i.e.*]{} it is completely spin-polarized by a Zeeman field of infinitesimal strength. In practice, of course, the field experienced by a 2DES in the quantum Hall regime is [*not*]{} infinitesimal; however the field’s Zeeman coupling to the electron spin is typically very weak compared to other energy scales. (In referring to these systems as ferromagnets we are emphasizing that they remain spin polarized in the limit of zero Zeeman splitting. In experimental systems,[@Barrett] typical values of the interaction and Zeeman energy scales are $\sim 160 $ K and $\sim 3$ K respectively. The spin-splitting produced by this bare Zeeman coupling is usually negligible in paramagnetic states.) The quantum Hall ferromagnet has a large gap for charge excitations, and hence has a robust quantum Hall effect. For typical Zeeman coupling strengths, its elementary charged excitations are topologically charged spin textures (Skyrmions) containing several flipped spins.[@general; @Sondhi; @Fertig] Large Skyrmions have a Hartree energy cost smaller that those of conventional quasiparticles and, because the spins align locally, only slightly higher exchange energy. [@Brey] Because Skyrmions are the lowest-energy charged excitations, the global electron spin polarization is expected to decrease rapidly as $|1-\nu|$ increases. This has been observed in nuclear magnetic resonance (NMR) experiments. [@general; @Barrett; @Fertig] On the other hand, the $\nu=1$ state of non-interacting disordered electrons differs qualitatively. The ground state is a compressible paramagnet with no Knight shift and no gap for charged excitations. For zero Zeeman coupling, quasiparticle states at the Fermi energy are quasi-extended and cause the Hall conductivity to suddenly jump by $2 e^2/h$ as this filling factor is crossed; $\nu=1$ is in the middle of a Hall ‘riser’, instead of being at the middle of a Hall plateau. The competition between disorder and interactions at $\nu=1$ can be addressed[@olderhfdis] using the Hartree-Fock approximation which has the virtue of being exact[@general] in both the non-interacting and the non-disordered limits. Experimental information on this competition comes primarily from transport and NMR studies. Early NMR studies[@Barrett] of weak-disorder quantum Hall ferromagnets, yielded relatively featureless lineshapes and Knight shifts in good agreement with Hartree-Fock (HF) theory estimates[@Fertig; @Brey; @Cote] of Zeeman-coupling and filling-factor dependent Skyrmion sizes. (Effective field theory estimates[@Sondhi; @Rajaraman] are not accurate in the case of typical Zeeman coupling strengths.) More recent experiments[@Barrett_private] paint a more complex picture, in part because the measurements were performed at lower temperatures where the signal is not motionally averaged.[@Jairo; @Yalegrouppapers] It is now clear that disorder plays a role in the interpretation of these experiments, even when it is weak. At stronger disorder, as the non-interacting limit is approached, the spin-polarization must eventually vanish. In our model calculations we find that as the interaction strength is increased relative to disorder at $\nu=1$, the 2DES ground state suffers a continuous phase transition a from paramagnetic to a ferromagnetic state. Depending on the details of the disorder model, a second continuous phase transition to a fully spin-polarized incompressible strong ferromagnet with a gap for charged excitations may occur at still stronger interactions. For the disorder models we use, the fully polarized state is reached when the Coulomb energy scale is approximately twice the Landau-level-broadening disorder energy scale. Away from $\nu=1$, screening by mobile charges reduces the importance of disorder and the system reaches maximal spin-polarization at smaller interaction strengths. The maximally polarized ground state at moderate interaction strengths is best described as a glass of localized conventional quasiparticles formed in the $\nu=1$ fully polarized vacuum. Only for stronger interactions do we find a phase transition to a state with non-collinear magnetization in which the localized particles have Skyrmionic character.
We organize this paper as follows. In Sec. II we summarize our implementation of finite-size HF theory in the lowest Landau level (LLL). In Sec. III and IV we present and discuss our numerical results for calculations at $\nu=1$ and $\nu\ne 1$. The possibility, discussed in recent work,[@Nederveen] that at $\nu=1$ disorder will induce reduced-size Skyrmion-anti-Skyrmion pairs in the ground state is specifically addressed in Sec. III. Finally, in Sec. V we present our conclusions.
Hartree-Fock Theory in the LLL
==============================
The HF approximation allows the interplay between disorder and interactions to be addressed while retaining a simple independent-particle picture of the many-body ground state. For the current study, the use of this approximation is underpinned by the fact that it reproduces the exact ground state at $\nu=1$ in both weak interaction and strong interaction[@general] limits. HF theory is a self-consistent mean field theory, and as such it has strengths and shortcomings, which we discuss later. In this section, we outline the basic formalism of HF approximation calculations in the LLL limit.
In a strong magnetic field, the Landau level splitting is very large and, since excitations to higher Landau levels are effectively forbidden at the low experimental temperatures, we follow the common practice of considering only LLL states. Neglecting the frozen kinetic-energy degree of freedom, the Hamiltonian in second quantization is written as $${\cal H}={\cal H}_I+{\cal H}_{dis}+{\cal H}_Z\,,$$ where ${\cal H}_I$ is the interaction part of the Hamiltonian $$\begin{aligned}
{\cal H}_I&=&\frac{1}{2}\int d{{\bf r}}\int d{{\bf r}}' \sum_{\sigma\, \sigma'}
v_I({{\bf r}}-{{\bf r}}') \hat\psi^\dagger_\sigma({{\bf r}})
\hat\psi^\dagger_{\sigma'}({{\bf r}}') \hat\psi_{\sigma'}({{\bf r}}') \hat\psi_\sigma({{\bf r}})\,\,,\end{aligned}$$ ${\cal H}_{dis}$ is the external disorder part of the Hamiltonian $$\begin{aligned}
{\cal H}_{dis}&=&\int d{{\bf r}}\sum_{\sigma}v_{E}({{\bf r}})
\hat\psi^\dagger_\sigma({{\bf r}})\hat\psi_\sigma({{\bf r}})\,,\end{aligned}$$ and ${\cal H}_Z$ is the Zeeman term $${\cal H}_Z=-\frac{1}{2}g\mu_B \int d{{\bf r}}\sum_{\sigma \sigma'}
\hat\psi^\dagger_{\sigma'}({{\bf r}}') \hat\psi_{\sigma}({{\bf r}}')
\vec{\tau}_{\sigma' \sigma}\cdot\vec{B}(\vec r)\,,$$ with $\sigma=\uparrow,\downarrow$, $v_I$ and $v_E$ being the Coulomb interaction and disorder potentials respectively, and $\tau_i$ being the Pauli matrices. Here we also define the Zeeman coupling strength as $\tilde g=g\mu_B B/(e^2/\epsilon l)$ for later reference. We chose the the Landau gauge elliptic theta functions as our basis $$\phi_m(x,y)=\frac{1}{\sqrt{L_y l \sqrt{\pi}}}
\sum_{s=-\infty}^\infty e^{i\frac{1}{l^2}x_{m,s}y}
e^{-\frac{1}{2 l^2}(x-x_{m,s})^2}\,\,,$$ where $x_{m,s}=\frac{2\pi m l^2}{L_y}+s L_x$, $m,m'=1,\dots,N_\phi$, $N_\phi=A/(2\pi l^2)$, and $l$ is the magnetic length which we set equal to $1$ for simplicity. These wavefunctions satisfy the semi-periodic boundary conditions $\phi_m(x,y)=\phi_m(x,y+L_y)$ and $\phi_m(x+L_x)=\exp(+iL_x y/l^2)\phi_m(x,y)$.
We consider the HF single particle states to be a linear combination of the up and down spin states of these basis functions $$|\alpha\rangle=\sum_{m,\sigma}\langle m \sigma|\alpha\rangle \,\,
|m \sigma\rangle\,\,.$$ Before writing down the Hamiltonian matrix in the HF approximation, we introduce several notation simplifying definitions, closely following previous HF studies. [@olderhfdis] The expectation value of the particle density (in momentum space) is given by $$\begin{aligned}
\langle \rho({{\bf q}})\rangle&=&\sum_\alpha n_F(\epsilon_F-\epsilon_\alpha)\langle \alpha
|e^{-i{{\bf q}}\cdot{{\bf r}}}|\alpha\rangle\\
&\equiv&N_\phi e^{-\frac{1}{4}q^2}
\sum_{\sigma,\sigma'} \delta_{\sigma,\sigma'} \Delta_{\sigma'\,\sigma}({{\bf q}})
e^{-i\frac{q_x q_y}{2}}\,\,,\end{aligned}$$ where we define $$\Delta_{\sigma'\,\sigma}({{\bf q}})\equiv\frac{1}{N_\phi}
\sum_{m,m'}\delta_{(x_{m'},x_m+q_y)}e^{-iq_x x_m}
\rho_{\sigma'\,\sigma}(x_{m'}|x_m),$$ and $$\begin{aligned}
\rho_{\sigma'\,\sigma}(x_{m'}|x_m)
&=&\sum_\alpha n_F(\epsilon_F-\epsilon_\alpha) \langle
m'\sigma'|\alpha\rangle\langle\alpha|m\sigma\rangle\,.\end{aligned}$$ Here $\delta_{(x_{m'},x_m+q_y)}$ is a periodic Kronig delta function, [*i.e.*]{} it is nonzero for $x_{m'}=x_m+q_y+s\,L_x$ for any integer s. With these definitions we can write the Hamiltonian matrix in the HF approximation in a compact form which is simple to diagonalize numerically $$\begin{aligned}
&&\langle m \sigma|{\cal H}|m' \sigma'\rangle_{HF} =
\sum_{{{\bf q}}\epsilon {\rm BZ}}
\left\{(\gamma\Delta_0({{\bf q}}){\rm U_H}({{\bf q}})+{\rm U_{\rm dis}}({{\bf q}})) {\rm I}+\right.
\nonumber\\
&&\left.\frac{\gamma}{2}(\Delta_0({{\bf q}}){\rm I}+\vec{\Delta}({{\bf q}})\cdot\vec{\tau})
{\rm U_F}({{\bf q}})\right\}
\delta_{(x_m,x_{m'}+q_y)}e^{+iq_x x_{m'}}\nonumber\\
&&-\frac{1}{2}\tilde g \hat B\cdot \vec{\tau}\,,\end{aligned}$$ where $\Delta_\alpha({{\bf q}})={\rm Tr}\{\Delta_{\sigma \sigma'}({{\bf q}})
\tau^{\alpha}\}$, and $\hat B$ specifies the orientation of the external magnetic field. The various effective potentials which appear here are defined as $${\rm U_{dis}}({{\bf q}})=\frac{1}{A}\sum_{{\bf G}}e^{-\frac{1}{4}|{{\bf q}}+{{\bf G}}|^2}v_E({{\bf q}}+{{\bf G}}) e^{
\frac{i}{2}(q_x+G_x)(q_y+G_y)}\,\,,
\label{Udis}$$ $${\rm U_H}({{\bf q}})=\frac{1}{2\pi}\sum_{{{\bf G}}} e^{-\frac{1}{2}|{{\bf q}}+{{\bf G}}|^2}
\frac{2\pi e^2}{|{{\bf q}}+{{\bf G}}|}(1-\delta_{{{\bf q}}+{{\bf G}},0})\,\,,$$ and $${\rm U_F}({{\bf q}})=-\frac{1}{A }\sum_{{{\bf q}}'}
e^{-\frac{1}{2}|{{\bf q}}'|^2}e^{i{q'}_x q_y-iq_x q'_y}
\frac{2\pi e^2}{|{{\bf q}}'|}(1-\delta_{{{\bf q}}',0})\,,
\label{U_F}$$ with ${{\bf G}}=(\frac{2\pi N_\phi}{L_x} n_x,\frac{2\pi N_\phi}{L_y} n_y)$.
For $v_E({{\bf r}})$, we choose a white noise potential without spatial correlation, $\langle\langle v_E({{\bf r}})v_E({{\bf r}}')\rangle\rangle=\sigma^2 \delta({{\bf r}}-{{\bf r}}')$. The density of states in the non-interacting limit has been calculated exactly by Wegner for this distribution of the disorder potential,[@Wegner] yielding a full width at half maximum of approximately $1.06 \sigma$. In our calculations the parameter $\gamma=e^2/\epsilon\sigma$ specifies the relative strength of interactions and disorder broadening. This type of disorder potential distribution is characterized by a single parameter $\sigma$, which we use as our unit of energy. As we discuss later, our results are insensitive to correlations in the disorder potential on length scales smaller than $l$, but would change in some respects for disorder potentials which are smooth on the magnetic length scale. The HF equations are solved by an iterative approach which can create difficulties which must be addressed. The HF equations generally have many solutions that correspond to different, usually metastable, extrema of the HF energy functional. The challenge is to locate the true global minimum. In particular, the iteration process will not break any symmetries of the Hamiltonian which are not broken by the starting charge and spin-densities, even though the global minimum of the HF energy functional frequently does break at least some of these symmetries. To counter such problems, it is usually a good idea to introduce small artificial terms in the Hamiltonian which break the continuous symmetries and help the iterative process to reach the lowest energy state. In this problem, the iterative process is also hampered by severe convergence problems at zero temperature connected with the localization of HF quasiparticle wavefunctions and the long range nature of the Coulomb interactions. A small change in the energy of a particular orbital may involve a substantial rearrangement of the charges. These problems can be mitigated by always working at a temperature which is comparable to the finite-size quasiparticle energy level spacing and which scales to zero as the system size increases. The Zeeman term in the Hamiltonian, for which we choose typical experimental values, reduces the SU(2) spin symmetry to a U(1) symmetry. In order to break the continuous U(1) symmetry we introduce an artificial (but very small) local magnetic field at the center of our simulation cells which points in the x-direction. It is this space-dependent field, required on purely technical grounds, which has motivated developing the formalism in a manner which permits non-constant Zeeman coupling strengths. To ensure that this artificially field and the finite temperature do not affect the final solution, we lower the magnitude of these terms until no change is seen in local charge and spin densities, or in HF quasiparticle energies.
The phase diagrams discussed in Secs. III and IV, were obtained by starting from the non-interacting case and incrementing the interaction strength $\gamma$, taking as the starting densities the self-consistent densities from the previous $\gamma$ value. There is, of course, some hysteresis involved in this process so we do a backwards sweep on $\gamma$ once we have reached the maximum interaction strength for a given run. If they differ, we use the smaller of the values obtained in upward and downward sweeps for the energy per particle. The energy per particle is obtained using the expression $$\begin{aligned}
&&\frac{E}{N}=\frac{1}{\nu}\sum_{{{\bf q}}\in {\rm BZ}}{\rm U}_{\rm dis}({{\bf q}}) {\Delta_0}^*({{\bf q}})
+\frac{\gamma}{2\nu}\sum_{{{\bf q}}\in {\rm BZ}}{\rm U}_{\rm H}({{\bf q}}) |\Delta _0({{\bf q}})|^2\\
&&+{\rm U}_{\rm F}({{\bf q}})
\left(|\Delta _{\uparrow\uparrow}({{\bf q}})|^2
+|\Delta _{\downarrow\downarrow}({{\bf q}})|^2+|\Delta_{\uparrow
\downarrow}({{\bf q}})|^2+| \Delta_{\downarrow\uparrow}({{\bf q}})|^2\right)\\
&&-\frac{\gamma}{2} \tilde g \hat B\cdot \vec{P}_{tot}\,,\end{aligned}$$ where $ \vec{P}_{tot}$ is the total global spin polarization. The local spin magnetization density, which we calculate as well, is given by $$\langle S_i({{\bf r}})\rangle=
\frac{\hbar}{2}
\sum_{m',m,\sigma,\sigma'} \rho_{\sigma \sigma'}(x_{m'}|x_m)\tau^i_{\sigma \sigma'}
\phi_m^*({{\bf r}})\phi_{m'}({{\bf r}})\,\,.$$ We define the local spin polarization as $\langle P_i({{\bf r}})\rangle=2\langle S_i({{\bf r}})\rangle/(\hbar
\langle \rho({{\bf r}})\rangle)$. Note that in this case $\langle |\vec P({{\bf r}})| \rangle$ does not have to be equal to $1$ since the system is compressible except in the limit of very large $\gamma$ where $\langle |\vec{P}({{\bf r}})|\rangle\rightarrow 1$ for all ${{\bf r}}$.
Our criteria for convergence is that $$\delta \Delta\equiv\frac{1}{N_\phi^2} \sum_{{{\bf q}}\in {\bf BZ}}
\sum_{\sigma \sigma'}
|\Delta_{\sigma \sigma'}^{i}({{\bf q}})-\Delta_{\sigma \sigma'}^{i-1}({{\bf q}})|^2< 1\times
10^{-6}\,,$$ where $i$ stands for the $i$th iteration. We have performed calculations for several disorder realizations at different values of $\nu$ for system sizes of $N_\phi=16-32$. The finite size effects come mainly from the effective exchange potential $U_F({{\bf q}})$ and have been studied in detail previously.[@Allan_finite] The main effect is on the interaction part of the energy per particle and it is well understood and easily corrected. We believe that the physics of the phase transitions observed in these calculations is not affected qualitatively by finite-size effects. Our qualitative conclusions are based on persistent features which are obtained for several different disorder realizations. The values of $\gamma$ at which the various transitions and cross overs we discuss below take place, do not change by more than $5\%$ for different realizations. The results we present here are for one particular disorder realization.
Results at $\nu=1$
==================
At $\nu=1$ the disorder free ($\gamma \to \infty$) 2DES has[@general] a $S =N/2$ ground state. The $S_z=S=N/2$ member of this multiplet is a single Slater determinant and can therefore be obtained by solving Hartree-Fock equations self-consistently. It is only in recent years that samples which are sufficiently clean to reach, or at least nearly reach, complete spin polarization have been grown.[@Barrett] The collective behavior producing such a ground state was not exhibited in earlier samples which had more disorder in the form of unintended impurities, interface dislocations, and, in modulation doped samples, the potential from remote ionized donors. Fig. \[phase\_diag2\] summarizes the HF theory results we have obtained for the dependence of the spin polarization on interaction strength. The calculations were performed for a realistic value of the Zeeman coupling strength, $\tilde g=0.015$, and at a very small value, $\tilde g=0.0018$. Extrapolating from these two values to $\tilde g =0$, allows us to identify parameter values for which spontaneous spin polarization occurs, [*i.e.,*]{} values for which the ground state is ferromagnetic. We find that ferromagnetism occurs for $\gamma \gtrsim 0.5$ in the Hartree-Fock approximation; at smaller values of $\gamma$ the single-particle disorder term dominates and yields a spin-singlet ground state. Notice that the spin susceptibility, which may be estimated from the difference between the spin-polarizations at the two $\tilde g$ values, is small in the singlet state, and becomes large as the phase transition to the ferromagnetic state is approached. For the specific finite-size disorder realization we have studied, complete spin polarization is reached at a finite value of $\gamma \sim 1.5$. At larger values of $\gamma$, the system has a finite gap for charge excitations. We must be aware, however, that the HF approximation overestimates the tendency of the system to order so the interaction strength at both transition points should be taken as lower limits. In addition, any physically realistic disorder potential is likely to have rare strong disorder regions which prevent the fully polarized state from being reached.
In our calculations, there is a wide region of interaction strengths $\gamma$ for which partially spin-polarized states occur. In this regime our HF ground states nearly always have non-collinear magnetic order. We show local spin polarization and charge density profiles of typical partially polarized states in Figs. \[local\_pol2\] and \[local\_dens2\]. The origin of the reduced spatially integrated spin polarization is partly due to variation of spin-orientation, but principally due to a reduction in the average value of the of the [*magnitude*]{} of the local spin polarization. This point is illustrated in Fig. 1 (open circles) and may be inferred from Fig. \[local\_pol2\] (a). The reduction in spin-polarization is due to the occurrence of doubly-occupied orbitals, [*i.e.*]{} to disorder induced charge fluctuations which cannot be accurately described in models which include only the spin degree of freedom. The charged excitations of the ground state in this regime are ungapped and involve population of localized quasiparticle states. We also remark that local density profiles at these relatively small $\gamma$ values, illustrated in in Fig. \[local\_dens2\] (a), follow the effective disorder potential smoothed by the form factor for lowest Landau level electrons. Rapid spatial variation components in the white noise model disorder potential have little effect on the electronic state. The relationship between electron number density and the Pontryagan index density of the local spin orientation, valid for slow spin-orientation variation and nearly constant charge density,[@general] is [*not*]{} valid in this regime. Still, the collective nature of the 2DES manifests itself in the nonzero spin polarization density perpendicular to the Zeeman field. As interactions strengthen further, the local charge density smoothes out favoring the minimization of Coulomb energy at a cost in disorder energy. (See Fig. 2 (b) and 3 (b) for $\gamma\approx 1.5$.)
Experimentally, the effects of disorder can be seen most directly in the NMR spectral line shape obtained at the lowest possible temperatures where the spin profile is frozen on the experimental time scale. [@Jairo; @Yalegrouppapers] The NMR intensity spectrum in this regime is given by $$I(f,\gamma)\propto \int d{\bf r} \rho_N(z)e^{
-\frac{1}{2\sigma^2}(2\pi f -2\pi K_s \rho_e(z) \langle \vec{S}({{\bf r}};\gamma)\rangle)}
\,,
\label{NMRspectra}$$ with $\sigma=9.34 {\rm ms}^{-1}$ and $K_s\sim 25{\rm KHz}$. Here $\rho_N(z)$ is the nuclear polarization density and $\rho_e(z)$ is the electron density envelope function in the quantum well. The evaluation of such spectra has been outlined elsewhere;[@Jairo; @Yalegrouppapers] here we simply show results for several interaction strengths in Fig. \[NMR2\]. The parameters used in Eq. \[NMRspectra\] are the same as the ones used in Ref. .
Note that the quantity usually identified experimentally as the Knight shift, the location of the peak in the NMR spectrum in Fig. \[NMR2\], does not match the global polarization. This Knight shift measurement always overestimates the global polarization. In order to obtain the global polarization of the system from the measured spectrum one has to extract the first moment of a normalized spectrum.[@Jairo] One sees from the NMR spectrum at $\gamma=1.25$ that disorder induced spin density variation leads to a broadening of the maximum peak and can even lead to secondary peaks at lower Knight shift frequencies. Note, however, that features corresponding to negative Knight shifts, corresponding to regions of reversed electronic spins, are unlikely because the typical size of such regions is small and because they are also obscured by the finite width of the quantum wells which trap the 2DES. Our calculations demonstrate that care must be taken in interpreting low temperature NMR data in the quantum Hall regime.
The partially polarized regime can also be studied experimentally by measuring the transport activation gap. Provided that weak Zeeman coupling can be ignored, the extended quasiparticle states are expected to be precisely at the Fermi level when the 2DES is in a paramagnetic state. The Hall conductivity should jump from $0$ to $2 e^2/h$ at $\nu=1$. In the ferromagnetic state, the majority-spin extended quasiparticle state will be below the Fermi level, the majority-spin extended state will be above the Fermi level and the Hall conductivity at $\nu=1$ should be quantized at $\sigma_{xy} = e^2/h$. The spontaneous splitting of the two extended state energies is experimentally accessible and should exhibit interesting power law critical behavior as the ferromagnetic state is entered. This transport gap should vary monotonically with the global spin polarization, although the precise relationship between these quantities is not trivial.
It is interesting to note that the Skyrmion-anti-Skyrmion pairs predicted recently by Nederveen and Narazov [@Nederveen] do not appear in our calculations. We do not conclude that these objects cannot appear at $\nu=1$; we would expect them, for example, if we choose a disorder model with relatively large potential variations, but only on a length scale much larger than the Skyrmion size. In this case the NL$\sigma$ model considerations in Ref. should be applicable. Our calculations demonstrate rather clearly however, that charge density variation at $\nu=1$ does not necessarily, or even usually, require the existence of well defined Skyrmion quasiparticles.
Results at $\nu\ne 1$
=====================
In clean (large $\gamma$) samples where full polarization is observed at $\nu=1$, the global polarization decays rapidly with $|1-\nu|$.[@Barrett] It is generally accepted that this property is a unique signature which experimentally establishes the thermodynamic stability of Skyrmion collective quasiparticles. In the strong disorder limit, on the other hand, spontaneous spin polarization does not occur at any filling factor near $\nu = 1$.
The global polarization results for $\nu \ne 1$ in Fig. \[phase\_diag3\], illustrate how the system interpolates between these two extrema. As the interaction strength $\gamma$ is increased from 0 to 2, the behavior is similar to the $\nu=1$ case. For strong disorder charge variation is dominant, and small spin polarizations occur primarily because many single particle orbitals are occupied by both up and down spin electrons. Charge variation is the dominant response to disorder, and it continues to play an important role at all interaction strengths. At sufficiently large $\gamma$, our finite size systems reach a state with the maximum spin polarization allowed by the Pauli exclusion principle. This maximally polarized state is reached earlier than in the case at $\nu=1$ ($\gamma\sim1.4-1.6$) because, we believe, a larger number of charged quasiparticles are available to screen the random potential. At this point the system forms what we refer to as a conventional quasiparticle glass (CQG). The conventional Laughlin quasiparticles are initially localized in the deepest minima (or maxima for $\nu<1$) of the effective disorder potential and as the interaction strength increases, or equivalently the depth of the disorder potential wells becomes smaller, the charged quasiparticles rearrange themselves locally into a quasi-triangular Wigner crystal pinned by the strongest of the disorder potential extrema. At larger $\gamma$ we observe a transition from a CQG to a Skyrmion glass. The location of this transition is marked by a reduction of the global polarization from its maximally polarized value. For a specific disorder realization the point of cross over from the CQG to the Skyrmion glass, as illustrated in Fig. \[phase\_diag3\], depends on filling factor and $\tilde g$. The dependence of the transition point on $\tilde{g}$ in this regime can be approximated by considering a simple model for a single Skyrmion trapped at a disorder potential extrema. We approximate its energy by $$E(K)=U(K-K_0)^2+g^*\mu_B B K+\sigma AK\,,
\label{model}$$ where $K$ is the number of spin flips per Skyrmion, $\sigma$ is the strength of the disorder potential and $A$ is a phenomenological parameter. The first two terms determine the optimal Skyrmion size in the absence of disorder.[@UKmodel_Allan] The form for the third term reflects the property that Skyrmions with smaller $K$ are smaller and will be able to concentrate more strongly close to the potential extrema. This simple model gives an estimate of the interaction strength at which $K > 0$ Skyrmions first become stable $$\gamma^*=\frac{A}{2K_0U/(e^2/\epsilon \ell)-\tilde{g}}\,.
\label{gammastar}$$ The parameters $U$ and $K_0$ can be estimated[@UKmodel_Allan] for filling factor $\nu=1.25$ as $U/(e^2/\epsilon \ell)\sim0.014$ and $K_0\sim 1$. Using our numerical result for where the transition occurs at $\tilde{g}=0.0018$, we estimate that $A\sim 0.1$. From this, one obtains an estimate of $\gamma^*\sim 7$ for the cross over point from conventional quasiparticles to Skyrmions at $\tilde{g}=0.015$. This is in reasonable agreement with the actual cross over point $\gamma^*\sim 10$ (see Fig. \[phase\_diag3\], the transition is out of scale in Fig. \[phasediag\]) given the simplicity of the model. These estimates of the maximum disorder strength at which Skyrmion physics is realized could be checked by performing NMR experiments in samples where electron density, and hence the interaction strength, is adjusted by the application of gate voltages.
For a particular realization of the disorder potential, particle-hole symmetry is broken in a finite system and is recovered only in the limit of very large $\gamma$. The particle-hole symmetry relation for the global spin polarization is i.e. $(1-\epsilon) P_z(\nu=1-\epsilon)=(1+\epsilon) P_z(\nu=1+\epsilon)$ where $\epsilon
<1$. At large $\gamma$ this relation is approximately satisfied. Also in this limit, the Pontryagan relation between the local density profile and the local spin density [@general] becomes accurate. In the clean limit, the Skyrmion system crystallizes in a square lattice for the filling factors considered here. (The Skyrmion crystal is triangular[@UKmodel_Allan] for $\nu$ very close to 1.) The disordered Skyrmion glass state has very smooth fluctuations of the local spin density, compared to the CQG, although both lattices are pinned by the disorder potential. We remark that quantum fluctuations in Skyrmion positions are not accounted for in HF theory, and it is quite possible that even in this limit the ground state is a liquid rather than a crystal. [@Juanjo] As noted in Ref. it is possible that the broken U(1) symmetry of the Skyrmions orientation order predicted by Hartree-Fock does not survive quantum fluctuations. We show an example of the CQG in Fig. \[local\_pol3\] (a) and \[local\_dens3\] (a), and of the quasi Skyrmion lattice state in Fig. \[local\_pol3\] (b) and \[local\_dens3\] (b). Note that we find, in agreement with Nederveen and Narazov, [@Nederveen] a shrinking of the Skyrmion size as disorder broadening increases. This effect may help explain the appearance of a “tilted plateau” centered around $\nu=1$ in the Knight shift vs. filling factor data. [@Barrett_private] Rare highly disorder regions in the sample may localize and reduce the effective size of the few Skyrmions present at these filling factors. This would allow the bulk of the sample to be fully polarized at $\nu\ne 1$ and give rise to a Knight shift equivalent to the one at $\nu=1$. The plateau is tilted because of the change in fully polarized density as pointed out in Ref. .
Discussion
==========
We have used the Hartree-Fock approximation to study the competition between interactions and disorder near Landau level filling factor $\nu=1$. At a qualitative level our results can be summarized by the schematic zero-temperature phase diagram shown in Fig. \[phasediag\], which is drawn for the case of small but non-zero Zeeman coupling. Distinct ground states can be distinguished by different values for the quantized Hall conductivity, $\sigma_{xy}$, by the presence or absence of spontaneous spin-polarization perpendicular to the direction of the Zeeman field, and by the presence or absence of a gap for spin-flip excitations. At small $\gamma$ (strong disorder), the electronic state is paramagnetic (denoted as PC in Fig. \[phasediag\]), there is no spin-polarization in the absence of Zeeman coupling, and the Hall conductivity is expected to jump from $0$ to $2 e^2/h$ as the filling factor $\nu$ crosses the $\nu=1$ line. For a small Zeeman coupling, there will be a small splitting between the majority-spin and minority-spin extended state energies and the zero-temperature Hall conductance should have a narrow intermediate $e^2/h$ plateau centered on $\nu=1$. However, we do not expect that this plateau will be observable at accessible temperatures, and have indicated this in Fig. \[phasediag\] by using a thick line to mark the $0$ to $2 e^2/h$ phase boundary. At somewhat larger $\gamma$ there is a phase transition at zero Zeeman energy between paramagnetic and ferromagnetic states (denoted as FC in Fig. \[phasediag\]). In our calculations this transition occurs at a larger value of $\gamma$ at $\nu=1$ than away from $\nu=1$. As $\gamma$ increases in the ferromagnetic state, we expect that the separation between majority-spin and minority-spin extended state levels will increase rapidly so that the $\nu=1$ integer quantum Hall plateau will broaden and become observable. At still larger $\gamma$, we find a transition to a state with the maximum spin polarization allowed by the Pauli exclusion principle. At $\nu \le 1$, this is full spin-polarization. In these states, marked ‘SG’ for spin-gap in Fig. \[phasediag\], the differential spin-susceptibility vanishes. For realistic disorder models, it seems likely that in the thermodynamic limit there will always be rare high-disorder regions in the sample which prevent maximal spin-polarization from being achieved. For this reason, the phase transition we find in our finite systems likely indicates a crossover from large to small differential spin-susceptibility in macroscopic systems; we have therefore marked this transition by a dashed line. Finally at the largest values of $\gamma$ (weakest disorder) the physics for $\nu$ near $1$ is dominated by Skyrmion quasiparticles which emerge from the $\nu=1$ ferromagnetic vacuum. In this regime, the system develops spontaneous spin-polarization in the plane perpendicular to the direction of the Zeeman field. In Fig. \[phasediag\] we have labelled this regime NCF for non-collinear ferromagnet.
This phase diagram is intended to represent the filling factor interval $0.85 \le \nu \le 1.15$, over which fractional quantum Hall effects are not normally observed and it appears likely that Hartree-Fock approximation calculations are able to represent interaction effects. Some of our findings may help explain the striking tilted plateau feature observed in the NMR spectra [@Barrett_private] near $\nu=1$. Nevertheless, we have found rich structure in the crossover between non-interacting and disorder-free limits of the $\nu=1$ quantum Hall effect which helps explain the difficulty experienced in attempting to construct a simple interpretation of low-temperature NMR spectra. Our calculations motivate experimental studies of $\nu=1$ transport activation energy studies near the paramagnetic to ferromagnetic phase transition.
Helpful conversations with S.E. Barrett, Luis Brey, and Tatsuya Nakajima are greatly acknowledged. This work was supported by the National Science Foundation under grants DMR-9714055 and DMR-9820816.
For an introduction to Skyrmions and related topics see S.M. Girvin, [*The Quantum Hall Effect: Novel Excitations and Broken Symmetries*]{} in Les Houches Summer School 1998 (to be published by Springer Verleg and Les Editions de Physique, 1999), and references therein. S.M. Girvin and A.H. MacDonald, in [*Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-Dimensional Semiconductor Structures*]{}, edited by S. Das Sarma and A. Pinczuk (Wiley, New York, 1997).
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\` Edinburgh 99/1\
\` OUTP-99-12P\
[**Monopole clusters, Z(2) vortices and confinement in SU(2).\
**]{}\
A. Hart$^1$ and M. Teper$^2$.\
[**Abstract.**]{}
We extend our previous study of magnetic monopole currents in the maximally Abelian gauge [@hart98] to larger lattices at small lattice spacings ($20^4$ at $\beta = 2.5$ and $32^4$ at $\beta = 2.5115$). We confirm that at these weak couplings there continues to be one monopole cluster that is very much longer than the rest and that the string tension, $K$, is entirely due to it. The remaining clusters are compact objects whose population as a function of radius follows a power law that deviates from the scale invariant form, but much too weakly to suggest a link with the analytically calculable size distribution of small instantons. We also search for traces of Z(2) vortices in the Abelian projected fields; either as closed loops of ‘magnetic’ flux or through appropriate correlations amongst the monopoles. We find, by direct calculation, that there is no confining condensate of such flux loops. We also find, through the calculation of doubly charged Wilson loops within the monopole fields, that there is no suppression of the $q=2$ effective string tension out to at distances of at least $r \simeq 1.6/ \sqrt{K}$, suggesting that if there are any vortices they are not encoded in the monopole fields.
PACS indices: 11.15.Ha, 12.38.Aw, 14.80.Hv.
Introduction {#sec_intro}
============
Many recent efforts to elucidate the mechanism of confinement in [qcd]{} and non–Abelian gauge theories have focused on isolating a reduced set of variables that are responsible for the confining behaviour. In the dual superconducting vacuum hypothesis [@mandelstam76; @thooft81] the crucial degrees of freedom are the magnetic monopoles revealed after Abelian projection. In the maximally Abelian gauge [@thooft81; @kronfeld87] one finds that the extracted U(1) fields possess a string tension that approximately equals the original SU(2) string tension (‘Abelian dominance’) [@suzuki90], and that this is almost entirely due to monopole currents in these Abelian fields (‘monopole dominance’) [@stack94; @bali96]. The magnetic currents observed in the maximally Abelian gauge are found to have non-trivial correlations with gauge-invariant quantities such as the action and topological charge densities (see for example [@feurstein97; @bakker98] and references therein) and this invites the hypothesis that the structures formed by the magnetic monopoles correspond to similar objects in the SU(2) vacuum, seen after gauge fixing and Abelian projection. If the magnetic monopoles truly reflect the otherwise unknown infrared physics of the SU(2) vacuum, analysis of these structures may provide important information about the confinement mechanism.
The main purpose of this paper is to extend our previous study [@hart98] of monopole currents to lattices that are larger in physical units at the smallest lattice spacings. As reviewed in Sec. \[sec\_cl\_str\], we obtained in [@hart98] a strikingly simple monopole picture at $\beta = 2.3$, 2.4. When the magnetic monopole currents are organised into separate clusters, one finds in each field configuration one and only one cluster which is very much larger than the rest and which percolates throughout the entire lattice volume. Moreover this largest cluster is alone responsible for infrared physics such as the string tension. The remaining clusters are compact objects with radii varying with length roughly as $r \propto \sqrt{l}$ and with a population that follows a power law as a function of length. We found the exponent of this power law to be consistent with a universal value of 3. This simple pattern became more confused at $\beta = 2.5$. The scaling relations for cluster size that we established in [@hart98] suggested that our $L=16$ lattice at $\beta = 2.5$ was simply too small. There was of course an alternative possibility: that the simple picture we found at lower $\beta$ failed as one approached the continuum limit. Clearly it is important to distinguish between these two possibilities, and this is what we propose to do in this paper. The cluster size scaling relations referred to above imply that an $L=32$ lattice at $\beta=2.5115$ should have a large enough volume. Such gauge fixed lattice fields were made available to us by G. Bali and we have used them, supplemented by calculations on an intermediate $L=20$ volume at $\beta=2.5$, to obtain evidence, as described in Secs. \[sec\_cl\_str\] and \[sec\_mon\_vor\_K\], that the monopole picture we found previously is in fact valid at these lattice spacings and that the deviations we found previously were due to too small a lattice size.
The fact that one has to go to space-time volumes that are ever larger, in physical units, as the lattice spacing decreases, hints at some kind of breakdown of ‘monopole dominance’ in the continuum limit. We finish Sec. \[sec\_cl\_str\], with a discussion of the form that this breakdown might take.
An attractive alternative to the dual superconducting vacuum as a mechanism for confinement is vortex condensation [@thooft79; @mack80; @nielsen79; @kovacs98; @deldebbio98; @ambjorn98]. Here the confining degrees of freedom are the vortices created by the ’t Hooft dual disorder loops [@thooft79] and the confining disorder is located in the centre Z(N) of the SU(N) gauge group. When such a vortex intertwines a Wilson loop, the fields along the loop undergo a gauge transformation that varies from unity to a non-trivial element of the centre as one goes once around the Wilson loop. For SU(2) this means that the Wilson loop acquires a factor of $-1$. A condensate of such vortices will therefore completely disorder the Wilson loop and will lead to linear confinement. At the centre of the vortex, which will be a line in $D=2+1$ and a sheet in $D=3+1$, the fields are clearly singular (multivalued) if we demand that the vortex correspond to a gauge transformation almost everywhere. In a properly regularised and renormalised theory, this singularity will be smoothed out [@thooft79] into a core of finite size in which there is a non-trivial but finite action density, and whose size will be $O(1)$ in units of the physical length scale of the theory. One can either try to study these vortices directly in the SU(2) gauge fields or one can go to the centre gauge [@deldebbio98; @ambjorn98], where one makes the gauge links as close to $+1$ or $-1$ as possible, and construct the corresponding fields where the link matrices take values in Z(2) (‘centre projection’) and where the only nontrivial fluctuations are singular Z(2) vortices. Just as a ’t Hooft–Polyakov monopole will appear as a singular Dirac monopole in the Abelian fields that one obtains after Abelian projection, one would expect the presence of a vortex in the SU(2) fields to appear as a singular Z(2) vortex after centre projection. This picture has received increasing attention recently and has, for example, proved successful in reproducing the static quark potential [@kovacs98; @deldebbio98] (‘centre dominance’). Our ability, in this paper, to address the question of how important are such vortices is constrained by the fact that we only work with Abelian projected SU(2) fields. So first we need to clarify how such vortices might be encoded in these Abelian fields and only then can we perform numerical tests to see whether there is any sign of their presence. This is the content of Sec. \[sec\_mon\_vor\_K\].
Finally there is a summary of the results in Sec. \[sec\_summ\].
Monopole cluster structure {#sec_cl_str}
==========================
Background
----------
Fixing to the maximally Abelian gauge of SU(2) amounts to maximising with respect to gauge transformations the Morse functional R = - \_[n,]{} ( U\_(n) i\_3 U\^\_(n) i\_3 ). \[eqn\_R\] It is easy to see that this maximises the matrix elements $|[U_\mu(n)]_{11}|^2$ summed over all links. That is to say, it is the gauge in which the SU(2) link matrices are made to look as diagonal, and as Abelian, as possible — hence the name. Having fixed to this gauge, the link matrices are then written in a factored form and the U(1) link angle (just the phase of $[U_\mu(n)]_{11}$) is extracted. The U(1) field contains integer valued monopole currents [@degrand80], $\{ j_\mu(n) \}$, which satisfy a continuity relation, $\Delta_\mu j_\mu(n) = 0$, and may be unambiguously assigned to one of a set of mutually disconnected closed networks, or ‘clusters.’
In [@hart98] we found that the clusters may be divided into two classes on the basis of their length, where the length is obtained by simply summing the current in the cluster l = \_[n,]{} | j\_(n) |. The first class comprises the largest cluster, which is physically the most interesting. It percolates the whole lattice volume and its length $l_{\max}$ is simply proportional to the volume $L^4$ (at least in the interval $2.3 \le \beta \le 2.5$) when these are re-expressed in physical units, [*i.e.*]{} $l_{\max}\sqrt{K} \propto (L\sqrt{K})^4$, where $K$ is the SU(2) lattice string tension in lattice units and we use $1/\sqrt{K}$ to set our physical length scale. We remark that over this range in $\beta$ there is a factor 2 change in the lattice spacing, and so one might have expected that the extra ultraviolet fluctuations on the finer lattice would lead to significant violations of the naïve scaling relation. That is to say, one might expect to need to coarse grain the currents at larger $\beta$ to obtain reasonable scaling. That this is not required is perhaps surprising.
The remaining clusters were found to be much shorter. Their population as a function of length (the ‘length spectrum’) is described by a power law N(l) = , \[eqn\_len\_spec\] where $\gamma \approx 3$ for all lattice spacings and sizes tested and the coefficient $c_l(\beta)$ is proportional to the lattice volume, $L^4$, and depends weakly on $\beta$. The radius of gyration of these clusters is small and approximately proportional to the square root of the cluster length, just like a random walk. When folded with the length spectrum, this suggests [@hart98] that the ‘radius spectrum’ should also be described by a power law N(r) = , \[eqn\_rad\_spec\] with $\eta \approx 5$ and $c_r(\beta)$ weakly dependent on $\beta$. Such a spectrum is close to the scale invariant spectrum of 4–dimensional balls of radius $\rho$, $N(\rho) d\rho \sim d\rho/\rho
\times 1/\rho^4$, and so one might try and relate these clusters to the SU(2) instantons in the theory, which classically also possess a scale-invariant spectrum. It is well known, however, that the inclusion of quantum corrections renders the spectrum of the latter far from scale invariant, at least for the small instantons where perturbation theory can be trusted, and so such a connexion does not seem to be possible [@hart98]. On sufficiently large volumes the difference in length between the largest and second largest cluster is very marked, and where this gulf is clear one finds that the long range physics such as the monopole string tension arises solely from the largest cluster. This is the case at $\beta = 2.3$, $L \ge 10$ and at $\beta = 2.4$, $L \ge 16$. On moving to a finer $L=16$ lattice at $\beta=2.5$ the gulf was found to disappear and the origin of the long range physics was no longer so clear cut. This could be a mere finite volume effect, or, much more seriously, it might signal the breakdown of this monopole picture in the weak coupling, continuum limit. Clearly this needs to be resolved and the only unambiguous way to do so is by performing the calculations on large enough lattices.
This calculation {#ssec_thiscalc}
----------------
The direct way to estimate the lattice size necessary at $\beta=2.5$ to restore (if that is possible) our picture is as follows. Suppose that the average size of the second largest cluster scales approximately as l\_ L\^()\^. We know that $l_{\max} \propto L^4 (\sqrt{K})^3$ to a good approximation for the largest cluster. So we will maintain the same ratio of lengths $l_{\second} / l_{\max}$, and a gulf between these, if = ( )\^[ -( )]{} \[eqn\_lK\_scale\] If we take our directly calculated values of $l_{\second}$, they seem to give roughly $\alpha \simeq 1$ and $\delta \simeq -2$. This suggests that we need to scale our lattice size with $\beta$ so as to keep $L(\sqrt{K})^{5/3}$ constant. This estimate is not entirely reliable because, on smaller lattices, the distributions of the ‘largest’ and ‘second largest’ clusters overlap so that they exchange [*rôles*]{}. An alternative estimate can be obtained from the tail of the distribution in eqn. \[eqn\_len\_spec\] that integrates to unity. Doing so [@hart98] one obtains $\alpha \simeq 2$ and $0 < \delta < 0.25 $. This suggests that we scale our lattice size so as to keep $L(\sqrt{K})^{\{1.4 \to 1.5\}}$ constant. This estimate is also not very reliable, since it assumes that the distribution of secondary cluster sizes on different field configurations fluctuates no more than mildly about the average distribution given in eqn. \[eqn\_len\_spec\]. In fact the fluctuations are very large. \[As we can see immediately when we try to calculate $\langle l^2 \rangle$ in order to obtain a standard fluctuation — it diverges for a length spectrum with $N(l)\propto dl/l^3$.\] Nonetheless, the two very different estimates we have given above produce a very similar final criterion: to maintain the same gap between the largest and second largest clusters as $\beta$ is varied, one should choose $L$ so as to keep $L(\sqrt{K})^{\sim 1.5}$ constant.
So if we wish to match the clear picture on an $L=10$ lattice at $\beta=2.3$ (where $K = 0.136$ (2)) we should work on a lattice that is roughly $L=28$ at $\beta=2.5$ (where $K = 0.0346$ (8)). In particular we note that an $L=32$ lattice at $\beta = 2.5115$ (where $K = 0.0324$ (10)) is more than large enough and an ensemble of 100 such configurations, already gauge fixed [@bali96], has been made available to us by the authors. The gauge fixing procedure used in obtaining these is somewhat different from the one we have used in our previous calculations (in its treatment of the Gribov copies — see below) and although this is not expected to affect the qualitative features that are our primary interest here, it will have some effect on detailed questions of scaling etc. We have therefore also performed a calculation on an ensemble of 100 gauge fixed $L=20$ field configurations at $\beta=2.5$. While the latter volume is not expected to be large enough to recreate a clear gulf between the largest and remaining clusters, we would expect to find smaller finite size corrections than with the $L=16$ lattice we used previously.
In gauge fixing a configuration we select a local maximum of the Morse functional, $R$, of which on lattices large enough to support non–perturbative physics there are typically a very large number [@hart97a]. These correspond to the (lattice) Gribov copies. Gauge dependent quantities appear to vary by ${\cal O}(10\%)$ depending upon the Gribov copy chosen; this is true not only of local quantities such as the magnetic current density [@hioki91] but also of supposedly long range, physical numbers such as the Abelian and monopole string tensions [@bali96; @hart97a]. Some criterion must be employed for the selection of the maxima of $R$, and in the absence of a clear understanding of which maximum, if any, is the most ‘physical’, one maximum was selected at random in [@hart98]. An alternative strategy, used in gauge fixing the $L=32$ lattices at $\beta = 2.5115$, is to pursue the global maximum of $R$ [@bali96]. Each field configuration is fixed to the maximally Abelian gauge 10 times using a simulated annealing algorithm that already weights the distribution of maxima so selected towards those of higher $R$. The solution with the largest $R$ from these is selected. Details of this method are discussed in [@bali96]. The difference in procedures invites caution in comparing exact numbers between this ensemble and those studied previously; for example a ${\cal O}(10\%)$ suppression in the string tension is observed. It is likely that cluster lengths will differ by a corresponding amount and this will prevent a quantitative scaling analysis using this ensemble. The power law indices do appear, however, to be robust [@hart97b] and it also seems likely that ratios of string tensions obtained on the same ensemble can be reliably compared with other ratios.
Cluster properties
------------------
The fact that the largest cluster does not belong to the same distribution as the smaller clusters is seen from the very different scaling properties of these clusters with volume [@hart98]. It is also apparent from the fact that the largest cluster is very much longer than the second largest cluster. Indeed for a large enough volume and for a reasonable size of the configuration ensemble, there will be a substantial gulf between the distribution of largest cluster lengths and that of the second largest clusters. By contrast the length distributions of the second and third largest clusters strongly overlap. This is the situation that prevailed for the larger lattices at $\beta=2.3$ and 2.4 but which broke down on the $L=16$ lattice at $\beta=2.5$. We can now compare what we find on our $L=20$ and $L=32$ lattices with the latter. This is done in Table \[table\_length\]. There we show the longest and shortest cluster lengths for the largest, second largest and third largest cluster respectively over the ensemble. The ensemble sizes are not exactly the same, but it is nonetheless clear that there is a real gulf between the largest and second largest clusters on the $L=32$ lattice while there is significant overlap in the $L=16$ case. The $L=20$ lattice is a marginal case. We conclude from this that the apparent loss of a well separated largest cluster as seen in [@hart98] at $\beta=2.5$ was in fact a finite volume effect, and that our scaling analysis has proved reliable in predicting what volume one needs to use in order to regain the simple picture.
In Figure \[fig\_curr\_dens\] we show how the length of the largest cluster varies with the lattice volume when both are expressed in physical units (set by $\sqrt{K}$). To be specific, we have divided $l_{\max}\sqrt{K}$ by $(L\sqrt{K})^4$ and plotted the resulting numbers against $L\sqrt{K}$ for both our new and our old calculations. The fact that at fixed $\beta$ the values fall on a horizontal line tells us that that the length of the largest cluster is proportional to the volume at fixed lattice spacing: $l_{\max}
\propto L^4$. The fact that the various horizontal lines almost coincide tells us that the current density in the largest cluster is consistent with scaling. That is to say, it has a finite non-zero value in the continuum limit. Thus the monopole whose world line traces out this largest cluster, percolates throughout the space–time volume and its world line is sufficiently smooth on short distance scales that its length does not show any sign of diverging as we take the continuum limit. We note that the $L=32$ lattice deviates by $\sim
10\%$ from the other values. This is consistent with what we might have expected from the different gauge fixing procedure used in that case.
Turning now to the secondary clusters, we display in Figure \[fig\_len\_spec\] the length spectrum that we obtain at $\beta=2.5115$. It is clearly well described by a power law as in eqn. \[eqn\_len\_spec\] and we fit the exponent to be $\gamma =
3.01$ (8). This is in accord with the universal value of 3 that was postulated in [@hart98] on the basis of calculations on coarser lattices. The value one fits to the spectrum obtained on the $L=20$ lattice at $\beta=2.5$ is $\gamma = 2.98$ (7) and is equally consistent. We also examine the dependence on $\beta$ of the coefficient $c_l(\beta)$ in eqn. \[eqn\_len\_spec\] adding to the older work our calculations at $\beta=2.5$ on the $L=20$ lattice. \[We do not use the $L=32$ lattice for this purpose because of the different gauge fixing procedure used.\] If we assume a constant power (which is approximately the case), then $c_l(\beta)$ is just proportional to the total length of the secondary clusters. At fixed $\beta$ we find this length to be proportional to $L^4$ just as one might expect. \[Small clusters in very different parts of a large volume are presumably independent.\] The dependence on $\beta$, on the other hand, is much less clear. Between $\beta=2.3$ and $\beta=2.4$ it varies weakly, roughly as $K^{0.12 \pm 0.13}$. Between $\beta=2.4$ and $\beta=2.5$ it varies more strongly, roughly as $K^{0.48 \pm 0.09}$. We can try to summarise this by saying that c\_l() = L\^4 \^ \[eqn\_coeff\_spec\] where $\zeta = 0.5 \pm 0.5$, which is consistent with what was found previously [@hart98]. The smaller clusters are compact objects in $d=4$, and having determined the cluster spectrum as a function of length we can then ask what is the spectrum when re-expressed as a function of the radius (of gyration) of the cluster. In [@hart98] we obtained this spectrum by determining the average radius as a function of length, and folding that in with the number density as a function of length. This is an approximate procedure (forced upon us by the fact that we did not foresee the interest of this spectrum during the processing of the clusters) and one can obtain the spectrum more accurately by calculating $r$ for each cluster and forming the spectrum directly. Doing so for the $L = 32$ lattice at $\beta =
2.5115$, also in Figure \[fig\_len\_spec\], we find a power law as in eqn. \[eqn\_rad\_spec\] with $\eta = 4.20$ (8). The spectrum on the $L=20$ lattice at $\beta=2.5$ yields $\eta = 4.27$ (6). We recall that in [@hart98] we claimed that the spectrum was consistent with the scale invariant result $dr/r \times 1/r^4$, [*i.e.*]{} $\eta = 5$. This followed from the fact that we found the the radius of the smaller clusters to vary with their length as $r(l) = s + t.l^{0.5}$ [*i.e.*]{} just what one would expect from a random walk. Folded with a length spectrum $N(l)
\sim 1/l^3$, this gives $\eta = 5$. On the $L=32$ lattice we still find that the random walk [*ansatz*]{} provides an acceptable fit but we also find that $r(l) = s + t.l^{0.65}$ works equally well over similar ranges. The latter, when folded with $\gamma = 3$, gives $\eta
= 4.2$. It is clear that the direct calculation of $N(r)$ is much more accurate than the indirect approach.
Treating the power as a free parameter in the fit, $r(l) = s + t.l^u$, we find $u = 0.57$ (3) on $L=32$ at $\beta = 2.5115$, consistent with $u = 0.58$ (4) on $L=20$ at $\beta=2.5$. Thus both $u=0.5$ and $u=0.65$ lie within about two standard deviations from the fitted value. Note that what the fitted powers $\gamma$ and $u$ parameterise are the means of the distributions of lengths and radii respectively. That combining these does not give the directly calculated value of $\eta$ is not unexpected, and reflects the importance of fluctuations around the mean in the distributions.
If the secondary monopole clusters can be associated with localised excitations of the full SU(2) vacuum (‘4–balls’), it would seem that such objects do not have an exactly scale invariant distribution in space–time, so that the number of larger radius objects is somewhat greater than would be expected were this the case. Now it is known that an isolated instanton (even with quantum fluctuations) is associated with a monopole cluster within its core (see [@hart96; @inst99] and references therein) and that the scale invariant semiclassical density of instantons acquires corrections due to quantum fluctuations. These corrections are, however, very large; in SU(2) the spectrum of small instantons (where perturbation theory is reliable) goes as $N(\rho) d\rho \propto
d\rho/\rho \times \rho^{10/3}$, where $\rho$ is the core size. The scale breaking we have observed for monopole clusters is negligible in comparison. Thus we cannot identify the ‘4–balls’ with instantons. Indeed, the fact that the monopole spectrum is so close to being scale invariant strongly suggests that these secondary clusters have no physical significance. In the next section we shall show explicitly that, in the large volume limit, they do not play any part in the long range confining physics.
Breakdown of ‘monopole dominance’?
----------------------------------
We finish this section by asking if there are hints from our cluster analysis that ‘monopole dominance’ might be breaking down as we approach the continuum limit. This question is motivated by the observation that the monopoles are identified by a gauge fixing procedure which involves making the bare SU(2) fields as diagonal as possible. Since the theory is renormalisable, the long distance physics increasingly decouples from the fluctuations of the ultraviolet bare fields as we approach the continuum limit. For example, the ultraviolet contribution to the action density is $O(1/\beta)$ while the long distance contribution is $O(e^{-c\beta})$. Thus as $a\to 0$ the maximally Abelian gauge will be overwhelmingly driven by ultraviolet rather than by physical fluctuations. Moreover at the location of the monopoles the Abelian fields are far from unity and so one would expect the SU(2) fields also to be far from unity. Thus the number of monopoles would seem to be constrained by the probability of finding corresponding clumps of SU(2) fields with large plaquette values. This probability depends on the detailed form of the SU(2) lattice action far from the Gaussian minimum and one could easily choose an action where it is completely suppressed and yet which one would expect to be in the usual universality class. None of the above arguments are completely compelling of course. In the Gaussian approximation, for example, the $O(1/\beta)$ ultraviolet fluctuations would not generate any monopoles at all, and in that case there would be no reason to expect any breakdown of monopole dominance. Nonetheless the arguments do suggest that it would be surprising if the long distance physics were to be usefully and simply encoded in the monopole structure (as defined on the smallest ultraviolet scales) all the way to the continuum limit.
There are different ways in which monopole dominance could be lost. The most extreme possibility is that as $a \to 0$ the fields simply cease to contain monopole clusters that are large enough to disorder large Wilson loops. That this is indeed so has been argued in [@grady98] where it has been claimed that the exponent $\gamma$ in our eqn. \[eqn\_len\_spec\] (but defined for loops rather than for clusters) increases rapidly with decreasing $a$. Of course this would not in itself preclude the existence of a large percolating cluster, as long as this cluster could be decomposed into a large number of small and correlated intersecting loops. Irrespective of this, we also note that the volumes used in [@grady98] are very small by the criterion given in eqn. \[eqn\_lK\_scale\]. For example, from our scaling relations we would expect to need an $L
\simeq 46$ lattice at $\beta=2.6$ and an $L \simeq 70$ lattice at $\beta=2.7$ in order to resolve our simple monopole picture, if it still holds at these values of $\beta$. This contrasts with the $L=12$ and $L=20$ lattices actually employed in [@grady98]. So it appears to us that while the claims in [@grady98] are certainly interesting, further calculations on much larger lattices are required.
Our work suggests a somewhat different form of the breakdown to the one above. We see from eqn. \[eqn\_coeff\_spec\] that the ratio of the (total) monopole current residing in the physically irrelevant, smaller clusters to that residing in the large percolating cluster, increases rapidly as $a\to 0$ as $1/\sqrt{K}^{3-\zeta} \propto 1/a^{3-\zeta}$. This suggests that as $a\to 0$ a calculation of Wilson loops will become increasingly dominated by the fluctuating contribution of the unphysical monopoles that are ever denser on physical length scales, and that this will eventually prevent us from extracting a potential or string tension. That is to say: calculations in the maximally Abelian gauge will eventually acquire a similar problem to that which typically afflicts Abelian projections using other gauges. In our case we can overcome this problem by going to a large enough volume that the physically relevant percolating cluster can be simply identified. \[The reason this cannot be done with other typical Abelian gauge fixings is that there the unphysical monopoles are dense on lattice scales making any meaningful separation into clusters impossible.\] We can then extract the string tension using, in our Wilson loop calculation, only this largest monopole cluster. The fact that the length of this cluster scales in physical units, with apparently no significant anomalous dimension, tells us that this calculation will not be drowned in ultraviolet ‘noise’ as we approach the continuum limit. Of course, the fact that we can only do this for volumes that diverge in physical units as $a\to 0$ is a symptom of the underlying breakdown of the Abelian projection.
The qualitative discussion in the previous paragraph over-estimates the effect of the secondary clusters; for example, the contribution that a cluster of fixed size in lattice units makes to a Wilson loop of a fixed physical size will clearly go to zero as $a\to 0$. So it is useful to ask how Wilson loops are affected by the secondary clusters, and to do so using approximations that underestimate the effect of these smaller clusters. Consider an $R\times R$ Wilson loop. A monopole cluster that has an extent $r$ that is smaller then $R$ will affect it only weakly through higher multipole fields which cannot on their own give rise to an area law decay and a string tension. So we neglect such clusters and consider only those larger than $R$. Let us first neglect the observed breaking of scale invariance and simply assume that $r \propto \sqrt{l}$ and that $\gamma = 3$. We then find, by integrating eqn. \[eqn\_len\_spec\] and using eqn. \[eqn\_coeff\_spec\], that the number of secondary clusters with $r>R$ is proportional to $L^4 \sqrt{K}^{\zeta}/R^4$. We further assume that such clusters must be within a distance $\xi$ from the minimal surface of the Wilson loop, where $\xi$ is the screening length, if they are to disorder that loop significantly. The lattice volume this encompasses is the area of the planar loop, $R^2$, multiplied by a factor of $\xi$ for each of the two orthogonal directions in $d=4$. So the probability for this Wilson loop to be disordered thus decreases with $R$ as $(R^2\xi^2/L^4 \times L^4
\sqrt{K}^{\zeta}/R^4)
\sim \sqrt{K}^{\zeta} (\xi/R)^2$. So if we look at a Wilson loop that is of a fixed size in physical units, [*i.e.*]{} $R / \xi$ fixed assuming $\xi$ scales as a physical quantity [@hart98], then the influence of the secondary clusters will decrease to zero as $a\to 0$ as long as $\zeta > 0$. If $\zeta < 0$, however, then we would have to go to Wilson loops that were ever larger in physical units as we approached the continuum limit, in order that the physical contribution from the percolating cluster should not be swamped by the unphysical contribution of the secondary clusters. Of course this calculation uses a scale invariant $dr/r \times 1/r^4$ spectrum, whereas, as we have seen, there is significant scale breaking and the actual spectrum is closer to $dr/r
\times 1/r^{3.2}$. If we redo the above analysis with the latter spectrum we see that we are only guaranteed to preserve this aspect of ‘monopole dominance’ if $\zeta >0.8$. As demonstrated in eqn. \[eqn\_coeff\_spec\], there is some evidence that $\zeta>0$ but it is not at all clear that $\zeta >0.8$. All this indicates that even in a calculation that errs on the side of neglecting the effect of the smaller clusters, they nonetheless will most likely dominate the values of Wilson loops on fixed physical length scales. It is only by separating the percolating cluster from the other smaller clusters, and calculating Wilson loops just using that largest cluster, that we can hope to be able to extract the string tension as $a\to 0$.
Monopoles, vortices and the string tension {#sec_mon_vor_K}
==========================================
In this section we begin by describing how we calculate the string tension from an arbitrary set of monopole currents. We then go on to show that even at the smallest lattice spacings, the string tension arises essentially entirely from the largest cluster, as long as we use a sufficiently large volume. We then calculate the string tension for sources that have a charge of $q=2$, 3 and 4 times the basic charge, and compare these results to a simple toy model calculation. Finally we discuss the implications of our calculations for the question whether it is really monopoles or vortices that drive the confining physics.
Monopole Wilson loops {#ssec_mon_wl}
---------------------
The monopole contribution to the string tension may be estimated using Wilson loops. If the magnetic flux due to the monopole currents through a surface spanning the Wilson loop, $\cal C$, (by default the minimal one) is $\Phi({\cal S})$, then the charge $q$ Wilson loop has value W([C]{}) = . We may obtain the static potential from the rectangular Wilson loops V(r) = \_[t ]{} V\_(r,t) \_[t ]{} . The string tension, $K$, may then be obtained from the long range behaviour of this potential, $V(r) \simeq Kr$. The string tension may also be found from the Creutz ratios K = \_[r ]{} K\_(r) \_[r ]{} . Square Creutz ratios at a given $r$ are useful because they provide a relatively precise probe for the existence of confining physics on that length scale. In addition Creutz ratios are useful where the quality of the ‘data’ precludes the double limit of the potential fit. This is so particularly when positivity is badly broken as it frequently is for our gauge dependent correlators.
The magnetic flux due to the monopole currents is found by solving a set of Maxwell equations with a dual vector potential reflecting the exclusively magnetic source terms. An iterative algorithm being prohibitively slow on $L=32$, we utilised a fast Fourier transform method to evaluate an approximate solution as the convolution of the periodic lattice Coulomb propagator and the magnetic current sources [@stack92]. The error in this solution was then reduced to an acceptable level by using it as the starting point for the over–relaxed, iterative method.
We may use any subset of the monopole currents as the source term to calculate the contribution to the Wilson loops and potential of those currents, provided that they i) are locally conserved and ii) have net zero winding number around the periodic lattice in all directions, [*e.g.*]{} Q\_[=4]{} \_[x,y,z]{} j\_4(x,y,z,t=1) = 0. If we choose complete clusters, then the first condition is always satisfied but the second condition is often not met (even though the winding number for all the clusters together must be zero). In such cases we introduce a ‘fix’ as follows. At random sites in the lattice we introduce a Polyakov–like straight line of magnetic current of corrective charge $-Q_\mu$ for each direction, and use these as sources for a dual vector potential. Such lines represent static monopoles and a random gas of these can lead to a string tension. This introduces a systematic error to the monopole string tension that we need to estimate. We do so by placing the same corrective loop on an otherwise empty lattice, along with a second loop of charge $+Q_\mu$ at another random site. From this new ensemble we calculate the string tension from Creutz ratios. One half of this is a crude estimate of the bias introduced in correcting the original configurations, and this is quoted as a second error on our string tension values, as appropriate.
The largest cluster {#ssec_largest}
-------------------
In [@hart98] we observed that at $\beta=2.3$, $L \ge 10$ and at $\beta=2.4$, $L \ge
14$ the $q=1$ monopole string tension was produced almost entirely by the largest cluster, and the other clusters had a string tension near zero. At $\beta=2.5$, $L=16$ the situation was more confused; the smaller, power law clusters still had a very low string tension, but that of the largest cluster alone was substantially less than the full monopole string tension. This suggested some kind of constructive correlation between the two sets of clusters. In our new calculation on an $L=20$ lattice at $\beta=2.5$, we still find a situation that is confused, although somewhat less so than on the $L=16$ lattice while on the $L=32$ lattice at $\beta = 2.5115$ the clear picture seen at $\beta=2.3$ re-emerges, with nearly all the string tension being produced by the largest cluster, and the remaining clusters having a negligible contribution. To illustrate this we display in Figure \[fig\_cr\_eff\] the effective string tensions as a function of $r$ for the lattices at $\beta=2.5$ and $\beta=2.5115$.
The confused [*rôles*]{} of the clusters on finer lattices [@hart98] is thus a finite volume effect and does not represent a breakdown of the monopole picture as we near the continuum limit. Due to the differing scaling relations for the lengths of the two largest clusters, it is not enough to maintain a constant lattice volume in physical units to reproduce the physics as we reduce the lattice spacing. Rather the lattice must actually become larger even in physical units, as discussed in subsection \[ssec\_thiscalc\].
The string tension arises from ‘disordering’ — [*i.e.*]{} switches in sign — of the Wilson loop by the monopoles. A monopole that is sufficiently close to a large Wilson loop will multiply the loop by $\exp [iq\pi]$ which would naïvely suggest that even–charged loops are not disordered and have no string tension. In a screened monopole plasma, however, as the monopole is moved away from the loop, the flux falls and the possibility for disorder and a string tension exists. \[This will also occur without screening, but only when the monopole is a distance away from the Wilson loop that is comparable to the size of the loop.\] Clearly the exact value of the string tension will depend upon the details of the screening mechanism, especially as we increase $q$. This can be calculated in the usual saddle point approximation [@polyakov77] where one finds that the string tension is proportional to $q$ [@ambjorn98]. One can obtain a crude model estimate with much less effort, and this we do in the next subsection. Returning to our lattice calculations, we list in Table \[tab\_sig\_mon\] the monopole string tensions that we obtain using charge $q$ Wilson loops at $\beta=2.5115$ on the $L=32$ lattice. We see that they are indeed consistent with a scaling relation $R(q) = q$, at least up to $q=4$.
A simple model {#ssec_simp_mod}
--------------
It is useful to consider here a simple model for the disordering of Abelian Wilson loops of various charges by monopoles. We consider only static monopoles in $d=4$, with a mean field type of screening, assuming that the macroscopic, exponential fall–off in the flux with screening length $\xi$ could be applied on the microscopic scale also. For numerical reasons we also impose a cut–off: beyond $N$ screening lengths the flux is set exactly zero. The magnetic field is thus B(d) = {
[ll]{} e\^[-d/]{} & d N\
0 & d > N .
. The flux from a monopole distance $z \le N\xi$ above a large (spacelike) Wilson loop through that loop is (N,z,) = \^1\_[z/N]{} dy ( - ). Considering a slab of monopoles and antimonopoles all distance $z$ above the loop (and similar below), the charge $q$ Wilson loop gives a string tension [@hart98] K(N,z,q) ( 1 - ). Integrating over all $|z| \le N\xi$, the ratio of string tensions calculated using charge $q$ and charge $q=1$ Wilson loops is R(N,q) = for this static monopole assumption. This may be evaluated numerically, and extrapolated as $N \to \infty$, where there is a well–defined limit, $R(q)$. The results for small $q$ are shown in Table \[tab\_sig\_rat\], where the error on $R(q)$ reflects the extrapolation uncertainty. Comparing these numbers to the actual ratio of string tensions, we find this simplistic model works remarkably well for $q=2$, but becomes less reliable as we increase $q$. This no doubt reflects the increasing importance of the neglected fluctuations of the flux away from the mean screened values.
Monopoles or vortices? {#sec_mon_or_vor}
----------------------
The fact that the Abelian fields that one extracts in the maximally Abelian gauge, and their corresponding monopoles, successfully reproduce the SU(2) fundamental string tension, provides some evidence for the dual superconducter model of confinement. As we remarked in the introduction, however, an attractive alternative picture exists, based on vortex condensation, and one has comparable evidence for that picture, obtained by going to maximal centre gauge and calculating Wilson loops using the singular vortices obtained after centre projection.
Since the Abelian projected fields seem to contain the full string tension, it is reasonable to assume that they encode all the significant confining fluctuations in the SU(2) fields, even if these are vortices. How would one expect a vortex to be encoded in the Abelian fields? And how can we test for their presence?
Recall that the kind of vortex we are interested in has a smooth core and flips the sign of any Wilson loop that it threads. Consider now a space-like Wilson loop in some time-slice of our Abelian projected lattice field. We observe that it will flip its sign if threaded by a loop of magnetic flux whose core contains a total flux equal to $\pi$. If the core size is not arbitrarily large, so that a (large enough) Wilson loop has negligible probability to overlap with the actual core, then a condensate of such fluxes will lead to linear confinement. Since the original SU(2) vortex has a smooth core, the simplest expectation is that this flux, if it reflects the vortex, should not have a singular monopole source; rather it should be a closed loop of magnetic flux. If its length is much larger than the size of the Wilson loop, it can easily thread the loop an odd number of times and can disorder it. So the natural way for a Z(2) vortex to be encoded in the Abelian projected fields is as a closed loop of magnetic flux, in roughly the same position, and with a smooth core of roughly the same size. If this is so, and if vortices are present in the SU(2) fields, we would expect that our Abelian fields contain two kind of confining fluctuations; singular magnetic monopoles and smooth closed loops of ($\pi$ units of) magnetic flux. Since these closed loops of flux are smooth they will be hard to identify individually in the midst of the magnetic fluxes generated by the monopoles. Their presence can however be easily tested for as follows. The flux in the U(1) fields is conserved and so any flux either originates on the monopoles or closes on itself as part of a closed flux loop. The monopoles are easy to identify and their flux can be calculated. So for any Wilson loop, $\cal{C}$, we can calculate the flux, $B_{\mon}(\cal{C})$, due to the monopoles and we can subtract it from the total flux, $B[\cal{C}]$, so as to obtain the remaining flux, B\_() B() - B\_(), \[eqn\_flux\_diff\] that comes from closed flux loops. The corresponding value of the Wilson loop will be $e^{-B_{\delta}(\cal{C})}$. In this way we can calculate the potential due to the non-monopole flux, and if we find a non-zero string tension this demonstrates the existence of a condensate of such flux loops and provides evidence for corresponding Z(2) vortices. If the flux loops carry $\pi$ units of flux, Wilson loops corresponding to sources with an even charge will have zero string tension.
We remark here that in U(1) lattice gauge theories, such loops of magnetic flux are not usually discussed as significant degrees of freedom. That is not because they cannot exist but rather that the dynamics is such that they usually play no significant [*rôle*]{}. \[One can always smoothly reduce the usual U(1) action by increasing the core size of such a loop. Ultimately they contribute a non-confining ‘spin wave’ contribution to the interaction.\] The Abelian projected fields, on the other hand, are not generated from some local U(1) action. They may possess any structures that are kinematically allowed.
Vortices can also be encoded in the Abelian fields in a more subtle way than the above. This involves long-distance correlations amongst the monopoles. In $d=2+1$ suppose that at least some of the monopoles lie along ‘lines’ in such a way that each monopole is followed by an antimonopole (and vice versa) as we follow the line. This will generate an alternating flux of $\pm\pi$ along the line [@deldebbio98]. So a Wilson loop threaded by this line will acquire a factor of $-1$. Such correlated ensembles can therefore encode the vortices in the original three dimensional SU(2) fields. A similar restriction of monopole current world lines to two dimensional sheets can be envisaged in $d=3+1$. In both cases, their presence would be signalled by the fact that they do not disorder Wilson loops corresponding to an even charge (unlike a plasma of monopoles). So if we calculate the string tension due to the monopoles, and if we find a significant suppression of the $q=2$ string tension, then this will indicate the significant presence of such correlations and hence of vortices.
This latter way of encoding vortices in the Abelian projected fields might seem less natural given the smoothness of the underlying Z(2) vortices. As pointed out in [@ambjorn98], however, such correlated monopole structures actually occur in what one usually regards as a standard example of a field theory that demonstrates linear confinement driven by monopole condensation: the Georgi-Glashow model in three dimensions. This model couples an SU(2) gauge field to a scalar Higgs field in the adjoint representation of the gauge group. The theory has a Higgs phase, and the Higgs field drives the gauge field into a vacuum state which has only U(1) gauge symmetry, save in the cores of extended topological objects. These ’t Hooft–Polyakov monopoles are magnetically charged with respect to the U(1) fields, and give rise to the linear confining potential, at least in the semi–classical approximation [@polyakov77] which holds good when the charged vector bosons are heavy. As pointed out in [@ambjorn98], however, this conventional picture cannot be true on large enough length scales since eventually the presence of the charged massive $W^{\pm}$ fields will lead to the breaking of strings between doubly charged sources (the $W^{\pm}$ possessing twice the fundamental unit of charge). A plasma of monopoles, on the other hand, will predict the linear confinement of such double charges. So it was argued that in this limit it is Z(2) vortices, which do not disorder doubly charged Wilson loops, that drive the confinement [@ambjorn98]. The crossover between the two pictures, it is argued, would occur beyond a certain length scale dictated by the $W^{\pm}$ mass, where the distribution of monopole flux would no longer be purely Coulombic, but would be collimated into structures of lower dimension — essentially strings of alternating monopoles and antimonopoles — that reflect the Z(2) vortices of the vacuum.
Of course one cannot carry this argument over in all its details to the case of the pure SU(2) gauge theory. Here there are no explicit Higgs or $W^{\pm}$ fields; any analogous objects would need to be composite. The theory also has only one scale, and so one would not expect an extended intermediate region between the onset of confining behaviour and the collimation of the flux signalling Z(2) disorder. But it does raise the possibility that the Z(2) vortices in the SU(2) fields might be encoded, after Abelian projection, in such correlations amongst the monopoles rather than in separate smooth closed loops of magnetic flux.
To probe for the presence of smooth loops of magnetic flux in the Abelian projected fields, we have calculated the ‘difference’ flux, as defined in eqn. \[eqn\_flux\_diff\], and the resulting string tension; and to probe for vortex-like ensembles of monopoles we have calculated the monopole string tension, $K(q)$, for various source charges, $q$.
We start with the latter. In Table \[tab\_sig\_mon\_eff\] we show the $q=1,2$ monopole effective string tensions that we have obtained from Creutz ratios on the $L=32$ lattice at $\beta = 2.5115$. We see that for $q=2$, just as for $q=1$, there are very few transients at small $r$, and the extraction of an asymptotic string tension appears to be unambiguous. We have accurate calculations out to a distance of $r=9a$ which corresponds to r=9a \~ \[eqn\_dist\_K\] in physical units at this $\beta$. Out to this distance there is absolutely no hint of any reduction in the $q=2$ effective string tension. It has been pointed out [@deldebbio98] that when the Wilson loop is not much larger than the typical vortex core, it is not completely unnatural to obtain an effective string tension comparable to the one from a monopole plasma. Here the size is beginning to be large compared to the natural scale of the theory, however, and it is hard not to view the lack of any variation at all in the $q=2$ effective string tension as pointing to the absence of the kind of correlations amongst the monopoles that might be encoding Z(2) vortices.
The second possibility is that the vortices might be encoded not in correlations amongst the monopoles but rather in closed loops carrying $\pi$ units of magnetic flux. Such loops would contribute to $K(q=1)$ but not to $K(q=2)$. We recall that there has been a calculation of $K(q)$, calculated within the full Abelian fields at $\beta = 2.5115$, and that there it was found [@bali96] that there is a finite $q=2$ effective string tension that extends out to at least as far as $r=9a$, and that the ratio of the U(1) string tensions is $K(q=2)/K(q=1) = 2.23$ (5). While this suggests that closed flux loops are not important, these string tensions necessarily include the contribution from monopoles, and it would be useful to have a calculation that excludes the latter. We have therefore calculated the effective string tension using only the flux that comes from closed flux loops, as defined in eqn. \[eqn\_flux\_diff\]. The results of this calculation are listed in Table \[tab\_sig\_mon\_eff\] for $q=1$. We see that, within small errors, there is asymptotically no string tension from such loops (a potential fit to the Wilson loops yields $K < 0.0025$). This shows in a direct way that there is no significant condensate of closed loops of flux in the Abelian projected fields.
In conclusion, our investigations here have shown no sign of vortices encoded in the Abelian projected fields in either of the two ways that one might plausibly have expected them to be.
It is worth stepping back at this point and reflecting upon the tentative nature of the above arguments. Our calculation of the monopole string tension takes each monopole to be a source of a simple Coulombic flux, as obtained by solving Maxwell’s equations. Treating the monopoles as being ‘isolated’ in this way, is the obvious starting point if one wishes to ask what is the physics ‘due to monopoles’. But it is no guarantee that such a question makes any sense. Indeed it is only in the Villain model that one has the exact factorisation of Wilson loops into monopole and non-monopole pieces that is needed for this question to be clearly unambiguous. For example, it is not [*a priori*]{} clear that the ensemble of monopoles one obtains in the maximally Abelian gauge is even qualitatively such as one would expect from a generic U(1) action. If it is not, then one must ask what are the fluctuations in the SU(2) fields that determine the nature of the monopole ensemble, and whether these features of the ensemble have a significant effect on the calculated string tension. If they do then the question we are asking, whether the string tension is ‘due to monopoles’, becomes intrinsically ambiguous. Our demonstration that there is no suppression of the $q=2$ monopole string tension may be regarded as a first step, but only a first step, towards showing that the monopole ensemble does not possess such features that require additional explanation. One should also mention that the Abelian fields are periodic in $2\pi$ (in the sense that the number density of plaquette angles peaks at multiples of $2\pi$), which is the requirement for Dirac strings to be invisible, and that they possess a screening length that is characteristic of plasmas [@hart98]. Equally, if we had found a significant flux loop condensate in the Abelian fields, we would have had to study carefully the (presumably) non-trivial correlations between the monopoles and flux loops in order to determine if there was any sense in claiming that some physics was ‘due to monopoles’. The fact that we have not found any sign of such a flux loop condensate, or of any anomalous features of the monopole plasma, means that we are not yet forced to confront this quite general problem. But this question clearly needs systematic exploration.
Summary {#sec_summ}
=======
We have studied the magnetic monopole currents obtained after fixing to the maximally Abelian gauge of SU(2), on lattices that are both large in physical units and have a relatively small lattice spacing. The monopole clusters are found to divide into two clear classes both on the basis of their lengths and their physical properties. The smaller clusters have a distribution of lengths which follows a power law, and the exponent is consistent with 3, as was previously seen on coarser lattices [@hart98]. These clusters are compact objects, and their radii also follow a power law whose exponent we found to be 4.2 (1). This is close to, but a little less than, the scale invariant value of 5 which indicates that if the smaller clusters correspond to objects in the SU(2) vacuum, these objects have a size distribution which yields slightly more large radius objects than would be expected in a purely scale invariant theory. This scale breaking is, however, far too weak to encourage the identification of such objects with the small instantons in the theory.
That is not to say that instantons are necessarily irrelevant; the correlations between the monopole currents and the action and topological charge densities ( [@feurstein97; @bakker98; @hart96; @inst99] and references therein) indicate some connexion. It would be interesting to measure the correlations separately using the largest cluster, and the remaining, power law clusters.
The small clusters do not appear relevant to the long range physics; they produce a zero, or at most very small, contribution to the string tension. Indeed the string tension is consistent with being produced by the largest cluster alone. The fact that there should be a large percolating monopole cluster associated with the long-distance physics is an old idea (see [@polikarpov93] for an early reference). The properties that we find for this cluster, however, are certainly not those associated with naïve percolation. In particular, as we approach the continuum limit the density of monopoles belonging to this cluster goes to zero. And indeed the fraction of the total monopole current that arises from this largest cluster also appears to go to zero. This is because this single very large cluster seems to percolate on physical and not on lattice length scales, while the physically unimportant secondary clusters have an approximatley constant density in lattice units. All this reproduces the properties that we previously obtained on coarser lattices, but which seemed to be lost when going to finer lattice spacings, albeit on volumes of a smaller physical extent. This study demonstrates that the breakdown was a finite volume effect, rather than a failure of the monopole picture in the weak coupling limit. The volume at which the picture was restored was as predicted by the scaling relations derived from the coarser lattices.
The fact that one has to go to volumes that are ever larger as $a \to 0$, can be interpreted as a breakdown of the Abelian projection. As we remarked, something like this is not unexpected: as $a\to 0$ the Abelian projection will presumably be increasingly driven by the irrelevant ultraviolet fluctuations of the SU(2) link matrices. This leads to an increasing fraction of the monopole current – that belonging to the smaller clusters – containing no physics and this contributes an increasing background ‘noise’ to attempts at extracting physical observables as we approach the continuum limit. Fortunately the unphysical gas of monopoles that one obtains by Abelian projection within the maximally Abelian gauge is sufficiently dilute that one can isolate the physically relevant ‘percolating’ cluster, even if the price is that one has to work with ever larger volumes.
We also calculated the monopole contribution to Wilson loops of higher charges, and found that the corresponding monopole string tensions appear to be simply proportional to the charge, at least up to $q=4$. This is what is predicted by a saddle point treatment of the U(1) lattice gauge theory [@ambjorn98] as can be seen more simply, if more approximately, within our simplistic charge plasma model.
Our main reason for studying these higher-$q$ string tensions, was to probe for any sign of a condensate of Z(2) vortices in the Abelian projected fields. It might, of course, be that such vortices are simply not encoded in the Abelian fields. It is plausible, however, to infer from the observed monopole and centre dominance that both when we force the SU(2) link matrices to be as Abelian as possible, and when we force them to be as close to $\pm 1$ as possible, the resulting Abelian and Z(2) fields capture essentially all the long range confining disorder present in the original SU(2) fields. In the case of Z(2) fields the disorder must be encoded by vortices (there is nothing else). In the Abelian case however the disorder can be carried either by monopoles or by closed loops of ‘magnetic’ flux. We argued that such a closed loop, carrying a net magnetic flux of $\pi$ units, provides a plausible way for the Abelian fields to encode the presence of an underlying Z(2) vortex. Our study of the monopole–U(1) ‘difference gas’ showed, however, that there is no significant contribution to confinement from such loops of magnetic flux. An alternative [@deldebbio98; @ambjorn98]. is that the Z(2) disorder is encoded in correlated strings of (anti)monopoles. If such correlations were important, however, they would lead to a significant suppression of the $q=2$ string tension, and this we do not observe. Instead we find that the effective monopole string tensions satisfies $K(q=2) = 2 K(q=1)$ very accurately to distances that are quite substantial in physical units. While there is a limit to what one can conclude about Z(2) vortices in a study that focuses solely on the Abelian projected fields, the fact that they do not manifest themselves in any of the ways that one might expect, must cast some doubt on their importance in the SU(2) vacuum.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The gauge fixed $L=32$ field configurations were crucial to the work of this paper and we are very grateful to Gunnar Bali for making them available to us. Our computations were performed on a UKQCD workstation and this work was supported in part by United Kingdom PPARC grant GR/L22744. The work of M.T. was supported in part by PPARC grant GR/K55752.
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[llr\*[4]{}[r@[ $-$ ]{}l]{}]{} & & & & & $l_{\max}$ & $l_{\second}$ &\
12 & 2.3 & 500 & 2358 & 3970 & 18 & 220 & 2172 & 3930\
14 & 2.4 & 500 & 894 & 3436 & 22 & 1112 & 28 & 3400\
16 & 2.5 & 500 & 268 & 2462 & 22 & 910 & 4 & 2414\
20 & 2.5 & 100 & 1718 & 5050 & 50 & 1644 & 318 & 4964 & 36 & 684\
32 & 2.5115 & 100 & 11872 & 20040 & 114 & 4676 & 9066 & 19886 & 92 & 2476\
$q$
-------------------------- ----- ------------- ----------------- ------------
$\beta = 2.5$, $L=20$ 1 0.035 (3) 0.026 (3) (1) $< 0.0015$
$\beta = 2.5115$, $L=32$ 1 0.0270 (10) 0.0240 (10) (3) $< 0.0010$
2 0.0520 (10) 0.0450 (10) (4) $< 0.0022$
3 0.075 (2)
4 0.103 (5)
: Monopole string tensions from Wilson loops of varying charge, using all current (‘all’), current from the largest cluster alone (‘lge’) and the remaining current (‘abl’).[]{data-label="tab_sig_mon"}
$q$
----- ----------- -----------
2 1.827 (1) 2.00 (9)
3 2.192 (1) 2.88 (14)
4 2.526 (1) 3.96 (25)
: Ratio of monopole string tensions from Wilson loops of varying charge, in the static plasma model and measured at $\beta = 2.5115$ on $L=32$.[]{data-label="tab_sig_rat"}
$r$
----- -------------- ------------- --------------
2 0.02572 (8) 0.0511 (2) 0.0799 (2)
3 0.02371 (9) 0.0497 (2) 0.0335 (2)
4 0.02355 (10) 0.0483 (3) 0.0176 (5)
5 0.02414 (14) 0.0492 (5) 0.0102 (9)
6 0.02487 (17) 0.0496 (7) 0.0082 (13)
7 0.02565 (20) 0.0501 (7) 0.0062 (21)
8 0.02628 (37) 0.0493 (25) 0.0076 (53)
9 0.02652 (25) 0.0556 (58) -0.0076 (27)
: Effective monopole string tensions from Creutz ratios for charges $q=1,2$ and from the difference of U(1) and monopole fluxes at $\beta = 2.5115$ on $L=32$.[]{data-label="tab_sig_mon_eff"}
| 0 |
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abstract: 'We propose a variant formulation of Hamiltonian systems by the use of variables including redundant degrees of freedom. We show that Hamiltonian systems can be described by extended dynamics whose master equation is the Nambu equation or its generalization. Partition functions associated with the extended dynamics in many degrees of freedom systems are given. Our formulation can also be applied to Hamiltonian systems with first class constraints.'
address: |
${}^\ast$\
${}^\dag$
author:
- and
title: |
Hidden Nambu mechanics:\
[A variant formulation of Hamiltonian systems ]{}
---
Introduction {#Introduction}
============
In general, we have a choice of variables describing a physical system. In most cases, we choose a set of variables whose number is same as the total number of degrees of freedom of the system so as to minimize the number of equations of motion. However, in some cases, it is quite useful to formulate the system by the use of variables including redundant ones. A system with gauge symmetry offers a typical example. To describe such a system, keeping the gauge symmetry manifest, we should employ a formulation that includes redundant variables. Although such a formulation is somewhat complicated, thanks to the symmetry, we can clearly understand the important properties of the system such as conservation laws and form of interactions, and can also calculate physical quantities in a systematic way [@W1; @W2].
Therefore, it is interesting to explore the general features of formulations including redundant degrees of freedom. Here we base this on a principle (or brief) that [*physics should be independent of the choice of variables to describe it*]{}, and make an attempt to formulate Hamiltonian systems (systems of Hamiltonian dynamics) in terms of new sets of variables including redundant ones. What kind of dynamics describes the time evolution of the new variables?
Our strategy and conjecture are as follows. Consider a Hamiltonian system described by a canonical doublet $(q, p)$. Take $N(\ge 3)$ variables $(x_1, \cdots, x_{N})$ that are functions of the canonical doublet, and deal with them as fundamental variables to describe the system. If they contain redundant variables, constraints between some variables must be induced. To handle the constraints, Dirac formalism [@D1; @D2] provides a helpful perspective, where constraints with Lagrange multipliers are added to the original Hamiltonian. The induced constraints play a similar role to the Hamiltonian. As for the dynamics of $N$ variables, Nambu mechanics [@N] is quite suggestive. In Nambu mechanics, fundamental variables form an $N$-plet, whose time evolution is generated by $N-1$ Hamiltonians according to the Nambu equations. Combining the advantages of the two theories, we conjecture that [*there is a formulation whose master equation has a form of the Nambu equation or its generalization, where the Hamiltonians consist of the original one and the induced constraints.*]{}
Nambu mechanics is a generalization of the Hamiltonian dynamics proposed by Nambu forty years ago [@N]. In his formulation, the dynamics of an $N$-plet is given by the Nambu equation, which is defined by $N-1$ Hamiltonians and the Nambu bracket, a generalization of the Poisson bracket. The structure of Nambu mechanics is so elegant that many authors have investigated its application. However, the applications have been limited to particular systems such as constrained systems, superintegrable systems, and hydrodynamic systems, because Nambu systems (systems of Nambu mechanics) should have multiple Hamiltonians or conserved quantities. For example, researchers have studied how Nambu mechanics can be embedded into constrained Hamiltonian systems [@BF; @CK; @R; @MS; @KT; @KT2] or how constrained systems can be described in terms of Nambu mechanics [@LJ].
In this article, we show that the structure of Nambu mechanics is, in general, hidden in systems of Hamiltonian dynamics. That is, Hamiltonian systems can be described by Nambu mechanics or its generalization by means of a change of variables from canonical doublets to multiplets. Our formulation can be generalized to many degrees of freedom systems, and the associated partition functions are given. We also apply our formulation to systems with first class constraints. Our approach can be regarded as a complementary one to the previous works [@BF; @CK; @R; @MS; @KT; @KT2; @LJ].
The outline of this article is as follows. In the next section, we give a formulation of Hamiltonian systems using Nambu mechanics and its generalizations. As an application, Hamiltonian systems with first class constraints are also formulated as Nambu systems in Sect. 3. In the last section, we give conclusions and discussions on the direction of future work. In Appendix A, we derive the Nambu equation from the least action principle. In Appendix B, we show that a Nambu system of an $N$-plet can be described by Nambu mechanics with an $N+r$-plet ($r\ge1$).
Nambu systems hidden in Hamiltonian systems {#Nambu systems hidden in Hamiltonian systems}
===========================================
Review {#Review}
------
We begin with a brief review of Hamiltonian systems and Nambu systems [@N]. A Hamiltonian system is a classical system described by a generalized coordinate $q=q(t)$ and its canonical conjugate momentum $p=p(t)$. These variables satisfy the Hamilton’s canonical equations of motion, $$\begin{aligned}
\frac{d q}{dt} = \frac{\partial H}{\partial p} ~,~~ \frac{d p}{dt}
= -\frac{\partial H}{\partial q}~,
\label{H-eq}\end{aligned}$$ where $H=H(q, p)$ is the Hamiltonian of this system. For any functions $A=A(q,p,t)$ and $B=B(q,p,t)$, the Poisson bracket is defined by means of the 2-dimensional Jacobian, $$\begin{aligned}
\{A, B\}_{\mbox{\tiny{PB}}} \equiv
\frac{\partial (A,B)}{\partial (q,p)}=
\frac{\partial A}{\partial q}\frac{\partial B}{\partial p}
- \frac{\partial A}{\partial p}\frac{\partial B}{\partial q}~.
\label{PB}\end{aligned}$$ In terms of the Poisson bracket, the Hamilton’s canonical equation of motion for any function $f=f(p,q)$ can be written as $$\begin{aligned}
\frac{d f}{dt} = \{f, H\}_{\mbox{\tiny{PB}}}~.
\label{H-eqf}\end{aligned}$$
On the other hand, a Nambu system is a classical system described by a multiplet. As the most simple example, let us consider a Nambu system described by a triplet $x=x(t)$, $y=y(t)$, and $z=z(t)$. These variables satisfy the Nambu equations $$\begin{aligned}
\frac{d x}{dt} = \frac{\partial ({H}_1, {H}_2)}{\partial (y, z)}~,~~
\frac{d y}{dt} = \frac{\partial ({H}_1, {H}_2)}{\partial (z, x)}~,~~
\frac{d z}{dt} = \frac{\partial ({H}_1, {H}_2)}{\partial (x, y)}~,
\label{N-eq}\end{aligned}$$ where ${H}_1(x, y, z)$ and ${H}_2(x, y, z)$ are $\lq\lq$Hamiltonians" of this system. For any functions $A={A}(x, y, z, t)$, $B={B}(x, y, z, t)$, and $C={C}(x, y, z, t)$, the Nambu bracket is defined by means of the 3-dimensional Jacobian, $$\begin{aligned}
\{{A}, {B}, {C}\}_{\mbox{\tiny{NB}}}
\equiv \frac{\partial ({A}, {B}, {C})}{\partial (x, y, z)}~.
\label{NB}\end{aligned}$$ In terms of the Nambu bracket, the Nambu equation for any function $f=f(x,y,z)$ can be written as $$\begin{aligned}
\frac{d {f}}{dt} = \{{f}, {H}_1, {H}_2\}_{\mbox{\tiny{NB}}}~.
\label{N-eqf}\end{aligned}$$
It is straightforward to extend the above formalism to a system described by an $N$-plet $x_i$ $(i=1, 2, \cdots, N)$. These variables satisfy the Nambu equations $$\begin{aligned}
\frac{d x_i}{dt} = \sum_{i_1, \cdots, i_{N-1}=1}^{N}
\varepsilon_{i i_1 \cdots i_{N-1}}
\frac{\partial {H}_1}{\partial x_{i_1}} \cdots
\frac{\partial {H}_{N-1}}{\partial x_{i_{N-1}}}~,
\label{N-eq-N}\end{aligned}$$ where ${H}_a={H}_a(x_1, x_2, \cdots, x_N)$ $(a=1, \cdots, N-1)$ are $\lq\lq$Hamiltonians" of this system and $\varepsilon_{i i_1 \cdots i_{N-1}}$ is the $N$-dimensional Levi–Civita symbol, the antisymmetric tensor with $\varepsilon_{12 \cdots N} =1$. For any functions ${A}_{\alpha}={A}_{\alpha}(x_1, x_2, \cdots, x_N, t)$ $(\alpha=1, \cdots, N)$, the Nambu bracket is defined by means of the $N$-dimensional Jacobian, $$\begin{aligned}
\{{A}_1, {A}_2, \cdots, {A}_N\}_{\mbox{\tiny{NB}}}
&\equiv&
\frac{\partial ({A}_1, {A}_2, \cdots, {A}_N)}{\partial (x_1, x_2, \cdots, x_N)}
\nonumber \\
&=& \sum_{i_1, i_2, \cdots, i_N=1}^{N} \varepsilon_{i_1 i_2 \cdots i_N}
\frac{\partial {A}_1}{\partial x_{i_1}}\frac{\partial {A}_2}{\partial x_{i_2}}
\cdots \frac{\partial {A}_{N}}{\partial x_{i_N}}~.
\label{NB-N}\end{aligned}$$ In terms of the Nambu bracket, the Nambu equation for any function ${f}={f}(x_1, x_2, \cdots, x_N)$ can be written as $$\begin{aligned}
\frac{d {f}}{dt} = \{{f}, {H}_1, {H}_2, \cdots, {H}_{N-1}\}_{\mbox{\tiny{NB}}}~.
\label{N-eqf-N}\end{aligned}$$
Hidden Nambu structure {#Hidden Nambu structure}
----------------------
Here let us describe a Hamiltonian system with a canonical doublet $(q, p)$ by means of $N$ variables $x_i=x_i(q, p)$ $(i=1, \cdots, N)$.
### Formulation {#Formulation}
First we study the case with $N=2$, for completeness. We assume that $x=x_1(q, p)$ and $y=x_2(q, p)$ satisfy $\{x, y\}_{\mbox{\tiny{PB}}} \ne 0$. In this case, the equation for a function $\tilde{f}(x, y) = f(q, p)$ is written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \frac{\partial(f, H)}{\partial(q, p)}
= \frac{\partial(\tilde{f}, \tilde{H})}{\partial(x, y)}
\frac{\partial(x, y)}{\partial(q, p)}
= \frac{\partial(\tilde{f}, \tilde{H})}{\partial(x, y)} \{x, y\}_{\mbox{\tiny{PB}}}~,
\label{H-eq(N=2)}\end{aligned}$$ where $\tilde{H}(x, y) = H(q, p)$. If $\{x, y\}_{\mbox{\tiny{PB}}}=1$, the transformation $(q, p) \to (x, y)$ is the canonical transformation, and $(x, y)$ are canonical variables.
Next we study the case with $N=3$. We assume that variables $x=x_1(q, p)$, $y=x_2(q, p)$, and $z=x_3(q, p)$ satisfy at least two of the conditions $\{x, y\}_{\mbox{\tiny{PB}}} \ne 0$, $\{y, z\}_{\mbox{\tiny{PB}}} \ne 0$, and $\{z, x\}_{\mbox{\tiny{PB}}} \ne 0$. In this case, the equation for a function $\tilde{f}(x, y, z) = f(q, p)$ is written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \frac{\partial(f, H)}{\partial(q, p)}
= \frac{\partial(\tilde{f}, \tilde{H})}{\partial(x, y)} \{x, y\}_{\mbox{\tiny{PB}}}
+ \frac{\partial(\tilde{f}, \tilde{H})}{\partial(y, z)} \{y, z\}_{\mbox{\tiny{PB}}}
+ \frac{\partial(\tilde{f}, \tilde{H})}{\partial(z, x)} \{z, x\}_{\mbox{\tiny{PB}}}~,
\label{H-eq(N=3)}\end{aligned}$$ where $\tilde{H}(x, y, z) = H(q, p)$. Note that $q$, $p$, and $H$ are, in general, not uniquely determined as functions of $x$, $y$, and $z$.
Introducing a function $\tilde{G}=\tilde{G}(x, y, z)$ that satisfies the conditions $$\begin{aligned}
\frac{\partial \tilde{G}}{\partial x} = \frac{\partial(y, z)}{\partial(q, p)}~,~~
\frac{\partial \tilde{G}}{\partial y} = \frac{\partial(z, x)}{\partial(q, p)}~,~~
\frac{\partial \tilde{G}}{\partial z} = \frac{\partial(x, y)}{\partial(q, p)}~,
\label{xyzG}\end{aligned}$$ Eq. (\[H-eq(N=3)\]) is rewritten as the Nambu equation in the form of Eq. (\[N-eqf\]), $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \{\tilde{f}, \tilde{H}, \tilde{G}\}_{\mbox{\tiny{NB}}}~,
\label{H-eqf(N=3)}\end{aligned}$$ where we use the formula $$\begin{aligned}
\frac{\partial (\tilde{A}, \tilde{B}, \tilde{C})}{\partial (x, y, z)}
= \frac{\partial (\tilde{A}, \tilde{B})}{\partial (x, y)}
\frac{\partial \tilde{C}}{\partial z}
+ \frac{\partial (\tilde{A}, \tilde{B})}{\partial (y, z)}
\frac{\partial \tilde{C}}{\partial x}
+ \frac{\partial (\tilde{A}, \tilde{B})}{\partial (z, x)}
\frac{\partial \tilde{C}}{\partial y}~.
\label{J(N=3)}\end{aligned}$$ The conditions (\[xyzG\]) are compactly expressed as $$\begin{aligned}
\frac{\partial \tilde{G}}{\partial x_i}
= \frac{1}{2} \sum_{j, k=1}^3 \varepsilon_{ijk}\{x_j, x_k\}_{\mbox{\tiny{PB}}}~~~
\mbox{or}~~~
\sum_{k=1}^3 \varepsilon_{ijk} \frac{\partial \tilde{G}}{\partial x_k}
= \{x_i, x_j\}_{\mbox{\tiny{PB}}}~.
\label{xyzG2}\end{aligned}$$ In Appendix A, the Nambu equations in the form of Eq. (\[N-eq\]) are also derived from a Hamiltonian system with a canonical doublet $(q, p)$ using the least action principle.
By the use of Eq. (\[xyzG2\]), it is shown that the Poisson bracket between $G(q,p)=\tilde{G}(x,y,z)$ and an arbitrary function $u(q,p)=\tilde{u}(x, y, z)$ vanishes such that $$\begin{aligned}
\{G, u\}_{\mbox{\tiny{PB}}}
&=& \frac{1}{2}
\sum_{i, j=1}^{3} \frac{\partial (\tilde{G}, \tilde{u})}
{\partial (x_i, x_j)} \{x_i, x_j\}_{\mbox{\tiny{PB}}}
= \frac{1}{2} \sum_{i, j, k=1}^{3} \varepsilon_{ijk}
\frac{\partial (\tilde{G}, \tilde{u})}{\partial (x_i, x_j)}
\frac{\partial \tilde{G}}{\partial x_k}\nonumber \\
&=& \frac{\partial (\tilde{G}, \tilde{u}, \tilde{G})}{\partial (x, y, z)} = 0~.
\label{Gu=0}\end{aligned}$$ This means that $G$ is a constant. We can eliminate the constant by redefining $G$, and the resulting $\tilde{G}(x,y,z)=0$ can be regarded as a $\it constraint$, which is induced by enlarging the phase space from $(q, p)$ to $(x, y, z)$.
Here we give two comments on the induced constraint $\tilde{G}(x,y,z)=0$. First, in the case in which $\partial \tilde{G}/\partial z \ne 0$, we can solve $\tilde{G}(x,y,z)=0$ for $z$ and obtain $z=z(x,y)$. Because the condition $\partial \tilde{G}/\partial z= \{x, y\}_{\mbox{\tiny{PB}}} \ne 0$ also enables us to express $q$ and $p$ as functions of $x$ and $y$, the expression $z=z(x,y)$ can also be obtained by inserting $q=q(x,y)$ and $p=p(x,y)$ into $z=z(q,p)$. Therefore the implicit form of the constraint $\tilde{G}(x,y,z)=0$ has an equivalent explicit form $z=z(x,y)$, which clearly shows that $z$ is a redundant variable in this case. Second, $\tilde{H}(x,y,z)$ is not uniquely determined as a function of $x$, $y$, and $z$, i.e., we can add a term $\tilde{\lambda}(x,y,z)\tilde{G}(x,y,z)$ to $\tilde{H}(x,y,z)$, where $\tilde{\lambda}(x,y,z)$ is some function. If a Hamiltonian $\tilde{H}(x,y,z)$ satisfies $\tilde{H}(x,y,z)=H(q,p)$ and Eq. (\[H-eqf(N=3)\]), another Hamiltonian $\tilde{H}(x,y,z)+\tilde{\lambda}(x,y,z)\tilde{G}(x,y,z)$ also satisfies them. This is because the additional term $\tilde{\lambda}(x,y,z)\tilde{G}(x,y,z)$ always vanishes on the Nambu bracket.
It is straightforward to extend the above formulation to the case with general $N(\ge 3)$. We assume that at least $N-1$ of $\{x_i, x_j\}_{\mbox{\tiny{PB}}}$ $(i, j=1, \cdots, N)$ do not vanish. In this case, the equation for any function $\tilde{f}(x_1, \cdots, x_N) = f(q, p)$ is written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \frac{\partial(f, H)}{\partial(q, p)}
= \frac{1}{2} \sum_{i, j=1}^{N}
\frac{\partial (\tilde{f}, \tilde{H})}{\partial (x_{i}, x_{j})}
\{x_{i}, x_{j}\}_{\mbox{\tiny{PB}}}~,
\label{H-eq(gN)}\end{aligned}$$ where $\tilde{H}(x_1, \cdots, x_N) = H(q, p)$.
Introducing functions $\tilde{G}_b=\tilde{G}_b(x_1, \cdots, x_N)$ $(b=1, \cdots, N-2)$ that satisfy the conditions $$\begin{aligned}
\frac{1}{(N-2)!} \sum_{i_3 \cdots i_{N}=1}^{N}
\varepsilon_{i_1 i_2 i_3 \cdots i_{N}}
\frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{N-2})}
{\partial (x_{i_3}, \cdots, x_{i_{N}})}
= \{x_{i_1}, x_{i_2}\}_{\mbox{\tiny{PB}}}~,
\label{xiGb}\end{aligned}$$ Eq. (\[H-eq(gN)\]) is rewritten as the Nambu equation in the form of Eq. (\[N-eqf-N\]), $$\begin{aligned}
\frac{d \tilde{f}}{dt} =
\{\tilde{f}, \tilde{H}, \tilde{G}_1, \cdots, \tilde{G}_{N-2}\}_{\mbox{\tiny{NB}}}~,
\label{H-eqf(gN)}\end{aligned}$$ where we use the formula concerning Jacobians, $$\begin{aligned}
\frac{\partial (\tilde{A}_1, \tilde{A}_2, \cdots, \tilde{A}_N)}
{\partial (x_1, x_2, \cdots, x_N)}
= \frac{1}{2(N-2)!} \sum_{i_1, i_2, i_3, \cdots i_{N}=1}^{N}
\varepsilon_{i_1 i_2 i_3 \cdots i_{N}}
\frac{\partial (\tilde{A}_1, \tilde{A}_2)}{\partial (x_{i_1}, x_{i_2})}
\frac{\partial (\tilde{A}_3, \cdots, \tilde{A}_{N})}
{\partial (x_{i_3}, \cdots, x_{i_{N}})}~.
\label{J(gN)}\end{aligned}$$
By the use of Eq. (\[xiGb\]), it is shown that the Poisson bracket between any of $N-2$ functions $G_b(q,p)=\tilde{G}_b(x_1, x_2, \cdots, x_{N})$ and an arbitrary function $u(q,p)=\tilde{u}(x_1, x_2, \cdots, x_{N})$ vanishes such that $$\begin{aligned}
\{G_b, u\}_{\mbox{\tiny{PB}}}
&=& \frac{1}{2} \sum_{i_1, i_2=1}^{N}
\frac{\partial (\tilde{G}_b, \tilde{u})}{\partial (x_{i_1}, x_{i_2})}
\{x_{i_1}, x_{i_2}\}_{\mbox{\tiny{PB}}}
\nonumber \\
&=& \frac{1}{2(N-2)!} \sum_{i_1, i_2, i_3, \cdots i_{N}=1}^{N}
\varepsilon_{i_1 i_2 i_3 \cdots i_{N}}
\frac{\partial (\tilde{G}_b, \tilde{u})}{\partial (x_{i_1}, x_{i_2})}
\frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{N-2})}
{\partial (x_{i_3}, \cdots, x_{i_{N}})}
\nonumber \\
&=& \frac{\partial (\tilde{G}_b, \tilde{u}, \tilde{G}_1, \cdots, \tilde{G}_{N-2})}
{\partial (x_{1}, x_{2}, x_{3}, \cdots, x_{N})} = 0~.
\label{Gbu=0}\end{aligned}$$ Hence $G_b$ are constants. We can eliminate the constants by redefining $G_b$, and the resulting $\tilde{G}_b(x_1,x_2,\cdots, x_{N})=0$ can be regarded as [*induced constraints*]{}, which are associated with enlarging the phase space from $(q, p)$ to $(x_1,x_2,\cdots, x_{N})$.
In this way, [*Hamiltonian systems can be formulated as Nambu systems by the use of $N$ variables $x_i=x_i(q, p)$ $(i=1, 2, \cdots, N)$. The variables form an $N$-plet, and the $N-1$ Hamiltonians are given by the original Hamiltonian $\tilde{H}(x_1,x_2,\cdots, x_{N})=H(q,p)$ and induced constraints $\tilde{G}_b(x_1,x_2,\cdots, x_{N})=0$ $(b=1, \cdots, N-2)$.*]{} Note that $\tilde{H}(x_1,x_2,\cdots, x_{N})$ is not uniquely determined, because of the freedom to add a term $\sum_b\tilde{\lambda}_b(x_1,x_2,\cdots, x_{N})\tilde{G}_b(x_1,x_2,\cdots, x_{N})$ to $\tilde{H}(x_1,x_2,\cdots, x_{N})$. Here $\tilde{\lambda}_b(x_1,x_2,\cdots, x_{N})$ are some functions.
### Examples {#Examples}
Here we present two simple examples to show how induced constraints are obtained for given multiplets.\
(a) $N=3$\
Consider composite variables, $$\begin{aligned}
x=\frac{1}{4}\left(q^2 - p^2\right)~,~~
y=\frac{1}{4}\left(q^2 + p^2\right)~,~~ z=\frac{1}{2}qp~,
\label{Ex1}\end{aligned}$$ which satisfy the following relations: $$\begin{aligned}
\{x, y\}_{\mbox{\tiny{PB}}} = z~,~~
\{y, z\}_{\mbox{\tiny{PB}}} = x~,~~ \{z, x\}_{\mbox{\tiny{PB}}} = -y~.
\label{Ex1-PB}\end{aligned}$$ Then the conditions (Eq. (\[xyzG\])) become $$\begin{aligned}
\frac{\partial \tilde{G}}{\partial x} = x~,~~
\frac{\partial \tilde{G}}{\partial y} = -y~,~~
\frac{\partial \tilde{G}}{\partial z} = z~,
\label{Ex1-dG}\end{aligned}$$ and $\tilde{G}$ is obtained by $$\begin{aligned}
\tilde{G}= \frac{1}{2}\left(x^2 - y^2 + z^2\right)+C~,
\label{Ex1-G}\end{aligned}$$ where $C$ is a constant. Redefining $\tilde{G}$ as $\tilde{G}-C$, we obtain the induced constraint $\tilde{G}(x,y,z)=G(q,p)=0$.
\
(b) $N=4$\
Consider variables including composite ones, $$\begin{aligned}
x_1=q~,~~ x_2=p~,~~ x_3=x_3(q, p)~,~~ x_4=x_4(q, p)~,
\label{Ex2}\end{aligned}$$ which satisfy the following relations: $$\begin{aligned}
&& \{x_1, x_2\}_{\mbox{\tiny{PB}}}
= 1~,~~ \{x_1, x_3\}_{\mbox{\tiny{PB}}} = \frac{\partial x_3}{\partial p}~,~~
\{x_1, x_4\}_{\mbox{\tiny{PB}}} = \frac{\partial x_4}{\partial p}~,~~
\nonumber \\
&& \{x_2, x_3\}_{\mbox{\tiny{PB}}} =
-\frac{\partial x_3}{\partial q}~,~~
\{x_2, x_4\}_{\mbox{\tiny{PB}}} = -\frac{\partial x_4}{\partial q}~,~~
\nonumber \\
&&\{x_3, x_4\}_{\mbox{\tiny{PB}}} =
\frac{\partial x_3}{\partial q} \frac{\partial x_4}{\partial p}
- \frac{\partial x_3}{\partial p} \frac{\partial x_4}{\partial q}~.
\label{Ex2-PB}\end{aligned}$$ Then the conditions (Eq. (\[xiGb\])) become $$\begin{aligned}
\sum_{i_3, i_4=1}^{4} \varepsilon_{i_1 i_2 i_3 i_4}
\frac{\partial \tilde{G}_1}{\partial x_{i_3}}
\frac{\partial \tilde{G}_2}{\partial x_{i_4}}
= \{x_{i_1}, x_{i_2}\}_{\mbox{\tiny{PB}}}~,
\label{Ex2-dG}\end{aligned}$$ and $\tilde{G}_1$ and $\tilde{G}_2$ are given by $$\begin{aligned}
\tilde{G}_1= x_3-x_3(x_1, x_2)+C_1~,~~ \tilde{G}_2= x_4-x_4(x_1, x_2)+C_2~.
\label{Ex2-G}\end{aligned}$$ where $C_1$ and $C_2$ are constants. By redefining $G_1$ and $G_2$ to eliminate the constants, we obtain the induced constraints $\tilde{G}_1(x_1,x_2,x_3,x_4)=G_1(q,p)=0$ and $\tilde{G}_2(x_1,x_2,x_3,x_4)=G_2(q,p)=0$.
Many degrees of freedom systems {#Many degrees of freedom systems}
-------------------------------
Let us extend our formulation to Hamiltonian systems with many degrees of freedom. Consider a Hamiltonian system described by $n$ sets of canonical doublets $(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})$ (k$=1, 2, \cdots, n$). As is the case with $n=1$ given in Sect. 2.2, hidden Nambu structure can also be found in this system. Here we present the $N=3$ case, i.e., the case with $n$ sets of triplets $x_{i{\mbox{\tiny(k)}}} = x_{i{\mbox{\tiny(k)}}}
(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})$ $(i=1,2,3)$. Generalization to the $N(\ge 3)$ cases is straightforward.
### Dynamics {#Dynamics}
In this system, the Poisson bracket of $A$ and $B$ is defined as $$\begin{aligned}
\{A, B\}_{\mbox{\tiny{PB}}}
\equiv \sum_{{\rm k}=1}^{n}
\left(\frac{\partial A}{\partial q_{\mbox{\tiny(k)}}}
\frac{\partial B}{\partial p_{\mbox{\tiny(k)}}}
- \frac{\partial A}{\partial p_{\mbox{\tiny(k)}}}
\frac{\partial B}{\partial q_{\mbox{\tiny(k)}}}\right)~,
\label{PB-n}\end{aligned}$$ and the Hamilton’s equation of motion for any function $f=f(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ can be written as $$\begin{aligned}
\frac{d f}{dt} = \{f, H\}_{\mbox{\tiny{PB}}}~,
\label{H-eqf-n}\end{aligned}$$ where $H=H(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ is the Hamiltonian of the system. On the other hand, the Nambu bracket of $\tilde{A}$, $\tilde{B}$, and $\tilde{C}$ is defined as $$\begin{aligned}
\{\tilde{A}, \tilde{B}, \tilde{C}\}_{\mbox{\tiny{NB}}}
\equiv \sum_{{\rm k}=1}^{n}
\frac{\partial (\tilde{A}, \tilde{B}, \tilde{C})}
{\partial (x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})}~,
\label{NB-n}\end{aligned}$$ where $x_{\mbox{\tiny(k)}} = x_{1\mbox{\tiny(k)}}$, $y_{\mbox{\tiny(k)}} = x_{2\mbox{\tiny(k)}}$, and $z_{\mbox{\tiny(k)}} = x_{3\mbox{\tiny(k)}}$. Then the Nambu equation for any function $\tilde{f}=
\tilde{f}(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}}, \cdots,
z_{\mbox{\tiny({\it n})}})$ can be written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \{\tilde{f}, \tilde{H}, \tilde{G}\}_{\mbox{\tiny{NB}}}~.
\label{N-eqf-n}\end{aligned}$$ Here $\tilde{H}=
\tilde{H}(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}},
\cdots, z_{\mbox{\tiny({\it n})}})
=H(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, \cdots,
q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ is the Hamiltonian and $\tilde{G}=\tilde{G}(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}},
\cdots, z_{\mbox{\tiny({\it n})}})
= \sum_{\rm k} \tilde{G}_{\mbox{\tiny(k)}}
(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})$ is the sum of the induced constraints that satisfy the conditions $$\begin{aligned}
\frac{\partial \tilde{G}_{\mbox{\tiny(k)}}}{\partial x_{\mbox{\tiny(k)}}} =
\frac{\partial(y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})}
{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}~,~~
\frac{\partial \tilde{G}_{\mbox{\tiny(k)}}}{\partial y_{\mbox{\tiny(k)}}} =
\frac{\partial(z_{\mbox{\tiny(k)}}, x_{\mbox{\tiny(k)}})}
{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}~,~~
\frac{\partial \tilde{G}_{\mbox{\tiny(k)}}}{\partial z_{\mbox{\tiny(k)}}} =
\frac{\partial(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}})}
{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}~.
\label{xyzG-n}\end{aligned}$$ Note that the induced constraints are defined so as to be zero, $\tilde{G}_{\mbox{\tiny(k)}}
(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})
=G_{\mbox{\tiny(k)}}(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})=0$, and the Hamiltonian is not uniquely determined because of the freedom to add a linear combination of $\tilde{G}_{\mbox{\tiny(k)}}$ to $\tilde{H}$.
The $3n$ variables $x_{i\mbox{\tiny(k)}}$ satisfy the relations $$\begin{aligned}
\{
x_{i_1{\mbox{\tiny($\rm k_1$)}}},
x_{i_2{\mbox{\tiny($\rm k_2$)}}},
x_{i_3{\mbox{\tiny($\rm k_3$)}}}
\}_{\mbox{\tiny{NB}}}
&=& \varepsilon_{i_1 i_2 i_3}~~~~\mbox{for ~$\rm k_1=k_2=k_3$}~,
\label{NB-n-rel1}\\
\{
x_{i_1{\mbox{\tiny($\rm k_1$)}}},
x_{i_2{\mbox{\tiny($\rm k_2$)}}},
x_{i_3{\mbox{\tiny($\rm k_3$)}}}
\}_{\mbox{\tiny{NB}}}
&=& 0~~~~~~~~~~\mbox{otherwise}.
\label{NB-n-rel2}\end{aligned}$$ The first type of relation (Eq. (\[NB-n-rel1\])) is invariant under the time evolution (Eq. (\[N-eqf-n\])) irrespective of the form of $\tilde{H}$. To be more specific, for infinitesimal transformations $x_{i\mbox{\tiny(k)}} \to x'_{i\mbox{\tiny(k)}}=
x_{i\mbox{\tiny(k)}}+(dx_{i\mbox{\tiny(k)}}/dt) dt$, $$\begin{aligned}
\{x'_{\mbox{\tiny(k)}}, y'_{\mbox{\tiny(k)}}, z'_{\mbox{\tiny(k)}}\}
_{\mbox{\tiny{NB}}}
= 1~\label{NB-n-rel'}\end{aligned}$$ hold. We can also show an important relation, $$\begin{aligned}
\frac{\partial(x'_{\mbox{\tiny(1)}}, y'_{\mbox{\tiny(1)}}, z'_{\mbox{\tiny(1)}},
\cdots,
x'_{\mbox{\tiny({\it n})}}, y'_{\mbox{\tiny({\it n})}}, z'_{\mbox{\tiny({\it n})}})}
{\partial(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}}, z_{\mbox{\tiny(1)}}, \cdots,
x_{\mbox{\tiny({\it n})}}, y_{\mbox{\tiny({\it n})}}, z_{\mbox{\tiny({\it n})}})}
= 1~,\label{LT}\end{aligned}$$ which guarantees the Liouville theorem, the conservation law of the phase space volume under time development. On the other hand, the second type of relation (Eq. (\[NB-n-rel2\])) does not always hold, unless there is no interaction between the $n$ subsystems, i.e., $\tilde{H}$ has a form such as $\tilde{H}(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}},
\cdots, z_{\mbox{\tiny({\it n})}})
= \sum_{\rm k} \tilde{H}_{\mbox{\tiny(k)}}
(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})$.
### Partition functions {#Partition functions}
It is well known that the partition function $Z_{\rm H}$ for a canonical ensemble of the Hamiltonian system $(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, \cdots,
q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ is defined as $$\begin{aligned}
Z_{\rm H} \equiv
\iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n} dq_{\mbox{\tiny(k)}} dp_{\mbox{\tiny(k)}} e^{-\beta H}~,
\label{ZH}\end{aligned}$$ where $\beta = 1/(k_BT)$ is the inverse temperature made up of the Boltzmann constant $k_B$ and the temperature $T$. Here we study the partition function $Z_{\rm N}$ for an ensemble of the Nambu system $(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}}, \cdots, z_{\mbox{\tiny({\it n})}})$ hidden in the Hamiltonian system.
First let us conjecture the form of $Z_{\rm N}$ on physical grounds. Since $\tilde{H}=H$, $Z_{\rm N}$ must contain the $\lq\lq$Boltzmann weight" such as $e^{-\beta \tilde{H}}$. The other Hamiltonian $\tilde{G}$ is the sum of the constraints $\tilde{G}_{\mbox{\tiny(k)}}
(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})
= G_{\mbox{\tiny(k)}} (q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})= 0$, and therefore there should be delta functions such as $\delta(\tilde{G}_{\mbox{\tiny(k)}})$ in $Z_{\rm N}$. Furthermore, $Z_{\rm N}$ must contain the volume element $\prod_{{\rm k}=1}^{n}
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}}$ from the Liouville theorem.
On the basis of the above observations, it is expected that $Z_{\rm N}$ should have a form such that $$\begin{aligned}
Z_{\rm N} &\equiv&
\iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n}
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}}
\delta(\tilde{G}_{\mbox{\tiny(k)}}) e^{-\beta \tilde{H}}
\label{ZN1}\\
&=& \iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n}
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}}
\int_{-\infty}^{\infty} \frac{d\gamma_{\mbox{\tiny(k)}}}{2\pi}
e^{-\beta \tilde{H}
-i \sum_{\rm k} \gamma_{\mbox{\tiny(k)}} \tilde{G}_{\mbox{\tiny(k)}}}~.
\label{ZN2}\end{aligned}$$ We can derive $Z_{\rm H}$ (Eq. (\[ZH\])) from this expression for $Z_{\rm N}$. For example, let us consider the case that $\partial \tilde{G}_{\mbox{\tiny(k)}}/\partial z_{\mbox{\tiny(k)}} \ne 0$. We assume that there are $N_{\mbox{\tiny k}}$ solutions of $\tilde{G}_{\mbox{\tiny(k)}}=0$, $z_{\mbox{\tiny(k)}}^{(a_{\mbox{\tiny k}})}$ $(a_{\mbox{\tiny k}} = 1, 2, \cdots, N_{\mbox{\tiny k}})$, and all of them satisfy the conditions (Eq. (\[xyzG-n\])). Then using the formula for the delta function and the change of variables, Eq. (\[ZN1\]) becomes $$\begin{aligned}
Z_{\rm N} &=&
\iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n}
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}}
\sum_{a_{\mbox{\tiny k}}=1}^{N_{\mbox{\tiny k}}}
\delta(z_{\mbox{\tiny(k)}}-
z_{\mbox{\tiny(k)}}^{(a_{\mbox{\tiny k}})}(x_{\mbox{\tiny(k)}},y_{\mbox{\tiny(k)}}))
\left|\frac{\partial \tilde{G}_{\mbox{\tiny(k)}}}
{\partial z_{\mbox{\tiny(k)}}}\right|^{-1} e^{-\beta \tilde{H}}
\nonumber \\
&=& \iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n}
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}}
\sum_{a_{\mbox{\tiny k}}=1}^{N_{\mbox{\tiny k}}}
\delta(z_{\mbox{\tiny(k)}}-
z_{\mbox{\tiny(k)}}^{(a_{\mbox{\tiny k}})}(x_{\mbox{\tiny(k)}},y_{\mbox{\tiny(k)}}))
\left|\frac{\partial (x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}})}
{\partial (q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}\right|^{-1}
e^{-\beta \tilde{H}}
\nonumber \\
&=& {\mathcal N}
\iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n} dq_{\mbox{\tiny(k)}} dp_{\mbox{\tiny(k)}} e^{-\beta H}
= {\mathcal N} Z_{\rm H}~,
\label{ZH=ZN}\end{aligned}$$ where ${\mathcal N}=\prod_{{\rm k}=1}^{n}N_{\mbox{\tiny k}}$ is a constant normalization factor. This factor is irrelevant to the evaluation of physical quantities.
It is natural to require that $Z_{\rm N}$ should agree with $Z_{\rm H}$ (up to some normalization factor), because we just describe the same physical system using different formulations. It should be noted here that both expressions for $Z_{\rm N}$ (Eq. (\[ZN1\]) or Eq. (\[ZN2\])) are different from that proposed in Ref. [@N]. This comes from the fact that the Nambu mechanics considered here is an effective one induced by the redundancy of the variables.
Finally, we just give the result for the case of general $N(\ge 3)$. The possible form of the partition function is given by $$\begin{aligned}
Z_{\rm N} = \iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n}
dx_{1\mbox{\tiny{(k)}}} dx_{2\mbox{\tiny{(k)}}}
\cdots dx_{N\mbox{\tiny{(k)}}}
\delta(\tilde{G}_{1\mbox{\tiny{(k)}}}) \delta(\tilde{G}_{2\mbox{\tiny{(k)}}})
\cdots \delta(\tilde{G}_{N-2\mbox{\tiny{(k)}}}) e^{-\beta \tilde{H}}~,
\label{ZN-N}\end{aligned}$$ where $\tilde{G}_{b\mbox{\tiny{(k)}}}=0$ $(b = 1, 2, \cdots, N-2)$ are induced constraints. This expression should agree with $Z_{\rm H}$ (Eq. (\[ZH\])) up to some constant normalization factor.
Generalized Nambu equations {#Generalized Nambu equations}
---------------------------
We generalize our formulation to include a specific case that all multiplets share some variables. In such a case, a generalization of the Nambu equation would be required.
Let us describe a Hamiltonian system with $n$ sets of canonical doublets $(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})$ $({\rm k}=1, \cdots, n)$ using $2n+m$ variables $w_{\ell}$ $(\ell=1, \cdots, 2n+m)$. We classify the variables $w_{\ell}$ into two groups, $x_{a}$ $(a=1, \cdots, 2n)$ and $z_s$ $(s=1, \cdots, m)$, where $x_a$ are assumed to satisfy $\det \{x_a, x_b\}_{\mbox{\tiny{PB}}} \ne 0$. Note that the classification of variables is not unique.
First we study the case with $m=0$ for completeness. In this case, the equation for any function $\tilde{f}(x_1, \cdots, x_{2n}) =
f(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ can be written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \sum_{{\rm k}=1}^{n}
\frac{\partial(f, H)}{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}
= \frac{1}{2} \sum_{{\rm k}=1}^{n}
\sum_{{a, b}=1}^{2n} \frac{\partial(\tilde{f}, \tilde{H})}{\partial(x_a, x_b)}
\frac{\partial(x_a, x_b)}{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}
= \sum_{{a, b}=1}^{2n} \tilde{g}_{ab}
\frac{\partial(\tilde{f}, \tilde{H})}{\partial(x_a, x_b)}~,
\label{H-eq(m=0)}\end{aligned}$$ where $\tilde{H}=\tilde{H}(x_1, \cdots, x_{2n}) =
H(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ and $\tilde{g}_{ab}$ is defined as $$\begin{aligned}
\tilde{g}_{ab}(x_1, \cdots, x_{2n})
= g_{ab}(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})
\equiv \frac{1}{2}\sum_{{\rm k}=1}^{n}
\frac{\partial(x_a, x_b)}{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}
= \frac{1}{2}\{x_a, x_b\}_{\mbox{\tiny{PB}}}~.
\label{Gab}\end{aligned}$$ The $\tilde{g}_{ab}$ plays the role of a metric tensor, because it transforms under a change of variables $x_a \to x'_a$ as follows: $$\begin{aligned}
\tilde{g}'_{ab}(x'_1, \cdots, x'_{2n})
= \sum_{{c, d}=1}^{2n} \frac{\partial x'_a}{\partial x_c}
\frac{\partial x'_b}{\partial x_d} ~\tilde{g}_{cd}(x_1, \cdots, x_{2n})~.
\label{G'ab}\end{aligned}$$ In the case in which $\tilde{g}_{ab}$ depends on $x_a$, neither the transformation $(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})
\to (x_1, \cdots, x_{2n})$ nor the time evolution of $x_a$ is a canonical transformation. The latter means that the Liouville theorem in general does not hold for the dynamics of $x_a$. This fact reminds us of the superiority of canonical variables.
Now let us proceed to the case with $m \ge 1$. The equation for a function $\tilde{f}(w_1, \cdots, w_{2n+m}) =
f(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ can be written as $$\begin{aligned}
\frac{d \tilde{f}}{dt}
&=& \frac{1}{2} \sum_{{a, b}=1}^{2n} \frac{\partial(\tilde{f}, \tilde{H})}
{\partial(x_a, x_b)}\{x_a, x_b\}_{\mbox{\tiny{PB}}} \nonumber \\
&&+ \sum_{a=1}^{2n}\sum_{s=1}^{m} \frac{\partial(\tilde{f}, \tilde{H})}
{\partial(x_a, z_s)}\{x_a, z_s\}_{\mbox{\tiny{PB}}}
+ \frac{1}{2} \sum_{s, t=1}^{m} \frac{\partial(\tilde{f}, \tilde{H})}
{\partial(z_s, z_t)}\{z_s, z_t\}_{\mbox{\tiny{PB}}}~,
\label{H-eq(m)}\end{aligned}$$ where $\tilde{H}= \tilde{H}(w_1, \cdots, w_{2n+m})
=H(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$. Introducing functions $\tilde{G}_s$ $(s=1, \cdots, m)$ and $\tilde{g}^{(m)}_{ab}$ that satisfy the following relations, $$\begin{aligned}
\frac{1}{2} \{x_a, x_b\}_{\mbox{\tiny{PB}}}
&=&
\tilde{g}^{(m)}_{ab} \frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{m})}
{\partial (z_{1}, \cdots, z_{m})}~,~~
\label{phi-2n+m1}\\
\frac{1}{2}\{x_a, z_{s}\}_{\mbox{\tiny{PB}}}
&=&
-\sum_{b=1}^{2n} \tilde{g}^{(m)}_{ab}
\frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{s-1},
\tilde{G}_{s}, \tilde{G}_{s+1}, \cdots, \tilde{G}_{m})}
{\partial (z_{1}, \cdots, z_{s-1}, x_b, z_{s+1}, \cdots,z_{m})}~,~~
\label{phi-2n+m2}\\
\{z_s, z_t\}_{\mbox{\tiny{PB}}}
&=&\!\!\!\!
\sum_{a,b=1}^{2n}\tilde{g}^{(m)}_{ab}
\frac{\partial (\tilde{G}_1, \cdots,
\tilde{G}_{s-1}, \tilde{G}_{s}, \tilde{G}_{s+1}, \cdots,
\tilde{G}_{t-1}, \tilde{G}_{t}, \tilde{G}_{t+1}, \cdots, \tilde{G}_{m})}
{\partial (z_{1}, \cdots, z_{s-1}, x_{a}, z_{s+1},
\cdots, z_{t-1}, x_{b}, z_{t+1}, \cdots, z_{m})}~,
\label{phi-2n+m3}\end{aligned}$$ where $s<t$, Eq. (\[H-eq(m)\]) can be rewritten as $$\begin{aligned}
\frac{d \tilde{f}}{dt}
= \sum_{a, b=1}^{2n} \tilde{g}^{(m)}_{ab}
\frac{\partial (\tilde{f}, \tilde{H}, \tilde{G}_1, \cdots, \tilde{G}_{m})}
{\partial (x_a, x_b, z_1, \cdots, z_m)}
\label{N-eqf(m)}\end{aligned}$$ using a formula concerning Jacobians.
By the use of Eqs. (\[phi-2n+m1\])–(\[phi-2n+m3\]), it is shown that the Poisson bracket between any of the $m$ functions $G_s(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})
=\tilde{G}_s(w_1, \cdots, w_{2n+m})$ and an arbitrary function $u(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})
=\tilde{u}(w_1, \cdots, w_{2n+m})$ vanishes such that $$\begin{aligned}
\{G_s, u\}_{\mbox{\tiny{PB}}}
&=& \frac{1}{2} \sum_{a, b=1}^{2n}
\frac{\partial(\tilde{G}_s, \tilde{u})}
{\partial(x_a, x_b)}\{x_a, x_b\}_{\mbox{\tiny{PB}}}
\nonumber \\
&&
+ \sum_{a=1}^{2n} \sum_{s=1}^{m}
\frac{\partial(\tilde{G}_s, \tilde{u})}
{\partial(x_a, z_s)}\{x_a, z_s\}_{\mbox{\tiny{PB}}}
+ \frac{1}{2} \sum_{s, t=1}^{m}
\frac{\partial(\tilde{G}_s, \tilde{u})}
{\partial(z_s, z_t)}\{z_s, z_t\}_{\mbox{\tiny{PB}}}
\nonumber \\
&=& \sum_{a, b=1}^{2n}
\tilde{g}^{(m)}_{ab}
\frac{\partial (\tilde{G}_s, \tilde{u}, \tilde{G}_1, \cdots, \tilde{G}_{m})}
{\partial (x_a, x_b, z_1, \cdots, z_m)} = 0~.
\label{phiu=0}\end{aligned}$$ Hence $G_s(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ are constants and, if necessary, we can define $G_s=\tilde{G}_s=0$ by shifting constants. We refer to Eq. (\[N-eqf(m)\]) as the generalized Nambu equation. Note that the Liouville theorem does not hold in general for the dynamics given by this equation. This unfavorable property is a result of two factors: Eq. (\[N-eqf(m)\]) has $x_{a}$-dependent $\tilde{g}^{(m)}_{ab}$ and multiplets in Eq. (\[N-eqf(m)\]) share common variables $z_{s}$. The latter means that it is difficult to define an appropriate phase space volume.
One of the non-vanishing components of $\tilde{g}^{(m)}_{ab}$ can be set to $\frac{1}{2}$ by redefinition of constraints $\tilde{G}_s$. For example, in the case in which $n=1$, we can set $\tilde{g}^{(m)}_{12}=\frac{1}{2}$ (and $\tilde{g}^{(m)}_{21}=-\frac{1}{2}$) by redefining $\tilde{G}_s$, and Eq. (\[N-eqf(m)\]) reduces to the Nambu equation (Eq. (\[H-eqf(gN)\])) with $N=2+m$.
Finally, we consider the case in which the variables $x_a$ and $z_s$ are further classified into $M$ $\lq\lq$irreducible“ sets, $\{x_{a^1}^{\mbox{\tiny{({\rm 1})}}},
z_{s^1}^{\mbox{\tiny{({\rm 1})}}}\}
\bigoplus
\{x_{a^2}^{\mbox{\tiny{({\rm 2})}}},
z_{s^2}^{\mbox{\tiny{({\rm 2})}}}\}
\bigoplus \cdots \bigoplus
\{x_{a^M}^{\mbox{\tiny{({\it M})}}},
z_{s^M}^{\mbox{\tiny{({\it M})}}}\}$, where $a^i=1,\cdots,2n^i$ ($\sum_{i=1}^{M}n^i=n$) and $s^i=1,\cdots,m^i$ ($\sum_{i=1}^{M}m^i=m$). Here $\lq\lq$irreducible” means that the Poisson bracket between any two elements that belong to different sets vanishes, i.e., $\{x_{a^i}^{\mbox{\tiny{({\it i})}}},
x_{a^j}^{\mbox{\tiny{({\it j})}}}\}_{\mbox{\tiny{PB}}}=0$, $\{x_{a^i}^{\mbox{\tiny{({\it i})}}},
z_{s^j}^{\mbox{\tiny{({\it j})}}}\}_{\mbox{\tiny{PB}}}=0$, and $\{z_{s^i}^{\mbox{\tiny{({\it i})}}},
z_{s^j}^{\mbox{\tiny{({\it j})}}}\}_{\mbox{\tiny{PB}}}=0$ for $i \ne j$. Note that this classification is not unique, either. The equation of motion for any function $\tilde{f}(w_1, \cdots, w_{2n+m})$ can be expressed in the form of the generalized Nambu equation, $$\begin{aligned}
\frac{d \tilde{f}}{dt}
= \sum_{i=1}^{M}
\sum_{a^i, b^i=1}^{2n^i} \tilde{g}^{(m^i)}_{a^i b^i}
\frac{\partial (\tilde{f}, \tilde{H},
\tilde{G}_{1}^{\mbox{\tiny{({\it i})}}},
\cdots, \tilde{G}_{m^i}^{\mbox{\tiny{({\it i})}}})}
{\partial (x_{a^i}^{\mbox{\tiny{({\it i})}}},
x_{b^i}^{\mbox{\tiny{({\it i})}}},
z_{1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{m^i}^{\mbox{\tiny{({\it i})}}})}~.
\label{N-eqf(m)2}\end{aligned}$$ Here $\tilde{G}_{s^i}^{\mbox{\tiny{({\it i})}}}$ and $\tilde{g}^{(m^i)}_{a^ib^i}$ should satisfy the following conditions: $$\begin{aligned}
\frac{1}{2} \{x_{a^i}^{\mbox{\tiny{({\it i})}}},
x_{b^i}^{\mbox{\tiny{({\it i})}}}\}_{\mbox{\tiny{PB}}}
&=&
\tilde{g}^{(m^i)}_{a^ib^i}
\frac{\partial (\tilde{G}_{1}^{\mbox{\tiny{({\it i})}}},
\cdots, \tilde{G}_{m^i}^{\mbox{\tiny{({\it i})}}})}
{\partial (z_{1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{m^i}^{\mbox{\tiny{({\it i})}}})}~,~~
\label{phi-2n+m4}\\
\frac{1}{2} \{x_{a^i}^{\mbox{\tiny{({\it i})}}},
z_{s^i}^{\mbox{\tiny{({\it i})}}}\}_{\mbox{\tiny{PB}}}
&=&
-\sum_{b^i=1}^{2n^i} \tilde{g}^{(m^i)}_{a^ib^i}
\frac{\partial (\tilde{G}_{1}^{\mbox{\tiny{({\it i})}}},
\cdots, \tilde{G}_{s^i-1}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{s^i}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{s^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
\tilde{G}_{m^i}^{\mbox{\tiny{({\it i})}}})}
{\partial (z_{1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{s^i-1}^{\mbox{\tiny{({\it i})}}},
x_{b^i}^{\mbox{\tiny{({\it i})}}},
z_{s^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{m^i}^{\mbox{\tiny{({\it i})}}})}~,~~
\label{phi-2n+m5}\\
\{z_{s^i}^{\mbox{\tiny{({\it i})}}},
z_{t^i}^{\mbox{\tiny{({\it i})}}}\}_{\mbox{\tiny{PB}}}&=& \nonumber \\
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\sum_{a^i,b^i=1}^{2n^i}\tilde{g}^{(m^i)}_{a^ib^i}
\frac{\partial (\tilde{G}_{1}^{\mbox{\tiny{({\it i})}}},
\cdots, \tilde{G}_{s^i-1}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{s^i}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{s^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
\tilde{G}_{t^i-1}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{t^i}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{t^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
\tilde{G}_{m^i}^{\mbox{\tiny{({\it i})}}})}
{\partial (z_{1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{s^i-1}^{\mbox{\tiny{({\it i})}}},
x_{a^i}^{\mbox{\tiny{({\it i})}}},
z_{s^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{t^i-1}^{\mbox{\tiny{({\it i})}}},
x_{b^i}^{\mbox{\tiny{({\it i})}}},
z_{t^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{m^i}^{\mbox{\tiny{({\it i})}}})}~,~~
\label{phi-2n+m6}\end{aligned}$$ where $s^i< t^i$. We refer to the systems where the master equations are given by Eq. (\[N-eqf(m)\]) or Eq. (\[N-eqf(m)2\]) as generalized Nambu systems.
Nambu systems in constrained Hamiltonian systems {#Nambu systems in constrained Hamiltonian systems}
================================================
Subject {#Subject}
-------
In the previous section, we found that a Hamiltonian system can be formulated as a Nambu system with multiplets including composite variables of $q$ and $p$. The main feature of our formulation is the existence of induced constraints that are required just for consistency between the variables. Together with the Hamiltonian of the original system, the induced constraints serve as Hamiltonians of the Nambu system. Therefore it is intriguing to study how constrained Hamiltonian systems, systems with [*physical*]{} constraints, are cast into Nambu systems in our formulation.
The relations between Nambu systems and constrained Hamiltonian systems have been investigated by many authors [@BF; @CK; @R; @MS; @KT; @KT2; @LJ]. To clarify the difference between previous works and our approach, here we give a brief summary of the results obtained so far. In most works, Nambu systems are treated as the original systems, and studies have been carried out to find appropriate constrained Hamiltonian systems into which the Nambu systems can be embedded [@BF; @CK; @R; @MS; @KT; @KT2]. Specifically, it has been shown that Nambu equations (Eq. (\[N-eq\])) are compatible with the following equations: $$\begin{aligned}
&& p_i = H_1 \frac{\partial H_2}{\partial x_i}~,\label{N-eq-con1}\\
&& \sum_{i=1}^3 \frac{\partial(H_1, H_2)}{\partial(x_i, x_j)}\frac{dx_i}{dt} = 0~.
\label{N-eq-con2}\end{aligned}$$ Here $p_i$ ($i=1,2,3$) are the canonical conjugate momenta defined as $p_i \equiv \partial L/\partial \dot{x}_i$ with the Lagrangian $$\begin{aligned}
L = H_1 \sum_{i=1}^3 \frac{\partial H_2}{\partial x_i}\frac{dx_i}{dt} ~.
\label{N-eq-conL}\end{aligned}$$ Equation (\[N-eq-con2\]) can be derived as the Euler–Lagrange equation from this Lagrangian, and Eq. (\[N-eq-con1\]) leads to the relations $\phi_i \equiv p_i - H_1 {\partial H_2}/{\partial x_i} = 0$, which can be regarded as constraints. In this way, Nambu systems can be interpreted as Hamiltonian systems with specific constraints.
On the other hand, researchers have studied whether constrained systems can be described as Nambu systems or not. Specifically, it has been shown that constrained Hamiltonian systems can be formulated in terms of (a generalized form of) Nambu mechanics by introducing an extra phase-space variable [@LJ]. For a system with canonical variables $(q_k, p_k)$ $(k=1, \cdots, n)$ and $m$ first class constraints $\phi_l(q_1, \cdots, p_n)=0$, the equations of motion are given by $$\begin{aligned}
\frac{d q_k}{d t} &=& \frac{\partial H}{\partial p_k}
+ \sum_{l=1}^m \left(\
\frac{\partial \lambda_l}{\partial p_k}\phi_l
+ \lambda_l \frac{\partial \phi_l}{\partial p_k}
\right)~,\label{eq-constraints1}\\~~
\frac{d p_k}{d t} &=& -\frac{\partial H}{\partial q_k}
- \sum_{l=1}^m \left(\
\frac{\partial \lambda_l}{\partial q_k}\phi_l
+ \lambda_l \frac{\partial \phi_l}{\partial q_k}
\right)~,
\label{eq-constraints2}\end{aligned}$$ where $\lambda_l$ are Lagrange multipliers. Equations (\[eq-constraints1\]) and (\[eq-constraints2\]) are derived from (a generalized form of) the Nambu equation $$\begin{aligned}
\frac{d f}{dt} = \sum_{k=1}^n \frac{\partial (f, H_1, H_2)}{\partial (q_k, p_k, r)}~,
\label{N-eq-constraints}\end{aligned}$$ where $f=f(q_1, \cdots, p_n)$, $r$ is an extra phase-space variable, and Hamiltonians are defined as $$\begin{aligned}
H_1 = H - r~,~~ H_2 = r+ \sum_{l=1}^m \lambda_l \phi_l~.
\label{H1H2-constraints}\end{aligned}$$ The equation for $r$ is given by $$\begin{aligned}
\frac{d r}{dt}= - \sum_{l=1}^m
\left(\lambda_l \{\phi_l, H\}_{\mbox{\tiny{PB}}}
+\phi_l \{\lambda_l, H\}_{\mbox{\tiny{PB}}}
\right)
= - \sum_{l=1}^m \lambda_l \frac{d \phi_l}{dt},
\label{r-eq-constraints}\end{aligned}$$ where the last equality holds after imposing constraints. Requiring the extra variable $r$ to decouple from the dynamics, i.e., $dr/dt = 0$, we obtain ${d \phi_l}/{dt} = 0$.
Our approach differs from these previous works. Our starting point is not Nambu systems but Hamiltonian systems with constraints, and we do not introduce extra variables but use redundant variables.
Nambu structure in constrained Hamiltonian systems {#Nambu structure in constrained Hamiltonian systems}
--------------------------------------------------
Here we demonstrate that systems with first class constraints can be formulated as Nambu systems or generalized Nambu systems, without introducing extra degrees of freedom.
As a warm-up, we consider a system of two canonical doublets $(q_1, p_1)$ and $(q_2, p_2)$ with one first class constraint $\phi(q_1, p_1, q_2, p_2) = 0$ that is time independent: $d\phi /dt = \{\phi, h\}_{\mbox{\tiny{PB}}}=0$. Here $h=h(q_1, p_1, q_2, p_2)$ is the Hamiltonian of this system. The constraint $\phi$ is associated with gauge degrees of freedom, and an auxiliary condition $\chi(q_1, p_1, q_2, p_2) = 0$ such that $\{\phi, \chi\}_{\mbox{\tiny{PB}}} \ne 0$ should be imposed to fix the freedom.
By an appropriate canonical transformation $(q_1, p_1, q_2, p_2) \to (Q_1, P_1, Q_2, P_2)$, we can eliminate one of the canonical variables. Here we show the case in which $P_2$ is eliminated as follows:[^1] $$\begin{aligned}
P_2 = \chi(q_1, p_1, q_2, p_2) = 0~.
\label{P2}\end{aligned}$$ The new Hamiltonian $K$ is given by $K(Q_1, P_1, Q_2, P_2)=h(q_1, p_1, q_2, p_2)$, and the original constraint $\phi$ is transformed as $\Phi(Q_1, P_1, Q_2, P_2)=\phi(q_1, p_1, q_2, p_2)$. From $\{\phi, \chi\}_{\mbox{\tiny{PB}}} = \partial \Phi/\partial Q_2 \ne 0$, the constraint $\Phi=0$ can be solved by $Q_2$ to give $Q_2=Q_2(Q_1,P_1)$. Then we obtain a constraint $\Psi \equiv Q_2 - Q_2(Q_1,P_1)=0$, which is equivalent to the original constraint $\phi=0$.
If we consider a system described by the variables $(Q_1, P_1, Q_2)$ with the constraint $\Psi =0$, it is easy to show that the equation of motion for any function $f=f(Q_1, P_1, Q_2)$ can be written in the form of the Nambu equation, $$\begin{aligned}
\frac{df}{dt} = \frac{\partial(f, H, \Psi)}{\partial(Q_1, P_1, Q_2)}~,
\label{N-eqQP}\end{aligned}$$ where $H=H(Q_1, P_1, Q_2)=K(Q_1, P_1, Q_2, P_2=0)$ is the Hamiltonian. In fact, for $f=Q_1$ and $f=P_1$, Hamilton’s canonical equations of motion $$\begin{aligned}
\frac{dQ_1}{dt} = \frac{\partial H(Q_1, P_1, Q_2(Q_1, P_1))}{\partial P_1} ~,
~~\frac{dP_1}{dt} = -\frac{\partial H(Q_1, P_1, Q_2(Q_1, P_1))}{\partial Q_1} ~,
\label{H-eqQP}\end{aligned}$$ are derived from Eq. (\[N-eqQP\]), and for $f=\Psi$, we obtain time independence of the constraint, $d\Psi/dt = 0$. On the other hand, for $f=Q_2$, the following equation is derived: $$\begin{aligned}
\frac{dQ_2}{dt} = \frac{\partial(H(Q_1, P_1, Q_2), {\Psi})}{\partial(Q_1, P_1)}
= \frac{\partial(Q_2(Q_1, P_1), H(Q_1, P_1, Q_2(Q_1, P_1)))}{\partial(Q_1, P_1)}~.
\label{N-eqQ2}\end{aligned}$$ Using $d\Psi/dt = 0$ and Eq. (\[N-eqQ2\]), we obtain Hamilton’s equation of motion for $Q_2(Q_1, P_1)$, $$\begin{aligned}
\frac{d Q_2(Q_1, P_1)}{d t} =
\frac{\partial(Q_2(Q_1, P_1), H(Q_1, P_1, Q_2(Q_1, P_1)))}{\partial(Q_1, P_1)}~.
\label{H-eqQ2}\end{aligned}$$
By referring to the results in Sect. 2.2, let us formulate this system by means of the composite triplet $X=X(Q_1, P_1)$, $Y=Y(Q_1, P_1)$, and $Z=Z(Q_1, P_1)$, imposing a constraint $\tilde{G}(X, Y, Z)$ that is equivalent to the original constraint $\phi$. We assume that $\partial(X, Y)/\partial(Q_1, P_1) \ne 0$, i.e., $Z$ is a redundant variable such that $Z=Z(X, Y)$. If the variables satisfy the conditions $$\begin{aligned}
\frac{\partial \tilde{G}}{\partial Z} = \frac{\partial(X,Y)}{\partial(Q_1,P_1)}~,~~
\frac{\partial \tilde{G}}{\partial X} = \frac{\partial(Y,Z)}{\partial(Q_1,P_1)}~,~~
\frac{\partial \tilde{G}}{\partial Y} = \frac{\partial(Z,X)}{\partial(Q_1,P_1)}~,
\label{dphi/dZ}\end{aligned}$$ then the time evolution of any function $\tilde{f}=\tilde{f}(X,Y,Z)$ is given by the Nambu equation, $$\begin{aligned}
\frac{d\tilde{f}}{dt} =
\frac{\partial(\tilde{f}, \tilde{H}, \tilde{G})}{\partial(X, Y, Z)}~,
\label{N-eqXYZ}\end{aligned}$$ where $\tilde{H}$ is equal to the original Hamiltonian, $\tilde{H}(X,Y,Z)=H(Q_1, P_1, Q_2)$. We can define various types of Nambu systems depending on the choice of variables and the constraint.
Here we present two simple examples. First, if we choose $X=Q_1$, $Y=P_1$, $Z=Q_2(Q_1,P_1)$, and $\tilde{G}(X,Y,Z)=\Psi(Q_1, P_1, Q_2)$, Eq. (\[N-eqXYZ\]) clearly holds from Eq. (\[N-eqQP\]). Next, let us choose $Y=P_1$, $Z=Q_2(Q_1,P_1)$, and $\tilde{G}(X,Y,Z)=\Phi(Q_1, P_1, Q_2)$. In this case, if the variable $X$ is given by $$\begin{aligned}
X = \int \frac{\partial \tilde{G}}{\partial Z}dQ_1
= \int \frac{\partial {\Phi}}{\partial Q_2}dQ_1~,~~
\label{XZ}\end{aligned}$$ then the variables satisfy the conditions (Eq. (\[dphi/dZ\])), and the system is described as a Nambu system.
It is straightforward to extend the above $\lq\lq$warm-up" discussion to many degrees of freedom systems. Consider a system of $n$ sets of canonical doublets $(q_k, p_k)$ $(k=1, \cdots, n)$ with $m$ kinds of first class constraints $\phi_s(q_1, p_1, \cdots, q_n, p_n) = 0$ $(s=1, \cdots, m)$. The Hamiltonian of this system is given by $h(q_1, p_1, \cdots, q_n, p_n)$. To fix the gauge degrees of freedom, $m$ kinds of auxiliary conditions $\chi_t(q_1, p_1, \cdots, q_n, p_n) = 0$ $(t=1, \cdots, m)$ that satisfy $\det \{\phi_s, \chi_t\}_{\mbox{\tiny{PB}}} \ne 0$ should be imposed.
By an appropriate canonical transformation $(q_1, p_1, \cdots, q_n, p_n) \to (Q_1, P_1, \cdots, Q_n, P_n)$, we can eliminate some of the canonical variables. Here we show the case in which $P_{n-m+t}$ are eliminated as follows: $$\begin{aligned}
P_{n-m+t} = \chi_t(q_1, p_1, \cdots, q_n, p_n) = 0~.
\label{Ps}\end{aligned}$$ The new Hamiltonian is given by $K=K(Q_1, P_1, \cdots, Q_n, P_n)=h(q_1, p_1, \cdots, q_n, p_n)$, and the original constraints $\phi_s$ are transformed as $\Phi_s(Q_1, P_1, \cdots, Q_n, P_n)=\phi_s(q_1, p_1, \cdots, q_n, p_n)$. From $\det \{\phi_s, \chi_t\}_{\mbox{\tiny{PB}}} =
\det (\partial \Phi_s/\partial Q_{n-m+t}) \ne 0$, the constraints $\Phi_{s}=0$ can be solved by $Q_{n-m+t}$ to give $Q_{n-m+t} = Q_{n-m+t}(Q_1,P_1, \cdots, Q_{n-m}, P_{n-m})$. By referring to the results in Sect. 2.4, let us formulate this system by means of composite variables $X_{a}=X_{a}(Q_1, P_1, \cdots, Q_{n-m}, P_{n-m})$ $(a = 1, \cdots, 2n-2m)$ and $Z_s=Z_s(Q_1, P_1, \cdots, Q_{n-m}, P_{n-m})$, imposing the constraints $\tilde{G}_s(X_{1}, \cdots, X_{2n-2m}, Z_{1}, \cdots, Z_{m})$ that are equivalent to the original constraints $\phi_s$. We assume that $\det\{X_a, X_b\}^{\prime}_{\mbox{\tiny{PB}}} \ne 0$, where the Poisson bracket is defined as $$\begin{aligned}
\{A, B\}^{\prime}_{\mbox{\tiny{PB}}}
\equiv \sum_{\alpha=1}^{n-m}
\frac{\partial (A, B)}{\partial (Q_{\alpha}, P_{\alpha})}~.
\label{PBQPs}\end{aligned}$$ This means that $Z_s$ are redundant variables such that $Z_s=Z_s(X_1, \cdots, X_{2n-2m})$.
If $\tilde{G}_s$ and $\tilde{g}^{(m)}_{ab}$ satisfy the following relations: $$\begin{aligned}
\frac{1}{2} \{X_a, X_b\}^{\prime}_{\mbox{\tiny{PB}}}
&=&
\tilde{g}^{(m)}_{ab} \frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{m})}
{\partial (Z_{1}, \cdots, Z_{m})}~,~~
\label{dphi/dZm1} \\
\frac{1}{2}\{X_a, Z_{s}\}^{\prime}_{\mbox{\tiny{PB}}}
&=&
-\sum_{b=1}^{2n-2m} \tilde{g}^{(m)}_{ab}
\frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{s-1},
\tilde{G}_{s}, \tilde{G}_{s+1}, \cdots, \tilde{G}_{m})}
{\partial (Z_{1}, \cdots, Z_{s-1}, X_b, Z_{s+1}, \cdots, Z_{m})}~,~~
\label{dphi/dZm2} \\
\{Z_s, Z_t\}^{\prime}_{\mbox{\tiny{PB}}}
&=&\!\!\!\!\!\!
\sum_{a,b=1}^{2n-2m}\tilde{g}^{(m)}_{ab}
\frac{\partial (\tilde{G}_1, \cdots,
\tilde{G}_{s-1}, \tilde{G}_{s}, \tilde{G}_{s+1}, \cdots,
\tilde{G}_{t-1}, \tilde{G}_{t}, \tilde{G}_{t+1}, \cdots, \tilde{G}_{m})}
{\partial (Z_{1}, \cdots, Z_{s-1}, X_{a}, Z_{s+1},
\cdots, Z_{t-1}, X_{b}, Z_{t+1}, \cdots, Z_{m})},
\label{dphi/dZm3}\end{aligned}$$ where $s<t$, then the time evolution of any function $\tilde{f}=\tilde{f}(X_{1}, \cdots, X_{2n-2m}, Z_{1}, \cdots, Z_{m})$ can be written as $$\begin{aligned}
\frac{d \tilde{f}}{dt}
= \sum_{a,b=1}^{2n-2m} \tilde{g}^{(m)}_{ab}
\frac{\partial (\tilde{f}, \tilde{H}, \tilde{G}_1, \cdots, \tilde{G}_{m})}
{\partial (X_{a}, X_{b}, Z_{1}, \cdots, Z_{m})}~,
\label{N-eqQPs}\end{aligned}$$ where $\tilde{H}$ is the Hamiltonian, $$\begin{aligned}
\tilde{H}(X_{1}, \cdots, X_{2n-2m}, Z_{1}, \cdots, Z_{m})&=&\nonumber\\
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
K(Q_1, P_1, \cdots, Q_{n-m}, P_{n-m}, Q_{n-m+1}, P_{n-m+1}=0,
\cdots, Q_{n}, P_{n}=0)~.
\label{Hamiltonian}\end{aligned}$$ We can define various types of Nambu systems depending on the choice of variables and the constraint. For example, in the case in which we take $X_{n-m+\alpha}=P_{\alpha}$ $(\alpha=1, \cdots, n-m)$, $Z_s=Q_{n-m+s}$, and $\tilde{G}_s=\Phi_s$, if the variables $X_{\alpha}$ are functions of $Q_{\alpha}$ that are given by $$\begin{aligned}
X_{\alpha} = X_{\alpha}(Q_{\alpha})&=&
2 \int \tilde{g}^{(m)}_{\alpha~n-m+\alpha}
\frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{m})}
{\partial (Z_{1}, \cdots, Z_{m})} dQ_{\alpha}
\nonumber \\
&=& 2 \int {g}^{(m)}_{\alpha~n-m+\alpha}
\frac{\partial ({\Phi}_1, \cdots, {\Phi}_{m})}
{\partial (Q_{n-m+1}, \cdots, Q_{n})} dQ_{\alpha}~,
\label{Xalpha}\end{aligned}$$ then the variables satisfy the conditions (Eqs. (\[dphi/dZm1\])–(\[dphi/dZm3\])), and the system is described as a Nambu system.
In this way, [*Hamiltonian systems with first class constraints can be described as Nambu systems where the master equations are Nambu equations or generalized ones.*]{} It is straightforward to formulate constrained Hamiltonian systems as Nambu systems where both the original constraints and the induced ones serve as Hamiltonians. Such systems can be realized by introducing many more redundant variables.
Example: relativistic free particle {#Example: relativistic free particle}
-----------------------------------
As an example, we consider a relativistic particle moving freely in 4-dimensional Minkowski space. The motion is expressed by the space-time 4-vector $q^{\mu}=q^{\mu}(\tau)$ and corresponding canonical momenta $p_{\mu}=p_{\mu}(\tau)$ $(\mu=0, 1, 2, 3)$, where $\tau$ is the proper time. Here we use the metric tensor $\eta_{\mu\nu} = \mbox{diag}(1, -1, -1, -1)$. The system has four sets of canonical pairs $(q^{\mu}, p_{\mu})$ and one first class constraint $$\begin{aligned}
\phi = p^{\mu} p_{\mu} - m^2 c^2 = 0~,
\label{p2}\end{aligned}$$ where $m$ is the mass of the particle, $c$ is the speed of light, and Einstein’s summation convention is used. We impose an auxiliary condition $\chi = q^0 - c \tau = 0$ to fix the gauge freedom. Performing a canonical transformation $$\begin{aligned}
&&q^0 \to Q^0=\chi,~~~ q^i \to Q^i=q^i~,\nonumber \\
&&p_0 \to P_0=p_0,~~~p_i \to P_i=p_i~,
\label{Can}\end{aligned}$$ where $i=1,2,3$, we can eliminate $Q^0$, and the system is described by three sets of canonical pairs $(Q^i, P_i)$ with the new Hamiltonian $K = -cP_{0}$. The original constraint $\phi$ is transformed as $$\begin{aligned}
\phi \to \Phi = P^{\mu} P_{\mu} - m^2 c^2 = 0~,
\label{p2t}\end{aligned}$$ which has an equivalent expression, $$\begin{aligned}
\Psi = P_{0} + \sqrt{{\mbox{\boldmath $P$}}^2 + m^2 c^2}=0~,
\label{explicit}\end{aligned}$$ where ${\mbox{\boldmath $P$}}^2 = \sum_{i} P_i^2$. Then Hamilton’s equations of motion for $Q^i$ and $P_i$ are given by $$\begin{aligned}
\frac{d Q^i}{dt} &=& \frac{\partial K}{\partial P_i}
= \frac{cP_i}{\sqrt{{\mbox{\boldmath $P$}}^2 + m^2 c^2}}~,
\label{Hamilton-rel1}\\
\frac{d P_i}{dt} &=& -\frac{\partial K}{\partial Q^i} = 0 ~.~~
\label{Hamilton-rel2}\end{aligned}$$
Using the results in Sect. 3.2, let us construct Nambu systems that are equivalent to this system. The target equation is Eq. (\[N-eqQPs\]) with $a,b = 1, \cdots, 6$ and $m = 1$. Here we present three types of Nambu systems. In each case the Hamiltonian is given by $\tilde{H}=K$.
\
(a) First we consider the simplest construction, $$\begin{aligned}
&~& X_i = Q^i~,~~ X_{i+3}=Y_i=P_i~,~~ Z=P_0~,~~
\nonumber\\
&~& \tilde{G}= \Psi = Z + \sqrt{{\mbox{\boldmath $Y$}}^2 + m^2 c^2}~,
\label{G-rel1}\end{aligned}$$ where ${\mbox{\boldmath $Y$}}^2 = \sum_{i} Y_i^2$. From Eq. (\[dphi/dZm1\]) we obtain $$\begin{aligned}
\tilde{g}^{(1)}_{ab}=\frac{1}{2}(\delta_{a,b-3}-\delta_{a-3,b})~,
\label{gab-rel1}\end{aligned}$$ and Eq. (\[N-eqQPs\]) becomes $$\begin{aligned}
\frac{d \tilde{f}}{dt}
= \sum_{i=1}^3 \frac{\partial (\tilde{f}, \tilde{H}, \tilde{G})}
{\partial (X_i, Y_i, Z)}~.
\label{N-eq-rel1}\end{aligned}$$ This equation reduces to Hamilton’s equations of motion (Eqs. (\[Hamilton-rel1\])–(\[Hamilton-rel2\])) and the energy conservation law .
\
(b) Next we consider a slightly different construction, $$\begin{aligned}
&~& X_i = Q^i~,~~ X_{i+3}=Y_i=P_i~,~~ Z=P_0^2~,~~
\nonumber\\
&~& \tilde{G}= \Phi = Z - {\mbox{\boldmath $Y$}}^2 - m^2 c^2~.
\label{G-rel2}\end{aligned}$$ In this case, $\tilde{g}^{(1)}_{ab}$ has the same form as Eq. (\[gab-rel1\]), because in both cases (a) and (b), $\{X_i, X_{i+3}\}^{\prime}_{\mbox{\tiny{PB}}}=1$ and ${\partial \tilde{G}}/{\partial Z}= 1$ holds. Therefore the resulting equation is same as Eq. (\[N-eq-rel1\]).
\
(c) Finally we consider a case in which ${\partial \tilde{G}}/{\partial Z}\ne 1$, $$\begin{aligned}
&~& X_i = 2P_0 Q^i~,~~ X_{i+3}=Y_i=P_i~,~~ Z=P_0~,~~
\nonumber\\
&~& \tilde{G}= \Phi = Z^{2} - {\mbox{\boldmath $Y$}}^2 - m^2 c^2~.
\label{G-rel3}\end{aligned}$$ From Eq. (\[dphi/dZm1\]) each component of the factor $\tilde{g}^{(1)}_{ab}$ is determined as follows: $$\begin{aligned}
&~& \tilde{g}^{(1)}_{ij}=-\frac{X_iY_j-X_jY_i}{2Z^2}~, \nonumber \\
&~& \tilde{g}^{(1)}_{il}=\frac{1}{2}\delta_{i+3,l}~,
~~\tilde{g}^{(1)}_{li}=-\frac{1}{2}\delta_{l,i+3}~, \nonumber \\
&~& \tilde{g}^{(1)}_{lm}=0~,
\label{g-rel3}\end{aligned}$$ where $i,j = 1, 2, 3$ and $l,m = 4, 5, 6$. Although this is different from Eq. (\[gab-rel1\]), we obtain the same equation as Eq. (\[N-eq-rel1\]) again. This is because ${\partial (\tilde{f}, \tilde{H}, \tilde{G})}/{\partial (X_i, X_j, Z)}=0$ holds in this case.
Conclusions and future work {#Conclusions and future work}
===========================
We have given a variant formulation of Hamiltonian systems in terms of variables including redundant degrees of freedom. By use of a non-canonical transformation that enlarges the phase space from $(q, p)$ to $(x_1, x_2, \cdots, x_{N})$, we can reveal the Nambu mechanical structure hidden in a Hamiltonian system. The Hamiltonians required for the Nambu mechanical description are given by the Hamiltonian of the original system and constraints induced due to the consistency between the variables. Our formulation can be extended to many degrees of freedom systems and systems with first class constraints. Generalized forms of Nambu equation (Eqs. (\[N-eqf(m)\]) and (\[N-eqf(m)2\])) are required in some cases. Our approach to constrained systems is different from the preceding works [@BF; @CK; @R; @MS; @KT; @KT2; @LJ], i.e., we treat Nambu mechanics as effective mechanics, and we introduce not extra degrees of freedom but redundant degrees of freedom.
Our formulation is not just a change of description, but gives a new insight into the statistical or quantum mechanical treatment of Hamiltonian systems. For example, the Nambu equation (Eq. (\[H-eqf(gN)\])) could give a basis for a novel quantization scheme for a Hamiltonian system. In the present work, the constraints $(\tilde{G}_{1}, \tilde{G}_{2}, \cdots, \tilde{G}_{N})$ are [*unphysical*]{} ones and they all are set to zero. However, if we give them some appropriate values, Eq. (\[H-eqf(gN)\]) could provide semi-classical equations for quantum-mechanical expectation values [@OVP; @AH]. The non-vanishing $\tilde{G}_{b}$ come from quantum fluctuations, e.g., if we take $x=\langle\hat{q}\rangle$, $y=\langle\hat{p}\rangle$, and $z=\langle\hat{q}^{2}\rangle$, then $\tilde{G}=z-x^2$ has a non-zero value in general. The same argument holds for statistical-mechanical expectation values. Therefore we expect that the Nambu equation (Eq. (\[H-eqf(gN)\])) with non-vanishing $\tilde{G}_{b}$ could be a master equation for the statistical or quantum mechanics of Hamiltonian systems. More detailed studies will be presented in a future publication, and they might provide important clues for handling the statistical or quantum mechanics of Nambu systems [@Takh; @Gaut; @Chat; @Dito1; @Dito2; @Hoppe; @Awata; @Minic; @Kawamura; @Curt1; @Zach].
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by scientific grants from the Ministry of Education, Culture, Sports, Science and Technology under Grant Nos. 22540272 and 21244036 (Y.K.).
Derivation of the Nambu equation from the least action principle
================================================================
Let us derive the Nambu equation (Eq. (\[H-eqf(gN)\])) from a Hamiltonian system using the least action principle. Here we show the case with $N=3$, where the Nambu equations are given in the form of Eq. (\[N-eq\]).
Our starting point is the action integral such that $$\begin{aligned}
S = \int \left(p \frac{dq}{dt} - H(q, p)\right) dt
= \int \left(p(x,y,z) \frac{d}{dt}q(x,y,z) - \tilde{H}(x, y, z)\right) dt~,
\label{S}\end{aligned}$$ where $x=x(q,p)$, $y=y(q,p)$, $z=z(q,p)$, and we assume that the Hamiltonian $H$ can be expressed by $$\begin{aligned}
H(q, p) = \tilde{H}(x, y, z)~.
\label{H}\end{aligned}$$ As mentioned in Sect. 2.2, in general, $q$, $p$, and $H$ are not uniquely determined as functions of $x$, $y$, and $z$. Our goal is to obtain the equations of motion that hold independently of the expressions of $q$ and $p$.
Let us regard $x=x(t)$, $y=y(t)$, and $z=z(t)$ as independent variables, and consider their variation $x \to x + \delta x$, $y \to y + \delta y$, and $z \to z + \delta z$. Integrating by parts and ignoring the surface terms, the variation of $S$ can be written as $$\begin{aligned}
\delta S &=& \int
\left(\delta p \frac{dq}{dt} - \delta q \frac{dp}{dt} - \delta \tilde{H}\right) dt
\nonumber \\
&=& \int \left[\left(\frac{\partial p}{\partial x} \delta x
+ \frac{\partial p}{\partial y} \delta y
+ \frac{\partial p}{\partial z} \delta z\right)\frac{dq}{dt}
- \left(\frac{\partial q}{\partial x} \delta x
+ \frac{\partial q}{\partial y} \delta y
+ \frac{\partial q}{\partial z} \delta z\right)\frac{dp}{dt}\right.
\nonumber \\
&& ~~~~~ - \left.\left(\frac{\partial \tilde{H}}{\partial x} \delta x
+ \frac{\partial \tilde{H}}{\partial y} \delta y
+ \frac{\partial \tilde{H}}{\partial z} \delta z\right)\right] dt
\nonumber \\
&=& \int \left[\left(\frac{\partial p}{\partial x} \frac{dq}{dt}
- \frac{\partial q}{\partial x} \frac{dp}{dt}
- \frac{\partial \tilde{H}}{\partial x}\right) \delta x
+ \left(\frac{\partial p}{\partial y} \frac{dq}{dt}
- \frac{\partial q}{\partial y} \frac{dp}{dt}
- \frac{\partial \tilde{H}}{\partial y}\right) \delta y \right.
\nonumber \\
&& ~~~~~ \left.
+ \left(\frac{\partial p}{\partial z} \frac{dq}{dt}
- \frac{\partial q}{\partial z} \frac{dp}{dt}
- \frac{\partial \tilde{H}}{\partial z}\right) \delta z \right] dt~.
\label{deltaS}\end{aligned}$$ Imposing the least action principle $\delta S = 0$, we obtain the equations of motion $$\begin{aligned}
- \frac{\partial(q, p)}{\partial(x, y)} \frac{dy}{dt}
+ \frac{\partial(q, p)}{\partial(z, x)} \frac{dz}{dt}
&=& \frac{\partial \tilde{H}}{\partial x}~,
\nonumber\\
- \frac{\partial(q, p)}{\partial(y, z)} \frac{dz}{dt}
+ \frac{\partial(q, p)}{\partial(x, y)} \frac{dx}{dt}
&=& \frac{\partial \tilde{H}}{\partial y}~,
\nonumber \\
- \frac{\partial(q, p)}{\partial(z, x)} \frac{dx}{dt}
+ \frac{\partial(q, p)}{\partial(y,z)} \frac{dy}{dt}
&=& \frac{\partial \tilde{H}}{\partial z}~,
\label{eqs-xyz}\end{aligned}$$ where we use the chain rule for the derivative of a function $\tilde{u}(x,y,z)$, $$\begin{aligned}
\frac{d}{dt}\tilde{u}(x,y,z) =
\frac{\partial \tilde{u}}{\partial x}\frac{dx}{dt}
+ \frac{\partial \tilde{u}}{\partial y}\frac{dy}{dt}
+ \frac{\partial \tilde{u}}{\partial z}\frac{dz}{dt}~.
\label{chain}\end{aligned}$$
In the case in which $\partial(x, y)/\partial(q, p) \ne 0$, $q$ and $p$ can be expressed by $x$ and $y$, and then the following equations are derived: $$\begin{aligned}
\frac{dx}{dt}=
\frac{\partial \tilde{H}}{\partial y} \frac{\partial(x, y)}{\partial(q, p)}~,~~
\frac{dy}{dt}=
-\frac{\partial \tilde{H}}{\partial x} \frac{\partial(x, y)}{\partial(q, p)}~,~~
\label{eqs-xy}\end{aligned}$$ using the relations $$\begin{aligned}
\frac{\partial(q, p)}{\partial(x, y)} =
\left(\frac{\partial(x, y)}{\partial(q, p)}\right)^{-1}~,~~
\frac{\partial(q, p)}{\partial(z, x)} = 0~,~~
\frac{\partial(q, p)}{\partial(y, z)} = 0~.
\label{qp-xy}\end{aligned}$$ In the same way, for the case of $\partial(y, z)/\partial(q, p) \ne 0$, $q$ and $p$ are expressed by $y$ and $z$, and then the following equations are derived: $$\begin{aligned}
\frac{dy}{dt}=
\frac{\partial \tilde{H}}{\partial z} \frac{\partial(y, z)}{\partial(q, p)}~,~~
\frac{dz}{dt}=
-\frac{\partial \tilde{H}}{\partial y} \frac{\partial(y, z)}{\partial(q, p)}~,~~
\label{eqs-yz}\end{aligned}$$ and for the case of $\partial(z, x)/\partial(q, p) \ne 0$, we obtain $$\begin{aligned}
\frac{dz}{dt}=
\frac{\partial \tilde{H}}{\partial x} \frac{\partial(z, x)}{\partial(q, p)}~,~~
\frac{dx}{dt}=
-\frac{\partial \tilde{H}}{\partial z} \frac{\partial(z, x)}{\partial(q, p)}~.
\label{eqs-zx}\end{aligned}$$ Combining Eqs. (\[eqs-xy\]), (\[eqs-yz\]), and (\[eqs-zx\]), we can write down a set of equations, $$\begin{aligned}
\frac{dx}{dt}&=&
\frac{\partial \tilde{H}}{\partial y} \frac{\partial(x, y)}{\partial(q, p)}
- \frac{\partial \tilde{H}}{\partial z} \frac{\partial(z, x)}{\partial(q, p)}~,~~
\label{eqs-xyz2x}\\
\frac{dy}{dt}&=&
\frac{\partial \tilde{H}}{\partial z} \frac{\partial(y, z)}{\partial(q, p)}
-\frac{\partial \tilde{H}}{\partial x} \frac{\partial(x, y)}{\partial(q, p)}~,~~
\label{eqs-xyz2y}\\
\frac{dz}{dt}&=&
\frac{\partial \tilde{H}}{\partial x} \frac{\partial(z, x)}{\partial(q, p)}
-\frac{\partial \tilde{H}}{\partial y} \frac{\partial(y, z)}{\partial(q, p)}~,
\label{eqs-xyz2z}\end{aligned}$$ which is consistent with every expression of $q$ and $p$. For example, in the case in which $q=q(x,y)$ and $p=p(x,y)$ ($\partial(x, y)/\partial(q, p) \ne 0$), Eqs. (\[eqs-xyz2x\])–(\[eqs-xyz2y\]) are reduced to Eq. (\[eqs-xy\]), and Eq. (\[eqs-xyz2z\]) is equivalent to Hamilton’s equation of motion.
Introducing a function $\tilde{G}=\tilde{G}(x, y, z)$ that satisfies the conditions (\[xyzG\]), Eqs. (\[eqs-xyz2x\])–(\[eqs-xyz2z\]) are rewritten as Nambu equations in the form of Eq. (\[N-eq\]), $$\begin{aligned}
\frac{d x}{dt} = \frac{\partial (\tilde{H}, \tilde{G})}{\partial (y, z)}~,~~
\frac{d y}{dt} = \frac{\partial (\tilde{H}, \tilde{G})}{\partial (z, x)}~,~~
\frac{d z}{dt} = \frac{\partial (\tilde{H}, \tilde{G})}{\partial (x, y)}~.
\label{N-eqS}\end{aligned}$$ These equations hold independently of the expression of $q$ and $p$.
Hidden Nambu systems in Nambu systems
=====================================
Let us formulate a Nambu mechanical system with an $N$-plet $x_i$ $(i=1, \cdots, N)$ using $N+r$ variables $y_j=y_j(x_1, \cdots, x_N)$ $(j=1, \cdots, N+r)$, where $r$ is a positive integer. We assume that at least $r+1$ of $\{y_{j_1}, \cdots, y_{j_{N}}\}_{\mbox{\tiny{NB}}}$ ($j_{1}, \cdots, j_{N}=1, \cdots, N+r$) do not vanish, where $\{y_{j_1}, \cdots, y_{j_{N}}\}_{\mbox{\tiny{NB}}}$ is the Nambu bracket defined by Eq. (\[NB-N\]). Then the equation for any function $\tilde{f}(y_1, \cdots, y_{N+r}) = f(x_1, \cdots, x_N)$ can be written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} &=&
\frac{\partial(f, H_1, \cdots, H_{N-1})}{\partial(x_1, x_2, \cdots, x_{N})}
\nonumber \\
&=& \frac{1}{N!} \sum_{j_1, j_2, \cdots, j_{N}=1}^{N+r}
\frac{\partial (\tilde{f}, \tilde{H}_1, \cdots, \tilde{H}_{N-1})}
{\partial (y_{j_1}, y_{j_2}, \cdots, y_{j_{N}})}
\{y_{j_1}, y_{j_2}, \cdots, y_{j_{N}}\}_{\mbox{\tiny{NB}}}~,
\label{N-eq(N)}\end{aligned}$$ where $\tilde{H}_a(y_1, \cdots, y_{N+r})
= H_a(x_1, \cdots, x_N)$ $(a=1, \cdots, N-1)$.
Introducing functions $\tilde{G}_c(y_1, \cdots, y_{N+r}) = G_c(x_1, \cdots, x_N)$ $(c=1, \cdots, r)$ that satisfy the relation $$\begin{aligned}
\frac{1}{r!}\sum_{j_{N+1}, \cdots, j_{N+r}=1}^{N+r}
\varepsilon_{j_1 j_2 \cdots j_N j_{N+1} \cdots j_{N+r}}
\frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_r)}
{\partial (y_{j_{N+1}}, \cdots, y_{j_{N+r}})}
=\{y_{j_1}, y_{j_2}, \cdots, y_{j_{N}}\}_{\mbox{\tiny{NB}}}~,
\label{yjG}\end{aligned}$$ Eq. (\[N-eq(N)\]) is rewritten in the form of the Nambu equation, $$\begin{aligned}
\frac{d \tilde{f}}{dt}
= \frac{\partial(\tilde{f}, \tilde{H}_1, \cdots, \tilde{H}_{N-1},
\tilde{G}_1, \cdots, \tilde{G}_r)}
{\partial(y_1, y_2, \cdots, y_N, y_{N+1}, \cdots, y_{N+r})}~,
\label{N-eqf(N+1)}\end{aligned}$$ where we use the formula for any functions $\tilde{A}_j=\tilde{A}_j(y_1, \cdots, y_{N+r})$, $$\begin{aligned}
\frac{\partial (\tilde{A}_1, \tilde{A}_2, \cdots, \tilde{A}_N,
\tilde{A}_{N+1}, \cdots, \tilde{A}_{N+r})}
{\partial (y_1, y_2, \cdots, y_N, y_{N+1}, \cdots, y_{N+r})}
&=& \!\!\!
\frac{1}{N!~r!}
\sum_{j_1, j_2, \cdots, j_{N}, j_{N+1}, \cdots, j_{N+r}=1}^{N+r}
\varepsilon_{j_1 j_2 \cdots j_N j_{N+1} \cdots j_{N+r}}
\nonumber \\
&&\!\!\!\!\!\!\!
\times~
\frac{\partial (\tilde{A}_1, \tilde{A}_2, \cdots, \tilde{A}_N)}
{\partial (y_{j_1}, y_{j_2}, \cdots, y_{j_N})}
\frac{\partial (\tilde{A}_{N+1}, \cdots, \tilde{A}_{N+r})}
{\partial (y_{j_{N+1}}, \cdots, y_{j_{N+r}})}~.
\label{J(N)}\end{aligned}$$ By the use of Eq. (\[yjG\]), it can be shown that the Nambu bracket between $G_c(x_1, \cdots, x_N)=\tilde{G}_c(y_1, \cdots, y_{N+r})$ and the arbitrary functions $u_a(x_1, \cdots, x_N)=\tilde{u}_a(y_1, \cdots, y_{N+r})$ $(a=1, \cdots, N-1)$ vanishes such that $$\begin{aligned}
\{G_c, u_1, \cdots, u_{N-1}\}_{\mbox{\tiny{NB}}}
&=& \frac{1}{N!} \sum_{j_1, j_2, \cdots, j_N=1}^{N+r}
\frac{\partial (\tilde{G}_c, \tilde{u}_1, \cdots, \tilde{u}_{N-1})}
{\partial (y_{j_1}, y_{j_2}, \cdots, y_{j_{N}})}
\{y_{j_1}, y_{j_2}, \cdots, y_{j_{N}}\}_{\mbox{\tiny{NB}}}
\nonumber \\
&=& \frac{1}{N!~r!}
\sum_{j_1, j_2, \cdots, j_N, j_{N+1}, \cdots, j_{N+r}=1}^{N+r}
\varepsilon_{j_1 j_2 \cdots j_N j_{N+1} \cdots j_{N+r}}
\nonumber \\
&&~~~~~~~~~~~~~
\times~
\frac{\partial (\tilde{G}_c, \tilde{u}_1, \cdots, \tilde{u}_{N-1})}
{\partial (y_{j_1}, y_{j_2}, \cdots, y_{j_{N}})}
\frac{\partial (\tilde{G}_{1}, \cdots, \tilde{G}_{r})}
{\partial (y_{j_{N+1}}, \cdots, y_{j_{N+r}})}
\nonumber \\
&=& \frac{\partial (\tilde{G}_c, \tilde{u}_1, \cdots, \tilde{u}_{N-1},
\tilde{G}_1, \cdots, \tilde{G}_r)}
{\partial (y_{1}, y_{2}, \cdots, y_{N}, y_{N+1}, \cdots, y_{N+r})}=0~.
\label{Gu=0-N}\end{aligned}$$ Hence $G_c$ are constants. We can eliminate the constants by redefining $G_c$, and the resulting $\tilde{G}_c(y_1,\cdots, y_{N+r})=0$ can be regarded as [*induced constraints*]{}, which are associated with enlarging the phase space from $(x_1, \cdots, x_{N})$ to $(y_1, \cdots, y_{N+r})$.
In this way, [*Nambu systems with an $N$-plet $x_i$ $(i=1, \cdots, N)$ can be formulated as Nambu systems with an $N+r$-plet $y_j=y_j(x_1, \cdots, x_N)$ $(j=1, \cdots, N+r)$.*]{}
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[^1]: The canonical transformation generated by the generator $G = \lambda \phi$ is the gauge transformation, and the infinitesimal one is given by $\delta q_r = ({\partial{G}}/{\partial p_r}) \delta \xi$ and $\delta p_r = -({\partial{G}}/{\partial q_r}) \delta \xi$ $(r=1, 2)$. Here $\lambda$ is an arbitrary function of the canonical variables and $\delta \xi$ is an infinitesimal parameter. If we take $\lambda = (p_2 - \chi)/(({\partial{\phi}}/{\partial q_2})\xi)$ using a finite parameter $\xi$, $p_2$ is transformed into $P_2 = \chi$ under the constraint $\phi=0$. In the same way, one of the canonical variables can be eliminated by an appropriate canonical transformation. The variable to be eliminated depends on the physical systems.
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abstract: 'In this work, we study the [A$_{x}$Fe$_{2-y}$Se$_2$]{} (A=K, Rb) superconductors using angle-resolved photoemission spectroscopy. In the low temperature state, we observe an orbital-dependent renormalization for the bands near the Fermi level in which the [$d_{xy}$]{} bands are heavily renormliazed compared to the [$d_{xz}$/$d_{yz}$]{} bands. Upon increasing temperature to above 150K, the system evolves into a state in which the [$d_{xy}$]{} bands have diminished spectral weight while the [$d_{xz}$/$d_{yz}$]{} bands remain metallic. Combined with theoretical calculations, our observations can be consistently understood as a temperature induced crossover from a metallic state at low temperature to an orbital-selective Mott phase (OSMP) at high temperatures. Furthermore, the fact that the superconducting state of [A$_{x}$Fe$_{2-y}$Se$_2$]{} is near the boundary of such an OSMP constraints the system to have sufficiently strong on-site Coulomb interactions and Hund’s coupling, and hence highlight the non-trivial role of electron correlation in this family of iron superconductors.'
author:
- 'M. Yi'
- 'D.H. Lu'
- 'R. Yu'
- 'S. C. Riggs'
- 'J.-H. Chu'
- 'B. Lv'
- 'Z. Liu'
- 'M. Lu'
- 'Y.-T. Cui'
- 'M. Hashimoto'
- 'S.-K. Mo'
- 'Z. Hussain'
- 'C. W. Chu'
- 'I.R. Fisher'
- 'Q. Si'
- 'Z.-X. Shen'
title: 'Observation of Temperature-Induced Crossover to an Orbital-Selective Mott Phase in [A$_{x}$Fe$_{2-y}$Se$_2$]{} (A=K, Rb) Superconductors'
---
Electron correlation remains a central focus in the study of high temperature superconductors. The strongly correlated cuprate superconductors are understood as doped Mott insulators (MI) [@1LeePA06] while the iron-based superconductors (FeSCs) have been found to be at most moderately correlated [@2LuDH08; @3YangWL09; @4Qazibash09]. Even though the low energy electronic structures of different FeSCs families share the common Fe $3d$ bands, there are systematic variations in their physical properties, such as ordered magnetic moment and effective mass [@5YinZP11]. In general, electron correlation is the weakest in iron phosphides with relatively low mass renormalization [@2LuDH08], and moderate in the more extensively studied iron arsenides [@2LuDH08; @3YangWL09]. The Fe(Te,Se) chalcogenide family, in comparison, seems to harbor stronger correlation as inferred from larger ordered moment, yet metallic resistivity is still observed [@6Liu10]. The newest iron chalcogenide superconductors, [A$_{x}$Fe$_{2-y}$Se$_2$]{} (A=alkali metal) [@7Guo10; @8Krzton11; @9Li11; @10Fang11; @11Wang11] (AFS) hints at even stronger correlation with a large observed moment of 3.3[$\mu_{\tiny{\textrm{B}}}$]{} [@12Wei11] and insulating transport behavior in the phase diagram.
Another important factor in understanding the FeSCs lies in their multi-orbital nature. In such a system, orbital-dependent behavior as well as competition between inter- and intra-orbital interactions could play a critical role in determining their physical properties. One example is the orbital anisotropy that onsets with the in-plane symmetry breaking structural transition as observed in the underdoped arsenides [@13Yi11]. Another example is the different pairing symmetry that could arise considering the relative strength of inter- and intra-orbital interactions extensively studied theoretically [@14chubukov12]. Theoretical models have considered correlation effects in the bad metal regime in terms of an incipient Mott picture [@5YinZP11; @15si08], and the proximity to the Mott transition may be orbital-dependent even for orbitally-independent Coulomb interactions [@16Yu11; @17Yu11b; @18Zhou11; @19craco11]. What arises from the model is an orbital selective Mott phase (OSMP), in which some orbitals are Mott-localized while others remain itinerant. First introduced in the context of the Ca$_{2-x}$Sr$_x$RuO$_4$ system, an OSMP may result from both the orbital-dependent kinetic energy and the combined effects of the Hund’s coupling and crystal level splittings [@20anisimov02; @21demedici]. An OSMP links naturally with models of coexisting itinerant and localized electrons that have been proposed to compensate for the shortcomings of both strong coupling and weak coupling approaches [@22You11; @23moon10]. However, to date, there has been no experimental evidence for OSMP in any FeSC.
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In this paper, we present angle-resolved photoemission spectroscopy (ARPES) data from two superconducting AFSs, [K$_{x}$Fe$_{2-y}$Se$_2$]{} (KFS) and [Rb$_{x}$Fe$_{2-y}$Se$_2$]{} (RFS), with $T_C$ of 32K and 31K, respectively, as well as insulating and intermediate dopings (see SI). We observe the superconducting AFSs undergoing a temperature-induced crossover from a metallic state in which all three $t_{2g}$ orbitals-[$d_{xy}$]{}, [$d_{xz}$]{} and [$d_{yz}$]{} are present near the Fermi level ([$E_F$]{}) to a state in which the [$d_{xy}$]{} bands has diminished spectral weight while the [$d_{xz}$/$d_{yz}$]{} bands remain metallic. In addition, the intermediate doping shows stronger correlation than the superconducting doping, as seen in the further renormalization of the [$d_{xz}$/$d_{yz}$]{} bands and the much more suppressed [$d_{xy}$]{} intensity, while the insulating doping has no spectral weight near [$E_F$]{} in any orbitals. From comparison with our theoretical calculations, the ensemble of our observations are most consistent with the understanding that the presence of strong Coulomb interactions and Hund’s coupling places the superconducting AFSs near an OSMP at low temperatures, and crosses over into the OSMP via raised temperature, while the intermediate and insulating compounds are on the boundary of the OSMP and in the MI phase, respectively, suggesting the importance of electron correlation in this family of FeSCs.
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The low temperature electronic structure of [K$_{x}$Fe$_{2-y}$Se$_2$]{}is shown in Fig. \[fig:fig1\]. The Fermi surface (FS) of KFS consists of large electron pockets at the Brillouin zone (BZ) corner-X point, and a small electron pocket at the BZ center-$\Gamma$ point (Fig. \[fig:fig1\](a)), consistent with previous ARPES reports [@24zhang11; @25qian11; @26mou11]. For the crystallographic 2-Fe unit cell, LDA calculations predict two electron pockets at the X point [@27nekrasov11] (Fig. \[fig:fig1\](i)). While FS map appears to show only one electron pocket at X, measurements under different polarizations (Fig. \[fig:fig1\](j)-(l)) reveal the expected two electron bands with nearly degenerate Fermi crossings ([$k_F$]{}) but different band bottom positions-a shallower one around -0.05eV and a deeper one that extends to the top of the [$d_{xz}$/$d_{yz}$]{} hole-like bands at -0.12eV. The Luttinger volume of the two electron pockets gives $\sim$0.16 electrons per Fe. Considering that the C$_4$ symmetry of the crystal dictates degeneracy of the [$d_{xz}$/$d_{yz}$]{} electron band bottom and hole band top at X, the shallower electron band that is not degenerate with the hole-like band at higher binding energy is most likely of [$d_{xy}$]{} character, and the deeper one [$d_{xz}$]{} along $\Gamma$-X and [$d_{yz}$]{} along the perpendicular direction. This observed orbital character assignment seems to contradict the LDA prediction of the shallower electron band as [$d_{xz}$/$d_{yz}$]{} and the deeper one [$d_{xy}$]{} in FeSC [@28graser10] (Fig. \[fig:fig1\](i)), as observed in Co-doped [BaFe$_2$As$_2$]{} (Fig. \[fig:fig1\](f)-(g)). However, this assignment can be understood if we consider the KFS band structure as a whole. Three filled hole bands are observed near the $\Gamma$ point (Figs. \[fig:fig1\](d)-(e)), where the two lower ones can be identified as [$d_{xz}$/$d_{yz}$]{} and the higher one [$d_{xy}$]{}. Interestingly, the [$d_{xy}$]{} band is far more renormalized ($\sim$a factor of 10) compared to LDA than the [$d_{xz}$/$d_{yz}$]{} bands ($\sim$a factor of 3), indicating stronger correlation for the [$d_{xy}$]{} orbital. This is in contrast to the Co-doped [BaFe$_2$As$_2$]{} band structure, in which all orbitals are renormalized by roughly the same factor ($\sim$2) compared to LDA. Hence, our assignment of [$d_{xy}$]{} character to the shallower electron band is consistent with strong orbital-dependent renormalization, which brings the deeper [$d_{xy}$]{} electron band at X predicted by LDA to be shallower than the [$d_{xz}$/$d_{yz}$]{} band. This orbital-dependent renormalization behavior also emerges from our theoretical calculations as will be discussed later.
As the electronic structure at [$E_F$]{} is dominated by the large electron pockets at X, we focus on these features. Fig. \[fig:fig2\] shows a temperature-dependent study of these bands taken in two polarization geometries. In Fig. \[fig:fig2\](a), the matrix element for [$d_{xy}$]{} is stronger than [$d_{xz}$]{}. At low temperatures, [$d_{xy}$]{} band is clearly resolved with weaker intensity for the [$d_{xz}$]{} band. With increasing temperature, the spectral weight of the [$d_{xy}$]{} band noticeably diminishes, eventually revealing the remaining [$d_{xz}$]{} band at high temperatures. In Fig. \[fig:fig2\](b), the deeper [$d_{yz}$]{} band has more intensity than the corresponding [$d_{xz}$]{} band in Fig. \[fig:fig2\](a) while the presence of the [$d_{xy}$]{} band is still very noticeable from the increased intensity where they significantly overlap, as well as from the energy distribution curves (EDCs) shown in Fig. \[fig:fig2\](c). With increasing temperature, again, the spectral weight of the [$d_{xy}$]{} band diminishes, leaving only the [$d_{yz}$]{} band at high temperatures, which clearly has a deeper band bottom than the shallow [$d_{xy}$]{} band (Fig. \[fig:fig2\](c)). In addition, we have artificially introduced a 210K thermal broadening to the 30K spectra as shown in the last column of Fig. \[fig:fig2\]. The clear contrast to the 210K data rules out a trivial thermal broadening as an origin for the observed diminishing of [$d_{xy}$]{} spectral weight.
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To capture this behavior quantitatively, we analyze the temperature dependence of the EDCs at the X-point, stacked in Fig. \[fig:fig3\](c). At all temperatures, there is a large hump background corresponding to the large hole-like dispersion at high binding energy. At low temperatures, there is an additional peak around -0.05eV corresponding to the [$d_{xy}$]{} band bottom. We fit these EDCs with a Gaussian background for the large hump feature and a Lorentzian peak for the [$d_{xy}$]{} band. The integrated spectral weight for the fitted [$d_{xy}$]{} peak is plotted in Fig. \[fig:fig3\](d), which decreases toward zero with increasing temperature, seen as a non-trivial drop between 100K and 200K. As an independent check against trivial thermal effect, we choose small regions in the raw spectral image (marked in Fig. \[fig:fig3\](a)-(b)) dominated by [$d_{xy}$]{} (blue), [$d_{yz}$]{} (green), and mixed [$d_{xy}$]{} and [$d_{xz}$/$d_{yz}$]{} (magenta/cyan) characters and plot their integrated intensities as a function of temperature (Fig. \[fig:fig3\](e)). The spectral weight from [$d_{xy}$]{}-dominated region rapidly decreases, consistent with the fitted result in Fig. \[fig:fig3\](d), while that of [$d_{yz}$]{}-dominated region does not drop in a similar manner. The regions of mixed orbitals show a slower diminishing spectral weight compared to that for [$d_{xy}$]{}, reflecting the combined contributions from both [$d_{xz}$/$d_{yz}$]{} and [$d_{xy}$]{} orbitals. Although this method has the uncertainty of small leakage of other orbitals into the chosen regions, which is the cause of the finite residual value for the [$d_{xy}$]{} curve, the contrasting behavior of the [$d_{xy}$]{} versus [$d_{xz}$/$d_{yz}$]{} orbitals is clearly demonstrated. A temperature cycle test was performed to exclude the possibility of sample aging (see SI). Measurements on the sister compound RFS reveal similar behavior (SI), suggesting the generality in this family of superconductors. This observation of a selected orbital that loses coherent spectral weight while the others remain metallic is reminiscent of a crossover into an OSMP in which selected orbitals become Mott localized while others remain metallic.
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To further understand this phenomenon, we perform theoretical calculations based on a five-orbital Hubbard model to study the metal-to-insulator transition (MIT) in the paramagnetic phase using a slave-spin mean-field method [@30yu; @31demedici05]. At commensurate electron filling n=6 per Fe (corresponding to Fe2$^+$ of the parent FeSC), we find the ground state of the system to be a metal, an OSMP or a MI depending on the intra-orbital repulsion U and the Hund’s coupling J. Furthermore, the MIT can be triggered by increasing temperature (Fig. \[fig:fig4\](a)) due to the larger entropy of the insulating phase. At a fixed interaction strength (within a certain range, Fig. \[fig:fig4\](a)), the system goes from a metal to an OSMP and then to a MI with increasing temperature. The MI phase is suppressed by electron doping. By contrast, the OSMP can survive, as shown in Fig. \[fig:fig4\](b) for n=6.15, which roughly corresponds to the filling of the superconducting state from ARPES measurements. From the evolution of the orbitally resolved quasiparticle spectral weight, $Z_{\alpha}$, as a function of the temperature (Fig. \[fig:fig4\](c)), we show that the OSMP corresponds to the [$d_{xy}$]{} orbital being Mott localized ($Z=0$) and the rest of the $3d$ orbitals remaining delocalized ($Z>0$). This result originates primarily from a combined effect of the orbital dependence of the projected density of states and the interplay between the Hund’s coupling and crystal level splitting (see SI and Ref. ).
To compare with the ARPES measurements, we have calculated the quasiparticle spectral function A(k,E) in the 2-Fe BZ. At low temperature and in the non-interacting limit U=0 (Fig. \[fig:fig4\](d)), the electronic structure of the model agrees well with that from LDA, with the [$d_{xy}$]{} band deeper than the [$d_{xz}$/$d_{yz}$]{} band at X. This order switches with sufficiently strong interaction. At U=3.75 eV (Fig. \[fig:fig4\](e)), the [$d_{xy}$]{}-dominated bands are pushed above their [$d_{xz}$/$d_{yz}$]{} counterparts by the strongly orbital-dependent mass renormalization, as reflected in the orbital dependent quasiparticle spectral weights (Fig. \[fig:fig4\](c)). The mass renormalization is the largest for the [$d_{xy}$]{} orbital ($\sim$10), and smaller for the [$d_{xz}$/$d_{yz}$]{} orbitals ($\sim$3), which is compatible with the low-temperature ARPES results (Fig. \[fig:fig1\](e)). The temperature-induced crossover to the OSMP is clearly seen from the suppression of the spectral weights in the [$d_{xy}$]{} orbital that accompanies the reduction of the weights in the other orbitals (Fig. \[fig:fig4\](c),(e)-(g)), in agreement with the ARPES results (Fig. \[fig:fig3\]).
The temperature-induced nature of the crossover constrains these AFS superconductors to be very close to the boundary of the OSMP in the zero temperature ground state, which is also the superconducting state. The best agreement between theory and experiments is achieved at U$\sim$3.75 eV. While the absolute value of this interaction is sensitive to the parameterization of the crystal levels and Hund’s coupling, it is instructive to make a qualitative comparison with the case of the iron pnictides. The enhanced correlation effects for AFS tracks the reduction of the width of the (U=0) [$d_{xy}$]{} band, which is about 0.7 of its counterpart in 1111 iron arsenides.
One known concern for the AFS materials is the existence of mesoscopic phase separation into superconducting and insulating regions [@32li12], which would both contribute spectral intensity in ARPES data. From measurement on an insulating RFS sample (Fig. \[fig:figsi4\](d)), we see negligible spectral weight and no well-defined dispersions within 0.1eV from [$E_F$]{}, as expected for an insulator. Hence, the insulating regions in the superconducting compounds do not contribute spectral weight to the near-[$E_F$]{} energy range, in which the temperature-induced crossover is observed. We have also measured a KFS sample whose resistivity is intermediate between superconducting and insulating (Fig. \[fig:figsi4\](b)), and was previously proposed to be semiconducting containing both metallic and insulating phases [@33chen11]. Interestingly, its [$d_{xz}$/$d_{yz}$]{} bands, which must come from the metallic phase, appear further renormalized by a factor of 1.3 compared with those of the superconducting compounds. In addition, we resolve small but finite spectral weight for a very shallow [$d_{xy}$]{} electron band near the X-point (Fig. \[fig:figsi5\](c)). As expected, the peak position is even closer to [$E_F$]{}, consistent with additional renormalization for the shallow band near X, which is harder to discern with temperature dependent study. These observations are consistent with the OSMP picture in that that the metallic phase in this KFS compound is likely even closer to the boundary of OSMP at low temperatures from the mass-diverging behavior of the [$d_{xy}$]{} bands. For the same interaction strength, calculation from our model also identifies the low temperature ground state of the superconducting, intermediate, and insulating phases to be located close to an OSMP, just at the boundary of an OSMP, and in a MI phase, respectively (see SI).
While we cannot completely rule out alternative explanations for the observations presented above, the consistency of the totality of the observations-including strongly orbital dependent band renormalization for [$d_{xy}$]{} versus [$d_{xz}$/$d_{yz}$]{} at both $\Gamma$ and X points in the low temperature metallic state, the non-trivial temperature-dependent spectral weight change for only the [$d_{xy}$]{} band, systematic doping dependence of the related effects in the intermediate and insulating compounds-and the theoretical calculations makes this understanding a most likely scenario, suggesting that the superconductivity in this AFS family exists in close proximity to Mott behavior.
We thank V. Brouet, W. Ku, B. Moritz and I. Mazin for helpful discussions. ARPES experiments were performed at the Stanford Synchrotron Radiation Lightsource and the Advanced Light Source, which are both operated by the Office of Basic Energy Science, U.S. Department of Energy. The work at Stanford is supported by DOE Office of Basic Energy Science, Division of Materials Science and Engineering, under contract DE-AC02-76SF00515. The work at Rice has been supported by NSF Grant DMR-1006985 and the Robert A. Welch Foundation Grant No. C-1411. The work at Houston is supported in part by US Air Force Office of Scientific Research contract FA9550-09-1-0656, and the state of Texas through the Texas Center for Superconductivity at the University of Houston. MY thanks the NSF Graduate Research Fellowship Program for financial support.
[99]{}
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Supplementary Information
=========================
Materials and methods
---------------------
Single crystals of KFS were grown by the following method. Precursor FeSe was synthesized using Se powder (Alfa 99.999$\%$) and Fe powder (99.998$\%$) in a 1:1 molar ratio. The reagents were weighed and placed in a 2mL alumina crucible. The quartz tube was sealed after being flushed with argon and evacuated. The sealed quartz tube was placed in a furnace and heated from room temperature to 1050$^\circ$C in 12 hours. The furnace remained at 1050$^\circ$C for 12 hours, then was shut off and cooled to room temperature. Single crystals of KFS were obtained by a self-flux method with mixtures of K (Alfa, 99.95$\%$), and FeSe in molar ratios of 1:3 respectively. Potassium has a low boiling point so a small amount of additional K needs to be incorporated into the growth during synthesis. The reagents were weighed and placed in a 2mL alumina crucible, which was then sealed in a 2mL quartz tube after being flushed with argon and evacuated. The 2mL quartz tube was then placed into a larger 5ml quartz tube and sealed after again being flushed with argon and evacuated. The double sealed quartz tube technique is employed because potassium attacks quartz. The sealed quartz tubes were placed in a furnace and heated from room temperature to 1040$^\circ$C over the course of 4 hours. After dwelling for 2 hours, the furnace was cooled to 850$^\circ$C. The quartz tube was then removed from the furnace, rotated 180$^\circ$C, and spun in a centrifuge for a few seconds to separate the self-flux from the single crystals. Crystals with dimensions up to approximately 3mm x 3mm x 50mm could readily be extracted from the crucible. The crystals have a platelike morphology, with the c-axis perpendicular to the plane. Two types of KFS were studied, one superconducting with TC onset of 32K and chemical composition as K$_{0.76}$Fe$_{1.72}$Se$_2$, and the other non-superconducting with composition as K$_{0.76}$Fe$_{1.78}$Se$_2$. Single crystals of RFS were grown as described elsewhere [@s1gooch]. Two types of RFS were studied: one superconducting with T$_C$ onset of 31K and chemical composition Rb$_{0.93}$Fe$_{1.70}$Se$_2$; and the other insulating with composition Rb$_{0.90}$Fe$_{1.78}$Se$_2$.
As the crystals are sensitive to air, all sample preparations were preformed inside a nitrogen-flushed glove box. ARPES measurements were carried out at beamline 10.0.1 of the Advanced Light Source and beamline 5-4 of Stanford Synchrotron Radiation Lightsource using SCIENTA R4000 electron analyzers. The total energy resolution used was 25 meV or better and the angular resolution was 0.3$^\circ$C. Single crystals were cleaved in situ at low temperatures and measured in an ultra high vacuum chamber with a base pressure better than $3x10^{-11}$ Torr.
Discussion of orbital assignment
--------------------------------
The common band structure for FeSC in the Brillouin zone (BZ) corresponding to the two-Fe unit cell consists of three hole bands near the $\Gamma$ point and two electron bands at the X point [@s2graser]. Under C$_4$ rotational symmetry, the [$d_{xz}$]{} and [$d_{yz}$]{} bands are degenerate at the in-plane high symmetry points of the BZ-$\Gamma$ and X points. At the $\Gamma$ point, this means that the [$d_{xz}$]{} and [$d_{yz}$]{} hole bands must be degenerate. Hence we have assigned the two lower degenerate hole bands in Fig. \[fig:fig1\]e to [$d_{xz}$/$d_{yz}$]{} and the higher one to [$d_{xy}$]{}. At the X point, C$_4$ symmetry dictates the bottom of the [$d_{xz}$/$d_{yz}$]{} electron band and the top of the [$d_{xz}$/$d_{yz}$]{} hole band be degenerate. Since the shallow electron band is separated from the hole band by more than 70meV, it is unlikely to be the [$d_{xz}$/$d_{yz}$]{} electron band, but is instead the [$d_{xy}$]{} band.
There is in principle the possibility of an alternative understanding. As described in Ref. [@s4Lin; @s4Brouet], the alternating arsenic atoms about the iron plane may induce parity-switching of certain orbitals when folding from the 1-Fe BZ to the 2-Fe BZ, which would swap the orbital characters of the [$d_{xz}$]{} electron band and the [$d_{xy}$]{} electron band at X along $\Gamma$-X direction, making the originally [$d_{xz}$]{} electron band to be more observable under an odd polarization (Fig. \[fig:fig1\]c). Under this understanding, the shallow band is still the [$d_{xz}$]{} band rather than the [$d_{xy}$]{} band. However, for this understanding to hold, the aforementioned degeneracy of the [$d_{xz}$/$d_{yz}$]{} bands must be lifted to account for the 70meV gap between the shallow electron band and the hole band at X.
There are two mechanisms that could lift this degeneracy. One is spin-orbit coupling. However, in this compound, the magnitude of the spin-orbit coupling is not strong enough to cause such a large gap. Moreover, the effect of the spin-orbit coupling is stronger at the $\Gamma$ than at the X point [@s3mazin], and should cause a larger gap between the [$d_{xz}$/$d_{yz}$]{} hole bands at $\Gamma$, which is not observed. The second mechanism is in-plane symmetry breaking, which causes anisotropic shift of the [$d_{xz}$/$d_{yz}$]{} bands and hence lift the degeneracy of these two orbitals, as has been observed in the orthorhombic state of [BaFe$_2$As$_2$]{} [@s4Yi11]. Up to date, there has not yet been any report of in-plane symmetry breaking in the AFS superconductors. In addition, one should note that the orthorhombicity causes the existence of natural twinning in the crystals. Hence in an unstressed crystal, the ARPES signal is a combination of dispersions from both types of domains, and would see bands from $\Gamma$-X and $\Gamma$-Y directions simultaneously, as was observed in [BaFe$_2$As$_2$]{} [@s4Yi11]. For the microscopically phase separated AFS superconductors, such twinning effect should be even more noticeable than [BaFe$_2$As$_2$]{}, but is not observed in the dispersions of AFS superconductors.
Another factor that should be considered is the comparison between $\Gamma$ and X. The flat [$d_{xy}$]{} hole band observed at $\Gamma$ is renormalized by a factor of $\sim$10 compared to bare LDA while the [$d_{xz}$/$d_{yz}$]{} hole bands are renormalized by a factor of $\sim$3. When such kind of orbital-dependent renormalization is considered for the bands at X, it would bring the [$d_{xy}$]{} electron band to be shallower than the [$d_{xz}$/$d_{yz}$]{} electron bands, which is more consistent with the assignment of the shallower electron band to [$d_{xy}$]{}. Further support for this assignment comes from the general consideration that a stronger mass renormalization would imply a stronger temperature dependence of the corresponding quasiparticle spectral weight, as has been shown in the main text.
To summarize, while the possibility in principle exists for other unknown mechanisms and alternative explanations, given the totality of the observations and considerations given above, the understanding of the shallow electron band as [$d_{xy}$]{} seems to be the most consistent overall.
Temperature cycle test
----------------------
To test that the disappearance of the [$d_{xy}$]{} orbital spectral weight with raised temperature is not due to sample aging, we have performed a temperature cycle test in which we cleaved the sample at 10K and took measurements as temperature is raised up to 300K and cooled back down again. Spectra images for selected temperatures as well as the EDC at X point in this temperature cycle are shown in Fig. \[fig:figsi1\], where we see that the spectral weight for the [$d_{xy}$]{} orbital dominated shallow electron band diminishes at higher temperatures, and recovers when cooled back down. This shows that the observed temperature-induced crossover is not due to any surface effects and is a robust bulk phenomenon.
![\[fig:figsi1\]Temperature cycle test on [K$_{x}$Fe$_{2-y}$Se$_2$]{} (a) Spectral images across X point measured at selected temperatures in the temperature cycle from 10K to 300K to 40K. Polarization geometry and photon energy is the same as that in Fig. \[fig:fig2\](a). (b) EDC at X point taken in the temperature cycle, where the peak around -0.05eV is the [$d_{xy}$]{}band bottom, whose spectral weight diminishes up to 300K and recovers when cooled back down.](FigSI1){width="45.00000%"}
Results on [Rb$_{x}$Fe$_{2-y}$Se$_2$]{}
---------------------------------------
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We have done similar measurements on superconducting RFS as discussed in the main text for superconducting KFS. Fig. \[fig:figsi2\] summarizes the main results. RFS has similar electronic structure as KFS, including Fermi surface, band dispersions and their orbital characters, specifically there is a shallow electron band at X that is [$d_{xy}$]{} and a deep electron band that is [$d_{xz}$/$d_{yz}$]{}. Fig. S2d-f shows the temperature dependence of the electron bands, equivalent to Fig. \[fig:fig2\](a) for the KFS. Here again we see that the shallow [$d_{xy}$]{} band is present at low temperatures, and its spectral weight diminishes with raised temperature. Similar quantitatively analysis is also done on the RFS. Fig. \[fig:figsi2\](g) shows the EDC at X point taken in the geometry of Fig. \[fig:figsi2\](d)-(f), taken from 10K to 200K. Similar to the case of KFS, the $\sim$-0.05eV hump shaded by blue is the [$d_{xy}$]{} band bottom, whose spectral weight diminishes with raised temperature. The EDCs are again fitted with a Gaussian background and a Lorentzian peak, whose integrated intensity as a function of temperature is plotted in Fig. S2h. The method of integrating intensity in boxed regions is also done on RFS. Here two regions were chosen (Fig. \[fig:figsi2\]d): blue box for a region dominated by [$d_{xy}$]{} band, and magenta box for a region of mixed [$d_{xy}$]{} and [$d_{xz}$]{}. The background for each region is taken as a box of same energy window away from dispersions, marked by dotted boxes in Fig. \[fig:figsi2\](d). The integrated intensity of these two regions as a function of temperature is plotted in Fig. \[fig:figsi2\](i), showing the diminishing trend of [$d_{xy}$]{}region and the slower trend of the mixed region. These behaviors in RFS are very similar to those in KFS in the main text.
Orbital-selective Mott phase of the five-orbital Hubbard model for [K$_{x}$Fe$_{2-y}$Se$_2$]{}
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The five-orbital Hubbard model is given by $H=H_{\mathrm{kin}} + H_{\mathrm{int}}$, where $H_{\mathrm{kin}}$ and $H_{\mathrm{int}}$ respectively denote the kinetic and the on-site interaction parts of the Hamiltonian. $H_{\mathrm{int}}$ contains the intra- and inter-orbital Coulomb repulsion, as well as the Hund’s rule coupling and the pair hoppings.[@s8yu] The corresponding coupling strengths are respectively $U$, $U^\prime$, and $J$, which satisfy $U^\prime=U-2J$. [@s5castellani] For simplicity, we consider only the density-density interactions and neglect the spin-flip and pair-hopping terms. The results including these terms are qualitatively the same. The kinetic part $H_{\mathrm{kin}}$ is a tight-binding Hamiltonian, and is conveniently specified in the momentum space. In FeSCs, each unit cell contains two Fe ions. Hence, ideally the tight-binding Hamiltonian must be defined in the BZ corresponding to this two-Fe unit cell. However, the lattice symmetry of the FeSC system allows us to work in an unfolded BZ corresponding to one-Fe unit cell. Following Ref. [@s6wen] and notice that the ions are invariant under the transformation $P_zT_x$ and $P_zT_y$, where $T_{x(y)}$ is the translation along $x(y)$ direction by one Fe-lattice spacing, and $P_z$ refers to the reflection about the Fe plane, we may define a pseudocrystal momentum $\bf{\tilde{k}}$ in the extended one-Fe BZ. This pseudocrystal momentum and the conventional momentum $\bf{k}$ are related by $\bf{\tilde{k}}=\bf{k}+\bf{Q}$ in [$d_{xz}$]{} and [$d_{yz}$]{} orbitals (where $\bf{Q}=(\pi,\pi)$), but $\bf{\tilde{k}}=\bf{k}$ in other orbitals. In the extended BZ, the tight-binding Hamiltonian reads
$$\label{eqn:eqn2}
H_{\mathrm{kin}}=\sum\limits_{\bf{\tilde k}\alpha\beta\sigma}[\xi_{\alpha\beta}(\bf{\tilde k})+(\Delta_\alpha-\mu)\delta_{\alpha\beta}]d^{\dagger}_{\bf{\tilde k}\alpha\sigma}d_{\bf{\tilde k}\beta\sigma}.$$
Here $\xi_{\alpha\beta}(\bf{\tilde{k}})$ is the hopping matrix in the momentum space associated with orbitals $\alpha$ and $\beta$, $\Delta_{\alpha}$ is the on-site energy reflecting the crystal field splitting, $\mu$ is the chemical potential, and $\delta_{\alpha\beta}$ is the Kronecker’s delta function. The expression of $\xi_{\alpha\beta}(\bf{\tilde{k}})$ is given in Ref. [@s7yu], and it has the same form as appeared in the appendix of Ref. [@s2graser]. We adopt the tight-binding parameters of Ref. [@s7yu], where they are obtained by fitting the LDA band structure of KFS. To better fit the LDA results, we have further tuned several hopping parameters by hand from their values in Ref. [@s7yu]. Using the same notation as in Ref. [@s2graser], the tight-binding parameters used in this paper for KFS are listed in Table \[Table:tab1\]. The chemical potential corresponding to the electron filling $n$=6.15 is $\mu$=-0.365 eV.
The five-orbital Hubbard model is studied using the recently developed U(1) slave-spin mean-field method [@s8yu]. In this method, a slave quantum S=1/2 spin is introduced to carry the charge degree of freedom, and the spin of the electron is carried by a fermionic spinon. This approach determines the quasiparticle spectral weight $Z_\alpha$ in each orbital. An orbital $\alpha$ is delocalized when $Z_\alpha>0$, but becomes Mott localized if $Z_\alpha=0$.
$\alpha=1$ $\alpha=2$ $\alpha=3$ $\alpha=4$ $\alpha=5$
------------------------ ------------ ------------ ------------ ------------ ------------ ----------- ------------
$\Delta_\alpha$ -0.36559 -0.36559 -0.56466 -0.05096 -0.91583
$t^{\alpha\alpha}_\mu$ $\mu=x$ $\mu=y$ $\mu=xy$ $\mu=xx$ $\mu=xxy$ $\mu=xyy$ $\mu=xxyy$
$\alpha=1$ -0.11475 -0.38868 0.20881 -0.04557 -0.00866 -0.03143 0.01899
$\alpha=3$ 0.32523 -0.09783 -0.00537
$\alpha=4$ 0.20633 0.09682 -0.07525 -0.02189 0.00423
$\alpha=5$ -0.0427 0.01117 0.00177 -0.01349
$t^{\alpha\beta}_\mu$ $\mu=x$ $\mu=xy$ $\mu=xxy$ $\mu=xxyy$
$\alpha\beta=12$ 0.10161 -0.02017 0.03273
$\alpha\beta=13$ -0.31447 0.06225 0.0103
$\alpha\beta=14$ 0.13785 -0.03105 0.0104
$\alpha\beta=15$ -0.04825 -0.10096 -0.01204
$\alpha\beta=34$ -0.04795
$\alpha\beta=35$ -0.30966 -0.01498
$\alpha\beta=45$ -0.08359 -0.00766
{width="90.00000%"}
To compare with the ARPES data, the coherent part of the orbital resolved spectral function $A_\alpha(\bf{k},E)$ in the folded BZ corresponding to the two-Fe unit cell is further calculated by convoluting the slave-spin and spinon Green’s functions and writing them in the Lehmann representation via the following formula
$$\label{eqn:eqn5}
A_\alpha(\bf{k},E) = \sum\limits_{\alpha\nu} Z_\alpha| U^{\alpha\nu}_{\bf{k}}|^2 \delta(E-\varepsilon_{\nu\bf{k}}).$$
Here, $\varepsilon_{\nu\bf{k}}$ and $U^{\alpha\nu}_{\bf{k}}$ are respectively the $\nu$’s eigenenergy and eigenvector of the hopping matrix $\xi_{\alpha\beta}(\bf{k})$. This allows us to determine the orbital character of each band near the Fermi level. In Fig. \[fig:figsi3\], we show the spectral functions near the Fermi level in the five-orbital model at n=6.15 and T=10 K for three different U values. Compared to the U=0 band structure, the bands are strongly renormalized by the interactions. Moreover, with increasing U, the spectral weight of the bands with a [$d_{xy}$]{} orbital character is reduced (Fig. \[fig:figsi3\](b)) and eventually goes away (Fig. \[fig:figsi3\](c)). The bands with the [$d_{xz}$/$d_{yz}$]{} character, on the other hand, still have nonzero spectral weights. This behavior clearly indicates an interaction driven transition to an OSMP.
Several factors favor stabilizing the OSMP.[@s15yu] Firstly, from the orbitally-projected density of states of the non-interacting bands, the width of the [$d_{xy}$]{} orbital is narrower than that of the other Fe 3d orbitals, especially the [$d_{xz}$/$d_{yz}$]{} orbitals. This factor recalls the mechanism for the OSMP initially proposed for the Ca$_{2-x}$Sr$_x$RuO$_4$ system [@s10anisimov]. Secondly, when the Hund’s coupling is sufficiently strong compared to the (nonzero) splitting between the [$d_{xy}$]{} and [$d_{xz}$/$d_{yz}$]{} orbitals, the high-spin configuration is favored, and, due to the crystal level splitting, the [$d_{xy}$]{} orbital is non-degenerate and located at a higher energy than other orbitals. As a result, for a range of densities, the [$d_{xy}$]{} orbital is kept at half-filling while the [$d_{xz}$/$d_{yz}$]{} orbitals are more than (though still close to) half filled. The non-degenerate [$d_{xy}$]{} orbital has a lower repulsion threshold for the Mott transition than the doubly degenerate [$d_{xz}$/$d_{yz}$]{} orbitals within a wide range of Hund’s coupling. This picture has some connection with the one studied in a different regime for a three-orbital model away from half-filling [@s11demedici]. As a combined effect of these factors, the [$d_{xy}$]{} orbital is more strongly localized than the [$d_{xz}$/$d_{yz}$]{} orbitals.
Comparison of insulating, intermediate and superconducting compounds
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{width="90.00000%"}
![\[fig:figsi5\]Comparison of superconducting and intermediate KFS (a) Calculated orbital-resolved quasiparticle spectral weight for U=3.75eV and T=30K as a function of electron filling, n, for [$d_{xz}$/$d_{yz}$]{} (red squares) and [$d_{xy}$]{} (blue triangles). (b) Measured EDC at X (black solid curve) from superconducting KFS from Fig. \[fig:figsi4\](a), fitted to a Gaussian plus a Lorentzian, plotted in red and blue, respectively. (c) Same as (b) but for the intermediate KFS sample from Fig. S4b, with the EDC from the superconducting sample renormalized by 1.3 in energy for comparison (gray dotted curve). Blue arrows point to the positions of the [$d_{xy}$]{} band bottom.](FigSI5){width="50.00000%"}
Here, we compare the measured electronic structure of three kinds of phases in KFS and RFS. The four compounds shown in Fig. \[fig:figsi4\] have compositions determined by energy dispersive X-ray spectroscopy to be (a) K$_{0.76}$Fe$_{1.72}$Se$_2$, (b) $K_{0.76}$Fe$_{1.78}$Se$_2$, (c) Rb$_{0.93}$Fe$_{1.70}$Se$_2$, and (d) K$_{0.90}$Fe$_{1.78}$Se$_2$. From resistivity data shown, (a) and (c) are superconducting, (d) is insulating, and (b) shows a behavior somewhat between superconducting and insulating, which we shall call intermediate. Its resistivity is most like the compound suggested to be semiconducting in a previous report [@s12chen], which suggests that both insulating and metallic regions exist in these samples. All compounds were measured in the same experimental geometry. The two superconducting samples exhibit similar dispersions as discussed in previous section. The insulating sample shows negligible spectral weight towards [$E_F$]{} and no well defined dispersions near [$E_F$]{}, as expected of an insulator. The interesting case is the intermediate sample, which has resolvable dispersions, but different from that of the superconducting samples. Firstly, we see that the most well-resolved band is the [$d_{yz}$]{} band, which is traced in green in the second derivative plot in the bottom panel. Its bandwidth is approximately renormalized by a factor of 1.3 compared to its equivalent in the superconducting samples. Secondly, from the EDC taken at the X point from both the superconducting KFS and the intermediate KFS (Fig. \[fig:figsi5\](b)-(c)), we see that the peak around -0.05eV indicating the [$d_{xy}$]{} electron band bottom in the superconducting sample (Fig. \[fig:figsi5\](b)) becomes a smaller but discernible shoulder closer to [$E_F$]{} in the intermediate sample (Fig. \[fig:figsi5\](c)). Even after a renormalization of 1.3 to account for the increased renormalization of the [$d_{yz}$]{} band (gray dotted line in Fig. \[fig:figsi5\](c)) as discussed above, we see that the [$d_{xy}$]{} band in the intermediate compound still needs a further renormalization going from the superconducting to the intermediate sample (Fig. \[fig:figsi5\](c)). Also, the relative spectral weight of the [$d_{xy}$]{} orbital to that of [$d_{yz}$]{} is much more reduced in the intermediate sample. Both the further renormalization of the bandwidth and the reduction of spectral weight of the [$d_{xy}$]{} band indicate that the intermediate sample is even closer to the OSMP than the superconducting compounds at low temperatures.
Assuming the same interaction strengths, the various phases in the KFS and RFS compounds can be understood in terms of an interplay between the electron doping and vacancy order. In the vacancy disordered case, our calculation (Fig. \[fig:figsi5\](a)) identifies a doping-induced transition to an OSMP near n=6.02 per Fe. Hence the undoped system with n=6 is already in an OSMP. The vacancy order further drives it through a Mott transition in all orbitals to a Mott insulator [@s13yu; @s14craco]. This accounts for the absence of spectral weights near [$E_F$]{} in the insulating RFS. For the superconducting samples and the intermediate KFS compound, the phases giving ARPES signals are likely to be vacancy disordered. We therefore interpret them as corresponding to two vacancy disordered phases at two different electron densities (n=6.15-6.25 for the superconducting samples, and n=6.05-6.10 for the metallic phase in the intermediate KFS). In addition, we emphasize that the high temperature state of the superconducting region is intrinsically different from the vacancy-ordered insulating phase. Rather, the superconducting, intermediate, and insulating phases likely have increasing correlation as they may be located close to an OSMP, just at the boundary of an OSMP, and in a Mott insulating phase, respectively.
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abstract: 'Intensity mapping is a promising technique for surveying the large scale structure of our Universe from $z=0$ to $z \sim 150$, using the brightness temperature field of spectral lines to directly observe previously unexplored portions of out cosmic timeline. Examples of targeted lines include the $21\,\textrm{cm}$ hyperfine transition of neutral hydrogen, rotational lines of carbon monoxide, and fine structure lines of singly ionized carbon. Recent efforts have focused on detections of the power spectrum of spatial fluctuations, but have been hindered by systematics such as foreground contamination. This has motivated the decomposition of data into Fourier modes perpendicular and parallel to the line-of-sight, which has been shown to be a particularly powerful way to diagnose systematics. However, such a method is well-defined only in the limit of a narrow-field, flat-sky approximation. This limits the sensitivity of intensity mapping experiments, as it means that wide surveys must be separately analyzed as a patchwork of smaller fields. In this paper, we develop a framework for analyzing intensity mapping data in a spherical Fourier-Bessel basis, which incorporates curved sky effects without difficulty. We use our framework to generalize a number of techniques in intensity mapping data analysis from the flat sky to the curved sky. These include visibility-based estimators for the power spectrum, treatments of interloper lines, and the “foreground wedge" signature of spectrally smooth foregrounds.'
author:
- 'Adrian Liu$^{\dagger}$, Yunfan Zhang, Aaron R. Parsons'
bibliography:
- 'biblio.bib'
title: Spherical Harmonic Analyses of Intensity Mapping Power Spectra
---
Introduction {#sec:Intro}
============
[[^1]]{} In recent years, intensity mapping has been hailed as a promising method for conducting cosmological surveys of unprecedented volume. In an intensity mapping survey, the brightness temperature of an optically thin spectral line is mapped over a three-dimensional volume, with radial distance information provided by the observed frequency (and thus redshift) of the line. By observing brightness temperature fluctuations on cosmologically relevant scales (without resolving individual sources responsible for the emission or absorption), intensity mapping provides a relatively cheap way to survey our Universe. In addition, with an appropriate choice of spectral line and a suitably designed instrument, the volume accessible to an intensity mapping survey is enormous. This allows measurements to be made over a large number of independent cosmological modes, providing highly precise constraints on both astrophysical and cosmological models. For example, intensity mapping experiments tracing the $21\,\textrm{cm}$ hyperfine transition of hydrogen can easily access $\sim 10^9$ independent modes, which is much greater than the $\sim 10^6$ accessible to the Cosmic Microwave Background, in principle unlocking a far greater portion of the available information in our observable Universe [@loeb_and_zaldarriaga2004; @mao_et_al2008; @tegmark_and_zaldarriaga2009; @ma_and_scott2016; @scott_et_al2016].
A large number of intensity mapping experiments are in operation, and more have been proposed. Post-reionization neutral hydrogen $21\,\textrm{cm}$ intensity mapping is being conducted by the Canadian Hydrogen Intensity Mapping Experiment [@bandura_et_al2014], the Green Bank Telescope [@masui_et_al2013], Tianlai telescope [@chen_et_al2012], Baryon Acoustic Oscillations from Integrated Neutral Gas Observations project [@battye_et_al2013], Hydrogen Intensity and Real-time Analysis eXperiment [@newburgh_et_al2016], and BAORadio [@ansari_et_al2012]. These experiments use neutral hydrogen as a tracer of the large scale density field, with a primary scientific goal of constraining dark energy via measurements of the baryon acoustic oscillation feature from $0 < z < 4$ [@wyithe_et_al2008; @chang_et_al2008; @pober_et_al2013a]. At $z \sim 2$ to $3.5$, data from the Sloan Digital Sky Survey have been used for Ly $\alpha$ intensity mapping [@croft_et_al2016]. Other experiments such as the CO Power Spectrum Survey [@keating_et_al2015; @keating_et_al2016] and the CO Mapping Array Pathfinder [@li_et_al2016] use CO as a tracer of molecular gas in the epoch of galaxy formation at roughly $z \sim 2$ to $3$. Using \[CII\] instead is the Spectroscopic Terahertz Airborne Receiver for Far-InfraRed Exploration (operating at $0.5 < z < 1.5$; @uzgil_et_al2014), and the Tomographic Ionized carbon Mapping Experiment (operating at $5 < z < 9$; @crites_et_al2014). The highest redshift bins of the latter encroach upon the Epoch of Reionization (EoR), when the first galaxies systematically reionized the hydrogen content of the intergalactic medium. Extending into the EoR, intensity mapping efforts are mainly focused around the $21\,\textrm{cm}$ line. The Donald C. Backer Precision Array for Probing the Epoch of Reionzation array (PAPER; @parsons_et_al2010), the Low Frequency Array [@van_haarlem_et_al2013], the Murchison Widefield Array [@bowman_et_al2012; @tingay_et_al2013], the Giant Metrewave Radio Telescope [@paciga_et_al2013], the Long Wavelength Array (M. W. Eastwood et al., in prep.), 21 Centimeter Array [@huang_et_al2016; @zheng_et_al2016], and the Hydrogen Epoch of Reionization Array [@deboer_et_al2016] are radio interferometers that aim to use the $21\,\textrm{cm}$ line to probe the density, ionization state, and temperature of hydrogen in the range $6 < z < 13$ and beyond. The future Square Kilometre Array [@mellema_et_al2015] will provide yet more collecting area for $21\,\textrm{cm}$ intensity mapping to complement the aforementioned experiments. With such a large suite of instruments covering an expansive range in redshift, tremendous opportunities exist for understanding the formation of the first stars and galaxies via direct measurements of the IGM during all the relevant epochs [@hogan_and_rees1979; @scott_and_rees1990; @madau_et_al1997; @tozzi_et_al2000], as well as fundamental cosmological parameters [@mcquinn_et_al2006; @mao_et_al2008; @visbal_et_al2009; @clesse_et_al2012; @liu_et_al2016] and exotic phenomena such as dark matter annihilations [@valdes_et_al2013; @evoli_et_al2014].
Despite its promise, intensity mapping is challenging, and to date the only positive detections have been tentative detections of Ly $\alpha$ at $z \sim 2$ to $3.5$ [@croft_et_al2016] and CO from $z\sim 2.3$ to $3.3$ [@keating_et_al2016], as well as detections of HI at $z\sim 0.8$ via cross-correlation with optical galaxies [@chang_et_al2010; @masui_et_al2013]. To realize the full potential of intensity mapping, it is necessary to overcome a large number of systematics. A prime example would be radiation from foreground astrophysical sources, which are particularly troublesome for HI intensity mapping. Especially at high redshifts, foregrounds add contaminant emission to the measurement that are orders of magnitude brighter than the cosmological signal [@dimatteo_et_al2002; @santos_et_al2005; @wang_et_al2006; @deOliveiraCosta_et_al2008; @sims_et_al2016]. Low frequency measurements (for instance, those targeting the $21\,\textrm{cm}$ EoR signal), are mainly contaminated by broadband foregrounds such as Galactic synchrotron emission or extragalactic point sources (whether they are bright and resolved or are part of a dim and unresolved continuum). These foregrounds are typically less dominant at the higher frequencies and are thus easier (though still challening) to handle for CO or \[CII\] intensity mapping experiments. However, such experiments must also contend with the problem of interloper lines, where two spectral lines of different rest wavelengths may redshift into the same observation band, leading to confusion as to which spectral line has been observed.
In addition to astrophysical foregrounds, instrumental systematics must be well-controlled for a successful measurement of the cosmological signal. Among others, these systematics include beam-forming errors [@neben_et_al2016b], radio frequency interference [@offringa_et_al2013; @offringa_et_al2015; @huang_et_al2016], polarization leakage [@geil_et_al2011; @moore_et_al2013; @shaw_et_al2014b; @sutinjo_et_al2015; @asad_et_al2015; @moore_et_al2015; @kohn_et_al2016], calibration errors [@newburgh_et_al2014; @trott_and_wayth2016; @barry_et_al2016; @patil_et_al2016], and instrumental reflections [@neben_et_al2016a; @ewall-wice_et_al2016a; @thyagarajan_et_al2016].
In this paper, we focus specifically on measurements of the power spectrum $P(k)$ of spatial fluctuations in brightness temperature, where roughly speaking, the temperature field is Fourier transformed and then squared. In diagnosing the aforementioned systematics as they pertain to spatial fluctuation experiments, it is helpful to decompose the fluctuations into modes that separate purely angular fluctuations from purely radial fluctuations from those that are a mixture of both. In recent years, for example, simulations and measured upper limits of the $21\,\textrm{cm}$ power spectrum have often been expressed as cylindrically binned power spectra. To form cylindrically binned power spectra, one begins with unbinned power spectra $P(\mathbf{k})$, where $\mathbf{k}$ is the three-dimensional wavevector of spatial Fourier modes. If the field of view is narrow, there exists a particular direction that can be identified as the line-of-sight (or radial) direction. One of the three components of $\mathbf{k}$ can then be chosen to lie along this direction and labeled $k_\parallel$ as a reminder that it is *parallel* to the line-of-sight. The remaining two components—which we arbitrarily designate $k_x$ and $k_y$ in this paper—describe transverse (i.e., angular fluctuations), and have a magnitude $k_\perp \equiv \sqrt{k_x^2 + k_y^2}$. Binning $P(\mathbf{k})$ along contours of constant $k_\perp$ gives $P(k_\perp, k_\parallel)$, the cylindrically binned power spectrum.
Expressing the power spectrum as a function of $k_\perp$ and $k_\parallel$ is a powerful diagnostic exercise because intensity mapping surveys probe line-of-sight fluctuations in a fundamentally different way than the way they probe angular fluctuations. Systematics are therefore usually anisotropic and have distinct signatures on the $k_\perp$-$k_\parallel$ plane [@morales_and_hewitt2004]. For example, cable reflections and bandpass calibration errors tend to appear as features parallel to the $k_\parallel$ axis [@dillon_et_al2015; @ewall-wice_et_al2016b; @jacobs_et_al2016]. Thus, the cylindrically binned power spectrum is a useful intermediate quantity to compute before one performs a final binning along constant $k \equiv \sqrt{k_\perp^2 + k_\parallel^2}$ to give an isotropic power spectrum $P(k)$.
The diagnostic capability of $P(k_\perp, k_\parallel)$ is particularly apparent when considering foregrounds. Assuming that they are spectrally smooth, foregrounds preferentially contaminate low $k_\parallel$ modes, since $k_\parallel$ is the Fourier conjugate to line-of-sight distance, which is probed by the frequency spectrum in intensity mapping experiments. The situation is more complicated for the (large) subset of intensity mapping measurements that are performed on interferometers. Interferometers are inherently chromatic in nature, causing intrinsically smooth spectrum foregrounds to acquire spectral structure, which results in leakage to higher $k_\parallel$ modes. Even this leakage, however, has been shown in recent years to have a predictable “wedge" signature on the $k_\perp$-$k_\parallel$ plane, limiting the contaminated region to a triangular-shaped region at high $k_\perp$ and low $k_\parallel$ [@Datta2010; @Vedantham2012; @Morales2012; @Parsons_et_al2012b; @Trott2012; @Thyagarajan2013; @pober_et_al2013b; @dillon_et_al2014; @Hazelton2013; @Thyagarajan_et_al2015a; @Thyagarajan_et_al2015b; @liu_et_al2014a; @liu_et_al2014b; @chapman_et_al2016; @pober_et_al2016; @seo_and_hirata2016; @jensen_et_al2016; @kohn_et_al2016]. In fact, the foreground wedge is considered sufficiently robust that some instruments have been designed around it [@pober_et_al2014; @deboer_et_al2016; @dillon_et_al2016; @neben_et_al2016a; @ewall-wice_et_al2016a; @thyagarajan_et_al2016], implicitly pursuing a strategy of foreground avoidance where the power spectrum can be measured in relatively uncontaminated Fourier modes outside the wedge. This mitigates the need for extremely detailed models of the foregrounds, providing a conservative path towards early detections of the power spectrum.
Despite its utility, the $k_\perp$-$k_\parallel$ power spectrum is limited in that it is ultimately a quantity that is only well-defined in the flat-sky, narrow field-of-view limit, where a single line-of-sight direction can be unambiguously defined. For surveys with wide fields of view, different portions of the survey have different lines of sight that point in different directions with respect to a cosmological reference frame. Note that this is a separate problem from that of wide-field imaging: even if one’s imaging software does not make any flat-sky approximations (so that the resulting images of emission within the survey volume are undistorted by any wide-field effects), the act of forming a power spectrum on a $k_\perp$-$k_\parallel$ invokes a narrow-field approximation. If one insists on forming $P(k_\perp, k_\parallel)$ as a diagnostic, the simplest way to do so is to split up the survey into multiple small patches that are individually small enough to warrant a narrow-field assumption. A separate power spectrum can then be formed from each patch by squaring the Fourier mode amplitudes, and the resulting collection of power spectra can then be averaged together. While correct, such a “square-then-average" procedure results in lower signal-to-noise than a hypothetical “average-then-square" procedure whereby a single power spectrum is formed out of the entire survey. The latter allows the spatial modes of a survey to be averaged together coherently, which allows instrumental noise to be averaged down very quickly. Roughly speaking, if $N$ patches of sky are averaged in a coherent fashion to constrain a particular spatial mode, the noise on the measured mode amplitude averages down as $1/\sqrt{N}$. Squaring this amplitude to form a power spectrum then results in a quicker $1/N$ scaling of noise. In contrast, a “square-then-average" method combines $N$ independent pieces of information after squaring, and thus the power spectrum noise scales more slowly[^2] as $1/\sqrt{N}$. The result is a less sensitive statistic, whether for the diagnosis of systematics or for a cosmological measurement. To be fair, one could recover the lost sensitivity by also computing all cross-correlations between a series of small overlapping patches. However, the necessary geometric adjustments for such high precision mosaicking will likely be computationally wasteful, and it quickly becomes preferable to adopt an approach that incorporates the curved sky from the beginning.
In this paper, we rectify the shortcomings of the $k_\perp$-$k_\parallel$ plane by introducing an alternative that is well-defined in the wide-field limit. Rather than expanding sky emission in a basis of rectilinear Fourier modes, we propose a spherical Fourier-Bessel basis. In this basis, the sky brightness temperature $T(\mathbf{r})$ of a survey (where $\mathbf{r}$ is the comoving position) is expressed in terms of $\overline{T}_{\ell m} (k)$, defined as[^3] $$\label{eq:TellmEverything}
\overline{T}_{\ell m} (k) \equiv \sqrt{\frac{2}{\pi}} \int \! d\Omega dr\, r^2 j_\ell (kr) Y_{\ell m}^* ({\hat{\mathbf{r}}}) T(\mathbf{r}),$$ where $k$ is the *total* wavenumber, $\ell$ and $m$ are the spherical harmonic indices, $Y_{\ell m}$ denotes the corresponding spherical harmonic, $r \equiv | \mathbf{r}|$ is the radial distance, $\mathbf{\hat{r}} \equiv \mathbf{r} / r$ is the angular direction unit vector[^4], and $j_\ell$ is the $\ell$th order spherical Bessel function of the first kind. The quantity $P(k_\perp, k_\parallel)$ is replaced by the analogous quantity $S_\ell (k)$, the spherical harmonic power spectrum, which roughly takes the form $$\label{eq:Sellkrough}
S_\ell (k) \propto \frac{1}{2 \ell + 1} \sum_{m = -\ell}^\ell |\overline{T}_{\ell m} (k)|^2,$$ where the sum over $m$ is analogous to the binning of $k_x$ and $k_y$ into $k_\perp$, and a more rigorous definition (with constants of proportionality) will be defined in Section \[sec:SphericalPspecFormalism\]. Instead of the $k_\perp$-$k_\parallel$ plane, power spectrum measurements are now expressed on an $\ell$-$k$ plane. Now, we will show in Section \[sec:SphericalPspecFormalism\] that in the limit of a translationally invariant cosmological field, $S_\ell (k)$ reduces to $P(k)$. Therefore, just as $P(k_\perp, k_\parallel)$ can be averaged along contours of constant $k$ to form $P(k)$ once systematic effects are under control, the same can be done for $S_\ell (k)$ to form $P(k)$ by averaging over all values of $\ell$ for a particular $k$.
Spherical Fourier-Bessel methods have been explored in the past within the galaxy survey literature [@binney_quinn1991; @lahav_et_al1994; @fisher_et_al1994; @fisher_et_al1995; @heavens_taylor1995; @zaroubi_et_al1995; @castro_et_al2005; @leistedt_et_al2012; @rassat_refregier2012; @shapiro_et_al2012; @pratten_munshi2013; @yoo_desjacques2013]. In this paper, we build upon these methods and present a framework for implementing them in an analysis of intensity mapping data. We emphasize the way in which intensity mapping surveys have unique geometric properties, and how these properties affect spherical Fourier-Bessel methods. For instance, we pay special attention to the fact that particularly for the highest redshift observations, intensity mapping experiments probe survey volumes that are radially compressed but angularly expansive (as illustrated in Figure \[fig:surveyGeom\]). In harmonic space, this expectation is reversed, and there is excellent spatial resolution along the line-of-sight (since high spectral resolution is relatively easy to achieve), but poor angular resolution. In addition to addressing these geometric peculiarities, we also show how interferometric data can be analyzed with spherical Fourier-Bessel methods. Importantly, we find that the foregrounds again appear as a wedge in interferometric measurements of $S_\ell (k)$, which suggests that the $\ell$-$k$ plane is at least as powerful a diagnostic tool as the $k_\perp$-$k_\parallel$ plane, particularly given the signal-to-noise advantages discussed above.[^5]
The rest of this paper is organized as follows. In Section \[sec:Notation\] we establish notational conventions for this paper. Section \[sec:SphericalPspecFormalism\] introduces spherical Fourier-Bessel methods for power spectrum estimation, with the complication of finite surveys (in both the angular and spectral directions) the subject of Section \[sec:FiniteVolume\]. In Section \[sec:Foregrounds\] we compute the signature of smooth spectrum foregrounds on the $\ell$-$k$ plane. Interloper lines are explored in Section \[sec:Interlopers\]. A framework for interferometric power spectrum estimation using spherical Fourier-Bessel methods (which includes a derivation of the foreground wedge) is presented in Section \[sec:Interferometry\]. To build intuition, we develop a parallel series of flat-sky, narrow field-of-view expressions in a series of Appendices. Our conclusions are summarized in Section \[sec:Conclusions\]. Because of the large number of mathematical quantities defined in this paper, we provide a glossary of important symbols for the reader’s convenience in Table \[tab:Definitions\].
Quantity Meaning/Definition Context
------------------------------------------------- ------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------
$\mathbf{r}$ Comoving position Section \[sec:Intro\]
$\mathbf{\hat{r}}$ Angular direction unit vector Section \[sec:Intro\]
$\mathbf{r}_\perp$ Comoving transverse distance Eq.
$r(\nu)$ or $r_\nu$ Comoving radial distance Eq.
$s(r)$ Incorrect radial distance assumed for true radial distance $r$ due to interloper lines Eq.
$\nu(r)$ or $\nu_r$ Observed frequency of radio emission Section \[sec:Notation\]
$\alpha$ Linearized conversion factor between frequency and radial comoving distance Eq.
$ \boldsymbol \theta$ Sky image angle Eq.
$\mathbf{k}$ Wavevector of rectilinear spatial Fourier modes Section \[sec:Intro\]
$k_\perp$ Magnitude of wavevector components perpendicular to line of sight Section \[sec:Intro\]
$k_\parallel$ Magnitude of wavevector components parallel to line of sight Section \[sec:Intro\]
$k$ Total wavenumber/wavevector magnitude of rectilinear spatial Fourier modes Section \[sec:Intro\]
$\phi(\mathbf{r})$ Survey volume selection function Section \[sec:FiniteVolume\]
$\phi(r)$ Radial survey profile or survey volume selection function assuming full-sky covarage Section \[sec:FiniteVolume\]
$\Phi(r)$ Radial survey profile centered on radial midpoint of survey Section \[sec:MostlyRadialNoInterferometry\]
$T(\mathbf{r})$ or $T({\hat{\mathbf{r}}}, \nu)$ Sky temperature in configuration space Eq.
$ \overline{T}_{\ell m} (k)$ Sky temperature in spherical Fourier-Bessel space Eq.
$ \overline{T}_{\ell m}^\textrm{meas} (k)$ Estimated sky temperature in spherical Fourier-Bessel space for finite-volume surveys Eq.
$\widetilde{T} (\mathbf{k})$ Sky temperature in rectilinear Fourier space Eq.
$\kappa (\nu)$ Frequency spectrum of foreground contaminants Eq.
$q_\ell (k)$ Frequency spectrum of foreground contaminants in radial spherical Bessel basis Eq.
$a_{\ell m} (\nu)$ Sky temperature in frequency/spherical harmonic space Eq.
$Y_{\ell m} $ Spherical harmonic function Section \[sec:SphericalPspecFormalism\]
$\psi_{\ell m} (k; {\hat{\mathbf{r}}}, \nu)$ Spherical Fourier-Bessel basis function in configuration space Eq.
$j_\ell (kr) $ $\ell$th order spherical Bessel function of the first kind Section \[sec:SphericalPspecFormalism\]
$C_\ell$ Angular power spectrum Section \[sec:RotationalInvarianceOnly\]
$P(\mathbf{k})$ Brightness temperature power spectrum Section \[sec:Intro\]
$P(k_\perp, k_\parallel)$ Brightness temperature power spectrum, assuming cylindrical symmetry Section \[sec:Intro\]
$P(k)$ Brightness temperature power spectrum, assuming isotropy Eq.
$S_\ell (k) $ Spherical harmonic power spectrum Eq.
$ \mathbf{b}$ Interferometer baseline vector Section \[sec:Interferometry\]
$\tau$ Interferometric time delay Eq.
$V(\mathbf{b}, \nu)$ Interferometric visibility Eq.
$\widetilde{V}(\mathbf{b}, \tau)$ Interferometric visibility in delay space Eq.
$A({\hat{\mathbf{r}}}, \nu)$ Primary beam of receiving elements of interferometer Eq.
$B({\hat{\mathbf{r}}}, \nu)$ Rescaled primary beam Eq.
$\overline{B^2}(\theta) $ Squared primary beam profile, averaged azimuthally about a baseline vector Eq.
$\gamma (\nu)$ Delay transform tapering function Eq.
$f_{\ell m} (\mathbf{b}, \nu)$ Response of baseline $\mathbf{b}$ at frequency $\nu$ to unit perturbation of spherical harmonic mode $Y_{\ell m}$ Eq.
$g_{\ell m} (\mathbf{b}, \tau)$ Response of baseline $\mathbf{b}$ at delay $\tau$ to unit perturbation of spherical harmonic mode $Y_{\ell m}$ Eq.
$W_\ell (k; \mathbf{b}, \tau)$ Spherical harmonic power spectrum window function for a single baseline delay-based Eq.
power spectrum estimate
$\Theta(\nu)$ Re-centered frequency profile of the foregrounds as seen in the data, with finite bandwidth Section \[sec:CurvedSkyWedge\]
and tapering effects
$D(\mathbf{r})$ Survey volume selection function including primary beam, bandwidth, and data analysis Appendix \[sec:RectilinearInterferometerPspecNorm\]
tapering effects
Notational preliminaries {#sec:Notation}
========================
Suppose an intensity mapping survey has surveyed the brightness temperature field $T({\hat{\mathbf{r}}}, \nu)$ of a particular spectral line as a function of angle (specified here in terms of unit vector ${\hat{\mathbf{r}}}$) and frequency $\nu$. Such a quantity represents a three-dimensional survey of our Universe, since different frequencies of a spectral line map to different redshifts, and thus different radial distances from the observer. Explicitly, the comoving radial distance $r$ is given by $$\label{eq:ComovingDistDef}
r (\nu) = \frac{c}{H_0} \int_0^{z(\nu)} \frac{dz^\prime}{E(z^\prime)},$$ where $c$ is the speed of light, $H_0$ is the present day Hubble parameter, with $$1 + z \equiv \frac{\nu_\textrm{rest}}{\nu}\quad \textrm{and} \quad E(z) \equiv \sqrt{\Omega_\Lambda + \Omega_m (1+z)^3},$$ where $\nu_\textrm{rest}$ is the rest frequency of the spectral line, $z$ is the redshift, $\Omega_\Lambda$ is the normalized dark energy density, and $\Omega_m$ is the normalized matter density. There is thus a one-to-one mapping between frequency and comoving radial distance, and as shorthand throughout this paper, we will adopt the notation $r_\nu \equiv r(\nu)$. Similarly, we will often use the symbol $\nu_r$ to denote frequency, with the subscript reminding us that the observed frequency is a function of the radial distance. Given a radial distance, transverse distances may also be computed given ${\hat{\mathbf{r}}}$ (or angle on the sky) using simple geometry.
If one’s survey occurs over a narrow radial range, the distance-frequency relation is often replaced by a linearized approximation where $$\label{eq:LinearDistanceApprox}
r - r_\textrm{ref} \approx - \alpha (\nu - \nu_\textrm{ref} ),$$ with $r_\textrm{ref}$ and $\nu_\textrm{ref}$ being a reference comoving radial distance and a reference frequency, respectively, with values constrained by Eq. , and $$\label{eq:AlphaConversion}
\alpha \equiv \frac{1}{\nu_\textrm{rest}} \frac{c}{H_0} \frac{(1+z_\textrm{ref})^2}{E(z_\textrm{ref})},$$ where $1 + z_\textrm{ref} = \nu_\textrm{rest} / \nu_\textrm{ref}$. In this paper, the symbols $\nu_r$ and $r_\nu$ will always refer to the exact nonlinear relations, and any invocations of the linearized approximations will be written out explicitly using Eq. . When using the linearized approximation for the radial distance, we will often (though not always) also make the small angle approximation for converting between the angle $\boldsymbol \theta$ and the transverse comoving position $\mathbf{r}_\perp$ from some reference direction, where $$\label{eq:AngularConversion}
\mathbf{r}_\perp = r \boldsymbol \theta.$$
Given the well-defined prescriptions for converting between instrument-centric parameters (such as frequency $\nu$ and direction on the sky ${\hat{\mathbf{r}}}$) and cosmology-centric ones (such as $r$ and $\mathbf{r}_\perp$), we will often use both sets of parameters to describe the same quantities. For example, we will sometimes write the brightness temperature field as $T({\hat{\mathbf{r}}}, \nu)$, whereas other times we will write the same quantity as $T(\mathbf{r})$, where $\mathbf{r}$ is the comoving position. We will additionally find it useful to exhibit similar flexibility in our notation even for quantities that are not cosmological in nature, such as the primary beam of a radio telescope.
Spherical Fourier-Bessel Formalism {#sec:SphericalPspecFormalism}
==================================
In this section we introduce the mathematical framework for describing the sky in terms of the spherical harmonic power spectrum. Our treatment here is essentially identical to that of @yoo_desjacques2013, albeit with different Fourier-Bessel transform conventions. No claims of originality are made in this section (except perhaps for Section \[sec:RotationalInvarianceOnly\]), and the formalism is included only for completeness. We will, however, occasionally provide previews of how various parts of the framework are particularly helpful for intensity mapping and interferometry. In the spherical Fourier-Bessel basis, angular fluctuations are expressed by expanding the temperature field $T({\hat{\mathbf{r}}}, \nu)$ in spherical harmonics, such that $$\label{eq:SHTdef}
a_{\ell m} (\nu) \equiv \int d\Omega Y_{\ell m}^* ({\hat{\mathbf{r}}}) T({\hat{\mathbf{r}}}, \nu).$$ To capture modes along the line-of-sight, we perform a Fourier-Bessel transform along the frequency direction, yielding $$\label{eq:FBdef}
\overline{T}_{\ell m} (k) \equiv \sqrt{\frac{2}{\pi}} \int_0^\infty \! dr\, r^2 j_\ell (kr) a_{\ell m} (\nu_r),$$ with these last two expressions of course combining to give Eq. . The temperature field of the sky may therefore be thought of as being a linear combination of a set of basis functions $\psi_{\ell m } (k; {\hat{\mathbf{r}}}, \nu)$ that are indexed by $(k,\ell,m)$, so that $$\label{eq:InverseTrans}
T({\hat{\mathbf{r}}}, \nu) = \sum_{\ell m} \int dk\, \psi_{\ell m } (k; {\hat{\mathbf{r}}}, \nu) \overline{T}_{\ell m} (k),$$ where $$\label{eq:BasisFcts}
\psi_{\ell m } (k; {\hat{\mathbf{r}}}, \nu) \equiv k^2 \sqrt{\frac{2}{\pi}} j_\ell (kr_\nu) Y_{\ell m} ({\hat{\mathbf{r}}}).$$ Eqs. and are the forward transforms into the harmonic basis, while Eqs. and define the inverse transforms back into configuration space. This can be verified by substituting Eq. into Eq. , and using orthonormality of spherical harmonics, as well as the analogous identity for spherical Bessel functions, given by $$\label{eq:BesselOrthog}
\int \! dr \,r^2 j_\ell (k r) j_\ell (k^\prime r) = \frac{\pi}{2 k k^\prime} \delta^D (k - k^\prime),$$ where $\delta^D$ is the Dirac delta function. Note that our convention for the radial transform differs from that of most works in the literature. From Eqs. and , one sees that our convention is symmetric in the following sense. Whether one is switching from $r$-space to $k$-space or vice versa, the prescription is always to multiply by $\sqrt{2 / \pi} j_\ell (kr)$ and the square of the coordinate (i.e., $r^2$ or $k^2$) of the original space before integrating over it. This makes our forward and backward transforms aesthetically and conveniently symmetric. Most previous works (e.g., @leistedt_et_al2012 [@rassat_refregier2012; @yoo_desjacques2013]), in contrast, opt for an asymmetric convention: an extra factor of $k$ is included in the forward transform from $r$ to $k$, and correspondingly there is one fewer factor of $k$ in the backwards transform.
Translationally invariant fields in the spherical Fourier-Bessel formalism {#eq:TransInvarFields}
--------------------------------------------------------------------------
In some sense, the decision to expand fluctuation modes along the line of sight in terms of spherical Bessel functions rather than some other set of basis functions is arbitrary. However, we will now show that spherical Bessel functions are a particularly good choice for describing temperature fields that are statistically translation invariant. Translation-invariant fields admit a representation in terms of their power spectrum $P(k)$, which we define implicitly via the equation[^6] $$\label{eq:RectilinearPspecDef}
\langle \widetilde{T} (\mathbf{k}) \widetilde{T}^* (\mathbf{k^\prime}) \rangle = (2 \pi)^3 \delta^D (\mathbf{k} - \mathbf{k}^\prime) P(k),$$ where the angled brackets $\langle \cdots \rangle$ signify an ensemble average over random realizations of the cosmological temperature field $T(\mathbf{r})$, whose Fourier transform $\widetilde{T} (\mathbf{k})$ we define by the convention $$\label{eq:forwardNormal}
\widetilde{T} (\mathbf{k}) = \int \! d^3 r \,e^{-i \mathbf{k} \cdot \mathbf{r}} T(\mathbf{r})$$ with the inverse transform given by $$\label{eq:inverseNormal}
T(\mathbf{r}) = \int \! \frac{d^3 k}{(2 \pi)^3} e^{i \mathbf{k} \cdot \mathbf{r}} \widetilde{T} (\mathbf{k}).$$ Unless otherwise stated, this Fourier convention for the temperature field will be the one used for all Fourier transforms in this paper. Ideally, our spherical Fourier-Bessel description should be directly relatable to $P(k)$, for it would be pointless if an estimation of the power spectrum required first returning to position space. We will now show that this requirement is met by our $\overline{T}_{\ell m} (k)$ modes.
To relate $\overline{T}_{\ell m} (k)$ to $P(k)$, we combine Eqs. , , and to obtain $$\label{eq:YetAnotherTellm}
\overline{T}_{\ell m} (k) = \sqrt{\frac{2}{\pi}} \int \! \frac{d^3 k^\prime}{(2 \pi)^3} \widetilde{T} (\mathbf{k}^\prime) \int \! d^3 r\, j_\ell (kr) Y_{\ell m}^*({\hat{\mathbf{r}}}) e^{i \mathbf{k}^\prime \cdot \mathbf{r}}.$$ To simplify this, we expand $e^{i \mathbf{k}^\prime \cdot \mathbf{r}}$ in spherical harmonics using the identity $$\label{eq:PlaneWaveSphericalHarmonicExpansion}
e^{i \mathbf{k} \cdot \mathbf{r}} = 4\pi \sum_{\ell m} i^\ell j_\ell (kr) Y_{\ell m}^* ({\hat{\mathbf{k}}}) Y_{\ell m} ({\hat{\mathbf{r}}}),$$ which leads to $$\label{eq:TlmTkConversion}
\overline{T}_{\ell m} (k) = \frac{i^\ell}{(2\pi)^{\frac{3}{2}}} \int \frac{d^3 k^\prime}{k k^\prime} Y_{\ell m}^* ({\hat{\mathbf{k}}}^\prime) \delta^D (k - k^\prime) \widetilde{T} (\mathbf{k}^\prime).$$ This provides a link between the temperature field as expressed in our $(k,\ell, m)$ basis, and the same field in the rectilinear Fourier basis. Taking a cue from Eq. , where the power spectrum is closely related to the two-point correlation between different rectilinear Fourier modes, we may form a two-point correlator between different modes in our spherical Fourier-Bessel basis, giving $$\begin{aligned}
\label{eq:CurvedPspecDef}
\langle \overline{T}_{\ell m} (k) \overline{T}_{\ell^\prime m^\prime}^* (k^\prime) \rangle && = \frac{i^\ell (-i)^{\ell^\prime}}{(2\pi)^3} \!\! \int \frac{d^3 k_1}{k k_1} \frac{d^3 k_2}{k^\prime k_2} \nonumber \\
&& \qquad \times Y_{\ell m}^* ({\hat{\mathbf{k}}}_1) Y_{\ell^\prime m^\prime} ({\hat{\mathbf{k}}}_2) \langle \widetilde{T} (\mathbf{k}_1) \widetilde{T} (\mathbf{k}_2)^* \rangle\nonumber \\
&& \qquad \times \delta^D (k - k_1) \delta^D (k^\prime - k_2) \nonumber \\
&& = \frac{\delta^D(k - k^\prime) }{k^2} \delta_{\ell \ell^\prime} \delta_{m m^\prime} P(k),\end{aligned}$$ where the last equality follows from Eq. and some algebraic simplifications. From this, we see that forming the power spectrum from $\overline{T}_{\ell m} (k)$ modes is remarkably similar to forming it from the rectilinear Fourier modes. Comparing Eqs. and , we see that if (roughly speaking) one can form $P(k)$ by squaring $\widetilde{T} (\mathbf{k})$ and normalizing appropriately, one can equally well form $P(k)$ by squaring $\overline{T}_{\ell m} (k)$ and normalizing (albeit with a different—and $k$ dependent—normalization that we will derive more explicitly in Section \[sec:FiniteVolume\]).
To understand why the squaring of $\overline{T}_{\ell m} (k)$ produces such a similar result to squaring $\widetilde{T} (k)$ (with both giving a result proportional to the power spectrum), notice that Eq. can be simplified to give $$\overline{T}_{\ell m} (k) = \frac{i^\ell}{(2\pi)^{\frac{3}{2}}} \int d\Omega_k Y^*_{\ell m} ({\hat{\mathbf{k}}}) \widetilde{T} (\mathbf{k})\bigg{|}_{|\mathbf{k}| = k},$$ where $ \widetilde{T} (\mathbf{k})$ is restricted to the shell where $|\mathbf{k}| = k$. In this form, one sees that an alternate way to understand our spherical harmonic Bessel modes is to view them as a spherical harmonic decomposition of $ \widetilde{T} (\mathbf{k})$ in Fourier space. In other words, going from the rectilinear Fourier modes to spherical harmonic Bessel modes is simply a change of basis—to spherical harmonics—in angular Fourier coordinates. Now, suppose one were to form an estimate of $P(k)$ in by squaring $ \widetilde{T} (\mathbf{k})$ and then averaging over a shell of constant $|\mathbf{k}| = k$. Parseval’s theorem ensures that such a squaring and averaging operation is basis-independent. Thus, it does not matter whether the Fourier amplitudes on the shell of constant $|\mathbf{k}| = k$ are expressed in a spherical harmonic basis. Squaring and averaging $\overline{T}_{\ell m} (k)$ must therefore also yield the power spectrum, up to some $k$-dependent conversion factors to account for the radius of shells in Fourier space. Note that Eq. also cements the interpretation (suggested by our notation) that the quantity $k$ of our Fourier-Bessel basis is the total magnitude of the wavevector $\mathbf{k}$, rather than some wavenumber that only pertains to radial fluctuations.
Rotationally invariant fields in the spherical Fourier-Bessel formalism {#sec:RotationalInvarianceOnly}
-----------------------------------------------------------------------
While the cosmological temperature field is expected to possess translationally invariant statistics, contaminants in an intensity mapping survey (such as foreground emission) will in general not possess such symmetry. This difference in symmetry will result in different signatures on the $\ell$-$k$ plane that can in principle be used to separate contaminants from the cosmological signal.
To elucidate the contrast in these signatures, suppose we discard the assumption (from previous derivations) of translationally invariant statistics. In general, the two-point correlator will cease to exhibit the diagonal form given by Eq. . As a concrete example of this, consider a random temperature field that is statistically isotropic but not homogeneous. In the radial direction, suppose this field has some fixed (non-random and angular position-independent) radial dependence. Such a field would be an appropriate description for a (hypothetical) population of unresolved point sources. Under these assumptions, Eq. reduces to $$\overline{T}_{\ell m} (k) = a_{\ell m} q_\ell (k),$$ where $$\label{eq:qellk}
q_\ell (k) \equiv \sqrt{\frac{2}{\pi}} \int_0^\infty dr r^2 j_\ell (kr) \kappa (\nu_r),$$ with $\kappa (\nu_r)$ specifying the spectral (and therefore radial) dependence of our hypothetical sky as it appears in our data. The two-point correlator then becomes $$\langle \overline{T}_{\ell m} (k) \overline{T}_{\ell^\prime m^\prime}^* (k^\prime) \rangle = C_\ell q_\ell(k) q_\ell (k^\prime) \delta_{\ell \ell^\prime} \delta_{m m^\prime},$$ where statistical rotation invariance of the field allows us to invoke relation $\langle a_{\ell m} a_{\ell^\prime m^\prime}^* \rangle \equiv C_\ell \delta_{\ell \ell^\prime} \delta_{m m^\prime}$, with $C_\ell$ signifying the angular power spectrum.
Our example illustrates the way in which the two-point correlator ceases to be diagonal in $k$ and $k^\prime$ once translation invariance is broken. In general, if the sky exhibits rotational invariance (in the statistical sense), the correlator takes the form $$\langle \overline{T}_{\ell m} (k) \overline{T}_{\ell^\prime m^\prime}^* (k^\prime) \rangle \equiv M_\ell (k, k^\prime) \delta_{\ell \ell^\prime} \delta_{m m^\prime},$$ for some function $M_\ell (k, k^\prime)$. In the limit that the sky is statistically homogeneous in addition to isotropic, $M_\ell (k, k^\prime)$ becomes $\ell$-independent and reduces to $P(k) \delta^D (k - k ^\prime) / k^2$, as demonstrated in Eq. . If one is simply squaring $\overline{T}_{\ell m} (k)$ measurements to estimate the power spectrum but there are non-statistically homogeneous contaminants in the data, one obtains $$\langle | \overline{T}_{\ell m} (k)|^2 \rangle \equiv M_\ell (k) \delta_{\ell \ell^\prime} \delta_{m m^\prime},$$ where $M_\ell (k)$ is a function of both $\ell$ and $k$ rather than just $k$ alone.
We thus see that the spherical Fourier-Bessel formulation fulfills the goals we laid out near the beginning of this section. In particular, the foreground contaminants appear differently on the $\ell$-$k$ plane than the cosmological signal does, owing to the translation-invariant statistics of the latter. This generalizes the symmetry arguments for foreground mitigation laid out in @morales_and_hewitt2004 in a way that is well-defined for wide fields of view. We note, however, that as the formalism currently stands, $M_\ell (k)$ and $P(k)$ are not directly comparable; indeed, they have different units. This arises because the two quantities scale differently with volume. For a random cosmological field described by $P(k)$, the magnitude of $\overline{T}_{\ell m} (k)$ scales as $\sqrt{V}$, where $V$ is the volume of a survey. On the other hand, contaminants may not be describable as random fields. In the case of foregrounds, for example, the signal is smooth and coherent along the radial/frequency direction. As a result, $\overline{T}_{\ell m} (k)$ scales more quickly than $\sqrt{V}$. Indeed, the difference between these scalings was proposed as a method for distinguishing between foreground contamination and cosmological signal in @cho_et_al2012. To derive a quantity for describing survey contaminants on the $\ell$-$k$ that is directly comparable to $P(k)$ it is necessary to specify a survey volume. In the following sections, we will depart from the idealized treatment considered in this section, where we imagined having access to a perfectly sampled field over an infinite volume.
Estimating the power spectrum from finite-volume surveys {#sec:FiniteVolume}
========================================================
In this section, we consider the effects of the necessarily finite extent of any real survey. Finite selection effects were considered in @rassat_refregier2012 and @leistedt_et_al2012, and here we provide a complementary treatment that is not only tailored for intensity mapping, but also provides explicit expressions for the power spectrum on the $\ell$-$k$ plane.
Suppose the extent of our survey is given by a function $\phi(\mathbf{r})$, such that $\phi(\mathbf{r})$ is zero everywhere beyond the boundaries of the survey. A survey with uniform sensitivity can then be modeled by setting $\phi(\mathbf{r}) = 1$ inside the survey. In what follows, however, we do not make this assumption, and we allow for spatially varying sensitivity within the survey. This permits the treatment of angular masks as well as radial selection functions. In general, the temperature field that is analyzed is $\phi(\mathbf{r}) T(\mathbf{r})$ rather than $T(\mathbf{r})$. A result, the measured spherical Fourier-Bessel modes $\overline{T}_{\ell m}^\textrm{meas}(k)$ are not described by Eq. , but instead are given by $$\begin{aligned}
\label{eq:Tellm^meas}
\overline{T}_{\ell m}^\textrm{meas} (k) = \frac{i^\ell}{(2\pi)^{\frac{3}{2}}} \int \frac{d^3 k^\prime}{k k^\prime} && \frac{d^3 k^{\prime \prime}}{(2\pi)^3} Y_{\ell m}^* ({\hat{\mathbf{k}}}^\prime) \delta^D (k - k^\prime) \nonumber \\
&& \times \widetilde{\phi} (\mathbf{k}^\prime - \mathbf{k}^{\prime \prime}) \widetilde{T} (\mathbf{k}^{\prime\prime}),\end{aligned}$$ where we have invoked the convolution theorem to write our expression in terms of $\widetilde{\phi}$, the Fourier transform of $\phi$.
Despite this revised expression, one might still expect the power spectrum to be closely related to $\overline{T}_{\ell m}^\textrm{meas} (k)$. Squaring and taking the ensemble average gives $$\begin{aligned}
\langle | \overline{T}_{\ell m}^\textrm{meas} (k) |^2 \rangle
= \frac{1}{(2\pi)^3} \int \frac{d^3 k_a}{k k_a} \frac{d^3 k_b}{k k_b} \frac{d^3 k_c}{(2\pi)^3} Y_{\ell m}^* ({\hat{\mathbf{k}}}_a) Y_{\ell m} ({\hat{\mathbf{k}}}_b) \nonumber \\
\times P(k_c) \widetilde{\phi} (\mathbf{k}_a - \mathbf{k}_c) \widetilde{\phi}^* (\mathbf{k}_b - \mathbf{k}_c) \delta^D (k - k_a) \delta^D (k - k_b),\qquad\end{aligned}$$ where we have again used the definition of the power spectrum from Eq. to simplify the ensemble average of the two factors of $\widetilde{T}$. Now, if the survey volume is reasonably large, $\phi(\mathbf{r})$ will tend to be a relatively broad function, and thus the two copies of $\widetilde{\phi}$ will be sharply peaked about $\mathbf{k}_a \approx \mathbf{k}_b \approx \mathbf{k}_c$. These then work in conjunction with the two Dirac delta functions to require $k \approx k_c$. With all these conditions, the only part of the integrand that contributes substantially to the integral is the part where $P(k_c) \approx P(k)$, allowing the power spectrum to be factored out of the integral (assuming it is a reasonably smooth function). Doing so and subsequently re-expressing $\widetilde{\phi}$ in terms of $\phi$, our expression simplifies to $$\begin{aligned}
\label{eq:TlmPkProportionality}
\langle | \overline{T}_{\ell m}^\textrm{meas} (k) |^2 \rangle && \approx \frac{P(k)}{(2\pi)^3} \int d^3 r \phi^2 (\mathbf{r}) \nonumber \\
&& \quad \times \Bigg{|} \int \frac{d^3 k_a}{k k_a}Y_{\ell m}^* ({\hat{\mathbf{k}}}_a) e^{-i \mathbf{k}_a \cdot \mathbf{r}} \delta^D (k - k_a) \Bigg{|}^2 \nonumber \\
&& = P(k) \frac{2}{\pi} \int d^3 r \phi^2 (\mathbf{r}) j_\ell^2 (kr) \big{|} Y_{\ell m} ({\hat{\mathbf{r}}}) \big{|}^2, \qquad\end{aligned}$$ where in the last equality we performed the integral over $k_a$ by inserting Eq. and invoking the orthonormality of spherical harmonics. The final result is a direct proportionality between the ensemble average of hypothetical noiseless measurements $ | \overline{T}_{\ell m}^\textrm{meas} (k) |^2$ and the power spectrum. Heuristically, this equation implies that the power spectrum can be estimated using any $(k,\ell, m)$ mode simply by taking $| \overline{T}_{\ell m}^\textrm{meas} (k) |^2$ and dividing out by everything on the right hand side[^7] after $P(k)$. A subsequent averaging of such estimates obtained from modes with the same $k$ but different $\ell$ and $m$ increases the signal-to-noise.
A similar proportionality exists within the framework of rectilinear Fourier modes for relating the squares of the measured Fourier amplitudes $\widetilde{T}^\textrm{meas} (\mathbf{k})$ and $P(k)$ (which we derive in Appendix \[sec:RectilinearFKP\] to facilitate the comparative discussion that follows). With rectilinear modes, $\langle |\widetilde{T}^\textrm{meas} (\mathbf{k}) |^2 \rangle$ is also proportional to $P(k)$, with the constant of proportionality also given by an integral that has units of volume. However, there exists a crucial difference between the volume integral seen here and the one for the rectilinear framework in Appendix \[sec:RectilinearFKP\]. With the rectilinear case, the volume factor is independent of the orientation of $\mathbf{k}$ (i.e., ${\hat{\mathbf{k}}}$), so that Fourier modes of all orientations are equally sensitive to the power spectrum. It follows that an optimal estimate of the power spectrum can be obtained by an average of $|\widetilde{T}^\textrm{meas} (\mathbf{k}) |^2$ over spheres of constant $| \mathbf{k}| = k$ with uniform weighting, as we show in Appendix \[sec:RectilinearFKP\].
In contrast, the volume integral in Eq. is a function of $\ell$ and $m$. For a particular $(k, \ell, m)$ mode, the value of $\ell$ determines how much the total wavenumber $k$ is comprised of angular fluctuations (as opposed to radial fluctuations), while the value of $m$ determines the orientation of the angular fluctuations. Putting these facts together, it follows that with $\overline{T}_{\ell m}^\textrm{meas} (k)$ modes, the sensitivity to the power spectrum does depend strongly to a mode’s orientation. As an example, suppose the survey’s sensitivity $\phi(\mathbf{r})$ is localized in small region around some radius $r_0$ away from the observer (illustrated in Figure \[fig:surveyGeom\]), as is typical for many high-redshift intensity mapping surveys. Now consider (as an extreme case), modes where $\ell \gg k r_0$. For such modes, the Bessel function in Eq. can be approximated by a power series as $$j_\ell (kr) \approx \frac{(kr)^\ell}{(2 \ell + 1)!!}.$$ The integral on the right hand side of Eq. thus becomes extremely suppressed by a $[(2 \ell + 1)!!]^2$ dependence, giving a small proportionality constant between $| \overline{T}_{\ell m}^\textrm{meas} (k) |^2$ and $P(k)$ for high $\ell$. Thus, high $\ell$ modes that satisfy $\ell \gg k r_0$ are not high signal-to-noise probes of the power spectrum. To understand this, consider instead the modes with $k \sim \ell / r_0$. Such modes are essentially constant in the radial direction, and describe fluctuations that are almost entirely in the angular direction. Temporarily invoking the language of the flat-sky approximation for the sake of intuition, we may say that in this regime, the total wavenumber $k$ is dominated by $k_\perp$. Increasing $\ell$ beyond this to get back to the case where $\ell \gg k r_0$, we have situation that approximately corresponds to having $k_\perp > k$. Such a scenario would be a mathematical impossibility in the flat-sky approximation, and formally the amplitude of the signal would go to zero. In our curved-sky treatment, however, we see that the cut-off for high $\ell$, while dramatic, is not precisely zero. This is due to projection effects, which cause any given $\ell$ mode to sample a spread of $k$ modes, in principle allowing arbitrarily high $\ell$ modes to have some (tiny) response to Fourier modes with very low $k$ values.
With such a strong dependence in power spectrum sensitivity to the values of $\ell$ and $m$, different modes should be weighted differently when averaged together. In principle, this weighting should depend on both $\ell$ and $m$. For simplicity, we will assume that different $m$ values are averaged together with uniform weights. This is a reasonable approximation for wide-field surveys, which is of course the regime that is being targeted in this paper. Indeed, for an all-sky survey, one can show that the integral in Eq. becomes independent of $m$, implying equal sensitivity to all $m$ modes and thus no reason to favor one specific mode over another. Performing the uniform average over Eq. and invoking Unsöld’s theorem then gives $$\label{eq:TotallyUnsold}
\frac{\sum_{m = -\ell}^\ell\langle | \overline{T}_{\ell m}^\textrm{meas} (k) |^2 \rangle}{2\ell + 1} \approx \frac{P(k)}{2 \pi^2} \int d^3 r \phi^2 (\mathbf{r}) j_\ell^2 (kr).$$ From this, it follows that given a set of modes with some particular $k$ and $\ell$ values, an estimator of the power spectrum can be formed by computing $$\label{eq:SlkDef}
S_\ell (k) \equiv 2 \pi^2 \left[\int d^3 r \phi^2 (\mathbf{r}) j_\ell^2 (kr)\right]^{-1} \frac{\sum_{m} | \overline{T}_{\ell m}^\textrm{meas} (k) |^2}{2 \ell + 1},$$ which we dub the spherical harmonic power spectrum. This is the quantity that we were seeking in Section \[sec:RotationalInvarianceOnly\], a curved sky analog to the cylindrical power spectrum $P(k_\perp, k_\parallel)$. If $\overline{T}_{\ell m}^\textrm{meas} (k)$ consists of contaminants to one’s measurement, $S_\ell (k)$ would essentially be the “power spectrum of contaminants", even though such a quantity is in principle not well-defined as the contaminants are typically not statistically translation-invariant. However, $S_\ell (k)$ and $P(k)$ can be directly compared since the two quantities have the same units, and in the limit of translation invariance, the ensemble average of $S_\ell(k)$ reduces to $P(k)$, by construction. We thus have a well-defined quantity that can be considered “the power spectrum of the signal on the $\ell$-$k$ plane", regardless of the relative ratios of cosmological signal and contaminants.[^8]
Once $S_\ell (k)$ has been computed for all $\ell$ values accessible to an experiment, different $\ell$ modes can be averaged together form a final estimate $\widehat{P} (k)$ of the power spectrum $P(k)$. Unlike with the average over $m$, uneven weights for the $\ell$ average are crucial since different $\ell$ modes can have very different sensitivities to the power spectrum, as our earlier example illustrated. The optimal weights $w_\ell$ for different $\ell$ values will in general depend on the details of one’s survey instrument. As a simple toy example, suppose an instrument has equal noise in all $\overline{T}_{\ell m}(k)$ modes (which is an impossibility in practice, since all instruments have finite angular resolution). An optimal signal-to-noise weighting of $| \overline{T}_{\ell m}^\textrm{meas} (k) |^2$ then reduces to a weighting by the strength of the signal, since the noise is constant. This is given by the integral in Eq. , which quantifies the extent to which the power spectrum is amplified (or depressed) in each $| \overline{T}_{\ell m}^\textrm{meas} (k) |^2$ mode. Forming a minimum variance estimator then requires a variance (i.e., squared) weighting by this factor, giving an estimator $\widehat{P} (k)$ of the power spectrum that takes the form $$\label{eq:WeightedPk}
\widehat{P} (k) \equiv \sum_\ell w_\ell S_\ell (k),$$ where $$\label{eq:MinVarEllWeights}
w_\ell \equiv \frac{ \left[ \int d^3 r \phi^2 (\mathbf{r}) j_{\ell}^2 (kr) \right]^2}{\sum_{\ell^\prime} \left[ \int d^3 r^\prime \phi^2 (\mathbf{r}^\prime) j_{\ell^\prime}^2 (kr^\prime) \right]^2}.$$
Foreground signatures in the spherical harmonic power spectrum {#sec:Foregrounds}
==============================================================
Having established $S_\ell (k)$ as a potential tool for separating contaminants from cosmological signal in a power spectrum measurement, we now specialize and consider the particular case of astrophysical foreground contamination. Our goal is to derive the signature of foreground contamination in $S_\ell (k)$, and to show that $S_\ell (k)$ is indeed a useful diagnostic for separating foregrounds from the cosmological signal. We will find that $S_\ell (k)$ performs this role for wide-field, curved-sky power spectrum analyses just as well as $P(k_\perp, k_\parallel)$ did for narrow fields of view. By this, we mean that in both cases the foregrounds are localized to predictable regions in the $\ell$-$k$ or $k_\perp$-$k_\parallel$ plane, enabling foregrounds to be mitigated by a few simple cuts to data.
![Example spherical Bessel functions $j_\ell (kr)$, arbitrarily normalized for ease of comparison. The grey band indicates the comoving radial extent of a $21\,\textrm{cm}$ intensity mapping survey operating from $145\,\textrm{MHz}$ to $155\,\textrm{MHz}$ (corresponding to a central redshift of 8.5, or a central radial distance of $r_0 \approx 6290h^{-1}$Mpc). The spherical Bessel functions enter in the radial transform from position space to the spherical Fourier-Bessel basis, and are integrated over the grey band with an $r^2$ weighting. Basis functions that describe fluctuations that are predominantly in the angular directions have $\ell \sim kr_0$ behave as power laws over the radial profile of the survey (red curve), and essentially average over the line-of-sight direction. Those whose fluctuations are oriented mainly in the radial direction have $\ell \lesssim kr_0$ behave like slowly modulated sinusoids (blue curve), and effectively take a Fourier transform along the line of sight. Modes with $\ell > kr_0$ (black curve) have very little response.[]{data-label="fig:bessels"}](bessels.pdf){width="48.00000%"}
When performing an intensity mapping survey with a spectral line, the cosmological component of the signal is expected to fluctuate rapidly as a function of frequency, since different frequencies probe different portions of our Universe. Foregrounds, on the other hand, are expected to be spectrally smooth [@dimatteo_et_al2002; @oh_and_mack2003; @deOliveiraCosta_et_al2008; @jelic_et_al2008; @liu_and_tegmark2012]. In principle, this allows foregrounds to be separated from the cosmological signal, for instance by fitting out a smooth spectral component [@wang_et_al2006; @liu_et_al2009a; @bowman_et_al2009; @liu_et_al2009b]. To take an even simpler approach, one expects spectrally smooth foregrounds to appear only at low $k_\parallel$, since $k_\parallel$ is the Fourier dual to line-of-sight distance, which is probed by the frequency spectrum. This is illustrated in the top left panel of Figure \[fig:fgSigs\], where we compute the $P(k_\perp, k_\parallel)$ signature of flat spectrum foregrounds for an intensity mapping survey with a radial profile given by $$\label{eq:CosineRadial}
\phi(r) = \cos \left[ \pi \left( \frac{r-r_0}{r_\textrm{max} - r_\textrm{min}} \right) \right],$$ within the comoving radial range of $r_\textrm{min} \approx 6230\,h^{-1}\textrm{Mpc}$ to $r_\textrm{max} \approx 6350\,h^{-1}\textrm{Mpc}$ and zero outside this range. This is representative of a $21\,\textrm{cm}$ intensity mapping survey with a $10\,\textrm{MHz}$ bandwidth centered around a frequency of $150\,\textrm{MHz}$ (corresponding roughly to $z \sim 8.5$). The precise form of the profile is arbitrary, and is only for illustrative purposes in this paper. In the angular direction we assume all-sky coverage. The foregrounds are assumed to have intrinsically flat (frequency-independent) spectra. One sees that their contribution to the power spectrum decreases in amplitude rapidly towards higher $k_\parallel$, suggesting that foregrounds can be mostly avoided by simply looking away from the lowest $k_\parallel$. Note that we have arbitrarily normalized the power to emphasize the morphology (rather than the absolute level) on the $k_\perp$-$k_\parallel$ plane.
{width="100.00000%"}
We now generalize the signature of foregrounds from the narrow-field to the curved sky using the spherical harmonic power spectrum. The foregrounds are again assumed to be independent of frequency, giving rise to a set of frequency-independent spherical harmonic coefficients $a^\textrm{fg}_{\ell m}$. The resulting $(k,\ell, m)$ modes are then given by $$\label{eq:fgTlm}
\overline{T}_{\ell m}^\textrm{fg} (k) = a_{\ell m}^\textrm{fg} \sqrt{\frac{2}{\pi}} \int_0^\infty \!dr\, r^2 j_\ell (kr) \phi(r),$$ which is simply Eq. but with the limitation of a survey volume $\phi$ and a flat spectrum assumption. Note that in this section, we will assume that the survey covers the entire angular extent of the sky (as depicted in Figure \[fig:surveyGeom\]), so that we have $\phi(r) $ rather than $\phi (\mathbf{r})$. In an analysis of real data this assumption may be inappropriate, but here we invoke it for the purposes of mathematical clarity. Inserting this expression into Eq. gives the spherical harmonic power spectrum of flat-spectrum foregrounds $$\label{eq:fgSlk}
S_\ell^\textrm{fg} (k) = 4 \pi C_\ell^\textrm{fg} \frac{\left[\int_0^\infty \!dr\, r^2 j_\ell (kr) \phi(r) \right]^2}{\int_0^\infty \!dr\, r^2 j_\ell^2 (kr) \phi^2(r)},$$ where $C_\ell^\textrm{fg}$ is the angular power spectrum of the foregrounds. For a given survey geometry and foreground model, one can evaluate this expression numerically to derive the signature of foregrounds as manifested in the spherical harmonic power spectrum. Before doing so, however, it is helpful to evaluate $S_\ell^\textrm{fg} (k)$ analytically in various limiting regimes on the $\ell$-$k$ plane to gain intuition for how the spherical harmonic power spectrum behaves. To identify these regimes (which demonstrate qualitatively different behavior), consider Fig. \[fig:bessels\], which shows $j_\ell (kr)$ for various choices of $\ell$ and $k$. Not all parts of these curves are relevant to the integrals in Eq. , since the radial extent of the survey $\phi(r)$ (indicated by the grey band) picks out only regions where $r \approx r_0$ to integrate over. Roughly speaking, there are two limiting regimes of interest. The first is where $\ell \sim k r_0$. In this regime, the Bessel functions behave like power laws that rise to a peak. The other regime is where $\ell \lesssim k r_0$. There, the Bessel functions are highly oscillatory, and the radial transform of Eq. is closely related to a Fourier transform along the line of sight. In principle, there exist modes with $\ell > kr_0$ exist, but as we argued in Section \[sec:FiniteVolume\], these modes have very low signal-to-noise, and we will not consider this regime further.
Mostly angular modes: $\ell \sim k r_0$ {#sec:MostlyAngular}
---------------------------------------
As discussed previously, the condition that $\ell \sim k r_0$ is synonymous with the statement that fluctuations are almost entirely in the angular direction. In this regime, the spherical Fourier-Bessel functions are not highly oscillatory, and are instead reasonably smooth. They are thus relatively broad compared to $\phi(r)$. To a good approximation, then, $r^2 j_\ell (kr)$ and $r^2 j_\ell^2 (kr)$ may be factored out of the integrals in Eq. , evaluating them at $r = r_0$. What remains is $$\begin{aligned}
\label{eq:FinalSlfgkHighEll}
S_\ell^\textrm{fg} (k) \Bigg{|}_{\ell \gtrsim k r_0} &\approx& 4 \pi C_\ell^\textrm{fg} r_0^2 \frac{\left[\int_0^\infty \!dr\, \phi(r) \right]^2}{\int_0^\infty \!dr\, \phi^2(r)} \nonumber \\
&\sim& 4 \pi C_\ell^\textrm{fg} r_0^2 \Delta r_\textrm{survey},\end{aligned}$$ where the final approximation is exact only for a survey that has a tophat profile in the radial direction, but still likely to be correct up to a factor of order unity otherwise. One sees that the $k$ dependence of $S_\ell^\textrm{fg} (k)$ drops out, and the measurement is essentially of the angular power spectrum of foregrounds because the radial Bessel transform effectively just averages all the radial fluctuations of the survey together.
Mostly radial modes: $\ell \ll k r_0$ {#sec:MostlyRadialNoInterferometry}
-------------------------------------
At low $\ell$ values, most of the spatial variations in one’s basis functions are along the line-of-sight. We enter this low $\ell$ regime when $\ell \ll kr_0$, in which case the Bessel functions may be approximated as $$j_\ell (kr) \approx \frac{1}{kr} \sin \left(kr-\frac{\pi \ell}{2} \right).$$ In this limit, the integral in the numerator of Eq. becomes $$\begin{aligned}
&& \int_0^\infty \!dr\, r^2 j_\ell (kr) \phi(r) =\frac{1}{k} \int_0^\infty \! dr\, r \sin \left(kr- \frac{\pi \ell}{2} \right) \phi(r) \nonumber \\
&& =- \frac{1}{k^2} \frac{\partial}{\partial \alpha} \left\{ \textrm{Re} \left[ \int_0^\infty \! dr \,e^{-i\alpha kr +i \pi \ell / 2} \phi(r) \right] \right\}_{\alpha = 1},\end{aligned}$$ where the “$\alpha = 1$" label signifies that $\alpha$ is to be set to unity after the partial derivative is taken. To proceed, we expand the definition of $\phi(r)$ to include the (unphysical) region of $r < 0$, declaring $\phi(r)$ to be zero when $r < 0$. This allows us to extend the integral to $-\infty$, which enables us to interpret it as a Fourier transform. Further defining $\Phi (r - r_0) \equiv \phi (r)$ to be a re-centered version of the radial profile of the survey for our convenience, we have $$\begin{aligned}
&&\int_0^\infty \!dr\, r^2 j_\ell (kr) \phi(r) \nonumber \\
&=& \frac{1}{k} \left[ r_0 \sin \left( k r_0- \frac{\pi \ell}{2} \right) \widetilde{\Phi} (k) - \cos\left( k r_0- \frac{\pi \ell}{2} \right) \widetilde{\Phi}^\prime (k) \right], \qquad\end{aligned}$$ where the $\widetilde{\Phi}^\prime \equiv \partial \widetilde{\Phi} / \partial k$. Using similar manipulations, the denominator of Eq. can be shown to be $$\begin{aligned}
\label{eq:DenomMostlyRadialFG}
&&\int_0^\infty \!dr\, r^2 j_\ell^2 (kr) \phi^2(r) \nonumber \\
&=& \frac{1}{2 k^2} \left[ \int_{-\infty}^\infty \!dr\, \Phi^2 (r) - \cos\left(2 k r_0- \pi \ell \right) \widetilde{\Phi} \star\widetilde{\Phi} (2 k) \right], \qquad\end{aligned}$$ where $\star$ denotes a convolution. To simplify matters, we may ignore the second term in this expression because it is small compared to the first. To see this, note that the first term can be written as $\widetilde{\Phi^2} (0)$. The relative size of the two terms is therefore determined by the relative magnitudes of $\widetilde{\Phi^2} (0)$ and $ \widetilde{\Phi} \star\widetilde{\Phi} (2 k) $. Now, $\widetilde{\Phi} (k)$ is a function that is reasonably sharply peaked about $k=0$, with a characteristic width given by $\sim 1/ \Delta r_\textrm{survey}$. We expect $\widetilde{\Phi^2}$ to be slightly broader; a back-of-the-envelope estimate would suggest that $\widetilde{\Phi^2}$ is roughly a factor of $\sqrt{2}$ broader than $\widetilde{\Phi}$. Continuing with our approximate line of reasoning, one would then expect $ \widetilde{\Phi} \star\widetilde{\Phi} (2 k)$ to be approximately the same size as $ \widetilde{\Phi} (\sqrt{2} k)$, which is likely to be small because typical $k$ values are of order $\sim 1/ \Delta r_\textrm{survey}$ or larger, placing one beyond the characteristic width of $\widetilde{\Phi}$, where the amplitude is much suppressed compared to the $k=0$ point. We thus conclude that the second term of Eq. may be neglected.
Putting everything together, we obtain $$\begin{aligned}
\label{eq:OscOsc}
S_\ell^\textrm{fg} (k) \approx \frac{8 \pi C_\ell^\textrm{fg}}{\int_{-\infty}^\infty \!dr\, \Phi^2 (r)} \bigg{[}&& r_0^2 \sin^2 \left( k r_0- \frac{\pi \ell}{2} \right) \widetilde{\Phi}^2 (k) \nonumber \\
&& -r_0 \sin \left( 2 k r_0- \pi \ell \right) \widetilde{\Phi} (k) \widetilde{\Phi}^\prime (k) \nonumber \\
&&+ \cos^2\left( k r_0- \frac{\pi \ell}{2} \right) \widetilde{\Phi}^{\prime 2} (k) \bigg{]}. \qquad\end{aligned}$$ This result can be further simplified by considering the length scales involved. Recall that that the key approximation of this subsection is that the spatial fluctuations are mostly along the radial direction. For a survey with radial resolution $\Delta r_\textrm{res}$ (determined by an instrument’s spectral resolution), a natural choice for a bin size in $k$ would be $\sim\!2 \pi / \Delta r_\textrm{res}$. Since the value of $k$ is multiplied by $r_0$ inside the oscillatory terms of Eq. , and $r_0 \gg \Delta r_\textrm{res}$, it follows that one goes through many cycles of the sinusoids within each bin in $k$ in any practical measurement. The middle term of Eq. thus averages to zero, while the squared sinusoids average to $1/2$. We thus have $$S_\ell^\textrm{fg} (k) \approx 4 \pi \frac{ C_\ell^\textrm{fg}}{\int_{-\infty}^\infty \!dr\, \Phi^2 (r)} \left[ r_0^2 \widetilde{\Phi}^2 (k) +\widetilde{\Phi}^{\prime 2} (k) \right]$$ Now, the two terms seen here that comprise $S_\ell^\textrm{fg} (k) $ are not of equal importance. Dimensional analysis suggests that the derivative of $\Phi$ is of order $\Phi^\prime \sim \Phi / \Delta r_\textrm{survey}$, while the derivative of its Fourier transform $\widetilde{\Phi}$ is of order $\widetilde{\Phi}^\prime \sim \widetilde{\Phi} \Delta r_\textrm{survey}$, a fact that can be verified by testing various functional forms for $\Phi$. The first term in our expression for $S_\ell^\textrm{fg} (k)$ is thus larger than the second term by a factor of $(r_0 / \Delta r_\textrm{survey})^2$, which greatly exceeds unity for high-redshift measurements. These simplifications yield the final expression $$\label{eq:FinalSlfgkLowEll}
S_\ell^\textrm{fg} (k) \Bigg{|}_{\ell \lesssim k r_0} \approx 4 \pi C_\ell^\textrm{fg} \frac{r_0^2 \widetilde{\Phi}^2 (k)}{\int_{-\infty}^\infty \!dr\, \Phi^2 (r)}.$$ This result is essentially identical to its flat-sky counterpart on the $k_\perp$-$k_\parallel$ plane. There, the foregrounds were seen to be confined mostly to low $k_\parallel$ values, with the characteristic width of the fall-off towards higher $k_\parallel$ of $\sim 1 / \Delta r_\textrm{survey}$, as expected from the Fourier transform of data that spans a length of $\Delta r_\textrm{survey}$. Here, in the regime where our modes are dominated by radial fluctuations, we have $k$ taking the place of $k_\parallel$. But the behavior is the same, since $\widetilde{\Phi} (k)$ falls off as $\sim 1 / \Delta r_\textrm{survey}$.
Numerical Results {#sec:Numerics}
-----------------
Summarizing the last two results, it is pleasing to note that the even though Eqs. and were derived as different limiting cases, the latter converges to the former when $k\rightarrow 0$. This suggests a rather smooth transition between the two regimes and a simple signature of foregrounds as a function of $\ell$ and $k$: at low $k$, the foregrounds are a strong contaminant, but their influence quickly falls off towards higher $k$.
We confirm this behavior in the top right panel of Figure \[fig:fgSigs\] by plotting a numerically computed $S_\ell^\textrm{fg} (k)$. The survey parameters are assumed to be the same as in Section \[sec:Foregrounds\]. There is a qualitative similarity between the flat-sky plot of $P(k_\perp, k_\parallel)$ in the top left panel, and the curved-sky plot of $S_\ell (k)$ in the top right. This suggests that the latter will be just as successful as the former in localizing foregrounds in their respective planes. Quantitatively, one sees a sharp drop-off towards higher $k$ (or $k_\parallel$), with some ringing due to our cosine radial profile. Admittedly, the drop-off is not quite as steep as one might hope, given that the foregrounds can easily be six to nine orders of magnitude brighter than the cosmological in power spectrum units [@santos_et_al2005; @jelic_et_al2008; @bernardi_et_al2009; @bernardi_et_al2010]. However, a large number of tools can be employed to further suppress foregrounds at high $k$ (or $k_\parallel$). For example, foregrounds can be filtered or directly subtracted, whether via the construction of foreground models or through blind methods [@wang_et_al2006; @gleser_et_al2008; @liu_et_al2009a; @bowman_et_al2009; @liu_et_al2009b; @harker_et_al2009; @petrovic_and_oh2011; @paciga_et_al2011; @Parsons_et_al2012b; @liu_and_tegmark2012; @chapman_et_al2012; @chapman_et_al2013; @wolz_et_al2014; @shaw_et_al2014a; @shaw_et_al2014b; @wolz_et_al2015]. Leakage of foregrounds from low $k$ to high $k$ can be mitigated by imposing tapering functions to apodize the radial profile $\phi(r)$ [@Thyagarajan2013]. This would, for instance, reduce the Fourier space ringing from the cosine form of Eq. , which causes the horizontal stripes that are visually obvious in the top row of Figure \[fig:fgSigs\]. Finally, statistical methods can be employed to selectively downweight foreground contaminated modes, whether prior to the squaring of temperature data in power spectrum estimation [@liu_and_tegmark2011; @liu_et_al2014a; @trott_et_al2016] or after [@dillon_et_al2014; @liu_et_al2014b]. Our goal here was only to show that $S_\ell (k)$ is just as viable a foreground diagnostic for the curved sky as $P(k_\perp, k_\parallel)$ is for the flat sky, and Figure \[fig:fgSigs\] shows that this is indeed the case.
Interloper lines in the spherical harmonic power spectrum {#sec:Interlopers}
=========================================================
Aside from broadband foregrounds that are spectrally smooth, some intensity mapping surveys must also deal with the problem of interloper lines, where emission from two different spectral lines that are sourced at different radial distances may nonetheless redshift into the same observing band. More concretely, an interloper line with a rest frequency of $\nu_\textrm{rest}^\prime$ emitted at redshift $z^\prime$ will appear at the same observed frequency as another line (say, the one targeted by an intensity mapping survey) with rest frequency $\nu_\textrm{rest}$ at redshift $z$ if $(1+z^\prime) / \nu_\textrm{rest}^\prime = (1+z) / \nu_\textrm{rest}$. The interloper line thus acts as an additional foreground contaminant. For $21\,\textrm{cm}$ intensity mapping this is typically not a problem, simply because there lack plausible spectral line candidates with appropriate rest frequencies. In contrast, \[CII\] and CO lines are both candidates for intensity mapping surveys, and can easily be confused with one another.
Since interloper lines may themselves trace cosmic structure (albeit at different redshifts), they are not spectrally smooth foreground contaminants, and thus cannot be mitigated by the methods described in the rest of this paper. To deal with this, a variety of techniques have been proposed in the literature, including source masking [@silva_et_al2015; @yue_et_al2015; @breysse_et_al2015], cross-correlation with external datasets [@visbal_and_loeb2010; @gong_et_al2012; @gong_et_al2014], comparison to companion lines [@kogut_et_al2015], and the exploitation of angular fluctuations to reconstruct three-dimensional source distributions [@dePutter_et_al2014]. Recently, @cheng_et_al2016 and @lidz_and_taylor2016 proposed a method for separating interloper lines by invoking the statistical isotropy of the cosmological signal. The key observation is that the rest frequency of a line enters the frequency-radial distance mapping of Eq. in a different way than it does in the angle-transverse distance conversion of Eq. . If emission from an interloper line is mistaken as the targeted line in a survey, it will be mapped to incorrect cosmological coordinates. As a result, the emission will no longer be statistically isotropic, in contrast to emission from the targeted line, which will have been mapped correctly and thus will be statistically isotropic. In terms of the power spectrum, emission from the targeted line will appear in the cylindrical power spectrum $P(k_\perp, k_\parallel)$ as a function of $k \equiv (k_\perp^2 + k_\parallel^2)^{1/2}$ only, while interloper emission will have a non-trivial dependence on $k_\perp$ and $k_\parallel$. This difference in $k_\perp$-$k_\parallel$ signature provides a way to identify interloper emission.
In this section, we build on the work of @cheng_et_al2016 and @lidz_and_taylor2016, generalizing their flat-sky treatment to the curved sky using the spherical harmonic power spectrum. Our goal will be to show that just as $P(k_\perp, k_\parallel)$ is no longer just a function of $k$ if the incorrect rest frequency $\nu_\textrm{inc}$ is assumed, $S_\ell (k)$ will similarly develop a dependence on $\ell$ under those circumstances. To begin, we note that Eq. is always exact, since it only relies on angular information, which does not require knowledge of the rest frequency of the spectral line. The assumption of an incorrect rest frequency enters only in Eq. , when one must map frequencies to radial distances. Suppose some emission originates from a comoving location $\mathbf{r} = r {\hat{\mathbf{r}}}$. If the incorrect frequency-radial distance relation is used due to a mistaken assumption about the rest frequency of the emission, this emission will be mapped to a location $\mathbf{r} \equiv s(r) {\hat{\mathbf{r}}}$ instead, where $s$ is the incorrect radial distance, which is a function of the correct distance $r$. As a result, Eq. becomes $\overline{T}_{\ell m}^\textrm{inc} (k)$, the incorrectly mapped version of $\overline{T}_{\ell m} (k)$, and take the form $$\overline{T}_{\ell m}^\textrm{inc} (k) \equiv \sqrt{\frac{2}{\pi}} \int d^3 r j_\ell (kr) Y_{\ell m}^* ({\hat{\mathbf{r}}}) T[s(r) {\hat{\mathbf{r}}}] \phi[s(r) {\hat{\mathbf{r}}}],$$ where we have included the finite volume of our survey via the function $\phi$, just as we did in the previous section. Writing the $T\phi$ term in terms of their Fourier transforms and repeating steps analogous to the ones used between Eqs. and , we obtain $$\begin{aligned}
\overline{T}_{\ell m}^\textrm{inc} (k) = i^\ell 4 \sqrt{2 \pi} \int \frac{d^3 k^\prime}{(2\pi)^3} \frac{d^3 k^{\prime \prime}}{(2\pi)^3} Y_{\ell m} ({\hat{\mathbf{k}}}^\prime ) \widetilde{\phi} (\mathbf{k}^\prime - \mathbf{k}^{\prime \prime}) \nonumber \\
\times \widetilde{T} (\mathbf{k}^{\prime \prime}) \int dr r^2 j_\ell (kr) j_\ell [ k^\prime s(r)]. \quad \end{aligned}$$ To relate this to the power spectrum, we square this expression, take the ensemble average, and average over $m$ values. Performing manipulations similar to those that led to Eq. results in $$\begin{aligned}
&&\frac{\sum_{m = -\ell}^\ell\langle | \overline{T}_{\ell m}^\textrm{meas} (k) |^2 \rangle}{2\ell + 1} \approx P(\overline{k}) \frac{2}{\pi^4} \int d^3 r \phi^2 (\mathbf{r}) \nonumber \\
&&\times \left( \int dr^\prime r^{\prime 2} d k^\prime k^{\prime 2} j_\ell (k^\prime r) j_\ell(k r^\prime) j_\ell [ k^\prime s (r^\prime)] \right)^2, \qquad\end{aligned}$$ where $\overline{k}$ is some wavenumber that is not necessarily equal to $k$. In other words, with an incorrect mapping of radial distances, we should not necessarily expect $\langle | \overline{T}_{\ell m}^\textrm{meas} (k) |^2 \rangle$ to probe a distribution of power that is sharply peaked around $k$. Any bias in the probed wavenumber, however, is irrelevant for our present purposes, which is simply to show that an $\ell$ dependence is acquired in our (no longer isotropic) estimate of the power spectrum. Performing the $k^\prime$ integral using Eq. (but with $r$ and $k$ swapping roles) and inserting the result into Eq. , one obtains $$\begin{aligned}
S_\ell^\textrm{inc} (k)=&&P(\overline{k}) \left[ \int d^3 r \phi^2 (\mathbf{r}) j_\ell^2 (kr) \right]^{-1} \nonumber \\
&&\times \int d^3 r \frac{\phi^2 (\mathbf{r})}{s^\prime [s^{-1} (r)]} \left(\frac{s^{-1} (r)}{r}\right)^2 j_\ell^2 [ks^{-1} (r)] \qquad\end{aligned}$$ for the estimated spherical harmonic power spectrum under the assumption of a mistaken rest frequency. Here, $s^\prime \equiv \partial s / \partial r$ (i.e., the derivative of the incorrectly mapped radial distance with respect to the true radial distance) and $s^{-1}$ denotes an inverse mapping, not a reciprocal. Notice that if the rest frequency is correct (i.e., one is dealing with emission from the targeted line rather than the interloper line), then $s$ is the identity function, $s^\prime$ is unity, and the two integrals cancel to leave a result that is $\ell$-independent. In general, however, the result will be $\ell$-dependent. We thus conclude that just as anisotropies in $P(k_\perp, k_\parallel)$ can be used to detect interloper lines within the flat-sky approximation, $S_\ell (k)$ can be used in the same way for a full curved-sky treatment.
Spherical Harmonic Power Spectrum Measurements with Interferometers {#sec:Interferometry}
===================================================================
In previous sections, we have focused on understanding the *intrinsic* spherical harmonic power spectrum $S_\ell (k)$ without the inclusion of any instrumental effects other than a selection function to account for survey geometry. For some intensity mapping efforts, the exclusion of these effects will not result in major differences in $S_\ell (k)$. For instance, at higher frequencies (say, those relevant to \[CII\] intensity mapping) it is common to perform intensity mapping with traditional single dish telescopes and spectrometers. With such systems, the equations derived so far in this paper are a reasonable approximation for what one might see in real data, perhaps with the addition of a high noise component at high $\ell$ and $k$ to reflect finite angular and spectral resolution. In contrast, at low frequencies it is common to perform intensity mapping using radio interferometers. In this section, we will show that with data from interferometers, $S_\ell (k)$ behaves qualitatively differently from what we have considered so far. Despite these differences, once the data (and any accompanying metrics for describing their statistical properties) are reduced to modes in the spherical Fourier-Bessel basis, it is irrelevant whether they were collecting using single dish telescopes or interferometers. The spherical Fourier-Bessel basis and the spherical harmonic power spectrum $S_\ell (k)$ may thus be a useful meeting point for cross-correlations between the $21\,\textrm{cm}$ and CO/\[CII\] lines (e.g., as proposed in @lidz_et_al2011).
Interferometers are frequently used for intensity mapping measurements because they are essentially Fourier-space instruments, with each baseline of an interferometer directly sampling a fringe pattern that approximates one of the spatial Fourier modes of interest. They are therefore a relatively inexpensive way to perform high-sensitivity measurements of the power spectrum. However, the picture of an interferometer as a Fourier-space instrument is precisely correct only in the limit that the sky is flat. This assumption is typically invoked in derivations of estimators for connecting interferometric measurements to power spectra [@hobson_et_al1995; @white_et_al1999; @padin_et_al2001; @halverson_et_al2002; @hobson_et_al2002; @myers_et_al2003; @parsons_et_al2012a; @parsons_et_al2014]. It is, however, explicitly violated by the wide-field nature of many instruments built for intensity mapping. In this section, we will address this shortcoming, using the spherical Fourier-Bessel formalism to relate interferometric data to the cosmological power spectrum in a way that fully respects curved sky effects.
For the purposes of three-dimensional intensity mapping experiments, interferometers come with the added complication of being inherently chromatic instruments. Consider, for example, the visibility measured by a single baseline of an interferometric array:
$$\begin{aligned}
V(\mathbf{b}, \nu) &=& \int d\Omega \phi(r_\nu) A({\hat{\mathbf{r}}}, \nu) I({\hat{\mathbf{r}}}, \nu)e^{ - i 2\pi \nu \mathbf{b} \cdot{\hat{\mathbf{r}}}/ c} \label{eq:VisDef}\\
&\equiv&\int d \Omega \phi(r_\nu) B({\hat{\mathbf{r}}}, \nu) T({\hat{\mathbf{r}}}, \nu) e^{ - i 2\pi \nu \mathbf{b} \cdot{\hat{\mathbf{r}}}/ c} \label{eq:CurvedVisibility}\\
&\approx& \int \frac{d^2 r_\perp}{r_\nu^2} \phi(r_\nu) B({\hat{\mathbf{r}}}, \nu) T({\hat{\mathbf{r}}}, \nu)e^{ - i 2\pi \nu \mathbf{b} \cdot \mathbf{r}_\perp / c r_\nu}, \label{eq:FlatVisibility}\qquad\quad \end{aligned}$$
where in the last line we invoked the narrow-field, flat-sky approximation, allowing a “line-of-sight" direction to be unambiguously identified and a position vector $\mathbf{r}_\perp$ transverse to this direction to be defined. In the penultimate line we used the Rayleigh-Jeans Law to convert from intensity to brightness temperature, defining a modified primary beam $$\label{eq:Bdef}
B({\hat{\mathbf{r}}}, \nu) \equiv \frac{2 k_B}{c^2} \nu^2 A({\hat{\mathbf{r}}}, \nu).$$ One sees that in the flat-sky limit, the complex exponential takes the form of $\exp \left( - i \mathbf{k}_\perp \cdot \mathbf{r}_\perp \right)$, and thus the baseline probes a spatial mode perpendicular to the line of sight with wavevector $\mathbf{k}_\perp = 2 \pi \nu \mathbf{b} / c r_\nu$. The key feature to note here is that this spatial scale is dependent on $\nu$. Interferometers are therefore inherently chromatic in the sense that the Fourier mode probed by a particular baseline depends on frequency, particularly if the baseline is long. This complicates the power spectrum measurement, for in order to access Fourier modes along the line of sight (characterized by wavenumber $k_\parallel$), it is necessary to perform a Fourier transform along the frequency axis. At least for data from a single baseline, the chromaticity means that $\mathbf{k}_\perp$ is not held constant during the line of sight Fourier transform. This causes couplings between $k_\parallel$ and $\mathbf{k}_\perp$ modes, and is responsible for the wedge feature that has been discussed extensively in the previous literature. The wedge arises when the chromaticity of an interferometer imprints this chromaticity on observed foregrounds. Being spectrally smooth, the foregrounds should in principle be localized to low $k_\parallel$ modes (as we saw in the top panels of Figure \[fig:fgSigs\]), but in practice the imprinted chromaticity causes them to appear at higher $k_\parallel$ modes in a wedge-like signature.
The wedge is both a problem and an opportunity. The wedge is a problem because it increases (compared to a theoretically ideal situation with no instrument chromaticity) the number of Fourier modes that are foreground-dominated and thus unavailable for a measurement of the cosmological signal. These unavailable modes are often the ones that are highest in signal-to-noise, resulting in a significant reduction in sensitivity [@pober_et_al2014; @chapman_et_al2016]. However, the wedge is also an opportunity because it can be shown (in a reasonably general manner) that it is limited in extent, i.e., the foreground contamination does not extend beyond the confines of the wedge shape. Observations can therefore be targeted at modes that are outside the wedge, and instruments may be designed conservatively to optimize such observations [@parsons_et_al2012a]. Indeed, this is the general principle behind the design of HERA [@deboer_et_al2016].
That smooth spectrum foregrounds have a well-defined signature in the form of the wedge is one of the reasons that recent works have espoused the $P(k_\perp,k_\parallel)$ power spectrum as a useful diagnostic for data analysis. In order for our proposed statistic $S_\ell (k)$ to be useful in the same way, it is necessary to show that the chromatic influence of an interferometer also gives a well-defined and well-localized signature $\ell$-$k$ space. We will do so in the following subsections once we have established the connection between curved sky power spectra and interferometeric data, finding that foregrounds are again localized to a wedge. We will focus on single-baselines analyses of the data, as this provides a conservative estimate for the extent of the foreground wedge in $S_\ell (k)$. Multi-baseline information can be used to alleviate wedge effects, because one can essentially combine data from different frequencies and different baselines that have the same ratio $\nu \mathbf{b} / r_\nu$, alleviating the chromatic effects that caused the wedge in the first place. There thus exist methods for reducing the extent of the wedge, and our single-baseline treatment should be considered a worst-case scenario.
Delay spectrum power spectrum estimation {#sec:DelayIntro}
----------------------------------------
To estimate the power spectrum from a single baseline, one begins by forming the *delay spectrum* of the baseline’s visibility. This is accomplished by Fourier transforming the visibility along the frequency axis to obtain $$\label{eq:DelayDef}
\widetilde{V} (\mathbf{b}, \tau) \equiv \int \!d\nu\, \gamma(\nu) e^{i 2\pi \nu \tau} V(\mathbf{b}, \nu),$$ where $\gamma (\nu)$ is an optional tapering function chosen by the data analyst. Given that $V$ approximates the sky brightness Fourier transformed in the axes perpendicular to the line of sight, $\widetilde{V}$ serves as an approximation for the $\widetilde{T}(\mathbf{k})$. The delay spectrum can then be squared and normalized to yield an estimator for the power spectrum $P(k)$.
As we discussed above, a single baseline probes different $\mathbf{k}_\perp$ scales at different frequencies. Power spectra estimated using delay spectra are therefore often considered mere approximations to “true" power spectra. However, an estimator formed from the delay spectrum represents a perfectly valid estimator, so long as error statistics are included in the final results. The quoted error statistics on a power spectrum estimate $\widehat{P}(k_*)$ at some spatial scale $k_*$ should include not only the error bars on the value of $\widehat{P}$ itself, but also window functions for describing the (sometimes broad) distribution of $k$ values that contribute to a power estimate that is centered on $k = k_*$. Because single-baseline estimators have a chromatic scale-dependence, their resulting window functions will be wider than what might be in principle achievable using a well-controlled multi-baseline approach. In general, however, the latter will still give windows of non-zero width (due to a combination of finite-volume and analysis pipeline effects), and in that sense a delay spectrum power spectrum with well-documented error statistics is not any more of an approximation than any other method.[^9]
In the following subsections we establish the framework for single-baseline analyses of the power spectrum in the curved sky. Section \[sec:WindowFcts\] computes the window functions associated with delay spectrum power spectrum estimation. Section \[sec:SingleBlNorm\] provides a rigorous derivation of power spectrum normalization, using our spherical harmonic formalism to incorporate curved sky treatments that have so far been neglected in the literature. Section \[sec:CurvedSkyWedge\] then demonstrates how the foreground wedge signature seen in $P(k_\perp, k_\parallel)$ spectra is preserved in $S_\ell (k)$.
Window functions of a delay-based power spectrum estimate {#sec:WindowFcts}
---------------------------------------------------------
As mentioned above, one estimates the power spectrum from a single baseline by first forming the delay spectrum $\widetilde{V}$, followed by a subsequent squaring of the result. Computing the window functions of such an estimate requires relating our measurements to a theoretical power spectrum. To do so, we take the definition of a single baseline’s visibility from Eq. and expand the temperature field in spherical harmonics, giving $$V(\mathbf{b}, \nu) = \sum_{\ell m} \phi (r_\nu) a_{\ell m} (\nu) f_{\ell m} (\mathbf{b}, \nu),$$ where we have defined $$\label{eq:flm}
f_{\ell m}(\mathbf{b}, \nu) \equiv \int d \Omega B({\hat{\mathbf{r}}}, \nu) Y_{\ell m} ({\hat{\mathbf{r}}}) e^{ - i 2\pi \nu \mathbf{b} \cdot{\hat{\mathbf{r}}}/ c}.$$ as the response of a baseline $\mathbf{b}$ to an excitation of the spherical harmonic with indices $\ell$ and $m$. The detailed properties of this response function have previously been explored in the literature [@shaw_et_al2014a; @zheng_et_al2014; @shaw_et_al2014b; @zhang_et_al2016a; @zhang_et_al2016b]. Here, we relate this response function to a delay spectrum approach. To proceed, we use Eq. (or rather, the inverse of the transformation it describes) to express $a_{\ell m}$ in terms of its spherical Fourier-Bessel expansion, giving $$\label{eq:VisTlmConnection}
V(\mathbf{b}, \nu) = \sqrt{\frac{2}{\pi}} \sum_{\ell m} \int \!dk\, k^2 j_\ell (k r_\nu) \overline{T}_{\ell m} (k) f_{\ell m} (\mathbf{b}, \nu) \phi(r_\nu).$$ Forming the delay spectrum $\widetilde{V}$ from this then yields $$\label{eq:VtildeInToverline}
\widetilde{V} (\mathbf{b}, \tau) = \sqrt{\frac{2}{\pi}} \sum_{\ell m} \int \!dk\, k^2 g_{\ell m} (k; \mathbf{b}, \tau) \overline{T}_{\ell m} (k),$$ where $$\label{eq:glm}
g_{\ell m}(k; \mathbf{b}, \tau) \equiv \int d\nu e^{i 2\pi \nu \tau} j_\ell (k r_\nu) f_{\ell m} (\mathbf{b}, \nu) \phi(r_\nu) \gamma (\nu).$$
Now suppose the measured sky consists only of the cosmological signal. The $\overline{T}_{\ell m} (k)$ modes are then directly related to the power spectrum via Eq. , and the ensemble average of the square of the delay spectrum reduces to $$\begin{aligned}
\label{eq:delayWindow}
\langle |\widetilde{V} (\mathbf{b}, \tau) |^2 \rangle && = \frac{2}{\pi} \sum_{\ell m \ell^\prime m^\prime} \int dk dk^\prime k^2 k^{\prime 2} \langle \overline{T}_{\ell m} (k) \overline{T}_{\ell^\prime m^\prime}^* (k^\prime)\rangle\nonumber \\
&& \qquad \times \, g_{\ell m} (k; \mathbf{b}, \tau) g_{\ell^\prime m^\prime}^* (k^\prime; \mathbf{b}, \tau) \nonumber \\
&& = \sum_{\ell} \int dk W_\ell^\textrm{unnorm} (k; \mathbf{b}, \tau) P(k), \qquad \end{aligned}$$ where $$\label{eq:DelayWindowFcts}
W_\ell^\textrm{unnorm} (k; \mathbf{b}, \tau) \equiv \frac{2k^2}{\pi} \sum_m | g_{\ell m} (k ; \mathbf{b}, \tau)|^2$$ are the (unnormalized) window functions. For given values of $\mathbf{b}$ and $\tau$, Eq. shows that the window function describes the linear combination of modes on the $\ell$-$k$ plane that are probed by the quantity $\langle |\widetilde{V} (\mathbf{b}, \tau) |^2 \rangle$. If $ |\widetilde{V} (\mathbf{b}, \tau)|^2$ is to be a good estimator of the power spectrum, the window function for each $(\mathbf{b},\tau)$ pair should satisfy two conditions. First, each window function should be reasonably sharply peaked around some location on the $\ell$-$k$, giving a precise measurement of the power spectrum on some scale rather than a broad combination of scales. Second, the window functions for different values of $(\mathbf{b},\tau)$ should be centered on different locations on the $\ell$-$k$ plane. In other words, the ideal collection of window functions should divide the $\ell$-$k$ plane into a set of mutually exclusive and collectively exhaustive cells [@tegmark_et_al1998].
![Example window functions on the $\ell$-$k$ plane, given by Eq. . Each set of contours describes the linear combination of modes on the $\ell$-$k$ plane sampled by a power spectrum estimator formed by a particular baseline-delay combination. From bottom to top, the three rows correspond to the windows for estimators with delay $\tau = 273\,\textrm{ns}$, $703\,\textrm{ns}$, and $1133\,\textrm{ns}$. From left to right, each column corresponds to windows for estimators with baseline lengths from $10\,\textrm{m}$ to $190\,\textrm{m}$ in $10\,\textrm{m}$ increments. To allow everything to be easily visualized on a common color scale, each window function is normalized to peak at unity. The boundary $\ell = k r_0$ is demarcated by the bold red line. Parts of the plane below this line are difficult to access, and all window functions are seen to taper off towards the line.[]{data-label="fig:windowFcts"}](windowFcts.pdf){width="48.00000%"}
In Figure \[fig:windowFcts\] we show example $\ell$-$k$ plane window functions for various choices of $(\mathbf{b},\tau)$, computed using the same survey parameters as in Section \[sec:Numerics\]. All the window functions tend to taper off towards the line $\ell = kr_0$, consistent with our previous argument that regions below this line are difficult to probe with any substantial signal-to-noise. We find that to a good approximation, the peaks of the window functions are located at $$\label{eq:PeakLocs}
k \approx 2 \pi \sqrt{\left(\frac{\tau}{\alpha_0}\right)^2 + \left(\frac{b \nu_0}{c r_0}\right)^2}; \quad \ell \approx \frac{2 \pi b \nu_0}{c},$$ where $\alpha_0$ is the radial distance-frequency conversion from Eq. evaluated with the reference frequency set to $\nu_0$, the frequency at the middle of our observational band. These expressions are what one would write down assuming a flat-sky mapping between interferometer parameters $(\mathbf{b}, \tau)$ and spatial fluctuation wavenumbers $\ell$ and $k$. Given this, it is unsurprising that the accuracy of these approximations goes down at low $\ell$, where curved sky effects are expected to be the most important. Nonetheless, the accuracy is reasonable throughout: we find that the $\ell$ location of the peaks predicted by Eq. to be good to $\sim 10\%$ at $\ell \sim 30$, improving to $5\%$ by $\ell \sim 50$ and with further improvements as $\ell$ increases. Nowhere in the $\ell$-$k$ range bracketed by the window functions shown in Figure \[fig:windowFcts\] do we find the errors to be larger than $10\%$. Our prediction for the $k$ location of the peaks is better yet, with the errors never exceeding $\sim 5\%$, and more typically at the sub-percent level. In any case, our approximations are meant for illustration purposes only. In a practical estimation of power spectra, one should compute the exact window functions (as we have done here by numerical means), and these window functions should accompany any power spectrum results that are presented.
For $|\widetilde{V} (\mathbf{b}, \tau)|^2$ to serve as a useful estimator of the power spectrum, its window functions must not only be centered on different parts of the $\ell$-$k$ plane for different values of $\mathbf{b}$ and $\tau$ (as we have just shown). The windows must also be relatively compact, and we see in Figure \[fig:windowFcts\] that this is indeed the case. A key feature, however, is that the window functions become elongated in the $k$ direction as one moves to higher $\ell$. This effect is exactly analogous to the $k_\parallel$ elongation of window functions at high $k_\perp$ in the flat-sky case examined in @liu_et_al2014a, and is due to the fact that the higher $\ell$ (or $k_\perp$) are probed by longer baselines, which (as we discussed in Section \[sec:Interferometry\]) exhibit a more chromatic response. The elongation seen here is our first hint of the foreground wedge, since an extended window function in $k$ (or $k_\parallel$) will pick up more foreground contamination from the lower portions of the $\ell$-$k$, where foregrounds intrinsically reside. This causes foregrounds to leak “upwards" on the plane, with the extent of the leakage tracking the increasingly exaggerated elongation towards higher $\ell$ (or $k_\perp$), thus resulting in a wedge-like feature. We will derive the $\ell$-$k$ plane foreground wedge more rigorously in Section \[sec:CurvedSkyWedge\]. For now, it suffices to say that since the window functions seen in Figure \[fig:windowFcts\] are reasonably compact, we have successfully demonstrated that $|\widetilde{V} (\mathbf{b}, \tau)|^2$ is just as potent an estimator of the power spectrum in our full curved-sky formalism as it is in the flat-sky.
Normalizing a delay-based power spectrum estimate {#sec:SingleBlNorm}
-------------------------------------------------
In the previous subsection, we showed that the $|\widetilde{V} (\mathbf{b}, \tau)|^2$ is a suitable estimator for the cosmological power spectrum. However, it is not yet properly normalized. Here, we derive the normalization factor that $|\widetilde{V} (\mathbf{b}, \tau)|^2$ must be divided by to obtain an unbiased estimate of the power spectrum.
From Eq. , we see that $\langle |\widetilde{V} (\mathbf{b}, \tau) |^2 \rangle$ measures a weighted sum/integral of the power spectrum. For our estimator to be properly normalized, the weighted sum/integral ought to be a weighted average instead. We can accomplish this by dividing $\langle |\widetilde{V} (\mathbf{b}, \tau) |^2 \rangle$ by the sum/integral of $W_\ell^\textrm{unnorm} (k; \mathbf{b}, \tau)$, which serves as a normalization factor. This normalization can be considerably simplified: $$\begin{aligned}
\label{eq:DelayNormSimplification}
\sum_{\ell m} \!\!\!&&\int dk W_\ell^\textrm{unnorm} (k; \mathbf{b}, \tau) \nonumber \\
&=& \frac{2}{\pi} \sum_{\ell m} \!\int dk \,k^2 |g_{\ell m} (k; \mathbf{b}, \tau)|^2 \nonumber \\
&=& \frac{2}{\pi} \int d\nu d\nu^\prime e^{i 2\pi (\nu-\nu^\prime) \tau} \int dk k^2 j_\ell (kr_\nu) j_\ell(k r_{\nu^\prime}) \nonumber \\
&& \qquad \phi(r_\nu) \phi(r_{\nu^\prime}) \gamma (\nu) \gamma (\nu^\prime) \sum_{\ell m} f_{\ell m} (\mathbf{b}, \nu) f_{\ell m}^* (\mathbf{b}, \nu^\prime) \qquad \nonumber \\
&=& \int \frac{d\nu}{r_\nu} \frac{d\nu^\prime}{r_{\nu^\prime}} e^{i 2\pi (\nu-\nu^\prime) \tau} \delta^D (r_\nu - r_{\nu^\prime}) \nonumber \\
&& \qquad \phi(r_\nu) \phi(r_{\nu^\prime}) \gamma (\nu) \gamma (\nu^\prime) \sum_{\ell m} f_{\ell m} (\mathbf{b}, \nu) f_{\ell m}^* (\mathbf{b}, \nu^\prime), \qquad\end{aligned}$$ where in the last line we invoked the orthnormality of spherical bessel functions with different arguments. Continuing, we have $$\begin{aligned}
&& \int \frac{d\nu}{r_\nu} \frac{d\nu^\prime}{r_{\nu^\prime}} e^{i 2\pi (\nu-\nu^\prime) \tau} \frac{\delta^D (\nu - \nu^\prime)}{\alpha_\nu} \nonumber \\
&& \qquad \phi(r_\nu) \phi(r_{\nu^\prime}) \gamma (\nu) \gamma (\nu^\prime) \sum_{\ell m} f_{\ell m} (\mathbf{b}, \nu) f_{\ell m}^* (\mathbf{b}, \nu^\prime) \nonumber \\
&=& \int \frac{d\nu}{r_\nu^2 \alpha_\nu} \phi^2(r_\nu) \gamma^2 (\nu) \sum_{\ell m} | f_{\ell m} (\mathbf{b}, \nu)|^2 \nonumber \\
&=& \int d\Omega d\nu \frac{B^2 ({\hat{\mathbf{r}}}, \nu) \phi^2(r_\nu) \gamma^2 (\nu)}{r_\nu^2 \alpha_\nu},
\end{aligned}$$ where in the last equality we used Eq. in conjunction with the fact that $\sum_{\ell m} Y_{\ell m} ({\hat{\mathbf{r}}}) Y_{\ell m}^* ({\hat{\mathbf{r}}}^\prime) = \delta^D({\hat{\mathbf{r}}}, {\hat{\mathbf{r}}}^\prime)$.
Putting everything together, a properly normalized estimator $\widehat{P}(k)$ of the power spectrum is given by $$\label{eq:curvedSkyNormFinalResult}
\widehat{P} (k) = \left( \frac{c^2}{2k_B} \right)^2 \frac{|\widetilde{V} (\mathbf{b}, \tau) |^2}{ \int d\Omega d\nu \nu^4 A^2 ({\hat{\mathbf{r}}}, \nu) \phi^2(r_\nu) \gamma^2 (\nu) / r_\nu^2 \alpha_\nu},$$ where it is understood that the copy of $k$ on the left hand side is tied to the values of $\mathbf{b}$ and $\tau$ on the right hand side via Eq. . Remarkably, this result is almost identical to the estimator previously derived in the literature with many more assumptions (chiefly the flat-sky approximation), reproduced in Appendix \[sec:RectilinearInterferometerPspecNorm\] for completeness. Comparing Eqs. and , one sees that the flat-sky approximation has only a minor effect on the result. The two expressions differ only in that with the curved sky case, $r_\nu$ and $\alpha_\nu$ appear inside a radial integral and are evaluated using their full nonlinear expressions, whereas in the flat-sky case, they appear outside the integral and are evaluated at the middle of the radial profile of our survey. Numerically, we find that for the PAPER primary beam, the difference between the Eqs. and is $\sim 0.1\%$. This rigorously justifies the previous use of flat-sky normalization factors in delay-spectrum-based estimates of the power spectrum [@pober_et_al2013b; @parsons_et_al2014; @ali_et_al2015; @jacobs_et_al2015], regardless of whether an instrument’s beam is narrow.
The foreground wedge in the spherical Fourier-Bessel formalism {#sec:CurvedSkyWedge}
--------------------------------------------------------------
In Section \[sec:WindowFcts\], we saw that our power spectrum window functions became elongated at high $\ell$, providing our first hints of the foreground wedge. However, these hints were not derived in an entirely rigorous fashion, since Section \[sec:WindowFcts\] and Section \[sec:SingleBlNorm\] both assumed that the sky temperature is comprised entirely of the cosmological signal. For the purposes of deriving a power spectrum normalization, this is the correct assumption to make. On the other hand, this is insufficient for a derivation of the foreground wedge, since we saw from Section \[sec:RotationalInvarianceOnly\] that foregrounds have different statistical properties than the cosmological signal.
When the sky consists of more than just the cosmological signal, Eq. becomes more complicated because the two-point correlator of $\overline{T}_{\ell m}(k)$ is no longer proportional to the cosmological power spectrum. Instead, foregrounds form an additive contribution to $\overline{T}_{\ell m}(k)$, and—since they are uncorrelated with the cosmological signal—an additive contribution to the two-point correlator. As a simple example, consider the foreground model discussed in Section \[sec:RotationalInvarianceOnly\], where the foregrounds possess (statistical) rotation invariance but not translation invariance along the radial/frequency direction. With these foregrounds, Eq. becomes $$\begin{aligned}
\langle |\widetilde{V} (\mathbf{b}, \tau) |^2 \rangle = \sum_{\ell} \int \! dk \,W_\ell^\textrm{unnorm} (k; \mathbf{b}, \tau) P(k) \nonumber \\
+ \frac{2}{\pi} \sum_{\ell m} C_\ell \Bigg{|} \int \! dk \,k^2 q_\ell (k) g_{\ell m} (k ; \mathbf{b}, \tau) \Bigg{|}^2,\end{aligned}$$ where $q_\ell (k)$ is the radial spherical Fourier-Bessel transform of the foreground spectrum, as defined by Eq. . Inserting explicit expressions for the $q_\ell$ and $g_{\ell m}$ results in considerable simplifications to the integral in the second term of our expression: $$\begin{aligned}
&& \int \! dk \, k^2 q_\ell (k) g_{\ell m} (k ; \mathbf{b}, \tau) \nonumber \\
&& = \sqrt{\frac{2}{\pi}} \int \! d\nu dr^\prime e^{i 2\pi \nu \tau} r^{\prime 2} f_{\ell m} (\mathbf{b}, \nu) \phi(r_\nu) \gamma (\nu) \kappa(\nu_{r^\prime}) \nonumber \\
&& \quad \times \int dk \,k^2 j_\ell (k r^\prime) j_\ell (k r_\nu) \nonumber \\
&& = \sqrt{\frac{\pi}{2}} \int \! d\nu\, e^{i 2\pi \nu \tau} f_{\ell m} (\mathbf{b}, \nu) \phi(r_\nu) \gamma (\nu) \kappa (\nu),\end{aligned}$$ where in the second equality we used the orthogonality of spherical Bessel functions from Eq. . Inserting $f_{\ell m }$ from Eq. then gives $$\sqrt{\frac{\pi}{2}} \int d\Omega B({\hat{\mathbf{r}}}) Y_{\ell m} ({\hat{\mathbf{r}}}) \int \! d\nu\, e^{i 2\pi \nu (\tau - \mathbf{b} \cdot {\hat{\mathbf{r}}}/ c)} \phi(r_\nu) \gamma (\nu) \kappa (\nu),$$ where in this section we are assuming that $B({\hat{\mathbf{r}}})$ is approximately frequency independent in order to highlight the interferometric phenomenology of the foreground wedge. Now, define for notational convenience the quantity $\Theta ( \nu - \nu_0) \equiv \phi(r_\nu) \gamma(\nu_r) \kappa(\nu_r)$ as a re-centered frequency profile of the foregrounds as seen in the data (i.e., including the finite bandwidth $\phi$ of the instrument and the tapering function $\gamma$ imposed by the data analyst). The foreground contribution to $\langle |\widetilde{V} (\mathbf{b}, \tau) |^2 \rangle$ then becomes $$\begin{aligned}
\label{eq:VtildeSqFgCell}
\langle |\widetilde{V} (\mathbf{b}, \tau) |^2 \rangle \Big{|}_\textrm{fg} = \sum_{\ell m} C_\ell && \Bigg{|} \int d\Omega B({\hat{\mathbf{r}}}) Y_{\ell m} e^{i 2\pi \nu_0 (\tau - \mathbf{b} \cdot {\hat{\mathbf{r}}}/ c)} \nonumber \\
&& \times \widetilde{\Theta} \left[ 2 \pi \left( \tau - \frac{\mathbf{b} \cdot {\hat{\mathbf{r}}}}{c} \right) \right] \Bigg{|}^2.\end{aligned}$$ To prevent any obscuration of our understanding of the foreground wedge in the spherical Fourier-Bessel formalism, we assume at this point that $C_\ell$ is a constant. As an extreme example of why this is necessary, consider the case where $C_\ell$ is zero everywhere except for one particular value of $\ell$. Clearly, the signature of foregrounds on the $\ell$-$k$ would then be dominated by the rather peculiar form for $C_\ell$, rather than the chromatic interferometric effects we wish to examine here. Setting $C_\ell$ to a constant, our expression reduces to $$\begin{aligned}
\label{eq:FinalWedgeEq}
\langle |\widetilde{V} (\mathbf{b}, \tau) |^2 \rangle \Big{|}_\textrm{fg} \propto \int d\Omega B^2 ({\hat{\mathbf{r}}}) \Bigg{|} \widetilde{\Theta} \left[ 2 \pi \left( \tau - \frac{\mathbf{b} \cdot {\hat{\mathbf{r}}}}{c} \right) \right] \Bigg{|}^2 \nonumber \\
= \frac{c}{b} \int_{2\pi ( \tau - b/c)}^{2\pi ( \tau + b/c)} ds\overline{B^2} \left[ \arcsin\left( \frac{c \tau}{b} - \frac{sc}{2\pi b} \right) \right] | \widetilde{\Theta} (s) |^2, \quad\end{aligned}$$ where we performed the polar integral by aligning our polar axis along the direction of the baseline. We then defined $\overline{B^2}$ to be the beam squared profile averaged azimuthally about the baseline axis. However, in the final form of the expression we assumed that the profile has a hemispherical reflection symmetry about the plane perpendicular to the baseline axis, and used this to express $\overline{B^2}$ in a more conventional coordinate system where the polar axis is pointed at zenith.
Eq. contains all the details of the foreground wedge. To make this clear, consider the long baseline limit, which we know from Eq. maps to the high $\ell$ portion of the power spectrum. In this regime, $\overline{B^2}$ is a very broad function of $s$ compared to $\widetilde{\Theta}$, which is compactly localized around $s\approx 0$ (since $\Theta$ is a centered spectral profile) for spectrally smooth foregrounds that are surveyed by an instrument with broad frequency coverage. We may thus factor $\overline{B^2}$ out of the integral, evaluating it at $s=0$ to give $$\label{eq:approxFinalWedgeEq}
\langle |\widetilde{V} (\mathbf{b}, \tau) |^2 \rangle \Big{|}_\textrm{fg} {\mathrel{\vcenter{
\offinterlineskip\halign{\hfil$##$\cr
\propto\cr\noalign{\kern2pt}\sim\cr\noalign{\kern-2pt}}}}}\frac{c}{b} \overline{B^2} \left[ \arcsin\left( \frac{c \tau}{b} \right) \right] \int_{2\pi ( \tau - b/c)}^{2\pi ( \tau + b/c)} ds| \widetilde{\Theta} (s) |^2.$$ There are two key features to this equation. The first is that $\langle |\widetilde{V} (\mathbf{b}, \tau) |^2 \rangle$ is zero if $\tau$ is not within $\pm b / c$ of zero, because $\widetilde{\Theta}$ is peaked around zero. This means that there will be no foreground emission beyond $\tau > b/c$. Inserting Eq. into this condition implies that foreground on the $\ell$-$k$ plane will be restricted to $$k < \ell \left( \frac{1}{\alpha_0^2 \nu_0^2}+\frac{1}{r_0^2}\right)^{\frac{1}{2}},$$ or in terms cosmological quantities, $$\label{eq:Finalkellwedge}
k < \ell \frac{H_0}{c} \left[\frac{E^2(z)}{(1+z)^2} + \left( \int_0^z \frac{dz^\prime}{E(z^\prime)}\right)^{-2}\right]^{\frac{1}{2}}.$$ We therefore have a wedge signature (beyond which there is minimal foreground contamination) similar to what is seen on the $k_\perp$-$k_\parallel$ plane. This is seen in the bottom right panel of Figure \[fig:fgSigs\], where we numerically evaluate Eq. for a flat intrinsic angular power spectrum for the foregrounds, with survey parameters kept the same as they were in previous sections.
The other key feature Eq. is the way in which foreground power drops off as one approaches the edge of the wedge. For regions of the $\ell$-$k$ plane that satisfy Eq. (i.e., “inside/below the wedge"), the integral in Eq. evaluates to a constant factor, leaving a spherical harmonic power spectrum signature $\widehat{S}_\ell^\textrm{fg} (k)$ of the form[^10] $$\begin{aligned}
\label{eq:Sellfgk}
\widehat{S}_\ell^\textrm{fg} (k) &\propto& \langle |\widetilde{V} (\mathbf{b}, \tau) |^2 \rangle \Big{|}_\textrm{fg} \nonumber \\
&{\mathrel{\vcenter{
\offinterlineskip\halign{\hfil$##$\cr
\propto\cr\noalign{\kern2pt}\sim\cr\noalign{\kern-2pt}}}}}&\frac{1}{\ell} \overline{B^2} \left[ \arcsin\left( \alpha_0 \nu_0 \sqrt{\frac{k^2}{\ell^2} - \frac{1}{r_0^2}} \right) \right].\end{aligned}$$ Ignoring the $1/\ell$ prefactor (which only weakly tilts the power profile), this expression shows that for regions within the wedge on the $k$-$\ell$ plane, contours of constant power take the form of straight lines where $k \propto \ell$. As $k$ increases, these contours decrease in power with a profile determined by the square of the beam, averaged along the direction perpendicular to the baseline.
Eq. does not hold for short baselines (i.e., at low $\ell$) because the approximations that led to Eq. no longer apply. In such a regime, the profile becomes proportional to $| \widetilde{\Theta} ( \alpha_0 k )|^2$, leading to the horizontally oriented power patterns seen at low $\ell$ in Figure \[fig:fgSigs\]. This contrast in behavior between low and high $\ell$ regions is familiar from the $k_\perp$-$k_\parallel$ plane: at low $\ell$ or low $k_\perp$, the leakage of flat-spectrum foregrounds towards the upper portions of the plane is driven by the radial extent of the survey, while at high $\ell$ or high $k_\perp$ the leakage is driven by the baseline chromaticity that causes the wedge. In intermediate regimes, Eq. is similar in form to a convolution. In fact, it would be precisely a convolution were it not for the $\arcsin$ and the some constant factors needed for unit conversions. This convolution-like operation enacts a smooth transition in behavior between the low- and high-$\ell$ regimes.
Fundamentally, the wedge signature arises because the chromaticity of an interferometer causes spectrally smooth foregrounds from low $k$ or $k_\parallel$ (as seen in the top row of Figure \[fig:fgSigs\]) to leak to higher $k$ and $k_\parallel$. In other words, power is smeared out along the $k$ or $k_\parallel$ axes. Though its most dominant effect is to cause the foreground wedge, this smearing affects all modes on the $k_\perp$-$k_\parallel$ and $\ell$-$k$ planes, particularly at high $k_\perp$ and high $\ell$ where chromatic effects are more prominent. This can be seen by examining Figure \[fig:windowFcts\], where the window functions for the cosmological signal are seen to vertically broaden at high $\ell$, regardless of location along the $k$ axis. (In principle, Figure \[fig:windowFcts\] only applies to signals that possess translation-invariant statistics, but the effects are qualitatively the same). The broadening with increasing $\ell$ can be seen by comparing the non-interferometric (top row) and interferometric (bottom row) results in Figure \[fig:fgSigs\]. As discussed in Section \[sec:Numerics\], the cosine radial profile given by Eq. causes ringing in Fourier space that gives horizontal stripes that are visually obvious in the non-interferometric case. For the interferometric case, the ringing is still present, but the peaks are smeared out, especially at high $\ell$. This reinforces what was found in @liu_et_al2014a, where it was argued that chromatic interferometric effects do not only cause the wedge, but also reduce the independence of different Fourier modes.
In summary, we have seen in this section that the spherical power spectrum provides the same foreground diagnostic capabilities on the $\ell$-$k$ plane as the rectlinear power spectrum did on the $k_\perp$-$k_\parallel$ plane. In the spherical Fourier-Bessel formalism, the foregrounds continue to be confined to a wedge. This is good news for analysts of wide-field intensity mapping data from interferometers, for it suggests that one’s intuition for the $k_\perp$-$k_\parallel$ plane can be easily transferred to the $\ell$-$k$ plane.
Conclusions {#sec:Conclusions}
===========
In this paper, we have established a framework for analyzing intensity mapping data using spherical Fourier-Bessel techniques. Such techniques easily incorporate the wide-field nature of many intensity mapping surveys, obviating the need to split up one’s field into several approximately flat fields during analysis. This builds sensitivity for science measurements as well as diagnostic tests, and additionally provides access to the largest angular scales on the sky.
Adapting spherical Fourier-Bessel techniques from galaxy surveys requires one to pay special attention to the unique properties of intensity mapping. For example, we saw in Figure \[fig:surveyGeom\] that intensity mapping surveys (particularly those that operate at high redshifts) tend to be compressed in the radial direction and have very fine radial resolution compared to angular resolution. Intensity mapping experiments must also contend with extremely bright foregrounds that overwhelm the cosmological signals of interest. A successful spherical Fourier-Bessel analysis framework must demonstrate that it is able to deal with such systematics at least as well as traditional rectilinear Fourier techniques can.
This paper demonstrates that spherical Fourier-Bessel modes are indeed a suitable basis for intensity mapping analyses. Focusing on power spectrum measurements, in Section \[sec:FiniteVolume\] we proposed that the cylindrically binned power spectrum $P(k_\perp, k_\parallel)$ be replaced by the spherical harmonic power spectrum $S_\ell (k)$. The quantity $S_\ell (k)$ is conveniently defined so that a weighted average of it over different $\ell$ values yields the spherically binned cosmological power spectrum $P(k)$. At the same time, by splitting up the measured power spectrum into a function of $\ell$ and $k$, angular fluctuations are separated from arbitrarily oriented spatial fluctuations. This separation of fluctuations into angular and non-angular modes provides a powerful diagnostic for systematics. This has historically been the motivation for viewing the power spectrum as a function of $k_\perp$ and $k_\parallel$, and $S_\ell (k)$ preserves this crucial property of $P(k_\perp, k_\parallel)$. Of course, this is not to say that the data should not also be examined in bases like $(\mathbf{b}, \tau)$ that are more closely related to the actual instrument’s measurement [@Vedantham2012; @Parsons_et_al2012b]. Doing so is particularly valuable prior to the squaring of the data to form power spectra, and both approaches can and should be used.
Chief amongst the systematics that may be discerningly diagnosed on the $k_\perp$-$k_\parallel$ plane are astrophysical foregrounds. Foregrounds are expected to have localized signatures in $P(k_\perp, k_\parallel)$, facilitating their removal. We have shown in this paper that the same is true for $S_\ell (k)$. For non-interferometric intensity mapping surveys, we have shown that the spectrally smooth nature of foregrounds results in their being sequestered at low $k$, and that interloper lines can be detected using $S_\ell (k)$ just as easily as they can be using $P(k_\perp, k_\parallel)$. For interferometric surveys, foregrounds tend to limited to a wedge-like feature on the $k_\perp$-$k_\parallel$ plane. Foregrounds are limited to a similar wedge on the $\ell$-$k$ plane. This suggests that $S_\ell(k)$ is just as capable a diagnostic quantity as $P(k_\perp, k_\parallel)$ for intensity mapping surveys, while simultaneously discarding unwarranted flat-sky approximations seen in previous papers. Another attractive property of our spherical Fourier-Bessel formulation is that many of the relevant formulae derived in this paper (such as the equation delineating the boundary of the foreground wedge) are very similar to their flat-sky counterparts. Intuition for the behavior of $P(k_\perp, k_\parallel)$ that has been built up in the prior literature is thus almost entirely transferrable to $S_\ell (k)$.
Our framework may be generalized in several ways in future work. For instance, we have thus far neglected to describe redshift space distortions, although the spherical formalism that we espouse here should be particularly well-suited for the purpose (C. J. Schmit et al., in prep.). A crucial area of investigation will be to determine whether cosmological redshift space distortions interfere with the signature of interloper lines. Another area of future development would be the incorporation of light-cone effects, since it has been shown that cosmological evolution cannot be neglected over the survey volume of a typical intensity mapping survey [@barkana_and_loeb2006; @datta_et_al2012; @datta_et_al2014; @laplante_et_al2014; @zawada_et_al2014; @ghara_et_al2015]. For now, however, this paper points to the promise of spherical Fourier-Bessel techniques for rigorous data analysis, providing yet another powerful diagnostic tool in the continuing progress of intensity mapping towards surveying an unprecedentedly large volume of our observable Universe.
The authors gratefully acknowledge delightful and helpful discussions with James Aguirre, Michael Eastwood, Aaron Ewall-Wice, Daniel Jacobs, Gregg Hallinan, Bryna Hazelton, Jacqueline Hewitt, Saul Kohn, Miguel Morales, Jonathan Pober, Jonathan Pritchard, Claude Schmit, Richard Shaw, and Nithya Thyagarajan. This research was completed as part of the University of California Cosmic Dawn Initiative. AL and ARP acknowledge support from the University of California Office of the President Multicampus Research Programs and Initiatives through award MR-15-328388, as well as from NSF CAREER award No. 1352519, NSF AST grant No.1129258, and NSF AST grant No. 1440343. AL acknowledges support for this work by NASA through Hubble Fellowship grant \#HST-HF2-51363.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
Estimating the power spectrum from rectilinear Fourier modes {#sec:RectilinearFKP}
============================================================
In this Appendix, we derive a relation between measured rectilinear Fourier amplitudes of the sky $\widetilde{T}^\textrm{meas} (k)$ and the power spectrum, analogous to Eq. for the spherical Fourier-Bessel modes. The derivation presented here is a standard one, and is only included to serve as an analogy to Eq. .
For a survey specified by the function $\phi (\mathbf{r})$—so that the measured temperature field is $\phi(r) T(\mathbf{r})$ rather than just $T(\mathbf{r})$—the measured Fourier amplitudes $\widetilde{T}^\textrm{meas} (\mathbf{k})$ are related to the true Fourier amplitudes $\widetilde{T} (\mathbf{k})$ by the convolution theorem, which gives $$\widetilde{T}^\textrm{meas} (\mathbf{k}) = \int \frac{d^3 k^\prime}{(2 \pi)^3} \widetilde{T} (\mathbf{k}^\prime) \widetilde{\phi} (\mathbf{k} - \mathbf{k}^\prime).$$ Squaring and ensemble averaging the result gives $$\langle | \widetilde{T}^\textrm{meas} (\mathbf{k}) |^2 \rangle = \int \frac{d^3 k^\prime}{(2 \pi)^3} \big{|} \widetilde{\phi} (\mathbf{k} - \mathbf{k}^\prime) \big{|}^2 P(k^\prime),$$ where we used Eq. to relate the ensemble average of the true Fourier amplitudes to the power spectrum. Assuming $|\widetilde{\phi}|^2$ is sharply peaked, $P(k^\prime)$ can be approximately factored out of the integral and evaluated at $k$. Changing integration variables from $\mathbf{k}^\prime$ to $\mathbf{k} - \mathbf{k}^\prime$ for the remaining integral and invoking Parseval’s theorem then yields $$\langle | \widetilde{T}^\textrm{meas} (\mathbf{k}) |^2 \rangle = P(k) \int d^3 r \phi^2 (\mathbf{r}).$$ This suggests that a power spectrum estimator $\widehat{P} (k)$ can be constructed by computing $$\widehat{P} (k) = \frac{\sum_{|\mathbf{k}| = k} | \widetilde{T}^\textrm{meas} (\mathbf{k}) |^2}{N_k \int d^3 r \phi^2 (\mathbf{r})},$$ where $N_k$ is the number of independent Fourier modes in the shell where $|\mathbf{k}| = k$. The rest of the normalization factor that comprises the denominator (i.e., the integral) is independent of $k$ and is a sensitivity-weighted volume factor. For a survey with uniform sensitivity, for example, $\phi(\mathbf{r}) = 1$ everywhere inside the survey and the integral is exactly the volume of the survey. Because this integral is independent of $\mathbf{k}$, it follows that the orientation of a Fourier mode (i.e., ${\hat{\mathbf{k}}}$) has no bearing on its sensitivity to the power spectrum, and all orientations are equally sensitive.
Delay spectrum normalization in the narrow-field flat-sky limit {#sec:RectilinearInterferometerPspecNorm}
===============================================================
For orientation, we now briefly review how visibility-based estimators of the power spectrum are usually derived. Again, the derivation that follows is not new to this paper, but we include it to provide a pedagogical comparison to Section \[sec:SingleBlNorm\], as well as to place a special emphasis on the approximations involved.
Our first approximation will be the flat-sky, narrow-field approximation. This makes Eq. the appropriate expression to use for our interferometric visibility. Next, we assume that the interferometric fringe in this visibility is frequency-independent, so that the factor of $\nu$ in the complex exponential term may be replaced by $\nu_0$, the median frequency of one’s observing volume. This is equivalent to the approximation that one has very short baselines (since the baseline vector $\mathbf{b}$ is multiplied by $\nu$), or alternatively, that most spectral structure comes from either the primary beam or the sky temperature. Correspondingly, we also replace all copies of $r_\nu$ with $r_0$ to yield[^11] $$V(\mathbf{b}, \nu) = \frac{1}{r_0^2}\int d^2 r_\perp \phi(r) B(\mathbf{r_\perp}, \nu) T(\mathbf{r_\perp}, \nu)e^{ - i 2\pi \nu_0 \mathbf{b} \cdot \mathbf{r}_\perp / c r_0}.$$ To access Fourier modes along the line-of-sight, we perform the delay transform defined by Eq. . Converting again to cosmological coordinates and defining $$\label{eq:kperpConversions}
\mathbf{k}_\perp \equiv \frac{2 \pi \nu_0}{r_0 c} \mathbf{b}; \qquad k_\parallel \equiv \frac{2 \pi \tau}{\alpha},$$ along with $\mathbf{k} \equiv (\mathbf{k}_\perp, k_\parallel)$, one obtains $$\begin{aligned}
\widetilde{V}(\mathbf{b}, \tau) & = & \frac{e^{i 2\pi \nu_0 \tau - i k_\parallel r_0}}{r_0^2 \alpha_0 } \int d^3r D(\mathbf{r})T(\mathbf{r}) \exp \left( - i \mathbf{k} \cdot \mathbf{r} \right) \nonumber \\
& = & \frac{e^{i 2\pi \nu_0 \tau - i k_\parallel r_0}}{r_0^2 \alpha_0 } \int \frac{d^3 k^\prime}{(2 \pi)^3} \widetilde{T} (\mathbf{k}^\prime) \widetilde{D} ( \mathbf{k} - \mathbf{k}^\prime),\end{aligned}$$ where $\gamma$ is the tapering functions used in Eq. , $\alpha_0$ is the distance-frequency conversion from Eq. evaluated at the redshift corresponding to the central frequency of the survey, and we have defined $D(\mathbf{r}) \equiv B(\mathbf{r}) \phi(r) \gamma(\nu_{r_\parallel}) $, with $\widetilde{D}$ denoting its Fourier transform. While survey geometry and tapering factors such as $\phi$ and $\gamma$ have not typically been included in literature derivations such as those in @parsons_et_al2012a and @parsons_et_al2014, they are crucial in practical analyses of the data (e.g., in @ali_et_al2015), and thus we include them here.
As suggested in Section \[sec:DelayIntro\], we may relate the delay-transformed visibility to the power spectrum by squaring $\widetilde{V}(\mathbf{b}, \tau)$ and taking an ensemble average over realizations of the random temperature field. This gives $$\begin{aligned}
\langle | \widetilde{V}(\mathbf{b}, \tau) |^2 \rangle &= & \left( \frac{1}{r_0^2 \alpha_0 } \right)^2 \int \frac{d^3 k^\prime}{(2 \pi)^3} \frac{d^3 k^{\prime \prime}}{(2 \pi)^3} \nonumber \\
&& \widetilde{D} ( \mathbf{k} - \mathbf{k}^\prime) \widetilde{D}^* ( \mathbf{k} - \mathbf{k}^{\prime \prime}) \langle \widetilde{T} (\mathbf{k}^\prime) \widetilde{T}^* (\mathbf{k}^{\prime \prime}) \rangle \nonumber \\
& = & \left( \frac{1}{r_0^2 \alpha_0 } \right)^2 \int \frac{d^3 k^\prime}{(2 \pi)^3} P(\mathbf{k}^\prime) |\widetilde{D} ( \mathbf{k} - \mathbf{k}^{\prime})|^2, \qquad\end{aligned}$$ where in the last equality we invoked the definition of the power spectrum, i.e., Eq. . At this point, we may make the approximation that the power spectrum is a relatively broad function, while $|\widetilde{D} ( \mathbf{k} - \mathbf{k}^{\prime})|^2$ is fairly sharply peaked at $\mathbf{k} = \mathbf{k}^\prime$. This allows $P (\mathbf{k})$ to be factored out of the integral, and by invoking Parseval’s theorem on what remains, we obtain $$\langle | \widetilde{V}(\mathbf{b}, \tau) |^2 \rangle \approx \left( \frac{1}{r_0^2 \alpha_0 } \right)^2 P (\mathbf{k}) \int d^3 r D^2 (\mathbf{r}).$$ A sensible estimator $\widehat{P} (\mathbf{k})$ for the power spectrum would thus be $$\begin{aligned}
\widehat{P} (\mathbf{k}) &=& r_0^4 \alpha^2 \left[ \int d^3 r D^2 (\mathbf{r}) \right]^{-1} | \widetilde{V}(\mathbf{b}, \tau) |^2 \nonumber \\
& =& \frac{\alpha r_0^2 }{ \int d^2\theta d\nu B^2 ({\boldsymbol \theta}, \nu) \gamma(\nu) \phi(r_\nu)} | \widetilde{V}(\mathbf{b}, \tau) |^2. \qquad\quad\end{aligned}$$ Now, even though this expression was derived using the flat-sky approximation, it has been applied to wide-field instruments in the past. The flat-sky approximation is crudely undone by promoting $d^2 \theta$ back to $d \Omega$, giving $$\label{eq:flatSkyNormFinalResult}
\widehat{P} (\mathbf{k}) = \left( \frac{ c^2}{2 k_B } \right)^2 \frac{r_0^2 \alpha | \widetilde{V}(\mathbf{b}, \tau) |^2 }{\int \!d\nu d\Omega \,\nu^4 A^2 ({\hat{\mathbf{r}}}, \nu)\gamma(\nu) \phi(r_\nu) },$$ where we have reinserted Eq. . It is implicitly assumed that the value of $\mathbf{k}$ on the left hand side of this equation is related to $\mathbf{b}$ and $\tau$ by Eq. .
The foreground wedge in the narrow-field limit
==============================================
In this Appendix, we work in the narrow-field limit and derive an analytic form for the signature of foregrounds in a power spectrum expressed in terms of rectilinear $k_\perp$-$k_\parallel$ Fourier modes (i.e., the “foreground wedge" on the $k_\perp$-$k_\parallel$ plane). Our starting point will be Eq. , but written in terms of angles on the sky and assuming a frequency-independent modified primary beam $B(\boldsymbol \theta)$: $$V(\mathbf{b}, \nu) = \int d^2 \theta \phi(r_\nu) B(\boldsymbol \theta) T(\boldsymbol \theta, \nu)e^{ - i 2\pi \nu \mathbf{b} \cdot \boldsymbol \theta / c}.$$ The delay-transformed visibility then takes the form $$\widetilde{V} (\mathbf{b}, \tau) = \int d\nu d^2 \theta \gamma(\nu) \phi(r_\nu) B(\boldsymbol \theta) T(\boldsymbol \theta, \nu) e^{i 2 \pi \nu ( \tau - \mathbf{b} \cdot \boldsymbol \theta / c)}.$$
In principle, our sky temperature $T(\boldsymbol \theta, \nu)$ should include contributions from both the cosmological signal and the foregrounds. However, if we assume that foregrounds and the cosmological signal are uncorrelated (as we did in Section \[sec:CurvedSkyWedge\]), we may derive the foreground wedge without including the cosmological signal. We thus assume in this Appendix that $T(\boldsymbol \theta, \nu)$ consists solely of foregrounds, taking the form $$T(\boldsymbol \theta, \nu) = T_\perp^\textrm{fg} (\boldsymbol \theta) \kappa(\nu),$$ and thus our delay-transformed visibility becomes $$\begin{aligned}
\widetilde{V} (\mathbf{b}, \tau) & = & \int d^2 \theta B(\boldsymbol \theta) T_\perp^\textrm{fg} (\boldsymbol \theta) e^{i 2 \pi \nu_0 ( \tau - \mathbf{b} \cdot \boldsymbol \theta / c)} \nonumber \\
&& \qquad \times \widetilde{\Theta} \left[ 2 \pi \left( \tau - \frac{\mathbf{b} \cdot \boldsymbol \theta}{c}\right)\right] \nonumber \\
&=& \int \frac{d^2 \ell}{(2\pi)^2} \widetilde{T}_\perp^\textrm{fg} (\boldsymbol \ell) \int d^2 \theta B(\boldsymbol \theta) e^{i \boldsymbol \ell \cdot \boldsymbol \theta} e^{i 2 \pi \nu_0 ( \tau - \mathbf{b} \cdot \boldsymbol \theta / c)} \nonumber \\
&& \qquad \times \widetilde{\Theta} \left[ 2 \pi \left( \tau - \frac{\mathbf{b} \cdot \boldsymbol \theta}{c}\right)\right],\end{aligned}$$ where $\widetilde{T}_\perp^\textrm{fg} (\boldsymbol \ell) \equiv \int d^2\theta e^{i \boldsymbol \ell \cdot \boldsymbol \theta} T_\perp^\textrm{fg} (\boldsymbol \theta)$ is the Fourier transform of $T_\perp^\textrm{fg}$, which we assume (as we did in Sections \[sec:RotationalInvarianceOnly\] and \[sec:CurvedSkyWedge\]) is a field with rotationally invariant statistics. This means that $$\langle \widetilde{T}_\perp^\textrm{fg} (\boldsymbol \ell) \widetilde{T}_\perp^\textrm{fg} (\boldsymbol \ell)^* \rangle = (2\pi)^2 \delta^D (\boldsymbol \ell - \boldsymbol \ell^\prime) C_\ell^\textrm{fg},$$ where we have suggestively chosen the symbol $C_\ell^\textrm{fg}$ on the right hand side because in the flat-sky limit, $C_\ell^\textrm{fg}$ can be shown to converge to the angular power spectrum [@hu2000].
Following Section \[sec:CurvedSkyWedge\], we form our estimator of the power spectrum by squaring the absolute magnitude of the delay-transformed visibility to obtain $$\begin{aligned}
\langle | \widetilde{V} (\mathbf{b}, \tau) |^2 \rangle = \int \frac{d^2 \ell}{(2\pi)^2} C_\ell^\textrm{fg} \Bigg{|} \int d^2 \theta e^{i (\boldsymbol \ell - 2 \pi \nu_0 \mathbf{b} / c) \cdot \boldsymbol \theta} \nonumber \\
\times B(\boldsymbol \theta) \widetilde{\Theta} \left[ 2 \pi \left( \tau - \frac{\mathbf{b} \cdot \boldsymbol \theta}{c}\right)\right] \Bigg{|}^2. \qquad\end{aligned}$$ Again, we may consider the special case where $C_\ell^\textrm{fg}$ is a constant in order to elucidate the effects of the foreground wedge. This yields $$\begin{aligned}
\langle | \widetilde{V} (\mathbf{b}, \tau) |^2 \rangle &\propto& \int d^2 \theta B^2 (\boldsymbol \theta) \Bigg{|} \widetilde{\Theta} \left[ 2 \pi \left( \tau - \frac{\mathbf{b} \cdot \boldsymbol \theta}{c} \right) \right] \Bigg{|}^2 \nonumber \\
&=& 2 \pi \int d \theta \overline{B^2} (\theta) \widetilde{\Theta} \left[ 2 \pi \left( \tau - \frac{b \theta}{c} \right) \right] \Bigg{|}^2 \nonumber \\
&=& \frac{c}{b} \int ds \overline{B^2} \left( \frac{c \tau}{b} - \frac{sc}{2\pi b} \right) | \widetilde{\Theta} (s) |^2 \nonumber \\
&\approx & \frac{c}{b} \overline{B^2} \left( \frac{c \tau}{b} \right) \int ds | \widetilde{\Theta} (s) |^2\end{aligned}$$ where $\overline{B^2} (\theta) \equiv (1/2\pi) \int d\theta^\prime B^2(\theta, \theta^\prime) $, and in the last line we assumed we were in the long baseline (or the high $k_\perp$) regime where the foreground wedge is relevant. This allowed $\overline{B^2}$ to be factored out of the integral.
Recalling that $\langle | \widetilde{V} (\mathbf{b}, \tau) |^2 $ serves as a good estimator for the power spectrum at Fourier coordinates given by Eq. , the foreground contamination $\widehat{P}^\textrm{fg}$ of our power spectrum estimate is thus given by $$\widehat{P}^\textrm{fg} (k_\perp, k_\parallel) \propto \frac{1}{k_\perp} \overline{B^2} \left( \frac{\alpha_0 \nu_0 k_\parallel}{r_0 k_\perp}\right).
\vspace{0.1cm}$$ Contours of constant foreground power are therefore along lines where $k_\parallel \propto k_\perp$, and if $\overline{B^2}$ is zero (or negligible) beyond some characteristic angle $\theta_c$ away from its peak, foreground emission will be confined to $$\label{eq:WedgeLineEquation}
k_\parallel < k_\perp \frac{H_0 r_0 E(z) \theta_c}{c (1+z)} ,$$ where we have written $\alpha_0$ in terms of cosmological parameters. Since $k = (k_\perp^2 + k_\parallel^2)^{1/2}$, we may also write this in terms of $k$ and $k_\perp$. This gives $$k < k_\perp r_0 \frac{H_0}{c} \left[\frac{\theta_c^2 E^2(z)}{(1+z)^2} + \left( \int_0^z \frac{dz^\prime}{E(z^\prime)}\right)^{-2}\right]^{\frac{1}{2}}.$$\
Recalling that $\ell \approx k_\perp r_0$ in the flat-sky approximation, this is essentially the same as the full curved-sky expression, Eq. . The only slight difference is that in the flat-sky approximation, the angular coordinates are rectilinear and formally go from $-\infty$ to $+\infty$, necessitating some arbitrary cut-off angle $\theta_c$ for the primary beam. In the curved sky treatment, a cutoff is naturally imposed by the horizon.
[^1]: $^{\dagger}$Hubble Fellow
[^2]: In @parsons_et_al2016 it was shown that in specialized situations it is possible to pre-filter visibility data from an interferometer to recover some of the loss of sensitivity from a square-then-average approach. However, such an approach does not recover large scale angular modes from a wide field of view.
[^3]: It is an unfortunate coincidence that the spherical harmonic indices are typically denoted by $\ell$ and $m$ in the cosmological literature, while in radio astronomy they are reserved for the direction cosines from zenith in the east-west and north-south directions, respectively. In this paper, $\ell$ and $m$ will always represent spherical harmonic indices, and never direction cosines.
[^4]: In this paper, we use hats for two different purposes. When placed above a vector (e.g., with ${\hat{\mathbf{r}}}$), the hat indicates that the vector is a unit vector. When placed above a scalar (e.g., with $\widehat{P}$), the hat indicates an estimator of the hatless quantity.
[^5]: This does not, of course, preclude the examination of systematics in other spaces. For example, though cable reflections may have well-defined signatures on the $k_\perp$-$k_\parallel$ or $\ell$-$k$ planes, they are an example of a systematic that can (and should) also be diagnosed in spaces appropriate for raw data coming off an instrument.
[^6]: We implicitly assume throughout this paper that we are dealing only with temperature *fluctuations*. In other words, we assume that that the mean sky temperature has already been subtracted off (or simply does not enter the measurement itself, as is the case with most interferometric measurements).
[^7]: Note that even though this was derived assuming that $P(k)$ is smooth (which does not necessarily hold when substantial foreground contaminants are involved; @liu_et_al2014b), the resulting normalization is still the correct one to use.
[^8]: As expected from Section \[sec:RotationalInvarianceOnly\], the definition of $S_\ell (k)$ depends on the survey geometry $\phi(\mathbf{r})$. This dependence cancels out for the cosmological signal, but not for contaminants. Thus, while two different surveys should give identical results for the cosmological power spectrum $P(k)$, the contaminant (e.g., foreground) contributions to the power are not directly comparable, and two surveys with identical contaminating influences but different sky coverage may measure different total power spectra. Note that this argument is due purely to the differences in scaling with survey volume discussed in Section \[sec:RotationalInvarianceOnly\]. It thus applies equally well to both $P(k_\perp, k_\parallel)$ and $S_\ell (k)$, and is not simply a peculiarity of the latter.
[^9]: The term “window function" is unfortunately rather overused. In various parts of the literature, it has been used to refer to what we have called the tapering function $\gamma$ in this paper, and in other parts of the literature it has been used to describe what we have called the survey profile $\phi$. In this paper, a window function will *always* refer to the function that describes the linear combination of true power spectrum probed by one’s statistical estimator of the power spectrum. A mathematically precise definition for the window functions of our particular estimator will be provided in Eqs. and .
[^10]: Note that while our results for the boundary of the wedge and its profile are qualitatively robust, minor differences can arise depending on the precise form of the power spectrum estimator that is employed. Consider, for example, the estimator used in @liu_et_al2014a where visibility data was convolved onto a Fourier-space grid using the primary beam as a gridding kernel. There, the profile of the wedge was shown to be primary beam convolved with itself, rather than the primary beam squared as we have it here for our estimator.
[^11]: In principle, converting $\nu_0$ to a radial distance does not yield $r_0$ because the distance-frequency relation is nonlinear. In practice, the radially compressed geometry of a typical intensity mapping survey (see Figure \[fig:surveyGeom\]) means that linearized distance-frequency relations such as Eq. are excellent approximations. We are thus justified in making the assumption that $\nu_0$ and $r_0$ roughly refer to the same radial distance.
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